Econ Assignment

Homework 4 (Practice Final)
Prof. Saki Bigio
Due Date: June 6, 2022
READ THESE INSTRUCTIONS before you start writing: The exam has 5 questions. All ve
are mandatory to recieve full credit for the exam. Your maximal score is 120 points. Use your time
wisely.
You have 180 minutes to nish the exam.
NOTE THAT POINTS VARY BETWEEN QUESTIONS.
DO NOT OPEN UNTIL THE EXAM BEGINS
Name:
UID:
Grade:
Q1 ……………..
Q2
Q3
Q4
Q5
TOTAL
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1. Multiple Choice Question (20 Points – 4 points each).
1.A Assume that a Peruvian company, DMB LLC, just reported its earnings this year. The reported revenue was $10 million and the reported cost was $9 million. The discount rate is
8%. Mark ALL the CORRECT statements. For this question, prot = revenue − cost. Hint:
Apply the Gordon Formula to the prots of the rm.
a) If the prot is not expected to be constant, the present value of all the company’s future prots
is $125 million.
b) If the prot is expected to grow 3% annually, the present value of all the company’s future prots
is $20 million.
c) If the prot is expected to grow 4% annually, the present value of all the company’s future prots
is $25.75 million.
d) If the prot is expected to grow 6% annually, the present value of all the company’s future prots
is $50 million.
e) If the prot is expected to grow 10% annually, the present value of all the company’s future
prots is negative.
1.B
Suppose the capital share in New Zealand is α = 3/5. Mark ALL of the CORRECT statements.
For this question, use the growth accounting formula given in class.
a) If capital increases by 5%, labor hours increase by 15%, and total output increases by 10%
relative to last year, then TFP should increase by 1%.
b) If capital increases by 15%, labor hours decrease by 10%, and TFP increases by 5% relative
to last year, then total output should increase by 5%.
c) If capital increases by 10%, TFP increases by 5%, and total output increases by 10% relative to last year, then labor hours should decrease by 1%.
d) If labor hours increase by 5%, TFP increases by 5%, and total output increases by 10% relative to last year, then capital should increase by 5%.
e) None of the above.
1.C
Suppose the Federal Reserve buys Treasury Bills and Treasury Bonds in the open market. Compared
to a setting without such open market operations, which of the following answers are CORRECT?
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a) Banks short of reserves end up with the same amount of loans.
b) Banks short of reserves end up with less deposits.
c) Banks short of reserves end up with less equities.
d) The federal funds rate is lower.
e) The money supply is lower.
1.D
Assume the money supply grows at a rate of µ > 0 initially. At some time later, an immediate
increase in the rate of growth of money supply is announced. In other words, the money supply will
grow at a higher rate µ
0 > µ. After that, a one time decrease in the money supply is announced.
Mark ALL the CORRECT graphs.
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2. Ricardian Equivalence VS Government with Money (30 Points).
This exercise will show you the dierence between a government that can attempt to stimulate the
economy by nancing through taxes and borrowing and a government that can print money.
a) (5 points, easy) Consider the following government budget constraint between two periods
where government could print the money
P1G1 + M1 + (1 + i)B1 = P1τ1 + B2 + M2.
What are the ways the goverenment could nance its decit in this economy?
b) (5 points, easy) Suppose the government plans to print money at a rate of µ:
M2 = (1 + µ)M1.
Re-write the government decit as a function of money demand where in equilibrium md
(Y, i) =
MS
P
. Since the government doesn’t plan to borrow, you could drop Bt from the budget constraint.
c) (7 points, moderate) Suppose that people make N trips to the bank each year. Each trip cost
them F for transportation cost. And the nominal interest rate is i. The average money balance is
M =
P Y
2N
.
Explain why people want to hold money in this economy. Derive the demand for money when
people want to minimize the cost of holding money by choosing the number of trips they make to
the bank each year.
min
N
P F N + iM
Is the demand for money increasing or decreasing in the following variables: income, nominal
interest rate?
d) (10 points, moderate) Now, suppose the household knows that the government will increase
the money supply every period at the rate of µ. Substitute the money demand that you derived in
(c) to the government budget constraint in (h) and replace i = r + µ.
e) (3points, moderate) If we assume a constant government decit (Gt − τt) for every period,
can the government stimulate consumption? What is the implication of government scal policy in
this model, and how does it compare to the Ricardian Equivalence result we established earlier in
this class?
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3. Analytic Question on Consumption and Labor decisions during the Pandemic
(30 Points).
In this question we will study how workers change their labor supply. Imagine there is a
consumer/worker with preference over consumption C and leisure ` given by the equation below:
U(C, `) = log(C) + log(`)
1. (2 points)[easy] Assume the consumer faces wage w and consumption prices P. She also has
one unit of available time to spend working or resting. Solve for the consumer problem by
choosing hours worked and consumption. Compute the elasticity of consumption with respect
to real wages.
2. (3 points)[easy] Now assume there is a shopping spree in the economy, wherein the consumer
receives more utility from consumption. We model this by changing the preferences to:
U(C, l) = ϕ log(C) + log(`)
with ϕ > 1. Solve for the hours worked and consumption under this new assumption. Compare your answer with the previous part (where ϕ = 1). Does the consumer wants to work
more or less?
3. (5 points)[moderate] The government fears that a virus will spread in the population, so it
decides to limit the number of hours the consumer can work. We model this by assuming that
the maximum hours worked can be ζ, with 0 < ζ < 1
2
. With words, explain why the optimal
hours worked for the consumer will be h = ζ. Solve for consumption and leisure, given h = ζ.
4. (5 points)[moderate] Now imagine that after the government’s new regulation occurs, a new
form of work becomes available to the consumer. We assume she can work online with no
restrictions because the government has no control on online hours worked. Explain why the
new budget constraint can be written as: P C = wt(1 − o − `) + woo , where wt is the wage
in the traditional sector, wo is the wage in the new online sector and o are the hours worked
online. Assuming w0 < wt < wo
ϕ(1+ζ)
ζ
, explain why the consumer would choose to work as
many hours as possible in the traditional sector and the remaining hours they work online.
5. (10 points)[hard] Following part d, solve for the hours worked in both sectors and the total
hours worked. Does this new form of work improve consumption? Compare consumption
with part c.
6. (5 points)[moderate] Following part d, what would be the optimal labor restriction ζ? Explain your answer based on the First Welfare Theorem, assuming the prices clear markets in
a competitive equilibrium.
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4. Real Business Cycles (40 Points) In this question we will study the Real Business Cycle
(RBC) model from the lecture. In this example, the economy is characterized by the following
information. There is a single household that is alive for two periods:
U =

c1 − θ
L
1+
1+
1−σ
1 − σ
+
βc1−σ
2
1 − σ
where c1and c2are consumption of the production good in each period and L is the household’s
labor supply in period 1. The household receives wage income from supplying labor and receives all
prots (if any) earned by rms. Households can also save by freely converting the production good
into capital, which can generate income by being rented out to rms in period 2. Thus, households
have the following budget constraints in each period:
c1 + K = wL + π1
c2 = r
KK + K + π2
Finally, production in each period is produced by rms operating the following technologies:
Y1 = A1L
α
Y2 = A2K1−α
For the purposes of this problem, we may assume σ > 1 and 0 < α < 1. The only restriction on
the other parameters (, θ, A1, and A2) is that these values are all positive (i.e. greater than zero).
1. (10 points)[hard] Solve for the equilibrium allocation of capital and labor. For capital, it is
sucient to provide an implicit function as found in the lecture notes. (Hint: You may use
the social planner’s problem to nd the equilibrium allocations.)
2. (5 points)[easy] Rewrite the household budget constraint as a life-time (intertemporal) budget constraint. What is the relationship between the period 2 rental rate and the real interest
rate in this problem?
3. (5 points)[easy] Solve for the equilibrium value of the period 1 wage rate and the period 2
rental rate of capital. (Hint: Recall that equilibrium prices must rationalize the rm’s input
allocation as prot maximizing choices.)
4. (10 points)[medium] Suppose there is an increase in productivity in period 1 (A1) while
productivity in period 2 remains constant. Explain how this aects the equilibrium values of
labor, capital, wages, and real interest rates (it is sucient to say if these values go up, down,
no change, or if the eect is ambiguous). For each variable, provide an intuition for the sign
of the eect that you nd.
5. (10 points)[hard] Suppose households anticipate an increase in productivity in period 2
(higherA2) while productivity in period 1 remains constant. Explain how this aects the
equilibrium values of labor, capital, wages, and real interest rates (it is sucient to say if
these values go up, down, no change, or if the eect is ambiguous). In particular, provide an
intuition for why capital accumulation adjusts in the way you describe.
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5. IS-LM model and the value of Commitment. This question will help you understand
the IS-LM model in great detail. We derive the IS-LM model. There is a single household that is
alive for two periods:
U =

c1 − θ
L
1+
1+
1−σ
1 − σ
+
βc1−σ
2
1 − σ
where c1and c2are consumption of the production good in each period and L is the household’s
labor supply in period 1. The budget constraints in both periods are:
c1 + K =
w
p1
L + Π1,
p2c2 = r
KK + K.
Here, Π1 stand for the prots of monopolistic behaving rms. Here households invest in capital K
only to invest in it and rent it out later.
Production in each period is produced by rms operating the following technologies:
Y1 = L,
Y2 = K1−α
.
Dierent from model’s we’ve seen before, now the household invests directly in capital, and then
rents it to rms at time t = 2.
(a) [easy] Demonstrate that the labor supply is given by
w
p1
= θL
.
(b) [easy] Show that the households optimal investment in K is given by (the Euler equation):

c1 − θ
L
1+
1 + 
−σ
= β
1 + r
K


(c2)
−σ
.
(c) [easy] Show that if rms at t = 2 behave competitively, the maximization of their prots
leads to the following equilibrium return on capital:
r
K = (1 − α) K−α → K =

(1 − α)
rK
1/α
.
(d) [easy] Assume that rms ideally want to charge a markup relative to marginal costs given
by:
pi = cq
η
η − 1
| {z }
markup
,
where the nominal marginal cost is simply the wage:
cq = w.