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Midterm Exam
Economics 514
Macroeconomic Analysis
November 15, 2004
Short Answer (10 points each)
1. The inflation rate of output goods price is 6% (i.e. π = .06) and the inflation rate
of investment goods is 8% (i.e. πI = .08). The nominal interest rate is 9% (i = .09).
The relative price of investment goods to output goods is ptI =.75 and the
depreciation rate is 10%. Solve for capital productivity when the marginal
productivity of capital equals the capital rental rate and production is given by the
Cobb-Douglass function.
1
2
Qt  Kt 3 Lt 3
2. A corporation has a book value of debt equal to 500 million. The firm has 10
million shares outstanding with a market price of $45. The replacement value of
the firms’ capital is 750 million. Calculate Tobin’s q for this firm. Under q-theory,
should this firm invest more than enough to replace depreciated capital or less?
Long Answer (20 points each)
3.
Assume that there are a number of identical households beginning that live for a
number of periods in an economy. Each maximizes utility
U  u (C0 )   u (C1 )   2u (C2 )   3u (C3 )  ...   T u (CT )
subject to the constraint that the present value of consumption equals the present
value of income.
C
C2
CT
Q
Q2
QT
C0  1 
 .... 
 Q0  1 
 .... 
2
T
2
1  r (1  r )
(1  r )
1  r (1  r )
(1  r )T
All of the households in the economy are identical and have constant
intertemporal elasticity of substitution felicity function u (c ) 
1 1
c
1 1
. If investment and
government spending are zero, the economy is closed and output growth is constant
and equal to 1+g = 1.02, what is the autarky interest rate when β = .98 and the
elasticity of substitution is ψ = 1. What is the autarky interest rate when the elasticity
of substitution is ψ = 5.
4. A household lives for two periods. The household begins with zero financial
wealth and earns Q0 = Q1 = 100. The household maximizes the constraint
U  u(C0 )   u(C1 ) subject to the present value of consumption being equal to
C
Q
the present value of output C0  1  Q0  1 .
1 r
1 r
a. Assume that the felicity function is quadratic u(c) = 1000 c –½ c2 so that
the marginal utility of consumption is u’(c) = 1000 – c. Draw a picture of
the marginal utility of consumption as a function of c. What is the
consumption and savings in period 0 if β∙(1+r) = 1. What is the marginal
utility of felicity in period 0?
b. Now, assume that income in period 1 is either 50 or 150 with equal
likelihood (Prob = .5). Calculate the expected marginal utility of
consumption in period 1 when savings is equal to that solved under
certainty in part a.
c. Is expected marginal utility in period 1 in part b larger than the marginal
utility of consumption in period 0, under period a. Is the household a
precautionary saver?
5. A household lives for 20 periods from period 0 to period T = 19. The household
operates according to the permanent income hypothesis so that it consumes the
annuity value of wealth.
a. Calculate the fraction of wealth, W, that will be consumed in each period
if the real interest rate is equal to 10% (i.e. r = .10).
b. Assume that the household starts life with zero financial wealth so that the
households’ wealth is the present value of income earned by producing
goods. Assume that the household’s income is always equal to some
permanent level Qt  Q . What is consumption and savings in period 0 if
Q  100 ?
c. Now assume that the household receives some temporary boost to income
in time 0, Y0. This temporary boost dissipates only slowly at a rate, ρ < 1,
so that Yt = ρ Yt-1 = ρt Y0. Calculate the human wealth of this household
when Qt  Q  Yt  Q   tY0 , ρ = ¾, r = .10 and Y0 = 100. Under the
permanent income hypothesis, what is consumption and savings?
d. Now assumes that ρ = ¼, r = .10 and Y0 = 100. Calculate human wealth,
consumption and savings under the permanent income hypothesis. Discuss
the relationship between savings and the persistence of the temporary
increase in income, ρ.
Midterm Exam
Economics 514
Macroeconomic Analysis
November 15, 2004
Short Answer (10 points each)
6. The inflation rate of output goods price is 6% (i.e. π = .06) and the inflation rate
of investment goods is 8% (i.e. πI = .08). The nominal interest rate is 9% (i = .09).
The relative price of investment goods to output goods is ptI =.75 and the
depreciation rate is 10%. Solve for capital productivity when the marginal
productivity of capital equals the capital rental rate and production is given by the
Cobb-Douglass function.
1
2
Qt  Kt 3 Lt 3
The marginal product of capital is

1   tI1  I  1  i
1   tI1  I
Q R
M PK t 1  13  t 1  1  r  (1   )
p


(1


)
 t 
 pt
K Pt 1 
1   t 1 
1   t 1 
1   t 1
1.08 
Q
1.09

 .9
.75  0.083490566  =0.250471698

1.06 
K
1.06
Or
M PKt 1  13
Q Rt 1

  r    { tI1   t 1} ptI  i   tI1    ptI
K Pt 1 
 .11.75  0.0825 
Q
=0.2475
K
7. A corporation has a book value of debt equal to 500 million. The firm has 10
million shares outstanding with a market price of $45. The replacement value of
the firms’ capital is 750 million. Calculate Tobin’s q for this firm. Under q-theory,
should this firm invest more than enough to replace depreciated capital or less?
tq 
950
I
 1.266666667>1   
750
K
Long Answer (20 points each)
8.
Assume that there are a number of identical households beginning that live for a
number of periods in an economy. Each maximizes utility
U  u (C0 )   u (C1 )   2u (C2 )   3u (C3 )  ...   T u (CT )
subject to the constraint that the present value of consumption equals the present
value of income.
C
C2
CT
Q
Q2
QT
C0  1 
 .... 
 Q0  1 
 .... 
2
T
2
1  r (1  r )
(1  r )
1  r (1  r )
(1  r )T
All of the households in the economy are identical and have constant
intertemporal elasticity of substitution felicity function u (c ) 
1 1
c
1 1
. If investment and
government spending are zero, the economy is closed and output growth is constant
and equal to 1+g = 1.02, what is the autarky interest rate when β = .98 and the
elasticity of substitution is ψ = 1. What is the autarky interest rate when the elasticity
of substitution is ψ = 5.
u '(Ct )   (1  r )u '(Ct 1 ), Ct  Qt
1
u '(Qt )
Qt 
1
 1 r A 


1
 u '(Qt 1 )  Qt 1

 Q  1  g  1.02 
  t 1  


.98
 Qt 
1
1
1
  1  1  r A  1.040816327,   5  1  r A  1.024457529
9. A household lives for two periods. The household begins with zero financial
wealth and earns Q0 = Q1 = 100. The household maximizes the constraint
U  u(C0 )   u(C1 ) subject to the present value of consumption being equal to
C
Q
the present value of output C0  1  Q0  1 .
1 r
1 r
a. Assume that the felicity function is quadratic u(c) = 1000 c –½ c2 so that
the marginal utility of consumption is u’(c) = 1000 – c. Draw a picture of
the marginal utility of consumption as a function of c.
1000-c
c
What is the consumption and savings in period 0 if β∙(1+r) = 1. What is the
marginal utility of felicity in period 0?
When β∙(1+r) = 1, consumption will be the same in each period. Since income
is the same in each period, consumption is equal to income (i.e. C = 100) and
savings is zero. The marginal felicity is 1000 – C = 900.
b. Now, assume that output in period 1 is either 50 or 150 with equal
likelihood (Prob = .5). Calculate the expected marginal utility of
consumption in period 1 when savings is equal to that solved under
certainty in part a. When savings is zero, consumption must be equal to
income which is either 50 or 150. Marginal utility is either (1000-50= 950)
or (1000-150) = 850. The average or expected value of these two
outcomes is 900.
c. Is expected marginal utility in period 1 in part b larger than the marginal
utility of consumption in period 0, under period a. Is the household a
precautionary saver?
Expected marginal utility in period 1 is equal to marginal utility in period
0 when savings are zero, even when future income is uncertain. The
houshehold is not a precautionary saver.
10. A household lives for 20 periods from period 0 to period T = 19. The household
operates according to the permanent income hypothesis so that it consumes the
annuity value of wealth.
a. Calculate the fraction of wealth, W, that will be consumed in each period
if the real interest rate is equal to 10% (i.e. r = .10).
The annuity value of wealth is a fraction of
 1 
 1 
1 
1  

 1 r  
 1.1   0.106781477
wealth
T 1
20
 1 
 1 
1 
1


 
 1 r 
 1.1 
b. Assume that the household starts life with zero financial wealth so that the
households’ wealth is the present value of income earned by producing
goods. Assume that the household’s income is always equal to some
permanent level Qt  Q . What is consumption and savings in period 0 if
Q  100 ?
T 1
20
 1 
 1 
1 
1  

1 r 
1.1
W =W  
Q     9.364920092*100=936.4920092
1


 1 
1 
1  

 1 r 
 1.1 
Consumption is 100 and savings is zero.
c. Now assume that the household receives some temporary boost to income
in time 0, Y0. This temporary boost dissipates only slowly at a rate, ρ < 1,
so that Yt = ρ Yt-1 = ρt Y0. Calculate the human wealth of this household
when ρ = ¾, r = .10 and Y0 = 100. Under the permanent income
hypothesis, what is consumption and savings?
W  Q0 
Q
Q19
Q1
Q2

 ... 

2
1  r (1  r )
(1  r )19
Y19
Y
Y2
Q
Q
Q

 ... 
 Y0  1 
 ... 

2
19
2
1  r (1  r )
(1  r )
1  r (1  r )
(1  r )19
T 1
 1 
1 

2
 19 
 1  r  Q  Y 1    

...

0 

2
(1  r )19 
 1 
 1  r (1  r )
1 

 1 r 
T 1
T 1
 1 
  
1 
1 


1 r 
1 r 
W 
Q 
Y0
 1 
  
1 
1 


 1 r 
 1 r 
T 1
 1 
 1 
  
1 
1 

 1 

1 r 
1 r 
1 r 



C
W Q
Y0
T 1
T 1
  
 1 
 1 
1 
1 
1 



 1 r 
 1 r 
 1 r 
T 1

 1 
   
1 

 1 
 

1 r 
1 r  


S  Q  Y0  C  1 
 Y0
T 1
   
 1  1 
1 




 1  r  
 1 r 
W  936.4920092  314.1375661=1250.629575
If ρ = ¾, r = .10 and Y0 = 100
C  133.5440733, S  200  C  66.45592669
d. Now assumes that ρ = ¼, r = .10 and Y0 = 100. Calculate human wealth,
consumption and savings under the permanent income hypothesis. Discuss
the relationship between savings and the persistence of the temporary
increase in income, ρ.
W  936.4920092  129.4117647=1065.903774
e.
C  113.8187794, S  200  C  86.18122061
Midterm 2
Economics 514
Macroeconomic Analysis
A household begins life with zero financial wealth and lives for two periods, t = 0 and
1. The household will earns y0 = 1 in real goods in period 0 and retire in period 1,
earning nothing. There are zero taxes in either period. The household’s preferences
in the two periods are given by the utility function
1 1
1
U t  c0 2  .8  c1 2 
2
a. Calculate the optimal consumption in period 0 and 1 when the real interest rate is
1+r = 1.25. Is the household, a borrower or a saver.
c
We can write the budget constraint as c0  1  1
1 r
Maximize utility subject to the budget
c 1
1
1 
max (1  1 ) 2  .8  c1 2 
c1
2
1 r

c 1 1
1
1
1
FOC : (1  1 ) 2
 .8  c1 2
4
1 r 1 r 4
c 1
1
1
constraint.  (1  1 ) 2  1  r  .8c1 2  (1.25)  .8c1 2  Since the household
1 r
2.25
5
1
c1  c1 
1.25
9
c
4
5
c0  1  1  c0   c0 
1.25
9
9
consumes less than income in period 0, it is a saver.
b. What is the intertemporal elasticity of substitution? Does a rise in the real interest
rate lead to an increase or a decrease in consumption in period 0? What is the
maximum intertemporal substitution at which a rise in the real interest rate would
lead to an increase in consumption in period 0?
Midterm 2
Economics 514
Macroeconomic Analysis
A household begins life with zero financial wealth and lives for two periods, t = 0 and
1. The household will earns y0 = 1 in real goods in period 0 and retire in period 1,
earning nothing. There are zero taxes in either period. The household’s preferences
in the two periods are given by the utility function
1 1
1
U t  c0 2  .8  c1 2 
2
c. Calculate the optimal consumption in period 0 and 1 when the real interest rate is
1+r = 1.25. Is the household, a borrower or a saver.
c
We can write the budget constraint as c0  1  1
1 r
Maximize utility subject to the budget
c 1
1
1 
max (1  1 ) 2  .8  c1 2 
c1
2
1 r

c 1 1
1
1
1
FOC : (1  1 ) 2
 .8  c1 2
4
1 r 1 r 4
c 1
1
1
constraint.  (1  1 ) 2  1  r  .8c1 2  (1.25)  .8c1 2  Since the household
1 r
2.25
5
1
c1  c1 
1.25
9
c
4
5
c0  1  1  c0   c0 
1.25
9
9
consumes less than income in period 0, it is a saver.
d. What is the intertemporal elasticity of substitution? Does a rise in the real interest
rate lead to an increase or a decrease in consumption in period 0? What is the
maximum intertemporal substitution at which a rise in the real interest rate would
lead to an increase in consumption in period 0?
We can write a general constant intertemporal elasticity of substitution utility
function and its accompanying first order condition as
1
1
1 
1  1

Ut 
c    c1  
1  0

1 


1

1
c0

  (1  r )
c1
Where the elasiticity is ψ This along wth the budget constraint would imply
c0    (1  r )c1  

   1  r  c0
c
1
. When ψ < 1, then a rise in
c0  1  c0  
 c0 
 1
1 r
1 r
1    1  r 
the interest rate will increase consumption and when ψ> 1, a rise in the interest
rate will reduce consumption. In this problem, ψ=2, so a rise in the interest rate
will reduce consumption.