Macro Qualifying Exam
January 14, 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 You are the “state economist” for Alaska, and you are tasked with determining the optimal
extraction path for the state’s oil reserves. In the current period, call it period “0”, there are R0 units of
oil. We assume that all oil extracted in a period is consumed within that period. We let rt denote the flow
rate of resource extraction in period t. By definition, Rt+1 = Rt − rt . The economic value to the state from
extracting “r” units of oil in period t is ξt P (r), where P (·) is strictly increasing and strictly concave, and ξt
is a stochastic process that follows a known Markov chain. The discount factor β accounts for the alternative
return on safe investments in the state treasury.
1. (10 points) What is the state variable (or state variables) for the problem?
2. (10 points) Write down the Bellman equation for the problem. Be sure to construct it in such a way
that the choice variable “today” is the value of the state variable the following period. Use primes to
indicate the value of variables in the following period: so if x were the value of a variable “today”, x0
would be the value the following period. If necessary, define any mathematical objects that are not
self explanatory.
3. (15 points) Derive the corresponding Euler equation. Be sure to indicate all the steps you take in doing
this.
4. (5 points) In words, give intuition for the Euler equation. What does it “say”?
For the remaining questions, assume that there is no uncertainty: ξt = 1 for all t ≥ 0. Also, assume
that the payoff function, P , is logarithmic.
5. (25 points) For this simplified problem, guess that the value function takes the form P (R) = Aln(R)
for some constant A. Use the usual steps to check if the guess is correct. If it is incorrect, then propose
an alternative guess and follow the same steps to prove that your guess is correct. Once you have
determined the correct guess, use it to derive an explicit formula for the value function.
6. (20 points) Provide an explicit formula for the policy function and plot it.
7. (15 points) Solve explicitly for the long-run value of the state variable.
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Exercise 2 Consider the following (relatively) simple model which might be used to evaluate climate change
policy. It is a modified version of the Brock-Mirman optimal growth example (i.e., the Neoclassical Growth
Model with log utility, Cobb Douglass production, and 100% depreciation). In the model, energy e is used as
an input to production. The amount of energy used in production is assumed to equal the flow of pollution
(greenhouse gases) into the atmosphere. Pollution in turn “accumulates” in a very simple way: yesterday’s
emission flow becomes tomorrow’s pollution stock. In particular, letting s denote the stock of pollution in
the atmosphere, st+1 = et . The pollution stock causes damages that have economic consequences. This is
indicated by a damage function, ω(s) = exp(−s), which enters the model by multiplying output. Output,
net of damages, takes the following form:
F (k, e, s) = k α e1−α ω(s).
Here, k is the capital stock, e is the energy input to production, and ω(s) are damages resulting from
pollution stock s (inherited from the prior period).
In sequence form, this optimal growth problem with a pollution externality can be stated as follows:
max ∞
{et ,kt+1 }t=0
∞
X
β t ln(ct )
t=0
subject to
kt+1 = F (kt , et , st ) − ct , for all t ≥ 0,
st+1 = et , for all t ≥ 0,
s0 ≥ 0 and k0 ≥ 0 given.
1. (20 points) Indicate the state variables for the problem, and write the Bellman equation. In doing so,
substitute the constraints for the endogenous variables and write the problem in such a way that the
“choice” variables today are the values of the state variables “tomorrow”.
2. (15 points) Guess that the value function takes the following form:
V (k, s) = A + B ln(k) − Ds,
for constants A, B and D. Using this guess, take the first-order conditions for both control variables.
Leave your answer in the “raw” form that you get when applying the standard derivative rules. Don’t
simplify.
3. (15 points) Leave the FOC for s0 in its raw form above. Then solve for k 0 from the FOC for k 0 .
Substitute the latter equation into the former, and in doing so, prove that the optimal choice of e in
any period is a constant that does not depend on the inherited values of the state variables. For the
remainder of the problem, refer to this constant level of e as e∗ .
4. (25 points) Verify that the guessed form of the value function in part 2 is correct.
5. (25 points) Derive an explicit formula for the policy function for physical capital. In general terms,
how does this compare to the policy function for physical capital in the standard Brock-Mirman model
discussed in class? (Do not solve for the optimal policy function in the standard model; rather, just
recall the general form of the policy function in that model if you can.)
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Exercise 3 Consider the following growth model. A final good is produced according to Y = AL1−α K α .
If we define k ≡ K/L, then Y = ALk α . The labor force L grows at a rate n. Capital in this model is made
only of information technology (IT) goods that do not depreciate and benefit from knowledge spillovers.
−β
Producing one additional unit of capital requires 1q = K
units of the final good. Both production of the
B
capital good and of the final good are competitive, and individual firms do not internalize spillovers (that
is, they take q as given). Consumers discount the future at a rate ρ > 0, and their instantaneous utility
function defined in units of the consumption good is u(ct ) = ct for all t. Because of the linear utility, r = ρ
always.
1. (20 points) What is the price p(t) of a unit of capital? Also, firms will invest in new capital goods up
to the point at which the cost of a unit of capital is equal to its benefit, that is up to the point where
Z ∞
∂Y
p(t) =
e−r(τ −t)
dτ
∂K
t
By differentiating this condition with respect to time, derive an asset equation for p similar to the ones
you are familiar with in optimal control theory.
2. (20 points) Use the asset equation you just derived to define the growth rate of the price of capital
goods, gp . Use this growth rate together with the demand for capital per worker as a function of its
price you got in question 1 to show that
gp = −
β
β
β(1 − α)
n, gk =
n, gc = α
n
1−α−β
1−α−β
1−α−β
where gk , gc denote growth rates of capital per worker and consumption per capita respectively.
1−α−β
3. (20 points)Let z = KL1−α . Differentiate z with respect to time to show that the evolution of z
over time satisfies the equation ż = −az + b, and determine the value of the two constants a, b. The
parameter a represents the speed of convergence toward a steady state. How is this parameter affected
by α, β? Comment.
4. (20 points) Introduce now a second capital good, also produced competitively, denoted by X such that
Y = AL1−α K α X γ = ALk α xγ , where x ≡ X/L. The good X is produced one for one with the final
good, and depreciates at a constant rate equal to δ. Hence, the gross rental rate for this good must
equal the interest rate plus the depreciation rate.
∗
∗
What is the demand for X? Use this demand to show that output can be written as Y = A∗ L1−α K α ,
and determine the value of A∗ , α∗ .
5. (20 points) Show that the previous analysis carries through this more general case when A∗ α∗ are used
instead of A, α. Is the value of the speed of convergence a affected by this new technology specification?
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Exercise 4 Consider the following 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. We consider the role of capital stock in the framework. The labor productivity parameter p
that you are familiar with in the basic matching model is reinterpreted as labor-augmenting technology,
exogenous, that converts labor into effective labor. Denoting aggregate capital and employment by K, N
respectively, the aggregate production function is F (K, pN ), with diminishing returns to capital and effective
labor, and constant returns to scale. Denote output per effective worker by f (k) ≡ F (K/(pN ), 1), which
satisfies the usual Inada conditions. Further, denote the depreciation rate on capital stock by δ > 0. Finally,
we assume that there is a perfect market for second-hand capital goods, so that it makes no difference to
own or rent capital from a firm’s perspective, as the rental price of capital is equal to its rate of return.
A vacancy with expected present discounted value V costs cp per unit time, c > 0. The difference that
capital stock makes is in the determination of the expected present discounted value of a matched job. In
fact, when a job match occurs, the firm hires capital k for each efficiency unit of labor. The capital stock
owned by the firm becomes part of the value of the job, so that the asset value of an occupied job is given
by J + pk.
The expected present discounted value of unemployment is denoted by U . Jobs are destroyed at an
exogenous rate λ. 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, and we
focus on steady state values of capital stock.
1. (10 points) (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. (10 points) (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. (30 points) (a) Write down asset equations for V and J + pk. (b) What must be the value of V in
equilibrium, and why? (c) By maximizing over capital stock per worker the equation describing the
value of an occupied job, show that the typical condition f 0 (k) = r + δ must hold. (d) Considering the
equilibrium value of vacancies, show that the job creation condition for this model is:
p[f (k) − (r + δ)k] − w =
(r + λ)pc
q(θ)
and provide an economic intuition for it.
4. (10 points) 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 λ, and the vacancy/unemployment ratio θ.
5. (10 points) 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. Since the market for capital goods
is perfect, the introduction of capital makes no difference as far as wage determination is concerned.
Solve the Nash bargaining problem and show that β is the share of workers in the total surplus created
by a job match.
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6. (15 points) (a) Show that the surplus-maximizing wage depends only on rU, β, p[f (k) − (r + δ)k], and
is therefore equal for all firms. (b) Solve for rU as a function of β, p[f (k) − (r + δ)k], c, and θ, and (c)
write down the equilibrium wage equation representing the wage as a function of θ, capital stock, and
the other parameters of the model.
7. (15 points) (a) Define in one sentence the equilibrium of the labor market, and (b) write down the four
equations that solve for an equilibrium. (c) Briefly explain what differences the introduction of capital
stock makes relative to the benchmark model with no capital.
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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. The three main empirical approaches for taking the Solow model to the data that we discussed in class
were (1) growth accounting, (2) Barro’s convergence test, and (3) the approach in Mankiw, Romer, Weil
(1992). Which of these does not require that one run a regression? Explain how the approach works.
What question is it intended to answer? What type of data is needed to implement the approach? Be
specific.
2. Explain what a state variable is in a dynamic economic model. Recall the stochastic growth model
with an endogenous labor-leisure decision that we discussed at the end of the semester. What is the
state variable (or state variables) in this model?
3. Suppose your research brings you up against a dynamic programming problem for which the guess and
verify method is unavailable. Explain in as much detail as you can (within the allowed space) how you
would proceed. Be specific and indicate all the necessary steps.
4. Define a contraction. Then use your definition to check if the following function is a contraction on
the set of positive real numbers:
T (y) = .8y + 1.
Use a graph to justify your answer.
5. Describe in words the basic reason why inequality is harmful for growth in political–economic equilibrium models such as the ones by Persson and Tabellini (1994) or Alesina and Rodrick (1994). What is
the role played by progressive taxation in this framework?
6. Explain how the human capital model first studied by Lucas (1988) 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?
7. Does the Shapiro-Stiglitz model explain frictional or structural unemployment? What type of unemployment is the focus of the Mortensen-Pissarides model? Do these models deal with the same type of
unemployment?
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. Briefly explain the debate between proponents of institutions as the fundamental cause of economic
growth on the one hand, and proponents of a human-capital based explanation. Further, another
influential take on the issue has to do with the role of geography as opposed to institutions. Briefly
summarize the main conclusions of an important AER paper by Acemoglu, Johnson and Robinson
that deals with the issue.
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