Download Economics 514

Name________________
Student ID ________________
Economics 514
Macroeconomic Analysis
Midterm Exam 2
November 15, 2007
Write all answers on the white exam paper. Do not turn in the blue books. 20 points each.
1. Interest Rates and Saving
There are three households each of who live for two periods (period 0 and period 1).
Each household begins the period with zero financial wealth and has a utility of the
form ln(C0 )  ln(C1 ) . Note that this utility function implies the household is patient
and has a subjective discount factor β = 1. Each household can save or borrow across
time to maximize utility and faces a real interest rate of zero, r = 0, 1+r = 1. The only
difference comes in terms of the time path of income Y0 and Y1 which is given in the
following table
Household Y0
Y1
i)
0
100
ii)
50
50
iii)
100
0
a. Calculate the present value of lifetime income for each household.
Present value of lifetime income is hw  W  Y0 
Y1
 100 for all households
1 r
b. Calculate consumption, C0, and savings, S0 = Y0 – C0 for each household.
Maximize utility, ln(C0 )  ln(C1 ) subject to hw  C0 
u '(C0 )  (1  r )  u '(C1 )
C1
the first order condition is
1 r
1
1
 (1  r ) 
(1  r )  1  C0  C1
C0
C1
Household C0
S0
i)
50
-50
ii)
50
0
iii)
50
50
1
c. Assume that the real interest rate goes up by 1% to (1+r) = 1.01. By what % will
C
the ratio 0 change (i.e. what is the intertemporal elasticity of substitution) for
C1
each household?
C 
 ln  0 
C 
C
1
 C1 
 0  ln  0    ln(1  r ) 
 1
 ln(1  r )
(1  r ) C1
 C1 
d. What will be consumption and saving after the interest rate rise for each
household
If r = .01, then the first order conditions suggests C1 = (1+r)C0 so
(1  r )C0
hw  C0 
 C0  C0 . This means that regardless of the interest rate,
1 r
households consume half of human wealth. Calculate the human wealth under the
different interest rates.
Household Hw
C0
S0
i)
99.0099 49.50495
-49.505
ii)
99.50495 49.75248 0.247525
iii)
100
50
50
2
e. Explain how the response of saving by household iii) to the rise in the interest rate
is qualitatively different than the response of households i) and ii). Explain why.
An interest rate rise has two effects on consumption. The higher relative price of
current consumption causes a shift of the consumption pattern toward the future. This
substitution effect tends to reduce current consumption. Second, the higher interest
rate, will change income that a household earns on its savings or must pay on its
borrowing. This income effect will give more funds to consume for savers and fewer
funds to consume for borrowers. For household 1, the income effect and the
substitution effect both work to reduce consumption and increase saving. For
household 2, there is no income effect and the substitution effect reduces
consumption and increases saving. For household 3, the substitution effect cancels
out the substitution effect.
3
2. Retirement and Savings
In our economy, households will begin period 0 of their life with no financial wealth
and live for 61 years until time T = 60. The household will work for 41 years and then
retire. The household has an income stream equal to Yt = 100 from period 0 to period
N = 40 and Yt = 0 from period 41 to period 60. Assume the real interest rate is
constant and the Permanent Income Hypothesis is true so the household borrows or
saves to choose a constant consumption level C from time 0 to time 60.
a. Calculate the (human) wealth of the household at time 0. Calculate the
consumption, C0, and savings, S0 = Y0 – C0, of the household in period 0.
The human wealth of the household is
Y3
Y
Y2
Y41
hw  Y0  1 

 ... 

2
3
1  r (1  r ) (1  r )
(1  r ) 41
1
100
100
100
100
(1  r ) 41
100 



...


100

1
1  r (1  r ) 2 (1  r )3
(1  r ) 41
1
1 r
The present value of consumption is equal to human wealth.
C3
C61
C
C2
hw  C0  1 

 ... 
2
3
1  r (1  r ) (1  r )
(1  r )61
1
1
C
C
C
C
(1  r )61
C



...


C
1
1  r (1  r ) 2 (1  r )3
(1  r )61
1
1 r
1
1
1
1
1

1
41
41
(1  r )
(1  r )
(1  r )61
1  r hw 
C 
100  S  100  C 
1
1
1
1
1
1
61
61
(1  r )
(1  r )
(1  r )61
1
4
b. Assume that there are a number of households equal to POP so that consumption
is C = C *POP. The GDP per capita of the economy is equal to the output of a
household that is working (i.e. Y = 100) multiplied by the fraction of the
GDP  C
population that is not retired. Calculate the savings rate
of this
GDP
economy if 75% of the households are employed and 25% are retired. Calculate
the savings rate of the household if 50% of the households are employed and 50%
are retired.
Define f as the share of households who are working.
1
GDP  C 100  f  POP  C  POP
C
1
(1  r ) 41

 1
 1
GDP
100  f  POP
100  f
f 1 1
(1  r )61
The savings rate is lower when retirees are a larger share of the population
1
5
c.
If the household is to continue consuming this level, C after retirement they will
need to have acquired financial wealth. From the standpoint of period 41, how
much wealth will they need to continue to consume C for the remaining 20 years
of their life, if they have no income during those years.
W  C41 
C43
C60
C42
C44


 ... 
2
3
1  r (1  r ) (1  r )
(1  r )19
1
1
1
20
C
C
C
C
(1  r )
(1  r ) 41
C



...


100
1
1
1  r (1  r ) 2 (1  r )3
(1  r )19
1
1
1 r
(1  r )61
1
6
3. Investment Prices and the Real Wage Rate
According to the Penn-World tables, in China, the average price of investment goods
relative to the price of output goods in year 2000 was pI = 1.53. The average price of
investment goods relative to in the USA in the same year was pI = .85. Assume that
the real interest rate was 4% and the depreciation rate was 12% and expected inflation
in output goods prices equaled the expected inflation in investment goods prices.
What would be the real capital rental rate for capital invested in year 2000? What
would be the marginal product of capital in year 2001 if firms chose a profit
maximizing level of investment in 2000? If both countries had a Cobb-Douglas
production function with equal technology levels, A = 1, and equal capital intensity
functions α = 1/3, what would be the capital labor ratio in each country in 2001?
What would be the real wage rate in each country?
Rt 1 
1   tI1  I
 1  rt  (1   )
 pt . If we have r =.04 and δ
Pt 1 
1   t 1 
R
=1, then we have t 1   rt    ptI  .16 ptI . The marginal product
Pt 1
The capital rental price is
=.12 and
1   tI1
1   t 1
1
L 
Y
.16 p  MPK t 1   t 1    t 1 
K t 1
 K t 1 
I
t
of capital should be set so that

2
 L 3
  t 1  
 K t 1 
1
3

 32
K t 1
 .48 ptI
Lt 1
. The real wage rate equals the marginal product of labor which is proportional to

labor productivity
K 
Wt 1
Y
 MPLt 1  (1   ) t 1  1     t 1  
Pt 1
Lt 1
 Lt 1 

2
3
.48 p 
I
t
pI
USA
China
1

2
3
.48 p  
I
t
 32
2
3
1
.72 ptI
Rt 1
Pt 1
K t 1
Lt 1
0.85
1.53
0.136
0.2448
7
Wt 1
Pt 1
3.837159
1.588916
1.043707
0.777933
4. Structural Employment
In an economy with a labor force of 1 million people, the average fraction of
unemployed people who find jobs is 50%. The average fraction of employed people
who lose their jobs is 2%. Solve for the number of unemployed people in steady state.
s
.02
1
ur SS 


s  f .5  .02 26
38461.54
8
5. Uncertainty and Saving
A household lives for two periods, 0 and 1, has zero financial wealth, and earns
income Y0 = 100 in the first period. The household maximizes a utility function U(C0,
C1) = ln(C0)+ln(C1). The real interest rate is zero, r = 0.
a. If output is, with certainty, equal to Y1 = 100, what would be consumption
and savings in period 0.
C0 = 100, S0 = 0
9
b. Assume that income in period 1 is uncertain and will be Y1,GOOD = 150
with a 50% probability or Y1,BAD = 50 with a 50% probability. Calculate
the expected marginal utility of consumption in period 1, if the savings
that the household does is equal to that solved for in part a. Is this greater
than or less than the marginal utility in period 0, if the savings were equal
to that solved for in part a. If output in the second period were uncertain,
would this household have positive savings? Explain your answer.
Marginal utility of consumption in time 0 is u '(C1 ) 
utility is E[u '(C1 )]  .5
1
 .5
1
1
Expected marginal
100
. IF S = 0, then this is
Y1,GOOD  S
Y1, BAD  S
1
1
1
E[u '(C1 )]  .5
 .5 
. The expected marginal utility of consumption
150
50 75
is higher in the future. The household should maximize expected utility by
shifting consumption through the future by saving.
10
c. [Warning: Save this for last] Solve for a level of savings which would set
the marginal utility of consumption in period 0 equal to the expected
marginal utility of consumption in period 1. Reminder: If ax2 + bx + c = 0
then
E[u '(C1 )]  .5
Y
Y
1, BAD
1
Y1,GOOD  S
 .5
 S   Y1,GOOD  S 
1,GOOD
 S Y1, BAD  S 

1
Y1, BAD  S

1
 u '(C0 )
Y0  S
2
Y0  S
 Y1, BAD  Y1,GOOD

S

2
Y0  S 
1
 

 
Y0  S Y1,GOOD  S Y1, BAD  S  Y1,GOOD  S Y1, BAD  S 
 Y1,GOOD  S Y1, BAD  S   Y0  S Y0  S 
 2 S 2  Y1,GOOD  Y1, BAD  S  (Y1,GOOD  Y1, BAD  Y0 2 )  0
 2 S 2  200 S  2500  S  100 S  1250  0
S
100  10000  5000
 50  25 6
2
S=11.23724
11