課程名稱︰總體經濟理論一
課程性質︰必修
課程教師︰蔡宜展
開課學院：社科院
開課系所︰經濟所
考試日期（年月日）︰103.11.10
考試時限（分鐘）：0910-1210 (180mins)
是否需發放獎勵金：是
（如未明確表示，則不予發放）
試題 :
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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