1. Hula hoop fabricators cost $100 each. The Hi-Ho hula
Hoop Company is trying to decide how many of these machines to buy. HHHHC
expects to produce the following numbers of hoops each year for each level of
capital stock shown.
Number of Fabricators
Number of Hoops
Produced per Year
0
0
1
100
2
150
3
180
4
195
5
205
6
210
Hula hoops have a real value of $1 each. HHHHC has no other costs
besides the cost of fabricators.
a.
Find the expected future marginal product of
capital (in terms of dollars) for each level of capital. The MPKf
for the third fabricator, for example, is the real value of the extra output
obtained when the third fabricator is added
b.
If the real interest rate is 12% per year and the
depreciation rate of capital is 20% per year, find the user cost of capital (in
dollars per fabricator per year). How many fabricators should HHHHC buy?
c.
Repeat Part (b) for a real interest rate of 8%
per year
d.
Repeat Part (b) for a 40% tax on HHHHC’s sales
revenues.
e.
A
technical innovation doubles the number of hoops a fabricator can produce. How
many fabricators should HHHHC buy when the real interest rate is 12% per year?
8% per year? Assume that there are no taxes and that the depreciation rate is
still 20% per year.
2. (a)
This chart shows the MPKf
as the increase in output from adding another fabricator:
|
#
Fabricators |
Output |
MPKf |
|
0 |
0 |
— |
|
1 |
100 |
100 |
|
2 |
150 |
50 |
|
3 |
180 |
30 |
|
4 |
195 |
15 |
|
5 |
205 |
10 |
|
6 |
210 |
5 |
(b)
uc
= (r
+
d)pK
= (0.12
+ 0.20)$100
= $32.
HHHHC should buy two fabricators,
since at two fabricators, MPKf
= 50
> 32 =
uc. But at three fabricators,
MPKf
= 30
< 32 =
uc. You want to add fabricators only
if the future marginal product of capital exceeds the user cost of capital. The
MPKf
of the third fabricator is less than its user cost, so it should not be added.
(c) When
r =
0.08, uc
= (0.08
+ 0.20)$100
= $28. Now they should buy three
fabricators, since
MPKf
=
30 >
28
=
uc
for the third fabricator and MPKf
=
15
<
28
=
uc
for the fourth fabricator.
(d) With taxes,
they should add additional fabricators as long as (1
-
t)MPKf
>
uc. Since
t
= 0.4,
1 -
t
= 0.6. They should buy just one
fabricator, since (1 -
t)MPKf
= 0.6
´
100 = 60
> 32
= uc. They shouldn’t buy two,
since then (1 -
τ)MPKf
= 0.6
´
50 = 30
< 32
= uc.
(e) When output
doubles, the MPKf doubles
as well. At r
= 0.12, they should buy three
fabricators, since then MPKf
= 60
> 32 =
uc; they shouldn’t buy four, since
then MPKf
= 30
< 32 =
uc.
At
r =
0.08, they should buy four fabricators, since then
MPKf
= 30
> 28 =
uc; they shouldn’t buy five, since
then MPKf
= 20
< 28 =
uc.
2. An economy has full-employment output of 6000. Government purchases,
G, are 1200. Desired consumptions and desired investment are
Cd =3600 – 2000r + 0.10Y, and
Id = 1200 – 4000r,
Where Y is output and r is the real interest rate.
a.
Find an equation relating desired national
saving, Sd, to r and Y
b.
Find the real interest rate that
clears the good market. Assume that output equals full-employment output.
c.
Government purchases rise to 1440. How does this
increase change the equation describing desired national saving? Show the change
graphically. What happens to the market-clearing real interest rate?
(a)
Sd
=
Y -
Cd
-
G
=
Y -
(3600 - 2000r
+ 0.1Y)
- 1200
= -4800
+ 2000r
+ 0.9Y
(b) (1) Using
Eq. (4.7): Y
=
Cd
+
Id
+
G
Y =
(3600 - 2000r
+ 0.1Y)
+ (1200
- 4000r)
+ 1200
= 6000
- 6000r
+ 0.1Y
So 0.9Y
= 6000
- 6000r
At full employment, Y = 6000.
Solving 0.9
´
6000 = 6000
- 6000r, we get r
= 0.10.
Sd =
Id
-4800
+
2000r
+
0.9Y
=
1200 -
4000r
0.9Y
=
6000 -
6000r
When Y
=
6000, r =
0.10.
So we can use either Eq. (4.7) or (4.8) to get to the same result.
(c) When
G =
1440, desired saving becomes Sd
=
Y -
Cd
-
G =
Y -
(3600 - 2000r
+ 0.1Y)
- 1440
=
-5040
+ 2000r
+ 0.9Y.
Sd is now 240 less for any
given r and
Y; this shows up as a shift in the
Sd line from
S1 to
S2 in Figure
Setting Sd = Id,
we get:
-5040
+ 2000r
+ 0.9Y
= 1200
- 4000r
6000r
+ 0.9Y
= 6240
At Y
= 6000, this is 6000r
= 6240
- (0.9
´
6000) = 840, so r
= 0.14. The market-clearing real
interest rate increases from 10% to 14%.
3. Suppose that the expected future marginal product of
capital is MPKf = 20 –
0.02K, where K is the future capital stock. The depreciation rate of capital,
d, is 20% per period. The current
capital stock is 900 units of capital. The price of a unit of capital is 1 unit
of output. Firms pay taxes equal to 50% of their output. The consumption
function in the economy is C= 100 + 0.5Y-200r, where C is consumption, Y is
output, and r is the real interest rate. Government purchases equal 200, and
full-employment output is 1000.
a. suppose that the real interest rate is 10% per period. What are the
values of the tax-adjusted user cost of capital, the desired future capital
stock, and the desired level of investment?
b. Now consider the real interest rate determined by goods market
equilibrium. This part of the problem will guide you to this interest rate.
i. Write the tax-adjusted user cost of capital as a function of the real
interest rate r. also write the desired future capital stock and desired
investment as functions of r.
ii. Use the investment function derived in Part (i) along with the
consumption function and government purchases, to calculate the real interest
rate that clears the goods market. What are the goods market-clearing values of
consumption, saving, and investment? What are the tax-adjusted user cost of
capital and the desired capital stock in this equilibrium?
(a)
r =
0.10
uc/(1
-
τ)
= (r
+
d)pK/(1
-
t)
= [(.1
+ 0.2)
´
1]/(1 - 0.5)
= 0.6.
MPKf
=
uc/(1
-
t),
so 20 - 0.02K
= 0.6; solving this gives
K
= 970.
Since K - K-1 = I - dK, I = K - K-1 + dK
I= 970 - 900 + (.2)(900) = 250
(b) i.
Solving for this in general:
uc/(1
-
t)
= (r
+
d)pK/(1
-
τ)
= [(r
+
.2)
´
1]/(1 - 0.5)
.
MPKf
=
uc/(1
-
t),
so .
I = K - K-1 + dK =
I = 980 - 100r - 900 + (.2)(900)
I = 260 - 100r
ii. Y
= C
+ I
+ G
1000 = [100
+ (.5
´
1000) - 200r]
+ ( 260 - 100r )
+ 200
1000 = 100 +500 - 200r +260 - 100r + 200
r = .2
C = 560
I =
uc/(1 - t) = .8
K = 1000
4.
Use the saving-investment diagram to analyze the effects of the following
on national saving, investment, and the real interest rate. Explain your
reasoning.
a.
Consumers become more future-oriented and thus
decide to save more.
b.
The government announces a large, one-time bonus
payment to veterans returning from a war. The bonus will be financed by
additional taxes levied on the general population over the next five years.
c.
The government introduces an investment tax
credit (offset by other types of taxes, so total tax collections remain
unchanged).
d.
A large number of accessible oil deposits are
discovered, which increases the expected future marginal product of oil rigs and
pipelines. It also causes an increases in expected future income.