Information and Distribution: The Role of Merchants in the Market Economy Under Uncertainty (New Frontiers in Regional Science: Asian Perspectives, 49) 9811001006, 9789811001000

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New Frontiers in Regional Science: Asian Perspectives 49

Yasuhiro Sakai Keisuke Sasaki

Information and Distribution The Role of Merchants in the Market Economy Under Uncertainty

New Frontiers in Regional Science: Asian Perspectives Volume 49 Editor-in-Chief Yoshiro Higano, University of Tsukuba, Tsukuba, Ibaraki, Japan

New Frontiers in Regional Science: Asian Perspectives This series is a constellation of works by scholars in the field of regional science and in related disciplines specifically focusing on dynamism in Asia. Asia is the most dynamic part of the world. Japan, Korea, Taiwan, and Singapore experienced rapid and miracle economic growth in the 1970s. Malaysia, Indonesia, and Thailand followed in the 1980s. China, India, and Vietnam are now rising countries in Asia and are even leading the world economy. Due to their rapid economic development and growth, Asian countries continue to face a variety of urgent issues including regional and institutional unbalanced growth, environmental problems, poverty amidst prosperity, an ageing society, the collapse of the bubble economy, and deflation, among others. Asian countries are diversified as they have their own cultural, historical, and geographical as well as political conditions. Due to this fact, scholars specializing in regional science as an inter- and multi-discipline have taken leading roles in providing mitigating policy proposals based on robust interdisciplinary analysis of multifaceted regional issues and subjects in Asia. This series not only will present unique research results from Asia that are unfamiliar in other parts of the world because of language barriers, but also will publish advanced research results from those regions that have focused on regional and urban issues in Asia from different perspectives. The series aims to expand the frontiers of regional science through diffusion of intrinsically developed and advanced modern regional science methodologies in Asia and other areas of the world. Readers will be inspired to realize that regional and urban issues in the world are so vast that their established methodologies still have space for development and refinement, and to understand the importance of the interdisciplinary and multidisciplinary approach that is inherent in regional science for analyzing and resolving urgent regional and urban issues in Asia. Topics under consideration in this series include the theory of social cost and benefit analysis and criteria of public investments, socio-economic vulnerability against disasters, food security and policy, agro-food systems in China, industrial clustering in Asia, comprehensive management of water environment and resources in a river basin, the international trade bloc and food security, migration and labor market in Asia, land policy and local property tax, Information and Communication Technology planning, consumer “shop-around” movements, and regeneration of downtowns, among others. Researchers who are interested in publishing their books in this Series should obtain a proposal form from Yoshiro Higano (Editor in Chief, [email protected]) and return the completed form to him.

Editor in Chief Yoshiro Higano, University of Tsukuba Managing Editors Makoto Tawada (General Managing Editor), Aichi Gakuin University Kiyoko Hagihara, Bukkyo University Lily Kiminami, Niigata University Editorial Board Yasuhiro Sakai (Advisor Chief Japan), Shiga University Yasuhide Okuyama, University of Kitakyushu Zheng Wang, Chinese Academy of Sciences Hiroyuki Shibusawa, Toyohashi University of Technology Saburo Saito, Fukuoka University Makoto Okamura, Hiroshima University Moriki Hosoe, Kumamoto Gakuen University Budy Prasetyo Resosudarmo, Crawford School of Public Policy, ANU Shin-Kun Peng, Academia Sinica Geoffrey John Dennis Hewings, University of Illinois Euijune Kim, Seoul National University Srijit Mishra, Indira Gandhi Institute of Development Research Amitrajeet A. Batabyal, Rochester Institute of Technology Yizhi Wang, Shanghai Academy of Social Sciences Daniel Shefer, Technion - Israel Institute of Technology Akira Kiminami, The University of Tokyo Jorge Serrano, National University of Mexico Binh Tran-Nam, UNSW Sydney, RMIT University Vietnam Ngoc Anh Nguyen, Development and Policies Research Center Thai-Ha Le, Fulbright University Vietnam Advisory Board Peter Nijkamp (Chair, Ex Officio Member of Editorial Board), Tinbergen Institute Rachel S. Franklin, Brown University Mark D. Partridge, Ohio State University Jacques Poot, University of Waikato Aura Reggiani, University of Bologna

More information about this series at http://www.springer.com/series/13039

Yasuhiro Sakai • Keisuke Sasaki

Information and Distribution The Role of Merchants in the Market Economy Under Uncertainty

Yasuhiro Sakai Shiga University Shiga, Japan

Keisuke Sasaki Toyo University Tokyo, Japan

ISSN 2199-5974 ISSN 2199-5982 (electronic) New Frontiers in Regional Science: Asian Perspectives ISBN 978-981-10-0100-0 ISBN 978-981-10-0101-7 (eBook) https://doi.org/10.1007/978-981-10-0101-7 © Springer Nature Singapore Pte Ltd. 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

This book is dedicated to the memory of Sir John Hicks Professor Michio Morishima Toko, Tora, Pin, and Ai Hitomi and Ami

Preface

There are two economics superstars who have greatly influenced our academic careers. They are Sir John Hicks (1904–1989) and Professor Michio Morishima (1923–2004). Hicks was a British economist, presumably one of the most influential economists in modern times. Although he received the Nobel Economics Prize in 1972 for his pioneering contributions to general equilibrium theory and welfare economics, he nevertheless had a sort of mixed feelings for the honor. He thought that those contributions were done a long time ago, thus in his later days, feeling himself to have outgrown from such big gaps between the ideal and the real. In a sense, the New Hicks should be different from the Old Hicks. Morishima was a Japanese economist, almost twenty years younger than Hicks. Morishima was perhaps the most famous economist Japan has ever produced in the postwar period: in fact, he has long been regarded as the Japanese scholar who was the closest to Nobel Economics Prize. During the Second World War, he once served as a naval cadet whose duty was to break secret codes of the enemies including the USA and the UK. When he left home toward a battlefront, he used to carry with him Hicks’ newly published book Value and Capital. Since then, Morishima paid his everlasting respect to the life and work of Hicks. Understandably, Hicks in turn played the role of a strong promoter of Morishima's career in the UK. In short, it would be no exaggeration to say that Hicks has been served as an academic master of Morishima. When Sakai was a student at Kobe University, he purchased a special Asian edition of Hicks' masterwork Value and Capital. The penetrating logic and lucid English seen in the book affected Sakai profoundly. At that time, Morishima at Osaka University was conspicuously a rising star in the Japanese economics circle, publishing so many articles in first-rate international journals. The ideal combination of Hicks and Morishima drove Sakai to have dream of going abroad for graduate study and eventually becoming a professor of economics someday in the future. Several years after Sakai received a Ph.D. degree in economics at the University of Rochester, with Professor Lionel W. McKenzie being his main thesis advisor, he had a chance to teach economic theory at the University of Tsukuba, a brand-new ix

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university built on the basis of very old institutions. Then, he was before long informed that McKenzie's supervisor at Oxford was once John Hicks, and that McKenzie and Morishima were really good friends doing research in the same field, namely the turnpike theory. Later, when Sakai was taking care of a sequence of graduate courses at Tsukuba, he happened to meet Keisuke Sasaki, an ambitious and hard-working student from Akita, a northern snowy part of Japan. Although Sasaki's major was originally Education Philosophy a la John Dewey, his encounter with Sakai caused him to change his research area from the philosophy of pragmatism to the economics of uncertainty, which was then Sakai's favorite subject. At Tsukuba, Sakai and Sasaki soon became close joint workers, successfully presenting their joint papers at many workshops and seminars, with publications in numerous international journals. It is on the basis of such Sakai-Sasaki cooperation over so many years that the basic framework of our present book Information and Distribution has been generated and gradually fermented. We would like to add that at Tsukuba and other areas in Japan and oversees, our talks on the theory of oligopoly and information with so many friends and colleagues have been very helpful. In this connection, we wish to mention the following names: Lionel W. McKenzie (Rochester), James Friedman (Rochester, North Carolina), Edward Zabel (Rochester, Florida), Hugh Rose (Rochester, Johns Hopkins), Hiroshi Atsumi (Rochester, Tsukuba), William Brock (Rochester, Chicago), Walter Oi (Rochester), Shigeo Minabe (Rochester, Albany), Michihiro Ohyama (Rochester, Keio), Masayoshi Hirota (Rochester, Tokyo Rika), Akira Takayama (Rochester, Southern Illinois), Asatoshi Maeshiro (Pittsburgh), Jack Ochs (Pittsburgh), Esther Gal-Or (Pittsburgh), Martin Bronfenbrenner (Carnegie-Mellon), Ryuzo Sato (New York), Paul A. Samuelson (M.I.T., New York), Martin Beckman (Brown), Xavier Vives (Barcelona, Pennsylvania), Ed Schidravsky (Budapest, Arizona), Hirofumi Uzawa (Tokyo), Takashi Negishi (Tokyo), Koji Okuguchi (Tokyo Metropolitan), Mitsuo Suzuki (Tokyo Kogyo), Kunio Kawamata (Keio), Masahiro Okuno (Tokyo), Takashi Kiritani (Tokyo Metropolitan), Tatsuo Hatta (Osaka), Masayoshi Maruyama (Okayama, Kobe), Fukukane Nikaido (Hitotsubashi, Tsukuba), Kotaro Suzumura (Hitotsubashi), Masamichi Kawano (Tsukuba, Kwansei Gakuin), Masahiro Abiru (Tsukuba, Fukuoka), Takehiko Yamato (Tsukuba, Tokyo Kogyo), Akihiko Yoshizumi (Tsukuba), Masashi Tajima (Shiga), and many others. For a long time, both Sakai and Sasaki had been enthusiastic readers of the books and articles written by Hicks and Morishima. When we happened to read the following sentence by our idol Hicks, our excitement rose at its peak: "Economics, if it is on the edge of the sciences is also on the edge of the history; facing both ways, it is in a key position" (Hicks, Causality in Economics, 1979, p. 4). So wrote John Hicks, one of the most respected economists in modern times. Besides, he paid special attention to the part of merchant played in the market economy. Theory and History—those two items should not be discussed separately, but rather are closely interconnected. Yes, before Hicks, there were very few predecessors including German philosopher-economist Karl Marx, who boldly

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attempted to combine theory and history into one, named the materialistic conception of history. In hindsight, Marx's classical doctrine, however, was too deterministic and even dogmatic to the modern mind: indeed, he completely failed to discuss the role of uncertainty and information in human behavior. It was a modern economist Hicks who had a very flexible mind to introduce those stochastic factors into the theory of economic history, thus escaping from dogmatic determinism wellillustrated by the "contradiction between production forces and human relations" or "the class struggle between the capitalist and the worker." Then wisely, Hicks pondered over the following question: "Where shall we start? There is a transformation which is antecedent to Marx's Rise of the Market and which, in terms of more recent economics, looks like being even more fundamental. This is the Rise of the Market, the Rise of the Exchange Economy" (Hicks, A Theory of Economic History , 1969, p. 7). According to Hicks, the key figure in the exchange economy is the merchants, wholesaler, or shopkeeper, who buys to sell again. We give wholehearted support to this view. As the Mercantile Economy grows, two trading centers are geographically more apart from each other, so that each center may have more comparative advantage in the collection of information. As Hicks has stressed, the trade between the seller and the buyer is not only advantageous for both merchants themselves, but also for the third party surrounding those people. In pre-modern Japan, the Ohmi merchants used to be very famous for engaging their trade on the basis of the "principle of three-way advantages": namely what is good for the seller and for the buyer should also be good for the society as well. We believe that such Ohmi principle is still alive in Japan today. In the above, we have argued our purpose or motivation for writing the present book. As saying goes, saying is one thing, but doing is another! So, it is necessary to write down the outline of the book here. Roughly speaking, this book consists of three parts. The first part contains the first two chapters, namely Chaps.1 and 2, dealing with Merchants in Historical Perspective. The second part contains the subsequent five chapters from Chaps. 3–7, turning to Information Exchange in Theoretical Perspective. The third part consists of the final two chapters, i.e., Chaps. 8 and 9, with a focus on Risk Aversion in Mathematical Perspective. We are now in a position to let the reader know the more detailed contents of this book. As we mentioned above, we follow the Hicks doctrine that History and Theory should be the inseparable parts of the whole structure named “the Theory of Economic History.” History should not separately be discussed from theory and vice versa. This book is presumably our ambitious attempt to integrate History and Theory into one entity. The first two chapters deal with History, and the remaining chapters are concerned with Theory, which first contains simple oligopoly models and gradually develops into more advanced and even mathematical contents. More specifically, Chap. 1 will intensively discuss and carefully compare the Liverpool Merchants of Britain and the Ohmi Merchants of Japan in a historical perspective. To the best of our knowledge, such comparison of Liverpool and Ohmi merchants has never been attempted in the academic world. It will be seen that those merchants were actively

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engaged in their respective Triangular Trade, producing their respective socioeconomic systems. Although those two merchants played very similar functions in the Exchange Economy, they were quite different entities as risk-managing players, reflecting basic cultural-historical differences between Britain and Japan. For instance, whereas ill-famed slave trade formed an integral part of the British triangular trade, such inhuman trade was nonexistent in Japanese history. Chapter 2 will adopt a historical approach to the role of merchants in the exchange economy. We will especially focus on the work of Werner Sombart, an almost forgotten German economist-historian. In the 1990s, however, we saw a remarkable comeback of Sombart, named the "Sombart Renaissance": in fact, his intensive work on the role of Capitalist Spirit played in the three stages of capitalism—namely, Early Capitalism, High Capitalism, and Late Capitalism—is gaining a deeper interest in recent times. We should add to say that the Ohmi merchants of Japan have long acted as lively players with such capitalist spirit a la Sombart. We can learn new lessons from old teachings of Sombart. From Chap. 3 and onward, we will successively pay our main attention to Theory, yet maintaining History as an integral background of Theory. Chapter 3 is expected to serve as an introductory guide of our rather bulky structure of Theory, also being hoped to make a good bridge between History and Theory. By working with simple equilibrium models of the industry and doing a sequence of comparative economic analyses, we intend to shed a new light on an important yet rather neglected area in the economics profession. Let us suppose that the demand side is subject to many changes and may be represented by a simple uniform distribution function with two parameters, i.e., mean and variance. Then, we can show that the entry of the informed distributor between the producer and the consumer would cause two opposing welfare effects: a negative Intermediation Effect and a positive Information Effect. If the degree of relative risk is large enough, then the Information Effect overpowers the Intermediation Effect. Therefore, the introduction of the distributor into the market economy will increase both producer and consumer surpluses: in other words, it will make all the parties better off. We believe that this theoretical result may endorse the principle of All-round Advantages in the Mercantile Economy, which has been repeatedly advocated by John Hicks' Theory of Economic History. The following three chapters taken together—Chaps. 4–6—can be regarded as a systematic investigation into “Information Exchange among Firms and their Welfare Implications.” Chapter 4 (or Part I) is concerned with the basic Dual Relations between the Cournot and Bertrand models. Chapter 5 (or Part II) begins to deal with the world of risk and uncertainty, with a discussion of the Cournot Duopoly with a Common Demand Risk as a starting point. It then deals with Other Types of Duopoly Models with a Common Risk. Chapter 6 (or Part III) discusses more complicated problems such as Private Risks and Oligopoly Models. The true motivation for writing such survey-like papers is to strive for a synthesis of the Economics of Imperfect Competition and the Economics of Imperfect Information. The problem at issue is how and to what extent information exchanges among firms influence the welfare of producers, consumers, and the whole society. In the real

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world, trade associations may be thought of as typical information-gathering mechanisms. Put it differently, we are asking the question of whether and to what extent the presence and development of trade associations contributes to the realization of “Principle of All-round Advantages” a la John Hicks. Here again, we may see a nice bridge built between Theory and History. It is in Chap. 7 that we turn our attention to the working and performance of Labor-Managed Firms in the Market Economy. Looking at the real world in a historical perspective, we understand that there have been a great variety of market economies. Presumably, at the one end of the spectrum of possible forms, there surely exists the typical American-type economy, which may be well-described by the traditional neoclassical profit-maximizing firm. Interestingly enough, however, nearly at the other end, there exists the once-admired Japanese-type economy, which seems to be different from the American-type economy in many respects. First of all, the Japanese firm can often be regarded as a large family, with the company president playing the role of a head of the family. Next, in Japan, decision-making and information flow are not exclusively monopolized by a selected number of top executives but are largely shared by all employees. As convincingly pointed out by Ryutaro Komiya, a leading Japanese economist, the typical Japanese firm seems to have many characteristics of the labor-managed firm as opposed to the neoclassical profit-maximizing firm. Since Japan has become a world economic power, it should be worthwhile to carefully compare the profit-maximizing firm and the labormanaged firm from an informational point of view. Here again, we can see a delicate relationship between Theory and History. When we discuss human behavior under risk and uncertainty, there appear to be two key concepts which play a very critical role. They are: risk aversion and expected utility. The purpose of the last two chapters of this book—Chaps. 8 and 9—is to scrutinize the relationship between these concepts, whence shedding a new light on the impact of risk and uncertainty on many economic activities. As K.J. Arrow, a great economist and Nobel Prize winner, convincingly remarked, individuals tend to display aversion to the taking of risks, and risk aversion in turn is an explanation for many observed phenomena to the economic world. Although this fact was well-recognized by many historians as shown in Chaps. 1 and 2, it is quite unfortunate that the operational theory of risk-averse oligopoly has been rather underdeveloped so far. One of the reasons for such underdevelopment is that we have to get into a mathematical jungle with no easy exit in sight. In Chap. 8, to get out of the mathematical jungle, we attempt to combine the constant-absolute-risk aversion function K.J. Arrow and J.W. Pratt, two great economists of the twentieth century, and the normal distribution function invented by K.F. Gauss, a mathematical genius of the nineteenth century: the resulting situation may be called the CARA-NORMAL case. We then have to invent a very powerful mathematical theorem for this specific yet important case and then apply it to the theory of risk-averse oligopoly. It is shown among other things that the comparative static results depend on the degree of risk aversion and the state of product differentiation. The final chapter, namely Chap. 9, continues to carry a very tough task of exploring the relationship between Risk Aversion and Information Exchange. In

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contrast to the situation of the last chapter, i.e., Chap. 8, in which the two firms face a Common Demand Risk, we now dare to tackle a more demanding situation under which each firm has to face its Own Cost Uncertainty. If firms display risk aversion and thus maximize the Expected Utility of Profit rather than the mere Expected Profit, then the exchange of cost information between them affects the Mean Values of outputs as well as their Variances. By employing a constant absolute risk aversion model, we are able to show the Variance Effect may sometimes overpower the Mean Effect, whence information sharing may possibly make firms even worse off. As our daily experience tell us, “going alone and ignorant” may sometimes be a better policy than “going together and informed.” In conclusion, History and Theory are inseparable components of the whole human behavior. History without Theory looks like a simple collection of unorganized data. Theory without History looks like a human body with a mere skeleton but no flesh and blood. As a perfect body requires bones as well as flesh and blood, so a perfect academic product requires Theory and History. The preparation of this book has been made possible by the Japanese Ministry of Education, Culture, Sports, Science and Technology through Grand-in-Aid for Scientific Research. Special thanks are due to the late Sir John Hicks (Oxford) and the late Professor Michio Morishima (Osaka, London) for their everlasting encouragement as well as the following institutions for their generous support: namely Shiga University, Toyo University, University of Tsukuba, Ryukoku University, Hiroshima University, Osaka University, Hiroshima University, Kobe University, New York University, University of Colorado, University of Pittsburgh, University of Rochester, and many other academic institutions and societies. We are deeply grateful to Professor Yoshiro Higano (Tsukuba), editor-in-chief of the series New Frontiers in Regional Science: Asian Perspectives, our generous guru Professor Hirotada Kono (Tsukuba), our good friends Professor Peter Nijkamp (Amsterdam) and Professor Makoto Tawada (Nagoya), and many other core members of the worldwide Regional Economic Association. And last but not least, we would like to say so many thanks to Mr. Yutaka Hirachi and other staff members of Springer Japan for their kind suggestions and heartwarming encouragement. Shiga‚ Japan Tokyo‚ Japan 1 December 2020

Yasuhiro Sakai Keisuke Sasaki

Contents

Part I 1 2

Liverpool Merchants Versus Ohmi Merchants: How and Why They Dealt with Risk and Insurance Differently . . . . . . . . . . . . . . .

3

The Role of Merchants in the Exchange Economy: A Historical Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

21

Part II 3 4

5

6 7

9

Information Exchange: Theoretical Perspective

A Theory of Information and Distribution: The Market Economy and Demand Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

43

Information Exchanges among Firms and Their Welfare Implications (Part I): The Dual Relations between the Cournot and Bertrand Models . . . . . . . . . . . . . . . . . . . . . . . . . . . .

67

Information Exchanges among Firms and Their Welfare Implications (Part II): Alternative Duopoly Models with Different Types of Risks . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Information Exchange among Firms and Their Welfare Implications (Part III): Private Risks and Oligopoly Models . . . . . .

109

The Profit-Maximizing and Labor-Managed Firms: A Unified Approach to the Role of Information . . . . . . . . . . . . . . .

139

Part III 8

Merchants: Historical Perspective

Risk Aversion: Mathematical Perspective

Risk Aversion and Information Exchange: The Constant Absolute Risk Aversion Function and its Applications . . . . . . . . . .

167

Information Sharing of Private Cost Information: An Application of the Cardano Cubic Formula . . . . . . . . . . . . . . . .

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Part I

Merchants: Historical Perspective

Chapter 1

Liverpool Merchants Versus Ohmi Merchants: How and Why They Dealt with Risk and Insurance Differently

Abstract The purpose of this paper is to intensively discuss and carefully compare the Liverpool merchants of Britain and the Ohmi merchants of Japan in a historical perspective. The question of much interest is how and why those two merchants dealt with risk and insurance differently. In his later years, John Hicks did a great contribution on the theory of economic history. He paid a special attention to the rise of the market in which the merchant played as the main actor of the history theater. According to the Hicks doctrine, the relation between theory and history should not be one-to-one, but rather flexible to a certain degree. Therefore, it would be quite interesting to carefully compare the Liverpool merchants of Britain and the Ohmi merchants of Japan. It will be seen that they were engaged in their respective triangular trade, producing their respective socioeconomic systems. In short, we have to take a pluralistic view in order to fully understand the concept of risk and insurance from the viewpoint of economic history. Keywords Liverpool merchants · Ohmi merchants · John Hicks · Risk · Insurance · Triangular trade · Slavery

1.1

The United Kingdom Versus Japan on Risk and Uncertainty: An Introduction

This paper intends to intensively discuss and carefully compare the Liverpool merchants of the United Kingdom and the Ohmi merchants of Japan in a historical perspective. The question of much interest is how and why those two merchants dealt with risk and insurance differently. The United Kingdom and Japan have many things in common. First of all, they are both typical island nations: the former is located off the western edge of the European Continent, and the latter off the eastern edge. Second, both nations are the constitutional monarchies which are “rare species” in the modern times. The British This chapter is largely written on the basis of Sakai (2018). © Springer Nature Singapore Pte Ltd. 2021 Y. Sakai, K. Sasaki, Information and Distribution, New Frontiers in Regional Science: Asian Perspectives 49, https://doi.org/10.1007/978-981-10-0101-7_1

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1 Liverpool Merchants Versus Ohmi Merchants: How and Why They Dealt with Risk. . .

people cheerfully sing the national anthem “God Save the Queen,” whereas the Japanese people sing the one “Long Reign the Emperor.” Third, both countries are the advanced nations which have long history and unique cultures, and are economically very strong and influential. Fourth, chivalry is not yet dead in the noble country of Union Jack, whereas the samurai spirits are still alive in the beautiful country of the Rising Sun. In an economic perspective, both the United Kingdom and Japan belong to the exclusive club of the Group of Seven. The two countries are proud of being “insurance superpowers.” It is surprising to see, however, that the ways people have dealt with risk and insurance are so different between the two. Since the British people tend to make their saving decision on the basis of cost and benefit, they purchase insurance as a mere item of portfolio selection. In contrast, the Japanese people regard insurance as a good mean of security against possible hazards: the way they buy life insurance is psychologically motivated rather than economically oriented. Although Japan today is accelerating the process of transition toward westernization and getting closer to Western Europe and the United America, the historical and cultural foundations of Japan remain the same as before. The unique culture that has been nurtured more than 1000 years cannot be changed by the transformation of the economic environment in recent 70 years or so. Kazuya Mizushima (1995), a leading authority on insurance, once remarked: If we take a close look at the Japanese insurance system from the viewpoint of the insurance thought which has been commonly shared by the Western people, then we have to reach the conclusion that ‘the insurance superpower’ does not necessarily mean ‘the insuranceadvanced nation’. More specifically, we must find a right answer to the question of whether and to what extent such ‘mismatch’ between culture and insurance exist in Japan. (Mizushima 1995, p. 3)

We think that Mizushima has posed us a very important problem. The question of whether and to what extent the Japanese insurance corresponds to the culture might be an interesting question. We wonder, however, if it is really a right question to ask. Mizushima prefers a variety of pluralistic views to a single definite viewpoint. If we adopt such a pluralistic view a la Mizushima, we may proceed to say that the correspondence between culture and insurance should not be one-to-one: many forms of insurances may be associated with one culture. In the present paper, we would like to focus on the role of merchants in the market economy and partially give an answer to the Mizushima problem aforementioned. More specifically, we pick up Liverpool merchants as a representative of the British system and Ohmi merchants as a good sample of the Japanese system. The problem of how and why they deal with risk and insurance differently is posed, and will be carefully examined. The contents of this paper are as follows. Section 1.2 will introduce and examine the view of John Hicks on the mercantile economy, on which the entire framework of our investigation here is built. Section 1.3 will study Liverpool merchants and their role in the classical triangular trade. The Zong events followed by the civil insurance trial will be a focal point of investigation. Section 1.4 will turn to Ohmi merchants and their role in the Japanese triangular trade. It will be seen that those

1.2 The Place of Economic History in the Work of John Hicks

5

merchants served well as main promoters of the mercantile economy in pre-modern Japan, and that their influence on the modern Japan remains a great deal. Section 1.5 aims to discuss and compare the unique insurance system adopted by Ohmi merchants and the modern one used by Liverpool merchants. Final concluding remarks will be made in the Sect. 1.6.

1.2 1.2.1

The Place of Economic History in the Work of John Hicks Two Great Economists: John Hicks and Michio Morishima

Michio Morishima (1923–2004) is perhaps the most famous economist Japan has ever produced after the Second World War. He once worked for the London School of Economics for the period 1970–1988 as the Sir John Hicks Professor of Economics. His personal and academic relation to John Hicks (1904–1989), a Nobel laureate and one of the greatest economists in the twentieth century, is quite interesting and illuminating, Throughout his long academic career, Morishima regarded Hicks as the mentor, thus constantly feeling great respect for the life and work of Hicks. In a small yet important booklet, Morishima (1993) once remarked: In his student days, Hicks dared to change his major from mathematics to economics. Within the field of economics, his study began with labor problems rather than pure theory. As a result, his research area became so wide and extensive, and published so many books in a great variety of fields. If I am allowed to say my favorite books among them, I would like to select the two books, A Theory of Economic History (1969) and A Market Theory of Money (1989). (Morishima 1993, p. 54)

In 1972, Hicks was awarded the Nobel memorial prize in economic science for his outstanding contribution to theoretical and welfare economics, which was successfully conducted in his much earlier work Value and Capital (1939). According to Morishima (1993), however, Hicks was not so impressed by the way in which the Nobel prize was given to him: in fact, his happiness would have been much greater if he had been awarded for his work on economic history rather than on abstract theory. In this connection, it is worthwhile to record his own words: It [the theoretical work of Hicks] was done a long time ago, and it was with mixed feelings that I found myself honored for that work, which I felt myself to have outgrown. (Hicks 1977, Preface and Survey, p. v)

It seems that Hicks was a many-sided person; at least what we have to do is to distinguish between the young Hicks and the old Hicks. Although the work of the young Hicks was well-represented by Value and Capital (1939), the old Hicks felt himself to have outgrown it. Indeed, Hicks in his later years placed a much higher value on his work on economic history than pure theory.

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1 Liverpool Merchants Versus Ohmi Merchants: How and Why They Dealt with Risk. . .

In short, those two great economists—Hicks and Morishima —gradually shifted their interests from theory to history when their ages advanced. More exactly, making a nice bridge between theory and history became the main target of their career goals. It is in line of such Hicks-Morishima tradition that we carefully conduct comparative studies of Liverpool and Ohmi merchants in a historical and theoretical perspective.

1.2.2

Hicks on the Mercantile Economy

Hicks has long been interested in economic history. Hicks has thought that the major function of economic history should be a forum in which economists and many other scholars including political scientists, lawyers, sociologists, and historians can meet and talk to one another. It is needless to say that the economic theory of Hicks completely differed from that of Karl Marx (1818–1883), a promoter of labor value theory and a noted socialist. The way in which Hicks dealt with economic history, however, had striking resemblance to the method of Marx. Hicks (1969) once remarked; It will be great deal nearer to the kind of thing that was attempted by Marx, who did take from his economics some general ideas which he applied to history, so that the pattern which he saw in history had some extra-historical support That is much more the kind of thing I want to try to do. (Hicks 1969, p. 2)

More than 100 years ago, Marx published Das Kapital (1867), a monumental work unifying theory and history. According to Hicks (1969), although there have been enormous developments in social sciences in those long years, so little should have been emerged. Marx may have been correct in his vision of logical processes at work in history. Added with the knowledge of fact and social logic which he did not possess, however, the modern mind should consider the nature of those processes in a distinctly different way. The Hicks doctrine mentioned above teaches us that theory and history do not necessarily have a one-to-one correspondence. It might be the case that different theories can adopt similar historical approaches. One of the nice examples for this is the delicate relation between Marx and Hicks. Hicks paid a special attention to the rise of the market or the one of the exchange economy. It is a great transformation which is antecedent to, and even more fundamental than, the rise of capitalism that was discussed with great energy by Marx. Remarkably, there should not be only one way of transformation: There are several possible ways by which we can historically deduce what must have occurred. If we are allowed to apply Hicks’ s approach to economic history, we would have a variety of ways of the rise of the market; namely, the British way, the Japanese way, the German, and so on. The core of the mercantile economy consists of a body of specialized traders engaged in external trade. The trading center is dependent on trade with the people

1.3 Liverpool Merchants and the Classical Triangular Trade

7

living far away from their local community. In other words, the mercantile economy as a whole depends on how the people inside it trade with the people outside it. According to Hicks (1969), the fact that one trading center has a different geographical location from another may give the former comparative advantage over the latter in the collection of information; by the trading of a locality with a center, these advantages can be utilized and the possible risks on the sides can significantly be reduced. The key figure in the mercantile economy is nothing but the merchant himself, wholesaler or shopkeeper, who buys in order to sell again. It is recalled that the first chapter of Hicks (1969) on history book began with a discussion on the relation between theory and history. The third chapter, which was regarded as the culmination of the book, lucidly discussed the rise of the market in which the merchant played a critical part. Unlike Marx, the industrial revolution followed by the emergence of capitalism did not occupy the central place in the Hicks framework. As was mentioned above, the relation between theory and history should not rigidly be one-to-one, but rather flexible enough. The sort of mechanical determinism à la Marx should not be applied here. In historical perspective, there have existed many kinds of merchants; different countries have different histories and cultures, with each country having produced their own merchants. Among those historically important merchants are the Liverpool merchants of Britain and the Ohmi merchants of Japan, whose comparison is expected to shed new light on the even bigger question of how and why Asians and Westerners think and act differently.1

1.3 1.3.1

Liverpool Merchants and the Classical Triangular Trade Liverpool as the Start and Goal of the Triangle

Liverpool is known as a major city and a metropolitan borough in northwest England. The city is proud of having a long history: it became a borough from 1207 and a city from 1880. It is really remarkable to see that the history of the city has two opposing sides, namely a bright side and a dark side. Let us begin our discussion with the bright side. Liverpool celebrated its 800th anniversary in 2007, and honorably held the European Capital of Culture title. Several areas of the city were designated the World Heritage Cite status by UNESCO in 2004. It is also nicknamed the World Capital City of Pop because it is the birthplace of the Beatles, a world renowned singer group. The popular song “Yesterday” was born in the port city Liverpool and later wide spread by ship or by air around the world.

1 For this point, see Nesbett (2003), a noted psychologist. For a general discussion on the economic thought of risk and uncertainty, see Sakai (2010).

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1 Liverpool Merchants Versus Ohmi Merchants: How and Why They Dealt with Risk. . .

We should remember, however, that Liverpool has a sad history as well. It has a big international port connected with any other place in the world. It is really such a geographically nice location that has contributed a great deal to its socioeconomic development and cultural diversity in many ways. Apart from the historic fact that it was originally the registered port of the tragic ocean liner Titanic, it has also been home to the oldest black African community in the country. In fact, Liverpool used to be the most important slaving port in Europe; its ships and merchants dominated the transatlantic slave trade in the second half of the eighteenth century. The world history tells us that around three-quarters of all European slaving ships in this period left from the port of Liverpool. It is a very sad historical fact that the mighty British Empire which once controlled the seven major seas and oceans in the world had long thrived on the foundation of the slave trade and many colonies overseas. In a neatly written book aforementioned, Hicks (1969) discussed the slavery system very seriously, and lamented as follows: The darkest episodes in the history of mercantile slavery (putting aside, as before, the horrors of slave-catching which always apply) are a matter of the large-scale employment of slaves: the employment of slaves in gangs, on plantation (such as the Roman latifundia, and the cotton and sugar plantations of Americas and the West Indies), in mines, and as galley-slaves on ships. (Hicks 1969, p. 126)

Although there have been so many books and articles on economic history, it is regrettable that so few writings on the slave trade have been available in the economics literature. In short, the word “the slave trade” was once regarded as a sort of taboo in the academic world. Hicks’ warm heart and professionalism to seriously discuss such an important topic should greatly be appreciated.

1.3.2

The Classic Triangular Trade: The Three Passages in the Atlantic Ocean

In general, the triangular trade is a historical term which indicates the trade among three ports, regions, or countries. The particular routes under question were historically formed by a number of weather factors such as currents and winds during the age of sailing ships. Among those important routes were the classical triangular trade starting with and getting back to Liverpool. The classical triangular trade in the eighteenth century is depicted in Fig. 1.1. In general, any triangle is supposed to have three points and three sides. In this specific case, Britain (Liverpool, Bristol), West Africa (Slave Coast), and the Caribbean (Jamaica) corresponded to those three points, whereas the Western Passage, the Middle Passage, and the Eastward Passage constituted those three sides. Let us explain such a historical triangle in greater details. As was shown in Fig. 1.1, the first leg of the triangle was the Westward Passage, sailing from a British port (Liverpool or Bristol) to West Africa (Slave Coast). Ships in this leg mainly

1.3 Liverpool Merchants and the Classical Triangular Trade

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Fig. 1.1 The classical triangular trade on the Atlantic

carried textiles, guns, and many other manufactured goods. When the ships reached West Africa, their cargos were sold for captured slaves. Those slaves were often the items for nonlife insurance because they were usually regarded as ordinary goods rather than human beings.2 On the second leg, many ships made painful journeys on the famous (or infamous) Middle Passage or the Transatlantic Passage from West Africa to the Caribbean (Jamaica). Understandably, so many slaves died of physical and mental diseases in the crowded holds of the slave ships. Once the ships arrived at the Caribbean or the West Indies, enslaved survivors were mainly sold for American plantations. The popular song Old Black Joe was composed by Stephen Foster (1826–1864), a famous American musician. It is said that the song’s sadness and melancholy indicated well the hard lives of slaves in an American plantation. Let us record here the first paragraph of Foster (1853): Gone are the days when my heart was young and gay, Gone are my friends from the cotton fields away, Gone from the earth to a better land I know, I hear their gentle voices calling ‘Old Black Joe’. (Foster 1853, song)

The slave trade via the Middle Passage gave Liverpool and many other European merchants a handsome amount of profits. It was little exaggeration to say that the mighty British Empire thrived on the transatlantic slave trade. Daniel Defoe (1719) was then very famous as the author of the popular novel The Life and Strange Surprising Adventures of Robinson Crusoe (1719). It is noted here that he was also the author of the interesting commerce book A Plan of the English Commerce (1728). In fact, in this enlightening book, Defoe (1728) described the way how the huge income and wealth was brought about to Britain by the slave trade: The trade carried on here, whether by the English, or other European nations consists in but three capital articles, viz. slaves, teeth, and gold; a very gainful advantageous commerce, especially as it was once carried on, when these were all purchased at low rates from the savages; and even those low rates paid in trifles, and toys, such as knives and scissors, kettles and clouts, glass beads, and cowries, things of the smallest value, and as we say next to nothing; but even this part of the trade is abated in its goodness since by the strife and envy

2 Kunta Kinte was one of the most famous slaves; he was described by Haley (1976), an American author, as the slave who bravely fought back. Once known as James Island in West Africa, Kunta Kinte Island was then a holding ground for captured slaves before shipping to America via the Middle Passage.

10

1 Liverpool Merchants Versus Ohmi Merchants: How and Why They Dealt with Risk. . . among the traders, we have had the folly to instruct the savages in the value of their own goods, and inform them of the cheapness of our own; endeavoring to supplant one another, by under selling and overbidding, by which we have taught the negroes to supplant both, by holding up the price of their own productions, and running down the rates of what we carry them for sale. (Defoe 1728, p. 329)

This is surely a very long sentence containing many semicolons. With his high spirits and eloquence, Defoe used racist words such as slaves and savages many times, and honestly said that the local African people were always treated extremely unfairly in their trade with the British merchants. Therefore, the Middle Passage served as the very core of the classical triangular trade, and thus greatly contributed to the prosperity of the mighty British Empire. The third leg of the triangle consisted of the return voyage to Liverpool or home ports in Britain from the Caribbean. The main export items were sugar, tobacco, and cotton. The ships got back to the Old World from the New World, thus completing the classical triangular trade.

1.3.3

The Zong Events and the Civil Insurance Trial

In 26 November 2006, Tristan Hunt (2006), a leading historian, wrote a very impressive article in the Guardian, a famous British newspaper. The article had the very appealing title: “Slavery: the long road to our historic ‘sorrow’,” The historical “sorrow” he referred to was associated with the inhuman slave trade and the strange civil insurance trial named “the Zong massacre case.” In our opinion, the historic “sorrow” was actually an understatement; it should correctly have been stated the historic “shame.” More than 200 years ago (more exactly, on 29 November 1781 and the following days), the Zong massacre took place. Captain Luke Collingwood of the slave ship Zong, which was then carrying 440 slaves from West Africa to Jamaica, coldheartedly ordered the crew to threw 133 slaves overboard into the sea in order to save the rest. The ship was owned by the Gregson syndicate, based on Liverpool, engaged in the Atlantic slave trade. Remarkably, the syndicate had engaged in marine insurance contract for the slaves onboard as cargo items. Unfortunately, the voyage of the Zong was not going well because of strong winds and poor navigation, and the cargo was beginning to deteriorate. After the ship struggled to arrive at Jamaica after a difficult and delayed voyage, the insured (the Gregson syndicate) made a huge claim to the insurer (the Gilbert insurance company) for the huge loss of slaves as the cargo. Understandably, the insurer refused to pay; and in 1783, the damage claim ended up in a London court, not as a murder trial but as a civil insurance case. After severe legal battles in the court, the presiding judge found in favor of the insured against the insurer. The judge had the general opinion that there should be some circumstances in which the deliberate killing of slaves was quite reasonable and thus legal, and specifically decided that the Zong case represented one of those circumstances: therefore, the insurer should pay the loss of slaves and its related

1.3 Liverpool Merchants and the Classical Triangular Trade

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damage to the insured. After such first trial, however, another ruling in the next court was announced against the slave-trading company; a set of new evidences were introduced to suggest that Captain Collingwood and his crew of the Zong were also at fault. Because of the harsh legal dispute, the detailed reports of the Zong events received increasing publicity, thus gradually stimulating the abolition of the slave trade in the late eighteenth century and the early nineteenth century. The movements eventually resulted in the British Slave Trade Act 1807, which announced the formal and final act for the abolition of the notorious African slave trade. It is in the memorial year of 2007 that marked the bicentenary of the British Slave Trade Act. As Hunt (2006) clearly stated, the 200th anniversary of slavery abolition should be a moment of pride as well as guilt. We can learn some important lessons from the Zong events and the following civil insurance trial. First, we would like to point out that seas and oceans are always very dangerous places to live; we have to take account of the physical and economical loss of cargo onboard as well as the damages caused by stranded ships or piracy. In short, whenever we are onboard, we are subject to a variety of risks which may not necessarily be measurable. Second, we have to learn the usefulness and limitations of insurances as the means of risk management. As the Zong case has taught us, it might be the case that both the insured and insurer are engaged in tiresome legal battles. When insurance premiums become abnormally high, many persons tend to escape from them. In his famous book, Adam Smith (1776) once observed: Moderate as the premium of insurance commonly is, many people despise the risk too much to care to pay it. Taking the whole kingdom at an average, nineteen houses in twenty or rather, perhaps, ninety-nine in a hundred, are not insured from fire. Sea risk is more alarming to the greater part of people, and the proportion of ships insured for those not insured is much greater. Many sail, however, at all seasons, and even in time of war, without any insurance. (Smith 1776, pp. 108–109)

Smith concluded that such neglect of insurance on houses and ships was not the result of cool calculation, but rather the one of mere thoughtless rashness and presumptuous contempt of the risk. Third, we have to respect high moral in economic activities. As we have said repeatedly, Liverpool once served as a major port city in the infamous slave trade, thus greatly contributing to the prosperity of the mighty British Empire. Someone might say that money and moral are in trade-off relations; in other words, both are seldom compatible with each other. In our opinion, however, the question of whether and to what extent people combine efficiency with equity depends on the culture of a country under question. In contrast to the Western culture, “the harmony of mankind and the coexistence with the nature” have constituted the core of the Eastern philosophy. In the light of those lessons we have learned so far, it would be a question of the greatest importance to carefully compare the Liverpool merchants of Britain and the Ohmi merchants of Japan in historical and cultural perspectives.

1 Liverpool Merchants Versus Ohmi Merchants: How and Why They Dealt with Risk. . .

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1.4 1.4.1

The Ohmi Merchants of Japan and Their Triangular Trade Ohmi Merchants: Their Role in the Japanese Mercantile Economy

Ohmi merchants were those merchants who were born in the Ohmi region encircling Mother Lake Biwa, Central Japan, and extensively engaged in distribution activities across the nation, even extending to Hokkaido, an almost foreign land in pre-modern Japan. Their business philosophy could be characterized as the principle of sampoyoshi or three-way advantage: the trade must be advantageous not only for sellers and buyers but also for the society as a whole. According to Eiichiro Ogura (1991), a leading authority on Ohmi merchants, those merchants must be designated “the National Cultural Asset” of Japan.3 Although the name of Ohmi merchants and the principle of three-way advantage have been very famous in Japan, it is quite unfortunate that they have rarely been heard and written outside Japan. Correcting such asymmetric treatment between the East and the West is one of our motivations of writing this paper. As mentioned above, Hicks (1969) paid special attention to the rise of the market in which merchants played as main actors. While he found much interest in Europe and the Mediterranean, his reference to Japan and the Sea of Japan seemed to less than satisfactory. For instance, he once remarked: The city state of Europe is a gift of the Mediterranean. The Mediterranean has been outstanding as a highway on contract, between countries of widely different productive capacities; further, it is rich in pockets and crannies, islands, promontories, and valleys. . . Asia has little to offer that is at all comparable. The Inland Sea of Japan is tiny in comparison with the Mediterranean (it is not even as large as the Aegean); the districts that surrounded it do not differ in natural resources as the Mediterranean countries do. (Hicks 1969, pp. 38–39)

Considering geographical and historical differences between Britain and Japan, we should not blame much on Hicks for this point. We would like to point out, however, that Asia has much to offer that is comparable to Europe; in fact, Ohmi merchants have engaged in extensive trading activities around the Sea of Japan and the Northwest Pacific, which is large enough in comparison with the Aegean or even the Mediterranean. Pre-modern Japan was proud of having three capitals. They were Kyoto as the imperial capital where the emperor (the formal head of the nation) led the ceremonial function, Osaka as the commercial capital in which merchants acted as the center of

3 For the principle of Sampo-yoshi and its related history of Ohmi merchants, see Ogura (1980). It is remarkable to see that in characterizing the merit of the mercantile economy, Hicks himself referred to the principle of all-round advantage, meaning the advantage of all parties; the merchants themselves and the “surrounding” peoples with whom they trade (see Hicks 1969, p. 51). It seems that those two principles, namely Sampo-yoshi and all-round advantage, are amazingly similar concepts.

1.4 The Ohmi Merchants of Japan and Their Triangular Trade

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business and commerce, and Yedo (modern Tokyo) as the political capital where the Shogun (the top of the Samurai class) carried out the executive function. Fortunately, the Ohmi district which was the birth place of Ohmi merchants was located in the center of Japan, thus being in the neighborhood of Kyoto and Osaka and easily connected with Tokyo via a number of main roads. While the Ohmi district contained Lake Biwa, the biggest lake of Japan, it was not suitable for agriculture and destined to find something else to do for its survival. So many local people had to engage in distribution activities; they dared to take pains to buy and sell many goods in Tokyo and many other remote areas across the nation. They had their own philosophy of business which could easily be summarized as the principle of Sampo-yoshi, or the one of three-way advantage. Let us quote some of famous family rules of conducting business: The purpose of the merchant is to meet the supply and demand of all goods, so that satisfying the needs of all the people. If the merchant forgets this philosophy and seeks only his private interest, then he acts against the teaching of God and will eventually destroy himself. . . . Doing business may be regarded as an act of worship for Buddha. It should always be respected in many ways; it is good not only for the seller and the buyer, but also good for the society, and may appeal to the compassionate heart of Buddha. The true virtue lies in benevolence and hard work. (Ogura 1991, pp. 12–13)

It is clear that the Ohmi merchants of Japan distinctly differ from the Liverpool merchants aforementioned. First of all, the slave trade has never existed in Japan. Second, the philosophy of Ohmi merchants seemed more appealing to the modern mind than Liverpool merchants. Specifically, the principle of Sampo-yoshi or the three-way advantage is still alive today, and will easily be applicable in foreign countries. Third, in Japan, doing business activities was closely related to doing religious activities, perhaps more so than in Europe. In fact, Ohmi merchants were very religious persons, and took much care of expensive Buddhist altars. In spite of those differences between Ohmi and Liverpool merchants, it should be pointed out that they have one important thing in common. As Liverpool merchants sustained the British economy by means of the classical triangular trade, so did Ohmi merchants contribute to the development of Japanese economy by actively promoting their own triangular trade.

1.4.2

The Japanese Triangular Trade

As in the classical triangular trade of Britain, the Japanese triangular trade consisted of three regions and three passages connecting any two of the passages. Let us take a close look at Fig. 1.2. Those three regions were Kamigata (Kyoto and Osaka), Yedo (modern Tokyo), and Yezo (modern Hokkaido and Aomori), whereas those three passages were the Middle or East-West Passage, the Eastward Passage, and the Westward Passage. Among those three passages, the Middle Passage connecting the two main regions (Kamigata and Yedo) were the most important one. The Ohmi district was

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1 Liverpool Merchants Versus Ohmi Merchants: How and Why They Dealt with Risk. . .

Fig. 1.2 Ohmi merchants and the Japanese triangular trade

located between those two. In order to show what a critical position Ohmi once occupied in pre-modern Japan, let us write down the following witty remark: The man who controls Ohmi may control All Japan. (folklore in pre-modern Japan)

In Fig. 1.2, there are two different kinds of lines or curves. Each solid line or curve indicates the land/lake route on which people have to travel on foot or take a small boat on the lake, whereas each dotted line or curve shows the sea route on which people have to charter a large-scale cargo ship. In old days, the first leg started from Kyoto or Osaka and took a long walk toward Tokyo. Concerning such Middle Passage, Ohmi merchants had the two options, the Naka Road or the Tokai Road. They preferred the Naka to the Tokai because the former promised travelers a bit longer yet more secure route. Ohmi merchants carried textiles, medicines, sake barrels, and mosquito nets to Tokyo, bringing back silk, seaweed, and dye materials to Kyoto and Osaka. In later days, however, a new and more efficient sea route by special Higaki cargo ships between Osaka and Tokyo was adopted by many merchants. It is noted that the sea route was largely more secure and more damage-free than the troublesome land route. In feudal times, Yezo (modern Hokkaido and Aomori) was thought of as a very remote area from the center of Japan, being so far away either from Kyoto or from Yedo (modern Tokyo). In spite of such geographical disadvantage, however, Yezo attracted so many merchants from Central Japan, especially Ohmi merchants, since it produced a great variety of marine products such as sermons, cods, crabs, kelps, and many other seaweeds. The Westward Passage from Hokkaido to Kyoto and Osaka was regarded as a passage of “high risk and high return.” Although the sea of Japan in winter season was very stormy and caused so many wrecks, those courageous seamen who dreamed the dream of making a fortune at one stroke quick dared to take on special Kitamae cargo ships. There were basically two sub-routes—the old sea-land combined route, with Tsuruga port or Obama port being as a transfer junction, and the new sea-only route through Shimono-seki port. In spite of the fact the new route was more roundabout than the old one, it guaranteed merchants the minimum loss of cargo in transit, thereby gradually becoming a preferred route. Likewise, the Eastward Passage from Hokkaido to Tokyo via the Northwest Pacific was also almost as dangerous as the Westward Passage. After all, Kyoto, Osaka, and Tokyo were the three greatest places for consumption of seafood. If

1.5 Alternative Ways of Risk Sharing: The East Versus the West

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anyone did a successful voyage from Hokkaido to those giant cities, he would have become a very rich man in one night.4 The modern mind might think that such a voyage via the Westward or Eastward Passage would have offered us a nice subject of scientific risk management. Someone with cool head might invent a special marine insurance to take care of this matter. The historical fact taught us, however, that the western style of insurance system in its strict sense did not exit in pre-modern Japan. Fortunately, the history did not end here. A kind of risk sharing whose function was similar to modern insurance was invented by a group of cargo ship users on the Westward and Eastward Passage of the Japanese trade triangle. Among those famous users were Ohmi merchants. In a rather small yet very influential book, Ogura (1980) once remarked: Big Ohmi merchants traded so many textiles, with the annual total amounting to several hundred thousand tons. They utilized as many transportation means as they could; they made efficient use of ships, boats, oxcarts, wagons and like. . . . Unfortunately, they were involved in so many accidents in sea traffic. So, they determined to unite together as a solid group. One of the results of such group activities was what they called kaijo-tsumikin, namely the marine reserve fund. This was nothing but the beginning of modern marine insurance system. (Ogura 1980, pp. 13–14)

We can interpret the marine reserve fund as the fund into which the same group members are asked to annually or monthly offer a fixed amount of money in preparation for marine accidents and contingencies. When a certain accident such as ship damages and/or cargo deterioration happens, a certain amount of money is to be paid out of the fund to the victim. It is in this sense that the fund can be thought of a reasonable way of risk sharing or risk spreading. We have no doubt that Ogura’s remark aforementioned interests us very much and surely requires further investigation. In the next section, we will explore the question of whether and to what extent Ogura’s historical remark is correct in an analytical framework.

1.5 1.5.1

Alternative Ways of Risk Sharing: The East Versus the West The Japanese Way of Risk Sharing: The Marine Reserve Fund

As mentioned above, in pre-modern Japan, the marine insurance system which is now commonly available in the world did not develop well. We would like to point out, however, that its nice alternative has grown among Ohmi merchants. Such

4

For a detailed history of Kitamae cargo ships, see Kato (2003).

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1 Liverpool Merchants Versus Ohmi Merchants: How and Why They Dealt with Risk. . .

Fig. 1.3 The Japanese way of risk sharing via the marine reserve fund: the joint council of representatives

alternative system was called the marine reserve fund. It is high time to explore its working and performance in an analytical framework. For the sake of simplicity, let us assume that the marine reserve fund consists of five trade units; namely, units 1, 2, 3, 4, and 5. The fund members are not necessarily individuals, but rather trading units. For instance, in pre-modern times, a typical shipping council in Osaka and Tokyo in pre-modern times was well known as a strong organization of many knit units with togetherness and solidarity. In Fig. 1.3, five trade units are plotted in the shape of pentagon. A solid line there indicates strong ties between the two units. In fact, those units share the same destiny, so that they think and act under the same banner. In the case of sea accidents, they are always ready to share the risks. Under such fellow feelings, it would be a very nice idea that each trade unit selects its own representative so that the joint council of those representatives may determine their joint duty as well as the allocation of risk sharing for each unit. In Fig. 1.3, such commitment as an entire group is shown by five dotted lines connecting the joint council at the center with the five units plotted at the periphery. As was seen in Kato (2003), the transportation of a large amount of goods by the Kitamae cargo ships was always difficult, and sometimes extremely dangerous, being involved in many serious accidents such as collision, wreck, pirate, loss of control, cargo damage, sick sailors, and so on. The introduction of special marine reserve fund into the hard voyage was certainly a very good device of risk sharing system. In a historical perspective, such special fund was thought of as a precedent of modern marine insurance; it thus acted as a very effective mediator for the modern capitalist system. It should be noticed that it based on very strong human ties among the member units. In the case of contingencies, it offered not only a pecuniary aid to the victim, but also a material and moral support in an extensive way.

1.5 Alternative Ways of Risk Sharing: The East Versus the West

17

We must remember that every coin has both sides—a bright side and a dark side. While the marine reserve fund worked so well in a pre-modern society, it had some limitations as well. First of all, since the size of the participating member units was limited, it could not handle well the accident which was very large. Second, the joint council of representatives was not necessarily operated democratically, possibly being controlled by a greedy big boss. Presumably, one of the ways of correcting those limitations was to invent a new large-scale insurance company in which the insured and the insurer were economically and morally independent of each other. According to Ogura (1980, 1991), the merchants of eighteenth century Ohmi already adopted the modern way of book keeping by double entry. Besides, as Hicks (1969) himself learned from Crawcour (1961), the merchants of the seventeenth century Osaka developed the very advanced mercantile dealings such as the establishment of future dealings. Those historical facts clearly show that Japan developed well a modern way of risk management and accounting before it opened its door to the West in the 1860s. In short, there should be many ways of risk sharing. The Japanese way is certainly one of them. The Western way is another, and will be discussed in the following subsection.

1.5.2

The Western Way of Risk Sharing: The Presence of Independent Insurers

Let us turn our attention to another way of risk sharing, namely a modern way of dealing with risk and insurance.5 As the saying goes, seeing is believing. Let us have a careful look at Fig. 1.4. What it makes different from the previous figure or Fig. 1.3 is the presence of independent insurers occupying the center position. As before, there are five trade units—units 1, 2, 3, 4, and 5. The relation between any pair of units is no longer direct and solid, but rather indirect as a dotted line between them may show. The units are only legally related with each other. They may not know each other, but they make the insurance contract with the independent insurer who is supposed to have information of all the insured. Comparison of Figs. 1.3 and 1.4 teaches us that the relation between a solid line and a dotted line should be just reversed; a solid line in Fig. 1.3 becomes a dotted line in Fig. 1.4, and vice versa. Evidently, this is due to the fact that the center circle is now occupied by the independent insurer, not by the joint council of representatives. The modern insurance system distinguishes itself from the pre-modern system of risk sharing by joint reserve fund in several aspects. First, the insurer as an independent entity plays the role of main actor with professional ability and determination. Second, the scale of insurance system per se may become very large.

5 Detailed discussions on risk, uncertainty, and information can be found in Sakai (1982, 2010). For the working and limitations of modern insurance system, also see Dionne (2000).

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1 Liverpool Merchants Versus Ohmi Merchants: How and Why They Dealt with Risk. . .

Fig. 1.4 The Western way of risk sharing: the presence of the independent insurer

Third, its operation must be done on the basis of mathematical probability. Apparently, so far so good. We must admit, however, that any coin has both sides; a positive side may easily be turned over to a negative one. In fact, in contrast to the traditional risk sharing system, we note that the good human factors including mutual trust and solidarity may become so weakened that the bad phenomena such as moral hazard and adverse selection may emerge and gain in strength. In short, a seemingly good medicine may produce unexpectedly bad side effects. The modern insurance system seems to have solid foundations. As the history teaches us, however, once the system cracks a bit, it might eventually degenerate into a white elephant.

1.6

Hicks on Economics as the Edges of Sciences and History: Concluding Remarks

In this chapter, we have intensively discussed and carefully compared the Liverpool merchants of Britain and the Ohmi merchants of Japan in both historical and analytical perspectives. Both merchants had at least one thing in common since they took advantage of their respective triangular trade, thus playing the role of main promoters in the market economy. It is noted, however, that they were different kinds of characters historically and culturally. First of all, Liverpool had a very sad past; it used to be the most important slaving port in Europe. In contrast, Ohmi was famous of the birth place of the philosophy of “Sampo-yoshi” or three-way advantages, which meant that the trade must be good for the seller, good for the buyer, and also good for the society. Second, while

References

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Liverpool developed the modern system of insurance on the basis of relationship between the insurer and the insured, Ohmi was content to invent a more conventional system of risk sharing by reserve fund donators. In our opinion, those differences between the two originated in historical and cultural factors. We cannot say that one culture is better than another. Likewise, we should not attempt to discuss the advantage of one economic system over another. After all, diversity really matters! Hicks (1969) has argued that economics is on the edge of sciences, and also on the edge of history; facing both ways, it is a key position. Putting it differently, economics may be regarded as the intersection of sciences and history. Provided the same scientific reasoning, different histories and different cultures may result in different economies and different risk sharing systems. It is in this sense that the diversity of economic histories has sustained until today and will continue tomorrow.6 Needless to say, there remain so much areas left untouched in this paper. Although we focused on the two types of merchants, Liverpool and Ohmi, there should be many other merchants in the West and the East. Besides, many types of risk management other than insurance and reserve fund have existed and worked well in economic histories. We believe that this paper may serve well as a good starting point for further research.

References Crawcour ES (1961) Development of a credit system in 17th century Japan. J Econ Hist 21 (3):342–360 Defoe D (1719) The life and strange surprising adventures of Robinson Crusoe. Charles Rivington, London Defoe D (1728) A plan of the English commerce being a complete prospect of the trade of this nation, as well the home trade as the foreign. Charles Rivington, London Dionne G (ed) (2000) Handbook of insurance. Kluwer Academic Publishers, Norwell Foster S (1853) Old black Joe. Firth & Pond & Company, New York Haley A (1976) Roots: the saga of an American family. Doubleday, New York Hicks J (1939) Value and capital. Oxford University Press, Oxford Hicks J (1969) A theory of economic history. Oxford University Press, Oxford Hicks J (1977) Economic perspectives: further essays on money and growth. Oxford University Press, Oxford Hicks J (1989) A market theory of money. Oxford University Press, Oxford Hunt T (2006) Slavery: the long road to our historic ‘sorrow’. The Guardian, London Kato T (2003) Kitamae cargo ships: their roles as general trading companies (in Japanese). Mumyo Publishers, Akita Marx K (1867) Das Kapital. Felix Meiner Verlag, London

6 In a separate paper, Sakai (2016) intensively discussed the view of Hicks on the relation between sciences and history. Also see Sakai (2019), Chap. 3, pp. 54–55.

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1 Liverpool Merchants Versus Ohmi Merchants: How and Why They Dealt with Risk. . .

Mizushima K (ed) (1995) Insurance culture: risk and the Japanese people (in Japanese). Chikura Publishers, Tokyo Morishima M (1993) Modern economics as economic thought (in Japanese). NHK Publishers, Tokyo Nesbett RE (2003) The geography of thought: how Asians and Westerners think differently . . . and why. Free Press, New York Ogura E (1980) Ohmi merchants: their origins and developments (in Japanese). Nihon Keizai Publishers, Tokyo Ogura E (1991) The ideas and thought of Ohmi merchants: important rules of the family business (in Japanese). Sunrise Publishers, Hikone Sakai Y (1982) The economics of uncertainty (in Japanese). Yuhikaku Publishers, Tokyo Sakai Y (2010) Economic thought of risk and information (in Japanese). Minerva Publishers, Kyoto Sakai Y (2016) J. M. Keynes on probability versus F.H. Knight on uncertainty: reflections on the miracle year of 1921. Evol Inst Econ Rev 13(1):1–21 Sakai Y (2018) Liverpool merchants versus Ohmi merchants: how and why they dealt with risk and insurance differently. Asia Pac J Reg Sci 2(1):15–33 Sakai Y (2019) J.M. Keynes versus F.H. Knight: risk, probability, and uncertainty. Springer, Singapore Smith A (1776) An inquiry into the nature and causes of the wealth of nations, the first Tuttle edition, 1979. Oxford University Press, New York

Chapter 2

The Role of Merchants in the Exchange Economy: A Historical Perspective

Abstract This paper aims to discuss the relationship between economic theory and market economy from a new historical angle. Historically speaking, Werner Sombart seems to be a man in paradox. Although he was once a famous professor at Berlin, he became an almost forgotten man after the Second World War. In the 1990s, however, we saw a remarkable comeback of Sombart, named the Sombart Renaissance; his work on the role of capitalist spirit played in the three stages of capitalism is now worth serious investigation. In contrast, John Hicks has mainly been regarded as an important theoretician of general equilibrium and welfare, but his later work on economic history is also worthy of vital consideration. Hicks pays special attention to the role of merchant played in the exchange economy. By comparing the works of Sombart and Hicks in many ways, we can shed new light on the immortal problem of the relationship between Theory and History. We strongly believe that Ohmi merchants of Japan give us very good examples of the Hicks-type mercantile economy. Hicks has emphasized that the trade should comply with the “principle of all-round advantage.” Certainly, it corresponds well to the “principle of sampoyoshi” obeyed by Ohmi merchants: namely, the principle of being good for a seller, good for a buyer, and good for the society.” We also would like to add that Ohmi merchants acted as lively players with capitalistic spirits a la Sombart. In short, we can learn a set of new lessons from the old teachings of Sombart and Hicks. Keywords Theory and history. W. Sombart · Capitalist spirit · John Hicks · Exchange economy · Ohmi merchant · Principle of all-round advantage · This chapter is a completely revised paper of Sakai (2019).

2.1

John Hicks on Theory and History: An Introduction

The purpose of this paper is to systematically discuss the role of the merchants in the exchange economy from a historical perspective. In particular, Werner Sombart and John Hicks will be taken out as two giants in studies of such relationship, and will critically be evaluated and carefully compared from various angles.

© Springer Nature Singapore Pte Ltd. 2021 Y. Sakai, K. Sasaki, Information and Distribution, New Frontiers in Regional Science: Asian Perspectives 49, https://doi.org/10.1007/978-981-10-0101-7_2

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John Hicks (1904–1989) was one of the most outstanding economists in the twentieth century. While he led a scholastic life somehow detached from the real world, he was definitely a man with wide knowledge and deep insight. While he was not an economic historian in its strict sense, he had long been interested in economic history. In fact, in his young days when he was working for a university in South Africa, he lectured on English medieval economic history. Back in the United Kingdom, Hicks (1941) changed his subject to labor economics, thus publishing a nicely written book The Theory of Wages (1941), in which useful theoretical tools including the concept of the elasticity of substitution were invented and applied. Just after the Second World War, Hicks (1946) published a masterly theoretical work Value and Capital (1946), which was widely regarded as one of landmarks in economic theory in the twentieth century.1 Even after Hicks succeeded in establishing himself as a world famous theorist, he seemed to never forget his “first love” for history. After all, the first love of a young man would be remembered until his death! In 1969, the year in which I myself was a graduate student majored in mathematical economics at the University of Rochester, Hicks decided to publish “a small book on a large subject―an enormously large subject” (Hicks, 1969, p. 1). It had a modest yet interesting title A Theory of Economic History. Ironically enough, in 1972, the year when I began to teach general equilibrium theory at the University of Pittsburgh just after receiving a Ph. D. from Rochester, he was given a Nobel Economic Science Prize for his classical work on general equilibrium and welfare economics. To tell the truth, he was not so happy to receive the Nobel Prize for his old subject of general equilibrium rather than for his new field of economic history. For this point, he once remarked: They gave me a Nobel prize (in 1972) for my work on ‘general equilibrium and welfare economics,’ no doubt referring to Value and Capital (1939) and to the papers onConsumers’ Surplus which I wrote soon after that date. ... But it was done a long time ago, and it was with mixed feelings that I found myself honored for that work, which I felt myself to have outgrown (Hicks 1977, Preface, p. v). If I am allowed to simplify the matter, the New Hicks was awarded the honorable Nobel Prize for the past work by the Old Hicks. So, a kind of mixed feeling derived from such a mismatch seemed to be occurred in his mind. The late Professor Michio Morishima (1923–2004) was one of the greatest Japanese economists after the Second World War. We had great respect for Morishima, who in turn had great respect for Hicks. Morishima once remarked: When I read A Theory of Economic History, I asked Professor Hicks, “Would you like to continue such a history work a la Max Weber from now on?” Taking a pause, he replied to me, “Well, I would not think so.” After several days, however, he confessed his honest opinion, “If I was given a Nobel Prize for my recent work on Economic History rather than Pure Theory, I surely would have felt much happier.”

1 For a detailed discussion on the life and work of Sombart, see Backhaus (1996a), Volume I, Introduction, pp. 13–18.

2.1 John Hicks on Theory and History: An Introduction

23

This may clearly demonstrate that he himself evaluated his work on History much higher than the one on Theory (Morishima 1994, p. 74). Personally speaking, Sakai has ever met Professor John Hicks on several occasions. Sakai’s last and most impressive meeting with him occurred in Summer 1988, when Sakai was in Bologna, Italy to present his technical papers first at the World Congress of Econometric Society and then at the Annual Meeting of the European Economics Society held at the University of Bologna, one of the oldest universities in the world. We should add that the Special Memorial Conference in Honor of John Hicks was also organized at a separate place in Bologna. Very fortunately, Sakai was also invited as a special guest at the Hicks memorial conference because of the kind invitation of the late Professor Hirofumi Uzawa (1928–2014), another great Japanese economist. When Sakai met with John Hicks at the conference, he was very old and used a wheelchair. Yet he looked lively and always in good spirits. One year after the conference was over, however, Sakai was informed that Hicks suddenly passed away. So, it seemed that Sakai was one of the last persons who could talk to Professor Hicks in an academic meeting. In retrospect, Sakai left his heart in the Hicks Conference aforementioned. Still keeping it in our fond memory, we would like to write this paper in order to carefully investigate the relation between History and Theory, a very favorite subject in the late years of Hicks. According to Toshihiro Fukuda (2011), both Max Weber (1904) and Werner Sombart (1902, 1911, 1912, 1938 ) were regarded as those shining stars in the famous German Historical School who did outstanding contributions to comparative economic studies. Although Weber is still academically alive, it is quite unfortunate that Sombart is now an almost forgotten scholar in the academic profession around the world. Sombart’s serial works such as Sombart (1902, 1911, 1912, 1938) became more or less neglected. It is my real intention here to make a bridge between the later work of Hicks and the forgotten work of Sombart. As far as we know, such a bridge has been never attempted to build. We strongly believe that it should be of much value at the Second Age of Uncertainty we face today.2 The contents of this paper are as follows. In Sect. 2, the important problem of “capitalism versus socialism” will be reinvestigated in new perspectives. Section 3 will explore the comparative economic theory of Sombart, with an intensive discussion of its important part to be played in modern times. Section 4 will turn to the synthesis of theory and history by Hicks, comparing the Hicks doctrine with the Sombart view. Final remarks on the relation between the Sombart-Hicks approach and the Ohmi merchant theory will be made in Sect. 5.

2

Toshihiro Fukuda is one of leading authorities on the German Historical School. In recent times, he has developed his own ideas in line with the so-called Third Way, namely the midway between Capitalism and Socialism. For details, see Fukuda (2011).

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2.2

2 The Role of Merchants in the Exchange Economy: A Historical Perspective

Capitalism Versus Socialism: The Powerful Rivals in the Twentieth Century

In historical perspective, the twentieth century could rightly be called the Century of Socialism. In 1917, Vladimir Lenin (1870–1924) and his company overturned the old Russian regime, thereby succeeding in establishing the socialist government first in human history. Since then, capitalism and socialism had become powerful rivals for a long time until 1989, when the Berlin Wall fell and immediately later the socialist Soviet Union disintegrated into the more market-oriented Russian Federation and many other countries.

2.2.1

The Two Different Views of Seiji Kaya and Shigeto Tsuru

In the 1960s when Sakai was a student at Kobe University, Japan, the world politicoeconomic map was by and large divided into the two power blocs. One bloc was called the “blue bloc” or the “capitalist bloc” containing Western Europe, North America, and Japan. The other bloc was named the “red bloc” or the “socialist bloc” consisting of the USSR, China, and East Europe. Capitalism versus socialism—this rival relation was also apparent in all Japanese universities. In Japan in the 1960s, Das Kapital (1867) written by a noted socialist Karl Marx (1867) was so powerful in Japanese academia, clearly overpowering any other economic books in terms of selling volumes and influential scopes. In this connection, let us recall that Professor Seiji Kaya, a noted natural scientist and then the Chairman of the Japan Science Council of Japan, wrote in a newspaper3: On reflection it is really ridiculous that mankind cannot live in this globe peacefully with each other when they possess the knowledge and know-how even of making a round-trip to the moon. The most important thing from now on seems to be to join our efforts in making the time nearer when we can all visit the moon as friendly tourists, instead of being involved in the clash between communism and capitalism (Quoted by Tsuru 1961, p. 1). In response to Kaya’s opinion, Professor Shigeto Tsuru, a famous social scientist and then the President of Hitotsubashi University, quickly wrote: There are some among the experts in the field who would regard such terms as “capitalism” and “socialism” as emotional expressions and prefer not to use them in technical discussion. I would not agree with them. ... The distinction between capitalism and socialism as a social system is not due to emotional antagonism of politicians or to doctrinaire rigidity of academic people. Dr. Kaya’s wish for a

3

The Asahi Shimbun, a leading Japanese newspaper, 6 November 1957. Incidentally, Professor Kaya also served as the President of the University of Tokyo.

2.2 Capitalism Versus Socialism: The Powerful Rivals in the Twentieth Century

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harmonious world is everybody’s wish; but he should be aware that there does exist here a scientific problem of differentiating different social systems by an objective criterion and that the difference between them cannot be wished away (Tsuru 1961, pp. 2–3). It was very likely that the difference in their views over the distinction between capitalism and socialism reflected the differences of their research areas. On the one hand, Kaya as a natural scientist always sought universal natural laws such as the universal law of gravitation which were applicable at any place and at any time, regardless to possible differences of histories, cultures, skin colors, and so on. In his mind, the heated debate on the choice of capitalism or socialism as a social system seemed so highly emotional that it had nothing to do with those technical discussions which were rather common among natural scientists. On the other hand, Professor Tsuru did not agree with Professor Kaya, and emphasized the influences of cultures and ideologies on individual behaviors. Although Tsuru personally might have wished to make a round-trip to the moon together with Kaya, the former believed that any theoretical compromise between capitalists and socialists was almost impossible because the gap between the two socio-economic systems could not be wished way overnight. In our opinion, John Hicks, a noted economist and Nobel prize winner, seemed to be in general agreement with Tsuru. Interestingly enough, Hicks (1979) took one more step forward to write a small yet important book titled Causality in Economics (1979) on a very fundamental problem on the cause-and-effect relationship. According to his opinion, unlike natural sciences, economic knowledge is far from perfect. There exist very few economic laws we can know with exact precision. These laws are by and large subject to errors and ambiguities, which would be thought of as intolerable nonsense by natural scientists. To sum up, Hicks came to the conclusion that economics gave us a leading example of uncertain knowledge.4

2.2.2

The Official Textbook of Economics by Soviet Science Academy

In the 1960s when Sakai was young and an ambitious student at Kobe University, the most fashionable topic among students was about the sustainability of the capitalist regime as a socio-economic system. Among our fellow students, there were a lot of debates over the pros and cons. Will capitalism survive for many years to come? When and how will socialism overtake capitalism? What is on the earth the best socio-economic system from the viewpoint of ethics and justice? To find the right answers for those difficult questions, the Japanese young students needed a set of nice guide books. Among those books was the official Economics Textbook, the third edition (1959) published by the Soviet Science 4

For a detailed discussion on this point, see Sakai (2016), pp. 16–17.

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Academy. (1959). It was a very bulky book nicknamed the “Red Text” by young and diligent students. It was often compared to the “Blue Text”, namely Economics: An Introduction, the seventh edition (1955) written by a very influential American economist Paul A. Samuelson (1955). When we carefully read and compared both books, we had to confess that we were overwhelmed by the strong messages and passions of the more exciting Red Text, thus more or less underestimating the cool logic and elegant exposition of the less exciting Blue Text. After all, in young days, human passion tends to overwhelm reasoning! More exactly speaking, the Red Text consisted of four volumes and totally around 800 pages, which far outnumbered any textbook of modern economics available. Its general outline is evidently seen in Table 2.1. We still remember how much we were moved by the very powerful sentence in the very first page of the Red Text: Since the Publication of the Second Edition of the Official Textbook of Economics, in the Soviet Union and other People’s Democratic Countries as well, the socialist mode of production has constantly promoted and thus attained uninterrupted growth until today: indeed, not only has the planned leadership of the nation economy greatly improved, but also both the administration methods and the open discussions by the general public have been made better than ever before. In contrast to the flourishing socialist bloc, the capitalist bloc is now moving into the process of the overall crisis of capitalism. While every colonial system is eroding more rapidly than ever before, both domestic conflicts and international contradictions are surfacing more drastically than ever before (Soviet Science Academy 1959, Preface, p. 1). The rivalry between capitalism and socialism is on the top agenda of the Red Text. The way in which one economic system is replaced by another must be determined in advance. It is one directional from pre-capitalism to capitalism, and further to socialism, and not other way around. Consequently, no matter how a capitalist society is thriving now, it gradually loses a growth power, and eventually being doomed to destruction. According to the Red Text, communism will eventually be reached as the culmination of socialism, thereby guaranteeing the final victory of socialism over capitalism. In this connection, the following sentence at the very end of the bulky text must be very impressive: In the above, we have carefully and thoroughly investigated all the processes of economic development of a society. Consequently, we have come to the most important conclusion of economics that in historical perspective, capitalism is doomed to collapse whereas the victory of communism over capitalism will be unavoidable. The historical tendency that a modern society is moving toward communism must have a very solid foundation from which the objective laws of social development is surely derived. The communism, which is led by the communist party and supported by the Marx-Leninism, must be produced as the natural outcome of consciously creative activities of almost 100 million working mass. Our society has a built-in mechanism of going forward communism. This is definitely the

2.3 The Sombart Renaissance Revisited

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historical tendency which cannot be changed whatever by any means around the world (Soviet Science Academy 1959, Conclusion, p. 1050). Both the first page and the final page of the Red Text are decorated by exactly the same message, namely the final victory of socialism over capitalism. We must remember, however, that a powerful trumpet may sometimes sound very hollow. Although the debate between capitalism and socialism has been hot and exciting, it has somehow sounded hollow and neglected the historical truth. It is true that as Karl Marx noticed, the rise of capitalism is important and must carefully be explored for full understanding of economy history. We would like to say, however, that there exists a more fundamental transformation in human history, which is even precedent to Marx’s concept of the Rise of Capitalism. This is what we may rightly call the Rise of the Market or the Rise of the Mercantile Economy. Although there are relatively few scholars who have pointed it out, there are some conspicuous exceptions. The Old Werner Sombart and the New John Hicks belong to the exceptional group, which should be the next topic of my investigation in this paper.5

2.3

The Sombart Renaissance Revisited

In hindsight, Werner Sombart seemed to be a man in paradox. Before the First World War, he was a famous professor at Berlin University, being widely regarded as a great economist belonging to the German Historical School. After the war was over, however, his fame fell down quickly and became an almost forgotten economist. Only recently, just after the Fall of Berlin Wall in 1989, his destiny took another turn. According to Jürgen Backhaus (1996a, 1996b, 1996c), the classical work of Sombart saw a considerable comeback: in fact in 1991, a distinguished group of many scholars around the world gathered in the City of Heilbronn, Germany, to intensively discuss the work of Sombart and its modern implications. In what follows, we will critically reevaluate the so-called Sombart renaissance and its true significance.6

5

This was already pointed out by John Hicks (1969). While we agree with him in this respect, we wonder why he failed to refer to Werner Sombart as one of his pioneers. Now, it is high time to do equal justice to ignored economic historians in the past. 6 At present, the three volumes evaluating the work of Werner Sombart are available. Backhaus (1996a, 1996b, 1996c) as an organizer of the Sombart Conference did an outstanding contribution to the timely return of Sombart to the world academia today.

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2.3.1

2 The Role of Merchants in the Exchange Economy: A Historical Perspective

Werner Sombart Versus Max Weber: Friends and Rivals

Both Werner Sombart (1863–1941) and Max Weber (1864–1920) were the two towering figures belonging to the German Historical School originated by Gustav von Schmoller (1838–1917). They were contemporaries; more exactly, Sombart was only one year older than Weber. They were close friends, serving as coeditors of the leading German journal Zeitschrift für National Ökonomie, yet sometimes becoming fierce rivals. The very significant difference between Sombart and Weber laid in the fact that Sombart outlived Weber by 21 years. On the one hand, Weber was periodically mentally ill for a long time and passed way in 1920 when he was 56 years old: this was the time when Germany was just defeated in the First World War, and the rise and dominance of the German dictator Adolf Hitler (1889–1945) was not be seen yet. On the other hand, after Sombart was appointed a full professor to the most important economic chair in Germany, he dared to write a very controversial book Deutschr Soecialismus in 1938, which was grossly interpreted as a work paying a tribute of praise to Nazis Germany. Without the 1938 book, Sombart would have somehow retained his prestige as a respected representative of the German Historical School. With its publication, however, he grossly seemed to be taken as a sort of war criminal, thereby being destined to be a defeated man.7 After the death of Sombart in 1941, so many years have passed. We are in the new twenty-first century. Since the Sombart conference held in 1991, Sombart’s classical work has seen a remarkable comeback. As Backhaus (1996a, 1996b, 1996c) rightly told us, Sombart may be thought of as one of founders of the economics of comparative systems. Now, the return of the old masters including Sombart should urgently be needed.

7

Before the Sombart Renaissance took place, Galbraith (1987) uncharacteristically criticized Sombart’s work in a very bitter fashion: “Its principal exponent was Werner Sombart (1863–1941), the German historian-economist, a diligent but not completely reliable scholar. Intuitively and perhaps even openly anti-Semitic, Sombart sought in his later years to give a measure of theoretical sanction to National Socialism” (Galbraith, 1987, Footnote 2, pp. 22–23). It seems that the “theoretical sanction to National Socialism (or Nazi)” and the “anti-Semitic” stance are a sort of taboos which must be avoided at all cost in the western world. Let us recall that Schumpeter (1954) said harsh things about the German Historical School, and especially spoke bitterly of Sombart. “The only work of Sombart that needs to be mentioned here, his Der Moderne Capitalismus (or Modern Capitalism, 1902) shocked professional historians by its often unsubstantial brilliance. They failed to see in it anything that would call real research—the material of the book is in fact wholly second hand—and they entered protests against its carelessness” (Schumpeter, 1954, Footnote, pp. 816–817). We belong to the oriental world, thus do not quite agree with Galbraith and Schumpeter. We would sincerely wish that the creative, but not dogmatic, Sombart would rightly come back into the academic world.

2.3 The Sombart Renaissance Revisited

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Fig. 2.1 Alternative theories of comparative economic systems: the demand side approach versus the supply side approach Table 2.1 The Soviet science academy (1959): the contents of the official textbook of economics

2.3.2

The textbook of economics: preface Chap. 1 the subject of economics Chap. 2 the production mode before capitalism The capitalist production mode Part 1 Capitalism before monopoly From Chaps. 3–14 Part 2 Monopoly capitalism: imperialism From Chaps. 15–19 The socialist production mode Part 1 The transition period from capitalism to socialism From Chaps. 20–23 Part 2 The national economic system of socialism From Chaps. 24–36 Conclusion

The Demand Side Approach Versus the Supply Side Approach

We are now in a position to draw a historical chart of interdependence among economists. As is seen in Fig. 2.1, there are fundamentally two different ways— horizontal and vertical ways. On the one hand, if we look at the figure horizontally, we can classify the economists into two groups; namely, the group of the “demand

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2 The Role of Merchants in the Exchange Economy: A Historical Perspective

side approach” and the one of the “supply side approach.” Needless to say, a market economy consists of two sides; namely, the demand or consumer side, and the supply or producer side. In historical perspective, the demand group, which emphasizes the demand side more than the supply side, contains Mercantilism (led by Daniel Defoe (1728) and James Steuart (1767)), Werner Sombart (1902, 1911, 1912, 1938), and J.M. Keynes (1936). The supply group, which attaches more importance to supply side than the demand side, includes Karl Marx (1867), Max Weber (1904), and Joseph Schumpeter (1926). Since the New Hicks is an open-minded man who intends to integrate the demand side and the supply side into a grand new system, his position should be located somewhere between the two groups. On the other hand, now by looking back at Fig. 1.1 vertically, we may classify the economists into four schools; namely, the Classical School, the Historical School, the Modern School, and the Contemporary School. In Fig. 2.1, the solid line arrow (A ! B) shows that A strongly influences B whereas the dotted line arrow (C ! D) means that the influence of C on D is rather weak. For instance, while Mercantilism as a demand side approach in the early days strongly influences Sombart and Keynes, the influence of Sombart on Keynes looks rather weak and requires a further investigation. Perhaps against the current stream of thinking, we are nevertheless inclined to believe that Sombart’s impact on Hicks is fairly substantial.8

2.3.3

Sombart on the Capitalist Spirit

The word “capitalism” is now very popular and so frequently used together with its rival word “socialism” . Remarkably, the former word was not seen at all in the Wealth of Nations by Adam Smith (1776), the Father of Economics. Although Karl Marx (1867) in his main work Das Kapital eloquently discussed the dominance of the “capitalist class” over the “worker class” along with the characteristics of the “capitalist production mode,” he never employed the keyword “capitalism.” It is Werner Sombart himself who first invented and regularly employed the word “capitalism” (or Kapitalismus) and its companion “capitalist spirit” (or kapitalistische Geist). Unfortunately, this historical fact has been almost forgotten in the economics history literature. We might to add that unlike his close friend Max Weber, Werner Sombart lived long enough (perhaps too long!) to be involved in the darkest episode of fascist Nazi Germany with anti-Semitism. A longer life may not guarantee a better life!

Keynes strongly argues that there is “the element of scientific truth in the mercantilist doctrine” (Keynes, 1936, p. 335) that is designed to maximize the export of a nation and minimize its import so as to increase the aggregate demand. 8

2.3 The Sombart Renaissance Revisited

31

In his most important book Der moderne Kapitalismus (or modern capitalism), Sombart (1902) gave very important characteristics of the capitalist economy in the following way: Capitalism is an exchange economic organization, in which normally two different groups of people exist. They are the people who own the means of production, being responsible for the management and thus being economic subjects, and the people who are mere workers (as economic subjects) and are united together and interconnected through the market, being motivated by the earning principle and economic rationalism (Sombart 1902, p. 319). Apparently, Sombart’s view of capitalism was different from Marx’s one. First of all, concerning the definition of capitalism, Sombart adopted a more flexible stance than Marx. Sombart differed from Marx in the sense that capitalism should be regarded as an exchange economy organization rather than a production economy organization. Second, while Marx argued that the rich capitalist who monopolized the means of production had a power to exploit the poor worker, Sombart argued that both capitalists and workers were engaged in reciprocal relations in the market economy. Third, all the people including capitalists and workers were economically rational men, whence being motivated by the earning principle and economic rationalism. To sum up, Sombart’s view of capitalism should be very akin to the market exchange economy, which could be interpreted widely enough to include any kind of commercial trading in the pre-capitalistic feudal economy. Der moderne Kapitalismus (or modern capitalism), consisting of huge three volumes, was a historical and systematic portrayal of the economic life of all Europe from its beginning to the present time, no doubt represented the life work of Werner Sombart. Those three volumes of Sombart (1902) could be compared to the bulky three volumes of Das Kapital (1867) by Karl Marx (1867). Although the views of Sombart and Marx looked similar, they were fundamentally different. Seeing is believing! Sombart’s view on the history of economic systems may be depicted in Fig. 2.2. The key concept played in the history of economic systems was the capitalist spirit, being located in the center, which intermediated between the pre-capitalist era and the era of capitalist economy. The importance of the capitalistic spirit for the Sombart world could not be overstated. Sombart without the capitalistic spirit looked like Hamlet without the prince. According to Sombart, the spirit of capitalism was describable by three factors: monetary transactions, competitive trading, and economic rationality. All trades were carried out by means of money, and must be free from outside regulations. Traders or merchants behaved as profit seekers in the sense that they maximized revenues and minimized losses.9 Historically speaking, any economic organization could largely be divided into the two: namely, the non-capitalist economy in an earlier era and the capitalist economy in a later era. Moreover, the non-capitalist economy consisted of the two subgroups; namely, the self-supporting economy with farming and local communities, and the circulation economy with handcraft masters and associates.

9

For this point, see Stehr and Grundman (2001), Chap. 1.

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Fig. 2.2 Sombart on the history of economic systems, with focus on the capitalist spirit

Unfortunately, the vital capitalistic spirit did not come to light yet. The critical factors which contributed to the transformation of the non-capitalist economy into the capitalist economy consisted of the three—the rise of the capitalist spirit, the capitalist functions, and the capitalist technology. In contrast to Marx who mechanically emphasized the irreconcilable contradictions between ever-socializing productive power and antagonistic private relations in big enterprises, Sombart shifted his interest to various human factors such as individualism, rationality, scientific knowledge, freedom of movement, and active marketing. In short, let the capitalist spirit decide! This was really the essence of Sombart’s concept of the capitalist economy. According to Sombart, the era of capitalist economy has so far developed through three stages. The first stage, called Frühkapitalismus (or early capitalism), covered the long period from trading activities in the thirteenth century to the Industrial Revolution in the 1760s. There emerged many specialized merchants with lively spirit on the scene. Although some were actively engaged in speculation and even

2.4 Hicks on the History of Economic Systems

33

military activities, they could be regarded by Sombart as the bearers of the capitalist spirit. The second stage, named Hochcapitalismus (or high capitalism), contained the period from the Industrial Revolution to the break of the First World War in 1914. This period was characterized by the high development of capitalist system all over the world. The third stage, called Spätkapitalismus (or late capitalism), corresponded to the period after 1914 until the present day. This late period is in a sense the beginning of the end of capitalist economy, in which Sombart bravely predicts the eventual return of the non-capitalist economy. Although this seems to be a bit strange argument, I believe that by and large, Sombart’s analysis on the capitalist spirit remains to be very important even today.

2.4

Hicks on the History of Economic Systems

When Hick’s new book A Theory of Economic History was published in 1969, Sakai was a graduate student at Rochester. Sakai was taking a sequence of courses in economic theory, with Professor Lionel McKenzie being the outstanding leader of the theory group. Although Richard Thaler, Sakai’s good friend at Rochester, recommended Sakai to attend econometric history classes of Professor Robert Fogel, Sakai himself was then much more impressed by the power and beauty of pure mathematics than the seemingly odd mixture of econometrics and history. To our deep regret, neither McKenzie nor Fogel seemed to refer to Hicks’ new approach to economic history, which will be the topic to be discussed in the following subsections.10

2.4.1

The Rise of the Market

Hicks was a multifaceted man. His research interests were wide enough to cover philosophy, theory, and history. Quite unfortunately, however, so many people tended to refer only to the Old Hicks as a theorist, thus thinking light of the New Hicks as a historian. We have to mend such unbalanced view of Hicks’ academic achievements. For this point, we recommend to see Hicks (1979). In our opinion, Hicks’ position in the theory of economic history is quite unique. First of all, he was not fond of employing rigid terms such as the rise of capitalism

10 Several years later, the great teacher Robert Fogel moved from Rochester to Harvard and was awarded Nobel Economics Prize for his contribution to econometric history. Sakai’s good friend Richard Thaler also moved from Rochester to Chicago, and later was awarded Nobel Economics Prize for his contribution to behavior economics.

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2 The Role of Merchants in the Exchange Economy: A Historical Perspective

and its transition to socialism, but rather felt a strong affection for gentler expressions such as “the rise of the market” and “the role of the merchant.” For this point, he once remarked: Where shall we start? There is a transformation which is antecedent to Marx’s Role of Capitalism, and which, in terms of more recent economics, looks like being even more fundamental. This is the Rise of the Market, the Rise of the Exchange Economy. It takes us back to a much earlier stage of history, at least for beginnings (Hicks 1969, p. 7). Hicks started his historic inquiry with the rise of the market or the emergence of the exchange economy. This could be compared very well to Sombart’s idea of capitalism as a distributive economic organization. Indeed, Sombart’s capitalism was very akin to Hicks’ market economy; both of them were well-illustrated in the activities of merchants in the Mediterranean trade and those at the Age of Discovery, covering for the long period from thirteenth century to the eighteenth century. Second, Hicks also avoided the use of hard idioms such as mercantilism and the Industrial Revolution, and instead liked to employ softer expressions such as mercantile economy and industrialism. According to Hicks, mercantilism sounded too political to be academically usable, and unlike the riotous French Revolution, the key players of the Industrial Revolution were far from well-defined. Third, he did not want his theory to rely on any form of Historical Determinism including Marx’s materialistic interpretation of history. He did not think that a socio-economic society was forced to change one-sidedly by way of the “conflict between expanding productive power and stiffening productive relations.” His historical stance was very flexible to accept the interactions between economic and non-economic factors, also observing the possibilities of backward movements, cycles, and many other non-regular movements. In my opinion, the second and third points aforementioned clearly demonstrate the flexibility and open-mindedness of the “Hicksian theory of economic history,” thus distinguishing itself from the German Historical School including Marx, Weber, and Sombart. Hicks’ view on the history of economic systems may be depicted in Fig. 2.3. The key concept played in the Hicks system is the rise of the market with regular trading, which is located in the center of the figure. According to Hicks, the history of economic systems began with the primitive non-market organization, which contained two types of economies; namely, the “custom economy” with local communities, and the “command economy” with military order such as the Mongol control by the strong man Genghis Khan (1162–1227). Historically speaking, the primitive non-market organization was destined to fade out and even vanish by the rise of the market, in which many specialized merchants appeared and engaged in regular and permanent trading. According to Hicks, the mercantile economy had the three phases of development. The “first phase” was characterized by the continuation and expansion of market trading in the city state. In the “second phase” of mercantile development, market centers emerged and flourished, with insurance and stocks being the items of trading. The “modern

2.4 Hicks on the History of Economic Systems

35

Fig. 2.3 Hicks on the history of economic systems, with focus on the rise of the market

phase” began with the Industrial Revolution with modern technology and industries. This modern phase of Hicks was quite analogous to the concept of “high capitalism” dealt with by Sombart in his analysis on the history of economic systems. As mentioned above, Sombart preferred to employ a light phrase of “industrialism” rather than a heavier idiom of “industrial revolution.” So, the “capitalist production mode,” which was heavily favored by Marx, was merely regarded by Hicks as “just one phase of the bigger framework of mercantile economy.” The question which would naturally arise here is how Sombart’s view on the history of economic systems is similar to, and different from, Hicks’ view. No doubt, it constitutes the core of this paper. Fortunately, a good answer to this question would be given by comparison of the last two figures, namely, Figs. 2.2 and 2.3. It is quite apparent from a structural viewpoint that those two figures look alike. So, they should have a lot of things in common. First of all, as far as the historical structure of economic systems is concerned, Sombart’s view is very akin to Hicks’ one. On the one hand, as is seen in Fig. 2.2, Sombart argues that the economic history starts with the pre-capitalist era with self-supporting and/or circulating economies, later

36

2 The Role of Merchants in the Exchange Economy: A Historical Perspective

acquires the strong capitalist spirit, functions, and technology, and finally gets into the era of capitalist economy which in turn consists of the three stages of early capitalism, high capitalism, and late capitalism. On the other hand, as Fig. 2.3 tells us, Hicks thinks that the economic history begins with primitive non-market organization with custom and/or command economies, later faces the remarkable rise of the market with regular trading, and finally reaches the mercantile economy which contains the three stages of first, middle, and modern phases. If we are allowed to identify Sombart’s capitalism with Hicks’ market economy, then the fundamental structures of Sombart and Hicks doctrines become very similar. Second, both economists—Sombart and Hicks—attach the greatest importance to the role of merchant in the market economy. On the one hand, Sombart observes the strong capitalist spirit in external activities of adventurous Italian and Dutch merchants. On the other hand, Hicks pays special attention to the rise of the market in which regular trading occurs among high-spirited merchants. In either case, both Sombart and Hicks adopt demand side economics, thus being different from Marx and Weber as the believers of supply side economics. We should add that comparing Sombart and Hicks, Hicks’ position is more balanced than Sombart. After all, the Hicks theory should be regarded as the magnificent synthesis of the demand side and supply side approaches. Third, those two economists do with all their strengths for the “grand integration of history and theory”, also taking account of economic and non-economic factors. The abstract-minded economists Marx and Weber are apt to think light of the real economy, thus dreaming of “ideal types” and “abstract economic men.” In contrast, the empiricists Sombart and Hicks attach the greatest importance to the real economy with ordinary men. To sum up, Sombart is an ambitious economic historian who has put his heart and soul into the project of “actualization of economic history.” In contrast, Hicks in his later years has done everything in his power to finish the job of establishing “theoretical economic history.” Hopefully, we would like to combine the works of Sombart and Hicks toward a new grand synthesis of history and theory.

2.4.2

Slavery in the Mercantile Economy

Hicks was known as a man of honesty and good conscience. In his history book, Hicks (1969) candidly remarked: The darkest episodes in the history of mercantile slavery (putting aside the horrors of slave-catching) are a matter of the large-scale employment of slaves such as the cotton and sugar plantations of Americas and the West Indies (Hicks 1969, p. 126). As was eloquently argued by Thomas (1997) and Sakai (2018, 2019), the mighty British Empire had long thrived on the foundation of slave trade and many colonies overseas. Its most famous trade route was known as the triangular trade route, connecting three ports; namely, Britain (Liverpool, Bristol), West Africa (Slave Coast), and the Caribbean (Jamaica). Among the three possible passages connecting

2.5 Koji Egashira on the Ohmi Merchant: Final Remarks

37

any two of those ports, the Transatlantic Passage from West Africa to the Caribbean was best-known among the traders. In his popular book, the novelist and polemic Daniel Defoe (1728) once described the enormous amount of money the greedy merchants brought about to Britain by the slave trade. And at present, in his bestseller, the French rising star Thomas Piketty (2013) eloquently discussed the historical importance of slavery in the New World and the Old World. I [Piketty] cannot conclude this examination of the metamorphoses of capital in Europe and the United States without examining the issue of slavery and the place of slaves in US fortune (Piketty 2013, p. 158). The importance of slave trade in the mercantile economy should not be underestimated. Now, it seems that Pikkety can be thought of as another Hicks. We would sincerely hope that, following Piketty’s lead, many other successors will cheerfully come out.

2.5

Koji Egashira on the Ohmi Merchant: Final Remarks

The late professor Koji Egashira (1900–1978) was a noted authority on Ohmi merchants. In his lifework, Egashira (1959) remarked: The controversy between Werner Sombart and Max Weber in Germany was once passionately introduced to Japan, becoming among economists one of fashionable topics in Japan. It remains to be unsolved even today, however. The Ohmi merchants, which have been best representatives of Commercial Capital in Japan. I have no doubt that thorough studies in those merchants may greatly contribute to an inquiry into the development of Japanese commercial capital. I have so far exerted all my energy to examine the famous Ohmi chants in many possible ways. As a result, I have reached the conclusion that I am in a position to fully support neither the Weber doctrine nor the Sombart doctrine (Egashira 1959, Preface). It was so remarkable to see that Egashira paid special attention to the controversy between Sombart and Weber on the part of merchants played in a capitalist economy. He also argued that Ohmi merchants greatly contributed to the development of Japanese commercial capital. He was not so sure, however, how and to what degree the economic activities of those merchants could be explained by either Sombart or Weber, or possibly both. We are inclined to support Sombart more than Weber. We believe that the capitalist spirit a la Sombart can be compared very well to the ethic and moral of Ohmi merchants. Such comparison is very important, requiring a further examination. The related question of importance is how and to what degree the old stage of “commercial capitalism” was transformed to the new stage of “industrial capitalism.” As was mentioned above, the demand side economist Sombart argued that such transformation should have been gradual and continuous because changes in people’s demand were always slow and steady. In contrast to Sombart, his contemporary economist Weber sided with the supply side approach, thus pointing out the “wide mental gulf” between commercial and industrial capitalisms, which was

38

2 The Role of Merchants in the Exchange Economy: A Historical Perspective

caused by the uplift of the Protestant ethics. Then, we would like to ask the following question: which should be the right doctrine, the continuous transformation doctrine of Sombart or the discontinuous transformation doctrine of Weber? In Japan in the 1950s and the 1960s, Weber supporters greatly outnumbered Sombart supporters. Thus, it was unfortunate to see that Sombart’s theory almost vanished along with the defeat of the Nazi Germany. Now, however, so many years have passed after the Second World War. We strongly believe that it is high time to say farewell to the defeatist doctrine, thereby reevaluating the Sombart-Weber controversies aforementioned from new angles. In our opinion, Hicks’ new approach to the theory of economic history can give us a useful guide to solve the Sombart-Weber antagonism. As was mentioned above, concerning the history of economic systems, Sombart’s doctrine is based on a demand side approach, whereas Weber’s view is a supply side one. Needless to say, the working and performance cannot correctly be explored by one side only: both demand side and supply side approaches must be integrated into a grand new synthesis. For such integration, we still require a further investigation. As Hicks (1969) has repeatedly stressed, the rise of the “mercantile economy” constitutes the very core of his inquiry. Now, the problem of choice between capitalism and socialism becomes a secondary issue because only capitalism, and not socialism, can be associated with the mercantile economy. We strongly believe that Ohmi merchants of Japan may give us good examples of the Hicks-type mercantile economy. To our deep regret, Hicks hardly referred to Ohmi merchants, or the Japanese merchants who had strong forward-looking natures. We also would like to add that those Ohmi merchants acted as lively players with capitalist spirits a la Sombart. For this point, see Ogura (1980). Remarkably, Hicks has emphasized that the trade should obey the “principle of all-round advantage.” In other words, the trade should produce the advantage of all the parties; namely, the advantages of merchants themselves and the “surrounding” people with whom they trade.11 Such principle reminds us of the “principle of sampo –yoshi” obeyed by Ohmi merchants of Japan; namely, the principle of “being good for a seller, good for a buyer, and good for the society.” How exactly and to what extent the Hicks principle and the Ohmi merchant principle are similar or different remains to be an open question. We need to do a further investigation.12

References Backhaus JG (1996a) Werner Sombart (1863–1941) - social scientist, volume: his life and work. Metropolis-Verlag, Marburg

11 12

See Hicks (1969), p. 51. For details of Ohmi merchants, see Egashira (1959) and Ogura (1980).

References

39

Backhaus JG (1996b) Werner Sombart (1863–1941) ― social scientist, volume: his theoretical approach reconsidered. Metropolis-Verlag, Marburg Backhaus JG (1996c) Werner Sombart (1863–1941) - social scientist, volume: then and now. Metropolis-Verlag, Marburg Defoe D (1728) A plan of the English commerce being a complete prospect of the trade of this nation, as well the home trade as the foreign. Charles Rivington, London Egashira K (1959) On Ohmi merchants (in Japanese). Shibundo Publications, Tokyo Galbraith JK (1987) A history of economics: the past as the present. Hamish Hamilton Fukuda T (2011) The economic thought of the third way: how to deal with the age of crisis (in Japanese). Kouyou Publishers, Kyoto Marx K (1867) Das Kapital (in Germany), vol I. Morishima M (1994) Modern economics as economic thought (in Japanese). Iwanami Publishers, Tokyo Hicks J (1941) The theory of wages. Macmillan Hicks J (1946) Value and capital, 2nd edn. Oxford University Press Hicks J (1969) A theory of economic theory. Oxford University Pres Hicks J (1977) Economic perspectives: further essays on money and growth. Oxford University Press Hicks J (1979) Causality in economics. Basic Blackwell, Oxford Keynes JM (1936) The general theory of employment, interest and money. Macmillan Ogura E (1980) Ohmi merchants: a historical perspective (in Japanese). Nihon Keizai Publishers, Tokyo Piketty T (2013) Le capital au XXI siècle (in French). Éditions du Seuil. English translation by Goldhammer A (2014) Capital in the twenty-first century. Harvard University Press Sakai Y (2016) J.M. Keynes on probability versus F.H. knight on uncertainty. Evol Inst Econ Rev 13(1):1–21 Sakai Y (2018) Liverpool merchants versus Ohmi merchants: how and why they dealt with risk and insurance differently. Asia-Pacific J Reg Sci 2(1):15–33 Sakai Y (2019) The role of merchants in the exchange economy: J.R. Hicks versus W.R. Sombart. In: A research paper read at the international workshop on regional economics, at the University of Tsukuba, in January 2019 Samuelson PA (1955) Economics: an introduction, the seventh edition. McGraw-Hill Schumpeter JA (1926) Theorie der Wirtschaften Entwicklung (in Germany) Schumpeter JA (1954) History of economic analysis. Allen & Unwin Smith A (1776) An inquiry into the nature and causes of the wealth of nations. U.K. Sombart W (1902) Der Moderne Kapitalismus (in Germany). Verlag von Duncker and Humblot, Berlin Sombart W (1911) Die Juden und das Wirtschatsleben (in Germany). Berlin Sombart W (1912) Liebe, Luxus und Kapitalismus (in Germany). Berlin Sombart W (1938) Deutscher Socialismus (in Germany). Berlin Stehr N, Grundman R (2001) Werner Sombart: economic life in the modern age. Selected papers of Werner Sombart, English translation. Transaction Publishers, Brunswick and London Steuart J (1767) An inquiry into the principles of political economy. UK Thomas H (1997) The slave trade: the story of the Atlantic slave trade, 1440–1870. Simon & Schuster, New York Tsuru S (1961) Has capitalism changed?: an international symposium on the nature of contemporary capitalism. Iwanami Publishers, Tokyo Soviet Union Science Academy (1959) The official textbook of economics, revised third version, Japanese translation of the Russian original. Godo Publishers, Tokyo Weber M (1904) Die Protestsntische Ethik under der ‘Geist’ des Kapitalismus (in Germany). Berlin

Part II

Information Exchange: Theoretical Perspective

Chapter 3

A Theory of Information and Distribution: The Market Economy and Demand Risk

Abstract This paper discusses the relationship between information and distribution, with special reference to the role of the merchant in the market economy. By working with simple equilibrium models of the industry and doing a sequence of comparative economic analyses, we intend to shed a new light on an important yet rather neglected area in the economics profession. Let us suppose that the demand side is subject to many changes and may be represented by a simple uniform distribution function with two parameters, i.e. mean μ and variations σ 2. Then, we can show that the entry of the informed distributor between the producer and the consumer would cause two opposing welfare effects: A negative intermediation effect and a positive information effect. If the degree of relative risk is large enough in the sense that the σ 2 - μ2ratio exceeds a certain threshold value, then the information effect becomes a dominant force. Therefore, the introduction of the distributor into the economy will increase both producer and consumer surpluses: It will make all the parties better off. In a historical perspective, the Ohmi merchant is known to have a good faith in Sampo Yoshi, or the principle of all-round advantages of trading. Hopefully, the result obtained in the paper will give some theoretical ground for such an old and new principle. Keywords Information · Distribution · Market economy · Role of merchant · Stackelberg · Leader-follower · Intermediation effect · Information effect · All-round advantages

The original theory part of this paper was first outlined by Sakai and Sasaki when they were jointly researching at the University of Tsukuba in the 1990s, and later greatly developed in Sakai and Sasaki (1996). The history part was much later added by Sakai in the 2000s. This chapter intends to combine both the theory and history parts into one entity, thus comprehensively exploring the informational role of merchants in the economy under uncertainty. © Springer Nature Singapore Pte Ltd. 2021 Y. Sakai, K. Sasaki, Information and Distribution, New Frontiers in Regional Science: Asian Perspectives 49, https://doi.org/10.1007/978-981-10-0101-7_3

43

44

3.1

3 A Theory of Information and Distribution: The Market Economy and Demand Risk

Konosuke Matsushita and Isao Nakauchi: An Introduction

This chapter deals with the relationship between information and distribution, with a focus on the role of the merchant in the market economy. Since the consumer and the producer act as two main players on the stage of economics profession, the intermediate role of the distributor has been undervalued as a minor supporting actor, or even completely ignored. The purpose of this paper aims to more or less mend such unfair treatment in economic science, thus shedding a new light on the informational role of a variety of merchants in a right perspective. Konosuke Matsushita (1894–1989), the famous founder of the powerful Matsushita group of manufacturers and a legendary figure who made his business career from rags to riches, once observed: When I [Matsushita] put my full energy into business, an unlimited amount of wisdom springs out of my mind. This is really a very important point. Many governmental officers and university professors, however, tend to forget such a plain truth in the business world. Even someone might say that wholesale merchants are useless entities engaging in intermediate exploitation. Being a practical man myself, I am sure that this is no more than a utterly nonsense story, regrettably being shared by naive men of no business experience (Matsushita 1974, p. 126). Isao Nakauchi (1922–2005), the founder of the giant Daiei group of supermarkets, may be regarded as a noted champion of the Japanese distribution system. In his well-sold book, he remarked: It is information that constitutes the origin of everything. The person who acquires the information is the consumer. In my opinion, the distributor should be regarded as an indispensable entity in collecting the information from the consumer (Nakauchi 1982, p. 140). As Nakauchi has rightly noticed, it seems that everything comes from information. The person who controls information may control the market economy. In a real world, the distributor is in the best position to collect the consumer data which is in turn transmitted to the producer, whence the market economy may smoothly work. This is the crux of our research, and will repeatedly be pointed out throughout this paper. The contents of this paper are as follows. The next session will non-technically discuss the three vital functions of merchants as active intermediaries in the economy, with a special focus on some historical examples. The third session will model the world without uncertainty as a reference point. Comparisons between the case of direct trade between the producer and the consumer and the case of the entry of the distributor as a “third man” will be done through the use of equilibrium analysis. The fourth session will introduce demand uncertainty and analyze its effects on equilibrium values. The non-symmetric situation in which the distributor collects the demand data but the producer is not so informed is worthy of intensive investigation. There will two different kinds of effects working in opposite directions. They are: the negative intermediation effect and the positive information effect. As far as the

3.2 The Role of Merchants as Vital Intermediaries

45

positive effect considerably overpowers the negative one, it will be likely that the presence of the distributor will rather enhance the welfare of all the parties of the society. Some final remarks will be made in the final session.

3.2

The Role of Merchants as Vital Intermediaries

This section will carefully discuss the role of merchants as vital intermediaries in the market economy. By illustrating some historical examples, we will show how and to what extent merchants really matter as the intermediaries between the producer and the consumer. Then, we will turn to a theoretical argument to systematically discuss three important functions of merchants.

3.2.1

Merchants Really Matter: Some Historical Examples

It would be no exaggeration to say that the history of merchants is almost as old as the history of civilization. In what follows, several examples will be picked up in order to demonstrate the historical fact that merchants really matter. Example 1 The first historical example was provided by Sima Qian (145 B.C.– 86 B.C.), a noted historian of Ancient China, in his great book entitled Shi ji, or Records of the Grand Historian (109 B.C.–91 B.C.), Volume 69. In this volume, he passionately discussed the success stories of wealthy persons. It is quite remarkable to see that Sima Qian began his story with the following sentence: Any common man with no government position is nevertheless able to find good opportunities to sell and buy goods so that he can increase his wealth. To be sure, this should not harm politics whatever. nor to people’s daily activities. A wise man of greater knowledge will gain much more from the trade of goods. This is precisely the reason why we have thus decided to write Volume 69, namely The Success Story of Wealthy Persons (Sima Qian n.d. 109 B.C.–91 B.C.; Japanese Translation 1975, p. 150). This clearly teaches us that Adam Smith(1723–1790), presumably claimed the Father of Economic Science, may not be the historically first person to discuss the wealth of persons and/or nations. Long before Jesus Christ (4 B.C.?–29 A.D.?) was born, the concept of wealth or moneymaking was already noticed and effectively discussed by Chinese people. In fact, the distributive role of merchants between sellers and buyers was clearly recognized by Sima Qian: We can eat food thanks to the labor of farmers. Timbers are first supplied by wood cutters, then transformed into finished items by craftsmen, which in turn will be distributed by merchants into any other place where the items are demanded.

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3 A Theory of Information and Distribution: The Market Economy and Demand Risk

The circulation of goods in the country are brought about by the combination of the powers of these ordinary people, not by the order of the government upon the private sectors. Each person who wants the good can really acquire it by means of his greatest possible talent and effort. The price of a cheap good might rise later whereas an expensive good might be cheaper tomorrow. As the water is naturally running down to a lower place, so each man is willing to do hard work day and night. People may gladly come to working places before they are ordered to do so, and trade goods among themselves without any kind of enforcement. No doubt, these acts accord with reason, showing the natural consequences of people’s free will (Sima Qian 109 B.C.–91 B.C.; Japanese Translation 1975, p. 152). Example 2 John Hicks (1904–1991) was among the greatest economists of the twentieth century, having had a long-standing influence on economic thought. As his age gradually advanced, he had underwent the great transformation from first-rate theoretician to outstanding historian. His later work on economics was well-presented in A Theory of Economic History (1969) in which the role of the merchant was singled out as a key concept for understanding the working and performance of the market economy. He remarked: The mere fact that one trading centre has a different geographical location from another gives it ‘comparative advantage’ in the collection of information; by trading between centers these advantages can be utilized and risks on both sides can be reduced (Hicks 1969, p. 49). According to Hicks, the merchant is making a profit by buying a commodity at a low price and selling it at a high price. There should be an advantage to the sellers of one commodity; and its buyers for just the same reason. Thus there is a profit to the merchants, and a gain to each of the parties with whom they trade. In technical language, the latter gain is called a consumer’s surplus. So long as the trade is carried out voluntarily, it must confer what Hicks names an “all-round advantage.”1 As the following example will show, the way how Hicks characterizes the working of the Mercantile Economy is quite similar to the way by which the Ohmi merchant of modern Japan has circulated many goods by land and/or by sea between different regions. Example 3 Eiichiro Ogura (1924–1992) was a leading scholar for the study of the Ohmi merchant, one of early pioneers of the Japanese capitalist economy. Ogura was well-known as the inventor of the catchy phrase “Sampo Yoshi,” or the principle of three-way advantages. In his well-read book (1991), he characterized Sampo Yoshi as the behavioral principle of the Ohmi merchant in the following way: Any trade of goods between sellers and buyers could be advantages to both parties.

1

For a more detailed discussion, see Hicks (1969). Also see Hicks (1977), Preface and (Survey).

3.2 The Role of Merchants as Vital Intermediaries

47

We would also say that customers should be regarded as kings, not as subordinates. Certainly this would be rather common sense shared by all the merchants. What Ohmi merchants distinguish themselves from other merchants is the adoption of additional advantage, which may be named the advantage to the society. What is good for the seller and for the buyer is also good for the whole society (Ogura 1991, p. 9). The Ohmi merchant took advantage of every transportation means to reach almost every place in Japan including Hokkaido. The merchant’s style of carrying a shouldering pole named “Tenbinbo” symbolized his dedication to very hard work day and night. In order to carry goods and items between Kyoto and Hokkaido, he periodically showed his courage to take the risky Sea of Japan route in stormy weather. His innovative power of economic development in remote regions was well-documented in history, thus greatly contributing to make a solid foundation of modern Japan.2

3.2.2

Three Vital Functions of Merchants: A Theoretical Argument

Any national economy contains three types of players. They are: the producer, the consumer, and the distributor. In daily language, the last player is called the merchant or the trader who puts himself between the producer and the consumer, thus playing a part of intermediation. If the size of the economy is so small and the trade of goods between the supplier and the demander can be carried out with no frictions and/or with no time lags, then there should be no real reasons whatever to justify the presence of intermediating merchants: They would be thought of as mere spoilers or even as the players of “intermediate exploitation.” Of course, the reality should entirely be different! The economy in question is not so small and face-to-face matching of the supplier and demander becomes very hard and time-consuming, indicating for the necessity of go-betweens. We have to establish a realistic theory of the trade on the basis of the interactions of the supplier, the demander, and the intermediary. In order to understand the proper functions of the merchant, it is quite important to recognize the existence of possible three gaps between the supplier and the demander. Let us discuss what these gaps are all about.3 The first gap stands for a “location gap,” which may be shown by a horizontal sequence of dots in Chart (A) of Fig. 3.1. For instance, good quality apples are produced in Aomori, the northern end of Honshu, Japan, whereas they are consumed in Tokyo or Osaka, central Japan. Many merchants are necessary to fill the location 2 3

For details, see Ogura (1991). For a detailed discussion of this point, see Ogura (1989).

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3 A Theory of Information and Distribution: The Market Economy and Demand Risk

Fig. 3.1 Three different kinds of gaps: the role of the distributor

gap between the production and consumption of apples. Transportation by trains and trucks may also be carried out by intermediaries. The second gap indicates for a “time gap,” as can be seen by a vertical sequence of dots in Chart (B). Although rice is perhaps the most preferred food in Japan and is eaten almost everyday by most people, its harvest time is quite limited to September or October. The vital function of rice merchants is to fill such time gap by the means of storage and inventory operations. Many weather factors including atmospheric temporature, rainfall, and typhoon should not be ignored. Therefore insurance operations must be needed to take care of probable damages caused by weather, and also by accompanying price fluctuations. The third gap is related to “information gap.” As is seen in Chart (C), the information gap is demonstrated by the wide spread of dotted half-circles from both the producer and consumer sides, which represents to a certain degree of non-perfect transmission of information between both parties. We live in the world of uncertainty in which customers’ tastes and fashions may change drastically and even unpredictably. The argument that convenient stores near main train stations are best located for collecting the demand data quickly and exactly might be quite convincing. It would be safe to say that the company president sitting in a comfortable chair at Tokyo office would be in no position to know what is happening in remote areas in Japan or foreign countries. This is because manufacturers and customers may be different not only geometrically but also culturally. In such a

3.3 The Working of the Market Economy without Uncertainty

49

situation, only the presence of the intermediate merchant would be helpful for gathering the necessary demand data. In short, the merchant is supposed to perform those three vital functions which correspond to filling the three different kinds of gaps: location, time, and information gaps. This paper will pay a special attention to the third information gap, a rather neglected area in the existing literature of economics.

3.3 3.3.1

The Working of the Market Economy without Uncertainty Face-to-Face Trade Between the Producer and the Consumer: Case O

Let us begin our discussion with a simple market economy under full certainty. It is supposed here that the producer and the consumer in an industry under question may trade face-to-face without causing any frictions, and that a “third man” called the distributive intermediary is not needed for trading. This simple case is named Case O. Concerning the demand side, let us assume that the market demand function is described by the following linear function: p ¼ α  βx,

ð3:1Þ

where x and p, respectively, denote the amount of trade and the unit price. The parameter α is a positive constant. Without loss of generality, we may assume that β is just equal to one, so that the demand function is simply written as p ¼ α - x. Regarding the supply side, let us consider the monopoly firm in which its marginal cost c is just constant. Then we may regard the unit price p as a “net price,” namely a “gross price” minus the unit cost c. The producer’s profit is then given by Π ¼ p x ¼ ðα  xÞ x:

ð3:2Þ

In Fig. 3.2, Chart (A) represents our “direct distribution model” under consideration. It is noted here that the producer and the consumer meet each other, literally face-to-face and without troubles or frictions. This means that every kind of distributive intermediary or middleman is conspicuously absent in trading. The producer aims to maximize its profit. If we maximize Π with respect to x, we obtain α-2x ¼ 0. This yieldsα- x0 ¼ x0 at equilibrium, so that the equilibrium profit is calculated as x0 ¼ α/2.It is easy to obtain the equilibrium price and the equilibrium profit:

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3 A Theory of Information and Distribution: The Market Economy and Demand Risk

Fig. 3.2 The economy without uncertainty

p0 ¼ α  x0 ¼ α=2:

2 Π 0 ¼ α  x0 x0 ¼ x0 ¼ α2 =4:

ð3:3Þ ð3:4Þ

We assume that the consumer’s welfare by trading may be well-measured by the amount of the consumer’s surplus. It is easy to see in Fig. 3.3 that such surplus is shown by the area of triangle to be formed by the demand line D over p0. . More exactly, by means of (3.3), we obtain

2 CS0 ¼ α  p0 x0 =2 ¼ x0 =2 ¼ α2 =8:

ð3:5Þ

Since the total surplus is the sum of the producer’s profit and the consumer’s surplus, it follows from (4) and (5) that

3.3 The Working of the Market Economy without Uncertainty

51

Fig. 3.3 Producer and consumer surpluses: Case O and Case I

Table 3.1 The economy without uncertainty: Case O and Case I

Equilibrium values x p q Π Ω PS CS TS

TS0 ¼ Π 0 þ CS0 ¼ 3α2 =8:

Case O α/2 α/2 – α2/4 – α2/4 α2/8 3α2/8

Case I α/4 3α/4 α/2 α2/8 α2/16 3α2/16 α2/32 7α2/32

ð3:6Þ

For the summary of these calculation results, see the second column of Table 3.1. We would like the reader to understand the importance of this table.

52

3 A Theory of Information and Distribution: The Market Economy and Demand Risk

3.3.2

The Entry of the Distributive Intermediary: Case I

We are now in a position to introduce a “third man” on the stage of trading: The distributive intermediary is supposed to enter between the producer and the consumer. Then the structure of the present case (Case I) must become more complicated than the previous case (Case O), as is indicated by Chart (B) of Fig. 3.2 above. Let us further assume that the trading structure is the one of Stackelberg-type, with the producer serving as a leader and the distributor as a follower. As is seen in Chart (B), the producer takes an initiative to set up the producer price q, and then the distributor who takes account of the demand schedule ( p ¼ α -x) determines the amount of trade x for any given q.4 When the distributor is present in the economy, it is important to distinguish two different kinds of prices. They are: the producer price q and the consumer price p. The difference between these two prices ( p-q) stands for what we usually call the “distribution margin.” If we completely ignore distribution costs for the sake of analytical simplicity, we can calculate the distributor’s profit in the following way: Ω ¼ ðp  qÞ x ¼ ðα  q  xÞ x:

ð3:7Þ

As mentioned above, the distributor as a follower takes the producer price q as a given and determines the amount of trade x to maximize its profit Ω. By using (3.7), the first-order condition for such maximization is provided by. α  q  2x ¼ 0:

ð3:8Þ

From this equation, the distributor’s reaction function is derived by. x ¼ ðα  qÞ=2:

ð3:9Þ

If we employ (3.9) as well as the demand eq. (3.1) (note β ¼ 1), we may find the relationship between the consumer price p and the producer price q: p ¼ α  x ¼ α  ðα  qÞ=2 ¼ ðα þ qÞ=2:

ð3:10Þ

The producer as a Stackelberg-type leader, taking account of the distributor’s reaction function (3.9), determines the producer price q so as to maximize its profit:

4

In May 2012, Sakai had a chance to talk to his friend Katuhito Iwai (University of Tokyo) about the functions of merchants in the market economy. Sakai then found that the three possible “gaps” between the producer and the consumer, which Sakai pointed out in this paper, were closely related to Iwai’s unique idea of thinking of “differences” as an indispensable part of the capitalist system. Unquestionably, “gaps” and “differences” seem to be both sides of the same coin. See Iwai (1985).

3.3 The Working of the Market Economy without Uncertainty

Π ¼ q x ¼ q ðα  qÞ=2:

53

ð3:11Þ

The first-order condition of such maximization is given by ðα  2qÞ=2 ¼ 0,

ð3:12Þ

from which immediately follows the producer price at equilibrium: qI ¼ α=2:

ð3:13Þ

The amount of trade and the consumer price at equilibrium are provided by: xI ¼ α=4

ð3:14Þ

pI ¼ α  xI ¼ 3α=4:

ð3:15Þ

On the one hand, it follows from (3.11), (3.13), and (3.14) that the producer’s profit at equilibrium is derived by. Π I ¼ qI xI ¼ α2 =8:

ð3:16Þ

On the other hand, by virtue of (3.7), (3.13), (3.14), and (3.15), the distributor’s profit at equilibrium can be calculated by.
ΩI ¼ pI  qI xI ¼ ½ð3α=4Þ  ðα=2Þ ðα=4Þ ¼ α2 =16:

ð3:17Þ

If we make use of equilibrium values mentioned above, it would be a rather easy job to find the values of the producer surplus PS, the consumer surplus CS, and the total surplus TS: PSI ¼ Π I þ ΩI ¼ 3α2 =16:  CSI ¼ α  pIÞ xI =2 ¼ α2 =32:

ð3:18Þ

TSI ¼ PSI þ CSI ¼ 7α2 =32:

ð3:20Þ

ð3:19Þ

The computation results of Case I will be summarized in the third column of Table 3.1 above.

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3 A Theory of Information and Distribution: The Market Economy and Demand Risk

3.3.3

Comparison between Cases O and I: The Effects of Distributive Intermediaries

We are dealing with the world without uncertainty. The question of interest is whether and to what extent the entry of the distributor between the producer and the consumer influences the working and performance of the market economy. We can find a definite answer by simply comparing the equilibrium values of the two cases: Case O without distributive intermediation and Case I in which the distributor is present. More specifically, making a series of comparisons of equilibrium between the second and third columns in Table 3.1 enables us to straightforwardly establish the following proposition: PROPOSITION 3.1 (Case O Versus Case I) 1. xI < xO, pI > pO. 2. PSI < PSO, CSI < CSO, TSI< TSO The messages of this proposition are quite clear. Property 1 shows that in the world without uncertainty, the entry of the distributor between the producer and the consumer makes the “product circulation pipe” more complicated than otherwise, thus yielding a rise in consumption price p and a decline in the amount of good x. As a result, as Property 2 indicates, the distributive intermediation makes both the producer and the consumer worse off, whence causing a decrease in the total welfare of the society. As can easily be seen, Fig. 3.3 above gives a good illustration of Proposition 1. There are two different kinds of trapezoids which are partially overlapping each other: The smaller dark-shadowed trapezoid, and the larger trapezoid that consists of the dark-shadowed area plus the light-shadowed area. This clearly shows that the total surplus TS is shrunk by the extent of the difference of these two areas through the entry of the distributor. It is also noted that the distance between p I and q I represents what we may call the “distribution margin,” an extra cost caused by the trade distribution.

3.4

Demand Risk and Non-symmetric Information

We are now in a position to introduce the new factor “demand risk” into the industry structure. Needless to say, we live in the world of risk and uncertainty. Main-stream economists, however, have had a tendency to underestimate or even neglect the risk factors. One of the main purposes of this paper is to do our best to somehow mend

3.4 Demand Risk and Non-symmetric Information

55

such “wrong tendency” so that we may make our economics get back to the “right track.”5 Let us suppose that the consumer’s taste is flexible and unpredictable, depending upon a change of his/her own preference and the fashion of the times where he/she lives. In the presence of such demand risk, the producer who lives in a long distance from the consumer may not be in the best position to collect the reliable demand data. There exist some situations in which the producer need to get a help from a “third man” serving as the distributive intermediary between the producer and the consumer, thus filling the gap of demand information. The market economy without the distribution channel would look like “Hamlet without Prince.”

3.4.1

Demand Risk and the Ignorant Producer: Case N

Let us suppose that the demand side is subject to many changes. Then can be seen in Fig. 3.4, it may be well-represented by a simple uniform distribution Φ(α): Φ(α) ¼ 1/2 for α ¼ H, L; Φ(α) ¼ 0 otherwise. For example, the state of business may be either “good” or “bad.” If it is good, the demand parameter takes on a high value, so that we obtainα ¼ H. If it is bad, the parameter takes on a low value, hence we have α ¼ L. Besides, we assume here that the probability of each state is a half, whence Prob (H) ¼ Prob (L ) ¼ 1/2. If we put H ¼ μ + σ and L ¼ μ-σ, then it is easy to see that α and μ, respectively, represent the mean and the standard deviation of the demand intercept α. We are concerned with two opposite cases as indicated in Table 3.2. They are: Case N in which the distributor is ignorant of the true value of α, and Case F where the distributor is fully informed. Let us begin our inquiry with the first case N. It should be noticed that the production strategy of the ignorant distributor must be the “routine action” in the sense that he/she takes on the same strategy regardless to a good or bad state of the economy. The ignorant distributor aims to maximize its expected or average profit: EΠ ¼ E p x ¼ E ðα  xÞ x ¼ x ðμ  xÞ:

ð3:21Þ

The first-order condition for such maximization yields.

5 In the present situation, the producer plays a leading actor and the distributor as a supporting actor on the Stackelberg trading theater. In other situations, however, the parts of both players can be interchanged: The distributor may have greater power than the producer, thus acting as a leader rather than as a follower. This equally important case will deserve a separate investigation.

56

3 A Theory of Information and Distribution: The Market Economy and Demand Risk

Fig. 3.4 The uniform distribution of α

Table 3.2 Demand risk and equilibrium values: Case N versus Case F

Equilibrium values x p q EΠ EΩ EPS ECS ETS

Case N μ/2 (2α-μ)/2 – μ2/4 – μ2/4 μ2/8 3μ2/8

x ðμ  2xÞ ¼ 0, from which follows the equilibrium amount of traded good:

Case F (2α-μ)/4 (2α + μ)/4 μ/2 μ2/8 μ2/16 + σ 2/4 3μ2/16 + σ 2/4 μ2/32 + σ 2/8 7μ2/32 + 3σ 2/8

ð3:22Þ

3.4 Demand Risk and Non-symmetric Information

57

xN ¼ μ=2:

ð3:23Þ

As long as the demand parameter α is a stochastic variable, the market price p is also subject to demand fluctuations. In the light of (3.23), the expected price at equilibrium is given by
E pN ¼ E α  xN ¼ μ=2:

ð3:24Þ

If we continue to adopt a calculation technique similar to the one used for Case O, we can find the expected profit and the expected consumer surplus at equilibrium in the following fashion: EΠ N ¼ xN ECSN ¼ μ—E p


2
N

¼ μ2 =4:

ð3:25Þ

xN =2 ¼ μ2 =8:

ð3:26Þ

These computation results may be summarized in the second column in Table 3.2. Clearly, there exist close similarities between Case O and Case N, the only yet important difference being that α is now replaced by μ, and equilibrium values by expected equilibrium values.

3.4.2

The Effective Entry of the Informed Distributor: Case F

In the world with demand risk, let us consider the entry of the distributor who acquires the demand information. As in Case I, let us suppose that the producer plays as a Stackelberg-leader to set up the production price q, whereas the distributor acts as a Stackelberg-follower to determine the amount of traded good x. Once the distributor is in a position to obtain the demand data of which the producer is ignorant, the distributor’s trade strategy has to undergo a drastic change. It is no longer the “routine action” but the “contingent action” in the sense that its trade volume should be flexible in response to the state of the economy: it increases or decreases the amount of trade x according to whether the prospect of its demand is good or bad. In contrast, the ignorant producer has to stick to the routine strategy as before. To begin with, let us consider the behavior of the informed distributor as a follower. Since the distributor acquires the demand information, it aims to determine the best contingent strategy x in response to the value of α so as to maximize its profit for a given the production price q set up by the producer as a leader. Ω ¼ px  qx ¼ ðα  q  xÞ x

ð3:27Þ

58

3 A Theory of Information and Distribution: The Market Economy and Demand Risk

The first-order condition for profit maximization leads to from which follows the reaction function of the distributor: α  q  2x ¼ 0,

ð3:28Þ

x ¼ ðα  qÞ=2:

ð3:29Þ

It is worthy of attention to see that the reaction function is now dependent on the value of α. This is clearly because the informed distributor can acquire the demand information. Let us turn to the behavior of the ignorant producer. The producer as a leader, taking account of the distributor’s reaction (3.29), aims to set up the production price q so as to maximize its expected profit: EΠ ¼ E qx ¼ E q ðα  qÞ=2 ¼ q ðμ  qÞ=2:

ð3:30Þ

The first-order condition for maximization yields from which follows the production price at equilibrium: ðμ  2qÞ=2 ¼ 0,

ð3:31Þ

qF ¼ μ=2:

ð3:32Þ

In the light of (3.29), we find the amount of traded good at equilibrium:
xF ¼ α  qF =2 ¼ ½α  ðμ=2Þ=2 ¼ ð2α  μÞ=4,

ð3:33Þ

whose expectation leads to the following: ExF ¼ ð2 μ  μÞ=4 ¼ μ=4:

ð3:34Þ

By making use of (3.33), we can obtain the consumption price at equilibrium: pF ¼ α  xF ¼ α  ð2α  μÞ=4 ¼ ð2α þ μÞ=4,

ð3:35Þ

whose expectation results in E pF ¼ ð2 μ þ μÞ=4 ¼ 3 μ=4:

ð3:36Þ

Let us find a series of expected surpluses at equilibrium for Case F. First of all, since by means of (3.32) and (3.33), we have.

3.4 Demand Risk and Non-symmetric Information

59

Π F ¼ qF xF ¼ ðμ=2Þ ½ð2α  μÞ=4 ¼ μð2α  μÞ=8,

ð3:37Þ

we obtain the expected producer’s surplus at equilibrium: EΠ F ¼ μð2 μ  μÞ=8 ¼ μ2 =8:

ð3:38Þ

Note that the distribution margin at equilibrium can be calculated by. pF  qF ¼ ð2α þ μÞ=4  ðμ=2Þ ¼ ð2α  μÞ=4:

ð3:39Þ

Therefore we obtain

ΩF ¼ pF  qF xF ¼ ð2α  μÞ2 =16 ¼ 4α2  4αμ þ μ2 =16,

ð3:40Þ

whose expectation leads to.

EΩ ¼ 4Eα2  4 μ2 þ μ2 =16 ¼ 4Eα2  3 μ2 =16:

ð3:41Þ

Let us recall the convenient formula: Eα2 ¼ μ2 þ σ 2 ,

ð3:42Þ

Then we can derive the expected distributor’s surplus at equilibrium: 

EΩF ¼ 4 μ2 þ σ 2  3μ2 =16 ¼ μ2 =16 þ σ 2 =4:

ð3:43Þ

Since the expected producer surplus is simply the sum of the expected producer’s surplus and the expected consumer’s surplus, it follows that EPSF ¼ EΠ F þ EΩF ¼ μ2 =8 þ μ2 =16 þ σ 2 =4




¼ 3μ2 =16 þ σ 2 =4:

ð3:44Þ

As said before, the consumer’s surplus is measured by the area of triangle formed by the demand line over the consumer price. Hence the expected consumer surplus at equilibrium is provided by ECSF ¼ E




 α  pF xF =2 ,

which in view of (3.33) and (3.35), may be transformed to

ð3:45Þ

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3 A Theory of Information and Distribution: The Market Economy and Demand Risk

ECSF ¼ E ½ð2α  μÞ=42 =2 ¼ E½2α  μÞ2 =32: Since we have E [(2α-μ)2] ¼ μ2+4σ 2 by means of (3.42), it follows that ECSF ¼ μ2 =32 þ σ 2 =8:

ð3:46Þ

Hence, the expected total surplus is computed as.

ETSF ¼ EPSF þ ECSF ¼ 3 μ2 =16 þ σ 2 =4 þ μ2 =32 þ σ 2 =8 ¼ 7 μ2 =32 þ 3σ 2 =8:

ð3:47Þ

These computation results for Case F may be summarized in the third column of Table 3.2. In contrast to Case N where the distributor is ignorant of the demand parameter α and only the value of mean μ is relevant, the value of variance σ 2 is newly added in Case F with the informed distributor being present, presumably playing a critical role in the assessment of our equilibrium values.

3.4.3

The Intermediation Effect Versus the Information Effect

We are interested in asking how and to what extent the entry of the distributor influences the whole picture of the market economy. As we discussed before, if the distributor is ignorant of the demand data, mere presence of the distributor between the producer and the consumer will make the distribution channel unnecessarily more complicated than otherwise, thus badly affecting the working of the economy: The producer, the consumer, and the society will be all worse off. This is really a Pareto-inferior situation. When the distributor acquires the demand information, however, the whole picture is expected to drastically change. Presumably, there would be two different effects working in opposing directions. They are: the negative effect of complication caused by distributive intermediation and the positive effect of caused by collection of the demand information. The first and second effects may, respectively, be called the intermediation effect and the information effect. The key question would be which one of the two effects becomes stronger. The answer should be like this. “It depends!” In some, but not all, probable situations, a kind of Pareto-improving situation could emerge by the introduction of the informed distributor into the economy. What we need to do is a more detailed analysis based on exact calculations.

3.4 Demand Risk and Non-symmetric Information Table 3.3 The entry of the informed distributor: its welfare implications

The Value of σ2/μ2 (1) σ 2/μ2 > 3/4 (2) 5/12 < σ2/μ2 < 3/4 (3) 1/4 < σ 2/μ2 < 5/12 (4) σ 2/μ2 < 1/4

61 ΔEPS + + + 

ΔECS +   

ΔETS + +  

If we compare corresponding equilibrium values on Case N and Case F, we can establish the following important proposition: PROPOSITION 3.2 (Case F versus Case N). 1. ExF < ExN, EpF > EpN. N 2 2 > 2. EPSF > < EPS , σ /μ < 1/4 . F > N 2 ECS < ECS , σ /μ2 > < 3/4. N 2 2 > ETSF > ETS , σ /μ < < 5/12. In the light of Table 3.2, the proof of this proposition is rather straightforward. First of all, comparing equilibrium values of x for the two cases, we find. ExF  ExN ¼ μ=4  μ=2 ¼ μ=4,

ð3:48Þ

which is negative. In a similar fashion, we obtain EpF  EpN ¼ 3 μ=4  μ=2 ¼ μ=4,

ð3:49Þ

which is positive. This completes Property 1. The meaning of this property is very clear. The introduction of the informed distributor causes a decline in the expected amount of good, and a rise in the expected consumer price. Interestingly enough, the mean μ is present in (3.48) and (3.49), but the variance σ 2 is not there. This implies that only the negative intermediation effect is working, thus yielding Property 1, Next, if we compare corresponding equilibrium values in Table 3.2, it is a rather easy job to derive.
EPSF  EPSN ¼ 3 μ2 =16 þ σ 2 =4  μ2 =4 ¼ μ2 =16 þ σ 2 =4:

ð3:50Þ

It is noted that the most right-hand side of the above equation consists of the two terms. They are: the negative intermediation effect associated with μ, and the positive information effect related to σ 2. The relative strength of these two effects is not one-sidedly determined, depending on the values of μ and σ 2. In a similar fashion, we can also obtain. ECSF  ECSN


¼ μ2 =32 þ σ 2 =8  μ2 =8 ¼ 3 μ2 =32 þ σ 2 =8:

ð3:51Þ

62

3 A Theory of Information and Distribution: The Market Economy and Demand Risk

ETSF  ETSN


¼ 7 μ2 =32 þ 3σ 2 =8  3 μ2 =8 ¼ 5 μ2 =32 þ 3σ 2 =8:

ð3:52Þ

As is clear enough, the most right-hand sides of (3.51) and (3.52) contain both the negative intermediation term and the positive information term. Here again, which one of those two terms is more powerful is very critical, depending on the values of μ and σ 2. In the light of (3.50), (3.51), and (3.52), it is a rather easy work to obtain Property 2. In our opinion, Property 2 of the above proposition is very important, perhaps representing the best result in this paper. In order to fully understand it in a deeper perspective, it is very useful to construct a comprehensive table like Table 3.3. A close look at Table 3.3 teaches us a very important lesson. The value of the ratio σ 2/μ2 plays a critical role in determining the welfare effects of the entry of the informed distributor. More precisely, this is the ratio of the variance of the demand parameter to its mean square, relatively measuring the state of spread of α around μ. The greater the ratio, the greater is the degree of demand risk. On the one hand, when the degree of the demand risk is large enough in the sense that the ratio (σ 2/μ2) exceeds 3/4, acquisition by the distributor of the demand information is so important. In this case, the positive information effect is expected to overpower the negative intermediation effect, whence the entry of the informed distributor makes all the parties involved better off: ECS, MPS, and ETS are all expected to rise. On the other hand, if the degree of the risk is small enough in that the ratio is less than 1/4, just the opposite results would happen: the intermediation effect overtakes the information effect. The presence of the informed distributor makes all the parties worse off. Between these opposite cases, there exists an intermediate range. If the variancemean square ratio is smaller than 3/4 but larger than 1/4, the producer is better off, but the consumer is worse off. This is neither a Pareto-superior situation nor a Pareto-inferior one, but something between. Even in that range, if the ratio is large enough to exceed 5/12 (yet less than 3/4), the distributor’s entry contributes to an increase in ETS, so that a possible side payment from the producer to the consumer would make all the parties better off. To sum up, in the world with demand risk, the introduction of the informed distributor into the economy would produce two mutually opposing effects: the negative intermediation and the positive information effects. If the degree of the risk is large enough, then the information effect would become a dominant force, so that all the parties would be better off. Presumably, such a situation would do justice to the existence of the distributive intermediary. We could say that this is like an arranged marriage. The entry of a good match maker with good experience could bring happiness to a hesitant couple.

3.5 John Hicks on Theory and History: Concluding Remarks

3.5

63

John Hicks on Theory and History: Concluding Remarks

As the saying goes, old memories die hard. In his long academic career, the year of 1972 stood up as a very special year when Sakai was teaching economic theory at the University of Pittsburgh, USA. Personally speaking, in that year, Sakai successively received a Ph.D. degree of economics from his alma mater, the University of Rochester, USA. It seemed to be a great accomplishment to him. Although around 50 years have passed since then, he still remains to be very grateful to many colleagues and friends for an unofficial celebration party in Pittsburgh. We should add to say, however, that 1972 meant a special year not only because of the personal reason aforementioned. In the same year, a Nobel Economic Prize ceremony took place in Sweden, and one of the prize winners was John Hicks, a towering economist in the twentieth century. Academically speaking, Hicks was a sort of Sakai’s grandfather in two ways: Hicks at Oxford was once the teacher of Lionel W. McKenzie, Sakai’s mentor at Rochester. Not only that, Hicks was also the guru of Michio Morishima whom both Sakai and Sasaki admired as an inspiring idol.6 In later years when we happened to read Hicks’ later essays Economic Perspectives (1977), we were really shocked like a thunder out of blue sky to see the following sentence in its Preface: They gave me〔John Hicks〕a Nobel prize (in 1972) for my work on ‘general equilibrium and welfare economics’, no doubt referring to Value and Capital (1939) and on the papers on Consumers’ Surplus which I wrote soon after that date. But it was done a long time ago, and it was with mixed feelings that I myself felt myself to have outgrown” (Hicks 1969, p. 6). The mixed feelings Hicks felt at that time was also recorded by Morishima on his later book Modern Economic as a Thought (1993): Hicks’ s research area has been very wide, covering so much topics. Among so many writings of Hicks, I myself〔Morishima himself」liked A Tory of Economic History and A Market Theory of Money so much. When I read the former book, I asked him [Hicks], “Would you like to switch your work to write books a la Max Weber?

6 It is in the 1980s and the 1990s that the theory of information and oligopoly was intensively discussed in the economics profession for the first time. Active researchers included Gal-Or (1985, 1986), Sakai (1982, 1984, 1985, 1987), Vives (1984, 1987), and others. While an interest in the theory had been a bit on wane since then, Sakai believes that a unifying work of imperfect competition and imperfect information remains unfinished, and its applications to distribution and regional problems will be left for further research. In fact, a group of research papers on oligopoly and information have come back in the 1990s and the 2000s. For instance, see Vives (1999, 2002, 2008), Sakai (1990a, 1990b, 1991, 1993, 2015), Sakai and Yoshizumi (1991a, 1991b), Sakai and Sasaki (1996), Demange and Laroque (1995), Raith (1996), Jin (1998), and many others. For a detailed discussion on risk and uncertainty, see Sakai (2010, 2019).

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3 A Theory of Information and Distribution: The Market Economy and Demand Risk

He then replied, “I would not think so.” A bit later on, however, he told me, “I would have felt a lot happier if I had been awarded a Nobel Prize for A Theory of Economic History. It really demonstrated his own evaluation by which his later work [on Economic History] was of greater importance than his earlier work [on Value and Capital] (Morishima 1993, p. 123). It seems to us that whereas Hicks’ contributions to economic science is wide and deep, the link between theory and history nevertheless remains a missing link to be filled. In this connection, it is remarkable to see that he wrote the following sentence: Where shall we start? There is a transformation which is antecedent to Marx’s Rise of Capitalism, and which, in terms of more recent economics, looks like being more fundamental. This is the Rise of the Market, the Rise of the Exchange Economy. It takes us back to a much earlier stage of history, at least for its beginnings: so far back indeed that on those beginnings (or first beginnings) we have little direct information. But there are several ways in which we can deduce, fairly reliably, what must be occurred” (Hicks 1969, p. 7). In our opinion, one of several ways to discuss the Rise of the Market is the way in which in this paper, we intensively discussed the informational role of the distributive intermediary in the economy. In standard economics textbooks, the role of the distributor or the merchant has been overly underestimated or completely ignored. It would be high time to correct such an unfair treatment in economic science. Needless to say, there would remain so many problems to be left over for future research. Hopefully, this paper would show us a right direction towards the New Economic Science unifying history and theory.

References Demange G, Laroque J (1995) Private information and the design of securities. J Econ Theor 65:233–257 Gal-Or E (1985) Information sharing in oligopoly. Econometrica 53:329–343 Gal-Or E (1986) Information transmission: cournot and bertrand equilibria. Rev Econ Stud 53:85–92 Hicks J (1969) A theory of economic history. Oxford University Press, Oxford Hicks J (1977) Economic perspectives: further essays on money and growth. Oxford University Press, Oxford Iwai K (1985) The Venice merchant’s theory of capital (in Japanese). Chikuma Publishers, Tokyo Jin JY (1998) Information sharing about a demand shock. J Econ 68(2):137–152 Matsushita K (1974) In: Ishiyama S, Koyanagi M (eds) Konosuke Matsushita remembers: the secret of business (in Japanese). Diamond Publishers, Tokyo Morishima M (1993) Modern economics as a thought (in Japanese). NHK Publishers, Tokyo Nakauchi I (1982) Inspiring speeches of Isao Nakauchi (in Japanese). Chuo Keizai Publishers, Tokyo Ogura E (1989) The innovative power of development by the Ohmi merchant (in Japanese). Chuo Keizai Publishers, Tokyo Ogura E (1991) The Ohmi merchant: new ideas from old teachings (in Japanese). Sunrise Publishers, Hikone Raith M (1996) A general model of information sharing in oligopoly. J Econ Theor 71:260–288

References

65

Sakai Y (1982) Economics of uncertainty (in Japanese). Yuhikaku Publishers, Tokyo Sakai Y (1984) The role of information in a Stackelberg-type duopolistic market. Discussion paper: 84 - 1, University of Tsukuba Sakai Y (1985) The value of information in a simple duopoly model. J Econ Theor 36 Sakai Y (1987) Cournot and Bertrand equilibria under imperfect information. J Econ 46:213–232 Sakai Y (1990a) Theory of oligopoly and information (in Japanese). Toyo Keizai Publishers, Tokyo Sakai Y (1990b) Information sharing in oligopoly: overview and evaluation. Part: alternative models with a common risk. Keio Econ Stud 27:17–41 Sakai Y (1991) Information sharing in oligopoly: overview and evaluation. Part: private risks and oligopoly models. Keio Econ Stud 28:51–71 Sakai Y (1993) The role of information in profit-maximizing and labor-managed duopoly models. Managerial Decision Econ 14:419–432 Sakai Y (2010) Economic thought of risk and uncertainty (in Japanese). Minerva Publishers, Tokyo Sakai Y (2015) Risk aversion and expected utility: the constant-absolute-risk aversion function and its application to oligopoly. Hikone Ronso 403:172–187 Sakai Y (2019) J.M. Keynes versus F.H. Knight: risk, probability, and uncertainty. Springer Nature Singapore, Singapore Sakai Y, Sasaki K (1996) Demand uncertainty and distribution systems: information acquisition and transmission. In: Sato R, Hori H, Ramachandran R (eds) Organization, performance, and equity. Kluwer Academic Press, New York Sakai Y, Yoshizumi A (1991a) The impact of risk aversion on information transmission between firms. J Econ 53:51–73 Sakai Y, Yoshizumi A (1991b) Risk aversion and duopoly: is information exchange always beneficial to firms? Pure Math Appl (Ser B) 2-2:129–145 Sima Qian (n.d.) (109B.C-91B.C.) Shi ji, or the records of the grand historian (in Chinese). Ancient China. Japanese Translation by Ogawa K, Imataka M, Hukushima Y (1975) Shiki. Iwanami Publishers, Tokyo Vives X (1984) Information equilibrium: cournot and bertrand. J Econ Theor 34:71–94 Vives X (1987) Trade association, disclosure rules, incentives to share information and welfare. Discussion paper 87-1. University of Pennsylvania Vives X (1999) Oligopoly pricing: old ideas and new tools. MIT Press, Cambridge Vives X (2002) Private information, strategic behavior, and efficiency in Cournot markets. RAND J Econ 33:361–376 Vives X (2008) Information and learning in markets: the impact of market microstructure. Princeton University Press, Princeton

Chapter 4

Information Exchanges among Firms and Their Welfare Implications (Part I): The Dual Relations between the Cournot and Bertrand Models

Abstract This long series of papers consist of three parts. Part I is concerned with the basic dual relations between the Cournot and Bertrand models. Part II begins to deal with the world of risk and uncertainty, with a discussion of the Cournot duopoly model with a common demand risk as a starting point. It then deals with other types of duopoly models with a common risk. Part III discusses more complicated problems such as private risks and oligopoly models. All these three parts taken together aim to carefully outline and critically evaluate the problem of information exchanges in oligopoly models, one of the most important topics in contemporary economics. The true motivation of writing such survey papers is to strive for a synthesis of the economics of imperfect competition and the economics of imperfect information. The problem at issue is how and to what extent the information exchanges among firms influence the welfare of producers, consumers, and the whole society. It is seen in the paper that a definite answer to the problem really depends on the following many factors. (1) The type of competitors (Cournot-type or Bertrand-type), (2) the nature of risk (a common value or private values; demand risk or cost risk), (3) the degree and direction of physical and stochastic interdependence among firms, and (4) the number of firms. If any set of those factors is specified in a given oligopoly model, the welfare and policy implications may very systematically be derived by way of their decomposition into the following four effects. That is, (i) own variation effects, (ii) cross variation effects, (iii) own efficiency effects, and (iv) cross efficiency effect. In the real world, trade associations may be regarded as typical information exchange mechanisms. Many welfare implications obtained in the papers will shed a new light to the effectiveness and limitations of those trading groups. Keywords Information exchange · oligopoly models · welfare implications · trade associations · Cournot · Bertrand

This paper is a completely revised version of the opening part of Sakai (1989). © Springer Nature Singapore Pte Ltd. 2021 Y. Sakai, K. Sasaki, Information and Distribution, New Frontiers in Regional Science: Asian Perspectives 49, https://doi.org/10.1007/978-981-10-0101-7_4

67

68

4.1

4 Information Exchanges among Firms and Their Welfare Implications (Part I): The. . .

Cournot as the Great Founder of Oligopoly Theory: An Introduction

There is a historical episode by Joseph A. Schumpeter (1883–1950), one of the greatest economists in the twentieth century. When Schumpeter was a distinguished professor at Harvard University in the 1930s, he surprised all the attending students, one of whom was the young Paul A. Samuelson (1915–2005), by saying the following remark1: Listen, everybody. In the light of the long history of economics, it seems to us that there exist four great economists. Believe or not, three out of those four are Frenchmen! Can you guess who they really are? (Schumpeter, quoted by Yasui 1979, p. 10). According to Schumpeter’s opinion, F. Quesnay (1694–1774), A. A. Cournot (1801–1877), and M. E. L. Walras (1834–1910) were strong candidates for such an exclusive economics club. Surely, Quesnay was very famous of newly inventing an economic table as the flow chart of all the economic activities in a national economy. Cournot was a born mathematician, later applying a mathematical approach to oligopoly theory. Walrus was regarded as a pioneer of modern general equilibrium theory, thus being admired so much by Lionel W. McKenzie (1923–2008) when Sakai himself was a graduate student at the University of Rochester in the 1960s. There remains one question unanswered: who should be the last of the four great economists according to Schumpeter’s preference? One thing is for certain. The last person should not be a Frenchman. He could have been Adam Smith (1723–1790), the author of the epoch-making book The Wealth of Nations (1776). Or possibly, Carl Menger (1840–1921), an important member of the influential Austrian School? Or J. G. K. Wicksell (1851–1926), the founder of the outstanding Swedish School? Or J. M. Keynes (1883–1946), the author of the revolutionary book The General Theory (1936)? Or perhaps Schumpeter himself? No one really knows. Fortunately or unfortunately, Schumpeter kept his mouth shut until his death, thus contributing to the creation of a great mystery in the history of economic thought. There was another interesting episode which connected Schumpeter with Cournot, when Schumpeter was teaching economics at the University of Bonn long before the Second World War. The following question was asked by Schumpeter to the young Nakayama, who was then a visiting foreign student at Bonn and later became a leading professor of modern economics in the Japanese academics. “Mr. Nakayama, I [Schumpeter] would like to ask you how you have managed to study economics before coming to Germany.” Nakayama’s answer was simple, yet gave Schumpeter a really nice surprise. “Yes, Sir. I [Nakayama] have carefully read the works of Cournot, Gossen and Walrus under the direction of my Japanese mentor” (Schumpeter and Nakayama, quoted by Yasui 1979, p. 13).

1

See Yasui (1979).

4.1 Cournot as the Great Founder of Oligopoly Theory: An Introduction

69

Honestly speaking, in spite of his monumental works, Cournot has been mostly underestimated with a few exceptions. One outstanding exception was Schumpeter, who as mentioned before, very highly evaluated Cournot. In such long survey of papers, we would like to gladly share this Schumpeter spirits, thus critically evaluating and freely extending Cournot’s work on oligopoly to the world of risk and uncertainty. In this connection, it is also worthy of attention to record the following sentence by John Hicks (1904–1994), one of the greatest economists in the twentieth century: The generally increased interest in mathematical economics during the last few years has naturally turned attention back to the work of Cournot, the great founder of the subject, and still one of best teachers. It was Cournot’s creation of elementary monopoly theory which was the first great triumph of mathematical economics; yet Cournot had left much undone. It is not surprising that the endeavor to complete his work have been an attractive occupation for his successors (Hicks 1935, p. 1). As Hicks noted in his survey paper (1935), Cournot was regarded as the great founder of the theory of monopoly and oligopoly, and still one of the best teachers in the 1930s. It was quite fortunate rather than unfortunate that Cournot had left much undone, thereby the endeavor to make his work complete has been continuously an attractive task for his successors until today. It is our sincere hope that I will be one of his good successors. More exactly speaking, more than 180 years have passed since the publication of a legendary book Recherches sur les principes mathématiques de la theorie des richesses by Cournot (1838). It is appropriate as well as important to see how and to what extent Cournot’s pioneering work has contributed to the economics profession. One of the main goals of this paper is to show that Cournot is academically alive and indeed very much alive, and continue to be so. Cournot used to be called an insolent founding father. Even before the Marginal Revolution in the 1870s, he invented marginal concepts such as marginal revenue, marginal cost, demand elasticity, and the like. More than 100 years before the appearance of Game Theory, he made full use of a very important concept of equilibrium in non-cooperative games―Nash equilibrium. It is also worth mentioning that it took as many as 45 years for Cournot’s great book to be reviewed by Bertrand (1884). The three chapters taken together
Chapters 4–6
aim to overview and evaluate the problem of information exchanges in oligopoly models, one of the most fashionable topics in contemporary economics. It is intended to discuss a synthesis of the two important fields, the economics of imperfect competition and the economics of imperfect information.2 2 The imperfect competition revolution took place in the 1930s, with J. Robinson (1933), Chamberlin (1933), and Stackelberg (1934) being its front runners. We would add to say that another equally important revolution
the imperfect information revolution
happened in the 1970s in which Arrow (1970), Akerlof (1970), Stiglitz (1975a, b), and Spence (1974) were primary promoters. The nature and significance of this new revolution was intensively discussed by Sakai (1982). For the economic thought of risk and uncertainty, see Sakai (2010).

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4 Information Exchanges among Firms and Their Welfare Implications (Part I): The. . .

Needless to say, the issue of information transmissions and exchanges among producers is important not only from a theoretical point of view, but also from antitrust policy implications. In the real economy, there exist several types of institutions in which producers exchange their private information with each other. Trade associations are among those information-pooling mechanisms. In order to determine under what conditions the information exchanges among producers should be encouraged or discouraged in terms of the consumer welfare or the total social welfare, it is first necessary to fully understand the working and performance of oligopoly markets under the conditions of imperfect information. Let us consider the market of either a homogenous or differentiated product. Then, this paper addresses to the following set of questions. 1. First of all, are firms with different demand and/or cost functions willing to reveal or share information about demand or cost? 2. Next, how and to what extent does such information transmission affect consumers and the whole society? 3. And finally, are the welfare implications of information exchanges sensitive to the number of participating firms? In the 1970s and the 1980s, we observed an explosion of papers discussing those questions. The line of research was initiated by Basar and Ho (1974) and Ponssard (1979), and continued by many works in the 1980s including Novshek and Sonnenschein (1982), Clarke (1983), Vives (1984), Okada (1982), Sakai (1985, 1986, 1987, 1989), Gal-Or (1985, 1986, 1987), Li (1985), Shapiro (1986), Nalebuff and Zwckhauser (1986), and many others. Besides, there have appeared a number of remarkable papers in the 1990s and even in the 2000s.3 At the first glance, there appear no definite answers in the existing literature, so that the antitrust implications of information sharing in oligopoly might be far from clear. In some papers, firms are assumed to behave as Cournot competitors whereas in others, they are regarded as Bertrand competitors. There may exist a common risk or private (or firm-specific) risks. Risk may be about the demand or cost side. Products may be homogenous or differentiated. Even if differentiated, they may be substitutes, independent or complements. When there exist more than two sources of risks, they may be positively or negatively correlated. Besides, the number of participation firms may be just two, three, four, ..., or any finite number. It is generally expected that different models lead to different consequences. The problem of information exchanges in oligopoly models has no exception for such a universal rule. Once a specific set of assumptions is made to describe an oligopoly model to work with, however, a definite set of answers will be obtained. The following set of items must be checked:

3 For instance, see Vives (1990, 1992, 1999, 2008). Kühn and Vives (1994), Sakai (1990), Sakai and Yamato (1990). All of those works on oligopoly and information were done on the mathematical basis of the theory of games with incomplete information (see Harsanyi (1967–1968)). And, they were also applied to the evaluation of industrial policies in the real economies (see Komiya (1975)).

4.1 Cournot as the Great Founder of Oligopoly Theory: An Introduction

71

1. 2. 3. 4.

The type of competitors (Cournot-type or Bertrand-type). The type of risks (demand or cost risks). The nature of risks (one common risk or several private risks). The degree of physical and stochastic correlation among firms (positively or negatively correlated). 5. The number of firms (two, three, four, or any finite number).

In this chapter, a wide variety of oligopoly models will successively be introduced, and the problem how a change in one of those assumptions may result in a corresponding change in some of welfare results will be the focus of investigation. For the sake of presentation and also subject to the space constraint, however, little or no attention will be paid to some other related issues such as those of risk aversion, measurement errors, partial sharing, garbling, first-mover versus secondmover advantages and the like. While there may exist many possible models regarding information exchanges in oligopoly models as mentioned above, it is remarkable to see that there is only one mathematical approach to such problems, namely the approach based on game theory. As is well-known, game theory has played a key role in integrating the two branches of economics into one, the Economics of Imperfect Competition and the Economics of Imperfect Information. In fact, a recent body of work in oligopoly has been associated with the application of many concepts borrowed from game theory, with the concept of Nash equilibrium continuing to be a dominant one.4 Even if each of various models aforementioned is set up on the stage, it is no easy task to systematically analyze all the welfare effects of information exchanges among firms, and to provide clear-cut and intuitive interpretations for the results. When the problem at issue is too complicated to seize the essence of the matter, it is a well-established wisdom to break it into several pats and to examine the welfare results piecewise before knitting them together. As will be seen, the consequences of information exchanges among firms can be classified under the four headings; namely, own and cross variation effects and own and cross efficiency effects. Interestingly enough, the first effect or the own variation effect represents how information flows affect the variability of each firm’s strategic variable (each output for Cournot models or each price for Bertrand models), whereas the second effect or the cross variation effect shows how it influences the degree of strategic interdependence among firms. The third and fourth effects demonstrate in which direction information exchanges contribute to the efficiency of resources on an industry-wide basis. In particular, whereas the own efficiency effect is related to a better or worse correspondence between each stochastic parameter (the demand intercept or the unit cost) and its associate strategic variable, the cross efficiency

The theory of games was first established as the joint product of a born mathematician and a noted economist in von Neumann and Morgenstern (1944), and later developed by the “man with beautiful mind,” namely, Nash (1951). For its applications to oligopoly problems, see Shubik (1982), J.M. Friedman (1986), and more recent works by many others. Also see Marschack and Radner (1972). 4

72

4 Information Exchanges among Firms and Their Welfare Implications (Part I): The. . .

effect is connected with a changed correspondence between the stochastic parameter of one firm and the strategic variable of the other firm. It will be seen that those four effects provide quite useful tools by which to trace the welfare implications of information exchanges among firms. As was seen in the detailed contents at the beginning, the long series of papers consist of three parts: namely, Part I, Part II, and Part III. Chap. 4 will correspond to Part I, and aims to set up a basic framework of differentiated duopoly in the absence of any risks. The dual relations between the Cournot and Bertrand models will carefully be investigated. More specifically, it is noted that the Cournot model with substitutable (or complementary) goods is the dual of the Bertrand model with complements (or substitutes). Part I also prepares for later discussions on extensive comparisons of more general oligopoly models on the basis of the type of competitors, the type of risks, and the number and nature of risks. Part II will be discussed in the Chap. 5. It begins to deal with the world of risk and uncertainty, with a discussion of the most fundamental model: the Cournot duopoly model with a common demand risk. It will be followed by the same Cournot model with a common cost and by the corresponding Bertrand models with a common demand or cost risk. Part III will be the target of Chap. 6. In this last part of the series, many possible extensions of the duopoly results to very general oligopoly situations will be carried out.

4.2 4.2.1

Alternative Duopoly Models with and without Risk Factors The Dual Relations Between the Cournot and Bertrand Duopoly Models: The World of Perfect Information

As was noted above, there exist two types of competition (Cournot or Bertrand), two more types of risk (demand or cost), and still two more types of information structures (a common value or private values). Therefore, when all the possible combinations are considered, eight different types of oligopoly models will have to be discussed. This could probably constitute a very repetitious and even tiresome task. Fortunately, there would be a great help from the duality argument! Indeed, as will be seen below, there exist the nice dual relations between the Cournot and Bertrand models in the world of perfect information. Let us pick up any two arbitrary models. Then if they share the same formal structure and differ only in the interpretation placed on variables and parameters, we say that they are dual. One natural consequence of such duality argument is that a proposition derived for one model can become a proposition for the other if variables and parameters are duly interchanged between the two models. In the light of the history of economic thought, it is Cournot himself who was close enough yet fell short of adopting what we may now call a dual approach to oligopoly theory.

4.2 Alternative Duopoly Models with and without Risk Factors

73

Cournot has established the important proposition that the output supplied under duopoly is greater than the output supplied under pure monopoly. Since the market demand curve is usually downward sloping, this implies that the price charged under duopoly is lower than the price charged under pure monopoly. While there is the wide range of physical interdependence between two outputs, Cournot discussed only the two extreme cases, namely, the case of perfect substitutes and the one of perfect complements. This clearly indicates the limitations of the original analysis of Cournot, thus showing the necessity to extend it to a wider range of intermediate cases between those of perfect substitutes and perfect complements.5 The model we are going to analyze here is the following non-stochastic duopoly model with differentiated products and/or cost differences. On the production side, we have a duopolistic sector with firms 1 and 2, each one producing a differentiated product, and a competitive numéraire sector. Let x0 be the output of the numéraire good, xi be the output of the i th firm, and pi be its unit price (i ¼ 1, 2). The unit price of x0 is of course unity, namely p0 ¼ 1. On the consumption side, we have a continuum of consumers of the same type with utility functions which are linear and separable in the numéraire good. For tractability, it is assumed that the utility function U of the representative consumer is quadratic:
 U ¼ x0 þ α1 x1 þ α2 x2
ð1=2Þ βx1 2 þ 2βθx1 x2 þ βx2 2 ,

ð4:1Þ

where αi and β are all positive, and the value of θ lies between
1 and 1. Let us assume that the consumer is supposed to maximize U subject to the budget constraint, x0 + p1x1 + p2 x2 ¼ m, where m denotes his given income. Then it can easily be seen that inverse demand functions are given by the following set of linear equations: p1 ¼ α1
βx1
βθx2 ,

ð4:2Þ

p2 ¼ α2
βx2
βθx1 :

ð4:3Þ

If we use matrix notation, we can summarize (4.2) and (4.3) as follows: 

p1 p2



 ¼

α1 α2




β

1

θ

θ

1



x1 x2

 :

Now, assuming that α1
α2θ > 0 and α2
α1θ > 0, let us put.

5 There is now a growing body of literature dealing with the working and performance of oligopoly markets under product differentiation, centering around the duality and efficiency comparison between Cournot and Bertrand equilibriums. For its earlier works, see Singh and Vives (1984), Vives (1984), Okuguchi (1986), and Sakai (1986). Also see Suzumura and Okuno-Fujiwara (1987).

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4 Information Exchanges among Firms and Their Welfare Implications (Part I): The. . .


 a1 ¼ ðα1
α2 θÞ=β 1
θ2 ,
 a2 ¼ ðα2
α1 θÞ=β 1
θ2 ,
 b ¼ 1=β 1
θ2 : It is then easy to see that these newly introduced parameters, a1, a2 and b, are all positive. In the light of (2) and (3), it is not hard to obtain the following set of direct demand equations: x1 ¼ a1
bp1 þ bθp2 ,

ð4:4Þ

x2 ¼ a2
bp2 þ bθp1 :

ð4:5Þ

Alternatively, in matrix notation, we have the following equation: 

x1



 ¼

x2

a1 a2




b

1


θ


θ

1



p1 p2

 :

It is noted that the value of θ stands for a good measure of the substitutability of the two products. In fact, x1 and x2 can be regarded as substitutes, independent, or complements according to whether θ is positive, zero, or negative. We assume that the technology exhibits constant returns to scale, so that firm i has constant unit cost κi. Profits of firm i are provided by Π i ¼ ( pi
κi) xi. It is noted that Π i is not symmetric in pi and xi unless κi vanishes. In general, the Π i functions treat ( pi
κi) and xi symmetrically. In order to make symmetric treatment clearer, it is instructive to reformulate (4.2)–(4.5) in the following way: p1
κ1 ¼ ðα1
κ1 Þ
βx1
βθx2 ,

ð4:6Þ

p2
κ2 ¼ ðα2
κ2 Þ
βx2
βθx1 ,

ð4:7Þ

x1 ¼ ða1
bκ1 þ bθκ 2 Þ
b ðp1
κ1 Þ þ bθðp2
κ2 Þ,

ð4:8Þ

x2 ¼ ða2
bκ2 þ bθκ 1 Þ
b ðp2
κ2 Þ þ bθ ðp1
κ1 Þ:

ð4:9Þ

In matrix notation, these four equations may be rewritten as follows: 

p1
κ 1 p2
κ 2



 ¼

α1
κ1 α2
κ2




β

1

θ

θ

1



x1 x2

 ,

4.2 Alternative Duopoly Models with and without Risk Factors

75

Table 4.1 The dual relations between the Cournot and Bertrand models Variables and parameters Strategic variables

Cournot model x1 x2 p1
κ1 p 2
κ2 α1
κ1 α2
κ2 β θ

Dependent variables Parameters



x1 x2





  a1
bκ 1 þ bθκ2 1 ¼
b
θ a2
bκ 2 þ bθκ1

Bertrand model p1
κ1 p2
κ2 x1 x2 a1
bκ 1 + bθκ2 a2
bκ 2 + bθκ1 b
θ


θ 1



 p1
κ 1 : p2
κ 2

As is well-known, the Cournot equilibrium is the Nash equilibrium in outputs, whereas the Bertrand equilibrium is the Nash equilibrium in prices. In view of eqs. (4.6–4.9) or their matrix notations aforementioned, the dual relations between the Cournot and Bertrand models are given in Table 4.1. There is an outstanding duality between the Cournot and Bertrand equilibriums: Cournot equilibrium with substitute (or complementary) outputs is the dual of Bertrand equilibrium with complements (or substitutes). Once the Cournot equilibrium strategies are determined, the Bertrand equilibrium strategies are also given by the duality argument. All we have to do is to replace xi with ( pi
κi), ( pi
κi) with xi, (αi
κi) with (ai
bκi
bθκj), β with b, and θ with (
θ) (i, j ¼ 1,2; i 6¼ j). More specifically, the equilibrium concept we are going to use in this paper is the application of Nash equilibrium to many oligopoly models of Cournot and Bertrand types. In the absence of any risks, we say that the pair (x1C, x2C) of output strategies is an equilibrium if the following conditions are met: x1 C ¼ Arg Maxx1 x2 C ¼ Arg Maxx2


 Π 1 x1 , x2 C ,
 Π 2 x1 C , x2 :

When the Cournot equilibrium is reached, no firm has an incentive to deviate from it. Since we find the reaction functions of firms 1 and 2 are provided by Π 1 ðx1 , x2 Þ ¼ ðα1
κ1
βx1
βθx2 Þ x1 , Π 2 ðx1 , x2 Þ ¼ ðα2
κ2
βx2
βθx1 Þ x2 , R1 C : x1 ¼ ð1=2βÞ ðα1
κ1
βθx2 Þ, R2 C : x2 ¼ ð1=2βÞ ðα2
κ2
βθx1 Þ:

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4 Information Exchanges among Firms and Their Welfare Implications (Part I): The. . .

(A) ȟ

ȟ< 0:

The case of substitutes

The case of complements

Fig. 4.1 Cournot duopoly equilibriums under no risks

The Cournot duopoly equilibrium under no risks can easily be depicted in Fig. 4.1. There are the two charts (A) and (B) in the figure. The left chart (A) indicates the case of substitutes (namely, θ > 0) in which the reaction curves are negatively sloping. In contrast, the right chart (B) shows the case of complements (i.e., θ < 0) where the reaction curves are positively sloping. As mentioned above, the Bertrand equilibrium with price strategies constitutes the dual of the Cournot equilibrium with output strategies. Therefore, we say that the ( p1B, p2B) of price strategies is an equilibrium if the following conditions are satisfied: p1 B ¼ Arg Maxp1 p2 B ¼ Arg Maxp2


 Π 1 p1 , p2 B ,
 Π 2 p1 B , p2 :

We note the following equations: Π 1 ð p1 , p2 Þ ¼ ðp1
κ1 Þ ða1
bp1 þ bθp2 Þ, Π 2 ð p1 , p2 Þ ¼ ðp2
κ2 Þ ða2
bp2 þ bθp1 Þ: Therefore, the reaction functions of firms 1 and 2 are given by

4.2 Alternative Duopoly Models with and without Risk Factors

(A) ȟ> 0:

(B) ȟ< 0:

The case of substitutes

The case of complements

77

Fig. 4.2 Bertrand duopoly equilibriums under no risks

R1 B : p1 ¼ ð1=2bÞ ða1 þ bκ1 þ bθp2 Þ, R2 B : p2 ¼ ð1=2bÞ ða2 þ bκ2 þ bθp1 Þ: We can depict the Bertrand duopoly equilibrium under no risks in Fig. 4.2. The left chart (A) stands for the case of substitutes (θ > 0), in which the reaction curves are positively sloping. The right chart (B) corresponds to the case of complements (θ < 0), where the reaction curves are negatively sloping. Comparison of Figs. 4.1 and 4.2 shows the existence of dual relations between Cournot and Bertrand. It is quite interesting to see that Charts (A) and (B) of Fig. 4.1, respectively, correspond to Charts (B) and (A) of Fig. 4.2 if price and output variables are interchanged. The duality argument is very convenient and really powerful. However, it should not be almighty. It may sometimes break down. In fact, as will be seen below, when we discuss consumer surplus, it surely breaks down! It is a rather common practice in economics that consumer surplus is measured by CS ¼ U
x0
∑i pi xi. Therefore, if we make use of (4.1)–(4.3), we find the following CS formula:

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4 Information Exchanges among Firms and Their Welfare Implications (Part I): The. . .

X CS ¼ ð1=2Þ i ðαi
pi Þ xi X ¼ ð1=2Þ i fαi
ðpi
κi Þ g xi
ð1=2Þ

X i

κ i xi :

ð4:10Þ

It is easy to see that the formula does not treat xi and ( pi
κi) symmetrically. Consequently, the duality argument applies only to profits and producer surplus, but not to consumer surplus and total surplus at all.

4.2.2

Introducing Risk Factors into Alternative Duopoly Models

We are now ready to introduce risk factors and investigate how the presence of demand or cost risk affect the working and performance of an oligopoly market. The problem here is that there are so many ways of introducing stochastic factors into our model, depending on the type of risk (demand or cost, a common value or private values) faced by firms.6 First of all, let us assume that risk is about the demand side. For simplicity, suppose that α1 and α2 are now random variables, and may be described in the following way: α1 ¼ α þ ε1 , α2 ¼ α þ ε2 :

ð4:11Þ

Here, α denotes a stochastic demand common to all the firms, whereas εi shows a stochastic demand specific to the i th firm (i ¼ 1,2). It is noted that ε1 and ε2 may be positively or negatively correlated. For instance, suppose that x1 and x2, respectively, represent “one week trip in New York and Boston” and “one week trip in California” as two attractive goods in the tourist industry. Then α may mean the fluctuations of the yen-dollar exchange, whereas ε1 andε2, respectively, show the weather in the Eastern Coast and the one in the Western Coast. It is recalled that the Cournot and the Bertrand models are dual. If α1 and α2 are stochastic parameters in the former model, so are a1 and a2 in the latter model. As was stated above, the relations between these two set of stochastic parameters must be indicated by the following formulas:

6 In the light of the history of economic thought, there have been so many ways of introducing risk and uncertainty into economic models. For details, see Sakai (2010).

4.2 Alternative Duopoly Models with and without Risk Factors

79


 a1 ¼ ðα1
α2 θÞ=β 1
θ2 ,
 a2 ¼ ðα2
α1 θÞ=β 1
θ2

ð4:12Þ

Now, let us turn our attention to the case in which risk is about the cost side. Assume that κ1 and κ2 are stochastic parameters, being written as follows: κ 1 ¼ κ þ τ1 , κ 2 ¼ κ þ τ2 :

ð4:13Þ

Here κ stands for a stochastic cost common to all the firms, whereas κi shows a stochastic cost specific to the i th firm (i ¼ 1, 2). It is noticed that τ1 andτ2 may be positively or negatively correlated. For instance, let us consider the fluctuations of oil prices in the world. The common parameter κ may represent the dollar/yen exchange rate which fluctuates frequently but influences every firm’s cost at the same ratio. Assume that τ1 andτ2, respectively, stand for the imported price of Iraq oil and the one of Venezuela oil. The Iraq oil and the Venezuela oil may rise or decline in the same direction or in opposite directions, depending on the domestic conditions of each country. The question which would naturally arise is whether or not the nice relationship between the Cournot and Bertrand models remain intact in the presence of risk. On the one hand, if risk is about the demand side, the parameters α1 and α2 are random in the Cournot model, the parameters a1 and a2 are random in the Bertrand model (see Table 4.1). As a result, the introduction of risk, whether it is common or firmspecific, does not change the dual relation between these two models. On the other hand, if risk is about the cost side, a completely new situation will emerge since the simple duality argument is no longer applicable. As can be seen in Table 4.1, when κ1 and κ2 are random variables, they affect not only parameters but also dependent variables in the Cournot model, whereas they influence parameters as well as strategic variables in the Bertrand Model. Therefore, the way how cost risk changes the relations between strategic and dependent variables in the Cournot model must be different from the way how it changes these relations in the Bertrand model. So when cost risk is introduced into an oligopoly model, the Cournot equilibriums with substitute (or complementary) outputs are no longer the dual of the Bertrand equilibriums with complements (or substitutes).7 In short, the duality argument is powerful, but not almighty. As common sense tells us, it may be helpful in some situations, it may not be so in other situations. This shows the necessity for differentiating the case of demand risk from the one of the cost risks.

7 For this point, see Sakai and Yamato (1989, 1990). The usefulness and limitations of the duality argument must always be kept in mind. Everything must have a sunny side as well as a shady side.

80

4.3

4 Information Exchanges among Firms and Their Welfare Implications (Part I): The. . .

Cournot Is Still Alive Today: Concluding Remarks

In the above, we have intensively discussed the information sharing in oligopoly and its welfare implications. It is hoped that such discussions will lead to a synthesis of the economics of imperfect competition and the economics of imperfect information. At the memorial Third Congress of the European Economic Association at the University of Bologna in 1988, Gray-Bobo as an invited speaker impressed so many people by saying the following8: A 150 years old book, written 15 years after Ricardo’s death by an almost entirely isolated man, can be so brilliantly argued that some of its parts are still discussed today (Gray-Bobo 1988, p. 2). More than 30 more years have passed since Gary-Bobo’s interesting remark. It is true that Augustin A. Cournot spent his isolated life as a first-rate mathematician, later applied differential and integral calculus to the problem of oligopoly. His courageous attempt to synthesize powerful mathematics and practical economics, however, may now be regarded as a towering landmark in the history of economic thought. It would be fair to say that Cournot is so great because his doctrine is still alive after 180 years of its first publication. In the absence of no risks, there exist the remarkable dual relations between Cournot and Bertrand oligopoly models. In fact, the Cournot equilibrium with substitutable (or complementary) outputs is the dual of the Bertrand equilibrium with complements (or substitutes). Once the Cournot equilibrium strategies are determined, so are the Bertrand equilibrium strategies by the duality argument. It is really one of the main purposes of this paper to discuss whether and to what extent introduction of risk factors into the Cournot or Bertrand models would influence such duality analogy. On the one hand, if risk is about the demand side, the introduction of risk, whether is of common type or of firm-specific type, does not change the dual relation between the two models. On the other side, if risk is about the cost side, a completely new situation has to emerge since the simple duality argument is no longer applicable. In conclusion, we can say that the duality argument is powerful, but not almighty. It may be useful in some situations, but it may not be so in other situations. We must distinguish accurately between the case of demand risk and the case of cost risk. In this chapter, we have worked with admittedly simple oligopoly models with or without conditions of risk. We do believe, however, that the results obtained in this paper are fundamentally robust. Much work remains to be left for future research.

8 See Gray-Bobo (1988), p. 20. Sakai is still remember how much he was excited when he read his paper on a new topic on the theory of oligopoly and information before the huge audience at the University of Bologna, Italy. Sakai really left his heart in the presumably oldest university in the world. We can surely learn new lessons from old teachings!

References

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References Akerlof GA (1970) The market for lemons: qualitative uncertainty and the market mechanism. Q J Econ 84:480–500 Arrow KJ (1970) Essays in the theory of risk-bearing. North Holland Basar T, Ho Y (1974) Informational properties of the Nash solution of the two stochastic non-zerosum games. J Econ Theory 7:370–384 Bertrand J (1884) Book review of Cournot (1838): Theorie mathematique de la richess social and of recherches Sur les principles mathematique de la theorie des richesses. J des Savants 24:400–508 Chamberlin EH (1933) The theory of monopolistic competition. Harvard University Press Clarke RN (1983) Collusion and the incentives for information sharing. Bell J Ecol 14:383–394 Cournot AA (1838) Recherches sur les principles mathematique de la theorie des richesses. Hachette, Paris Friedman JM (1986) Game theory with applications to economics. Oxford University Press Gal-Or E (1985) Information sharing in oligopoly. Econometrica 53:329–343 Gal-Or E (1986) Information transmission
Cournot and Bertrand equilibria. Rev Econ Stud 54:279–292 Gal-Or E (1987) First mover disadvantages with private information. Rev Econ Stud 54:259–270 Gray-Bobo R (1988) Cournot: a great precursor of mathematical economics. In: Paper presented at the third congress of the European economic association, Bologna, Italy Harsanyi JC (1967–1968) Games with incomplete information played by ‘Bayesian players.’ Manag Sci 14(Part I): 159–182, (Part II), 320–334, (Part III), 486–502 Hicks J (1935) Annual survey of economic theory: the theory of monopoly. Econometrica 3 (1):1–20 Komiya R (1975) Planning in Japan. In: Bernstein M (ed) Economic planning: east and west. Ballinger, New York, pp 189–227 Kühn KU, Vives X (1994) Information exchanges among firms and their impact on competition. Discussion paper, Institute of Analytical Economics, University of Barcelona, 94-1: 1–126 Li L (1985) Cournot oligopoly with information sharing. Rand J Econ 16:521–536 Marschack J, Radner R (1972) Economic theory of teams. Yale University Press Nalebuff B , Zwckhauser R (1986) The ambiguous implications of information sharing. Discussion Paper , Kennedy School of Government, Harvard University, 86–14: 1–36 Nash JF (1951) Non-cooperative games. Ann Math 54:286–295 Novshek W, Sonnenschein H (1982) Fulfilled expectations Cournot duopoly with information acquisition and release. Bell J Econ 13:214–218 Okada A (1982) Information exchange between duopolistic firms. J Oper Res Soc Jpn 25:58–76 Okuguchi K (1986) Equilibrium prices in the Bertrand and Cournot oligopolies. J Econ Theory 42:128–139 Ponssard JP (1979) Strategic role of information in demand function in an oligopolistic market. Manag Sci 25:240–250 Robinson J (1933) The economics of imperfect competition. Macmillan, London Sakai Y (1982) The economics of uncertainty (in Japanese). Yuhikaku Publishers, Tokyo Sakai Y (1985) The value of information in a simple duopoly model. J Econ Theory 36:56–74 Sakai Y (1986) Cournot and Bertrand equilibriums under imperfect information. J Econ 46:213–232 Sakai Y (1987) Cournot and Stakelberg equilibria under product differentiation: first mover and second-mover advantages. Tsukuba Econ Rev 18:1–33 Sakai Y (1989) Information sharing in oligopoly: overview and evaluation. Discussion paper 89-1, Social Sciences, University of Tsukuba, pp 1–87 Sakai Y (1990) Theory of oligopoly and information (in Japanese). Toyo Keizai Publishers, Tokyo Sakai Y (2010) Economic thought of risk and uncertainty (in Japanese). Minerva Publishers, Kyoto Sakai Y, Yamato T (1989) Oligopoly, information and welfare. J Econ 49:3–24

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Sakai Y, Yamato T (1990) On the exchange of cost information in a Bertrand-type duopoly model. Econ Stud Q 41:48–64 Shapiro C (1986) Exchange of cost information in oligopoly. Rev Econ Stud 53:433–466 Shubik M (1982) Game theory in the social sciences: concepts and solutions. MIT Press Singh N, Vives X (1984) Price and quantity competition in a differentiated duopoly. Rand J Econ 15:546–554 Spence M (1974) Market signaling: information transfer in hiring and related screening processes. Harvard University Press Stackelberg H von (1934) Marktform und Gleichgewicht (in Germany). Julius Springer, Berlin Stiglitz JE (1975a) The theory of ‘screening,’ education, and the distribution of income. Am Econ Rev 63:283–300 Stiglitz JE (1975b) Incentives, rsk and information: notes toward a theory of hierarchy. Bell J Econ 6:552–579 Suzumura K, Okuno-Fujiwara M (1987) Industiral policy in Japan: overview and evaluation. In: Sato R, Wachtel P (eds) Trade friction and economic policy: problems and prospects for Japan and the United States. Cambridge University Press, New York, pp 50–79 Vives X (1984) Duopoly information equilibrium: Cournot and Bertrand. J Econ Theory 34 (1):71–94 Vives X (1990) Information and comparative advantage. Int J Ind Org 8:33–52 Vives X (1992) Trade association disclosure rules, incentives to share information, and welfare. Rand J Econ 21(3):409–430 Vives X (1999) Oligopoly pricing: old ideas and new tools. MIT Press Vives X (2008) Information and learning in markets: the impact of microstructure. Princeton University Press Von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton Yasui T (1979) Economics and my life (in Japanese). Bokutaku Publishers, Tokyo

Chapter 5

Information Exchanges among Firms and Their Welfare Implications (Part II): Alternative Duopoly Models with Different Types of Risks

Abstract The purpose of this paper is to overview and evaluate the problem of information exchanges in oligopoly, an important topic in contemporary economics. It is intended as a synthesis of the two streams of economic theories: the economics of imperfect competition and the economics of risk and information. This long series of papers consists of three parts. The previous chapter which dealt with Part I discussed the dual relations between the Cournot and Bertrand duopoly models in the absence of risk. This paper turns to Part II, focusing on many duopoly models in which a common risk is present. The starting point of discussion is the Cournot duopoly model with an industry-wide common demand risk. Many other duopoly models such as the Cournot duopoly with cost risk and the Bertrand duopoly with demand or cost risk are successively discussed. It will be seen that the existence of various risk factors and the informational exchanges between Cournot or Bertrand firms influence the welfare implications on consumers and the society in many complicated ways. The next paper which deals with Part III will be concerned with more complicated problems such as private risks and/or oligopoly models. Keywords Duopoly · Oligopoly · Cournot · Bertrand · Common risk · Information exchanges

5.1

Introduction

This paper explores the working and performance of a Cournot duopoly model when the firms face a common demand risk. It will serve as a starting point for our later discussions of all types of oligopoly models under conditions of various risks. William Shakespeare (1564–1616), a great English dramatist, once remarked: “All’s well that ends well.” There should be no objections against such a maxim. We wish to add that another maxim is also valid: “All’s well that begins well.” This chapter is a greatly revised article of Sakai (1989). © Springer Nature Singapore Pte Ltd. 2021 Y. Sakai, K. Sasaki, Information and Distribution, New Frontiers in Regional Science: Asian Perspectives 49, https://doi.org/10.1007/978-981-10-0101-7_5

83

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5 Information Exchanges among Firms and Their Welfare Implications (Part II):. . .

In historical perspective, the problem of information exchanges in oligopoly was initiated with this simple type of duopoly by Basar and Ho (1974) and Ponssard (1979) and later developed by Novshek and Sonnenschein (1982), Clark (1983b), Sakai (1984a), Li (1985), Gal-Or (1985), and many others. While they all assumed that goods are just homogeneous (namely θ ¼ 1), Vives (1984) extended their results to cover the more general case of product differentiation (i.e., 1 ≦θ≦ 1). Moreover, there is a growing body of works in the 1990s and even the 2000s.1 The long series of Chaps. 4, 5, and 6 correspond to three parts. The last Chapter 4, which corresponded to Part I, dealt with alternative models of oligopoly in the absence of risks. We focused on the nice relations between the Cournot duopoly model with output strategies and the Bertrand duopoly model with price strategies. More specifically, the Cournot model where goods are substitutable (or complementary) and the Bertrand model where goods are complements (or substitutes) are really dual in the following sense: The welfare results obtained in one system may nicely be applicable to those in the other system. Presumably, they can also be extended to the leader–follower model of Stackelberg (1934). The present Chapter 5 turns to Part II, which systematically discusses many duopoly models in which a common risk is present. As a starting point of discussions, we pick up the Cournot duopoly model with a common demand risk. To this end, we newly introduce the two different effects, namely the variation and efficiency effects. Those effects are quite useful in analyzing the issue of information sharing in oligopoly theory. We are also concerned with other duopoly models such as the Cournot duopoly cost risk as well as the Bertrand duopoly with demand or cost risk. Part III, which is the target of Chapter 6, will aim to first discuss many duopoly models with private risks and later extend the welfare results obtained for those duopoly models to the very general case of oligopoly where there are any finite number of firms. Let us increase the number of firms from 2 to 3, 4, ..., n. Then it is expected that the welfare of “consumers as outsiders” may increase as the number of “firms as insiders” increases. Such a kind of “spill over effect” will have very interesting reactions. Besides, some policy implications of information sharing among firms will also be our consistent concern.

5.1.1

Four Information Structures: Game-Theoretic Interpretations

There are two Cournot types of firmsfirm 1 and firm 2. We assume that each firm is confronted with the common demand risk which is indicated by the value of the

1 For instance, see Kühn and Vives (1994), Sakai (1990), Vives (1990, 1992, 1999, 2008), and others.

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85

Fig. 5.1 The ordering of information structures by means of fineness (!)

ȞN1 = [1,0]

ȞS = [ 1,1]

ȞO = [㸮,0]

ȞN2 = [ 0,1]

demand parameter α. It must determine the optimal level of output on an ex ante basis, namely on the basis of its estimate of α. In line with Marschack and Radner (1972), we will find it useful to represent the information structure as a vector η¼ [η1, η2 ] in the following manner: For each i, ηi ¼ 1 if firm i is informed of the realized value of α, ηi ¼ 0 if it is not so informed. Note that ηi takes on either 1 or 0. Therefore there exist four information structures conceivable: (i) ηO ¼ [0,0]: Neither firm 1 nor firm 2 has information about α. (ii) ηN1 ¼ [1,0]: Firm 1 is informed of α, but firm 2 remains to be ignorant. (iii) ηN2 ¼ [0,1]: In contrast to (ii), only firm 2 is informed of α. (iv) ηS ¼ [1,1]: The two firms agree to share information about α, so that both of them are well informed of α. As later discussions will show, it is convenient to treat ηO ¼ [0,0] as a reference point. We may order these four information structures by means of “fineness.” Let us take a look at Fig. 5.1. For any two information structures, the notation “ηA ! ηB” means that “ηB is finer than ηA” As can easily be seen, ηS ¼ [1,1] is finer than ηN1 ¼ [1,0] or ηN2 ¼ [0,1], each of which in turn is finer than ηO ¼ [0,0]. However, ηN1 ¼ [1,0] and ηN2 ¼ [0,1] are not comparable by fineness. In this paper, we are especially interested in comparing the two structures, ηN1 ¼ [1,0] and ηS ¼ [1,1]. Regarding those four information structures aforementioned, we will find it quite convenient to give game-theoretic interpretations. More specifically, the extensive forms of games will be very instructive in understanding the similarities and differences among the information systems. In Fig. 5.2, chart (A) illustrates the extensive form of the Cournot duopoly game with no information, ηO. The point Po is regarded as the “Nature” who behaves like a person and selects, from the start point O1, two alternatives (αH or αL) with certain combinations. A simple case for this situation would be the case in which they are evenly distributed: Prob (αH) ¼ Prob (αL) ¼ 1/2. The point P1 indicates player 1 (namely firm 1) and P2 player 2 (namely firm 2). Assume that the two players, P1 and P2, must choose either a high (H ) or low (L ) level of output. Since P1 does not know α in advance, it cannot distinguish between the points O2 and O3, whence these two points belong to the same information set U1. In a similar fashion, since P2

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5 Information Exchanges among Firms and Their Welfare Implications (Part II):. . .

(A) ȞO = [0. 0]

(C) ȞN2 = [0. 1]

(B) ȞN1 = [1. 0]

(D) ȞS = [1. 1]

Fig. 5.2 Cournot duopoly with a common demand risk: extensive game presentations

is not informed of α, the four points from O4 through O7 belong to the same information set U2. Charts (B) and (C), respectively, correspond to the Cournot duopoly games under non-symmetric information, ηN1 andηN2. The case in which only P1 has information about α is depicted in chart (B). Such a case is radically different from the previous case of no information, ηO. Since P1 can now distinguish between O2 and O3, its information structure comprises the two information sets: U11 ¼ {O2} and U12 ¼{O3}. However, because no information is available to P2, its information structure continues to contain only one information set: U2 ¼{O4, O5, O6, O7}. In contrast to this, the case where only P2 is informed of α is shown in chart (C). Since P1 is now ignorant, the two points O2 and O3 belong to the same information structure U1: U1 ¼ {O2, O3}. Chart (D) represents the Cournot duopoly game under shared information, ηS. On the one hand, because P1 is informed of α, it can distinguish the two points O2 and O3. As a result, there exist two information sets like chart (B): U11 ¼{O2} and U12 ¼ {O3}. On the other hand, since P2 is informed of α, the number of information structures is also two like panel (C): U21 ¼ {O4, O5} and U22 ¼ {O6, O7}. It should be noted here that the duopoly game we are dealing with is a sort of simultaneous game in the sense that both players make their moves simultaneously. In other words, unlike a sequential game with the first and second movers being present, P1 cannot still distinguish the points O4 and O5 nor P2 the points O6 and O7. It is also

5.1 Introduction

87

worthy of attention that the game under shared information is a refinement of the game under non-symmetric information.2 In this paper, we are particularly eager to pick up and compare the two cases represented by charts (B) and (D). Such a comparison enables us to systematically analyze how and to what extent the information transmission from one player (namely one firm in the duopoly game) to the other player (i.e., the other firm) affects the welfare of each player and the one of a third party such as consumers, as well as the welfare of the whole society.3

5.1.2

The Cournot Equilibriums Under Different Information Structures

The equilibrium concept we are going to employ throughout this paper is the Cournot equilibrium, which can be regarded as a predecessor of Nash equilibrium. There are essentially two types of information structures: A pair of symmetrical cases, η0 ¼ [0,0] and ηS ¼ [1,1], and another pair of non-symmetrical cases, ηN1 ¼ [1,0] and ηN2 ¼ [0,1]. While the former cases are easier to handle, the latter cases require a special care for computation.4 First of all, given η0 ¼ [0,0], we say that the pair (x10, x20) of output strategies is an equilibrium pair under η0 if the following equations hold: 
 x1 0 ¼ Arg Max x1 Eα Π 1 x1, x2 0 , α , 
 x2 0 ¼ Arg Max x2 E α Π 2 x1 0 , x2 , α : Therefore, when an equilibrium is reached, no firm has an incentive to deviate from it. In order to find the concrete values of Ex1 ¼ x10 and Ex2 ¼ x20, we first note that

2 For detailed discussions on simultaneous and sequential games, see Von Neumann and Morgenstern (1944), Harsannyi (1967–1968), Basar and Ho (1974), Okada (1982), Friedman (1986), Suzuki (1999), and others. Also see Marschack and Radner (1972). 3 If such transformation transmission benefits all the parties concerned, we are in the “win–win–win situation” in the sense that it is good for the information transmitter, good for the information receiver, and also good for the whole society. Needless to say, it should be an ideal world that can hardly be attainable in the real world. 4 It is remarkable to see that the pioneering work of Cournot (1838) was printed around 180 years ago. Then it was reviewed by the outstanding article of Bertrand (1884) after 46 years. This indicates a bad state of information transmission in the academic world of nineteenth century. Fortunately, the transmission was significantly speed up in the twentieth century. Nash (1951) succeeded in generalizing the concept of Cournot equilibrium in a game-theoretic framework. The extension of Cournot-Nash equilibrium to the situation of imperfect information was then done by Harsanyi (1967–1968), Selten (1973, 1978), and others.

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5 Information Exchanges among Firms and Their Welfare Implications (Part II):. . .

Π 1 ¼ ð p1  κ 1 Þ x 1 , where p1 ¼ αβx1 βθx2. By maximizing EΠ 1 ¼ E f½ðα  κ1 Þ  βx1  βθx2  x1 g with respect to x1, we find that 2βx1 0 þ βθx2 0 ¼ μ  κ 1 :

ð5:1Þ

In a similar fashion, if we maximize EΠ 2 ¼ [(ακ 2)  βx2  βθx1] x2 with respect to x2, we obtain the following: 2βx2 0 þ βθx1 0 ¼ μ  κ 2 :

ð5:2Þ

Equations (5.1) and (5.2) can be combined in matrix notation:  β

2

θ

θ

2



x1 0 x2 0



 ¼

μ  κ1 μ  κ2

 :

ð5:3Þ

Solving for x10 and x20, we obtain the following equations:
 x1 0 ¼ ½μð2  θÞ  2κ 1 þ θκ2 =β 4  θ2 ,
 x2 0 ¼ ½μð2  θÞ  2κ2 þ θκ 1 =β 4  θ2 :

ð5:4Þ ð5:5Þ

Once a firm acquires information about α, its strategy becomes a contingent action, meaning that its output strategy now depends on the realized value of α. So given ηN1 = [1,0], the pair (x1N1 (α), x2N1) is called an equilibrium under ηN1 if the following conditions are met:
 x1 N1 ðαÞ ¼ Arg Maxx1 Π 1 x1 , x2 N1 , α for any given α, 
 x2 N1 ¼ Arg Maxx2 E α Π 2 x1 N1 ðαÞ, x2 , α This non-symmetrical case where only firm has information about α requires a special care for handling. Since we have Π 1 = (p1κ1)x1, where p1 = αβx1βθx2, maximization of Π 1 with respect to x1 results in the following: 2β x1 N1 ðαÞ þ βθx2 N1 ¼ α  κ 1 :

ð5:6Þ

Remarkably, the term x1N1 (α) in the above equation shows that the equilibrium value of x1 depends on each α. In other words, firm 1 as an informed player must take a contingent action with the contingency related to the value of α.

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89

In contrast to firm 1, firm 2 is an ignorant player whose action should not be contingent but rather routine. If we note that EΠ 2 = [(ακ 2)βx2βθx1]x2 , its maximization with respect to x2 yields 2βx2 N1 þ βθEx1 N1 ðαÞ ¼ μ  κ2 :

ð5:7Þ

Here again, we note that the first variable, x1, is a function of the demand parameter α under ηN1. We need to do a small trick to solve for (x1N1 (α), x2N1) from the two Eqs. (5.6) and (5.7). To this end, let us take expectations of both sides of (5.6). Then we have the following equation: 2βEx1 N1 ðαÞ þ βθx2 N1 ¼ μ  κ1 :

ð5:8Þ

We can easily combine (5.8) with (5.7) in matrix notation:  β

2 θ θ 2



   μ  κ1 Ex1 N1 ðαÞ ¼ μ  κ2 x2 N1

ð5:9Þ

Comparison of (5.9) and (5.3) enables us to obtain Ex1N1 (α) = x10 and x2N1 = x20. If we take care of (5.6) and (5.8), then it is not a difficult job to derive the following: 2βx1 N1 ðαÞ  2βEx1 N1 ðαÞ ¼ α  μ: It immediately follows from this equation that x1 N ðαÞ ¼ Ex1 N1 ðαÞ þ ðα  μÞ=2β ¼ x1 0 þ ðα  μÞ=2β:

ð5:10Þ

When firm 1 decides to reveal its information to firm 2, the latter firm’s strategy becomes a contingent action as well. Therefore, given ηS ¼ [1,1], the pair (x1S (α), x2S (α)) is named an equilibrium under ηs if the following conditions are satisfied: x1 S (α) ¼ Arg Maxx1 Π 1 (x1, x2S(α), α) for any given α, x2 S (α) ¼ Arg Maxx2 Π 2 (x1 S (α), x2, α) for any given α. In this symmetric case, ηs, we can derive the equilibrium values of x1 and x2 in a way similar to another symmetric case, η0. What we have to do is to simply replace μ by α. So we now have the following set of equations in matrix form:  β

2

θ

θ

2



x1 S ðαÞ x2 S ðαÞ



 ¼

α  κ1 α  κ2

 :

Solving for x1 S (α) and x2 S (α), we find the following: x1 S ðαÞ
 ¼ ½μð2  θÞ  2κ1 þ θκ2 =β 4  θ2

ð5:11Þ

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5 Information Exchanges among Firms and Their Welfare Implications (Part II):. . .

Table 5.1 Equilibrium output strategies under ηO, ηN1, and ηS: the Cournot duopoly with a common demand risk:

q

q

q

q

q

q

q q

¼ x1 0 þ ðα  μÞ=βð2 þ θÞ,

ð5:12Þ

x2 S ðαÞ
 ¼ ½μð2  θÞ  2κ2 þ θκ1 =β 4  θ2 ¼ x2 0 þ ðα  μÞ=βð2 þ θÞ:

ð5:13Þ

All the computational results derived above can systematically be summarized in Table 5.1. As may naturally be expected, the output pair (x10, x20) serves as a reference point for all the Cournot equilibriums: indeed, the whole analytical structure has been built on the base of no information, ηO. All is well that begins well!

5.1.3

Welfare Formulas

The purpose of this subsection is to compute and compare the following set of equilibrium values under alternative information structures: Π i: firm i’s expected profit (i ¼ 1, 2), EPS: expected producer surplus, ECS: expected consumer surplus, ETS: expected total surplus.

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91

In order to carry out such a hard task, it is quite useful to make use of a group of welfare formulas. To this end, we have to newly invent a set of small parts, the combinations of which will later lead to a big architecture. More specifically, we are interested in comparing the equilibrium values under non-shared information, ηN1, and those under shared information, ηS. It is expected that such comparison is quite helpful in analyzing the welfare effects of an information transmission agreement on an ex ante basis if the timing structures of the two firms are to be carried out in the following four stages. (i) At the first stage, both firms have the opportunity to make a certain ex ante agreement concerning the transmission of demand information from one firm to the other. Such an agreement can be made either by a binding contract or through a third independent agency such as a trade association. (ii) At the second stage, firm 1 observes the realized value of a random demand parameter, α, whereas firm 2 remains to be ignorant. (iii) Then at the third stage, firm 1 transmits its information to firm 2 according to the ex ante agreement made at the first stage. Garbling or cheating on the part of the informed firm (i.e., firm 1) is not permitted. In other words, both firms are supposed to be honest players in the information exchange agreement: implementation of the agreement must be done correctly and thoroughly. (iv) At the fourth and final stage, each firm makes its production decision, thus selecting the optical level of its own output. Now, let us try to express all the relevant welfare quantities in statistical terms, more exactly, in terms of variances and co-variances relative to strategic variables and stochastic parameters. Recalling that Π i ¼ ( pi κi ) xi by definition, it is not difficult to show that for i ¼ 1,2, the equilibrium value of firm i’s expected profit is provided by EΠ i ¼ E ½ðpi  κi Þ xi  ¼ Eðpi  κ i Þ E ðxi Þ þ Cov ðpi  κi , xi Þ ¼ EΠ i O þ Cov ðpi , xi Þ,

ð5:14Þ

where EΠ i O ¼ (E ( pi )  κi ) E (xi ). Because expected total surplus is the sum of expected profits across firms, it is given by EPS ¼ EPSO þ

X i

Cov ðpi , xi Þ,

ð5:15Þ

where EPSO ¼ ∑i EΠ i O. In the case of a common demand risk (α), consumer surplus is simply given by CS ¼ ð1=2ÞΣ i ðα  pi Þ xi : If we take the expectation of both sides of this equation, we obtain the following:

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5 Information Exchanges among Firms and Their Welfare Implications (Part II):. . .

ECS ¼ ð1=2Þ Σ i E ðαxi Þ ð1=2Þ Σ i E ðpi xi Þ ¼ ð1=2Þ Σ i fE ðαÞ E ðxi Þ þ Cov ðα, xi Þ g ð1=2Þ Σi fEðpi ÞE ðxi Þ þ Cov ðpi , xi Þ g X ¼ ECSO  ð1=2Þ Cov ðpi , xi Þ i X þð1=2Þ i Cov ðα, xi Þ,

ð5:16Þ

where we have ECSO ¼ (1/2) ∑i {E(α)E( pi)}E(pi). The welfare level of the whole society can be measured by expected total surplus. Since it is the sum of expected producer surplus and expected consumer surplus, it is provided by ETS ¼ EPS þ ECS X ¼ EPSO þ Cov ðpi , xi Þ i þECSO  ð1=2Þ

X i

Cov ðpi , xi Þ

X þð1=2Þ i Cov ðα, xi Þ ¼ ETSO þ ð1=2Þ þð1=2Þ

X i

X i

Cov ðpi , xi Þ

Cov ðα, xi Þ:

ð5:17Þ

Now, let us break up the term (Cov ( pi, xi )) into several parts. Since pi ¼ αβxi  βθxj (i, j ¼ 1,2; i 6¼ j ), we can derive the following:
 Cov ðpi , xi Þ ¼ Cov α  βxi  βθx j , xi ¼ βVar ðxi Þ  βθCov ðx1 , x2 Þ þ Cov ðα, xi Þ

ð5:18Þ

ði, j ¼ 1, 2; i 6¼ jÞ Consequently, by inserting (5.18) into (5.14), (5.15), (5.16), and (5.17), we can obtain the following set of welfare formulas: EΠ i ¼ EΠ i O þ βVar ðxi Þ  βθCov ðx1 , x2 Þ þ Cov ðα, xi Þ:

ð5:19Þ

5.1 Introduction

EPS ¼ EPSO þ

93

X

fβVar ðxi Þ  βθCov ðx1 , x2 Þ þ Cov ðα, xi Þg X X ¼ EPSO  β i Var ðxi Þ  2βθCov ðx1 , x2 Þ þ Cov ðα, xi Þ, i X ECS ¼ ECSO  ð1=2Þ fβVar ðxi Þ  βθCov ðx1 , x2 Þ þ Cov ðα, xi Þg i X þ ð1=2Þ i Cov ðα, xi Þ X ¼ ECSO þ ðβ=2Þ Var ðxi Þ þ βθCov ðx1 , x2 Þ, i X X Var ð x Þ  βθCov ð x , x Þ þ Cov ðα, xi Þ: ETS ¼ ETSO  ðβ=2Þ i 1 2 i i i

ð5:20Þ

ð5:21Þ ð5:22Þ

These formulas teach us that the relative strength of the following four component parts plays a critical role in evaluating the welfare of producers, consumers, and the whole society: (i) Var (xi ), (ii) Cov (x1, x2 ), (iii) Cov (α, xi ), and (iv) θ. If we compare (5.18) and (5.19), then we immediately see that an increase in the variance of each output affects the welfare of producers and the one of consumers in opposite directions: increased variability of each output, ceteris paribus, makes producers worse off but consumers better off. This is due to the fact that firm 1’s profit is a concave function of xi, and producer surplus a concave function of x1 and x2, but that consumer surplus a convex function of x1 and x2. The sign and value of θ are very important and play a critical part in understanding the welfare implications of our oligopoly models with risks. On the one hand, it measures the degree of technical substitutability or complementary relationship between the two goods, x1 and x2. On the other hand, it also demonstrates how the demands for these two goods are stochastically correlated. If x1 and x2 are substitutes (or complements), then firms’ reaction curves are negatively (or positively) sloping, so that the value of Cov (x1, x2) must be negative (or positive). Therefore, the quantity (θCov (x1, x2 )) can measure the degree of combined interaction between x1 and x2, taking account of both physical and stochastic interaction. As can naturally be expected, the greater the value of this quantity, the more advantageous is the position of “producers as insiders,” and the more disadvantages is the position of “consumers as outsiders.”5 So far, we have intensively discussed how the variability of each firm’s strategic variable or the interaction between the two strategic variables influences the welfare of producers, consumers, and the whole society. This effect may be called the variation effect. There is another sort of effect, however. Such a new effect is represented by the value of Cov (α, xi ), which shows how and to what extent the value of stochastic parameter α and the value of each strategic variable xi are correlated. The better the correspondence between these values, the greater is the welfare of producers. This effect can be named the efficiency effect. It is noted that consumers are not directly 5 For the properties of reaction curves in the case of differentiated products, see Gal-Or (1985), Li (1985), Sakai (1985,1986,1987, 1989, 1990). Also see Shapiro (1986).

94

5 Information Exchanges among Firms and Their Welfare Implications (Part II):. . .

affected by this effect although they could indirectly be affected via corresponding changes in x1 and x2. These two effectsthe variation and efficiency effectsmight appear to be somehow interlocked, but must be separated for an exact and detailed investigation. Only after reasonable separations of things at an early stage, a full unification at a later stage will be feasible and truly effective!6

5.1.4

The Impact of Informational Transmission on Various Welfare Components

We are now in a position to compare the non-shared information equilibrium (with only firm 1 being informed) and the shared information equilibrium on an ex ante basis. Suppose that the two firms make an arrangement of information transfer from firm 1 to firm 2 before the market demand is realized. The question of interest is how much and in what direction such an arrangement contributes to the welfare of producers, consumers, and the whole society. We assume here that each firm truthfully reveals its information by a binding contract or an unwritten rule, thus ignoring the problem of possible garbing and information manipulation.7 As was shown above, there are several component parts which enter into formulas for each firm’s expected profit, expected producer surplus, expected consumer surplus, and expected total surplus. So, it would be a very good idea to separately analyze the impact of information transmission on each of the components and then unite them together rather than to merely gloss over such impact on the whole entity. Taking advantage of Table 5.1, we can easily make such computations. Table 5.2 demonstrates the results obtained for the two information structures, ηN1 and ηS. With regard to Table 5.2, it is noted that the following notations are employed for the sake of simplification: V1 ¼ Var (x1 ) ¼ the variance of x1, V2 ¼ Var (x2 ) ¼ the variance of x2, CV ¼ θCov (x1, x2 ) ¼ the product of the substitution coefficient θ and the covariance of x1 and x2, E1 ¼ Cov (α, x1 ) ¼ the covariance of α and x1, E2 ¼ Cov (α, x2 ) ¼ the covariance of α and x2. Let us pay due attention to the values in the last row starting with the difference term (ηS  ηN1). They clearly indicate exactly how the information transmission from firm 1 to firm 2 affects each welfare component.

The term “the variation and efficient effects” was first introduced and intensively discussed by Sakai and Yamato (1988, 1989, 1990). 7 For the problem of garbling and information manipulation, see Marschack and Radner (1972), Okuno-Fujiwara et al. (1986), and others. 6

5.1 Introduction

95

Table 5.2 The equilibrium values of variation and efficiency components: the Cournot duopoly with a common demand risk (α)

1. Such transmission decreases (or increases) the variability of x1 if goods are substitutes (or complements), whereas it does increase the variability of x2 regardless of the degree of technical substitutability between x1 and x2. 2. As can naturally be expected, it tends to reinforce the degree of interaction between the output strategies of the two firms, which is represented by the difference (Cov (x1S, x2S)  Cov (x1N1, x2N1) ). 3. Whereas it decreases (or increases) the covariance of α and x1 whenever goods are substitutes (or complements), it always increases the covariance of α and x2. As can rightfully be expected, there is a nice correspondence between Property 1 and Property 3 mentioned above.

5.1.5

Visual Explanations by Means of Diagrams

We are concerned with the effects of information transmission on various welfare components. The situations we are facing appear rather complicated. As the saying goes, seeing is believing! It is certain that visual explanations by means of diagrams would be a great help. Let us take a close look at Fig. 5.3. For simplicity, assume that the common demand intercept (α) can be two equally likely valuesa high value (H ) or a low value (L ). The reaction functions, or the best response functions, are depicted in Fig. 5.3. If goods are substitutes (or complements), then the reaction lines are negatively (or positively) sloping. Suppose that both firms get information about α. When the demand is high (namely α¼ H ), firm 1’s reaction curve for its rival’s choice x2 is shown as R1H. It is actually

96

5 Information Exchanges among Firms and Their Welfare Implications (Part II):. . .

(A) ȟ㸼㸮

(B) ȟ 㸺㸮

Fig. 5.3 Graphical illustrations of the Cournot duopoly equilibriums under η0 , ηN1, and ηS: the case of a common demand risk

linear since we assume linear demand and constant unit cost. When the demand is low (i.e., α ¼ L ), firm 1’s reaction curve for x2 is drawn as R1L, which lies lower than R1H due to a fall in demand. A dotted line R1O denotes the average of these two reaction curves for firm 1. In a similar fashion, we can draw the two reaction curves R2H and R2L together with their average R2O for firm 2. In Fig. 5.3, we are able to find the Cournot-Nash equilibriums under various information structures. When both firms are ignorant of α, QO represents an equilibrium point, with (x1O, x2O) being the pair of equilibrium output strategies. When only firm 1 knows α, the equilibrium will be represented by the pair of the two points, QHO and QLO, with (x1HO x1LO; x2O) being the vector of equilibrium output strategies. This is because x1HO and x1LO, respectively, represent firm1’s best responses to x2O for the demands H and L, while x2O remains firm2’s best response to the average of these two demand values. In case both firms can know α, the equilibrium will be shown by the pair of the two points, QHH and QLL. In this symmetric case, it is quite clear that the vector (x1HH, x1LL; x2HH, x2LL) stands for the equilibrium output strategies of the two firms.8 We are ready to see diagrammatically how the information transmission from firm 1 to firm 2 influences various welfare components. First, take a look at chart (A). In the case of substitutable goods (θ > 0), it is readily seen that QHH lies west of QHO and QLL east of QLO. As a result, the information transmission makes both Var (x1) and Cov (α, x1 ) smaller. Next, see chart (B). In the case of complementary For a diagrammatic representation of Cournot-Nash equilibriums under ηN1 and ηS, see OkunoFujiwara et al. (1986).

8

5.1 Introduction

97

goods (θ < 0), QHH lies east of QHO and QLL west of QLO, so that the information transmission makes both Var (x1 ) and Cov (α, x1 ) larger. Although such a visual approach is quite useful, we must bear in mind its inescapable limitations as well. For instance, by merely looking at chart (A) only, we cannot determine the sign of the sum term ∑i Var (xi ), which comprises one of the key components in the welfare formulas (19)(22) aforementioned.

5.1.6

Comparisons Between the Equilibrium Values Under Non-symmetric Information and Those Under Symmetric Information

Let us make a sequence of comparisons between the equilibrium values of each firm’s profit, producer surplus, consumer surplus, and total surplus under the two information structures: (i) non-symmetric information,ηN1, and symmetric information, ηS. To this end, for any arbitrary variable Z, let us denote by ΔZ the difference between the equilibrium value under ηS and the one underηN1. Then in the light of (5.19)–(5.22), it is a straightforward job to derive the following set of equations: ΔEΠ i ¼ EΠ i S  EΠ i N1




 ¼ EΠ i O þ βVar xi S  βθCov xi S , x j S þ Cov α, xi S 


  EΠ i O þ βVar xi N1  βθCov xi N1 , x j N1 þ Cov α, xi N1 ¼ βΔVar ðxi Þ  βθΔCov ðx1 , x2 Þ þ ΔCov ðα, xi Þ, ð5:23Þ

ΔEPS ¼ EPSS  EPSN1 X X ¼ β i ΔVar ðxi Þ  2βθΔCov ðx1 , x2 Þ þ ΔCov ðα, xi Þ, i ΔECS ¼ ECSS  ECSN1 ¼ ðβ=2Þ

X i

ΔVar ðxi Þ þ βθΔCov ðx1 , x2 Þ,

ΔETS ¼ ETSS  ETSN1 X X ¼ ðβ=2Þ i ΔVarðxi Þ  βθΔCov ðx1 , x2 Þ þ ΔCov ðα, xi Þ: i

ð5:24Þ ð5:25Þ

ð5:26Þ

The welfare effects of information transmission through variation and efficiency channels are carefully summarized in Table 5.3. For example, the information transmission from firm 1 to firm 2 leads to a decrease or an increase in ∑i Var (xi ) according to whether θ is larger or smaller than θ*, where that θ* is a larger root of the quadratic equation:

98

5 Information Exchanges among Firms and Their Welfare Implications (Part II):. . .

Table 5.3 The welfare impact of information transmission through variation and efficiency channels: the Cournot duopoly with a common demand risk (α)

D

D

D

D

D

Remark. ȟ* = 2(Ҁ2̾㸯㸧Ҹ 0.8284

4  4θ  θ2 ¼ 0: Solving it for θ, we should have θ ¼ 2 (√2–1) ¼ 0.8284. If we observe a mosaic-type diagram enchased with many plus and minus signs in Table 5.3, we would immediately see that it is no easy job to systematically analyze the welfare effects of the information transmission from firm 1 to firm 2. First of all, there are various (own and cross) variation and efficiency channels through which such information transmission influences expected profits, expected producer surplus, expected consumer surplus, and expected total surplus. Besides, in most of these channels, the direction of influence (a positive or negative sign) cannot uniquely be determined, depending on the value of θ. One of the few exceptions for this is the efficiency impact on EPS and ECS: Whereas the information transmission contributes positively to EPS through the efficiency channel, regardless of the value of θ, there is no efficiency effect present on the part of ECS. The last column of Table 5.4 indicates the total welfare impact of information transmission combining variation and efficiency effects: ΔETS ¼ ΔEPS + ΔECS. There are three critical values of θ for the determination of the total impact: θ ¼ θ*, 0, θ*. It is recalled here that θ* is a larger root of the quadratic equation:

5.1 Introduction

99

Table 5.4 The degree of technical substitution and the welfare impact of information transmission: the Cournot duopoly with a common demand risk (α)

Remark. ȟ* Ҹ 0.8284

4  4θ  θ2 ¼ 0, Solving it for θ, we should have θ ¼ 2 (√2–1) ¼ 0.8284. The relationship between the degree of technical substitution and the information transmission from firm 1 to firm 2 may systematically be shown in Table 5.4. If we carefully observe Table 5.4, then we are able to obtain the following welfare results. 1. If goods are substitutes (namely θ> 0), then we find EΠ 1 < 0, so that firm 1 does not wish to reveal information. In particular, when goods are strong substitutes and nearly homogeneous (i.e., θ >θ* ), the loss of firm 1 from the information agreement overpowers the benefit of firm 2, with the result that expected producer surplus must decline: Therefore, ΔEPS ¼ EPSS  EPSN1 < 0. It should also be noted that in this case of strong substitutes, the information transmission surely increases ETS although it decreases EPS. Therefore, when implementing industrial policies for information flows, the government authority should be encouraged to somehow mix them with other supplementary measures.9 2. In case goods are weak complements (i.e., θ* < θ < 0), we find ΔEΠ 1, ΔEΠ 2, and ΔECS are all positive. For this case, the revealing case is Pareto-superior to

The significance of this point was first emphasized by Ponssard (1979) and Clarke (1983) for the special case of perfect substitutes (namely θ¼ 1). However, these results are no longer valid if goods are complements (i.e., θ< 0). 9

100

5 Information Exchanges among Firms and Their Welfare Implications (Part II):. . .

the non-revealing case. This possibility is clearly indicated by a solid enclosure in Table 5.4. 3. In a wide range of intermediate (either substitutable or complementary) cases in which θ* < θ θ* ).

11

In order to save the space, detailed tables showing the welfare effects through variation and efficiency channels for the present and following cases are omitted in this paper. See Sakai (1986, 1989).

5.2 Other Duopoly Models with a Common Risk

103

Table 5.5 The Bertrand duopoly with a common demand risk (α): various degrees of technical substitution

Remark.

ȟ* Ҹ 0.8284

4. In the case of strong complements (viz., θ< θ* ), we observe EPS, ECS, and ETS all decreasing. Therefore, the information transmission is harmful to the welfare of producers and of consumers. See a double-dotted enclosure in the lower left corner in Table 5.5. Clearly, this is the worst situation we could imagine regarding the information transmission. It is quite unfortunate that such possibility has drawn little attention in the existing literature on oligopoly and information.

5.2.3

The Bertrand Duopoly with a Common Cost Risk

Let us assume that Bertrand competitors face a common cost risk such that the common unit cost (κ) is a stochastic variable. By introducing cost risk into the Bertrand duopoly, as was noted above, a completely new situation would come out and the simple duality argument could no longer be applicable. The combination of Bertrand and cost risk would turn out to be very alarming!12 A set of welfare formulas we are going to use for the Bertrand duopoly with a common cost risk are as follows: 12

It seems to be a rather common misunderstanding that when we move from the world of a common demand risk into the world of a common cost risk, the Cournot and Bertrand models continue to have nice dual relations. This is perhaps the reason why so few papers on the Bertrand duopoly with a common cost risk have been published so far. Filling in such a gap is really the goal of this paper.

5 Information Exchanges among Firms and Their Welfare Implications (Part II):. . .

104

Table 5.6 The Bertrand duopoly with a common cost risk (κ): various degrees of technical substitution

Remark.

ȟ* Ҹ 0.8284,

̾ȟ** Ҹ ̾ 0.8393

ΔEΠ i ¼ b ΔVar ðpi Þ þ bθΔCov ðp1 , p2 Þ þ bΔCov ðκ, pi Þ
 bθΔCov κ, p j ði 6¼ jÞ, ð5:31Þ X X ΔEPS ¼ b i ΔVar ðpi Þ þ 2bθΔCov ðp1 , p2 Þ þ bð1  θÞ i ΔCov ðκ, pi Þ, X

ð5:32Þ

ΔECS ¼ ðb=2Þ i ΔVar ðpi Þ  bθΔCov ðp1 , p2 Þ, ð5:33Þ X X ΔETS ¼ ðb=2Þ i ΔVar ðpi Þ þ bθΔCov ðp1 , p2 Þ þ bð1  θÞ i ΔCov ðκ, pi Þ: ð5:34Þ As is seen in (5.31), regarding the welfare impact of firm i’s expected profit, there is a cross efficiency term associating κ with pj ( j 6¼ i ). For example, the information transmission from firm 1 to firm 2 changes not only the value of Cov (κ,p1) but also the value of Cov (κ,p2 ). This is a completely new situation we have never seen for other duopoly cases. The sensitivity of the welfare impact to the value of θ is well represented by Table 5.6. It is noted here that there emerges a new critical value of θ, denoted by θ** ≒  0.8393, which is the only real root of the following cubic equation: 2  2θ2 þ θ3 ¼ 0:

5.2 Other Duopoly Models with a Common Risk

105

As is seen in (5.31), regarding the welfare impact of firm i’s expected profit, there is a cross efficiency term associating κ with pj ( j 6¼ i ). For example, the information transmission from firm 1 to firm 2 is expected to change not only the value of Cov (κ, p1) but also the value of Cov (κ, p2 ). This is a completely new situation we have never seen for other duopoly cases. The sensitivity of the welfare impact to the value of θ is well represented by Table 5.6. It is noted here that there emerges a new critical value of θ, denoted by θ** ≒  0.8393, which is the only real root of the following cubic equation: 2  2θ2 þ θ3 ¼ 0: It is noted that the value of (θ** ) is slightly less than the value of (θ*). This is because, as stated above, θ* ¼  2(√2  1) ≒ 0.8284. By taking a close look at Table 5.6, we are able to derive the following welfare implications: 1. Concerning the sign pattern ofΔEΠ 1, Table 5.6 resembles Table 5.4 although there is now a cross efficiency effect working behind the scene. When goods are complements (or substitutes), firm 1 wishes (or does not wish) to reveal the information to firm 2. In contrast to the previous cases, however, there emerges the new possibility that the value of receiving information is amazingly negative. Indeed, when goods are strong complements (viz., θ< θ** ), the welfare of firm 2 must go down by acquiring the information from its rival firm: namely ΔEΠ 2 < 0. As the saying goes, ignorance may sometimes be bliss! 2. Independently of the value of θ, the information transmission increases EPS. If a side payment is feasible between the firms, the transmission may make both firms better off. Concerning the impact on ECS, the sign pattern in Table 5.6 is just the opposite of the one in Table 5.4. Unless goods are strong substitutes, the information revelation is beneficial to consumers as outsiders. 3. If goods are weak complements in the sense that θ** 0 and ρ ¼ 0), her analysis corresponds very well to Panel (A) in Fig. 6.2. 4

6.1 The Case of Private Risks: An Introduction

115

Table 6.1 The Cournot duopoly with private demand risks (α1 ,α2 ) Own Variation The Welfare Impact

Cross Variation

Own Efficiency

Cross Efficiancy

OV1

OV2

CV

OE1

OE2

CE1

CE2

㻗

㻗

㻙

㻗

㻗

㻜

㻜

D EΠ1

㻙

㻜

㻗

㻗

㻜

㻜

㻜

㻗

D EΠ 2

㻜

㻙

㻗

㻜

㻗

㻜

㻜

㻗

D EPS

㻙

㻙

㻗

㻗

㻗

㻜

㻜

㻗

D ECS

㻗

㻗

㻙

㻜

㻜

㻜

㻜

㻙

D ETS

㻙

㻙

㻗

㻗

㻗

㻜

㻜

㻗

Total

Remark. OV1 ¼ ΔVar (x 1 ), OV2 ¼ ΔVar (x2 ); CV ¼θΔ Cov (x1 , x2 ) ; OE1¼ ΔCov (α1, x 1 ), OE2 ¼ ΔCov (α2 , x 2 ) ; CE1 ¼ ΔCov (α1 , x 2) , CE2 ¼ ΔCov (α2 , x 1)

Let us compare the two systems: namely, the present system (6.1) - (6.4) for private demand risks, and the previous system (5.23) - (5.26) for a common demand risk. Then, we are able to find that these two systems are very similar, the only difference being that there are now firm-specific parameters αi (i ¼ 1,2) instead of a industry-wide parameter α. As in the previous situations, there exist two channels through which the information sharing between the two firms affects the equilibrium values of welfare quantities: the variation and efficiency channels. A good summary of the welfare effects of information pooling for Cournot duopoly with private demand risks is provided in Table 6.1. The following shorthand notations are employed here: OV1 ¼ ΔVar (x1 ) ¼ an increment in the variance of x1 , OV2 ¼ ΔVar (x2 ) ¼ an increment in the variance of x2 , CV ¼ θ ΔCov (x1, x2 ) ¼ the product of the substitution coefficient θ and an increment in the covariance of x1 and x2 , OE1 ¼ ΔCov (α1, x1 ) ¼ an increment in the covariance of α1 and x1, OE2 ¼ ΔCov (α2, x2 ) ¼ an increment in the covariance ofα2 and x2, CE1 ¼ ΔCov (α1, x2 ) ¼ an increment in the covariance ofα1 and x2, CE2 ¼ ΔCov (α2, x1 ) ¼ an increment in the covariance ofα2 and x1, Let us compare the two tables: Table 6.1 of Part III for private demand risks, and Table 6.3 of Part II for a common demand risk. Then we immediately see that a mosaic-type diagram enchased with plus, minus, and zero signs becomes much

116

6 Information Exchange among Firms and Their Welfare Implications (Part III):. . .

simpler in the sense that only one sign is attached to each block regardless of the value of θ. The reason for it is that the transmission of information between the two firms is now “a two-way street” instead of “a one-way street,” and thus both firms may be treated very symmetrically. Surely, symmetry makes everything simple and beautiful! By taking a close look at Table 6.1, we are able to obtain the following welfare results: 1. The exchange of private demand information between the two Cournot firms makes each firm’s production activity more responsive to a change in demand, so that it increases the variance of each output (the own variation effect). This yields a fall in expected producer surplus as well as a rise in expected consumer surplus. 2. The information sharing has a tendency to reinforce the degree of (negative or positive) interaction between the strategies of the two firms (the cross variation effect). Since the reaction curves of firms are negatively (or positively) sloped whenever goods are substitutes (or complements) as is clearly seen in Fig. 6.1, the information pooling always increases the product of θ and ( Cov (x1, x2 ) ). The greater the strategic interaction between both firms, the more advantageous is the position of “producers as insiders” and the more disadvantageous the position of “consumers as outsiders.” 3. The information pooling contributes to the efficiency allocation of resources (the efficiency effect). In fact, it increases the value of Cov (αi , xi ), meaning that the firm facing greater (or smaller) demand is likely to have a larger market share. A better correspondence between demands and outputs means an additional gain in the welfare of producers. It is noted here that consumers are not directly affected by such reallocation, even if it could be indirectly influenced through corresponding changes in outputs. 4. The last column indicates the total welfare impact which combines the own and cross variation effects and the efficiency effects. The information sharing between the firms increases the producer welfare as well as the overall welfare, but decreases the consumer welfare. These implications would clearly agree with common sense.

6.1.2

Other Duopoly Models with Private Risks

Let us continue to assume that firms act as Cournot competitors and thus employ quantities as their strategic variables. Then as we noted in Part II, whether private risks are about demands or costs does not matter at all. If we discuss the situation under which a stochastic vector under question is the vector (κ1, κ 2 ) of cost parameters rather than vector (α1 , α2 ) of demand parameters, we are able to draw a table analogous to Table 6.1, only the difference being that we must now compute the value of (ΔCov (κi , xi ) ) instead of that of ΔCov (αi , xi ). Therefore, the welfare results obtained for the case of private demands can be applied to the present case of private costs with appropriate modifications. Now, let us turn our attention to the situation under which firms play as Bertrand competitors and thus use prices as their strategic variables. In such a case of Bertrand

6.1 The Case of Private Risks: An Introduction

117

duopoly, the question of whether private risks are about demands or costs becomes very important, may significantly affect the concluding part of welfare implications of information sharing in oligopoly. Let us assume that each of Bertrand competitors faces its own demand risk. Specifically, we assume that the demand parameters α1 and α2 are random variables whose joint distribution is a bivariate normal distribution. We are concerned with comparing non-sharing information and sharing information equilibriums on an ex ante basis. Table 6.2 gives us a summary of such comparison. It is noted there that the following shorthand notations are conveniently used:5) OV1 ¼ ΔVar ( p1 ) ¼ an increment in the variance of p1 , OV2 ¼ ΔVar (p2 ) ¼ an increment in the variance of p2 , CV ¼ θΔCov ( p1, p2 ) ¼ the product of the substitution coefficient θ and an increment in the covariance of p1 and p2 , OE1 ¼ ΔCov (a1 , p1 ) ¼ an increment in the covariance of a1 and p1 , OE2 ¼ ΔCov (a2 , p2 ) ¼ an increment in the covariance of a2 and p2 , CE1 ¼ ΔCov (a1 , p2 ) ¼ an increment in the covariance of a1 and p2, CE2 ¼ ΔCov (a2 , p1) ¼ an increment in the covariance of a2 and p1. A careful look at Table 6.2 enables us to obtain the following results concerning the welfare impact of information pooling through variation and efficiency channels:6 1. If the Bertrand firms to agree to exchange private demand information with each other, then each firm’s price level becomes more responsive to a change in its private demand (the own variation effect). As a result, expected producer surplus and expected total surplus fall while expected consumer rises. 2. The information pooling has an effect of reinforcing the strategic interaction between the two firms (the cross variation effect). The greater such an interaction, the stronger will be the position of producers, and thus the weaker will be the position of consumers. 3. A better correspondence between demands and prices is now possible by the information exchange between the two Bertrand firms (the own efficiency effect). This is not only beneficial to producers, but is now definitely harmful to consumers as well, which is a new feature of the Bertrand model with private demand risks. This may be contrasted with the previous Cournot world in which the own efficiency effect is not working for the interest of consumers. 4. In order to investigate the welfare implications of the information sharing, we must take into consideration those three effects mentioned above. In so doing, we can take advantage of the dual relationship between the Bertrand equilibrium with

5

See Sakai (1990a, 1990b,1991) for detailed derivations. It is noted that, only in the 1980s, there emerged a number of papers dealing with the Bertrand model with demand/cost risks. More specifically, the Bertrand model with private demand risks was first intensively studied by Sakai (1987). It is unfortunate, however, that the welfare analysis of information sharing was not complete in this earlier paper. It not only failed to investigate the welfare impact on consumers and the whole society, but also neglected the decomposition into variation and efficiency channels. 6

6 Information Exchange among Firms and Their Welfare Implications (Part III):. . .

118

Table 6.2 The Bertrand Duopoly with private demand risks (α1 ,α2 ) Own Variation The Welfare Impact

Cross Variation

Own Efficiency

Cross Efficiancy

OV1

OV2

CV

OE1

OE2

CE1

CE2

㻗

㻗

㻗

㻗

㻗

㻜

㻜

Total

DE

1

㻙

㻜

㻗

㻗

㻜

㻜

㻜

㻗

DE

2

㻜

㻙

㻗

㻜

㻗

㻜

㻜

㻗

D EPS

㻙

㻙

㻗

㻗

㻗

㻜

㻜

㻗

D ECS

㻗

㻗

㻙

㻙

㻙

㻜

㻜

㻙

D ETS

㻙

㻙

㻗

㻜

㻜

㻜

㻜

㻗

Remark. OV1 ¼ ΔVar ( p1 ) , OV2 ¼ ΔVar ( p2 ) ; CV ¼ θΔCov ( p1, p2 ); OE1 ¼ ΔCov (a1 , p1 ), OE2 ¼ ΔCov (a2 , p2 ); CE1 ¼ ΔCov (a1 , p2 ), CE2 ¼ ΔCov (a2 , p1)

substitutes (or complements) and the Cournot equilibrium with complements (or substitutes). Such a duality can be confirmed by comparing the sign pattern of ΔEΠ 1, ΔEΠ 2 , and ΔEPS in Table 6.2 and the corresponding sign pattern in Table 6.1: in fact, these two patterns are the same. 5. In a sharp contrast to the Cournot case, however, there emerges a new sign pattern for the welfare impact on ECS and ETS through the own efficiency channel. It is noted that a gain in EPS and a loss in ECS via this route are just counterbalanced, so that ETS remains unaffected. 6. In spite of the appearance of the own efficiency effect on the part of consumers, it is remarkable to see that the total welfare impact of the private demand information sharing between the Bertrand firms is the same as the one between the Cournot firms. As in the Cournot case, the information pooling increases the welfare of producers and the total welfare, but it decreases the welfare of consumers. Finally, let us discuss the situation under which the Bertrand firms are subject to private cost risks. Among the four cases of private risks, this constitutes the most delicate case in order to derive the welfare results. If we carry out our task of

6.1 The Case of Private Risks: An Introduction

119

Table 6.3 The Bertrand duopoly with private cost risks (κ 1 ,κ 2 ) Own Variation The Welfare Impact

Cross Variation

Own Efficiency

Cross Efficiency

OV1

OV2

CV

OE1

OE2

CE1

CE2

㻗

㻗

㻗

㻗

㻗

㻗

㻗

D

㻙

㻜

㻗

㻗

㻜

㻙

㻜

㼼㻔㻖㻕

D

㻜

㻙

㻗

㻜

㻗

㻜

㻙

㼼㻔㻖㻕

D EPS

㻙

㻙

㻗

㻗

㻗

㻙

㻙

㼼㻔㻖㻕

D ECS

㻗

㻗

㻙

㻜

㻜

㻜

㻜

㻙

D ETS

㻙

㻙

㻗

㻗

㻗

㻙

㻙

㻙

Total

2

Remark: (*) ΔEΠ i (i¼1,2), ΔEPS ⋛ 0 , θρ⋛ 2 43θ ð2θ2 Þ OV1 ¼ΔVar ( p1), OV2 ¼ΔVar ( p2); CV ¼ θΔCov ( p1, p2 ) ; OE1 ¼ ΔCov (κ 1, p1), OE2 ¼ ΔCov (κ 2 , p2 ) ; CE1 ¼ ΔCov (κ 1 , p2 ), CE2 ¼ ΔCov (κ 2 , p1)

computation, we will be able to obtain Table 6.3 which summarizes the final results. The following shorthand notations are employed here:7 OV1 ¼ ΔVar ( p1 ) ¼ an increment in the variance of p1 , OV2 ¼ ΔVar ( p2 ) ¼ an increment in the variance of p2 , CV ¼ θ ΔCov ( p1, p2 ) ¼ the product of the substitution coefficient θ and an increment in the covariance of p1 and p2 OE1 ¼ ΔCov (α1 , p1 ) ¼ an increment in the covariance ofα1 and p1 , OE2 ¼ ΔCov (α2 , p2 ) ¼ an increment in the covariance ofα2 and p2 , CE1 ¼ ΔCov (α1 , p2 ) ¼ an increment in the covariance ofα1 and p2 , CE2 ¼ ΔCov (α2 , p1) ¼ an increment in the covariance ofα2 and p1 . Let us have a very careful look at Table 6.3. Then we are able to have the following welfare results for this case: 1. As in the previous cases, the pooling of private cost information between the Bertrand firms tends to increase the variance of each firm’s price (the own variation effect) and to strengthen the degree of interaction between the two prices (the cross variation effect).

7

To save the space, we omit those detailed tables which indicate the welfare impact through variation and efficiency channels for the present and following cases. For more detailed explanations, see Sakai (1989). Also see Sakai (1990a,1990b,1991).

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6 Information Exchange among Firms and Their Welfare Implications (Part III):. . .

2. The own variation effect contributes negatively to the welfare of producers and the whole society, and positively to the welfare of consumers. It is interesting to see that such cross variation effect has exactly opposite welfare implications from the own variation effect. 3. Switching our attention from variation channels to efficiency channels, the information exchange yields an improved correspondence between the cost and price of each firm (the own efficiency effect). Therefore, just as in the case of private demand risks, this has a beneficial effect on the welfare of producers and the whole society. However, contrary to the situation of private demand risks, it has no effect whatever on the welfare of consumers. 4. Remarkably, there is another kind of allocation repercussion across firms, which is represented by Cov (αi , xj ) (i 6¼ j ). It can be shown that if goods are substitutes (or complements) then the information pooling increases (or decreases ) the covariance between the cost of one firm and the price of the other. Such repercussions have a disturbing impact of resource allocation across firms, regardless of the technical substitution between goods. Presence of such cross efficiency effect distinguishes the welfare analysis of the Bertrand duopoly with private cost risks from all other duopoly cases with private risks. In short, the cross allocation is literally the crossing factor that disturbs our welfare analysis! 5. The final column indicates the total welfare impact taking account of the four effects—the own and cross variation effects and the own and cross efficiency effects. The information sharing may benefit firms in some situations but it may hurt them in other situations, depending on the degree of substitutability, θ, and the degree of correlation, ρ. The product (θρ)of these two parameters measures the degree of combined interaction between the two firms, joining together the physical and stochastic factors. In general, the cross variation and efficiency effects operate in mutually opposing directions. If, and only if, the combined interaction is large enough (more exactly, θρ > [(4 3θ2) / [2(2 θ2)]], the cross variation effect would dominate the cross efficiency effect, so that the exchange of private cost information would benefit the participating firms. This should really be a very important point. So, let us discuss it in more rigorous ways, both mathematically and graphically. In fact, we can obtain the following equation:8

8 The Betrand duopoly under private cost risks was studied by Gal-Or (1986) for the special case where goods are substitutes and costs are not correlated (i.e.,θ> 0 andρ ¼ 0). However, the welfare impact on consumers and the whole society was not discussed in her otherwise excellent work. A more complete welfare analysis which allows for complementary goods and also for positively or negatively correlated costs was independently and thoroughly carried out by Sakai & Yamato (1988).

6.1 The Case of Private Risks: An Introduction

121

Fig. 6.3 The ( θ, ρ ) diagram for ΔEPS : the Bertrand duopoly with private cost risks (κ 1 ,κ 2 )

ΔEPS ¼




 2bσ 2 θ2 ð1  ρ2 Þ  2θρ 2  θ2  4  3θ2 ,  2 2 2 4  θ ð2  θρÞ

ð6:5Þ

First of all, it is noted that this equation tells us that ΔEPS vanishes whenever θ ¼ 0 or ρ ¼  1. Next, we easily find the following interesting relation: ΔEPS ⋛ 0 according to whether θρ ⋛

4  3θ2
, 2 2  θ2

ð6:6Þ

hence ΔEPS is positive if the product of θ and ρ is greater than the fractional quantity (4  3θ2)/ [2(2 θ2)]. Those results aforementioned may graphically be summarized in Fig. 6.3, which shows how the sign of the quantity ΔEPS is sensitive to the combination of θ and ρ. In the interior of the shaded areas CDE and FGH, this quantity takes on positive values, so that the information sharing is beneficial to firms. It really vanishes on the curves CE and FH where the equation θ ρ ¼ (4  3θ2) / [2(2 θ2 )] holds, and also

122

6 Information Exchange among Firms and Their Welfare Implications (Part III):. . .

on the horizontal line segments AD and GJ, and on the vertical one BL, meaning that the firms’ gains due to the information pooling is then nil. Moreover, any point in the remaining blank area which is quite large represents the situation under which the information pooling is harmful to firms. More information may mean less benefit! 6. Concerning the consumer side, only the cross variation effect is operating against the consumer surplus, whereas there is no (own or cross) efficiency effect present. The result is that the information sharing is detrimental to consumers. 7. In sharp contrast to all previous cases with private cases, the information pooling is not socially desirable. More significantly, except when the combined interaction is positive and strong, the pooling case must be Pareto inferior to the non-pooling case. This is presumably the worst possible situation we could imagine among all types of duopoly under private risks. In conclusion, as we have often been told, everything has two sides—a bright side and a dark side. In almost all cases, the information sharing is good for producers, and may also be good for consumers if a side payment from producers to consumers is appropriately accompanied. There is an exception to this general rule, however. We bear in mind that as the implication (7) above indicates, the information pooling may make every member of the society too sensitive to fluctuations, thus possibly making all of them worse off.

6.2

Oligopoly Models

In the above, we have carried out a detailed analysis of welfare implications of the information transmission between firms. We have found that those welfare implications are sensitive to strategic variables (outputs versus prices), the source of risk (demands versus costs), and the type of risks (an industry-wide common risk versus firm-specific private risks). What we are going to do in this section is to show that the implications are also very sensitive to the number of firms in an industry. In particular, we must be very careful of extending the consumer welfare analysis from the simple case of duopoly to the general case of oligopoly with more than two firms. This is because the possibility that the information sharing among firms benefits “consumers as outsiders” would arise and gradually grow as the number of “producers as insiders” increases. As can naturally be expected, such “insideroutsider story” or “spillover story” in oligopoly under risks may emerge and become more complicated in a more general framework. While we aim to extend our welfare analysis to the general case of oligopoly, we limit our attention to the situation under which Cournot or Bertrand firms face private cost risks. In the light of the previous discussions on many types of duopoly, we believe that this case constitutes the most interesting one in the world of oligopoly, and that all other cases may be handled in a more or less analogous fashion.

6.2 Oligopoly Models

6.2.1

123

The Basic Model

The generalization of a duopoly model to an oligopoly model is rather straightforward if each firm is treated symmetrically. On the production side, we have an oligopoly sector with n firms, with firm i producing a differentiated output x i (i ¼ 1, 2 ,..., n ), a competitive sector producing a numeraire good xo. Let pi be the unit price of x i (i ¼ 1,2,..., n ). On the consumption side, we have a continuum of consumers of the same type such that the utility function of the representative consumer is of the following form: X X XX U ¼ xo þ α i xi  ð1=2Þ β x2þθ i i i

j6¼i

 xi x j ;

ð6:7Þ

where both αand β are positive. Without loss of generality, we assume that β is unity. If the utility function U is to be concave, the following matrix must be positive definite: This implies that the value of θ must lie between (1)/(n1) and 1 . For instance, 1 < θ < 1 for n ¼ 2 ; 1/2 0 , ρ >

2ðn þ 15Þ ðn  1Þðn  3Þ

ð6:28Þ

Let us put ρ* ¼ 2(n +15)/ (n 1)(n 3). Clearly, the amount of thisρ* represents a critical value on which consumers can enjoy the benefit of a third party. Specifically speaking, se can show the following results: (i) For n ¼ 10, ρ* ¼ 50/63 ≒ 0.7937 ; (ii) For n ¼ 20, ρ* ¼ 70/323 ≒ 0.2167 ; (iii) For n ¼ 50, ρ* ¼ 130/2303 ≒ 0.05645 . These results clearly demonstrate the possibility that the information pooling among “producers as insiders” benefits “consumers as outsiders” becomes greater as the number of producers becomes greater. As social psychology teaches us, the “insider-outsider” story is both complicated and intriguing!

6.3

Concluding Remarks: Laboremus!

It is true that this paper is mainly a theory-oriented work. We believe, however, the welfare results obtained so far are expected to have interesting policy implications regarding the effectiveness and limits of information-sharing agreements among firms. On the one hand, trade associations may be regarded as those nice examples of the institutions in which the information transmission between firms takes place and is properly organized. Any kind of information-sharing agreement is seen to be double-edged: it may strengthen the power of coalition among firms, whereas it

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6 Information Exchange among Firms and Their Welfare Implications (Part III):. . .

enhances the efficiency of resource allocation across firms. In the light of those mutually opposing welfare effects working behind, antitrust authorities in the USA have not taken a clear-cut position on the agreements on information pooling. This is admittedly an ambiguous and even confusing fact.17 On the other hand, there are many economists who think that, among a set of industrial policies undertaken by the Japanese authorities, those policies which explicitly or implicitly contribute to the improvement of flows of industrial information have been very successful measures. In short, there are some industrial policies which may be effective in Japan but may not be so in the USA. This may in part reflect cultural and historical differences between the two countries.18 It is strongly hoped that our theoretical investigation of the information transmission among firms sheds new light both on the effectiveness or limitations of trade associations and on the merits or demerits of industrial policies. It seems that we can derive the following set of policy implications from our theoretical analysis conducted above. 1. The most important thing we must bear in mind is that the welfare implications of the information transmission among firms are sensitive to many factors. They are enumerated as follows: the type of competition (Cournot or Bertrand), the nature of risk (demand or cost), the character of information (a common value or private values), and the number of participating firms (two, three or any finite number). Even if every one of those factors is specified, the welfare results may as well depend on the degree of technical substitution between any two outputs and the value and direction of stochastic interdependence between any two demand or cost parameters. 2. It goes without saying that the policy implications are closely linked to the welfare results, given a certain criterion of social welfare. Even if we regard the expected sum of the producer and consumer surpluses as a good measure of social welfare, we should be very careful of which kind of oligopoly we are discussing, and of which sort of risk and information we are talking about. As can naturally be expected, different assumptions on oligopoly, risk, and information are likely to lead to different policy implications. 3. In order to have a clear-cut conclusion on the merits or demerits of the information transmission agreements, it is first necessary to determine whether the risk each firm is confronted with is of a common industry-wide type or a firm-specific type. Suppose that every Cournot or Bertrand firm belonging to the same industry is subject to the same demand or cost risk. Then, as our welfare analyses aforementioned have shown, the information flow from one firm to others results in an increase in expected social surplus, with the exception of the case in which firms are Betrand competitors facing a common demand risk and goods are not 17

For the trade association laws and antitrust laws, see Vives (1992, 1999, 2008) and other papers. Also see Yasui (1979). 18 For the evaluation of industrial policies in the post-war Japan, see Komiya (1975) and Suzumura & Okuno-Fujiwara (1987).

6.3 Concluding Remarks: Laboremus!

135

strong substitutes. Besides, in all those favorable cases, if side payments are permitted between firms and goods are moderately substitutable or complementary, such information transmission is most likely to represent a Pareto improvement in the sense that it makes both producers and consumers better off. Therefore, except the situation of Betrand oligopoly with a common demand risk, the government authority should pursue a policy which encourages the spreading of information among firms. If such a policy happens to harm consumers although it does increase total surplus, it appears that we are a sort of dilemma, since consumer protection is often regarded as antitrust policy makers as their main objective. It follows that public policies for information transmission should be supplemented with income distribution policies, so that some of the increased social surplus may be shifted to consumers, for instance, through taxes and subsidies. 4. The most troublesome case rests with the situation under which firms are Bertrand competitors facing a common demand risk. Unless goods are strong substitutes, the demand transmission among firms has a negative effect on social welfare. In such a case, the authority should be discouraged from engaging in the information transfer. 5. Let us turn our attention to the more interesting case where each firm faces its own demand risk or cost risk. In the case of such private firm-specific risk, the number of participating firms plays an important role in deciding the effect of the information sharing among producers on the welfare consumers. Apart from the Bertrand oligopoly with cost risk, any information pooling agreement yields an increase in expected producer surplus and in expected total surplus, whatever the degree of technical substitution and the value of stochastic correlation. Regarding the effect on consumers, there appears a dividing line between “a few firms” and “many firms.” When the number of firms is “small,” the information pooling among producers is always harmful to consumers, showing the need of introduction of supplementary income redistribution policies. If, however, the number becomes “large,” then the situation would change completely. Then unless goods are homogeneous (which is unlikely in today’s business circle), the shared information case is most likely to be Pareto superior to the non-shared information case. This is no doubt the most fortunate case we can imagine, so that we may ask the public authority to positively interfere information flows in private sectors. 6. If firms are Bertrand competitors facing private cost risk, the more information means less social benefit in the sense that the information pooling makes the “economic pie” smaller. This is presumably the most unfortunate situation among possible combinations of oligopoly and risk. Although the authority is not recommended to help diffuse private cost risk across firms, it might do so under the pressure of business circle because the information sharing is likely to increase the share of producers in social surplus if the number of producers is sufficient large. To make the problem even more complicated, there are some other circumstances in which the information pooling among producers may increase the welfare of consumers if the number of firms is “large.”

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6 Information Exchange among Firms and Their Welfare Implications (Part III):. . .

7. To sum up, policy implications of an information transmission agreement among firms depends on whether risk is of an industry-wide type or of a firm-specific type, whether information is about demand or cost, and on whether inter-firm competition is of the Cournot quantity type or the Bertrand price type. Moreover, those implications are also sensitive to the degree of technical substitution among goods, the value and direction of stochastic correlation among demand or cost parameters, and the number of participating firms. The above considerations seem to lead to making a case-by-case analysis quite effective if we have to take much care of adopting a Pareto-improving policy. If, however, we allow for a certain kind of side payment among firms, the scheme of welfare-enhancing policy becomes much simpler. This is due to the fact that unless the oligopoly in question is Bertrand oligopoly with a common demand risk or private cost risks, any government policy of promoting the information flows among firms has an effect on increasing total welfare although it might decrease the welfare of certain members of the society. Since the economic pie per se gets larger by the information transmission among firms, it is possible to make every member better off if an information-flow-promoting policy is supplemented by a series of income redistribution policies. On the other hand, there are a limited number of cases in which the information transmission or sharing among firms does indeed hurt total welfare. Those unfortunate cases are only two: namely, the Bertrand oligopoly with a common demand risk and the same-type of oligopoly with private cost risks. Besides, there are more possible cases where the information pooling is harmful to consumers as outsiders if the number of producers as insiders is rather small. What we have learned from our detailed analysis so far is that these “bad cases” may clearly be identified and should be distinguished from many other “good cases.” The government agencies should have sharp eyes to select only “good cases” and, if necessary, should supplement policies for information transfer with policies for income redistribution. Needless to say, how much effective these policies really is solely based on the social trust by the people for their democratic government. No good democracy, no good policies! It should be noted that there remain some limitations in our welfare analysis and many other directions in which the analysis may be further extended. First of all, we have been working with a simple oligopoly model with explicit functional forms assumed for the utility functions of consumers, the cost functions of producers, and the density functions of stochastic variables. It is our strong belief that simplification is the essence of science and may be justified if it straightforwardly leads us to the heart of the matter. Second, we have intentionally ignored the problem of risk aversion on the part of producers and/or consumers along with the problem of information cost. Needless to say, we are quite aware of the fact that people in the street tend to avoid any risk, and that information per se is sometimes a very expensive good.19 19 For the effect of risk aversion on the information sharing in oligopoly, see Sakai & Yoshizumi (1991a, 1991b).

References

137

Third, the question of partial information sharing and possible garbling has not been discussed at all in this paper. We know, however, that any kind of partial commitment and any degree of cheating are conceivable in every aspect of people’s behaviors. These problems aforementioned remain unsolved and will be the target of future research. And finally, we have paid no attention to the leader–follower model of Stackelberg (1934) in this paper. Stackelberg competition could employ either quantities or prices as their strategic variables. Besides, in the Stackelberg framework, the risk in question may be of an industry-wide type or a firm-specific type, and the information in question may be about demand or cost parameters. Taking these factors into account, we would have so many Stackelberg models to work with. Then there would be a certain class of circumstances under which a less informed firm is willing to act as a follower, with a more informed firm playing the role of a leader. Such an analysis would throw new light on the long-standing problem of the first mover advantage versus the second mover advantage.20 In conclusion, we believe that economists should share any kind of information with each other through oral discussions or written papers or even e-mails, with the strong faith that information is power in our academic circle. Let us recall of the wise maxim followed by J.H. Fabre (1823-1915), a legendary French entomologist: Laboremus!

References Basar T, Ho Y (1974) Informational properties of the Nash solution of the two stochastic non-zerosum games. J Econ Theory 7:370–384 Bertrand J (1884) Book review: Theorie mathematique de la richess social and of Recherches sur les principles mathematique de la theorie des richesses. J Savants:400–508 Clarke RN (1983) Collusion and the incentives for information sharing. Bell J Econ 14:383–394 Cournot AA (1838) Recherches sur les principles mathematique de la theorie des richesses. Hachette, Paris Fried D (1984) Incentives for information production and disclosure in a duopolistic environment. Quarter J Econ 99:367–381 Friedman JM (1986) Game theory with applications to economics. Oxford University Press Gal-Or E (1985) Information sharing in oligopoly. Econometrica 53:329–343 Gal-Or E (1986) Information transmission — Cournot and Bertrand equilibria. Rev Econ Studies 53:279–292 Gal-Or E (1987) First mover disadvantages with private information. Rev Econ Studies 54:259–273 Harsanyi JC (1967-68) Games with incomplete information played by ‘Bayesian players.’ Manag Sci 14: (Part I ) 159-182, (Part II) 320-334, (Part III), 486-502 Jin JM (1998) Information sharing about a demand shock. J Econ 68(2):137–152 Komiya R (1975) Planning in Japan. In: Bernstein M (ed) Economic planning: the East and the West. Ballinger, New York, pp 189–227

20 For the information sharing and welfare in a Stackelberg-type leader-follower model, see Gal-Or (1987), Sakai (1987), and others.

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Kühn KU, Vives X (1994) Information exchanges among firms and their impact on competition. Discussion paper, Institute of Analytical Economics, Barcelona 94-1: 1-126 Li L (1985) Cournot oligopoly with information sharing. Rand J Econ 16:521–536 Nalebuff B, Zeckhauser R (1986) The ambiguous implications of information sharing. Discussion paper, Kennedy School of Government, Harvard University 147: 1-31 Nash JF (1951) Non-cooperative games. Ann Math 54:286–295 Okada A (1982) Information exchange between duopolistic firms. J Oper Res Soc Jpn 25:58–76 Ponssard JP (1979) Strategic role of information in demand function in an oligopolistic market. Manag Sci 25:240–250 Raith M (1996) A general model of information sharing in oligopoly. J Econ Theory 71:260–288 Sakai Y (1985) The value of information in a simple duopoly model. J Econ Theory 36:56–74 Sakai Y (1986) Cournot and Bertrand equilibriums under imperfect information. J Econ 46:213–232 Sakai Y (1987) Cournot and Stakelberg equilibria under product differentiation: first mover and second-mover advantages. Tsukuba Econ Rev 18:1–33 Sakai Y (1989) Information sharing in oligopoly: overview and evaluation. Discussion Paper 89-1 , Institute of Social Sciences, University of Tsukuba, pp 1-92 Sakai Y (1990a) Theory of oligopoly and information. (in Japanese). Toyo Keizai Publishers, Tokyo Sakai Y (1990b) Information sharing in oligopoly: overview and evaluation, Part I: alternative models with a common risk. Keio Econ Studies 27-2:17–41 Sakai Y (1991) Information sharing in oligopoly: overview and evaluation, Part II: private risks and oligopoly models. Keio Econ Studies 28-1:51–70 Sakai Y, Yamato T (1988) Information sharing in Bertrand oligopoly. Discussion Paper 88-1, Social Sciences, University of Tsukuba, pp 1-31 Sakai Y, Yamato T (1989) Oligopoly, information and welfare. J Econ 49-1:34–51 Sakai Y, Yamato T (1990) On the exchange of cost information in a Bertrand-type duopoly model. Econ Stud Quarter 41-1:48–64 Sakai Y, Yoshizumi A (1991a) The impact of risk aversion and information transmission between firms. J Econ 53:51–73 Sakai Y, Yoshizumi A (1991b) Risk aversion and duopoly: is information exchange always beneficial to firms? Pure Math Appl 2:129–145 Scherer FM (1981) Industrial market structure and economic performance, 2nd edn. Rand McNally, Chicago Selten R (1973) A simple model of imperfect competition where 4 are few and 6 are many. Int J Game Theory 2:141–201 Shapiro C (1986) Exchange of cost information in oligopoly. Rev Econ Stud 53:433–466 Suzumura K, Okuno-Fujiwara M (1987) Industrial policy in Japan: overview and evaluation. In: Sato R, Wachtel P (eds.) Trade friction and economic policy: problems and prospects for Japan and the United States. Cambridge University Press, New York Vives X (1984) Duopoly information equilibrium: Cournot and Bertrand. J Econ Theory 34 (1):71–94 Vives X (1985) On the efficiency of Bertrand and Cournot equilibria with product differentiation. J Econ Theory 38(1):166–175 Vives X (1992) Trade association disclosure rules, incentives to share information, and welfare. RAND J Econ 21(3):409–430 Vives X (1999) Oligopoly pricing: old ideas and new tools. MIT Press, Boston Vives X (2008) Information and learning in markets: the impact of microstructure. Princeton University Press, Princeton von Neumann J, Morgenstern O (1944) Theory of games and economic behavior. Princeton University Press, Princeton von Stackelberg H (1934) Marktform und Gleichgewicht (in German). Julius Springer, Berlin Yasui T (1979) Economics and my academic life (in Japanese). Bokutaku Publishers, Tokyo

Chapter 7

The Profit-Maximizing and Labor-Managed Firms: A Unified Approach to the Role of Information

Abstract This paper analyzes the welfare implications of acquiring information for profit-maximizing and labor-managed firms (in short, PMF and LMF). We invent a unified method of exploring the role of information in a two-person game under uncertainty on the basis of comparative static analysis, and then apply the method to both the PMF and LMF. It is shown that whereas the LMF’s behavior is analogous to that of the PMF in some circumstances, the former may be entirely different from the latter in others. Because a special status is accorded to labor as a variable factor of production, the informational analysis of the LM economy requires special care for both computation and interpretation. Looking carefully at reality, there are a variety of capitalist firms, presumably forming a sort of spectrum with PMF at one end and the LMF at the other end. We would strongly believe that LMF also matters and should be worthy of due investigation. Keywords Profit-maximizing firms · Labor-managed firms · American-type economy · Japanese-type economy · Creating shared value · Three-way advantages

7.1

A Variety of Market Economies: An Introduction

This chapter aims to compare the working and performance of profit-maximizing and labor-managed firms from an informational point of view. In what follows, those two different types of firms are, respectively, shortened to PMF and LMF. While there are many intriguing problems in modern oligopoly theory, it is unquestionably correct to say that the issue of the impact of the information acquisition and transmission on the activities and welfare of firms operating under uncertainty constitutes a most, if not the most, important problem we have to investigate today. In reality, there exist a variety of institutions through which

This paper is the most newly revised version of Sakai (1995). The revision work has been done on the basis of the recent development of related areas of research. © Springer Nature Singapore Pte Ltd. 2021 Y. Sakai, K. Sasaki, Information and Distribution, New Frontiers in Regional Science: Asian Perspectives 49, https://doi.org/10.1007/978-981-10-0101-7_7

139

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7 The Profit-Maximizing and Labor-Managed Firms: A Unified Approach to the Role. . .

firms in an industry are able to obtain information about demand, cost, or whatever. No doubt, government agencies and trade associations are among those informationgathering organizations. The question of much interest is whether and to what extent the establishment contributes to the welfare of producers, consumers, and the whole society. Looking at the real world in a historical perspective, we understand that there have been a great variety of market economies. Presumably, at the one end of the spectrum of possible forms, there surely exists the typical American-type economy, which as can often be seen in many standard micro-economics tests, may be welldescribed by the traditional, neoclassical profit-maximizing firm. Interestingly enough, however, nearly at the other end, there exists the once-admired Japanesetype economy, which seems to be different from the American economy in many respects. First of all, the Japanese firm can often be regarded as a large family, with the company president playing the role of a head of the family. Second, a small group of company managers are no longer thought of mere agents of stockholders, but rather they are more likely the representatives of the whole employees. Third, in Japan, decision making and information flow are not exclusively monopolized by a selected number of top executives but are largely shared by all employees, including even ordinary salaried workers. Fourth, and most importantly, a considerable portion of total profit goes into the pockets of all workers through extra payment of summer and winter bonuses, provision of sport and leisure facilities and the like. In short, as convincingly pointed out by Komiya (1988a, 1988b) and others, the typical Japanese firm seems to have many characteristic of the LMF as opposed to the neoclassical PMF. Since Japan has become a world economic power, it should be worthwhile to carefully compare the PM and LM economies from an informational point of view.1 The issue of information transmission and exchange among firms was initiated by Basar and Ho (1974) and Ponssard (1979) as applications of stochastic nonzero-sum games to an oligopoly market with PMFs. While the literature on that issue has been extensive since then, it is quite unfortunate that little attention has been paid to the role of information in oligopoly with LMFs. In fact, although the working and performance of the LM economy under uncertainty has been discussed by Fukuda (1980), Hey (1981), Hey and Suckling (1980), Muzondo (1982) and others, it appears that the role of information has not drawn due attention. The purpose of this paper is to fill in such a gap by exploring the effects of information acquisition on LMFs.2 It is well-known that comparative static results for the LMF are usually different from, and sometimes even opposite to, those for the PMFs. For instance, as contrasted with the standard PMF situation, the output of each LMF responds negatively to a rise in the product price and positively to an increase in the final cost. Those and other “perverse results” were first noticed by Ward (1958) and have

1

For the critical role of information in the Japanese economy, see Imai (1992). For an overview and evaluation of information sharing in the PM economy, see Sakai (1990, 1991). 2

7.1 A Variety of Market Economies: An Introduction

141

subsequently provided the focus for much of the literature on the LMF. The question we would like to ask in this paper is whether and to what extent the role of information in the emerging LM economy really differs from that in the traditional PM economy.3 We find quite useful to newly invent a unified method of systematically exploring the role of information in a two-person game under uncertainty. We develop the general framework on the basis of comparative static analysis, and proceed to apply it two specific models, namely the PM and LM duopolies. When there is no information available, the optimal strategy of each player must be a “routine action” because it cannot know any specific value of a stochastic (demand or cost or whatever) parameter. In plain English, an ignorant walking man with no guide maps has no option but simply walk forward on a narrow road in front of his very eyes. If a certain amount of precious information becomes available, however, the optimal strategy of the player becomes no longer routine, but rather a more flexible “contingent action” in the sense that it should be dependent on each realized value. In other words, a cautious and well-equipped climber chooses his best combination of climbing routes taking account of weather and road conditions. For instance, if the mountain whether happens to change, his climbing route might change accordingly. Thus, the important question of whether additional information is beneficial or harmful to a player can simply be reduced to the straightforward one of whether by taking a contingent action rather than a routine action the player is better off or worse off. As can easily be expected, in most cases knowledge is indeed valuable. The real world where many imperfect men like us live, the answer should not be simple like that. Indeed, there exist some other circumstances in which less knowledge may be better than more knowledge: possibly, ignorance produces courage, thus becoming unexpected bliss. We are really interested in comparing the welfare results of information acquisition and transmission on the LM duopoly with those of its PM twin. By making a sequence of such comparisons, we will succeed in obtaining the following set of comparative static results: 1. In general, the effect of acquiring demand information on the expected output of both the LMF and the PMF cannot be determined unless a set of restrictions are placed on the form of the demand and (inverse) production function. If the demand function is specifically linear and each (inverse) production is quadratic, however, the expected output of each LMF surely increases by gathering demand information. This specific result may be contrasted with the corresponding PMF case in which the expected output remains unaffected by such information transfer. 2. Generally speaking, the acquisition of demand information may positively or negatively contribute to both the expected profit per worker of each LMF and the expected profit of each PMF, depending again on the form of demand and 3 An excellent summary of the literature on the LM economy was given by Bonin and Putterman (1987).

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production functions. In the specific yet important case of linear demand and quadratic (inverse) production, whereas acquisition of information definitely makes the PMF better off, it may make the LPF better off or worse off, depending on the specific form of the production function of a stochastic (demand or production) parameter. Remarkably, this demonstrates the possibility that in some circumstances, ignorance is bliss to the LMF. 3. Let us turn to the case when uncertainty is about fixed cost. Then the effects of the acquisition of (fixed) cost information on both the expected output and the welfare of each LMF are indeterminate in sign. These results are definitely positive on the specific case mentioned above. Such welfare results are in sharp contrast to the traditional PMF situation in which both the expected output and the welfare of each PMF remain unscathed by such information transmission. To summarize, it is a more intricate task to explore the welfare impact of information transmission on the LMF than on the traditional PMF. In the simple yet important case of linear demand and quadratic production, the welfare results for the LFM are clearly different from those for the PMF. This indicates the intriguing properties characteristic of the LM economy. The paper is organized as follows. In the next section we introduce a unified method of analyzing the role on information in a two-person game under uncertainty. This method is applied to the PM economy on the third section and to the LM economy in the fourth section. A detailed analysis of the simple yet important case where the demand function is linear and each production function is quadratic is carried out in the fifth section. And conclusions are made in the final section.

7.2

Comparative Statics and the Role of Information: The General Framework and its Applications

This section will introduce a unified method of investigating the role of information in a two-person game under uncertainty. The general framework will be developed on the basis of comparative static analysis, and will be applied in the following sections to two specific models, namely PM and LM duopolies.

7.2.1

The General Framework

Our basic framework is the following two-person game under uncertainty. There are two players in our model―players 1 and 2. Let Zi (yi, yj , α) be the objective function of player i (i ¼ 1, 2; i 6¼ j), where yi represents the strategic variable of player i andαis a common stochastic parameter. As is usual, it is assumed that Zi is an increasing and concave function of yi.

7.2 Comparative Statics and the Role of Information: The General Framework and its. . .

143

It should be noted that the parameter αtis subject to a certain probability density function φ(α). Each player may or may not know the realized value of αa before making his decision. Concerning the information structure of our model, we are content to limit our attention to the following two opposite cases: 1. The case of no information whatever, denoted by η0, in which each player is completely ignorant of α. 2. The case of complete information, written as ηc, in which each player can get information about α presumably through a third party such as a government agency or an independent research association. In this paper, we are also content to make an additional set of assumptions. First of all, information is neither costly nor noisy. Next, we take into account of no possibility of telling a lie or cheating. Moreover, other problems relating to non-symmetric information and risk aversion are not considered here. Although we understand the importance of those problems, we dare to omit them here as a first approximation. Under no information, e0 , each player aims to maximize the expected value of his objective function, where the expectation is taken over α. The player makes a Cournot-Nash type of conjecture about the rival, that is, he chooses his best strategy on the assumption that his rival’s strategy is fixed, which gives rise to a CournotNash equilibrium. More specifically, the pair (y10, y20) of strategies is said to be an equilibrium pair under η0 if the following set relations holds: 
 yi 0 ¼ arg max yi E Z i yi , yj 0 , α

ð7:1Þ

ði ¼ 1, 2; i 6¼ jÞ At equilibrium, the optimal strategy of each player stands for a “routine action” since it depends on any specific value of α. In plain English, a man sticks to the same action for all possible changes of α. In what follows, we will introduce the following linearity assumption: Assumption (L) The objective function Zi of each player is a linear function of α (i ¼ 1, 2). At the first glance, this assumption may look rather restrictive. However, it is quite convenient for our analytical purpose. Besides, as will be seen later, it is satisfied in most of the standard duopoly models under uncertainty.4 If we make Assumption (L), then E [Zi (yi, yj, α)] is mathematically equivalent to Zi (yi, yj, E), whence Eq. (7.1) can be reformulated as follows: yi 0 ¼ arg max yi Z i




 yi , yj 0 , Eα :

ð7:2Þ

ði ¼ 1, 2; i 6¼ jÞ

4

For instance, Assumption (L) is met in the pioneering models of Basar and Ho (1974) and Ponssard (1979). Also see Sakai (1990, 1991).

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Let us assume that the function Zi is continuously differentiable to a desired degree. Then, a set of sufficient conditions for maximization are given by the following relations: ∂Z i =∂yi ¼ 0:ði ¼ 1, 2Þ

ð7:3Þ

2

ð7:4Þ

∂ Z i =∂yi 2 < 0:ði ¼ 1, 2Þ

In the light of (7.3), it is not a difficult job to derive the following set of reaction functions:
 yi ¼ Ri 0 yj , Eα ði, j ¼ 1, 2; i 6¼ jÞ

ð7:5Þ

Clearly, an equilibrium pair (y10, y20), if it exists, is a pair of strategies which satisfy Eq. (7.5). Along with some other regularity conditions to be imposed on the objective function, those conditions aforementioned will ensure the existence of a Cournot-Nash equilibrium under η0 . However, they are not sufficient to ensure the stability of that equilibrium. Following Samuelson (1946), for the sake of comparative static analysis, we have to require stability. This will be guaranteed if the absolute value of the slope of each reaction curve is less than unity, so that the following set of equations must hold: j d Ri 0 =d yj j< 1:ði, j ¼ 1, 2; i 6¼ jÞ

ð7:6Þ

where the derivative is evaluated at equilibrium.5 Let us turn to the case of complete information, tc. In this case, each player can acquire information about αn, and for any given α he chooses a best strategy against some optimal strategy of his rival. Therefore, at equilibrium each optimal strategy is regarded as a “contingent action,” meaning that it is dependent on each realized value of α. Mathematically speaking, the pair (y1c (α), y2c (α)) of strategies is called an equilibrium pair under rc if for each given αf, the following conditions are met:
 yi c ðαÞ ¼ arg max yi :Z i yi , yj c ðαÞ, α :

ð7:7Þ

ði, j ¼ 1, 2; i 6¼ jÞ As is usual, sufficient conditions for maximization are provided by the following equations: ∂Z i =∂yi ¼ 0:ði ¼ 1, 2Þ

ð7:8Þ

2

ð7:9Þ

∂ Z i =∂yi 2 < 0:ði ¼ 1, 2Þ By virtue of (7.8), we obtain a pair of reaction functions as follows: 5

For the stability analysis of a two-person game and its application to duopoly, see Friedman (1977) and Okuguchi (1978).

7.2 Comparative Statics and the Role of Information: The General Framework and its. . .


 yi ¼ Ri c yj , α ði, j ¼ 1, 2; i 6¼ jÞ

145

ð7:10Þ

It is noted here that on appearance, the reaction function Ric under ηc has the same functional form as the reaction function Ri0 under η0, the only difference being thatα is now present instead of Es. We would like to stress that such difference is more than mere appearance: it should be very substantial indeed. Under some regularity conditions on the objective function, there should exist an equilibrium pair (y1c (α), y2c (α)) such that yi ¼ Ric (yj (α), (α)). In addition, stability will be ensured under ηc whenever, for each given α, the following relations hold: j d Ri c =d yj j< 1, ði, j ¼ 1, 2; i 6¼ jÞ

ð7:11Þ

where the derivative is evaluated at equilibrium (see Samuelson (1946)). We are in a position to explore the role of information in our two-person game under uncertainty. We can do such a task by making a sequence of comparisons between equilibrium values underη0 and those underac . We are especially interested in comparing yi0 with E yic (α), and Z 0 with E [Zi (α)]. Under ηc, both yi and Z are functions of α (i ¼ 1, 2). These functions may be or may not be convex (or concave) in α. Suppose that yi (α) is convex (or concave) in α. Then, it follows from Jensen’s inequality formula that for any possible probability density function ϕ(α) of α, E [yi (α)] is greater than (or less than) yi (Eα), where the expectation is taken for ϕ(α) . Under Assumption (L) above, we find that yic (Eα) ¼ yi0 . and Zic (Eα) ¼ Zi0. We can thus establish the following result: Proposition 7.1 Under Assumption (L), information acquisition increases (or decreases) E [yi] if yi is a convex (or concave) function of α. A similar result also holds for E [Zi] (i ¼ 1, 2). In order to investigate the impact of information acquisition on E [Zi], it is necessary to examine the convexity (or concavity) of Zi with respect to α. If we differentiate Zi ¼ Zi (yi (α), yj (α), α) with respect to α, we have the following equation:

 d Z i =dα ¼ ð∂Z i =∂yi Þ ðd yi =dαÞ þ ∂Z i =∂yj d yj =dα þð∂Z i =∂αÞ: ði, j ¼ 1, 2; i 6¼ jÞ

ð7:12Þ

The first, second, and third terms on the right-hand side of (7.12), respectively, denote the “indirect own effect,” the “indirect cross effect,” and the “direct effect” of a rise in α on Zi . Since (∂Zi/∂yi) must vanish by the fist-order condition, (7.12) can be simplified to the following:

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d Z i =dα ¼ ∂Z i =∂yj dyj =dα þ ð ∂Z i =∂αÞ: ði 6¼ jÞ

ð7:13Þ

If we further differentiate both sides of (7.13) with respect toα, we can determine the sign of the second-order derivative d2Zi/dα2. In fact, we have the following equation: d2 Z i =dα2 h  2 ¼ ∂ Z i =∂yi ∂yj ðd yi =dαÞ  
i
 2 þ ∂ Z i =∂yj 2 d yj =dα dyj =dα 

  2 þ ∂Z i =∂yj d2 yj =dα2 þ ∂ Z i =∂α2 : ði 6¼ jÞ If the game in question is symmetric, then the resulting Cournot-Nash equilibrium is also symmetric, so that the equilibrium value of yi and yj should be just equal. Consequently, (7.13) can be rewritten as follows:

h

d2 Z i =dα2

 2 ∂ Z i =∂yi ∂yj  i 2 þ ∂ Z i =∂yj 2 ðd yi =dαÞ2 

  2 þ ∂Z i =∂yj d2 yj =dα2 þ ∂ Z i =∂α2 ¼

:ði 6¼ jÞ

ð7:14Þ

It is noted that the right-side of (7.14) consists of three terms. The first term is ambiguous in sign, depending on the value and sign of the two second-order derivatives (∂2Zi/∂yi∂yj) and (∂2 Zi/∂yj2). While the second term is positive (or negative) if yi is convex (or concave) in α, the sign of the third term is indeterminate. Needless to say, the total impact of information acquisition is a combination of these three components each of which might go either direction. Therefore, we should be extremely careful before reaching any definite conclusion. As the saying goes, Rome was not built in a day. Likewise, a unified approach to oligopoly and information cannot be established so easily, requiring a detailed caseby-case analysis.6 Let us safely get out of such a “blind ally of ambiguity.” We believe that a diagrammatic explanation would be very instructive in understanding the meaning 6

We may say that y i and y i are strategic substitutes (or strategic complements) if and only if the second cross derivative ∂2Zi /∂yi ∂yj is negative (or positive). See Bulow et al. (1985).

7.2 Comparative Statics and the Role of Information: The General Framework and its. . .

147

Fig. 7.1 The simple uniform distribution: φ (α) Remark. φ (H ) ¼ φ(L ) ¼ 1/2; φ(α) ¼ 0 otherwise

Fig. 7.2 Equilibrium under uncertainty: η0 versus ηc. Remark. Q0 = (y10, y20); QH ¼ ( y1 (H ), y2 (H ), QL ¼ ( y1 (L ), y2 (L )

of Proposition 7.1. For simplicity, as is indicated in Fig. 7.1, let us assume that the stochastic parameter αotakes on one of two equally likely values―high (H) or low (L). It is noted that Eα ¼ (H + L)/2. In this simple two-valued distribution case, the relevant reaction curves may be positively or negatively sloping, depending on the functional form of Z . For α ¼ H, player 1’ s reaction curve for player 2’s choice of y2 is shown as R1H, whereas for α ¼ L, it is drawn as R1L . Note that R1L lies west of R1H because L is numerically less than H. A dotted curve R10 denotes the average of these two reaction curves for player 1. In a similar fashion, we are able to draw the reaction curves R2H and R2L together with R20 for player 2. In Fig. 7.2, we may easily find Cournot-Nash equilibriums under η0 and those under mc. On the one hand, when both players are ignorant ofαn, the point represents a stable equilibrium under 0, with (y10, y20) being a pair of equilibrium strategies. On

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7 The Profit-Maximizing and Labor-Managed Firms: A Unified Approach to the Role. . .

Fig. 7.3 The convexity of yi (α)

the other hand, when both players can know α, a pair of the two points, QH and QL, show stable equilibriums. In this latter case, the vector of two pairs, ((y1 (H), y2 (H)) ; (y1 (L), y2 (L)) stands for a pair of equilibrium strategies of players 1 and 2 under ηc. We are in a position to see by means of a diagram how information acquisition by both players affects E yi, the average of yi s. Suppose that yi (ppose that of ion to see α ppose that of ion to see, (H, yi (H)), (L, yi (L)) and (E) ai0), are located as in Fig. 7.3, where E ¼ (H + L)/2. Then, we can easily find the following relations: ðyi ðH Þ þ yi ðLÞÞ=2 ¼ E yi ðαÞ > yi ðEαÞ ¼ yi 0 : Consequently, information acquisition, on average, makes both players more active and lively.

7.2.2

Applications to Profit-Maximizing Duopoly

We are ready to apply the general framework developed in the previous section to two important classes of two-person games under uncertainty, namely, profitmaximizing and labor managed duopolies facing demand or cost risk. This section will deal with the impact of information acquisition on the PM economy. Let us consider an industry with two firms producing a homogeneous product. Firm i produces output xi with the help of labor li and other unspecified fixed factors. The relation between xi and li is described by the inverse production function li ¼ g (xi) , in which g' (xi) > 0 and g''' (xi) > 0 (i ¼ 1, 2). Each firm faces a price pi for its product, being given by the inverse demand function p ¼ b  h (X) , where h' (X) > 0 and X ¼ x1 + x2 . Let w be the competitively

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149

given wage rate and k the fixed cost. Then, firm i’ s profit is defined by the following:7 Πi ¼ ðb  h ðX Þ Þ xi  w g ðxi Þ  k:ði ¼ 1, 2Þ

ð7:15Þ

The profit function Πi is a linear function of b or k , thus satisfying Assumption (L) stated above. Under some circumstances, the parameter b may be a stochastic parameter showing demand uncertainty. Under others, the parameter k may be a stochastic parameter representing fixed cost uncertainty. For instance, if xi stands for the amount of beer production by the i th brewery, then b may represent the fluctuations of GDP or the state of the weather whereas k may be related to the variations of various rents or the breakdown of machinery. Sufficient conditions for profit maximization are given as follows: ∂Πi =∂xi  b  h  h0 xi  w g0 ¼ 0, ði ¼ 1, 2Þ

ð7:16Þ

∂ Πi =∂xi 2  ð2 h0 þ h0 xi þ w g00 Þ 2

< 0:ði ¼ 1, 2Þ

ð7:17Þ

In the light of (7.16), we find that, at equilibrium, xi is a function of xj, which is nothing but firm i’ s reaction function for firm j , being simply denoted by Ri (xj). Indeed, it is not a difficult job to derive the first-order derivatives of Ri (xj) as follows: d Ri =dxj ¼ ½h þ h00 xi =½2h þ h00 xi þ w g00 :

ð7:18Þ

ð i, j ¼ 1, 2; i 6¼ j Þ Now, recall the stability condition (6) implies the following equation: j d Ri =d xj j< 1:ði, j ¼ 1, 2; i 6¼ jÞ This together with the second-order condition (7.17) above implies that [2h + h" xi + w g"] > | h + h"xi |, which in turn implies the following: 3h0 þ 2h} xi þ w g} > 0

ð7:19Þ

Let us carry out comparative static analysis and explore the impact of information acquisition by PMF on equilibrium values. If we differentiate (7.16) with respect to b, then we obtain the following first derivative:

In general, the inverse demand function should be written as p ¼ H (X, b), where b represents a shift parameter. To make our analysis manageable, we assume here that the function H is separable such that H (X, b) ¼ b h(X) 7

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7 The Profit-Maximizing and Labor-Managed Firms: A Unified Approach to the Role. . .


 d xi= d b ¼ 1= 3 h0 þ 2 h} xi þ w g} ,

ð7:20Þ

which is positive by virtue of (7.19). Further differentiation of (7.20) with respect to b results in the following second derivative: d2 xi= d b2 ¼ ð8h0 þ 4h000 xi þ w g000 Þ=ð3h0 þ 2h00 xi þ w g00 Þ : 3

ð7:21Þ

Mathematically speaking, the second derivative d 2xi / d b 2 may go in either direction, depending on the sign of the quantity (8h' + 4h"xi + w g''') . Therefore, we can establish the following proposition: Proposition 7.2 d2 xi =d b2 ⋛0 , 8h000 þ 4h000 xi þ w g000 ⋛0: In the light of Proposition 7.1 above, this proposition tells us that information acquisition about b may increase or decrease the expected output of each PMF, depending on the sign and value of h and g. Now, let us focus on the special yet interesting case in which the demand function is linear, and the production function is linear or quadratic, implying that h" ¼ h"' ¼ 0 and g"' ¼ 0. Then, since the quantity (8h" + 4h"' xi + wg"') vanishes in this special case, it follows from Proposition 7.2 that d 2xi / d b 2 ¼0, so that the expected output of each PMF remains unscathed by information acquisition about b. Interesting enough, such a special and simple case has long been a focal point of investigation in the literature on oligopoly with uncertainty (see Ponssard (1979), Sakai (1990, 1991), for instance). Besides, a much more detailed analysis of this simple case will be given in the next section. Now, let us consider the impact of changes in b on profits. By making use of (7.15) , (7.20), and (7.21) above, we have the following first and second derivatives: dΠi =db ¼ xi ð 1  h0 ðd xi =d b Þ Þ ¼ xi ð 2 h0 þ 2h00 xi þ w g00 Þ=ð 3 h0 þ 2h00 xi þ w g00 Þ
 d2 Πi =db2 ¼ ðd xi =d b Þ  ðh0 þ 2h00 xi Þ ðd xi =db Þ2  h0 xi d2 xi =d b2 ¼ ð2h0 þ wg00 Þ=ð3h0 þ 2h00 xi þ w g00 Þ

ð7:22Þ

2

þ h0 xi ð 8h00 þ 4 h000 xi þ w g000 Þ=ð3h0 þ 2h00 xi þ wg00 Þ

3

ð7:23Þ

Consequently, in view of (7.23), we can establish the following proposition: Proposition 7.3 d2 Πi =db2 ⋛0 , ð2h0 þ w g00 Þ ð3h0 þ 2h00 xi þ w g00 Þ þ h0 xi ð 8h00 þ 4 h000 xi þ w g000 Þ⋛0 ði ¼ 1:2Þ

7.2 Comparative Statics and the Role of Information: The General Framework and its. . .

151

The sign of the second derivative d2Πi / db 2 cannot be determined unless some specific conditions are imposed on demand and production. In general, the acquisition of demand information may positively or negatively contribute to the expected profit of each PMF, depending to the sign and value of h and g . In the special case of linear demand and quadratic production, (7.23) is reduced to the following: d2 Πi =db2 ¼ ð2h0 þ w g00 Þ ð3h0 þ w g00 Þ , 2

which is clearly positive. Therefore, in this simple case, the acquisition of demand information makes each PMF better off, which agrees common sense.8 Now, suppose that there is uncertainty about k, the fixed cost. Since there are no terms associated with k present in (7.16), it follows that d2xi / d k 2 ¼ 0. Moreover, by means of (7.15), we have d2Πi / d k 2 ¼ 0. We can summarize these observations as follows: Proposition 7.4 d2 xi =d k 2 ¼ 0 and d2 Πi =d k 2 ¼ 0:ði ¼ 1, 2Þ This proposition says that, as can be expected, the acquisition of fixed cost information does not affect the expected output and expected profit of each PHF in any way.

7.2.3

Applications to Labor-Managed Duopoly

In this section, we will examine the role of information in the LM economy, the main theme of this chapter. It is well-known that the comparative static analysis of the LMF leads to some “perverse results.” We employ the term “perverse” to indicate behavior opposite to that of the PMF. The question of interest is how and to what extent the LMF’s response to information acquisition is different from that of the PMF.9 Using the same notation as in the previous sections and following the tradition of Ward (1958), we assume that the LMF’s objection is to maximize its profit per

8 It is Oi (1961), one of Sakai’s respected teachers at Rochester, who first noticed the convexity of the profit function with respect to price under perfect competition, arguing that price instability makes the PMF better off. Later, Rothenberg and Smith (1971) extended Oi’s analysis to cover a general competitive model where the feedback for other sector is also allowed for. In their analysis, the PMF is expected to instantaneously adjust its output decision ex post for a random demand, which , in effect, means the PMF takes a contingent action on an ex ante basis. In the present paper, by reformulating the ex post Oi-Rothenberg-Smith analysis as an ex ante one, we shed new light on the value of information in duopoly. 9 See Miyamoto (1980a, 1980b), Bonin and Putterman (1987), and others.

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7 The Profit-Maximizing and Labor-Managed Firms: A Unified Approach to the Role. . .

worker rather than profit per se. Specifically, firm i’s profit per worker is given as follows: Si  Πi =li ¼ ½ðb  h Þ xi  w g  k =g:ði ¼ 1, 2Þ

ð7:24Þ

We can write the sufficient first-order and second-order conditions as follows: ∂Si =xi  b  h  h0 xi  ðw þ Si Þ g0 =g ¼ 0, ð i ¼ 1, 2Þ

ð7:25Þ

∂ Si =∂xi 2  ½ ð2 h0 þ h00 xi þ ðw þ Si Þ g00 =g < 0:ð i ¼ 1, 2Þ

ð7:26Þ

2

Observation of (7.25) tells us that, at equilibrium, xi is a function of xj, which is nothing but firm i’s reaction function for firm j, being simply denoted by Ri (xj). Indeed, it is a rather routine task to derive the first-order derivatives of each LMF as follows: g0 h0 xi  g ðh0 þ h00 xi Þ d Ri ¼ d xj g ½ð2 h0 þ h00 xi þ ðw þ Si Þ g00 :

ð7:27Þ

ð i, j ¼ 1, 2; i 6¼ jÞ The stability condition (7.6) above in conjunction with the second-order condition (7.26) implies that | g' h' xi  g (h' + h" xi) | < g [(2 h' + h" xi + (w + Si) g"] , which in turn leads to the following:10 Di  g ½ 3 h0 þ 2h00 xi þ ðw þ Si Þ g00   g0 h0 xi > 0:

ð7:28Þ

The question to ask is how and to what extent the acquisition of information of demand or cost affects the expected output and expected profit per worker of each LMF. First, we wish to explore the LMF’s response to information acquisition about b , the demand intercept. By differentiating (7.25) with respect to b, we obtain the first-order and second order derivatives: d xi =db ¼ ðg0 xi  gÞ=Di ,

ð7:29Þ

d2 xi =db2 ¼ ðg0 xi  gÞ½2g00 xi Di  ðg0 xi  g Þ Ei =Di 3 ,

ð7:30Þ

where the quantity Ei is defined as follows: E i ¼ 2h ð g0  g00 xi Þ þðg g000 þ g0 g00 Þ ðw þ Si Þ

10 It is noted that the reaction function may be positively or negatively sloped, depending on the form of the demand and production functions.

7.2 Comparative Statics and the Role of Information: The General Framework and its. . .

153

þ4g ð 2h0 þ h00 xi Þ ði ¼ 1, 2Þ Note that the quantity (g'  g" xi) is positive whenever g is convex in xi. Therefore, it is seen in (7.29) that an increase in demand really decreases the output level, confirming a famous “perverse behavior” of the LMF. In the light of (7.30), we immediately establish the following proposition: Proposition 7.5 d2 xi =db2 ⋛0 , 2g00 xi Di  ðg0 xi  g Þ Ei ⋛0 ði ¼ 1, 2Þ If we compare this proposition with Proposition 7.2 above, then we readily see that it is generally a more demanding job to analyze the informational implications for the LM duopoly than those for the PM duopoly. Indeed, it is not easy to determine whether the quantity 2g"xi Di is greater or less than the quantity (g' xi  g) Ei. Therefore, the effect of obtaining demand information on the expected output of each LMF is ambiguous unless further restrictions are placed on the form of the demand and production functions. It is in the next section, we will carry out a detailed analysis for the simple yet important case where h is linear and g is quadratic. The impact of acquiring demand information on the welfare of each LMF can be measured by the following first-order and second-order derivatives: d Si =db ¼ ðxi =gÞ ½1  h0 ðd xi =dbÞ ¼ xi ½ Di þ h0 ð g0 xi  gÞ=g Di   d2 Si =db2 ¼ ðg0 x  gÞ=g2 ðd xi =dbÞ ½ 1  h0 ðdxi =dbÞ h
i  ðxi =gÞ 2h00 ðd xi =dbÞ2 þ h0 d2 xi =db2

ð7:31Þ

ð7:32Þ

ði ¼ 1, 2Þ Inspection of (7.32) indicates that the question of determining the sign of the second derivative d2Si/d b2 is rather involved. While the first term of the right-hand side of (7.32) is positive since d xi/db is negative (see (7.29)), the second term is ambiguous in sign since the signs of h" and d2xi/db2 are indeterminate. By inserting (7.29) and (7.40) into (7.32) and performing some calculations, we find the following: d2 Si =db2 ¼ ð g0 xi  g Þ F i =g2 Di 3 , where Fi denotes the quantity defined as follows:

ð7:33Þ

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  F i ¼ Di ð g0 xi  g Þ Di þ h0 ð g0 xi  g Þ  2 g h} xi    g h0 xi 2g} xi Di  ð g0 xi  g Þ E i :ði ¼ 1, 2Þ

ð7:34Þ

We can thus establish the following proposition: Proposition 7.6 d2 Si =db2 ⋛0 , Fi ⋛0:ði ¼ 1, 2Þ This proposition shows the effect of acquiring demand information on each LMF’s expected profit per worker. As expected, it is generally ambiguous unless further restrictions are placed on the demand and production functions. With some conditions on h and g , the second-order derivative d2Si / db2 may be negative, implying that ignorance may be bliss. Since we believe that this is an intriguing result, we will conduct a more detailed analysis in the next section. Let us turn to the case of fixed cost uncertainty. By differentiating the first-order condition (7.25) with respect to k , we now have the following results: d xi =d k ¼ g0 =Di ; ði ¼ 1, 2Þ

ð7:35Þ

d2 xi =d k 2 ¼ g0 ð 2g00 Di  g0 E i Þ=Di 3 ,

ð7:36Þ

By making use of (7.36), it is fairly easy to establish the following proposition: Proposition 7.7 d2 xi =d k2 ⋛0 , 2g00 Di  g0 Ei ⋛0

ði ¼ 1, 2Þ

As it is seen in this proposition, the expected output of each LMF may respond positively or negatively to the acquisition of fixed cost information: indeed, it depends on the form of the demand and production functions. This result is in sharp contrast to the PMF situation in which xi is linear in k, and hence the expected output of each PMF remains unaffected by the acquisition of fixed cost information (see Proposition 7.4). If we consider the simple case where the demand function is linear and the production function is quadratic, then we may show that the production activity of each LMF, on average, increases by the information acquisition.11 Finally, let us explore the welfare impact of fixed cost information on the LMF. In the same manner as above, we can measure the impact by the following first-order and second-order derivatives:

11 In passing, we note that dxi / dk is positive (see (7.35). Therefore, output and labor respond positively to increase in fixed cost. Interestingly, this is another “perverse behavior” of the LMF.

7.3 A Simple Case of Linear Demand and Quadratic Production

d Si =d k ¼ ð1=gÞ ½1 þ h0 xi ð d xi =dk Þ  ðh0 g0 xi þ Di Þ=g Di ;
 d2 Si =d k 2 ¼ g0 =g2 ð d xi =dk Þ ½ 1 þ h0 xi ð d xi =d k Þ  h
i  ð1=gÞ ð2h00 xi þ h0 Þ ð d xi =dk Þ2 þ h0 xi d2 xi =dk2

 ¼ g0 =g2 Gi =Di 3 , ð i ¼ 1, 2Þ

155

ð7:37Þ

ð7:38Þ

where Gi is the quantity newly defined as follows:  
 Gi ¼ g0 Di ðDi þ h0 g0 xi Þ  g Di g0 2h} xi þ h0 þ h0 xi ð 2g0 Di  g0 E i Þ We can thereby establish the following proposition: Proposition 7.8 d2 Si =d k 2 ⋛0 , Gi ⋛0:ð i ¼ 1, 2Þ In order to see how fixed cost information affects the welfare of each LMF, it is necessary to examine the convexity or concavity of Si with respect to k (see Proposition 7.1). According to Proposition 7.8, we cannot say whether the secondorder derivative d2Si / d k 2 is positive or negative unless we specify the exact form for the demand and production functions. Consequently, the effect of acquiring fixed cost information on each LMF’s expected profit per worker may go in either direction. It will be shown in the next section, however, that when h is linear and g is quadratic, the welfare of each LMF increases by the acquisition of fixed cost information, which would agree with common sense.12

7.3

A Simple Case of Linear Demand and Quadratic Production

Whereas the informational properties of the LM economy are more or less similar to those of its PM twin in some circumstances, the former may be entirely different from the latter in others. In order to make the distinction between these two regimes much clearer, in this section we will carry out a very detailed analysis of the simple yet important case in which the demand function is linear and each firm’s production function is quadratic.

12 This result is sensitive to the assumption that the PMF is a risk-neutral player in the sense that it seeks maximal expected profit. If we instead assume that the PMF displays risk aversion and hence maximizes its expected utility of profit, then the acquisition of cost information is expected to have a significant influence on the welfare of the PMF. The same reservation should be kept for the LMF world. In short, risk aversion really matters! For this point, see Sakai and Yoshizumi (1991a, 1991b). Also see Chapters 8 and 9 of this book.

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7 The Profit-Maximizing and Labor-Managed Firms: A Unified Approach to the Role. . .

Fig. 7.4 Profit-maximizing (PM) duopoly: a simple case

For simplicity, let us make the following assumption: Assumption (DL-PQ) 1. (Demand: Linear) h (X) = x1 + x2 . 2. (Production: Quadratic) g (xi ) = xi2 ( i = 1, 2) First, we will deal with the PM economy. Clearly, firm i’s profit function is written as Πi ¼ (b  h (X)) xi  w g (xi)  k ¼ (b  xi  xj) xi  w xi2  k . So, if firm i maximizes Πi under the Cournot-Nash assumption on its opponent’s output, the following first-order condition must be met: ∂Πi =∂xi  b  2 ð1 þ w Þ xi  xj ¼ 0

ð7:39Þ

Since ∂2Πi / xi2   2 (1 + w) < 0, the second-order condition is always satisfied. In the light of (7.39), we can derive each PMF’s reaction function as follows:
 ½PMF Ri : xi ¼ ½1=2 ð1 þ wÞ b  xj :ði 6¼ jÞ The two reaction curves, R1 and R2, are shown in Fig. 7.4. As is expected, both curves are downward-sloping straight lines. Point Q* represents a unique CournotNash equilibrium which is apparently (globally) stable. The equilibrium output of each PMF is then provided as follows: xi  ¼ b=ð3 þ 2 w Þ:ðxi ¼ 1, 2Þ

ð7:40Þ

And, by virtue of (7.39), firm i’s equilibrium profit can be rewritten as follows: Πi  ¼ ð1 þ w Þ ðxi Þ2  k:

ð7:41Þ

7.3 A Simple Case of Linear Demand and Quadratic Production

157

We must determine the signs of the second-order derivatives with respect to the parameters in question in order to explore the impact of information acquisition by PMFs on equilibrium values. If we differentiate (7.40) and (7.41) with respect to b twice, then we find the following: d2 xi  =db2 ¼ 0; d2 Πi  =db2 ¼ 2ð1 þ wÞ=ð3 þ 2w Þ2 > 0: In this simple case, xi* is a linear function of b . As a result, information acquisition about b has no impact on each PMF’s expected output, but does increase its expected profit. Similarly, we can have the following: d2 xi  =d k 2 ¼ 0; d2 Πi  =d k2 ¼ 0: Since both xi* and Π i* are linear functions of k , it follows that information about k does not affect the expected outputs and profits of PMF s in any way. Now, let us turn to the role of information in the LM economy. Note that under Assumption (DL-PQ), we have Si  Πi / li ¼ [(b  xi  xj) xi w xi2  k ] / xi2 . by means of (7.24) . So, if firm i maximizes Si under the Cournot-Nash assumption on its rival’s output, the following first-order conditions should be satisfied:
 ∂Si =xi  dexi xj  b þ 2 k =xi 3 ¼ 0: ð i, j ¼ 1, 2; Þ i 6¼ j Þ

ð7:42Þ

Making use of (7.42), firm i’s equilibrium profit per worker is expressed by the following equation: Si  ¼ k=ð xi Þ2  1  w:

ð7:43Þ

The second-order condition for profit per worker requires the following: 2 ∂ Si 2 ½xi ð b  xi Þ  3k   < 0: xi 4 ∂xi 2

Because xi (b xi)  3k ¼  k < 0 by means of (7.42), the second-order condition is satisfied. From (7.42), each LMF’s reaction function is given as follows: ½LMF  Ri : xi ¼ 2k=ð b  xi Þ:ði 6¼ jÞ The two reaction curves for the LM economy are shown in Fig. 7.5. For convenience, assume that b2 is greater than 8 k . Then, as is clear from the figure,

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7 The Profit-Maximizing and Labor-Managed Firms: A Unified Approach to the Role. . .

Fig. 7.5 Labor-managed (LM) duopoly: a simple linear case

there exist two equilibrium points, Q* and Q**. The corresponding equilibrium outputs are then given as follows:
1=2  b  b2  8k =2  
1=2 Q   : xi  ¼ b þ b2  8k =2 Q : xi  ¼



ð7:44Þ ð7:45Þ

Let us compare this Fig. 7.5 with the last Fig. 7.4. Then, we see that even in the simple case of linear demand and quadratic production, each LMF’s reaction curve is no longer a straight line but a rectangular hyperbola. Thus, it is a more difficult job to investigate the working of the LM economy than that of its PM twin. In Fig. 7.5, Point Q* is stable but Point Q** is not so. Therefore, for our comparative static analysis to be valid, we have to focus only on the stable point, namely Point Q*. As a result, xi* stands for the only relevant equilibrium output of firm i . We are ready to examine the informational implications for the LM economy. By repeatedly differentiating (7.44) and (7.43) with respect to b, we obtain the following second derivatives: d2 x i  2 x i  ð b  x i Þ 4k ¼ ¼
3=2 > 0; 2 3 2 db ð b  2 x i Þ b  8k

7.3 A Simple Case of Linear Demand and Quadratic Production

159

Fig. 7.6 The relationship between h and Si*

d2 Si  ð b  x i Þ ð b  4 x i Þ ¼ db2 x i  ð b  2 x i Þ 3 i h
1=2 ih
2 1=2 b þ b2  8k b 2 b  8k h ¼ ⋛0:
1=2 i
2 3=2 b  b2  8k b  8k

ð7:46Þ

It is seen that xi* is a convex function of b . Thus, for the simple case of linear demand and quadratic production, we can unequivocally determine the impact of information acquisition about b on E xi . Obtaining demand information increases the expected output of each LMF. This result may be contrasted with the PMF case of linear demand and quadratic production, where as stated above, the expected output remains unaffected by such information acquisition. In the present simple case of PMF, we have shown that e i* is a convex function of b , implying that the welfare of each PMF increases by the acquisition of demand function. It is seen in (7.46), however, Si* is neither convex nor concave in b, whence the impact of the demand information acquisition on the welfare of each LMF is ambiguous. In fact, the second-order derivative d2 Si*/db 2 depends on the sign of the quantity 【2 (b 28k) 1/2 b 】, which is positive or negative according to whether b is greater than or less than the quantity (32 k/3) 1/2 . Figure 7.6 demonstrates the relationship between b and Si*. Apparently, Curve Si* is inverse-S shaped with M being a reflection point. For instance, let us consider the following discrete uniform distribution of b , that is, ϕ(b) ¼ 1/2 for b ¼ (32 k/3) 1/2 + m , (32 k/3) 1/2 + 2 m ; and ϕ(b) ¼ 0 otherwise. (Note that m is a given positive integer.)

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7 The Profit-Maximizing and Labor-Managed Firms: A Unified Approach to the Role. . .

Table 7.1 The impact of information acquisition: a simple case of linear demand and quadratic production Uncertainty Demand Fixed Cost

PM duopoly d2xi*/db2 = 0 d2Πi*/db2 > 0 d2xi*/db2 = 0 d2Π i*/db2 = 0

LM duopoly d2xi*/db2 > 0 d 2Si*/db2 ⋛ 0 (*) d2xi*/db2 > 0 d2Si*/db2 > 0

Remark: Exactly speaking, the following relations hold (*) d2Si*/db2 ⋛ 0 , b ⋛ (32k/3)1/2

Then, the information transmission makes each LMF better off (or worse off) whenever m is positive (or negative). Even working with the case of linear demand and quadratic production, we have thus been led to a “perverse result,” characteristic of the LM economy. Once again, we have learned that the value of information may be negative and ignorance may be bliss! Finally, let us analyze the effects of obtaining fixed cost information on the equilibrium values. In view of (7.44) and (7.46), it is not difficult to derive the following second-order derivatives: d2 x i  8 8 ¼ ¼
3=2 > 0; 2 d k2 ð b  2 x i Þ 3 b  8k h
i 1=2 16b 3 b2  8k b 4b ðb  3 xi Þ d2 Si  ¼ ¼h
1=2 i3  2 3=2 > 0: d k2 ðxi Þ3 ðb  2 xi Þ3 b  b2  8k b  8k It is noted by (7.42) and (7.46) that at equilibrium, b  3 xi* ¼ 2 [k  (xi*) 2]/xi* ¼ xi* (w + Si*) > 0. Therefore, the second-order derivative d2 Si*/dk2 must be positive as shown above. Since both xi* and Si* are convex functions of k, it follows from Proposition above that the acquisition of fixed cost information increases the expected output and expected profit per worker of each LMF. These results are in marked contrast to the PMF world where the information acquisition has no influence on the expected output and expected profit f each PMF. Table 7.1 summarizes the effects of obtaining demand and cost information on the equilibrium values for the PM and LM duopolies. While we limit our attention to the simple case of linear demand and quadratic production, we may easily understand that the information implications for the LM economy are different from those for the PM economy. In fact, it appears that information plays a greater role in the LM economy than in the PM economy. This is because the sign pattern for the former economy looks more complicated and more intriguing than for the latter economy.

7.4 The Labor-Managed Economy Also Matters : Conclusions

7.4

161

The Labor-Managed Economy Also Matters : Conclusions

In this paper, we have been concerned with the role of information played in PM and LM duopoly models. We are especially interested in seeing how and to what extent the LMF’s response to the acquisition of demand or cost information is different from the traditional PMF twin. To this end, we have first developed a unified approach to the role of information in a two-person game under uncertainty, and then applied the approach to the PM and LM duopolies. Generally, it is a more demanding task to explore the informational implications for the LM duopolies than those for the PM duopolies. This is because in contrast to the equal treatment of labor and other factors of production in the conventional PM economy, a special status is accorded to labor as a vital factor of production. As can easily be understood, such non-symmetric treatment in the LM economy requires special care in the computations and interpretations of welfare results. In short, human beings play special status at the LM firm, thus being fundamentally different from other raw materials and machinery. This distinction should clearly be recognized in the discussion of the LM economy. If we want to discuss the working and performance of the capitalist economy, it is correct to say that the PMF matters as one of the conventional forms. We want to assert, however, that the LMF also matters as an alternative form. Michael E. Porter (1947-) is a distinguished American academic who is wellknown for his theories on economicscience, business incentives, and social relations. In a very influential article in Harvard Business Review, Porter and his fellow worker Mark R. Kramer have once remarked: The capitalist system is under siege. In recent years business increasingly has been criticized as a major cause of social, environmental, and economic problems. Companies are wisely thought to be prospering at the expense of their communities (Porter & Kramer 2011, p. 11). We think that Porter’s recognition of the capitalist system as being “the system under siege” is of the utmost importance. We agree with Porter that a big business should be viewed as a major cause of social, environmental, and economic problems: Companies may be prospering at the expense of other players such as smaller businesses, non-profit organizations, and the general public. According to Porter, companies could bring business and the society back together if we redefine their purpose as “creating shared value” rather than “producing surplus value,” meaning that what is good for business is also good for the society. This reminds us of the traditional philosophy of Japanese merchants of “three-way advantages,” implying that what is good for the seller is good for the buyer and also good for the society as a whole. In other words, the seller, buyer, and the society should create shared value. In this respect, it is recalled that Ryutaro Komiya, an influential Japanese economist, has once regarded the Japanese firm as a sort of LMF. Besides, Masahiko Aoki

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7 The Profit-Maximizing and Labor-Managed Firms: A Unified Approach to the Role. . .

(1990), another distinguished economist, has worked hard toward an economic model of the Japanese firm within the framework of a “cooperative game theory.”13 In conclusion, in reality, there exist a variety of capitalist firms, forming a wide spectrum containing the American-type PMF at one end and the Japanese-type LMF at the other end. We would like to stress that LMF also matters: it is worthy of serious investigation. A few final words. There are many directions in which we can extend our analysis of the role of information in the LM duopoly markets. First, in this paper, we have not fully examined the impact of information acquisition on the welfare of the consumer as a third party. Second, no attention was not paid to the possibility that information acquisition may favorably affect the group solidarity that presumably distinguishes the LM economy from the PM economy. Third, we have ignored the problem of risk aversion and firm-specific risks. As shown by Sakai and Yoshizumi (1991a, 1991b), the presence of risk aversion has an effect on increasing the degree of concavity of the objective function (the whole profit or per capita profit function) of each firm, this enhances the possibility that information is harmful rather than beneficial. Fourth, we could apply our analysis to the case of differentiated products and/or the Bertrand-type situation where the strategy of each LMF is not its quantity but its price. As the work of Okuguchi (1986) indicates, the comparison between Bertrand and Cournot equilibriums for the LM economy under product information is very important. Finally, in this paper, the number of firms in an industry is limited to only two. We believe that the generalization of our analysis to an oligopoly market would be a challenging problem to tackle. Those and other related problems will be left for further research. So far so good. However, so many unsolved problems are waiting for us!

References Aoki M (1990) Toward an economic model of the Japanese firm. J Econ Literature 28- 1:1–27 Basar T, Ho Y (1974) Informational properties of the Nash solution of two stochastic nonzero-sum games. J Econ Theory 7-2: 370-384 Bonin P, Putterman L (1987) Economics of cooperation and labor-managed economy. Harwood Academic Publishers, –New York Bulow J, et al. (1985) Multimarket oligopoly: strategic substitutes and complements. J Polit Econ 93-3: 488-511 Friedman JW (1977) Oligopoly and the theory of games. North-Holland, Amsterdam Fukuda W (1980) The theory of the labor-managed firm under uncertainty. Kobe Econ Rev 26-4:46–61 Hey JD (1981) A unified theory of the behavior of profit-maximizing, labour-managed and jointstock firms operating under uncertainty. Econ J 91-3:364–374

The Ohmi merchants of Japan give us a good example of the traditional philosophy of “three-way advantages,” namely, advantages for the seller, the buyer, and the society. It seems that the philosophy is still alive in Japan today. See Ogura (1980, 1991). 13

References

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Hey JD, Suckling J (1980) On the theory of the competitive labor-managed firm under price uncertainty: comment. J Comparat Econ 4-3:338–342 Imai K (1992) Competition among different systems in capitalism. Chikura Publishers, Tokyo Komiya R (1988a) The structural and behavioral characteristics of the Japanese firm: Part I. Tokyo University Econ Rev 54-3:2–16 Komiya R (1988b) The structural and behavioral characteristics of the Japanese firm: Part II. Tokyo Univ Econ Rev 54-4:54–66 Miyamoto Y (1980a) The labor-managed firm and oligopoly. Osaka City University Econ Review 16:17–31 Miyamoto Y (1980b) The labor-managed firm’s reaction function reconsidered. In: Working paper, economics, Osaka City University, pp 27-42 Muzondo Y (1982) On the theory of competitive labor-managed firm under price uncertainty. J Comparat Econ 3-2: 127-144 Ogura E (1980) Ohmi merchants: their origins and developments (in Japanese). Nihon Keizai Publishers, Tokyo Ogura E (1991) The ideas and thought of Ohmi merchants (in Japanese). Sunrise Publishers, Hikone Oi WY (1961) The desirability of price instability under perfect competition. Econometrica 29-1:58–61 Okuguchi K (1978) The stability of price-adjusting oligopoly with conjectural variations. J Econ 38-1:55–60 Okuguchi K (1986) Labor-managed Bertrand and Cournot oligopolies. J Econ 46-2:115–122 Ponssard JP (1979) The strategic role of information on the demand function in an oligopolistic market. Manag Sci 25-2:243–250 Porter ME, Kramer MR (2011) Creating shared value. Harvard Bus Rev 89-1:–2-77 Rothenberg TI, Smith KR (1971) The effect of uncertainty on resource allocation. Quarterly J of Econ 85-3:440–453 Sakai Y (1990) Information sharing in oligopoly; overview and evaluation: Part . alternative models with a common risk, I, Keio Econ Studies. 27-3: 17-41 Sakai Y (1991) Information sharing in oligopoly; overview and evaluation: Part II, , private risks and oligopoly Models. Keio Econ Studies 28-1: 51-71 Sakai Y, Yoshizumi A (1991a) The impact of risk aversion on information transmission between firms. J Econ 53-1:51–73 Sakai Y, Yoshizumi A (1991b) Risk aversion and duopoly: is information exchange always beneficial to firms? Pure Math Appl 2-2:129–145 Samuelson PA (1946) The foundations of economic analysis. MIT Press, New York Ward B (1958) The firm in Illyria: market syndicalism. Am Econ Rev 68:566–589

Part III

Risk Aversion: Mathematical Perspective

Chapter 8

Risk Aversion and Information Exchange: The Constant Absolute Risk Aversion Function and its Applications

Abstract The purpose of this paper is to carefully investigate the relationship between the concepts of risk aversion and expected utility, with a focus on the constant-risk-aversion function and its application to oligopoly theory. Whereas there is now a growing literature on risk, uncertainty, and the market, the operational theory of risk-averse oligopoly has been rather underdeveloped so far.One of the reasons for such underdevelopment is that the established concept of risk aversion remains too abstract rather than reasonably operational, whence very few economists have dared to study the economic consequences of a change of risk aversion by firms. In this paper, we attempt to combine the constant absolute risk aversion function developed by K. J. Arrow and J. W. Pratt, two great economists of the 20th century, and the normal distribution function invented by K.F. Gauss, a mathematical genius of the 19th century: The resulting situation may be called the CARA-NORMAL case. We intend to invent a very useful mathematical theorem for this specific yet important case, and then apply it to the theory of risk-averse oligopoly. In particular, the impact of increasing risk aversion on the outputs of duopolies is carefully examined. It is shown among other things that the comparative static results depend on the degree of risk aversion and the state of product differentiation. Keywords Risk aversion · Expected utility · Oligopoly · Arrow · Pratt · Gauss · Constant absolute risk aversion function

This chapter is largely written on the basis of Sakai (2015). © Springer Nature Singapore Pte Ltd. 2021 Y. Sakai, K. Sasaki, Information and Distribution, New Frontiers in Regional Science: Asian Perspectives 49, https://doi.org/10.1007/978-981-10-0101-7_8

167

168

8.1

8 Risk Aversion and Information Exchange: The Constant Absolute Risk Aversion. . .

Discussing Risk Aversion: From Daniel Bernoulli to Kenneth J. Arrow

We live in the world of risk and uncertainty. In this world, a single act may not necessarily yield a single outcome. It is very common that one act results in many outcomes: Which one really happens among those outcomes depends on the state of the world where we live in. Presumably, a farmer is the biggest gambler in the world. Let us consider how and to what extent risk and uncertainty affect agriculture. As a matter of fact, whether or not rice harvest in fall is good or bad is determined by a variety of weather conditions in spring and summer such as temperature, sunlight hours, rainfall, typhoon, and injurious insects. Besides, the business condition of sightseeing industry is more or less affected by unknown factors including economic and political affairs. When we discuss human behavior under risk and uncertainty, there are two key concepts which play a very important role. They are: risk aversion and expected utility. The purpose of this paper is to scrutinize the relationship between these concepts, whence shedding a new light on the impact of risk and uncertainty on many economic activities. In reality, risk aversion and human behavior are closely intermingled. In his remarkable book, Kenneth J. Arrow (1970), a great economist and Nobel Prize winner, once remarked:1 From the time of Bernoulli on, it has been common to argue that individuals tend to display aversion to the taking of risks, and that risk aversion in turn is an explanation for many observed phenomena in the economic world. (Arrow 1970, p. 3) In this chapter, we would like to discuss more specifically the measure of absolute risk aversion and demonstrate how, in connection with the expected-utility hypothesis, it may be employed to obtain concrete and useful results in economic theory. As was pointed out above, Daniel Bernoulli (1700~82) who was a member of the famous Bernoulli family of mathematical geniuses, wrote an epoch-making paper on decision making under risk. He was the first scholar to introduce the expected utility hypothesis to solve the St. Petersburg paradox in the game of tossing coins. From the time of Daniel Bernoulli to modern times, there have been rather irregular rises and declines in the economics of risk and uncertainty until the 1970s when many economists as a group emerged in the economics profession. Among those economists were K.J. Arrow (1965, 1970) and J. W. Pratt (1964), who intensively

1 In the light of the history of economic thought, the year of 1970 is regarded as the birth year of the modern economics of risk and uncertainty. It is in that year that K.J. Arrows’ outstanding essays in risk-bearing and G. Akerlof’s famous paper on the lemons market were both published. See Akerlof (1970), Sakai (1982, 1990, 2010, 2014, 2015), and Sinn (1983).

8.2 How to Measure Risk Aversion

169

discussed how and to what degree individuals displayed risk aversion in economic phenomena.2 More specifically, we would like to combine the constant absolute risk aversion function developed by economists including K.J. Arrow and J.W. Pratt in the latter half of the 20th century, and the normal distribution function mainly invented by K.F. Gauss, a mathematical genius of the 19th century. Hopefully, such combination will produce a set of nice results in the 21st century. The resulting situation aforementioned may be called the KARA-NORMAL case. We intend to introduce a very powerful mathematical theorem for this specific yet important case, and then apply it to the theory of risk-averse oligopoly. In particular, the impact of increasing risk aversion on the outputs of duopolies is carefully examined. It is shown among other things that the comparative static results depend on the degree of risk aversion and the state of product differentiation.3 The contents of this paper are as follows. The next section will discuss how to measure risk aversion, and then focus on the CARA-NORMAL case. This constitutes the core of the paper. Section 2 will be concerned with its applications to oligopoly theory. Concluding remarks will be made in the final section 4.

8.2 8.2.1

How to Measure Risk Aversion Risk Aversion

“He that fights and runs away may live to fight another day.” “Do not put all eggs in one basket.” As the saying goes, clever people tend to keep clear of any possible risk. When they find that the risk in question is unavoidable, they tend to minimize it by means of risk spreading or purchasing insurance. In literature, to stay or to run away, that is the question. This is certainly the simple world of black and white. The real world where we live is more complex、 however, presumably composing a delicate layer of grey zones. Even if people face the same risk, the degree to which they avoid it may belong to a personal matter: Jack may display stronger risk aversion than Betty. The question to ask is how to measure the personal degree of risk aversion in conjunction with the traditional theory of expected utility. In what follows, I will summarize the theory of risk aversion already developed by K.J. Arrow, J. Pratt and their followers, and then attempt to invent a set of new techniques for the purpose of applications to oligopoly and many other problems.

2

See Arrow (1965,70) and Pratt(1964). There is now a vast literature on the working and performance of oligopoly under information. See Sakai (1990). Unfortunately, very few papers have ever discussed the CARA-NORMAL case, however. This paper intends to further develop this special yet important case. 3

170

8 Risk Aversion and Information Exchange: The Constant Absolute Risk Aversion. . .

In order to discuss the question of risk aversion, it is very convenient to assume that a person in question is asked to choose one out of the following prospects: Prospect (A): income x with probability 1. Prospect (B): income x-h with probability 1/2, income x + h with probability 1/2. Prospect (A) stands for a fixed amount of income x, whereas Prospect (B) indicates a random income (x
h ) with an equal chance of gaining or losing h. Observe that both prospects guarantee the same amount of average income x. It is naturally expected that a man in the street is risk averse, whence he tends to prefer fixed income (A) to random income (B). According to the traditional expected theory, this observation leads us to the following inequality: U ðxÞ > ð1=2Þ U ðx  hÞ þ ð1=2Þ U ðx þ hÞ:

ð8:1Þ

Graphically speaking, this shows that a risk averse person has a concave utility function, and vice versa. In a similar way, it would be an easy job to see that a risk loving person has a convex utility function. Concerning the choice between two prospects, let us keep away from fifty-fifty chance for a while, and turn our eyes to a third prospect. Prospect (C) : income x -h with probability 1 -ρ, income x + h with probability ρ. As is easily seen, Prospect (C) is of a more general form than Prospect (B) since ρ can take any value of the unit interval [0, 1]. Now consider the choice between Prospects (A) and (C). If the value of ρ is near zero, an ordinary person likes Prospect (C) better. If it is near unity, he or she likes Prospect (A) better. Now consider some intermediate values of ρ. When ρ ¼ 1/2, a risk averse person still prefers (A) to (C) as was seen above. If we assume that the utility function is continuous and smooth, we may find a certain value ρ* between 1/2 and 1 so that the following equality holds. U ðxÞ ¼ ð1  ρÞU ðx  hÞ þ ρ  U ðx þ hÞ:

ð8:2Þ

The value of ρ* relative to (1-ρ*) represents how much weight a risk averse person must give to a pair of gain (x + h) and loss (x-h) so that both prospects (A) and (B) may have just the same value. It is naturally expected that a more risk averse person has a more strongly concave utility function, and thus a lager value of ρ*. The graphic illustration of (2) can clearly be seen in Fig. 8.1. In calculus, the concept of the Taylor expansion is generally a very powerful tool. There is no exception for the case of risk aversion. If we employ the theorem of Taylor expansion series, it follows from (2) that 
  U ðxÞ ¼ ð1  ρÞ U ðxÞ  h U 0 ðxÞ þ h2 =2 U 00 ðxÞ þ . . . 
  þ ρ U ðxÞ þ h U 0 ðxÞ þ h2 =2 U 00 ðxÞ þ . . . : If we do some calculations, it follows from (3) that

ð8:3Þ

8.2 How to Measure Risk Aversion

171

Fig. 8.1 Making the two prospects equal values: U (x ) ¼ (1-ρ*)U (x-h )+ρ*U (x + h )


 0 ¼ h U 0 ðxÞ ð2ρ  1Þ þ h2 =2 U } ðxÞ þ . . . , which leads us to obtain the following: ρ ¼ ð1=2Þ þ ðh=4Þ R  þterms of higher order in h,

ð8:4Þ

in which we have R ¼ U } ðxÞ=U 0 ðxÞ:

ð8:5Þ

The value of R* represents a measure of risk aversion, being specifically named the absolute risk aversion. It is noted that the value of R * is closely connected with the minus of U" (x) which in turn shows the degree of concavity of the utility function: Namely, a stronger risk averter has a stronger concave utility. As the saying goes, “so many men, so many minds.” This is a famous maxim showing that the degree of risk aversion varies person to person. Someone may be very prudent and tend to keep the maximum distance from a possible danger, whereas others may be less careful and sometimes dare to challenge the danger. There is clearly a wide intermediate range between these two extremes. Besides a man’s mind is subject to change, and may change situation to situation: Usually prudent man may suddenly change his mind and display a more brave behavior than before. If we keep these facts in mind, we need to invent a new approach to human behavior. It is high time for us to invent an operational theory of risk aversion, so that we may discuss quantitatively rather than qualitatively the welfare implications of a possible change of risk aversion. This is the problem we are going to turn to in the next section.

172

8 Risk Aversion and Information Exchange: The Constant Absolute Risk Aversion. . .

Fig. 8.2 The constant absolute risk aversion function: U (x ) ¼ a - b exp (-Rx )

8.2.2

The Cara-Normal Case

In this section, we would like to combine the two functions that are important in quantitative analysis. They are: the constant absolute risk aversion function developed by K.J. Arrow (1965, 1970) and J.W. Pratt (1964), two great social scientists of the 20th century; and the normal distribution function invented by K.F. Gauss, a mathematical genius of the 19th century. Such an academic combination across the two centuries may be named the CARA-NORMAL case, and is expected to produce a series of nice quantitative properties.4 As was seen in (8.5), the absolute risk aversion function is provided by R *(x) ¼ U" (x) / U 0 (x) . If we put R * (x) ¼ R where R denotes a positive constant, and integrate both sides of this equation, we immediately obtain the following equation: U ðxÞ ¼ a  b exp ðR xÞ:

ð8:6Þ

This is clearly what we may call the constant absolute risk aversion function, or in short the CARA function. The CARA function is a sort of exponential function, and depicted in Fig. 8.2. It is so simple and beautiful: it is increasing, concave, and bounded above, with the upper bound being indicated by a. Let us remind the reader of the historical fact that a simple and beautiful function was first introduced by K.F. Gauss (1777-1855), one of the greatest mathematicians we have ever seen: namely, the normal or Gaussian distribution function N (μ, σ 2)

The CARA-NORMAL case was first introduced to oligopoly theory by Sakai and Yoshizumi (1991a, b), and later developed in related areas by several papers including Sakai and Sasaki (1996). For a systematic discussion on statistics, see Mood, Graybill and Boes (1974).

4

8.2 How to Measure Risk Aversion

173

with average μ and variance σ 2. More specifically, the probability density function Φ(e α) of a stochastic variable e α is written as follows:   ðe α  μÞ2 1 Φðe αÞ ¼ pffiffiffiffiffi exp  2σ 2 2π σ

ð8:7Þ

Now, consider the case where the utility function is given by (8.6) and the probability density function by (8.7). Then in this special CARA-NORMAL case, we can skillfully escape from the computational jungle in which we would usually be involved and got lost, and quantitatively find the concrete value of the expected utility: Z EU ðx; e αÞ ¼

1

1

U ðx; e αÞΦðe αÞde α

We are now in a position to establish the following mathematical results for the CARA-NORMAL case. Theorem 8.1 (The CARA-NORMAL Case): Let e α~N (μ、σ 2) and k be a constant. Then we obtain the following properties: 2 2 (i) E exp [ ke α] ¼ exp [ kμ + (1/2) h k σ2 ]i,

kμ 1 (ii) exp ke α2 ¼ pffiffiffiffiffiffiffiffiffiffiffi exp  1þ2kσ 2 : 1þ2kσ 2 It seems that this theorem per se is not novel in an advanced statistics book such as Mood & Graybill & Boes (1974). To our surprise, however, it has rarely been discussed in our economic science. Under such circumstances, we think that it is worthwhile to give a detailed proof of the theorem here. To begin with, it is noted that the process to prove Property (i) corresponds well to the one to obtain the moment-generating function that is commonly used in statistics. By view of (87), we can immediately following:  have the 2 R e α μ ð Þ 1 1 E exp [ke α] ¼ pffiffiffiffi exp ½ke α exp  2σ2 de α 2π σ 1 1 ¼ pffiffiffiffiffi 2π σ

 2  e α þ μ2  2σ 2 ke α α  2μe exp  de α: 2σ 2 1

Z

1

ð8:8Þ

We note the following:


e α þ μ2  2σ 2 ke α¼ e α  μ þ kσ 2 2  2σ 2 kμ þ ð1=2Þk2 σ 2 : α2  2μe If we substitute (8.9) into (8.8), we obtain the following equation: E exp [ke α]

ð8:9Þ

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8 Risk Aversion and Information Exchange: The Constant Absolute Risk Aversion. . .

( )  Z 1 2 ½e α  ðμ þ kσ 2 Þ 1 k2 σ2 de α: ¼ pffiffiffiffiffi exp kμ þ exp  2 2σ 2 2π σ 1

ð8:10Þ

Taking into account of the property of the normal distribution N(μ+kσ 2, σ 2), we find that 1 pffiffiffiffiffi 2π σ

Z

(

1

½e α  ðμ þ kσ 2 Þ exp  2σ 2 1

2

) de α¼1

ð8:11Þ

By substituting (8.11) into (8.10), we immediately obtain the following: E exp [ ke α] ¼ exp [ kμ + (1/2) k2 σ 2], which proves Property (i). Property (ii) will be proved in a similar fashion. If we do integral calculation, it is not hard to obtain the following:   2

R1 ðeαμÞ 2 1 E exp [-ke α2] ¼ pffiffiffiffi exp ke α de α exp  2σ 2 2π σ 1 Z

h i D exp  2 de α, 2σ 1

1 ¼ pffiffiffiffiffi 2π σ

1

ð8:12Þ

where we have α2 D ¼ (e α-μ)2 + 2σ 2 ke 
 ¼ 1 þ 2kσ 2 e α

μ 1 þ 2kσ 2

2 þ

2kσ 2 μ2 : 1 þ 2kσ 2

ð8:13Þ

If we substitute (8.13) into (8.12), we obtain E exp [-ke α 2] 

1 kμ2 ¼ pffiffiffiffiffi exp  1 þ 2kσ 2 2π σ

2 2 3 μ e α  1þ2kσ 2 7 6 exp 4 2  5de α σ 1 2 1þ2kσ 2

Z

1

ð8:14Þ

If we carefully notice the property of the normal distribution N (μ/( 1 + 2kσ 2), σ 2/ (1 + 2kσ 2 ) ), the following equality obviously holds. 2 2 3 μ e α  1þ2kσ 2 7 1 6

 5de exp 4 α¼1 pffiffiffiffiffi σ  σ2 1 2 2π pffiffiffiffiffiffiffiffiffiffiffi 2 2 1þ2kσ 1þ2kσ Z

1

ð8:15Þ

Therefore, if we take care of both equations (8.14) and (8.15), we can find

8.2 How to Measure Risk Aversion

175

Fig. 8.3 The CARA function and its indifference curves: μ¼(1/2 ) Rσ 2 + e (e is a constant)



E exp ke α

2



  1 kμ2 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp  1 þ 2kσ 2 1 þ 2kσ 2

which is just the same as Property (ii). This completes the proof of Theorem 1. Let us get back to the CARA -NORMAL case in which the utility function U(x) is given by the constant absolute risk aversion function, and the stochastic variable e α follows the normal distribution function. For loss of generality, we may assume a ¼ b ¼ 1, so that (6) above simply reduces to the following: U ðxÞ ¼ 1  exp ðRxÞ:

ð8:16Þ

Then, applying Theorem 1(i) to (8.16), we immediately have

EU ðx Þ ¼ 1  E exp ðR xÞ ¼ 1  exp Rμ þ ð1=2ÞR2 σ 2

ð8:17Þ

In order to find the indifference curve for the CARA-NORMAL case, let us put EU(x) ¼ a constant. Then we immediately obtain -Rμ+ (1/2)R 2σ 2 ¼ a constant. We may let this constant term equal to (-eR ) where e is a constant. Thus it is easy to find that μ ¼ ð1=2ÞRσ 2 þ e ðe is a constantÞ:

ð8:18Þ

For this special case, Fig. 8.3 depicts a class of indifference curves when e ¼ 0, 1, 2, 3. Each indifference curve is upward-sloped, and in fact a straight line with its slope R / 2. When a man displays stronger risk aversion, he is expected to have steeper indifference lines. In general, the utility curve of a risk averse man is concave, whence the following Jensen’s inequality holds.

176

8 Risk Aversion and Information Exchange: The Constant Absolute Risk Aversion. . .

Fig. 8.4 Risk aversion and risk premium: EU (x ) ¼ U (μ-π* )

EU ðxÞ < U ðExÞ ¼ U ðμÞ:

ð8:19Þ

Let us introduce a risk premium π* so that the inequality mentioned above may become an equality. EU ðxÞ ¼ U ðμ  π Þ:

ð8:20Þ

The value of π* indicates the maximum amount of extra money a risk averter is willing to pay for a 100% sure income in exchange for a risky income. The relation between risk aversion and risk premium ρ* can easily be understood in Fig. 8.4. In the special case of the CARA-NORMAL case, we can proceed more and do direct computations. In fact, if we make use of (8.17), we obtain 1  exp Rμ þ ð1=2ÞR2 σ 2 ¼ 1  exp ½Rðμ  π Þ , from which clearly follows π ¼ ð1=2Þ Rσ 2 :

ð8:21Þ

This equation has significant meaning. In the case of the CARA-NORMAL case, the amount of risk premium π* is just equal to the half of the product of the degree of absolute risk aversion R and the value of variance σ 2. As is naturally expected, an increase in R or σ 2 corresponds well to a rise in π*. It should be noticed that such a correspondence holds not only locally, but also very globally. Surely, such a global property represents one of the nicest results we can derive from the CARANORMAL specification.

8.3 Applications to Oligopoly

8.3

177

Applications to Oligopoly

As the saying goes, seeing is believing. In general, if we focus on the CARA-NORMAL case, we are able to simplify the computation process very significantly with much loss of generality, and sometimes help get out of the mathematical jungle in which we would be miserably involved. This claim will be confirmed in the application of the results associated with the CORA-NORMAL specification into oligopoly theory operating under demand risk. In order to make the matter clear, let us take care of simple duopoly. There are two firms in an industry: Firms 1 and 2. Let x i ≧ 0 be the output level firm i, and pi ≧ 0 its unit price level ( i ¼ 1, 2). Suppose that the demand equations are given by linear equations: p1 ¼ α  βð x1 þ θx2 Þ,

ð8:22Þ

p2 ¼ α  βð x2 þ θx1 Þ,

ð8:23Þ

in which α stands for a common demand intercept. Besides, we assume that β is a positive constant and θtakes any value out of the unit interval [-1, 1]. More specifically, the value of θ is a measure of the substitutability of the two goods: Namely, the goods are substitutes, complements or independent according to whether θ is positive, negative, or zero. We may assume without loss of generality that the value ofβ equals unity.5 Suppose that the cost functions of the firms are provided by linear equations: Ci ðxi Þ ¼ ci xi ði ¼ 1, 2Þ:

ð8:24Þ

We assume here that the unit cost c i of each firm is a constant , and ignore the existence of fixed costs. Then in the light of (8.23) and (8.24), the profit Π i of firm i is shown by
 Π i ¼ pi xi  ci xi ¼ αi  ci  xi  θx j xi ði, j ¼ 1, 2; i 6¼ j Þ:

ð8:25Þ

Now consider the situation in which the two firms are subject to the same demand risk. In other words, when each firm determines its production plan, it cannot foresee ex ante how well its product will be sold. For instance, a beer producer cannot exactly predict the amount of beer sales in the coming summer, since its production critically depends upon many weather conditions such as temperature, sunshine hours, rainfall, and the like. Besides, it should be noted that product differentiation is rather common in the beer industry: The two brands of beer may be competitive, complementary, or independent.

5

For a detailed discussion on this point, see Sakai (1990).

178

8 Risk Aversion and Information Exchange: The Constant Absolute Risk Aversion. . .

In order to introduce such demand risk, it is convenient to assume that the common demand intercept α is now a stochastic variable : Therefore , we will write e α with a wave attached. In particular, let us suppose that this e α follows the normal distribution N (μ, σ 2 ) with meanμ and varianceσ 2. Now consider the case in which each firm displays risk aversion and its utility function is given by the constant absolute risk aversion function. In other words, we will make full use of CARA-NORMAL specification. In this important case, the utility of the profit of firm i may be expressed by the following exponential function: U i ðΠ i Þ ¼ 1  exp ½Ri Π i  ðRi > 0; i ¼ 1, 2Þ:

ð8:26Þ

As we discussed in the last section, the important feature of the utility function (8.26) is that the degree of absolute risk aversion is a constant R i . A greater (or a smaller) value of R i represents a greater (or a smaller) degree of risk aversion on the part of firm i . We are in a position to define the Cournot-Nash equilibrium (x1*, x2*) under demand risk in the following way: x1  ¼ arg max x1 Eea U ½Π 1 ðx1 , x2  ; e αÞ x2  ¼ arg max x2 Eea U ½Π 2 ðx1  , x2 ; e αÞ Once the Cournot-Nash equilibrium is reached, each firm has no incentive to deviate from it. In what follows, we will attempt to derive the numerical values of the equilibrium solution under demand risk. In the light of (8.25) and (8.26), we can find the following: E U 1 ðΠ 1 Þ ¼ 1  E exp ½R1 x1 ðe α  c1 x1  θx2 Þ  α exp ½ R1 x1 ðc1 ¼ 1  E exp ½R1 x1 e þx1 þ θx2 Þ:

ð8:27Þ

We recall that e α per se is a stochastic variable following the normal distribution N (μ,σ 2). Then we can make use of Theorem 8.1 above to obtain E exp ½R1 x1 e α ¼ exp ½R1 x1 μþ ð1=2ÞR1 2 x1 2 σ 2 :

ð8:28Þ

By substituting (8.28) into (8.27), we obtain E U 1 ðΠ 1 Þ ¼ 1  exp fR1 x1 ½μ  c1
  1 þ R1 σ 2 =2 x1  θx2 g:

ð8:29Þ

8.3 Applications to Oligopoly

179

If we differentiate (8.29) with respect to x 1 and put the resulting partial derivative just zero, we find that
 μ  c1  2 þ R1 σ 2 x1  θx2 = 0:

ð8:30Þ

In a similar fashion, if we now differentiate (8.29) with respect to x2, and put the resulting partial derivative just zero, it is not difficult for us to derive
 μ  c2  2 þ R2 σ 2 x2  θx1 = 0:

ð8:31Þ

The twin equations (8.30) and (8.31), respectively, indicate firm 1’s and firm 2’s reaction functions under the present CARA-NORMAL specification. If we think of these equations as a system of simultaneous equations, and attempt to solve for the pair of solutions (x1*, x2*), then we are able to find the Cournot-Nash equilibrium pair in the following way: x1  ¼

ð2 þ R2 σ 2 Þðμ  c1 Þ  θðμ  c2 Þ : ð2 þ R1 σ 2 Þð2 þ R2 σ 2 Þ  θ2

x2  ¼

ð2 þ R1 σ 2 Þðμ  c2 Þ  θðμ  c1 Þ : ð2 þ R1 σ 2 Þð2 þ R2 σ 2 Þ  θ2

Therefore, the amounts of equilibrium outputs depend on the following five factors: 1. 2. 3. 4. 5.

The value of μ, namely the average value of changeable demand; The values of c1 and c2 that represent the cost conditions of the two firms; The value of θ, or the degree of production differentiation; The value of σ 2, or the degree of the demand risk; The values of R1 and R2 , or the degrees of risk aversion on the part of the two firms.

Presumably, there are a variety of comparative static analyses we are able to carry out. In this paper, however, we would like to investigate Case 5 only. In other words, we are interested in discussing how and to what extent a change in R1 or R2 affects the values of x1* and x2*. We are facing the situation of product differentiation. The two goods may be substitutes (θ> 0), complements (θ< 0), or independent (θ¼ 0 ). Let us begin our inquiry with the case of substitutes. Then the Cournot duopoly equilibrium is shown by Point Q in Fig. 8.5. The straight lines H1 and H2, respectively, stand for the reaction line of firms 1 and 2 and are negatively sloped. The question to ask is how a change in R i (i ¼ 1, 2) influences the position of Point Q. Suppose that because of some reasons, firm 1 displays a stronger risk aversion than before. Then as the value of R1 becomes greater, the reaction line H1 evolves clockwise to H10 , thus shifting the equilibrium point from Q to Q 0 . These changes result in a decrease in x1 and an increase in x2. Therefore, a stronger risk averse firm must decrease in its own output whereas the output of the other firm

180

8 Risk Aversion and Information Exchange: The Constant Absolute Risk Aversion. . .

Fig. 8.5 Cournot duopoly equilibrium: the two goods are substitutes (θ> 0)

Fig. 8.6 The case of complementary goods (θ< 0)

declines. In short, the two risk-averse firms producing substitutable goods are competitive rivals: One firm’s gain is achievable only at the expense of the other firm. Let us turn to the case of complements. As is seen in Fig. 8.6, reaction lines H1 and H2 of firms are positively sloped. The equilibrium point is indicated by the intersection of the reaction lines, i.e. Point Q . An increase in R1 will cause a shift of reaction line H1 counter-clockwise to H10 . Hence an increase (or decrease) in risk aversion on the part of one firm will decrease (or increase) not only the output of that firm, but also the one of the other firm. In short, in case the products are

8.4 Concluding Remarks: In Memory of Professor Kiyoshi Oka

181

Fig. 8.7 The case of independent goods (θ¼ 0)

complements, the firms are in cooperative relation: All firms in an industry will thrive or decline together. Now consider the third case in which the two goods are independent. Then as is seen in Fig. 8.7, the reaction line H1 of firm 1 is vertical, whereas the reaction line H2 of firm 2 is horizontal. When there is an increase in R1, the equilibrium point will shift to the left from Q to Q 0 along the horizontal line H2. In plain English, when firm 1 becomes more bearish in the face of demand risk, the production activity of the firm has to shrink, whereas the one of firm 2 remains to be the same as before. This is because x1 and x2 are now independent: Hence firm 1’s psychological mind of being bearish or bullish has no effects at all on the production plan of firm 2. To sum up, the comparative static results of oligopoly models under risk are largely dependent on the state of product differentiation and the degree of risk aversion. We have shown that the CARA-NORMAL specification is powerful enough to effectively derive quantitative results.

8.4

Concluding Remarks: In Memory of Professor Kiyoshi Oka

Gone are the days when my heart was young and gay (Foster 1860, 1968, p. 61) This is the first phase of a very popular song written and composed by Stephan Foster (1826-1864), a great American composer. When Sakai was born in Osaka, Japan, his heart was surely very young and gay. As the war broke between Japan and the USA, however, the everyday life of his family became harder and more miserable. Sakai still remembers the tragic days when a group of American B29 bombers dropped so many firebombs so many times over Osaka. Fortunately, his family has

182

8 Risk Aversion and Information Exchange: The Constant Absolute Risk Aversion. . .

survived during the air attacks, yet our painful wartime memory still lingers. After the war, people lost almost everything, hence they had nothing to be afraid of. In other words, they displayed no risk aversion at all for making a living. Since then, turbulent twenty years had passed before Sakai became a still young and fairly ambitious student at Kobe University. In the 1960s, the Japanese society was very unstable and often disturbed by railway strikes, mining shutdowns, street demonstrations, and so on. One day, the National Diet Building in Tokyo was surrounded by so many active students, thus ceasing to do its proper function which is required to do. Those students seemed to display a sort of risk preference for political reform. In those restless days, Sakai happened to read a collection of nice essays written by Kiyoshi Oka (1970), a great Japanese mathematician. It was really a great inspiration for me. He once remarked: In the world of mathematics after the Second World War, there has been a new research direction toward hhextreme abstractionii emerged . According to this direction, very general results were welcomed for the sake of generality, whence more specific yet more fruitful discussions were underestimated or even neglected. I am afraid that such unfortunate tendency still continues and is growing. I then felt as if mathematicians became no longer human: Merely being wandering from place to place in winter wilderness, those people could not see charming green leaves, nor very lovely flowers. Now I am firmly determined to drastically change the mathematical trend from the “chilly winter wilderness” to the “warm spring mildness” So recently, I have written a series of mathematical papers with warm spring flavor. (Oka 1970, p.57) It was no wonder that the teachings of legend Oka gave Sakai an incentive to drastically change my lifestyle: Sakai decided to go abroad for graduate study, looking for the spring warmth. When Sakai took graduate courses at the University of Rochester, however, he was really shocked to see that many economics professors were eager to solve very abstract questions with no reference to warm human heart: They seemed to wander from place to place in the chilly winter wilderness. General equilibrium theory promoted by many Rochester professors represented the culmination of mathematical abstraction and generalization with no human heart.6 After finishing his doctor thesis in mathematical economics, Sakai got a chance to teach economic theory at the University of Pittsburgh. Fortunately, Pittsburgh was a nice place to live: Both professors and students had warm human heart. It was at that

6

When Sakai began graduate courses in the USA, his life was really guided by Professor Oka’s invaluable teachings: Hence he had due respect for mathematics, but had no fear for American culture whatever. We still remember the following inspiring words by Oka (1970): “When I [Oka] began my course work at the Department of Science, Kyoto University, I had too much fear for mathematics to specialize in it. I have never thought, however, that foreign cultures were fearsome and overwhelming. To my regret, the ordinary Japanese people tend to look at this matter from the opposite point of view: Although they do not show due respect for mathematics, they are so afraid of foreign cultures. I would like to emphasize that this is nothing but a terrible mistake.” As of 2020, our respect for Oka’s important teachings still remains, and is really growing larger.

References

183

time that he said good-bye to the winter wilderness and attempted to write economics papers with lovely spring flavor. In our opinion, as pointed out by late Professor Oka, there are two different kinds of problems in every science including mathematics and economics. They are: The problems of the chilly winter color, and those of the warm spring color. Hopefully, this paper will help change the direction of economic science toward more human flavor. To sum up, this paper aims to investigate the relationship between risk aversion and expected utility, with a focus on the constant-risk-aversion function and its application to oligopoly theory. Whereas there is a growing literature in risk, uncertainty, and the market, the operational theory of risk-averse oligopoly has been rather underdeveloped so far. One of the reasons for such underdevelopment is that the established concept of risk aversion remains too abstract rather than operational, whence very few economists have dared to study the consequences of a risk aversion change on oligopoly under imperfect information. Needless to say, there remain so many unsolved problems in the related area of research. One of those problems is the relationship between theory and history. As was wisely pointed out by John Hicks (1969), economics is surely on the intersection of abstract mathematical theory and concrete human history. The problem of risk aversion per se is quite human, and should not be discussed by mathematics only. It is our sincere hope that our discussion in this chapter will give a spring board to step up for the promotion of risk aversion theory and its application in economic science.

References Akerlof GA (1970) The market for lemons: qualitative uncertainty and the market mechanism. Quart J Econ 84:488–500 Arrow KJ (1965) Aspects of the theory of risk-bearing. Yrjo Janssonin Saatio, Helsinki Arrow KJ (1970) Essays in the theory of risk-bearing. North-Holland, Amsterdam Foster S (1860) Old Black Joe. Firth, Pond & Co. Earhart W, Birge EB (eds.) Songs of Stephen foster. University of Pittsburgh Press, Pittsburgh, pp 61-62. Hicks JR (1969) A theory of economic history. Oxford University Press, Oxford Mood AM, Graybill FA, Boes DC (1974) Introduction to the theory of statistics, 3rd edn. McGrawHill, New York Oka K (1970) Ten tales on spring evenings (in Japanese). Mainichi Shinbun Publishers, Tokyo Pratt JW (1964) Risk aversion in the small and in the large. Econometrica 32:61–75 Sakai Y (1982) The economics of uncertainty (in Japanese). Yuhikaku Publishers, Tokyo Sakai Y (1990) The theory of oligopoly and information (in Japanese). Toyo Keizai Publishers, Tokyo Sakai Y (2010) Economic thought of risk and uncertainty (in Japanese). Mineruva Publishers, Kyoto Sakai Y (2014) Information and distribution: the role of merchants in the market economy with demand risk. The Hikone Ronso 399:66–81 Sakai Y (2015) Risk aversion and expected utility : the constant-absolute-risk aversion function and its application to oligopoly. The Hikone Ronso 402:172–187

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Sakai Y, Sasaki K (1996) Demand uncertainty and distribution systems: information acquisition and transmission. Contained in Sato R, Ramachandran R, Hori H (eds.) Organization, performance, and equity: perspectives on the Japanese economy: 59-91. Kluwer Academic Publishers, Norwell, Massachusetts Sakai Y, Yoshizumi A (1991a) Risk aversion and duopoly: is information exchange always beneficial to firms? Pure Math Appl 2:129–145 Sakai Y, Yoshizumi A (1991b) The impact of risk aversion on information transmission between firms. J Econ 53:51–73 Sinn H-W (1983) Economic decisions under uncertainty. North-Holland, Amsterdam

Chapter 9

Information Sharing of Private Cost Information: An Application of the Cardano Cubic Formula

Abstract There have been so many papers on the theory of oligopoly and information. In spite of growing literature on this subject, however, we believe that there is nevertheless a conspicuously missing link in it. To our surprise, very few papers have ever attempted to investigate an important subject of “information exchange and risk aversion.” Although such a subject seems to demand very tough computations and psychological pains, we strongly believe that someone must take up a challenge. So, the main purpose of this paper is to do our best for filling in such a missing gap, thus hoping to do a contribution to the important subject of oligopoly and information. More specifically, this paper aims to discuss the value of additional information in Cournot duopoly when each firm faces its own cost uncertainty. If firms display risk aversion and thus maximize the expected utility of profits, the exchange of cost information between them affects the mean values of outputs as well as their variances. By employing a constant absolute risk aversion model, we are able to show the variance effect may sometimes overpower the mean effect, whence information sharing may possibly make firms worse off. As our daily experience shows, “going together” may sometimes be a better policy than “going alone.” Keywords Information exchange · Oligopoly · Welfare implications · Trade association · Risk aversion · Variance effect · Mean effect

This chapter is a completely revised version of Sakai-Yoshizumi (1991a). Sakai has exerted all his energy for revitalizing it in line with more recent developments of oligopoly theory under imperfect information. We wish to dedicate this article to the fond memory of our old friend Mr. Akihito Yoshizumi, who unfortunately retired from active academic work some time ago. © Springer Nature Singapore Pte Ltd. 2021 Y. Sakai, K. Sasaki, Information and Distribution, New Frontiers in Regional Science: Asian Perspectives 49, https://doi.org/10.1007/978-981-10-0101-7_9

185

186

9.1

9 Information Sharing of Private Cost Information: An Application of the Cardano. . .

A Missing Link in Duopoly and Information: An Introduction

There have been so many papers on the theory of oligopoly and information. In spite of a growing literature on this subject, however, we believe that there is nevertheless a conspicuously missing link in it. To our surprise, very few papers have ever attempted to investigate an important subject of “information exchange and risk aversion.” Although such a subject seems to demand very tough computations and psychological pains, we strongly believe that someone must take up a challenge. So, the main purpose of this paper is to do our best for filling in such a gap, thus contributing to the important subject of oligopoly and information. As the saying goes, there is a will, there is a way! In his monumental essays, Arrow (1970) once wisely remarked:1 From the time of Bernoulli on, it has been common to argue that (a) individuals tend to display aversion to the taking of risks, and (b) that risk aversion in turn is an explanation for many observed phenomena in the economic world, In this essay, I [Arrow] wish to discuss more specifically the measures of risk aversion and to show how, in conjunction with the expected-utility hypothesis, they can be used to derive quantitative rather than merely qualitative results in economic theory (Arrow 1970, p. 90). Although around fifty long years have passed since then, it looks to us that Arrow’s remark is still alive even today. To be honest, we are really in full agreement with his view. As was correctly pointed out by Arrow, from the old time of Bernoulli (1738) on, there should have been a two-way street between economic behavior and risk aversion. On the one hand, observing people’s economic behavior in their daily lives, they tend to display risk aversion. On the other, risk aversion explains very well many observed phenomena to the economic world. So far so good. If we turn to the issue of “oligopoly and information,” however, it is quite unfortunate that such a two-way street has hardly been usable presumably because of some technical and computational difficulties.2

1 The year of 1970, when Arrow’s Essays were first published, can be regarded as a very “memorial year” in the sense that it well represents the dawn of a new era named “Risk and Uncertainty.” Together with Akerlof, Spence, and Stiglitz, the Great Figure Arrow was a good representative of the “A-S Age,” which was so-called by collecting “A” and “S,” the initials of those four pioneers. Arrow’s Essays have been given Sakai a great shock until today. Personally speaking, we are so happy to say that both Sakai and Sasaki have the lucky initial “S.” For details, see Sakai (1982). 2 Bernoulli (1738) was first published in Latin as a mathematical paper at St. Petersburg, the capital of the Russian Empire, and more than two hundred years later translated in English. In this epochmaking and long standing paper, he boldly introduced the Law of Decreasing Marginal Utilities, which was an outstanding achievement in the history of economic thought, being far ahead of the times of the Marginal Revolution in the 1980s. This clearly tells as that Bernoulli was also historically first scholar who introduced the concept of Risk Aversion in the Theory of Decision Making under Risk.

9.1 A Missing Link in Duopoly and Information: An Introduction

187

In this chapter, we are concerned with the value of additional information in a duopoly model in which risk-averse firms are confronted with cost uncertainty. More specifically, we would like to investigate the question of whether and how much the exchange of information between firms is beneficial, or possibly harmful, to them when they display risk aversion. One of the most fashionable topics in modern oligopoly theory is centered around the welfare implications of information sharing among firms. The line of research was already initiated in the 1970s, continued by the explosion of works in the 1980s and the 1990s. And, even in the new century, we have seen the continuation and further development of research on the theory of oligopoly and information.3 A great variety of oligopoly models under risks have been studied so far. In some papers, firms are assumed to behave as Cournot competitors with output strategies, whereas in other papers, they are instead regarded as Bertrand competitors with price strategies. They may exist a common risk or else private (i.e., firm-specific) risks. Uncertainty may be about the demand side or the cost side. Besides, products may be homogeneous or differentiated. While all the existing papers explore the problem of information sharing in oligopoly theory in a very extensive way, it is quite unfortunate that they all have one conspicuous defect in common. This is because they merely assume that firms behave as expected-profit maximizers, implying that firms are risk neutral players. The aim of this paper is to mend such a grave deficiency by making the assumption that firms display a certain degree of risk aversion. To see how and to what degree the introduction of risk aversion into an oligopoly model influences the welfare results of information sharing, we consider here the most standard model of duopoly
Cournot duopoly with cost uncertainty. The basic idea behind our model should be simple and clear. There are two firms in an industry that produce homogeneous products. Each firm has information about its own cost, but not its rival’s. At the starting point, suppose that both firms make a certain agreement concerning the exchange of cost information between them. Such an agreement may be made either by a binding contract or through a third independent agency such as a trade association. The question to ask here is how such an exchange agreement contributes to the welfare of participating firms. It is now well-known that within the framework aforementioned, information pooling between risk neutral firms leads to an increase in expected profits. So far so good: It agrees with our common sense indeed! If, however, firms are risk averse players and thus maximize the expected utility of profits rather than the mere profits, the situation must change drastically. The exchange of cost information between risk-averse firms affects expected outputs in two distinct ways. Indeed, it increases both the mean and variances of outputs. 3

For those long years from the 1970s to the present, there have been a vast volume of papers on oligopoly and information. See Basar and Ho (1974), Ponssard (1979a, 1979b), Clark (1985), Vives (1984, 1999, 2002, 2008), Okada (1984), Sakai (1985, 1990, 1991, 1993, 2015), Sakai and Yoshizumi (1991a, 1991b), Shapiro (1986), Gal-Or (1985), Demange and Laroque (1995), Raith (1996), Jin (1998), and many others.

188

9 Information Sharing of Private Cost Information: An Application of the Cardano. . .

Presumably, the first mean effect constitutes a plus factor for the expected utility of profits, whereas the second variance effect serves as a minus factor for the welfare of firms. It can naturally be conjectured that there would be the case in which the variance effect dominates the mean effect and thus information sharing makes firms worse off. In their remarkable paper, Newbery and Stiglitz (1984) have demonstrated that free trade may be Pareto inferior to no trade whatever. Possibly, no information may be better than noisy information! Our result is consistent with the Newberry–Stiglitz result if we regard the flow of information as one form of trade. In historical perspective, closing one’s country to outsiders may sometimes be better than the complete opening of a country to the outside world. It really depends! The remainder of this chapter is organized as follows: In Section 2, we set up our analytical framework for the Cournot-type duopoly model in which each firm faces its own risk about the cost side. In Section 3, we focus on a simple yet interesting case, involving constant absolute risk aversion utility functions and normally distributed random variables. It is in this specific model that we are able to compute various equilibrium values under private and shared information. Section 4 is devoted to exploring the impact of information sharing on the welfare of producers. Computations may not be so easy. We do hope, however, that our efforts should really be rewarding. Section 5 provides some concluding remarks.

9.2

A Stochastic Model of Duopoly under Private Cost Information

Let us deal with an ordinary Cournot duopoly game where each player treats an output as its strategic variable and is confronted with its own cost risk. We consider an industry in which there are two firms, namely firm 1 and firm 2, which produce homogeneous products. More specifically, let x i be the output of firm i (i ¼ 1,2) and p its unit price. Whenever we build an economic model, we like to obey the golden maxim: “Simple is best!” Undoubtedly, linear equations are simple and beautiful and so are exponential and logarithm functions. Besides, in a stochastic world, normal distribution functions, also known as Gaussian functions, are equally plain and cool! Let the (inverse) demand function in the market be written as follows: p ¼ F ðx1 þ x2 Þ, where F is a differentiable, decreasing function, so that F ' < 0. Now, let the cost function of firm i be denoted in the following way:

ð9:1Þ

9.2 A Stochastic Model of Duopoly under Private Cost Information

Ci ¼ Ci ðxi ; ki Þ ði ¼ 1, 2Þ,

189

ð9:2Þ

where the cost parameters k 1 and k 2 stand for random parameters. Let Φ (k 1, k 2) be the joint distribution function of k 1 and k 2. In particular, we assume that the form of the function Φ per se is common knowledge for the two firms. If we utilize (9.1) and (9.2), we can express the profit of firm i in the following fashion: Πi ¼ Πi ðx1 , x2 ; ki Þ ¼ F ðx1 þ x2 Þ xi
C i ðxi : ki Þ ði ¼ 1, 2Þ:

ð9:3Þ

Here, we assume that each firm has a von-Neumann-Morgenstern utility functionU i (Πi). We especially suppose that U i is an increasing and concave function, meaning that each firm displays risk aversion. We are in a position to pay attention to the information structure we are dealing with. We are content to focus on the following two types: 1. Private information, written as ηp, in which each firm knows its own cost, but not its rival’s cost. 2. Shared information, denoted by ηs, in which both firms share cost information with each other. We would like to compare the two Cournot-Nash equilibriums, namely the one under private information, ηp, and the other under shared information, ηs. On the one hand, a pair (x 1 p (k 1), x 2 p (k 2)) of output strategies is said to be an equilibrium pair under ηp if for each k 1 and each k 2, the following pair of equations simultaneously hold: x1 p ðk1 Þ ¼ arg max x1 E ½U 1 ðΠ1 ðx1 , x2 p ðk 2 ÞÞ j k1 ,

ð9:4Þ

x2 ðk2 Þ ¼ arg Maxx2 E ½U 2 ðΠ2 ðx1 ðk 1 Þ, x2 Þ j k2 :

ð9:5Þ

p

p

On the other hand, a pair (x 1 p (k1, k2), x 2 p (k1, k2)) of output strategies is called an equilibrium pair under ηs if for each (k1, k2), the following pair of equations simultaneously hold: x1 p ðk1, k 2 Þ ¼ arg max E x1 ½U 1 ðΠ1 ðx1 , x2 p ðk1 , k2 ÞÞ j ðk 1, k2 Þ ,

ð9:6Þ

x2 p ðk1 , k2 Þ ¼ arg max E x2 ½U 2 ðΠ2 ðx1 p ðk 1 , k 2 Þ, x2 Þ j ðk1, k 2 Þ :

ð9:7Þ

It is assumed here that each firm wants to maximize expected utility of profits subject to information available to it. On the one hand, under ηp, the optimal output level of firm i is contingent on k i (i ¼ 1, 2). On the other hand, under ηs, it is contingent on both k1 and k2. Note that Cournot-Nash equilibrium represents a sort of “passive equilibrium” in the sense that once it is reached, no player has an “active incentive” to change its strategy unilaterally.

190

9 Information Sharing of Private Cost Information: An Application of the Cardano. . .

Fig. 9.1 The private cost risk depicted like a bellshaped figure

9.3 9.3.1

The Case of Constant Absolute Risk Aversion A Cournot Duopoly Model with Simplifying Assumptions

The model we are going to work with is a simple Cournot duopoly model where each firm is subject to its own cost uncertainty. More specifically, let us make the following set of simplifying assumptions: First of all, we suppose that the demand function each firm faces is linear: F ðx1 , x2 Þ ¼ a
b ðx1 þ x2 Þ, a, b > 0,

ð9:8Þ

where we assume without loss of generality that b is unity. Next, we also assume that the cost function of each firm is simple and linear: C i ðxi ; ki :Þ ¼ k i xi ði ¼ 1, 2Þ:

ð9:9Þ

Concerning the cost uncertainty of the firms, we assume that the pair (k1, k2) of unit costs has a standard symmetric normal distribution of two variables with E (k i) ¼ μ, Var (k i) ¼ σ 2, and Cov (k 1, k2) ¼ ρ σ 2 (i ¼ 1 , 2). In other words, the relevant variance matrix should be of the following simple form:
Σ¼σ

2

1

ρ

ρ

1

 ð9:10Þ

The aforementioned normal distribution may be depicted like a bell-shaped figure. As can easily be seen in Fig. 9.1, the top of the bell is reached at (k 1, k2) ¼ (μ, μ) (see Mood and Graybill (1963)).

9.3 The Case of Constant Absolute Risk Aversion

191

Fig. 9.2 The utility function as an exponential function

In what follows, we suppose that both firms have the same utility function, thereby occasionally dropping the subscript i. For the sake of simplicity, we assume that the utility function is exponential: U i ðΠi Þ ¼ β
γ exp ½
R Πi , β, γ, R > 0,

ð9:11Þ

in which R represents the coefficient of absolute risk aversion. When we put β ¼ γ ¼ 1 for convenience, (9.11) becomes a simpler equation: U i ðΠi Þ ¼ 1
exp ½
R Πi , R > 0,

ð9:12Þ

whose figure is depicted in Fig. 9.

9.3.2

Equilibrium under Private Information

Let us begin our investigation with the case of private information, ηp. First of all, since b is assumed to be unity, we can write the profit of firm i as follows:    Πi ¼ a
b xiþ xj xi
 k i xi ði, j ¼ 1, 2; i 6¼ jÞ ¼ xi a
k i
xi
xj

ð9:13Þ

In the light of (12), given the value of ki and the rival’s output strategy xjp (kj), firm i is supposed to choose its output xi so as to maximize the following equation:

192

9 Information Sharing of Private Cost Information: An Application of the Cardano. . .

              E U i Πi xi , xj p k j ; ki j ki ¼ 1
E exp
R Πi xi , xj p kj ; k i j ki        ¼ 1
E exp
R xi a
ki
xi
xj p k j j ki : ð9:14Þ Under those simplifying linearity assumptions aforementioned, we should expect to find the result that under ηp, the equilibrium output of firm i is also linear in k i: xi p ðk i Þ ¼
A ðki
μÞ þ B

ði ¼ 1, 2Þ,

ð9:15Þ

where A and B represent the quantities to be determined as below. Since both firm 1 and firm 2 are symmetrically treated in this paper, x 1 p (k 1) and x 2 p (k 2) should have the same functional form indicated by (15), so that we also obtain     xj p kj ¼
A k j
μ þ B

ð j ¼ 1, 2Þ,

ð9:16Þ

So, in the light of (16), (14) becomes       E U i Πi xi , xj p k j j ki        ¼ 1
E exp
R xi a
ki
xi þ A k j
μ
B j k i :

ð9:17Þ

We note that the conditional density of the quantity (k j
μ), given k i, is normal with mean ρ(k i
μ) and variance σ 2(1
ρ2). Consequently, the conditional density of the quantity (a
k i
x i + A (k j
μ)
B), given k i, is normal with mean (a
k i
x i + Aρ(k i
μ)
B) and variance A2σ 2(1
ρ2).4 Besides, we find it very helpful to use the following mathematical lemma: LEMMA (Expectation of Exponential Function over Normal Distribution) Let a stochastic variable y be normally distributed with mean μ and variance σ 2 and γ be a constant. Then we obtain the following properties:   1: E exp ½ γy  ¼ exp γμ þ ð1=2Þ γ2 σ 2 :   h  1=2 i    2: E exp
γy2 ¼ 1= 1 þ 2γσ2 exp
γμ2 = 1 þ 2γσ2 :

ð9:18Þ ð9:19Þ

The proof of this lemma is rather straightforward, but perhaps will cause some psychological pain. So, we would like the reader to find detailed proof in a different article.5

4

For the properties of the conditional probability of the multivariate normal distribution, see Mood and Graybill (1963). More generally, let the two-dimensional random variable (x, y) have the bivariate normal distribution with mean (μx, μy) and variance (σx 2, σy 2). Then the conditional density of y, given x, is normal with mean μx + (ρσy /σx) (x
μx) and variance σy 2 (1
ρ2). 5 For detailed proof of this lemma, see Sakai (2015), p. 177. Also see Mood and Graybill (1963).

9.3 The Case of Constant Absolute Risk Aversion

193

In what follows, we want to stress the applicability of the lemma in many ways. Indeed, if we make use of the two properties 1 and 2, we may clearly transform (9.17) into the following equation:       E U i Π i x i , xj p k j j k j h  i ¼ 1
exp
R xi ða
k i
xi þ Aρðk i
μÞ
BÞ þ ð1=2Þ R2 ðxi Þ2 A2 σ 2 1
ρ2      ¼ 1
exp
R xi 2 a
k i
1 þ ð1=2Þ Q A2 þ Aρðk i
μÞ
B , ð9:20Þ

where Q is defined by Q ¼ R σ2 (1
ρ2). Hence, the first-order condition for expected utility-maximization with respect to x i is clearly provided in the following way.   a
ki
2 þ Q A2 xi p ðki Þ þ A ρðki
μÞ
B ¼ 0,

ð9:21Þ

from which follows the equation:    xi p ðk i Þ ¼
ð1
AρÞ= 2 þ Q A2 ðk i
μÞ   þ ða
μ
BÞ= 2 þ Q A2 :

ð9:22Þ

Now, we are ready to compare the two equations, (9.15) and (9.22). Since they should unquestionably be identical equations, we obtain the following results:   A ¼ ð1
AρÞ= 2 þ Q A2 ,   B = ða
μ
BÞ= 2 þ Q A2 :

ð9:23Þ ð9:24Þ

If we rearrange (9.22) and (9.23), then we may easily derive the following equations:   R σ 2 1
ρ2 A3 þ ð2 þ ρÞ A
1 ¼ 0,     B ¼ ða
μÞ= 3 þ Rσ 2 1
ρ2 A2 :

ð9:25Þ ð9:26Þ

It is noted that Eq. (9.25) represents a cubic equation with respect to A except for the perfect correlation case ρ ¼ 1, where the equation becomes just linear. Although the A 2 - term is not present in (9.25), the equation per se is nevertheless fairly complicated and very hard to solve for A. In order to inquire into the properties of such cubic equation more deeply, let us newly introduce the function g (A) as follows:   g ðAÞ ¼ R σ 2 1
ρ2 A3 þ ð2 þ ρÞ A
1:

ð9:27Þ

Then, differentiating (9.26) with respect to A, we find the following derivative:

194

9 Information Sharing of Private Cost Information: An Application of the Cardano. . .

Fig. 9.3 The graphical representation of g (A)

  g0 ðAÞ ¼ 3 R σ 2 1
ρ2 A2 þ ð2 þ ρÞ,

ð9:28Þ

which is always positive. Consequently, g (A) is an increasing function. If we differentiate (9.28) once more, we immediately obtain the following second derivative:   g00 ðAÞ ¼ 6 R σ 2 1
ρ2 A,

ð9:29Þ

implying that g00 (A) ⋛ 0 if and only if A ⋛ 0. As is easily shown, g (0) ¼
1 and g (1 / (2 + ρ)) > 0. Since g (A) is increasing by (9.28), it follows from Fig. 9.3 that letting g (A) ¼ 0 gives only one real root, A *, whose value must lie between zero and 1 / (2 +ρ). Besides, by virtue of (9.29), g (A) is concave (or convex) if A is negative (or positive). So, the point (0,
1) gives us the only one reflection point of the cubic curve g (A). Very long time ago, Gerolamo Cardano (1501-1576), a spirited mathematician from Pre-modern Italy, successfully attempted to find a general formula to solve

9.3 The Case of Constant Absolute Risk Aversion

195

cubic equations. Today, modern mathematicians are a bit smarter than Cardano, thus cleverly offering us a better formula than his old formula. Specifically, we are now ready to use the following improved formula for cubic equations:6 THEOREM (The Cubic Formula of Cardano, with Drawbacks Corrected) Let us consider the cubic equation ax 3 + b x 2 + c ¼ 0. Then we have the solution as follows: h  1=2 i1=3 h  1=2 i1=3 x ¼ p þ p2 þ q3 þ p
p2 þ q3
ðb=3aÞ,

ð9:30Þ

where     p ¼
b2 =27a3 þ b c=6 a2
ðd=2 aÞ,   q ¼ ðc=3aÞ
b2 =9a2 :

ð9:31Þ ð9:32Þ

The exact proof of the Cardano Formula is so complicated that it is wisely omitted here, letting other mathematical papers handle it (See footnote 6). Now, let us dare to apply the Cardano Formula (9.30) to the cubic function g (A) ¼ 0. Then after some calculations, we will find the following solution: h i
h i1=3 h i1=3  þ 1
ð1 þ LÞ1=2 1 þ ð1 þ LÞ1=2 , A ¼ 1=ð2 QÞ1=3

ð9:33Þ

where   Q ¼ R σ 2 1
ρ2 ,

  L ¼ 4 2 þ ρ3 =27Q:

ð9:34Þ

If we insert (9.33) and (9.34) into (9.26), we can calculate the value of B in the following way:   B ¼ ða
μÞ= 3 þ Q A2 :

ð9:35Þ

In the light of (9.15), we can compute the expected value E and variance V of x i p (k i) as follows:

6

  Exi p ¼ B ¼ ða
μÞ= 3 þ Q A2 ,

ð9:36Þ

V xi p ¼ E ð xi p
E xi p Þ 2 ¼ σ 2 A 2 :

ð9:37Þ

For the work of Cardano and his solution of cubic equations, see Dorsey, Downie, and Huber (2020). Also see Wikipedia Web (2020). As far as we know, the cubic formula a la Cardano was first applied to economics in Sakai and Yoshizumi (1991b).

196

9 Information Sharing of Private Cost Information: An Application of the Cardano. . .

We are ready to compute the expected utility of equilibrium profits, EUip. In the light of Eq. (9.16) above, we find it convenient to start with the following conditional expectation over ki:       E j U i Π i xi , x j p k j j k i       ¼ 1
Ej exp
R xi p ðki Þ a
ki
xi p ðk i Þ þ A kj
μ
BÞ j ki : ð9:38Þ Now, it is the right time to employ the mathematical lemma aforementioned. Then, we obtain the following: i h   h    E j U i Πi xi , xj p kj j ki ¼ 1
exp
R xi p ðki Þ a
ki
xi p ðki Þ     i þ Aρ k j
μ
B þ ð1=2Þ R2 ðxi p ðki ÞÞ2 A2 σ 2 1
ρ2     ¼ 1
exp
R xi p ðki Þ a
k i
1 þ ð1=2Þ Q A2 xi p ðki Þ   ð9:39Þ þ Aρ kj
μ
B : In the light of (9.21), we note the following relation:     a
ki
1 þ ð1=2Þ Q A2 xi p ðki Þ þ A ρ k j
μ
B   ¼ 1 þ ð1=2Þ Q A2 xi p ðki Þ:

ð9:40Þ

If we take account of (9.40), (9.39) can be transformed to the following: h   h i      E j U i Πi xi , xj p k j j ki ¼ 1
exp
R 1 þ ð1=2Þ Q A2 ðxi p ðki ÞÞ2 : ð9:41Þ Now, If we take once again the expectation over k i of the conditional expectation (9.41), and later employ (9.18), then we obtain the following equation: h        EU i p ¼ E i E j U i Πi xi , xj p k j j ki ¼ 1
1=F 1=2 exp ½
G , ð9:42Þ where F and G represent the quantities defined by the following:   F ¼ 1 þ 2 R 1 þ ð1=2Þ Q A2 V xi p ,   1 þ ð1=2Þ Q A2 ðE xi p Þ2    G¼ 1 þ 2 R 1 þ ð1=2Þ Q A2 V xi p :

ð9:43Þ ð9:44Þ

9.3 The Case of Constant Absolute Risk Aversion

9.3.3

197

Equilibrium Under Shared Information

Let us turn to the case of shared information, ηs, in which each firm can know both values of k 1 and k2. In this case, for a given k i, firm i chooses its output x i so as to maximize the following utility of profit:       U i Πi xi , xj s ðk 1, k2 Þ; k i ¼ 1
exp
R xi a
ki
xi
xj s ðk1, k2 Þ : ð9:45Þ The first-order condition gives us the following equation: a
k i
2xi s ðk1, k 2 Þ
xj s ðk1, k2 Þ ¼ 0 ði, j ¼ 1, 2; i 6¼ jÞ:

ð9:46Þ

If we solve a set of equations for x1s and x2s, then we obtain the following:   xi s ðk 1, k2 Þ ¼ ð1=3Þ a
2ki þ k j

ði, j ¼ 1, 2; i 6¼ jÞ,

ð9:47Þ

from which we can easily compute the following: E xi s  E ½xi s ðk1, k 2 Þ  ¼ ð1=3Þða
μÞ,    V xi s  E ðxi s ðk1, k2 Þ
E xi s  2 ¼ ð1=9Þ ð5
4ρÞ σ 2 :

ð9:48Þ ð9:49Þ

Using (9.45) and (9.46), we immediately find h i    U i Πi xi s ðk1, k 2 Þ, xj s ðk 1, k2 Þ; k i ¼ 1
exp
R ðxi s ðk 1, k2 ÞÞ2 :

ð9:50Þ

Note that by virtue of (9.47), xis (k1, k2) is a normally distributed random variable whose mean and variance are respectively given by (9.48) and (9.49). Hence, taking the expectation of (9.41) over both k1 and k2, and later using (9.18), we find h i  EU i s ¼ 1
E exp
R ðxi s ðk1, k2 ÞÞ2 ¼ 1
1=H 1=2 exp ½
K ,

ð9:51Þ

where H and K stand for the quantities given by the following: H ¼ 1 þ 2R V xi s, K¼

s 2

R ð E xi Þ 1 þ 2R V xi s

ð9:52Þ ð9:53Þ

9 Information Sharing of Private Cost Information: An Application of the Cardano. . .

198

9.4

The Impact of Information Exchange: The Mean and Variance Effects

The problem we now wish to ask is how and to what extent the information exchange between firms affects their output levels in terms of expected values and variances. While the exchange tends to increase the expected outputs of firms, it is also likely to increase their variances. The first effect may be called the mean effect and the second the variation effect. Which one of the two effects is a dominating actor on the stage of information exchange should be of utmost importance.

9.4.1

The Impact of Information Sharing on Outputs

We are in a position to establish the following proposition of great importance: PROPOSITION 9.1 Under the set of linearity-normality assumptions aforementioned, we obtain the following results: 1:ðthe mean effectÞ E xi s ≧E xi p ,

ð9:54Þ

where the equality holds if and only if ρ ¼ 1. 2:ðthe variance effectÞ V xi s ≧V xi p ,

ð9:55Þ

where the equality holds if and only if ρ ¼ 1. To prove Property 1 of this proposition, let us write again here the mean and variance of xip and xis as follows (see Eqs. (9.36) and (9.48)):   Exi p ¼ ða
μÞ= 3 þ Q A2 ,

E xi s ¼ ða
μÞ=3;

  Q ¼ R σ 2 1
ρ2

which clearly shows that E xis is at least as great as E xip, and that they are just equal if and only if ρ2 ¼ 1. This proves Property 1. To prove Property 2, let us rewrite the mean and variance of xip and xis as follows (see Eqs. (9.37) and (9.49)): V xi p ¼ σ 2 A 2 ,

V xi s ¼ σ 2 ½ ð5
4ρÞ=9;

 2 0 < A2 < 1= 2 þ ρ2 :

We can show that V x i s is at least as great as V x i p, and that they are just equal if and only if ρ2 ¼ 1. This proves Property 2. At this point, a graphical explanation of Proposition 1 would be very instructive. To this end, let us consider the relevant reaction functions under ηp and ηs. In Fig. 9.4, the horizontal axis measures the expected output of firm i, and the vertical

9.4 The Impact of Information Exchange: The Mean and Variance Effects

199

Fig. 9.4 The impact of information sharing on outputs: E x i s ≧ E x i p

axis the expected output of firm j. On the one hand, it is not a difficult job to find that under ηp, a pair of derived reaction functions in terms of E x 1 and E x2 is provided in the following way:     a
μ
2 þ Rσ 2 1
ρ2 A2 E xi p
E xj p ¼ 0 ði, j ¼ 1, 2; i 6¼ jÞ:

ð9:56Þ

On the other hand, in a similar fashion, a pair of derived reaction functions under ηs is given follows: a
μ
2 Ei E j xi p
E i E j xj p ¼ 0 ði, j ¼ 1, 2; i 6¼ jÞ:

ð9:57Þ

or more simply, a
μ
2 E xi p
E xj p ¼ 0 ði, j ¼ 1, 2; i 6¼ jÞ:

ð9:58Þ

9 Information Sharing of Private Cost Information: An Application of the Cardano. . .

200

In Fig. 9.4, points Q p and Q s, respectively, represent Cournot-Nash equilibrium under ηp and ηs. Unless k 1 and k 2 are perfectly (positively or negatively) correlated, point Q s lies northeast of Q p. Clearly, this implies that E x i s is greater than E x i p. Let us take a more careful look again at Proposition 2. Interestingly, the exchange of cost information between risk averse firms affects outputs in two distinctive ways. First, it leads to an increase in the expected value of each output. This may be called the mean effect. Second, it results in a rise in the variance of each output as well. This is because information sharing makes the production activities of firms more responsive to changes in cost conditions: it can be named the variation effect. Which plays a dominant part, the mean effect or the variance effect? In short, when we consider the impact of private information between risk-averse players, this must be a very critical question to ask.

9.4.2

The Impact on the Welfare of Firms

We are now interested in the welfare aspect of information exchange. To this end, we are ready to establish the following welfare proposition of utmost importance: PROPOSITION 9.2 Under the set of linearity-normality assumptions aforementioned, we have the following results: E U i s ⋛E U i p according to whether ð1=2RT Þ log ðH=F Þ⋛ða
μÞ2 ,

ð9:59Þ

where   h  2   i T ¼ 2 þ Q A2 = 2 3 þ Q A2 1 þ Rσ 2 A2 2 þ Q A2  
1= 9 þ 2 ð5
4ρÞ R σ 2 :

ð9:60Þ

To prove this proposition, we note by virtue of (9.42) and (9.51) the following equivalence relations:   E U i s ⋛E U i p , 1
1=H 1=2 exp ð
K Þ⋛1
1=F 1=2 exp ð
GÞ , ðH=F Þ1=2 ⋛ exp ðG
K Þ , ð1=2Þ log ðH=F Þ⋛G
K

ð9:61Þ

Now we have to prove that (1) H is greater than F, and that (2) T, defined by (60), is positive. To this end, if we substitute (9.10), (9.11), (9.14), and (9.15) into the definitions of G and K, we find the following:

9.4 The Impact of Information Exchange: The Mean and Variance Effects

G
K ¼ R T ða
μ Þ2 :

201

ð9:62Þ

To prove (1), let y ¼ Rσ2 A 2 > 0. Then from (9.24), we immediately have 

 1
ρ2 y þ 2 þ ρ ¼ 1=A,

ð9:63Þ

so that Rσ 2 ¼ y=A2 ¼



 2 1
ρ2 y þ 2 þ ρ y:

ð9:64Þ

Substituting these equations into (9.60), we obtain T ¼ I=J,

ð9:65Þ

where h   2 i    I ¼ 9 þ 2ð5
4ρÞ 1
ρ2 y þ 2 þ ρ y 2 þ 1
ρ2 y    2     
2 3 þ 1
ρ2 y 1 þ y 2 þ 1
ρ2 y   h  2 ¼ 1
ρ2 y 8 1
ρ2 ð1
ρÞ y3 þ 4 ð1
ρÞ2 ð1 þ ρÞð11 þ 4ρÞ y2 i   þ2 ð1
ρÞ 38 þ 30ρ þ 4ρ2 y þ ð41 þ 16ρÞ , ð9:66Þ    2       J ¼ 2 3 þ 1
ρ2 y 1 þ 2 þ 1
ρ2 y y h   2 i  9 þ 2ð5
4ρÞ 1
ρ2 y þ 2 þ ρ y :

ð9:67Þ

Cleary, we see that both I and J are positive, so that T should be positive. To prove (2), we adopt a method similar to (1). Then, we find the following:   2     H
F ¼ ½2ð5
4ρÞ=9 1
ρ2 y þ 2 y
2 þ 1
ρ2 y y     ¼ y 1
ρ2 =9        2ð5
4ρÞ 1
ρ2 y2 þ 31
12ρ
16ρ2 y þ 2 ð11 þ 4ρÞ , ð9:68Þ which must be positive. Hence, we should find H greater than F. Now, the proof is complete. In what follows, we would like to consider economic implications of the second proposition in many possible ways. As the proposition shows, the impact of information sharing on the welfare of risk-averse producers may go in either direction, depending upon many factors to be listed below: (i) the expected values of net demand intercept, a
μ,

202

9 Information Sharing of Private Cost Information: An Application of the Cardano. . .

Fig. 9.5 The impact of information sharing on the expected utility of profits: changes in (a
μ)

(ii) the degree of risk aversion, R, (iii) the degree of risk per se. σ 2, (iv) the value of correlation, ρ. Clearly, there are so many combinations of those four factors conceivable. In some situations, firms may gain from exchanging their cost information with each other. In other situations, however, they may not gain from such exchange. To find more definite answers, we must have the courage to go several steps further, First of all, we would like to examine how welfare gain or loss is related to factor (i) above. As we can see from the first proposition above, there exist two different channels through which information exchange affects the welfare of risk averse firms, namely the mean effect and the variance effect. If the factor (a
μ) is sufficiently small, the mean effect presumably dominates the variance effect, so that information sharing makes both firms better off. When the factor (a
μ) is sufficiently large, however, the situation must change drastically. Risk aversion now plays such a critical role in determining the welfare impact of information sharing between firms: possibly, information sharing is even harmful rather than beneficial.

9.4 The Impact of Information Exchange: The Mean and Variance Effects

203

Fig. 9.6 The impact on the welfare of firms: changes in ρ

Figure 9.5 indicates the significance of the second proposition in a more visible fashion. The horizontal axis measures the quantity (a
μ) and the vertical axis the quantity (EU s
EU p). For convenience, we assume that R is ten and σ 2, is unity. Let us suppose that ρ takes on five different values, namely ρ ¼
0.95,
0.7, 0, 0.7, 0.95. Then, as is seen in Fig. 9.5, we have five U-shaped curves corresponding to those values of ρ. When any one of those curves lies above the horizontal line OH, we are in the normal situation in which information exchange is beneficial to firms. If, however, a U-shaped curve happens to lie below the line OH, we enter the anomalous world in which non-sharing is even better than sharing. As our experience teaches us, “going alone” is sometimes better than “going together”! Figure 9.6 gives us another look at the problem of how the welfare gain or loss from information is sensitive to factor (iv), namely the value of correlation, ρ. Note that as before, R is ten and σ 2 is unity, and that (a
μ) is now fixed at two. When ρ is negative, the two firms are stochastically in a complementary relation, so that the exchange of information is likely to contribute positively to the welfare of firms. By contrast, in case ρ is positive, the conflict of interests between rival firms arises so seriously that information pooling may be rather harmful to them. In short, between friends, “going together” is always helpful. Between rivals, however, “going alone” may sometimes be a better policy. Human relation are really complicated indeed! Finally, we can see the relationship between the degree of risk aversion and the value of information. Note that the horizontal axis now measures the value of R. Just for the sake of convenience, we assume that σ 2 ,¼ 0.8, ρ ¼ 0, and a
μ ¼ 1.9. Then, we can draw a unique figure like a playground slide in Fig. 9.7. When the value of R is sufficiently small (in fact, between one and four in the present case), EU s exceeds EU p. When the value of R becomes sufficiently large,

204

9 Information Sharing of Private Cost Information: An Application of the Cardano. . .

Fig. 9.7 The relationship between the degree of risk aversion and the value of information

however, the non-normal situation under which information exchange is rather harmful may emerge. This clearly demonstrates that an increasing degree of risk aversion has a negative impact on the information exchange between firms. As the saying goes, a wise man keeps clear of danger!7

9.5

Is Information Beneficial or Hurtful? Concluding Results

In this chapter, we have been intensively concerned with the important issue of risk aversion and duopoly. The fundamental question to ask is simple like this: “Is the information exchange between the two firms always beneficial to them? Or possibly, is it rather hurtful to them?” In our daily life, we tend to be guided, and perhaps perplexed, by the following two opposing kinds of proverbs. As one kind of proverbs teaches us the following: “Knowledge is power,” “Two heads are better than one.” 7 Among topics closely relating to our research in this paper are informational non-efficient markets and Pareto inferior trades. See Grossman and Stiglitz (1980) and Newbery and Stiglitz (1984).

References

205

There is another kind of proverbs, however, that says exactly the opposite: “Ignorance is bliss.” “What the eye doesn’t see, the heart doesn’t grieve at.” What we have shown in this chapter is that both kinds of proverbs are not contradictory as they seem, and that their validity depends on the real circumstances we are confronting with. More specifically, we have devoted all our energy into an investigation of how and to what extent the presence of risk aversion affects the outputs and welfare of producers. On the one hand. information pooling tends to increase the means and variances of output simultaneously. On the other hand, information sharing may sometimes be harmful rather than beneficial to firms. Therefore, the welfare results when the firms display risk aversion are different from those of many existing papers, in which the firms engaging in information sharing are usually supposed to be risk neutral. Admittedly, our duopoly model with private cost exchange is a very simple one, involving constant absolute risk aversion utility functions, normally distributed random variables, together with linear demand and cost functions. It is true that weakening some of those specific assumptions could make our model more general. In theory, we have no objection against this generalization whatever. It would make our task of deriving equilibrium values under alternative information structures, however, which already requires fairly troublesome computations, an even more formidable from a pragmatic point of view. Besides, we believe that even if we work with a more general framework than we have done in this paper, the possibility that the value of acquiring additional information may be negative still remain. We must bear in mind that both good and bad coins are in circulation and that bad coins may sometimes drive out good ones.

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