ECON-UA 12 — Intermediate Macroeconomics
Professor: Jaroslav Borovicˇka
TA: Hyein Han
Fall 2025
Sample midterm exam 1A — not for examination
Suggested solution
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1. This is a 70 minute midterm exam. It is worth 100 points. This booklet contains
15 pages.
2. At the beginning, you have 3 minutes to look through the exam. This time is not
counted in the examination time above. No writing or note-taking is allowed during
this period — only reading.
3. When the time is up, you have to stop writing and return the booklet to me within
exactly 1 minute. This is your responsibility.
4. The exam is closed book, no notes.
5. No electronic devices are allowed.
6. Use empty pages at the end as scratch paper. Material written on scratch paper will
not be considered for grading.
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1 True or false? [24 points total]
For each of the following statements, decide whether it is true or false, and explain your
answer in at most two sentences using the material you learned in the course. No points
will be given without an appropriate explanation!
Question 1.1 [3 points] The fast growth in the two most populous world economies, China
and India, is evidence of increasing global income inequality.
Answer FALSE. While inequality within many individual countries is increas-
ing, the global distribution of income exhibits rather the opposite tendency
during the last decades. The fast growth in China and India allows incomes in
those countries to catch up with the more developed world.
Question 1.2 [3 points] Total GDP in a country must necessarily grow faster than GDP per
capita.
Answer FALSE. Intuitively, total GDP will grow faster that GDP per capita
if and only if population growth is positive. Mathematically, we have shown
that the growth rate of GDP per capita, gy is related to the growth rate of total
GDP gY and the population growth rate gL as gy = gY − gL. Again, the same
conclusion follows — gY can be bigger than gy if gL is positive.
Question 1.3 [3 points] We have used the Cobb–Douglas production function because that
is the only production function with constant returns to scale.
Answer FALSE. We have analyzed other production functions that exhibited
constant returns to scale—linear, Leontief or the more general constant elastic-
ity of substitution production functions. The main reason (apart from math-
ematical tractability) for the use of the Cobb–Douglas production function is
that it generates constant factor shares—something which we argued is a good
approximation of empirical evidence.
Question 1.4 [3 points] The failure of our production model to explain differences in in-
come per capita across countries is largely due to the incorrect assumption of perfect com-
petition.
Answer FALSE. The production technology Y = AKαL1−α that we used in our
model did not rely on perfect competition. We only used the argument of per-
fect competition in order to derive the income shares for a profit maximizing
competitive firm. The failure of our production model stems largely from the
assumption of identical TFP parameter across countries.
Question 1.5 [3 points] Corporate income taxes increase the user cost of capital and there-
fore (other things equal) also increase the rental rate charged by capital owners.
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Answer TRUE. The user cost of capital increases, and the arbitrage equation
then implies that in order to keep investors indifferent between different uses
of their money, the rental rate they charge has to increase.
Question 1.6 [3 points] The Gordon growth formula
ps
d
=
1
R− gd
implies that an increase in the risk premium, perhaps due to an increase in the riskiness of
the economy, will lead to a fall in the stock market valuation.
Answer TRUE. The interest rate R is the properly risk-adjusted interested rate
which includes the risk premium. When the risk premium (and thus R) in-
creases, we should see a fall in the price-dividend ratio ps/d.
Question 1.7 [3 points] In the Solow growth model, a high saving rate is unambiguously
good for the society because it allows the country to accumulate a large amount of capital
and thus produce a large amount of output.
Answer FALSE. When a country starts from a given level of capital K0, a
higher saving rate implies more investment, and thus less consumption today.
It is true that an economy with a higher saving rate will ultimately accumulate
more capital and this may (although also may not!) lead to a higher consump-
tion level in the long run. However, there is a tradeoff between a lower con-
sumption today and a (potentially) higher consumption in the future which
needs to be considered.
Question 1.8 [3 points] In the Solow growth model, net investment can be negative.
Answer TRUE. Net investment corresponds to the change in the capital stock.
Net investment is negative when the economy decumulates capital.
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2 Growth rate of output and output per capita [22 points total]
Question 2.1 [5 points] Consider the Cobb–Douglas production function
Yt = AtKαt L
1−α
t (2.1)
Derive (show at least two or three intermediate steps!) the growth rate of output gY as a
function of the growth rates of TFP gA, capital gK and labor gL.
Answer The growth rate of output gY is defined through
1 + gY =
Yt+1
Yt
=
At+1Kαt+1L
α
t+1
AtKαt L
1−α
t
=
At+1
At
(
Kt+1
Kt
)α (Lt+1
Lt
)1−α
=
= (1 + gA) (1 + gK)
α (1 + gL)
1−α
and thus, using the logarithmic approximation
gY = gA + αgK + (1− α) gL. (2.2)
Question 2.2 [3 points] In the U.S. over the last fifty years, output grew at 3.5% per year,
capital at 3% per year and population at 1.5% per year. What is the implied annual growth
rate of TFP in the U.S. economy? In your calculation in this question, use the usual as-
sumption that the capital share is equal to 13 .
Answer Plugging in the numbers, we have
3.5% = gA +
1
3
× 3% + 2
3
× 1.5% = gA + 2%
and thus gA = 1.5%.
Question 2.3 [3 points] Define the output per capita as yt = Yt/Lt and the capital-labor
ratio as kt = Kt/Lt. Divide equation (2.1) by Lt to derive the relationship between yt and
kt (this relationship will also depend on At but not directly on Kt and Lt).
Answer We have
Yt
Lt
=
AtKαt L
1−α
t
Lt
= At
(
Kt
Lt
)α
and thus
yt = Atkαt .
Question 2.4 [3 points] Use this derived relationship between yt, At and kt to compute
the growth rate of output per capita gy as a function of the growth rate of TFP gA and the
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growth rate of the capital-labor ratio gk (notice that gy and gk are different growth rates
than gY and gK above).
In this and the following questions, you do not need to provide the intermediate steps
again if you do not want to.
Answer Using the same calculation as in the first question
gy = gA + αgk.
Question 2.5 [4 points] Let us return to the definitions of output per capita as yt = Yt/Lt
and the capital-labor ratio as kt = Kt/Lt. Use these formulas to derive
1. the growth rate of output per capita gy as a function of the growth rate of output gY
and the growth rate of labor gL;
2. the growth rate of capital per capita gk as a function of the growth rate of capital gK
and the growth rate of labor gL.
Answer Again, using calculations analogous to those before
gy = gY − gL
gk = gK − gL.
Question 2.6 [4 points] We now want to show that you derived the same expression in two
different ways.
1. Take the expression from Question 2.4 that shows the growth rate of output per
capita gy as a function of the growth rate of TFP gA and the growth rate of the capital-
labor ratio gk;
2. Substitute in the expressions for gy and gk derived in Question 2.5;
3. Reorganize the resulting expression to show that what you obtained is the same re-
lationship as the one that you derived in Question 2.1.
Answer From Question 2.4, we have
gy = gA + αgk
and thus, substituting results from Question 2.5, we obtain
gY − gL = gA + α (gK − gL) .
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Reorganizing, we obtain
gY = gA + α (gK − gL) + gL = gA + αgK + (1− α) gL
which is the same as formula (2.2).
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3 Arbitrage and a construction boom [14 points total]
In this problem, we analyze how optimistic beliefs about the future price of housing can
lead to construction boom.
We start with an investor who decides whether to invest into building new houses or
saving the money in the savings account. Her decision problem leads to the arbitrage
equation
R = r− δ+ ∆ph
ph
where R is the interest rate on the savings account, r is the rental rate that she can collect
on renting out the house, δ is the depreciation rate and ∆phph is the capital gain on the house
purchase.
We will assume in this question that output can be produced using houses. Houses
represent here structures like office or factory buildings. Consider the production technol-
ogy
Y = AHα α ∈ (0, 1)
where Y is total output, H is the amount of houses and A is a TFP parameter (we can
abstract from labor in this problem). The profit-maximizing competitive firm that uses
this production technology chooses to rent the amount of houses H from house owners
(investors) at a given rental rate r.
Question 3.1 [5 points] Set up the firm’s maximization problem. The profit of the firm is
given by its output minus the rental cost rH. Take the first order condition with respect to
the housing choice.
Answer The firm’s problem is
max
H
{AHα − rH}
and the first-order condition
αAHα−1 = r
Question 3.2 [4 points] Solve the first-order condition for the number of houses H chosen
by the firm. Is the H chosen by the firm an increasing or decreasing function of r? Show it
(a verbal argument using the formula is enough).
Answer Solving the first-order condition, we get
H =
(
αA
r
) 1
1−α
An increase in r in the denominator will decrease H, so H is a decreasing func-
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tion of r. Formally
dH
dr
= − 1
1− α
(
αA
r
) α
1−α αA
r2
< 0.
Question 3.3 [5 points] We now want to show that optimism about the capital gains in the
housing market will lead to a construction boom.
Imagine that investors start believing that the growth rate of house prices will be higher
than before. Assume that the interest rate R and the depreciation rate δ remain unchanged.
Use the arbitrage equation to argue how the rental rate r has to adjust.
Consequently, using the results from the optimization problem of the firm, what will
happen to the quantity of houses in the market? Can you interpret this as a construction
boom?
Answer When ∆phph increases while R and δ remain unchanged, then the rental
rate r has to decrease. Consequently, because demand for houses is a decreas-
ing function of r, the equilibrium amount of houses will increase. Since these
houses have to be built, this corresponds to a construction boom.
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Figure 1: The Solow diagram for Question 4.2. The steady state level of capital K∗ corre-
sponds to the level of capital at the intersection of the saving (investment) and depreciation
curves.
4 Experiments in the Solow growth model [40 points total]
In the questions in this section, we will use the Solow growth model, described by equa-
tions
Yt = AKαt L
1−α (4.1)
∆Kt+1 = sYt − δKt
with a given initial value K0.
Question 4.1 [3 points] What are empirically plausible values for α, s and δ?
Answer We used in class α = 13 , δ = 0.1 and s = 0.2− 0.5 (the last depending
on the country). I would tolerate if you answered with any numbers in the
range α = 0.25− 0.4, δ = 0.05− 0.2, and s = 0.1− 0.5.
Question 4.2 [5 points] Sketch the Solow diagram (with the stock of capital on the hori-
zontal axis), including curves for output, depreciation and saving. Provide appropriate
notation for axes and the curves. Depict the steady state value for capital, K∗.
Answer See Figure 1.
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Figure 2: The Solow diagram for Question 4.3. The steady state level of capital K∗ corre-
sponds to the level of capital at the intersection of the saving (investment) and depreciation
curves.
Question 4.3 [3 points] Into a separate graph, plot the net investment curve as a function
of the stock of capital. Clearly depict the steady state level of capital in this graph.
Answer The net investment curve is the difference between investment and
depreciation. It can be directly read off the Solow diagram as the distance
between the investment and depreciation curves. It starts at zero for the zero
level of capital, then it is positive (first increasing and then decreasing), and it
crosses zero at K∗, being negative for K > K∗. Also, it is concave everywhere.
See Figure 2.
4.1 Experiment 1: Productivity increase
Question 4.4 [5 points] Use a new Solow diagram to capture the following experiment.
The economy starts in a steady state (with steady state level of capital K∗). Then at time t0
the TFP parameter A increases.
Plot the depreciation, investment, and output functions before and after the change in
A, clearly denoting each curve. Also clearly denote the old steady state K∗ and the new
steady state K∗∗.
Answer See Figure 3.
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Figure 3: The Solow diagram for Question 4.4. The steady state levels of capital K∗ and
K∗∗ correspond to the levels of capital at the intersections of the saving (investment) and
depreciation curves.
Question 4.5 [5 points] In two separate graphs (underneath each other), plot the time tra-
jectories of the following quantities for the experiment described in the preceding ques-
tion:
1. Capital;
2. Output;
In each of the graphs, clearly depict the time t0, and the original and new steady states
of each of the quantities.
Answer See Figure 4. The increase in A occurs at time t0 = 0. Notice that
while capital starts increasing smoothly (because it is a stock variable and ac-
cumulation takes time), output experiences an initial jump due to the sudden
increase in A. I did not denote steady state levels of the quantities in this graph
but they are clearly visible from the shape of the graph.
Question 4.6 [4 points] Set up the problem of a profit-maximizing firm. The firm uses
production function (4.1) to produce output, pays each worker a competitive wage wt and
rents each unit of capital at a competitive rental rate rt. Take the first-order condition with
respect to capital to derive the relationship between the marginal product of capital and
the rental rate.
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Figure 4: The time paths for Question 4.5. The initial and terminal steady state levels of the
variables should be denoted on the vertical axis. The time t0 corresponds to the moment
when the economy departs away from the original steady state.
Answer The firm’s problem is
max
K,L
AKαt L
1−α − wtLt − rtKt.
The first-order condition with respect to capital gives
MPK =
∂Yt
∂Kt
= αAKα−1t L
1−α = rt
Question 4.7 [3 points] What is the steady-state level of the rental rate r∗? How does it
depend on the productivity parameter A?
Hint: The following expression for the steady-state level of capital will be useful:
K∗ =
(
sA
δ
) 1
1−α
L.
All you have to do is to use this expression to substitute out capital in the first-order con-
dition for the capital choice.
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Figure 5: The Solow diagram for Question 4.8. The steady state levels of capital K∗ and
K∗∗ correspond to the levels of capital at the intersections of the saving (investment) and
depreciation curves.
Answer Using the first-order condition evaluated at the steady state, we have
r∗ = αA (K∗)α−1 L1−α = αA
((
sA
δ
) 1
1−α
L
)α−1
L1−α =
= αA
(
sA
δ
)−1
= α
δ
s
which is independent of the parameter A.
Question 4.8 [4 points] Plot the time trajectory for the rental rate r. Clearly depict the time
t0, and the original and new steady state for the rental rate. Is the new steady-state rental
rate r∗∗ lower or higher than the original steady-state rental rate r∗?
Answer Since the steady-state rental rate is independent of A, the two steady-
state rental rates are equal, r∗ = r∗∗. See Figure 5. Notice that the rental rate
increases on impact because for the higher level of TFP, there is not enough
capital in the economy (capital is scarce, firms would like to rent more of it at
the original rental rate, and therefore the rental rate has to increase).
Question 4.9 [3 points] The expenditure approach to GDP calculates GDP by adding up
components that are determined by how income is used (what is it spent on). On the other
hand, the income approach to GDP calculates GDP by adding up components that are
determined by how income is distributed (who earns it).
When we studied the model of production and the Solow growth model, we have seen
equations that capture each of these approaches. So for each of the two approaches, write
down the particular equation and write in one sentence what this equation represents.
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Hint: One of these equations was related to the zero-profit result of a firm with Cobb–
Douglas technology, while the other was a resource constraint.
Answer For the expenditure approach, we have
Yt = Ct + It
which is the resource constraint that says that output can be used either for
consumption or for investment but you cannot consume and invest more than
what you produce.
For the income approach, we have shown that for the Cobb–Douglas pro-
duction technology,
Yt = wtLt︸︷︷︸
(1− α)Yt
+ rtKt︸︷︷︸
αYt
.
so that output, in our case, is distributed in constant shares between workers
and capital owners.
4.2 Experiment 2: Decline in saving rate
Question 4.10 [5 points] Assume that the economy starts in a steady state with capital K∗.
Imagine that at time t0 the saving rate in the economy declines to zero.
What will be the new steady state level of capital K∗∗? What will be the new steady
state level of wages w∗∗? Provide an economic justification of your answers.
Hint: The wage is equal to the marginal product of labor, which you can compute from
the production function.
Answer When the saving rate declines to zero, s = 0, the steady state equation
implies
sY∗ = δK∗
Because the left-hand side is equal to zero, then K∗ = 0, and therefore also
Y∗ = A (K∗)α L1−α = A (0)α L1−α = 0.
The wage is equal to the marginal product of labor:
w =
∂Y
∂L
= (1− α) AKαL−α
and because K∗ = 0, then also w∗ = 0.
Economically, the economy decumulates all capital. Every period, a frac-
tion δ of the capital depreciates but none is replaced through new investment.
The Cobb–Douglas production function implies that with zero capital, output
is equal to zero.
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References
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