课程名称︰总体经济理论一 课程性质︰必修 课程教师︰蔡宜展 开课学院：社科院 开课系所︰经济所 考试日期（年月日）︰103.11.10 考试时限（分钟）：0910-1210 (180mins) 是否需发放奖励金：是 （如未明确表示，则不予发放） 试题 : Instrctions: You have hours to complete this examination. Please number each Question, underline your final answers, and present your work as clearly as possible. 1.Consider a version of the one-sector growth model where households' period utility is u(c)=-e^(-γc), γ>0, and the subjective discount factor 0<β<1. Households have a unit endowment of time, which may be supplied either as labor or leisure. Production takes the Cobb-Douglas form, F(k,n)=A(k^α)N^(1-α) with A>0 and 0<α<1. The output can be used either as consumption or investment. The depreciation rate of capital is 0<δ<1 and the initial capital stock is k0>0. Each of thr questions below involves the Social Planner's problum. Further, all questions are to be answered using the utility and production functions listed here. (a) (5 points) State the sequence problem. (b) (5 points) Derive the necessary and sufficient conditions for the optimal solution of this sequence problem. (c) (5 points) Formulate the planner's optimal problem as a dynamic programming problem. What are the state variable(s)? What are the choice variable(s)? (d) (5 points) Derive the first-order and Benveniste-Scheinkman conditions and show that the optimal conditions for sequence problem are identical to that of functional equation. (e) (5 points) Solve for the steady state values for capital, k* and consumption, c*. 2.Consider the sequential competitive equilibrium of the neoclassical growth model described in question (1). (a) (10 points) Define a sequential competitive equilibrium. (b) (5 points) Derive the household's Euler equation involving ct, c(t+1) and the real return in period t+1 to savings, R(t+1) (c) (5 points) Using the firm's profit maximization, replace R(t+1) to show that the sequential competitive equilibrium reproduces the Social Planner's Euler equation. Explain the significance of this result. (d) (10 points) Define a recursive competitive equilibrium. 3.Consider the infinite horizon Ak growth model. An infinitely-lived representative household values consumption by u(c)=log c in each period. Capital, the sole factor of production, is owned by household. The marginal product of capital is constant and, given its initial stock, the social planner solves the following problem. V(k)= max (log(Ak-k')+βV(k')), (1) 0≦k'≦Ak For this problem there is a unique function V satisfied (1). Conjecture that the value function takes the form V(k)=E+Flog k, (2) where E, F are a function of the parameters of the problem: A, and β. (a) (5 points) Use the method of undetermined coefficients to solve E and F. (b) (10 points) Next, define the operator T:C(R)→C(R), TV(k)=max(log(Ak-k')+βV(k')). (3) Use the method of successive approximation to solve for V, the fixed point of (3). Let V0=0 and use the operator T to define V1 and V2, where V1=TV0 and V2=TV1. Derive V≡ lim (T^N)V0. N→∞ (c) (5 points) Having found the value function, provide a solution for k'=g(k) in terms of the primitives of the problem (A,β). Describe the growth rate of output (AKt) as γy and the saving rate (K(t+1)/Akt) as s. How does a rise in A change each? Explain your answer. 4.Consider a neoclassical growth model with two sector, one producing consumption goods and the other producing investment goods. Consumption is given by Ct=(Kct^α)(Lct^(1-α)) and investment is given by It=(Kit^α)(Lit^(1-α)) where Kjt is the amount of capital in sector j at the beginning of period t and Ljt is the amount of labor used in sector j in period t. The total amount of labor in each period is equal to L (leisure is not valued). Labor can be freely allocated in each period between the two sectors: L=Lct+Lit. Capital, by contrast, is sector-specific: once it is installed in a given sector, it cannot be moved to the other sector. Investment goods, however, can be used to augment the capital stock in either sector. In particular, the capital stock in two sectors evolve according to K(j,t+1)=(1-δ)Kjt+Ijt, j=c, i, where It=Ict+Iit. The social ∞ planner seeks to maximize Σ(β^t)u(Ct), given Kc0 and Ki0, subject to the t=0 constraints on technology. Note that although leisure is not valued (i.e., the total amount of labor supply does not appear in the planner's objective), the planner must nonetheless decide in each period how to allocate L across the two sectors. (a) (5 points) Formulate the planner's optimization problem as a sequence problem. (b) (5 points) Find the necessary and sufficient for the optimal solution of the sequence problem. (c) (5 points) Formulate the planner's optimal problem as a dynamic programming problem. What are the state variable(s)? What are the choice variable(s)? (d) (5 points) Derive the first-order and envelope conditions that an optimal solution to planning problem must satisfy. (e) (5 points) Find a set of equations that determine the steady-state values for capital and labor in the economy. --

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