課程名稱︰總體經濟理論一 課程性質︰必修(經濟碩博/財金博/會計博) 課程教師︰毛慶生(與蔡宜展合開) 開課學院：社會科學院 開課系所︰經濟學研究所 考試日期（年月日）︰2015年1月某日 考試時限（分鐘）：180分鐘 試題 : Final Examination: Macroeconomic Theory(Fall 2014) [110 pts.total] "以下小t,t-1,t+1,t+j...等皆為下標，代表第t,t-1,t+1,t+j...期" "d0及E0則代表第0期" 1.Lucas Tree Model[50 pts.total] Consider a deterministic version of the Lucas tree model. The representative consumer is assumed to solve the following optimization problem(notations and model setup follow my class lecture): ∞ max Σ β^t*u(ct),0<β<1 {ct,zt,bt} t=0 subject to ct+bt+(qt*zt)=(1+rt-1)*(bt-1)+(qt+dt)*(zt-1),∀t Equilibrium requires ct=dt, zt=1 and bt=0 for all t (1)[10 pts.] Please write down the pricing equations for bond and stock(i.e.,first order conditions together with market clearing conditions). Interpret briefly. (2)[10 pts.] The time t 'stock demand' (zt)^d and the 'bond demand' (bt)^d are functions of {rt+j,qt+j,dt+j}j=0 to ∞. Please discuss the effect on (zt)^d and (bt)^d of a change in {rt+j,qt+j,dt+j}j=0 to ∞, respectively. (3)[10 pts.] Assume dt=[(1+μ)^t]*d0, μ>0 and the utility function is of CRRA form: u(c)=(c^1-γ)/(1-γ) if γ≠1 and u(c)=ln(c) if γ=1. Please derive a closed-form solution for the equilibrim stock price qt and the real interest rate rt.[Assume β[(1+g)^1-γ]<1] (4)[10 pts.] Based on your solution in(3), what is the effect of a rise in μ on qt and rt? Justify your arguments.[no discussion, no points] (5)[10 pts.] Suppose the divident stream {dt}t=0 to ∞ is stochatic with a known probability distribution. Let 1+(rt)^s=[(qt+1)+(dt+1)]/qt be the ex-post rate of return on stock. Please show that the expected equity premium satisfies the following relationship: Et[(rt)^s]-rt is a ratio of Covt[dt+1,1+(rt)^s]/Et[u'(dt+1)] Why is the 'riskiness'of the stock positively related to the correlation between dt+1 and (rt)^s? Discuss intuition behind. 2.Stochastic Ramsey Model[60 pts. total] Consider a version of CKR model with exogenous government purchase. The representative household is assumed to solve the following problem (Gt is government purchase at time t, other variables are standard): ∞ max E0[Σβ^t*u(ct)] {ct,kt+1} t=0 subject to ct+[(kt+1)-(1-δ)kt]+Gt=yt=λt*f(kt), {Gt,λt}~stationary stochatic process. (1)[10 pts.] Please write down the Bellman's equation and derive the equilibrium conditions of the model. (2)[10 pts.] Please use 'phase diagram'to analyze the effects of a permanent rise in "λt" and draw the response functions of capital, consumption and the implicit real interest rate. Discuss intuition behind. (3)[10 pts.] Please use 'phase diagram'to analyze the effects of a permanent increase in "Gt" and draw the response functions of capital, consumption and the implicit real interest rate. Discuss intuition behind. (4)[10 pts.] Let u(ct)=ln(ct), yt=(λt)*(kt)^α, α∈(0,1), δ=1, Gt=(gt)*(yt),gt∈(0,1). Further, assume λt and gt are both i.i.d with constant mean. Please derive a closed-form solution for consumption ct and capital kt+1. The solution is a function of kt, λt and gt. (5)[10 pts.] Suppose you are asked to introduce a stock market into the economy, where shares of the representative firm are competitively traded. Let qt be the stock price and dt=yt-wt-[kt+1-(1-δ)kt] the dividend remitted to consumers at time t. Note that, to obtain dividend, one has to deduct from revenue the investment expenditures and the wage cost, which is wt=MPLt in CKR model because labor is fixed at one. Using the specification in (4), please derive a closed form solution for qt. Again, the solution should be expressd as a function of state variables. (6)[10 pts.] In general, the closed-form solution for stock price does not exist. Please describe a numerical procedure that wil deliver a solution for qt. Why do you think your procedure would work? --

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