Econ 3022: Macroeconomics Final Exam

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
(c) The economic hypothesis that taxation must be equal every
period.
(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
(c) Decrease
(d) Stay constant
1
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
(c) MRSl,l0 =
w(1+r)
w0
(d) All of the above
Question 4 If total factor productivity tomorrow, z0, increases.
What should happen to

investment?
(a) Unclear
(b) Increase
(c) Decrease
(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 = ↵
(c) The savings rate that leads to a steady state with the highest
level of income per capita
(d) The savings rate that leads to a steady state with the lowest
level of income per capita
2
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
(c) (5 pts) Find an expression for the level of per capita capital
stock today as a function
of per capita capital stock tomorrow. Clearly show your work.
(d) (5 pts) Does the model converge to a steady? Discuss.
3
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).
(c) (5 points) How does you answer to part (a) change? You do
not need to redefine the
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
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
, t̂ L, t̂
0
L
, t̂ H, 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.
(c) (5 points) Find the Competitive Equilibrium allocation (ĉL,
ĉ
0
L
, ĉH, ĉ
0
H
, t̂ L, t̂
0
L
, t̂ H, 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
, t̂ L, t̂
0
L
, t̂ H, 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

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?
(c) What impact do you think these loans will have on the
choices of the firm as well as the
other endogenous variables? Make sure you are defending these
with the model.
6

Econ 3022: Macroeconomics Final Exam

Econ 3022: Macroeconomics Final Exam

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