| An important research issue in the field of macroeconomics is the reason why economic system periodically shrinks and expands. The idea and corresponding approaches to designing a macro-economic model proposed by Kydland and Prescott, who studied the impact of technology shock on economic fluctuation by means of developing the dynamic general equilibrium model, have deeply influenced modern macroeconomics. Moreover, their theory of real business cycle has become the microcosmic base of the theory of modern business cycle, especially for the method of dynamic general equilibrium has been widely used by most of macroeconomists. In that, the theory of real business cycle has been considered as the one of significant and notable progresses on macro-economics.According to the theory, it is not by unexpected monetary policy, but mainly by persistent realistic shocks that economic fluctuation is caused, so technology shocks are the fundamental reason for economic fluctuation. And it is why rational agents maximize their utility via adjusting the support of labor and consumption when confronted with stochastic technology shocks. Thus, economic fluctuation is the result of optimal behavior for producer and consumer to deal with technology shocks in competitive market.The main purpose of this paper is to investigate how rational agentsgain their maximum utilities during the finite cycles, and to verify the capability of the model proposed by this thesis, which explains the economic fluctuation of China, using simulative data obtained from the model. It adopted the representative agent model and assumed that agent would supply the same number of labors in different stages according to the framework of the basic RBC model and to the consideration for the facts of abundant labor resource in China. It also assumed that rational agents, who predicted the emergence of a new-coming technology or upgrading of industrial structure, would positively push the real production after it appears in the light of the hypothesis proposed by Cochrane. It presented sufficient condition of solution existing and the necessary condition of optimal control by using the controllability of discrete systems and the maximum principle.Consider the following optimal problem:(?) (1)where U(·) : Rn→R is a continuously differentiate and increasing concave function, Fk(·) = (Fk1(·),…,Fki(·),…,Fkn(·))T, Fki(·) : Rn→R is a continuously differentiable and increasing concave function,β∈(0,1), and the beginning point x0∈Rn is given.In this paper, we solve this model by finding a sequence get to xN after N steps, starting from x0, to maximize the objective function max(?). For convenience, we transform the initial model into the following model:state equation: Xk+1=(?), (?): Rn×Rn→Rn,δis a, discountrate, 0≤δ≤1;control vector: uk∈U, U is a bounded open set in Rn;boundary condition: (?) = x0 = xN;objective function:(?).Our main results is as follows:DefineSn(0) = {x|x∈Rn, exist a sequence u∈△,s.t.GN(x, u) - x = 0}.then we have the following theorems.Theorem 1 If△is a bounded open set in RN,the function J(x,u)≡(GN(x,u) - x)∈C1 (RN×△,Rn), and x0∈RN \{0} where 0∈Rn \ J{x0,α△) and deg(J(x0, u),△, 0)≠0, then x0 is a interior point of SN(0).Definition 1 Letσ(x) be a real and finite function in X and Zσ= {z|z= (σ(x),x),x∈X}.The real function S(x) is called a support function of Zσin the point (?), if there exists a real numberαwhich satisfies the following conditions:If Sk(xk|yk-1) is a real and finite function in Xk(xk-1), then there exists a real numberαk which satisfies the following conditions: where Sk{xk|yk-1) is called theαk support function of the curved sufaceΠk(yk-1) at the point (?).Theorem 2 (General maximum principle of a discrete system)Let {uk*},{xk*} be the optimal control sequence and its corresponding trajectory of the model we considered, and satisfy the following conditions:(1) U(xk,uk) and (?)(xk,uk) is differentiable at the point xk ;(2) M Pk(xk) is differentiable in a neighborhood of xk* ;(3) xk* is a interior point of set Xk.Then there exist a support function Sk(xk|yk-1*) at the point xk* in the curved surfaceΠk(yk-1*) such that following equations hold:1)2)Theorem 3 Let {uk*} , {xk*} be the optimal control sequence and its corresponding trajectory of model. Under the hypotheses of theorem 2, if mpk(xk|yk-1*) is differentiable in B(xk*,δk), and S{xk|yk-1*) is differentiable in B(xk*,δk), then conclusion 1) and 2) of the theorem 2 hold at the point xk* ofΠk(yk-1*) for any support function S{xk|yk-1*).Definition 2 Let z∈Rn+1. The set X is convex in z direction if and only if there existsγ≥0 such thatλx1 + (1 -λ)x2 -γz∈X holds for all x1,x2∈X (?) Rn+1 and anyλ∈[0,1] If there exist a linear support function at a point (?) of curved surfaceΠk(yk-1) ,then the hamilton function isTheorem 4 Let {uk*},{xk*} be the optimal control sequence and its corresponding trajectory of the model we considered, and satisfy the following conditions:(1) U(xk,uk) and (?)(xk,uk) have derivatives at the point xk;(2)MPk(xk) is differentiable in a neighborhood B(xk*,δk);(3)mpk(xk|yk-1*) is differentiable in B(xk*,δk);(4:)xk* is a interior point of set Xk*(xk*),(?)∈Yk-1,the set Yk(yk-1)is convex in e direction.Then there existφk*∈Rn,k = 0,1,…, N - 1 such that the following equations hold:1)2)...
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