Market structure and competitive equilibrium in dynamic economic models

Recently theorists have analyzed economies which potentially contain both finite and infinite horizon overlapping generations, using “Arrow-Debreu” (complete) markets. Typically, applied models assume recursive spot and contingent securities markets, implying a different equilibrium concept. Indeed, if infinite horizon agents are present recursive equilibria cannot exist without some side conditions on debt. With the right side conditions, we show that every recursive market equilibrium allocation is a complete market equilibrium allocation and vice versa. This bridges a gap between theory and applications, and extends existing equivalence results on market structure.     

Existence and local indeterminacy of periodic equilibrium paths in infinite horizon models with external effects

Summary. It is expected that every periodic equilibrium path may exist even under standard assumptions such as low discounting and the concavity of utility functions in infinite horizon models with external effects. Nevertheless, until now no such example has been presented. In this note we will first construct a bounded growth model that has an external effect and every periodic equilibrium path under any discount factor. Next we will study the conditions under which periodic equilibrium paths have a local indeterminacy. Received: December 23, 1998; revised version: April 19, 1999     

Construction of a state space for interrelated securities with an application to temporary equilibrium theory

Summary We construct an endogenous state space in an exchange economy with possibly infinite horizon. Every period agents trade securities whose payoffs depend on future dividends and asset prices. We reject the perfect foresight assumption on the ground that agents have not only limited knowledge of other individuals' endowments and preferences, but also limited capacity to compute equilibria. We choose instead absence of arbitrage as the principle which allows agents to determine if a system of future prices is possible. We give an alogrithm to compute the set of nonarbitrage prices every period, with both finite and infinite horizon. We then apply this endogenous structure of uncertainty to an infinite horizon temporary equilibrium model.I would like to thank Professor Donald Brown for his constant help and guidance. I have also greatly benefited from helpful discussions with Professors Jacques Drèze, Bernard Dumas, Mordecai Kurz, Carsten Nielsen, Jan Werner, and Ho-Mou Wu.     

Spanning and completeness in markets with contingent claims

We characterize financial markets that are “complete” or that contain portfolios which “span” all measurable functions of a particular asset payoffs, either finite and infinite dimensional. These results are then employed to describe the extent to which options trading is sufficient to complete markets, to investigate the existence of “efficient funds,” and to establish the extent of market completeness required to ensure the unanimity and irrelevance results of modern corporation finance.     

Characterizing the Common Prior Assumption

Logical characterizations of the common prior assumption (CPA) are investigated. Two approaches are considered. The first is called frame distinguishability, and is similar in spirit to the approaches considered in the economics literature. Results similar to those obtained in the economics literature are proved here as well, namely, that we can distinguish finite spaces that satisfy the CPA from those that do not in terms of disagreements in expectation. However, it is shown that, for the language used here, no formulas can distinguish infinite spaces satisfying the CPA from those that do not. The second approach considered is that of finding a sound and complete axiomatization. Such an axiomatization is provided; again, the key axiom involves disagreements in expectation. The same axiom system is shown to be sound and complete both in the finite and the infinite case. Thus, the two approaches to characterizing the CPA behave quite differently in the case of infinite spaces. Journal of Economic Literature Classification Numbers: C70, D80.     

How to gamble against all odds

We compare the power of betting strategies (aka martingales) whose wagers take values in different sets of reals. A martingale whose wagers take values in a set A is called an A-martingale. A set of reals B anticipates a set A, if for every A-martingale there is a countable set of B-martingales, such that on every binary sequence on which the A-martingale gains an infinite amount at least one of the B-martingales gains an infinite amount, too.We show that for two important classes of pairs of sets A and B, B anticipates A if and only if the closure of B contains rA, for some positive r. One class is when A is bounded and B is bounded away from zero; the other class is when B is well ordered. Our results generalize several recent results in algorithmic randomness and answer a question posed by Chalcraft et al. (2012).     

Limited observability as a constraint in contract design

Limited observability is the assumption that economic agents can only observe a finite amount of information. Given this constraint, contracts among agents are necessarily finite and incomplete in comparison to the ideal complete contract that we model as infinite in detail. We consider the extent that finite contracts can approximate a complete contract. The objectives of the paper are: (i) to identify properties of agents’ preferences that determine whether or not finiteness of contracts causes significant inefficiency; (ii) to evaluate the performance of finite contracts against the ideal optimal contract in a bilateral bargaining model.     

Optimality in stochastic OLG models: Theory for tests

We consider general OLG economies under uncertainty, with short maturity assets and with dividend paying assets of infinite maturity and fiat money, and study the optimality properties of equilibria with a sequence of asset markets that are sequentially complete. We provide necessary and sufficient conditions, in terms of asset prices and dividends, for equilibria to be conditionally Pareto optimal. These results provide a theoretical basis for empirical investigation.     

Valuation in infinite-horizon sequential markets with portfolio constraints

Summary. We develop a theory of valuation of assets in sequential markets over an infinite horizon and discuss implications of this theory for equilibrium under various portfolio constraints. We characterize a class of constraints under which sublinear valuation and a modified present value rule hold on the set of non-negative payoff streams in the absence of feasible arbitrage. We provide an example in which valuation is non-linear and the standard present value rule fails in incomplete markets. We show that linearity and countable additivity of valuation hold when markets are complete. We present a transversality constraint under which valuation is linear and countably additive on the set of all payoff streams regardless of whether markets are complete or incomplete. Received: March 9, 2000; revised version: February 13, 2001     

Sustainable reputations with rating systems

In a product choice game played between a long lived seller and an infinite sequence of buyers, we assume that buyers cannot observe past signals. To facilitate the analysis of applications such as online auctions (e.g. eBay), online shopping search engines (e.g. BizRate.com) and consumer reports, we assume that a central mechanism observes all past signals, and makes public announcements every period. The set of announcements and the mapping from observed signals to the set of announcements is called a rating system. We show that, absent reputation effects, information censoring cannot improve attainable payoffs. However, if there is an initial probability that the seller is a commitment type that plays a particular strategy every period, then there exists a finite rating system and an equilibrium of the resulting game such that, the expected present discounted payoff of the seller is almost his Stackelberg payoff after every history. This is in contrast to Cripps, Mailath and Samuelson (2004) [5], where it is shown that reputation effects do not last forever in such games if buyers can observe all past signals. We also construct finite rating systems that increase payoffs of almost all buyers, while decreasing the seller?s payoff.     

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.     

Uniqueness of asset prices in an exchange economy with unbounded utility

Summary. This paper studies conditions under which the price of an asset is uniquely determined by its fundamental value – i.e., no bubbles can arise – in Lucas-type asset pricing models with unbounded utility. After discussing Gilles and LeRoy's (1992) example, we construct an example of a two-period, representative agent economy to demonstrate that bubbles can arise in a standard model if utility is unbounded below, in which case the stochastic Euler equation may be violated. In an infinite horizon framework, we show that bubbles cannot arise if the optimal sequence of asset holdings can be lowered uniformly without incurring an infinite utility loss. Using this result, we develop conditions for the nonexistence of bubbles. The conditions depend exclusively on the asymptotic behavior of marginal utility at zero and infinity. They are satisfied by many unbounded utility functions, including the entire CRRA (constant relative risk aversion) class. The Appendix provides a complete market version of our two-period example. Received: January 22, 1996; revised version: February 18, 1997     

Limit theorems for markets with sequential bargaining

This paper models trade as a non-coopertative, strategic game played at an infinite sequence of dates. A single, indivisible commodity is traded. Buyers and sellers have transferable utility and are characterized by their reservation utilities. They meet at random and “bargain” over the price at which a single unit of the good will be exchanged. Under a variety of circumstances it is shown that as the costs of search and bargaining become negligible, the outcome of the game converges to the competitve (flow) equilibrium, even when there is complete information.     

货币模型中时间偏好与经济稳定性的分析

传统经济理论假设经济人是完全理性的,具有完备记忆、稳定的偏好,然而这样的理性假设实际上超越了人的内在特性。本文建立了一个货币管理机构采取利率反馈规则的货币模型,并假定个体的时间偏好依赖于整个经济的收入、平均消费和货币量,运用动态法研究了具有社会属性的时间偏好是如何影响经济长期均衡处的稳定性的。结论是异质的时间偏好对于货币政策的实现效果起着重要作用,不同的货币政策会对市场能否恢复到均衡状态产生影响。只有当个体的效用函数满足本文给出的充分条件时,调节利率的经济政策才不会对均衡处的稳定性产生影响,否则政策会失效甚至起到副作用。     

The Leontief Two-sector Model and Undiscounted Optimal Growth with Irreversible Investment: The Case of Labor-intensive Consumption Goods

This essay reports results on optimal growth in a Leontief two-sector model originally formulated by Bruno and analyzed in discrete time by Nishimura-Yano. Applying geometric methods pioneered by Khan-Mitra, we derive a complete characterization of optimal policy. Assuming that the consumption goods sector is more labor-intensive, our results depart from those in the capital-intensive case: (i) every optimal program is monotonic, (ii) every optimal program converges to the golden-rule stock in a finite number of periods, and (iii) every optimal program undergoes either unemployment or excess capacity of capital.     

Trading with a common agent under complete information: A characterization of Nash equilibria

We analyze an abstract model of trading where N principals submit quantity-payment schedules that describe the contracts they offer to an agent, and the agent then chooses how much to trade with every principal. This represents a special class of common agency games with complete information. We study all the subgame perfect Nash equilibria of these games, not only truthful ones, providing a complete characterization of equilibrium payoffs. In particular, we show that the equilibrium that is Pareto-dominant for the principals is not truthful when there are more than two of them. We also provide a partial characterization of equilibrium strategies.     

Existence of equilibria in the overlapping generations model: the nontransitive case

Summary This paper investigates the existence of competitive equilibria in dynamic exchange models with countably many periods and countably many agents. At each period the commodity space can be finite or infinite dimensional. The preferences of agents are not assumed to be transitive or complete. A first equilibrium existence theorem is established under the classical assumption that there exists a finite set of non-negligible agents. In the particular case of an overlapping generations model, a second existence theorem allows simultaneously for finite-lived assets and infinite-lived assets and limits the previous assumption to infinite-lived assets. This theorem covers obviously the standard case of an overlapping generations model where the agents have no endowment outside their lifetime.     

An Elementary Approach to Approximate Equilibria with Infinitely Many Commodities

We prove the existence of approximate equilibria in a finite exchange economy with a countably infinite number of commodities and nonconvex preferences, when every trader has an excess demand set that is finitely spannable, i.e., that could be covered by a union of its convex subsets in finitely many steps. We show that the bound on the norm of the per capita aggregate excess demand is reciprocally related to the square root of the population. Extensions are also made to the case where countably many commodities are indivisible. The proofs are elementary. Journal of Economic Literature Classification Numbers: D50, C62, D52.     

Robust equilibria under non-common priors

This paper considers the robustness of equilibria to a small amount of incomplete information, where players are allowed to have heterogeneous priors. An equilibrium of a complete information game is robust to incomplete information under non-common priors if for every incomplete information game where each player's prior assigns high probability on the event that the players know at arbitrarily high order that the payoffs are given by the complete information game, there exists a Bayesian Nash equilibrium that generates behavior close to the equilibrium in consideration. It is shown that for generic games, an equilibrium is robust under non-common priors if and only if it is the unique rationalizable action profile. Set-valued concepts are also introduced, and for generic games, a smallest robust set is shown to exist and coincide with the set of a posteriori equilibria.     

A characterization of the distributions that imply mean—Variance utility functions

If there is a riskless asset, then the distribution of every portfolio is determined by its mean and variance if and only if the random returns are a linear transformation of a spherically distributed random vector. If there is no riskless asset, then the spherically distributed random vector is replaced by a random vector in which the last n ? 1 components are spherically distributed conditional on the first component, which has an arbitrary distribution. If the number of assets is infinite, then there must exist random variables m, v, y, where the distribution of y conditional on m and v is standard normal, such that every portfolio is distributed as some linear combination of m and vy. If there is a riskless asset, then m has zero variance. These distributions exhibit two-fund separability even if the utility function is not concave.