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Econ 306 (Spring 2008)
Final Exam
Instructor: Chao Wei
The exam starts at 6:10 pm and ends at 9:00 pm.
There are 50 points on this exam.
There are four questions. You are REQUIRED to
answer question 1, but can choose to answer TWO out
of the three remaining questions. Please specify which
question you choose to answer. If you fail to do that
and answer all the questions, I will grade the first three
questions. The first question is worth 20 points. The
other three questions are worth 15 points each.
Read through the entire test before answering any
questions. Do not use your book or notes.
Please keep your answers clear and concise. Be sure
to identify important results. Put a box around any
critical mathematical results.
Correct but irrelevant material will generate no credit.
Problem 1 Indivisible Labor Real Business Cycle Model (20 points). The economy is populated by a large number of infinitely lived agents whose expected utility is defined as
E 0
X^ ∞
t=
βt^ [log (Ct) − θNt]
where Ct represents consumption and Nt represents hours worked per capita. The production function is
Yt = Aαt Kt^1 −αNtα ,
where At stands for technology, Yt for output and Kt for capital. The capital accumulation process is:
Kt+1 = (1 − δ) Kt + Yt − Ct.
We now look for a steady-state or balanced growth path of this model, in which technology, capital, output, and consumption all grow at a constant common rate. We use the notation G for this growth rate: G ≡ A At+1t in the steady state.
(a) Derive the first order conditions of the model and interpret their eco- nomic implications.
(b) Derive the steady state values of (^) KAtt , (^) KYtt , C Ytt , and Nt.
(c) Log-linearize the intertemporal first-order condition with respect to Kt+1, that is, write Et 4 ct+1 as a function of Etat+1, Etnt+1, and Etkt+1, where lower-case letters represent the log deviations from the steady state of the corresponding variables.
(d) Describe the heuristic labor supply curve and the labor demand curve generated by the model.
(e) Use the graphic analysis of the labor market behavior to describe the responses of the real wage and labor hours to a bad technology shock.
period.
(b). Compute the "long-term" (two-period) interest rate, i.e, the interest rate on a discount bond which promises one unit of consumption good in two periods.
(c) Note that the log of the growth rate of consumption is given by
log ct+1 − log ct = α 0 − α 1 st + εt+1.
Thus, the conditional expectations of this growth rate is just α 0 − α 1 st+ μ. Note that when st = 0, growth is high, and when st = 1, growth is low. thus, loosely speaking, we can identify st = 0 with the peak of the cycle (or good times) and st = 1 with the trough of the cycle (or bad times). Given the above conditions, are short term rates pro- or countercyclical?
(d) Are long rates pro- or countercyclical? If you cannot give a definite answer to this question, find conditions under which they are either pro- or countercyclical, and interpret your conditions in terms of the "permanence" (you get to define this) of the cycle.
Problem 3 The real business cycle model assumes that investment can be negative. This implies that existing capital can be costlessly converted into the consumption good. In this problem, we impose a non-negativity constraint on investment. Assume that households maximize
E 0
X^ ∞
t=
βtU (ct) (1)
subject to
yt = ztkαt n^1 t −α, (2) kt+1 = (1 − δ) kt + it, (3) yt = ct + it, (4) it ≥ 0. (5)
(a) Derive the Bellman equation, the first-order conditions and the Envelop condition when there is a non-negativity constraint on investment.
(b) What is the Euler equation when the non-negativity constraint is not binding? What are the solutions to kt+1 and ct in periods when the non- negativity constraint is binding?
(c) Now we further assume that
ct + it
1 + h
μ it kt
≤ ztkαt n^1 t −α, (6)
where h
it kt
describes the resources required to install new capital, where
h (0) = 0, h^0 >^0 ,and^2 h^0 (x) +^ xh^00 (x)^ >^0 for all^ x >^0 ,^ Write down the Bellman equation, first-order conditions and the envelop condition in this case.
(d) Define xt = i ktt , show that if 1 + h (x) + xh^0 (x) = 1 and h^0 (x) = 0, the solutions of the model are equivalent to those of (a).
Problem 4 Consider a representative household who faces the following prob- lem:
max {Ct,Kt+1}∞ t=
X^ ∞
t=
βt^
Ct^1 −γ 1 − γ
, γ > 1 (7)
subject to
0 ≤ Kt+1 ≤ (1 − δ) Kt + (1 − τ ) F (Kt) − Ct + ψt, (8) K 0 > 0
Here τ is the tax rate levied on all gross output. We assume that the gov- ernment gives back all the tax revenues in the form of lump-sum rebate ψt, which is equal to τ F (Kt). Note that by the definition of lump-sum rebate, the derivative of ψt with respect to control variables are zero. In other words, the representative household does not take into account the impact of their decisions on ψt. Assume that yt = F (Kt) = Atkt,
