On the rationalizability of observed consumers’ choices when preferences depend on budget sets and (potentially) on anything else

We prove that defining consumers’ preferences over budget sets is both necessary and sufficient to make every fully informative and finite set of observed consumption choices rationalizable by a collection of preferences which are transitive, complete, and monotone with respect to own consumption. Our finding has two important theoretical consequences. First, assuming that preferences depend on budget sets is illegitimate under the scientific commitments of revealed preference theory. Second, as long as consumers’ preferences are not defined over budget sets, we can assume that preferences depend on observable objects other than own consumption without compromising the logical possibility to reject the model against observation. We however point out that, despite this logical possibility, in practice it can be almost impossible to reject a model where preferences are defined over objects that depend on budget sets. As an example of this we show that if preferences are defined over consumption choices of other individuals then rationalization fails only in cases of negligible practical interest.     

Budget Balancedness and Optimal Income Taxation

We make two contributions to the theory of optimal income taxation. First, assuming conditions sufficient for existence of a Pareto optimal income tax and public goods mechanism, we show that if agents' preferences satisfy an extended notion of single crossing called capacity constrained single crossing, then there exists a Pareto optimal income tax and public goods mechanism that is budget balancing. Second, we show that, even without capacity constrained single crossing, existence of a budget balancing Pareto optimal income tax and public goods mechanism is guaranteed if the set of agent types contains no atoms.     

Preference for flexibility in infinite horizon problems

Summary. We consider the extension of the classical problem of preference for flexibility to many periods. Preferences are defined over sets of infinite paths of choices. The main result provides a set of axioms on preferences that yield an additive representation over a subjective state space. This space is the set of preferences over choice today and feasible set tomorrow. The main new axiom, stochastic dominance, is a stronger form of the assumption of monotonicity. Received: September 11 2000; revised version: December 18, 2001     

Generalized monotonicity analysis

Complex economic models often lack the structure for the application of standard techniques in monotone comparative statics. Generalized monotonicity analysis (GMA) extends the available methods in several directions. First, it provides a way of finding parameter moves that yield monotonicity of model solutions. Second, it allows studying the monotonicity of functions or subsets of variables. Third, GMA naturally provides bounds on the sensitivity of variables to parameter changes. Fourth, GMA may be used to derive conditions under which monotonicity obtains with respect to functions of parameters, corresponding to imposed parameter moves. Fifth, GMA contributes insights into the theory of comparative statics, for example, with respect to dealing with constraints or exploiting additional information about the model structure. Several applications of GMA are presented, including constrained optimization, nonsupermodular games, aggregation, robust inference, and monotone comparative dynamics.     

Weak and strong monotone comparative statics

Summary This paper develops necessary and sufficient conditions for the set of solutions to an optimization problem to be nondecreasing in a weak sense still strong enough to guarantee the existence of an increasing selection, and thus strong enough to guarantee monotonicity when the solution is unique, as well as necessary and sufficient conditions for the set of optimizers to be nondecreasing in a strong sense which is strong enough to rule out the possibility of a decreasing selection. These necessary and sufficient conditions are variations of quasisupermodularity and the single crossing property introduced in Milgrom-Shannon [13]. Moreover, to determine when an objective function satisfies these conditions, this paper develops several characterizations of quasisupermodularity and the single crossing property and their variants, both in terms of differential conditions and in terms of restrictions on the structure of the level sets of these functions. Several examples are given to choice theory under loss aversion and to an auction problem.I am grateful to Don Brown and Paul Milgrom for numerous helpful conversations concerning earlier versions of this paper. This paper is a revised and expanded version of the paper An Ordinal Theory of Games with Strategic Complementarities (Shannon [15]). I am pleased to acknowledge the financial support of the National Science Foundation, and an Alfred P. Sloan Foundation Doctoral Dissertation Fellowship.     

Ex post implementation

We analyze the problem of fully implementing a social choice set in ex post equilibrium. We identify an ex post monotonicity condition that is necessary and—in economic environments—sufficient for full implementation in ex post equilibrium. We also identify an ex post monotonicity no veto condition that is sufficient. Ex post monotonicity is satisfied in all single crossing environments with strict ex post incentive constraints.We show by means of two classic examples that ex post monotonicity does not imply nor is it implied by Maskin monotonicity. The single unit auction with interdependent valuations is shown to satisfy ex post monotonicity but not Maskin monotonicity. We further describe a Pareto correspondence that fails ex post monotonicity but satisfies Maskin monotonicity.     

An additively separable representation in the Savage framework

This paper proposes necessary and sufficient conditions for an additively separable representation of preferences in the Savage framework (where the objects of choice are acts: measurable functions from an infinite set of states to a potentially finite set of consequences). A preference relation over acts is represented by the integral over the subset of the product of the state space and the consequence space which corresponds to the act, where this integral is calculated with respect to an evaluation measure on this space. The result requires neither Savage's P3 (monotonicity) nor his P4 (weak comparative probability). Nevertheless, the representation it provides is as useful as Savage's for many economic applications.     

Endogenous tastes and stable long-run choice

Suppose that short-run preferences depend upon consumption one period earlier. Then there is an acyclic long-run strict preference relation iff, for every finite set, every conservative choice sequence converges. If long-run preferences are acyclic, then a unique long-run choice from a compact set is globally stable. If the long-run choice set includes multiple choices, there is a weaker stability property. Under special assumptions these results are extended to cases when the short-run consumption set is endogenous, and when more previous periods affect the present.     

Reallocation of an infinitely divisible good

Summary. We consider the problem of reallocating the total initial endowments of an infinitely divisible commodity among agents with single-peaked preferences. With the uniform reallocation rule we propose a solution which satisfies many appealing properties, describing the effect of population and endowment variations on the outcome. The central properties which are studied in this context are population monotonicity, bilateral consistency, (endowment) monotonicity and (endowment) strategy-proofness. Furthermore, the uniform reallocation rule is Pareto optimal and satisfies several equity conditions, e.g., equal-treatment and envy-freeness. We study the trade-off between properties concerning variation and properties concerning equity. Furthermore, we provide several characterizations of the uniform reallocation rule based on these properties. Received: August 29, 1995; revised version June 26, 1996     

Revealed preferences in macroeconomic policy decisions

This paper extends the usual instrument-target framework by directly estimating macroeconomic policy preferences over a number of policy targets using a derived reduced form and then solving the nonlinear optimality conditions for optimal instrument choice. Solving the optimality equations using observations on predetermined variables yields the implicit set of preference weights which induced the observed choices. The derivation format parallels the integrability discussions about consumer preferences implicitly expressed through demand functions. The approach is applied to a simple macro model estimated over the period from 1955 through 1972.     

Optimal Fiscal Policy in an Economy Facing Sociopolitical Instability

The paper presents a model of optimal government policy when policy choices may exacerbate sociopolitical instability (SPI). The authors show that optimal policy that takes into account SPI transforms a standard concave growth model into a model with both a poverty trap and endogenous growth. The resulting equilibrium dynamics inherit the properties of government policies and need not be monotone. Indeed, for a broad set of conditions, government policy is unable to eliminate the poverty trap; when these conditions do not hold, “most” countries eventually reach a balanced growth path. The predictions of the model are tested by developing three new measures of SPI for a panel of 58 countries. Estimating optimal policies and the growth equation derived from the model reveals strong support for the theory.     

Present values and internal rates of return

This paper resolves the question of finding the complete set of conditions under which the present value of a return stream decreases as interest rates rise. Previous work has shown that if a return stream can be costlessly abandoned, or truncated, at any date prior to its natural termination, then its value is monotone. We show that truncation is the same as monotonicity. First, monotonicity is equivalent to the value function being identical to the truncation set value function. Second, monotonicity is equivalent to the truncation set being contained within the convex hull, augmented by free (value) disposal.     

Profit maximization and supermodular technology

A dataset is a list of observed factor inputs and prices for a technology; profits and production levels are unobserved. We obtain necessary and sufficient conditions for a dataset to be consistent with profit maximization under a monotone and concave revenue based on the notion of cyclic monotonicity. Our result implies that monotonicity and concavity cannot be tested, and that one cannot decide if a firm is competitive based on factor demands. We also introduce a condition, cyclic supermodularity, which is both necessary and sufficient for data to be consistent with a supermodular technology. Cyclic supermodularity provides a test for complementarity of production factors. We are very grateful to two anonymous referees for suggestions, comments, and corrections. We also thank Kim Border for his suggestions on an earlier draft.     

Empirical Analysis of Limit Order Markets

We provide empirical restrictions of a model of optimal order submissions in a limit order market. A trader's optimal order submission depends on the trader's valuation for the asset and the trade-offs between order prices, execution probabilities and picking off risks. The optimal order submission strategy is a monotone function of a trader's valuation for the asset. We test the monotonicity restriction in a sample of order submissions and their realized outcomes from the Stockholm Stock Exchange. We do not reject the monotonicity restriction for buy orders or sell orders considered separately, but reject the monotonicity restriction for buy and sell orders considered jointly.     

The principal-agent problem with a satisficing agent

The principal-agent problem is formulated with an expected utility maximizing principal and a satisficing agent. If the outcome set is finite and if there is only one aspiration level, then there are always optimal incentive schemes which can be represented by vectors having only the unique aspiration level and zero as coordinate values. The relation between the monotone likelihood ratio property and the monotonicity of optimal incentive schemes is investigated. This is done not only in the above model, but also in a continuous model with outcomes in an interval, which is much more interesting.     

Optimal Contracting of Separable Production Technologies

We analyze the class of agency problems with a risk-neutral principal and a risk-averse agent, in which hidden actions and hidden information (on the agent's efficiency) are jointly present. Technological assumptions such as monotonicity of likelihood are no longer sufficient for the optimal contract to be monotone in the sharing rule. We show that these regularity properties obtain for the rather wide class of production technologies that are separable in the agent's inputs. Together with ordinal properties such as monotone differences and affiliation of inputs, separability yields the monotonicity in efficiency of the agent's actions and indirect utility.     

Robust implementation in general mechanisms

A social choice function is robustly implemented if every equilibrium on every type space achieves outcomes consistent with it. We identify a robust monotonicity condition that is necessary and (with mild extra assumptions) sufficient for robust implementation.Robust monotonicity is strictly stronger than both Maskin monotonicity (necessary and almost sufficient for complete information implementation) and ex post monotonicity (necessary and almost sufficient for ex post implementation). It is equivalent to Bayesian monotonicity on all type spaces.     

The Effect of Work Experience on Female Wages and Labour Supply

This paper develops and implements a semiparametric estimator for investigating, with panel data, the importance of human capital and time nonseparable preferences to females when aggregate shocks are present. It provides a set of conditions for making statistical inferences about agents' expectations of their correlated future choices, from a short panel. Under the assumption that observed allocations are Pareto optimal, a dynamic model of female labour supply and participation is estimated, in which experience on the job raises future wages, and time spent off the job in the past directly affects current utility (or, indirectly through productivity in the no-market sector).     

Nash Equilibrium and Welfare Optimality

If A is a set of social alternatives, a social choice rule (SCR) assigns a subset of A to each potential profile of individuals' preferences over A , where the subset is interpreted as the set of "welfare optima". A game form (or "mechanism") implements the social choice rule if, for any potential profile of preferences, (i) any welfare optimum can arise as a Nash equilibrium of the game form (implying, in particular, that a Nash equilibrium exists) and, (ii) all Nash equilibria are welfare optimal. The main result of this paper establishes that any SCR that satisfies two properties—monotonicity and no veto power—can be implemented by a game form if there are three or more individuals. The proof is constructive.     

Envy and Altruism: Contrasting Bivariate and Univariate Prospect Preferences

We compare prospect ordering with and without envy and altruism. We find that envy can induce a violation of the univariate first‐degree stochastic dominance (FSD), and thus a violation of the classic expected utility monotonicity axiom. Surprisingly, altruism can also violate FSD preferences. The intuitive explanation of the result in the case of altruism hinges on the sign of the mixed derivative of the bivariate preference: the individual might prefer a certain correlation between her wealth and her peer group's wealth, and is therefore willing to violate FSD as long as the outcomes of the two parties are ordered according to her preferences. When investments are considered, envy and altruism can distort not only preferences but also actual choices.