Affective Decision Making Under Uncertainty: Risk, Ambiguity and Black Swans 3030595110, 9783030595111

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Lecture Notes in Economics and Mathematical Systems 691

Donald J. Brown

Affective Decision Making Under Uncertainty Risk, Ambiguity and Black Swans

Lecture Notes in Economics and Mathematical Systems Volume 691

Founding Editors M. Beckmann, Heidelberg, Germany H. P. Künzi, Heidelberg, Germany Editors-in-Chief Günter Fandel, Faculty of Economics, University of Hagen, Hagen, Germany Walter Trockel, Murat Sertel Institute for Advanced Economic Research, Istanbul Bilgi University, Istanbul, Turkey; Institute of Mathematical Economics (IMW), Bielefeld University, Bielefeld, Germany Series Editors Herbert Dawid, Department of Business Administration and Economics, Bielefeld University, Bielefeld, Germany Dinko Dimitrov, Chair of Economic Theory, Saarland University, Saarbrücken, Germany Anke Gerber, Department of Business and Economics, University of Hamburg, Hamburg, Germany Claus-Jochen Haake, Fakultät für Wirtschaftswissenschaften, Universität Paderborn, Paderborn, Germany Christian Hofmann, München, Germany Thomas Pfeiffer, Betriebswirtschaftliches Zentrum, Universität Wien, Wien, Austria Roman Slowiński, Institute of Computing Science, Poznan University of Technology, Poznan, Poland W. H. M. Zijm, Department of Behavioural, Management and Social Sciences, University of Twente, Enschede, The Netherlands

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Donald J. Brown

Affective Decision Making Under Uncertainty Risk, Ambiguity and Black Swans

123

Donald J. Brown Emeritus Yale Professor of Economics Yale University New Haven, CT, USA

ISSN 0075-8442 ISSN 2196-9957 (electronic) Lecture Notes in Economics and Mathematical Systems ISBN 978-3-030-59511-1 ISBN 978-3-030-59512-8 (eBook) https://doi.org/10.1007/978-3-030-59512-8 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are solely and exclusively licensed 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Dedicated to the Memory of My Beloved Wife, Elizabeth Morgan Brown. December 7, 1937 to June 28, 1998. I am indebted to former graduate students and colleagues for their insightful commentary, during my tenure as Professor of Economics at the Cowles Foundation for Research in Economics at Yale.

Preface

This manuscript is an exploration of the ubiquity of ambiguity in decision-making under uncertainty. The text consists of two parts, Part I on Affective Moods and Part II on Affective Equilibrium in Markets for Risky and Ambiguous Assets, where the author’s published CFDPs comprise the mathematics in Parts I and II. The prerequisites for reading and understanding Parts I and II are the standard introductory graduate courses in economic theory, computational complexity and convex optimization. The intended readers are precocious undergraduates, graduate students and postdocs specializing in behavioural economics, empirical finance or applied portfolio analysis. The recommended readings, listed in the back matter, are reader-friendly and opiniated, funny and wise; full of pictures, graphs, charts, spreadsheets and occasional cartoons. These readings are intended as a nontechnical introduction to the affective principles in this manuscript. Give Part I to your kids. The mathematics in Part I is high-school algebra and the differential calculus for functions on R2. The suggested titles are Kindle e-books, so they are an inexpensive addition to their libraries. Enjoy! New Haven, USA

Donald J. Brown

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Contents

Affective Moods Introduction . . . . . . . . . . . . . . . . . . Smooth Affective Utilities . . . . . . . . . Non-smooth Affective Utilities . . . . . Portfolio Analysis . . . . . . . . . . . . . . Positive Linear Utilities . . . . . . . . . . Approximation Testing for Feasibility SEC Internet Investment Advisers . . . Black Swan-Assets . . . . . . . . . . . . . .

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The Ambiguity of Black Swans . . . . . . . . . . Prospect Theory . . . . . . . . . . . . . . . . . . . . . . . Revealed Preference Analysis . . . . . . . . . . . . . Induced Value Theory . . . . . . . . . . . . . . . . . . Independence of Risk Aversion and Ambiguity Black Swans . . . . . . . . . . . . . . . . . . . . . . . . . Stable Distributions . . . . . . . . . . . . . . . . . . . .

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Affective Equilibrium in Markets for Risky and Ambiguous Assets Affective Portfolio Analysis: Risk, Ambiguity and (IR)rationality Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approximation Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prospect Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diversification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Smooth APT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Affective Fourfold Pattern of Decision-Making Under Risk and Ambiguity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nonsmooth APT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Affective Utility Functions . . . . . . . . . . . Smooth APT . . . . . . . . . . . . . . . . . . . . . Linear Rationalizations of Affective Asset Induced Value Theory . . . . . . . . . . . . . . Non-Smooth Affective Portfolio Theory . Post Script . . . . . . . . . . . . . . . . . . . . . . .

Contents

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Approximate Solutions of Walrasian and Gorman Polar Form Equilibrium Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Dual Walrasian Equilibrium Inequalities . . . . . . . . . . . . . . . . Uniform Bounds on the Marginal Utilities of Income . . . . . . . . . . Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Gorman Polar Form Equilibrium Inequalities . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Approximate Solutions of Walrasian Equilibrium Inequalities with Bounded Marginal Utilities of Income . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Dual Walrasian Equilibrium Inequalities . . . . . . . . . . . . . . . . Uniform Bounds on the Marginal Utilities of Income . . . . . . . . . . Approximation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Gorman Polar Form Equilibrium Inequalities . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Computational Complexity of the Walrasian Equilibrium Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Solving the Walrasian Equilibrium Inequalities . . . . . . . . . . An Approximation Theorem for Walrasian Markets . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Figures

The Ambiguity of Black Swans Fig. 1 Fig. 2 Fig. 3

Euler Diagram 1 (Brown 2020) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euler Diagram 2 (Mandelbrot 1975) . . . . . . . . . . . . . . . . . . . . . . . . Euler Diagram 3 (Brown 2020) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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List of Tables

The Ambiguity of Black Swans Table 1 Table 2 Table 3

Fourfold pattern under risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourfold pattern under risk and ambiguity affective mood: optimism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fourfold pattern under risk and ambiguity affective mood: patience . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Affective Moods

Introduction

(Markowitz 1955) introduced the mean-variance analysis of the efficient tradeoff between risk and return in diversified portfolios of risky assets, that was later extended by (Tobin 1957) to allow riskless assets as a proxy for fiat money. These papers are the beginning of modern portfolio theory or MPT. Investors may choose to invest in portfolios of risky and ambiguous assets, such as volatility indices, bitcoin, or IPOs, to hedge uncertainties of future portfolio returns that are not risks. Ambiguous assets are assets where the objective probabilities of future asset-returns are ill defined or may not exist. A property shared by the stated examples. Ambiguous assets are also described by financial advisers as “safe” assets since their returns are uncorrelated with the standard market indices. In fact, bitcoin is sometimes described as “digital” gold. The behavioral economists explain investing in these examples as reflecting “loss aversion.” Black swan assets are defined in this manuscript as a family of assets where the distributions of future returns are unpredictable. What does this mean? Is the distribution of future returns of bitcoin, volatility indices or IPOs unpredictable? Who knew? (Mandelbrot 1974) knew! At least, he knew for risky assets. Part II extends Mandelbrot’s operational notion of random unpredictable events in markets for risky assets to markets for ambiguous assets. Investment advisors may elicit a client’s risk tolerance and loss aversion for portfolios of risky assets with questionnaires, framed as a series of hypothetical investing prospects. The method of elicitation proposed in this manuscript is a revealed preference analysis of the revealed, recent history of the portfolios chosen by the client subject to budget constraints. In this manuscript, we consider pairwise affective moods that are common in the economic literature on investing, e.g., optimism and pessimism or patience and impatience. An equivalent representation of the client’s affective moods is the client’s affective expected utility function, computed with respect to the distribution of affective state probabilities. SEC registration as an: Internet Investment Adviser requires the Internet Investment Adviser to provide investment advice to clients exclusively through an “interactive website.” If a client wishes to hold a portfolio of risky and ambiguous assets, then internet investment advisers may use © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. J. Brown, Affective Decision Making Under Uncertainty, Lecture Notes in Economics and Mathematical Systems 691, https://doi.org/10.1007/978-3-030-59512-8_1

3

4

Introduction

revealed preference analysis, to elicit the client’s affective mood as a function of risk aversion and ambiguity aversion. The comparable stated preference methods are not permissible under the SEC’s requirements for an appointment as an Internet Investment Adviser, since these methods rely on statistical estimates of diverse populations of potential clients without regard for ambiguity aversion or controlling for random unobservable aspects of investing in risky assets such as race, sexual orientation, credit scores and political affiliation.

Smooth Affective Utilities If affective utility functions are smooth, then the smooth affective Afriat inequalities are defined as the first order conditions for maximizing the composite utility function V(x) = J(U(x)) subject to budget constraints, where x is a state contingent claim, y = U(x) is a state- utility vector. Both y and x, are in the interior of X = Y = RN and the gradient of V(x), denoted DV(x), is computed using the chain rule. DV(x) is bilinear, hence solving the smooth affective Afriat inequalities for the existence of a smooth affective utility function is NP-hard, where in the worst case the smooth affective Afriat inequalities for smooth utility functions are exponential in the number of inequalities and unknowns. Affective expectations of future returns of ambiguous assets are computed with respect to the distribution of affective state probabilities. The family of smooth affective expected utilities are derived from affective smooth utilities using the Legendre transform for smooth convex functions, where the gradient of V(x) is 1 to 1 on the interior of the positive orthant of RN .

Non-smooth Affective Utilities For non- smooth affective utility functions, such as piecewise linear solutions of the non-smooth Afriat inequalities, the Legendre-Fenchel transform is used in lieu of the Fenchel transform to derive equivalent representations of non-smooth affective preferences as non-smooth affective expected utility functions, without invoking the chain rule. If V(x) is a convex function on RN , then the Legendre transform of V(x), V**(x), is the sup of convex functions majorized by V(x), hence convex. If V(x) is a concave function on RN , then the Legendre transform of V(x), −V**(x) is the inf of concave functions minorized by −V(x), hence concave. If VLB (x) := V**(x) and VUB (x) := −V**(x), then VLB (x) < V(x) < VUB (x); where VLB (x) is convex and VUB (x) is concave. If V(x) is convex then V(x) = V**(x). If −V(x) is concave, then −V(x) = −V**(x). Unfortunately, V(x) is unknown, and unobservable, but given a finite number of observations of a client’s revealed optimal portfolio choices, approximations of VLB (x), and VUB (x) are computable, using the Legendre transform

Non-smooth Affective Utilities

5

for smooth convex functions, and the Fenchel Legendre transform for non-smooth convex functions.

Portfolio Analysis This section prescribes a refutable, affective portfolio theory for rationalizing an investor’s revealed holdings of portfolios of risky, and ambiguous assets. The mathematics used in the commentaries are applications of theorems and concepts from three fundamental subfields of convex analysis: Convex Optimization, Disciplined Convex Programming, and Variational Analysis. Polynomial-time approximation theorems derived using revealed preference analysis produce problematic bounds on the degree of approximation error, even for the simplistic linear approximation model of V**(x), the Legendre transform of V(x). If V**(x) is the intended computable rationalization for the unknown and unobservable This section prescribes a refutable, affective portfolio theory for rationalizing an investor’s revealed holdings of portfolios of risky, and ambiguous assets. The mathematics used in the commentaries are applications of theorems and concepts from three fundamental subfields of convex analysis: Convex Optimization, Disciplined Convex Programming, and Variational Analysis. Polynomial-time approximation theorems derived using revealed preference analysis produce problematic bounds on the degree of approximation error, even for the simplistic linear approximation model of V**(x), the Legendre transform of V(x). If V**(x) is the intended This theorem derived using revealed preference analysis produce problematic bounds on the degree of approximation error, even for the simplistic linear approximation model of V**(x), the Legendre transform of V(x). If V**(x) is the intended computable rationalization for the unknown and unobservable computable rationalization for the unknown and unobservable.

Positive Linear Utilities DV(x) is bilinear, hence solving the smooth affective Afriat inequalities for the existence of a smooth affective utility function is NP-hard, where Arbitrary systems of linear inequalities can be solved in polynomial time as a function of the number of inequalities and unknowns, using interior-point algorithms. This observation suggests approximation theorems, where NP-hard systems of affective Afriat inequalities are approximated by linear systems of inequalities.

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Introduction

Approximation Testing for Feasibility To derive the approximation theorems for testing the feasibility of the convex and concave affective Afriat inequalities, we define two families of relaxed linear affective Afriat inequalities, indexed by uniform, scalar slack variables t and s, where t > 0 and s < 0. To derive the approximation theorems for testing the feasibility of the convex and concave affective Afriat inequalities, we define two families of relaxed linear affective Afriat inequalities, indexed by uniform, scalar slack variables t and s, where t > 0 and s < 0. If VLB(x) := V**(x) and VUB(x): = −V**(x), then VLB(x) is a lower bound for V(x), and VUB(x) is an upper bound for V(x). An approximation for the investor’s unobservable true preferences approximation for the investor’s unobservable true preferences over assets is the piecewise linear Afriat utility function which rationalizes the investor’s revealed asset-demands. Notice, it is not assumed that the investor’s true preferences are represented by smooth or non-smooth affective utility functions. To test the feasibility of the convex affective Afriat inequalities for VLB(x), consider the relaxed families of possible, convex affective Afriat Inequalities, by adding uniform slack variables t > 0 to the right sides of the affective Afriat inequalities. Theorem (1) If t* is the minimum of t subject to the relaxed convex, affective Afriat inequalities, then the convex Affective Afriat inequalities are feasible if and only if t* = 0. To test the feasibility of the concave affective Afriat inequalities for VUB(x), consider the relaxed families of possible, concave affective Afriat Inequalities, by adding uniform slack variables s < 0 to the right sides of the affective Afriat inequalities. Theorem (2) If s* is the maximum of s subject to the relaxed concave, affective Afriat inequalities, then the concave affective Afriat inequalities are feasible if and only if s* = 0.

SEC Internet Investment Advisers The SEC requires an Internet Investment Adviser to provide investment advice to clients exclusively through an “interactive website.” Internet Investment advisors can elicit a client’s risk tolerance and loss aversion for portfolios of risky assets with questionnaires, framed as a series of hypothetical investing prospects. This is an instance of stated preference analysis. As such it lacks a firm theoretical basis, absent statistical proxies for relevant but unobserved random variation in the population of potential clients, such as sexual orientation, political affiliation, or income. If a client wishes to hold a portfolio of risky and ambiguous assets, then internet investment advisers may use revealed preference analysis, to elicit the client’s affective mood as a function of risk aversion and ambiguity aversion.

SEC Internet Investment Advisers

7

The comparable stated preference methods are not permissible under the SEC’s requirements for an appointment as an Internet Investment Adviser, since these methods rely on statistical estimates of diverse populations of potential clients without regard for ambiguity aversion or controlling for the random unobservable aspects of investing in risky assets, as previously cited.

Black Swan-Assets Black Swan-assets are defined in this manuscript as a family of assets where the distributions of future returns are unpredictable. What does this mean? Is the distribution of future returns of bitcoin, volatility indices or IPOs unpredictable? Who knew? (Mandelbrot 1974) knew! At least, he knew for risky assets. Part I is an extension of Mandelbrot’s operational notion of random unpredictable events in markets for risky assets to markets for ambiguous assets.

The Ambiguity of Black Swans

Prospect Theory The fourfold pattern of decision-making under risk, discussed in (Kahneman 2011), is described by the author as “one of the core achievements of prospect theory”. It is a 2 × 2 contingency table, where the columns are high probability, the certainty effect, and low probability, the possibility effect, and the rows are gains and losses from a given status quo. The entries in the four cells are illustrative prospects. One cell is a surprise, where in the high probability/losses cell. Kahneman and Tversky observe risk seeking for with negative prospects, commonly referred to as loss aversion. Kahneman identifies “three cognitive principles at the core of prospect theory. They play an essential role in the evaluation of financial outcomes. The third principle is loss aversion (Table 1). Prospect theory and its generalization cumulative prospect theory are empirical, psychological theories of decision making under risk, inspired by Allais’s paradox. Affective Portfolio Theory or APT extends the fourfold pattern of decision-making under risk to a fourfold pattern of decision-making under risk and ambiguity. This fourfold pattern is also a 2 × 2 contingency table. The columns are Risk Averse and Risk Seeking and the rows are Ambiguity Averse and Ambiguity Seeking. Entries in the cells are affective moods derived from sufficient conditions for the composition of convex and concave functions to be convex or concave as specified in the theory of disciplined convex programming. The mood in Table 2 is optimism and the mood in Table 3 is patience. Impatience or intertemporal myopia is characterized as an affective mood where present consumption is preferred to future consumption and the taste for future consumption diminishes as the time for consumption recedes into the future.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. J. Brown, Affective Decision Making Under Uncertainty, Lecture Notes in Economics and Mathematical Systems 691, https://doi.org/10.1007/978-3-030-59512-8_2

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The Ambiguity of Black Swans

Table 1 Fourfold pattern under risk Gains

Losses

Ambiguity averse

Risk

Risk

Averse

Seeking

Ambiguity seeking

Risk

Risk

Seeking

Averse

Table 2 Fourfold pattern under risk and ambiguity affective mood: optimism Risk Ambiguity averse Ambiguity seeking

Risk

Averse

Seeking

Pessimistic

Pessimistic

Preferences

Preferences

Optimistic Preferences

Optimistic Preferences

Table 3 Fourfold pattern under risk and ambiguity affective mood: patience

Ambiguity averse Ambiguity seeking

Risk

Risk

Averse

Seeking

Impatient

Impatient

Preferences

Preferences

Patient

Patient

Preferences

Preferences

The insight that stable distributions for random processes provide the richest examples of random unpredictable aspects of market risk is due to Mandelbrot in Anderson and Mandelbrot (2010). Table 3 and the 4 Euler diagrams illustrate his early appreciation that asset prices in the real world are unpredictable, hence the risk in financial markets cannot be managed. Two non-Gaussian distributions: the Euler and the Cauchy distributions demonstrate that MPT, APT, ARCH, GARCH are unable to hedge unpredictable asset prices.

Prospect Theory

11

This Photo by Unknown Author is licensed under CC BY-SA

The Gaussian Probability Distribution A Two Parameter Normal Distribution

This Photo by Unknown Author is licensed under CC BY-SA

The Levy Distribution A Two Parameter Non-Normal Distribution

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The Ambiguity of Black Swans

This Photo by Unknown Author is licensed under

The Cauchy Distribution A Two Parameter Non-Normal Distribution This is a model of atemporal affective choice, for investors endowed with affective utility functions in intertemporal economies with known finite horizons (Fig. 1). The analysis in this paper is predicated on the notion of affective utility functions, V(x),where V(x) = J(U(x)); x is a state-contingent claim; U(x) is a representation of the client’s preferences for risk aversion; y = U(x) is a state-utility vector; J(y) is a representation of the client’s preferences for ambiguity aversion. The composite affective utility function, V(x) is a representation of the client’s

Fig. 1 Euler Diagram 1 (Brown 2020)

Prospect Theory

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affective moods, where the crucial assumption is the independence of risk aversion and ambiguity aversion. Recently, cognitive psychologists, using fMRI, found that the neural mechanisms which govern economic decision-making under risk and decision-making under ambiguity are independent. Their finding is consistent with the model of affective decision-making presented in this manuscript.

Revealed Preference Analysis This area of microeconomics is well surveyed in the first year on graduate school, one of our perquisites, but for completeness we recall the two families of axioms in revealed preference analysis, used in this manuscript. GARP; The Generalized Axiom of Revealed Preferences. (Varian1982). SSARP; The Strong Strong Axiom of Revealed Preference. (Chiappori and Rochet 1987). Invoking quite a bit of literary license, we refer to SARP as the non-smooth affective Afriat inequalities and refer to SSARP as the smooth affective Afriat inequalities. As the gentle reader has guessed, smooth affective utility functions are solutions to the smooth affective Afriat inequalities and non-smooth affective-utility functions are solutions to the non-smooth affective Afriat Inequalities.

Induced Value Theory Vernon Smith shared the Nobel prize in Economics in 2002 with Daniel Kahneman for their seminal contributions to the method of experimental economics. Kahneman’s well known contribution is his joint work with Amos Tversky on Prospect Theory, discussed in Kahneman (2010). Smith’s contribution is summarized in the following quotation from Smith (1976): “(Smith 1973) depends upon the postulate of non-satiation: “Given a costless choice between two alternatives, identical except that the first yields more of the reward medium (usually currency) than the second, the first will always be chosen (preferred) over the second, by an autonomous individual, i.e., utility is a monotone increasing function of the monetary reward; U(M), DU/DM > 0.” Smith then induces demand functions for consumers, endowed with smooth, concave, monotone increasing, utility functions, and induces supply functions for producers endowed with smooth, convex, monotone decreasing cost functions. and supply schedules that are independent. In effect, a one good model for several different goods. As is well known, under these assumptions, a producer ‘s behavior in competitive markets is characterized by the profit function,

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The Ambiguity of Black Swans

where the prices of inputs are fixed and prices of outputs the intersection of the market supply and market demand curves define the competitive equilibrium prices. Less well known, is that the profit function is the Legendre transform of the cost function. Hence the [Legendre transform of [the Legendre transform of V(x)], the so-called biconjugate of V(x), is computable in polynomial-time without invoking the chain rule, thereby ending the need to solve an NP-hard family of affective Afriat inequalities.

Independence of Risk Aversion and Ambiguity Aversion Huettel et al. (2006) reported the independence of the neural mechanism which governs economic decision-making under risk and the neural mechanism which governs economic decision- making under ambiguity. Levy et al. (2010) using a different experimental design were unable to duplicate the outcome of the Huettel experiment. To resolve the inconsistent findings of the two experiments, Brown et al. (2010) using a third experimental design, estimated a parametric mixed logit model. They were unable to reject the null hypothesis that economic preferences for risk and ambiguity are independent. Although, they did find that risk aversion and ambiguity aversion are uncorrelated. Using Fisher’s nonparametric exact test, they showed that ambiguity and risk are independent. A necessary condition for computing the gradient of a composite function, using the chain rule.

Black Swans The Theory of Black Swans published in Taleb (2010) argues for a distinction between unprecedented events in our past and unpredictable events in our future. The defining property of Black Swan random events is that they are unpredictable, i.e., small probabilities of unlikely random events COVID-19 was not unprecedented, but it was unpredictable. Black Swan-assets are assets, where the cumulative probability distribution or CPD of the probability distribution of random future asset returns is a power distribution. More precisely if X and Y are real-valued random variables. k and α are real numbers then Y = k Xα , in our case Y = CPD (X) and X = future random asset-returns. There are two disjoint sets of Black Swan-assets: risky Black Swan-assets and ambiguous Black Swan-assets. Consequently, there are two disjoint sets of risky assets, where X is the distribution of future risky returns and CPD (X) is a Gaussian probability distribution or CPD (X) is a power distribution. An observation originally attributed to Mandelbrot (1972) and later credited to Tableb (2010). Risky and ambiguous assets are disjoint asset classes, Consequently, there are two disjoint classes of ambiguous assets, where X is the distribution of future affective asset returns; CPD (X) is a Gaussian probability distribution or CPD (X) is a power

Black Swans

15

distribution. The characterization of these two disjoint classes of ambiguous assets, defines a new asset class of ambiguous assets, the ambiguous Black Swan-assets. This is a new asset class with unpredictable random future outcomes.

Stable Distributions The Normal Distribution, The Cauchy distribution, and the Levy Distribution are the only  2-Paramater Stable Distributions. The parameters are the Mean μ and the variance of the respective distributions, but the asymptotic behavior of the sample means, and variances are vastly different. In fact, only the normal distribution is predictable, i.e., the sample means and the sample variances converge to the mean and the variance of the population mean and variance (Figs. 2 and 3). Fig. 2 Euler Diagram 2 (Mandelbrot 1975)

16

The Ambiguity of Black Swans

Fig. 3 Euler Diagram 3 (Brown 2020)

Gaussian Distribution

Affective Equilibrium in Markets for Risky and Ambiguous Assets

Affective Portfolio Analysis: Risk, Ambiguity and (IR)rationality

Introduction In the theory of decision-making under uncertainty, ambiguous assets are assets where objective and subjective probabilities of tomorrow’s asset-returns are illdefined or may not exist. If so, then tomorrow’s uncertain payoffs are characterized by (IR)rational state probabilities which depend on the investor’s (IR)rational state of mind. (IR)rational probabilities are computable moments of the distribution of returns for ambiguous assets. (IR)rational probabilities are computable alternative descriptions of the distribution of returns for ambiguous assets. (IR)rational probabilities may be used to define an investor’s (IR)rational expected utility function in the class of non-expected utilities. Investors may choose to diversify portfolios of fiat money, stocks and bonds by investing in ambiguous assets to hedge the uncertainties of future returns that are not risks. Investors select optimal portfolios of fiat money, stocks, bonds and ambiguous assets by rationalyzing recent portfolio investments with (IR)rational expected utilities and hedging forecasts of future losses of the chosen optimal portfolios by purchasing minimum-cost portfolio insurance. The theory of (IR)rational portfolio analysis differs significantly from the mean-variance analysis of the efficient trade-off between risk and return in diversified portfolios of risky assets. See Chap. 1 in Lam (2016), where investment advisors implement the elicitation of investor’s risk tolerance and loss aversion with questionnaires, framed as a series of hypothetical investing scenarios, often lacking demographic controls. This is an instance of stated preference analysis. The method of elicitation proposed in this paper is revealed preference analysis which is predicated on the history of investor’s portfolio choices in asset markets. As is now well known, the refutable implications of market equilibria can be derived from revealed preference analysis.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. J. Brown, Affective Decision Making Under Uncertainty, Lecture Notes in Economics and Mathematical Systems 691, https://doi.org/10.1007/978-3-030-59512-8_3

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Affective Portfolio Analysis: Risk, Ambiguity and (IR)rationality

The origin of (IR)rational portfolio analysis is the Keynesian notion of (IR)rational equilibrium in asset markets. Keynes viewed equilibrium prices in asset markets as a balance of the sales of bears, the pessimists, and the purchases of bulls, the optimists. Subjective expected utility theory, originally proposed by Savage as the foundation of Bayesian statistics, is a theory of decision-making under uncertainty that “… does not leave room for optimism or pessimism to play any role in the person’s judgment” (Savage 1954, p. 68). This viewpoint is not the perspective of Keynes. That is, “equilibrium prices in asset markets will be fixed at the point at which the sales of the bears and the purchases of the bulls are balanced” (Keynes 1930). In Keynes, equilibrium in asset markets is an (IR)rational notion. Keynes argued that It is the optimism and pessimism of investors not the risk and return of assets that determine future asset-returns. The equilibration of optimistic and pessimistic beliefs of investors is rationalized by investors maximizing (IR)rational expected utility functions subject to budget constraints defined by asset-prices and expenditures of investors. The family of (IR)rational expected utilities is a subclass of non-expected utility functions in the theory of decision-making under uncertainty. (IR)rational expected utility functions represent the preferences of investors for optimism defined as the composition of the investor’s preferences for risk and preferences for ambiguity. That is, an investor may be risk averse or risk seeking and ambiguity averse or ambiguity seeking and optimistic or pessimistic. If U(x) is a representation of the investor’s preferences for risk, and J(y) is a representation of the investor’s preferences for ambiguity, where the state-utility vector y = U(x) for some limited liability state-contingent claim x, then V(x) = J(U(x)), the composition of U(x) and J(y), represents the investor’s preferences for optimism. In the decision-theoretic literature, averse preferences are represented by strictly concave utility representations; and seeking preferences are represented by strictly convex utility representations. This convention is followed in this manuscript to describe Keynes’s notion of how bulls and bears invest in asset-markets. Talking heads on cable TV often summarize today’s financial news as a “bear market” or a “bull market”. If (IR)rational utility functions are smooth, then the (IR)rational Afriat inequalities are defined as the first order conditions for maximizing the composite utility function, V(x), subject to a budget constraint, where the gradient of V is computed using the chain rule. Solving the (IR)rational Afriat inequalities for smooth (IR)rational utility functions is, in general, NP-hard. That is, in the worst case the (IR)rational Afriat inequalities are exponential in the number of inequalities and unknowns. Suppose V(x) = J(U(x)), where U:X → Y, J: Y → R. X is the family of limited liability assets or state-contingent claims, and Y is a family of state-utility vectors, where X and Y are N dimensional linear vector spaces. If U is a diagonal N × N matrix, then DV(x) = DU(x) [J(y)] is the pointwise product of DU(x) and [J(y)]. That is, in general, DV(x) is bilinear, hence the ensuing NP-hard computational complexity.

Introduction

21

The family of positive linear functions is a family of utility functions that are closed under composition. L(x) is a positive linear function if L(x) = d·x, for some fixed d > 0 and all x > 0 in (R)N . If the utility functions for risk and ambiguity are positive linear functions, then their composition, the utility function for optimism, is also a positive linear function. Suppose U(x) = b…x and J(k) = a…k, where a and b are positive, then V(x) = J(U(x)) is also a positive linear utility function, where V(x) = c…x and c is the pointwise product of a and b. Hence the marginal utility of expenditures in the affective Afriat inequalities for V can be normalized to one for all elicited optimal choices of the investor. Arbitrary systems of linear inequalities can be solved in polynomial time as a function of the number of inequalities and unknowns, using interior-point algorithms.

Approximation Theorems This observation suggests approximation theorems, where NP-hard systems of (IR)rational Afriat inequalities are approximated by linear systems of inequalities. The family of smooth (IR)rational expected utilities are derived from smooth (IR)rational utilities using the Legendre duality theorem for smooth convex functions, assuming that the gradient of V(x) is 1 to 1 on the interior of X, the positive orthant of RN . In the nonsmooth case, the Legendre-Fenchel duality theorem can be used in lieu of Fenchel’s duality theorem to derive an equivalent family of representations of nonsmooth (IR)rational preferences as a family of (IR)rational expected utility functions, without invoking the chain rule. For any function V(x), the bi-conjugate, denoted V**(x), is the sup of all the convex functions majorized by V**(x), hence convex, and the bi-conjugate of –V (x) is the inf of all the concave functions minorized by –V**(x), hence concave. Theorem (1) If VLB (x): = V**(x) and VUB (x): = –V**(x), then VLB (x) < V**(x) < VUB (x) VLB (x). To derive an approximation theorem for testing the feasibility of the convex (IR)rational Afriat inequalities, we define the family of relaxed linear (IR)rational Afriat inequalities, indexed by the scalar t > 0. The relaxed (IR)rational linear Afriat inequalities are feasible for sufficiently large t. Minimizing t with respect to the observations defines the optimal linear approximation, where the shadow prices for the dual linear program are proxies for the degree of approximation. A proxy for the investor’s unobservable true preferences over assets is the piece-wise, linear Afriat function that approximately rationalizes the optimal observed individual asset-demands. Note, it is not assumed that the investor’s true preferences are represented by (IR)rational utility functions.

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Affective Portfolio Analysis: Risk, Ambiguity and (IR)rationality

To test the feasibility of convex (IR)rational Afriat inequalities for VLB (x), consider the relaxed convex, (IR)rational Afriat Inequalities and solve the following linear program: (P)

  t∗ = Max tj : s.t. 0 ≤ tj     VLB (xi ) − VLB xj ≤ ßj pj · xi − xj + tj Theorem (2) t* = 0 iff the convex IR(rational) Afriat inequalities are feasible. To test the feasibility of concave (IR)rational Afriat inequalities for VLB (x), consider the relaxed convex/concave (IR)rational Afriat Inequalities and solve the following linear program: (Q)

  s∗ = Max sj : s.t. 0 ≤ sj     ßj pj · xi − xj + sj ≤ VLB (xi ) − VLB xj Theorem (3) s* = 0 iff the concave (IR)rational Afriat inequalities are feasible. (P) and (Q) are linear systems of inequalities that can be solved in polynomial time. Using Afriat’s construction we construct the piecewise linear convex functions:      V#LB (x) = max 1 < j : VLB xj + ßj pj · x − xj + tj Using Afriat’s construction we construct the piecewise linear concave functions:      V#UB (x) = min 1 < j : sj + ßj pj · x − xj + VLB xj Theorem 4: There exists functions that bound the unobserved VLB (x), the biconjugate of the (IR)rational utility function V(x).These functions are computable in polynomial time.

Prospect Theory The fourfold pattern of preferences discussed in chapter 29 of Thinking Fast and Slow (2011) by Daniel Kahneman is described as “one of the core achievements of prospect theory”. In a 2 × 2 contingency table, where the columns are high probability. (certainty effect) and low probability (possibility effect).and the rows are gains and losses from the status quo. The entries in the four cells are illustrative

Prospect Theory

23

prospects. One cell is a surprise, where in the high probability/losses cell. Kahneman and Tversky observe risk seeking with negative prospects, commonly referred to as loss aversion. In his insightful monograph, Kahneman identifies “three cognitive principles at the core of prospect theory. They play an essential role in the evaluation of financial outcomes…. The third principle is loss aversion.” Prospect theory and its generalization cumulative prospect theory are empirical, psychological theories of decision making under risk, inspired by the Allais paradox. (IR)rational portfolio analysis, theory, extends the fourfold pattern of decision-making under risk to a fourfold pattern of decision-making under risk and ambiguity. (IR)rational portfolio analysis is an empirical, psychological theory of decision making under risk and ambiguity, inspired by the Ellsberg’s paradox. The fourfold pattern of (IR)rational decision-making under risk and ambiguity is also a 2 × 2 contingency table, where the columns are Risk Averse and Risk Seeking and the rows are Ambiguity Averse and Ambiguity Seeking. Entries in the cells are preferences for optimism derived from sufficient conditions for the composition of convex and concave functions as specified in the theory of disciplined convex programming. See Lemma 1.in Grant, et al. (2006). Composition Theorem for Convex/Concave Functions If f: R → R is convex and nondecreasing and g: RN → R is convex, then h = fog is convex. If f: R → R is convex and nonincreasing and g: RN → R is concave, then fog is convex. If f: R → R is concave and nondecreasing and g: RN → RN is concave, then f o g is concave. If: R → R is concave and nonincreasing and g: RN → RN is convex, then f o g is concave. f For (IR)rational utilities the Composition theorem implies: If J is concave and nonincreasing and U is convex, then the investor is pessimistic. If J is convex and nondecreasing and U is convex, then the investor is optimistic. If J is concave and decreasing and U is concave, then the investor is pessimistic. If J is convex and nonincreasing and U is concave, then the investor is optimistic. The Fourfold Pattern of (IR)rational decision-making under risk and ambiguity is a 2 × 2 contingency table, where the columns are Risk Averse and Risk Seeking and the rows are Ambiguity Averse and Ambiguity Seeking. Entries in the cells are preferences for optimism derived from sufficient conditions for the composition of convex and concave functions as specified in the Composition theorem. The Fourfold Pattern of (IR)rational Decision-Making under Risk and (IR)rational Portfolio Analysis is an empirical, psychological theory of investing under risk and ambiguity, inspired by the Ellsberg paradox.

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Affective Portfolio Analysis: Risk, Ambiguity and (IR)rationality

Ambiguity RISK

RISK

AVERSE

SEEKING

PESSIMISTIC

PESSIMISTIC

AVERSE

PREFERENCES

PREFERENCES

AMBIGUITY

OPTIMISTIC

OPTIMISTIC

SEEKING

PREFERENCES

PREFERENCES

AMBIGUITY

IR(rational) state probabilities differ from subjective state probabilities in that they may depend on the outcomes in different states of the world. In the Foundations of Statistics (1954) Savage, in postulate P2, explicitly excludes (IR)rational probabilities from his axiomatic derivation of subjective expected utility theory. In his seminal analysis of subjective probability theory, Risk, Ambiguity, and The Savage Axioms (1961), Daniel Ellsberg introduces the notion of ambiguity as an alternative to the notion of risk in decision making under uncertainty. That is, uncertainties that are not risks, where the state probability of future outcomes are unknown or may not exist. In this case, non-expected utility models by Huriwitz (1957) and Ellsberg (1962) provide an alternative characterization of the investor’s attitudes regarding risk, ambiguity and optimism. Their models are the provenance of (IR)rational utility functions. In a series of thought experiments using urns with known and unknown distributions of colored balls, he conjectured that some individuals may violate, Savage’s Postulate the so-called SURE THING PRINCIPLE. These thought experiments have been conducted many times in many classrooms and Ellsberg’s conjecture has been confirmed.

Diversification This paper has 6 technical appendices comprised of 12 Cowles Foundation Discussion Papers (CFDP’s). The appendices are listed as prior art in my pending non-provisional (utility) patent application: AFFECTIVE PORTFOLIO THEORY; Application/Control Number: 16/501,575; Filing Date: 05/02/2019. The appendices extend the benefits of diversification as a hedge against risk in portfolios of stocks and bonds, i.e., portfolios of risky assets, for investors endowed with objective or subjective state probabilities of asset-payoffs tomorrow. If these state probabilities are ill-defined or non-existent then investors may choose to invest in ambiguous assets where tomorrow’s uncertain payoffs are characterized by (IR)rational state probabilities.

Diversification

25

Nonsmooth affective portfolio theory, or nonsmooth APT, is a sequel to smooth affective portfolio theory, or smooth APT. This section prescribes a refutable generalization of smooth APT, for rationalizing a history of, elicited, optimal portfolios of risky and ambiguous assets of investors endowed with nonsmooth, affective utilities. The approximation theorem for NP-hard rationalizations of elicited portfolio data in this section subsumes the linear approximation theorem for Np-hard rationalizations of investors endowed with smooth affective utilities. The technical results are derived from two methodologies in convex analysis: (a) Revealed Preference Analysis (b) Legendre-Fenchel Duality Theory The analysis in this section is an abridged summary of the specifications in my nonprovisional (utility) patent application, Affective Portfolio Theory, patent pending May 23, 2019.

Smooth APT The origin of smooth APT is the Keynesian notion of affective equilibrium in financial markets. Keynes viewed the equilibrium prices in asset markets as a balance of the sales of bears, the pessimists, and the purchases of bulls, the optimists. That is, “equilibrium prices in asset markets will be fixed at the point at which the sales of the bears and the purchases of the bulls are balanced” (Keynes 1930). Keynes believed that It is the optimism and pessimism of investors not the risk and return of assets that determine equilibrium in financial markets. This is a theory of affective investing, where the prices of assets today equilibrate the optimism and pessimism of bulls and bears regarding future asset-payoffs. In smooth APT, the equilibration of optimistic and pessimistic beliefs of investors is rationalized by investors maximizing affective utilities subject to budget constraints, defined by asset prices and the expenditures of investors. Affective utilities represent the preferences of investors for optimism or pessimism, defined as the composition of the investor’s preferences for risk and preferences for ambiguity. That is, an investor may be risk averse or risk seeking and ambiguity averse or ambiguity seeking and optimistic or pessimistic. If U(x) is a representation of the investor’s preferences for risk, and J(y) is a representation of the investor’s preferences for ambiguity, where the state-utility vector y = U(x) for some limited liability state-contingent claim x, then V(x) = J(U(x)), the composition of U(x) and J(y), is a representation of the investor’s preferences for optimism. We follow the decision-theoretic literature, where averse preferences have strictly concave utility representations and seeking preferences have strictly convex utility representations. In addition, smooth APT assumes all representations of preferences are smooth. Following Keynes, smooth APT assumes that optimistic preferences

26

Affective Portfolio Analysis: Risk, Ambiguity and (IR)rationality

have strictly convex utility representations and pessimistic preferences have strictly concave utility representations. The fourfold pattern of affective decision making under risk and ambiguity is a 2 × 2 contingency table, where the columns are Risk Averse and Risk Seeking and the rows are Ambiguity Averse and Ambiguity Seeking. Entries in the cells are preferences for optimism derived from sufficient conditions for the composition of convex and concave functions, in the Composition Theorem for Convex/Concave function proved in Disciplined Convex Programming. The affective Afriat inequalities in smooth APT are defined as the first order conditions for maximizing the composite utility function, V(x), subject to a budget constraint, where the gradient of V is computed with the chain rule. Solving the affective Afriat inequalities for rationalizing asset demands of investors endowed with smooth affective utility functions is, in general, NP-hard. That is, in the worst case, the time it takes to solve a system of affective Afriat inequalities is exponential in the number of inequalities and unknowns. If U is a diagonal N × N matrix, then DV(x) = DU(x) [J(y)] is the pointwise product of DU(x) and [J(y)]. That is, in general, DV(x) is bilinear, hence the ensuing NP-hard computational complexity. The family of positive linear functions is a family of utility functions that are closed under composition, where L(x) is a positive linear function if L(x) = d·x, for some fixed d > 0 and all x > 0 in (R)N . If the utility functions for risk and ambiguity are positive linear functions, then their composition, the utility function for optimism, is also a positive linear function. Suppose U(x) = b…x and J(k) = a…k, where a and b are positive, then V(x) = J(U(x)) is also a positive linear utility function, where V(x) = c…x and c is the pointwise product of a and b. Hence the marginal utility of expenditures in the affective Afriat inequalities for V can be normalized to 1 for all the investor’s elicited optimal choices. Arbitrary systems of linear inequalities can be solved in polynomial time as a function of the number of inequalities and unknowns, using interior-point algorithms. This observation suggests approximation theorems for NP-hard systems of affective Afriat inequalities, where linear systems of inequalities are used for the approximations.

The Affective Fourfold Pattern of Decision-Making Under Risk and Ambiguity To derive the Affective Fourfold Pattern of Decision-Making under Risk and Ambiguity, we cite the Composition theorem on Convex/Concave Functions introduced in Disciplined Convex Programming. Theorem (Boyd et al.). If f: R → (R U + oo) is convex and nondecreasing and

The Affective Fourfold Pattern of Decision-Making …

27

g: RN → (R U + oo) is convex, then h = fog is convex. If f: R → (R U + oo) is convex and nonincreasing and g: RN → (R U + oo) is concave, then fog is convex. If f: R → (R U + oo) is concave and nondecreasing and g: RN → (RN U + oo) is concave, then f o g is concave. If f: R → (R U + oo) is concave and nonincreasing and g: RN → (RN U + oo) is convex, then f o g is concave. For affective utilities their theorem implies: If J is concave and nondecreasing and U is concave, then the investor is pessimistic. If J is concave and nonincreasing and U is convex, then the investor is pessimistic. If J is convex and nondecreasing and U is convex, then the investor is optimistic. If J is convex and nonincreasing and U is concave, then the investor is optimistic. The Fourfold Pattern of Decision-Making under Risk and Ambiguity in smooth APT derives from the Fourfold Pattern for Decision-Making under Risk in Prospect Theory Fourfold Pattern of Decision-Making under Risk and Ambiguity RISK

RISK

AVERSE

SEEKING

AMBIGUITY

PESSIMISTIC

PESSIMISTIC

AVERSE

PREFERENCES

PREFERENCES

AMBIGUITY

OPTIMISTIC

OPTIMISTIC

SEEKING

PREFERENCES

PREFERENCES

In smooth APT, equivalent representations of smooth affective utilities, are smooth affective expected utilities, derived using the Legendre duality theorem for smooth convex functions. Assuming that the gradient of V(x) is 1 to 1 on the interior of X, the positive orthant of RN , the chain rule is used to compute the gradient of V(x) = J(U(x)), hence the NP- hard complexity of solving the affective Afriat inequalities.

Nonsmooth APT Legendre-Fenchel Duality is an alternative theory of duality for nonsmooth affective utilities, V(x), where the bi-conjugate of V(x), denoted V**(x), is the sup of all the convex functions majorized by V(x) and the bi-conjugate of –V (x) is the inf of all the concave functions minorized by –V(x). That is, sup {f(x) < V(x), where f(x) is convex} < V(x) < inf{g(x) > V(x), where g(x) is a concave}. Denote the LHS of the inequality as VLB (X) and the RHS of the inequality as VUB (x). Then VLB (x) < V(x) < VUB (x) where VLB (x) is convex, hence a Bull and VUB (x) is concave, hence a Bear. These are affective utility bounds, in the sense of Keynes that

28

Affective Portfolio Analysis: Risk, Ambiguity and (IR)rationality

“best” approximate the investor’s true tolerances for risk, ambiguity and optimism, denoted V(x), as a Bull or Bear. Unfortunately V(x) is unknown. A computable proxy for V(x) is W(x), a solution of a system of relaxed convex Afriat Inequalities, where the marginal utility of income for W(x) is 1 in every observation. W(x) minimizes the l1 error of approximation subject to the investor’s elicited optimal choices over systems of relaxed convex Afriat inequalities, indexed by the nonnegative scalar variable t. This model defines an infinite family of feasible linear Program Pt for the data set D = {(x1 ,p1 ),(x2 ,p2 ),…(xN , pN )}, where pk are the asset prices in period k. and < pk ,xk > is the investor’s expenditure in period t∗ = inf t S.T 0 ≤ t     W(xi ) − W xj < pj · xi − xj + tj t* = 0 iff the convex, relaxed affective Afriat inequalities are feasible and W(xk ) = V(xk ) for k = 1, 2, … N. To test feasibility of concave, relaxed affective Afriat inequalities for Z(x), we solve for each s, the linear program Qs s∗ = sup s = −inf − s S.T. 0 ≤ si     p1 · xi − xj − si ≤ Z(xi ) − Z xj where s* = 0 iff the concave, affective Afriat inequalities are feasible. (Pt ) and (Qt ) are linear systems of inequalities solvable in polynomial time, with interior point algorithms. Using Afriat’s construction we construct a convex function WLB (x) = max {1 < k < N}: W(xk ) + p · (x -xk )} + t* Using Afriat’s construction we construct a concave function ZUB (x) = min{1 < k < N} : V(xk ) + p · (x − xk )} + s∗ These are the Keynesian approximating linear affective utility functions, with explicit bounds on the approximation errors as solutions of the dual linear programs.

Affective Utility Functions

29

Affective Utility Functions The set of affective utility functions is a new class of non-expected utility functions representing preferences of investors for optimism or pessimism, defined as the composition of the investor’s preferences for risk and her preferences for ambiguity. Bulls and bears are defined respectively as optimistic and pessimistic investors. Simply put, bulls are investing optimists who believe that asset prices will go up tomorrow, and bears are investing pessimists who believe that asset prices will go down tomorrow. The fourfold pattern of preferences discussed in chapter 29 of Thinking Fast and Slow (2011) by Daniel Kahneman is described as “one of the core achievements of prospect theory”. In a 2 × 2 contingency table, where the columns are high probability. (certainty effect) and low probability (possibility effect).and the rows are gains and losses from the status quo. The entries in the four cells are illustrative prospects. One cell is a surprise, where in the high probability/losses cell. Kahneman and Tversky observe risk seeking with negative prospects, commonly referred to as loss aversion. In his insightful monograph, Kahneman identifies “three cognitive features at the heart of prospect theory. They play an essential role in the evaluation of financial outcomes…. The third principle is loss aversion.” Prospect theory and its generalization cumulative prospect theory are descriptive, psychological theories of decision making under risk, inspired by the Allais paradox. In the social sciences they are the preferred alternatives to the normative, axiomatic expected utility model of decision making under risk in Theory of Games (1944) by Von Neumann and Morgenstern. In this paper, Affective Portfolio Theory or APT is a, descriptive, psychological theory of investing under, risk and ambiguity, where investors maximize affective expected utility, using affective probabilities. These probabilities differ from objective or subjective probabilities, since they may depend on affective outcomes in different states of the world. In the Foundations of Statistics (1954) Savage, in postulate P2, explicitly excludes affective probabilities from his axiomatic derivation of subjective expected utility theory. In his seminal analysis of subjective probability theory, Risk, Ambiguity, and The Savage Axioms (1961), Daniel Ellsberg introduces the notion of ambiguity as an alternative to the notion of risk in decision making under uncertainty. That is, uncertainties that are not risks, where the probability of outcomes tomorrow are unknown or may not exist. In this case, non-expected utility models by Huriwitz (1957) and Ellsberg (1962) provide an alternative characterization of the investor’s attitudes regarding risk, ambiguity and optimism. Their models are the origins of affective utility functions.

30

Affective Portfolio Analysis: Risk, Ambiguity and (IR)rationality

Smooth APT Smooth Affective Portfolio Theory, or Smooth APT, extends the mean-variance model for optimizing portfolios of risky assets to optimizing portfolios of risky and ambiguous assets, such as bitcoin, digital currencies, volatility indices or any IPO, where the uncertainties regarding the portfolio’s future payoffs are not risks. That is, ambiguous assets are characterized by affective states of the world, where objective or subjective probabilities of future returns are ill-defined and may not exist. This generalization prescribes affective interactive web sites defined by the SEC as Robo-advisors, that are programmed with affective portfolio theory in a suite of three personalized apps allowing investors, based on their affective preferences for risk, ambiguity and optimism, to hold optimal portfolios of risky and ambiguous assets spanned by mutual funds of bonds, stocks, and bitcoin. Investors with loss aversion can hedge losses in their optimal portfolios with minimum—cost portfolio insurance, where the unrealistic assumption of complete asset markets in MPT is replaced by the weaker assumption of complete derivative markets In A. In this paper Affective Portfolio Theory or APT is an alternative, descriptive, psychological theory of investing under risk and ambiguity. Savage in the Foundations of Statistics (1954), in postulate P2, explicitly excludes affective probabilities from his axiomatic derivation of subjective expected utility theory. In his seminal analysis of subjective probability theory, Risk, Ambiguity, and The Savage Axioms (1961) Daniel Ellsberg introduces the notion of ambiguity as an alternative to the notion of risk in decision making under uncertainty, that is, uncertainties that are not risks, where the probability of outcomes are unknown or may not exist. In a series of thought experiments using urns with known and unknown distributions of colored balls, he conjectured that some individuals may violate, Savage’s Postulate 2, the so-called SURE THING PRINCIPLE. These thought experiments have now been conducted many times in many classrooms and Ellsberg’s conjecture has been confirmed. To fully appreciate Ellsberg’s paradigm changing contribution to decision making under uncertainty, read his recently published Ph.D. dissertation: Risk, Ambiguity, and Decision (1962), This paper prescribes a suite of three personalized digital investment apps, programmed with affective portfolio theory which advise investors who wish to hedge uncertainties of ambiguous assets, such as bitcoin or volatility indices, where the uncertainties regarding returns in future states of the world are not risks. The first app, for each of the four types of quasilinear approximations to the investor’s true affective preferences, rationalizes a stated history of the investor’s past optimal portfolio selections and selects the best “quasilinear” approximation of the investor’s true preferences. Unfortunately, the composition of quasilinear utility functions for risk and ambiguity need not be quasilinear. The example presented in this paper illustrate polynomial time approximations to NP-hard affective Afriat inequalities where utility functions for risk and ambiguity are linear functions, a special class of quasilinear utility functions, that are closed under composition. L(x) is said to be linear if L(x) = b…x, where for fixed a ≥ 0 and arbitrary x ≥ 0 in (R)N . Suppose U(x) = r…x and J(k) = a…x, then V(x) = J(U(x))

Smooth APT

31

is also a linear utility function, where V(x) = c…x and c = a*r, the pointwise product of a and r. Hence the marginal utility of income in the affective Afriat inequalities for V is one for all observed optimal choices. That is µp = p =  V(x). The second app selects the optimal portfolio from a stated menu of the investor’s potential future investments, using the output of the first app, the best quasilinear approximation. The third app, given the investor’s loss aversion, a stated lower bound on the losses of chosen optimal portfolio, using the output of the second app, hedges the investor’s losses by computing the premium for minimum-cost portfolio insurance, The three apps are Android apps, cited as “the world’s most popular operating system”, by Walter and Sherman in Learning MIT App Inventor, (2015). MIT App Inventor is a visual programming language. MIT App Inventor is the suggested programming language for the suite of apps. A Google account gives the inventor of an app the opportunity to use Google Services, Google Data Bases and upload Android apps to Google Play Store for distribution. Affective utility functions are defined as the composition of an investor’s preferences for risk, her preferences for ambiguity, and her preferences for optimism That is, an investor may be risk averse or risk seeking and ambiguity averse or ambiguity seeking and optimistic or pessimistic. U(x) is a representation of the investors preferences for risk, and J(y) is a representation of the investors preferences for ambiguity, where y = U(x) for some limited liability state-contingent claim x. V(x) = J(U(x)), the composition of U(x) and J(y), is a representation of the investor’s preferences for optimism. In the decision-theoretic literature, averse preferences have strictly concave utility representations; seeking preferences have strictly convex utility representations. Following Keynes’s characterization of bulls and bears, optimistic preferences have strictly convex utility representations; pessimistic preferences have strictly concave utility representations. This specification defines 4 types of affective utility functions that are consistent with affective decision making. The fourfold pattern of affective decision -making under risk and ambiguity is a 2 × 2 contingency table, where the columns are Risk Averse and Risk Seeking and the rows are Ambiguity Averse and Ambiguity Seeking. Entries in the cells are preferences for optimism derived from sufficient conditions, as specified in Lemma 1.in Grant, et al. (2006), for the compositions of convex/concave functions to be convex or concave. If f: R → (RU + oo) is convex and nondecreasing and g: RN → (RU + oo) is convex, then h = fog is convex. If f: R → (R U + oo) is convex and nonincreasing and g: RN → (RU + oo) is concave, then fog is convex. If f: R → (RU + oo) is concave and nondecreasing and g: RN → (RN U + oo) is concave, then f o g is concave. If f: R → (RU + oo) is concave and nonincreasing and g: RN → (RN U + oo) is convex, then f o g is concave. In addition, similar rules are described for functions with multiple arguments. Let f = J and g = U. If J is concave and nondecreasing and U is concave, then the investor is pessimistic.

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Affective Portfolio Analysis: Risk, Ambiguity and (IR)rationality

If J is concave and nonincreasing and U is convex, then the investor is pessimistic. If J is convex and nondecreasing and U is convex, then the investor is optimistic. If J is convex and nonincreasing and U is concave, then the investor is optimistic. The Fourfold Pattern of Affective Decision-Making under Risk and Ambiguity RISK

RISK

AVERSE

SEEKING

AMBIGUITY

PESSIMISTIC

PESSIMISTIC

AVERSE

PREFERENCE

PREFERENCE

AMBIGUITY

OPTIMISTIC

OPTIMISTIC

SEEKING

PREFERENCE

PREFERENCE

Linear Rationalizations of Affective Asset Demands Solving the affective Afriat inequalities for rationalizing asset demands of investors endowed with an affective utility functions is, in general, NP-hard. That is, in the worst case, the time it takes to solve a system of affective Afriat inequalities is exponential in the number of inequalities and unknowns. Arbitrary systems of linear inequalities can be solved in polynomial time as a function of the number of inequalities and unknowns, using interior -point algorithms. This observation suggests approximation theorems where NP-hard systems of inequalities are approximated by linear systems of inequalities, with a prior computable degree of approximation. The computational complexity of solving systems of affective Afriat inequalities is a consequence of the first order conditions for maximizing a composite utility function subject to a budget constraint and the chain rule. Assuming V(x) = J(U(x)), where U:X → Y, J: Y → R. X is the family of limited liability assets or statecontingent claims, and Y is a family of state-utility vectors. If U is a diagonal N × N matrix, then DV(x) = DU(x) [J(y)] is the pointwise product of diag[DU(x)] and [J(y)]. That is, in general, DV(x) is bilinear, hence the ensuing computational complexity. To approximate the bilinear Afriat inequalities with a system of linear inequalities, assume the scalar Bernoulli state-utility functions wj (xj ), and J(y), the ambiguity utility function, are linear utility functions. If the space of limited liability state-contingent claims state space is X = (RN+1 )+ then U: X → R is linear, if U(x) = a · x for a ≥ 0, and x = (x1 , …, xs , …, x N +1 ) is in X. Choose the N + 1 state-contingent claim as numeraire, which is a = (a1 , a2 , …, aN , 1). If J: Y → R is linear, where J(y) = b · y for b ≥ 0, and y = (y1 , …, ys , …, y N +1 ). A test of the feasibility of the affective Afriat inequalities, is the relaxed affective Afriat inequalities defining the convex optimization problem:

Linear Rationalizations of Affective Asset Demands

33

t∗ = Min t S.T. 0 ≤ t     V(xi ) − V xj ≤ p · xi − xj + t        w xi,s − w xi,r ≤ dw xi,r xi,s − xi,r + t∗   =1       J(U(xi )) − J U xj ≤ pj diag dw xj,r U(xi ) − U xj + t       2 pj diag dw xj,r ]=1 − J U xj ≤t: This is a quadratic program, hence solvable in polynomial time in CVX t* is a measure of the degree of approximation. That is, t* = 0 if and only if the affective Afriat inequalities are feasible.

Induced Value Theory The principal references are Experimental Economics: Induced Value Theory by V.L. Smith (1976) and An Experimental Study of Competitive Market Behavior by V.L. Smith (1962). Smith shared the Nobel prize in Economics in 2002 with Daniel Kahneman for their seminal contributions to the methodology of experimental economics. Kahneman’s well known contribution is his joint work with Amos Tversky on Prospect Theory, discussed in chapter 1. Smith’s contribution is summarized in the following quotation:from Smith’s (1976) paper, pg.275.” The concept of induced valuation (Smith 1973) depends upon the postulate of non-satiation: Given a costless choice between two alternatives, identical except that the first yields more of the reward medium (usually currency) than the second, the first will always be chosen (preferred)over the second, by an autonomous individual, i.e., utility is a monotone increasing function of the monetary reward, U(M), U’ > 0.[pg 22–23]” Smith then induces demand functions for consumers, endowed with smooth, concave, monotone increasing, utility functions, and induces supply functions for producers endowed with smooth, convex, monotone decreasing cost functions. As is well known, under these assumptions, a producer ‘s behavior in competitive markets is characterized by the profit function, where the prices of inputs are fixed and prices of outputs the intersection of the market supply and market demand curves define the competitive equilibrium prices. Smith induces individual demand and supply schedules that are independents affect, a 1 good model for several different goods. Less well known, is that the profit function is the Legendre transform of the cost function. This suggests that the biconjugate of V(x) = J(U(x)) can be induced,

34

Affective Portfolio Analysis: Risk, Ambiguity and (IR)rationality

eliminating the need to approximate theoretical affective utility functions by solving the affective Afriat inequalities as first order conditions for maximizing V(x) subject to budget constraints. Conditions where the computational complexity is Np-Hard, as a consequence of applying the chain rule to compute the first order conditions. For a composite function. Moreover, the polynomial-time approximation theorem derived using revealed preference analysis produces problematic bounds on the degree of approximation error even for the simplistic linear approximation model of V”(x), the Legendre bi-conjugate of V(x). If V” (x) is the intended efficiently computable proxy for the unknown and unobservable V(x), then the portfolios chosen using the linear approximation may be poor approximations to the counterfactual portfolios selected by the true V(x). Bottom Line: Revealed Preference Analysis approximates V” (x);Induced Value Theory induces V” (x), Now let’s consider the non-smooth case.

Non-Smooth Affective Portfolio Theory Nonsmooth affective portfolio theory, or nonsmooth APT, is a sequel to smooth affective portfolio theory, or smooth APT. This paper prescribes a refutable generalization of smooth APT, for rationalizing the recent, elicited, optimal portfolios of risky and ambiguous assets of investors endowed with nonsmooth, affective utilities. The approximation theorem for NP-hard rationalizations of elicited portfolio data in this paper subsumes the linear approximation theorem for Np-hard rationalizations of investors endowed with smooth affective utilities. The technical results are derived from three methodologies in convex analysis: (a) Revealed Preference Analysis (b) Legendre-Fenchel Duality Theory The analysis in this section is an abridged summary of the specifications in the nonprovisional (utility) patent application, Affective Portfolio Theory, patent pending May 23, 2019. The origin of smooth APT is the Keynesian notion of affective equilibrium in financial markets. Keynes viewed the equilibrium prices in asset markets as a balance of the sales of bears, the pessimists, and the purchases of bulls, the optimists. That is, “equilibrium prices in asset markets will be fixed at the point at which the sales of the bears and the purchases of the bulls are balanced” (Keynes 1930). Keynes believed that It is the optimism and pessimism of investors not the risk and return of assets that determine equilibrium in financial markets. This is a theory of affective investing, where the prices of assets today equilibrate the optimism and pessimism of bulls and bears regarding future asset-payoffs In smooth APT, the equilibration of optimistic and pessimistic beliefs of investors is rationalized by investors maximizing affective utilities subject to budget constraints, defined by asset prices and the expenditures of investors.

Non-Smooth Affective Portfolio Theory

35

Affective utilities represent the preferences of investors for optimism or pessimism, defined as the composition of the investor’s preferences for risk and preferences for ambiguity. That is, an investor may be risk averse or risk seeking and ambiguity averse or ambiguity seeking and optimistic or pessimistic. If U(x) is a representation of the investor’s preferences for risk, and J(y) is a representation of the investor’s preferences for ambiguity, where the state-utility vector y = U(x) for some limited liability state-contingent claim x, then V(x) = J(U(x)), the composition of U(x) and J(y), is a representation of the investor’s preferences for optimism. We follow the decision-theoretic literature, where averse preferences have strictly concave utility representations and seeking preferences have strictly convex utility representations. In addition, smooth APT assumes all representations of preferences are smooth. Following Keynes, smooth APT assumes that optimistic preferences have strictly convex utility representations and pessimistic preferences have strictly concave utility representations. The fourfold pattern of affective decision making under risk and ambiguity is a 2 × 2 contingency table, where the columns are Risk Averse and Risk Seeking and the rows are Ambiguity Averse and Ambiguity Seeking. Entries in the cells are preferences for optimism derived from sufficient conditions for the composition of convex and concave functions, in the Composition Theorem for Convex/Concave function proved in Disciplined Convex Programming. The affective Afriat inequalities in smooth APT are defined as the first order conditions for maximizing the composite utility function, V(x), subject to a budget constraint, where the gradient of V is computed with the chain rule. Solving the affective Afriat inequalities for rationalizing asset demands of investors endowed with smooth affective utility functions is, in general, NP-hard. That is, in the worst case, the time it takes to solve a system of affective Afriat inequalities is exponential in the number of inequalities and unknowns. If U is a diagonal N × N matrix, then DV(x) = DU(x) [J(y)] is the pointwise product of DU(x) and [J(y)]. That is, in general, DV(x) is bilinear, hence the ensuing NP-hard computational complexity. The family of positive linear functions is a family of utility functions that are closed under composition, where L(x) is a positive linear function if L(x) = d · x, for some fixed d > 0 and all x > 0 in (R)N . If the utility functions for risk and ambiguity are positive linear functions, then their composition, the utility function for optimism, is also a positive linear function. Suppose U(x) = b…x and J(k) = a…k, where a and b are positive, then V(x) = J(U(x)) is also a positive linear utility function, where V(x) = c…x and c is the pointwise product of a and b. Hence the marginal utility of expenditures in the affective Afriat inequalities for V can be normalized to 1 for all the investor’s elicited optimal choices. Arbitrary systems of linear inequalities can be solved in polynomial time as a function of the number of inequalities and unknowns, using interior-point algorithms. This observation suggests approximation theorems for NP-hard systems of affective Afriat inequalities, where linear systems of inequalities are used for the approximations.

36

Affective Portfolio Analysis: Risk, Ambiguity and (IR)rationality

The Affective Fourfold Pattern of Decision-Making under Risk and Ambiguity, is derived from the Composition theorem on Convex/Concave Functions, introduced in Disciplined Convex Programming. Theorem (Boyd, et al.). If f: R → (R U + oo) is convex and nondecreasing and g: RN → (R U + oo) is convex, then h = fog is convex. If f: R → (R U + oo) is convex and nonincreasing and g: RN → (R U + oo) is concave, then fog is convex. If f: R → (R U + oo) is concave and nondecreasing and g: RN → (RN U + oo) is concave, then f o g is concave. If f: R → (R U + oo) is concave and nonincreasing and g: RN → (RN U + oo) is convex, then f o g is concave. For affective utilities their theorem implies: If J is concave and nondecreasing and U is concave, then the investor is pessimistic. If J is concave and nonincreasing and U is convex, then the investor is pessimistic. If J is convex and nondecreasing and U is convex, then the investor is optimistic. If J is convex and nonincreasing and U is concave, then the investor is optimistic. The Fourfold Pattern of Decision-Making under Risk and Ambiguity in smooth APT derives from the Fourfold Pattern for Decision-Making under Risk in Prospect Theory. Fourfold Pattern of Decision-Making under Risk and Ambiguity RISK

RISK

AVERSE

SEEKING

PESSIMISTIC

PESSIMISTIC

AVERSE

PREFERENCE

PREFERENCE

AMBIGUITY

OPTIMISTIC

OPTIMISTIC

SEEKING

PREFERENCE

PREFERENCE

AMBIGUITY

In smooth APT, equivalent representations of smooth affective utilities, are smooth affective expected utilities, derived using the Legendre duality theorem for smooth convex functions. Assuming that the gradient of V(x) is 1 to 1 on the interior of X, the positive orthant of RN , the chain rule is used to compute the gradient of V(x) = J(U(x)), hence the NP- hard complexity of solving the affective Afriat inequalities. Legendre-Fenchel Duality is an alternative theory of duality for nonsmooth affective utilities, V(x), where the bi-conjugate of V(x), denoted V** (x),is the sup of all the convex functions majorized by V(x) and the bi-conjugate of -V (x) is the inf of all the concave functions minorized by - V(x). That is, sup {f(x) < V(x), where f(x) is convex} < V(x) < inf {g(x) > V(x), where g(x) is a concave} Denote the LHS of the inequality as VLB (X) and the RHS of the inequality as VUB (x) Then VLB (x) < V(x) < VUB (x) where VLB (x) is convex, hence a Bull and VUB (x) is concave, hence a Bear. These are affective utility bounds, in the sense of Keynes that “best approximate” the investor’s true tolerances for risk, ambiguity and optimism, denoted V(x), as a

Non-Smooth Affective Portfolio Theory

37

Bull or Bear. Unfortunately V(x) is unknown. A computable proxy for V(x) is W(x), a solution of a system of relaxed convex Afriat Inequalities, where the marginal utility of income for W(x) is 1 in every observation. W(x) minimizes the l1 error of approximation subject to the investor’s elicited optimal choices over systems of relaxed convex Afriat inequalities, indexed by the nonnegative scalar variable t. This model defines an infinite family of feasible linear Program Pt for the data set D = {(x1 ,p1 ),(x2 ,p2 ),…(xN ,pN )}, where pk are the asset prices in period k and < pk ,xk > is the investor’s expenditure in period k. t∗ = inf t S.T 0 ≤ t     W(xi ) − W xj < p · xi − xj + tj t* = 0 iff the convex, relaxed affective Afriat inequalities are feasible 7 and W(xk ) = V(xk ) for k = 1, 2, … N. To test feasibility of concave, relaxed affective Afriat inequalities for Z(x), we solve for each s, the linear program Qs s∗ = sup s = −inf − s S.T. 0 ≤ si     p1 · xi − xj − si ≤ Z(xi ) − Z xj where s* = 0 iff the concave, affective Afriat inequalities are feasible. (Pt ) and (Qt ) are linear systems of inequalities that can be solved in polynomial time, with interior point algorithms. Using Afriat’s construction we construct a convex function WLB (x) = max {1 < k < N}: W(xk ) + p(x -xk )} + t* Using Afriat’s construction we construct a concave function ZUB (x) = min{1 < k < N} : V(xk ) + p · (x − xk )} + s∗ These are the Keynesian approximating piecewise linear affective utility functions, with explicit bounds on the approximation errors as solutions of the dual linear programs. Subjective expected utility theory, originally proposed by Savage as the foundation of Bayesian statistics, is a theory of decision-making under uncertainty that “… does not leave room for optimism or pessimism to play any role in the person’s judgment” (Savage 1954, p. 68). This viewpoint is not the perspective of Keynes who viewed the equilibrium prices in asset markets as a balance of the sales of bears, the pessimists,

38

Affective Portfolio Analysis: Risk, Ambiguity and (IR)rationality

and the purchases of bulls, the optimists. That is, “equilibrium prices in asset markets will be fixed at the point at which the sales of the bears and the purchases of the bulls are balanced” (Keynes, 1930). In Keynes, equilibrium in asset markets is an affective notion. It is the optimism and pessimism of investors. The set of affective utility functions is a new class of non-expected utility functions representing preferences of investors for optimism or pessimism, defined as the composition of the investor’s preferences for risk and her preferences for ambiguity. Bulls and bears are defined respectively as optimistic and pessimistic investors. Simply put, bulls are investing optimists who believe that asset prices will go up tomorrow, and bears are investing pessimists who believe that asset prices will go down tomorrow. The fourfold pattern of preferences discussed in chapter 29 of Thinking Fast and Slow (2011) by Daniel Kahneman is described as “one of the core achievements of prospect theory”. In a 2 × 2 contingency table, where the columns are high probability. (certainty effect) and low probability (possibility effect).and the rows are gains and losses from the status quo. The entries in the four cells are illustrative prospects. One cell is a surprise, where in the high probability/losses cell. Kahneman and Tversky observe risk seeking with negative prospects, commonly referred to as loss aversion. In his insightful monograph, Kahneman identifies “three cognitive features at the heart of prospect theory. They play an essential role in the evaluation of financial outcomes…. The third principle is loss aversion.” Prospect theory and its generalization cumulative prospect theory are descriptive, psychological theories of decision making under risk, inspired by the Allais paradox. In the social sciences they are the preferred alternatives to the normative, axiomatic expected utility model of decision making under risk in Theory of Games (1944) by Von Neumann and Morgenstern. Affective Portfolio Theory or APT is a, descriptive, psychological theory of investing under, risk and ambiguity. state-contingent claims chosen by the rational self. Affective probabilities differ from subjective probabilities in that they may depend on the outcomes in different states of the world. In the Foundations of Statistics (1954) Savage, in postulate P2, explicitly excludes affective probabilities from his axiomatic derivation of subjective expected utility theory. In his seminal analysis of subjective probability theory, Risk, Ambiguity, and The Savage Axioms (1961), Daniel Ellsberg introduces the notion of ambiguity as an alternative to the notion of risk in decision making under uncertainty. That is, uncertainties that are not risks, where the probability of outcomes tomorrow are unknown or may not exist. In this case, non-expected utility models by Huriwitz (1957) and Ellsberg (1962) provide an alternative characterization of the investor’s attitudes regarding risk, ambiguity and optimism. Their models are the provenance of affective utility functions. If the objective or subjective state probabilities that define objective and subjective distributions of returns. Knight, Keynes and Fisher recognized the importance and existence of uncertainties in the market prices of commodities and financial assets that are not risks. The intellectual provenance of this manuscript is the recently published Harvard PH.D dissertation of Ellsberg, Risk, Ambiguity and Decision, where the affective state of mind is optimism or pessimism, anticipated by Keynes.

Non-Smooth Affective Portfolio Theory

39

The analogous affective state of mind in Fisher is patience and impatience, also anticipated by Keynes. In the Theory of Games and Economic Behavior by Von Neumann and Morgenstern, an axiomatic theory of decision-making under objective risk is introduced, where players maximize objective expected utility. In The Foundations of Statistics by Savage, an axiomatic theory of decision-making under subjective risk is introduced, where Bayesian decision-makers maximize subjective expected utility. Savage’s axioms explicitly preclude affective state probabilities. In Risk, Ambiguity and Decision, Ellsberg presents a theory of decision-making under risk and ambiguity, where decision-makers maximize affective expected utility. Both the Theory of Games and The Foundations of Statistics have an associated “paradox” due respectively to Allais and Ellsberg that violate the stated axioms. Recently, cognitive psychologists, using fMRI, found that the neural mechanisms which govern decision-making under risk and decision-making under ambiguity are independent and are therefore consistent with the model of affective decision-making presented in this manuscript. In general, experimental economics has confirmed the “Ellsberg paradox” that decision-makers are often ambiguity averse or ambiguity seeking in decision-making under uncertainty. Consequently they violate the Savage axioms in The Foundations of Statistics.

Post Script Robo-Advisors: A Portfolio Management Perspective, 2016, Lam (A Yale Senior Essay advised by David Swensen) Risk, Ambiguity and Decision, 2016, Ellsberg Thinking Fast and Slow, 2010, Kahnman The Black Swan, 2010, Taleb Prospect Theory for Risk and Ambiguity, 2010,Wakker Refutable Theories of Value, 2008, Brown and Kubler Nudge, 2008, Thaler Irrational Exuberance, 2000, Shiller, The Theory of Unemployment, 1936, Keynes The Theory of Profit, 1921, Knight The Theory of Interest, 1907, Fisher Appendix 1 Affective Decision-Making (ADM) [1.a] CFDP 1898R

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Affective Portfolio Analysis: Risk, Ambiguity and (IR)rationality

[1.b] CFDP 1891 Appendix 2 Revealed Preference Analysis [2.a] CFDP 1507 [2.b] CFDP 1399R Appendix 3 Computational Complexity [3.a] CFDP 1865 [3.b] CFDP 1395R2 Appendix 4 Approximation Theorems [4.a] CFDP 1955R2 [4.b] CFDP 1955R Appendix 5 Experimental Economics [5.a] CFDP 1890R [5.b] CFDP 1774 Appendix 6 Econometrics [6.a] CFDP 1526 [6.b] CFDP 1518

Approximate Solutions of Walrasian and Gorman Polar Form Equilibrium Inequalities

Introduction The Brown and Matzkin (1996) theory of rationalizing market data with Walrasian markets, where consumers are price-taking, utility maximizers subject to budget constraints, consists of market data sets and the Walrasian equilibrium inequalities. A market data set is a finite number of observations on market prices, income distributions and social endowments. The Walrasian equilibrium inequalities are the Afriat inequalities for each consumer, the budget constraints for each consumer and the market clearing conditions in each observation. The unknowns in the Walrasian equilibrium inequalities are the utility levels, the marginal utilities of income and the Marshallian demands of individual consumers in each observation. The parameters are the observable market data: market prices, income distributions and social endowments in each observation. The Walrasian equilibrium inequalities are said to rationalize the observable market data if the Walrasian equilibrium inequalities are solvable for some family of utility levels, marginal utilities of income and Marshallian demands of individual consumers, where aggregate Marshallian demands are equal to the social endowments in every observation. Brown and Matzkin show that the observed market data is consistent with the Walrasian paradigm, as articulated by Arrow and Debreu (1954), if the Walrasian equilibrium inequalities rationalize the observed market data. As such, the Brown–Matzkin theory of rationalizing market data with Walrasian markets requires an efficient algorithm for solving the Walrasian equilibrium inequalities. The Walrasian equilibrium inequalities are multivariate polynomial inequalities. The Tarski–Seidenberg theorem, Tarski (1951), provides an algorithm, “quantifier elimination,” that can be used to derive a finite family of multivariate polynomial inequalities, i.e., the “revealed Walrasian equilibrium inequalities” from the Walrasian equilibrium inequalities, where the unknowns are the observable market

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. J. Brown, Affective Decision Making Under Uncertainty, Lecture Notes in Economics and Mathematical Systems 691, https://doi.org/10.1007/978-3-030-59512-8_4

41

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Approximate Solutions of Walrasian …

data: market prices, income distributions and the social endowments in each observation. It follows from the Tarski-Seidenberg theorem that the revealed Walrasian equilibrium inequalities are solvable for the observed market data iff the Walrasian equilibrium inequalities are solvable for some family of utility levels, marginal utilities of income and Marshallian demands of consumers. An important example is the special case of the Walrasian equilibrium inequalities, recently introduced by Brown and Calsamiglia (2014). They propose necessary and sufficient conditions on observable market data to rationalize the market data with consumers endowed with utility functions, where the marginal utilities of income are constant: the so-called “strong law of demand”. The strong law of demand is a finite family of linear inequalities on the observed market data, hence solvable in polynomial time. See their paper for details. Unfortunately, in general, the computational complexity of the Tarski-Seidenberg algorithm, is known to be doubly exponential in the worse case. See Basu (2011) for a discussion of the Tarski–Seidenberg theorem and the computational complexity of quantifier elimination. Hence we are forced to consider approximate solutions of the Walrasian equilibrium inequalities. A decision problem in computer science is a problem where the answer is “yes” or “no.” In this paper, the decision problem is: Can the observed market data set be rationalized with Walrasian equilibrium inequalities? That is, are the Walrasian equilibrium inequalities solvable if the values of the parameters are derived from the observed market data? A decision problem is said to have polynomial complexity, i.e., the problem is in class P, if there exists an algorithm that solves each instance of the problem in time that is polynomial in some measure of the size of the problem instance. In the literature on computational complexity, polynomial time algorithms are referred to as “efficient” algorithms. A decision problem is said to be in NP, if there exists an algorithm that verifies, in polynomial time, if a proposal is a solution of the problem instance. Clearly, P ⊂ NP but it is widely conjectured by computer scientists that P = NP The decision problem A is said to be NP-hard, if every problem in NP can be reduced in polynomial time to A. That is, if we can decide the NP-hard problem A in polynomial time then we can decide every NP problem in polynomial time. In this case, contrary to the current beliefs of computer scientists, P = NP. What is the computational complexity of solving the Walrasian equilibrium inequalities? This important question was first addressed by Cherchye et al. (2011). They proved that solving the Walrasian equilibrium inequalities, reformulated as an

Introduction

43

integer programming problem, is NP-hard. We show that approximate solutions of the Walrasian equilibrium inequalities, reformulated as the dual Walrasian equilibrium inequalities introduced by Brown and Shannon (2000), can be computed in polynomial time. In the Brown-Shannon theory of rationalizing market data with Walrasian markets, the Afriat inequalities are replaced by the dual Afriat inequalities for minimizing the consumer’s smooth, monotone, strictly convex, indirect utility function over prices subject to her budget constraint, defined by her Marshallian demand at the equilibrium market prices. The dual Walrasian equilibrium inequalities are said to rationalize the observed market data if the inequalities are solvable for some family of indirect utility levels, marginal indirect utilities and Marshallian demands of individual consumers, derived from Roy’s identity, where the aggregate Marshallian demands are equal to the social endowments in every observation. Brown and Shannon proved that the Walrasian equilibrium inequalities are solvable iff the dual Walrasian equilibrium inequalities are solvable. We show that solving the dual Walrasian equilibrium inequalities is equivalent to solving a NP-hard minimization problem. Approximation theorems are polynomial time algorithms for computing approximate solutions of a NP-hard minimization problem, where there are explicit a priori bounds on the degree of approximation. The primary contribution of this paper is an approximation theorem for a NP-hard minimization problem equivalent to solving Walrasian equilibrium inequalities with uniformly bounded marginal utilities of income. Gorman introduced his polar form indirect utility functions in (1961). Recall that quasilinear, homothetic and CES indirect utility functions are all special cases of Gorman polar form indirect utility functions, and that endowing consumers with Gorman polar form indirect utility functions is a necessary and sufficient condition for the existence of a representative consumer. The second contribution of this paper is the derivation of the Gorman polar form equilibrium inequalities, where we prove an approximation theorem for a NP-hard minimization problem equivalent to solving the Gorman polar form equilibrium inequalities with uniformly bounded marginal utilities of income. Using the two approximation theorems, we test two simple hypotheses: (1) The null hypothesis H 0,w : The observed market data is rationalized by the Walrasian equilibrium inequalities with uniformly bounded marginal utilities of income, where the alternative hypothesis is H A,w : The Walrasian equilibrium inequalities with uniformly bounded marginal utilities of income are refuted by the observed market data. (2) The null hypothesis H 0,G : The observed market data is rationalized by the Gorman polar form equilibrium inequalities with uniformly bounded marginal utilities of income, where the alternative hypothesis is H A,G : The Gorman polar form equilibrium inequalities with uniformly bounded marginal utilities of income are refuted by the observed market data. There are four logical outcomes of testing two simple hypotheses, but only the following three outcomes are possible in our model: (a) We reject both null hypotheses and accept both alternative hypotheses. (b) We fail to reject the null

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Approximate Solutions of Walrasian …

hypothesis H 0,w , but reject the null hypothesis H 0,G and accept the alternative hypothesis H A,G (c) We fail to reject either null hypothesis. Hence (b) is the most interesting and important outcome. That is, failing to reject the null hypothesis H 0,w and accepting the alternative hypothesis H A,G means the observed market data cannot be rationalized by a representative consumer economy, but may be rationalized by an exchange economy with heterogeneous consumers.

The Dual Walrasian Equilibrium Inequalities In this section, we review and summarize the dual Walrasian equilibrium inequalities proposed by Brown and Shannon. We consider an exchange economy, with i ∈ {1, 2, …, M} consumers. For each observation j ∈ {1, 2, …, N}, pj is a vector of prices in RL++ , ηj is a vector of social endowments of commodities in RL++ and {I 1,j , I 2,j , …, I M,j } is the distribution of positive incomes of consumers in observation j, where i=M i=1 Ii,j =pj ·ηj for j = 1, 2, …, N. Brown and Shannonshow  that there exist smooth, monotone, strictly convex indirect utility functions Vi pI for the ith consumer and Marshallian demand vectors xij ∈ RL++ for the ith consumer in the jth observation that constitute a competitive equilibrium in the jth observation with respect to the observed data iff there exists numbers V i,j > 0 and λi,j > 0 and vectors qi,j Vi,j + qi,j · i=M  i=1

 1 pk − pj Dual Afriat Inequalities Ii,k Ii,j 1

−1 qi,j ≤ ηj λi,j Ii,j

Market Clearing

pj · −qi,j = λi,j FOC 2 Ii,j

(1)

(2) (3)

for all i ∈ {1, 2, …, M} and for all j, k ∈ {1, 2, …, N}, j = k, where the expression for qi,j follows the Marshallian demand vector of consumer i in observation j: xij = λ−1 i,j Ii,j from Roy’s identity. The intuition of this specification is immediate: V i,j is the ith consumer’s utility of x i,j in observation j; λi,j is her marginal utility of income  in observation j; qi,j is the p

gradient of her indirect utility function with respect to Ii,jj in observation j; Eq. (1) is the dual Afriat inequalities for minimizing her smooth, monotone, strictly convex, indirect utility function subject to her budget constraint in each observation; Eq. (2) are the market clearing conditions in observation j; Eq. (3) is the first order conditions for the minimization problem of consumer i in observation j, where she minimizes her smooth, monotone, strictly convex indirect utility function subject to her budget

The Dual Walrasian Equilibrium Inequalities

45

constraint, defined by market prices, her income and her Marshallian demand at the given market prices and her income. The system of inequalities defined by Eqs. (1) and (3) are linear in the unknown utility levels V i,j , marginal utilities of income λi,j and marginal indirect utilities qi,j . Unfortunately, Eq. (2) is nonconvex in λi,j and qi,j . In fact, this nonconvexity is the cause of the NP-hard computational complexity first observed by Cherchye et al.

Uniform Bounds on the Marginal Utilities of Income There is a special case of the dual Walrasian equilibrium inequalities where the computational complexity is polynomial. If we restrict attention to quasilinear exchange economies where λi,j = 1 for all i and j, as in Brown and Calsamiglia,  −1 then Eq. (2) can be rewritten as i=M i=1 Ii,j qi,j ≤ ηj . In this case, the dual Walrasian equilibrium inequalities: Eqs. are linear inequalities in the unknowns: ω ∈  (1)–(3), 

 , where Ω ≡ Vi,j , qi,j |Vi pI is a smooth, monotone, strictly convex, indirect utility function} and Eq. (1) holds for i = 1, 2,…, M; j = 1, 2, … N. Hence the dual Walrasian equilibrium inequalities for quasilinear exchange economies are solvable in polynomial time. We next normalize the indirect utility functions that are NOT quasilinear, i.e., indirect utility functions where the marginal utilities of income vary over the observed incomes, I i,j , and market prices, pj . For each such indirect utility     function, V pI , we compute the 2-norm of the gradient, with respect to pI at Ipr,ss ,   −∇ p V Ips −1 ( T ) r,s for r = 1, 2, …, M; s = 1, 2, …, N. If maxr,s is the normalizing Ir,s 2   constant for V pI , then the corresponding normalized indirect utility function If  ⎤ −∇ p V ps −1   ( ) Ir,s I ⎦ V p ≡ ⎣max V r,s I Ir,s I 2   qi,j pj = ∇p Vi Ii,j Ii,j   pj · −qi,j pj −qi,j pj λi,j = ∂I Vi = = · 2 Ii,j Ii,j Ii,j Ii,j p

then the gradient of Vi  ∇p Vi

pj Ii,j



p I

⎡

 at

pj Ii,j

is < ∇p Vi

  pj pj , ∂ V l i Ii,j >, where Ii,j

      −qr,s −1 −qr,s −1 qi,j pj = max = max ∇ V p i r,s Ir,s r,s Ir,s Ii,j Ii,j 2 2

46

Approximate Solutions of Walrasian …

 λi,j = ∂l Vi

pj Ii,j



      −qr,s −1 −qr,s −1 pj = max = max ∂I Vi λi,j = r,s r,s Ir,s 2 Ii,j Ir,s 2     −qr,s −1 pj · −qi,j −qr,s −1 pj −qi,j = max · max r,s I r,s I I I I2 r,s

2 2

r,s

i,j

2

i,j

i,j

Theorem 1 If the observed data on market prices, income distribution and social endowments are described by β, where   ps W (β) ≡ max 1, max r,s I r,s

1

and       −qr,s −1 −1 pj −q −q λi,j ≡ max λ = max · i,j r,s 2 i,j r,s Ir,s r,s Ii,j 2 then W (β) ≥ max{1, λi,j : i = 1, 2, . . . , M ; j = 1, 2, . . . , N } and   ps ≡ W (β) Upper Bound , λj ≤ max 1, max r,s I r,s

(4)

1

That is, W (β) is a uniform upper bound on the normalized marginal utilities of income, λi,j , for the normalized indirect utility functions, where the λi,j vary over the observed market prices and income distributions. Moreover, W (β) is an upper bound on the constant marginal utility of income for quasilinear utility functions, where λi,j = 1 for all i and j.

Approximation Theorem Definition 2 An approximation theorem for a NP-hard minimization problem, with optimal value OPT (β) for each input β, is a polynomial time algorithm for computing OPT (β), the optimal value of the approximating minimization problem for the input β, and the approximation ratio α (β) > 1 



OPT (β) ≤ OPT (β) ≤ α(β)OPT (β), This definition was taken from the survey paper by Arora (1998) on the theory and application of approximation theorems in combinatorial optimization now prove

Approximation Theorem

47

an approximation theorem for the dual Walrasian equilibrium    inequalities. In the

 Walrasian model α (β) = Θ W (β), and Ω ≡ Vi,j , qi,j |Vi pI is a smooth, monotone, strictly convex, indirect utility function}, where we approximate the nonconvex family of dual Walrasian equilibrium inequalities with a family of linear equilibrium inequalities derived from an exchange economy where consumers are endowed with quasilinear utility functions. Theorem 3 If  ≥ W ≥ 1 and W is the optimal value of the nonconvex program S W , where j=N i=M  1  −1 { ≡ min sj : Eqs. (1), (3), (4) hold and qi,j ≤ sj ηj ω∈Ω,sj ≥1 N I λ (ω) j=1 i=1 i,ji i,j

W

for 1 ≤ j ≤ N } : SW

(5)

W is the optimal value of the approximating linear program RW , where ΓW ≡ min

ω∈Ω,rj ≥1

j=N i=M  −1 1  rj : Eqs.(1), (3), (4) hold and qi,j ≤ rj ηj { N j=1 I i=1 i,j

for 1 ≤ j ≤ N } : RW

(6)

 W is the optimal value of the nonconvex program T W , where ΨW ≡ min

ω∈Ω,tj ≥1

j=N i=M  1  −1 { tj : Eqs.(1), (3), (4) hold and  qi,j ≤ tj ηj N j=1 I λ (ω) i=1 i,j i,j

for 1 ≤ j ≤ N } : TW

(7)

then W ≥ ΓW ≥ W

(8)

ΨW = W

(9)

and

Hence W ≥ ΓW ≥ W ⇔ ΓW ≥ W ≥

ΓW 

(10)

Proof Equation 6 is a special case of Eq. 5, with the additional constraint that λi,j (ω) = 1. Hence if r j is feasible in RW , then r j is feasible in S W . Since λ (ω) ≥ 1, i,j it follows that if t j is feasible in T W then t j is feasible in RW . This proves (8). If

48

Approximate Solutions of Walrasian … j=N i=M  −1 tj ΨW 1  tj : Eqs.(1) , (3), (4) hold and qi,j ≤ ηj ≡ min { ω∈Ω,tj ≥1 N   I λ (ω) j=1 i=1 ij i,j

for 1 ≤ j ≤ N } and W ≡ min

ω∈Ω,sj ≥1

j=N i=M  1  −1 { sj : Eqs.(1) , (3), (4) hold and qi,j ≤ sj ηj N j=1 I λ (ω) i=1 i,j i,j

for 1 ≤ j ≤ N } Then ΨW = ΓW . 

This proves (9).

Corollary 4 (a) W = 1, iff the Walrasian equilibrium inequalities with constant marginal utilities of income rationalize the observed market data β (b) It follows from the Brown and Calsamiglia paper that W = 1 iff the observed market data β satisfies the strong law of demand.

The Gorman Polar Form Equilibrium Inequalities In Gorman’s seminal (1961) paper on the existence of a representative consumer in an exchange economy with a finite number of consumers, he derived necessary and sufficient conditions that a representative consumer exists iff all consumers in the  exchange economy are endowed with indirect utility functions, Vi pI in polar form:   i (p) Vi pI = I −a where ai (p) and b(p) are concave and homogeneous of degree one b(p) functions of the market prices. Moreover, he assumed that the marginal utilities of income are the same for all consumers and only depend on the market prices. That is, λi,j = b 1p for all i. As suggested by Varian (1992), we represent Gorman polar ( j) form indirect utility functions as (G)Vi

p I

= Ie(p) + fi (p),

1 i (p) and fi (p) ≡ ab(p) as convex functions of p, where e(p) is We define e(p) ≡ b(p) homogeneous of degree minus one and f (p) is homogeneous of degree zero.

Theorem 5 The Gorman polar form equilibrium inequalities: Eqs. (G1)–(G6) and (11)–(13) are necessary and sufficient conditions for rationalizing the observed

The Gorman Polar Form Equilibrium Inequalities

49

market data with an exchange economywhere  consumers are endowed with Gorman polar form indirect utility functions, Vi pI ≡ Ie(p) + fi (p), where e(p) and f i (p) are smooth, monotone and strictly convex. For k = j and for 1 ≤ i ≤ M and 1 ≤ j, k ≤ N:   e(pk ) > e pj + ∇p e(pj )] · (pk − pj )

(G1)

  fi (pk ) > fi pj + ∇p fi (pj )] · (pk − pj )

(G2)

    [Ie (pk ) + fi (pk )] > [Ie pj + fi (pj )] + ∇p [Ie pj + fi (pj )] · (pk − pj )   + e pj (Ii,k − Ii,j )]

(G3)

    Vi,j = Ii,j e pj + fi pj

(G4)

     qi,j = ∇p Ii,j e pj + fi pj Ii,j   λi,j = e pj = λj Vi,k > Vi,j −

p I

i=M −1 qi,j ≤ ηj Market Clearing i=1 λi,j

(12)

pj · −qi,j = λi,j FOC 2 Ii,j

(13)

≡ max[[Ie(pk ) + fi (pk )] + ∇p [Ie(pk ) + fi (pk )] · (p − pk )] 1≤k≤N

+ e(pk )(I − Ii,k )] then

(G6)

 pj · −qi,j   qi,j  · pk − pj + Ii,k − Ii,j Dual Afriat Inequalities (11) 2 Ii,j Ii,j

Proof Necessity is obvious. For sufficiency, if Wi

(G5)

50

Approximate Solutions of Walrasian …

 Wi

pj Ii,j



     = Ii,j e pj + fi pj

If (p, I) satisfies the budget constraint defined by (pj , I i,j ): pj · x i,j = I i,j , where xi,j =

       −qi,j 1 = −∇p Ii,j e pj + fi pj /e pj . Ii,j λi,j

then        −pi,j · ∇p Ii,j e pj + fi pj = Ii,j e pj and        −p · ∇p Ii,j e pj + fi pj ≤ Ie pj Hence          ∇p Ie pj + fi pj · p − pj + e(pk ) I − Ii,j ≥ 0 and Wi

p I

     ≥ Ii,j e pj + fi pj = Wi



pj Ii,j



Recall that each consumer minimizes her indirect utility function subject to her P budget constraint. That is, Wi Ii,jj rationalizes the observed market data: (pj , I i,j ).     Wi PI is a lower bound for the Gorman polar form indirect utility function: Vi pI ≡        max I e(pk ) + ∇p e(pk ) · (p − pk ) + e(pk ) I − Ii,k + max fi (pk ) + ∇p fi (pk ) · 1≤k≤N 1≤k≤N   p p (p − pk ) and Wi Ii,jj = Vi Ii,jj . Hence the Gorman polar form indirect utility   function Vi PI rationalizes the observed market data: (pj , I i,j ). That is, Vi

p I

≥ Wi

p I

 ≥ Wi

pj Ii,j



 = Vi

pj Ii,j



     r,s −1 −qr,s −1 −qi,j · λ = max Theorem 6 If λj ≡ maxr,s −q j r,s Ir,s 2 2

  ps λj ≤ max 1, max I ≡ G (β) UpperBound, r,s r,s 1

pj Ii,j



 then

(14)

The Gorman Polar Form Equilibrium Inequalities

51

That is, G (β) is a uniform upper bound on the normalized marginal utilities of income, λi,j , for normalized indirect utility functions, where the λi,j vary over the observed market prices and income distributions. Moreover, G (β) is an upper bound on the constant marginal utility of income for quasilinear utility functions, where λi,j = 1 for all i and j. Proof See the argument preceding Theorem 1.  We now prove an approximation theorem for the Gorman polar form equilibrium   inequalities. In the Gorman model α(β) = G (β);where  = { Vi,j , qi,j |Vi pI is a smooth, monotone, strictly convex, indirect utility function} Theorem 7 If  ≥ G ≥ 1 and G is the optimal value of the nonconvex program S G , where G ≡ min

ω∈Ω,sj ≥1

j=N 1  { sj : Eqs. (G1) to G(6) and (11), (13) and (14) N j=1

and hold

i=M 1  −1 qi,j ≤ sj ηj for 1 ≤ j ≤ N : SG } λi (ω) i=1 Ii,j

(15)

G is the optimal value of the approximating linear program RG , where j=N 1  { ΓG ≡ min rj : Eqs. (G1) to G(6) and (11), (13) and (14) ω∈Ω,rj ≥1 N j=1

and hold

i=M 1  −1 ≤ rj ηj for 1 ≤ j ≤ N } : RG λi (ω) I qi,j

(16)

ij

 G is the optimal value of the nonconvex program T G , where ΨG ≡ min ω∈Ω,tj ≥1

j=N 1  { tj : Eqs. (G1) to G(6) and (11), (13) and (14) N j=1

and hold



i=M 

λj (ω)

i=1

−1 qi,j ≤ tj ηj for 1 ≤ j ≤ N } : TG Ii,j

(17)

then ΨG ≥ ΓG ≥ G and

(18)

52

Approximate Solutions of Walrasian …

ΨG = G

(19)

Hence G ≥ ΓG ≥ G ⇔ ΓG ≥ G ≥ Proof See the proof of Theorem 3

ΓG 

(20) 

Corollary 8 (a) G =1, iff the Gorman Polar Form equilibrium inequalities with constant marginal utilities of income rationalize the observed market data β. (b) It follows from the Brown and Calsamiglia paper that G =1 iff the observed market data satisfies the strong law of demand. The Gorman polar form equilibrium inequalities are a superset of the Walrasian equilibrium inequalities. As such, W = G . Since W = G , it follows that ΓW ≤ ΓGG . To test the two hypotheses, described in the introduction, we define the W “confidence intervals” CW and CG , where     ΓW ΓG CW ≡ , ΓW and CG ≡ , ΓG W G It follows from the approximation theorems that: W ∈ CW and G ∈ CG  (a) If 1 ∈ / CW CG , then we reject both null hypotheses and accept both alternative hypotheses. CGc , then we fail to reject the null hypothesis H 0,W , but reject the (b) If 1 ∈ CW null hypothesis  H 0,G and accept the alternative hypothesis H A,G (c) If 1 ∈ CW CG , then we fail to reject either null hypothesis.

Discussion In this final section of the paper, we describe our contribution to the literature on Algorithmic Game Theory (AGT) or more precisely our contribution to the literature on Algorithmic General Equilibrium (AGE), that predates AGT. AGE begins with Scarf’s seminal (1967) article on computing approximate fixed points, followed by his classic (1973) monograph: The Computation of Economic Equilibria. Codenotti and Varadarajan (2007) review the literature on polynomial time algorithms for computing competitive equilibria of restricted classes of exchange economies, where the set of competitive equilibria is a convex set. It is the convexity of the equilibrium set that allows the use of polynomial time algorithms devised for solving convex optimization problems. The authors conclude that the computational complexity of

Discussion

53

general equilibrium models, where the set of equilibria is nonconvex, is unlikely to be polynomial. Scarf does not explicitly address the issue of computational complexity of what is now called the Scarf algorithm, for computing competitive equilibria. His primary research agenda is the computation of economic equilibria in real world economies. This project is best illustrated by the (1992) monograph of Shoven and Whalley, two of Scarf’s graduate students, on Computable General Equilibrium (CGE) models. CGE models are now the primary models for counterfactual economic policy analysis, used by policy makers for estimating the economic impact of proposed taxes, quotas, tariffs, price controls, global warming, agricultural subsidies,… See the (2012) Handbook of Computable General Equilibrium Modeling edited by Dixon and Jorgenson. CGE models use parametric specifications of utility functions and production functions. The parameters are often estimated using a method called “calibration”. That is, choosing parameter values such that the CGE model replicates the observed equilibrium prices and observed market demands in a single benchmark data set. As you might expect there is some debate about the efficacy of this methodology among academic economists. In response to the obvious limitations of calibration and parametric specification of tastes and technology, Brown and Matzkin proposed the Walrasian equilibrium inequalities as a methodology for nonparametric estimation of CGE models, using several benchmark data sets. That is, Refutable General Equilibrium (RGE) models—see Brown and Kubler (2008). Brown and Kannan (2008) initiated the complexity analysis of searching for solutions of the Walrasian equilibrium inequalities. Subsequently, Cherchye et al. (2011) showed that solving the Walrasian equilibrium inequalities, formulated as an integer programming problem is NP-hard. Hence AGE consists of two computable classes of general equilibrium models. The parametric CGE models of Scarf-Shoven-Whalley and the nonparametric RGE models of Brown-Matzkin-Shannon. Both classes of models admit counterfactual policy analysis. Both classes of models contain special cases solvable in polynomial time. In general, both classes of models lack the polynomial time algorithms necessary for the efficient computation of solutions, hence they require approximation theorems to carry out effective counterfactual policy analysis. The contribution of this paper to the literature on refutable general equilibrium models are the proposed approximation theorems for the Walrasian and Gorman Polar Form equilibrium inequalities. Acknowledgements Dan Spielman recommended several excellent references on the computational complexity of algorithms. The remarks of Felix Kubler on an earlier draft of this paper were very helpful. I thank them, the referees and Karl Schmedders for their comments and advice. Finally, I wish to acknowledge the innumerable editorial contributions to my working papers and manuscripts suggested by Glena Ames, a dear friend and colleague.

54

Approximate Solutions of Walrasian …

References Afriat, S. (1967). The construction of a utility function from demand data. International Economic Review, 66–77. Arora, S. (1998). The approximability of NP-hard problems. In Plenary Lecture at ACM Symposium on Theory of Computing. Arrow, K., & Debreu, G. (1954). The existence of an equilibrium for a competitive economy. Econometrica, 22, 265–290. Aubin, J.-P. (1998). Optima and equilibria. Springer. Basu, S. (2011). Algorithms in real algebraic geometry: Survey. West Lafayette: Department of Mathematics, Purdue University. Brown, D. J., & Calsamiglia, C. (2014). Alfred Marshall’s cardinal theory of value: The strong law of demand. Economic Theory Bulletin, 2(1): 65–75. Brown, D. J., & Kannan, R. (2008). Two algorithms for solving the Walrasian equilibrium inequalities. In Brown & Kubler (Eds.), Computational aspects of general equilibrium theory: Refutable theories of value (pp. 69–97). Springer. Brown, D. J., & Kubler, F. (2008). Computational aspects of general equilibrium theory: Refutable theories of value (pp. 69–97). Springer. Brown, D. J., & Matzkin, R. L. (1996). Testable restrictions on the equilibrium manifold. Econometrica, 64, 1249–1262. Brown, D. J., & Shannon, C. (2000). Uniqueness, stability, and comparative statics in rationalizable Walrasian markets. Econometrica, 68, 1529–1539. Cherchye, L., et al. (2011). Testable implications of general equilibrium models: An integer programming approach. Journal of Mathematical Economics, 47, 564–575. Codenotti, B., & Varadarajan, K. (2007). Computation of market equilibria by convex programming. In Nisnan et al. (Eds.), Algorithmic game theory (pp. 69–77). Cambridge: Cambridge University Press. Tarski, A. (1951). A decision method for elementary and geometry (2nd rev. ed.). Berkeley and Los Angles: Rand Corporation. Varian, H. (1992). Microeconomic analysis (3rd ed.). W.W. Norton and Company.

Approximate Solutions of Walrasian Equilibrium Inequalities with Bounded Marginal Utilities of Income

Introduction The Brown-Matzkin (1996) theory of rationalizing market data with Walrasian markets, where consumers are price-taking, utility maximizers subject to budget constraints, consists of market data sets and the Walrasian equilibrium inequalities. A market data set is a finite number of observations on market prices, income distributions and social endowments. The Walrasian equilibrium inequalities are the Afriat inequalities for each consumer, the budget constraints for each consumer and the market clearing conditions in each observation. The unknowns in the Walrasian equilibrium inequalities are the utility levels, the marginal utilities of income and the Marshallian demands of individual consumers in each observation. The parameters are the observable market data: market prices, income distributions and social endowments in each observation. The Walrasian equilibrium inequalities are said to rationalize the observable market data if the Walrasian equilibrium inequalities are solvable for some family of utility levels, marginal utilities of income and Marshallian demands of individual consumers, where aggregate Marshallian demands are equal to the social endowments in every observation. Brown and Matzkin show that the observed market data is consistent with the Walrasian paradigm, as articulated by Arrow and Debreu (1954), iff the Walrasian equilibrium inequalities rationalize the observed market data. As such, the Brown-Matzkin theory of rationalizing market data with Walrasian markets requires an efficient algorithm for solving the Walrasian equilibrium inequalities. The Walrasian equilibrium inequalities are multivariate polynomial inequalities. The Tarski–Seidenberg theorem, Tarski (1951), provides an algorithm, “quantifier elimination,” that can be used to derive a finite family of multivariate polynomial inequalities, i.e., the “revealed Walrasian equilibrium inequalities” from the Walrasian equilibrium inequalities, where the unknowns are the observable market

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. J. Brown, Affective Decision Making Under Uncertainty, Lecture Notes in Economics and Mathematical Systems 691, https://doi.org/10.1007/978-3-030-59512-8_5

55

56

Approximate Solutions of Walrasian Equilibrium …

data: market prices, income distributions and the social endowments in each observation. It follows from the Tarski–Seidenberg theorem that the revealed Walrasian equilibrium inequalities are solvable for the observed market data iff the Walrasian equilibrium inequalities are solvable for some family of utility levels, marginal utilities of income and Marshallian demands of consumers. An important example is the special case of the Walrasian equilibrium inequalities, recently introduced by Brown and Calsamiglia (2014). They propose necessary and sufficient conditions on observable market data to rationalize the market data with consumers endowed with utility functions, where the marginal utilities of income are constant: the so-called “strong law of demand”. The strong law of demand is a finite family of linear inequalities on the observed market data, hence solvable in polynomial time. See their paper for details. Unfortunately, in general, the computational complexity of the Tarski–Seidenberg algorithm, is known to be doubly exponential in the worse case. See Basu (2011) for a discussion of the Tarski–Seidenberg theorem and the computational complexity of quantifier elimination. Hence we are forced to consider approximate solutions of the Walrasian equilibrium inequalities. A decision problem in computer science is a problem where the answer is “yes” or “no.” In this paper, the decision problem is: Can the observed market data set be rationalized with Walrasian equilibrium inequalities? That is, are the Walrasian equilibrium inequalities solvable if the values of the parameters are derived from the observed market data? A decision problem is said to have polynomial complexity, i.e., the problem is in class P, if there exists an algorithm that solves each instance of the problem in time that is polynomial in some measure of the size of the problem instance. In the literature on computational complexity, polynomial time algorithms are referred to as “efficient” algorithms. A decision problem is said to be in NP, if there exists an algorithm that verifies, in polynomial time, if a proposal is a solution of the problem instance Clearly, P ⊂ NP but it is widely conjectured by computer scientists that P = N P The decision problem A is said to be NP-hard, if every problem in NP can be reduced in polynomial time to A. That is, if we can decide the NP-hard problem A in polynomial time then we can decide every NP problem in polynomial time. In this case, contrary to the current beliefs of computer scientists, P = N P. What is the computational complexity of solving the Walrasian equilibrium inequalities? This important question was first addressed by Cherchye et al. (2011). They proved that solving the Walrasian equilibrium inequalities, reformulated as

Introduction

57

an integer programming problem, is NP-hard. We show that approximate solutions of the Walrasian equilibrium inequalities, reformulated as the dual Walrasian equilibrium inequalities introduced by Shannon and Brown (2000), can be computed in polynomial time. In the Brown-Shannon theory of rationalizing market data with Walrasian markets, the Afriat inequalities are replaced by the dual Afriat inequalities for minimizing the consumer’s monotone, strictly convex, indirect utility function over prices subject to her budget constraint, defined by her Marshallian demand at the equilibrium market prices. The dual Walrasian equilibrium inequalities are said to rationalize the observed market data if the inequalities are solvable for some family of indirect utility levels, marginal indirect utilities and Marshallian demands of individual consumers, derived from Roy’s identity, where the aggregate Marshallian demands are equal to the social endowments in every observation. Brown and Shannon proved that the Walrasian equilibrium inequalities are solvable iff the dual Walrasian equilibrium inequalities are solvable. We show that solving the dual Walrasian equilibrium inequalities is equivalent to solving a NP-hard minimization problem. Approximation theorems are polynomial time algorithms for computing approximate solutions of a NP-hard minimization problem, where there are explicit a priori bounds on the degree of approximation. The primary contribution of this paper is an approximation theorem for a NP-hard minimization problem equivalent to solving Walrasian equilibrium inequalities with uniformly bounded marginal utilities of income. A related contribution is the derivation of the family of Gorman Polar Form equilibrium inequalities, where consumers are endowed with indirect utility functions in Gorman Polar Form, with uniformly bounded marginal utilities of income. Gorman introduced his Polar Form indirect utility functions in (1961). Following the proof of our approximation theorem for the Walrasian equilibrium inequalities, we derive the Gorman Polar Form inequalities and prove an approximation theorem for the Gorman Polar Form equilibrium inequalities. Using the two approximation theorems, we test two simple hypotheses: (1) The null hypothesis H 0,W : The observed market data is rationalized by the Walrasian equilibrium inequalities with uniformly bounded marginal utilities of income, where the alternative hypothesis is H A,W : The Walrasian equilibrium inequalities with uniformly bounded marginal utilities of income are refuted by the observed market data. (2) The null hypothesis H 0,G : The observed market data is rationalized by the Gorman Polar Form equilibrium inequalities with uniformly bounded marginal utilities of income, where the alternative hypothesis is H A,G : The Gorman Polar Form equilibrium inequalities with uniformly bounded marginal utilities of income are refuted by the observed market data. There are four logical outcomes of testing two simple hypotheses, but only the following three outcomes are possible in our model: (a) We reject both null hypotheses and accept both alternative hypotheses. (b) We fail to reject the null hypothesis H 0,W , but reject the null hypothesis H 0,G and accept the alternative hypothesis H A, G (c) We fail to reject either both hypotheses. Recall that quasilinear, homothetic and CES indirect utility functions are all special cases of Gorman Polar Form indirect utility functions, commonly assumed in computable general equilibrium models, and that endowing consumers with Gorman Polar Form indirect utility

58

Approximate Solutions of Walrasian Equilibrium …

functions is a necessary and sufficient condition for the existence of a representative agent. Hence (b) is the most interesting and important outcome. That is, failing to reject the null hypothesis H 0,W and accepting the alternative hypothesis H A,G means the observed market data cannot be rationalized by a representative agent economy, but may be rationalized by an exchange economy with heterogeneous consumers.

The Dual Walrasian Equilibrium Inequalities In this section, we review and summarize the dual Walrasian equilibrium inequalities proposed by Brown and Shannon. We consider an exchange economy, with i ∈ {1, 2, …, M} consumers. For each observation j ∈ {1, 2, …, N}, pj is a vector of L L , ηj is a vector of social endowments of commodities in R++ and {I 1,j , prices in R++ I 2,j ,…, I M,,j } is the distribution of positive incomes of consumers in observation j,  where i=M i=1 Ii, j = p j · η j for j = 1, 2, …, N. Brown and Shannon show   that there exist smooth, monotone, strictly convex indirect utility functions Vi pI for the ith L for the ith consumer in the consumer and Marshallian demand vectors xi j ∈ R++ jth observation that constitute a competitive equilibrium in the jth observation with respect to the observed data iff there exists numbers V i,j > 0 and λi,j > 0 and vectors qi,j Vi, j + qi, j · i=M  i=1

 1 1 pk − p j Dual Afriat Inequalities Ii,k Ii, j

−1 qi, j ≤ η j λi, j Ii, j

Market Clearing

p j · −qi, j = λi, j FOC Ii,2 j

(1)

(2) (3)

for all i ∈ {1, 2, …, M} and for all j, k ∈ {1, 2, …, N}, j = k, where the expression qi, j for the Marshallian demand vector of consumer i in observation j: xi j = λi,−1 j Ii, j follows from Roy’s identity. The intuition of this specification is immediate: V i,j is the ith consumer’s utility of x i,j in observation j; λi, j is her marginal utility of income observation j; qi,j is the  in p

gradient of her indirect utility function with respect to Ii,jj in observation j; Eq. (1) is the dual Afriat inequalities for minimizing her smooth, monotone, strictly convex, indirect utility function subject to her budget constraint in each observation; Eq. (2) are the market clearing conditions in observation j; Eq. (3) is the first order conditions for the minimization problem of consumer i in observation j, where she minimizes her smooth, monotone, strictly convex indirect utility function subject to her budget constraint, defined by market prices, her income and her Marshallian demand.

The Dual Walrasian Equilibrium Inequalities

59

The system of inequalities defined by Eqs. (1) and (3) are linear in the unknown utility levels V i,j , marginal utilities of income λi, j and marginal indirect utilities qi,j . Unfortunately, Eq. (2) is nonconvex in λi, j and qi,j . In fact, this nonconvexity is the cause of the NP-hard computational complexity first observed by Cherchye et al.

Uniform Bounds on the Marginal Utilities of Income There is a special case of the dual Walrasian equilibrium inequalities where the computational complexity is polynomial. If we restrict attention to quasilinear exchange economies where λi,j = 1 for all i and j, as in Brown and Calsamiglia,  −1 then Eq. (2) can be rewritten as i=M i=1 Ii, j qi, j ≤ η j . In this case, the dual Walrasian equilibrium inequalities: Eqs. (1), (2) and (3), are linear inequalities in the unknowns  ≡ {(V i,j , qi,j ): Eq. (1) holds and i = 1, 2,…, M; j = 1, 2,…,N}. Hence solvable in polynomial time. We next normalize the indirect utility functions that are NOT quasilinear, i.e., indirect utility functions where the marginal utilities of income vary over the observed incomes, Ii,j, and market prices, pj . For each such indirect utility     function, V pI , we compute the 2-norm of the gradient, with respect to pI at Ipr,ss , 

   −∇ p V Ips  −1  ( T ) r,s  for r = 1, 2, …, M; s = 1, 2, …, N. If maxr,s  is the normalizing  Ir,s   2   constant for V pI , then the corresponding normalized indirect utility function

V

 p I

  ⎤   −∇ p V ps  −1    (I) Ir,s   ⎦ V p ≡ ⎣max  r,s  Ir,s I   ⎡

2

If

λi, j

then the gradient of Vi

 p I

 at

pj Ii, j



is < ∇ p Vi



pj Ii, j



, ∂l Vi



pj Ii, j



>, where

           −qr,s  −1  −qr,s  pj    − qi, j = max = max ∇ V p i r,s  Ir,s  r,s  Ir,s  I Ii, j i, j 2 2   −1   −1        −qr,s   −qr,s  pj pj   = max = max = ∂l Vi ∂ I Vi    I  r,s r,s Ii, j Ir,s 2 Ii, j r,s 2 

∇ p Vi λi, j

  qi, j pj = ∇ p Vi Ii, j Ii, j   pj p j · −qi, j p j −qi, j = = ∂ I Vi = · 2 Ii, j I Ii, j Ii, j i, j

pj Ii, j



60

Approximate Solutions of Walrasian Equilibrium …

        −qr,s  −1 p j · −qi, j  −qr,s  −1 p j −qi, j     = max = max · r,s r,s Ir,s 2 2 Ir,s 2 Ii, j Ii, j Ii,2 j 

λi, j

Theorem 1 If      ps    ΘW (β) ≡ max 1, max r,s Ir,s 1 then ΘW (β) ≥ max{1, λi, j : i = 1, 2, . . . , M; j = 1, 2, . . . , N }      

   −qr,s  −1  −qr,s  −1 p j · −qi, j   λi, j = max λi, j = max ≤ r,s  Ir,s  r,s  Ir,s  Ii,2j 2 2          ps       ≤ max ps  ≤ max 1, max ps  ≡ ΘW Upper Bound max      r,s r,s r,s Ir,s 2 Ir,s 1 Ir,s 1

Proof

 That is, W is a uniform upper bound on the normalized marginal utilities of income, λi, j , for normalized indirect utility functions, where the λi, j vary over the observed market prices and income distributions. Moreover, W is an upper bound on the constant marginal utility of income for quasilinear utility functions, where λi, j = 1 for all i and j.

Approximation Theorem Definition 2 An approximation theorem for a NP-hard minimization problem, with optimal value OPT (β) for each input β, is a polynomial time algorithm for computing O P T (β), the optimal value of the approximating minimization problem for the input β, and the approximation ratio α(β)≥ 1 



O P T (β) ≤ O P T (β) ≤ α(β)O P T (β), This definition was taken from the survey paper by Arora (1998) on the theory and application of approximation theorems in combinatorial optimization. Theorem 3 If  ≥ W ≥ 1 and W is the optimal value of the nonconvex program S W , where

Approximation Theorem

W ≡ min

ω∈Ω,s j ≥1

61

j=N i=M  1  −1 { s j : Eqs. (1), (3), (4) holdand qi, j ≤ s j η j N j=1 I λ (ω) i=1 i, ji i, j

for 1 ≤ j ≤ N } : SW

(5)

W is the optimal value of the approximating linear program RW , where ΓW

j=N i=M  −1 1  { ≡ min r j : Eqs. (1), (3), (4) holdand qi, j ≤ r j η j ω∈Ω,r j ≥1 N I j=1 i=1 i, j

for 1 ≤ j ≤ N } : RW

(6)

 W is the optimal value of the nonconvex program T W , where ΨW

j=N i=M  1  −1 ≡ min t j : Eqs. (1), (3), (4) holdand Θ qi, j ≤ t j η j { ω∈Ω,t j ≥1 N I λ (ω) j=1 i=1 i, j i, j

for 1 ≤ j ≤ N } : TW

(7)

Then ΨW ≥ ΓW ≥ W

(8)

ΨW = Θ W

(9)

and

Hence Θ W ≥ ΓW ≥ W ⇔ ΓW ≥ W ≥

ΓW Θ

(10)

Proof (i) If r j is feasible in RW , then r j is feasible in S W and (ii) if t j is feasible in T W then t j is feasible in RW . To prove (ii) note the 1–1 correspondence between sj and t j , where tj →

tj ≡ s j and s j → s j Θ ≡ t j Θ

That is, ⎧ j=N i=M  tj ΨW 1 ⎨ −1 ≡ min t j : Eqs. (1), (3), (4) holdand qi, j ≤ η j Θ Θ ω∈Ω,t j ≥1 N ⎩ I λ (ω) j=1 i=1 i j i, j for 1 ≤ j ≤ N }

62

Approximate Solutions of Walrasian Equilibrium …

⎧ j=N i=M  1 ⎨ −1 qi, j ≤ s j η j

W ≡ min s j : Eqs. (1), (3), (4) holdand ω∈Ω,t j ≥1 N ⎩ I λ (ω) j=1 i=1 i, j i, j for 1 ≤ j ≤ N }

Hence ΨW = ΘΓW .  Corollary 4 (a)  W = 1, iff the Walrasian equilibrium inequalities with constant marginal utilities of income rationalize the observed market data. (b) It follows from the Brown and Calsamiglia paper that  W = 1 iff the observed market data satisfies the strong law of demand.

The Gorman Polar Form Equilibrium Inequalities In Gorman’s seminal (1961) paper on the existence of representative agents, he derived the necessary and sufficient condition that a representative agent exists iff consumers are endowed with indirect utility functions, V i (p, I) in polar form: i ( p) , where ai (p) and b(p) are concave and homogeneous of degree Vi ( p, I ) = I −a b( p) one functions of the market prices. The marginal utilities of income are the same for all consumers and only depend on the market prices. That is, λi, j = b 1p for all ( j) i. As suggested by Varian (1992), we represent Gorman polar form indirect utility functions as (G)Vi ( p, I ) = I e( p) + f i ( p), where we assume e( p) ≡

1 b( p)

and f i ( p) ≡

ai ( p) b( p)

are convex functions of p.

Theorem 5 The Gorman Polar Form Equilibrium Inequalities: Eqs. (G1) to (G5) and (11), (13) (14) are necessary and sufficient conditions for rationalizing the observed market data with an exchange economy where consumers are endowed with Gorman polar form indirect utility functions. If e(p) and f i (p) are smooth and strictly convex, then for k = j and for 1 ≤ i ≤ M and 1 ≤ j ≤ N: e( pk ) > e( p j ) + ∇ p e( p j ) · ( pk − p j )

(G1)

f i ( pk ) > f i ( p j ) + ∇ p f i ( p j ) · ( pk − p j )

(G2)

The Gorman Polar Form Equilibrium Inequalities

63

Vi, j = Ii, j e( p) + f i ( p)

(G3)

di, j = Ii, j ∇ p e( p j ) + ∇ p f i ( p j )

(G4)

λi, j = e( p j )

(G5)

Vi,k > Vi, j + di, j · ( pk − p j ) + λ j (Ii,k − Ii, j )Dual A f riat I nequalities (11) i=M −1 i=1

λj

di, j ≤ η j Mar ket Clearing

p j · −di, j = λ j F OC Ii, j          ps    −1 pj   ≡ ΘG   −di, j · ≤ max 1, max λ j = max −dr,s 2 r,s r,s Ii, j Ir,s 1

(12) (13)

(14)

Upper Bound, That is, G is a uniform upper bound on the normalized marginal utilities of income, λi, j , for normalized indirect utility functions, where the λi, j vary over the observed market prices and income distributions. Moreover, G is an upper bound on the constant marginal utility of income for quasilinear utility functions, where λi, j = 1 for all i and j. Proof Necessity is obvious. To prove sufficiency, we use Afriat’s construction to derive piecewise linear convex indirect utility functions V i (p,I) satisfying Eqs. (11), (13) and (14) and piecewise linear convex functions e(p) and f i (p) satisfying Eqs. (G1)–(G5),—see Afriat (1967). If W i (p,I) = Ie(p) + f i (p) and ∂g(p) denotes the subdifferential of g(·) at p, then ∂W i (p, I) = I∂e(p) + ∂f i (p)—see Corollary 4.3 in Aubin et al. (1998). Hence W i (p,I) and V i (p,I) have the same subdifferential—see Exercise 8.31 in Rockafellar and Wets (1998). Hence they differ at most by a constant—see Theorem 24.9 in Rockafellar (1970). It follows that V i (p,I) and W i (p,I) define the same family of indifference curves.  We now prove an approximation theorem for the Gorman Polar Form Equilibrium Inequalities Theorem 6 If  ≥ G ≥ 1 and G is the optimal value of the nonconvex program S G , where ⎧ j=N 1 ⎨ s j : Eqs. (G1) to G(5) and (11) to(14) hold and

G ≡ min ω∈Ω,s j ≥1 N ⎩ j=1

64

Approximate Solutions of Walrasian Equilibrium …

1

i=M 

λ j (ω)

i=1

−1 ≤ s j η j for 1 ≤ j ≤ N } : SG Ii, j

(15)

G is the optimal value of the nonconvex program RG , where ΓG ≡

min

⎧ j=N 1 ⎨

ω∈Ω,r j ≥1 i=M  i=1

N ⎩ j=1

r j : Eqs. (G1) to (G5) and (11) to(14) hold and

−1 di, j ≤ r j η j for 1 ≤ j ≤ N } : RG Ii, j

(16)

 G is the optimal value of the nonconvex program T G , where ΨG ≡

min

ω∈Ω,t j ≥1

⎧ j=N 1 ⎨ t j : Eqs. (G1) to (G5) and (11) to(14) hold and N ⎩ j=1 Θ

i=M 

λ j (ω)

i=1

−1 di, j ≤ t j η j for 1 ≤ j ≤ N } : TG Ii, j

(17)

Then ΨG ≥ ΓG ≥ G

(18)

ΨG = Θ G

(19)

G Θ G ≥ ΓG ≥ G ⇔ ΓG ≥ G ≥ Θ

(20)

and

Hence

Proof See the proof of Theorem 3



Corollary 7 (a)  G = 1, iff the Gorman Polar Form equilibrium inequalities with constant marginal utilities of income rationalize the observed market data. (b) It follows from the Brown and Calsamiglia paper that  G =1 iff the observed market data satisfies the strong law of demand. The Gorman Polar Form equlibrium inequalities are a subset of the Walrasian equilibrium inequalities. As such, W ≤ G . Since W = G , it follows that ΘΓWW ≤ ΘΓGG . To test the two hypotheses, described in the introduction, we define the “confidence intervals” C W and C G , where

The Gorman Polar Form Equilibrium Inequalities

 CW ≡

65

   ΓW ΓG , ΓW and C G ≡ , ΓG ΘW ΘG

It follows from the approximation theorems that:

W ∈ C W and G ∈ C G  (a) If 1 ∈ / C W C G , then we reject both null hypotheses and accept both alternative hypotheses. C Gc , then we fail to reject the null hypothesis H 0,W , but reject the (b) If 1 ∈ C W null hypothesis  H 0,G and accept the alternative hypothesis H A,G C G , then we fail to reject both hypotheses. (c) If 1 ∈ C W

Discussion In this final section of the paper, we describe our contribution to the growing literature on Algorithmic Game Theory (AGT) or more precisely our contribution to the literature on Algorithmic General Equilibrium (AGE) that predates AGT. AGE begins with Scarf’s seminal (1967) article on computing approximate fixed points, followed by his classic (1973) monograph: The Computation of Economic Equilibria. Codenotti and Varadarajan (2007) review the literature on polynomial time algorithms for computing competitive equilibria of restricted classes of exchange economies, where the set of competitive equilibria is a convex set. It is the convexity of the equilibrium set that allows the use of polynomial time algorithms devised for solving convex optimization problems. The authors conclude that the computational complexity of general equilibrium models, where the set of equilibria is nonconvex, is unlikely to be polynomial. Scarf does not explicitly address the issue of computational complexity of what is now called the Scarf algorithm, for computing competitive equilibria. His primary research agenda is the computation of economic equilibria in real world economies. This project is best illustrated by the (1992) monograph of Shoven and Whalley, two of Scarf’s graduate students, on Computable General Equilibrium (CGE) models. CGE models are now the primary models for counterfactual economic policy analysis, used by policy makers for estimating the economic impact of proposed taxes, quotas, tariffs, price controls, global warming, agricultural subsidies,… See the (2012) Handbook of Computable General Equilibrium Modeling edited by Dixon and Jorgenson. CGE models use parametric specifications of utility functions and production functions. The parameters are often estimated using a method called “calibration”. That is, choosing parameter values such that the CGE model replicates the observed equilibrium prices and observed market demands in a single benchmark data set. As you might expect there is some debate about the efficacy of this methodology among academic economists. In response to the obvious limitations of calibration and parametric specification of tastes and technology, Brown and Matzkin proposed the Walrasian equilibrium inequalities as a methodology for nonparametric estimation

66

Approximate Solutions of Walrasian Equilibrium …

of CGE models, using several benchmark data sets. That is, Refutable General Equilibrium (RGE) models—see Brown and Kubler (2008). Brown and Kannan (2008) initiated the complexity analysis of searching for solutions of the Walrasian equilibrium inequalities. Subsequently, Cherchye et al. (2011) showed that solving the Walrasian equilibrium inequalities, formulated as an integer programming problem is NP-hard. Hence AGE consists of two computable classes of general equilibrium models. The parametric CGE models of Scarf-Shoven-Whalley and the nonparametric RGE models of Brown-Matzkin-Shannon. Both classes of models admit counterfactual policy analysis. Both classes of models contain special cases solvable in polynomial time. In general, both classes of models lack the polynomial time algorithms necessary for the efficient computation of solutions, hence they require approximation theorems to carry out effective counterfactual policy analysis. The contribution of this paper to the literature on refutable general equilibrium models are the proposed approximation theorems for the Walrasian and Gorman Polar Form equilibrium inequalities. Acknowledgments Dan Spielman recommended several excellent references on the computational complexity of algorithms. The remarks of Felix Kubler on an earlier draft of this paper were very helpful. I thank them for their comments. Finally, I wish to acknowledge the innumerable editorial contributions to my working papers and manuscripts suggested by Glena Ames, a dear friend and colleague.

References Afriat, S. (1967). The construction of a utility function from demand data. International Economic Review, 66–77. Arora, S. (1998). The approximability of NP-hard problems. In Plenary Lecture at ACM Symposium on Theory of Computing. Arrow, K., & Debreu, G. (1954). The existence of an equilibrium for a competitive economy. Econometrica, 22, 265–290. Aubin, J. -P. (1998). Optima and equilibria. Springer-Verlag. Basu, S. (2011). Algorithms in real algebraic geometry: Survey. West Lafayette: Department of Mathematics, Purdue University. Brown, D.J., & Calsamiglia, C. (2014) Alfred Marshall’s cardinal theory of value: The strong law of demand, Economic Theory Bulletin, 2(1), 65–75. Brown, D.J., & Kannan, R. (2008). Two algorithms for solving the walrasian equilibrium inequalities. In Brown & Kubler (eds.) Computational aspects of general equilibrium theory: Refutable theories of value (pp. 69–97). Springer-Verlag. Brown, D.J., & Kubler, F. (2008). Computational aspects of general equilibrium theory: Refutable theories of value (pp. 69–97). Springer-Verlag. Brown, D. J., & Matzkin, R. L. (1996). Testable restrictions on the equilibrium manifold. Econometrica, 64, 1249–1262. Brown, D. J., & Shannon, C. (2000). Uniqueness, stability, and comparative statics in rationalizable walrasian markets. Econometrica, 68, 1529–1539. Cheryche, L., et al. (2011). Testable implications of general equilibrium models: An integer programming approach. Journal of Mathematical Economics, 47, 564–575.

References

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Codenotti, B., & Varadarajan, K. (2007). Computation of market equilibria by convex programming. In Nisnan et al. (eds.) Algorithmic game theory, (pp. 69–77). Cambridge Diversity Press. Rockafellar, R.T. (1970). Convex analysis. Princeton University Press. Rockafellar, R.T., & Wets, R. (1998). Variational analysis. Springer-Verlag. Tarski, A. (1951). A decision method for elementary and geometry, 2nd rev. edn. Rand Corporation, Berkeley and Los Angles. Varian, H. (1992). Microeconomic analysis, 3rd edn. W.W. Norton and Company.

Computational Complexity of the Walrasian Equilibrium Inequalities

Introduction The Brown-Matzkin (1996) theory of rationalizing market data with Walrasian markets, where consumers are price-taking, utility maximizers subject to a budget constraint, consists of market data sets and the Walrasian equilibrium inequalities. A market data set is a finite number of observations on market prices, income distributions and social endowments. The Walrasian equilibrium inequalities are the Afriat inequalities for each consumer; the budget constraints for each consumer and the market clearing conditions in each observation. The unknowns in the Walrasian equilibrium inequalities are the utility levels, the marginal utilities of income and the Marshallian demands of individual consumers in each observation. The parameters are the observable market data: market prices, income distributions and social endowments in each observation. The Walrasian equilibrium inequalities are said to rationalize the observable market data if the Walrasian equilibrium inequalities are solvable for some family of utility levels, marginal utilities of income and Marshallian demands of individual consumers, where aggregate Marshallian demands are equal to the social endowments in every observation. This theory is intended to provide an empirical, nonparametric foundation for equilibrium in market economies, consistent with the Walrasian paradigm, as articulated by Arrow and Debreu (1954). As such, the Brown–Matzkin theory of rationalizing market data with Walrasian markets requires an efficient algorithm for solving the Walrasian equilibrium inequalities. The Tarski–Seidenberg theorem, Tarski (1951), proposes an algorithm, “quantifier elimination,” for deriving a finite family of multivariate polynomial inequalities from the Walrasian equilibrium inequalities, where the unknowns are the observable market data: market prices, income distributions and the social endowments in each observation. We define these inequalities as the “revealed Walrasian equilibrium inequalities.” It follows from the Tarski–Seidenberg theorem that the revealed Walrasian equilibrium inequalities are solvable for the given market data iff the equilibrium inequalities are solvable for some family of utility levels, marginal utilities © The Author(s), under exclusive license to Springer Nature Switzerland AG 2020 D. J. Brown, Affective Decision Making Under Uncertainty, Lecture Notes in Economics and Mathematical Systems 691, https://doi.org/10.1007/978-3-030-59512-8_6

69

70

Computational Complexity of the Walrasian Equilibrium …

of income and Marshallian demands of consumers. The revealed Walrasian equilibrium inequalities exhaust the empirical content of equilibrium in the Walrasian model of market economies. An example is the special case of Afriat’s (1967) seminal theorem on revealed consumer demand, where consumer’s demands are observable. The Afriat inequalities are linear inequalities, hence they can be solved in polynomial time, using interior point methods. In Afriat’s theory, GARP, introduced by Varian (1982), is the revealed Afriat equilibrium inequalities. Unfortunately, in general, the computational complexity of the Tarski–Seidenberg algorithm, is known to be doubly exponential in the worse case. See Basu (2011) for a discussion of the Tarski–Seidenberg theorem and the computational complexity of quantifier elimination. A decision problem is a problem where the answer is “yes” or “no.” In this paper, the decision problem is: Can the observed market data set be rationalized with Walrasian markets? That is, are the Walrasian equilibrium inequalities solvable if the values of the parameters are derived from the observed market data? A decision problem is said to have polynomial complexity, i.e., the problem is in class P, if there exists an algorithm that solves each instance of the problem in time that is polynomial in some measure of the size of the problem instance. For solving linear programming, interior-point methods are polynomial time algorithms, but the simplex method solves the worse case linear program in time that is exponential in some measure of the size of the worse case linear program. In the literature on computational complexity, polynomial time algorithms are referred to as “efficient” algorithms. A decision problem is said to be in NP, if there exists an algorithm that verifies, in polynomial time, if a proposal is a solution of the problem instance Clearly, P ⊂ NP but it is widely conjectured by computer scientists that P = N P What is the computational complexity of solving the Walrasian equilibrium inequalities? This important question was first addressed by Cheryche et al. (2011). They proved that solving the Walrasian equilibrium inequalities, reformulated as an integer programming problem is NP-hard. The decision problem A is said to be NPhard, if every problem in NP can be reduced in polynomial time to A. That is, if we can decide the NP-hard problem A in polynomial time then we can decide every NP problem in polynomial time. In this case, contrary to the current beliefs of computer scientists, P = N P.

Introduction

71

Brown and Shannon (2000) proposed a “dual” theory of rationalizing market data with Walrasian markets, where demands of individual consumers are not observed, and the Afriat inequalities are replaced by the dual Afriat inequalities for minimizing the consumer’s monotone, convex, indirect utility function over prices subject to her budget constraint. This “dual” family of Walrasian equilibrium inequalities is called the Hicksian equilibrium inequalities in this paper. The Hicksian equilibrium inequalities are said to rationalize the observed market data if the Hicksian equilibrium inequalities are solvable for some family of indirect utility levels, marginal indirect utilities and Marshallian demands of individual consumers, derived from Roy’s identity, where the aggregate Marshallian demands are equal to the social endowments in every observation. Brown and Shannon proved that the Walrasian equilibrium inequalities are solvable iff the Hicksian equilibrium inequalities are solvable. The Hicksian equilibrium inequalities are unsolvable iff for very solution of the consumer’s dual Afriat inequalities there is excess aggregate Marshallian demand in some observation. Of course, if for some family of consumer’s indirect utility functions “markets almost clear” in every observation, then the market data is “almost rationalized” by Walrasian markets. Theorems of this kind for NP-hard minimization problems, where the degree of approximation is explicit are called “approximation theorems” in the literature on computational complexity. The contribution of this paper is an approximation theorem for the NP-hard minimization, over indirect utility functions of consumers, of the maximum distance, over observations, between social endowments and aggregate Marshallian demands. In this theorem, we propose a polynomial time algorithm for computing an approximate solution of the Walrasian equilibrium inequalities, where explicit bounds on the degree of approximation are determined by observable market data.

Solving the Walrasian Equilibrium Inequalities For completeness, we recall the (strict) Afriat inequalities (1967) and the (strict) dual Afriat inequalities, introduced by Brown and Shannon (2000). Given solutions of the Afriat inequalities, we also recall Afriat’s construction of a piece-wise linear, monotone, concave utility function for rationalizing Marshallian demands. This is the same construction used by Brown and Shannon to derive a piece-wise linear, monotone, convex indirect utility function, from solutions of the dual Afriat inequalities, to rationalize Marshallian demand. Proposition 1 If the market data set is  N D ≡ p j , x j , I j j=1 where

72

Computational Complexity of the Walrasian Equilibrium …

  u x j = max u(y) p j ·y≤I j

and  V

pj Ij



 = min V p I j ·x j ≤1

p Ij



then the (strict) Afriat inequalities for N observations are:   u i < u j + λ j p j · xi − x j for i, j ≤ N and the dual (strict) Afriat inequalities for N observations are:  Vi − V j > ∇ Lp V

pj Ij

   pj pi · − Ii Ij

where (i) ui and uj are positive utility levels, (ii) λj are positive marginal utilities of income, (iii) pj are the market prices in observation, (iv) I j is the consumer’s income budget in observation j, (v) x i and x j are the Marshallian demands in the respective   sets, (vi) V i and V j are positive indirect utility level, (vii) q j ≡ ∇ Lp V p j /I j , and (viii) where qj  0. Proposition 2 If the (strict) Afriat inequalities are solvable for utility levels, uj , marginal utilities of income λj and Marshallian demands x j for the given market data N  D ≡ p j , x j , I j j=1 then   j=N  u(x) ˆ ≡ max u j + λ j p j · x − x j j=1

  j=N is a piece-wise linear, monotone, concave utility function that rationalizes x j j=1 . That is,     ˆ : pj · x ≤ Ij uˆ x j = max u(x) Proposition 3 If the (strict) dual Afriat inequalities are solvable for indirect utility levels, V j and marginal indirect utilities

Solving the Walrasian Equilibrium Inequalities

73

 q j ≡ ∇ Lp V

pj Ij



for the given market data  N D ≡ p j , x j , I j j=1 then    

p  j=N pj pj p · ≡ min ∇ Lp V − Vˆ j=1 I Ij I Ij is a piece-wise linear, monotone, convex indirect utility function that rationalizes   j=N p j /I j j=1 . That is, Vˆ



pj Ij





p p  = min Vˆ : · xj ≤ 1 I I

. In addition to the Afriat and the dual Afriat inequalities, our analysis is predicated on Roy’s identity. Proposition 4 Roy’s Identity [Theorem 22.3 in Simon and Blume (1994)] Let U(x) be a C 2 utility function that satisfies monotonicity and strict concavity. Let ξ (p, I) be the Marshallian demand function for U and V (p, I) the corresponding indirect utility function. If (p, I) and ξ (p, I) are all strictly positive, then ξi ( p, I ) = −

∂ V ( p,I ) ∂ pi ∂ V ( p,I ) ∂I

M N  Definition 5 If the market data set D = p j , ω j , Ii j i j then, the Hicksian equilibrium inequalities are defined as:(1) The strict, dual Afriat inequalities for each consumer.(2) Market clearing in each observation, where it follows from Roy’s identity that the Marshallian demands are:

xi j ≡ −

that is,

∇ p Vi ∇ I Vi

 pj Ii j

 = pj Ii j



q − Iii jj −

p j ·qi j Ii j

  −qi j I i j qi j = = . p j · qi j Ii j λi j

74

Computational Complexity of the Walrasian Equilibrium …

xi j =

I i j qi j p j · qi j

. Hence markets clear in each observation iff for j = 1, 2,…, N t=M 

xi j =

i=1

t=M  i=1

I i j qi j ≤ ωj. p j · qi j

Rationalizability of the Hicksian equilibrium inequalities was established by Brown and Shannon (2000). Theorem 6 (Brown and Shannon) If  M N D ≡ p j , ω j , Ii j i j is the given market data set, where for 1 ≤ j ≤ N i=M 

Ii j = p j · ω j ,

i=1

then the following statements are equivalent: (a) There exists a strictly convex, monotone, smooth indirect utility function V i (p/I) that rationalizes D (b) There exists numbers  Vi j ≡ Vi

pj Ii j



and vectors  qi j ≡ ∇ Vi p I

pj Ii j

 ∈ RL

for j, k = 1, 2, …, N such that k = j (i)  Vik − Vi j > qi j · for 1 ≤ i ≤ M; 1 ≤ j ≤ N. where (ii)

pj pk − Iik Ii j



Solving the Walrasian Equilibrium Inequalities

−

75

p j · qi j ≡ λi j > 0, qi j ν0 Ii2j

. Proof. See Theorem 1 and Lemma 1 in Brown and Shannon. Corollary 7 (Brown and Shannon) An equivalent family of dual Afriat inequalities, where prices are not normalized by incomes, is the following: for j, k = 1, 2, …, N such that k = j. Vik − Vi j >

 p j · qi j   qi j  · pk − p j − Iik − Ii j 2 Ii j Ii j

for 1 ≤ i ≤ M; 1 ≤ j ≤ N. where  ∇ p Vi

pj Ii j

 =

qi j Ii j

for 1 ≤ i ≤ M; 1 ≤ j ≤ N., where  ∇ I Vi

pj Ii j

 =−

p j · qi j ≡ λi j . Iii2

In Lemma 2, Brown and Shannon show that Hicksian equilibrium inequalities are equivalent to the Walrasian equilibrium inequalities. That is, the Walrasian equilibrium inequalities are solvable iff the Hicksian equilibrium inequalities are solvable.

An Approximation Theorem for Walrasian Markets The best known uniform bound on the marginal utilities of income is the assumption that consumers are endowed with quasilinear utilities. In this case, we restrict attention to rationalizing market data with Hicksian economies where λij = 1. That is, we assume the market data can be rationalized by a Hicksian quasilinear economy, where each consumer is endowed with a smooth, monotone, convex, indirect quasilinear utility function. The Hicksian quasilinear equilibrium inequalities consist of the Hicksian equilibrium inequalities and the linear equalities:   p j · −qi j = Ii2j , The Hicksian quasilinear equilibrium inequalities is a family of linear inequalities in qij , where

76

Computational Complexity of the Walrasian Equilibrium … i=M 

xi j ≡ −

i=1

∇ p Vi ∇ I Vi

 pj Ii j

i=M 

 q − Iii jj

pj Ii j

i=1

−

 =

p j ·qi j Ii2j

=

i=M  i=1

  −qi j . Ii j

Hence , the optimal value of the linear program R, can be computed in polynomial time, where ⎧ ⎨

⎫   ⎬  −qi j  ≡ min r : ≤ rωj : R ⎩1≤r ;qi j ≤0 ⎭ Ii j λ =1 ij

 = 1 iff the market data is rationalized by a Hicksian quasilinear economy. The importance of Hicksian quasilinear economies is that they allow us to approximate market economies, where we assume the consumer’s marginal utilities of income λij ≥ 1. The empirical justification for this assumption follows from restricting incomes of consumers to to be bounded above and the empirical finding of Laydard et al. (2008) and Elsas and Assman (2012) that the marginal utility of income diminishes with income, as conjectured by the classical economists. Diminishing marginal utility of income is one of the theoretical justifications for progressive taxation. Hence w.o.l.o.g. we assume that the lower bound on the marginal utilities of income for each consumer is one. In this case, the Hicksian equilibrium inequalities are augmented by the linear inequalities:   p j · −qi j ≥ Ii2j . If we recast the dual Afriat inequalities as the first order conditions for minimizing a smooth, monotone, convex indirect utility function subject to a budget constraint defined by the Marshallian demand, then we can invoke Gauvin’s (1977) theorem that the set of Lagrange multipliers for the budget constraint is bounded iff the consumers optimization problem satisfies the Mangasarian–Fromovitz constraint qualification, derived in Mangasarian–Fromovitz (1967). That is, for fixed (x ij , I ij ) the ith consumer solves the following optimization problem (Qij ):  

min  Vi K p∈R++ | I p ·xi j ≤1

p Ii j



 = Vi

 pj . Ii j

ij

The Lagrangian for Qij is   L p : μi j ≡ Vi

where



p Ii j



 + μi j

p · xi j − 1 Ii j



An Approximation Theorem for Walrasian Markets

xi j ≡ −

∇ p Vi ∇ I Vi

 pj Ii j

 = pj Ii j

qi j Ijj − p j ·qi j − Ii j

77

 q − Iii jj I i j qi j = = . p j · qi j λi j

Hence the first order conditions are: q   qi j pj qi j Ii j = −μi j xi j = μi j = ∇ p Vi = iff μi j = λi j . Ii j Ii j λi j Ii j

The Mangasarian–Fromovitz constraint qualification follows from Slater’s constraint qualification for convex inequality constraints, e.g., the budget constraint. See the Fritz John Theorem, Theorem 19.1, in Simon and Blume (1994) for the Mangasarian–Fromovitz constraint qualification and Theorem 19.12 in Simon and Blume for Slater’s constraint qualification. Gauvin computes a bound Uij on the Lagrange multipliers μij by solving the following linear program:  μi j ≤ ϒi j ≡

min ∇ p Vi { y∈R K |xi j ·y≤−1

pj Ii j



 · y = ∇ p Vi

pj Ii j

 · yi j .

We compute a universal upper bound on the μij , independent of p V (pj /I ij ) and x ij , as the optimal value of the maxmin optimization problem, where i is the family of dual Afriat inequalities for consumer i, ≡

max min { y∈R K |ω j·y≤−1}

qi j Ii j ∈i

  −qi j · y. Ii j

If ωj is the social endowment in observation j and x ij < ωj then Uij < . The results in Brown and Calsamiglia (2007) imply that the quasilinear solutions of the dual Afriat inequalities are characterized by the family of linear inequalities: λij = 1, hence 1≤ (−qij )/I ij and y lie in separable polyhedral constraint sets and the objective function is bilinear. Gosh and Boyd (2003) — see Sect. 2 on bilinear problems — show that maximin problems with these properties can be reduced to a linear program and solved in polynomial time. Since μij = λij , it follows that λij ≤ . For 1 ≤ j ≤ M: Definition 8 Approximation Theorems If OPT (β) is the optimal value of a NPhard minimization problem for the input β and O P T (β)is the optimal value of the O P T (β) approximating minimization problem for the input β, then the ratio O P T (β) is bounded above by the “approximation ratio” α(β) ≥ 1. Hence 



78

Computational Complexity of the Walrasian Equilibrium … 

O P T (β) ≤ O P T (β) ≤ α(β)O P T (β), 

where O P T (β)and α(β) can be computed in time polynomial in β. Lemma 9 [Approximation Lemma]: If ≥ λij ≥ 1, then

  i=M   −qi j xi j ≥ ≥ xi j f or 1 ≤ j ≤ N . Ii j λ >1 i=1

i=M  i=1

Proof.

i=M 

ij

xi j ≡ −

i=1

∇ p Vi ∇ I Vi

 pj Ii j

i=M 

 q − Iii jj

pj Ii j

i=1

−

 =

=

p j ·qi j Ii2j

i=M  i=1

  −qi j . Ii j λi j

If ≥ λij ≥ 1, then        −qi j  −qi j  −qi j ≥ > . I λ Ii j I λ λ >1 i j i j λ >1 λ >1 i j i j ij

ij

ij

That is,

i=M  i=1

  i=M   −qi j xi j ≥ ≥ xi j for 1 ≤ j ≤ N . Ii j λ >1 i=1 ij

 Theorem 10 [Approximation Theorem] If (1) is the optimal value of the nonconvex program S, where ⎧ ⎨

⎫   ⎬  −qi j

≡ min s : ≤ sω j : S. ⎩1≤s;qi j ≤0 ⎭ I λ λ >1 i j i j ij

(2)  is the optimal value of the linear program R, where ⎫   ⎬  −qi j ≤ r ω j : R.  ≡ min r : ⎭ 1≤r ;qi j ≤0 Ii j λ >1 ij

(3)  is the optimal value of the nonconvex program T, where ⎧ ⎨

⎫   ⎬  −qi j ≤ tω j : T.  ≡ min t : ⎩1≤t;qi j ≤0 ⎭ I λ λ >1 i j i j ij

An Approximation Theorem for Walrasian Markets

79

then (a)  ≥  ≥ . and (b)  = . that is, (c) ≥  ≥ . Proof. (a) follows from the approximation lemma. To prove (b) note the 1 – 1 correspondence between s and t, where t→

t ≡ s and s → s ≡ t

That is, ⎧ ⎫ ⎧ ⎫     ⎨ ⎬ ⎨ ⎬  −qi j  −qi j  t = min t : ≤ ω j = min s : ≤ sω j = . ⎭ ⎩1≤t;qi j ≤0 λ >1 Ii j λi j ⎭ ⎩1≤s;qi j ≤0 λ >1 Ii j λi j ij

ij

Hence  = .



= 1 iff the market data is rationalized by a Hicksian economy, where all consumers are endowed with indirect utility functions with marginal utilities of income, λij > 1.  = 1 iff the market data is rationalized by a Hicksian quasilinear economy, where all consumers are endowed with quasilinear utility functions, i.e., the marginal utilities of income, λij = 1. and  are optimal values of linear programs, hence they can be computed in polynomial time using interior point methods. is an NP – hard minimization problem.. Utility functions with marginal utilities of income bounded below by one are interesting in their own right, e.g., in fields like public finance and development economics. We close with an extension of the Brown-Calsamiglia characterization of quasilinear utility functions, where the marginal utilities of income equal one, to utility functions where the marginal utilities of income are bounded below by one.  j=N  Proposition 11 The market data p j , I j j=1 is rationalized by an indirect utility function V i (p, I), where the marginal utilities of income λi j ≡

  p j · −qi j Ii2j

are bounded below by one, iff the following family of linear inequalities are solvable,

80

Computational Complexity of the Walrasian Equilibrium …

 p j · qi j   qi j  Iik − Ii j , · pk − p j − Ii j Ii2j   (2) λi j ≥ 1 or equivalently p j · −qi j ≥ Ii2j .

(1) Vik − Vi j >

Proof. Necessity is immediate. We use Afriat’s construction for sufficiency:  Vi ( p, I ) ≡

j=N ∇ j=1

  p j · qi j   qi j  I − Ii j · p − pj − Vi j + Ii j Ii2j

It follows from convex analysis that the subgradient of the max of a finite family of convex functions is a convex combination of the subgradients of the component functions. Since the marginal utilities of income of the component functions are bounded below by one, the marginal utility of income of V i (p, I) is bounded below by one.  Acknowledgements Dan Spielman suggested several excellent references on the computational complexity of algorithms. The comments of Felix Kubler on an earlier draft of this paper were very helpful. I thank them both.

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