Macro Qualifying Exam
July 30, 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 Consider the problem of a social planner who wants to choose the optimal level of greenhouse
gas emission reductions (i.e., “abatement”) for an economy. The problem unfolds over a long period of time,
and for simplicity I take the time horizon to be infinite. Each period, z is the stock of greenhouse gases
in the atmosphere and a is the amount of emissions to be abated (abated emissions reduce the cumulative
stock of emissions in the atmosphere). Environmental damages are given by the damage function dz 2 , where
d > 0, and the cost of abatement is given by the cost function ca2 , where c > 0. Finally, greenhouse gases
accumulate according to the following transition equation:
zt+1 = δzt − at ,
where 0 < δ < 1. I don’t impose any other restrictions on the variables.
In sequence form, the planner’s problem can be written as follows:
max
{at }
∞
X
β t −dzt2 − ca2t ,
0 < β < 1,
t=0
subject to
zt+1 = δzt − at , for all t ≥ 0,
z0 > 0 given.
1. (15 points)
Indicate what the state variable is for the problem, then write the corresponding Bellman equation.
Do so in such a way that the choice variable “today” is the value of the state variable “tomorrow”.
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.
2. (15 points)
Use the Bellman equation to derive the Euler equation for the problem. Be sure to indicate clearly the
steps you take in doing so.
3. (15 points)
Determine the steady state value of z along the optimal trajectory.
4. (15 points)
Guess that the value function that solves the Bellman equation takes the form v(x) = Gx2 , for some
constant G > 0. (x in this case is just a placeholder. In particular, you should substitute the state
variable for the problem in place of x.) Use the method of guess and verify to determine if this guess
is correct. Justify your answer carefully.
5. (15 points)
Guess that the value function that solves the Bellman equation takes the form v(x) = Hx3 , for some
constant H > 0. (Again, x is just a placeholder; substitute the state variable in place of x.) Use guess
and verify to determine if this guess is correct. Justify your answer carefully.
6. (25 points)
Solve for the policy function. The algebra is messy, so be sure to work out the problem using scratch
paper first. Also, if you clearly indicate the steps you take, you may get partial credit even if the
algebra is imperfect.
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Exercise 2 Consider a modified version of the cake-eating problem: “cake-eating with a sneaky roommate”.
Each night, while you sleep, your roommate sneaks a constant fraction δ of the cake left over from the prior
day. The deterministic, infinite horizon version of the problem can be posed as the following sequence
problem:
max
{ct }
∞
X
β t u(ct ),
0 < β < 1,
t=0
subject to
Wt+1 = (1 − δ)Wt − ct , for all t ≥ 0,
W0 > 0 given.
1. (10 points)
Indicate what the state variable is, then write the corresponding Bellman equation. Do so in such a
way that the choice variable “today” is the value of the state variable “tomorrow”. 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. Be sure to indicate all relevant constraints.
2. (25 points)
Next, assume that the utility function is logarithmic. It turns out that the value function in this case
takes the following form: V (W ) = C + D ln(w), for some positive constants C and D. Use this fact
to solve for the policy function.
3. (15 points)
Using your results above, explain how your roommates theft (i.e., the depreciation rate δ) affects the
optimal policy. What happens to the amount you save from one period to the next as δ increases?
Justify your answer and be as quantitatively specific as possible.
4. (15 points)
Next, suppose that you want to modify the problem to better reflect the fact that you sometimes
feel “well” and that you sometimes feel “unwell”. On days when you feel well, you derive 20 percent
more utility from any given amount of cake than you would on an “average” day; on days when you
feel unwell, you derive 20 percent less utility from any given amount of cake than on an average day.
Suppose moreover that your sick feelings tend to persist. In particular, if you feel unwell on a given
day, there is an 80 percent chance you will continue to feel unwell on the following day. If you feel well
on a given day, there is a 95 percent chance that you will continue to feel well on the following day.
What mathematical objects can you use to describe this information? Describe them in detail.
5. (10 points)
Using the mathematical objects described in part 4, write down the Bellman equation for the modified
problem.
6. (15 points)
Derive the stochastic Euler equation for the modified problem. Carefully explain the steps you take.
Explain the intuition
7. (10 points)
Carefully explain the intuition implied in the Euler equation derived in part 6.
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Exercise 3 Consider the following overlapping generation (OLG) setting, based on Galor and Zeira (1993).
Individual agents have one unit of labor they supply inelastically each period they work. Individuals live
for two periods, and each individual has one parent and one child. Population is constant, and in each
generation there is a continuum of individuals indexed by j ∈ [0, L]. Agents only save in the first period,
and consume only in the second period. Each agent j’s utility function for period 2 is
uj (cj , bj ) = α log cj + (1 − α) log bj
(1)
where cj , bj denote consumption and the bequest left to future generations. The intertemporal budget
constraint requires consumption and bequests not to exceed the household’s total income yj over two periods.
As far as production is concerned, output can be produced using either of two technologies, one using
skilled labor and capital, and the other one using unskilled labor only. The two technologies satisfy:
Yts = F (Kt , LSt ) concave and CRS
Ytn
=
wn Lnt
(2)
(3)
where wn is the marginal product of unskilled labor in production. Capital is perfectly mobile across sectors,
and the rate of return on capital stock is constant and denoted by r.
1. (20 points)
(a) State and solve the individual’s utility maximization problem. Show that utility-maximizing consumption and bequest are both a constant fraction of individual income. (b) Derive the agents’ indirect
utility function u∗ (yj ), and show that it is of the form
u∗ (yj ) = + log yj ∀j ∈ [0, L]
The second choice made by individuals is whether to invest in acquiring human capital. An individual
inherits an amount xj from her parents, and can acquire human capital in period 1 only by not working and
paying a fixed sum denoted by h > 0, to earn a wage ws in period 2. Otherwise, an individual can work for
two periods, earning wn each period. Both ws , wn are constant. Hereafter, we will refer to educated workers
as skilled and to everyone else as unskilled. Savings in period 1 yields a rate of return denoted by r in period
2. On the other hand, if the amount inherited falls short of the education cost, an individual can borrow
funds at a rate i 6= r.
2. (20 points)
Using the results you obtained in part 1, write down the indirect utility and the bequest of: (a) an
individual choosing not to acquire education; (b) an individual who inherits more than h, and (c) an
individual inheriting less than h. (d) Show that the requirement
ws − h(1 + r) ≥ wn (2 + r)
must hold in order to ensure that rich people always choose to educate.
Next, we look at credit markets for education loans, which are characterized by a moral hazard problem. A
borrower who borrows an amount d can default on her loan payments, but lenders can monitor them, so that
default is costly. If a lender spends an amount z monitoring borrowers, a borrower can decide to default by
paying βz, β > 1. There is no solvency problem in capital markets.
3. (20 points)
(a) Write down the borrowers’ incentive-compatibility constraint, which ensures that borrowers are
at least as well off by repaying the loan as they are by defaulting; (b) write down the participation
constraint for lenders, which ensures that both risk-bearing and risk free loans are made in equilibrium.
Using both constraints, show that the borrowing rate i and the lending rate r fulfill:
i=
1 + βr
>r
β−1
(4)
We are now ready to characterize the equilibrium wealth distribution, how it impacts the economy’s macroeconomic performance, and the dynamics of the distribution of wealth in the economy.
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4. (10 points)
Comparing the indirect utility of a skilled borrower to that of an unskilled lender, derive the smallest
amount f an individual must inherit in order to acquire human capital in the economy. Write down
the inequalities that characterize poor, middle class, and rich in the economy in terms of xj , f, h.
5. (30 points)
(a) Using the fact that the inheritance received by an individual at t + 1 is the bequest left to her by
her parent at time t, write down the three differential equations describing the evolution of wealth in
the economy for the three classes. (b) Derive steady state distributions of wealth for the three classes,
and show that polarized outcomes are the only stable steady states. (c) Illustrate these findings in a
graph, and explain why this model speaks to the possibility of poverty traps as wealth distribution
influences macroeconomic performance and the future distribution of wealth.
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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 good
1
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.
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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. Define a balanced growth path. Is balanced growth possible in the Solow model in which labor and
technology are constant? Why? Justify your answer.
2. In simple language, what does the Contraction Mapping Theorem say? Why is it useful for developing
dynamic programming—both the theory and its application?
3. Provide a mathematical expression for the Solow residual. Explain term by term what the expression
says. What is captured by the residual?
4. Empirical fact: China’s richest provinces have caught up to the leading developed countries in terms
of income per capita. What is this an example of? Which of the empirical approaches developed in
class can be used to test for the existence of this type of “catch up”? Explain how the approach works.
5. Illustrate the important critique to first-generation endogenous growth models made by Jones (1995),
referring explicitly to which type of data was used to challenge the empirical predictions of such models.
Describe how the Jones’ criticism has been overcome in second-generation endogenous growth models.
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 how the human capital model first studied by Lucas (1988) also can be seen as providing a justification for AK-type growth. How does accumulation of human capital at a decentralized equilibrium
compare with the optimal path of human capital accumulation? Why is the decentralized equilibrium
path different from the optimal path?
8. Why is there equilibrium unemployment in the Shapiro and Stiglitz (1984) model? What is the source
of equilibrium unemployment in the Mortensen-Pissarides matching model? Are these explanations
related or not?
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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