No full-text available
To read the full-text of this research,
you can request a copy directly from the authors.
A test for risk-averse expected utility
Abstract
Chambers gratefully acknowledges the support of the National Science Foundation through grant SES-1426867. We are grateful to Federico Echenique, John Quah and Marciano Siniscalchi for comments and suggestions. Two anonymous referees also provided comments and corrections which substantially improved the paper. All errors are our own.
... Next, Theorem 9 is a composite of contributions by Varian (1983), Green and Srivastava (1986), and Chambers, Liu, and Martinez (2016). 20 Theorem 9. ...
... The equivalence between (3) and (4) is due to Varian (1983) and Green and Srivastava (1986). The rest of Theorem 9 is due to Chambers, Liu, and Martinez (2016). ...
... Since µ * s = k ′ s ′ h k s (k ′ , s ′ ), the function h k s can be thought of as providing mean preserving spreads. Then (1) can be reduced to (1): see Chambers, Liu, and Martinez (2016) for details. The equivalence between (2) and (3) is a matter of Farkas lemma (see for an exposition of this methodology). ...
The chapter reviews recent developments in revealed preference theory. It discusses the testable implications of theories of choice that are germane to specific economic environments. The focus is on expected utility in risky environments; subjected expected utility and maxmin expected utility in the presence of uncertainty; and exponentially discounted utility for intertemporal choice. The testable implications of these theories for data on choice from classical linear budget sets are described, and shown to follow a common thread. The theories all imply an inverse relation between prices and quantities, with different qualifications depending on the functional forms in the theory under consideration.
... The property in Statement 6 results from applying linear programming duality to the system of Afriat inequalities in Statement 7. Since μ * s = k s h k s (k , s ), the function h k s can be thought of as providing mean-preserving spreads. Then Statement 5 can be reduced to Statement 6 (see Chambers et al. 2016b for details). The equivalence between Statement 6 and Statement 7 is a matter of Farkas lemma (see Chambers & Echenique 2016 for an exposition of this methodology as it applies to revealed preference theory). ...
... The equivalence between Statement 7 and Statement 8 is due to Varian (1983b) and Green & Srivastava (1986). The rest of Theorem 9 is due to Chambers et al. (2016b). ...
This article reviews recent developments in revealed preference theory. It discusses the testable implications of theories of choice that are germane to specific economic environments. The focus is on expected utility in risky environments, subjected expected utility and maxmin expected utility in the presence of uncertainty, and exponentially discounted utility for intertemporal choice. The testable implications of these theories for data on choice from classical linear budget sets are described and shown to follow a common thread. The theories all imply an inverse relation between prices and quantities, with different qualifications depending on the functional forms in the theory under consideration.
Expected final online publication date for the Annual Review of Economics, Volume 12 is August 3, 2020. Please see http://www.annualreviews.org/page/journal/pubdates for revised estimates.
... Our results contribute to a long-standing active area of research in revealed preference theory (Green, Lau, and Polemarchakis, 1979;Green and Srivastava, 1986;Epstein, 2000;Heufer, 2014;Kübler, Selden, and Wei, 2014;Echenique and Saito, 2015;Chambers, Echenique, and Saito, 2016a;Chambers, Liu, and Martinez, 2016b;Kübler and Polemarchakis, 2017;Polisson, Quah, and Renou, 2020;Chambers, Echenique, and Lambert, 2021;Kübler, Malhotra, and Polemarchakis, 2021;Blow, Crawford, and Crawford, 2022). This literature focuses on the identification of testable necessary and sufficient conditions for observable demand data to be compatible with various kinds of (subjective) expected utility or other models of choice under certainty, risk and uncertainty. ...
Although portfolio diversification is the typical strategy followed by risk-averse investors, extreme portfolios that allocate all funds to a single asset/state of the world are common too. Such asset-demand behavior is compatible with risk-averse subjective expected utility maximization under beliefs that assign a strictly positive probability to every state. We show that whenever finitely many extreme asset demands are rationalizable in this way under such beliefs, they are simultaneously rationalizable under the same beliefs by: (i) constant absolute risk aversion; decreasing absolute risk aversion/increasing relative risk aversion (DARA/IRRA); risk-neutral; and ris-kseeking utility indices at all wealth levels; (ii) a distinct class of DARA/IRRA utility indices at some strictly positive fixed initial wealth; and (iii) decreasing relative risk aversion utility indices under bounded wealth. We also show that, in such situations, the observable data allow for sharp bounds to be given for the relevant parameters in each of the above classes of risk-averse preferences.
... The first generation of revealed-preference papers, examined in the first six chapters of , addresses cases in which there is no uncertainty. A next generation of papers examines settings in which an agent is choosing under uncertainty, but objective probabilities are given (Border (1992), Echenique (2016, Chapter 8), Chambers, Liu, andMartinez (2014), Fishburn (1975), Kubler, Selden, and Wei (2014), Lin (2019), Polisson, Quah, and Renou (2020, Section IB)). Fishburn (1975), focusing on expected utility maximization, highlights the usefulness of separation theorems, which implicitly lie behind our arguments, while the criterion derived by Border (1992) is closest to the spirit of our results. ...
Decision theory can be used to test the logic of decision making—one may ask whether a given set of decisions can be justified by a decision‐theoretic model. Indeed, in principal–agent settings, such justifications may be required—a manager of an investment fund may be asked what beliefs she used when valuing assets and a government may be asked whether a portfolio of rules and regulations is coherent. In this paper we ask which collections of uncertain‐act evaluations can be simultaneously justified under the maxmin expected utility criterion by a single set of probabilities. We draw connections to the fundamental theorem of finance (for the special case of a Bayesian agent) and revealed‐preference results.
... We use the lattice method and results from Polisson et al. (2020). Some other papers that provide the testable restrictions for market choices are Varian (1983); Green and Srivastava (1986); Diewert (2012); Geanakoplos (2013); Echenique and Saito (2015); Cosaert and Demuynck (2015), and Chambers et al. (2016). ...
In this paper, due to limited attention and lack of focus, an agent has tendency toward framing of available information that drives her to see the state space as a collection of the events or “small worlds” related to a particular ambience, which we call a state aggregation. This work axiomatizes the model of state aggregation, shows how to recover the structure of the “small worlds” together with beliefs and utility from the individual demand for assets, and provides comparative statics and revealed preference restrictions.
We develop a revealed preference test for the Choquet expected utility model with ambiguity aversion, which does not rely on specific functional form assumptions on the utility index. It is computationally efficient if the number of states is not too large, even for a large number of observations. This is a nice feature compared to other existing revealed preference tests for decision models with ambiguity. We illustrate the usefulness of our results by implementing our test on two experimental datasets from the literature, and we compare the empirical fit of this model to the subjective expected utility model. (JEL C91, D81, D91, G41)
We propose a new measure of deviations from the expected utility theory. For any positive number e, we give a characterization of the datasets with a rationalization that is within e (in beliefs, utility, or perceived prices) of expected utility theory, under the assumption of risk aversion. The number e can then be used as a measure of how far the data is to the expected utility theory. We apply our methodology to data from three large-scale experiments. Many subjects in these experiments are consistent with utility maximization, but not with expected utility maximization. Our measure of distance to expected utility is correlated with the subjects’ demographic characteristics.
We develop a nonparametric method, called Generalized Restriction of Infinite Domains (GRID), for testing the consistency of budgetary choice data with models of choice under risk and under uncertainty. Our test can allow for risk-loving and elation-seeking attitudes, or it can require risk aversion. It can also be used to calculate, via Afriat’s efficiency index, the magnitude of violations from a particular model. We evaluate the performance of various models under risk (expected utility, disappointment aversion, rank-dependent utility, and stochastically monotone utility) using data collected from several recent portfolio choice experiments. (JEL C14, D11, D12, D81)
Assuming the classic contingent claim setting, a number of financial asset demand tests of Expected Utility have been developed and implemented in experimental settings. However, the domain of preferences of these asset demand tests differ from the mixture space of distributions assumed in the traditional binary lottery laboratory tests of von Neumann–Morgenstern Expected Utility preferences. We derive new sets of axioms for preferences over contingent claims to be representable by an Expected Utility function. We also indicate the additional axioms required to extend the representation to the more general case of preferences over risky prospects.
Assuming the classic contingent claim setting, a number of financial asset demand tests of Expected Utility have been developed and implemented in experimental settings. However, the domain of preferences of these asset demand tests differ from the mixture space of distributions assumed in the traditional binary lottery laboratory tests of von Neumann–Morgenstern Expected Utility preferences. We derive new sets of axioms for preferences over contingent claims to be representable by an Expected Utility function. We also indicate the additional axioms required to extend the representation to the more general case of preferences over risky prospects.
In this paper, we provide a formal framework for studying the empirical content of a given theory. We define the falsifiable closure of a theory to be the least weakening of the theory that makes only falsifiable claims. The falsifiable closure is our notion of empirical content. We prove that the empirical content of a theory can be exactly captured by a certain kind of axiomatization, one that uses axioms which are universal negations of conjunctions of atomic formulas. The falsifiable closure operator has the structure of a topological closure, which has implications, for example, for the behavior of joint vis a vis single hypotheses.
The ideas here are useful for understanding theories whose empirical content is well-understood (for example, we apply our framework to revealed preference theory, and Afriat's theorem), but they can also be applied to theories with no known axiomatization. We present an application to the theory of multiple selves, with a fixed finite set of selves and where selves are aggregated according to a neutral rule satisfying independence of irrelevant alternatives. We show that multiple selves theories are fully falsifiable, in the sense that they are equivalent to their empirical content.
We develop a behavioral axiomatic characterization of subjective expected utility (SEU) under risk aversion. Given is an individual agent's behavior in the market: assume a finite collection of asset purchases with corresponding prices. We show that such behavior satisfies a “revealed preference axiom” if and only if there exists a SEU model (a subjective probability over states and a concave utility function over money) that accounts for the given asset purchases.
In the neoclassical model of consumer behavior, considerable work has been done investigating when a consumer's demand behavior can be described as having been derived from utility maximization. However, most discussions are in a certainty world. We expand on prior analyses in an uncertainty setting by providing conditions under which contingent claim and asset demands will be consistent with state independent Expected Utility maximization. The question is addressed using two different traditional approaches. First given the analytical form of the demand functions, we derive necessary and sufficient conditions such that the consumer's behavior can be rationalized by an Expected Utility function. Second, we provide a necessary and sufficient condition for a finite set of observations on prices, probabilities and quantities to be consistent with Expected Utility maximization for the case of a single commodity in each state. This condition is shown to be analogous to the strong axiom of revealed preference in classical certainty demand theory. For both approaches, we consider the complete and incomplete asset market cases.
The standard models of consumer behavior under uncertainty are the expected utility model and the mean-variance model. As with any models involving unobservables one might well ask about the empirical content of these hypotheses: what restrictions on observed behavior do these models impose? How can one test observed behavior for consistency with these models? How can one recover the underlying utility function and forecast behavior in new situations?
The primary aim of the paper is to place current methodological discussions in macroeconometric modeling contrasting the ‘theory first’ versus the ‘data first’ perspectives in the context of a broader methodological framework with a view to constructively appraise them. In particular, the paper focuses on Colander’s argument in his paper “Economists, Incentives, Judgement, and the European CVAR Approach to Macroeconometrics” contrasting two different perspectives in Europe and the US that are currently dominating empirical macroeconometric modeling and delves deeper into their methodological/philosophical underpinnings. It is argued that the key to establishing a constructive dialogue between them is provided by a better understanding of the role of data in modern statistical inference, and how that relates to the centuries old issue of the realisticness of economic theories.
This paper attempts to answer the question: When is a random variable Y “more variable” than another random variable X?
The goal of choice-theoretic derivations of subjective probability is to separate a decisionmaker's underlying beliefs (subjective probabilities of events) from their preferences (attitudes toward risk). Classical derivations have all relied upon some form of the Marschak-Samuelson "Independence Axiom" or the Savage "Sure-Thing Principle," which imply that preferences over lotteries conform to the expected utility hypothesis. This paper presents a choice-theoretic derivation of subjective probability in a Savage-type setting of purely subjective uncertainty, which neither assumes nor implies that the decisionmaker's preferences over lotteries necessarily conform to the expected utility hypothesis. Copyright 1992 by The Econometric Society.
Working in a complete-markets setting, a property of asset demands is identi...ed that is inconsistent with the investor's preference being based on probabilities. In this way, a market counterpart of the Ellsberg Paradox is provided. 1. INTRODUCTION The Ellsberg Paradox and related evidence (see [1] for a survey) shows that decision-makers may not use probabilities to represent likelihoods and guide choice in settings where information is imprecise. In such environments, where ambiguity prevails in addition to risk, nonindierence (aversion, for example) to ambiguity is inconsistent with the global reliance on probabilities. Though this evidence is con- ...ned to experimental settings, it is highly intuitive. For this reason and because it is intuitive also that the phenomenon of ambiguity aversion may be important in market settings, the Ellsberg Paradox has stimulated the development of models of preference that are not based on probabilities as well as applications of these mod...