Stochastic utility theorem

This paper analyzes individual decision making. It is assumed that an individual does not have a preference relation on the set of lotteries. Instead, the primitive of choice is a choice probability that captures the likelihood of one lottery being chosen over the other. Choice probabilities have a stochastic utility representation if they can be written as a non-decreasing function of the difference in expected utilities of the lotteries. Choice probabilities admit a stochastic utility representation if and only if they are complete, strongly transitive, continuous, independent of common consequences and interchangeable. Axioms of stochastic utility are consistent with systematic violations of betweenness and a common ratio effect but not with a common consequence effect. Special cases of stochastic utility include the Fechner model of random errors, Luce choice model and a tremble model of [Harless, D., Camerer, C., 1994. The predictive utility of generalized expected utility theories. Econometrica 62, 1251–1289].     

Equilibrium theory with unbounded consumption sets and non-ordered preferences : Part I. Non-satiation

A new condition is introduced for the existence of equilibrium for an economy where preferences need not be transitive or complete and the consumption set of each agent need not be bounded from below. The new condition allows us to extend the literature in two ways. First, the result of the paper can cover the case where the utility set for individually rational allocations may not be compact. As illustrated in Page et al. [Page Jr., F.H., Wooders, M.H., Monteiro, P.K., 2000. Inconsequential arbitrage. Journal of Mathematical Economics 34, 439–469], the no arbitrage conditions do not apply to an economy with a non-compact utility set. Second, we generalize the arbitrage-based equilibrium theory to the case of non-transitive preferences.     

Ordinal notions of submodularity

We consider several ordinal formulations of submodularity, defined for arbitrary binary relations on lattices. Two of these formulations are essentially due to Kreps [Kreps, D.M., 1979. A representation theorem for “Preference for Flexibility”. Econometrica 47 (3), 565–578] and one is a weakening of a notion due to Milgrom and Shannon [Milgrom, P., Shannon, C., 1994. Monotone comparative statics. Econometrica 62 (1), 157–180]. We show that any reflexive binary relation satisfying either of Kreps’s definitions also satisfies Milgrom and Shannon’s definition, and that any transitive and monotonic binary relation satisfying the Milgrom and Shannon’s condition satisfies both of Kreps’s conditions.     

Risk aversion in RDEU

Various notions of risk aversion can be distinguished for the class of rank-dependent expected utility (RDEU) preferences. We discuss the relationships amongst five of these, and describe simple (testable) characterizations in terms of elementary probability transformations for all but the weakest notion. The paper also provides the first complete characterization of the RDEU orderings that are risk-averse in the sense of Jewitt [Jewitt, I., 1989. Choosing between risky prospects: the characterization of comparative static results and location independent risk. Management Science 35, 60–70]. We also extend Chew et al.’s [Chew, S.H., Karni, E., Safra, Z., 1987. Risk aversion in the theory of utility with rank-dependent probabilities. Journal of Economic Theory 42, 370–381] important characterization of strong risk aversion [Rothschild, M., Stiglitz, J.E., 1970. Increasing risk: I. A definition. Journal of Economic Theory 2, 225–243] by relaxing strict monotonicity and differentiability assumptions, and allowing for discontinuities in the probability transformation function. The important special case of maximin choice falls within this relaxed RDEU class. It is shown that any strongly risk-averse RDEU order is a convex combination of maximin and another RDEU order with concave utility and continuous, concave probability transformation. Our proof of the result on strong risk aversion is also simpler (as well as more general) than that of Chew et al. [Chew, S.H., Karni, E., Safra, Z., 1987. Risk aversion in the theory of utility with rank-dependent probabilities. Journal of Economic Theory 42, 370–381].     

Demand properties of concavifiable preferences

The purpose of the present paper is to clarify the relation between choice theory for individual consumers, i.e., the observed demand behavior, and the preference ordering ?$?$ of that individual. Specifically, we study how concavifiability (i.e., representability of ?$?$ by a concave utility function) is expressed by quantities (cross-coefficients) appearing in revealed preferences theory. We present a sequence of rather explicit necessary conditions for concavifiability. All these conditions are quantitative asymptotic strengthenings of the strong axiom of revealed preference. The results and concepts are illustrated by means of examples in which an expenditure data is defined by providing its generating utility function.     

Are CEOs expected utility maximizers?

Are individuals expected utility maximizers? This question represents much more than academic curiosity. In a normative sense, at stake are the fundamental underpinnings of the bulk of the last half-century’s models of choice under uncertainty. From a positive perspective, the ubiquitous use of benefit-cost analysis across government agencies renders the expected utility maximization paradigm literally the only game in town. In this study, we advance the literature by exploring CEO’s preferences over small probability, high loss lotteries. Using undergraduate students as our experimental control group, we find that both our CEO and student subject pools exhibit frequent and large departures from expected utility theory. In addition, as the extreme payoffs become more likely CEOs exhibit greater aversion to risk. Our results suggest that use of the expected utility paradigm in decision making substantially underestimates society’s willingness to pay to reduce risk in small probability, high loss events.     

Consistency of assessments in infinite signaling games

In this paper we investigate possible ways to define consistency of assessments in infinite signaling games, i.e. signaling games in which the sets of types, messages and answers are complete, separable metric spaces. Roughly speaking, a consistency concept is called appropriate if it implies Bayesian consistency and copies the original idea of consistency in finite extensive form games as introduced by ^{Kreps and Wilson (Econometrica 1982, 50, 863–894)}. We present a particular appropriate consistency concept, which we call strong consistency and give a characterization of strongly consistent assessments. It turns out that all appropriate consistency concepts are refinements of strong consistency. Finally, we define and characterize structurally consistent assessments in infinite signaling games.     

A topological approach to Wilson’s impossibility theorem

We will present a topological approach to Wilson’s impossibility theorem [Wilson, R.B., 1972. Social choice theory without the Pareto principle. Journal of Economic Theory 5, 478–486] that there exists no non-null binary social choice rule which satisfies transitivity, independence of irrelevant alternatives, non-imposition and has no dictator nor inverse dictator. Our research is in line with the studies of topological approaches to discrete social choice problems initiated by [Baryshnikov, Y., 1993. Unifying impossibility theorems: a topological approach. Advances in Applied Mathematics 14, 404–415]. This paper extends the result about the Arrow impossibility theorem shown in [Tanaka, Y., 2006. A topological approach to the Arrow impossibility theorem when individual preferences are weak orders. Applied Mathematics and Computation 174, 961–981] to Wilson’s theorem.     

Patience in some non-additive models

Our aim is to give an axiomatization of preferences over infinite consumption streams. At first we adopt the additive case, and give a characterization of preferences which satisfy patience [Marinacci, M., 1998. An axiomatic approach to complete patience and time invariance. Journal of Economic Theory 83, 105–144] or equivalently what Diamond [Diamond, P.A., 1965. The evaluation of infinite utility streams. Econometrica 33, 170–177] named equal treatment of all generations and then, focus on stationary additive preferences. It appears that this class of functionals contains the discounting functionals axiomatized in Koopmans [Koopmans, T.C., 1972. In: McGuire, C.B., Radner, R. (Eds.), Representations of Preference Orderings Over Time. Decision and Organization, North-Holland, Amsterdam] and what is known as Banach-Mazur limit functionals. These results are extended to non-additives preferences where similar results are generalized and naive patience receives a positive treatement through the liminf criterion.     

Revealed preference theory on the choice of lotteries

The choice behavior of a decision-maker is said to be consistent with expected utility maximization if there exists a utility function defined on the set of prizes such that the decision-maker chooses lotteries with the highest expected utility. We present a revealed preference characterization of choice behavior that is consistent with expected utility maximization. A necessary and sufficient condition for expected utility maximization is that there does not exist a way to compound lotteries such that the probability distribution over the final prizes generated by the chosen lotteries of each observation is equal to that generated by the rejected lotteries of each observation. Our result is quite general and can be applied to any compact set of prizes and any choice correspondence.     

The topology of fear

For many years experimental observations have raised questions about the rationality of economic agents—for example, the Allais Paradox or the Equity Premium Puzzle. The problem is a narrow notion of rationality that disregards fear. This article extends the notion of rationality with new axioms of choice under uncertainty and the decision criteria they imply (Chichilnisky, G., 1996a. An axiomatic approach to sustainable development. Social Choice andWelfare 13, 257–321; Chichilnisky, G., 2000. An axiomatic approach to choice under uncertainty with Catastrophic risks. Resource and Energy Economics; Chichilnisky, G., 2002. Catastrophical Risk. Encyclopedia of Environmetrics, vol. 1. John Wiley & Sons, Ltd., Chicester). In the absence of catastrophes, the old and the new approach coincide, and both lead to standard expected utility. A sharp difference emerges when facing rare events with important consequences, or catastrophes. Theorem 1 establishes that a classic axiom of choice under uncertainty – Arrow’s Monotone Continuity axiom, or its relatives introduced by DeGroot, Villegas, Hernstein and Milnor – postulate rational behavior that is ‘insensitive’ to rare events as defined in (Chichilnisky, G., 1996a. An axiomatic approach to sustainable development. Social Choice andWelfare 13, 257–321; Chichilnisky, G., 2000. An axiomatic approach to choice under uncertainty with Catastrophic risks. Resource and Energy Economics; Chichilnisky, G., 2002. Catastrophical Risk. Encyclopedia of Environmetrics, vol. 1. John Wiley & Sons, Ltd., Chicester). Theorem 2 replaces this axiom with another that allows extreme responses to extreme events, and characterizes the implied decision criteria as a combination of expected utility with extremal responses. Theorems 1 and 2 offer a new understanding of rationality consistent with previously unexplained observations about decisions involving rare and catastrophic events, decisions involving fear, the Equity Premium Puzzle, ‘jump diffusion’ processes and ‘heavy tails’, and it agrees with (Debreu, G., 1953. Valuation equilibrium and Pareto optimum. Proceedings of the National Academy of Sciences, 40, 588–592) formulation of market behavior and his proof of Adam Smith’s Invisible Hand theorem.     

Intertemporal recursive utility and an equilibrium asset pricing model in the presence of Lévy jumps

This paper presents an equilibrium formulation of asset pricing in an environment of mixed Poisson–Brownian information with recursive utility. The optimal portfolio choice problem is studied together with a derivation of Euler equation as necessary condition for optimality. It is further shown that the price processes governed by the Euler equation, together with the market clearing conditions, constitute the equilibrium price processes. Closed form formulas are derived for European call options and for other derivative securities in a particular parameterization of the economy. The derived option pricing formula contain many existing models as special cases, and is potentially useful in explaining the moneyness biasedness associated with Black–Scholes model.     

Reconciling introspective utility with revealed preference: Experimental arguments based on prospect theory

In an experiment, choice-based (revealed-preference) utility of money is derived from choices under risk, and choiceless (non-revealed-preference) utility from introspective strength-of-preference judgments. The well-known inconsistencies of risky utility under expected utility are resolved under prospect theory, yielding one consistent cardinal utility index for risky choice. Remarkably, however, this cardinal index also agrees well with the choiceless utilities, suggesting a relation between a choice-based and a choiceless concept. Such a relation implies that introspective judgments can provide useful data for economics, and can reinforce the revealed-preference paradigm. This finding sheds new light on the classical debate on ordinal versus cardinal utility.     

A characterization and an impossibility of finite length anonymity for infinite generations

In the context of ranking infinite utility streams, the impartiality axiom of finite length anonymity requires the equal ranking of any two utility streams that are equal up to a finite length permutation ( Fleurbaey and Michel, 2003). We first characterize any finite length permutation as a composition of a fixed step permutation and an “almost” fixed step permutation. We then show that if a binary relation satisfies finite length anonymity, then it violates all the distributional axioms that are based on a segment-wise comparison. Examples of those axioms include the weak Pareto principle and the weak Pigou-Dalton principle.     

Discrete choice modeling with nonstationary panels applied to exchange rate regime choice

This paper develops a regression limit theory for discrete choice nonstationary panels with large cross section (N) and time series (T) dimensions. Some results emerging from this theory are directly applicable in the wider context of M-estimation. This includes an extension of work by Wooldridge [Wooldridge, J.M., 1994. Estimation and Inference for Dependent Processes. In: Engle, R.F., McFadden, D.L. (Eds.). Handbook of Econometrics, vol. 4, North-Holland, Amsterdam] on the limit theory of local extremum estimators to multi-indexed processes in nonlinear nonstationary panel data models.It is shown that the maximum likelihood (ML) estimator is consistent without an incidental parameters problem and has a limit theory with a fast rate of convergence N^1/2T^3/4 (in the stationary case, the rate is N^1/2T^1/2) for the regression coefficients and thresholds, and a normal limit distribution. In contrast, the limit distribution is known to be mixed normal in time series modeling, as shown in [Park, J.Y., Phillips, P.C.B., 2000, Nonstationary binary choice. Econometrica, 68, 1249–1280] (hereafter PP), and [Phillips, P.C.B., Jin, S., Hu, L., 2007. Nonstationary discrete choice: A corrigendum and addendum. Journal of Econometrics 141(2), 1115–1130] (hereafter, PJH).The approach is applied to exchange rate regime choice by monetary authorities, and we provide an analysis of the empirical phenomenon known as “fear of floating”.     

Rank-dependent preferences without ranking axioms

A key feature of the rank-dependent model for decision making under risk is that the weighting of an outcome depends on its relative rank. This theory received numerous axiomatizations, however, all these sets of axioms need to make an explicit reference to the ranking of the outcomes. This situation is unsatisfactory, as it seems to be desirable to get the ranking property of this model as a consequence of the model, rather than as an assumption. [Yaari, M.E., 1987. The dual theory of choice under risk. Econometrica 55, 95–115] offered a special version of this model (called dual theory), where the utility function is linear. This paper offers a set of axioms implying Yaari’s dual theory without making any reference to the order of the outcomes. The main axiom is called semi betweenness, which, unlike the usual case, is made on random variables rather than on distribution functions.     

‘Stochastically more risk averse:’ A contextual theory of stochastic discrete choice under risk

Microeconometric treatments of discrete choice under risk are typically homoscedastic latent variable models. Specifically, choice probabilities are given by preference functional differences (given by expected utility, rank-dependent utility, etc.) embedded in cumulative distribution functions. This approach has a problem: Estimated utility function parameters meant to represent agents’ degree of risk aversion in the sense of Pratt (1964) do not imply a suggested “stochastically more risk averse” relation within such models. A new heteroscedastic model called “contextual utility” remedies this, and estimates in one data set suggest it explains (and especially predicts) as well as or better than other stochastic models.     

No unbounded arbitrage,weak no market arbitrage and no arbitrage price system conditions; Equivalent conditions

Page and Wooders [Page Jr., F.H., Wooders, M., 1996. A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium. Economics Letters 52, 153–162] prove that the no unbounded arbitrage (NUBA), a special case of a condition in Page [Page, F.H., 1987. On equilibrium in Hart’s securities exchange model. Journal of Economic Theory 41, 392–404], is equivalent to the existence of a no arbitrage price system (NAPS) when no agent has non-null useless vectors. Allouch et al. [Allouch, N., Le Van, C., Page F.H., 2002. The geometry of arbitrage and the existence of competitive equilibrium. Journal of Mathematical Economics 38, 373–391] extend the NAPS introduced by Werner [Werner, J., 1987. Arbitrage and the existence of competitive equilibrium. Econometrica 55, 1403–1418] and show that this condition is equivalent to the weak no market arbitrage (WNMA) of Hart [Hart, O., 1974. On the existence of an equilibrium in a securities model. Journal of Economic Theory 9, 293–311]. They mention that this result implies the one given by Page and Wooders [Page Jr., F.H., Wooders, M., 1996. A necessary and sufficient condition for compactness of individually rational and feasible outcomes and existence of an equilibrium. Economics Letters 52, 153–162]. In this note, we show that all these conditions are equivalent.