Indeterminacy of equilibrium in stochastic OLG models

Summary. This paper studies the equilibria of a stochastic OLG exchange economies consisting of identical agents living for two periods, and having the opportunity to trade a single infinitely-lived asset in constant supply. The agents have uncertain endowments and the stochastic process determining the endowments is Markovian. For such economies, the literature has focused on studying strongly stationary equilibria in which quantities and prices are functions of the exogenous states of nature which describe the uncertainty: such equilibria are generalizations of deterministic steady states, and this paper investigates if they have the same special status as asymptotic limits of other equilibrium paths. The difficulty in extending the analysis of equilibria beyond the class of strongly stationary equilibria comes from the presence of indeterminacy: we propose a procedure for overcoming this difficulty which can be decomposed into two steps. First backward induction arguments are used to restrict the domain of possible prices; then if some indeterminacy is left, expectation functions are introduced to make the forward equilibrium equations determinate. The properties of the resulting trajectories, in particular their asymptotic properties, can then be studied. For the class of models that we study this procedure provides a justification for focusing on strongly stationary equilibria. For the model with positive dividends (equity or land) the justification is complete, since we show that the strongly stationary equilibrium is the unique equilibrium. For the model with zero dividends (money) there is a continuum of self-fulfilling expectation functions resulting in a continuum of equilibrium paths starting from any admissible initial condition: under conditions given in the paper, these equilibrium paths converge almost surely to one of the strongly stationary equilibria-either autarchy or the stochastic analogue of the Golden Rule. Received: November 19, 2001; revised version: March 22, 2002 RID="*" ID="*" We are grateful for the stimulating environment and research support provided by the Cowles Foundation at Yale University during the Fall 2000 when this paper was first conceived. We are also grateful to the participants of the SITE Workshop at Stanford University and the Incomplete Markets Workshop at SUNY Stony Brook during the summer 2001 for helpful discussions. Correspondence to: M. Magill     

Real Indeterminacy of Stationary Monetary Equilibria in Centralized Economies

We show that real indeterminacy of stationary equilibria, by which the set of stationary equilibria is a continuum and the real allocation varies among equilibria, may arise in some general equilibrium models with fiat money. The conditions under which such equilibria arise are: (i) each household optimally saves a constant amount of money; and (ii) at least two households face different budget constraints. We present various models, including a decentralized money search model and a centralized model with a monopoly firm, to explain how these conditions lead to real indeterminacy. Finally, we present a policy that uniquely implements any desirable outcome.     

Microeconomic uncertainty and macroeconomic indeterminacy

We propose a stylized intertemporal macroeconomic model wherein the combination of decentralized trading and microeconomic uncertainty (taking the form of privately observed and uninsured idiosyncratic shocks) creates an information problem between agents and generates indeterminacy of the macroeconomic equilibrium. For a given value of the economic fundamentals, the economy admits a continuum of equilibria that can be indexed by the sales expectations of firms at the time of investment. The Walrasian equilibrium is one of these possible equilibria but it is reached only if firms are optimistic enough. With a weaker degree of optimism, equilibrium output, employment and real wages will be lower than in the Walrasian equilibrium. Moreover, the range of possible equilibria will depend positively on the wage elasticity of the labour supply and on the magnitude of the information problem between buyers and sellers (in our case, the variance of the idiosyncratic shocks).Stochastic simulations performed on a calibrated version of the model show that pure demand expectation shocks may generate business cycle statistics that are not inconsistent with the observed ones.     

Does commodity money eliminate the indeterminacy of equilibrium?

Previous studies have shown that a random-matching model with divisible fiat money and without constraint on agents’ money inventories possesses a continuum of stationary single-price equilibria. Wallace (J. Econom. Theory 81 (1998) 223) conjectures that the indeterminacy can be eliminated by the use of commodity money. Instead, I find that in a similar random-matching model with dividend-yielding commodity money, a continuum of stationary single-price equilibria exists when the utility of dividend is not too high. This result casts doubt on the conventional belief that the indeterminacy of monetary equilibrium is be caused only by the nominal nature of money.     

Multiplicity of Cournot Equilibria and Involuntary Unemployment

In an overlapping generations model with Cournot competition on the goods market it is shown that a continuum of stationary states and perfect foresight trajectories exists with unemployment at arbitrary low wages. Decisive for this is the influence that different forecast functions have on the objective demand curve, even though they are consistent with perfect foresight. With an example it is shown that simple adaptive and constant memory forecast rules generate such unemployment equilibria. The corresponding temporary equilibrium dynamics may display stable unemployment and unstable full employment equilibria.Journal of Economic LiteratureClassification Numbers: D43, D51, E24     

Perfectly competitive markets as the limits of cournot markets

A perfectly competitive, partial equilibrium market for a single homogeneous good with a (bounded) continuum of infinitesimal firms is considered. Cost functions are essentially unrestricted and are allowed to vary smoothly across firms. A sequence (net) of Cournot markets (each with a finite number of firms) which converge smoothly to the perfectly competitive limit in terms of both the inverse demand functions and the distributioon of firm technologies is introduced and it is shown that all markets sufficiently far along the sequence have a Cournot equilibrium and all the Cournot equilibria converge to the perfectly competitive equilibrium of the limit market.     

Wage–vacancy contracts and multiplicity of equilibria in a directed search model of the labour market

This paper studies a directed search model of the labour market, which is standard in all aspects except two. First, we allow firms to post wage–vacancy contracts advertising the number of workers they would pay as well as the payment all will receive. Second, we consider two cases: one where workers are risk neutral and one where workers are risk averse, both in finite and large economies. Our paper shows that when firms post wage–vacancy contracts, whether workers are modelled as risk neutral or risk averse matters: the types of symmetric equilibria and the nature of multiplicity of equilibria are different. Somewhat surprisingly, when there are finite numbers of risk‐neutral workers and firms, we obtain a finite number of symmetric equilibria, but when workers are risk averse, we obtain a continuum of equilibria. Furthermore, our paper sounds a cautionary note on using large economies as an approximation of finite economies: when workers are risk neutral, the nature of equilibrium is preserved going from a finite to a large economy, but the nature of equilibrium is different when workers are risk averse.     

A THEORY OF MONEY AND MARKETPLACES*

This article considers an infinitely repeated economy with divisible fiat money. The economy has many marketplaces that agents choose to visit. In each marketplace, agents are randomly matched to trade goods. There exist a variety of stationary equilibria. In some equilibrium, each good is traded at a single price, whereas in another, every good is traded at two different prices. There is a continuum of such equilibria, which differ from each other in price and welfare levels. However, it is shown that only the efficient single‐price equilibrium is evolutionarily stable.     

Multiplicity in general financial equilibrium with portfolio constraints

This paper explores the role of portfolio constraints in generating multiplicity of equilibrium. We present a simple financial market economy with two goods and two households, households who face constraints on their ability to take unbounded positions in risky stocks. Absent such constraints, equilibrium allocation is unique and is Pareto efficient. With one portfolio constraint in place, the efficient equilibrium is still possible; however, additional inefficient equilibria in which the constraint is binding may emerge. We show further that with portfolio constraints cum incomplete markets, there may be a continuum of equilibria; adding incomplete markets may lead to real indeterminacy.     

A model of insurance markets with incomplete information

The paper investigates how a competitive market would allocate insurance policies if firms were not able to determine the riskiness of individual consumers. It is demonstrated that if all firms have static expectations with regard to the policy offers of other firms, no stationary equilibrium may exist. A second equilibrium concept is then introduced which incorporates a different expectation rule. Each firm assumes that any policy will be immediately withdrawn which becomes unprofitable after that firm makes its own policy offer. This equilibrium is shown to exist and some of its welfare properties are investigated.     

Firm Scale and the Endogenous Timing of Entry: a Choice between Commitment and Flexibility

This paper presents a model of stochastic oligopoly with demand uncertainty where firms endogenously choose entry timing. We examine two extreme types of market structure and show that the equilibrium correspondence that connects them is continous. With two identically sized firms, there are symmetric, Cournot type equilibria where the probability of early entry declines with greater uncertainty, and for low uncertainty two asymmetric equilibria. With one large firm with a continuum of nonatomic firms, there is a unique Stackelberg equilibrium. We conclude that the behavior of a dominant firm with a finite fringe can be approximated by Stackelberg equilibrium.Journal of Economic LiteratureClassification Numbers?: D21, L11.     

Multiple transitional growth paths in endogenously growing open economies

This paper extends research in indeterminacy literature to a small open economy, two-sector endogenous growth model. It shows that a continuum of equilibria exist in two situations, (i) the production functions exhibit social constant returns to scale with very small or even negligible externalities; and (ii) the production functions are standard constant returns to scale without externalities but the government implements asymmetric tax policies across sectors.     

Individual and Aggregate Real Balances in a Random-Matching Model

This article investigates the characteristics of stationary single-price equilibrium in a monetary random-matching model where agents can hold an arbitrary amount of divisible money and where production is costly. At such an equilibrium, agents' money holdings are endogenously determined and uniformly bounded. A refinement of weakly undominated strategies is argued to be necessary. It is shown that a continuum of single-price equilibria indexed by the aggregate real-money balance exists if one such equilibrium exists. Equilibria with different money-holdings upper bounds, hence different distributions, but with identical aggregate real-money balances can coexist.     

Growth in economies with non convexities: sunspots and lottery equilibria

Summary. We investigate the relation between lotteries and sunspot allocations in a dynamic economy where the utility functions are not concave. In an intertemporal competitive economy, the household consumption set is identified with the set of lotteries, while in the intertemporal sunspot economy it is the set of measurable allocations in the given probability space of sunspots. Sunspot intertemporal equilibria whenever they exist are efficient, independently of the sunspot space specification. If feasibility is, at each point in time, a restriction over the average value of the lotteries, competitive equilibrium prices are linear in basic commodities and intertemporal sunspot and competitive equilibria are equivalent. Two models have this feature: Large economies and economies with semi-linear technologies. We provide examples showing that in general, intertemporal competitive equilibrium prices are non-linear in basic commodities and, hence, intertemporal sunspot equilibria do not exist. The competitive static equilibrium allocations are stationary, intertemporal equilibrium allocations, but the static sunspot equilibria need not to be stationary, intertemporal sunspot equilibria. We construct examples of non-convex economies with indeterminate and Pareto ranked static sunspot equilibrium allocations associated to distinct specifications of the sunspot probability space.Received: 25 August 2003, Revised: 16 March 2004, JEL Classification Numbers: D84, D90.Correspondence to: Paolo SiconolfiWe thank Herakles Polemarchakis for helpful conversations on the topic. The research of Aldo Rustichini was supported by the NSF grant NSF/SES-0136556.     

Markov perfect equilibria in industries with complementarities

Summary.   This paper considers the existence and computation of Markov perfect equilibria in games with a “monotone” structure. Specifically, it provides a constructive proof of the existence of Markov perfect equilibria for a class of games in which a) there is a continuum of players, b) each player has the same per period payoff function and c) these per period payoff functions are supermodular in the player's current and past action and have increasing differences in the player's current action and the entire distribution of actions chosen by other players. The Markov perfect equilibria that are analyzed are symmetric, not in the sense that each player adopts the same action in any period, but rather in the sense that each player uses the same policy function. Since agents are typically distributed across many states they will typically take different actions. The formal environment considered has particular application to models of industries (or economies) in which firms face costs of price adjustment. It is in this context that the results are developed. Received: November 9, 1999; revised version: February 10, 2000     

ON THE ROLE OF TAX SUBSIDY SCHEME IN MONEY SEARCH MODELS*

This article investigates the roles of policy in search models with divisible money. Recently, real indeterminacy of stationary equilibria has been found in both specific and general search models with divisible money. Thus if we assume the divisibility of money, it would be quite difficult to make accurate predictions of the effects of simple monetary policies. Therefore, in this article, we introduce a certain tax subsidy scheme and show that this is effective in selecting a determinate efficient equilibrium. In other words, for a given efficient equilibrium and for any real number δ > 0 , a certain tax subsidy scheme induces a locally determinate equilibrium within the δ‐neighborhood of the given equilibrium. Moreover, the size of the tax subsidy can be arbitrarily small.     

Existence, uniqueness and efficiency of equilibrium in hedonic markets with multidimensional types

We study equilibrium in hedonic markets, when consumers and suppliers have reservation utilities, and the utility functions are separable with respect to price. There is one indivisible good, which comes in different qualities; each consumer buys 0 or 1 unit, and each supplier sells 0 or 1 unit. Consumer types, supplier types and qualities can be either discrete of continuous, in which case they are allowed to be multidimensional. Prices play a double role: they keep some agents out of the market, and they match the remaining ones pairwise. We define equilibrium prices and equilibrium distributions, and we prove that equilibria exist, we investigate to what extend equilibrium prices and distributions are unique, and we prove that equilibria are efficient. In the particular case when there is a continuum of types, and a generalized Spence–Mirrlees condition is satisfied, we prove the existence of a pure equilibrium, where demand distributions are in fact demand functions, and we show to what extent it is unique. The proofs rely on convex analysis, and care has been given to illustrate the theory with examples.     

Notes on optimal transportation

We study equilibrium in hedonic markets, when consumers and suppliers have reservation utilities, and the utility functions are separable with respect to price. There is one indivisible good, which comes in different qualities; each consumer buys 0 or 1 unit, and each supplier sells 0 or 1 unit. Consumer types, supplier types and qualities can be either discrete of continuous, in which case they are allowed to be multidimensional. Prices play a double role: they keep some agents out of the market, and they match the remaining ones pairwise. We define equilibrium prices and equilibrium distributions, and we prove that equilibria exist, we investigate to what extend equilibrium prices and distributions are unique, and we prove that equilibria are efficient. In the particular case when there is a continuum of types, and a generalized Spence–Mirrlees condition is satisfied, we prove the existence of a pure equilibrium, where demand distributions are in fact demand functions, and we show to what extent it is unique. The proofs rely on convex analysis, and care has been given to illustrate the theory with examples.     

Nonminimal product differentiation as a bargaining outcome

It is well known that in a duopoly model of product choice with quadratic transportation cost, the firms locate at the extreme endpoints of the market. Jehiel (1992, Int. J. Ind. Organ, 10, 633–641) has examined this model in an infinite horizon setting where in the initial period the firms choose location and in subsequent periods charge the Nash bargaining solution prices. Then, in the unique equilibrium both firms locate at the center of the market. This paper examines the case when the firms instead charge the prices determined by either the egalitarian bargaining solution or the Kalai–Smorodinski bargaining solution. It is shown that central agglomeration is an equilibrium. Furthermore, there is a continuum of symmetric equilibria in addition where the firms locate apart from each other. So the products are not necessarily minimally differentiated. Thus different bargaining solutions provide quite different outcomes.     

QUANTITY VERSUS PRICE IN A HOMOGENEOUS PRODUCT DUOPOLY*

In a simple homogeneous product setting, the paper looks at the debate on whether firms should choose quantity or price as their strategic variable. It examines a two-stage game between firms with symmetric costs in which the firms choose the strategic mode of operation in the first period and then, in the second period, price or output are chosen simultaneously according to the mode chosen in the first stage. In this game it is possible to have two Nash equilibria where either both play in quantities or both play in prices. One firm choosing price and the other quantity can never be a Nash equilibrium in the two-stage game. Both choosing quantity is always a Nash equilibrium. Both choosing prices may be a Nash equilibrium only in some situations: the structure of the cost functions decides this issue.