A Model of Simple Competition [Reprint 2014 ed.] 9780674430495, 9780674430488

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THE ANNALS OF THE COMPUTATION LABORATORY OF HARVARD UNIVERSITY VOLUME XLI A MODEL OF SIMPLE COMPETITION

THE ANNALS OF THE COMPUTATION LABORATORY OF HARVARD UNIVERSITY I II III IV V VI VII VIII IX X XI XII XIII XIV XVI XVII XVIII XIX XX XXI XXII XXIII XXIV XXV XXVI XXVII XXIX XXX XXXI XXXV XL

A Manual of Operation for t h e Automatic Sequence Controlled Calculator Tables of t h e Modified Hankel Functions of Order One-Third and of Their Derivatives . . . . . . . . . . . . Tables of the Bessel Functions of the First Kind of Orders Zero and One Tables of the Bessel Functions of t h e First Kind of Orders Two and Three . Tables of t h e Bessel Functions of t h e First Kind of Orders Four, Five, and Six Tables of the Bessel Functions of t h e First Kind of Orders Seven, Eight, and Nine . . . . . . . . . . . . . Tables of the Bessel Functions of t h e First Kind of Orders Ten, Eleven, and Twelve . . . . . . . . . . . . Tables of the Bessel Functions of the First Kind of Orders Thirteen, Fourteen, and Fifteen . . . . . . . . . . . . Tables of the Bessel Functions of the First Kind of Orders Sixteen through Twenty-Seven . . . . . . . . . . . Tables of the Bessel Functions of the First Kind of Orders Twenty-Eight through Thirty-Nine . . . . . . . . . . Tables of the Bessel Functions of t h e First Kind of Orders F o r t y through Fifty-One Tables of the Bessel Functions of t h e First Kind of Orders Fifty-Two through Sixty-Three Tables of the Bessel Functions of the First Kind of Orders Sixty-Four through Seventy-Eight . . . . . . . . . . . Tables of the Bessel Functions of the First Kind of Orders Seventy-Nine through One Hundred Thirty-Five . . . . . . . . Proceedings of a Symposium on Large-Scale Digital Calculating Machinery . Tables for t h e Design of Missiles . . . . . . . . Tables of Generalized Sine- and Cosine-Integral Functions: P a r t I Tables of Generalized Sine- and Cosine-Integral Functions: P a r t I I Tables of Inverse Hyperbolic Functions . . . . . . . Tables of Generalized Exponential-Integral Functions . . . . sin (' Tables of the Function • and of Its First Eleven Derivatives

1946

Tables of the Error Function and of I t s First Twenty Derivatives Description of a Relay Calculator . . . . . . . . Description of a Magnetic D r u m Calculator . . . . . . Proceedings of a Second Symposium on Large-Scale Digital Calculating Machinery . . . . . . . . . . . . Synthesis of Electronic Computing and Control Circuits . . . . Proceedings of an International Symposium on t h e Theory of Switching: Part I Proceedings of an International Symposium on the Theory of Switching: Part II Proceedings of a H a r v a r d Symposium on Digital Computers and Their Applications . . . . . . . . . . . Tables of the Cumulative Binomial Probability Distribution Tables of the Function arc sin ζ . . . . . . .

1952 1949 1952

1945 1947 1947 1947 1947 1947 1947 1948 1948 1948 1949 1949 1951 1948 1948 1949 1949 1949 1949 1949

1951 1951 1959 1959 1962 1955 1956

A M O D E L OF S I M P L E COMPETITION J O E L E. C O H E N

CAMBRIDGE, MASSACHUSETTS

HARVARD UNIVERSITY PRESS 1966

© Copyright 1966 by the PRESIDENT AND FELLOWS OF HARVARD COLLEGE All rights reserved Distributed in Great Britain by OXFORD UNIVERSITY PRESS, LONDON

Library of Congress Catalog Card Number 66-23470 Printed in Great Britain

PREFACE

Bookies and sports fans have long known that one important aspect of any competition is its outcome and that it is entertaining and possibly profitable to try to define a class of competitions whose outcomes follow some pattern. It may also be entertaining and useful to try to understand the pattern. This essay points out that in certain attractively simple, but nevertheless real-world, economic situations, if the success of competing sellers is measured by their capacity to produce or by their sales, then, ranked in order of size, these measures follow a pattern. Published data on the United States Portland cement, steel, oil refining, synthetic rubber, and aluminum industries are presented as evidence for this claim. The assertion that the relation between firm size and rank order can be stated mathematically is nothing new in economics. What is new here is the class of economic situations specified, the form of the mathematical relation (the pattern) proposed, and the kind of explanation offered for it. Some biologists seem earlier to have found the same pattern in the outcomes of certain biological competitions among related species of animals. On the assumption that the biologists have found a real regularity, this essay tries to offer them an improved explanation of it. From natural parsimony, it then tries to see what use economics can make of an economic translation of this explanation. No extravagant claim is made that economic and biological competition have therefore been shown to be "the same." Rather, this essay attempts, in a consciously experimental and incomplete way, simply to see whether it is possible to define some formally equivalent aspects of certain competitive situations in economics and biology and to use one mathematical model for both kinds. To do so requires ignoring many aspects of economic and biological competitions that are important to economists and biologists, respectively. But such abstraction has a definite heuristic value: it led to the discovery of the economic regularity which this essay presents; and it has a possible, but still unsettled, theoretical value: it may lead to a validated, wider-ranging way to conceive competition. I hope this essay will provoke economists and biologists to determine the uses and the limitations of such abstractions in their own fields and that it will interest others who like to try to understand social phenomena in unified, but testable, ways. If the economic ideas of this essay should turn out to be reliable, they may even provide businessmen and those who influence economic policies with a reasonable guide for normative judgments about economic concentration and market share. In the exposition, at the cost of some prolixity, I have tried to minimize the number of substantively important statements which might be inaccessible to an intelligent nonspecialist, so that economists, biologists, and any others interested could read all the way through. For those parts of the exposition which are boring or still baffling, I apologize. ν

PREFACE

Nearly all the mathematics in the essay has been relegated to appendices. The results of the appendices have been summarized as nonmathematically as possible in the text. Anyone who wishes to follow through the appendices will find that two years of the calculus and a solid year's course in probability theory are more than sufficient. Thus the mathematical level of the whole essay is quite elementary. However, I hope some mathematicians or mathematical statisticians will be interested by the unsolved problems left behind. For the solution of these, listed in the last chapter, some powerful mathematics will probably be required. Professor Anthony G. Oettinger, of Harvard University's Division of Engineering and Applied Physics, rescued this piece of research from dissipation in its perilous youth, made available every resource I needed to pursue it, and faced the onslaught of my early drafts with determination to make me write comprehensibly if possible. His many suggestions have been incorporated in the text without specific citation; his continuing support is recorded in the essay's very existence. Professors Carl Kaysen of the Department of Economics and E. 0 . Wilson of the Department of Biology at Harvard guided me to relevant literature, repeatedly raised pertinent and helpful questions, and—rarest of all—encouraged my presumption to deal competently with their fields. I also received various kinds of assistance in preparing early drafts from D. J . Bartholomew, J . Brode, P. C. Fischer, J . H. Klotz, R. G. Leahy, S. M. Loescher, S. A. Sussman, C. Trozzo, and H. C. White. In revision, I have benefited much from criticisms and comments by many of the above and also by N. G. Hairston, C. E. King, R. H. MacArthur, B. Mandelbrot, F. Mosteller, T. W. Schoener, H. A. Simon, and D. A. Skolnik. The generosity of these scholars with their time and their ideas impressed me deeply, and I am grateful for it. I alone must be held to account where I have persisted in error. The costs of computer time and of preparing the manuscript for publication were paid in part by the National Science Foundation under Grant GP-2723 and by Harvard's Milton Fund, respectively. Four years of liberal support from the National Merit Scholarship Corporation assured me the time and freedom needed to conceive and execute this work. I dedicate this book to my parents. Among other things, they have always encouraged me, whatever the problem, to figure it out for myself. J.E.C. Bangalore, India 12 January 1966

vi

CONTENTS CHAPTER

1. Introduction

PAKE

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1.1. Concepts of Competition . . . . . . . . 1.2. Biological Origins of t h e Ordered-Random-Intervals Model .

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2. T h e Ordered-Random-Intervals Model 2.1. Whitworth's Theorem 2.2. The Threshold

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3. The Balls-and-Boxes Model 3.1. T h e G a m e 3.2. A Concrete E x a m p l e

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3.3. Biological Interpretation of the Game 3.4. An Economic Question . . .

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17 17 18

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21 24

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3.5. Interpreting the Threshold . . . . . . . . . 3.6. A Possible Dynamics of the Game . . . . . . . . 3.7. Tropical Diversity: Niche Overlap and Niche Specialization . . . Mathematical Appendix 3A. Player r's Expected Share . . . . . Mathematical Appendix 3B. The Distribution in Detail . . . . Mathematical Appendix 3C. Answer t o an Economic Question . . . .

4. An Application to Economics . . . . . . 4.1. Introduction . . . . . . . . 4.2. Conditions on the Real World . . . . . 4.3. Stability of t h e Theoretical Size Distribution . . 4.4. T h e D a t a 4.5. P o r t l a n d Cement . . . . . . . 4.6. Steel 4.7. New York Daily Newspapers . . . . . 4.8. Oil Refining 4.9. SBR Synthetic R u b b e r 4.10. Aluminum . . . . . . . . 4.11. Concentration in American Industries . . . 4.12. Overview of the D a t a . . . . . . vii

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8 8 9 . 1 2 . 1 4 . 15

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Mathematical Appendix 2A. Variance . . . . . . Mathematical Appendix 2B. Hairston's "Variance R a t i o " . . . Mathematical Appendix 2C. Least-Squares E s t i m a t e of the Threshold

1

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25 26 28 34 . 3 7 40

42 42 44 48 49 . 5 1 53 54 55 57 60 61 65

CONTENTS CHAPTER

5. Relation to the Lognormal and the Pareto Size Distributions 5.1. Introduction . . . . . . . 5.2. The Lognormal Distribution . . . . 5.3. The Pareto Distribution . . . . . Mathematical Appendix 5A. The Frequency Function 6. Critique and Conclusion . . . . 6.1. Mathematical Results and Limitations 6.2. Economic Results and Limitations 6.3. Biological Results and Limitations 6.4. A Note on Method

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References

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Appendix. The Economic Data: Figs. A1-A47 Index

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viii

FIGURES PAGE

1.4. Illustration of the ordered-random-intervals model . . . . . . 2.1. Illustration of the ordered-random-intervals model with threshold . . . 2B.1. Illustration of why the Hairston "variance ratio" does not measure goodness of fit to the ordered-random-intervals model . . . . . . . 3:1. Illustration of the balls-and-boxes model . . . . . . . . &B.1. Diagram of the random variables involved in the distribution of balls of a single player in the balls-and-boxes model . . . . . . . . . 4.1< Largest firm's observed and predicted market share in East Coast oil refinery market, 1926-1951 5.1. Test of the lognormal law on the United States cement industry . . . Test of the lognormal law on the United States steel industry . . . . 5.3. Test of the lognormal law on the New York daily newspapers . . . . &Ά. Test of the lognormal law on simulated ordered-random-intervals distribution . Test of the Pareto law on the United States cement industry . . . . 5.6. Test of the Pareto law on the United States steel industry . . . . . 5.7. Test of the Pareto law on the New York daily newspapers . . . . 5.8. Test of the Pareto law on simulated ordered-random-intervals distribution . . 5.9. Some theoretical curves according to Mandelbrot . . . . . .

ix

3 1 0 1 5 1 9 38 58 68 69 69 70 73 7 4 74 75 75

TABLES PAGE

3.1. Comparison of the standard deviations of the ordered-random-intervals and the balls-and-boxes models, for ra = 10 . . . . . . . . . 2 1 3.2. Comparison of the expected size distributions of the balls-and-boxes model under two different formalizations of the principle of competitive exclusion . . . 32 4.1. Stability of predictions in a market with ten firms when only the eight largest are reported . . . . . . . . . . . . . 49 4.2. Selected American industries with national markets and number of companies in each in 1954 63 4.3. Observed and predicted frequencies of industry structures among selected American industries with national markets . . . . . . . . . 64

x

A MODEL OF SIMPLE COMPETITION

CHAPTER 1

INTRODUCTION

. . . b e w a r e of m a t h e m a t i c i a n s a n d all t h o s e who m a k e e m p t y prophecies. T h e d a n g e r a l r e a d y exists t h a t t h e m a t h e m a t i c i a n s h a v e m a d e a c o v e n a n t w i t h t h e devil t o d a r k e n t h e spirit a n d t o confine m a n in t h e b o n d s of Hell. —St. Augustine

1.1. Concepts of Competition The word "competition" is a mistress to many disciplines, and she serves their pleasures variously. Economics qualifies "competition" as "perfect," "pure," "oligopolistic," and in other ways. Milne (1061) distinguished a dozen different uses of the term in a recent 25-year span of biological literature. Sociology, psychoanalysis, and international relations have their own uses of "competition." The word has become a nearly ubiquitous label of which it is true that plus c'est la meme chose, plus ςα change. The available definitions of "competition" are usually either so vague that they include all "competitive" situations, in which case the concept predicts and explains little, or so narrow that they are applicable to only one type of situation. The purpose of this study is to describe, develop, apply, and evaluate a mathematical model of competition that falls between these extremes. The model's simplicity gives it generality of application. Yet the mathematical hypotheses of the model must be interpreted as definite restrictions on the part of the real world to which the model might apply. In the remainder of this chapter, we (1) describe the original biological applications of the ordered-random-intervals (ORI) model and state the conditions which the real world must satisfy, apparently, for that model to be applicable. The following chapters (2) give a simple argument for Whitworth's result and introduce a new concept, that of "threshold," or minimum size of random interval; (3) derive these and other results from the different point of view of a ball-throwing game; (4) apply the theoretical size distribution of the ORI and ball-throwing models in detail to some industries in which it works, and to some in which it does not; (5) derive a limiting form of the ORI size distribution and relate it to the lognormal and Pareto functions, previously used to describe economic size distributions; and (6) review and evaluate the results presented.

1

A MODEL OF SIMPLE COMPETITION 1.2. Biological

Origins

of the Ordered-Random-Intervals

Model

Noting that a variety of empirical equations had been fitted to data on the relative frequencies of different animal species in natural environments, MacArthur (1957) proposed instead to account for the observed frequency distributions by simple hypotheses about animals and their environments. Of the three hypotheses MacArthur offered, the one he and others found most useful in describing data is the ordered-random-intervals model. Suppose η — 1 distinct points are thrown at random on a line of unit length, where is a whole number. These η — 1 points divide the line into η nonoverlapping intervals. (If two or more points coincide, some of the intervals will be of length 0; however, this event is of probability 0 and causes no trouble.) The expected (or average) value of the rth smallest of these intervals is n-1

2 (η + 1 -

i=l

i)'1.

(1.1)

For example, suppose we divide the line into three intervals (see Fig. 1.1). We find some numbers from a uniform distribution between 0 and 1: 0.20, 0.45, 0.65, 0.94, 0.01, 0.77, . . . The first two numbers divide the line from 0 to 1 into intervals of length 0.20 = 0.20 — 0, 0.25 = 0.45 — 0.20, and 0.55 = 1.00 — 0.45. These three intervals are already arranged in order of increasing size. The next two numbers divide the line from 0 to 1 into intervals of length 0.65 = 0.65 — 0, 0.29 = 0.94 — 0.65, and 0.06 = 1.00 - 0.94. Arranged in order of increasing size, these are 0.06, 0.29, and 0.65. The next two numbers, 0.01 and 0.77, give intervals of 0.01, 0.76, and 0.23 or 0.01, 0.23, and 0.76 when arranged in order of increasing size. On the basis of these three observations, the average value of the smallest of the three intervals is then (0.20 + 0.06 + 0.01)/3 = 0.09 to the nearest 0.01; similarly the average values after three observations of the second smallest and third smallest of the three intervals are 0.26 and 0.65. Using a larger and larger number of pairs of numbers from a uniform distribution, instead of only three pairs, would give averages calculated in the same manner which would approach the values obtained from Eq. (1.1) by substituting η = 3 and r = 1, 2, and 3, namely 0.11, 0.28, and 0.61 (to the nearest 0.01). If instead of a unit line we have a line s units long, the expected value Gr of the rth smallest interval is Gr = ~ 2 — ^ η i=i η + 1 — ι

(1.2)

Notice that the expected value Gr increases with r. The smallest expected value is G1 = s/n2. MacArthur proposed that among s individuals from sympatric populations of η different species the abundance of the rth rarest species should be Gr. ("Sympatry of populations" means "the occurrence of two or more populations in the same area; more precisely, the existence of a population in breeding condition within the cruising range of individuals of another population" (Mayr, 1963: 672).)

2

INTRODUCTION

MacArthur reasoned that the line of unit length corresponded to some critical factor (such as food or territory) which was in limited supply and had to be distributed disjointly among the species. The abundance of each species, MacArthur argued, was proportional to the amount of this critical factor which the species could get for itself. One simple way to divide

First trial

0

0.20

0.45 0.25

Second trial

Third trial

0.55

0

0.65

0.94 1

0.65

0.29

0.01

-0.06

0.77

0.01

0.76

1 0.23

(o)

First trial Second trial Third trial (b) 0.09

0.26

0.65 (c)

0.11

, T2 3

— ι/ι ι ι \ 3· 0. (After one ball, it is impossible not to have exactly one nonempty box; after a very large number of balls, it becomes virtually certain that every box is occupied.) k

k

Thus economic players who took rules A, B, and C as prescriptive rather than descriptive could find out from Eqs. (3C.3) and (3C.4) how many balls to buy. Economic players desirous of more detailed information can turn to well-known results concerning the "classical occupancy problem" which we shall just mention here. The problem

40

THE BALLS-AND-BOXES MODEL is this: if k balls are randomly distributed into η boxes, and if each arrangement has probability n~k, what is the probability pm(k,n) of finding exactly m cells empty ? A variety of ways exist (Feller, 1957:58-59, 91-92, 277) to show that this probability is

It is clear from Eq. (3C.5) that for every m there is some k that will make pm(k,n) arbitrarily small. David and Barton (1962) offer a comprehensive discussion of occupancy problems.

41

CHAPTER 4

AN APPLICATION TO ECONOMICS

It's past the size of dreaming. —Shakespeare

4.1.

Introduction

The purpose of this chapter is to show t h a t the size distributions of firms of certain real industries may be closely approximated by the theoretical size distribution of Eq. (2.3), which is derived from the ordered-random-intervals model and from the balls-and-boxes model. We will try to distinguish between the cases in which Eq. (2.3) does and does not work by offering interpretations of the mathematical assumptions of both the BAB and ORI models. I n economics, a size distribution of firms is a function which specifies a magnitude, absolute or relative, for each firm in some specified set when the firms are ranked by size. The set may be geographically defined, e.g., to include firms in a state or country, or economically defined, e.g., to include all firms above a certain size producing or consuming certain products; or the set may be defined by a combination of these ways. At the end of this chapter, we will be left with an unresolved choice: Given that Eq. (2.3) approximates a certain category of firm size distributions, is it more useful to think that the competition which produced these size distributions can be "explained" in terms of the ORI model, which is essentially a null hypothesis about mechanisms of competition; or is it more useful to think that ecological principles like rules A, B, and C determine the outcome of economic competition, regardless of its mechanisms ? This chapter attempts only to establish theoretically plausible connections between a class of economic situations and a class of firm size distributions. We will first introduce some economic terms and explain where the work we are doing fits into economics. We then translate into economic terms the conditions on the real biological world which King (1964) found correlated with the successes of the ORI model there. We will relate these post hoc conditions to the mathematical assumptions of the ORI and BAB models. The purpose of this prefatory discussion will be to determine the economic situations, that is, the sets of firms, to which the theoretical size distribution may reasonably be expected to apply. Before proceeding to data, we will note some difficulties in finding situations that match our abstract ideal. Although we must compromise our ideal conditions to apply the theoretical

42

AN APPLICATION TO ECONOMICS

size distribution, the compromise may be profitable if the real situations we can then study are illuminated by our models and are well approximated by our size distribution. We shall compare the theoretical size distribution of Eq. (2.3) with observed size distributions in six industries: Portland cement, steel, New York daily newspapers, oil refining, synthetic rubber, and aluminum. I n the first three, we will look at various submarkets and subindustries at a given, fixed time. In the last three, we will look at time series as well. This division of the industries into two groups simply reflects the available data. After gaining some confidence in the theoretical size distribution as an approximation to a particular, simple economic reality, we will look briefly at a much more complex situation. We will compare the frequencies of different levels of concentration in American industries as observed by Kaysen and Turner (1959) with the frequencies of these different levels of concentration that the distribution (2.3) would lead us to expect. For definitions and general orientation within economics, we rely on the easily obtainable and eminently digestible introduction by Caves (1964:2): Studying the behavior of all the individual business units in the nation at once amounts to nothing less than studying the whole economy. Studying them one by one, we promptly lose sight of the forest for the trees. The subject of "industrial organization" was conceived in an effort to split the difference between these two extremes. Individual business units come in contact with one another in markets. A market includes a group of buyers and sellers of a particular product engaged in settling the terms of sale of that product. The sellers participating in a given product are called collectively the industry producing that product. (The jokers concealed in these clear definitions will become apparent soon enough.) The subject matter of industrial organization is the structure and behavior of industries and markets, and in what follows we rely heavily on existing descriptive studies of this subject matter. Industrial organization has been studied from various theoretical viewpoints in the past, and we should like eventually to be able to interpret the usable results of those viewpoints in terms of our models. The question that arises when the balls become costly in the BAB model, Sec. 3.4, may point the way to accounting for the results of price theory, for example. But we go no further in that direction now. Our models should also be related eventually to descriptive studies of concentration. For example, many economic studies find the concentration ratio of an industry: the percentage of an industry's total size accounted for by the k largest companies in the industry (Caves, 1964:8). Usually k is 4, sometimes 8 or 20. Now according to the theoretical size distribution with threshold, Eq. (2.3), the predicted concentration ratio of the k largest firms is, by easy algebra, (4.1) where gT is defined in Eq. (2.3). Thus, given s, n, and Δ, one can compare Rk calculated from Eq. (4.1) with an observed concentration ratio pk. 43

A MODEL OF SIMPLE COMPETITION

Note t h a t if a negative estimated threshold of sufficient magnitude is plugged into the right-hand member of Eq. (4.1), R k will be greater than 100 percent. This is actually observed in the tables under Figs. A41 and A42 (in the Appendix at the end of the book) and simply means that the observed distributions are much more skewed than would be predicted by Eq. (2.1). I t is also easy to see that, if the negative estimated threshold is of sufficient magnitude, the sum of the four smallest predicted shares will be negative. This too is observed in the Appendix and again signifies a poor fit to Eq. (2.1). Conversely, given s, n, and an observed concentration ratio pk, one can, by solving Eq. (4.1), get an estimate of Δ, ? _ Gn_k + (sjn) - (s/k)Pk (njs)Gn_lc

(4.2)

for comparison with that given by Eq. (2C.4). (There is no little gr in Eq. (4.2) because all terms containing Δ have been eliminated from the right-hand side.) The concentration ratio just defined is only one, and not necessarily the best, of many measures of economic concentration (which, incidentally, biologists ought to examine in their search for adequate measures of community dominance and diversity). The relations of these other measures of concentration to our models might also be explored. 4.2. Conditions on the Real World I n Sec. 1.2, we reviewed the characteristics which perhaps are shared by successful applications of the ORI distribution in biology and not shared by unsuccessful applications. We now suggest economic translations of these characteristics. The translations we suggest are not the only possibilities, and it is never possible to define exhaustively in advance the domain in which a theory will prove useful. These translations simply seem plausible; their real justification is that, used as a guide to data, they do yield interesting results. First, King found that the ORI distribution works when the species involved are "equilibrial" species, that is, when (1) the species have long life cycles; (2) they reproduce annually (as opposed to more often) and synchronously, so that each generation is born facing the same seasonal conditions; (3) they face a more or less constant environment in the course of succeeding generations; and (4) their individual members are relatively large. These conditions are clearly not independent. For example, annual reproduction (2) is more frequent among larger animals (4). Hence our proposed translation for (4) is a partial translation of (2) as well. A naive, literal translation of condition (2) into economic terms is not obvious, although stability of the organization during changes in personnel might do. The other conditions suggest that in an industry of equilibrial firms, (1) each firm should have a long life span, (unlike, say, retail gasoline outlets or neighborhood grocery stores, which, King has privately suggested, may be good economic analogs of Hutchinson's (1958) "fugitive species"); (3) the demand for the industry's output and the supply of the industry's inputs should not vary

44

AN APPLICATION

TO

ECONOMICS

wildly from year to year (unlike the hula hoop and entertainment industries, which depend on novelty); (4) the individual chunks of investment involved in each firm should be substantial (like part of a steel mill, and unlike part of a hot-dog stand). The requirement that investment be substantial might also be considered a translation of (2), since larger, annually reproducing animals represent larger "investments" of biomass or stored energy. There are plausible mathematical interpretations of the three empirically derived conditions (1), (3), and (4) and hence of (2) also. The size distribution derived from both the ORI and B A B models is a set of expected values of random variables. In any given throw of η — 1 points on a line, or in any given game of balls and boxes, the rth smallest interval or the number of balls thrown by the rth player can differ greatly from the distributions specified by Eqs. (2.1) and (3A.13), respectively. Only averages of many independent throws of points on a line or games of balls and boxes will approach the given size distributions. Strictly speaking, therefore, it is not legitimate to test one observation of some arbitrary industry against the theoretical size distribution. One should like to have the industry "repeated" independently many times; averages could then be legitimately compared with the expected size distribution. 1 Obviously such a demand is impossible to satisfy. One way to satisfy it approximately might be to choose industries whose present state represents a cumulation of many, small, independent games of balls and boxes in their past; if it is assumed that the number of boxes occupied by a given firm is the same at the end of each game, then at the end of many games the total number of balls it has thrown will be proportional to the expected number for a single game. Concretely, one would think of generation after generation of investors supplying lumps of capital (balls) for the long-lasting firm managements to throw; the firms would play a complete game for each generation of investors but would add the results of each game to the sum of its predecessors. A generation of investors might last only a few years, say for the duration of a cyclic upswing. Industries would have to be fairly stable and long lived just to last long enough for this kind of cumulation to take place. I think similar reasoning in biology about size and cumulation, mutatis mutandis, would account for King's empirical generalizations there. Simon's (1955) argument about stability also gives reasons why it would be plausible for single samples to approximate expected size distributions. King found, second, that the ORI model predicted more successfully when the taxonomic affinity (the relatedness) was high in the group of species considered. High taxonomic affinity, he said, was concomitant with close similarity in ecological requirements. In economic terms, this would perhaps mean that we should expect a better fit in industries whose firms competed with respect to a single, uniform product. The reason for the circumlocution "with respect t o " is that the original biological statement and the mathematical justification come to in a moment say nothing about whether the uniformity in product needs to be 0ft the buying or on the selling side of the industry. For example, oil refineries VWi ^.i·..« 1

I iJianl? H&rrisöti White, Department of Social Relations, Harvard University, for pointing out this fact.

45

A MODEL OF SIMPLE COMPETITION

produce a great variety of products, from gasoline to plastics, but they all buy a fairly uniform crude oil. Here firms must divide up a uniform input. Portland cement companies, on the other hand, produce an undifferentiated product requiring several inputs; brand loyalty on the part of construction companies in the face of price differentials is practically unheard of. Here the cement firms compete for demand for their uniform output. In terms of the ORI model, this requirement means that any little piece of the line being divided up is the same as any other little piece. The line is uniform with respect to its probabilistic properties, and the points are uniformly distributed on it. (Incidentally, Whitworth (1901) considers some distributions of intervals that arise if the probability density of points on the line is not uniform.) In terms of the BAB model, the second requirement means that a ball of any one player lands on the target in the same way that a ball of any other player would; the balls are undifferentiated in behavior. In economic terms, this requirement means that a barrel of crude oil purchased by one refinery is a barrel of crude oil not supplied to the others; a bag of cement sold by one plant is one bag less needed from the others. This perfect substitutability of the inputs or outputs of the firms guarantees that their market shares are disjoint (though customers may very well buy from more than one producer); the line segment or set of balls corresponding to one firm does not overlap with that corresponding to any other firm. If the biological principle of competitive exclusion has an economic analog, that analog would have to declare perfect substitutability among firms to be unstable and would have to predict that the firms would try to differentiate their inputs or outputs or otherwise vary their means of survival. Says Caves (1964:21), referring to Bain (1959:218-235) for further details: We can attempt a crude summary of those sectors of the American economy where product differentiation is important. . . . most manufacturing industries which sell to other producers are free of differentiation. The exceptions to this rule appear in such industries as typewriters and farm tractors, which sell to commercial buyers who may not be well-informed. The heaviest product differentiation falls in consumer-goods industries. . . . Outside of the manufacturing sector, local retail and service trades usually show significant differentiation based on location and qualities of service. Hence in applying our theoretical size distribution we look to manufacturing industries producing undifferentiated producers' goods; the firms in such industries have a high "taxonomic affinity." Third, King (1964) found that the ORI model succeeded when the domain sampled was small and uniform and failed when the domain was large and diverse. In terms of the ORI model, this condition means that the observed samples of different species must correspond to intervals of owe line, not intervals of two or more lines pooled after random division. In terms of the BAB model, we may think of the set of η boxes in the target as being selected out of a vast array of possible boxes. These boxes should be so located within the array that,

46

AN APPLICATION TO ECONOMICS

when the players throw, their balls will be as likely to land in any one of the η boxes as in any other. If, for the sake of concreteness, we think of the players all together leaning over a balcony dropping balls onto this vast array of boxes, then we would want the target set to be contiguous; if part of the target were right under the balcony and part of it were far away, the boxes would not be equally likely to receive a ball. Of course, as long as the boxes are in fact equally likely to receive the balls, there is no need for the boxes to be physically adjacent. The problem raised by this third condition, that of fixing an appropriate domain for study, is well known in economics as the problem of market boundaries. Here the slack in Caves' definitions of industry, market, and product becomes worrisome. Caves cites the example of two physically different products, steel and aluminum, which are partially substitutable for each other as "basic metals." Does one define one market around aluminum and another around steel or a common market around both? (Caves doesn't answer.) The same problem exists on a more refined level within the "steel market"; the market for hot-rolled ingots is not the same as the market for pipes and tubes. The industry, we said, is the set of sellers of a particular product. But a firm which makes a variety of products can devote more or less of its capacity to making and selling a particular product; an oil refinery might devote most of its capacity to making gasoline and also produce a small line of plastics. Gasoline would then be its primary product, plastics a secondary product. Would one include the oil refinery in the plastics industry ? For each "industry" defined by its Census of Manufactures, the U. S. government publishes an index of "primary product specialization" and an index of "coverage." "The 'primary product specialization index' measures the extent to which plants classified in the industry specialize' in making products regarded as primary to the industry" (U. S. Senate, 1957:313). It gives the value of shipments of primary products of plants in the industry as a fraction of the total shipments of all products made by these establishments excluding some "miscellaneous receipts." On the other hand, "the 'coverage index' measures the extent to which all shipments of primary products of an industry are made by plants classified in the industry, as distinguished from secondary producers elsewhere" (U. S. Senate, 1957:313). I t gives the value of shipments of the primary products made by plants classified in the industry as a fraction of the total shipments of primary products made by all producers, both in and out of the specified industry. An industry perfect for the purposes of our models would have both indices equal to one (although this condition would not guarantee that all firms were competing directly). Fortunately, the Census and other economists define many industries which have both indices at 0.75 or higher. The spatial distribution of firms further complicates matters. While any two of the very few aluminum plants in the country are necessarily in the same aluminum market, cement plants making uniform cement may or may not affect each other, depending on how close together they are. Grocery stores on opposite sides of the tracks in a city may be in totally disjoint local markets, and both may compete with the A & Ρ or other chain stores.

47

A MODEL OF SIMPLE COMPETITION

What is desired of industries defined in the real world is, as Caves (1964:7) puts it, "that all participants in a market should be highly sensitive to changes in the terms of transactions offered by the other participants. And they should not be sensitive to such moves by outsiders located in other markets." Caves offers no algorithm for producing such markets, and neither can we. Nationwide summary data do little to help. The markets studied below are largely those determined by the judgments of economists, who have studied industries in detail and have made available data on them. 4.3. Stability of the Theoretical Size

Distribution

To test the size distribution, Eq. (2.3), on a market, exhaustive data are required, namely, the size (by some reasonable measure) of every company in the industry. A partial test can be made with the total number η of all companies in the industry and the size of some of them. The sources of data used in this paper claim to include the number and sizes of all firms in the industries they cover. But caution compels us to check as to whether the predictions of our models would be radically affected if the number n' of firms reported in data differed from the true number η of firms in the industry because some small fraction δ of the total industry size s had escaped the economist's notice. The answer is that for reasonably small values of δ and for reasonably large values of n, the predictions are stable. After setting up the problem algebraically, we will give a numerical example. I t is enough to consider relative shares of a market of unit size (s = 1) with zero threshold. Let η be the true number of firms in the industry, and let H e a small positive number less than one. As usual the firm size Gr increases with r. Let q' be the largest integer q such that

I G r < «5. r = 1

(That is, the q' smallest firms add up to less than δ of the market, but the q' + 1 smallest firms add up to more than δ.) We suppose that the number of firms reported is η' = η — q' and that these are the largest n' firms in the industry, accounting for slightly more than 1 — δ of the whole market size. Then the reported rank r' of a firm is related to its true rank r by r' = r — q'. Now the expected market proportions generated by the true values of r and η are, for r = q' -f- 1, . . . , n, G'r =

Ρ

Ρ = ———

which may be approximated by Gr 1- t 48

,

(4.3)

AN APPLICATION TO ECONOMICS

Requiring our predictions t o be stable means t h a t we want G* t o be close t o G'r, calculated from the reported parameters of the m a r k e t as 1 r' 1 G'r· = Λ Σ η + 1 — ι:· η i=χ

Through substitution and algebra it would be possible to derive a n expression for t h e difference between E q . (4.3) and E q . (4.4), b u t it seems more useful to do a numerical calculation in a typical case. Table 4.1 presents such a calculation for η = 10 and x} =

• j ",

χ > x0>

α> 0

(5.4)

and Pr {X ^ x0} = 0. Subtracting Eq. (5.4) from 1 and differentiating with respect to χ gives the density function f(x) = + - ( - ) ~ " - 1 ,

x>x0;

f{x) = 0,

(5.5)

By taking logarithms of both sides of Eq. (5.4) we get log Pr {X > χ} = —a log χ + [a log a;0],

α> Ν "(Λ

.Ε D

100 _

1 1 1 111 X

50

-

1

1

χ > x0,

1 1 11 1 1 I

α > 0. 1

(5.6)

I 1I Mil. -

X X

C Ο

-

X

—

-

X X

f 20 α> σ α> CT φ 10 Γ Ε ^ 5

X X X X

-

ο

-

-

α)

χι Ε Ζ

2

X

-

1

0.1

ι ι ι 1 1 1 1 1 ι ι 1 1 1ι ι ι I 1 10 Size of firm, u (10® bbl/yr)

ι „ ι ι 100

Fig. 5.5. Graphical test of the Pareto law on the United States cement industry, 1956. Data distributed according to Pareto's law should fall on a straight line. If the random variable U representing the size of a firm has the Pareto distribution, then we should obtain a straight line with negative slope by plotting on doubly logarithmic graph paper the number of firms (the number being proportional to the probability) of size greater than each particular size u against the size u. The data on the total U. S. cement industry, the total U. S. steel industry, and the New York daily newspapers are replotted in this way in Figs. 5.5, 5.6, and 5.7. (Again, only a sample of the data has been plotted.) For comparison, the theoretical ORI distribution with % = 100 is plotted in Fig. 5.8, and Fig. 5.9 reproduces a figure from Mandelbrot (1963). If

73

A MODEL ΟΓ SIMPLE COMPETITION too

II,

50

χ 1

-

20

-

1 1 11 1 11I 1 1 1 1 1 111I1 X X X X X X X X

-

-

X

σ

α> 10 σ> "> κ5 -Ε

Γ _

11 ι π ιη

X

-

χ χ χ χ

-

-

χ

-

-

χ

α> 2 .ο 1m l 10

5

1

1 1 I 1 nil 100

1 1 ι ι ι ιι ι 1 1000

1

11

10000

-

ι

S i z e of firm, u ( I 0 3 net t o n s / y r )

Fig. 5.6. Graphical test of the Pareto law on the United States steel industry, 1960. Data distributed according to Pareto's law should fall on a straight line. one were in a generous mood, one could grant that for large firm sizes the data points do seem to fall along a straight line perhaps. Suppose that the data for individual markets are approximated by our ORI and BAB distribution and that the pooled data for an industry are approximated by a Pareto distribu100

' I I II I II

I

ι ι Mill

I I I Mill 10

I

I I I IΜ II 100

Ί

1—I I I I I IJ

in Ε ~

20 10 — 5

-

ο Ο) _α I

ζ

2" I

J I

1000

S i z e of f i r m , u ( I 0 3 copies/day)

Fig. 5.7. Graphical test of the Pareto law on the New York daily newspapers, 1959. Data distributed according to Pareto's law should fall on a straight line.

74

100*

1—I I I I I I (x

50

1

1—ι I I II I I χ X X X

1

1—I II I II. I

I

I in I I ι ι I 0.1

20 10

1

0.0001

J

I I I II I I I I ' I II H I 0.001 0.0t Length u of interval on a unit line

Fig. 5.8. Graphical test of the Pareto law on 100 simulated firms whose sizes match the ordered-random-intervals distribution with zero threshold. Sizes distributed according to Pareto's law should fall on a straight line.

Fig. 5.9. Some theoretical curves according to Mandelbrot (1963: Fig. 1): "Five doubly logarithmic plots: (a) two exponential distributions (very curved solid lines), having very different means; (b) two distributions satisfying Pareto's law from u = 1 on, and having the exponents 1 / 2 and 1; (c) a distribution having asymptotically a Paretian exponent of 4. I hope that the relations between these laws demonstrate graphically that distributions similar to (c) can readily be confused with the exponential but that small alpha exponents are reliable."

A MODEL OE SIMPLE COMPETITION

tion. Then the two must somehow be related. The following crude argument is intended to display one possible relation between the two. The argument may not be factually correct, but its assumptions are testable, and it shows that at least one line of derivation exists. Let U be a random variable, representing the size of a firm in the total pooled industry or total region. Let Vv V2, . • . be a family of random variables representing the size of a firm's capacity in disjoint submarkets 1, 2, . . . of the total industry. We assume, on the basis of the evidence presented in this paper, that one Vit say F 1; has the ORI distribution. Our problem is to define some function U = a0 + a^

+ a2V2 + . . .

(5.7)

and some relation on the F 4 s so that U has a Pareto distribution. We view U as the aggregate size of a firm, and we want to know how the parts, one of which behaves according to Eq. (5A.12), have to be added together. I t is convenient to convert Eq. (5A.12) into a form like Eq. (5.4). Straightforward integration gives f(s/tt) log η n 2

I

_ e-nx/,

Pr {71 >v}~

dx

mg—nv/S

J

= — j — ·

(5.8)

• e~nx's dx

Jo

For ν fixed and njs fixed, but η and s increasing indefinitely, the right-hand member is asymptotically e~nv/s, that is Pr {V1 > ν) ~ e~nv/s.

(5.9)

(This result should not surprise anyone familiar with the Poisson distribution.) Now we specify that the individual market shares Vt must all be related to V1 and that the coefficients a t be defined as follows: Vi = (Pi)!>

u β4 = t £ l!

u0 > 0,

1= 0,1,2,....

(5.10)

Under this assumption, V1 is the only Vi which has the frequency function, Eq. (5A.12), of the ORI and BAB models. Substituting Eq. (5.10) into (5.7) gives U = u0{l

+ V1+'^

+•

•

(5.11)

Then for ν > 0 and u = u0ev, λ = njs, Pr { U > u } = Pr {Fx > ν} ~ β"'" = (β")-; = ( - ) "

76

(5.12)

LOGNORMAL A N D PARETO SIZE

DISTRIBUTIONS

so that, over certain ranges, U has approximately a Pareto distribution. Because of the level of rigor of this whole discussion, we are not going to examine in detail the justification for the limiting operations we have used or the restrictions on the range in which Eq. (5.12) is valid. What we have done in essence is to perform a well-known exponential transformation of variables from an asymptotically exponential distribution (the ORI distribution) to a Pareto distribution. The only new thing here is an interpretation of the terms in the power-series expansion of the exponential function. This interpretation can be tested by examining the actual sizes of the plants of a given firm in different markets and seeing whether they conform to Eq. (5.10), repeating this test for many firms. The ORI distribution, the lognormal distribution, and the Pareto distribution are all "laws" that are valid in certain domains. On the basis of Chapter 4, the ORI distribution usually seems to be valid in certain "small" industries, although we have not compared its accuracy there with the lognormal and the Pareto distributions; on the basis of purely graphical, eyeball tests, the lognormal and the Pareto distributions seem more valid in "larger" or more loosely constituted industries, 1 though neither one is perfect in this domain, and we do not attempt to make any choice between them. This chapter has tried to show that the three laws may be plausibly related in a way that illuminates the limitations and special strengths of each. MATHEMATICAL APPENDIX 5A THE FREQUENCY FUNCTION The function gr in Eq. (2.3) gives a set of η values for r = 1, . . . , n. We now look upon these values as the lengths of η fixed intervals. Let / ( x ) δχ be the number of intervals whose length falls between χ and χ + δχ. As does Feller (1957), we define the symbol ~ to mean that the ratio of the quantities on either side approaches one as η approaches infinity. Then we shall show that in the limit as δχ —> 0, Μ2

f{x)~-e~nxis,

(5A.1)

s

and in the limit as δχ —> 1, where χ has the dimensionality of s, f(x)

~ n(l

-

e'nls)

e~nx's.

(5A.2)

Let r(x) be the smallest integer r such that χ