课程名称︰总体经济理论一
课程性质︰必修(经济硕博/财金博/会计博)
课程教师︰毛庆生(与蔡宜展合开)
开课学院：社会科学院
开课系所︰经济学研究所
考试日期（年月日）︰2015年1月某日
考试时限（分钟）：180分钟
试题 :
Final Examination: Macroeconomic Theory(Fall 2014)
[110 pts.total]
"以下小t,t-1,t+1,t+j...等皆为下标，代表第t,t-1,t+1,t+j...期"
"d0及E0则代表第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?