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
(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. 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?
(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. Repeat
Part (b) for a real interest rate of 8% per year
(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. Repeat
Part (b) for a 40% tax on HHHHC’s sales revenues.
(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. 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.
(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
(a)
Sd
=
Y -
Cd
-
G
=
Y -
(3600 - 2000r
+ 0.1Y)
- 1200
= -4800
+ 2000r
+ 0.9Y
b. Using
both versions of the goods market equilibrium conditions, Eqs. (4.7) and (4.8),
find the real interest rate that clears the good market. Assume that output
equals full-employment output.
(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.
(2) Using
Eq. (4.8):
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. 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?
(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 the Figure
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 economywide 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?
Some were confused about this problem. The issue is that in this answer the tax rate is 15% as compared to 50% as stated in the problem. I will let you walk through the problem to prove this to yourself. It is much too difficult for a 50 min exam, but the process is useful to walk through.
(a)
r =
0.10
uc/(1
-
τ)
= (r
+
d)pK/(1
-
t)
= [(.1
+ 0.2)
´
1]/(1 - 0.15)
= 0.35.
MPKf
=
uc/(1
-
t),
so 20 - 0.02K
= 0.35; solving this gives
K
= 982.5.
Since
K
-
K-1
=
I
-
dK, I
=
K -
K-1
+
dK = 982.5
- 900
+ (.2
´
900) = 262.5.
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?
(b) i.
Solving for this in general:
uc/(1
-
t)
= (r
+
d)pK/(1
-
τ)
= [(r
+
.2)
´
1]/(1 - 0.15)
= .235
+ 1.176r.
MPKf
=
uc/(1
-
t),
so 20 - 0.02K
= 0.235
+ 1.176r;
solving this gives
K
= 988.25
- 58.8r.
I =
K -
K-1
+
dK = 988.25
- 58.8r
- 900
+ (0.2
´
900) = 268.25
- 58.8r.
ii. Y
= C
+ I
+ G
1000 = [100
+ (.5
´
1000) - 200r]
+ (268.25
- 58.8r)
+ 200
1000 = 1068.25
- 258.8r,
so 258.8r
= 68.25
r =
0.264
C =
100 + (0.5
´
1000) - (200 × 0.264) = 547.2
I = 268.25
- (58.8 × 0.264) = 252.7 =
S
uc/(1
-
t)
= 0.235 + (1.176
´
0.264) = 0.545
K
= 988.25 - (58.8 × 0.264) = 972.7
(a) To find the equilibrium
interest rate (rw), we must first calculate the current
account for each country as a function of rw. Then we can find
the value of rw that clears the goods market, that is, where
CA + CAFor
= 0.
Home:
Cd
= 320
+ 0.4(1000 - 200)
- 200rw
= 320 + 320
- 200 rw
= 640 - 200 rw
CA
= NX
= Sd – Id
= Y – (Cd
+ Id
+ G)
= 1000 – (640 – 200 rw
+ 150 – 200rw
+ 275)
= –65 + 400 rw
Foreign:
CAFor
= NXFor
= SdFor
- IdFor
= YFor
-
(CdFor
+ IdFor
+ GFor)
= 1500 - (960
- 300rw
+ 225
- 300rw + 300)
= 15 + 600 rw
At
equilibrium, CA + CAFor
= 0, so:
–65
+ 400 rw
+ 15
+ 600 rw = 0
–50
+ 1000 rw
= 0
rw
= 0.05
C
= 640
- 200 rw = 630
CFor
= 960
- 300 rw = 945
S
= Y
- C
- G
= 1000
- 630
- 275 = 95
SFor
= YFor
- CFor
- GFor
= 1500
- 945
- 300 = 255
I
= 150
- 200 rw = 140
IFor
= 225
- 300 rw = 210
CA
= S
- I
= 95
- 140 =
-45
CAFor
= SFor
- IFor
= 255
- 210 = 45
(b) Cd
= 320
+ 0.4(1000 - 250)
- 200 rw
= 320 + 300
- 200 rw
= 620 - 200 rw
CA
= NX
= Sd
- Id
= Y
- (Cd
+ Id
+ G)
= 1000 - (620
- 200 rw
+ 150
- 200 rw +
325)
= -95
+ 400 rw
At
equilibrium, CA + CAFor
= 0, so:
-95 +
400 rw + 15
+ 600 rw
= 0
-80 +
1000 rw = 0
rw
= 0.08
C
= 620
- 200 rw = 604
CFor
= 960
- 300 rw = 936
S
= Y
- C
- G
= 1000
- 604
- 325 = 71
SFor
= YFor
- CFor
- GFor
= 1500
- 936
- 300 = 264
I
= 150
- 200 rw = 134
IFor
= 225
- 300 rw = 201
CA
= S
- I
= 71
- 134 =
-63
CAFor
= SFor
- IFor
= 264
- 201 = 63
So a
balanced-budget increase in government spending increases the home country’s
current account deficit.
(a) The home country’s saving
curve shifts to the right, from S1 to S2 in
Figure 5.5. The real world interest rate falls, so that the current account
surplus in the home country equals the current account deficit in the foreign
country. From Figure 5.5, S rises, I rises, CA rises, rw
falls.
Figure 5.5
(b)
The foreign country’s saving curve shifts to the right, from
to
in Figure 5.6. The real world interest rate must fall, so the
current account surplus in the foreign country equals the current account
deficit in the home country. As shown in the figure, S falls, I
rises, CA falls,
rw falls.
Figure 5.6
(c)
The foreign country’s saving curve shifts to the left, from
to
in Figure 5.7. The real world interest rate must rise, so the
current account deficit in the foreign country equals the current account
surplus in the home country. As shown in the figure, S rises, I
falls, CA rises,
rw rises.
Figure 5.7
(d)
If Ricardian equivalence holds, there is no effect. If Ricardian
equivalence does not hold, then
the result is the same as in part (b), as the foreign country’s saving curve
shifts to the right.