We show that there is a broad range of systems of simultaneous equations that arise in economics as descriptions of equilibrium that can be solved in elementary fashion via degree theory. Some of these systems are not susceptible to analysis by standard Brouwer fixed point methods. Two of our applications are to general equilibrium with incomplete markets, and to nonconvex production with noncompetitive pricing rules.
Observability of an individual’s excess demand function for assets and commodities as all prices and revenue vary suffices in order to recover his von Neumann-Morgenstern utility function. This is generically the case, even when the asset market is incomplete and the cardinal utility indices state dependent, as long as there are at least two commodities traded in spot markets at each state of nature. On the contrary, if the response of individuals’ excess demand for assets as prices in spot commodity markets vary is not observable, recoverability fails when the asset market is incomplete. In particular, it is not possible to contradict the claim that the competitive allocation is fully optimal in spite of the incompleteness of the asset market.
Observability of an individual’s excess demand function for assets and commodities as all prices and revenue vary suffices in order to recover his von Neumann-Morgenstern utility function. This is generically the case, even when the asset market is incomplete and the cardinal utility indices state dependent, as long as there are at least two commodities traded in spot markets at each state of nature. On the contrary, if the response of individuals’ excess demand for assets as prices in spot commodity markets vary is not observable, recoverability fails when the asset market is incomplete. In particular, it is not possible to contradict the claim that the competitive allocation is fully optimal in spite of the incompleteness of the asset market.
We show that there is a broad range of systems of simultaneous equations that arise in economics as descriptions of equilibrium that can be solved in elementary fashion via degree theory. Some of these systems are not susceptible to analysis by standard Brouwer fixed point methods. Two of our applications are to general equilibrium with incomplete markets, and to nonconvex production with noncompetitive pricing rules.
I survey the major results in the theory of general equilibrium with incomplete asset markets. I also introduce the papers in this volume and offer a few suggestions for further work.
We derive the existence of a Walras equilibrium directly from Nash’s theorem on noncooperative games. No price player is involved, nor are generalized games.
I survey the major results in the theory of general equilibrium with incomplete asset markets. I also introduce the papers in this volume and offer a few suggestions for further work.
Keywords: Asset markets, incomplete markets, general equilibrium theory
We derive the existence of a Walras equilibrium directly from Nash’s theorem on noncooperative games. No price player is involved, nor are generalized games.
JEL Classification: 026
Keywords: Walras equilibrium, game theory, Nash model, noncoperative games
Decision theory and game theory are extended to allow for information processing errors. This extended theory is then used to reexamine market speculation and consensus, both when all actions (opinions) are common knowledge and when they may not be. Five axioms of information processing are shown to be especially important to speculation and consensus. They are called nondelusion, knowing that you know, nested, balanced, and positively balanced. We show that it is necessary and sufficient that each agent’s information processing errors be nondeluded and (1) balanced so that the agents cannot agree to disagree, (2) positively balanced so that it cannot be common knowledge that they are speculating, and (3) KTYK and nested so that agents cannot speculate in equilibrium. Each condition is strictly weaker than the next one, and the last is strictly weaker than partition information.
We recast the capital asset pricing model (CAPM) in the broader context of general equilibrium with incomplete markets (GEI). In this setting we give proofs of three properties of CAPM equilibria: they are efficient, asset prices lie on a “security market line,” and all agents hold the same two mutual funds. The first property requires a riskless asset, the latter two do not. We show that across all GEI only one of these three properties of equilibrium is generally valid: asset prices depend on covariances, not variances. We extend CAPM to many consumption goods in such a way that all three properties hold. But now the definition of a riskless asset depends on preferences and endowments, and so cannot be specified a priori.