The Computation Of Equilibria For GEI On A Grassman Manifold

The model of general equilibrium with incomplete asset markets is the extension of Arrow-Debreu model. This model is mainly used to research how the price of financial assets and commodities are decided by the interaction of perfectly competitive asset markets and commodity markets under the circumstance of lack of some financial products, and how these prices influence the consumption and investment behavior of economic participants. The GEI analysis gave us a deep insight into the behavior of market economy. It studies many phenomena that the traditional Arrow-Debreu model of complete markets cannot describe, for example, it has demonstrated a significant difference between nominal assets and real assets; in the general equilibrium model with incomplete assets market, makets are inefficient despite being perfectly competitiye,financial instruments and money significantly influence equilibrium allocations.The existence of the general equilibrium with incomplete assets market had become the focus of study since the creating of the model of GEI by Radner in 1972; Radner just proved the existence of equilibrium when the short sales are permitted and bounded below. But whether the Equilibrium of GEI exist when there is no constrains to short sales of assets remained unsolved in the next over 10 years. The main problem in mathematics is that the Brouwer fixed-point theorem does not work anymore. To prove the existence of Equilibrium of GEI has required the use of new mathematical tools in economics. One leading work was done by Duffie-Shafer in 1985. They created fixed-point theorem of subspaces on a Grassmanian manifold and used Grassmanian manifold to describe subspace of income transfer, which represents the level of income transfer between consumers. They transform the problem of the existence of pseudo-equilibrium, which weaker than Equilibrium of GEI, into general fixed-point problem of nonlinear mapping on Grassmanian manifold and they proved the existence of pseudo-equilibrium. Furthermore , they pointed out that except a set of measure zero of initial endowment and asset structures the pseudo-equilibrium is the same as equilibrium, so they proved the existence of the equilibrium.Following work of Duffie-Shafer(1985), Brown, Demarzo and Eaves( 1996a) discussed the existence of fixed-point of non-linear mapping on Grassmanian manifold using the the famous algorithm about homotopy-path-following and relocalization This method is mainly based on the existence of a route of zeros of a homotopy to find a zero of a function or a fixed point by path-following. In this paper we use this method to discuss equilibrium of GEI problem in spot-real asset market. Because in the model of GEI of real asset the dimension of subspace of income transfer is not a continuous function of the spot prices and this lead to the discontinuous of excess demand function. There's some example in which the equilibrium of GEI does't exist when excess demand function is discontinuous. Therefore, we use Grassmanian manifold to describe the subspace of income transfer and the concept of pseudo-equilibrium to define a continuous excess demand function. Under the common smoothness condition assumptions(namely the utility function of consumers is smooth, strictly increasing and strictly quasi-concave) the excess demand function we defined holds good character of excess demand function in complete market, namely smooth, bounded below and Walrasian.The domain of the. homotopy not only include the price simplex P, but also Grassmanian manifold G. Consequently, we must transform the problem to Euclidean space to do a path-following to a route of zeros W, that is by following finite diffeomorphic pieces W_βin Euclidean space to follow the route W. In factas showed in figureConsider homotopy H = (E,F) : P×G×I→Q×(?)^SJ where P is the price simplex, G is the Grassmanian manifold, I is the parametric unit interval. Q = e^⊥, e = (1,1,…, 1)^?∈(?)^M, T_p is a linear bijection. Weconstruct T_p as Therefore, we not only proved the existence of pseudo-equilibrium of spot-real asset market by the algorithm about homotopy-path-following, but also proved the number of equilibria must be an odd. Futhermore, we gived detailed process to prove the consistency between the equilibrium and the pseudo-equilibrium.To explain how to select location index in relocalization we construct a specific example according to the theory of Eaves (1993) to demonstrate the computation steps and method. We haven't programmed to compute the equilibrium using method discussed in this paper, but we gave a detailed discussion about the steps and focal points by theory and specific method. Furthermore, we gave some figure to explain vividly.The paper is organized as follows. In first chapter, we gave a brief overview to the origin and development of the model of general equilibrium with incomplete asset markets. Besides, we in detail introduced GEI theory and its model description of spot-real asset market. The introduction of algorithm about homotopy-path-following and other some important concepts and theorem are included in this chapter too. In the second chapter, we define excess demand function firstly, then we constructed homotopy H, and obtained the character of H by a atlas of a Grassmanian manifold G and a family of homotopy . Finally, we proved the existence of pseudo-equilibrium. In the third chapter, we proved the consistency between of equilibrium and pseudo-equilibrium and introduced numerical method to compute general equilibrium using the algorithm about homotopy-path-following and relocalization .