Economic Development
CAS EC 320 Problem Set 1
September 23, 2010
Answers
Each question is worth 10 points.
1. What is
economic convergence? Use a diagram to
explain your answer. What is the
evidence on convergence over the past 30-50 years? What are the implications of this evidence
for economic policy in developing countries?
Economic convergence occurs when the difference in income
between the richest and the poorest countries narrows. This would occur if the poorest countries
grew faster than the richest countries.
The diagram has the growth rate in per capita income over some period of
time on the vertical axis and per capita income at the beginning of the period
on the horizontal axis. Convergence
would occur if the dots representing different countries were along a
negatively sloped line. This would
indicate higher growth rates for poor countries than for rich countries.
Evidence on per capita income does not indicate that there
has been convergence over the past 30 or 50 years. There has been little connection between
growth rates in per capita income and the initial level of per capita income. However, some poor countries have grown much
faster than rich countries. These
countries have generally followed certain economic policies that allowed them
to catch up with the rich countries. The
implication is that if other developing countries adopted these policies, they
also could grow faster than the rich countries.
2. What is
purchasing power parity (PPP)? How is it
calculated? What are the PPP per capita
GDP and nominal per capita GDP of your country or your parentsâ country? If you and your parents were born in America,
pick a country that begins with the same letter as your last name.
To calculate per capita GDP using purchasing power parity,
statisticians calculate (a) the number of units of a domestic currency needed
to buy a market basket of goods, and
(b) the number of US dollars needed to buy that market basket in the
US. This market basket includes the
kinds of goods bought by average people in developing countries. They then divide per capita GDP in domestic
currency by (a)/(b) to find PPP domestic
per capita income in US dollars. For
developing countries, PPP per capita income is usually much higher than per
capita income in nominal terms, where domestic per capita income in local
currency is converted into dollars at the nominal exchange rate.
3. What is
the Human Development Index? How is it
constructed? What are the HDI rank and
per capita GDP rank of âyourâ country?
The HDI combines measures of life expectancy at birth, the
percent of the population 15 years and older that is literate, the percent of
school age children who are actually enrolled in school, and per capita income
measured in PPP. Each of these measures
in used to calculate an index, and these indexes are added together to
calculate a countryâs Human Development Index.
Countries are then ranked according to the HDIs.
4. Suppose
Chinaâs GDP is growing by 9% a year and its population grows by 1% a year. Also suppose that US GDP grows by 3% a year
and its population grows by 1% a year, and that in 2010 US GDP is twice as
large as Chinaâs while its population is one quarter of Chinaâs. If these growth rates continue
a. By what
year will Chinaâs GDP double?
Rule of 72. 72/9 = 8
years to doubling. 1.098 = 1.99
b. By what
year will its population double?
Rule of 72. 72/1 = 72
years to doubling 1.0172 = 2.04
c. By what
year will its per capita GDP double?
Per capita income growth =
9 â 1 = 8. 72/8 = 9 years 1.089 =
2.0
d. In what
year will Chinaâs GDP equal US GDP?
Let Chinese GDP = Y and US GDP = 2Y
Then Y * 1.09n = 2Y * 1.03n
Divide by Y 1.09n =
2 * 1.03n
Take logs n log
1.09 =
log 2 + n log 1.03
Solve for n
n = log 2/(log 1.09 â log 1.03) = 12.54 years
e. In what
year will Chinaâs GDP per capita equal US GDP per capita?
Let Chinaâs per capita income = Y/P and
US per capita income = 2Y/(P/4) = 8 Y/P
Then (Y/P) * 1.08n
= 8(Y/P) * 1.02n
Solve for n
n = log 8/(log 1.08 â log 1.02)
= 36.38 years
5. A
Harrod-Domar economy has a capital stock of 3000, a capital/output ratio of 2,
and saving rate of 20% and a depreciation rate of 5%.
a. Does this
economy have increasing, decreasing, or constant returns to scale?
Constant
returns. A doubling of K results in a
doubling of Y
b. What is
its income today? Y= K/2 =
3000/2 = 1500
c. What
will its income be in 10 years?
Its
growth rate = g = s/v â d = .2/2 – .05 = .05
If Y =
1500 today and grows by 5% for 10 years, Y10 = 1500*(1.05)10 = 2443
d. What will
its income be in 10 years if its saving rate increases to 30%?
g = s/v
â d = .3/2 – .05 = .10. Then Y10 = 1500*(1.10)10 = 3891
6. A Solow
economy has a production function y = 4 x k0.5 . If the capital stock per worker = 36
a. What is
output per worker? y = 4 x 360.5
= 24
b. If s = saving and investment rate = .25 and
d = depreciation rate = .05, what are
steady state k and y?
Steady state occurs where ?k = sy â (d+n)k = 0 or sy =
(d+n)k
Since sy = .25 x 4 k0.5 = k0.5 =
(d+n)k = .05k, you can find k*
by solving
Solve for k: k0.5 =
1/.05 = 20 and k* = 400.
And y* = 4 x k0.5 = 80.
c. If labor
force growth rises from 0 to 5% a year, what are steady state k and y?
Now ?k = 0 implies
that k0.5 = (.05 + .05)k or k0.5 = 1/.10 = 10
Solving k* = 100 and
y* = 40
d. If labor
force growth stays at 5% but s rises to 50%, what are k* and y*
Now ?k = 0 implies
that .5 x 4k0.5 = (.05 + .05)k or k0.5 = 2/.10 = 20
Solving k* = 400 and
y* = 80.
7. Draw a
diagram with y on the vertical axis and k on the horizontal axis to show your
answers to 6 a, b, c, and d. You can use
a 2nd diagram if the 1st one gets too crowded.
Explain everything, either in this question or in 6.
a. The
production function starts at the origin, has a positive slope, and it concave,
ie, it becomes flatter and flatter as k increases, but it never becomes
entirely flat. This is because the
marginal productivity of capital is positive but declining. Output per worker (y) continues to increase
as capital per worker (k) increases, but at a decreasing rate.
b. The curve
for saving and investment also starts at the origin and has a positive but
declining slope. Each point on the
saving curve is âsâ of the production function curve, because saving is a
constant percent of output.
The curve for depreciation per worker (d) and labor force
growth (n) is a straight line coming out of the origin. The amount of depreciation in the economy and
the amount of capital needed to keep k constant increase linearly with the
amount of capital per worker (k).
The economy reaches a steady state where the amount of
saving equals the amount of depreciation plus the extra capital needed to
supply new workers with the amount of capital used by existing workers. When sy = (d+n)k, the capital stock per
worker will no longer grow, and therefore output per worker will no longer grow. This point (k*, y*) is shown on the Solow
model diagram by the intersection of the saving curve and the d+n curve. For kk*, saving is less than (d+n)k, and the capital stock per
worker will decrease.
c. If labor
force growth n rises from 0 to .05, the (d+n) curve will become steeper, since
each increase in k will now require more capital when a new worker joins the
labor force. This steeper (d+n) curve
will intersect the saving curve to the left of the old equilibrium point. Therefore the new k* and y* are lower than
before.
d. If the
saving rate s rises from .25 to .50, the s curve will shift up, because a
higher fraction of income is now being saved and invested. The new equilibrium between saving and (d+n)
will be to the right of the old equilibrium, so the new k* and y* will be
higher than before. The economy will
still reach a steady state, but at a higher level of capital and output per
worker than before. For economies that
are below their steady states, the growth rate will increase with this increase
in the saving rate.
8. What does
the Solow model say about
a. Convergence?
Poor countries with small capital per worker but the same
production functions, saving, depreciation, and labor force growth rates, can
grow faster than rich countries with more capital per worker. Because poor countries can grow faster than
rich countries, the incomes of the two will tend to converge.
b. The
importance of improving technology on long term economic growth?
According to the Solow model, with constant technology,
countries will eventually reach a steady state equilibrium in which saving just
equals the depreciation of the capital stock and the amount of new investment
needed to provide any increase in the labor force with the same amount of
capital that other workers use. In the
very long run, growth in output per worker can only occur because technology or
other sources of efficiency improve.
Even before an economy reaches the steady state, the
contribution of additional capital per worker will decline as it grows
richer. While per capita income growth
may not have reached zero, in rich countries technological improvements make
large contributions to growth compared with the contribution of additional
capital per worker.
c. The
effect of population (= labor force) growth on growth in GDP and per capita
GDP?
An increase in the labor force, according to the Solow
model, will increase total output, since the marginal product of labor never
goes to zero, especially if capital is also increasing. With constant returns to scale, if labor and
capital increase by a certain percent, so will total output. However, an increase in the labor force will
lower the growth of per capita income, since some or all of the saving
generated by the economy will have to go to providing capital for the
additional new workers.
d. The
effect of the saving rate on per capita GDP in the short term and long term?
In the short term, an increase in the saving rate will
increase per capita income and the growth rate of per capita income, because
the faster the capital stock increases, the faster income will increase.
In the long run, the steady state level of per capita GDP
will also increase if the economyâs saving rate increases. The economy will still reach a steady state,
but at a higher level of per capita income and per capita capital than in an
economy with a lower saving rate.
9. What is
the relationship between happiness (as reported on household surveys) and
income
a. Within a
country in a specific year?
On average, the level of happiness reported by individuals
increases with their incomes. A person
with twice the income of another typically claims to be considerably happier.
b. On
average in a country like the US or Japan over 40 or 50 years?
Developed countries have experienced very large increases in
average incomes over long period (40-50 years) but average happiness reported
on household surveys has increased very little.
Even when the income of the average person doubles over these long
periods, average happiness increases very little.
c. On
average across countries, from the poorest to the richest, in a specific year?
The average person in very poor countries reports a lower
level of happiness than the average person in middle income countries, who
reports approximately the same level of happiness as the average person in rich
countries.
d. If your
answers are different, explain why the relationship may differ.
Being very poor, having too little food or medical care to
be healthy and to live a long life, makes people unhappy. Once countries rise above extreme poverty, it
may be that happiness depends on being richer than your neighbors. Therefore, if a countryâs average income
rises from very low to middle income, the level of happiness reported by the
average rises, but further increases in average income have a much smaller
effect on average happiness. However, in
any year, people in rich and middle income countries are happier if they are
richer than their neighbors. Over time,
as average income rises further, neighbors also become richer, and average
happiness doesnât rise by much.