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 re- ported revenue was $10 million and the reported cost was $9 million. The discount rate is 8%. Mark ALL the CORRECT statements. For this question, pro�t = revenue − cost. Hint: Apply the Gordon Formula to the pro�ts of the �rm.

a) If the pro�t is not expected to be constant, the present value of all the company's future pro�ts is $125 million.

b) If the pro�t is expected to grow 3% annually, the present value of all the company's future pro�ts is $20 million.

c) If the pro�t is expected to grow 4% annually, the present value of all the company's future pro�ts is $25.75 million.

d) If the pro�t is expected to grow 6% annually, the present value of all the company's future pro�ts is $50 million.

e) If the pro�t is expected to grow 10% annually, the present value of all the company's future pro�ts 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% rela- tive 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% rel- ative 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′ > 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 di�erence 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 de�cit 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 de�cit 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 = PY

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

PFN + 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 de�cit (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. Com- pare 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: PC = 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? Ex- plain 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 −θL

1+�

1+�

)1−σ 1 −σ

+ βc1−σ2 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 pro�ts (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 = A2K 1−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 su�cient 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) bud- get 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 pro�t 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 a�ects the equilibrium values of labor, capital, wages, and real interest rates (it is su�cient to say if these values go up, down, no change, or if the e�ect is ambiguous). For each variable, provide an intuition for the sign of the e�ect 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 a�ects the equilibrium values of labor, capital, wages, and real interest rates (it is su�cient to say if these values go up, down, no change, or if the e�ect 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 −θL

1+�

1+�

)1−σ 1 −σ

+ βc1−σ2 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 pro�ts 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 = K 1−α.

Di�erent 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

L1+�

1 + �

)−σ = β

( 1 + rK

) (c2)

−σ .

(c) [easy] Show that if �rms at t = 2 behave competitively, the maximization of their pro�ts leads to the following equilibrium return on capital:

rK = (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︸ ︷︷ ︸ markup

,

where the nominal marginal cost is simply the wage:

cq = w.

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We assume that η/ (η − 1) > 1. Show that the target individual price is given by:

pi = p1θL � ·

η

η − 1 .

(e) [easy] Assume that the Central Bank, that is, the Federal Reserve can chose a quantity of money M, to target an interest interest rate i. Furthermore, it guarantees and in�ation rate between t = 1 and t = 2 given by π2, which we can treat as exogenous. We won't care about how it achieves this policy at t = 2. Argue that in that case, the Central Bank determines the real return on capital:

rK = i−π2,

thereby a�ecting real investment in the economy. (f) [Intermediate] We now derive the IS curve. We assume that all �rms �x their prices in

advance, prior to the governments choice. Use the production function

Y = L

to show that the I-S equation in this version of the new-Keynesian model is given by the following relationship between output and the nominal interest rate:

Y −θ Y 1/(1+ε)

1 + ε =

[ 1 −α i−π2

]1/α + β

[ 1

(1 + i−π2) σ

][ 1 −α i−π2

](1−α)/α .

Hint: use that Y = K + c1, and substitute this condition into the Euler equation. Demonstrate that the relationship between these Y and i is negative, that is, for a higher interest rate, output is declining. Draw this curve. What are the e�ects of increases in interest rates today? What happens if the private sector expects higher in�ation in the future? Is higher future in�ation expansionary or recessionary?

Extra Credit Questions. Value of Commitment (will not be tested in the exam)

(g) [Intermediate] From now on, we demonstrate the value of commitment to low in�ation. Argue that if there was no monopolistic power, output would be e�cient and equivalent to:

Y ∗ = [1/θ] 1/ε

.

Hint: setup the Lagrangian of the planner's problem and take the �rst order condition with respect to L and the replace the production function.

(h) [Hard] For this question, you ust read chapter 15.2. We now consider a game between the government and the private sector. We assume that µ �rms can chose the ideal price:

pf = w η

η − 1

the other set of �rms is stuck at a given price, chosen in advance, given the expectation of p1, that is E [p1]:

ps = E [p1] θL� · η

η − 1 .

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Recall that: p1 = µp

f + (1 −µ) ps.

Show that if the government wants to target Y ∗, we have that:

ps = E [p1] · η

η − 1

And using: p1 = µp

f + (1 −µ) ps,

show that:

ps = µ η η−1

1 − (1 −µ) η η−1

E [w] ·

(i) [Hard] Show that for any choice of E [w], the government would want to surprises �rms with even higher wages. What are the consequences of this? Do you think this type of behavior can lead to an ever increasing spiral of in�ation. What would be the e�ects on output if an in�ationary spiral is expected?

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