Advances in Mathematical Economics

Four instances of fundamental theorems of economics established by means of central results in various branches of mathematical analysis are listed.

We show in this paper how, in a model of assets exchange in complete competitive markets, heterogeneity of the agents’ subjective probabilities generates aggregate expenditures for Arrow-Debreu securities that have the gross substitutability property, with the consequences that competitive equilibrium is unique, stable in any tatônnement process, and that the weak axiom of revealed preferences is satisfied in the aggregate. For this result, heterogeneity is required to be highest among people who have the largest risk aversion.

We present new modes of convergences for bounded sequences in the space L _X ¹ (μ) of Bochner integrable functions over a complete probability space (Ω, F, μ) with values in Banach space X via the convergence of its truncated subsequences as well as we give several characterizations of weak compactness and conditionally weak compactness in L _X ¹ (μ). New results involving subsets in L _X ¹ (μ) which are closed in measure are obtained and also the characterizations of the Banach space X in terms of these modes of convergence.

Assuming symmetry across firms and constant unit costs Perloff and Salop (1985) show: If product differentiation increases, prices rise in a symmetric equilibrium. This raises the question of whether, in general, more product differentiation leads to higher market prices. Giving up the symmetry and the constant unit costs assumptions we present examples in which at least one firm lowers its equilibrium price when product differentiation increases. We formulate a model of product differentiation and state and discuss, within the theory of supermodular games, conditions ensuring that all firms raise their prices in a Nash equilibrium if product differentiation increases.

We study some mathematical models on default risk. First, we study a “standard model” which is an abstract setting widely used in parctice. Then we study how the hazard rates changes, if we change a basic probability measure. We show that the usual assumptions on hazard rates hold in a standard model, but do not hold in general if we change a basic measure. Finally we study a filtering model.

We study the rational evaluation of yield spread for defaultable credit with fixed maturity. The default occurs when the asset value hits a given fraction of the nominal credit value. The yield spread is continuously accumulated to the initial credit as an insurance fee for future default. By the rational credit pricing, we prove the unique existence of equilibrium yield spread which satisfies the arbitrage free property. Furthermore we show that this spread yield is independent of the choice of interest rate process. For the quantitative study of rational yield spread, we derive an explicit analytic formula for the equilibrium and show numerical example for various parameters.

Young measures theory is applied to the understanding of weak convergence without strong convergence in L ² spaces. The two-scale Young measures permit also to analyse, when it happens, a “modulated periodical” behavior and, in the general case, to get a kind of orthogonal decomposition. Some new examples are discussed.

In Nishimura and Yano (1996), we demonstrate that chaos may emerge as a solution to a dynamic linear programming (LP) problem. That result is closely related to a result of Nishimura and Yano (1995), which establishes the possibility of chaotic optimal accumulation in a two-sector model of optimal capital accumulation. This study intends to survey those results and explain the basic relationship between them.

This paper examines the determinacy of equilibria in an exchange economy with money and a nominal bond where the only source of uncertainty comes from fluctuations in the money supply. Money plays the role of medium of exchange and, through a cash-in-advance constraint, affects the real allocation. We show that the monetary economy can be transformed into a standard Arrow-Debreu economy, and these two economies have the same equilibrium allocations. Applying the theorem on the finiteness of equilibria by Debreu [3], we prove that the set of monetary equilibria is locally unique, generically for every level of money supply.