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.     

A theory of dynamic oligopoly,III: Cournot competition

We study the Markov perfect equilibrium (MPE) of an alternating move, infinite horizon duopoly model where the strategic variable is quantity. We exhibit a pair of difference-differential equations that, when they exist, differentiable MPE strategies satisfy. For quadratic payoff functions, we solve these equations in closed form and demonstrate that the MPE corresponding to the solution is the limit of the finite horizon equilibrium as the horizon tends to infinity. We conclude with a discussion of adjustment costs and endogenization of the timing.     

Simple Reputation Systems*

We develop a model of simple “reputation systems” that monitor and publish information about the behavior of sellers in a search market with asymmetric information. The reputations created influence the equilibrium search patterns of buyers and thus provide for market‐based “punishment” of bad behavior. We show that a reputation system that rewards honesty can enhance welfare by allowing good sellers to truthfully signal their type. However, the same system can be prone to strategic manipulation by sellers who always have low‐quality products. In that case, we show that an alternative system which assigns reputations based on product quality rather than honesty can be superior.     

WHO SEARCHES?*

We consider a directed search model with buyers and sellers and determine whether buyers look for sellers or vice versa. The buyers and sellers can choose to search or wait; what they do in equilibrium depends on the relative size of the two populations and the price formation mechanism. We study bargaining and auctions and find that when one population is much larger than the other the former searches and the latter waits. Under auction with roughly equal populations some buyers and sellers search and some wait. Our results challenge the practice of postulating who searches and who waits.     

Competing auctions with endogenous quantities

Two sellers decide on their discrete supply of a homogenous good. There is a finite number of buyers with unit demand and privately known valuations. In the first model, there is a centralized market place where a uniform auction takes place. In the second, there are two distinct auction sites, each with one seller, and buyers decide where to bid. Using the theory of potential games, we show that in the one-site auction model there is always an equilibrium in pure-strategies. In contrast, if the distribution of buyers values has an increasing failure rate, and if the marginal cost of production is relatively low, there is no pure-strategy equilibrium where both sellers make positive profits in the competing sites model. We also identify conditions under which an equilibrium with a unique active site exists. We deal with the finite and discrete models by using several results about order statistics developed by Richard Barlow and Frank Proschan [R. Barlow, F. Proschan, Mathematical Theory of Reliability, Wiley, New York, 1965; R. Barlow, F. Proschan, Inequalities for linear combinations of order statistics from restricted families, Ann. Math. Statist. 37 (1966) 1593-1601; R. Barlow, F. Proschan, Statistical Theory of Reliability and Life Testing, McArdle Press, Silver Spring, 1975].     

Existence and efficiency of equilibrium in infinite economies with finite aggregate wealth

Summary. We prove the existence and efficiency of equilibrium in economies with infinitely many consumers in which there are finitely many agents who own a positive portion of the aggregate endowment. We prove existence for commodity spaces which are employed in the general equilibrium asset pricing models and use incomplete and intransitive preferences. We discuss the importance of existence of finitely many agents who own a positive portion of the aggregate endowment in obtaining efficient equilibrium. For general equilibrium asset pricing applications we require forward properness only at individually rational Pareto optimal allocations. We provide an Arrow-Debreu model for these economies. We also give an application of our approach and results by employing Stochastic Differential Utility as the utility of each consumer in an infinite horizon model.     

WHEN DOES DETERMINACY IMPLY EXPECTATIONAL STABILITY?

Since the introduction of rational expectations, there have been issues with multiple equilibria and equilibrium selection. We study the connections between determinacy of rational expectations equilibrium and learnability of that equilibrium in a general class of purely forward‐looking models. Our framework is sufficiently flexible to encompass lags in agents' information and either finite horizon or infinite horizon approaches to learning. We are able to isolate conditions under which determinacy does and does not imply learnability and also conditions under which long‐horizon forecasts make a clear difference for learnability. Finally, we apply our result to a relatively general New Keynesian model.     

Asset prices, debt constraints and inefficiency

We consider (possibly non-stationary) economies with endogenous solvency constraints under uncertainty over an infinite horizon, as in Alvarez and Jermann (2000) [5]. A sort of Cass Criterion (Cass, 1972 [10]) completely characterizes constrained inefficiency under the hypothesis of uniform gains from risk-sharing (which is always satisfied in stationary economies when the autarchy is constrained inefficient). Uniform gains from risk-sharing also guarantee a finite value of the intertemporal aggregate endowment at a constrained optimum. Hence, no equilibrium exhibits a null interest rate in the long run. Finally, constrained inefficiency occurs if and only if there exists a feasible redistribution producing a welfare improvement at all contingencies.     

Equilibrium locations in a mixed duopoly with sequential entry in real time

We investigate a mixed duopoly model where a public firm and a private firm enter a market sequentially over an infinite time horizon, with and without uncertainty over the follower's entry date. We assume that there is a unit-length linear city and show that, if the public firm moves first, equilibrium location falls inside the second and third quartiles. The later the follower is expected to enter, the closer the two firms are. However, if the private firm acts first, it moves aggressively to locate at the middle point (one-half), forcing the public firm to locate nearer the periphery (one-sixth), to minimize consumers' transportation cost. In addition, social welfare is strictly greater when the public firm moves as the leader.     

社会保障基金最优持股比例研究

在不考虑信息不对称和外部性的条件下,在无穷期生命期限模型中,中央计划经济和竞争性均衡经济的结果是完全等价的;但是在有限生命期限模型中(如代际交叠模型等),中央计划经济和竞争性均衡经济的等价关系就不一定成立,而且在"动态无效率"的情况下,竞争性均衡是帕累托无效的,这时就有必要引入政府来调节经济效率。本文在代际交叠模型中引入国有股形式的社会保障来研究最优社会保障基金持有国有股的比例。研究发现:通过调节社会保障基金的持股比例可以实现中央计划经济和竞争性均衡经济的等价,实现最大的社会福利,从这个角度出发就可以确定最优的社会保障基金的持股比例。本文的结论对全国社会保障基金在未来的实际运作有指导意义。     

Convergence to perfect competition of a dynamic matching and bargaining market with two-sided incomplete information and exogenous exit rate

Consider a decentralized, dynamic market with an infinite horizon and incomplete information in which buyers and sellers' values for the traded good are private and independently drawn. Time is discrete, each period has length δ, and each unit of time a large number of new buyers and sellers enter the market. Within a period each buyer is matched with a seller and each seller is matched with zero, one, or more buyers. Every seller runs a first price auction with a reservation price and, if trade occurs, the seller and winning buyer exit with their realized utility. Traders who fail to trade either continue in the market to be rematched or exit at an exogenous rate. We show that in all steady state, perfect Bayesian equilibria, as δ approaches zero, equilibrium prices converge to the Walrasian price and realized allocations converge to the competitive allocation.     

Low price signals high capacity

We study pricing in a model where buyers are homogeneous and sellers have either capacity one or two. We show that if buyers observe prices but not capacities then an equilibrium exists where sellers of capacity two post lower prices than sellers of capacity one. The equilibrium satisfies the intuitive criterion.     

Trading mechanism selection with directed search when buyers are risk averse

We endogenize the trading mechanism selection in a model of directed search with risk averse buyers and show that the unique symmetric equilibrium entails all sellers using fixed price trading. Mechanisms that prescribe the sale price as a function of the realized demand (auctions, bargaining, discount pricing, etc.) expose buyers to the “price risk”, the uncertainty of not knowing how much to pay in advance. Fixed price trading eliminates the price risk, which is why risk averse customers accept paying more to shop at such stores.     

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.     

Intertemporal price discrimination in infinite horizon

In infinite horizon, a credible durable-good monopolist may resort to intertemporal price discrimination. We provide an analytical characterization of his optimal price policy when consumers and the monopolist have different values for the trade because of distinct discount factors.     

Sorting versus screening: Search frictions and competing mechanisms

In a market where sellers compete by posting trading mechanisms, we allow for a general search technology and show that its features crucially affect the equilibrium mechanism. Price posting prevails when meetings are rival, i.e., when a meeting by one buyer reduces another buyer's meeting probability. Under price posting buyers reveal their type by sorting ex-ante. Only if the meeting technology is sufficiently non-rival, price posting is not an equilibrium. Multiple buyer types then visit the same sellers who screen ex-post through auctions.     

Implementing the Mortensen rule in a frictional labor market

We show that, in settings where meetings can be multilateral, the allocation rule proposed by Mortensen (1982) can be relatively straightforward to implement: as a local auction conducted by sellers. The implications of using this mechanism in a simple model of the labor market are then explored. We characterize the equilibrium properties of this model, which include wage dispersion, and examine its implied Beveridge curve. A dynamic version of the model is calibrated to the US labor market, and we show that the model can account for observed vacancy rates, given parameters that are chosen to match the average wages and the natural rate of unemployment, although the implied wage dispersion is quite small. Finally, in the limit, as the time between offer rounds in the model approaches zero, the equilibrium approaches the Walrasian competitive equilibrium.     

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.     

Search and endogenous concentration of liquidity in asset markets

We develop a search-based model of asset trading, in which investors of different horizons can invest in two assets with identical payoffs. The asset markets are partially segmented: buyers can search for only one asset, but can decide which one. We show the existence of a “clientele’’ equilibrium where all short-horizon investors search for the same asset. This asset has more buyers and sellers, lower search times, and trades at a higher price relative to its identical-payoff counterpart. The clientele equilibrium dominates the one where all investor types split equally across assets, implying that the concentration of liquidity is socially desirable.     

A non-cooperative bargaining game with risk averse players and an uncertain finite horizon

For an alternating offer bargaining game with an uncertain horizon, the equilibrium payoff to a player is greater, the less risk averse is he relative to his opponent. As the horizon becomes infinite in length, the equilibrium partition converges to the Nash bargaining solution.