Macro Qualifying Exam
January 10, 2011
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 quesion 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 stongly encouraged to work out your initial algebra attempts on scrap paper so your …nal answer
is clean and easy to grade; your …nal 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 di¤erent 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 o¤ moving on and trying to return to it later.
Good Luck!
Part I: Modeling Exercises. Answer 3 out of 4 questions.
Exercise 1 The lone survivor of a ship wreck, Rousseau, is stranded on an island with a cargo load of canned
food. There is plenty of water on the island, but no other food source. The island is in the middle of nowhere
and there is no chance of being rescued, so she makes plans to allocate her food into the inde…nite future
(her planning horizon is in…nite). Rousseau discounts the future using a constant discount rate 0 < < 1
and her utility over food consumption in each period is logarithmic. Finally, the canned food is perfectly
nonperishable; this means that if Ft is the stock of food in period t, and she consumes ct units, then there
will be
Ft+1 = Ft ct
units of food leftover in the following period.
1. Formulate Rousseau’s optimization problem in sequence form. Your answer should clearly state what
her objective function is, what she is maximizing over, and what the relevant constraints are.
2. Next, translate the sequence problem into an equivalent dynamic programming problem. In doing this,
formulate the problem so there are two variables in each period: F and c. Throughout the exercise
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.
3. Specify which are the state variables and which are the control variables in your answer to part b.
4. Rewrite the Bellman equation in part b so that the control variable is the value of the state variable
"tomorrow".
5. It turns out that the value function in the (correctly stated) Bellman equation associated with question
d takes the following simple form:
V (x) = a + b ln(x);
where x is a placeholder for the state variable. Use this information to derive an explicit expression
for the policy function.
6. Plot the policy function.
7. Verify that the "guessed" form of the value function is correct.
8. Explain in 10 sentences or less how you how would go about solving the problem numerically if you
did not know the "guessed" form of the value function from part 5.
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Exercise 2 In sequence form, the nonstochastic growth problem has the planner choose a consumption
sequence fct g to maximize
1
X
t
u (ct ) , 0 < < 1,
t=0
subject to
kt+1
k0
= f (kt ) + (1
)kt
ct
> 0.
It is assumed that u ( ) and f ( ) are both strictly increasing and strictly concave.
1. How many state variables are there in this problem? Name them/it.
2. Construct the associated Bellman equation, doing so in such a way that the choice variable "today"
is the value of the state variable "tomorrow". For notation, 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.
3. Use your Bellman equation to derive the Euler equation for the problem. Your answer should clearly
explain the steps that you take to do this.
4. Interpret the Euler equation derived in part (c). What does it say?
For the remaining questions, suppose the utility and production functions take the following forms:
c1
1
,
(1)
f (k) = Ak .
(2)
u(c) =
5. Solve explicitly for the steady state level of capital.
6. Solve explicitly for the steady state level of consumption.
7. Building on the problem above, suppose that the level of technology follows an AR(1) process:
At+1 = At + "t ,
where 0 < < 1 and "t is a normally distributed random variable with known variance. Write
down the Bellman equation for the augmented problem. Be sure to clearly de…ne any additional
mathematical objects that you introduce.
8. Derive the stochastic Euler equation for this problem. Be sure to use the speci…ed functional forms
(1) and (2) above, and clearly explain the steps that you take to get to your answer.
9. Interpret the stochastic Euler equation— what does it say?
10. Suppose you were going to numerically solve the functional equation de…ned in your answer to question
2.7. Explain why the form of the random variable A might pose a problem, and explain how you would
propose to get around this di¢ culty. Your answer should be no more than 3 or 4 sentences.
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Exercise 3 Consider the following multisector framework, set in discrete time. The …nal good sector produces competitively using labor and a continuum of specialized intermediate products i 2 [0; 1], according to
the following Cobb-Douglas production function:
Z 1
Yt = L1
A1it xit di;
2 (0; 1)
(3)
0
The labor force is constant and normalized to 1. We assume that the …nal good is storable in the form of
capital, and that intermediate goods are produced using capital: xit = Kit for all i 2 [0; 1], where Kit is the
amount of capital in sector i used as input to produce xit at each time t. The market for capital consists
of the historically
predetermined supply of capital stock Kt , and aggregate demand as the sum of all sectors’
R1
demands 0 Kit di. Intermediate producers are monopolists in each sector, and using a unit of capital in
production of the intermediate good costs rkt , equal for all sectors.
1. Write and solve for xit the pro…t maximization problem for the intermediate producer. Denote the
intermediate producer’s pro…t by it .
R1
2. Let At
Ait di, and let kt
Kt =At , which denotes the capital stock per e¤ective worker since
0
L = 1. From the equilibrium in the market for capital, (i) show that the rental rate satis…es:
rkt =
2
kt
1
(4)
and (ii) solve for xit as a function of the sector-speci…c productivity parameter and kt only.
3. Write the pro…t function, that is the value of maximized pro…ts, for each intermediate good monopolist,
and show that the productivity-adjusted pro…t function ~ (kt )
it =Ait is increasing in kt . Provide
intuition for why this is the case (Hint: use equation (4) as a starting point).
4. Use the pro…t-maximizing value of intermediate products you found in 2.(ii) and integrate across sectors
to show that the resulting aggregate production function is the same as in the basic Neoclassical model:
Yt = At kt .
Next, we model the innovation sectors. In each sector i there is an entrepreneur who can possibly
innovate. If successful, she will become the local monopolist next period, achieving a productivity parameter
Ait = Ai;t 1 ; > 1. The probability of success depends on the expenditure on R&D, denoted by Rit
relative to the target productivity level Ait
Ait . Letting nit Rit =Ait , we assume that the probability
of innovation satis…es (nt )
nit ; 2 (0; 1). The innovating entrepreneur chooses how much to spend on
R&D to maximize expected pro…ts.
5. Write the innovator’s pro…t maximization problem, and solve the R&D arbitrage condition for nit .
Show that nit is equal for all i, and show it is increasing in the capital stock per e¤ective worker.
Provide intuition for this result.
6. Write the expected growth rate gt for this economy, and show that it is increasing in capital per e¤ective
worker. Denote gt = g~(kt ); g~0 > 0.
The …nal step is to determine the evolution of capital stock over time. For simplicity, assume that the
savings rate is constant and equal to s, as in the Solow model. Capital depreciates at the constant rate
> 0.
7. (i) Write down a …rst order di¤erence equation describing the evolution of aggregate capital stock
(remember that the model is set in discrete time). (ii) Write down a …rst order di¤erence equation
describing the evolution of the productivity parameter over time.
8. Use your …ndings in answering question 3.7 to show that the following holds: kt+1 [1 + g~(kt+1 )] =
skt + (1
)kt
9. De…ne a steady state of the model, and solve explicitly for it.
10. Plot the two sides of equation (??) as separate functions, and show that this model generates convergence to a steady value of capital stock.
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Exercise 4 Consider the following basic matching model, set in continuous time. The process of trade
between workers looking for jobs and …rms with vacancies is described by the following iso-elastic matching
function:
M (uL; vL) = (uL) (vL)1
where 2 (0; 1), L denotes total labor supply, uL is the number of unemployed workers and vL is the number
of vacancies. A matched job with expected present discounted value J produces output per worker per period
equal to A, and a vacancy with expected present discounted value V costs cA per unit time, c > 0. The
expected present discounted value of unemployment is denoted by U . Jobs are destroyed at an exogenous
rate b. The vacancy/unemployment ratio is considered as a separate variable and denoted by 2 [0; 1].
Employed workers receive a wage w for their labor services, and the interest rate is …xed exogenously at r.
The unemployment compensation is denoted by z 2 (0; w). Di¤ erently from the basic matching framework,
we assume that V; U; J are all time-varying. As usual, a dot over a variable denotes its time derivative.
1. (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 , and
provide an economic intuition behind the signs of these elasticities.
2. (a) Write down a di¤erential 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; ) plane? and how does it slope in the (u; v) plane?
3. (a) Write down asset equations for V and J, recalling that they are now time-varying. (b) What must
be the value of V and V_ in equilibrium, and why? Use this observation to solve for J as a function
of hiring costs and mean duration of a vacancy, and to …nd a job creation condition. (c) Write the
di¤erential equation describing the evolution of J over time after taking into account the equilibrium
condition on the value of vacancies.
4. Let W denote the time-varying present discounted value of expected income streams for an employed
worker. (a) Write asset equations for U and W , and (b) give steady-state solutions for rU; rW in terms
of the wage rate w, the job destruction rate b, and the vacancy/unemployment ratio . (c) Show that
employed workers enjoy a pure rent over unemployed workers, and brie‡y provide intuition for this
result.
5. We now turn to wage determination. The asset equations for J and W are …rm-speci…c, because they
depend on the …rm-speci…c wage wi , i = 1; : : : ; N being an index for …rms. Workers and …rms 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 ) (J(wi )
V )1
where is workers’bargaining power, while U is independent from wi and satis…es the equation you
found above in which w is the average wage in the labor market. Solve the Nash bargaining problem
and show that is the share of workers in the total surplus created by a job match. We assume that
wages are renegotiated continuously. This implies that the solution for the bargaining problem holds
both in levels and in rates of change.
6. (a) Show that the surplus-maximizing wage depends only on rU; ; A, and is therefore equal for all
…rms. (b) Solve for rU as a function of ; A; c, and , and (c) write down the equilibrium wage equation
representing the wage as a function of and the parameters of the model. Although this can be shown
from the results of question 4.5, we assume that the wage equation holds in and out of the steady state.
7. Allowing for time-varying asset values enables us to study the dynamics of the Beveridge curve you
found in answering 4.2. Use the …rst-order condition on wage-determination and the job creation
condition to derive a di¤erential equation in the vacancy-unemployment ratio.
8. Study analytically the stability properties of the dynamical system in (u; )-space described by the
equation you found answering 4.2 and the equation you found answering 4.7.
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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 20 lines.
1. Suppose you are studying economic growth in Ireland for the 30 year period leading up to the recent
crisis, and suppose that you would like to understand the extent to which physical capital accumulation
has contributed to observed growth. Describe the empirical strategy that you would adopt to answer
this research question. What kind of data would you need? What theoretical structure would you use
to interpret the data? Be as speci…c as possible.
2. Mankiw, Romer, and Weil (1992) estimated the following regression equation:
log yj
= constant +
+
1
log (sk;j )
1
log (sh;j )
1
1
log (nj + g +
log (nj + g +
h)
k)
+ "j ,
where j indicates a speci…c country. Describe the question the authors were interested in addressing,
and explain (speci…cally) how one would use this regression equation to answer their question— for
example, your answer should discuss what data you would need to collect to run your regression and
how you would use this data to construct the relevant series to run the regression.
3. State the two Neoclassical technology assumptions discussed in class.
4. In as much detail as you can, explain where the regression equation used by Barro (in "Barro regressions") came from. What research question are these regressions typically used to try to answer?
5. Explain why, although it provides a …rst useful framework to explain persistently positive growth rates
of GDP per capita, the basic AK model cannot explain cross-country convergence in growth rates.
Illustrate brie‡y the recent, terms of trade-based argument put forward by Acemoglu and Ventura
(2002) according to which an AK open economy framework can explain convergence.
6. Explain the fundamental di¤erence between endogenous growth models based on product variety and
Schumpeterian models. Is there any creative destruction in the Romer (1990) model? Why or why
not?
7. Describe the fundamental features of General Purpose Technologies (GPT). Make sure you explain
why: GPT usually generate: i) productivity slowdowns; ii) increasing wage inequality; iii) creative
destruction.
8. Illustrate the logic of using European settler mortality rates in colonized countries as an instrument
for explaining the e¤ect of economic institutions on economic growth in those countries.
9. What is the source of long-run unemployment in e¢ ciency wage-models such as the one by Shapiro
and Stiglitz (1984)? What type of frictions cause long-run unemployment in matching models à la
Mortensen and Pissarides (Pissarides, 2000, Chapter 1)? Why does an increase in workers’non-labor
income cause higher unemployment in both?
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