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Economics 514
Macroeconomic Analysis
Mid-term Exam 1
Tuesday, October 12th , 2010 9-11am
1. Relative Technology Levels
We observe two countries Argentina and Brazil. The investment rate in Brazil is
30% (s = .3) and the investment rate in Argentina is 15% (s = .15). In each
country, the population/labor force growth rate is 3%. In both countries, the
depreciation rate and technology growth rate are assumed to be identical though
the possibility exists that the level of technology, At, is different in the two
economies. Real wages in Argentina and Brazil are both 4 US dollars per hour.
Assume that output is given by a Cobb-Douglas production function with a capital
intensity parameter of α = ⅓.
Yt  K t ( At Lt )1
a. In which country, will capital productivity be highest when both countries are on
their balanced growth path. What is the ratio of steady state capital productivity,
SS
Y  in Brazil to that in Argentina?
 K
Steady state capital productivity for country j is
n   
. The ratio of capital
sj
n   
SS
Y 
sBRZ
s
BRZ
productivity in the two countries is  K SS

 ARG
1
sBRZ
2
n




Y 
 K  ARG
s ARG
Please write your answers on this exam paper.
b. In which country is labor productivity highest? What is the ratio of labor
productivity in Brazil to that in Argentina?
Both countries have equal real wage rates and equal marginal product of labor. With
Cobb-Douglas, this means both have equal average product of labor. The ratio is one
to 1.
c. In which country is technology highest. Assume that both countries are on their
balanced growth path. Calculate the ratio of technology in Brazil to Argentina.

BGP
yBRZ
BGP
y ARG
 Y  BGP   1 A

  K  BRZ 
BRZ
 s ARG   1 ABRZ


1




AARG
 sBRZ 
 Y  BGP   1 A
  K  ARG 
ARG



ABRZ  s ARG 


AARG  sBRZ 
1

1
2
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2. Cost of Capital and Real Wages
There is a fixed supply of labor in the economy, L=1000. Assume that output is given
by a Cobb-Douglas production function with a capital intensity parameter of ⅓.
Yt  K t ( Lt )1
Firms sell goods at a dollar price, Pt. Assume for the sake of simplicity that the
price level is Pt = 1. Price-taking firms rent labor, Lt, at wage rate, Wt, and capital,
Kt, at a rental rate Rt. Firms act to maximize profits, ∏t.
 t  Yt  Wt Lt  Rt Kt  Kt ( Lt )1  Wt Lt  Rt K t
Capital can be rented freely in the international market at a real capital rental rate,
R =.12.
a. One of the first order conditions is that the firm will hire workers until the
Y
marginal product of labor is equal to the wage. MPLt   t   Kt Lt   Wt
Lt
The other describes the demand for capital. What is the demand curve for
capital?
MPK t  
Yt
K  (R )
1

Kt
( 1)
  K t 1 L1t  Rt
L
b. With a fixed supply of labor, LS = 1000, and an elastic capital supply, R =.12,
solve for the profit maximizing level of capital and real wage when supply
equals demand in both markets.
K  (.36)
1
 (2/3)
1000  (.36)
3
2
1000
 462.96
 L
W  (1   ) K

 2   4.63 3  1.111
3
1
 L
 (1   ) K
1
3
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c. Now, assume that the government gives a monopoly franchise to a capital
importer. The capital importer will be able to rent capital from international
markets at a fixed rate of Q = .12. Then, they can charge a price R to the
production firms to maximize profits Profit  R  K  Q  K . Remember that
the demand for capital is a function of R, K = K(R) which was solved for in
part a. Calculate the monopoly profit maximizing price, R. Calculate the level
of capital that the firm will demand at that price with a labor supply of L =
1000. Calculate the equilibrium wage rate
max R R  K ( R)  QK ( R)
K ( R)  R  K '( R)  Q  K '( R)
K ( R)
)Q
R  K '( R)
1
R
Q
K ( R)
(1 
)
R  K '( R)
R  (1 
K ( R)  ( R )
1

( 1)
 L, K '( R)  1
K ( R)

R  K '( R ) R  1
1
(R )

(1 )
( 1)
1
(  1)
(1 )

L
( 1)
1
( R)
( 1)
1
(  1) 
1
1
1
R
Q
Q
Q  3Q  .36
1
(1    1)
( )
3
K  (R )
1

 (1.08)
W 2
3
2
( 1)
 L  (.36
1
)
3
2
L
3
1000  89.10
3  L
K
1
3
2
.891 3  .6145
3
1
1
L
( 1)

1
( R)
( 1)
1
1
(  1)
1
L
 (  1)
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3. Labor Taxes and Labor Supply A worker chooses how much labor to supply to
achieve the optimal trade-off between capital and labor. The worker has an
amount of time which must be spilt between labor, Lt and leisure, lst.
TIME  Lt  lst
The worker only has an income, wtLt, from working at a real wage rate, wt
normalized to 1. The worker must pay taxes and those taxes are a fraction of
income, TAXt =τ∙wtLt . This leaves income available for consumption of Ct =(1τ)∙wt Lt : The cost (in terms of consumption) to the worker of taking each extra
unit of leisure time is the lost after-tax earnings, (1-τ)∙wt. Assume that workers
have preferences toward consumption and leisure given by a logarithmic utility
function
U t  log(Ct )  log(lst )
so the Marginal Rate of substitution between consumption and leisure is given by
U
1
MU ls
C

ls
MRS 

 ls 
1
MU C U
ls
C
C
Normalize the time to be equal to TIME = 1. Calculate the optimal share of time
supplied as labor, Lt, when the tax rate, τ = 0. Calculate the labor supplied when
the tax rate, τ = ½. Explain the effect of the change in the tax rate on labor supply.
At optimum, the marginal rate of substitution equals the cost of leisure equal the
C
MRS   (1   )  w
ls
opportunity cost of leisure C  (1   )  w  L  (1   )  w  (TIME  ls )
(1   )  w  (TIME  ls )
 (1   )  w  ls  1 2 TIME
ls
The household spends half their time working and half their time in leisure regardless
of what is the real wage and what is the tax rate. If wages go up, taking leisure
becomes relatively more expensive and the household has the tendency to substitute
consumption for leisure (the substitution effect) but the household also can afford
more consumption without working harder and can afford more leisure without
cutting back on consumption (the income effect). In this situation, the substitution
effect cancels out the income effect and leisure/labor choices are unaffected by the
real wage. The same holds for the labor tax. When the tax goes up, leisure becomes
cheaper inducing the worker to substitute leisure for consumption. However, the
worker can afford less consumption and less leisure. The income and substitution
effects cancel out.
Please write your answers on this exam paper.
4. Human Capital and Endogenous Growth Output is a function of capital, Kt, and
skilled labor. Skilled labor is the product of human capital, Ht, and labor, Lt.
Normalize the fixed labor supply to be equal to Lt = 1. Skilled labor can be used
either to produce goods or to produce education. The share of human capital used to
produce education is sH and the share used for producing goods is (1-sH).
Yt  Kt ([1  s H ]  H t )1
The growth rate of output is a weighted average of the growth rate of capital and
the growth rate of human capital.
gtY   gtK  (1   ) gtH
Education is used to invest in human capital
H t 1  EDUCATIONt  H t
Human capital is used to produce education. .
a. Assume production of education is done without diminishing
EDUCATIONt  b  s H  H t
Solve for the long-term growth rate of human capital. Assume that investment is a
constant fraction of output.
Kt 1  Kt  sYt   Kt
Calculate the steady state capital productivity level. Calculate the long-term
growth rate of output along the balanced growth path
The growth rate of human capital is constant.
H t 1  H t  EDUCATIONt  b  s H  H t 
H t 1  H t
 gtH1  b  s H
Ht
The growth rate of capital is a function of capital productivity.
K  Kt
Y
Kt 1  K t  sYt   K t  t 1
 s t 
Kt
Kt
If capital growth is higher than the growth rate of human capital, the
growth rate of output is less than the growth rate of capital and capital
productivity is falling. Capital productivity will stop falling when the
growth rate of capital is equal to the growth rate of human capital and
thereby equal to the growth rate of output.
This balanced growth path is obtained when capital productivity is
H
Y
 Y  bs 
b  sH  s t      
Kt
s
K 
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b. Assume that education is produced with diminishing returns.
EDUCATIONt  b  s H   Ht 
0   1
Calculate the long-term growth rate of human capital. Calculate the steady state
capital productivity level. Calculate the long-term growth rate of productivity
along the balanced growth path.

The growth rate of human capital is a negative function of the level of human
H t 1  H t  EDUCATIONt  b  s H  H t  
capital. H t 1  H t
As human capital grows to
b  sH
 gtH1  1
Ht
Ht
infinity, the level of human capital will get so large that the growth rate will
Y
Y  
be zero. 0  s t       and the the growth rate of labor
Kt
K  s
productivity will be zero.