Macro Qualifying Exam
January 9, 2012
Instructions. The exam consists of two parts. Please answer 3 out of 4 questions in Part I, and 8 out
of 9 questions in Part II. Start your answer to each question on a fresh sheet of paper. Clearly label the
problem number and your assigned ID at the top of each page.
Try to answer as many parts of each question as possible. It is OK to skip part of a question and still
try to answer later parts to the extent this is possible. Nevertheless, verbal answers that do not engage the
math (when math is expected) will receive little or no credit.
You are strongly encouraged to work out your initial algebra attempts on scrap paper so your final
answer is clean and easy to grade; your final answer should nevertheless include all relevant steps. Messy or
confusing answers will be marked down.
Keep in mind, you will not receive any credit for answering a different question than the one being asked.
For this reason, it is very important that you read each question carefully. Be as precise in your answers as
possible.
You are encouraged to read over all questions for each part before choosing which ones to answer.
You have 5 hours to complete the exam. Part I will count for two-thirds of the total points and part II
will count for one third, so please allocate your time wisely. Try not to spend too much time bogged down
on any one question; you are better off moving on and trying to return to it later.
Good Luck!
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Part I: Modeling Exercises. Answer 3 out of 4 questions.
Exercise 1 (Hint: In answering this question, you should keep in mind that part 5 is worth half of the total
points. It can be answered without relying on answers to the prior questions.)
Consider the non-stochastic growth problem augmented to include a labor-leisure decision. Each period,
the representative household allocates a fixed endowment of time (normalized to one) between labor (n)
and leisure (l). In sequence form, the planner chooses a consumption sequence {ct } and labor and leisure
sequences, {nt } and {lt }, to maximize
∞
X
β t u (ct , lt ) ,
0 < β < 1,
t=0
where for each t ≥ 0,
nt + lt = 1,
and
kt+1 = f (kt , nt ) + (1 − δ)kt − ct
with
k0 > 0 given.
Note that u (c, n) and f (k, n) are both real-valued functions of two variables. They are each strictly
increasing in each variable and strictly concave. In your answers, you should use the following notation to
indicate partial derivatives: for an arbitrary real-valued function g(x, y),
gx (x, y) ≡
∂
g(x, y)
∂x
gy (x, y) ≡
∂
g(x, y).
∂y
and
1. (20 points) Construct the associated Bellman equation. Write the Bellman equation in such a way
that the decision today is to choose the level of capital stock tomorrow and the level of labor today.
For notation, use “primes” to indicate the value of variables in the following period; thus, if x were the
value of a variable today, x0 would be the value of the same variable tomorrow. (Hint: There is only
one state variable in this problem.)
2. (10 points) Find the first-order conditions associated with this problem. (Hint: To answer this question
you will need to apply the chain rule correctly. Utility is a function of consumption and leisure, so if
you were taking the partial derivative with respect to the second argument of the utility function, you
would need to write ul (·, ·), where the dots indicate the relevant arguments at which the function is
evaluated.)
3. (10 points) Find the relevant envelope condition associated with this problem. (Hint: The envelope
theorem applies in an analogous way when the maximization problem is taken over two variables.)
4. (10 points) Combine your answers to (2) and (3) above to derive the Euler equation(s) for the problem.
Your answer should clearly explain the steps that you take to do this.
5. (50 points) Next, suppose that labor is supplied inelastically, with n = 1 for all periods (of course, this
implies l = 0 for all periods). Also, assume that depreciation is 100% (δ = 1). With the level of labor
fixed at n = 1, the utility and production functions are further assumed to take the following forms:
u(c, 0) = ln(c),
(1)
f (k, 1) = k α .
(2)
and
Using the method of guess and verify, derive an explicit expression for the policy function. Start by
writing down the Bellman equation for this simplified problem. Next, indicate the functional form
that you intend to use for your guess. To get to an expression for the policy function, you will need
to go through a number of steps. Get as far as you can and partial credit will be assigned.
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Exercise 2 Consider the continuous-time Solow model where population, L, grows at constant rate a and
TFP, B, grows at constant rate b. Let K denote the aggregate stock of capital, and let
F (K, L, B) = (BL)1−α K α
be the aggregate production function.
growth theory is then
The fundamental (aggregate) differential equation in Neoclassical
K̇ = s(BL)1−α K α − δK.
K
. Showing all relevant steps, derive the corresponding differential equation
1. (15 points) Define κ = BL
governing the evolution of κ.
2. (15 points) What happens to κ in the long run? Justify your answer fully.
3. (15 points) Solve explicitly for the long run level of κ.
4. (15 points) What happens to capital intensity ( K
L ) over time? Be specific and justify your answer.
5. (10 points) What does the model predict about the ability of poor countries to catch up to rich
countries? Be specific and justify your answer.
6. (10 points) Define a balanced growth path. Does the model generate balanced growth? Be specific
and justify your answer.
7. (10 points) What happens to the model economy in the long run if a > 0 and b = 0?
a = b = 0? Be specific and justify your answer.
What if
8. (10 points) What does the model have to say about the fundamental determinants of economic growth?
Be specific and justify your answer.
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Exercise 3 Consider a mass 1 of identical consumers with preferences given by the utility function
Z ∞
C(t)1−θ − 1
U=
dt
1−θ
0
where
1
C(t) = (Csρ + Cuρ ) ρ , 0 < ρ < 1
is an aggregator of consumption over two final goods produced according to:
Ys = L1−α
s
Ns
Z
xα
i,s di
0
Yu = L1−α
u
Nu
Z
xα
i,u di
0
where s, u denote skilled and unskilled sector respectively, Lj , j = s, u is the type of labor of which sector j is
intensive in, and Nj , j = s, u is the number of varieties of intermediate products for sector-type j. Final goods
are produced competitively, but intermediate goods are produced by monopolists as in the Romer (1990)
model. For both the skilled intensive and the unskilled intensive sectors, one unit of the respective final good
is required to produce one unit of the intermediate good. By spending Z units of final good, a monopolist
entering the R&D business faces a probability λj Z of discovering a new variety of the intermediate good in
the sector intensive in Lj , j = s, u. The R&D sector is competitive (free entry). Finally, we normalize prices
of Cs , Cu so that the price of the consumption good is the numéraire:
ρ
ρ
psρ−1 + puρ−1
ρ−1
ρ
=1
(Note: All questions below are worth equal points.)
1. Compute the profit function (that is the value of maximized profits) for an intermediate good as a
function of pj , the price of Yj , j = s, u.
2. Compute the relative price between the two final goods ps /pu as the marginal rate of substitution
between the two consumption goods. Market clearing in consumption requires Cj = Yj , j = s, u. Use
this fact to express the skill premium ws /wu as a function of Ns , Nu .
3. State the research arbitrage equation for each sector j = s, u. Find the value of an innovation Vj in
sector j = s, u when the appreciation/depreciation term in the pricing equation of an innovation is
equal to zero, similarly to the Romer (1990) model.
4. Solve the ratio Vs /Vu as a function of relative varieties Ns /Nu and relative supply of skills Ls /Lu .
Isolate the price effect and the market size effect.
5. It can be shown that the elasticity of substitution σ between Ns and Nu satisfies σ =
an increase in the relative supply of skills going to increase the skill premium?
6. Is there a scale effect in this model? Why or why not?
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1−αρ
1−ρ .
When is
Exercise 4 Consider the following basic matching model. The process of trade between workers looking for
jobs and firms with vacancies is described by the following iso-elastic matching function:
M (uL, vL) = (uL)η (vL)1−η
where η ∈ (0, 1), L denotes total labor supply, uL is the number of unemployed workers and vL is the
number of vacancies. A matched job with expected present discounted value J produces output per worker
per period equal to A, and a vacancy with expected present discounted value V costs cA per unit time,
c > 0. The expected present discounted value of unemployment is denoted by U . Jobs are destroyed at
an exogenous rate b. The vacancy/unemployment ratio is considered as a separate variable and denoted
by θ ∈ [0, 1]. Employed workers receive a wage w for their labor services, and the interest rate is fixed
exogenously at r. There is no unemployment compensation. We assume that V, U, J are all stationary over
time.
(Note: All questions below are worth equal points.)
1. (a) Write down the rate per unit time at which job matches occur, and the rate at which workers
transition out of unemployment. (b) Calculate the elasticity of both rates with respect to θ, and
provide an economic intuition behind the signs of these elasticities.
2. (a) Write down a differential equation describing the evolution of the unemployment rate over time, and
solve for the steady state value of the unemployment rate. (b) How does steady state unemployment
slope in the (w, θ) plane? and how does it slope in the (u, v) plane?
3. (a) Write down asset equations for V and J. (b) What must be the value of V in equilibrium, and
why? Use this observation to solve for J as a function of hiring costs and mean duration of a vacancy,
and to find a job creation condition.
4. Let W denote the stationary present discounted value of expected income streams for an employed
worker. (a) Write asset equations for U and W , and (b) give a solution for rU, rW in terms of the wage
rate w, the job destruction rate b, and the vacancy/unemployment ratio θ. (c) Show that employed
workers enjoy a pure rent over unemployed workers, and briefly provide intuition for this result.
5. We now turn to wage determination. The asset equations for J and W are firm-specific in that they
depend on the firm-specific wage wi , i = 1, . . . , N being an index for firms. Workers and firms that are
matched and create a job share the equilibrium rents arising from job creation by agreeing on a wage
that maximizes the Nash product of their respective gains from a job match:
(W (wi ) − U )β (J(wi ) − V )1−β
where β is workers’ bargaining power, while U is independent from wi and satisfies the equation you
found above in which w is the average wage in the labor market. Solve the Nash bargaining problem
and show that β is the share of workers in the total surplus created by a job match.
6. (a) Show that the surplus-maximizing wage depends only on rU, β, A, and is therefore equal for all
firms. (b) Solve for rU as a function of β, A, c, and θ, and (c) write down the equilibrium wage equation
representing the wage as a function of θ and the parameters of the model.
7. (a) Define in one sentence the equilibrium of the labor market. (b) Represent the labor market equilibrium in the (w, θ) plane and in the (u, v) plane, and show the comparative statics effect of: (c) a
shock to labor productivity, (d) a shock to the job destruction rate, (e) a shock to the unemployment
compensation.
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Part 2: Short Essay Questions. Answer 8 out of 9 questions.
Your answer to each question in part 2 should not exceed 15 lines.
1. Consider a dynamic programming problem for which an analytical solution does not exist. Explain
the general steps you would take to solve the problem numerically.
2. Describe a typical research question that could be addressed using growth accounting.
would you need to implement the approach?
What data
3. Describe a typical research question that could be addressed using a Barro regression.
would you need to implement the approach?
What data
4. What is the left hand side variable in the regression equation estimated by Mankiw, Romer and Weil
(1992)? What are the right hand side variables? What research question did the authors seek to
answer? What data is needed to implement their approach?
5. Along the lines of Jones and Romer (2010), enumerate the “new” Kaldor facts of economic growth,
and illustrate the main elements of growth models that try to explain these facts.
6. Describe in words how a model in which government’s provision of public goods enters the production
possibilities of an economy can provide a justification for an AK-type endogenous growth framework.
Explain why zero taxation cannot be optimal in such a framework.
7. Explain why: (a) in the presence of credit constraints, wealth inequality might be harmful for growth,
and (b) how can you partially challenge this conclusion by appealing to heterogeneous individual
abilities. Make sure to include sketched policy recommendations for both (a) and (b). Is inequality
harmful for growth when the most able individuals are also the poorest? Explain.
8. Explain the main difference between the Romer (1990) model and the Schumpeterian (Aghion and
Howitt, 1992) growth model in terms of their respective welfare analyses. Is the laissez faire growth
rate smaller or larger than the optimal one? Is one of the two frameworks able to accommodate
excessive growth under laissez faire, and if so, why?
9. Is proximity to the Atlantic Ocean enough to explain differences in economic performance across
European countries during the centuries preceding the industrial revolution?
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