Chris Shannon

Monotone Comparative Statics

The authors derive a necessary and sufficient condition for the solution set of an optimization problem to be monotonic in the parameters of the problem. In addition, they develop practical methods for checking the condition and demonstrate its applications to the classical theories of the competitive firm, the monopolist, the Bertrand oligopolist, consumer and growth theory, game theory, and general equilibrium analysis. Copyright 1994 by The Econometric Society.

Subjective Beliefs and ex ante Trade

We study a definition of subjective beliefs applicable to preferences that allow for the perception of ambiguity, and provide a characterization of such beliefs in terms of market behavior. Using this definition, we derive necessary and sufficient conditions for the efficiency of ex ante trade and show that these conditions follow from the fundamental welfare theorems. When aggregate uncertainty is absent, our results show that full insurance is efficient if and only if agents share some common subjective beliefs. Our results hold for a general class of convex preferences, which contains many functional forms used in applications involving ambiguity and ambiguity aversion. We show how our results can be articulated in the language of these functional forms, confirming results existing in the literature, generating new results, and providing a useful tool for applications.

Weak and strong monotone comparative statics

SummaryThis 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.

Uniqueness, Stability, and Comparative Statics in Rationalizable Walrasian Markets

This paper studies the extent to which qualitative features of Walrasian equilibria are refutable given a finite data set. In particular, we consider the hypothesis that the observed data are Walrasian equilibria in which each price vector is locally stable under tatonnement. Our main result shows that a finite set of observations of prices, individual incomes and aggregate consumption vectors is rationalizable in an economy with smooth characteristics if and only if it is rationalizable in an economy in which each observed price vector is locally unique and stable under tatonnement. Moreover, the equilibrium correspondence is locally monotone in a neighborhood of each observed equilibrium in these economies. Thus the hypotheses that equilibria are locally stable under tatonnement, equilibrium prices are locally unique and equilibrium comparative statics are locally monotone are not refutable with a finite data set.

Quadratic Concavity and Determinacy of Equilibrium

One of the central features of classical models of competitive markets is the generic determinacy of competitive equilibria. For smooth economies with a finite number of commodities and a finite number of consumers, almost all initial endowments admit only a finite number of competitive equilibria, and these equilibria vary (locally) smoothly with endowments; thus equilibrium comparative statics are locally determinate. This paper establishes parallel results for economies with finitely many consumers and infinitely many commodities. The most important new condition we introduce, quadratic concavity, rules out preferences in which goods are perfect substitutes globally, locally, or asymptotically. Our framework is sufficiently general to encompass many of the models that have proved important in the study of continuous-time trading in financial markets, trading over an infinite time horizon, and trading of finely differentiated commodities.

Strict single crossing and the strict Spence-Mirrlees condition : A comment on monotone comparative statics

A handle for a hand-carried article such as a handbag, pocketbook or the like; the handle consisting of two straps, one of which is secured to each opposite side of the article, each strap being twisted to form a loop through which a persons hand can be inserted, and each loop being stretchable so to fit different sizes of hands.

Sharing risk and ambiguity

We study the market implications of ambiguity in common models. We show that generic determinacy is a robust feature in general equilibrium models that allow a distinction between ambiguity and risk.

Knightian Uncertainty and Moral Hazard

This paper presents a principal-agent model in which the agent has imprecise beliefs. We model this situation formally by assuming the agent[modifier letter apostrophe]s preferences are incomplete as in Bewley (1986) [2]. In this setting, incentives must be robust to Knightian uncertainty. We study the implications of robustness for the form of the resulting optimal contracts. We give conditions under which there is a unique optimal contract, and show that it must have a simple flat payment plus bonus structure. That is, output levels are divided into two sets, and the optimal contract pays the same wage for all output levels in each set. We derive this result for the case in which the agent[modifier letter apostrophe]s utility function is linear and then show it also holds if this utility function has some limited curvature.

Regular nonsmooth equations

Abstract This paper uses notions of regular points and values for nonsmooth functions introduced by Rader (1973) to establish local uniqueness and sensitivity results for solutions to equations involving nonsmooth functions, and as the basis for an approach to degree theory for nonsmooth functions. This paper shows that if y is a regular value of a continuous function ⨍, then the solutions to the equation ⨍(x) = y are locally unique, and the degree of ⨍ on an open set A at y is given by the formula d(⨍, A, y) = Σ x∈⨍ -1 (y) sgn det D ⨍(x) . These results are used to extend Dierkers (1972) results on the number of equilibria in regular exchange economies to nonsmooth exchange economics.

Verifiability and Group Formation in Markets

We consider group formation in markets with asymmetric information. Our model nests standard matching problems, including one-to-one, many-to-one, and many-to-many matching, as well as matching with salaries or contracts and matching with incomplete information. Prices for group positions and private goods as well as the groups that form are determined endogenously in equilibrium, as a result of demand and supply forces. The setup includes problems as diverse as moral hazard in teams, screening on ability, and mechanism design. Our analysis, including the definition of equilibrium and existence, revolves around the randomness in matching. Our main results characterize the limits on efficiency in such a general equilibrium, and show that a sufficiently rich set of group types can ensure the existence of an efficient equilibrium.