Macro Qualifying Exam
July 18, 2014
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 continuous-time Solow model where population, N , grows at constant rate n > 0,
total factor productivity, A, grows at constant rate g > 0, and households save a constant fraction, s ∈ (0, 1),
of disposable income each period. Let K denote the aggregate stock of capital, and let
F (A, K, N ) = K α (AN )1−α
be the aggregate production function. The law of motion for aggregate capital is
K̇ = sK α (AN )1−α − δK.
Finally, suppose that the economy starts with initial capital stock K0 , initial TFP level A0 and initial
population N0 .
1. (10 points) Define a balanced growth path.
2. (30 points) Prove that this economy converges to a balanced growth path. Show all relevant steps in
your derivation. A correct answer will include an algebraic derivation, at least one appropriate graph,
and supporting discussion.
3. (20 points) Modify your analysis of the last question for the case in which g = 0. Show all relevant
steps in your derivation.
4. (15 points) What is the growth rate of income per capita in the long run? Justify your answer and
show all relevant steps in your derivation.
5. (15 points) What is the interest rate in the long run? How does it vary over time? Justify your answer
and show all relevant steps in your derivation.
6. (10 points) Would the above economy converge to a balanced growth path if the production function
were changed to
F (A, K, N ) = (AK)α (N )1−α .
Justify your answer.
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Exercise 2 Consider the following stochastic cake eating problem in recursive form:
V (W, δ) =
max
0≤W 0 ≤δW
ln(δW − W 0 ) + βEδ0 |δ V (W 0 , δ 0 ),
where δ is stochastic and described by the following Markov process:
ln δ 0 = ρ ln δ + , where E = 0.
(1)
(Hint: It may be useful at some point to use the fact, implied by (??), that Eδ0 |δ ln δ 0 = ρ ln δ.)
1. (20 points) Derive the stochastic Euler equation. Indicate all necessary steps.
2. (15 points) Guess that the value function takes the form V (W, δ) = a + b ln(W ). Prove if this guess is
correct or not.
3. (15 points) Guess that the value function takes the form V (W, δ) = a + b ln(W ) + c ln(δ). Prove if this
guess is correct or not.
4. (20 points) Derive an explicit expression for the value function.
5. (15 points) Derive an expression for the policy function.
6. (15 points) Comparing the policy function with the stochastic Euler equation, is there any way to tell
that these different formulas are characterizing the answer to the same question?
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Exercise 3 Consider the following growth model. There are j = 1 . . . , N competitive firms in the economy. Each firm produces output Yj according to the following technology: Yj = ĀKjα L1−α
. Output is
j
homogeneous with capital stock, and homogeneous across sectors. Labor is also homogeneous across sectors.
Depreciation is zero, and population grows at a rate n > 0. Denoting
intensity in sector j as
P thecapital
η
N
kj ≡ Kj /Lj , the stock of technology is assumed to satisfy: Ā ≡ A0
k
,
which
captures the fact that
j=1 j
the more workers are endowed with capital stock, the more productive the system becomes. The individual
firm takes Ā as given, even though capital accumulation in each firm generates knowledge spillovers across
firms. The parameter η describes the extent of the knowledge spillovers.
1. (10 points) Denote aggregate capital stock per worker by k and total output per worker by y. Show
that kj = k/N for all j.
2. (10 points) What is the implication of what you just found for the aggregate technology parameter Ā?
Show that output per worker in the economy is y = Ak α+η , where A depends directly on the number
of firms in the economy.
We now turn to households. There is a representative dynasty of consumer households with CRRA(θ)
preferences defined over consumption streams. Households receive both wage and profit income, and
can accumulate assets a(t) ≡ k(t) + b(t), where b(t) denotes household debt.
3. (10 points) Write the accumulation constraint for the representative household, and the usual No Ponzi
Game condition that must be fulfilled for household debt not to explode over time.
4. (20 points) Find the Euler equation for consumption.
5. (15 points) Define a market equilibrium for the economy. Write down the Euler equation and the
accumulation equation for capital stock, both evaluated at a market equilibrium.
6. (15 points) Define a balanced growth path (BGP) for the economy. By looking at the growth rate along
a BGP, show that this model encompasses both the Neoclassical Growth model and the AK model as
special cases, depending on the magnitude of the parameter η with respect to (1 − α). Interpret your
results.
7. (20 points) Solve the problem faced by a benevolent planner that internalizes the knowledge externality.
Show that the resulting efficient path involves a growth rate that is strictly above the growth rate in
the decentralized economy. Provide intuition.
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Exercise 4 Consider the following one sector growth model. Assume that individuals have linear intertemporal preferences
over consumption streams, so that the present discounted value of utility over an infinite
R∞
horizon is 0 e−ρτ yτ dτ , where τ is an index for (continuous) time. Each of the L individuals in an economy
is endowed with one unit flow of labor which s/he provided inelastically, so that L is also total labor supply.
Population is constant.
Output of the consumption good is produced according to the technology y = Axα , α ∈ (0, 1), where
x is the amount of the intermediate good used. An innovation develops a new quality of the intermediate
good that replaces the old one, and raises the productivity parameter by an amount γ > 1. Subsequent
intermediate product generations are indexed by t = {0, 1, 2, ...} =
6 τ ∈ [0, ∞).
The economy’s labor supply L has two competing uses. It can be utilized in manufacturing one for one,
or in R&D. Denote R&D workers by n.
In the basic Schumpeterian model, it is assumed implicitly that the intermediate good monopolist can
charge any price to the final good producers without having to fear entry by potential competitors. Suppose
instead that there is a competitive fringe of firms that produces an intermediate good perfectly substitutable
with that of the intermediate monopolist. This competitive good requires χ > 1 units of final output to
produce.
1. (5 points) Write the labor market clearing condition that must hold at any period.
When n workers are employed in R&D, innovations arrive at a Poisson rate equal to λn, where λ >
0 indicates the productivity of the research technology. A firm which is successful in innovating
monopolizes the intermediate goods sector.
2. (5 points) Derive, and explain why the amount of labor devoted to R&D is determined by, the following
arbitrage condition:
wt = λVt+1
where t denotes product generations, w denotes the wage, and Vt+1 denotes the value of the next
generation innovation.
3. (10 points) What is the asset equation determining Vt+1 ? Provide an interpretation.
Next, Consider the role of the competitive fringe in the model.
4. (5 points) What is the maximal price that the intermediate monopolist can charge for its product as
compared to the competitive fringe’s production cost χ? Is this limit price binding if χ ≥ 1/α? Is it
binding if χ < 1/α? Explain.
5. (15 points) Consider the case where the limit price is binding (this is called in the literature the
non–drastic innovation case). Show that the monopolist’s equilibrium profits at time t fulfill:
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Πt = (χ − 1)(α/χ) 1−α
and show that profits are increasing in the competitive fringe’s price χ.
6. (10 points) Write down a system of difference equations describing the evolution of researchers and the
evolution of the productivity-adjusted wage over time.
7. (10 points) Show that the system above has a unique steady-state equilibrium in which the productivity
adjusted wage and the number of researchers are constant over time.
8. (5 points) (a) Considering the consumption good at time τ , by how much is its logarithm increased
each time an innovation occurs? (b) What is the average growth rate of the economy in steady state?
9. (15 points) Determine the effect on the equilibrium growth rate of the economy and on the steady
state population of researchers of i) an increase in labor supply; ii) a decrease in the interest rate;
iii) a decrease in the degree of market competition; iv) an increase in the innovation size γ and v) an
increase in the productivity of R&D λ.
10. (20 points) State and solve the problem of a social planner choosing x, n to maximize the steady
state utility of a representative consumer in the economy. Make sure you identify the main sources of
difference between the decentralized allocation and the optimal one.
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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. A key stylized fact regarding the growth experience of the United States over the past 200 years is
that it has grown consistently over the long run. What assumption or assumptions is/are needed in
neoclassical growth theory to generate a model consistent with this observation? Justify your answer
with words.
2. Describe the calibration approach used in the Real Business Cycle literature. Be specific and be sure
to clearly explain how the predictions from the stochastic growth model are compared to real data in
this literature.
3. Suppose you want to determine the portion of recent growth in China that is attributable to capital
accumulation. What empirical approach would you use? What data would you need to implement
this?
4. Use words to describe the regression equation estimated by Mankiw, Romer and Weil (1992)? What
research question did the authors seek to answer? What data is needed to implement their approach?
5. Consider the elasticity of GDP to physical capital, denoted by α. (i) Is there a direct or inverse
relationship between α and the speed of convergence to the steady state in the Solow model? (ii)
Suppose you estimated α to be in the range [.7, 1]. Is this problematic for the Solow model? Why?
(iii) How can you ‘augment’ the Solow model to make sense of the issue?
6. By focusing on the role of the saving rate in the growth process, illustrate the main differences between
the learning by doing–based AK growth model by Frankel (1962, with constant saving rate) on the one
hand, and the Solow (1956) growth model on the other.
7. It is often argued that endogenous growth is incompatible with perfectly competitive markets and
constant returns to scale (CRS) in production. (i) Explain the reason why this is so; (ii) give an
example of a production function that violates CRS but can generate endogenous growth even with
perfectly competitive markets.
8. Why is the market equilibrium in the Romer (1990) model not Pareto–efficient? Explain.
9. Why is the market equilibrium in the basic Pissarides (2000, Chapter 1) matching model in general
not Pareto–efficient? What are the key parameters to consider in this case?
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