# Economics 5140

Economics 5140
Macroeconomic Analysis
Mid-term Exam AM
Thursday, October 17th , 2013 9:30-11:30am
1. A household gets utility from consumption and leisure, Ct, and leisure, lst ,
ln(Ct )  2ln lst
Consumption is a function of wage and non-wage income. Ct  wt Lt  t .
Available time, TIME, can be used for leisure or working, Lt.
a. Write Consumption as a function of leisure. Write the first order condition that
describes optimal leisure.
Ct  wt TIME  lst   t
ln( wt TIME  lst    t )  2ln lst
w
wt
w
dU
2
0 t
 t 
dls
Ct wt TIME  lst    t Ct lst
b. Assume TIME = 24, wages are wt =1 and t = 6. Solve for optimal leisure, labor
and consumption.
 
2
wt lst  2 wt TIME  lst    t   lst  TIMEt   t   20
3
wt 
L  4, C  10
1
c. Suppose wages double. Solve for optimal labor. Explain the qualitative change in
the level of labor.
wt lst  2 wt TIME  lst    t   lst 
t 
2
TIMEt     18
3
wt 
L  6, C  18
The elasticity of substitution is 1. A 100% increase in wages (i.e. the price of leisure) will
reduce optimal leisure to consumption/income by 100%. Since income does not go up 1
for 1 with the wage rate, so the substitution effect is larger than the income effect and
labor increases.
d. Suppose both wages and non labor income doubles. Solve for optimal labor.
Explain why the answer to c. is qualitatively different than the answer to d.
 
2
wt lst  2 wt TIME  lst    t   lst  TIMEt   t   20
3
wt 
L  4, C  10
Elasticity of substitution is 1. In percentage terms, the relative price and income change
by the same degree so substitution effect is equal to the income effect.
2
2. Japan is implementing a tax on consumption that they have announced will be
implemented next year. What impact will this have on current consumption and
savings? A household lives for two periods, period 0 and period 1. The household
earns Y0 = 100, Y1 =110. 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  1.12 and   1 . Solve for consumption in each period.
1.1
1
C1  110  1.1100  C0   C1  1.21 C0  1.1C0
1.1
C1  110, C0  100, B0  0
3
c. On Graph 1, represent the solution to question b.
C1
Autarky
Y1
C0
Y0
d. The government informs the public they must also pay a tax in period 1.
C1  Y1  (1  r ) B0  TAX1 . The tax will be proportional to consumer spending,
TAX1    C1 so that (1   )C1  Y1  (1  r ) B0 . Write future consumption as a
function of current consumption.
1
C1 
Y1  (1  r ) Y0  C0 
(1   )
4
e. Assume that taxes are set at  =.1. Solve for future consumption, current
consumption, and savings, B0.
1
Y1  (1  r ) Y0  C0 
(1   )
1
ln(C0 )   ln(
Y1  (1  r ) Y0  C0 )
(1   )
dU
1
(1  r ) 
(1  r ) 
(1  r )
0


 C1 
 C0
dC0
C0 Y1  (1  r ) Y0  C0  (1   )C1
(1   )
C1 
C
1.21 1
1
C0  C0 
Y1  (1  r ) Y0  C0  
(1.1) (1.1)
(1   )
C0  100  C1  B0  0
C1 
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 is accumulated through
investment and depreciates at rate .08. Population grows at a rate of n = .02.
a. Calculate the average productivity of capital if the golden rule investment rate is
implemented.
y
y
y (n   )   .11 1
g k  s  (n   )    s  (n   )    


k
k
k
s
.33 3
b.
If labor productivity is 27, calculate the level of technology A, Total Factor
Productivity.
y
y  .33
yt  (kt ).33 ( At ).67  ( yt ).33 ( yt ).67  ( yt ).67  ( ) .33 ( At ).67  yt  ( ) .67 ( At )
k
k
.33
.67
2
.34
.67
yt  (3) ( At )  ( At )  3
15.71679
TFPt = TFPt  At1  At .67  6.33
5
4.
An economy is along its balanced growth path. The production function is
Yt  ( K t ).5 ( At (1  s R ) Lt ).5 . Capital is accumulated through investment and depreciates at
rate .06. Population grows at a rate of n = .03. Technology grows according to the
function At 1  At  Bs R Lt where s R  .36 and B = .25. The investment rate is
I
s  t =.48.
Yt
a.
Calculate the level of capital productivity, APKBDP.
2n   .12
APK 

 .25
s
.48
b.
L
Calculate the ratio  t A 
t

BGP
.
At 1  At
L
L
n
.03 1
 n  Bs R t  t  R 

At
At
At Bs
.09 3
c.
Calculate the level of labor productivity yt 
workers.
y
yt  ( ) 1 ((1  s R ) At )  4  .64  300  778
k
6
Yt
Lt
when the economy has 100