# CONCORDIA UNIVERSITY ECON 498/598: Homework Assignment 3

November 24, 2016

100 points
1. (70 points) Accounting for Income Differences Across Countries
The data for this problem is provided in ?le jones-tablec2-HW3.xls (taken from the textbook
Appendix, Table C2).
According to the Solow growth model with technological progress and human capital, if all
countries are on their BGPs and face identical growth rates of technology, their relative incomes per capita stay constant over time, and they can be decomposed into the contribution
from capital, human capital and technology according to equation (3.8) on page 58 in the
textbook:
?

y =
ˆ

sK
ˆ
x
ˆ

?
1??

ˆˆ
hA

(3.8)

where
y?
j
?
yUS is per capita income of country
sK, j
sˆ = sK,US is relative saving rate,
K

• y? =
ˆ

• x=
ˆ

n j +g+?
nUS +g+?

ˆ
• A=

A j (t)
AUS (t)

j relative to per capita income of U.S.A.,

is relative level of technology at time t.

Assume that ? =

1
3

and g + ? = 0.075.

(a) Suppose human capital h is only a function of schooling u: h = e0.1u . Using equation
(3.9), predict the following values for the entire sample of countries (except for those
with missing data):
ˆ
1. relative per capita incomes if A = 1.
Solution:
For Kenya:
ˆ
h = e0.1(uKenya ?uUS ) = 0.523, x = 1.186, s = 0.75
ˆ
ˆ
?

y =
ˆ

sK
ˆ
x
ˆ

?
1??

ˆ
h = 0.416

ˆ
2. relative levels of technology A that explain the differences in the relative per capita
incomes given the human capital measure h = e0.1u .
Solution:
ˆ ˆ
For Kenya: A = y?

x
ˆ
sK
ˆ

?
1??

ˆ
h?1 = 0.072

Plot these values against countries’ relative per capita income in 2008 (column y08) and
interpret your results from the BGP point of view. Mark country Kenya on your plots
and use it as an example in your discussion. Provide your calculations for Kenya (do not
submit the entire Excel worksheet; one country is suf?cient).
Note: Adding a 45-degree line to your plots in (a.1) is useful for interpreting the results.
Your plots should be similar to Figures 3.1 and 3.2 in the textbook (you do not need to
use log scale).
(b) Now suppose human capital h is determined by schooling u and quality q as follows:
h = q0.2 e0.1u . Also assume that schooling quality is proportional to the income per capita:
q = 0.05y. For example, q could stand for public education expenditure which is on
ˆ
average 5% of income per capita. Predict relative levels of technology A given the human
capital measure with quality of education. Plot these values against countries’ relative per
capita income in 2008 (column y08) and discuss the differences in your results in parts
(a) and (b) using Kenya as an example.
Solution:
For Kenya:
ˆ
h=

0.05yKenya 0.2 0.1(uKenya ?uUS )
e
0.05yUS

= 0.259

ˆ
A = 0.145
Conclusions:
1. Differences in years of schooling may understate true differences in human
capital across countries. In this example, human capital measure that accounts
for quality of schooling is about half of the human capital measure that uses
years of schooling only (0.259 vs 0.523).
2. The measure of human capital affects the role technology plays in explaining
differences in incomes per capita across countries. In particular, when we try to
account for quality of education across countries, the differences in technology
across countries shrink, i.e. technology plays a smaller role while human capital plays a larger role in explaining why some countries are poor. For Kenya,
income accounting that uses human capital with quality of education produces
an estimate of the relative level of technology twice as high as in the case without quality of education (0.145 vs 0.072).
2. (30 points) Growth Accounting
Page 2

Consider the following production function:
?
Y = BKIT KnonIT (hL)1????
?

where KIT is the stock of information-technology equipment (like computers), KnonIT is the
rest of the capital stock. Human capital is de?ned as a function of per capita expenditure on
education q and the average years of schooling u as follows:
h = q? e?u .
Assume ? = 0.05, ? = 0.3, ? = 0.2, ? = 0.1, and u = 12.
(a) Derive a growth accounting equation that corresponds to the above production function.
Solution:
gY = gB + ?gKIT + ?gKnonIT + (1 ? ? ? ?)gL + (1 ? ? ? ?)gh
where gh = ? gq .
(b) Using the data for a hypothetical economy in Table 1, compute the growth rates and
percentage contributions of each factor of production (KIT , KnonIT , h) and technology (B)
to the growth in aggregate output. Use the following formula to compute an annual growth
?Xt?1
rate: gX = Xt Xt?1 .
Solution:
growth rates
gL
gq
gKnonIT

gh

gB

0.1

0.0976

gY

gKIT

0.1538

0.1

KIT
gKIT
? gY

KnonIT
g
? KnonIT
gY

percentage contributions to gY
L
h
gL
gh
(1 ? ? ? ?) gY (1 ? ? ? ?) gY

3.25%

-19.5%

-0.1

0.025

10.56%

0.5

B

42.25%

63.44%

Table 1: Data for Problem 2.
Year

Y

KIT

KnonIT

L

q

2011
2012

\$260,000
\$300,000

\$800
\$880

\$2500
\$2250

1,200
1,230

\$x
\$1.5x

Page 3

total

gB
gY

100%

(c) Suppose we made no distinction between the two capital stocks, replacing them in the
production function with an aggregate capital stock K = KIT + KnonIT , and using a combined capital income share of 0.35.
Y = BK ?+? (hL)1????
What would be the contributions of K, h and B to growth in Y in this case? Comment on
the difference in the results in (b) and (c).
Solution:
growth rates
gL
gq

gY

gK

0.1538

-0.0515

0.025

0.5

gh

gB

0.1

0.0906

percentage contributions to gY
K
L
h
B
total
gL
gh
gB
K
(? + ?) gY (1 ? ? ? ?) gY (1 ? ? ? ?) gY
g
gY
-11.72%
10.56%
42.25%
58.91% 100%
The second accounting gives equal weight to both types capital. As a result, the
aggregate capital stock declined, but by a smaller amount than the non-IT capital
stock. Moreover, the residual growth, attributed to the growth in TFP, is also lower
(9.1% vs 9.8%). Hence, distinguishing between the two types of capital affects the
contribution of technology to the growth of output.

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