macro test, needed finish in 4 hours

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Econ 3022: Macroeconomics

Spring 2020

Final Exam - Due April 24th 11:59pm

1 Multiple Choice Questions (5 points each)

Question 1 What is Ricardian Equivalence?

(a) The economic hypothesis that agents’ decisions are una↵ected by the timing of taxation

and government spending

(b) The economic hypothesis that agents’ decisions are a↵ected by the timing of taxation

and government spending

(d) The economic hypothesis that it is impossible to individually identify taxation today

and taxation tomorrow.

Question 2 Consider the consumer problem from the microeconomic foundations we dis-

cussed in class. Suppose the wage decreases. What do we expect to happen to house-

hold labor supply?

(a) Unclear

(b) Increase

(d) Stay constant

Question 3 Consider the consumer problem from the real intertemporal model. Which of

the following conditions must be satisfied at the solution?

(a) MRSl,c = w

(b) MRSc0,l0 = 1 w0

(d) All of the above

Question 4 If total factor productivity tomorrow, z0, increases. What should happen to

investment?

(a) Unclear

(b) Increase

(d) Stay constant

Question 5 Consider the standard Solow model from class where the production function

is zF (K, N) = zK↵N1�↵. What is the golden rule savings rate?

(a) sgr = 1 � ↵

(b) sgr = ↵

(d) The savings rate that leads to a steady state with the lowest level of income per capita

2 Economic Growth (20 points)

Consider the Solow Growth Model seen in class where the production function is Cobb-

Douglas and given by:

Y = zK↵ (N) 1�↵

where 0 < ↵ < 1 and z is a constant. Let s be the savings rate of this economy, so that

aggregate savings is just a constant fraction of aggregate output: S = sY . Let n be the rate

of population growth, so N 0

N = 1 + n. Finally, let d be the depreciation rate, and assume the

law of motion for aggregate capital is given by:

K 0 = (1 � d) K + I

(a) (5 pts) Find an expression for the steady state level of capital per capita (k⇤) that only

depends on parameters of the model. Clearly show your work.

(b) (5 pts) Discuss how per capita variables (consumption and income) as well as aggregate

variables (consumption, capital stock, output, and savings) behave in steady state.

Now, suppose that we have a linear production function given by

Y = zK

where z is a constant. Let s be the savings rate of this economy, so that aggregate savings

is just a constant fraction of aggregate output: S = sY . Let n be the rate of population

growth, so N 0

N = 1 + n. Finally, let d be the depreciation rate, and assume the law of motion

for aggregate capital is given by:

K 0 = (1 � d) K + I

of per capita capital stock tomorrow. Clearly show your work.

(d) (5 pts) Does the model converge to a steady? Discuss.

3 Two Period Model with Investment (30 points)

Consider a two period economy model with a representative consumer who has lifetime

utility:

U (c, l) + �U (c0, l0)

The consumer works and consumes in each period, and he is able to save or borrow at the

interest rate, r. He has h hours available each period to divide between leisure and work.

He is the owner of the representative firm and receives dividends (profits) from the firm in

each period.

There is a representative firm with production function zF � K, N

d � that produces output in

each period. Capital, K, is given in the first period. The firm can choose how much to invest

for the future period capital, K0. Capital depreciates at a rate d each period. At the end of

the last period, after production, the firm can sell the undepreciated capital and distributes

the proceeds as profits to the consumer.

Finally, there is a government that imposes lump sum taxes, (T, T 0), on the consumers. The

revenue from the taxes are used to finance government spending (G, G0). The government

can also borrow or lend at the interest rate, r.

(a) (10 points) Define the Competitive Equilibrium

(b) (5 points) Characterize the equilibrium as much as possible. You should end up with 5

equations which are functions of 5 unknowns (c, c0, N, N 0, K0).

Suppose, now, that instead of the lump sum taxes (T, T 0), the government imposes two taxes

on the consumers and a tax on the firm in each period. First, consumers pay taxes on labor

income, (⌧l, ⌧ 0 l ), in each period and a tax on the dividend income from the firm, (⌧⇡, ⌧

0 ⇡ ), in

both periods. The firm pays a payroll tax on the amount it pays to workers, � ⌧p, ⌧

0 p

� , in each

period. The revenue from the taxes finance government spending, (G, G0).

equilibrium, but you must make clear exactly what will be di↵erent.

(d) (5 points) Set up the Social Planner’s problem for this economy.

(e) (5 points) Is the Competitive Equilibrium with taxes Pareto Optimal? Why or why

not?

4 Heterogeneity and Taxation (25 points)

This question is an application of the Competitive Equilibrium in the model with exogenous

income to a situation with heterogeneous consumers.

Consider an economy with two consumers (N = 2) indexed by i 2 {H, L} with utility functions given by

Ui(ci, c 0 i ) = ln(ci) + �i ln(c

0 i )

Consumer H is high income and consumer L is low income. Accordingly, assume that yH = 9

and yL = 7; Suppose that both consumers discount the future at rate � = 0.5.

Suppose government spending is 4 in both periods, G = G0 = 4. Additionally, assume

that initially both types pay the same taxes in both periods and taxes in the first period

cover spending in the first period, tL = tH, t 0 L = t0

H , and T = G.

(a) (5 points) Define a Competitive Equilibrium for this environment.

(b) (5 points) Find the Competitive Equilibrium allocation (ĉL, ĉ 0 L , ĉH, ĉ

0 H , t̂L, t̂

0 L , t̂H, t̂

0 H , r̂).

Suppose that everything remains the same except now only the high income individuals

are required to pay taxes. So, tL = t 0 L = 0. However, it is still true that T = G.

0 H , t̂L, t̂

0 L , t̂H, t̂

0 H , r̂).

Suppose instead that only the low income individuals are required to pay taxes. So,

tH = t 0 H = 0. However, it is still true that T = G.

(d) (5 points) Find the Competitive Equilibrium allocation (ĉL, ĉ 0 L , ĉH, ĉ

0 H , t̂L, t̂

0 L , t̂H, t̂

0 H , r̂).

(e) (5 points) Discuss how changing who pays taxes impact the equilibrium values. How

does this relate to policy debates on taxation?

5 EXTRA CREDIT (+10 points)

On Friday, March 27, 2020, President Trump signed the CARE Act into law. This Act pro-

vided over $2 trillion in economic stimulus to help Americans through the crisis caused by

COVID-19. In addition to the checks to American families, the act also provides assistance

to small businesses (fewer than 500 employees) in the form of forgiveable loans.

(a) Discus the details of these loans. Be sure to include details on how much a firm is eligible

for and what requirements they must meet for the loan to be forgiven.

(b) In the context of the two-period model with investment, how will these loans show up

in the model. How will they show up di↵erently if they are or are not forgiven?

other endogenous variables? Make sure you are defending these with the model.