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.     

Dynamic optimization with a nonsmooth, nonconvex technology: the case of a linear objective function

This paper studies a one-sector optimal growth model with linear utility in which the production function is only required to be increasing and upper semicontinuous. The model also allows for a general form of irreversible investment. We show that every optimal capital path is strictly monotone until it reaches a steady state; further, it either converges to zero, or reaches a positive steady state in finite time and possibly jumps among different steady states afterwards. We establish conditions for extinction (convergence to zero), survival (boundedness away from zero), and the existence of a critical capital stock below which extinction is possible and above which survival is ensured. These conditions generalize those known for the case of S-shaped production functions. We also show that as the discount factor approaches one, optimal paths converge to a small neighborhood of the capital stock that maximizes sustainable consumption.This paper is dedicated to Professor Mukul Majumdar on his 60th birthday. His research with various co-authors in the late 70s and the 80s pioneered innovative techniques for the analysis of nonconvex dynamic optimization models – both deterministic and stochastic. Roy considers himself particularly fortunate for having had the opportunity to learn economic theory and mathematical economics from Professor Majumdar. This paper has benefited from helpful comments and suggestions by an anonymous referee. Financial support from the 21st Century COE Program at GSE and RIEB, Kobe University, is gratefully acknowledged.     

Equilibrium theory with asymmetric information and with infinitely many commodities

The traditional deterministic general equilibrium theory with infinitely many commodities cannot cover economies with private information constraints on the consumption sets. We bring the level of asymmetric information equilibrium theory at par with that of the deterministic one. In particular, we establish results on equilibrium existence for exchange economies with asymmetric (differential) information and with an infinite dimensional commodity space. Our new equilibrium existence theorems include, as a special case, classical results, e.g. Bewley [Existence of equilibria in economies with infinitely many commodities, J. Econ. Theory 4 (1972) 514-540] or Mas-Colell [The price equilibrium existence problem in topological vector lattices, Econometrica 54 (1986) 1039-1053].     

Linear exchange economies with a continuum of agents

The purpose of this paper is to study how the equilibrium prices vary with respect to the initial endowments in a linear exchange economy with a continuum of agents. We first state the model and give conditions of an increasing strength for existence, uniqueness and continuity of equilibrium prices. Then, if we restrict ourselves to economies with essentially bounded initial endowments and if we assume that there is, from the point of view of preferences, only a finite number of types of agents, we show that, on an open dense subset of the space of initial endowments, the equilibrium price vector is an infinitely differentiable function of the initial endowments. The proof of this claim is based on a formula allowing to compute the equilibrium price vector around a so-called “regular” endowment where it is known.     

Nonstandard foundations of equilibrium search models

We develop an equilibrium sequential search model which includes most of the literature as special cases. In particular, the model can accommodate heterogeneity in buyers’ search costs and demand functions and firms’ cost functions, with general demand and cost functions. We identify conditions which ensure existence of equilibrium in pure strategies, utilizing recent progress in the theory of large games by Khan and Sun. These conditions elucidate the essential structure of equilibrium search models. Although we focus on sequential search, our methodology can be used for other classes of equilibrium search models as well.     

Non-existence of monotone equilibria in games with correlated signals

Participation in economic games such as auctions is typically costly, which means that players (potential bidders) must consider whether to participate. Such decisions may be no less crucial than how to bid, and yet the literature has been mostly concerned with bidding, assuming an exogenously given number of bidders; see Krishna [Auction Theory, Academic Press, New York, 2002]. Landsberger and Tsirelson [Correlated signals against monotone equilibria, preprint SSRN 222308, Social Science Research Network Electronic Library, May 2000. Available online: http://papers.ssrn.com/abstract=222308] have shown that fundamental results established in symmetric auction theory with correlated signals, such as the existence of a monotone equilibrium, may not hold if participation decisions are part of equilibrium. The major goal of this paper is to illustrate and explain this result that may be considered counter intuitive given the emphasis placed on monotone equilibria in the auction literature.     

Multidimensional cheap talk with sequential messages

This paper studies a two-dimensional cheap talk game with two senders and one receiver. The senders possess the same information and sequentially send messages about that information. In one-dimensional sequential message cheap talk games where the state space is unbounded, the information is fully transmitted under the self-serving belief, as suggested by Krishna and Morgan (2001b). However, this result depends crucially on the structure of the one-dimensional model. It generally does not hold in two-dimensional models. We consider the extended self-serving belief, which implies full information transmission even if the self-serving belief cannot work. Then, we show that the necessary and sufficient condition for the existence of the fully revealing equilibrium is that the senders have opposing-biased preferences.     

Equilibrium existence and approximation of regular discontinuous games

Many conditions have been introduced to ensure equilibrium existence in games with discontinuous payoff functions. This paper introduces a new condition, called regularity, that is simple and easy to verify. Regularity requires that if there is a sequence of strategies converging to s* such that the players’ payoffs along the sequence converge to the best-reply payoffs at s*, then s* is an equilibrium. We show that regularity is implied both by Reny’s better-reply security and Simon and Zame’s endogenous sharing rule approach. This allows us to explore a link between these two distinct methods. Although regularity implies that the limits of e{\epsilon}-equilibria are equilibria, it is in general too weak for implying equilibrium existence. However, we are able to identify extra conditions that, together with regularity, are sufficient for equilibrium existence. In particular, we show how regularity allows the technique of approximating games both by payoff functions and space of strategies.     

Stability, global dynamics and Markov equilibrium in models of endogenous economic growth

This paper studies the local and global dynamics of two-sector models of endogenous growth with economy-wide external effects and taxes on capital and labor. The local analysis classifies the parameter space depending on the number of stationary solutions and local stability of equilibria. The global analysis shows that if taxes are within certain bounds and the size of the external effects on the average level of human capital is smaller than the share of physical capital, the equilibrium path is monotone and therefore a continuous Markov equilibrium can be constructed.     

On the consistency of stationary Markov equilibria with an exogenous distribution

A technique to show the existence of stationary Markov equilibria is provided by Duffie, Geanakoplos, Mas-Colell and McLennan Econometrica 62 (1994) 745-781. In many applications, a state space is constructed as the product of a space of exogenous variables and a space of endogenous variables. In addition, the exogenous shock is given and is a Markov process with a transition probability and an initial distribution. Then their theorem does not ensure the existence of a stationary Markov equilibrium that is consistent with the exogenous distribution. In this paper, we give a sufficient condition for settling the inconsistency problem.     

Restricted feedback in long term relationships

This paper studies long term relationships, modeled as repeated games, with restricted feedback. Players condition current play on summary statistics of past play rather than the entire history, as may be the case in online markets. Our state strategy equilibrium framework allows for arbitrary restrictions on strategies. We derive a recursive characterization for the set of equilibrium payoffs similar to that of Abreu, Pearce, and Stacchetti (1986, 1990) [2], [3] for perfect public equilibria and show that the set of equilibrium payoffs is the largest fixed point of a monotone operator. We use our characterization to derive necessary and sufficient conditions for efficient trade in a repeated product choice game where costumers condition their purchase decisions only on the last performance signal.     

Strong comparative statics of equilibria

Some results in the monotone comparative statics literature tell us that if a parameter increases, some old equilibria are smaller than some new equilibria. We give a sufficient condition such that at a new parameter value every old equilibrium is smaller than every new equilibrium. We also adapt a standard algorithm to compute a minimal such newer parameter value and apply this algorithm to a game of network externalities. Our results are independent of a theory of equilibrium selection and are valid for games of strategic complementarities.     

Controlling a biological invasion: a non-classical dynamic economic model

This paper analyzes the optimal intertemporal control of a biological invasion. The invasion growth function is non-convex and control costs depend on the invasion size, resulting in a non-classical dynamic optimization problem. We characterize the long run dynamic behavior of an optimally controlled invasion and the corresponding implications for public policy. Both control and the next-period invasion size may be non-monotone functions of the current invasion size; the related optimal time paths may not be monotone or convergent. We provide conditions under which eradication, maintenance control, and no control are optimal policies.      

Star-shapedness of Richter-Aumann integral on a measure space with atoms: theory and economic applications

This paper provides sufficient (as well as necessary) conditions for the integral of a correspondence defined on a measure space with atoms to exhibit star-shaped values. This result is used to analyze the existence of a Nash equilibrium in games with a measure space of agents with atoms and of a competitive equilibrium in economies with mixed markets. In either case it is shown that an exact equilibrium exists whenever atoms are “small enough”.     

Auctions with both common-value and private-value bidders

This paper shows the existence of monotone pure-strategy equilibrium in auctions with both common-value bidders and private-value ones. In equilibrium, the common-value bidders bid less aggressively when there are more private-value bidders. Further, resale is discussed as an application.     

Non-convex aggregate technology and optimal economic growth

This paper examines a model of optimal growth where the aggregation of two separate well behaved and concave production technologies exhibits a basic non-convexity. First, we consider the case of strictly concave utility function: when the discount rate is either low enough or high enough, there will be one steady state toward which the convergence of the optimal path is monotone and asymptotic. When the discount rate is in some intermediate range, we will find sufficient conditions for having either one equilibrium or multiple equilibria steady state. Depending to whether the initial capital per capita is located with respect to a critical value, we show that the optimal paths monotonically converge to one single appropriate equilibrium steady state. Second, we consider the case of linear utility and provide sufficient conditions to have either unique or two steady states when the discount rate is in some intermediate range. In this range, we give conditions under which the above critical value might not exist, and the economy attains one steady state in finite time, then stays at the other steady state afterward. P. Michel passed away when this research was completed. This paper is dedicated to his memory as a friend and colleague. N. M. Hung and C. Le Van thank the referee for vey helpful remarks and criticisms. They are grateful to Takashi Kamihigashi for very fruitful discussions. They also thank J.-F. Leclerc for editing the final version of this paper.     

On the importance of public choice in migration models

"Recent general equilibrium Tiebout models that use single-crossing indifference curves to establish a migration equilibrium incorporate both voting and migration decisions. These models have concentrated on the important question of the existence of an equilibrium, but the incorporation of both voting and migration has important efficiency consequences as well. This paper demonstrates the important effect of the interaction of voting and migration on efficiency conditions by showing that head taxes (which are generally considered to provide efficient migration incentives with constant marginal congestion costs) do not provide incentives for efficient migration decisions unless the public choice mechanism results in optimal public service provision within jurisdictions."     

Equilibrium among local jurisdictions: toward an integrated treatment of voting and residential choice

In models of local public goods with mobile consumers, existence of equilibrium is problematic. Difficulties with existence of equilibrium that arise in models with discrete locations and in models with voting are compounded when both features are introduced into the same model. We present conditions under which equilibrium exists in a model where freely mobile households choose community of residence and amount of housing consumption, and vote on the level of public goods provision. These conditions involve restrictions on preferences and the technology of public goods supply. At least some of these conditions appear consistent with empirical observations. We discuss the implications of the conditions, and their role in assuring existence of equilibrium. A series of computational examples provide illustrations of the way these conditions interact, and the difficulties that must be confronted if they are to be relaxed.     

Nash equilibrium and party polarization in an electoral model with mixed motivations

We study the existence problem of Nash equilibrium as well as the patterns of equilibrium policy outcomes in an electoral competition model with mixed motivations. Each party maximizes a sum of party members’ expected utility and office rent. The inclusion of office rent renders the payoff of each party discontinuous. This makes it difficult to apply usually fixed point arguments to prove the existence of Nash equilibria. By using a recently developed concept, multiple restrictional security (MR‐security) we provide conditions under which a pure‐strategy Nash equilibrium exists within fairly general settings, and further the analysis by presenting conditions under which various patterns of policy choices, including polarization, arise in equilibrium.     

Existence of a Condorcet Winner When Voters Have Other‐Regarding Preferences

In standard political economy models, voters are “self‐interested” that is, care only about “own” utility. However, the emerging evidence indicates that voters often have “other‐regarding preferences” (ORP), that is, in deciding among alternative policies voters care about their payoffs relative to others. We extend a widely used general equilibrium framework in political economy to allow for voters with ORP, as in Fehr and Schmidt (1999) . In line with the evidence, these preferences allow voters to exhibit “envy” and “altruism,” in addition to the standard concern for “own utility.” We give sufficient conditions for the existence of a Condorcet winner when voters have ORP. This could open the way for an incorporation of ORP in a variety of political economy models. Furthermore, as a corollary, we give more general conditions for the existence of a Condorcet winner when voters have purely selfish preferences.