# CONCORDIA UNIVERSITY ECON 498/598: Homework Assignment 3

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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