Download Midterm Exam

Please write your answers on this exam paper.
Name____________________
Student ID____________________
Midterm Exam
Economics 514
Macroeconomic Analysis
November 13, 2008
Each Question 14 2 7 Points Each.
1. Permanent Income Hypothesis Two households born at time 0 lives through
time T = ∞. The household begins period zero with zero financial wealth Thus,
the present value of lifetime consumption is equal to the present value of lifetime
income.


Ct
Yt



t
t
t 0 1  r 
t 0 1  r 
the interest rate is 20% (i.e. r = .2). Each household has initial income Y0.and
chooses consumption according to the permanent income hypothesis. In each
case, we can write consumption as proportional to initial income C0 = mpc∙Y0.
A. The first household experiences continuously declining income in all future
periods. The level of income in each period is a fraction of the last period. Yt =
ρ∙Yt-1. Solve for mpc when ρ = .9

C0 
t 0
C0 
1  r 

1 r
1 r
  

C0  Y0  
Y0
 
r
r  (1   )
t 0  1  r 
t
1
t
r
1 r
2

Y0  Y0
1  r r  (1   )
3
1
B. The second household experiences constant growth in output through period N,
when the household retires from work. Yt = (1+g)∙Yt-1 if t  N At time N+1 and
for all periods after, the household has zero income. Solve for mpc when N = 9
and g = .2.

C0 
t 0
1  r 
T
1 r
 1 g 
C0   
 Y0
r
t 0  1  r 
t
1
t

t
 1 g 
g  r  
 Y0  TY0
t 0  1  r 
r
10
C0 
 T  Y0
1 r
6
T
2
Please write your answers on this exam paper.
C. Explain in words why mpc is an increasing function of ρ in part A. and an
increasing function of N in part B
The household attempts to smooth consumption by consuming a weighted average of
current and future income. A more permanent level of income indicated by a high ρ is
also consistent with a higher future income; A longer working life, consistent with a
high N also indicates a higher average.
2. Implied Capital Rental Rate The inflation rate of output goods price is 6% (i.e.
π = .06). The nominal interest rate is 9% (i = .09). The relative price of
investment goods to output goods is always ptI =.75 and the depreciation rate is
9%. Solve for capital productivity and the capital labor ratio when the marginal
productivity of capital equals the capital rental rate and production is given by the
Cobb-Douglass function.
1 Y
MPK  ( )  (r   )  p I  .09 
2 K
Y
K
 .18   
K
L
 12

K
 .182  30.86
L
Yt  Kt 2 Lt
1
3
1
2
3. Precautionary Savings A household lives for two periods. The household begins
with zero financial wealth and earns Y0 = 100. The household is perfectly patient
with discount rate β =1 and faces an interest rate (1+r) = 1. The household faces
uncertainty about future income. The household maximizes expected utility
U  u(C0 )   E0 u(C1 ) where the felicity function is u(C) = 1000 C –½ C2 and
marginal utility of consumption is U’(C) = 1000 – C subject to the present value
of consumption being equal to the present value of output.
A. The household has a 1 3 chance of receiving Y1 = 150 and a 2 3 chance of receiving
Y1 = 60. Calculate the expected value of Y1 and the saving in the first period.
E[Y1 ]  13 150  32 60  50  40  90
The household maximizes
U  u (C0 )  13 u (150  (100  C0 ))   23 u (60  (100  C0 ))  
u '(C0 )  13 u '(150  (100  C0 ))   23 u '(60  (100  C0 ))   E0 u '(C1 ) 
1000  C0  E[1000  C1 ]  C0  E C1   E[Y1  (Y0  C0 )]  E[Y1 ]  (Y0  C0 ) 
C0 
Y0  E[Y1 ] 100  90

 95
2
2
S=5
B. Calculate the saving in period 0 if the household knows with certainty that there
income will be equal to the expected value of output calculated in the section A.
Is the household a precautionary saver?
u '(C0 )  u '(C1 )
1000  C0  1000  C1  90  (100  C0 ) 
C0  95
Not a precautionary saver.
4
5
Please write your answers on this exam paper.
4.
Autarky and the Interest Rate Assume a patient household with a discount rate
β = 1.lives for 2 periods Each maximizes utility
U
11
C0
1 1
 1 C1   1
 1 1
1 1


subject to the constraint that the present value of consumption equals the present
value of income.
C
Y
C0  1  Y0  1
1 r
1 r
The income in Y0 = 100 and income in period 2 is Y1 = 121.
A.
What is the level of the interest rate such that the household would neither
borrow nor save at the end of time 0 when ψ = 2.
C
1
1
C0   (1  r )  C1   1  (1  r )  2  1.21  r  .1
C0
B.
Do you think the household would be a borrower or a saver if the actual
interest rate were r = .11. Explain your intuition.
If the interest rate were at the level that would cause the household to save
nothing, (i.e. r = .1) and it rose a little bit, we would think that the substitution
effect would be present, but the income effect would be absent, therefore the
household would save more (i.e. a posiive amount).
6
5. Excise Taxes on Capital Assume the production function of a firm is given by
.1
Y  K  K 2 so that the marginal product of capital was MPK  1  .1K .
2
A. The firm can rent capital at a capital rental rate of R  .1 . Calculate the
P
optimal level of capital. Calculate output at that level of capital. Calculate the
capital bill. Calculate the surplus.
MPK  1  .1K  R
P
 .1  K  9
Y = 4.95. Capital bill = .9. Surplus 4.05
B. Now, the government imposes a direct tax, tw, on hiring capital so that the
after-tax cost of hiring capital is R  tw and profits are given
P
.1 2 R
 tw)  K . Calculate the tax bill when tw = .1 and
by Profits  K  K  (
P
2
.8. Explain which regime produces the largest tax bill and why.
MPK  1  .1K  R  tw  .2  K  8  tw  K  .8
P
MPK  1  .1K  R  tw  .9  K  1  tw  K  .8
P
The higher capital tax reduces the optimal capital and reduces the tax base. This
means the same amount is collected as at a lower tax rate.
7
C. Calculate the deadweight loss when tw = .8.
Tw=.8 implies K = 1 implies Y = .95. and the capital bill is .1 and the tax bill is .8.
This means the surplus is .05. The deadweight loss is the change in surplus less the
tax bill = 4 -.8 = 3.2.
6. Spending and Medical Bills A household begins period zero with zero financial
wealth, lives for T periods and earns Y0 = Y1 = 100 in each period. The household
faces a zero net interest rate, r=0 and maximizes utility subject to the present
value of consumption being equal to present value of income.
T
T
Ct
Yt



t
t
t 0 1  r 
t 0 1  r 
The household is extremely patient and has a subjective discount factor of β = 1.
In each period, the household faces some medical expenses MBt which generate
no utility but require some consumption expenditure. The felicity function is
1
u (Ct )  ln(Ct  MBt ) u '(C ) 
so the household only gets utility from
(Ct  MBt )
consumption beyond the medical bill.
A.
Set T = 1. Solve for C0 and C1 when MB0 = 0 and MB1 = 0.
1
1
1
1



 C0  C1
(C0  MB0 ) (C1  MB1 )
(C0 ) (C1 )
2C0  200  C0  100
8
9
B.
Set T = 1. Solve for C0 and C1 when MB0 = 50 and MB1 = 0. Explain the
impact on consumption in each period.
1
1
1
1



 C1  C0  MB0
(C0  MB0 ) (C1 )
(C0 ) (C1 )
2C0  MB0  200 
MB0
MB1
; C1  100 
2
2
C0 = 75; C1 = 125
C0  100 
C.
C19.
Set T = 19 and MB0 =….=MB18 = 0 and MB19 = 1000. Solve for C0 and
1
1
1
1
1


 ... 


(C0 ) (C1 ) (C2 )
(C18 ) (C19  MB19 )
C0  C1  C2  ...  C18
C19  C0  1000
20  C0  1000  Y0  ...  Y19  2000
C0  50, C19  1050
10
Please write your answers on this exam paper.
7. Investment in the A-K Model {A little harder} A firm has an extremely capital
intensive production technology so that Yt  AKt where we assume for simplicity
that MPKt = A = 1. The firm pays no wages but must buy dividends at a price of
pI = 1. Assume that there is a zero depreciation rate and a firm increases its
capital stock by installing investment Kt 1  Kt  I t . However, if they adjust
investment quickly in either direction they pay some adjustment costs. The
1
2

amount of investment goods they must buy to install It is  I t    I t  
2


The value of the firm at time 0 is the discounted sum of the firm’s profits:
1
2

Kt   I t    I t  

V0
2



t
P0 t 0
(1  r )
a. Assume that the managers of the firm choose It and Kt+1 to maximize the
value of the firm to the shareholders. Use the Lagrangian method to write the
first order conditions that describe the optimum.
1
2

Kt  ptI   I t    I t    qt  K t 1  K t  I t 

V0
2




P0 t 0
(1  r )t
1  I t  qt  I t  qt  1
qt 
1  qt 1
1 r
11
b. Suppose the real interest rate is r = .5. Calculate the level of investment at

x
time t = 0. (Hint: If x < 1, then  x t 
)
1 x
t 1

1  qt 1
1
x
1
qt 
,x 
 q0   x t 
 2
1 r
1 r
1 x r
t 1
I0  2 1  1
12