H. M. Polemarchakis

We can't disagree forever

Under the assumption of common priors, if the information partitions of two agents are finite, then simply by communicating back and forth and revising their posteriors the two agents will converge to a common equilibrium posterior, even though they may base their posteriors on quite different information. Furthermore, given any integer, n, one can construct an example in which the revision process not only takes n steps to converge, but no evident revision occurs n for (n-1) steps both agents repeat the same conflicting posteriors n until the last step when the two agents decide to agree. Common knowledge of each others posterior does not necessarily lead agents to the posterior they would have agreed upon had information been directly exchanged. On the other hand, the examples that are characterized by a discrepancy between the direct and indirect communication equilibrium are rare: with probability 1, the revision process constructed here leads the two agents in one step to the direct communication equilibrium.

Walrasian indeterminacy and Keynesian macroeconomics

Overlapping generations models with or without production or a portfolio demand for money display a fundamental indeterminacy. Expectations matter; and they are not, in the short run, constrained by the hypotheses of agent optimization, rational expectations, and market clearing. No short run policy analysis is possible without some explicit understanding of how agents expect the economy to respond to the policy. In this framework of perfect foresight and market clearing prices, it is possible to make Keynesian assumptions about the rigidity of money wages and the exogeneity of animal spirits of investors, to use the standard IS-LM apparatus, and to derive Keynesian conclusions about the short run effectiveness of policy. Alternatively, starting from different but no less rational expectations, one can derive the new classical neutrality propositions.

On the structure of the set of fixed price equilibria

Working in the framework suggested by Dreze, this paper studies the number of fixed price equilibria and their continuity with respect to the price system. In an exchange economy, the concept of a rationing scheme is introduced, which specifies how shortages are shared among agents. For given utility functions and a given rationing scheme, under standard assumptions, an existence theorem is recalled, and it is shown that the graph of the equilibrium correspondence, when prices and initial endowments vary, is a piecewise continuously differentiable manifold. Moreover, generically, the number of equilibria for an economy, at given prices, is finite and the set of equilibria varies continuously with the price system and the initial endowments.

Recovering Cardinal Utility

Individual choice under uncertainty depends on an agents attitudes towards risk, typically represented by his von Neumann-Morgenstern cardinal utility function, u. This function is, of course, an unobservable characteristic. What is, at least in principal, observable is the agents demand for alternative assets. Problems of prediction and welfare often require conclusions to be drawn concerning the unobservable characteristics based on observable behaviour. To predict the response to a change in the stochastic production plan of a firm, or to determine the compensation necessary so as to avoid a loss in welfare due to a change in the capital tax structure, knowledge of an agents attitude towards risk is required. It is the purpose of the present paper to demonstrate that, as long as the joint distribution of returns of the available assets is known, the cardinal utility function can be recovered without ambiguity from asset demands, provided that the set of available assets contains a riskless asset. The distinguishing characteristic of the recoverability problem as posed here is the possible incompleteness of markets. In a framework of complete markets, if nominal income and prices vary independently, the range of the demand correspondence can be assumed to cover an open subset of the consumption set. The utility function can then be immediately recovered, up to a monotone transformation, over this range, given mild regularity conditions on demand behaviour. Given incomplete markets, however, this argument fails. Since the choice set is a lower dimensional subspace of state space, we learn only about the agents preferences within this subspace. We show that the existence of a riskless asset (and some risky asset) implies recoverability of von NeumannMorgenstern preferences, even if markets are incomplete. In the presence of a riskless asset, any positive, riskless level, a, of wealth is demanded for some asset prices. Using arguments from Pratt (1964), the ratio u(a)/u(a) can be recovered without ambiguity for all positive (resp. non-negative) values of a, and hence, an integration argument can be used to recover the cardinal utility function, u, up to a positive affine transformation. Beyond the argument based on knowledge of the asset demand correspondence, we explore an alternative approach to recoverability. We demonstrate that knowledge of the portfolio indifference surfaces permits recovery of the cardinal utility function, provided there is a riskless asset. The intuition here is that the agents risk aversion is directly related to the curvature of the indifference curve around the riskless point and the variance of the risky asset(s). That this approach is in general equivalent to the demand correspondence approach is well known-we give a brief proof for twice continuously differentiable and strictly monotone utility functions. We point out two results concerning the informational requirements of the argument for recoverability. Specifically, the argument does not require complete knowledge of the distribution of returns independently of the asset demands. Instead, it suffices to know the mean and variance of the returns to just one asset other than the riskless, and the return of the riskless asset. Furthermore, if the demand for assets is known only on a subset of the

Portfolio choice, exchange rates, and indeterminacy

Consider an exchange economy under uncertainty. The asset market is incomplete: there are fewer assets, A, than states of nature, S. Assets are nominal and are denominated in N distinct “currencies.” Competitive equilibria are distinct if the associated allocations are distinct. An economy displays K degrees of indeterminacy if the set of distinct competitive equilibria contains a K-dimensional open set. Generically, the economy displays NS−A(N−1)−N degrees of indeterminacy.

Monopolistic Quantity Rationing

In this paper we address the question of whether a price-setting monopolist can improve his welfare by imposing quantity constraints on buyers. We show first that if all buyers are identical, quantity rations are not useful for the monopolist. We then show by means of an example that if buyers are diverse, quantity rations may be desirable. It is shown that if there are only two commodities, the only constraint that may be useful to the monopolist is a zero constraint on one of the two commodities. An example shows that this property does not hold for more than two commodities.

Implicit Contracts and Employment Theory

In a Walrasian economy, differences among agents in their attitudes towards risk do not constitute an inducement to trade. This is an outcome of the assumption of existence of a complete system of markets in contingent commodities.1 The assumption is, however, empirically unjustified. It is this observation that underlies the implicit contract theory of employment introduced by Baily (1974) and Azariadis (1975) and further elaborated by Baily (1975), Sargent (1975), Feldstein (1976), Negishi (1976) and Varian (1976). The assumptions and conclusions of the implicit contract theory can be briefly summarized as follows: It is assumed that a firm has a certain pool of workers associated with it and faces an uncertain price for its output. It is further postulated that the firms objective function is the expected value of its profits, while workers desire to maximize the expected utility of their income, the latter characterized by risk aversion. The firm chooses the employment contract which maximizes its expected profit subject to the constraint that it provide the workers with a minimum of expected utility. The latter is supposed to reflect the opportunities open for workers elsewhere in the economy. Under these assumptions it can be demonstrated that the optimal contract involves full employment of the firms labour pool at all states of nature, and a constant wage rate-i.e. a wage rate independent of the contingency realized. The above formulation involves a number of conceptual and empirical problems: It is assumed there is unanimity concerning the probability of occurrence of the various states of nature. Furthermore, not only are workers identical, but firms know the exact form of their utility function. The argument depends crucially on the risk neutrality of firms. In the absence of markets for contingent securities this assumption can only be justified in the very special case of perfect negative correlation between the profits of different firms. Otherwise one has to rely either on the superiority of firms vis-a-vis workers concerning the accessibility to capital markets, or in some Knightian distinction between the innate risk neutrality of entrepreneurs and risk aversion of workers. Workers are assumed to have an indirect utility function separable in income and prices. If this is not the case, even though firms behave parametrically with respect to the prices of goods, they must take into account portfolio-theoretic considerations on the part of workers. Since the only constraint faced by a firm is that the expected utility provided by the contract it offers be greater than or equal to some competitively determined level, it is implicitly assumed that workers cannot abandon the firm after the state of nature has been realized. Equivalently, the costs of movement for workers past the initial contracting period are assumed to be infinitely high. For otherwise, the constraints faced by a firm would take the form of a minimum wage to be paid at each state of nature. In the extreme case of costless labour mobility, this implies that firms are wage takers in the labour market.

Identifiability of the von Neumann–Morgenstern Utility Function from Asset Demands

Publisher Summary This chapter discusses the identifiability of the von Neumann−Morgenstern utility function from asset demands. The question of identifiability acquires additional complexity in the case of assets demanded by an investor who maximizes a von Neumann−Morgenstern utility function. As the prices of assets vary and, as is likely to be the case in a framework of incomplete markets, the number of states of nature exceeds the number of assets, only a lower dimensional subspace of the space of state-dependent values of terminal wealth is attainable. No direct observations can be made on preferences for state-contingent wealth patterns outside this subspace. The standard argument for identifiability, thus, fails. On the other hand, the utility function can be assumed to have the special additively separable form implied by the axioms of expected utility theory. Knowledge of the asset demands is sufficient to identify the von Neumann−Morgenstern utility function, provided the latter can be assumed to be independent of the state of nature and analytic over the nonnegative real line. The problem of identifiability of the utility function is of empirical and theoretical interest.

Convexity and sunspots: A remark

Abstract Economies in which competitive equilibrium allocations are optimal and individual preferences are strictly convex are immune to sunspots; this is so even in the presence of non-convexities in production.

Approximate aggregation under uncertainty

Abstract For a collection of agents with von Neumann-Morgenstern preferences, a price-independent income distribution, and identical probability beliefs, there exists a von Neumann-Morgenstern approximate aggregator. The risk tolerance of the approximate aggregator is equal to the sum of the individual agent risk tolerances at prices which yield constant, “risk-free”, contingent consumption. The application of the approximate aggregator to standard asset pricing models in finance is discussed briefly.