Please write your answers on this exam paper.
Economics 5140
Macroeconomic Analysis
Mid-term Exam PM
Thursday, October 17th , 2013 4-6:00 pm
1. A household must consume some fixed amount of food, FOOD, and gets utility
from any non-food consumption, Ct, beyond the quantity of food
ln(Ct )  ln lst
Available time, TIME, can be used for leisure or working, Lt, at a wage of wt = 1.
The budget of the household is Ct  FOOD  TAX t  wt Lt .
a. Write consumption as a function of leisure. Write the first order condition that
describes optimal leisure.
Ct  TIME  lst  (1   ) FOOD
1
1
1



lst Ct TIME  lst  (1   ) FOOD
TIME  lst  TAX  lst 
1
TIME  TAX 
2
b. Consider if taxes are set proportional to food spending, TAX t    FOOD .
Assume TIME = 24 and FOOD = 10. Solve for optimal leisure as a linear
function of the tax rate, ls = A + B τ. What are A and B?
1
1
1



lst Ct TIME  lst  (1   ) FOOD
1
1
1
TIME  (1   ) FOOD   TIME  FOOD    FOOD   7  5
2
2
2
A = 7, B = -5
lst 
c. Now suppose only spending on non-food spending is taxed, so TAX t    Ct .
Write consumption as a function of leisure. Continue to assume TIME = 24 and
FOOD = 10. Solve for optimal leisure as a linear function of the tax rate, ls = G +
H τ. What are G and H?
1
(1   )Ct  TIME  lst  FOOD  Ct 
TIME  lst  FOOD 
(1   )
1 (1   )
(1   )



lst
Ct
(1   )(TIME  FOOD  lst )
1
1
TIME  FOOD   TIME  FOOD   7  G
2
2
H 0
lst 
1
Please write your answers on this exam paper.
d. Explain why B and H have different signs (i.e. positive, negative or zero).
Taxing food reduces the income but does not affect the relative price of consumption
and leisure on the margin. The income effect dominates and higher taxes means less
leisure. Taxing consumption changes both the income and relative price in equal
measure so the income and substitution effect cancel out.
2.
A household lives for two periods period 0 and period 1. The household earns Y0
= 200, Y1 =220. The household can consume in either period and gets utility
U = ln(C0 )   ln(C1 ) .
The household can save their income B0  Y0  C0 and earn real interest rt :
C1  Y1  (1  r ) B0 .
a. Write future consumption as a function of current consumption. Write the Euler
equation that describes optimal consumption.
C1  Y1  (1  r ) Y0  C0 
ln(C0 )   ln(Y1  (1  r ) Y0  C0 )
dU
1
(1  r ) 
(1  r ) 
0


 C1  (1  r )  C0
dC0
C0 Y1  (1  r ) Y0  C0 
C1
b. Assume (1  r )  1.21  (1.1)2 and   1 , solve for consumption in each period.
1.1
1
C1  220  1.21 200  C0   C1  1.21 C0  1.1C0
1.1
C1  220, C0  200, B0  0
c. On Graph 1, represent the solution to question b.
C1
Autarky
Y1
C0
Y0
2
Please write your answers on this exam paper.
d. Assume that real interest rates go to r =.1. Solve for future consumption, current
consumption, and savings, B0.
1
C1  220  1.1 200  C0   C1  1.1 C0  C0
1.1
440
440
440
C1 
 209.52, C0 
 209.52, B0  200 
 9.52  0
2.1
2.1
2.1
Draw the effect of the change in interest rate on the budget constraint on graph below.
C1
Autarky
Y1
C0
Y0
3. An economy is along its balanced growth path. The production function is
Yt  ( K t ).33 ( At Lt ).67 . Technology grows at rate η = .01. Capital depreciates at
rate .08. Population grows at a rate of n = .01. The real interest rate r = .03.
a. Calculate the average productivity of capital if the modified golden rule
investment rate is implemented.
y
y (r   ) .11 1
  (r   )  


k
k

.33 3
b. If labor productivity is 27, calculate the level of technology A.
y
y  .33
yt  (kt ).33 ( At ).67  ( yt ).33 ( yt ).67  ( yt ).67  ( ) .33 ( At ).67  yt  ( ) .67 ( At )
k
k
.33
2
yt  (3) .67 ( At )  ( At )  3
.34
.67
 15.71
3
Please write your answers on this exam paper.
4.
The production function is Yt  ( Kt ).5 ( At (1  s R ) L).5 . Capital is accumulated
through investment and depreciates at rate .06. Population is constant, n = 0. Normalize L
= 1. Technology grows according to the function At 1  At  Bt s R L where s R  .4 . The
efficiency of the R&D sector is due in part to spillovers from past inventions
I
Bt  B0  B1 At where B0 = 1, and B1 = .2. The investment rate is s  t =.48. The
Yt
a.
Calculate the growth rate of technology when At = 10.
At 1  At  Bt s R L   B0  B1 At  s R  s R B0  s R B1 At
B

1

.4
gtA1   0  B1  s R    .2 .4  .08 
At
 At

 At

A = 10 implies .012
b.
Calculate the growth rate of technology in the long run when technology becomes
infinitely large.
A = ∞ means g = .08
4