Markov equilibria in dynamic matching and bargaining games

Rubinstein and Wolinsky [Rev. Econ. Stud. 57 (1990) 63] show that a simple homogeneous market with exogenous matching has a continuum of (non-competitive) perfect equilibria; however, the unique Markov-perfect equilibrium of this model is competitive. By contrast, in the more general case of heterogeneous markets, even the Markov property is not enough to guarantee the perfectly competitive outcome. We define a market game that allows for heterogeneous values on both sides of the market and exhibit a number of examples of (non-competitive) Markov-perfect equilibria, with and without discounting. Unlike the homogeneous case, these equilibria allow for inefficient trades and for trade at non-uniform prices. The non-competitive equilibrium may be unique.     

Markov-perfect Nash equilibria in a class of resource games

Summary. A general model of non-cooperating agents exploiting a renewable resource is considered. Assuming that the resource is sufficiently productive we prove that there exists a continuum of Markov-perfect Nash equilibria (MPNE). Although these equilibria lead to over-exploitation one can approximate the efficient solution by MPNE both in the state space and the payoff space. Furthermore, we derive a necessary and sufficient condition for maximal exploitation of the resource to qualify as a MPNE. This condition is satisfied if there are sufficiently many players, or if the players are sufficiently impatient, or if the capacity of each player is sufficiently high.Received: November 1, 1996This revised version was published online in February 2005 with corrections to the cover date.     

Global Newton Method for stochastic games

The Global Newton Method for games in normal form and in extensive form is shown to have a natural extension to computing Markov-perfect equilibria of stochastic games.     

A second chance at success: A political economy perspective

This paper characterizes a stationary Markov-perfect political equilibrium where agents vote over income taxation that distorts educational investment. Agents become rich or poor through educational investment, and the poor have a second chance at success. The results show the following concerning the cost of a second chance. First, when the cost is low, the economy is characterized by high levels of upward mobility and inequality, and a low tax burden supported by the poor with prospects for upward mobility. Second, when the cost is high, there are multiple equilibria with various patterns of upward mobility, inequality and redistribution. Numerical examples show that the shift from a high-cost economy to a low-cost economy may reduce social welfare.     

Uniqueness of stationary equilibria in a one-dimensional model of bargaining

We prove uniqueness of stationary equilibria in a one-dimensional model of bargaining with quadratic utilities, for an arbitrary common discount factor. For general concave utilities, we prove existence and uniqueness of a “minimal” stationary equilibrium and of a “maximal” stationary equilibrium. We provide an example of multiple stationary equilibria with concave (nonquadratic) utilities.     

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.     

Evolutionary Drift and Equilibrium Selection

This paper develops an approach to equilibrium selection in game theory based on studying the learning process through which equilibrium is achieved. The differential equations derived from models of interactive learning typically have stationary states that are not isolated. Instead, Nash equilibria that specify different out-of-equilibrium behaviour appear in connected components of stationary states. The stability properties of these components can depend critically on the perturbations to which the system is subjected. We argue that it is then important to incorporate such drift into the model. A sufficient condition is provided for drift to create stationary states, with strong stability properties, near a component of equilibria. Applications to questions of forward and backward induction are developed.     

Stationary equilibria in stochastic games: structure, selection, and computation

This paper introduces an algorithm to compute stationary equilibria in stochastic games that is guaranteed to converge for almost all such games. Since in general the number of stationary equilibria is overwhelming, we pay attention to the issue of equilibrium selection. We do this by extending the linear tracing procedure to the class of stochastic games, called the stochastic tracing procedure. As a by-product of our results, we extend a recent result on the generic finiteness of stationary equilibria in stochastic games to oddness of equilibria.     

Global sunspots in OLG models: Theory and computational analysis

This paper discusses sunspots equilibria in a context that is general in the sense that: (i) the evolution of the system takes place in a general state space (i.e., a space which is not necessarily finite or even countable); and (ii) the orbits of the unperturbed, deterministic component of the system converge to subsets of the state space which can be more complicated than a stationary state or a periodic orbit, i.e., they can be aperiodic or chaotic. This problem is represented mathematically as a system of stochastic difference equations the invariant probability distributions of which correspond to stationary sunspots equilibria. The conditions for stochastic stability are recalled and the theoretical results are applied to a model of overlapping generations with individuals living three periods. A computational analysis of this model is provided, covering the basic different cases suggested by the theory.     

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     

The robustness of optimal equilibrium among overlapping generations

Summary. We combine and strengthen optimality and robustness theorems for the overlapping-generations model of money. Roughly, we find a Pareto-optimal monetary equilibrium of a generic stationary economy that is near an optimal monetary equilibrium of each nearby non-stationary economy. Since the nearby equilibria are monetary, the general problem of macroeconomic stabilization reduces to maintaining the money supply. And since the nearby equilibria are optimal, stabilization is socially desirable. Received: October 27, 1997; revised version: March 25, 1998     

On the Multi-Country Version of the Solow–Swan Model

I study a multi-country version of the Solow–Swan model of capital accumulation. Capital is perfectly mobile and flows instantaneously to countries providing the highest return. I show that, in general, the model possesses infinitely many stationary equilibria that differ from each other in terms of the world interest rate and world output. Analysing the dynamics of the model, I find that multiple equilibria exist for any given initial allocation of capital. Finally, I discuss a generalization of the golden rule to the multi-country version of the Solow–Swan model and show that it satisfies a Nash equilibrium property.
JEL Classification Numbers: F20, O41     

A Spatial-Temporal Model of Human Capital Accumulation

We develop an overlapping generations model of human capital accumulation and analyze its spatial and temporal properties. The interplay between local spillovers of human capital and global market participation partitions the society into socio-economic classes. We focus upon stationary equilibria and show that a large class of them are locally stable. Stationary equilibria can be homogeneous (where the human capital distribution is degenerate) or stratified (with a distinct spatial pattern of human capital distribution). We provide several examples and specify conditions under which equilibria are sensitive to the spatial structure of the society. We illustrate our results by numerical simulations. Journal of Economic Literature Classification Numbers: D90, I21, J24, O15.     

Manifolds of golden rule and balanced steady state equilibria

Summary. We show that the set of balanced steady state (resp. golden rule) equilibria, parameterized by endowments, of stationary overlapping-generations economies are smooth manifolds diffeomorphic to Euclidean spaces. These properties extend similar properties of the Walrasian equilibria and enable one to apply the natural projection approach to the study of these equilibria. Received: October 30, 1995; revised version: October 10, 1996     

Non-parametric counterfactual analysis in dynamic general equilibrium

In this paper, we examine non-parametric restrictions on counterfactual analysis in a dynamic stochastic general equilibrium model. Under the assumption of time-separable expected utility and complete markets all equilibria in this model are stationary. The Arrow-Debreu prices uniquely reveal the probabilities and discount factor. The equilibrium correspondence, defined as the map from endowments to stationary (probability-free) state prices, is identical to the equilibrium correspondence in a standard Arrow-Debreu exchange economy with additively separable utility. We examine possible restriction on this correspondence and give necessary as well as sufficient conditions on profiles of individual endowments that ensure that associated equilibrium prices cannot be arbitrary. Although restrictions on possible price changes often exist, we show that results from a representative-agent economy usually do not carry over to a setting with heterogeneous agents.     

Quota-Induced Cycles

We present a new framework to compare the dynamic effect of tariffs and quotas in the presence of oligopoly. Suppose that the domestic and the foreign firm play a quantity-setting game over time in a perfectly stationary economy. A Markov-perfect equilibrium has the foreign firm exporting at the constant rate under a tariff. In contrast, under the quota the rate of exports changes monotonically over the course of each year, causing seasonal fluctuations in domestic production. Quota-induced cycles can make dynamic market segmentation possible and raise profits for both the firms above what they earn under the equal-import tariff.     

Multiple equilibria and hysteresis in simple exchange models

This paper presents a simple two-good general equilibrium model of exchange in which demands are linear functions of relative prices. The solution of the model is represented by a cubic equation which can have either a single root which corresponds to a unique equilibrium price or three roots corresponding to multiple equilibria. The major properties of the model are that the relative price can make large discrete jumps in response to small parameter changes, and can display hysteresis. The paper shows how recent developments in non-linear methods can be used to examine the equilibrium properties of the model and establish conditions under which multiple equilibria can arise. The model is compared with that used by Shapley and Shubik.     

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.     

Time-Consistent Public Policy

In this paper we study how a benevolent government that cannot commit to future policy should trade off the costs and benefits of public expenditure. We characterize and solve for Markov-perfect equilibria of the dynamic game between successive governments. The characterization consists of an inter-temporal first-order condition (a "generalized Euler equation") for the government, and we use it both to gain insight into the nature of the equilibrium and as a basis for computations. For a calibrated economy, we find that when the only tax base available to the government is capital income—an inelastic source of funds at any point in time—the government still refrains from taxing at confiscatory rates. We also find that when the only tax base is labour income the Markov equilibrium features less public expenditure and lower tax rates than the Ramsey equilibrium.     

On the stability of non-Walrasian equilibria

We present a dynamic disequilibrium macro model which permits non-Walrasian or Keynesian equilibria as stationary states. Unlike the stability results obtained by Varian (1977), and Eckalbar (1979,1980,1981) in a series of papers in the same model, we show that such equilibria have the saddle point property when the real wage rate, the real money balances and expected output demand, all change in disequilibrium.