Macro Qualifying Exam
May 2, 2011
Instructions. The exam consists of two parts. Please answer 3 out of 4 questions in Part I, and 7 out
of 8 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, 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!
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Part I: Modeling Exercises. Answer 3 out of 4 questions.
Exercise 1 Consider the following problem in sequence form. The problem is to maximize:
1
X
t
[ln (ct ) + ln (ct
1 )] ,
0<
< 1,
>0
t=0
subject to
kt+1 + ct = Akt , for all t
k0 > 0 given and c
0,
1
0<
< 1,
A>0
> 0 given.
Here ct is consumption at period t, and kt is capital stock at the beginning of period t. The current utility
function ln (ct ) + ln (ct 1 ) is designed to represent habit persistence in consumption. The idea is that the
utility agents derive from consumption depends not only upon how much they consume today but also upon
how big current consumption is relative to last period’s consumption. In order to represent this problem
recursively, it is necessary to have two state variables: capital and one-period-lagged consumption.
P1
1. Let v(k; c 1 ) be the value of t=0 t [ln (ct ) + ln (ct 1 )] for a consumer who begins period 0 with
capital stock k and lagged consumption c 1 and behaves optimally. Formulate the Bellman equation
in terms of v(k; c 1 ). (Note that "c" would then be lagged consumption "tomorrow", while k0 would
be capital tomorrow.) You should formulate the problem so the choice variable "today" is k0.
2. Derive the Euler equation. Be clear about the steps you take in doing this.
3. "Guess" that the value function takes the form v(k) = A, where A is a constant in R. Go through the
standard guess and verify steps and argue that the solution to the Bellman equation cannot take this
form. Be speci…c and justify your answer.
4. It turns out that the true value function takes the form v(k; c 1 ) = A + B ln(k) + C ln(c 1 ), where A;
B and C are constants in R. Use this information to compute the policy function for k0, given k and
c 1 , all as a function of A, B, and C. So, in particular, to answer this question, you do not need to
solve for A, B, and C.
5. Next, "guess" that the value function takes the form given in part 4. Verify that your guess is correct.
In doing so, you should come up with explicit values for A; B and C in terms of fundamental parameters
of the model.
6. Use your answer from part 5, together with your answer from part 4, to derive an explicit expression
for the policy function k0(k; c 1 )
in terms of fundamental parameters of the model.
7. Explain in 10 sentences or less how you 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 Consider the problem of maximizing
1
X
t
ln (ct ) ,
0<
< 1,
t=0
subject to
kt+1 + ct = Akt , for all t
0,
0<
< 1,
k0 > 0 given.
1. Write down the corresponding Bellman equation. Do 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.
2. De…ne the associated Bellman operator. Explain what it does: in particular, what is the input and
what is the output. Be speci…c.
3. How is this operator used to "solve" the Bellman equation? What is the justi…cation for using it
in this way? Be speci…c and ground your answer in the logic and justi…cation developed in class.
A correct answer will include the terms "contraction", "Contraction Mapping Theorem", and "…xed
point", among others.
4. Starting from an initial guess v (0) (k) = 0 iterate on the Bellman operator analytically (with algebra)
to get v (1) (k).
5. Iterate a second time to get v (2) (k).
6. Next, changing direction, "guess" that the value function takes the form v(k) = A, where A is a
constant in R. Go through the standard guess and verify steps and argue that the solution to the
Bellman equation cannot take this form. Be speci…c and justify your answer.
7. Now guess (correctly) that the value function takes the form v(k) = + ln(k), where and are
constants in R. Verify that the "guessed" form of the value function is correct. Be speci…c and justify
your answer.
8. Finally, using your answer to the last question, derive an expression for the policy function.
answer should be written in terms of fundamental model parameters (not and ).
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Your
Exercise 3 Consider the following endogenous growth model. The …nal (consumption) good is produced
according to the following technology:
Z Mt
Yt = L1Y
xi;t di
0
where LY denotes workers in the …nal sector, M is the number of varieties (designs) of intermediate products,
and xi are the intermediate products used for producing the …nal good. One unit of intermediate good is
produced using one unit of the …nal good, and each intermediate goods producer is a monopolist in its sector.
The monopolist …rm pays the interest rate r to each unit of input utilized. Workers can be employed either
in producing the …nal 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( ) preferences and maximize the present value of
utility over an in…nite horizon, taking into account the typical asset equation. There is no government, nor
foreign sector. Suppose that the production function for innovations is:
M_ = (M LM 1 )LM ;
2 (0; 1);
2 (0; 1)
where captures the amount of spillovers generated by past discoveries, and is meant to capture the fact
that researchers might engage in useless duplication activity while doing research.
1. 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=
1
L_ M
LM
2. In a decentralized (laissez-faire) economy, …rms take the productivity of each researcher, that is
(M LM 1 ) 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. 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 satis…es
r1
LY
=
LM
g
4. 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
r 1
sector. Show that s = 1+1 , where
g . Using the constancy of the ratio between workers employed
in the two di¤erent sectors, show that the growth rate of designs is proportional to the growth rate of
population, and not to the number of researchers.
5. Using the current-value approach, derive the Euler equation for a consumer discounting utility streams
at a rate equal to
n and maximizing the present value of utility streams under the typical asset
accumulation constraint. Use the Euler equation for consumption to …nd an expression for in terms
of the parameters of the model only.
6. Is economic policy aimed at in‡uencing the growth rate through increasing the number of researchers
e¤ective in this model? Use the expression you found for the growth rate of the economy to brie‡y
discuss.
7. 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 in…nite horizon under
the accumulation and the innovation constraint. What are the e¤ects 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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Exercise 4 Consider the following unemployment model. Workers maximize over an in…nite horizon the
utility function u(w; e) = w e. Labor e¤ ort e can assume only two values, e = 0 or e = 1. The interest rate
is denoted by r. There is an exogenous probability of the contract being terminated given by b 2 (0; 1), and
there is a costless monitoring technology that gives a probability of a worker supplying zero e¤ ort to be …red.
Denote this probability by q 2 (0; 1). Let VSE be the expected lifetime utility of a shirking employed worker,
VNE the expected lifetime utility of a non-shirking employed worker, and VU the expected lifetime utility of
an unemployed worker. Employees take VU as an exogenous variable. There is also a job acquisition rate
a 2 (0; 1), exogenous for the moment. Denote as z the unemployment compensation.
1. Using a limit argument, show that the in…nite horizon utility maximization for the shirking worker
yields the following asset equation: rVSE = w + (b + q) VU VSE
2. Using a limit argument, derive the asset equation for the non-shirking worker (Hint: keep the e¤ort
notation as e).
3. Solve the system formed by the two equations just derived for VNE and VSE .
4. Derive a no-shirking condition, and use it to obtain the wage the …rm has to o¤er to its employees. How
does the critical wage vary with (i) e¤ort? (ii) VU ? (iii) q? (iv) the interest rate? (v) b? Comment.
We now turn to aggregation. There are M identical …rms in this model. Production of output requires
only labor:
f (Li ); when e = 1
yi =
0;
when e = 0
Firms compete in o¤ering wage packages, subject to the no-shirking condition derived above. Assume
that the unemployment P
compensation is zero for simplicity. The total output produced in the economy
will be Y (L) = max(Li ) yi .
5. Derive the asset equation for the unemployed worker, using the fact that in equilibrium the expected
utility of an employed worker must equal VNE . Solve the system formed by the equation in Exercise
4.2 and this equation for the ‡ows of utility rVNE and rVu .
6. Derive the aggregate no-shirking condition relating the wage to e¤ort and all the parameters in the
model, namely a; b; q; r. Plot this condition in the plane (employment, wage). Comment on how the
critical wage reacts to the variables determining it.
7. Denote the total labor force as N
L. (i) Give a de…nition for u, the unemployment rate for this
economy. Assume that the job-acquisition rate is now endogenous and given by a = bL=(N L). (ii)
Derive the aggregate no-shirking condition using the unemployment rate you just de…ned.
8. (i) Show using the de…nition of a that no-shirking is inconsistent with full employment. (ii) Identify
the equilibrium wage for the economy by equating aggregate labor demand and aggregate labor supply
(You may want to ask yourself what is labor demand in this model …rst).
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Part 2: Short Essay Questions. Answer 7 out of 8 questions.
Your answer to each question in part 2 should not exceed 15 lines.
1. Use simple calculus to derive the precursor expression to Solow’s growth accounting relationship. By
“precursor expression," I mean the one that includes the elasticity of production with respect to capital
and the elasticity of production with respect to labor. Next, explain how this expression simpli…es when
the assumption is added that factor markets are perfectly competitive. Why is this step important
for being able to empirically implement the relationship? Give your …nal expression for the Solow
residual (the proposed estimate of TFP).
2. Explain the crucial di¤erences between Barro regressions and the regression equation of Mankiw,
Romer, and Weil (1992) in terms of how the equations were derived, what data they use, and what
question they are meant to answer.
3. Suppose the aggregate production function is given by F (K (t) ; L (t) ; A (t)) = AK (t), as in the AK
model; in particular, A is constant. Embedding this assumption in the standard Solow model with
constant population growth rate n, derive an expression for the growth rate of per capita income
(y) in this model. Suppose the parameters are such that sA
> n. Does this economy exhibit
sustained growth? If so, how can you reconcile this with our earlier observation that sustained growth
is impossible in the neoclassical model without exogenous technical change?
4. De…ne a balanced growth path and explain what a balanced growth path “looks like" in the Solow
model with exogenous growth in TFP.
5. Describe the main market failure in the Human Capital model studied by Lucas (1988). 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 di¤erent from the optimal path?
6. (a) Enumerate the basic Kaldor facts of economic growth, and (b) explain why the Neoclassical Growth
Model (NGM) provides a good explanation for these basic facts, but is not suitable to explain sustained
long-run growth in real GDP per capita. In particular, (c) using the familiar Euler Equation that holds
at a competitive equilibrium in economy i with population growing exponentially at a rate ni and zero
(or equal) capital depreciation for all countries, explain what can be the only sources of cross-country
income di¤erences according to the NGM.
7. Explain why: (a) in the presence of credit constraints, wealth inequality might be harmful for growth,
and (b) how can you partially challenge this conclusion by appealing to heterogeneous individual
abilities. Make sure to include sketched policy recommendations for both (a) and (b). Is inequality
harmful for growth when the most able individuals are also the poorest? Explain.
8. Illustrate the important critique to …rst-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.
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