Equilibria of a stationary economy with recursive preferences

We consider an intertemporal stationary economy in discrete time, where agents have recursive preferences. Using dynamic programming, we show that equilibrium consumption trajectories from a capital stock are interior Pareto optima and are characterized by a strictly positive parameter in ^n–1, the set of agents' initial weights. We then exhibit prices that support the Pareto optima and use the Negishi method to characterize the parameters corresponding to equilibria. Finally, we prove the existence of equilibria and show that the number of regular equilibria is odd.     

Endogenous Fluctuations in Two-Sector Models: Role of Preferences

We consider a discrete-time two-sector CES (constant elasticity of substitution) economy with sector specific external effects and nonlinear preferences. Our goal is to examine carefully the influence of the utility curvature on the occurrence of multiple equilibria. We show that local indeterminacy depends on an interplay between factor substitutability and the elasticity of intertemporal substitution in consumption. Moreover, considering that, when the external effects are set equal to zero, we get a two-sector optimal growth model, we study also the role of the utility curvature on the occurrence of competitive equilibrium cycles. We show that persistent endogenous fluctuations and macroeconomic volatility require a strong enough elasticity of intertemporal substitution in consumption. We thank two anonymous referees for helpful comments and suggestions. External effects are feedbacks from the other agents in the economy who face also identical maximizing problems. See Benhabib and Farmer (Ref. 1) for a survey. Global indeterminacy based on a finite number of equilibria is associated with the existence of thresholds and multiple steady states. See Deissenberg, Feichtinger, Semmler, and Wirl (Ref. 2).     

Asymptotic behavior in chemical reaction-diffusion systems with boundary equilibria

We consider the asymptotic behavior for large time of solutions to reaction-diffusion systems modeling reversible chemical reactions. We focus on the case where multiple equilibria exist. In this case, due to the existence of so-called "boundary equilibria", the analysis of the asymptotic behavior is not obvious. The solution is understood in a weak sense as a limit of adequate approximate solutions. We prove that this solution converges in L^1 toward an equilibrium as time goes to infinity and that the convergence is exponential if the limit is strictly positive.     

Supercritical Spatially Homogeneous Branching in Admits no Equilibria

At the beginning of investigations in spatially homogeneous branching processes in Euclidean space (Liemant [1]) it seemed to be obvious that the existence of equilibria implies criticality of branching. This prejudice was disproved by the example [2] of a subcritical homogeneous branching equilibrium in dimension one. We prove that supercritical homogeneous branching processes in Euclidean space and, more general, in a broad class of topological groups have no (non – void, homogeneous) equilibria.     

Non-additive anonymous games

This paper introduces a class of non-additive anonymous games where agents are assumed to be uncertain (in the sense of Knight) about opponents’ strategies and about the initial distribution over players’ characteristics in the game. We model uncertainty by non-additive measures or capacities and prove the Cournot–Nash equilibrium existence theorem for this class of games. Equilibrium distribution can be symmetrized under milder conditions than in the case of additive games. In particular, it is not required for the space characteristics to be atomless under capacities. The set-valued map of the Cournot–Nash equilibria is upper-semicontinuous as a function of initial beliefs of the players for non-additive anonymous games.     

Existence of stationary solutions and local minima for 2D models of fine structure dynamics

In this paper we study two-dimensional models for the motion of a viscoelastic material with a non-monotone stress-strain relationship. We prove existence of infinitely many stationary solutions to two model problems. This is achieved by constructing sequences of increasingly oscillatory functions, whose limit is a stationary solution. These equilibria may have arbitrarily small energy. We also prove that it is always possible to construct paths in phase space that strictly decrease the energy. This result negates the existence of local minima for the energy and asymptotically stable equilibria. These results are important first steps towards understanding the dynamics of fine structure in more than one dimension.     

Asymmetric Supply Function Equilibria with Forward Contracts

We consider markets in which firms offer supply functions, rather than a quantity or price alone: the most important examples are wholesale electricity markets. The equilibria in such markets can be hard to characterize. In many cases, whole families of supply function equilibria occur so there are difficulties in determining which equilibrium will be chosen. In this paper, we consider supply function equilibria, when firms hold forward contracts, which is common in electricity markets. Under the assumption that contract positions have been fixed in advance, we characterize the families of supply function equilibria in a duopoly. The existence of forward contracts implies a tightening of the conditions for an equilibrium, and a greater likelihood that no equilibrium solution exists. In the case of three firms, there can be at most one supply function equilibrium, provided that the lowest demand be small enough.     

Financial Networks with Intermediation and Transportation Network Equilibria: A Supernetwork Equivalence and Reinterpretation of the Equilibrium Conditions with Computations

In this paper, we consider two distinct classes of network problems – financial networks with intermediation and with electronic transactions and transportation network equilibrium problems, which have been modeled and studied independently. We then prove that the former problem can be reformulated as the latter problem through an appropriately constructed abstract network i.e., a supernetwork. The established equivalence allows one to then transfer the methodological tools, in particular, algorithms, that have been developed for transportation network equilibria to the financial network domain. In addition, this connection provides us with a novel interpretation of the financial network equilibrium conditions in terms of paths and path flows and a direct existence result. We further show how the theoretical results obtained in this paper can be exploited computationally through several numerical examples.      

Computing equilibria of Cournot oligopoly models with mixed-integer quantities

We consider Cournot oligopoly models in which some variables represent indivisible quantities. These models can be addressed by computing equilibria of Nash equilibrium problems in which the players solve mixed-integer nonlinear problems. In the literature there are no methods to compute equilibria of this type of Nash games. We propose a Jacobi-type method for computing solutions of Nash equilibrium problems with mixed-integer variables. This algorithm is a generalization of a recently proposed method for the solution of discrete so-called “2-groups partitionable” Nash equilibrium problems. We prove that our algorithm converges in a finite number of iterations to approximate equilibria under reasonable conditions. Moreover, we give conditions for the existence of approximate equilibria. Finally, we give numerical results to show the effectiveness of the proposed method.     

Financial market equilibria with heterogeneous agents: CAPM and market segmentation

We consider a single-period financial market model with normally distributed returns and heterogeneous agents. Specifically, some investors are classical expected utility maximizers whereas some others follow cumulative prospect theory. Using well-known functional forms for the preferences, we analytically prove that a Security Market Line Theorem holds. This implies that capital asset pricing model is a necessary (though not sufficient) requirement in equilibria with positive prices. We prove that equilibria may not exist and we give explicit sufficient conditions for an equilibrium to exist. To circumvent the complexity arising from the interaction of heterogeneous agents, we propose a segmented-market equilibrium model where segmentation is endogenously determined.     

A generalized Nash equilibrium problem arising in banking regulation: An existence result with Tarski's theorem

When hit with an adverse shock, banks that do not comply with capital regulation sell risky assets to satisfy their solvency constraint. When financial markets are imperfectly competitive, this naturally gives rise to a GNEP. We consider a new framework with an arbitrary number of banks and assets, and show that Tarski's theorem can be used to prove the existence of a Nash equilibrium when markets are sufficiently competitive. We also prove the existence of ?-Nash equilibria.     

An extension of Krugman’s core–periphery model to the case of a continuous domain: Existence and uniqueness of solutions of a system of nonlinear integral equations in spatial economics

We consider a model that is an extension of Krugman’s core–periphery model to the case of a bounded closed domain included in a Euclidean space. We can describe the relation of the density of workers, the density of nominal wages, and the density of real wages by the system of nonlinear integral equations of the model. If we obtain a solution of the system under the condition that the density of workers is given, then the solution is called a short-run equilibrium. In this paper we prove that this model has a short-run equilibrium, and we obtain a sufficient condition for its uniqueness. Moreover we obtain upper and lower estimates for short-run equilibria, and we construct a useful iteration scheme to numerically obtain short-run equilibria.     

Finitely additive and epsilon Nash equilibria

We prove the existence of a mixed strategy Nash equilibrium in normal form games when the space of mixed strategies consists of finitely additive probability measures. It is then proved that from this result an existence result for epsilon equilibria with countably additive mixed strategies can be obtained. These results are applied to the classic Cournot game.     

Coverage Properties of One-Sided Intervals in the Discrete Case and Application to Matching Priors

We consider asymptotic coverage properties of one-sided posterior confidence intervals for discrete distributions, with a unidimensional parameter of interest and a nuisance parameter of arbitrary dimension. In this case, no higher order asymptotic expansion of the frequentist coverage for these intervals is established, unless some randomization is added. We study here the existence of such frequentist expansions and propose simple continuity corrections based on a uniform random vector. This helps in determining a family of matching priors for one sided intervals in the discrete case.     

Stationary focusing mean-field games

We consider stationary viscous mean-field games (MFG) systems in the case of local, decreasing and unbounded coupling. These systems arise in ergodic MFG theory and describe Nash equilibria of games with a large number of agents aiming at aggregation. We show how the dimension of the state space, the behavior of the coupling, and the Hamiltonian at infinity affect the existence and nonexistence of regular solutions. Our approach relies on the study of Sobolev regularity of the invariant measure and a blow-up procedure that is calibrated on the scaling properties of the system. In very special cases, we observe uniqueness of solutions. Finally, we apply our methods to obtain new existence results for MFG systems with competition, namely, when the coupling is local and increasing.     

Irregular dynamics and homoclinic orbits to Hamiltonian saddle centers

We consider 4-dimensional, real, analytic Hamiltonian systems with a saddle center equilibrium (related to a pair of real and a pair of imaginary eigenvalues) and a homoclinic orbit to it. We find conditions for the existence of transversal homoclinic orbits to periodic orbits of long period in every energy level sufficiently close to the energy level of the saddle center equilibrium. We also consider one-parameter families of reversible, 4-dimensional Hamiltonian systems. We prove that the set of parameter values where the system has homoclinic orbits to a saddle center equilibrium has no isolated points. We also present similar results for systems with heteroclinic orbits to saddle center equilibria. © 1997 John Wiley & Sons, Inc.     

Probabilities of Pure Nash Equilibria in Matrix Games when the Payoff Entries of One Player Are Randomly Selected

The Nash equilibrium in pure strategies represents an important solution concept in nonzero sum matrix games. Existence of Nash equilibria in games with known and with randomly selected payoff entries have been studied extensively. In many real games, however, a player may know his own payoff entries but not the payoff entries of the other player. In this paper, we consider nonzero sum matrix games where the payoff entries of one player are known, but the payoff entries of the other player are assumed to be randomly selected. We are interested in determining the probabilities of existence of pure Nash equilibria in such games. We characterize these probabilities by first determining the finite space of ordinal matrix games that corresponds to the infinite space of matrix games with random entries for only one player. We then partition this space into mutually exclusive spaces that correspond to games with no Nash equilibria and with r Nash equilibria. In order to effectively compute the sizes of these spaces, we introduce the concept of top-rated preferences minimal ordinal games. We then present a theorem which provides a mechanism for computing the number of games in each of these mutually exclusive spaces, which then can be used to determine the probabilities. Finally, we summarize the results by deriving the probabilities of existence of unique, nonunique, and no Nash equilibria, and we present an illustrative example.     

Robust Equilibria in Indefinite Linear-Quadratic Differential Games

Equilibria in dynamic games are formulated often under the assumption that the players have full knowledge of the dynamics to which they are subject. Here, we formulate equilibria in which players are looking for robustness and take model uncertainty explicitly into account in their decisions. Specifically, we consider feedback Nash equilibria in indefinite linear-quadratic differential games on an infinite time horizon. Model uncertainty is represented by a malevolent input which is subject to a cost penalty or to a direct bound. We derive conditions for the existence of robust equilibria in terms of solutions of sets of algebraic Riccati equations.     

Existence, uniqueness and ergodicity of Markov branching processes with immigration and instantaneous resurrection

We consider a modified Markov branching process incorporating with both state-independent immigration and instantaneous resurrection. The existence criterion of the process is firstly considered. We prove that if the sum of the resurrection rates is finite, then there does not exist any process. An existence criterion is then established when the sum of the resurrection rates is infinite. Some equivalent criteria, possessing the advantage of being easily checked, are obtained for the latter case. The uniqueness criterion for such process is also investigated. We prove that although there exist infinitely many of them, there always exists a unique honest process for a given q-matrix. This unique honest process is then constructed. The ergodicity property of this honest process is analysed in detail. We prove that this honest process is always ergodic and the explicit expression for the equilibrium distribution is established.