A Homotopy Algorithm And General Equilibrium Model With Incomplete Markets

In this paper,we mainly illustrate the background of general equilibrium model with incomplete markets in the first chapter.The analysis of the general equilibrium model with incomplete markets(GEI model) has led to fundamental insight into the behavior of market economies.The GEI model studies many phenomena that the traditional Arrow-Debreu model of complete markets cannot describe.The basic GEI Model describe an exchange economy over two time periods(T=0,1) with uncertainty over the state of nature in period 1.At time T=0 the economy is in state s=0 which is known by each of the I agents(i=1,2,3...,I,I<∞) participating in the economy. But it is not know which of the S possible state(s=1,2,3,...,S,S <∞) at time T=1 will occur.In each state s=1,2,3,...,S there are L goods(L <∞). For each of the goods, there exists a spot market in every state. The vector of spot price is denoted bywhere M = L(S + 1) is the total number of goods. Each agent i chooses a consumption bundlewhere x_sⁱ∈R₊^L denote the consumption bundle for state s. Agent i is characterized by an initial endowment of goods,,and her preferences which are represented by a utility function uⁱ : R₊₊^Mâ†'R having standard properties(1)uⁱ is smooth(u_i∈C^∞)(2)Strictly monotone Duⁱ(x)∈R₊₊^M for all the x∈R₊₊^M (3)Satisfies a boundary condition{x∈R₊₊^M: uⁱ(x)≥uⁱ(x)}is closed inx∈R₊₊^M(4)Represents differentiably strictly convex preferences h^T D²u_i(x)h < 0for all h≠0 such that Duⁱ(x)h = 0Agent face a separate budget constraint in every state of nature. In order to transfer income between periods and states of nature,agents have to hold asset. There is a finite number N of real asset traded on financial markets at asset pricesq = (q₁, q₂,..., q_N)∈R^N.The payoff of an asset A_j = (a₁^j,a₂^j, ...,a_S^j)^T∈R₊₊^SL,j = 1,2, ...,N;s = 1,2, ...,S.There are no asset payoffs in state 0. The matrix A = (A₁,A₂,...,A_N)∈R^SL*N whose columns are the asset payoffs is called the asset matrix of the economy. Also define the matrix:Denote by Q_-L the matrix Q with its first L columns deleted. Then the vector denotes the nominal return of asset j in period 1. Define the nominal asset return matrix of economy byA portfolioθ∈R^N of the N assets has a nominal return of R(p)θin period 1.The market price of this portfolio in period 0 isqθ(we need the following notational con-vention:The gradient vectorsDuⁱ(x)and the price vectors p and q are row vectors.All other vectors are column vectors .)Let (u,ω) = ((uⁱ)_i=1,2,...,I, (ωⁱ)i=1,2,...,I)The I agents with their characteristics (u,ω)and the asset matrix A constitute the economy(E) = (E(u,ω), A)We assume S≥N, If Rank(R(p)) = s,then markets are called complete.If Rank(R(p)) < s,markets are called incomplete.Now,the agent cannot transfer the income between periods and states of nature,moreover,it effects the optional utility of the agent.In the second chapter,we carefully introduce penalties methods and homo-topy algorithm, Specially,we elaborate the path-following methods in detail.The core issues of studying of GEI model is the existence of GEI model .The earliest results of the existence is given by Rander in 1972.He prove that the economy have equilibrium when the asset short selling have lower bound ,but there exist no answer to the problem when there is no constrain for the asset short selling. Agent face a utility maximization problem as follow.By the penalties methods,It turn toWith t as homotopy parameter. Building homotopy mapping as followsIn the third chapter,we apply homotopy algorithm to the computation of general equilibrium model with incomplete markets.The crucial idea of our approach is to obtain smooth demand functions by introducing penalties asset sales and gradually lifting this restriction as the algorithm proceeds.The penalties lead to implicit bounds on the optimal asset transactions,eliminating any incentive for agents to inflate their portfolios of assets.Using this algorithm and applying a homotopy in-variance theorem,we proof the Index Theorem for the GEI model.we will see the validity of the algorithm through calculates the example.