Please write your answers on this exam paper.
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Mid-Term Exam # 1
Economics 514
Macroeconomic Analysis
Tuesday, October 18th , 2011
Write your answers on this white exam paper. Do not write your
answers on the blue book! Do not hand in your blue book!
1.
Population Growth and Investment
1
2
Price taking firms produce goods with the production function Yt  Kt 3 Lt 3 , where Lt is
labor and Kt is capital. Firms produce in period t and period t+1. Real cash flow in every
period is CFt = Yt - wtLt. The firm’s dividends at time t are cash flow less investment
(CFt –It) and the dividends at time t+1 are cash flow plus liquidated capital (CFt +Kt+1) .
The firm maximizes the present value of dividends. There is no depreciation.
CF  K t 1
max CFt  I t  t 1
Lt , Lt 1 , Kt 1
1 r
A.
Solve for the first order conditions that determine optimal hiring and investment.
1

(1   ) K t ( Lt )

 Kt  3
2
 3    MPLt 1  wt
 Lt 
1

(1   ) K t 1 ( Lt 1 )

 K 3
  t 1   MPLt 1  wt 1
 Lt 1 
2
3
r  MPK t 1   K t11 ( Lt 1 )1
K 
 13  t 1 
 Lt 1 
1
 23
 K t 1   3r  2 Lt 1
3
In period t, normalize initial labor and capital to 1, Kt = Lt = 1.
B.
Solve for equilibrium real wages in period t.
1
 K 3
wt  23  t  
 Lt 
2
3
Assume that we can write Lt+1 =1+n > 1, so the population is growing. Further, the real
interest rate is 33.33%, r=⅓,
C.
of n.
Solve for optimal wages at time t+1 and optimal investment at time t as functions
K t 1   3r  2 Lt 1  Lt 1  1  n
3
1
 K t 1  3 2
2
wt 1  3 
 3
 Lt 1 
I t  K t 1  K t  1  n  1  n
2
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D.
Explain in words, the relationship between population growth and the growth rate
of real wages. Explain the relationship between population growth and investment.
Wages are invariant to the number of workers. When the population grows,
capital grows proportionally. Since capital and labor are scaling up
proportionally and returns to scale are proportionate, productivity and labor are
unaffected.
E.
Suppose that consumption is equal to total wage income, Ct.= wtLt. Calculate the
current account as a function of n.
wtLt=
2
3
, , Yt = 1. St = 13 CAt = St - It =
1
3
-n
3
Please write your answers on this exam paper.
2.
Growth and the Current Account
The representative consumer in a country has a utility function ut + βut+1 where felicity,
ut, is the natural log of consumption: ut =ln Ct,. The household starts with zero financial
wealth and earns income from producing goods, Yt and Yt+1. The household can borrow
or deposit funds at the end of time t, at interest rate, 1+r = 1.1. Normalize income in the
first period, Yt = 1, and Yt+1 = 1+g. Assume zero investment and zero government
spending. Also assume the real interest rate is equal to the inverse of the subjective
discount factor, β(1+r)=1. Write the current account as a function of growth rate of
income.
Ct = Ct+1
C
(1  g )Yt 2  r  g
2r
Ct  t  Ct
 Yt 

1 r
1 r
1 r
1 r
2r  g
g
Ct 
 1
2r
2r
g
CA  Yt  Ct  
2r
4
3. Economic Growth and R & D
In an economy, there is a fixed supply of labor (which we will normalize to 1), Lt = 1, so
the growth rate of the labor supply is n = 0. The household can divide the labor supply
between producing goods or to doing R & D. The amount of time spent on research is sR
∙Lt = sR and the time spent on producing goods is equal to (1- sR) ∙ L = (1-sr) . The
amount of goods produced are given by a Cobb-Douglas production function where Kt is
the capital stock and At is the technology produced.
1
2
1
2
Yt  Kt 3 ( At  (1  s R )  Lt ) 3  Kt 3 ( At  (1  s R )) 3
Technological spillovers mean that the ability to produce new research is an increasing
function of the technology level. The amount of new technology produced is given by
At 1  At  At  s R  Lt  At  s R
The share of time spent in R & D is 3% (i.e. sR = .03). The accumulation of capital is
done according to the function
Kt 1  (1   )  Kt  I t
Assume that the depreciation rate of capital is 7% (i.e. δ = .07). Investment is a
constant fraction of output It = s∙Yt. Assume that the investment rate, s = .2.
A.
What is the growth rate of technology?
A A
  t 1 t  s R  .03
At
B.
Calculate the growth rate of the capital-labor ratio and the growth rate of labor
productivity when capital productivity is .65.
K t 1  K t
Y
 s t    .2 *.65  .07  .06
Kt
Kt
gy 
C.
1 K 2 A
g  g  .02  .02  .04
3
3
Calculate the steady state capital productivity level.
  gA n
s

.1
  .5 .5
.2
5
4.
Perfect Substitution
In an economy, the growth rate of the population is n= 0 and the population level is
I
normalized to Lt = 1. Assume that the investment rate, s  t  .2 . Capital is
Yt
accumulated according to Kt 1  (1   )  Kt  I t with a depreciation rate of  = .07.
Assume that the production function is given with a modified version of the CobbDouglass production function.
Yt  .6( Kt  Lt )
A.
Show that the production function has constant returns to scale by writing down
Y
the productivity function. That is solve for yt  t as a function of the capital labor
Lt
K
ratio, kt  t .
Lt
y
1
yt  .6(kt  1)  t  .6(1  )
kt
kt
y
y
gtk1  s t    .2 t  .07
kt
kt
B.
Calculate capital productivity and the growth rate of the capital-labor ratio when
the level of the capital labor ratio is 1, 27, 125, 1000, and ∞. Suppose the interest rate
k
Y
gk
K
1
1.2
.17
10
.66
.062
100
.606
.0512
.6
.05
∞
Explain in 6 sentences or less why capital can act as the long-term engine of growth in
this model.
Capital displays diminishing average product in this model. However, there is a
lower bound, below which the productivity of capital does not drop. As long as
capital productivity stays sufficiently high, capital based growth can continue.
6
7
5.
Capital Substitution Assume that production occurs with a Cobb-Douglas
production function with α = .5. Assume a constant technology and labor level (i.e. gA =
n = 0). To make things simple set Lt = At = 1 so labor productivity equals output Yt  yt
and the capital labor ratio equal capital: Kt  kt
Yt   Kt
A.

1
2
Total factor productivity is a geometrically weighted average of labor productivity
1
Y 
1
and capital productivity TFPt   t   yt 2 . Calculate total factor productivity in this
 Kt 
case.
2
y  y 
  y   y   k   1 
k 
1
1
yt
1
2
t
t
1
2
2
2
t
t
t
1
2
t
1
2
y  y

 
k 
1
2
t
1
t
2
t
1
2
y 
 t 
 kt 
1
2
y 
t
1
2
 TFP
Now assume output is made with two different capital stocks, i) equipment and ii)
structures: K tQ and K tS each of which has a different depreciation rate, δQ = .16 and δS =
.04.
   
Yt  KtQ
1
4
KtS
1
4
The different types of capital are accumulated according to the standard equation
KtQ1  (1  Q ) KtQ  ItQ
KtS1  (1   S ) KtS  ItS
The investment rate for capital type Q is sQ, ItQ  sQYt while the investment rate of type
B is sS, I tS  sS Yt .
8
B.
Equal investment case. First, assume sQ  sS =.16.
i.
Solve for the steady state productivity of capital of both types,
SS
SS
 Y 
 Y 
Q
S
APK SS
  Q  and APK SS
 S .
K 
K 
Q
Q
Q
S
K t 1  K t
It
K t 1  K tS
I tS








Q
S
K tQ
K tQ
K tS
K tS
K tQ1  K tQ
 Y 
 0   Q  sQ  Q 
Q
Kt
K 
K tS1  K tS
 0   S  sS
K tS
SS
Q  Y 
KtQ1  KtQ

0


1
KtQ
sQ  K Q 
Solve for labor productivity, y.
KtS1  KtS

Y 
0 S  S 
S
Kt
sS  K 
SS
ii.
k  k 
  y   y   k  k    y  
y 
1
yt
1
2
t
2
t
S
t
1
4
Q
t
1
1
4
S
t
2
1
Q
t
4
t
1
2
t
 
yt
iii.
 yS 
  tS 
 kt 
1
2
 ytQ 
 Q
 kt 
1
2
 Y 
KS 


1
 s S  2  sQ 
 S   Q 
   
1
2

s
1
 S 2 Q
1

2
1
4
 yS 
  tS 
 kt 
.16
S Q
1

4
SS
SS
 .25
 ytQ 
 Q
 kt 
1
.16 .16

2
.2  .4 .08
Define total capital as Kt  KtS  KtQ . Solve for the steady state
SS
productivity of total capital APK SS
Y 
   . Solve for steady state output.
K 
1
Y 
1
Calculate steady state TFPt   t   yt 2
 Kt 
Y
K Q  Y K S  4Y K  5Y   15
K
1
1
1
TFPt     2 2  .4  1
5
2
9
2
4