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Homework 1
1. In 1900 GDP per capita in Japan (measured in 2000 dollars) was $1,433.
In 2000 it was $26,375. Calculate the growth rate of income per capita
in Japan over this century. Now suppose that Japan grows at the same
rate for 21st century. What will Japanese GDP per capita be in the year
2100?
The average growth rate was
y2000
g
= y1900 (1 + g)100
¶1/100
µ
y2000
=
− 1 = 0.03 = 3%.
y1900
(You can use natural logarithms and, in this case, the answer is the same
since the period is long enough.)
If Japan continues growing at the same rate, income in 2100 will be
y2100 = 26375 · 1.03100 = $485, 443.
2. In 2000 GDP per capita in the United States was $35,587 while GDP per
capita in Sri Lanka was $3,527. Income per capita in the United States
has been growing at a constant rate of 1.9% per year. Calculate the year
in which income per capita in the United States was equal to year 2000
income per capita in Sri Lanka.
We asked to calculate n in the following formula;
35587 = 3527 · 1.019n .
Taking natural logarithms
ln 35587 = ln 3527 + n ln 1.019
and solving for n
n=
ln 35587 − ln 3527
= 122 years;
ln 1.019
i.e., the United States were at this level of income in 1878.
3. Relation between productivity (output per worker) and income
per capita. Call L the number of workers (labour force), F the number
of people of working age, and N total population. We usually assume no
unemployment in the long run. Productivity, y, equals output per worker
(y = Y /L) while income per capita, let us call it i, equals output per
person (i = Y /N ). The labour force participation ratio, l, is the proportion
of people in the labour force to people of working age (l = L/F ) while the
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dependency ratio, d, is the proportion of total population to population
of working age (d = N/F ). Therefore,
Y
N
=
i =
Y LF
LFN
y·l
d
(a) What is the relation between an increase in productivity and an increase in income per capita?
The natural log of income per capita equals
ln i = ln y + ln l − ln d.
Taking its derivative with respect to time, we obtain its growth rate
i̇
ẏ l˙ d˙
= + −
i
y l d
Thus, in growth terms the growth rate of income (measured output)
per capita will be the growth rate of productivity (output per worker)
plus the growth rate of the labour force participation ratio minus the
growth rate of the dependency ratio; or the growth rate depends
positively in productivity and labor force participation growth and
negatively on dependency growth.
(b) Suppose that population growth in Ameropa decreases from 2% to
1%. As a consequence, productivity increases by 4%. The population
growth does not affect the labour force participation ratio but it
affects the dependency ratio. Suppose that the dependency ratio
decreases by 10% because there are fewer children. Calculate the
total effect on income per capita of the demographic change; i.e.,
calculate the combined effect of the increase in productivity and of
the change in the dependency ratio.
In this case, both the increase in productivity and the decrease in the
dependency ratio have the same effect of increasing measured income
per capita: 0.04 − (−0.10) = 0.14 = 14%.
(c) Suppose again that population growth decreases and, as a consequence, productivity increases by 4%. However, suppose instead that
the population becomes older and the dependency ratio increases by
7%. What is in this case the total effect on income per capita of
the demographic change: i.e., the combined effect of the increase in
productivity and of the change in the dependency ratio?
Income per capita decreases because the increase in productivity is
not enough to compensate for the increase in the dependency ratio:
0.04 − 0.07 = −0.03 = 3%.
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(d) Suppose now that in case c. the government, worried about the aging
of the population, encourages women to work outside the home with
the effect that the labour force participation ratio increases by 5%.
What is the total effect on the measured income per capita of the
demographic change plus the government action?
The combined effect of the actions equals 0.04 + 0.05 − 0.07 = 0.02 =
2%; i.e., the government can partially compensate the effect of the
aging of the population on income per capita by trying to increase
the labour force participation ratio or increasing the retirement age
which, hopefully, will have the effect of decreasing the dependency
ratio.
4. The growth rate of aggregate productivity (sectoral composition). Suppose that the economy is composed of two sectors, agriculture
and manufacture, whose production functions are, respectively,
YA = BA LA
and
YM = BM LM
where LA and LM (LA + LM = L) are the quantities of labour used in
each sector and BA and BM the productivities of each sector. The total
quantity of output is Y = YA + YM . The aggregate level of productivity
in the economy is defined as
B=
Y
BA LA BM LM
=
+
.
L
L
L
Show that the growth rate of aggregate productivity is a weighted average
of the growth rates of the productivity in the two sectors where the weights
are the sectors’ share of total output. (Hint: first, calculate Ḃ keeping
quantities of labour constant; second, divide by B to obtain Ḃ/B; finally,
multiply and divide each term by the productivity in each sector.)
To obtain the change in aggregate productivity we take the derivative with
respect to time; we assume that the quantities of labour are constant;
therefore,
LA
LM
Ḃ = ḂA
+ ḂM
;
L
L
the change in aggregate productivity is a weighted average of the changes
of productivity in the two sectors, but n this case the weights are given
by the sectors’s share of total labour, not of total output.
If we divide both sides of the previous equation by B,
LA
LM
Ḃ
= ḂA
+ ḂM
.
B
BL
BL
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Notice that, by definition, BL = Y . Now if we multiply and divide each
term by the productivity in the sector
Ḃ
B
=
ḂA BA LA
ḂM BM LM
+
=
BA BL
BM BL
ḂM YM
ḂA YA
+
;
BA Y
BM Y
i.e., growth rate of aggregate productivity is a weighted average of the
growth rates of the productivity in the two sectors and the weights in this
case are the sectors’ share of total output.
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