Computing Economic Equilibria by a Homotopy Method

Economic equilibria are usually solutions to fixed point problems [6], and the fixed point theory is the basis of constructive theorems, which give conditions for existence of economic equilibria and provide algorithms for computing these points. These algorithms, called simplicial algorithms, have a finite convergence property. However, equilibria of the large-scale economic problems may be difficult to compute with these algorithms [12]. Alternative methods are developed, which perform well on high dimensional economic equilibrium problems. The fast development of computer science and powerful computers enable equilibria computation of extremely large-scale and nonlinear economic models. Unfortunately, powerful computers are not sufficient for solving these models. Inefficient algorithms can increase the computation time and even provide inaccurate results. As one can see, reliable and good computation algorithms are important ingredients in solving and analyzing economic models. In literature, there are a lot of proposed algorithms which compute equilibria in general equilibrium models. Algorithms based on homotopy method take an important place among this class of computational methods. Homotopy methods for solving fixed point problems were introduced in [2]. As fixed points play an important role in the theory of economic equilibria, homotopy methods have been intensively used in this field. The homotopy method is a class of techniques for solving systems of nonlinear equations. As opposed to Newton's and Newton-like methods, which are locally convergent, homotopy methods are globally convergent starting from an arbitrary initial point.

The market equilibrium problem consists of finding a set of prices and allocations of goods to economic agents such that each agent maximizes her utility, subject to her budget constraints, and the market clears. The equilibrium equations, which are satisfied under mild assumptions [7], express a static condition characterized by the fact that the market demand for each good equals its market supply. These equations are defined by the excess demand function :

, where D is finite commodity space, and commodity prices are strictly positive. By allowing that the price of certain good is equal to zero, one obtains a more general definition of the equilibrium. A vector of prices ( ) ( )

Problem ( 1) is known in literature as the nonlinear complementarity problem (NCP).

In this article the possibilities of solving NCP by homotopy methods will be investigated. By deriving the robust homotopy algorithm for solving a NCP, one obtains an efficient computational method for computing economic equilibria. The rest of the paper is organized as follows: Section 2 describes homotopy method; Section 3 is devoted to the basic concept of the equilibrium; Section 4 presents numerical results obtained applying the algorithm on well-known examples, while Section 5 contains the summary and conclusion.

Homotopy methods provide a useful approach to find the zeros of smooth mapping :

n n F ℜ → ℜ in a globally convergent way. Such methods have been used to constructively prove the existence of solutions to many economic and engineering problems. The idea is to transform a difficult problem into a simpler one with easily calculated zeros and then gradually deform this simpler problem into the original one computing the zeros of the intervening problems and eventually ending with a zero of the original problem. Deformation of the difficult problem is done by a homotopy function :

 is a smooth map with known zero points. There are various homotopy functions that are generally used, like the convex homotopy such as )

and the Newton homotopy defined by

Once the homotopy function is defined, path following (continuation) methods are applied to track all paths starting at 1 λ = i.e. at the known solutions of ( ,1) 0 H x = and ending at 0 λ = , converging to the solution of the initial equation ( , 0) ( ) 0 H x F x ≡ = . The path obtained by the sequential solving system of the nonlinear equation The solution path can be obtained by successively decreasing the parameter λ by a fixed, small increment and solving 0 ( ) , H x λ = . The drawback of this idea is that it will fail, if turning points of the curve with respect to λ are encountered. The remedy to this problem is to parameterize the curve with respect to the arclength s . In this way the solution path is defined by the parameterized curve ( ) ( ) ( ) ( )

. Concerning homotopy methods, several questions arise:

1. When is it assured that a curve 1 (

)

x lying in the curve does exist and is smooth?

2. If such a curve exists, when is it assured that it will intersect the target homotopy level in a finite length? 3. How can such a curve be traced?

The first question is answered by the implicit function theorem, namely, if the Jacobian ( ,1)

and tangent 0 (0) c ′ ≠ will exist at least lo- cally i.e. on some open interval around zero [1]

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