Geodesics on the equilibrium manifold

We show the existence of a Riemannian metric on the equilibrium manifold such that a minimal geodesic connecting two (sufficiently close) regular equilibria intersects the set of critical equilibria in a finite number of points. This metric represents a solution to the following problem: given two (sufficiently close) regular equilibria, find the shortest path connecting them which encounters the set of critical equilibria in a finite number of points. Furthermore, this metric can be constructed in such a way to agree, outside an arbitrary small neighborhood of the set of critical equilibria, to any given metric with economic meaning.     

Evolution paths on the equilibrium manifold

In a pure exchange smooth economy with fixed total resources, we define the length between two regular equilibria belonging to the equilibrium manifold as the number of intersection points of the evolution path connecting them with the set of critical equilibria. We show that there exists a minimal path according to this definition of length.     

On the structure of the equilibrium manifold

An elementary proof is given of Y. Balasko's theorem that the equilibrium manifold is diffeomorphic to R^lm. Our proof includes the case (l,m)=(2,2), which the original proof does not cover.     

A note on the structural stability of the equilibrium manifold

In a smooth pure exchange economy with fixed total resources we investigate whether the smooth selection property holds when endowments are redistributed across consumers through a continuous (non-local) redistribution policy. We show that if the policy is regular then there exists a unique continuous path of equilibrium prices which support it.     

Tests on the validity of static equilibrium models

This paper provides a framework to test the validity of static cost minimizing equilibrium assumptions that form the basis for much of the empirical literature on industrial production. The point of departure in our model is to allow the observed technology to be at a short-run equilibrium where firms minimize variable costs while being constrained by the utilization levels of quasi-fixed factors. The long-run equilibrium is then inferred by minimizing total costs with the optimal adjustment of quasi-fixed factor levels.We use results from the optimization problem to form tests with respect to quantities and prices. In the quantity space version, departures between the actual and the optimal long-run levels of quasi-fixed factors are tested for statistical significance. A significant non-zero departure implies the rejection of a static equilibrium specification. In the price space version, the test is cast as a comparison of the market price and the long-run shadow value of a quasi-fixed factor. Although the two versions would give identical results in the non-stochastic case, the rejection powers of these two tests are found to depend on the particular functional form chosen to represent the production process (i.e. cost function).In an application based on aggregate U.S. manufacturing where capital is taken to be quasi-fixed, we were able to reject the static equilibrium specification. These results cast doubt on the validity of a number of previous empirical studies.     

Computing equilibria of GEI by relocalization on a Grassmann manifold

An algorithm is described to compute equilibria of the general economic model with incomplete asset markets, that is, of GEI. The algorithm is based on the existence of a route of zeros of a homotopy whose domain includes the price simplex and a Grassmann Manifold. This route is followed, in effect, by localizing and following diffeomorphic pieces in Euclidean space, and by relocalizing as is necessary.     

A mathematical approach to the boolean minimization problem

Any minimization problem involves a computer algorithm. Many such algorithms have been developed for the boolean minimizations, in diverse areas from computer science to social sciences (with the famous QCA algorithm). For a small number of entries (causal conditions in the QCA) any such algorithm will find a minimal solution, especially with the aid of the modern computers. However, for a large number of conditions a quick and complete solution is not easy to find using an algorithmic approach, due to the extremely large space of possible combinations to search in. In this article I will demonstrate a simple alternative solution, a mathematical method to obtain all possible minimized prime implicants. This method is not only easier to understand than other complex algorithms, but it proves to be a faster method to obtain an exact and complete boolean solution.     

纳什均衡与一般均衡的关系研究

经济均衡构成了现代经济学的基础。本文运用数学方法对纳什均衡与一般均衡的关系进行研究,说明了一般均衡只是纳什均衡多个解中的一个特例。如果一个可自由交换的竞争经济能够实现无限而且重复交换的纳什均衡,那么这个纳什均衡解就是一般均衡解。因此,纳什均衡比一般均衡更具一般性,是现代经济的普适解,一般均衡则是经济系统的最优解。     

Existence of equilibrium actions and of equilibrium : A note on the ‘new’ existence theorems

A lemma by Ky Fan, which asserts the existence of a maximal element relative to some ordering (an equilibrium action) in a compact choice set, is extended to a larger class of orderings. This extension allows one to generalize slightly several existence theorems and seems to be useful for the applications. Kakutani's fixed-point theorem and the existence of an equilibrium in a generalized game are used to exemplify this. In the last case the method of proof may be of some independent interest involving, as it does, the reduction of an n-person game to a corresponding 1-person game.     

Some results on general assumptions about the existence of economic equilibrium

In the present paper we maximally use the possibilities provided by the Nash approach and the Kakutani fixed point theorem for proving the existence of an economic equilibrium. We obtain a general existence theorem which does not require a special form for income distribution functions and producer's objectives, independence of consumers' tastes, ordered preference and zero degree homogeneous price dependence. The role of the non-satiation assumption becomes more clear.     

General competetive equilibrium of the spatial economy

The spatial economy is analyzed in the general equilibrium framework by considering the production and the utility functions depending on spatial distribution and sets. The geometric constraint between the location and the extension of goods supplies the equilibrium condition for space. The existence of an equilibrium is demonstrated by extending the Gale-Nikaido theorem to the case under examination. Consequently, the competetive equilibrium exists, under the assumptions of the theorem, although each point of space is heterogeneous with any other point (because of the different location): the existence is allowed by the perfect partibility of space.     

Utility perturbations and the equilibrium price set

For pure exchange economies in which agents are described by a compact smooth manifold of smooth strictly monotonic and strictly concave utilities, it is shown that, at least generically, the equilibrium price set is a smooth manifold of the same dimension. Given any smooth selection from the equilibrium price manifold and any sufficiently close smooth function, the function is a selection from the equilibrium price correspondence for some manifold of economies close to the original one. In particular, the set of equilibria corresponding to any open neighborhood of an economy contains an open subset of the price simplex.     

Impatience and equilibrium indeterminacy

This paper extends the neoclassical one-sector optimal growth model by postulating that individual agents’ time preference (impatience) depends on the economy-wide average consumption and average income, which are social factors taken as external by individual agents. The paper shows that with the socially determined individual time preference local equilibrium indeterminacy can arise. Moreover, local indeterminacy can also be associated with global indeterminacy in many cases. The results hold in models with bounded and unbounded growth.     

Piecewise linear exchange equilibrium

It is known that if a pure exchange model in which all consumers have linear utility functions has an equilibrium it has one which is rational in the initial data. Examples show on the other hand that this is no longer true in general if utility functions are merely piecewise linear. In this note we observe that for the case of two traders the rationality result carries over and we sketch a finite procedure for calculating the equilibrium.