ECON-UA 12 — Intermediate Macroeconomics
Professor: Jaroslav Borovicˇka
TA: Hyein Han
Fall 2025
Sample midterm exam 1A — not for examination
Do not turn this page until instructed!
Read the instructions below first.
Name:
Signature:
1. This is a 70 minute midterm exam. It is worth 100 points. This booklet contains
21 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.
1
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.
Question 1.2 [3 points] Total GDP in a country must necessarily grow faster than GDP per
capita.
2
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.
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.
3
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.
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.
4
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.
Question 1.8 [3 points] In the Solow growth model, net investment can be negative.
5
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.
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 .
6
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).
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
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.
7
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.
8
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.
9
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.
10
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).
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?
11
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?
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∗.
12
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.
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∗∗.
13
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.
14
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.
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.
15
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∗?
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.
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.
16
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.
17
References
Scratch paper — material on this page will not be considered for grading.
18
Scratch paper — material on this page will not be considered for grading.
19
Scratch paper — material on this page will not be considered for grading.
20
Scratch paper — material on this page will not be considered for grading.
21

学霸联盟