Homework Assignment 2
Economics 215
Intermediate Macroeconomics
Assigned: March 22, 2000
Due March 28, 2000
1. Investment, the Interest Rate, and Technology. A firm has hired 100 labor
units for next period (L+1 = 100). The firm uses the production technology
Y1  T1  K 1  L1  MPK 1  12 
Y
K
Assume the technology level T+1 = 1.
a.
What is the marginal product of capital when K+1 = 900. What is the
marginal product of capital when K+1 = 1600. Draw the marginal
product of capital schedule. What is the marginal product of when K+1 =
300 1
K  900  Y1  30  10  300  MPK 1  12 

900 6
100.
400 1
K  1600  Y1  40  10  400  MPK 1  12 

1600 8
b.
Assume 0 taxes and a depreciation rate of capital of 10% (d = .1).
Calculate the marginal product of capital at which the net present value
of an investment project would be 0 if the real interest rate were r  16 .
Calculate the marginal product of capital at which the NPV of an
investment project would be 0 if the real interest rate were r = .25.
Calculate the marginal product of capital at which NPV of an
investment project is 0 when r = .1. With zero taxes, the net present
value of an investment project is zero when the marginal product of
capital is equal to net interest costs + depreciation costs.
10
r  d  MPK 1  12 
K*
2
rd  
 75 
 K     351.5625
 4

4
15
 .2666  MPK 1
r  d  14  101 
7
20
 100 
 .35  MPK 1  K *  
  204.0816
 7 
1
6
1
10
*
2
2
rd 
c.
1
10

1
10

2
10
 .2  MPK 1
 25 
 K     625
 1 
*
Calculate the target capital stock when r  16 , r = .25, and r = .1. d.
Now, assume a huge technological breakthrough doubles the expected
value of technology, T+1 = 2. Solve for the target capital stock when
r  16 , r = .25, and r = .1.
r  d  MPK 1  12 
20
K*
2
r  d  .2666  MPK 1
 75 
 K  4     1406.25
 4
*
2
r  d  14  101 
7
20
 100 
 .35  MPK 1  K *  4  
  816.34
 7 
2
 25 
r  d  101  101  102  .2  MPK 1  K *  4     2500
 1 
2. Investment and Taxes. A Taxi company considers buying taxis for next period.
The taxi company buys taxis every period and then resells them. They set the
quantity of their capital (the value of their taxis) such that the after tax net present
value of their taxis is zero:
 P1  MPK 1  P1  (1  d )  TAXESt  TAXCREDITS t

 Pt   0 Assume

1 i


that taxes are a constant fraction the extra revenue of any investment project
TAXES+1 =  MPK+1. Tax credits are a deduction for depreciation costs
TAXCREDITS+1 =   d. Solve for the marginal product of capital that sets the
after tax net present value of an investment project equal to zero when  = .25, d =
.1 and the real interest rate r = .1. Solve for the marginal product of capital that
sets the after tax net present value of an investment project equal to zero when  =
.5, d = .1 and r = .1. Under this tax regime what effect does a raise in  have on
the target capital stock (i.e. will a raise in taxes increase or reduce the target
capital stock)?


  MPK 1  (1  d )    MPK 1  d  1     (1   )  MPK 1  (1   )  d  r   0
t


1 i

1 r


P1


P

(1   )  MPK 1  (1   )  d  r  MPK  d  (1r )
 = .25, d = .1 r = .1 MPK.2333
 = .5, d = .1 r = .1 MPK.5
A rise in taxes reduces after tax revenues from capital and also reduces
the after tax depreciation costs. However, the rise in taxes does not reduce
interest costs. Thus, a tax rise reduces revenues relative to costs and reduces the
target stock of capital at any interest rate.
3. Permanent Income and Savings. Assume that the real interest rate is r = .1.
Consider two individuals, Kelly and Andy who live for two periods. Both start off
with zero wealth. Kelly will earn QK = 200 in this period and QK+1 = 110 next
period. Andy will earn QA = 100 in this period and QA+1 = 220 next period.
a.
b.
c.
Calculate the lifetime net present value of Kelly and Andy’s earnings.
Q
Lifetime net present of income are: W  Q  1
1 r
110
220
W K  200 
 300
W A  100 
 300
1.1
1. 1
Assume that Kelly and Andy have preferences such that they will
want to have the same consumption today as in the future period (CK =
CK+1 and CA = CA+1).
Calculate CK and CA. (i.e. calculate the permanent income of each
individual). Calculate the current saving of each individual.
C
C
1 r
W  C  1
C  C 1  C 
W  C 
W
1 r
1 r
2r
1.1
r  .1, W  1  C 
300  157.14
2.1
SK=200-157.14=42.86
SA=100-157.14=-.57.14.
Now, assume that both Kelly and Andy target a certain consumption
level in the second period. Both will act to set CK+1 = CA+1 = 150
regardless of the interest rate. Calculate the current saving necessary to
hit that future target level when r = .1. C1  Q1  (1  r ) S
40
150  110  (1  r ) S K  S K 
1 r
r  .1  S K  36.36
r  .2  S K  33.33
150  220  (1  r ) S A  S A 
r  .1  S  63.63
A
 70
1 r
Calculate the current
r  .2  S  -58.33
A
saving necessary to hit that target level when r = .2. Whose current
consumption will rise and who’s will fall. A rise in the interest rate will
reduce the necessary savings of Kelly and thus increase Kelly’s current
consumption. By contrast, a rise in the interest rate will reduce the
amount of borrowing that Andy can do and hit his target future
consumption. Thus, he must cut back on current consumption.
4. The World Interest Rate. There are two countries in the world economy,
Albania and Zimbabwe. The saving function of Albania is written as SA = 400 +
10 r and the investment function is IA = 500 – 20 r. The saving function of
Zimbabwe is SZ = 200 + 10r and the investment function is IZ = 300 – 10 r .
a.
b.
Solve for the world interest rate. What is the trade balance of Albania
and Zimbabwe at this interest rate? World Saving must equal world
investment
400+10r+200+10r=500-20r+300-10r
or
600+20r = 800-30r
or
50r = 200 or r = 4.
A
S = 440 IA = 420. NXA = 20 SZ = 240 IZ = 260 NXZ = -20.
A technology expansion in Zimbabwe shifts the investment function to
IZ = 400 – 10 r. What is the new interest rate? Solve for the new
savings, investment, and trade balance in Albania and Zimbabwe.
50r = 300 or r = 6.
SA = 460 IA = 380. NXA = 80 SZ = 260 IZ = 340 NXZ = -80.