Macro Qualifying Exam
January 15-16, 2015
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 an infinitely lived consumer with instantaneous utility function u(c),
which as usual is strictly increasing and strictly concave in c. The consumer discounts future utility with
discount factor β ∈ (0, 1). In addition, they start out in the initial period with a0 units of an asset, they
earn constant interest rate r on savings, and they also earn a constant income of w units in each period.
The consumer’s utility maximization problem can be written:
max ∞
∞
X
{ct ,at+1 }t=0
β t u(ct )
t=0
subject to
at+1 = (1 + r)at + w − ct , all t
a0 > 0 given.
(We will further assume that an appropriate transversality condition has been imposed to rule out “no Ponzi
schemes”, but we won’t worry about the technical details, so you should ignore this point if you find it
confusing.)
1. (20 points) Define stationarity. Is this problem stationary? Justify your answer.
2. (15 points) Write down the corresponding Bellman equation. Do so in such a way that the choice
variable today is the value of the state variable next period.
3. (20 points) Derive the Euler equation for the problem. Be sure to indicate all steps taken in doing this.
4. (15 points) Note that r and β are exogenous parameters in the model. Use the Euler equation to derive
a condition on the relative size of r and β under which the optimal path of consumption is constant.
5. (15 points) Use the Euler equation to derive a condition on the relative size of r and β under which
the optimal path of consumption is increasing over time: i.e., ct+1 > ct for all t.
6. (15 points) Suppose you were going to solve this problem numerically. To proceed, would you need to
make any additional assumptions? If so, suggest what assumptions you would make.
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Exercise 2 Consider a two period (non-stochastic) cake eating problem. Time s runs from s = 1, 2. In the
first period, you start with W1 units of cake, of which you consume c1 units. Each period you derive utility
u(c) = ln(c) from consuming cake. In the initial period, you discount utility in period 2 using the discount
factor β ∈ (0, 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) Write down a maximization problem that describes the decision in the second period. Define
this by a value function (i.e., the “continuation value” starting in period 2). Then write a Bellman
equation that uses the second period continuation value as an input to the first period decision.
3. (20 points) Use dynamic programming to solve for the optimal policy function in the first period.
For the remainder of the exercise, assume that the time horizon for the cake eating problem (with log
utility) is infinite.
4. (10 points) Write the infinite horizon problem in sequence form.
5. (10 points) Write the corresponding Bellman equation. Do so in such a way that the choice variable
today is the value of the state variable next period.
6. (10 points) Guess that the value function takes the form v(W ) = A ln(W ). Check to see if the guess
is correct.
7. (10 points) Guess that the value function takes the form v(W ) = A ln(W ) + B. Check to see if the
guess is correct.
8. (20 points) Use the correct guess to derive an expression for the optimal policy function. Use a graph
to compare the policy function for the infinite horizon problem with the policy function derived above
in part 3.
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Exercise 3 Consider the following Neoclassical growth model, where all lower-case variables are defined in
per-capita terms. There is a representative household with instantaneous preferences defined by the following
utility function:

 (c − c̄)1−θ − 1
, θ > 0, θ 6= 1
u(c) =
1−θ
 ln(c −
c̄)
θ=1
where c̄ > 0 denotes a subsistence level of consumption that is constant over time. The representative household does not accumulate debt, so that its assets at every moment in time are made up only of capital stock
(per capita) k(t), and discounts the future at a constant rate ρ > 0. Further, the representative household has
one unit of labor, supplied inelastically at the going wage w(t). Competitive firms produce a single good, homogeneous with capital stock, according to the intensive production function y(t) = Ak(t)α , A > 0, α ∈ (0, 1).
Capital stock is priced at r(t), and depreciates at a constant rate δ > 0. Population grows at a rate n > 0.
There is no technological change. We are interested in analyzing the implication of subsistence consumption
for a standard Neoclassical growth model.
1. (10 points) Show that the relative risk aversion coefficient −c[u00 (·)/u0 (·)] of the utility function is not
constant. How does relative risk aversion depend on current consumption and subsistence consumption?
Can consumption at any time t be below the subsistence level? Would relative risk aversion be constant
if c̄ = 0?
2. (20 points) Write down the optimal control problem for the representative consumer using the current–
value Hamiltonian approach, and find the Euler equation for consumption per capita.
3. (30 points) (i) Define a competitive equilibrium (CE) for the economy, and write down the dynamical
system describing the evolution of consumption per capita and capital stock per capita along the
CE. (ii) Show that one of the steady states of this model involves consumption per capita to be at
its subsistence level. (iii) How are the combinations of subsistence consumption, and corresponding
subsistence capital stock (c̄, k̄) such that k̇ = 0 = ċ, related to each other? Can you provide intuition
for this result?
4. (15 points) Illustrate qualitatively (that is, without explicitly linearizing the system) the dynamics
of the subsistence steady state of the model (c̄, k̄) using a phase diagram with capital stock on the
horizontal axis and consumption on the vertical axis. What does the phase diagram suggest regarding
the stability of the subsistence steady state?
5. (15 points) Calculate the steady state css , kss such that css > c̄. Show that subsistence consumption
does not matter in this case: that is, a model where c̄ = 0 would deliver the same result. Draw the
phase diagram and, as before, analyze qualitatively the dynamical properties of this steady state.
6. (10 points) Is the equilibrium path of this model Pareto-efficient? Why or why not?
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Exercise 4 Consider the following modified AK framework. Output Y is produced using capital stock and
a flow of an extracted, non-renewable resource R, according to the following production function Y (t) =
AK(t)R(t)φ , with φ ∈ (0, 1). Denote the resource stock as S, which is constrained to remain non-negative at
all times. There is no market for exchanging debt among consumers, capital accumulation follows the typical
transition equation, and we assume depreciation to be zero: K̇(t) = Y (t) − C(t). On the other hand, the
resource stock is depleted through extraction, so that Ṡ(t) = −R(t). Further, we assume total population
to be normalized to 1 and constant. The representative household discounts the future at a rate ρ > 0, and
has a CRRA(θ) utility function.
1. (20 points) State the representative household’s intertemporal utility maximization problem. What
are the control variables? What are the state variables? Find the first order necessary conditions for
an optimal control.
2. (10 points) Find the Euler equation for consumption. What is the growth rate of the shadow price of
the resource? Can you provide an economic interpretation for this result?
3. (20 points) Is it possible for this economy to sustain positive consumption growth indefinitely? What
is the role played by the stock of resource in the answer? What is the consumption growth rate onto
which this economy will eventually settle? We will refer to this consumption path as asymptotic.
4. (20 points) Is the asymptotic consumption path of this economy equitable from an intergenerational
standpoint? Explain.
5. (20 points) Would your conclusions change if you considered a production function of the kind Y =
AK α Rφ , with α ∈ (0, 1)? Derive your answer analytically.
6. (10 points) What are the intergenerational equity features of the asymptotic consumption path of this
alternative economic environment?
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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, not including equations.
1. State Euler’s theorem for Y = F (K, L), a generic constant returns to scale function. Next, suppose
F (K, L) = K α L1−α , where α ∈ (0, 1); show that Euler’s theorem holds for this function.
2. State the Contraction Mapping Theorem. Consider the mapping (or function) T : R → R defined by
T (x) = 0.7x − 1. Is T a contraction? Justify your answer.
3. Explain three key differences between the application in Mankiw, Romer, and Weil (1992) and the
application in Barro (1991).
4. List the so-called Kaldor facts. Build on your answer to define a Balanced Growth Path.
5. Consider a simple AK growth model with no depreciation, no public sector, no population growth,
CRRA(θ) preferences for the representative consumer, and constant discount rate equal to ρ < A. We
know that in this case the consumption Euler equation is ċ/c = θ−1 (A − ρ), and that kk̇ = A − c/k.
Using the very definition of the saving rate s ≡ (Y − C)/Y , what can you say about the role played
by s on the growth rate of the economy along a balanced growth path (BGP)? Would the saving rate
play a role in the BGP of a corresponding Neoclassical or Solow growth model?
6. Illustrate the role of transitional dynamics in explaining conditional convergence in Neoclassical growth
models. Do endogenous growth models such as the one by Romer (1990) or Jones (1995) feature either
conditional, or even absolute convergence? Why or why not?
7. Explain what do growth economists mean when they refer to the market size effect as opposed to the
price effect, and why the difference matters to explain the features of skill–biased technical change
(SBTC).
8. Give two examples of ‘market failures’ that give rise to models of long run growth that are directly
comparable to an AK model, and briefly illustrate the welfare properties of these models’ decentralized
equilibrium paths.
9. You developed an m–equation model in continuous time based on dynamic optimization. The linearization of the model’s dynamical system returns n eigenvalues with real parts that are greater than zero.
How many forward–looking (or ‘jump’) variables do you need for the model to be conditionally (that
is, saddle–path) stable? Does the textbook matching model meet this criterion? Explain by explicitly
referring to the forward–looking and the backward–looking variables in the model.
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