Economic Growth and Development
EC 375
Prof. Murphy
Problem Set 1 Answers
Chapter 1 #2, 3, 4, 5, 6, 7 (on pages 24-25) and Appendix problems A.1 and A.2 (on
pages 28-29).
2.
Let g be the rate of growth. The rule of 72 says that 72/g ! 9. So g ! 8%.
3.
Using the rule of 72, we know that GDP per capita will double every 72/g years, where g is
the annual growth rate of GDP per capita. Working backwards, if we start in the year 1900
with a GDP per capita of $1,000, to reach $4,000 by the year 1948, GDP per capita must
have doubled twice. To see this, note that after doubling once, GDP per capita would be
$2,000 in some year, and doubling again, GDP per capita would be $4,000, exactly the GDP
per capita in year 1948. Using the fact that GDP doubled twice within 48 years and
assuming a constant annual growth rate, we conclude that GDP per capita doubles every 24
years. Solving for the equation, 72/g = 24, we get g, the annual growth rate, to be three
percent per year.
4.
Between-country inequality is the inequality associated with average incomes of different
countries. Country A’s average income is given by adding Alfred’s Income and Doris’s
Income and then dividing by 2. This yields an average income of 2,500 for Country A.
Similar calculations reveal that Country B’s average income is 2,500. Because the average
income for Country A is equal to that of Country B, there is no between-country inequality
in this world.
Within-country inequality is the inequality associated with incomes of people in the same
country. In Country A, Alfred earns 1,000 while Doris earns 4,000, making it an income
disparity of 3,000. In Country B, the income disparity is 1,000. Therefore, we see withincountry income inequality in both Country A and Country B. Because there is no betweencountry inequality, world inequality can be entirely attributed to within-country inequality.
5.
We can solve for the average annual growth rate, g, by substituting the appropriate values
into the equation:
(Y1900) × (1 + g)100 = Y2000.
Letting Y1900 = $1,433, Y2000 = $23,971, and rearranging to solve for g, we get:
g = ($23,971/$1,433)(1/100) – 1,
g ! 0.0286.
Converting g into a percent, we conclude that the growth rate of income per capita in Japan
over this period was approximately 2.86 percent per year.
To find the income per capita of Japan 100 years from now, in 2100, we solve
(Y2000) × (1 + g)100 = Y2100.
Letting Y2000 = $23,971 and g = 0.0286,
($23,971) × (1 + 0.0286)100 = Y2100,
2
Y2100 = $402,103.76.
That is, if Japan grew at the average growth rate of 2.86 percent per year, we would find the
income per capita of Japan in 2100 to be about $402,103.76.
6.
In order to calculate the year in which income per capita in the United States was equal to
income per capita in Sri Lanka, we need to find t, the number of years that passed between
the year 2005 and the year U.S. income per capita equaled that of 2005 Sri Lanka income
per capita. Equating income per capita of Sri Lanka in year 2005 to income per capita of the
United States in year 2005 – t, we now write an equation for the United States as
(YU.S., 2005 – t) × (1 + g)t = YU.S., 2005.
Since YU.S., 2005 – t = YSri Lanka, 2005 = $4,650, YU.S., 2005 = $36,806, and g = 0.019, we then substitute
in these values and solve for t.
($4,650) × (1 + 0.019)t = $36,806.
(1 + 0.019)t = ($36,806/$4,650).
One can solve for t by simply trying out different values on a calculator. Alternatively,
taking the natural log of both sides, and noting that ln(x y ) = y ln(x), we get
t ln(1 + 0.019) = ln($36,806/$4,650)
t = 109.92.
That is, 109.92 years ago, the income per capita of the United States equaled that of Sri
Lanka’s income in the year 2005. This year was roughly 2005 – t, i.e., the year 1895.
7.
In order to calculate the year in which income per capita in China will overtake the income
per capita in the United States, we first need to find t, the number of years it will take for the
income per capita in both countries to be equal. That is,
(YU.S., 2005) × (1 + .022)t = (YChina, 2005) × (1 + .073)t.
Since YU.S., 2005 = $36,806, YChina, 2005 = $5,955, we then substitute in these values and solve for
t.
(1 + 0.073/1 + .022)t = ($36,806/$5,955).
We can solve for t by trying out different values on a calculator. Alternatively, taking the
Natural Log of both sides, and noting that ln(x y ) = y ln(x), we get
t ln(1.05) = ln($36,806/$5,955)
t = 37.33.
That is, in 37.33 years, assuming they grow at the current growth rates, the income per
capita of China will surpass that of the United States. This year will roughly be 2005 + t, i.e.,
the year 2042.
3
Appendix Questions
A.1.
The number of people living on less than a dollar a day will be larger if we calculate it
using market exchange rates instead of purchasing power exchange rates because market
exchange rates only take into account the relative value of traded goods, which are
relatively more expensive in poorer countries. Individuals in these countries will have low
purchasing power for traded goods. By using the market exchange rate, we are assuming
that traded goods and non-traded goods are the same price, and therefore individuals in
poor countries will have low purchasing power for non-traded goods as well, which will
make them appear poorer than they actually are.
A.2.
a. The level of GDP per capita in each country, measured in its own currency is
(Computers per capita × Price) + (Ice Cream per capita × Price) = GDP per capita.
Therefore, Richland’s GDP per capita is 40 and Poorland’s GDP per capita is 4.
b. The market exchange rate is determined by the law of one price. As computers are the
only traded good, the price of computers should be the same. Consequently, the
exchange rate must be 2 Richland dollars to 1 Poorland dollar.
c. To find the ratio of GDP per capita between Richland and Poorland, we must first
convert GDP denominations into the same currency. In the analysis that follows, I
choose to convert GDP denominations into Poorland dollars, but converting to
Richland dollars is equally correct, similar, and will yield the same result. From Part
(a), we convert Richland’s GDP per capita, denominated in Richland dollars, into
Poorland dollars by multiplying GDP per capita with the market exchange rate. Since
from Part (b), we know 2 Richland dollars equals 1 Poorland dollar, we multiply 1/2 to
Richland’s GDP per capita, yielding 20 Poorland dollars. Thus, the ratio of Richland
GDP per capita to Poorland GDP per capita is 5:1.
d. A natural basket to use is the world consumption basket: 3 computers and 1 ice cream.
The cost of this basket in Richland is 10 Richland dollars. The cost of this basket in
Poorland is 4 Poorland dollars. Equating the costs of baskets to be one price, the
purchasing power parity exchange rate must be 10 Richland dollars: 4 Poorland dollars.
e. To find the ratio of GDP per capita between Richland and Poorland, we must first
convert GDP denominations into the same currency. In the analysis that follows, I
choose to convert GDP denominations into Poorland dollars, but converting to
Richland dollars is equally correct, similar, and will yield the same result. From Part
(a), we convert Richland’s GDP per capita, denominated in Richland dollars, into
Poorland dollars by multiplying GDP per capita with the PPP exchange rate. Since
from Part (d), we know 10 Richland dollars equals 4 Poorland dollars, we multiply
4/10 to Richland’s GDP per capita, yielding 16 Poorland dollars. Thus the ratio of
Richland GDP per capita to Poorland GDP per capita is 4:1.
4
Chapter 2 # 1, 2, 3, 4, 7, 8 (on page 46).
1.
Proximate causes are causes that directly affect the variable of interest. Low levels of
physical and human capital, technology, and efficiency are all examples of a proximate
cause of low GDP per capita.
2.
Fundamental causes are causes that indirectly affect the variable of interest by systematically
affecting one or many other causes that in turn affect the variable of interest. Possible
fundamental causes may be government, culture, ethnic composition, rule of law,
geography, climate, resources, and so forth. These causes affect GDP per capita by affecting
the proximate causes of low GDP per capita.
3.
To show different levels of factors of production, the figures must not intersect at the same
level of output. To show different levels of productivity, the figures must have different
slopes. In the figure below, Country 1 and Country 2 have the same level of output per
worker. However, Country 1 has a higher level of factors of production than does Country 2,
and Country 1 has a lower level of productivity than does Country 2.
4.
In the long run, the two countries would be expected to have the same levels (and thus
growth rates) of income, because they have the same fundamentals. In the short run Country
B would be expected to have faster growth because the two countries are moving toward
having similar income levels, but Country B is starting out with a lower level.
7.
a.
Although the majority of right-wing voters may live longer, the inference that being a
political conservative is good for you is incorrect because correlation does not imply
causation. A majority of right-wing voters may live longer, not because they are
conservative but rather, because they lead healthier lifestyles that right-wing policies
promote. Thus, we have an omitted third variable affecting both the choice of party
affiliation and the length of life.
b. Although people in hospitals are generally less healthy than those outside hospitals, the
inference that one should avoid hospitals is incorrect because of reverse causation. That
is, a majority of people go to the hospital because they are unhealthy in contrast to the
reverse inference, whereby going to the hospital makes one unhealthy.
5
8.
a.
Positive Correlation. It is reasonable to assume that higher (lower) GDP per capita
increases (decreases) available expenditure for printing books. Moreover, it is also
reasonable to assume that a greater (smaller) number of books printed per capita
increases (decreases) the level of education within a country, translating into higher
(lower) levels of GDP per capita.
b. Negative Correlation. The higher GDP is per capita, the more likely it is that basic
nutrition needs of the population will be met, and the smaller the number of people
suffering from malnutrition, the more likely it is that there will be a healthier labor force
to produce higher levels of GDP per capita. Hence, higher GDP per capita should be
correlated with lower fractions of people suffering from malnutrition and vice versa.
c. No Correlation or Positive Correlation. There are two things to consider. First, does
eyesight progressively deteriorate with age? Second, does the level of GDP positively
affect both one’s ability to diagnose and correct vision problems and one’s life
expectancy through access to better nutrition, health care, and so on? If one does not
assume the above to be true, then there should be no correlation between life expectancy
and the population that wears eyeglasses. On the other hand, if one does assume the
above to be true, then one should see high life expectancy figures when one sees a high
fraction of people wearing eyeglasses, for the simple reason that there is a large elderly
population with poor vision able to afford glasses.
d. No Correlation. There is no obvious relationship between the number of letters in a
country’s name and the number of automobiles per capita.
SOLUTIONS CHAP 3 WEIL 2nd ed
2. In the steady state, the growth rate of capital must be zero because investment in
capital is exactly offset by depreciation in capital. (Note: there is no population growth
here). If we let the investment rate be given by γ, then the investment level is equal to
1
γy = γk 2 . If capital depreciates at rate δ, then the steady state capital stock (k ss ) is given
by the following equality:
1
γk ss 2 = δk ss .
With γ = 0.5 and δ = 0.05, we have k ss = 102 = 100. At 400, the present capital stock thus
exceeds the steady-state stock. This means that the stock will go down over time. Indeed,
we can verify this with the following:
1
1
∆k = γk 2 − δk = 0.5 ∗ (400) 2 − 0.05 ∗ 400 = −10 < 0.
At kt = 400, depreciation exceeds investment.
3. An example in biology is that of the deer population on an island. The quantity of deers
that can be supported by the island is limited by the food available on it. If there are very few
deers, the food is abundant and their population will grow fast, i.e. births numbers exceed
deaths numbers. Conversely, if there are very many deers suddenly brought on the island,
food availability per deer will be low and deaths numbers will exceed births; population size
goes down. Between these two extremes, there must be a long-run equilibrium number of
deers that can be supported indefinitely into the future as the numbers of deaths and births
are equal. This is another instance of a steady-state equilibrium in a dynamic setting.
4. Assuming that output per capita can be represented by a Cobb-Douglas functional
form, i.e. y = Ak α , we have, in the steady-state:
γAk α = δk.
Which yields the following steady-state capital stock:
1
µ ¶ 1−α
γA
ss
.
k =
δ
Inserting this value in the output function, we get the following SS:
α
³ γ ´ 1−α
1
ss
ssα
1−α
y = Ak = A
.
δ
If two countries differ solely by their investment rate:
¢ α
α
1 ¡
µ
¶ 1/3
µ ¶ 1−α
A 1−α γδi 1−α
0.05 1−1/3
γi
yiss
=
=
= 0.5.
=
1 ¡γ ¢ α
yjss
γj
0.2
A 1−α δj 1−α
In the long run, i.e. at the steady-state, income per capita in country j will be twice that of
country i because the latter’s savings rate is four times lower.
1
2
But if α = 2/3, we have
yiss
==
yjss
µ
γi
γj
α
¶ 1−α
=
µ
0.05
0.2
2/3
¶ 1−2/3
= 0.0625 =
1
.
16
In the long run, income per capita in country j will now be sixteen times that of country i
because the latter’s savings rate is four times lower.
5.a) If we follow the same procedure as that of the preceding problem, the Solow model
predicts that:
α
µ ¶ 1−α
¶ 1/3
µ
γT
0.303 1−1/3
yTss
=
=
= 1.75.
yBss
γB
0.099
In reality, the income per capita ratios is:
14260
= 2.06,
6912
which is somewhat close to the Solow model prediction.
5.b) In this case, the Solow model predicts:
α
µ ¶ 1−α
¶ 1/3
µ
ss
γN
yN
0.075 1−1/3
=
=
= 0.717.
yTss
γT
0.146
While in reality, the income per capita ratios is:
3648
= 0.209,
17491
which is quite far from the Solow model’s predictions.
5.c) In this case, the Solow model predicts:
α
µ
µ ¶ 1−α
¶ 1/3
0.313 1−1/3
yJss
γJ
=
= 1.23.
=
ss
yN
γN
0.207
While in reality, the income per capita ratios is:
48389
= 1.116,
43360
which is somewhat close to the Solow model’s predictions.
6. The fact that output per capita grows in country X suggests that its capital stock
is now below its SS value, and conversely for country Y . According to the Solow model,
income per capita and capital per capita at the SS both increase with the savings rate. The
fact that both countries now have the same income per capita suggests that the investment
rate in country X is higher than in country Y .
3
7.a) The per capita level of capital (k ss ) in SS must respect: γk ss1/2 = δk ss . Hence
¡ ¢2
1
k ss = γδ = 25 and y SS = 25 2 = 5.
7.b) In period 2, you should get: k = 16.2, y = 4.02, γy = 1.005, δk = 0.81, ∆k = 0.195.
Hence, the period 3 capital stock is k = 16.395. And so on. In period 8, you should get:
k = 17.33, y = 4.16, γy = 1.041, δk = 0.87, ∆k = 0.174.
7.c) The growth rate between years 1 and 2 is:
X2 − X1
4.02 − 4
g=
=
= 0.005 = 0.5%.
X1
4
7.d) The growth rate between years 7 and 8 is:
4.16 − 4.14
g=
= 0.0048 = 0.48%.
4.14
6.e) The growth rate goes down the closer is the economy to its steady-state value.
8. (See accompanying graphic.)
If y = c∗ , then investment is i = 0. If y > c∗ then i = γ(y − c∗ ) = γ(f (k) − c∗ ).
The output and the depreciation curves are not affected by this. But the investment curves
shifts down as per the accompanying figure. There is a strictly positive income level below
which investment becomes nil. This income level is referred to as the subsistence income
level.
If the depreciation rate is not too high, it crosses the investment curve at two places.
There are thus two possible steady-states, which we refer to as k0ss et k1ss , with k0ss < k1ss .
However, only k1ss denotes a stable steady-state. Indeed, any deviation around that value
will bring the economy back to it. In the case of k0ss , a deviation to the right-hand side will
send the economy over to k1ss in the long run, while a deviation to the left-hand side will lead
to an eventual disappearance of capital. Indeed, to the LHS of k0ss , depreciation is always
above investment, while the converse holds to the RHS.
Finally, if the depreciation rate were too high, then there is no crossing between the
investment and the depreciation curves, the latter being always above the former. In the
long run, capital always disappears.
SOLUTIONS CHAP 4
1. As seen in chapter one, the formula is
g=
(
Xt+n
Xt
) n1
−1=
(
6400000000
2
1
) 100000
− 1 = 0.000218888 = 0.0218%.
2. The graphs accompanying this answer are in the file solutions_graph_chap4b.pdf.
2.a) The initial equilibrium is at point A, where population does not grow. With
the new seed variety, each worker can produce more; curve Z shifts to the right at
Z ′ . In the short run, output jumps to point B, resulting in a higher per capita
output. This higher output per capita causes the population to grow, as seen on the
lower graphic. In the long run, the new steady-state equilibrium is at point C, where
consumption per capita is the same as before the introduction of the new seed variety.
′
At LSS , the total population size is however larger than initially.
2.b) The initial equilibrium is at point A, where population does not grow. Population size suddenly drops by half: the economy jumps to point B suddenly. This
increases per capita output because there is more land available per worker. As a
result, population starts to grow, as indicated by the lower graph. In the long run,
the economy returns to its initial point A.
2.c) The initial equilibrium is at point A, where population does not grow. With
the destruction of half of the land, output per capita reduces by half for a given
population size (assuming constant returns to land). This displaces curve Z to the
′
left to Z ′ . But population size also decreases by half simultaneously, from LSS à LSS .
The economy thus jumps from one steady-state to another one without any transition
period. Income per capita is the same at the new steady-state but the population
size has reduced by half.
3. The initial equilibrium is at point A, where population does not grow. The
population growth curve suddenly shifts from V to V ′ : at each given income level,
people want more kids. Population growth suddenly jumps to point B and becomes
positive. As population size increases, income per capita decreases. In the long run,
′
income per capita reaches level y SS , which is lower than initially and population size
′
is larger at LSS .
5. The ratio of incomes per capita in the steady-state is given by:
1
2
1
1−α
(
γi
δ+ni
α
) 1−α
A
ỹi
=
=
α
(
) 1−α
1
ỹj
γj
1−α
A
δ+nj
(
γi
δ+ni
γj
δ+nj
α
) 1−α
=
(
0.20
0.05+0
0.05
0.05+0.04
1/3
) 1−1/3
= 2.683.
6. The graphs accompanying this answer are in the file solutions_graph_chap4a.pdf.
In this problem, the population growth rate is endogenous, i.e. it depends on the
income per worker. More specifically, the line (n + δ)k is given by (n1 + δ)k when
income per capita is below f (k̄), and given by (n2 + δ)k when income per capita is
above f (k̄). Note that with n2 < n1 , we simply represent the fact that population
growth decreases with income.
There are two possible steady-state equilibria: one at k1SS with a low income per
capita; the other at k2SS with a high income per capita. This is another instance of
a development trap: A country that starts off poor has a higher population growth
and therefore stays poor because of the capital dilution effect. A country that start
off rich stays rich because of its lower population growth. In order to sustainably
improve the standards of living in the poor country, we would need to find a way
to make its capital stock jump above the threshold k̄ for a little while with outside
development aid for instance.
7. We saw that all else equal, the higher the population growth rate, the lower the
income per capita in the steady-state. Hence, at the steady-state, income per capita
in country A is lower than in country B. Because both countries have presently the
same income per capita level, this indicates that country B is “farther away” from
its steady-state than country A. For this reason, income per capita in country B is
expected to grow faster in country B than A.
This conclusion has intuitive appeal. Indeed, the only thing that differentiates the
two countries is the fact that country A has a larger population growth than B. It is
thus not surprising to see that B will have a larger income growth than A since the
burden of the population growth is less severe for B than A.
9) The graphs accompanying this answer are in the file solutions_graph_chap4a.pdf.
9.a) See graph.
9.b) According to the problem’s data, we have:
1
1
1
Y
L2 X 2
X2
y=
=
= 1.
L
L
L2
3
This implies
1000000
X
=
.
2
y
y2
This relation appears on graph 7.b).
9.c) At the steady-state, the growth rate of population must be zero, i.e. L̂ = 0.
Hence y = 100. Substituting the value of y found above, we have:
L=
1
y=
or
X2
= 100,
1
L2
1
1000000 2
= 100.
1
L2
This implies that the population at the steady-state is L = 100.
SOLUTIONS CHAPTER 6
HUMAN CAPITAL
6.1 Effects of the invention of a vaccine against malaria. (See
accompanying fig 6.1 in solutions-graphs-chap6.pdf.)
As we have seen, the causal link between health and income goes
both ways. Curve h(y) denotes the positive effect of income on health.
But a better health also allows for a higher income; this is represented
by curve y(h).
With a newly invented vaccine, people will achieve better health
levels all else equal. This means that for any given income level, people
will be healthier: curve h(y) shifts up to, say, h′ (y). Now before the
new vaccine is introduced, the economy is in equilibrium at point A.
The first, direct effect is that health jumps to hB . But with higher
health level hB , people will be able to work better, hence income goes
up to yB . And so on to the final equilibrium at point C, where both
health and income have increased taking into account the multiplier
effect.
6.2 Effects of a more polluted country. (See accompanying fig
6.2 in solutions-graphs-chap6.pdf.)
Suppose that country B is more polluted than country A. For a
given income level, people in country A are thus in better health. Curve
hB (y) is thus below curve hA (y). If nothing else differed between the
two countries, then income in country A would be above that of country
B, i.e. yA > yB at points A and B respectively. In order to reconcile
this with the fact that both countries have the same income level, it
must be the case that curve y B (h) > y A (h), i.e. y B (h) is to the righthand side of y A (h). This means that for any given health level, workers
in country B can produce more than workers in A. This could be due
to either higher physical and human capital levels or better technology,
or both. The equilibrium for country B is at point B ′ , where yA = yB′ .
6.3 Welfare effects of education
We suppose that education has no impact on people’s ability to produce more wealth, but it does allow those who are more educated to
steal part of the output produced by those who are less educated. In
other words, education does not increase the size of the pie but simply
increases the share of the pie for those who have more of it. If this were
1
2
true, then people would still invest in education in order to receive a
larger share of the pie. Now such investments lower the output level
because students are not producing any output. As a result, countries
with higher human capital would, all else equal, be actually poorer than
countries with less human capital in per capita terms. Since the data
indicates quite the opposite, the initial assumption does not appear to
be valid.
6.4 Measuring the returns to education
Using the numbers from page 164, a worker with nine years of education has a salary of
(1.134)4 ∗ (1.101)4 ∗ 1.068 = 2.595w0 ,
where w0 is the salary of a worker without any schooling. The share of
wages due to education is thus
2.595 − 1
= 61.5%.
2.595
6.5 (2nd Ed.) Average wage is 17.45$/hr and the salary for “raw
work” is 5.85$/hr. This implies that human capital receives 17.45−5.85
=
17.45
66.5% of all salaries paid, leaving 33.5% for raw work.
6.5 (3rd Ed.) Suppose that the total size of the population in the
USA is L = 1000 and the salary of a worker without any schooling is 1.
Raw work thus receives an aggregate payment of 1000. Total salaries
are
schooling no of workers
0
4
4
8
8
19
10
67
12
362
14
224
16
316
TOTAL:
1000
salary
total
1
4
4
1.134 = 1.65
13.2
1.65 ∗ 1.1014 = 2.43 46.17
2.43 ∗ 1.0682 = 2.77 185.59
2.77 ∗ 1.0682 = 3.16 1143.92
3.16 ∗ 1.0682 = 3.61 808.64
3.61 ∗ 1.0682 = 4.11 1298.76
3500.28
The share of salaries due to human capital is thus
3500 − 1000
= 71.4%.
3500
3
6.6 We saw that if two countries differ only by the average level of
education per worker, then the ratio of income per capita in the long
run will be given by
hi
yiSS
= ,
SS
hj
yj
where hi and hj denote the human capital of each country respectively.
Human capital levels are calculated using the returns to education according to data on schooling. Consequently, for country i, we have,
hi = (1.134)4 × (1.101)4 × (1.068)2 = 2.77.
And for country j, we have
hj = (1.134)4 = 1.65.
This implies that
hi
2.77
yiSS
=
=
= 1.67.
SS
hj
1.65
yj
In the long run, we would expect country i to have an income per
capita 67% larger than that of country j based solely on differences in
average education level.
6.7 With twelve years of education on average, the average salary in
year 2000 will be
w2000 = (1.134)4 × (1.101)4 × (1.068)4 = 3.16.
In year 1900 it was
w1900 = (1.134)2 = 1.28.
If, through the 20th century, the average annual growth in income was
given by g, we would have:
w2000 = w1900 (1 + g)100 .
As a result, we have
3.16 = 1.28(1 + g)100 ,
and thus
) 1
3.16 100
− 1 = 0, 0091 = 0, 91%.
g=
1.28
The accumulation of human capital would be responsible for an annual
growth of income per capita of 0.91%. Note that although this number
may appear to be small, once compounded over 100 years, it causes
people’s income to increase by a factor of 2.47!
(
Economic Growth and Development
EC 375
Prof. Murphy
Problem Set 3 Answers
Chapter 6 # 1, 2, 5, 7, 8, 10 (on pages 182-183).
1. Assuming the presence and prevalence of malaria within a given country, the invention of an
effective vaccine would shift upward the h(y) curve. The implication is that for any given
level of income per capita, the vaccine will increase the health of the population. Therefore,
the level of income per capita remains constant and the level of health rises by the
magnitude of the shift (Movement from Points A to B). The improvement in workers’ health
will allow further increases in output that will feed back into further improvements in health.
That is, the vaccine will demonstrate a multiplier effect by increasing output and further
increasing health. Graphically, this is shown by movement along the new h(y) curve from
Points B to C. At this point (Point C), the economy settles into equilibrium.
2. Because for any given level of income Country A will generally be healthier than Country
B, we can determine that the hA (y) curve for Country A will be positioned above the hB (y)
curve for Country B. In addition, we observe that the income levels of both countries are
identical. (The graphical depiction of this information is given by the first figure.) This
implies that Country A achieves the same level of income as Country B, even though its
workers are healthier. Consequently, it must be the case that the impact of health on income
in Country A must be less, for any given level of health, than the impact on Country B. That
is, yB (h) will be positioned to the right of yA (h). The equilibrium configuration is depicted
in the second figure.
2
5. As in Problem 4, the payment to human capital is the difference between the total wage and
the wage for raw labor. Given the wage for raw labor to be $5.85 and the total wage to be
$17.45, we first compute the share of wages paid to raw labor as,
Payment to Raw Labor/Total Payment =
$5.85
= 0.335.
$17.45
The fraction of wages paid to raw labor is 33.5 percent. For the fraction of wages paid to
human capital, similar calculations reveal,
Payment to Human Capital/Total Payment =
$17.45 ! $5.85
= 0.665.
$17.45
The fraction of wages paid to human capital is 66.5 percent. (Equivalently, one could just as
easily subtract the first number from 1.)
3
7. The relative return to 10 years of schooling is 2.77, and the relative return to four years of
schooling is 1.65 (using Figure 6.6 and Table 6.2 of the text). Denoting hi = 2.77 and
h j = 1.65, we can solve for the steady-state ratio for two countries identical in every respect
expect for education as follows:
yi,ss
y j,ss
=
hi 2.77
=
= 1.68.
h j 1.65
Thus, the ratio of output per worker in the steady state is 1.68.
8. The relative return to 12 years of schooling is 3.16, and the relative return to two years of
schooling is (1.134)2 or 1.29 (using Figure 6.6 and Table 6.2 of the text). Writing the steadystate ratio for one country over time and denoting h1900 = 1.29 and h2000 = 3.16, we get:
y2000, ss
y1900, ss
=
h2000 3.16
=
= 2.45.
h1900 1.29
Thus, the ratio of steady-state output per worker for this country over time is 2.45. If over
100 years, the steady-state output has increased by a factor of 2.45, we can solve for the
growth rate, g, by the following calculation.
1
g = (2.45)100 ! 1 = 0.009.
We conclude that the annual average growth rate of output per worker due to the increase in
education is 0.9 percent.
10. Because Country A has a higher rate of growth than Country B, and because both countries
are at the same level of income, the Solow model predicts that Country A is farther from its
steady-state level. That is, yA,ss > yB,ss . Furthermore, we are told that the countries are
identical in every respect except for the level of human capital. Therefore, we can correctly
say that
yA,ss > yB,ss implies hA > hB .
The level of human capital must be higher in Country A, suggesting that investment in
human capital is higher in Country A as well.
4
Chapter 7 # 1, 2, 3, 5, 6 (on pages 206-208).
1.
a.
There are three essential components to answering this question. First, the level of
output for Country 1 must be above the level of output for Country 2. Second, the level
of factors used in production for Country 1 must be to the left of the level of factors
used in production for Country 2. (Essentially, Country 1 should be positioned
northwest of Country 2.) Finally, the production function for Country 1 must be steeper
than the production function for Country 2. The diagram is given below. Any slight
variation of the diagram below, with the three essential properties, is a correct answer.
b. Again, there are three essential components to answering this question. First, the level of
output for Country 1 must be above the level of output for Country 2. Second, the level
of factors used in production for Country 1 must be to the right of the level of factors
used in production for Country 2. (Essentially, Country 1 should be positioned northeast
of Country 2.) Finally, the production function for Country 2 must be steeper than the
production function for Country 1. The diagram is given below. Any slight variation of
the diagram below, with the three essential properties, is a correct answer.
5
2.
The equation for output at year t:
yt = At kt! ht1" ! .
And the ratio of output between years t and t + 100
1( '
! yt +100 $ At +100 kt'+100 ht +100
.
=
#" y &%
At kt' ht1( '
t
Rearranging the equation, using the given information and taking α = 1/3, we get
At +100
At
= 8/2 = 4.
Thus, productivity increased by a factor of 4.
3.
a.
In order to calculate the relative levels of productivity in Freedonia and Sylvania, we
must first find the levels of factor accumulation for each country. Denoting a subscript F
for Freedonia, a subscript S for Sylvania, and substituting in the appropriate values, we
get:
Factors F = kF! hF1" ! = (100)1/2 (64)1/2 = (10)(8) = 80
Factors S = kS! hS1" ! = (100)1/2 (25)1/2 = (10)(5) = 50.
The level of factors of production for Freedonia and Sylvania are 80 and 50,
respectively. The ratio of factors of production is given by:
Factors F 80
=
= 1.6.
Factors S 50
That is, level of factors of production is 1.6 times greater in Freedonia than in Sylvania.
Next, we find the ratio of output per worker by,
Output F 200
=
= 2.
Output S 100
The level of output per worker in Freedonia is twice as large as that in Sylvania. This
allows us to use the following equation:
Productivity F ! Output F $ ! Factors F $
=
'
.
Productivity S #" Output S &% #" Factors S &%
Solving for the above equation, we get 2/1.6 = 1.25. Therefore, the productivity level in
Freedonia is 1.25 times the productivity level in Sylvania.
6
b. If all differences in output were due to differences in productivity, then,
Factors F = Factors S .
Thus, the differences in output would be exactly the difference in productivity, and
Freedonia’s output would be 1.25 times greater than Sylvania’s output.
c.
If all differences in output were due to differences in factor accumulation, then,
Productivity F = Productivity S .
Thus, the differences in output would be exactly the difference in factor accumulation,
and Freedonia’s output would be 1.6 times greater than Sylvania’s output.
5.
We assume α = 1/3. Then, in order to calculate the ratio of factor accumulation relative to
the United States for each country, we utilize the following equation:
Factory Accumulation = k ! h (1" ! ) ,
where the ratio of physical capital relative to the U.S., k, and the ratio of human capital
relative to the U.S., h, are given for each country. In order to calculate the ratio of
productivity relative to the United States for each country, we utilize the following equation:
Productivity = y/ (k ! h (1" ! ) ),
where the denominator is simply the previous ratio of factor accumulation and y is the ratio
of output per worker relative to the United States for each country. The results of each of
these calculations for each country are listed in fourth and fifth columns of the table below.
1
y
Country
Sweden
Mauritius
Jordan
k
h
Output per Physical Capital Human Capital
Worker
per Worker
per Worker
0.76
0.55
0.18
0.80
0.32
0.13
0.93
0.70
0.86
2
k3 ! h3
A
Factor
Accumulation
Productivity
0.88
0.54
0.46
0.86
1.02
0.39
7
The table also shows that Sweden has the highest level of factor accumulation relative to the
United States among all the countries. In fact, the level of factor accumulation in Sweden is
nearly that of the United States. This allows us to conclude that differences in factor
accumulation cannot be responsible for the difference in output per worker. Instead, the
relative level of productivity which is closer to the relative output level plays the largest role
in explaining income relative to the United States. Similarly, the relative level of
productivity in Mauritius is by far the largest among all countries, and this level of
productivity exceeds the U.S. level of productivity. In conclusion, productivity differences
cannot explain the difference in output per capita. Rather, it is the relative level of factor
accumulation that is largely responsible for the differences in output per worker relative to
the United States. As for the country of Jordan, productivity plays a role smaller than in
Sweden and larger than in Mauritius, and factors play a role smaller than in Mauritius and
larger than in Sweden. In summary, productivity in Sweden plays the largest role in
explaining income relative to the United States, whereas factor accumulation in Mauritius
plays the largest role in explaining income relative to the United States.
6.
We assume α = 1/3. Then, in order to calculate the growth rate of factors for each country,
we utilize the following equation:
ˆ
Growth Rate of Factors = ! kˆ + (1 " ! )h,
ˆ and the growth rate of human
where the growth rate of physical capital per worker, k,
ˆ
capital per worker, h, are given for each country. In order to calculate the growth rate of
productivity for each country, we utilize the following equation:
Growth Rate of Productivity = Â = ŷ ! (" k̂ + (1 ! " )ĥ),
where the second term is simply the previous growth rate of factor accumulation and ŷ is the
growth rate of output per worker. The results of each of these calculations for each country are
listed in fourth and fifth columns of the table below. The country with the most growth due
to factor accumulation is Austria. The country with the most growth due to productivity
growth is Chile.
ŷ(%)
Country
ˆ
k(%)
ˆ
h(%)
Growth Rate Growth Rate of Growth Rate of
of Output per Physical Capital Human Capital
Worker
per Worker
per Worker
(1/ 3)kˆ + (2 / 3)hˆ
Â(%)
Growth Rate
of Factors
Growth Rate of
Productivity
−0.42
−0.02
0.46
0.37
0.4
Austria
1.9
2.82
0.26
1.11
0.79
Chile
1.54
0.85
0.63
0.70
0.84
Argentina
8
Chapter 8 # 1, 3, 4, 5, 7 (on pages 237-238).
1.
a.
Nonrival. Nonexcludable. One’s consumption of National Defense does not diminish
another’s consumption of National Defense, and within a given country’s borders, it is
difficult to selectively exclude others from consuming National Defense.
b. Rival. Excludable. Once a cookie is consumed, no one can consume that cookie.
Furthermore, one can easily prevent another from consuming the cookie.
c. Non-Rival. Excludable. My authorized use of a website does not diminish another’s use
of the same website. However, this good is excludable because a password is required,
and so only those selected can access the website.
d. Rival. Nonexcludable. The consumption of a piece of fruit ensures that no other person
can consume that same piece of fruit. However, because the fruit grows in a public
square, anyone
is able to consume the fruit.
3.
With no change in the fraction of workers devoted to research and development,
productivity and output per worker would continue to grow indefinitely at the previous rate.
That is,
ŷ = Â =
!L
,
µ
where ! denotes the fraction of workers devoted to R&D. With an increase in ! to ! ", we
know from the following equation that,
! < ! " implies ŷ = Â =
!L
" !L
.
< ŷ ! = Â =
µ
µ
Therefore, at the time of change, denoted as t1 in the graph below, the growth rate of output
per worker and productivity, ŷ ! and !ˆ " respectively, will be greater than before. However,
the increase in the rate of growth will be accompanied by a decrease in the level of output
per worker. Simply put,
y = A(1 ! " ) < y != A(1 " # !).
This amounts to an increase in the slope of, and a drop in, the level of output per worker. For
productivity, there will not be a drop in the level of A, but the slope will rise at the time of
change.
Because the change in ! to ! " is temporary, at the time t 2 , that ! " returns to the original
fraction ! , the process will reverse itself. Output per worker will jump up to a new level as
workers move out of the R&D sector. But the jump up is larger in magnitude than the jump
down. Intuitively, the level of productivity has risen during the temporary increase in the
number of workers devoted to R&D. Thus, with the same number of people moving to the
non-R&D sector as that moving to the R&D sector from before, the same number of workers
can now be more productive. Mathematically, the first and second jumps are given as
At t1: !y = y " y # = A(1 " $ ) " A(1 " $ #)= A($ #" $ ) = A(!$ ),
At t2: !y = y% " y # = A#(1 " $ ) " A#(1 " $ # ) = A#($ #" $ ) = A#(!$ ).
Because the change in ! is the same and A! > A, we know that the absolute difference in the
jump up is greater than the jump down. The level of A! is dependent on the length of the
temporary increase in the fraction of workers devoted to R&D. Regardless of the level, the
9
new growth path of output per worker will be the same rate before the temporary change, but
it will start out at a higher level than if no change had occurred. As for productivity, the
change in ! will return the growth rate of productivity to its original level, without any jumps.
The figures are given below.
4.
The given parameters of the model are, L = 1, µ = 5, and ! = 0.5. To calculate the growth
rate of output per worker, we substitute in these values into the following equation and solve
to get:
ŷ =
! L 0.5
=
= 0.1.
µ
5
The growth rate of output per worker is 10% per year. Similarly, with ! " = 0.75, we get:
ŷ ! =
" !L 0.75
=
= 0.15.
µ
5
In this case, output per worker grows at 15 percent a year. However, the level of output per
worker has dropped. The level before the drop can be found by substituting the previous
parameter values into the production function to get:
10
y = A(1 ! " ) = A(1 ! 0.5) = A(0.5).
Similarly, we can find the new level of output per worker to be:
y ! = A(1 " # ! ) = A(1 " 0.75) = A(0.25).
Therefore, the original level of output per worker is A(0.5) and was growing at a rate of 10
percent per year. The new level of output per worker is A(0.25) and is growing at a rate of
15 percent. We use the standard growth equation, substitute and solve for when the levels of
output per worker are equal.
y !(1 + ŷ !)t = y(1 + ŷ)t ,
A(0.25)(1 + 0.15)t = A(0.5)(1 + 0.1)t .
Dividing both sides by A(0.25), then dividing both sides by (1 + 0.1)t we get:
(1.15 / 1.1)t = 0.5 / 0.25
and taking logs, we get:
t ln(1.15/1.1) = ln(2).
Solving for t, we get:
t=
ln(2)
= 15.59 ! 15.5.
ln(1.15 /1.1)
That is, it will take approximately 15.5 years for the level of output per worker to surpass the
level it would have reached before the change in ! .
5.
In the diagram below, the increase in the fraction of labor devoted to R&D in Country 1 will
create a drop in the level of output per worker but an increase in the growth rate of
productivity as well as output per worker. Country 1 behaves in accordance with the onecountry model. However, the speed of growth in productivity in Country 1 raises the steady
state A1/A2 ratio (the second diagram). Consequently, µc , the cost of copying falls for
Country 2. The fall in the cost of copying will raise productivity in Country 2, and so, the
growth rate of output per worker in Country 2 will also rise. There will be no jump, down or
up, in output for Country 2, but the fall in the cost of copying will place Country 2 on a
higher growth path, as illustrated in the figure below. In the long run, the growth rates of
output for both Country 1 and Country 2 will be equal, with Country 1 at a higher level.
11
7.
a.
In the steady state, Â1 = Â2 . Therefore,
! A,1 L
µi
=
! A,2 L
µc
.
Rearranging and solving for µc , we get,
"! %
µc = $ A,2 ' µi .
# ! A,1 &
Setting the above steady-state condition equation to the specified cost-of-copying
function,
12
"! %
"A %
µc = $ A,2 ' µi = $ 1 '
# A2 &
# ! A,1 &
()
µi .
Rearranging, we find out solution to be:
1
! A1 $ ! ' A,1 $ (
#" A &% = # ' & .
" A,2 %
2
Without the exponent, the ratio of technology in Country 1 to Country 2 would be
determined proportionally by the ratio of the fraction of the labor force employed in
R&D. This is the case when ! = 1. Because we assume 0 < ! < 1, with the exponent, the
ratios will not be proportional. That is, as the value of β falls to zero, the proportional
difference in the level of technology between the two countries grows extremely large, and
as the value of β rises to one, the proportional difference in the level of technology between
the two countries matches the proportional difference in the fraction of worker devoted to
R&D.
b. If we assume ! = 1/2, µi = 10, ! A,1 = 0.2, and ! A,2 = 0.1, we can solve the previous
equation to get:
1
2
! A1 $ ! ' A,1 $ ( ! 0.2 $
=
=
& = 4.
#"
#" A &% # ' &
0.1 %
" A,2 %
2
That is, the steady-state ratio of technology in Country 1 to technology in Country 2 is
4. Note that you don’t need a value for µi to solve this problem, although the problem
gives you one!
The Role of Technologyin Growth
l. (a) Nonrivai.
Noncxclud$le,
Onet co.sndprio.ofNatiodalDefense,loes
nor
dininish dothefs consmption ofNalion.t Defense,md within a givencouniry.s
bordes,il is dimcdl b selectivetycxcludeorheNfiom consuningNalionatDetense.
(b)
Rival,Excludable.Oncea cookieis co.suoed,no onecanconsune$ar co6kie.
FunhenoE, oncce e$ily preven!mothd fron consumjngthecoo*ie.
(c)
Non-Rival.Excludable.My .urhori4d useof a websiedoesnot,to r cerrlin
ert.t bsauss ofweb nlttc, dir.inlsb dorher,s useofrhe samewebsile. However,this
d .r d ) o o n t yr h o 6\.e h ' e d c u r 4 c +
3 o o ' l i 6 ( r c u d d b h b e . 4 s c a o . $ \ o ' d , r c oc u
(d)
Rivol.Nodexcludable.Theco.sumplionofa pime ofliuit insfts thd no othcr
pe6oncm conslne ihll smc pieceofilit. Hoveler, bmuse $c nuir groNsin a
public square,.nyo.e h ablclo consunetie tun.
**ffisx'm****e$**n
jfrrt l
t-->
t fi i t l
:#
f
1
Itlf
TTIi
I
r-ill
t+]ll
]r-ljl
L'A!-,$t
EdI
] -i:Ffi
l=.l L
I
l Il nt-f lIl l
tl
l
r l
l
ri
t
ratio
scale
ratio
scale
!
f]l
1
L t*]^l
'Fz{l
rl
r-+l '
irEr
f=-'
'
l l