Letter Code:
Macro Qualifying Exam
August 2, 2013
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 Consider the following generic problem. The goal is to choose a sequence of control variables
{zt }∞
t=0 to maximize the following objective function
∞
X
β t (Ryt2 + Qzt2 ),
t=0
where yt is a stock variable that evolves according to the following transition equation:
yt+1 = Ayt + Bzt , t = 0, 1, 2, . . . ,
with y0 given. β, R, Q, A, and B are known scalar constants. 0 < β < 1 and we assume that R < 0 and
Q < 0 to ensure the problem is concave.
1. (10 points) Write down the Bellman equation for this problem.
2. (10 points) Guess that the value function takes the following form:
V (y) = P y.
Determine if the guess is correct. Explain why.
3. (10 points) Guess that the value function takes the following form:
V (y) = P y 2 .
Determine if the guess is correct. Explain why.
4. (10 points) Guess that the value function takes the following form:
V (y) = P y 3 .
Determine if the guess is correct. Explain why.
5. (10 points) Using the correct guess, derive a quadratic equation for P that the correct guess must
satisfy. (Note: I will not ask you to determine which root of the quadratic equation is the correct one.
It can be shown that the only one of the two roots is negative and that the negative root is the correct
one for our purposes.)
6. (10 points) Assuming you have solved for the correct value of P , derive an expression for the optimal
policy function as a function of P . Given my statement above that P < 0, plot the policy function.
7. (15 points) Next, suppose that the transition equation is instead given by
yt+1 = Ayt + Bzt + t ,
where t ∼ N (0, 1) is a normal random variable.
Write the Bellman equation for the modified problem. Clearly define all mathematical objects used in
your equation.
8. (10 points) Indicate what restrictions on uncertainty are needed in order to include uncertainty in a
dynamic programming problem and explain why these restrictions are satisfied for this problem
9. (15 points) Derive the stochastic Euler equation. Indicate all steps in your derivation. Give intuition
for the derived equation.
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Exercise 2 Consider the following three period pizza problem. Time runs from t = 1, 2, 3. In the first
period, you and your roommates start with P1 square inches of tomato and artichoke pizza. Your task is to
choose how much to consume each period to maximize the discounted sum of future utility, where u(c) is
your utility function over consumption, c, in each period. Your discount rate is Γ, where 0 < Γ < 1.
1. (10 points) Write the relevant maximization problem in sequence form. Be sure to indicate all relevant
mathematical objects, including relevant constraints, etc.
2. (10 points) Rewrite the same problem as a sequence of two Bellman equations.
3. (20 points) Assume that utility is logarithmic and use finite horizon dynamic programming to derive
an explicit expression for the policy function in the initial period.
Next, assume the time horizon is infinite.
4. (5 points) Write the infinite horizon problem in sequence form.
5. (10 points) Write the corresponding Bellman equation. Indicate what “stationarity assumptions” are
needed to write the problem in this way.
6. (20 points) Use guess and verify to derive an explicit expression for the value function. Be sure to
clearly indicate all relevant steps.
7. (10 points) Derive an expression for the policy function.
8. (15 points) Compare the policy function derived in 3 with the policy function derived in 7. Plot both
policy functions on the same set of axes. Indicate what assumptions are needed to draw the plot in
this way. How are they similar? How are they different?
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Exercise 3 Consider the following one–sector growth model. Output Y is produced according to the production function
Y = AK + BK α L1−α , α ∈ (0, 1).
Time is continuous, and capital stock depreciates at a rate δ > 0. There is a representative consumer with
CRRA(θ) utility function with argument c, consumption per capita. Given an initial value N (0), the fully
employed labor force grows at a rate n > 0, and the discount rate is constant and equal to ρ > n. We also
assume that A > δ + n. There are no externalities nor public goods. Further, all markets are competitive,
and there is full information about current and future prices and quantities.
1. (10 points) Denote capital stock per worker by k, and write down the intensive production function.
Are the Inada conditions on the intensive production function satisfied? What is the implication of
your answer for the long–run growth path of the economy?
2. (20 points) Write the infinite horizon problem solved by a social planner maximizing the economy’s
aggregate utility, under the accumulation constraint, and find the first order conditions for an optimal control using the current–value approach. Would the first–order conditions be different in the
decentralized utility maximization problem? Why or why not?
3. (20 points) Write down a dynamical system describing the time paths of capital stock per worker and
consumption per worker. Show that this system has no singular point such that ċ = k̇ = 0.
4. (10 points) Define an asymptotic path as a sequence {c(t), k(t)} such that k(t) → ∞ at all t. Show
that the growth rate of consumption along an asymptotic path is a constant function of A, δ, ρ, θ.
5. (10 points) Although you are not required to show it, it turns out that this system has a balanced
growth path (BGP) where c, k grow at the same rate. Show that, along the BGP:
k(t) = (c(0)/φ) exp{(1/θ)(A − δ − ρ)}, φ ≡ (A − δ)(θ − 1)/θ + ρ/θ − n.
6. (25 points) Because there is no singular point in the (c, k) space, we can do a change of variables as
follows:
z ≡ f (k)/k, χ ≡ c/k.
which are constant at a BGP. Rewrite the dynamical system in terms of z, χ only, and show that its
steady state features zss = A, χss = φ.
7. (5 points) Is long–run growth endogenous in this model? Why or why not?
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Exercise 4 Consider the following growth model. The final (consumption) good is produced according to
the following technology:
Z Mt
Yt = L1−α
xα
i,t di
Y
0
where LY denotes workers in the final sector, M is the number of varieties (designs) of intermediate products,
and xi are the intermediate products used for producing the final good. One unit of intermediate good is
produced using one unit of the final good, and each intermediate goods producer is a monopolist in its sector.
The monopolist firm pays the interest rate r to each unit of input utilized. Workers can be employed either
in producing the final good or in R&D in order to expand the number of varieties available. Denoting by
LM the number of workers in the R&D sector, the following constraint must hold:
L ≥ LY + LM
where L is the total labor supply. Both types of workers supply their labor services inelastically to the wage.
The labor force grows at a rate n. Consumers have CRRA(σ) preferences and maximize the present value
of utility over an infinite horizon, taking into account the typical asset equation. There is no government,
nor foreign sector. Suppose that the production function for innovations is:
γ−1
Ṁ = λ(M φ LM
)LM , φ ∈ (0, 1), γ ∈ (0, 1)
where φ captures the amount of spillovers generated by past discoveries, and γ is meant to capture the fact
that researchers might engage in useless duplication activity while doing research.
1. (10 points) What is the growth rate g of new designs implied by the innovation technology? Along a
balanced growth path, this growth rate must be constant. Show that
g=
γ L̇M
1 − φ LM
2. (20 points) In a decentralized (laissez-faire) economy, firms take the productivity of each researcher,
that is λ(M φ Lγ−1
M ) as given. What is the quantity of intermediate goods produced by monopolists
in manufacturing? What is the arbitrage equation for the R&D sector? What is the value of an
innovation?
3. (10 points) Use the results you obtained above to show that, along a balanced growth path, the ratio
between workers employed in the consumption sector and workers in the R&D sector satisfies
LY
r1
=
LM
αg
4. (10 points) The implication of the result you just obtained is that a constant fraction, say s of the
working population L will be employed in R&D, and the remaining (1−s) will work in the consumption
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good sector. Show that s = 1+χ
, where χ ≡ αr g1 . Using the constancy of the ratio between workers
employed in the two different sectors, show that the growth rate of designs is proportional to the growth
rate of population, and not to the number of researchers.
5. (10 points) Using the current-value approach, derive the Euler equation for a consumer discounting
utility streams at a rate equal to ρ − n and maximizing the present value of utility streams under the
typical asset accumulation constraint. Use the Euler equation for consumption to find an expression
for χ in terms of the parameters of the model only.
6. (10 points) Is economic policy, aimed at influencing the growth rate through increasing the number of
researchers, effective in this model? Use the expression you found for the growth rate of the economy
to briefly discuss.
7. (30 points) Considering the intermediate inputs as accumulating capital goods, state the problem of a
social planner maximizing the present discounted value of utility streams over an infinite horizon under
the accumulation and the innovation constraint. What are the effects that the planner internalizes
compared to the laissez-faire economy? Can you say whether the decentralized allocation involves
more or less R&D than the allocation targeted by the social planner?
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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. Compare and contrast a Mankiw-Romer-Weil regression with a Barro regression. Your answer should
indicate two key differences and two key similarities.
2. Use bullet points to indicate the 8 to 10 most important pieces of a Matlab program that could be
used to solve a dynamic programming problem that can’t be solved analytically. You don’t need to
provide specific code, but you should clearly indicate the function of each component of your “pseudo
code”.
3. Determine if the following function is a contraction:
G(x) = 2x + 3.
Justify your answer. Then explain what happens if you iterate on G starting from an arbitrary initial
condition. Is this consistent with the Contraction Mapping Theorem? Be specific.
4. List the Kaldor facts and explain what data you would need to determine if they hold in a particular
economy. Then explain the relationship between the Kaldor facts and a balanced growth path.
5. 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.
6. Illustrate similarities and differences between the aggregate No Shirking Condition (NSC) in the
Shapiro/Stiglitz model and the aggregate labor supply curve that would follow from a labor–leisure
choice by workers. What is the main implication of the shape of the NSC relative to that of the labor
supply curve, and why does it matter for equilibrium unemployment?
7. What are the main equations describing the Mortensen–Pissarides model in reduced–form? What are
the endogenous variables in the model? What type of unemployment is the model meant to explain?
8. Suppose you want to use a framework building on the Lucas (1988) growth model in order to explain
the process of economic development in an economy with a large rural sector and a growing urban
sector with significant unemployment. What assumptions about the labor market would you need to
modify relative to the benchmark model? What will be the endogenous distributional variable needed
to ‘close’ the model?
9. Illustrate the skill-premium puzzle, and explain how contemporary endogenous growth models of directed (skill-biased) technical change (SBTC) can make sense of it. Some authors have argued that
the famous QJE paper by Piketty and Saez (2003) provides contrary evidence to SBTC predictions.
Why?
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