Finding The Optimal Investment And Consumption Solutions For A Kind Of Discrete Time Economic Growth Model

Economist Ramsey introduced the dynamic optimal method to the economic growth model, thus he made possible the dynamic analysis and research of the economy. For example, the neo-classical growth theorem represented by Solow in 1950's, the new growth theorem represented by Romer and Lucas in 1980's, linear endogenous growth theorem represented by Rebelo and Jones as well as the other present economic growth theorems are very important economic results developed by using the dynamic optimal method as a strong tool of analysis. Thus the dynamic optimal method has become indispensible for the developing of the economics science.Cass-Koopmans developed a new economic model called the Ramsey-Cass-Koopmans Model by introducing an endogenous saving rate variable in the Ramsey model, and Jones developed the Ramsey-Cass-Koopmans Model as a linear technology endogenous growth model. On the basis of the linear technology endogenous growth model, we discuss the optimal investment and consumption per capita, we are introducing a new production function that takes the investment per capita as the only variable of this function. We take into account not only the present investment per capita but also the preceding one period investment per captia as the variables of the new production function. In order to solve the new model, first we develop the dynamic analysis method for the infinite discrete time called value method, that means we can find the solutions of this infinite discontinous time optimal problem by finding the fixed point of the continuous function's operator. We also analyse the monotonous , convex, differetial characteristics of the value function under some assumptions, and prove the existence of the optimal solutions, we discuss the dynamic characteristics of the optimal solutions and the steady characteristics of the equilibrium point. Furthermore we get the optimal investment and consumption solutions of this new model by using the value method , When comparing the optimal solutions with the real economic data from 1995-2004, we get better results : the optimal solution of model takes both the preceding one period and present investment per captia as the variable of the production function giving closer results to the the real data, meanwhile we justify the value method as practical. Considering the following optimal problems.t.x_t+2∈Γ(x_t,x_t+1),Where F: R×R×R(?)X×X×X→R is continuous function,Γ: R×R(?)X×X→R is unempty and compact,β∈(0,1). The beginning point(x₀,x₁)∈X×X is given.Solving (4.20) is to find a sequence {x_t:x_t+2∈Γ(x_t, x_t+1} whose beginnigpoint is (x₀,x₁) in order to make the objective function reach a maximum.In this paper we have following assumptions Assumption 4.11 Feasible setΠ[(x₀,x₁) is unempty . Assumption 4.12 For and lim exists.(Including infinite case).Definition 4.9 Value function beginning with point (x₁,x₁) iss.t.x_t+2∈Γ(x_t,X_t+1).We get the following function according to the above value function :Theorem 4.18 The solution of (4.20)Ï…^* meets (4.21) , that means thesolutions of (4.20) is the solution of (4.21).Theorem 4.19 If v is the solution of (4.21) , and for then we haveÏ…=Ï…^*.Theorem 4.20 If is the solution of (4.20), then (?) meets thefollowing optimal conditionÏ…(x_t^*,x_t+1^*) = F(x_t^*,x_t+1^*,x_t+2^*)+βυ^*(x_t+1^*,x_t+2^*),t =0,1,2,... (4.22)Theorem 4.21 If meets (4.22) and the following bounded conditionthen is the optimal solution of (4.20).Definition 4.10 If for ,wecallthe optimal strategy respond. If G : X×X→X is the single map, then we call it policy function. Assumption 4.13 F(x,y,z) : A→R is bounded that means|F(x, y, z)| nÏ…₀-Ï…||≤βⁿ||Ï…₀-Ï…||3.The policy respond andÏ…are compact and uppercontinuous.Assumption 4.14 For and y,z, function F(.,y,z) is increasing strictly withrespect to x, meanwhile for any x,z, function F(x,.,z) is increasing strictlywith respect to y.Assumption 4.15 T is monotonous, that means for any x≤x',y≤y' , weTheorem 4.23 Under assumption 4.13, 4.14, 4.15, if value functionÏ…(x,y)isthe only one solution of (4.20), thenÏ…(x,y) is increasing strictly with respectto x, y respectively.Assumption 4.16 F is strict concave function, that means for any (x, y, z)∈A, (x', y', z')∈A,λ∈[0,1], we haveF(λx+(1-λ)x' ,λy+(1-λ)y',λz+(1-λ)z')≥λF(x,y, z) + (1-λ)F(x' , y' , z'),if only if (x, y, z)≠(x' , y' , z' ),λ∈(0,1) the equation is equal.Assumption 4.17Γis convex, that means for anyθ∈[0,1], (x,y, z)∈A, (x' , y',z')∈A, we haveθz + (1-θ)z'∈Γ(θx + (1-θ)x',θy + (1-θ)y'). Theorem 4.24 Under assumptions 4.13, 4.16, 4.17, the value function v is strict concave function, policy respond G is continuous single value function. Theorem 4.25 Under assumptions 4.13, 4.16, 4.17, C'(X×X) (?) C(X×X) is the set for bounded continuous concave functions,υ∈C' (X×X), we construct {Ï…_n},{g_n} as followingÏ…_n+1 = T_Ï…n,n = 0,1,2,...then for any x,y∈X, we have g_n(x,y)→g(x,y), if X is compact, then theconvergence is identical .When the objective function is unbounded we haveTheorem 4.26 Assumpt there existsÏ…|^: X×X→R to meet the followingconditions:1. TÏ…|^≤υ|^;2. For any (x₀,x₁)∈X×X,x|^∈II(x₀,x₁), we have u(x|⁾≤υ|^(x₀,X₁);3. IfÏ…: X→R is the fixed point of operator for any (x₀,x₁)∈X×X,x|^∈Π(x₀,x₁), we have thenÏ…=Ï…^*.Definition 4.11 If the point x meets the function F_z(x|-,x|-,x|-) +βF_y(x|-,x|-, x|-)+β²F_x(x|-,x|-,x|-) = 0, then we call x|- the equilibrium point of (4.20).Definition 4.12 If for any beginning point (x₀,X₁)∈X×X , the solutionof the (4.20) {x_t}₀^∞meets , then we call this equilibrium pointthe globally approximately steady point .Theorem 4.27 If X∈R is compact , g(x, y) : X×X→X is continuousfunction , x|- is the equilibrium point, if there exists a function L(x,y) meetsthe following conditions1 L(x,y) is continuous inside X×X , 2 L(x, y)≤L(x, x|-) if only if y =x|-, the equation is equal,3 L(x, g(x, y))≥L(x, y) if only if y = g(x, y) = x|- the equation is equal , then x|- is the globally approximately steady point .Now we solve the following new model by the new value function methods.t.k_t+1≤f(k_t-1,k_t) -c_t,β∈(0,1),k_t is the investment at t time, c_t is the consumption at t time, f(k_t-1,k_t) = A(θk_t-1 + k_t) is the production function at t time, where 0≤θ< 1 is the coefficient , and u(c_t) = ln c_t is the utility function at t time .We solve this economic model by the new value function method and get the optimal investment and consumption solutions.z(x,y) = Aβθx + AByÏ…(x,y) = D + 1/(1-β)ln(Aθx + ACy)1.θ= 0 that means the present investment per capita is the only variable of the production function, we call this the first case of the model, the optimal solution for investment isz⁰ = Aβy = 0.99y, the optimal solution for consumption is reachedc⁰ = A(1-β)y = 0.11y.2. Whenθ≠0, that means not only the present investment per capita but also the preceding one period investment per capita are the variables of the production function, we call this the second case of the model, the optimal investment solution isthe optimal solution for consumption is reachedAfter the analysis for the model we get the following results: the second case of this model is better than the first one.