Time and Uncertainty in Overlapping Generations Economies

This paper shows the general reversibility of every perfect foresight equilibrium of an overlapping generations economy. It then shows and characterizes the existence of reversible sunspot equilibria in these economies as well, which seems to be at odds with our intuition about the irreversibility of a tree of events. Although the paper establishes also that such reversible stochastic equilibria constitute a negligible subset of all the equilibria of their class, their mere existence may be considered somewhat puzzling for this intuition. Journal of Economic Literature Classification Numbers: D50, D80, D90.     

Determinacy and Stability under Learning of Rational Expectations Equilibria

This paper studies relationships between the local determinacy of a stationary equilibrium in the perfect foresight dynamics, and its local stability in dynamics arising from econometric learning procedures. Attention is focused on linear scalar economies where agents forecast only one period ahead, and with an arbitrary, but fixed, number of predetermined variables. In such a framework, it is well known that there are no clear links between the determinacy of the stationary state in the perfect foresight dynamics on the levels of the state variable, and its stability under learning. The paper emphasizes, however, that this is not the right perfect foresight dynamics to look at whenever agents try to learn the coefficients of the perfect foresight dynamics restricted to an eigenspace of lower dimension. Indeed the paper introduces a growth rate perfect foresight dynamics on these coefficients and proves equivalence between determinacy in that dynamics and stability under learning provided that a simple sign condition is satisfied. Journal of Economic Literature Classification Numbers: E32, D83.     

Learning to Play Limited Forecast Equilibria

This paper provides a learning justification for limited forecast equilibria, i.e., strategy profiles such that (1) players choose their actions in order to maximize the discounted average payoff over their horizon of foresight as given by their forecasts and (2) forecasts are correct on and off the equilibrium path. The limited forecast equilibria appear to be the stochastically stable outcomes of a simple learning process involving (vanishing) trembles.Journal of Economic LiteratureClassification Numbers: C72, D83.     

A new model of equilibrium involuntary unemployment

Summary. We show that equilibrium involuntary unemployment emerges in a multi-stage game model where all market power resides with firms, on both the labour and the output market. Firms decide wages, employment, output and prices, and under constant returns there exists a continuum of subgame perfect Nash equilibria involving unemployment and positive profits. A firm does not undercut the equilibrium wage since then high wage firms would attract its workers, thus forcing the undercutting firm out of both markets. Full employment equilibria are payoff dominated by unemployment equilibria, and the arguments are robust to decreasing returns.Received: 21 May 2001, Revised: 15 April 2003, JEL Classification Numbers: D43, E24.Correspondence to: Leo KaasWe thank an anonymous referee, Woojin Lee, Klaus Ritzberger and seminar participants in Konstanz, Manchester, Milan, Prague, Vienna, and Warwick for helpful comments. Financial support from the Economic and Social Research Council (UK) under grant L138251030 and from the Manchester School Visiting Fellowship Scheme is gratefully acknowledged.     

Indeterminacy of intertemporal equilibria under imperfect competition

Summary. It is known that overlapping generations models with imperfectly competitive firms may exhibit a continuum of stationary equilibria. The reason of this indeterminacy is that different price expectation functions of consumers lead to different objective demand functions against which firms maximize. All these expectation functions fulfill perfect foresight in the equilibrium, but they can be arbitrary off the equilibrium. In this paper it is shown that it is not this arbitrariness which is responsible for the indeterminacy, but that the continuum of stationary equilibria emerges even if expectation functions are rational. Received: March 25, 1999; revised version: February 16, 2000     

On the structure of Ramsey equilibrium: Cycles,indeterminacy, and sunspots

Summary We consider a one-sector neoclassical capital accumulation model under borrowing constraints with infinitely-lived heterogeneous households. Under the standard assumptions of strictly concave and time-additive utility functionals and a strictly concave production function we show that perfect foresight equilibria can be non-unique, even locally non-unique (indeterminate), and periodic of arbitrary long periodp. Moreover, we prove that there can exist non-trivial rational expectations (sunspot) equilibria when the agent's expectations about future factor prices depend on extrinsic uncertainty.Remarks made by an anonymous referee were extremely helpful in preparing the final draft of this paper.     

Ramsey Equilibrium in a Two-Sector Model with Heterogeneous Households

We prove an existence theorem for a stationary perfect foresight equilibrium under borrowing constraints in a two-sector model with infinitely lived heterogeneous agents. The most patient agent holds all the capital in this solution. We also show that if the capital goods sector is capital intensive and capital income is increasing in the aggregate capital stock, then the aggregate capital stock eventually is monotonic and converges to the steady state stock. If the consumption goods sector is more capital intensive and capital income is increasing in aggregate capital we prove convergence to the steady state under more restrictive conditions. Periodic equilibria are shown to exist under weaker hypotheses. Journal of Economic Literature Classification Numbers: D52, D90, E13.     

Stability of cycles and expectations

The actual dynamics of an economy depends on how agents forecast the future at every date as a function of their information on the past, while possibly learning the structure of their environment. We show in the case of a one-dimensional state variable that under mild conditions on expectations functions, a given cycle with perfect foresight that is stable in the actual dynamics is stable in a fictitious backward perfect foresight dynamics. We exhibit a restricted class of expectations functions for which the converse is true.     

Rationalizable foresight dynamics

This paper proposes and studies the rationalizable foresight dynamics. A normal form game is repeatedly played in a random matching fashion by a continuum of agents who make decisions at stochastic points in time. A rationalizable foresight path is a feasible path of action distribution along which each agent takes an action that maximizes his expected discounted payoff against another path which is in turn a rationalizable foresight path. We consider a set-valued stability concept under this dynamics and compare it with the corresponding concept under the perfect foresight dynamics.     

How to Select a Dual Nash Equilibrium

The paper analyzes situations, generalizing the duopoly problem, where two identical players are allowed with two control variables each, all of them linked through two non-strategic private constraints. Four dual equilibria are then obtained when each agent selects one leading variable to optimize and adjusts the other, and these equilibria are compared in a meta-game. For a simplified class of continuous games with linear constraints, it is shown that one symmetric dual equilibrium dominates the others and is the only perfect equilibrium of the metagame. The latter result holds locally for all quasi-concave utility functions and globally for all homogeneous ones, always keeping linear constraints. However, it is no longer valid in discrete games where the implicit constraints are not linear. Journal of Economic Literature Classification Numbers: C72, D43.     

Limited Foresight May Force Cooperation

This paper considers discounted repeated games with boundedly rational players. In each period, player i chooses his current action on the basis of his forecast about the forthcoming n _i action profiles; his assessment of the payoffs he will obtain next depends on his state of mind, which is non-deterministic. A limited forecast equilibrium is such that after every history the limited horizon forecasts formed by the players are correct. The set of all limited forecast equilibria is characterized and constructed. Application to the repeated prisoner's dilemma shows that limited foresight may sometimes induce purely cooperative paths while purely non-cooperative paths cannot arise.     

Growth in stages

Existing North–South growth models generally ignore the possibility that the South becomes an innovating high-wage country. The present paper presents an analytically tractable North–South growth model in which the North innovates all the time, while the South is at first engaged in imitation and potentially starts to innovate too, later on. Three interesting results emerge from the analysis. First, a perfect foresight growth equilibrium may fail to exist. Second, there may be global indeterminacy in that both convergence to the steady state of the regime with imitation in the South and switching to the regime with innovation in the South represent perfect foresight equilibria. Third, technology policies in the South may have hysteresis effects: a temporary policy may lead the South permanently from imitation-driven to innovation-driven growth.     

Perfect foresight models and the dynamic instability problem from a higher viewpoint

Dynamic macroeconomic models incorporating perfect foresight expectations can display a dynamic instability of the saddle point type. So that unless the initial values happen to place the system on the stable arm of the saddle point, the economic variables will diverge ever more from the equilibrium. We consider the dynamic instability problem in a simple model of monetary dynamics which is non-linear and assumes adaptive expectations which are characterized by an expectations time lag. This model is shown to have a stable limit cycle. By considering perfect foresight as the limit as the expectations time lag tends to zero we are able to view the perfect foresight model from a dimension higher than that from which is it is normally viewed. We are thus able to see that the stable limit cycle continues to exist for the perfect foresight model as well. In this framework there is no longer a dynamic instability problem since whatever the intial values time paths are tending to the stable limit cycle.     

Iterated potential and robustness of equilibria

For any given set-valued solution concept, it is possible to consider iterative elimination of actions outside the solution set. This paper applies such a procedure to define the concept of iterated monotone potential maximizer (iterated MP-maximizer). It is shown that under some monotonicity conditions, an iterated MP-maximizer is robust to incomplete information [A. Kajii, S. Morris, The robustness of equilibria to incomplete information, Econometrica 65 (1997) 1283-1309] and absorbing and globally accessible under perfect foresight dynamics for a small friction [A. Matsui, K. Matsuyama, An approach to equilibrium selection, J. Econ. Theory 65 (1995) 415-434]. Several simple sufficient conditions under which a game has an iterated MP-maximizer are also provided.     

Union games: technological unemployment

Summary. Given a production economy, we define union games by considering strategic behavior of the suppliers of factors. We refer to the Nash equilibria of this game as union equilibria. We analyze situations where the unemployment of factors is supported as a union equilibrium. The degree of unemployment depends on technological conditions. This allows us to model a source of unemployment which differs from the usual sources provided in the literature. We state a limit result that demonstrates that, as the market power of unions decreases, the corresponding sequence of union equilibria converges to the Walrasian equilibrium, that is, to full employment of factors. We also provide some examples that illustrate the main results.Received: 21 October 2004, Revised: 14 December 2004, JEL Classification Numbers: D51, C72. Correspondence to: Emma Moreno-GarcíaE. Moreno acknowledges financial support from the Research Grant BEC2000-1388-C04-01 (Ministerio de Ciencia y Tecnología and FEDER). G. Fernández de Córdoba and E. Moreno acknowledge financial support from the Research Grant SA091/02 from Junta de Castilla y León. We are indebted to C. Alós-Ferrer, C. Pita, D. Anisi, J. A. Ortega, F. Jimeno, J. P. Torres-Martínez, M. Steinert and C. Hervés for helpful comments and insights. We are particularly grateful to T. Kehoe and an anonymous referee for suggestions that improved this paper.     

p-Dominance and Equilibrium Selection under Perfect Foresight Dynamics

This paper studies equilibrium selection based on a class of perfect foresight dynamics and relates it to the notion of p-dominance. A continuum of rational players is repeatedly and randomly matched to play a symmetric n×n game. There are frictions: opportunities to revise actions follow independent Poisson processes. The dynamics has stationary states, each of which corresponds to a Nash equilibrium of the static game. A strict Nash equilibrium is linearly stable under the perfect foresight dynamics if, independent of the current action distribution, there exists a consistent belief that any player necessarily plays the Nash equilibrium action at every revision opportunity. It is shown that a strict Nash equilibrium is linearly stable under the perfect foresight dynamics with a small degree of friction if and only if it is the p-dominant equilibrium with p<1/2. It is also shown that if a strict Nash equilibrium is the p-dominant equilibrium with p<1/2, then it is uniquely absorbing (and globally accessible) for a small friction (but not vice versa). Set-valued stability concepts are introduced and their existence is shown. Journal of Economic Literature Classification Numbers: C72, C73.     

Endogenous Debt Constraints in Lifecycle Economies

We characterize competitive equilibria with perfect foresight in a deterministic, three-period pure-exchange overlapping generations economy with perfect information and no commitment to loan contracts. Commitment is replaced by an enforcement mechanism that excludes defaulters from asset markets for one period. For hump-shaped endowment profiles, young individuals face endogenous debt constraints that ration current consumption. Changes in current and future yields affect these constraints, inducing an additional income effect on rationed household demand that makes current and future consumption complements. This mechanism can lead to multiple steady states, persistent indeterminacy and regime switching. We show that sensitivity to shocks and complex dynamic behaviour are consistent with endogenous debt limits but not with exogenous liquidity constraints.     

Capital income taxation and perfect foresight

This paper develops a perfect foresight model of capital income taxation in a representative household neoclassical one-sector framework. The household sector is based on intertemporal utility maximization. The main result is the equivalence of a perfect foresight solution and an artificial central planning solution; if there is no tax, this is the equivalence of a market equilibrium and social optinum. The equivalence result is used to deduce the implications of a partial tax reform policy. It is shown that a partial reform increases capital at each time following the initial period of the program.     

Meaning and Credibility in Cheap-Talk Games

I define neologism-proofness, a refinement of perfect Bayesian equilibrium in cheap-talk games. It applies when players have a preexisting common language, so that an unexpected message′s literal meaning is clear, and only credibility restricts communication. I show that certain implausible equilibria are not neologism-proof; in some games, no equilibrium is. Journal of Economic Literature classification numbers: D83 D82 C73.     

Commitment Robust Equilibria and Endogenous Timing

This paper investigates which equilibria of a game are still viable when players have the opportunity to commit themselves. To that end we study a model of endogenous timing in which players face the trade-off between committing early and moving late. It is shown that mixed (resp. pure) equilibria of the original game are subgame perfect (resp. persistent) in the timing game only when no player has an incentive to move first. Consequently, mixed equilibria are viable only if no player has an incentive to move first. One needs strong evolutionary solution concepts to draw that conclusion for pure equilibria.Journal of Economic LiteratureClassification Number: C72.