Answer for Homework 3: Modern Macroeconomics I

1. If two countries have the same saving rate, the same population growth and the sametechnology, then a Solow model predicts that both countries converge to the samesteady state and attain the same GDP per capita in the long run. Explain economic(intuitive) reason behind this prediction.
Answer Suppose that two countries have the same saving rate, the same population
growth, and the same technology and di¤er only in the initial economic conditions
such as initial capital stock and initial population. Suppose country A has a lesscapital stock per worker than country B (that is, kA < kB ). Since the marginal
productivity of capital is decreasing in capital stock per worker, country A has ahigher marginal productivity of capital than country B (that is, MPKA = f 0 (kA ) >f (kB ) = MPKB ). In this case, the growth rate of capital stock per worker incountry A is higher than the one in country B. Intuitively, the less developed
country accumulate capital stock faster than the developed one and these growth
rates eventually converge to the rate of population growth.
2. Answer the following questions.
(a) Kaldor found 6 robust facts for the long run behavior of aggregate data. What
are the Kaldor’s Stylized facts?
Answer Kaldor’s stylized facts are as follows:
1 The growth rate of GDP per capita is nearly constant.
2 The growth rate of capital per capita is nearly constant.
3 The growth rate of output per worker di¤ers substantially across countries.
4 The rate of return to capital is nearly constant.
5 The ratio of physical capital to output is nearly constant.
6 The shares of labor and physical capital are nearly constant.
(b) Show how the neoclassical growth model in the steady state explains the Kaldor’s
Stylized Facts.
Answer (Revised) Note that on the steady state, ke is constant and therefore
ye = f (ke ) and f 0 (ke ) are constantI thank to Hiroshi Kitamura and Wataru Tamura who made these sample answers.

1

1 Note thatgy = gye T

gye + gT = gT = g:

Since g is constant, the growth rate of GDP per capita is constant in theneoclassical growth model.
AlternativeGDP per capita is written as yt = F (KtL;Tt t Lt ) = Tt f (ket ). Since the capitalstock per e¤ective worker is constant in the steady state (ket = ke ), thegrowth rate of yt is given bygy =yt+1 yt Tt+1 f (ke ) Tt f (ke )=yt
Tt f (ke )Tt+1 Tt= g:=Tt

Hence we observe that the growth rate of GDP per capita is constant in themodel.
2 Note that
gk = gke T gke + gT = gT = g:
Since g is constant, the growth rate of capital per capita is constant in theneoclassical growth model.
Alternative= Tt ket . Since the capital stock
Capital stock per capita is written as kt = KL
per e¤ective worker is constant in the steady state (ket = ke ), the growth rateof kt is given by
gk =kt+1 kt Tt+1 ke Tt ke
=kt
Tt ke
Tt+1 Tt== g:
Tt

Hence we observe that the growth rate of capital per capita is constant in themodel.
3 We have observed that, in steady state, the growth rate of output per capita
coincides with the growth rate of technology. Hence the model explains thedi¤erence in the growth rate of output per capita by the di¤erence in theexogenous growth rate of technology.
4 From the …rm’s pro…t maximization problem1 , the capital stock per e¤ective

1

Recall that the pro…t maximization problem ismax fP F (K; T L)

RKW Lgmax P T L ff (ke )rke

w=T g :K;L)

ke ;L2

worker and the real rental price should satis…esf 0 (ket ) = rt :In the steady state, the capital per e¤ective worker is constant (ket = ke ),the rate of return to capital (real rental price) is constant in the neoclassicalgrowth model.
5 Note thatKtk Tt Nt
k= e= eYt
ye Tt NtyeSince both ke and ye are constant in the steady state, the ratio of physicalcapital to output is constant in the neoclassical growth model.
6 Note that rt = f (ke ) in steady state. Then,r t Ktket= f 0 (ke )= f 0 (ke )ke =ye = constant
Ytyt
Since
1=
we have

wt Lt
=1
Yt

rt Kt + wt Lt;
Yt
r t Kt= constantYt(c) Describe 5 development facts stated by Parente and Prescott(1993) and Durlaufand Quah(1998).
Answer 5 development facts stated by Parente and Prescott(1993) and Durlaufand Quah(1998) are as follows:
1 Income di¤erence across countries is large.
2 Wealth distribution has shifted up.
3 Relative Income distribution does not show convergence.
4 There have been development miracles and disasters.
5 Relative Income distribution across countries shows two peaks.
3. Consider the following production function
Yt = At Kt (Nt )1
where Yt is GDP, Kt is capital stock, Nt is the number of population and At is Solow
residual at date t. Assume that a …rm maximizes its pro…t and every market iscompetitive. Assume that we can obtain data about Yt , Kt , Nt . I also assume thatit is possible to obtain data about the rental price of capital and the wage rate.
(a) Show how gA can be estimated?
Answer Note thatgY = gA + gK + (13) gN

Since a …rm maximizes its pro…ts in the competitive market,
r t KtYt
wt Nt=Yt=1Hence,r t KtgKYt

gA = gYwt NtgNYt(b) Suppose that it is di¢ cult to estimate the rental price of capital. Can you stillestimate gA ? Show your estimation strategy.
Answer Yes, we can estimate gA even when it is di¢ cult to estimate rt . Since
the production function is constant return to scale, we have
Yt = rt Kt + wt LtThen we haver t Kt=1
Yt

wt Lt
YtHence, we have
wt Lt
wt NtgN
gKYt
Yt
wt Lt
gK +(gK gN )Yt

gA = gY

1= gY

4. Suppose that Japan and the US has the same production function
Yt = Kt (Tt Nt )(1)

where Yt is output, Kt is capital stock, Tt is the labor augmenting technology and Nt
is the number of workers. Assume that the growth rate of technology, g, and thedepreciation rate, , and the growth rate of the number of workers, n, in Japan are thesame as those in the US. Assume that = 31 .
(a) Derive the steady state value of GDP per workers as a function of the saving rate,s, the growth rate of technology, g, the depreciation rate, and the the growthrate of the number of workers, n, and the current level of technology, Tt .
Answer Refer to lecture slide. You can derive the following dynamics of capitalper unit of e¤ective labor.
ke;t+1 =syet + (1) ket(1 + g) (1 + n)(1)twhere yet = TtYNt t and ket = TK
. It is easy to see that yet = ket and equationt Nt

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(1) becomes
ke;t+1 =) ket
sket + (1:(1 + g) (1 + n)

In the steady state, ket+1 = ket = ke and we haveke =s (ket ) + (1) ke.
(1 + g) (1 + n)

By solving this equation with respect to keske =g + n + + gn11.

YLet Nt be the steady state value of GDP per workers. Since yet = TtYNt t andtYt
Nt= (ke ) Ttsg + n + + gn

=Since1

Tt := 31 , we haveYts=
Ntg + n + + gn

12

Tt(b) Suppose that Japan has 2 times larger saving rate than US and they have thesame current level of technology. How much does income per capita di¤er on thesteady state?
Answer Since Japan has 2 times higher saving rate,hi12
YtsJP
Ttg+n+ +gn
Nt JP=hi 12YtsU S
Nt U S
Tt
g+n+ +gnsJP=sU S

121= 22
1

Therefore, Japan has 2 2 times higher GDP per capita.
(c) Assume that Japan and the US have the same saving rate, but the current levelof technology in the US is 1.5 times larger than that in Japan. How much does
income per capita di¤er on the steady state?

5Answer Since US has 2 times larger productivity than Japan,
hi12
Yts
TtU S
g+n+ +gn
Nt U S=hi 12
Yts
Nt JP
TtJP
g+n+ +gn

TtU S
TtJP= 1:5=

Therefore, the US has 1:5 times larger GDP per capita than Japan.
(d) A developed country has a saving rate of 30 percent and a population growthrate of 2 percent per year. Suppose that a developing country has a saving rateof 10 percent and a population growth rate of 5 percent. Suppose that initialtechnology of the developed country is 10 times higher than that of the developing
country and that both country has the same productivity growth and depreciation
rate: g = 0:02 and = 0:03. Assume that = 13 : How much is GDP per capita
in the developed country larger than that in the developing country in the steadystate?
Answer (Revised) GDP per capita in developed country is
Yt=
Nt Developedsg + n + + gn

1

TtDeveloped=

0:3
0:02 + 0:02 + 0:03 + 0:02 0:02

=

3000
704

12

12(1 + g)t T0developed(1 + g)t T0Developed

On the other hand, GDP per capita in developing country is
Yt=
Nt Developing

sg + n + + gn

1

TtDeveloping=

0:1
0:02 + 0:05 + 0:03 + 0:02 0:05

=

100
101

12

12(1 + g)t T0Developing(1 + g)t T0Developing

YtNt Developed
Yt
Nt Developing

5=4r3030
1120:75

Therefore, GDP per capita in the developed country is 20:75 times larger
than that in the developing country in the steady state.
(e) Derive a regression equation that shows log YL as a functions of log s and6

log (n + g + + ng).
Answer In the equilibrium, Lt = Nt . Then,Yt
log= log
Ltsg + n + + gn

= log Tt +71log s
2

12

Tt!

1log (n + g + + ng) :2