Economies with observable types

We study economies of asymmetric information with observable types. Trade takes place in lotteries. Individuals face a standard budget constraint, while the incentive compatibility constraints are imposed on the production set of the intermediaries. This formalization encompasses moral hazard, as in [Jerez, 2003] and [Jerez, 2005], and private information economies. Equilibrium allocations are constrained efficient, but, contrary to what stated for example in Jerez (2005), the set of equilibrium allocations may be empty and the Second Welfare Theorem may fail. This happens for two reasons. First, constrained efficient allocations may violate the necessary and sufficient conditions of price supportability for the individuals. Second, even when constrained efficient allocation are price supportable, they may fail to be a profit maximizing choice of the firm at the individual supporting prices. To restore existence of an equilibrium the firm has to be restricted to supply allocations with support in the set of incentive compatible contracts.     

Characterizing C2 aggregate excess demand functions

Summary. We integrate and sharpen two characterizations of aggregate excess demand functions: we obtain Mas-Colell's (1977) equilibrium invariance, and strengthen Geanakoplos' (1984) weakly concave utility functions to strictly concave ones. Our proofs modify and extend Geanakoplos' utility-construction. We note two applications: a sharper characterization of equilibrium price sets (cf. Mas-Colell (1977)), and a basis for the studies of computable economies (Richter and Wong (1996)). Received: July 29, 1996; revised version: February 20, 1997     

Network formation and stable equilibrium

We study the non-cooperative formation of a (directed) network that builds upon one-way immediate links. Our novel feature is that we require circular paths of links for productive activities. This captures more clearly the higher level of coordination that is often required for network formations than models such as those of Jackson and Wolinsky [A strategic model of social and economic networks, J. Econ. Theory 71 (1996) 44-74] and Bala and Goyal [A non-cooperative model of network formation, Econometrica 68 (2000) 1181-1229], in that in our model a single agent's unilateral action is limited in affecting the network structure. We completely characterize the class of Nash equilibria by showing that the minimality of a graph is both necessary and sufficient (Theorem 1). We also propose the concept of a stable network that is similar to the pairwise stability of Jackson and Wolinsky (1996). We show that stable equilibria are a wheel, a trivial network, or a sub-wheel partitioned network (Theorem 2). We modify the Bala and Goyal (2000) dynamic process with inertia, and obtain the convergence to stable equilibria regardless of the initial state (Theorem 3).     

On the uniqueness of competitive equilibrium

Summary . The paper is concerned with the following question: in addition to local uniqueness, what other conditions must be imposed to ensure global uniqueness of competitive equilibrium? The answer is provided within a standard framework involving excess demand functions. Conditions are identified which have the nice property that they are true close to a regular equilibrium. Uniqueness is established by considering an adjustment process and showing that under the mentioned conditions, every equilibrium is locally asymptotically stable and moreover the process itself is globally stable; uniqueness follows from an Arrow and Hahn (1971) result. Alternatively, the paper may be seen as identifying conditions under which there is a vectorfield satisfying the requirements of an uniqueness result due to Dierker (1974). Received: July 1, 1996; revised version October 7, 1996     

A qualitative approach to Markovian equilibrium in infinite horizon economies with capital

Using lattice programming and order theoretic fixpoint theory, we develop a new class of monotone iterative methods that provide a qualitative theory of Markovian equilibrium decision processes for a large class of infinite horizon economies with capital. The class of economies includes models with public policy, valued fiat money, monopolistic competition, production externalities, and various other nonconvexities in the production sets. The results can be adapted to construct symmetric Markov equilibrium in models with many agents and market incompleteness. As the methods are constructive, they provide the foundations for a rigorous analysis of numerical approximation schemes that study extremal Markovian equilibrium. Equilibrium comparative statics results relative to the space of economies are available. Of independent interest, we provide new conditions for preserving complementarity under maximization, and new generalized envelope theorems for nonconcave dynamic programming problems. Our fixed point algorithms are sharp, and are able to distinguish sufficient conditions under which Markovian equilibrium form a complete lattice of Lipschitz continuous, uniformly continuous and semicontinuous monotone functions as well as unique continuously differentiable equilibrium.     

Asset price fluctuations without aggregate shocks

We analyze the pricing of a productive asset in a class of dynamic exchange economies with heterogeneous, infinitely-lived agents, and self-enforcing intertemporal trades. Individual incomes fluctuate and are correlated; preferences, dividends and aggregate income are fixed. Almost all economies in this class have a unique stationary Markovian equilibrium with fluctuations in asset prices. As the set of unrationed households changes over time and states, excess demand functions shift, asset returns fluctuate, and some households are shut out of asset markets. Examples suggest that the amplitude of these movements is negatively correlated with the productivity of the asset and with the penalty for default.     

The existence of fully rational expectations approximate equilibria with noisy price observations

Nonrevealing fully rational expectations approximate equilibria exist in microeconomic pure exchange economies in which uninformed agents have suitably dispersed noisy price observations. Such traders maximize a state-dependent expected utility conditional on the price vector they observe, the distributions of noisy price observations, and the correct equilibrium relationship between states of the world and prices. In equilibrium, aggregate excess demand is small with high probability in every state of the world (and its expectation is also small); this magnitude diminishes as the noisy price observations become more accurate. Equilibria are obtained by applying a fixed point argument to state-dependent excess demand functions which are smooth because of the noisy price observations.     

A general existence theorem for equilibrium price dispersions

This paper presents sufficient conditions for the existence of non-degenerate equilibrium price dispersions in economies with sequentially searching consumers. This existence result is obtained by fully utilizing the fixed-point property of equilibrium price dispersions. Compared with the existing literature, the formal framework used in this paper is very general. Consequently, the results of the paper are valid for a large class of economic models.     

Nash and Walras equilibrium via Brouwer

Summary. The existence of Nash and Walras equilibrium is proved via Brouwer's Fixed Point Theorem, without recourse to Kakutani's Fixed Point Theorem for correspondences. The domain of the Walras fixed point map is confined to the price simplex, even when there is production and weakly quasi-convex preferences. The key idea is to replace optimization with “satisficing improvement,” i.e., to replace the Maximum Principle with the “Satisficing Principle.” Received: July 9, 2001; revised version: February 25, 2002 RID="*" ID="*" I wish to thank Ken Arrow, Don Brown, and Andreu Mas-Colell for helpful comments. I first thought about using Brouwer's theorem without Kakutani's extension when I heard Herb Scarf's lectures on mathematical economics as an undergraduate in 1974, and then again when I read Tim Kehoe's 1980 Ph.D dissertation under Herb Scarf, but I did not resolve my confusion until I had to discuss Kehoe's presentation at the celebration for Herb Scarf's 65th birthday in September, 1995. RID="*" ID="*"Correspondence to: C. D. Aliprantis     

Consumption dynamics in general equilibrium: A characterisation when markets are incomplete

We introduce a methodology for analysing infinite horizon economies with two agents, one good, and incomplete markets. We provide an example in which an agent's equilibrium consumption is zero eventually with probability one even if she has correct beliefs and is marginally more patient. We then prove the following general result: if markets are effectively incomplete forever then on any equilibrium path on which some agent's consumption is bounded away from zero eventually, the other agent's consumption is zero eventually—so either some agent vanishes, in that she consumes zero eventually, or the consumption of both agents is arbitrarily close to zero infinitely often. Later we show that (a) for most economies in which individual endowments are finite state time homogeneous Markov processes, the consumption of an agent who has a uniformly positive endowment cannot converge to zero and (b) the possibility that an agent vanishes is a robust outcome since for a wide class of economies with incomplete markets, there are equilibria in which an agent's consumption is zero eventually with probability one even though she has correct beliefs as in the example. In sharp contrast to the results in the case studied by Sandroni (2000) [29] and Blume and Easley (2006) [8] where markets are complete, our results show that when markets are incomplete not only can the more patient agent (or the one with more accurate beliefs) be eliminated but there are situations in which neither agent is eliminated.     

General equilibrium with nonconvexities and money

In a general-equilibrium economy with nonconvexities, there are sunspot equilibria with good welfare properties; sunspots can ameliorate the effects of the nonconvexities. For these economies, we show that agents act as if they have quasi-linear utility functions. We use this result to construct a new model of monetary exchange along the lines of Lagos and Wright, where trade occurs in both centralized and decentralized markets, but instead of quasi-linear preferences we assume general preferences but with indivisible labor. This suggests that modern monetary theory is more robust than one might have thought. It also constitutes progress on the classic problem of integrating monetary economics and general-equilibrium theory.     

A testable model of consumption with externalities

This paper presents a nonparametric model of interdependent preferences, where an individual's consumption may act as an externality on the preferences of other consumers. We assume that individual price consumption data is observed for all consumers. It is known that the general consumption model with externalities imposes few restrictions on the observed data, where the consistency requirement is Nash rationalizability. We motivate potential games as an important sub class of games where the family of concave potential games is refutable and imposes stronger restrictions on observed data. We use this framework to extend the analysis of Brown and Matzkin [D. Brown, R. Matzkin, Testable restrictions on the equilibrium manifold, Econometrica 64 (1996) 1249-1262] on refutable pure exchange economies to pure exchange economies with externalities. Finally, we discuss an application of this model to inter-household consumption data.     

Edgeworth's conjecture with infinitely many commodities: commodity differentiation

Summary. Convergence of the cores of finite economies to the set of Walrasian allocations as the number of agents grows has long been taken as one of the basic tests of perfect competition. The present paper examines this test in the most natural model of commodity differentiation: the commodity space is the space of nonnegative measures, endowed with the topology of weak convergence. In Anderson and Zame [12], we gave counterexamples to core convergence in L ¹, a space in which core convergence holds for replica economies and core equivalence holds for continuum economies; in addition, we gave a core convergence theorem under the assumption that traders' utility functions exhibit uniformly vanishing marginal utility at infinity. In this paper, we provide two core convergence results for the commodity differentiation model. A key technical virtue of this space is that relatively large sets (in particular, closed norm-bounded sets) are compact. This permits us to invoke a version of the Shapley-Folkman Theorem for compact subsets of an infinite-dimensional space. We show that, for sufficiently large economies in which endowments come from a norm bounded set, preferences satisfy an equidesirability condition, and either (i) preferences exhibit uniformly bounded marginal rates of substitution or (ii) endowments come from an order-bounded set, core allocations can be approximately decentralized by prices. Received: July 29, 1996; revised version: January 14, 1997     

General equilibrium models of monopolistic competition: A new approach

We analyze a class of ‘large group’ Chamberlinian monopolistic competition models by applying different concepts of functional separability to the same set of first-order conditions for utility maximization. We show that multiplicatively quasi-separable (MQS) functions yield ‘constant relative risk aversion’ (CRRA), and, therefore ‘constant elasticity of substitution’ (CES), functions, whereas additively quasi-separable (AQS) functions yield ‘constant absolute risk aversion’ (CARA) functions. We then show that the CARA specification sheds new light on: (i) pro-competitive effects, i.e., profit-maximizing prices are decreasing in the mass of competing firms; and (ii) a competitive limit, i.e., profit-maximizing prices converge to marginal costs when the mass of competing firms becomes arbitrarily large.     

Fundamentals, information, and international capital flows: A welfare analysis

In the recent discussion surrounding the design of a new international financial architecture, enhancing transparency has widely been proposed as a policy essential for increasing the efficiency of international capital markets. This paper uses a simple two-country (two-agent) general equilibrium model with incomplete markets and production to explore the welfare consequences of an increase in public information about country-specific fundamentals (increase in transparency). An improvement in the quality of information has two effects on the ex ante welfare of individual countries: A direct effect that increases the efficiency of global capital allocation and welfare, and an indirect general equilibrium effect that increases asset price volatility and may decrease welfare. When the degree of risk-aversion is low, at least one country will gain from an increase in information quality. If the degree of risk-aversion is high, then there are robust examples of economies for which an increase in information hurts all countries. The paper also discusses how certain institutional arrangements (international derivative markets, international agency) could ensure that all countries gain from better information by providing insurance against information-induced asset price risk.     

A Comment on Yin, Xiangkang and Yew-Kwang Ng: Quantity Precommitment and Bertrand Competition Yield Cournot Outcomes: A Case with Product Differentiation

This comment points out a gap in the proof of Theorem 2 contained in the paper by Yin and Ng. It is shown that the price reaction functions in their modelling approach are not continuous for all capacity subgames, contradicting the claim of Lemma 1. Possible remedies are suggested.     

Non-computability of competitive equilibrium

Summary. We provide a “computable counterexample” to the Arrow-Debreu competitive equilibrium existence theorem [2]. In particular, we find an exchange economy in which all components are (Turing) computable, but in which no competitive equilibrium is computable. This result can be interpreted as an impossibility result in both computability-bounded rationality (cf. Binmore [5], Richter and Wong [35]) and computational economics (cf. Scarf [39]). To prove the theorem, we establish a “computable counterexample” to Brouwer's Fixed Point Theorem (similar to Orevkov [32]) and a computable analogue of a characterization of excess demand functions (cf. Mas-Colell [26], Geanakoplos [16], Wong [50]). Received: September 9, 1997; revised version: December 17, 1997     

Excess demand functions with incomplete markets—a global result

The purpose of this paper is to give a global characterization of excess demand functions in a two-period exchange economy with incomplete real asset markets. We show that continuity, homogeneity and Walras’ law characterize the aggregate excess demand functions on any compact price set which maintains the dimension of the budget set.     

Endogenous market power

In this paper we develop a framework to study markets with heterogeneous atomic traders. The competitive model is augmented as we provide traders with correct beliefs about their price impacts to define equilibrium with endogenously determined market power and show that such equilibrium exists in economies with smooth utility and cost functions and is generically determinate. Traders? price impacts depend positively on the convexity of preferences or cost functions of the trading partners and are subject to mutual reinforcement. Compared to the competitive model, the volume of trade is reduced, and hence is Pareto inefficient. The price effects of non-competitive trading depend on the convexity of marginal utility or cost function.