Spring 2022 Econ 102 – Practice Mid-term Exam – Homework 3

Prof. Saki Bigio

May, 2022

READ THESE INSTRUCTIONS before you start writing: The exam has 4 questions. Your maximal score is 100
points. You MUST answer the Mandatory Multiple Choice Question (as the name implies, it is mandatory).
Among the three other questions, you can CHOOSE two of the three non-mandatory to be answered. You can get
partial credit on all questions. As such, you have a total of 20 mandatory points the mandatory question, plus 80

1. Mandatory Multiple Choice Question (20 Points – 5 points each).

1.A Consider the Ricardian Equivalence Proposition learned in class. Suppose that a government increases its
de�cit B. In doing so, it can either lower taxes τ1 while keeping government expenditures {G1, G2} �xed over
time, or it can �x τ1 while increasing G1. According to the Ricardian Equivalence Proposition, which of the
following answers are FALSE (mark all of them):

a) The increase in the government de�cit by more spending will lead the government to raise more taxes in
the future.

b) The increase in the government de�cit by lowering taxes will lead to an increase in private consumption
today because it stimulates the economy.

c) The increase in the government de�cit by lowering taxes will lead to an identical increase in private savings
that exactly o�sets the reduction in taxes, thereby having no e�ect on private consumption.

d) The increase in the de�cit by spending more will lead to a reduction in private expenditures.

e) The increase in the de�cit will have an e�ect on output, regardless of the format.

1.B Recently, �Pop Economist� Noah Smith recently criticized the New York Times saying: �Jesus Christ, New
York Times. What kind of economic journalism is this? IMPORTS DO NOT SUBTRACT FROM GDP!
Imports subtract from exports but they add to consumption!! They’re simply neutral for GDP!�

See the tweet here…

The tweet provoked a lot of discussion. Mark the following correct statements:

a) Indeed, if an import is consumed at cost: GDP does not increase.

b) Indeed, we cannot claim that GDP has fallen because M increased.

c) If imports are stored at cost as inventory: GDP does not increase and I increases.

d) If imports are combined with local labor to produce more output: imports will increase GDP increasing
labor.

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1.C Fatih lives and works in the US, and is planning to go on a trip to Australia for 1 month. Fatih believes that
the GDP per capita in Australia is US\$ 45,000, the exact same amount as in the US. Fatih had planned to take
US\$ 5000 dollars for his trip, but he reads an article in The Economist magazine and realizes (to his surprise)
that GDP per capita in Australia is only US\$ 20,000 once you translate nominal GDP into purchasing power
parity (PPP) � a number that is much lower than what Fatih initially thought. Fatih also learned that capital
per worker in Australia is the same as in the US. Which of the following reactions ARE CONSISTENT with
what Fatih learned in class:

a) �Australia is much more expensive than I thought. I should try to take a bit more of money or stay a
shorter period because my money will be worth much less over there. I will only be able to buy half the stu�
I was planning to.�

b) �I earn much more than I thought in the US, relative to what Australians earn. I should go and spend
more because Australia is cheaper.�

c) �The amount of physical goods produced in Australia is probably much lower than what is produced in the
US.�

d) �Wow, productivity in Australia must be much higher than in the US. I should expect to �nd a much better
educated population.�

e) �GDP per capita in PPP terms provides no additional information. The fact that both economies have the
same capital per worker and GDP per capita in nominal terms implies that, for the purpose of my trip, both
economies are identical.�

1.D The capital share in Peru is α = 2/3. Relative to last year, capital increased by 30%, labor hours increased by
10%, and total output increased by 10%. Which is the closest approximate number for the growth in Total
Factor Productivity, A? For this question, use the growth accounting formula given in class to deduce the
change in A.

a) 12% b) 6% c) 0% d) -6% e) -12%

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2. Numerical Question on the Solow Model (40 Points, easy). Let’s consider the Solow model with
population and technology growth. The steady state for capital per e�ective labor k̃ss is given by

k̃ss =

(
s

δ + n + x̃ + nx̃

) 1
1−α

,

a. How will the steady state value k̃ss change in response to a 50% decrease in the saving rate? To answer this
question, let the old and new steady state values be denoted by k̃oss and k̃

n
ss, respectively. Then, express k̃

n
ss in

terms of k̃oss and you should be able to obtain a concrete expression for their ratio. (10 points)
b. Let’s assume α = 0.5, s = 0.05, δ = 0.15, n = 0.05 and x̃ = 0.01 . Find an exact value for capital in steady
state. (10 points)
c. If the savings rate is increased to s = 0.1, what is capital in the new steady state? Is it higher or lower than
in the original steady state? Provide the economic intuition behind your result. (20 points)

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3. Analytic Question on Durable Consumption (40 Points).
It is known that durable consumption is more sensitive to interest rates and expected income. In this question,

we investigate if this is the case in the context of the model we have been studying. We investigate the following
decision:

U (c1, c2) = max
{c1,c2}

1

2

c1 + x0 +

β

2

c2

subject to the following budget constraints:
The time 1 budget constraint:

a = y1 − c1
and the time 2 budget constraint:

c2 = y2 + (1 + r) a.

The novelty is that x0 is now a variable to represents past purchases of goods 1. You can think of this as a car.
In turn, we have that x1 = c1 so x1 is consumption derived utility in period 2 derived from past purchases. For this
question assume that β (1 + r) = 1.

Answer the following questions. (40 points)
a. (5 points, easy) Substitute out a from both budget constraints, the one at time 1 and time 2, to write a single
intertemporal budget constraint.
b. (5 points, easy) Show that the Euler equation (the equation that relates the marginal utility of consumption
to the interest rate and the discount factor β), implies:

c1 + x0 = (β (1 + r))
−2c2.

Hint: form the Lagrangean of this problem, using your answer in (a). Then, obtain the �rst-order conditions and
combine them.

c. (5 points, easy) Suppose that x0 = 0. Then, argue that:

c2 = (β (1 + r))
2
c1

and substitute the condition into the intertemporal budget constraint to obtain the consumption shares:

c1 =
1

1 + β2 (1 + r)

(
y1 +

y2
1 + r

)
c2

1 + r
=

β2 (1 + r)

1 + β2 (1 + r)

(
y1 +

y2
1 + r

)
.

c. (10 points, moderate) Follow the same steps, to obtain an expression for the case where x0 > 0.
d. (15 points, harder) Consider now the case where y1 = y2 = ȳ. We now study an increase in ȳ by the amount
∆. Compute the rate of change of expenditures with respect to ∆ for the case where x0 = 0 and x0 > 0. For
this question, we assume that x0 < ȳ.

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4. Analytic Question on Good and Bad Governments. (40 points)
For the purpose of this question you can assume NO population growth NOR technology growth (namely, you
should solve the question with n = 0 and x = 0). Consider a version of the Solow model with energy. The law of
motion for aggregate capital in the Solow model is given by

Kt+1 = It + (1 − δ)Kt,

where It is total investment in this economy. Private investment is:

It = sY
D
t

where Y Dt is disposable income, given by:
Y Dt = Yt − Et.

Assume that Et = εYt, with ε ∈ (0, 1) is a parameter that captures how much the economy uses of energy. The
goal of this question is to study the uses of energy.

a. (5 points, easy) Provide an expression for Investment It only as a function of total income Yt (as opposed to
Y Dt ). Hint: replace Et and Y

D
t out from the previous equations given to you.

b. (5 points, easy) Now suppose we have a good government and that uses the energy resources Et for production.
In particular, assume that the production of goods now requires energy. Assume that the production function
is now:

Yt = AssE
1−γ
t K

γα
t L

γ(1−α)
ss

The idea is that 1 − γ is the parameter that governs the decreasing returns to energy. Substitute Et = εYt into
the equation above. Show that GDP can be re-written as:

Yt = A
1
γ
ssε

1−γ
γ Kαt L

(1−α)
ss .

c. (10 points, harder) Using the steps you’ve learned in class to obtain the steady-state in the standard Solow
model, �nd steady state output in this version. Explain how you would obtain the optimal expenditure of ε
that maximized GDP per capita in the long-run. Hint: this relates to the golden rule.
d. (10 points, harder) Instead, now assume that a fraction of the government’s investment in energy is diverted
to corrupt activities. In particular, assume that

Y Dt = Yt − Et

but now, only a fraction λ < 1 of the energy investments are actually spent in energy:

Et = λεYt.

For any given level of ε, what is the e�ect of λ on steady-state output?
e.(10 points, harder) Assume now that we learn that energy causes pollution and environmental damage. Namely,
assume that the production function changes to

Yt = AssE
1−γ
t K

γα
t L

γ(1−α)
ss X

−β
t

where Xt is pollution given by:
Xt = Et.

Find steady-state output and argue that β > 0 will reduce steady-state output. If pollution is unobservable,
would it appear as less TFP?

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