Problem Set 5
Microeconomics: 33001
Professor Owen Zidar
1. Present Discounted Value
Please give a numerical answer to each question. You may round off your answer to
two decimal points. Assume that the interest rate is r = 0.05 and that there is no
inflation. (Hint: it may be helpful to use a calculator or an Excel spreadsheet to solve
this problem.)
(a) Compute the present value of $100 one year from today. (In other words: suppose
that you will receive $100 exactly one year from today. What is the present value
of those $100?)
SOLUTION:
PV =
100
≈ 95.24
1 + 0.05
(b) Compute the present value of $100 two years from today.
SOLUTION:
PV =
100
≈ 90.70
(1 + 0.05)2
(c) Compute the present value of the following income stream: $100 one year from
today and then an additional $100 two years from today.
SOLUTION:
PV =
100
100
+
= 185.94
1 + 0.05 (1 + 0.05)2
2. Rental Price of Capital Services
Consider a capital good with capital price P0 = 100. The nominal interest rate is
r = 0.1 and the depreciation rate is δ = 0.2.
(a) Compute the capital rental rate R0 if P1 = 120.
SOLUTION:
R0 = 100 −
(1 − 0.2)120
0.8 · 120
= 100 −
≈ 12.73
1 + 0.1
1.1
(b) Compute the capital rental rate R0 if P1 = 150.
SOLUTION:
R0 = 100 −
(1 − 0.2)150
≈ −9.09
1 + 0.1
3. Housing Market
Let the stock of housing be denoted by H, let the rental price of housing be denoted
by R and let the capital price of housing be denoted by P . The demand for housing is
given by D(R). Assume that D(R) is decreasing in R.
(a) Draw a figure depicting equilibrium in the (rental) market for housing. Put H on
the x-axis and R on the y-axis. (Hint: see slide 27 from week 6.)
(b) Suppose that the demand for housing increases. (The demand curve D(R) shifts
outwards.) What is the immediate impact on the rental price of housing? What
is the immediate impact on the stock of housing? For each of the two variables,
please indicate whether it will increase, decrease, or remain unchanged. Illustrate
in the figure. (Hint: remember that the stock of housing is fixed in the short run.)
SOLUTION: See figure.
(c) What is the immediate impact on the capital price of housing? Will it increase,
decrease, or remain unchanged? Please provide an explanation. (Hint: see slide
24 from week 6.)
SOLUTION: Since the rental price increases, the capital price will also increase.
The capital price is equal to the present value of the future flow of rental prices
(adjusted for depreciation). If the rental price increases, the capital price will
increase.
(d) What is the immediate impact on investments in housing? Draw a figure illustrating the investment market for housing and explain. (Hint: see slide 29 from
week 6.)
SOLUTION: See figure.
(e) What will happen to the stock of housing over time? Illustrate in a figure.
SOLUTION: See the figure of the rental market. Note the movement in the stock
of housing.
4. Firm Investment
A firm faces an investment decision problem. Suppose that there are two periods t = 1
and t = 2. The firm starts with a capital stock of K1 = 100. The capital stock in the
second period is given by
K2 = (1 − δ)K1 + I
The price of the firm’s output good is p = 1. The prices of capital goods are P1 and
P2 = (1 + ν)P1 in each period.. The firm’s production function is given by
1/2
Yt = At Kt ,
t = 1, 2
Assume that A1 = 1 and A2 = 36. The cost of using capital (the rental price of capital)
is R1 = 1. Assume that the interest rate is r = 0.5 and that the depreciation rate of
capital is δ = 0.1.
(a) What is the optimal capital stock in the second period, K2∗ ?
SOLUTION: The formula for optimal capital stock (from slide 70, week 6) is
K2∗
=
2
1
A
2 2
R1 (1 + r)
In our case, we have
K2∗
1
36
2
=
1(1 + 0.5)
2
= 144
(b) What is the optimal investment I ∗ ?
SOLUTION:
1
A
2 2
2
R1 (1 + r)
− (1 − δ)100 = I ∗
144 − (1 − 0.1)100 = I ∗
I ∗ = 144 − 90 = 54
5. Wine and Art
Note: this is an optional bonus problem. You can receive full credit without turning in
this problem.
Recall from the lecture (week 6, slide 19) that the equilibrium rental cost of using an
asset is the cost of buying the good and re-selling it after one period:
Rt = Pt −
(1 − δ)Pt+1
1+r
(1)
Define the growth rate of the capital price as
Pt+1
≡ 1 + gP
Pt
(2)
Suppose that we are paid to hold the capital, i.e. Rt < 0. Using the two equations
above, we get
Rt < 0 ⇔ Pt −
1+r
Pt+1
(1 − δ)Pt+1
<0⇔
<
= 1 + gP ⇔ 1 + r < (1 + gP )(1 − δ)
1+r
1−δ
Pt
and finally
1 + r < 1 − δ + gP + δgP ⇒ r + δ < gP + δgP
The last term δgP is likely to be very small relative to the other terms, hence we can
approximate δgP ≈ 0 and obtain the following rule of thumb:
Rt < 0 ⇔ gP > r + δ
This says that the rental price is going to be negative whenever the growth rate of the
capital price is greater than the sum of the interest rate and the depreciation rate.
Your task:
Analyze how the growth rate of the capital price of wine relates to the interest rate
and the depreciation rate of wine. How does the the growth rate of the capital price
of paintings relate to the interest rate and the depreciation rate of paintings?
SOLUTION: The growth rate of the capital price of wine is likely to be approximately
equal to r + δ or slightly greater than r + δ. There is no enjoyment from just holding
the wine without consuming it. If anything, we would like to be paid to hold the wine
(Rt < 0). For paintings we would certainly expect gp < r + δ, otherwise we would get
paid to hang the paintings on our walls. Instead, museums often have to pay to rent
a painting for an exhibition.