Mid-Term Exam # 2
Economics 514
Macroeconomic Analysis
Tuesday, November 20th, 2012
1. (25 points) Taxes and Unemployment
All workers in the economy are either employed or unemployed. In every period, 5% of
employed workers lose their job, so the job separation rate is s = .04. In every period, all
unemployed workers receive one job offer. That job offers them a wage per labor unit that
ranges uniformly over the range 0 to 100. The cdf of a uniform distribution over this range is
x
F ( x) 
.
100
Assume that the utility of a household is given by the function
1
1
(1.1)
Ut  Ct 2  2lst 2
If the household has a job, they must spend all of their time at work. Normalize total time to
one, TIME = 1. If they don’t have a job, lst = 1; if they do have a job, lst = 0, Lt = 1. If they do
not have a job, they will be offered an unemployment benefit, b = 36.
a. Calculate the reservation wage, the job finding rate and the steady state unemployment
rate
The utility of an unemployed person is UU = U U  b  2 ls  36  2 1  8 . The utility
of an employed person is U E  wt where Ct = wtLt = wt . If the wage were wR = 64 then
the household would have equal utitilty as unemployed or employed. Only 36% of the
wage offers are above the reservation wage so f = .36. The unemployment rate is
s
.04
ur 

 .1
s  f .04  .36
.
b. Suppose the household must pay a tax on any wage income equal  
17
they receive,
81
64
). Calculate the reservation
81
wage, the job finding rate and the steady state unemployment rate.
so they will keep only a fraction of their pay (i.e. 1   
The utility of an unemployed person is UU = U U  b  3 ls  36  3 1  9 . The utility
64
 wt where Ct = wtLt = wt . If the wage
89
were wR = 81 then the household would have equal utitilty as unemployed or employed.
Only 19% of the wage offers are above the reservation wage so f = .19. The unemployment
s
.04
4
rate is ur 


s  f .04  .19 23
of an employed person is U E  (1   )  wt 
2. (25 points) Interest Rates and The Student Household
Consider a household that lives for two periods, beginning period 0 with zero financial wealth.
The household faces a real interest rate of 10%, (1+r = 1.10). The household maximizes a
discounted utility function of u(C0 )    u(C1 ) . The household earns Y0 from producing goods
in period 0 and Y1 from producing goods in period 1. The first period, the household is a
student and earns no income. In the second period, the household earns $242. The household
will need to borrow some money in order to finance consumption today.
a. Calculate the present value of lifetime income. Calculate period 0 consumption and
10
savings under the permanent income hypothesis (i.e.   ).
11
Y
242
 220 . With Permanent Income
Present value of lifetime income is W = 1 
1  r 1.1
hypothesis
C0 = C1. According to the budget constraint
C
C
1 
21
11

W  C0  1  C0  0  C0  1 
 C0   C0  W 115.2381 .

1 r
1 r
11
21
 1 r 
Savings is -115.2381
b. Assume that the interest rate rises to 15% (but the discount factor stays
10
constant,   ). Calculate the wealth effect. In other words, calculate how much the
11
present value of lifetime income decreases? Draw a graph indicating how the budget
constraint would change if the interest rate were to rise.
120.476
C1
C0
W
Y1
242

 210.44 . Human wealth declines by 9.56.
1  r 1.15
c. Assume that instead of maximizing lifetime utility, the household merely saves in order
to have a consumption level of C1 = 110 while working. Calculate how much consumption
the household can afford before and after the interest rate rises.
If the household wants to spend 110 in period 1, they can pay back a debt of 132. At a 10%
interest rate, they could borrow 120 so they could consume 120 in period 0. If the interest
rate was 15%, they could pay back borrow 132/1.15 =114.78 which they could consume in
period 0. .
[C .95  1]
.95
calculate the intertemporal elasticity of substitution for each case. In other words, in both
cases, the first order conditions of the utility maximizing consumer could be written in the
C
form: ln( 0 )  d  e  ln(1  r ) . Solve for e for Case A and Case B. In which case is relative
C1
consumption more responsive to changes in the interest rate.
d. Consider two possible cases: A) u(C )  20  [C .05  1] ; and B) u (C ) 
1
When
C
1

1
1

1

 .05,   20 . The first order condition is

C0

1
1
 .95,   1.052 and in Case B,
 u '(C )  C  . In Case A,
1

1


 u '(C0 )    1  r   u '(C1 )    1  r   C1
 C0     1  r  

 C1 
1


C0
    1  r  
C1
C 

 ln  0   ln(    1  r   )    ln     ln 1  r 
 C1 
e  
e. Calculate the optimal consumption level when the real interest rate is 10% and when
the real interest rate is 15% under Case A) and Case B).In which case does a rise in the
interest rate have the strongest impact on borrowing? Explain the difference between the
two cases in terms of the income, wealth, and substitution effect.
There are 3 reasons why a rise in the interest rate would lead to a decline in borrowing (an
increase in saving). First, the rise in the interest rate will reduce the present value of future
income. Second, the rise in the interest rate will reduce the amount of borrowing that the
can be done. Third, when the relative price of current consumption goes up, the student
would shift away from current consumption and toward future consumption and they will
borrow less. The first, present value effect and the second income effect will be the same
under both cases. The third substitution effect will be stronger in Case B.
3.
(25 points) Precautionary Saving
A household lives in period 0 and period 1. The household begins with zero financial worth and
will earn an income from producing goods in period 0 equal to Y0 = 1 and will earn income in
period 1 equal to either Y1GOOD = 1-x or Y1BAD = 1+x depending on the state of the economy. The
economy will be in good times or bad times in period 1 with equal probability (i.e. probability of
Pr(Good) = Pr(Bad) = ½). The household is patient and has no discount on the future (β = 1) and
faces a zero net real interest rate (1+r =1). The household chooses consumption (C0) today which
will also determine saving (S = Y0 – C0 <:>0)
S = Y0 - C0 (1
C1GOOD  Y1GOOD  S (2
C1BAD  Y1BAD  S (3
The household maximizes the sum of expected lifetime felicity u(C). In this case, we
can write expected utility as
1
1

u (C0 )  E[u (C1 )]  u (C0 )    u (C1Good )   u (C1Bad ) 
2
2

The felicity function is a polynomial: u (C )  10  10C  2C 2  13 C 3
f ( x)  d  e  x  f  x 2  g  x3 
[Hint
f '( x)  e  2 f  x  3 g  x 2
f ''( x)  2 f  6 g  x
]
f '''( x)  6 g
a. Calculate the optimal level of consumption and savings in period 0 if x = 0 so future
income was known with certainty. Calculate marginal utility in period 0, MU0 = u’(C).
If we know income with certainty next period, x = 0, then Y1 = Y0 = 1 we set u’(C1 ) =
u’(C0 ) so C1 = C0 = Y1 = Y0 = 1. Then u’(C0) = 10 – 4C+C2 = 7
b. Now, assume that x = ½ but the household chooses the consumption level that solved
part a. Calculate expected marginal utility E[u’(C1)]. Is expected marginal utility in
time 1 greater or less than marginal utility in time 0? Should the household save more
for the future than they would under certainty or less? Explain in 1 paragraph and 1
graph.
u'(C)
u'(C1BAD)
E[u'(C1)]
u'(C0)
u'(C1GOOD)
C
.5
1
1.5
If the household saves nothing, then their consumption in the second period would equal
income
C1GOOD  Y1GOOD  1.5
C1BAD  Y1BAD  .5
u '(C1GOOD )  10  4C1GOOD   C1GOOD 
2
and u '(C1BAD )  10  4C1BAD   C1BAD 
u '(Y1GOOD )  10  4Y1GOOD  Y1GOOD   6.25
2
2
u '(Y1BAD )  10  4Y1BAD  Y1BAD   8.25
2
So E u '(C1 )   .5  8.25  .5  6.25  7.25  7
The (average) marginal benefit of saving is greater than the marginal cost. Save more.
c. [Hard] Calculate the optimal level of consumption and savings in period 0 if x = ½ .
u '(C0 )  E u '(C1 ) 
u '(Y0  S0 )  E u '(Y1  S0 ) 
10  4  (1  S )  (1  S ) 2  10  4  (1  S )  (1  2 S  S 2 )
 7  2S  S 2
 1 2  u '(Y1GOOD  S0 )   1 2  u '(Y1BAD  S0 ) 
 1 2  10  4  ( 1 2  S )  ( 1 2  S ) 2   1 2  10  4  ( 3 2  S )  ( 3 2  S ) 2 
 10  4(1  S )  1 2  ( 3 2  S ) 2  ( 1 2  S ) 2  
 10  4(1  S )  1 2  ( 9 4  3S  S 2 )  ( 1 4  S  S 2 ) 
 10  4(1  S )  1 2  ( 9 4  3S  S 2 )  ( 1 4  S  S 2 ) 
 6  4 S  1 2  10 4  4 S  2 S 2   7.25  2 S  S 2
 7  S  S 2  7.25  2 S  S 2  S  .0625, C0  .9375
4.
(25 points) Capital Taxes and Production
Firms produce goods sold at price P with a constant technology Cobb-Douglass technology
1
1
Yt  Kt 2 ( Lt ) 2 with workers, Lt, hired at a competitive wage, Wt and capital accumulated
through investment, It, purchased at a price of PI. This means that the flow of income for the
1
1
firm is t  P  Kt 2 ( Lt ) 2  Wt  Lt  P I  It   P I  Kt . Assume that there is perfect price
stability (i.e zero inflation in either the goods price or the investment price) and that the relative
I
price is 1 (i.e. P
= 1). The interest rate is 5% (i = r = .05 ) and the depreciation rate is 10%
P
(  =.1).
a. Assume that managers of the firms choose a number of workers, Lt; investment, It; and
future capital, Kt+1 to maximize the discounted present value of the income stream generated
T
t
by the firm: 
subject to the capital accumulation equation Kt 1  (1   ) Kt  It . Write
t
t 0 1  i 
down the first order conditions describing the optimal decision of the firm.
Kt 2 ( Lt ) 2 
1
T
V0

P t 0
1
Maximize

P
 Lt 
Kt 2 ( Lt ) 2 
1
T
Wt
1
Pt I
1  i 
Wt
P
P
 It  
1) qt = 1, 2)
P
Kt
t
Kt 2 ( Lt ) 2 
1
T

t 0
1
Wt
 L  I   Kt
P t t
t
1  r 
 Lt  I t   Kt  qt  Kt 1  (1   ) Kt  I t 
1  r 
t
t 0
1
Pt I
1
2
Kt ( Lt )
2
 12

Wt
P
, 3) qt 
1
K t 1 2 ( Lt 1 ) 2  (1   )qt 1  
1  r 
1
2
1
Combining 1) and 3) 1  r   1 2 Kt 1 2 ( Lt 1 ) 2  (1     )
1
1
b. Calculate the profit maximizing capital-labor ratio for this firm when the tax rate is equal to
τ = 0, .05, and .1. Assume that labor supply is constant and equal to L = 1. What would be the
real wage that a firm would be willing to pay a single worker when the tax rate is equal to τ =
0, .05, and .1.
τ
Kt 1
Wt 1
P
0
11.11111
0.05
6.25
0.1
4
1
 12
1
Combining 1) and 3) 1  r   (1   ) 2 Kt 1 ( Lt 1 ) 2  (1   )
 K t 1 
1
2

 Lt 1 
L
1.666667
1.25
1
 12
 r    
K t 1  1 


Lt 1  .3  2 
1
2
 Kt 1 
 1 
W
1
2
  t P 

 .6  4 
 Lt 1 
K
1
1
1
Combine with L = 1, Kt+1 = t 1 , Yt  Kt 2 ( Lt ) 2  Kt 2
Lt 1
c. What would be the profit maximizing quantity of capital that the firm would want to have
at time t+1 if L = 1 and τ = 0, .05, and .1. Calculate the level of output that the firm would have
when the tax was set at zero. What is the level of output if the tax is raised to 5%. How much is
the level of output if the tax is raised to 10%?
d.
τ
Yt+1
Kt 1
0
0.05
0.1
2
2.777778
1.5625
1
3.333333
2.5
2