Economic Growth - David N. Weil - RESPOSTAS

Prévia do material em texto

Chapter 1 
The Facts to Be Explained 
Note: Special icons in the margin identify problems requiring a computer or calculator . 
 Solutions to Problems 
1. A ratio scale transforms absolute differences in the variable of interest to proportional differences. 
For instance, the GDP of Country X, whose GDP is 10 times greater than Country Y, will be the 
same distance apart as a Country Z whose GDP is 10 times smaller than Country Y ’s GDP, i.e., 
the distance between X, Y, and Z will be the same. On a common linear scale, the distance between 
X and Y would be 10 times greater than the distance between Y and Z. As a result, transforming 
Figure 1.1 into a ratio scale will convey the absolute differences in the height of marchers into 
proportional differences. 
2. Let g be the rate of growth. The rule of 72 says that 72/ 9.g ≈ 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 within-country income inequality in 
both Country A and Country B. Because there is no between-country inequality, world inequality 
can be entirely attributed to within-country inequality. 
2 Weil • Economic Growth, Second Edition 
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, 
 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(xy) = 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 + .075)t. 
Chapter 1 The Facts to Be Explained 3 
 Since YU.S., 2005 = $36,806, YChina, 2005 = $5,955, we then substitute in these values and solve for t. 
(1 + 0.075/1 + .022)t = ($36,806/$5,955). 
4 Weil • Economic Growth, Second Edition 
 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(xy) = 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. 
 Solutions to Appendix Problems 
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 
(CPUs per capita × Price) + (IC 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 CPUs 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 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 thatfollows, 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. 
Chapter 2 
A Framework for Analysis 
 Solutions to Problems 
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. 
Chapter 2 A Framework for Analysis 5 
5. A correct solution is a variable that systematically rises or falls only because of a rise or fall in 
GDP per capita. Take for instance expenditures on unproductive luxury items. It is reasonable to 
conclude that as GDP per capita rises, individuals allocate a larger amount of money to purchasing 
unproductive luxury items. Hence, with a rise in GDP per capita, we should see an increase in 
expenditures on unproductive luxury items because of increased income. However, it is unreasonable 
to assume that a rise in expenditures on unproductive luxury items raises GDP per capita. 
 An example of an incorrect solution would be health. While it is reasonable to assume that a rise in 
GDP per capita will raise the health level of a country, it is also reasonable to say that a rise in health 
will result in a rise in GDP per capita. As income increases, there will be a larger amount of money to 
increase the health of the population. However, a healthier population is also able to work harder and 
increase income levels as well. In this situation, we have both direct and reverse causation. 
6. Simply finding a correlation between being overweight and having a heart attack does not imply 
causation. The correlation could be due to a missing variable like genetics which may be a factor in 
a person’s weight as well as put him or her at risk for a heart attack. Also, reverse causation may be 
the reason for the correlation if heart disease has incapacitated a person, thus making him or her 
unable to exercise which leads to obesity. 
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. 
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. 
 
 
6 Weil • Economic Growth, Second Edition 
Chapter 3 
Physical Capital 
Note: Special icons in the margin identify problems requiring a computer or calculator and those 
requiring calculus . 
 Solutions to Problems 
1. The key characteristics of physical capital are that it is productive, it is produced, its use is limited, 
it can earn a return, and it wears out. 
a. A delivery truck is physical capital because it is productive, i.e., by allowing a delivery man to 
drive instead of walk it increases his output (deliveries made); it has been produced itself; it is 
rival in its use, i.e., only one person can drive it to use it at a time; it can earn a return if rented 
out; and it suffers wear and tear (depreciation) for the period of its use. 
b. Milk is not physical capital because it is not productive. (To be slightly technical, milk is used in 
making other things—for example, cheese. But in this case milk is a raw material rather than a 
factor of production. In economic terms, “cheese production” is the value added resulting from 
using capital and labor to turn milk into cheese.) 
c. Farmland is not physical capital because even though it allows a worker to produce more output, 
it has not been produced itself. 
d. The Pythagorean Theorem is not physical capital because it is non-rival in its use. That is, an 
unlimited amount of people can use it at the same time. 
2. To find the steady-state value of the country, we refer to Equation (3.3) on page 63. 
1 1
1 .ssy A
α
α
α
γ
δ
−
−
 =  
 
 
 Plugging in values: A = 1, α = 0.5, γ = 0.5, and δ = 0.05, we get: 
0.5
1 1 0.5
1 0.5
0.51 .
0.05ss
y−
−
 =  
 
 
 Simplifying the above equation, we get 10.ssy = 
 To find the current output per worker, we substitute in k = 400 into the production function to get: 
1 1
2 2400 20.y k= = = 
 That is, the current output is 20 whereas the steady-state output level is 10. Therefore, we conclude 
that ssy y> so the country is above its steady-state level of output per worker. 
Chapter 2 A Framework for Analysis 7 
4. Denoting each variable by the appropriate country subscript, we write Equation (3.3) from page 63 in 
ratio form. That is, 
1 1
1
,
, 11
1
.
i
i
ii ss
j ss
j
j
j
A
y
y
A
α
α
α
α
α
α
γ
δ
γ
δ
−
−
−
−
 
 
 =
 
  
 
 
 Since productivity, A, and depreciation, δ, are the same, we can cancel them and rewrite the previous 
ratio with the appropriate values: 1 20.05, 0.2,γ γ= = and setting 1/ 3.α = 
1(1/ 3) 1 11 31, 1 2
2, 2
0.05 (0.25) 0.5.
0.2
ss
ss
y
y
α
αγ
γ
  −  −      = = = =   
  
 
 For 1/ 2,α = we get, 
1(1/ 2) 11 21, 1 1
2, 2
0.05 (0.25) 0.25.
0.2
ss
ss
y
y
α
αγ
γ
  −  −      = = = =   
  
 
 Therefore, when 1/ 3,α = the ratio is 0.5 or 1 to 2 and when 1/ 2,α = the ratio is 0.25 or 1 to 4. 
5. Since we know productivity, A, and depreciation, δ , are the same, we know that they will cancel out 
in our steady state ratio analysis. Therefore, with α =1/3, our equation of interest boils down to 
1
1 2
1, 1 1
2, 2 2
,ss
ss
y
y
α
αγ γ
γ γ
−   
= =   
   
 
 for all three pairs of countries. 
a. Using a subscript T for Thailand and a subscript B for Bolivia, we rewrite the previous equation 
for Thailand and Bolivia as 
1
2
,
,
.T ss T
B ss B
y
y
γ
γ
 
=  
 
 
 Substituting in 0.303Tγ = and 0.099,Bγ = we get the steady state ratio to be: 
1
2,
,
0.303 1.75.
0.099
T ss
B ss
y
y
 = ≈ 
 
 
8 Weil • Economic Growth, Second Edition 
 The actual ratio is, 
$14,260 2.06.
$6,912
T
B
y
y
 = ≈ 
 
 
 Therefore, the Solow Model does a good job in predicting relative income for Thailand and 
Bolivia. 
b. Using a subscript N for Nigeria and a subscript T for Turkey, we rewrite the previous equation, 
with 0.075Nγ = and 0.146Tγ = to get, 
1 1
2 2,
,
0.075 0.72.
0.146
N ss N
T ss T
y
y
γ
γ
   = = ≈   
  
 
 The actual ratio is, 
$3,648 0.21.
$17,491
N
T
y
y
 = ≈ 
 
 
 Therefore, the Solow Model does a poor job in predicting relative income for Nigeria and Turkey. 
c. Using a subscript J for Japan and a subscript N for New Zealand, we rewrite the previous 
equation, with 0.313Jγ = and 0.207Nγ = to get, 
1 1
2 2,
,
0.313 1.23.
0.207
J ss J
N ss N
y
y
γ
γ
   = = ≈   
  
 
 The actual ratio is, 
$48,389 1.12.
$43,360
J
N
y
y
 = ≈ 
 
 
 Therefore, the Solow Model does a good job in predicting relative income for Japan and 
New Zealand. 
6. If output per worker is rising in Country X and output per worker is falling in Country Y, we can be 
assured that both countries are not in their respective steady states. Instead, they are converging to 
their respective steady states. In addition, for Country X and Country Y, we are given information 
that depreciation, productivity, and output per worker are identical. By the process of elimination, the 
only difference between the countries can and must be the level of capital stocks. Capital stock levels 
follow the process: 
( ) .i i i i ik f k kγ δ∆ = − 
Chapter 2 A Framework for Analysis 9 
 As such, we can conclude that differences in investment rates are responsible for the divergence in 
output per worker. Specifically, a rise in output per worker for Country X and a fall in output per 
worker for Country Y imply that the ratio of steady-state output per worker is, 
1
,
,
1; therefore, 1; implying, .X ss X X Y
Y ss Y
y
y
α
αγ
γ γ
γ
− 
> > > 
 
 
 Consequently, we can determine that Country X has a higher investment rate than that of Country Y, 
and these differences account for the falling and rising levels of output per worker. 
7. a. First we find the steady-state level of capital per worker. Using the values for investment, γ = 0.25, 
depreciation, δ = 0.05, productivity, A = 1, and α = 0.5, we get, 
1 1
1 1 0.5
2(1)(0.25) 5 25.
0.05ss
Ak
αγ
δ
− −   = = = =   
   
 
 That is, the steady-state level of capital per worker is 25. Plugging in ssk into the production 
function we get the steady-state level of output per worker to be: 
1 1
2 2(25) 5.ss ssy k= = = 
 That is, the steady-state level of output per worker is 5. 
b. For year 2, using 16.2 as the value for capital per worker, calculate output, y, followed by 
investment γ y, depreciation δ k, and then change in capital stock. Add the value for change in 
capital stock to 16.2, the value for capital per worker in year 2, to get capital per worker for 
year 3. Use year 3 capital to obtain all the values for year 3 and continue up to year 8. The filled 
in table is below. 
 
Year 
 
Capital 
 
Output 
 
Investment 
 
Depreciation 
Change in Capital 
Stock 
1 16.00 4.00 1.00 0.08 0.20 
2 16.20 4.02 1.01 0.81 0.20 
3 16.40 4.05 1.01 0.82 0.19 
4 16.59 4.07 1.02 0.83 0.19 
5 16.78 4.10 1.02 0.84 0.19 
6 16.96 4.12 1.03 0.85 0.18 
7 17.14 4.14 1.04 0.86 0.18 
8 17.32 4.16 1.04 0.87 0.17 
c. The growth rate of output between years 1 and 2 is given by: 
2
1
4.021 1 0.005.
4
yg
y
   = − = − =   
  
 
 That is, output per worker grew at a rate of 0.5 percent between years 1 and 2. (Using exact 
values, the growth rate is approximately 0.62 percent for years 1 and 2.) 
10 Weil • Economic Growth, Second Edition 
d. The growth rate of output between years 7 and 8 is given by: 
8
7
4.161 1 0.0048.
4.14
yg
y
   = − = − =   
  
 
 That is, output per worker grew at a rate of 0.48 percent between years 7 and 8. (Using exact 
values, the growth rate is approximately 0.52 percent for years 7 and 8.) 
e. The speed of growth has changed from 0.50 percent to 0.48 percent implying that growth has 
slowed down at a rate of 4 percent. Thus, as a country reaches their steady-state value, the rate 
of growth slows. 
8. Before beginning the analysis, we define two new variables. The level of capital per worker 
necessary to achieve consumption level c* is denoted k*. Technically, k* is given by 1 *( ).f c− 
Therefore, if the initial level of capital, ki is above k*, savings will be positive, and if ki is below k*, 
savings will 0. The second variable I is denoted k (refer to second figure). It is the level of capital at 
which depreciation is equal to savings and distinct from the steady-state level of capital, if it exists. 
We are now ready to begin our analysis. There are two cases. 
 Case 1. Depreciation is always greater than Saving; ( ),k f k kδ γ> ∀ 
 In the figure below, at any initial level of ,ik depreciation is always greater than savings. The level 
of capital falls over time, as does the level of income per worker. Consequently, the economy will 
continue to stagnate until the level of income and the capital stock are zero. 
 
 Case 2. Depreciation is not always greater than Saving; ( ),k f kδ γ≤ for some k 
 The figure below shows two possible scenarios. If the initial level of capital, ,ik is equal to or below 
k*, then savings in the economy will be zero. The level of capital in the economy falls due to 
depreciation and we achieve the same resultas in the first case. On the other hand, if ,ik k≥ then 
the level of savings exceeds or will exceed the level of depreciation and the capital stock rises over 
time. The capital stock will reach a state-state value as will income. If * ,ik k k< < then the amount 
of savings does not exceed the amount of depreciation. The level of capital stock begins to fall and 
we are in the first case where both income and capital go to zero levels. In the end, the ultimate 
determinant of where the economy rests is determined by the initial level of capital, commensurate 
with the initial level of income. 
Chapter 2 A Framework for Analysis 11 
 
9. First, in a steady-state level that maximizes consumption per worker, the change in capital stock will 
be zero. That is, 
0 ( ) .k f k kγ δ∆ = = − 
 Rearranging the previous equation, we know that investment must equal depreciation. 
( ) .f k kγ δ= 
 Second, given that any output not saved is consumed, we can write an equation for consumption as, 
( ) ( ) .C y f k A k kαγ δ= − = − 
 In the last part of the previous equation, we replace savings with depreciation and write output in 
functional form. In this form, we are able to take the derivative to find the necessary condition that 
will guarantee consumption maximization. Taking the derivative with respect to k and rearranging, 
1( ) ( ) .d dC A k k A k k
dk dk
α αδ α −= − = − 
 1( ) .aA k α δ− = 
 That is, the marginal product of capital must equal the rate of depreciation. Combining the consumption 
maximization condition 1( ( ) .)aA k α δ− = with the steady-state condition ( ( ) .),f k kγ δ= we get: 
1saving ( ) ( ) ( ) ( ) .y f k k A k k A k f k yα αγ γ δ α α α α−= = = = × = = = 
 Therefore, it is easy to see the γ must equal α by the above string of equalities. In any steady-state 
level of consumption per worker, the investment/saving rate must equal the value α. 
 
Chapter 4 
Population and Economic Growth 
Note: Special icons in the margin identify problems requiring a computer or calculator and those 
requiring calculus . 
 Solutions to Problems 
1. To find the average growth rate of the population, we use the following equation: 
(1 ) ,nt t nL g L ++ = 
 where tL is the population at time t, g is the growth rate, and n is the duration of growth. Substituting 
in n = 100,000; 2;tL = and 6.7t nL + = billion, we can rewrite and solve the equation for the growth rate. 
100,000
1
100,000
2(1 ) 6,700,000,000,
6,700,000,000 1 0.000219.
2
g+ =
  − = 
 
 
 Therefore, the average growth rate of the population has been roughly 0.0219 percent per year. 
2. a. We begin from Point A, where the population size is stable with no growth. With the discovery 
of a new strain of wheat that is twice as productive, the curve relating population size and income 
per capita shifts outward. At this point, we are at Point B. Here, population growth will be positive 
because of the high level of income per capita. As population grows and income per capita falls, 
we move along both curves as shown by the arrows and approach the long run steady-state level, 
Point C. At Point C, we are at a higher population but with no growth in population. 
 
Chapter 4 Population and Economic Growth 13 
b. We begin from Point A, where the population size is stable with no growth. With war killing half 
the population, no curve is shifted. Instead, we jump to Point B along the original curve. At this 
point, we have half the population with a higher income per capita level allowing population 
growth. As the population grows and income per capita falls, we move along both curves to reach 
the long run equilibrium Point A. That is, the death of half the population results in temporary 
gains and temporary population growth with no long-lasting impact on the ultimate steady-state 
population. 
 
c. We begin from Point A, where the population size is stable with no growth. With a volcanic 
eruption that kills half the people, we are faced with the same scenario as in Part (b). However, 
the volcanic eruption destroys half the land, shifting the curve relating population size and 
income per capita inward. The magnitude of the shift is such that, at the new population size 
and income per capita, growth in the population is zero. This is illustrated to be Point B. The 
short-run equilibrium and the long-run equilibrium are identical in this scenario. 
 
14 Weil • Economic Growth, Second Edition 
3. At Point A in time, the population size is stable with no growth. With a sudden change in cultural 
attitudes, the curve relating the population growth rate and income per capita shifts upward. The 
sudden shift, denoted by Point and Time B, implies that population growth will suddenly be positive. 
As the population size grows and as income per capita falls, the growth rate of population will fall. 
This dynamic is illustrated by movement along the curves from Points B to C. At Point and Time C, 
the country will be in a Malthusian steady-state population level with no growth. 
 
4. To calculate the steady-state level ratio of income per capita, we first find the steady-state level for 
each country and then divide. The steady-state level ratio for Country X to Y is given by: 
1
1
1
,
,
X
X ss X X X
YY ss Y
Y Y
y A n
y A
n
α
α
α
γ
δ
γ
δ
−
−
 
   + =  
  
 + 
 
 We now substitute in the values 20%, 0,X Xnγ = = and 5%Xδ = for Country X, and for Country Y, 
we use the values 5%, 4%,Y Ynγ = = and 5%.Yδ = Also, set 1/ 3α = and .X YA A= This yields, 
11(1/ 3) 1 23 1
2,
,
0.2
4 360 0.05 2.68.0.05 5 5
0.04 0.05 9
X ss
Y ss
y
y
  −  
     
    +  = = = ≈            +    
 
 Therefore, we conclude the ratio of Country X to Y in their steady-state levels of income per capita to 
be near 2.68. 
Chapter 4 Population and Economic Growth 15 
5. Because the equation for capital accumulation suggests that in the steady state, 
( ) ( ) ,f k n kγ δ= + 
 multiple values of population growth will yield multiple steady states under some conditions. In the 
diagram, Points A and B are the multiple capital per capita steady-state values and hence the resulting 
incomes per capita steady-state values. In addition, for a country with an initial capital stock below ,k 
the dynamics will move the country to Point A only. For a country with an initial capital stock 
above ,k the dynamics will move the country to Point B only. That is, in this specific case of multiple 
steady-states, initial capital levels completely determine whether a country achieves a high capital 
and low population growth equilibrium or a low capital and high population growth equilibrium. It is 
important to note that a country cannot transition from A to B or vice versa. 
 
6. Country A and B are identical in every respect but for their population growth rates. That is, .A Bn n> 
However, this implies that their respective steady states are not equal. Writing the reduced ratio 
equation, as on page 98, we get: 
1
,
,
.A ss B B
B ss A A
y n
y n
α
αδ
δ
− +
=  + 
 
 Utilizing the fact that A Bn n> and ,A Bδ δ= we can conclude that , , .A ss B ssy y< Because both countries 
currently have the same income per capita, we now know that Country B is farther away from its 
respective steady-state level. Therefore, Country B must have a higher growth rate of output per 
worker than does Country A. 
7. a. TFR = 4. 
 NRR = (1/2) × [(1 child) × (Probability of reaching age 25) + (1 child) × (Probability of reaching 
age 28) + (1 child) × (Probability of reaching age 32) + (1 child) × (Probability of reaching 
age 35)] 
 Substituting in the given information,we get 
 NRR= (1/2) × [(2/3) + (2/3) + (1/3) + (1/3)] = 1 
b. NRR = (1/2) × [(1) + (1) + (1/2) + (1/2)] = 1.5 
c. TFR = 2 
 NRR = (1/2) × [(1/2) + (1/2) + (1/2) × (1/2) + (1/2) × (1/2)] = .75 
16 Weil • Economic Growth, Second Edition 
8. a. We graph the equation, ˆ 100,L y= − in the figure below. 
 
b. First, we divide both sides of the production function by L and rearrange to get: 
1 1 1
2 2 2
,Y L X Xy
L L L
 = = =  
 
 
 Therefore, 
2
.XL
y
= 
 For X = 1,000,000 the figure is shown below. 
 
c. In the steady state, the growth rate of population is zero, ˆ 0.L = Using this value and rearranging 
the first equation, we solve for the steady-state value of income per capita: 
ˆ 0 100,
100ss
L y
y
= = −
=
 
Chapter 4 Population and Economic Growth 17 
 Substituting in this value into the production function, we back out the value of ssL as follows: 
2 2
1,000,000 100.
(100)ss ss
XL
y
= = = 
 The steady-state population is 100. 
9. a. The steady-state level of income per worker is characterized by ˆ 0.y = Hence, we must first find 
the relationship among the growth rate of income per worker, productivity, and population. 
By taking natural logs of the production function and taking the derivative with respect to time, 
we get: 
ln( ) ln( ) ln ln( ) ln( ) ln( ),
ln( ) ln( ) ln( ) ln( ),
ˆ ˆˆ .
AXy Ax A X L
L
d dy A X L
dt dt
y A X L
 = = = + − 
 
= + −
= + −
 
 Therefore, the growth rate of income per worker must equal the growth rate of productivity plus 
the growth rate of land minus the population growth rate. Since land X, and productivity A, are 
constant, ˆ 0.X A= = These values tell us that ˆˆ ,y L= and because we are interested in the steady-
state value of income per capita, ˆˆ 0 .y L= = Now, we plug the growth rate of population into the 
equation that relates population growth to income-per-capita to arrive at our solution. 
100ˆ 0 ,
100
100.ss
yL
y
−
= =
=
 
b. Referring back to our equation relating the growth rate of income per capita, productivity, land, 
and population, we can set ˆ 0X = and ˆ 0.1.A = 
ˆ ˆ ˆˆ 0.1 .y A X L L= + − = − 
 In the steady-state level of income per worker, yˆ must equal zero. Thus, we have that Lˆ must 
equal 0.1. Using this value to solve for ssy in the equation relating population growth and income 
per capita, 
100ˆ 0.1 ,
100
110.ss
yL
y
−
= =
=
 
 The steady-state value is higher in this scenario. Due to consistent productivity growth of 
10 percent, the population can grow as well, leading to a higher level of income per capita. 
18 Weil • Economic Growth, Second Edition 
 Solutions to Appendix Problems 
A.1. a. To calculate life expectancy at birth, we must find the area under the survivorship function. 
This amounts to solving the equation: 
30 80
0 30
1 0.5 30 25 55.dx dx+ = + =∫ ∫ 
 Equivalently, one could find the area using geometry. 
(40 20)(1) 20.B H× = − = 
 In discrete time analysis, we can extrapolate that the probability of being alive from age 0 to 29 
is 1; the probability of being alive from age 30 to 79 is 0.5; and the probability of being alive 
from age 80 to infinity is 0. Summing these probabilities, we get: 
29 79
0 30 80
( ) 1 0.5 0 30 25 0 55.
i
iπ
∞
= + + = + + =∑ ∑ ∑ ∑ 
 Therefore, the life expectancy at birth is 55 years. 
 b. To calculate the total fertility rate, we must find the area under the age-specific fertility rate 
function. This amounts to solving equation: 
40
20
1 20.dx =∫ 
 Equivalently, one could find the area using geometry. 
(40 20)(1) 20.B H× = − = 
 In discrete time analysis, we can extrapolate that the average number of children per woman 
from age 20 to 39 is one and the average number of children per woman for any other age is 
zero. Summing these probabilities, we get: 
19 39
0 20 40
( ) 0 1 0 0 20 0 20.
i
F i
∞
= + + = + + =∑ ∑ ∑ ∑ 
 Therefore, the total fertility rate is 20. 
 c. The net rate of reproduction is found by multiplying the number of girls that each girl born can 
be expected to give birth to. First noting that the probability of being alive from age 20 to 29 is 
one with the age-specific fertility rate at one child per woman and the probability of being alive 
from age 30 to 39 is 0.5 with the age-specific fertility rate at one child per woman, we solve the 
following equation: 
29 39
20 30
( ) ( ) (1 1) (0.5 1) 10 5 15.
i
i F iπ = × + × = + =∑ ∑ ∑ 
 That is, the rate of reproduction is 15. Adjusting this value by β, the fraction of live births that 
are girls, we conclude that the net rate of reproduction is 15β. 
Chapter 4 Population and Economic Growth 19 
A.2. For Country X and Country Y assume that the survivorship function is that given in Problem 1. The 
total fertility rates for both countries are given below. 
 
 The total fertility rate is the same for both countries. However, the rate of reproduction differs. 
For Country X, 
29 39
20 30
( ) ( ) (1 0.2) (0.5 0) 2 0 2.
i
i F iπ = × + × = + =∑ ∑ ∑ 
 Adjusting for β, the net rate of reproduction for Country X is 2β. For Country Y, 
29 39
20 30
( ) ( ) (1 0) (0.5 2.0) 0 1 1.
i
i F iπ = × + × = + =∑ ∑ ∑ 
 Adjusting for β, the net rate of reproduction for Country Y is 1β. Therefore, Country X has a net 
rate of reproduction twice as large as Country Y, but the survivorship function for both countries 
is identical, as well as the total fertility rate for both countries. This happens because in Country Y 
everyone decides to have the same number of children 10 years later than in Country X. However, 
because the probability of being alive changes in those 10 years, we have a difference in the net rate 
of reproduction. 
 
 
 
 
 
 
 
 
20 Weil • Economic Growth, Second Edition 
Chapter 5 
Future Population Trends 
Note: Special icons in the margin identify problems requiring a computer or calculator . 
 Solutions to Problems 
1. To calculate the population of Fantasia in the year 2001, we must first find the population of each age 
group. Given that of 100 zero-year-olds in the year 2000, and the probability of surviving to age one 
is 1, we conclude that in the year 2001, there will be 100 one-year-olds. Similar calculations reveal 
that in the year 2001, there will be 100 two-year-olds (100 × 1); 100 three-year-olds (100 × 1); 
50 four-year-olds (100 × 0.5); and no five-year-olds (100 × 0). As for the population of newborn 
children (i.e., zero-year-olds), we multiply the number of people by the number of children that each 
person births. Of 100 one-year-olds, each person gives birth to 0.8 children. Similarly, of 100 two-
year-olds, each person gives birth to 0.8 children. Therefore, the number of children born in 2001 will 
be (2) × (100) × (0.8) = 160. There will be 160 zero-year-olds in 2001. Summing the population over 
each age group we get the total population in 2001. 
160 + 100 + 100 + 100 + 50 = Total 2001 Population = 510. 
2. Libya currently has high fertility, rapid population growth, and a population age structure heavily 
weighted toward young people. Japan and has low fertility, almost zero population growth, and one 
of the oldest populations in the world. If the two countries had the same TFR, Libya would have 
higher population growth because of demographic momentum, that is, a larger fraction of its 
population are in their childbearing years. 
3. Population growth in 1950–2000 (denoted by subscript A) for the more developed country group, ,An 
is 0.8 percent, and predicted population growth in 2000–2050 (denoted by subscript B) for the same 
country group, ,Bn is 0.0 percent. Writing the steady-state levelof output per worker in a time 
difference ratio, and assuming that no other factor changes over the given time period, we can 
calculate the effects on the steady state as follows: 
1
,
,
.B ss A
A ss B
y n
y n
α
αδ
δ
− +
=  + 
 
 If given a value for α and δ, we can solve the equation above to find an exact ratio for the differences 
in the steady state. Assuming that α = 1/3 and δ = 0.05, then 
1
2,
,
0.05 0.008 1.077.
0.05 0
B ss
A ss
y
y
+ = = + 
 
 This means that with nothing but population growth changed, output per worker will be 7.7 percent 
greater in 2000–2050 for the more-developed country group. 
Chapter 4 Population and Economic Growth 21 
4. In 2025: each of the 60 million 0–20 year-olds from 2005 will have had 1 girl and moved on to the 
next age bracket; half of the 40 million 21–40 year-olds will have died; and all the 41–60 year-olds 
will have died. So, the new female population structure will be 60 million 0–20 year-olds, 60 million 
21–40 year olds and 20 million 41–60 year-olds. In 2045, this process continues, so we have 60 million 
0–20 year-olds, 60 million 21–40 year-olds, and 30 million 41–60 year-olds. In 2065, the same 
structure as 2045 exists. 
5. Immediately after fertility declines to zero, the working age fraction of the population will begin to 
rise. This is because the fraction of the population made up of children will fall (as some children 
become adults and are not replaced by new births). The fraction of the population made up of 
working-age adults will peak 20 years after the decline in fertility. At that point, the population will 
be composed solely of working-age and old people. From this point onward, the working-age fraction 
will fall as working age people grow old and are not replaced. 65 years after the decline in fertility, 
the working-age fraction of the population will reach zero. 
6. Calculating the growth rate of the percentage of the population that is of working age, g, the equation 
becomes, 
1
15
1965
1950
Working Age Fraction 1 .
Working Age Fraction
g
 − −
− = − − 
 
 Plugging in the appropriate values, we rewrite and solve the previous equation. 
1
1551.27 1 ( 0.00817) ( 0.817%).
57.9
g  = − = − = − 
 
 
 That is, the growth rate of the percentage of the population that is of working age is negative 
0.817 percent. According to Equation (5.2) on page 145, this demographic change lowers GDP 
per capita by 0.817 percentage points. 
7. The statement is not necessarily true. Higher fertility will lower the fraction of the population made 
up of working-age adults until the newly born children reach working age. Thus, in the short run 
the country’s dependency burden will be higher, and its standard of living lower. In the long run, a 
country with very low fertility may indeed be better off with an increase in fertility. 
8. a. To calculate the world population in any year, we add each country’s respective populations 
for that year. Therefore, the world population in 2000 is 2,000,000. For the following year, 
Country A’s 2001 population will be 1,000,000 × 1.02 = 1,020,000. For Country B, 2001 
population remains at 1,000,000. Thus, world population in 2001 is 2,020,000. The growth 
rate of world population, g, will be, 
2001
2000
Population 2,020,0001 1 0.01.
Population 2,000,000
g
   = − = − =   
  
 
22 Weil • Economic Growth, Second Edition 
 The growth rate is 1 percent. However, the growth rate of population will rise over time moving 
to a growth rate of 2 percent per year. The growth rate of population is graphed below. 
 
 Intuitively, Country B’s population continues to grow, making Country A’s population a miniscule 
fraction of the world’s population. Therefore, Country B’s growth rate determines the world 
population growth rate in the long run. 
b. Below is a graph showing the growth rate of total world GDP (not per capita). 
 
c. Below is a graph showing the growth rate of average GDP per capita. 
Chapter 4 Population and Economic Growth 23 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
24 Weil • Economic Growth, Second Edition 
Chapter 6 
Human Capital 
Note: Special icons in the margin identify problems requiring a computer or calculator . 
 Solutions to Problems 
 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 ( )Ah y curve for Country A will be positioned above the ( )Bh 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, ( )By h will be positioned to the right of ( ).Ay h The equilibrium configuration is 
depicted in the second figure. 
Chapter 4 Population and Economic Growth 25 
 
 3. In the case that education has no effect on productivity. That is, an additional year of education does 
not raise the productive ability of an individual. Therefore, the average level of education does not 
affect the total labor input of a country. The population again, determines total labor input of a country. 
The subsequent effect on the analysis of education differences among countries is that schooling 
differences have no ability to explain why income differs among countries. Simply put, human capital 
per worker h is identical for all countries, and these terms cancel out in the ratio of steady-state levels 
of output. 
 4. The wage for an individual with nine years of education, relative to one with no education, is 
(1.134)4 × (1.101)4 × (1.068)1 = 2.60. 
 In other words, if the wage of an individual with no education is W, the wage paid to this individual 
is 2.6W. Therefore, we can conclude that the payment to raw labor is W and the payment to human 
capital is the difference between the total wage and the wage for raw labor. To find the fraction, we 
compute, 
Payment to Human Capital/Total Payment = − =2.6 1
2.6
0.615. 
 The fraction of wages paid to human capital is 61.5 percent. 
26 Weil • Economic Growth, Second Edition 
 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
$17.45
0.335. 
 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
$17.45
0.665. 
 The fraction of wages paid to human capital is 66.5 percent. (Equivalently, one could just as easily 
subtract the first number from 1.) 
 6. In order to find the fraction of wages paid to human capital, we follow the approach outlined in 
Problem 4. We first subtract from each relative wage the relative wage paid for raw labor. This amounts 
to subtracting one from the relative wage for every category. Then, we scale this value, multiplying it 
by the percentage of people in each category. The resulting number is the relative wage paid to human 
capital for each group. The sum of these numbers over each category, therefore, is the relative wage 
paid to human capital for the United States. Because the total relative wage is simply the sum of 
the relative wage of each category multiplied by the appropriate percent, we can now calculate the 
fraction of the wage paid to human capital. 
Relative Wage Paid to Human Capital/Total Relative Wage = =2.3
3.3
 0.70. 
 Therefore the fraction of wages paid to human capital is nearly 70 percent. For the fraction of wages 
paid to raw labor, we subtract our previous figure from one to get: 1 − 0.70 = 0.30. That is, the fraction 
of wages paid to raw labor is 30 percent. 
 
 
Years of 
Schooling* 
 
Wage Relative 
to No Schooling* 
 
Percentage of 
People 
Relative Wage 
Paid to Human 
Capital 
Total 
Relative 
Wage 
No Schooling 0 1 0.8% 0 0.008 
Partial Primary 4 1.65 4.3% 0.02795 0.07095 
Complete Primary 8 2.43 3.9% 0.05577 0.09477 
Incomplete Secondary 10 2.77 22.9% 0.40533 0.63433 
Complete Secondary 12 3.16 20.0% 0.432 0.632 
Incomplete Higher 14 3.61 23.6% 0.61596 0.85196 
Complete Higher 16 4.11 24.5% 0.76195 1.00695 
 Sum 2.29896 3.29896 
 *The first two columns of the table above are replicated from Table 6.2. 
 
 
Chapter 4 Population and Economic Growth 27 
 7. The relative return to 10 years of schooling is 2.77, and the relative return to four years of schooling 
is 1.65 (from the table above). Denoting ih = 2.77 and jh = 1.65, we can solve for the steady-state 
ratio for two countries identical in every respect expect for education as follows: 
,
,
2.77 1.68.
1.65
i ss i
j ss j
y h
y h
= = = 
 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. Writing the steady-state ratio for one country over time and denoting 1900h = 1.29 
and 2000h = 3.16, we get: 
2000, 2000
1900, 1900
3.16 2.45.
1.29
ss
ss
y h
y h
= = = 
 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
100(2.45) 1 0.009.g 
 We conclude that the annual average growth rate of output per worker is 0.9 percent. 
 9. There are many examples of positive externalities associated with health. For instance, being 
surrounded by healthy people reduces the probability of contracting disease. 
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, , , .A ss B ssy y> 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 
, , implies .A ss B ss A By y h h> > 
 The level of human capital must be higher in Country A, suggesting that investment in human capital 
is higher in Country A as well. 
 
 
 
 
 
 
Chapter 7 
Measuring Productivity 
Note: Special icons in the margin identify problems requiring a computer or calculator . 
 Solutions to Problems 
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. 
28 Weil • Economic Growth, Second Edition 
 
2. The equation for output at year t: 
1 .t t t ty A k hα α−= 
 And the ratio of output between years t and t + 100 
1
100 100 100 100
1
.t t t t
t t t t
y A k h
y A k h
α α
α α
−
+ + + +
−
 
= 
 
 
 Rearranging the equation, using the given information and taking α = 1/3, we get 
100 8/2 4.t
t
A
A
+ = = 
 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: 
1 1/2 1/2
1 1/2 1/2
Factors (100) (64) (10)(8) 80
Factors (100) (25) (10)(5) 50.
F F F
S S S
k h
k h
α α
α α
−
−
= = = =
= = = =
 
 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 80 1.6.
Factors 50
F
S
= = 
 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 200 2.
Output 100
F
S
= = 
Chapter 7 Measuring Productivity 29 
 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 Output Factors .
Productivity Output Factors
F F F
S S 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. 
b. If all differences in output were due to differences in productivity, then, 
Factors Factors .F 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 Productivity .F 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. 
4. Physical capital per worker grew at the same rate in the two countries. This is clear because it rose 
27-fold over this period (this corresponds to an annual growth rate of 8.6% per year, but it was not 
necessary to calculate this). Similarly, human capital per worker grew at the same rate in the two 
countries. However, output grew faster in Z than in CountryX (in Country X, output rose 12-fold, 
while in Country Z it rose 24-fold). Therefore, productivity grew faster in Country Z. 
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: 
(1 )Factory Accumulation ,k hα α−= 
 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: 
(1 )Productivity /( ),y k hα α−= 
 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. 
 y k h 
1 2
3 3k h× A 
 
Country 
Output per 
Worker 
Physical Capital 
per Worker 
Human Capital 
per Worker 
Factor 
Accumulation 
 
Productivity 
Sweden 0.76 0.80 0.93 0.88 0.86 
Mauritius 0.55 0.32 0.70 0.54 1.02 
Jordan 0.18 0.13 0.86 0.46 0.39 
30 Weil • Economic Growth, Second Edition 
 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 ˆGrowth Rateof Factors (1 ) ,k h= + −α α 
 where the growth rate of physical capital per worker, ˆ,k and the growth rate of human capital per 
worker, ˆ,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 = ˆ ˆˆ ˆ= ( (1 ) ),A y k hα α− + − 
 where the second term is simply the previous growth rate of factor accumulation and yˆ 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. 
 ˆ(%)y ˆ(%)k ˆ(%)h ˆ ˆ(1/3) (2/3)k + h ˆ (%)A 
 
 
Country 
Growth Rate 
of Output per 
Worker 
Growth Rate of 
Physical Capital 
per Worker 
Growth Rate of 
Human Capital 
per Worker 
 
Growth Rate 
of Factors 
 
Growth Rate of 
Productivity 
Argentina −0.02 0.46 0.37 0.4 −0.42 
Austria 1.9 2.82 0.26 1.11 0.79 
Chile 1.54 0.85 0.63 0.70 0.84 
7. The effect of using data on school days increases the level of education for richer countries and lessens 
the level of education for poorer countries. This implies that the total return to education will be higher 
in richer countries than the total return to education based on years of schooling. Oppositely, the total 
return to education will be lower in poorer countries than the total return to education based on years 
of schooling. Not only does this ultimately increase the effect that human capital has on the output 
per worker of a country, the inclusion of days instead of years would more accurately capture the 
level of human capital in a country. As a result, there will be a stronger correlation between levels of 
human capital, as measured by the days of schooling, and output per worker. The stronger correlation 
ensures a greater emphasis on the factors of production by magnifying the variable h. This increase 
in the ability of factors of production to explain the variability of output per worker, ultimately, 
lowers the residual unexplained portion of the production function—that is, productivity A will fall. 
Hence the role of productivity in explaining variations in output per worker among countries will 
diminish. 
Chapter 7 Measuring Productivity 31 
Chapter 8 
The Role of Technology in Growth 
Note: Special icons in the margin identify problems requiring a computer or calculator . 
 Solutions to Problems 
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. 
2. The advantage to having patent laws in the creation of life-saving prescription drugs is that the resulting 
monopoly pricing will actually provide an incentive for companies to undertake that the research and 
development leading to the creation of the drug. On the other hand, monopoly pricing will cause the 
price to be higher than under competitive pricing. So the drugs may not be affordable to some people 
who need them. An alternative would be to charge differentially. For example, sell the drug for cheaper 
prices in developing countries. This way, poorer segments of society would have access to the drugs. 
However, this will reduce the profits of the drug companies. Also, a black market may develop, as 
people ship the drug from poor to rich countries. 
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, 
ˆˆ ,Ly A γµ= = 
 where γ denotes the fraction of workers devoted to R&D. With an increase in γ to ,γ ′ we know from 
the following equation that, 
γ <γ ′ implies ˆˆ Ly A γ
µ
= = < ˆˆ .Ly A γµ
′
′= = 
 Therefore, at the time of change, denoted as 1t in the graph below, the growth rate of output per 
worker and productivity, yˆ ′ and Αˆ′ 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, 
γ= −(1 )y A < (1 ).y A γ′= − ′ 
32 Weil • Economic Growth, Second Edition 
 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 inγ to γ ′ is temporary, at the time 2,t that γ ′ returns to the original fraction γ , 
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 
(1 ) (1 ) ( ) ( ),
(1 ) (1 ) ( ) ( ).
y y y A A A A
y y y A A A A
γ γ γ γ γ
γ γ γ γ γ
∆ = − ′= − − − ′ = ′− = ∆
′ ′ ′ ′∆ = ″ − ′= − − − ′ = ′− = ∆
1
2
:
:
At t
At t
 
 Because the change inγ 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 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 inγ will return the growth rate of productivity 
to its original level, without any jumps. The figures are given below. 
 
Chapter 7 Measuring Productivity 33 
4. The given parameters of the model are, L = 1, µ = 5, and γ = 0.5. To calculate the growth rate of 
output per worker, we substitute in these values into the following equation and solve to get: 
γ
µ
= = =
0.5ˆ 0.1.
5
Ly 
 The growth rate of output per worker is 10% per year. Similarly, with γ ′ = 0.75, we get: 
0.75ˆ 0.15.
5
Ly γ
µ
′
′= = = 
 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: 
γ= − = − =(1 ) (1 0.5) (0.5).y A A A 
 Similarly, we can find the new level of output per worker to be: 
(1 ) (1 0.75) (0.25).y A A Aγ′= − ′ = − = 
 Therefore, the original level of output per worker is A(0.5), and we need to find how long it will 
take to reach this level, starting from a level of A(0.25) with a growth rate of 15 percent. We use the 
standard growth equation, substitute and solve. 
(1 ) ,
(0.25)(1 0.15) (0.5).
t
t
y g y
A A
′ + =
+ =
 
 Dividing both sides by A(0.25) and taking logs, we get: 
t ln(1.15) = ln(2). 
 Solving for t, we get: 
= = ≈
ln(2) 4.96 5.
ln(1.15)
t 
 That is, it will take approximately five years for the level of output per worker to return to its 
previous level before the change inγ . 
34 Weil • Economic Growth, Second Edition 
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 one-country model. However, the 
speed of growth in productivity in Country 1 raises the steady state A1/A2 ratio (the second diagram). 
Consequently, µ ,c 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. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Chapter 7 Measuring Productivity 35 
6. We are given a scenario where the follower, Country 2, raises the fraction of the labor force devoted 
to R&D beyond the fraction in the leader, Country 1. The implication in the long run is for Country 2 
to become the technological leader. The process by which this occurs is illustrated in the diagrams 
of the growth rates of productivity and output per worker below. The initial rise in the fraction of 
workers devoted to R&D in Country 2 will raise the productivity of Country 2. In accordance, 
Country 2 will experience a drop in the level of output per worker but also achieve a faster rate of 
growth in output per worker. The timing of these events are denoted as t1. As productivity growth in 
Country 2 continues, the technological gap will lessen. That is A1/A2 will approach one, and the cost 
of copying will approach the cost of invention. Because the fraction devoted to R&D in Country 2 
exceeds that of Country 1, A2 will eventually equal A1. This point in time is denoted as t2. A2 will 
eventually overtake A1 and Country 2 will then become the technological leader and Country 1 will 
become the follower. Therefore, Country 1 is no longer faced with the cost of invention. Instead, 
Country 1 faces a declining cost of copying, thereby raising the growth rate of output per worker and 
the growth rate of productivity. The technology gap between both countries will increase until the 
steady-state value of A2/A1 is reached. This point in time is denoted as t3. At this time, both output per 
worker and productivity of Country 1 and Country 2 will be growing at an equal and constant rate. 
This is the final long run equilibrium. 
 
36 Weil • Economic Growth, Second Edition 
7. a. In the steady state, 1 2ˆ ˆ .A A= Therefore, 
γ γ
µ µ
=,1 ,2 .A A
i c
L L
 
 Rearranging and solving for µ ,c we get, 
γ
µ µ
γ
 
=   
 
,2
,1
.Ac i
A
 
 Setting the above steady-state condition equation to the specified cost-of-copying function, 
β
γ
µ µ µ
γ
−   
= =       
,2 1
,1 2
.Ac i i
A
A
A
 
 Rearranging, we find out solution to be: 
βγ
γ
  
=        
1
,11
2 ,2
.A
A
A
A
 
 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 β = 1. Because we assume 0 1,β< < 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,β = µ = 10,i γ =,1 0.2,A and γ =,2 0.1,A we can solve the previous equation 
to get: 
βγ
γ
    = = =          
1
2
,11
2 ,2
0.2 4.
0.1
A
A
A
A
 
 That is, the steady-state ratio of technology in Country 1 to technology in Country 2 is 4. 
 
Chapter 9 
The Cutting Edge of Technology 
Note: Special icons in the margin identify problems requiring a computer or calculator and those 
requiring calculus . 
 Solutions to Problems 
1. The annual growth rate of productivity is given by the following equation: 
ˆ ˆˆ .A y Lβ= + 
 We are given a value of 1/3 for β and 0 for ˆ,y leaving the growth rate of the population, ˆ,L as the only 
unknown. To solve for ˆ,L we use the standard growth equation with the initial population as 4 million 
and the final population after 10,000 years as 170 million. The equation is: 
10,000
(1/10,000)
ˆ4m.(1 ) 170m.
ˆ (170m./ 4m.) 1
0.000375.
L
L
+ =
= −
=
 
 Now we substitute to find our growth rate of productivity over this period: 
ˆ 0 (1/ 3)(0.000375) 0.000125.A = + = 
 That is, the growth rate of productivity over this period was roughly 0.0125 percent per year. 
2. For the relative prices, note that the high rate of productivity in the farming sector will create a fall in 
the real price of wheat; whereas, the lack of significant change in productivity in the hair-cutting sector 
will maintain the real price for haircuts. Thus, the fall in the price of wheat implies that the relative 
price of hair cuts has risen, and the relative price of wheat has, of course, fallen. 
 The improvement in technology for growing wheat raises the marginal productivity of labor for 
farmers, and because workers are always paid their marginal products, the real wage for farmers will 
rise, ceteris paribus. Similarly, the absence of technological improvements in cutting hair maintains 
the marginal productivity of barbers and the real wage for barbers does not change, ceteris paribus. 
However, because workers are free to move from the low marginal productivity hair cutting sector to 
the high marginal productivity farming sector, the equilibrium wage for both sets of workers will be 
equal. Ultimately, the real wages of both farmers and barbers will rise because of higher productivity 
in the farming sector (simply, more output is now produced). 
38 Weil • Economic Growth, Second Edition 
3. Technological progress in the production of a good will reduce the price of that good. Moreover, the 
reduction in price of the good will change the quantity demand for that good. The resulting change in 
the quantity demanded, or equivalently, the economy’s total spending on that good, is directly governed 
by the value of the price elasticity of demand. If the price elasticity of demand is large, a reduction in 
the price of a good will result in a larger increase in the demand for that good. Conversely, if the price 
elasticity of demand is small, a reduction in the price of a good will result in a smaller increase in the 
demand for that good. Thus, technological advance determines the change in the price, and the price 
elasticity of demand determines the subsequent demand change in response to the price change. 
4. a. In any given year, the production of bread must equal the production of cheese in this economy. 
That is, ,b cY Y= always. Knowing that the productivity of each good is equal at this point in 
time, we can solve for the quantity of labor devoted to each sector as follows. 
,
,
.
b c
b b c c
b c
Y Y
A L A L
L L
=
=
=
 
 Since ,b cL L L+ = 
/ 2.b cL L L= = 
 The labor force will be equally split between the two sectors. 
b. To calculate the growth rate of total output, we first calculate the growth rates of each sector by 
taking the natural log of both sides and differentiating with respect to time. For the bread sector: 
ln( ) ln( ) ln( ),
ln( ) ln( ) ln( ),
ˆˆ ˆ .
b b b
b b b
b b b
Y A L
d dY A L
dt dt
Y A L
= −
= −
= −
 
 Similarly, for the cheese sector, we get, 
ˆˆ ˆ .c c cY A L= − 
 We know the value for the growth rate of productivity in both sectors. Furthermore, in our answer 
to Part (a), we found that labor is currently equally divided among the two sectors. Thus, the 
growth of labor in one sector must be offset by the growth of labor in the other— ˆ ˆ( ).b cL L= − 
We substitute in these values and get, 
ˆ ˆ2% .b bY L= − 
 And, 
ˆ ˆ ˆ1% 1% .c c bY L L= − = + 
 Setting the growth rate of output in the bread sector to the growth rate of output in the cheese 
sector, we find that ˆ 0.5%bL = and ˆ ( 0.5%).cL = − So, ˆ ˆ ˆ 1.5%.b cY Y Y= = = 
Chapter 9 The Cutting Edge of Technology 39 
c. The figure below depicts the growth rate of output over time. From Part (b), we know that the 
growth rate of output is equal to 1.5 percent. But, productivity in the bread sector rises by a greater 
percent than in the cheese sector. Bread production would rise faster than cheese production and 
because one piece of bread is consumed with one piece of cheese, labor resources are continually 
shifted into the cheese sector. Over time, nearly the whole of the economy’s resources will be 
shifted to the cheese sector with a minimal amount devoted to the high productivity bread sector. 
Therefore, growth will be limited by the growth and productivity of the cheese sector. As the 
figure shows, the economy’s growth rate nears 1 percent, the growth rate of productivity in the 
cheese sector. 
40 Weil • Economic Growth, Second Edition 
 
5. There is no one correct answer to this question. However, be sure to explain your reasoning for the 
future pace of technological progress using as many examples from the text as necessary. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Chapter 9 The Cutting Edge of Technology 41 
Chapter 10 
Efficiency 
Note: Special icons in the margin identify problems requiring a computer or calculator and those 
requiring calculus . 
 Solutions to Problems 
1. Denoting each country by the appropriate subscript, we are given that, 
2X
Z
A
A
= and 4.X
Z
T
T
= 
 Since, productivity is defined as T × E, we can divide the productivity level of Country X by the 
productivity level of Country Z to obtain the ratio of efficiency in both countries. 
,
2 4.
X X X
Z Z Z
X
Z
A E T
A E T
E
E
= ×
= ×
 
 Therefore, 1 .
2
X
Z
E
E
= The level of efficiency in Country X is half the level of efficiency in Country 
Z. 
2. The level of productivity in Country X, relative to the United States is given as 
U.S.
1 .
2
XA
A
= 
 The level of technology in Country X, relative to the United States is, 
( )
U.S.
(1 ) ,X GT g
T
−= + 
 where g is the growth rate of technological progress and G is the number of years that Country X 
lags behind the United States. Given values of g = 1% and G = 20, we substitute and solve for the 
ratio of technology differences. 
( 20)
U.S.
(1 0.01) 0.82.XT
T
−= + = 
42 Weil • Economic Growth, Second Edition 
 Since, productivity is defined as T × E, we can divide the productivity level of Country X by the 
productivity level of the United States to obtain the ratio of efficiency in both countries. 
U.S. U.S. U.S.
U.S.
,
0.5 0.82.
X X X
X
A E T
A E T
E
E
= ×
= ×
 
 Solving the previous equation yields a value of 0.61 for the ratio of efficiency in Country X 
relative to the United States. Stated differently, Country X has 61% of the efficiency of the United 
States. 
3. Assuming that India United States India United States/ 1, / 0.35,E E A A= = and that the growth rate of productivity is 
0.66%, we write the functional productivity equation of each country in ratio form and substitute in 
the necessary values to obtain: 
U.S. U.S. U.S.
( ) ( )
U.S.
,
0.35 1 1 (1 ) (1.0066) .
I I I
I G G
A E T
A E T
T g
T
− −
= ×
= × = × + =
 
 Hence, we have an equation that we can solve for to find the magnitude of the technology gap in 
years between India and the United States. Taking the log of both sides and rearranging to solve 
yields, 
( )0.35 (1.0066)
In(0.35) ( ) In (1.0066),
ln(0.35) 159.59.
ln(1.0066)
G
G
G
−=
= −
−
= =
 
 The level of technology in India is, thus, approximately 160 years behind that of the United States. 
4. Brief examples of real-world inefficiencies for each category are given below. 
 Unproductive Activities: In addition to theft and political lobby activities, wars are generally extremely 
unproductive. Other examples are black marketactivities such as drug use, drug dealing, and so forth. 
 Idle Resources: Many economic fluctuations reduce efficiency and make people and capital lay 
unused. Recessions, currency crises, and labor strikes are instances that result in idle resources. 
 Misallocation of Factors among Sectors: Often, labor mobility is impeded across borders because 
work permits, licenses and credentials do not extend universally. Marginal products will not equalize, 
resulting in the misallocation of factors among sectors. The minimum wage is another example that 
results in the misallocation of factors among sectors. 
Chapter 9 The Cutting Edge of Technology 43 
 Misallocation of Factors among Firms: Many government subsidized or financed industries throughout 
the world regularly exhibit this form of inefficiency. In addition, certain industries such as the airline 
industry, exhibit oligopolistic behavior, preventing more efficient firms from entering. 
 Technology Blocking: The recent ban on stem cell research in the United States is a form of technology 
blocking. Many examples are present today and even throughout history. 
5. A correct answer to the problem will include specific examples within a country. In addition, be 
certain to categorize each type of inefficiency. 
6. The figure below depicts an inefficient allocation of labor between the urban and rural sector. An 
efficient allocation is given by Point A, the point where the marginal productivity of labor equalizes 
in both sectors. However, with a minimum wage of the urban sector set above the optimal wage 
(where the marginal products are equal), the economy rests at Point B. At this point, the marginal 
productivity of urban workers is higher than that of rural workers. The urban sector would ideally 
employ workers from the low productivity rural sector but the marginal product of an additional 
worker would be lower than the wage rate, as the additional worker must be paid the minimum wage. 
Consequently, the minimum wage restricts movement of workers from the rural sector to the urban 
sector, and so the marginal productivity of workers do not equalize. Instead, the rural sector employs 
a greater number of workers at a lower marginal productivity and wage than that found in the urban 
sector. The net loss in output from this inefficient allocation is depicted by the area of the shaded 
portion. 
 
7. The marginal product of Sector 1 is found by differentiating the production function of Sector 1. 
That is, 
Marginal Product = 
1
12
1 1 2
1 1
1 1
1 wage .
2
dY dL L
dL dL
 − 
 = = = 
 The average product of Sector 2 is found by dividing total output of Sector 2 by the amount of 
workers. Simply, 
Average Product = 
1
12
2 2 2
2 2
2 2
wage .Y L L
L L
 − 
 = = = 
44 Weil • Economic Growth, Second Edition 
 In equilibrium, the wage of both sectors must be equal, as workers are free to move between 
sectors. Therefore, we set 1 2wage wage= to solve for the number of workers in each sector. 
1 1
2 2
2 2 1 2
2 1
1wage wage .
2
4 .
L L
L L
   − −   
   = = =
=
 
 The number of workers in Sector 2 will be 4 times greater than the number of workers in Sector 1. 
 The figure below depicts, graphically, the scenario above with the assumption that L = 10. As predicted, 
the intersection of wage curves intersect where two workers are employed in Sector 1, where marginal 
products are paid, and four times that number of workers (eight workers) are employed in Sector 2, 
where average products are paid. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Chapter 9 The Cutting Edge of Technology 45 
Chapter 11 
Growth in the Open Economy 
 Solutions to Problems 
1. Wages of a U.S. citizen, earned in France, will be counted towards U.S. GNP but not in U.S. GDP. 
However, the wages will be counted towards France GDP and not in France GNP. 
2. In a closed economy, the usual mechanism of the Solow model operates: higher saving means more 
capital, higher GDP per worker, and higher wages. In an open economy, the level of capital is 
determined by the world interest rate. Thus, higher saving will not affect the level of capital in a small 
open economy, and will only raise the level of capital in a large open economy to the extent that the 
world interest rate is reduced. (Of course, higher saving will raise GNP in a small open economy by 
raising income from capital held abroad.) Whether being in an open or closed economy is better for 
an individual who saves a lot is not clear. If the economy as a whole has low savings, then the high-
saving individual is better off with the economy being closed; that way, he can earn a high return 
on his savings. If the economy has high savings, then the high-saving individual is better off with it 
being open. 
3. In the closed economy Solow model, an increase in the growth rate of the population will reduce the 
steady-state level of income per capita by reducing the steady-state level of capital per capita. To see 
this, recall that the accumulation of capital is positively related to the level of savings and negatively 
related to the growth rate of the population and depreciation. As a result, population growth will 
lower the income per capita of the country. 
 In an open economy, the change in the growth rate of the population does not affect the world rate of 
return, wr nor the steady-state level of capital within the country. The steady-state level of capital is 
given as, 
1
1
.ss
w
Ak
r
αα − 
=  
 
 
 Therefore, the growth rate of the population, n, does not enter into the determination of the steady 
state. There will be no change to GDP, although for GNP, the increase in n will lower the domestic 
level of capital per capita. The marginal product of capital will rise, inducing investment from abroad. 
The level of capital per capita will rise until the marginal product of capital falls to meet the world 
rental rate, .wr Thus, GNP falls. 
46 Weil • Economic Growth, Second Edition 
4. In an economy perfectly open to the world capital market, the steady-state level of output per capita is 
given by the following equation as reproduced from the chapter. 
1 1
1 .ss
w
y A
r
α
α
α
α −
−
 
=  
 
 
 Given that the world rental price of capital doubles to ,wr′ which is exactly 2 ,wr we can write the new 
steady-state level of output per worker in terms of the previous rental rate. 
1 1 11 1 1 1 1
1 1 1
1 1 .
2 2 2ss ssw w w
y A A A y
r r r
α α α α α
α α α α α
α α α
α α α− − − − −
− − −
         ′ = = = =         ′         
 
 And with a value of 1/2 for α, 
1(1/ 2) 1
21 1
2 2ss ss ss
y y y
  −  
     ′ = =   
   
 
 That is, the new steady-state level of income per capita falls by half when the world rental rate of 
capital doubles in an economy open to the world capital market. 
5. In a closed economy where one slice of cheese is consumed with one slice of bread, we can write a 
production function for each sector as 
c cY L= -and- (1/ 2) ,b bY L= 
 where bY and cY is the production of bread and cheese, respectively, with bL and cL as the number of 
labor hours devoted to each sector. In equilibrium, bY will equal .cY Therefore, 
,
(1/ 2) .
c b
c b
Y Y
L L
=
=
 
 We know that the total number of labor hours, ,c bL L+ available is 60. Combining this fact with the 
previous equation, we find that, 
(1/ 2) (3 / 2) 60,
40,
c b b b b
b
L L L L L
L
+ = + = =
=
 
 and, 
40 60,
20.
c b c
c
L L L
L
+ = + =
=
 
 As for output in each sector, 
20c cY L= = -and- (1/ 2) (1/ 2)(40) 20.b bYL= = = 
 Therefore, the allocation of labor hours towards the bread sector will be 40, resulting in an output of 
20 slices of bread, and the allocation of labor hours towards the cheese sector will be 20, resulting in 
an output of 20 slices of cheese. 
Chapter 9 The Cutting Edge of Technology 47 
 With the economy opened up to trade where the price of bread equals the price of cheese on the world 
market, the economy is strictly better off when they specialize in the good that they have a comparative 
advantage in and trade for the other. Because the economy can produce two slices of cheese for every 
slice of bread produced, the economy can specialize in cheese production. Sixty labor hours devoted 
to cheese production would lead to 60 slices of cheese and a dormant bread sector. In turn, 30 of 
those slices of cheese can be traded for 30 slices of bread on the world market. The economy can 
now consume 30 slices of cheese and 30 slices of bread, a consumption level strictly greater 
than in the closed economy scenario. The total amount of labor hours does not increase, but the 
allocation is 60cL = and 0.bL = 
6. If there is only one auto factory in each country under autarky, then each auto factory would have a 
local monopoly, and therefore there will be no competition driving the auto factory to become 
efficient. Upon the opening of trade, all the auto factories will begin competing with each other, and 
therefore will have an incentive to lower costs and become as efficient as possible. Cars will become 
relatively less expensive since they now cost less to produce and they are sold in a competitive 
market. 
 The pizza industry in each country, however, already was competitive because there were no size 
barriers to entry. Therefore, trade in pizza is likely to be unchanged, since the pizza industry already 
had high efficiency due to tight competition. 
 Whether the decline in auto prices leads factors to flow into or out of the auto industry depends on the 
price elasticity of demand for autos. If demand is inelastic, then as prices fall due to more efficient 
auto production, factors will flow out of the industry and the pizza industry will grow in size. 
 It is interesting to note that although each country was identical in terms of efficiency and technology 
under autarky, and trade did not affect the technology level in any county, the overall level of efficiency 
rose due to more competition. 
7. A tariff on the import of widgets is a form of trade protection, insulating the domestic widget industry 
from international competition and pressure. Although the result is a higher wage for those members 
of the widget industry, the higher wage comes at a cost to consumers of widgets and the economy in 
general. Trade liberalization can result in greater technological progress and higher levels of efficiency 
through competitive pressure. Consequently, the suggestion to raise the level of import tariffs on all 
industries, based on the findings of the study, does not incorporate the drawbacks to trade protection 
policy, and is harmful for the economy as a whole in the long run. 
 
 
Chapter 12 
Government 
Note: Special icons in the margin identify problems requiring calculus . 
 Solutions to Problems 
1. a. Standardization of the length of axles on carts is a form of a public good. Everyone benefits from 
improvements in travel, and improvements in travel are beneficial for growth. 
b. A stable currency is a public good. 
c. The creation of a national car in Indonesia, the Timor, is an instance of government failure. 
Competition is stunted by providing monopoly power to the Suharto company through protection 
policies and as a result, incorrectly aligning incentives. Ultimately, productivity falls and only the 
Suharto family benefits. 
d. Failure to get vaccinated imposes a negative externality, since a sick person is likely to spread 
disease to others. 
e. If one assumes that mail delivery services is a natural monopoly, as it may be inefficient for 
multiple firms to be able to route to all houses, government regulation can address the market 
failure present in monopolies. By operating mail delivery services, the government can insure 
that an inefficiently high price for mail delivery is not charged. However, this alone does not 
explain why the government has to forbid private competitors. The reason for this is that mail 
delivery often involves a cross-subsidy: In the United States, for example, a first-class letter 
traveling two blocks in Manhattan costs the same as a first-class letter going from rural Montana 
to rural Maine. The former subsidizes the latter. A private mail service could “skim the cream” 
by charging a lower price for short-distance or high-volume mail deliveries, thus unraveling the 
cross-subsidy implicit in national mail. 
f. The answer is unclear. On one hand, the imposition of a minimum wage can be an example of a 
government failure. The minimum wage can result in the misallocation of factors among firms 
and sectors, and in this case, it serves those who do receive the minimum wage. Growth may be 
hindered. On the other hand, the minimum wage may satisfy normative goals of government, that 
being equality and general well-being. In this case, the minimum wage is intended to serve 
everyone by eliminating low wage abuses by firms. Growth may be positive. 
g. The failure of African governments to maintain roadways is a government failure. The case for 
government intervention in the provision of public goods such as roadways is economically 
valid, as the private market cannot optimally provide the appropriate quantity of a public good. 
The result is that everyone is worse off and these failures are detrimental to growth. 
h. One justification for this policy might be externalities. Highly educated people might have positive 
externalities for the rest of the country (by importing new technologies, improving the quality of 
government, etc.). On the other hand, this may be an example of income redistribution from the 
poor to the middle class and wealthy, since they are the groups most likely to send their children 
to college. 
48 Weil • Economic Growth, Second Edition 
2. In the first figure below, we represent two countries with varying quality levels of government and 
varying incomes. In this figure, the horizontal axis measures output and the vertical axis measures 
the quality of government. A country with a low quality of government will be positioned vertically 
below a country with a high quality of government. As such, Country A has both more income and a 
better quality government than does Country B. 
 In the second figure, we graphically depict each channel of causation. Both curves are positively 
sloped, as both income and quality of government are positively correlated. However, the channel of 
income affecting the quality of government, termed the G(y) curve, is “curvier”. Intuitively, as income 
within a country rises, the effects of additional income will increase the quality of government but it 
will not increase the quality of government proportionally. That is, the effect of income on determining 
government quality is diminishing. The y(G) curve is, instead, more positively sloped. As government 
quality increases, the effects on output increases are greater than income’s effect on determining 
government quality. We use this framework to illustrate the government view and the income view. 
 The third figure illustrates the government view for the differences between Country A and B. The 
government view assumes that Country A and Country B have quality differences in governments for 
reasons other than income. For this reason, the ( )AG y curve that represents the quality of government 
for any given level of income, is positioned above the ( ).BG yAt any given level of income, Country A 
will have a higher quality of government than Country B. Also, you will notice that both countries 
share the same y(G) curve. Government quality affects income in the same manner in both countries. 
Therefore, the root cause of the differences in Country A and B is explained through root differences 
in the government “environment.” 
 The fourth and final figure illustrates the income view as the primary source for differences between 
Country A and B. Here, both countries share the same G(y) curve. That is, income affects the quality 
of government in the same manner. However, the differences are a result of fundamental income 
differences that are not affected by the quality of government. Graphically, the ( )Ay G curve is 
positioned to the right of the ( )By G curve. In this view, for any given level of government, Country A 
will have higher income than Country B, and it is precisely for this reason we see the differences in 
income and government quality. 
 
Chapter 12 Government 49 
 
3. The economist understands that the only way to determine whether more income causes better 
government or whether better government causes more income is to find an event in which we know 
that one determinant changed, and which cannot plausibly have caused a direct change to the other, 
and then see if the other determinant changed. In statistics, this is known as having a valid instrument. 
 In this example, the change in the prices of the different beans is obviously going to have an effect on 
income in the two countries. In this case, we know that the income change was not due to government 
quality. Therefore, if government quality rises due to this increase in income, we know that income 
causally affects government quality. It still could be the case that better government quality also affects 
income, but at least we have determined causality for the converse. 
 If the researcher only looked at the simple correlation between quality of government and income, he 
or she would not be able to determine causality. Also, it there is dual causation (causality running in 
both directions), then the magnitude of the correlation could be much different than simple correlation 
due to multiplier effects, for example. 
4. a. Denoting dQ as the quantity demanded and sQ as the quantity supplied, at the market-clearing 
price or equilibrium price of this model, the quantity supplied must equal the quantity demanded 
for a good. That is, 
,
100
50.
d sQ Q
P P
P
=
− =
=
 
 Therefore, for markets to clear, the equilibrium price in the absence of a tax is 50 for the good. 
b. At a tax rate ofτ for each good, the quantity demanded for any given price does not change 
because the price paid by the consumer remains unaffected. However, the quantity supplied for 
any given price decreases to a factor of (1 ),τ− as government collects from the supplier. 
Specifically, 
100 and (1 ) .d sQ P Q Pτ= − = − 
50 Weil • Economic Growth, Second Edition 
 In equilibrium, the price must be set such that the quantity demand meets the quantity supplied. 
Setting ,d sQ Q= and solving for P, we get: 
eq
100 (1 )
100 (1 1)
100 /(2 ).
P P
P
P
τ
τ
τ
− = −
= − +
= −
 
 The equilibrium price is the value given above. To find the equilibrium quantity for the price, 
we substitute and get, 
eq eq
100(1 )(1 ) .
(2 )
Q P ττ
τ
−
= − =
−
 
 These are the market-clearing values in the presence of a tax. 
c. In order to solve for the tax rate that will maximize government revenue, we first must write the 
revenue function as a function of the tax rate. Government revenue for each good is eq .Pτ Since 
the amount of the good sold is eq ,Q we know that government revenue is equal to eq eq .P Qτ × 
In Part (b), we solved for the equilibrium quantity and price as a function of the tax rate. We 
substitute in these values to finalize our first step. This yields the following equation that we then 
differentiate with respect to the tax rate to arrive at our solution. (With this given functional form, 
the tax rate at which the derivative of the function is equal to zero maximizes the revenue function.) 
2
eq eq 2
(1 )100Max Max .
(2 )
Q P
τ τ
τ τ
τ
τ
−
=
−
 
 Recall that 
2
2 2 2
2
2 4
( ) ( ) ( ) ( ) ( ) .
( ) ( )
(1 )100 (1 2 )(2 ) ( )( 2)(2 )(100) .
(2 ) (2 )
d f x f x g x f x g x
dx g x g x
d
d
τ τ τ τ τ τ τ
τ τ τ
′ ′−
=
− − − − − − − 
=  − − 
 
 Setting the above expression equal to zero, rearranging, and dividing out common terms, 
2(1 2 )(2 ) ( 2)( ) .τ τ τ τ− − = − − 
 Therefore, solving the above equation gives us: (2 / 3).τ = At this tax rate, government revenue 
will be maximized. 
5. a. In the steady-state level of output per worker, the quantities of government capital per worker 
and physical capital worker will not change over time. Therefore, 
0 , and
0 (1 ) .
x Ak x x
k Ak x k
α β
α β
τ δ
γ τ δ
∆ = = −
∆ = = − −
 
Chapter 12 Government 51 
 Using the values, (1/ 3),α β= = we now have two equations with two unknowns. Working 
through the algebra and solving for the steady-state values of ssx and ,ssk we get: 
3 2
3
3 2 2
3
(1 ) , and
(1 )
ss
ss
Ax
Ak
γ τ τ
δ
γ τ τ
δ
−
=
−
=
 
 Plugging in our steady-state values into the production function will now yield the steady-state 
level of output per worker. 
1 1
1 1 3 2 2 3 2 33 3
3 3
3 3 2
(1 ) (1 ) ( )(1 ).ss
A A Ay Ak x A γ τ τ γ τ τ γ τ τ
δ δ δ
− −   = = = −      
 
b. The value of τ that will maximize output per worker is the same value that will maximize 
( )(1 ),τ τ− as 3 2/A γ δ is a constant. Therefore, (1/ 2),τ = a tax rate of 50 percent. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
52 Weil • Economic Growth, Second Edition 
Chapter 13 
Income Inequality 
 Solutions to Problems 
1. a. The Lorenz curve for the economy is drawn below. The data for the curve are as follows. Total 
wealth in the economy is ($1)(5) + ($3)(5) = $20. The poorest 10 percent of the people own 
$1/$20, or 5 percent of the wealth. The poorest 20 percent own 10 percent and so forth until the 
poorest 50 percent. The poorest 60 percent own ($1)(5) + ($3)(1) = $8 dollars. That is, 8/20 or 
40 percent of the wealth. The poorest 70 percent own 55 percent; poorest 80 percent own 
70 percent; poorest 90 percent own 85 percent; and finally the entire economy owns 100 percent 
of total wealth. 
Lorenz Curve
0%
10%
20%
30%
40%
50%
60%
70%
80%
90%
100%
0% 10% 20% 30% 40% 50% 60% 70% 80% 90% 100%
Cumulative percentage of households
C
um
ul
at
iv
e 
pe
rc
en
ta
ge
 o
f h
ou
se
ho
ld
 in
co
m
e Line of Perfect Equality
A
B
C
D
Lorenz Curve
 
Chapter 12 Government 53 
b. The Gini is constructed by dividing the area between the line of perfect equality and the Lorenz 
curve by the entire area under the line of perfect equality. In our case, 
Gini Coefficient A
A B C D
=
+ + +
 
c. To calculate the Gini, we must find the Area of A, B, C and D. For the Area of B, C, and D, we 
apply the area formula for triangles. 
Area of B = (0.5)(0.25)(0.5) = 0.0625 
Area of C = (0.5)(0.75)(0.5) = 0.1875 
 Area of D = (0.5)(0.25) = 0.125 
 In order to find the Area of A, we first calculate the area under the line of perfect equality. This is 
simply a 1 by 1 right triangle implying that the area is 0.5. Following, we can subtract from this 
the area under the Lorenz curve to find the Area of A. Since the area under the Lorenz curve is 
B + C + D = 0.0625 + 0.1875 + 0.125 = 0.375, we find the Area of A to be,Area of A = (Area Under Line of Perfect Equality) − (Area Under Lorenz Curve) 
 = (A + B + C + D) − (B + C + D) = (0.5) − (0.375) = 0.125. 
 Now we substitute in these values into our equation from Part (b). 
0.125Gini Coefficient 0.25.
0.5
A
A B C D
= = =
+ + +
 
2. The perfection of “distance learning” serves to create a superstar effect for professors, thereby raising 
the level of inequality among professors. Because a single professor will be able to teach many 
students at any given time, only a few qualified professors will be needed, and those professors will 
earn a high wage. 
3. In the text, we see that higher inequality is good for growth via the channel of physical capital 
accumulation but bad for growth via the channel of human capital accumulation. Availability of 
student loans implies that at any given level of inequality, the accumulation of human capital will 
be higher. So the cost of inequality in terms of reducing human capital accumulation is mitigated. 
Thus the growth-maximizing level of inequality will be higher in the presence of student loans than 
in their absence. 
4. The more economic mobility a poor person perceives that there is, the less he or she will be in favor 
of redistributive taxation. Comparing two countries with the same level of inequality, redistributive 
taxation would be lower in a country with higher mobility. 
5. In the table, the probability that a mother in the middle-earning third will also have a daughter in the 
middle earning third is given by the exact center cell, and is 0.5. When this process is iterated for two 
generations, of the 50 percent daughters who are in the middle, 50 percent of their daughters will also 
be in the middle third. Therefore, 0.5*0.5 = 25 percent of all grandchildren will have both mothers 
and grandmothers who were middle class. 
54 Weil • Economic Growth, Second Edition 
 However, it could be the case that the grandmother was middle class, but her daughter was upper or 
lower class, and that her daughter was again middle class. So we need to do this calculation for three 
sets of people, and then sum the probabilities: 
a. P(Middle third grandmother has a bottom third daughter)*P(Bottom third mother has a middle 
third daughter) = 0.25*0.25 = 0.0625 
b. P(Middle third grandmother has a middle third daughter)*P(Middle third mother has a middle 
third daughter) = 0.5*0.5 = 0.25 
c. P(Middle third grandmother has an upper third daughter)*P(Upper third mother has a middle 
third daughter) = 0.25*0.25 = 0.0625 
0.0625 + 0.0625 + 0.25 = 0.375 or 37.5 percent. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Chapter 12 Government 55 
Chapter 14 
Culture 
 Solutions to Problems 
1. Many countries, regions, ethnic groups, organizations, and other existing groups can be analyzed to 
provide an acceptable answer to this question. As such, this solution is only an example. The following 
is taken from the culture of a professional sports team. 
a. Teamwork and trust are cultural attributes that are necessary to the success of a professional 
sports team. In professional American football, a play of the game is only successful if each 
member of the team is successful in accomplishing their respective tasks of blocking, rushing, 
passing, receiving, tackling, and so on. Each member of the team is dependent on this interplay, 
and trusts that each individual will accomplish their task. In this manner, the team is able to 
achieve a win. 
b. Unhealthy competition and “superstar” dynamics (not being a team player) are cultural attributes 
that prevent a team from being successful. Professional sports are characterized by a highly 
competitive environment, and often times, the level of competition is too extreme, breeding 
unhealthy competition among teams, coaches, and athletes. Also, self-interested superstars can 
arise. Unhealthy competition may breed distrust. Coupled with “superstar” dynamics, the important 
concepts of teamwork and trust are compromised. As a result, the success of the team is at risk. 
c. General discipline is an example of a cultural attitude that has been intentionally instilled in the 
members of a sports team. For instance, in baseball, it is discipline at the plate, discipline in 
attending practice, discipline in maintaining one’s health, and so forth. This attitude is instilled 
by the coaches in meetings, practice, and workouts. 
2. A residual for social capability is the difference between the actual value of social capability and the 
fitted value. The fitted value is obtained from a fitted line that gives a value of social capability for 
every value of GDP per capita. In Figure 14.3, Mexico and Greece have approximately the same level 
of GDP per capita. Thus, Mexico and Greece will have the same fitted value. However, Greece’s social 
capability index is greater than Mexico’s social capability index. This implies that the residual for 
Greece will be larger than that for Mexico. Equivalently, one may draw a best fit line and compute 
the residual, yielding the same result. 
3. In Figure 14.3, the two countries have roughly the same level of social capability. However, Thailand 
is significantly poorer. Since we would expect countries with the same levels of social capability to 
end up with similar incomes in the long run, we would expect Thailand to grow faster. 
4. A country’s index of ethnic fractionalization is, 
2
1
1 ,
I
i
i
n
=
− ∑ 
56 Weil • Economic Growth, Second Edition 
 where in is the fraction of the population belonging to group i. Given that there are three ethnic 
populations with respective fractions of 0.5, 0.25, and 0.25, we substitute and solve. 
1 − [(0.5)2 + (0.25)2 + (0.25)2] = 1 − [0.375] = 0.625. 
 That is, the index of ethnic fractionalization is 0.625. 
5. In the figure below, we begin at Point A and move to Point B after a rightward shift in the Y(M ) 
curve. From Point B, the final equilibrium position will be either Point C or Point D. If the economy 
is described by a relatively flatter MC(Y ) curve, the multiplier effect on the modernization of culture 
will be less pronounced, as changes in culture are less sensitive to changes in income. The economy 
will settle at Point C. On the other hand, if the economy is described by a steeper MD(Y ) curve, the 
multiplier effect on modernization of culture will be greater, as the change in culture is more sensitive 
to a change in income. Hence, for any given shift in the Y(M ) curve, a steeper M(Y ) curve will result 
in both a higher level of income per capita and a higher level of modernization of culture than in a 
flatter M(Y ) curve. 
 
6. The starting point of the economy is in an equilibrium position, Point A in the figure below. With an 
exogenous increase in the modernization of culture, the M(Y ) curve shifts upward. For any given 
level of income, the level of modernization of culture is higher than the previous level as a result of 
the exogenous shock. However, the economy’s level of income has not changed. This point is labeled 
Point B. Over time, the level of income will adjust to the new level of modernization, and the economy 
will move along the once-shifted M(Y ) curve to Point C, the long run position of the economy. 
 
Chapter 12 Government 57 
7. This is an exercise to be done by the student. Results will vary. 
8. Again, this is an exercise to be done by the student. Results will vary. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
58 Weil • Economic Growth, Second Edition 
Chapter 14 
Culture 
 Solutions to Problems 
1. Many countries, regions, ethnic groups, organizations, and other existing groups can be analyzed toprovide an acceptable answer to this question. As such, this solution is only an example. The following 
is taken from the culture of a professional sports team. 
a. Teamwork and trust are cultural attributes that are necessary to the success of a professional 
sports team. In professional American football, a play of the game is only successful if each 
member of the team is successful in accomplishing their respective tasks of blocking, rushing, 
passing, receiving, tackling, and so on. Each member of the team is dependent on this interplay, 
and trusts that each individual will accomplish their task. In this manner, the team is able to 
achieve a win. 
b. Unhealthy competition and “superstar” dynamics (not being a team player) are cultural attributes 
that prevent a team from being successful. Professional sports are characterized by a highly 
competitive environment, and often times, the level of competition is too extreme, breeding 
unhealthy competition among teams, coaches, and athletes. Also, self-interested superstars can 
arise. Unhealthy competition may breed distrust. Coupled with “superstar” dynamics, the important 
concepts of teamwork and trust are compromised. As a result, the success of the team is at risk. 
c. General discipline is an example of a cultural attitude that has been intentionally instilled in the 
members of a sports team. For instance, in baseball, it is discipline at the plate, discipline in 
attending practice, discipline in maintaining one’s health, and so forth. This attitude is instilled 
by the coaches in meetings, practice, and workouts. 
2. A residual for social capability is the difference between the actual value of social capability and the 
fitted value. The fitted value is obtained from a fitted line that gives a value of social capability for 
every value of GDP per capita. In Figure 14.3, Mexico and Greece have approximately the same level 
of GDP per capita. Thus, Mexico and Greece will have the same fitted value. However, Greece’s social 
capability index is greater than Mexico’s social capability index. This implies that the residual for 
Greece will be larger than that for Mexico. Equivalently, one may draw a best fit line and compute 
the residual, yielding the same result. 
3. In Figure 14.3, the two countries have roughly the same level of social capability. However, Thailand 
is significantly poorer. Since we would expect countries with the same levels of social capability to 
end up with similar incomes in the long run, we would expect Thailand to grow faster. 
4. A country’s index of ethnic fractionalization is, 
2
1
1 ,
I
i
i
n
=
− ∑ 
Chapter 12 Government 59 
 where in is the fraction of the population belonging to group i. Given that there are three ethnic 
populations with respective fractions of 0.5, 0.25, and 0.25, we substitute and solve. 
1 − [(0.5)2 + (0.25)2 + (0.25)2] = 1 − [0.375] = 0.625. 
 That is, the index of ethnic fractionalization is 0.625. 
5. In the figure below, we begin at Point A and move to Point B after a rightward shift in the Y(M ) 
curve. From Point B, the final equilibrium position will be either Point C or Point D. If the economy 
is described by a relatively flatter MC(Y ) curve, the multiplier effect on the modernization of culture 
will be less pronounced, as changes in culture are less sensitive to changes in income. The economy 
will settle at Point C. On the other hand, if the economy is described by a steeper MD(Y ) curve, the 
multiplier effect on modernization of culture will be greater, as the change in culture is more sensitive 
to a change in income. Hence, for any given shift in the Y(M ) curve, a steeper M(Y ) curve will result 
in both a higher level of income per capita and a higher level of modernization of culture than in a 
flatter M(Y ) curve. 
 
6. The starting point of the economy is in an equilibrium position, Point A in the figure below. With an 
exogenous increase in the modernization of culture, the M(Y ) curve shifts upward. For any given 
level of income, the level of modernization of culture is higher than the previous level as a result of 
the exogenous shock. However, the economy’s level of income has not changed. This point is labeled 
Point B. Over time, the level of income will adjust to the new level of modernization, and the economy 
will move along the once-shifted M(Y ) curve to Point C, the long run position of the economy. 
 
60 Weil • Economic Growth, Second Edition 
7. This is an exercise to be done by the student. Results will vary. 
8. Again, this is an exercise to be done by the student. Results will vary. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Chapter 12 Government 61 
Chapter 16 
Resources and the Environment 
at the Global Level 
Note: Special icons in the margin identify problems requiring a computer or calculator and those 
requiring calculus . 
 Solutions to Problems 
1. To find the annual growth rate of energy intensity of output over this period, we utilize the following 
equation, where a hat over the variable denotes its growth rate. 
ˆ ˆ ˆˆ ,I R y L= − − 
 where I is the energy intensity of output, R is energy consumption, y is GDP per capita, and L is the 
population. Thus, we must derive the right-hand side values from the table. Upon inspection of the 
table over the 35 year period, each variable has exactly doubled. Using the rule of 72, the growth 
rate of each variable is 72/35 or equivalently, two percent per year. That is, 
ˆ ˆˆ 2%.R y L= = = 
 Therefore, 
ˆ 2% 2% 2% 2%.I = − − = − 
 The energy intensity fell by two percent per year over this period. 
2. a. Because the stock of fish tS at time t is 20, we use the equation for the growth in the quantity of 
fish to determine .tG 
(100 ) 20 (100 20) 16.
100 100
t t
t
S SG × − × −= = = 
 Additionally, the stock of fish and the size of the harvest have been constant for a long period of 
time. Mathematically, this implies that, 
1 0.t t t t tS S S G H+∆ = − = − = 
 Thus, tG must equal ,tH implying that 16.tH = 
62 Weil • Economic Growth, Second Edition 
b. To find the optimal stock of fish in the lake, we differentiate tG with respect to .tS 
(100 ) 100 2 1 .
100 100 50
t t t t t
t t
dG S S S Sd
dS dS
× − −
= = = − 
 Setting the above equation to 0 and solving for ,tS we get 50.tS = The optimal stock of fish that 
maximizes the growth in the stock of fish will be 50. 
 At this stock level of fish, we repeat the analysis in Part (a) to arrive at the maximum sustainable 
yield. 
(100 ) 50 (100 50) 25.
100 100
t t
t
S SG × − × −= = = 
1 0.t t t t tS S S G H+∆ = − = − = 
 Thus, tG must equal ,tH implying that 25.tH = 
3. Current total energy use is calculated by summing over all country groups, the product of the population 
and energy use per capita. The results are shown in the “Current Energy Use” column in millions. 
The total energy use calculated with a country-common usage rate of the Upper Middle Income group 
is similarly calculated with the results in the final column. Total world energy use would rise by 
approximately 27% under this scenario. 
 
Country 
Income Group 
 
Population 
(millions) 
Commercial 
Energy Use 
per Capita 
Upper Mid 
Energy Use 
per Capita 
Current 
Energy Use 
(millions) 
Hypothetical 
Energy Use 
(millions) 
Low 2,319 510 2,221 1,182,690 5,150,499 
Lower Mid 2,236 1,159 2,221 2,591,524 4,966,156 
Upper Mid 798 2,221 2,221 1,772,358 1,772,358 
High 1,017 5,502 2,221 5,595,534 2,258,757 
Sum 11,142,106 14,147,770 
4. For a given rate of use, the number of years a given supply will last is given by the following 
equation (on page 475): 
(0)ln 1
(0) ,Sg
UT
g
 
+ 
 = 
 where g is the growth rate of use, U(0) is the annual usage of the resource at time 0, S(0) is the 
available stock of the resource at time 0, and T is the time at which the resource runs out. We are 
given g to be two percent. U(0) is not given, but we can write S(0) to be equal to 1,000 U(0). That is, 
at the current rate of use, U(0), the stock of the resource will last 1,000 years. Therefore, S(0) must 
equal 1,000 × U(0). Using these values in the previous equation, we solve for T. 
1,000 (0)ln 1
(0) ln[1 (0.02)(1,000)] ln(21) 152.23.
0.02 0.02
Ug
UT
g
 
+  + = = = ≈ 
 Therefore, the resource will run out in approximately 152.23 years at the current growth rate of usage. 
Chapter 12 Government 63 
5. In the first year after the price increase of gasoline by 100 percent to two dollars per gallon, the quantity 
of gasoline consumed will decrease in both the short-term increase and long-term increase scenarios. 
The first year is a short-run analysis where the economy cannot adjust to the rise in gasoline prices. 
However, four years after the price increase, the quantity of gasoline consumption will show further 
declines in the long-term price-increase scenario. Intuitively, sustained high gasoline prices will 
provide enough incentives for the economy to substitute away from costly gasoline usage and/or 
create a mix of factors that minimize gasoline usage. Thus, gasoline consumption falls. On the other 
hand, the short-term price-increase scenario does not provide enough incentives for the economy to 
substitute away from and/or rearrange the mix of factors to minimize gasoline usage. Gasoline 
consumption will not continue to fall. The source of these different outcomes arises from expectations, 
as a return to previous prices does not induce innovation and structural changes, but a sustained rate 
of two dollars per gallon does optimally induce innovation and structural changes. 
6. a. Since individuals know that there will be extra oil coming online in five years, they know the 
price will be lower then it otherwise would have been in the future. Therefore, they will try to 
sell more oil now while the price is still relatively higher. Supplies of oil rise, and price falls. 
b. Since people know that there will be more oil in the future, they know the price will be lower in 
the future. Therefore, when they choose capital goods (such as cars) they will figure in a low 
average price for oil and buy goods that are not energy efficient. Therefore, demand for oil rises. 
c. Since both supply and demand rise, the effect on price is ambiguous. 
7. Other types of pollution discussed in the text are examples of negative externalities that arise as a 
result of poorly defined property rights. In the case of air pollution, a firm will not internalize the cost 
of pollution and will not produce the socially efficient/optimal quantity, as property rights are poorly 
defined for air. However, in the case of indoor air pollution, property rights over the air consumed are 
clearly defined. Furthermore, the quantity of indoor air pollution is self-determined. Assuming that 
those who live and consume the indoor air are also responsible for the pollution of the air, the producer 
of indoor air pollution is directly affected by the quality of indoor air. In this scenario, the amount of 
indoor air pollution should be optimal, as the individual should internalize the externality. But if an 
individual is unaware of the quantity of and harmful effect of indoor air pollution, the pollution 
emitted will be greater than the optimal quantity. In this manner, indoor air pollution differs from 
other examples of pollution. 
8. In the short run, a firm is unable to substantially change the mix of factors of production. Thus a 
pollution tax will not lower pollution much, but will raise a lot of revenue. However, in the long run, 
the firm is able to adjust the factors of production and possibly innovate processes to increase output 
and reduce pollution. Firms will substitute away from pollution-heavy and thus tax burden-heavy 
factors to a pollution-minimizing and tax burden-minimizing mix of factors. Pollution will fall, as 
well as government revenues from the tax. 
 Solutions to Appendix Problems 
A.1. The rate of extraction, ,ε is one percent per year and the rate of total output growth for 36 years, ˆ,Y 
is two percent per year (using the rule of 72, growth = 72/doubling time). Utilizing the following 
equation relating the rate of extraction and the rate of output growth to the rate of technological 
growth, we solve for the rate of technological progress, g. 
ˆ ,
ˆ 2% 1% 3%.
Y g
g Y
ε
ε
= −
= + = + =
 
64 Weil • Economic Growth, Second Edition 
 As a result, the rate of technological progress is three percent per year. 
Chapter 12 Government 65 
 To find the number of years it will take for the stock of the resource to be reduced to one-eighth of 
its initial level, we first define an arbitrary initial level as X(0) and utilize the equation for the stock 
of the resource at any time t. 
( ) (0) .tX t X e ε−= 
 The stock of the resource must be one-eighth of its initial level, X(0). We can represent this by 
writing X(t) = (1/8)X(0). Substituting in this value, along with our given rate of extraction, 1%,ε = 
we can solve for t. 
1 (0) (0) .
8
tX X e ε−= 
 Taking the log of both sides and rearranging, 
1ln ( )( ),
8
ln(8) ln(8) 208.
0.01
t
t
ε
ε
 − = 
 
= = ≈
 
 Therefore, in approximately 208 years, the stock of the resource will be reduced to one-eighth of its 
initial level. Alternatively, we could just use the rule of 72, which says that if something shrinks 
at one percent per year, it will halve after 72 years. One-eighth represents three halvings, so will 
take approximately 216 years. 
 
 
 
	Weil_solutions_cap1
	Chapter 1 The Facts to Be Explained
	( Solutions to Problems
	1. A ratio scale transforms absolute differences in the variable of interest to proportional differences. For instance, the GDP of Country X, whose GDP is 10 times greater than Country Y, will be the same distance apart as a Country Z whose GDP is 10...
	2. Let g be the rate of growth. The rule of 72 says that So g8%.
	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,...
	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 fo...
	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. There...
	5. We can solve for the average annual growth rate, g, by substituting the appropriate values into the equation:
	Letting Y1900  $1,433, Y2000  $23,971, and rearranging to solve for g, we get:
	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
	Letting Y2000  $23,971 and g  0.0286,
	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 thatpassed between the year 2005 and the year U.S. income per capita equaled that o...
	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.
	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(xy)  yln(x), we get
	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,
	Since YU.S., 2005 $36,806, YChina, 2005  $5,955, we then substitute in these values and solve for t.
	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(xy)  yln(x), we get
	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.
	( Solutions to Appendix Problems
	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 goo...
	A.2. a. The level of GDP per capita in each country, measured in its own currency is
	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 CPUs 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 Richlan...
	d. A natural basket to use is 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 par...
	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 Richlan...
	Weil_solutions_cap2 3
	Chapter 2 A Framework for Analysis
	( Solutions to Problems
	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 composit...
	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 s...
	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 ar...
	5. A correct solution is a variable that systematically rises or falls only because of a rise or fall in GDP per capita. Take for instance expenditures on unproductive luxury items. It is reasonable to conclude that as GDP per capita rises, individu...
	An example of an incorrect solution would be health. While it is reasonable to assume that a rise in GDP per capita will raise the health level of a country, it is also reasonable to say that a rise in health will result in a rise in GDP per capita. ...
	6. Simply finding a correlation between being overweight and having a heart attack does not imply causation. The correlation could be due to a missing variable like genetics which may be a factor in a person’s weight as well as put him or her at risk...
	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...
	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 unhealt...
	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 ca...
	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 healthi...
	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 ...
	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.
	Chapter 3 Physical Capital
	( Solutions to Problems
	1. The key characteristics of physical capital are that it is productive, it is produced, its use is limited, it can earn a return, and it wears out.
	a. A delivery truck is physical capital because it is productive, i.e., by allowing a delivery man to drive instead of walk it increases his output (deliveries made); it has been produced itself; it is rival in its use, i.e., only one person can drive...
	b. Milk is not physical capital because it is not productive. (To be slightly technical, milk is used in making other things—for example, cheese. But in this case milk is a raw material rather than a factor of production. In economic terms, “cheese pr...
	c. Farmland is not physical capital because even though it allows a worker to produce more output, it has not been produced itself.
	d. The Pythagorean Theorem is not physical capital because it is non-rival in its use. That is, an unlimited amount of people can use it at the same time.
	2. To find the steady-state value of the country, we refer to Equation (3.3) on page 63.
	Plugging in values: A  1, (  0.5, (  0.5, and (  0.05, we get:
	Simplifying the above equation, we get
	To find the current output per worker, we substitute in k  400 into the production function to get:
	That is, the current output is 20 whereas the steady-state output level is 10. Therefore, we conclude thatso the country is above its steady-state level of output per worker.
	4. Denoting each variable by the appropriate country subscript, we write Equation (3.3) from page 63 in ratio form. That is,
	Since productivity, A, and depreciation, (, are the same, we can cancel them and rewrite the previous ratio with the appropriate values: and setting
	Forwe get,
	Therefore, when the ratio is 0.5 or 1 to 2 and when the ratio is 0.25 or 1 to 4.
	5. Since we know productivity, A, and depreciation, (, are the same, we know that they will cancel out in our steady state ratio analysis.Therefore, with ( 1/3, our equation of interest boils down to
	for all three pairs of countries.
	a. Using a subscript T for Thailand and a subscript B for Bolivia, we rewrite the previous equation for Thailand and Bolivia as
	Substituting in and we get the steady state ratio to be:
	The actual ratio is,
	Therefore, the Solow Model does a good job in predicting relative income for Thailand and Bolivia.
	b. Using a subscript N for Nigeria and a subscript T for Turkey, we rewrite the previous equation, with and to get,
	The actual ratio is,
	Therefore, the Solow Model does a poor job in predicting relative income for Nigeria and Turkey.
	c. Using a subscript J for Japan and a subscript N for New Zealand, we rewrite the previous equation, with and to get,
	The actual ratio is,
	Therefore, the Solow Model does a good job in predicting relative income for Japan and New Zealand.
	6. If output per worker is rising in Country X and output per worker is falling in Country Y, we can be assured that both countries are not in their respective steady states. Instead, they are converging to their respective steady states. In addition,...
	As such, we can conclude that differences in investment rates are responsible for the divergence in output per worker. Specifically, a rise in output per worker for Country X and a fall in output per worker for Country Y imply that the ratio of stea...
	Consequently, we can determine that Country X has a higher investment rate than that of Country Y, and these differences account for the falling and rising levels of output per worker.
	7. a. First we find the steady-state level of capital per worker. Using the values for investment, (  0.25, depreciation, (  0.05, productivity, A  1, and (  0.5, we get,
	That is, the steady-state level of capital per worker is 25. Plugging in into the production function we get the steady-state level of output per worker to be:
	That is, the steady-state level of output per worker is 5.
	b. For year 2, using 16.2 as the value for capital per worker, calculate output, y, followed by investment (y, depreciation (k, and then change in capital stock. Add the value for change in capital stock to 16.2, the value for capital per worker in ye...
	c. The growth rate of output between years 1 and 2 is given by:
	That is, output per worker grew at a rate of 0.5 percent between years 1 and 2. (Using exact values, the growth rate is approximately 0.62 percent for years 1 and 2.)
	d. The growth rate of output between years 7 and 8 is given by:
	That is, output per worker grew at a rate of 0.48 percent between years 7 and 8. (Using exact values, the growth rate is approximately 0.52 percent for years 7 and 8.)
	e. The speed of growth has changed from 0.50 percent to 0.48 percent implying that growth has slowed down at a rate of 4 percent. Thus, as a country reaches their steady-state value, the rate of growth slows.
	8. Before beginning the analysis, we define two new variables. The level of capital per worker necessary to achieve consumption level c* is denoted k*. Technically, k* is given by Therefore, if the initial level of capital, ki is above k*, savings wil...
	Case 1. Depreciation is always greater than Saving;
	In the figure below, at any initial level of depreciation is always greater than savings. The level of capital falls over time, as does the level of income per worker. Consequently, the economy will continue to stagnate until the level of income and...
	Case 2. Depreciation is not always greater than Saving; for some k
	The figure below shows two possible scenarios. If the initial level of capital, is equal to or below k*, then savings in the economy will be zero. The level of capital in the economy falls due to depreciation and we achieve the same result as in the ...
	9. First, in a steady-state level that maximizes consumption per worker, the change in capital stock will be zero. That is,
	Rearranging the previous equation, we know that investment must equal depreciation.
	Second, given that any output not saved is consumed, we can write an equation for consumption as,
	In the last part of the previous equation, we replace savings with depreciation and write output in functional form. In this form, we are able to take the derivative to find the necessary condition that will guarantee consumption maximization. Taking...
	That is, the marginal product of capital must equal the rate of depreciation. Combining the consumption maximization condition with the steady-state condition we get:
	Therefore, it is easy to see the ( must equal ( by the above string of equalities. In any steady-state level of consumption per worker, the investment/saving rate must equal the value (.
	Weil_solutions_cap4 5 6
	Chapter 4 Population and Economic Growth
	( Solutions to Problems
	1. To find the average growth rate of the population, we use the following equation:
	whereis the population at time t, g is the growth rate, and n is the duration of growth. Substituting in n  100,000; and billion, we can rewrite and solve the equation for the growth rate.
	Therefore, the average growth rate of the population has been roughly 0.0219 percent per year.
	2. a. We begin from Point A, where the population size is stable with no growth. With the discovery of a new strain of wheat that is twice as productive, the curve relating population size and income per capita shifts outward. At this point, we are at...
	b. We begin from Point A, where the population size is stable with no growth. With war killing half the population, no curve is shifted. Instead, we jump to Point B along the original curve. At this point, we have half the population with a higher in...
	c. We begin from Point A, where the population size is stable with no growth. With a volcanic eruption that kills half the people, we are faced with the same scenario as in Part (b). However, the volcanic eruption destroys half the land, shifting the ...
	3. At Point A in time, the population size is stable with no growth. With a sudden change in cultural attitudes, the curve relating the population growth rate and income per capita shifts upward. The sudden shift, denoted by Point and Time B, implies...
	4. To calculate the steady-state level ratio of income per capita, we first find the steady-state level for each country and then divide. The steady-state level ratio for Country X to Y is given by:
	We now substitute in the values and for Country X, and for Country Y, we use the values and Also, set and This yields,
	Therefore, we conclude the ratio of Country X to Y in their steady-state levels of income per capita to be near 2.68.
	5. Because the equation for capital accumulation suggests that in the steady state,
	multiple values of population growth will yield multiple steady states under some conditions. In the diagram, Points A and B are the multiple capital per capita steady-state values and hence the resulting incomes per capita steady-state values. In ad...
	6. Country A and B are identical in every respect but for their population growth rates. That is, However, this implies that their respective steady states are not equal. Writing the reduced ratio equation, as on page 98, we get:
	Utilizing the fact that and we can conclude that Because both countries currently have the same income per capita, we now know that Country B is farther away from its respective steady-state level. Therefore, Country B must have a higher growth rat...
	7. a. TFR 4.
	NRR  (1/2) ( [(1 child) ( (Probability of reaching age 25)  (1 child) ( (Probability of reaching age 28) (1 child) ( (Probability of reaching age 32) (1 child) ( (Probability of reaching age 35)]
	Substituting in the given information, we get
	NRR (1/2) ( [(2/3)  (2/3)  (1/3)  (1/3)]  1
	b. NRR  (1/2) ( [(1) (1) (1/2) (1/2)]  1.5
	c. TFR  2
	NRR  (1/2) ( [(1/2) (1/2) (1/2) ( (1/2) (1/2) ( (1/2)]  .75
	8. a. We graph the equation, in the figurebelow.
	b. First, we divide both sides of the production function by L and rearrange to get:
	Therefore,
	For X  1,000,000 the figure is shown below.
	c. In the steady state, the growth rate of population is zero, Using this value and rearranging the first equation, we solve for the steady-state value of income per capita:
	Substituting in this value into the production function, we back out the value of as follows:
	The steady-state population is 100.
	9. a. The steady-state level of income per worker is characterized by Hence, we must first find the relationship among the growth rate of income per worker, productivity, and population. By taking natural logs of the production function and taking th...
	Therefore, the growth rate of income per worker must equal the growth rate of productivity plus the growth rate of land minus the population growth rate. Since land X, and productivity A, are constant, These values tell us that and because we are in...
	b. Referring back to our equation relating the growth rate of income per capita, productivity, land, and population, we can set and
	In the steady-state level of income per worker, must equal zero. Thus, we have that must equal 0.1. Using this value to solve forin the equation relating population growth and income per capita,
	The steady-state value is higher in this scenario. Due to consistent productivity growth of 10 percent, the population can grow as well, leading to a higher level of income per capita.
	( Solutions to Appendix Problems
	A.1. a. To calculate life expectancy at birth, we must find the area under the survivorship function. This amounts to solving the equation:
	Equivalently, one could find the area using geometry.
	In discrete time analysis, we can extrapolate that the probability of being alive from age 0 to 29 is 1; the probability of being alive from age 30 to 79 is 0.5; and the probability of being alive from age 80 to infinity is 0. Summing these probabil...
	Therefore, the life expectancy at birth is 55 years.
	b. To calculate the total fertility rate, we must find the area under the age-specific fertility rate function. This amounts to solving equation:
	Equivalently, one could find the area using geometry.
	In discrete time analysis, we can extrapolate that the average number of children per woman from age 20 to 39 is one and the average number of children per woman for any other age is zero. Summing these probabilities, we get:
	Therefore, the total fertility rate is 20.
	c. The net rate of reproduction is found by multiplying the number of girls that each girl born can be expected to give birth to. First noting that the probability of being alive from age 20 to 29 is one with the age-specific fertility rate at one ch...
	That is, the rate of reproduction is 15. Adjusting this value by (, the fraction of live births that are girls, we conclude that the net rate of reproduction is 15(.
	A.2. For Country X and Country Y assume that the survivorship function is that given in Problem 1. The total fertility rates for both countries are given below.
	The total fertility rate is the same for both countries. However, the rate of reproduction differs. For Country X,
	Adjusting for (, the net rate of reproduction for Country X is 2(. For Country Y,
	Adjusting for (, the net rate of reproduction for Country Y is 1(. Therefore, Country X has a net rate of reproduction twice as large as Country Y, but the survivorship function for both countries is identical, as well as the total fertility rate fo...
	Chapter 5 Future Population Trends
	( Solutions to Problems
	1. To calculate the population of Fantasia in the year 2001, we must first find the population of each age group. Given that of 100 zero-year-olds in the year 2000, and the probability of surviving to age one is 1, we conclude that in the year 2001, t...
	2. Libya currently has high fertility, rapid population growth, and a population age structure heavily weighted toward young people. Japan and has low fertility, almost zero population growth, and one of the oldest populations in the world. If the two...
	3. Population growth in 1950–2000 (denoted by subscript A) for the more developed country group, is 0.8 percent, and predicted population growth in 2000–2050 (denoted by subscript B) for the same country group,is 0.0 percent. Writing the steady-state...
	If given a value for ( and (, we can solve the equation above to find an exact ratio for the differences in the steady state. Assuming that (  1/3 and (  0.05, then
	This means that with nothing but population growth changed, output per worker will be 7.7 percent greater in 2000–2050 for the more-developed country group.
	4. In 2025: each of the 60 million 0–20 year-olds from 2005 will have had 1 girl and moved on to the next age bracket; half of the 40 million 21–40 year-olds will have died; and all the 41–60 year-olds will have died. So, the new female population str...
	5. Immediately after fertility declines to zero, the working age fraction of the population will begin to rise. This is because the fraction of the population made up of children will fall (as some children become adults and are not replaced by new bi...
	6. Calculating the growth rate of the percentage of the population that is of working age, g, the equation becomes,
	Plugging in the appropriate values, we rewrite and solve the previous equation.
	That is, the growth rate of the percentage of the population that is of working age is negative 0.817 percent. According to Equation (5.2) on page 145, this demographic change lowers GDP per capita by 0.817 percentage points.
	7. The statement is not necessarily true. Higher fertility will lower the fraction of the population made up of working-age adults until the newly born children reach working age. Thus, in the short run the country’s dependency burden will be higher,...
	8. a. To calculate the world population in any year, we add each country’s respective populations for that year. Therefore, the world population in 2000 is 2,000,000. For the following year, Country A’s 2001 population will be 1,000,000 ( 1.02  1,02...
	The growth rate is 1 percent. However, the growth rate of population will rise over time moving to a growth rate of 2 percent per year. The growth rate of population is graphed below.
	Intuitively, Country B’s population continues to grow, making Country A’s population a miniscule fraction of the world’s population. Therefore, Country B’s growth rate determines the world population growth rate in the long run.
	b. Below is a graph showing the growth rate of total world GDP (not per capita).
	c. Below is a graph showing the growth rate of average GDP per capita.
	Chapter 6 Human Capital
	( Solutions to Problems
	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...
	2. Because for any given level of income Country A will generally be healthier than Country B, we can determine that the curve for Country A will be positioned above the curve for Country B. In addition, we observe that the income levels of both cou...
	3. In the case that education has no effect on productivity. That is, an additional year of education does not raise the productive ability of an individual. Therefore, the average level of education does not affect the total labor input of a country...
	4. The wage for an individual with nine years of education, relative to one with no education, is
	In other words, if the wage of an individual with no education is W, the wage paid to this individual is 2.6W. Therefore, we can conclude that the payment to raw labor is W and the payment to human capital is the difference between the total wage an...
	The fraction of wages paid to human capital is 61.5 percent.
	5. As in Problem 4, the payment to humancapital 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,
	The fraction of wages paid to raw labor is 33.5 percent. For the fraction of wages paid to human capital, similar calculations reveal,
	The fraction of wages paid to human capital is 66.5 percent. (Equivalently, one could just as easily subtract the first number from 1.)
	6. In order to find the fraction of wages paid to human capital, we follow the approach outlined in Problem 4. We first subtract from each relative wage the relative wage paid for raw labor. This amounts to subtracting one from the relative wage for ...
	Therefore the fraction of wages paid to human capital is nearly 70 percent. For the fraction of wages paid to raw labor, we subtract our previous figure from one to get: 1  0.70  0.30. That is, the fraction of wages paid to raw labor is 30 percent.
	*The first two columns of the table above are replicated from Table 6.2.
	7. The relative return to 10 years of schooling is 2.77, and the relative return to four years of schooling is 1.65 (from the table above). Denoting 2.77 and 1.65, we can solve for the steady-state ratio for two countries identical in every respect e...
	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. Writing the steady-state ratio for one country over time and denoting 1.29 and 3.16, we get:
	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.
	We conclude that the annual average growth rate of output per worker is 0.9 percent.
	9. There are many examples of positive externalities associated with health. For instance, being surrounded by healthy people reduces the probability of contracting disease.
	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, Furthermore, we are told that the co...
	The level of human capital must be higher in Country A, suggesting that investment in human capital is higher in Country A as well.
	Weil_solutions_cap7 8
	Chapter 7 Measuring Productivity
	( Solutions to Problems
	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 ...
	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 ...
	2. The equation for output at year t:
	And the ratio of output between years t and t  100
	Rearranging the equation, using the given information and taking ( 1/3, we get
	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...
	The level of factors of production for Freedonia and Sylvania are 80 and 50, respectively. The ratio of factors of production is given by:
	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,
	The level of output per worker in Freedonia is twice as large as that in Sylvania. This allows us to use the following equation:
	Solving for the above equation, we get Therefore, the productivity level in Freedonia is 1.25 times the productivity level in Sylvania.
	b. If all differences in output were due to differences in productivity, then,
	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,
	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.
	4. Physical capital per worker grew at the same rate in the two countries. This is clear because it rose 27-fold over this period (this corresponds to an annual growth rate of 8.6% per year, but it was not necessary to calculate this). Similarly, huma...
	5. We assume (  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:
	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 utili...
	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 fift...
	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...
	6. We assume (  1/3. Then, in order to calculate the growth rate of factors for each country, we utilize the following equation:
	where the growth rate of physical capital per worker, and the growth rate of human capital per worker, are given for each country. In order to calculate the growth rate of productivity for each country, we utilize the following equation:
	where the second term is simply the previous growth rate of factor accumulation and 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 cou...
	7. The effect of using data on school days increases the level of education for richer countries and lessens the level of education for poorer countries. This implies that the total return to education will be higher in richer countries than the total...
	Chapter 8 The Role of Technology in Growth
	( Solutions to Problems
	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.
	2. The advantage to having patent laws in the creation of life-saving prescription drugs is that the resulting monopoly pricing will actually provide an incentive for companies to undertake that the research and development leading to the creation of ...
	3. With no change in the fraction of workers devoted to research and development, productivity and output per worker would continue to grow indefinitelyat the previous rate. That is,
	where denotes the fraction of workers devoted to R&D. With an increase in to we know from the following equation that,
	Therefore, at the time of change, denoted as in the graph below, the growth rate of output per worker and productivity,and respectively, will be greater than before. However, the increase in the rate of growth will be accompanied by a decrease in th...
	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 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 tha...
	Because the change inis the same and  A, we know that the absolute difference in the jump up is greater than the jump down. The level ofis dependent on the length of the temporary increase in the fraction of workers devoted to R&D. Regardless of the...
	4. The given parameters of the model are, L  1, 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:
	The growth rate of output per worker is 10% per year. Similarly, with  0.75, we get:
	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:
	Similarly, we can find the new level of output per worker to be:
	Therefore, the original level of output per worker is A(0.5), and we need to find how long it will take to reach this level, starting from a level of A(0.25) with a growth rate of 15 percent. We use the standard growth equation, substitute and solve.
	Dividing both sides by A(0.25) and taking logs, we get:
	Solving for t, we get:
	That is, it will take approximately five years for the level of output per worker to return to its previous level 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 acc...
	6. We are given a scenario where the follower, Country 2, raises the fraction of the labor force devoted to R&D beyond the fraction in the leader, Country 1. The implication in the long run is for Country 2 to become the technological leader. The pro...
	7. a. In the steady state, Therefore,
	Rearranging and solving forwe get,
	Setting the above steady-state condition equation to the specified cost-of-copying function,
	Rearranging, we find out solution to be:
	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 Because we assume with the exponent, the ratios will n...
	b. If we assume and we can solve the previous equation to get:
	That is, the steady-state ratio of technology in Country 1 to technology in Country 2 is 4.
	Weil_solutions_cap9 10 11
	Chapter 9 The Cutting Edge of Technology
	( Solutions to Problems
	1. The annual growth rate of productivity is given by the following equation:
	We are given a value of 1/3 for ( and 0 for leaving the growth rate of the population, as the only unknown. To solve forwe use the standard growth equation with the initial population as 4 million and the final population after 10,000 years as 170 mi...
	Now we substitute to find our growth rate of productivity over this period:
	That is, the growth rate of productivity over this period was roughly 0.0125 percent per year.
	2. For the relative prices, note that the high rate of productivity in the farming sector will create a fall in the real price of wheat; whereas, the lack of significant change in productivity in the hair-cutting sector will maintain the real price fo...
	The improvement in technology for growing wheat raises the marginal productivity of labor for farmers, and because workers are always paid their marginal products, the real wage for farmers will rise, ceteris paribus. Similarly, the absence of techno...
	3. Technological progress in the production of a good will reduce the price of that good. Moreover, the reduction in price of the good will change the quantity demand for that good. The resulting change in the quantity demanded, or equivalently, the ...
	4. a. In any given year, the production of bread must equal the production of cheese in this economy. That is, always. Knowing that the productivity of each good is equal at this point in time, we can solve for the quantity of labor devoted to each s...
	Since
	The labor force will be equally split between the two sectors.
	b. To calculate the growth rate of total output, we first calculate the growth rates of each sector by taking the natural log of both sides and differentiating with respect to time. For the bread sector:
	Similarly, for the cheese sector, we get,
	We know the value for the growth rate of productivity in both sectors. Furthermore, in our answer to Part (a), we found that labor is currently equally divided among the two sectors. Thus, the growth of labor in one sector must be offset by the growt...
	And,
	Setting the growth rate of output in the bread sector to the growth rate of output in the cheese sector, we find that and So,
	c. The figure below depicts the growth rate of output over time. From Part (b), we know that the growth rate of output is equal to 1.5 percent. But, productivity in the bread sector rises by a greater percent than in the cheese sector. Bread producti...
	5. There is no one correct answer to this question. However, be sure to explain your reasoning for the future pace of technological progress using as many examples from the text as necessary.
	Chapter 10 Efficiency
	( Solutions to Problems
	1. Denoting each country by the appropriate subscript, we are given that,
	Since, productivity is defined as T ( E, we can divide the productivity level of Country X by the productivity level of Country Z to obtain the ratio of efficiency in both countries.
	Therefore, The level of efficiency in Country X is half the level of efficiency in Country Z.
	2. The level of productivity in Country X, relative to the United States is given as
	The level of technology in Country X, relative to the United States is,
	where g is the growth rate of technological progress and G is the number of years that Country X lags behind the United States. Given values of g  1% and G  20, we substitute and solve for the ratio of technology differences.
	Since, productivity is defined as T ( E, we can divide the productivity level of Country X by the productivity level of the United States to obtain the ratio of efficiency in both countries.
	Solving the previous equation yields a value of 0.61 for the ratio of efficiency in Country X relative to the United States. Stated differently, Country X has 61% of the efficiency of the United States.
	3. Assuming that and that the growth rate of productivity is 0.66%, we write the functional productivity equation of each country in ratio form and substitute in the necessary values to obtain:
	Hence, we have an equation that we can solve for to find the magnitude of the technology gap in years between India and the United States. Taking the log of both sides and rearranging to solve yields,
	The level of technology in India is, thus, approximately 160 years behind that of the United States.
	4. Brief examples of real-world inefficiencies for each category are given below.
	Unproductive Activities: In addition to theft and political lobby activities, wars are generally extremely unproductive.Other examples are black market activities such as drug use, drug dealing, and so forth.
	Idle Resources: Many economic fluctuations reduce efficiency and make people and capital lay unused. Recessions, currency crises, and labor strikes are instances that result in idle resources.
	Misallocation of Factors among Sectors: Often, labor mobility is impeded across borders because work permits, licenses and credentials do not extend universally. Marginal products will not equalize, resulting in the misallocation of factors among se...
	Misallocation of Factors among Firms: Many government subsidized or financed industries throughout the world regularly exhibit this form of inefficiency. In addition, certain industries such as the airline industry, exhibit oligopolistic behavior, ...
	Technology Blocking: The recent ban on stem cell research in the United States is a form of technology blocking. Many examples are present today and even throughout history.
	5. A correct answer to the problem will include specific examples within a country. In addition, be certain to categorize each type of inefficiency.
	6. The figure below depicts an inefficient allocation of labor between the urban and rural sector. An efficient allocation is given by Point A, the point where the marginal productivity of labor equalizes in both sectors. However, with a minimum wage...
	7. The marginal product of Sector 1 is found by differentiating the production function of Sector 1. That is,
	The average product of Sector 2 is found by dividing total output of Sector 2 by the amount of workers. Simply,
	In equilibrium, the wage of both sectors must be equal, as workers are free to move between sectors. Therefore, we set to solve for the number of workers in each sector.
	The number of workers in Sector 2 will be 4 times greater than the number of workers in Sector 1.
	The figure below depicts, graphically, the scenario above with the assumption that L  10. As predicted, the intersection of wage curves intersect where two workers are employed in Sector 1, where marginal products are paid, and four times that numb...
	Chapter 11 Growth in the Open Economy
	( Solutions to Problems
	1. Wages of a U.S. citizen, earned in France, will be counted towards U.S. GNP but not in U.S. GDP. However, the wages will be counted towards France GDP and not in France GNP.
	2. In a closed economy, the usual mechanism of the Solow model operates: higher saving means more capital, higher GDP per worker, and higher wages. In an open economy, the level of capital is determined by the world interest rate. Thus, higher saving ...
	3. In the closed economy Solow model, an increase in the growth rate of the population will reduce the steady-state level of income per capita by reducing the steady-state level of capital per capita. To see this, recall that the accumulation of capit...
	In an open economy, the change in the growth rate of the population does not affect the world rate of return,nor the steady-state level of capital within the country. The steady-state level of capital is given as,
	Therefore, the growth rate of the population, n, does not enter into the determination of the steady state. There will be no change to GDP, although for GNP, the increase in n will lower the domestic level of capital per capita. The marginal product ...
	4. In an economy perfectly open to the world capital market, the steady-state level of output per capita is given by the following equation as reproduced from the chapter.
	Given that the world rental price of capital doubles to which is exactly we can write the new steady-state level of output per worker in terms of the previous rental rate.
	And with a value of 1/2 for (,
	That is, the new steady-state level of income per capita falls by half when the world rental rate of capital doubles in an economy open to the world capital market.
	5. In a closed economy where one slice of cheese is consumed with one slice of bread, we can write a production function for each sector as
	where and is the production of bread and cheese, respectively, withandas the number of labor hours devoted to each sector. In equilibrium, will equal Therefore,
	We know that the total number of labor hours,available is 60. Combining this fact with the previous equation, we find that,
	and,
	As for output in each sector,
	Therefore, the allocation of labor hours towards the bread sector will be 40, resulting in an output of 20 slices of bread, and the allocation of labor hours towards the cheese sector will be 20, resulting in an output of 20 slices of cheese.
	With the economy opened up to trade where the price of bread equals the price of cheese on the world market, the economy is strictly better off when they specialize in the good that they have a comparative advantage in and trade for the other. Becaus...
	6. If there is only one auto factory in each country under autarky, then each auto factory would have a local monopoly, and therefore there will be no competition driving the auto factory to become efficient. Upon the opening of trade, all the auto fa...
	The pizza industry in each country, however, already was competitive because there were no size barriers to entry. Therefore, trade in pizza is likely to be unchanged, since the pizza industry already had high efficiency due to tight competition.
	Whether the decline in auto prices leads factors to flow into or out of the auto industry depends on the price elasticity of demand for autos. If demand is inelastic, then as prices fall due to more efficient auto production, factors will flow out of...
	It is interesting to note that although each country was identical in terms of efficiency and technology under autarky, and trade did not affect the technology level in any county, the overall level of efficiency rose due to more competition.
	7. A tariff on the import of widgets is a form of trade protection, insulating the domestic widget industry from international competition and pressure. Although the result is a higher wage for those members of the widget industry, the higher wage com...
	Weil_solutions_cap12 13 14 15 16
	Chapter 12 Government
	( Solutions to Problems
	1. a. Standardization of the length of axles on carts is a form of a public good. Everyone benefits from improvements in travel, and improvements in travel are beneficial for growth.
	b. A stable currency is a public good.
	c. The creation of a national car in Indonesia, the Timor, is an instance of government failure. Competition is stunted by providing monopoly power to the Suharto company through protection policies and as a result, incorrectly aligning incentives. Ul...
	d. Failure to get vaccinated imposes a negative externality, since a sick person is likely to spread disease to others.
	e. If one assumes that mail delivery services is a natural monopoly, as it may be inefficient for multiple firms to be able to route to all houses, government regulation can address the market failure present in monopolies. By operating mail delivery ...
	f. The answer is unclear. On one hand, the imposition of a minimum wage can be an example of a government failure. The minimum wage can result in the misallocation of factors among firms and sectors, and in this case, it serves those who do receive th...
	g. The failure of African governments to maintain roadways is a government failure. The case for government intervention in the provision of public goods such as roadways is economically valid, as the private market cannot optimally provide the approp...
	h. One justification for this policy might be externalities. Highly educated people might have positive externalities for the rest of the country (by importing new technologies, improving the quality of government, etc.). On the other hand, this may b...
	2. In the first figure below, we represent two countries with varying quality levels of government and varying incomes. In this figure, the horizontal axis measures outputand the vertical axis measures the quality of government. A country with a lo...
	In the second figure, we graphically depict each channel of causation. Both curves are positively sloped, as both income and quality of government are positively correlated. However, the channel of income affecting the quality of government, termed t...
	The third figure illustrates the government view for the differences between Country A and B. The government view assumes that Country A and Country B have quality differences in governments for reasons other than income. For this reason, the curve ...
	The fourth and final figure illustrates the income view as the primary source for differences between Country A and B. Here, both countries share the same G(y) curve. That is, income affects the quality of government in the same manner. However, the ...
	3. The economist understands that the only way to determine whether more income causes better government or whether better government causes more income is to find an event in which we know that one determinant changed, and which cannot plausibly have...
	In this example, the change in the prices of the different beans is obviously going to have an effect on income in the two countries. In this case, we know that the income change was not due to government quality. Therefore, if government quality ris...
	If the researcher only looked at the simple correlation between quality of government and income, he or she would not be able to determine causality. Also, it there is dual causation (causality running in both directions), then the magnitude of the c...
	4. a. Denoting as the quantity demanded and as the quantity supplied, at the market-clearing price or equilibrium price of this model, the quantity supplied must equal the quantity demanded for a good. That is,
	Therefore, for markets to clear, the equilibrium price in the absence of a tax is 50 for the good.
	b. At a tax rate offor each good, the quantity demanded for any given price does not change because the price paid by the consumer remains unaffected. However, the quantity supplied for any given price decreases to a factor of as government collects ...
	In equilibrium, the price must be set such that the quantity demand meets the quantity supplied. Setting and solving for P, we get:
	The equilibrium price is the value given above. To find the equilibrium quantity for the price, we substitute and get,
	These are the market-clearing values in the presence of a tax.
	c. In order to solve for the tax rate that will maximize government revenue, we first must write the revenue function as a function of the tax rate. Government revenue for each good is Since the amount of the good sold is we know that government rev...
	Recall that
	Setting the above expression equal to zero, rearranging, and dividing out common terms,
	Therefore, solving the above equation gives us: At this tax rate, government revenue will be maximized.
	5. a. In the steady-state level of output per worker, the quantities of government capital per worker and physical capital worker will not change over time. Therefore,
	Using the values, we now have two equations with two unknowns. Working through the algebra and solving for the steady-state values ofand we get:
	Plugging in our steady-state values into the production function will now yield the steady-state level of output per worker.
	b. The value of that will maximize output per worker is the same value that will maximize as is a constant. Therefore, a tax rate of 50 percent.
	Chapter 13 Income Inequality
	( Solutions to Problems
	1. a. The Lorenz curve for the economy is drawn below. The data for the curve are as follows. Total wealth in the economy is ($1)(5)  ($3)(5)  $20. The poorest 10 percent of the people own $1/$20, or 5 percent of the wealth. The poorest 20 percent o...
	b. The Gini is constructed by dividing the area between the line of perfect equality and the Lorenz curve by the entire area under the line of perfect equality. In our case,
	c. To calculate the Gini, we must find the Area of A, B, C and D. For the Area of B, C, and D, we apply the area formula for triangles.
	In order to find the Area of A, we first calculate the area under the line of perfect equality. This is simply a 1 by 1 right triangle implying that the area is 0.5. Following, we can subtract from this the area under the Lorenz curve to find the Are...
	Now we substitute in these values into our equation from Part (b).
	2. The perfection of “distance learning” serves to create a superstar effect for professors, thereby raising the level of inequality among professors. Because a single professor will be able to teach many students at any given time, only a few qualifi...
	3. In the text, we see that higher inequality is good for growth via the channel of physical capital accumulation but bad for growth via the channel of human capital accumulation. Availability of student loans implies that at any given level of inequa...
	4. The more economic mobility a poor person perceives that there is, the less he or she will be in favor of redistributive taxation. Comparing two countries with the same level of inequality, redistributive taxation would be lower in a country with hi...
	5. In the table, the probability that a mother in the middle-earning third will also have a daughter in the middle earning third is given by the exact center cell, and is 0.5. When this process is iterated for two generations, of the 50 percent daught...
	However, it could be the case that the grandmother was middle class, but her daughter was upper or lower class, and that her daughter was again middle class. So we need to do this calculation for three sets of people, and then sum the probabilities:
	a. P(Middle third grandmother has a bottom third daughter)*P(Bottom third mother has a middle third daughter) 0.25*0.25  0.0625
	b. P(Middle third grandmother has a middle third daughter)*P(Middle third mother has a middle third daughter) 0.5*0.5 0.25
	c. P(Middle third grandmother has an upper third daughter)*P(Upper third mother has a middle third daughter) 0.25*0.25  0.0625
	Chapter 14 Culture
	( Solutions to Problems
	1. Many countries, regions, ethnic groups, organizations, and other existing groups can be analyzed to provide an acceptable answer to this question. As such, this solution is only an example. The following is taken from the culture of a professional ...
	a. Teamwork and trust are cultural attributes that are necessary to the success of a professional sports team. In professional American football, a play of the game is only successful if each member of the team is successful in accomplishing their res...
	b. Unhealthy competition and “superstar” dynamics (not being a team player) are cultural attributes that prevent a team from being successful. Professional sports are characterized by a highly competitive environment, and often times, the level of com...
	c. General discipline is an example of a cultural attitude that has been intentionally instilled in the members of a sports team. For instance, in baseball, it is discipline at the plate, discipline in attending practice, discipline in maintaining one...
	2. A residual for social capability is the difference between the actual value of social capability and the fitted value. The fitted value is obtained from a fitted line that gives a value of social capability for every value of GDP per capita. In Fig...
	3. In Figure 14.3, the two countries have roughly the same level of social capability. However, Thailand is significantly poorer. Since we would expect countries with the same levels of social capability to end up with similar incomes in the long run,...
	4. A country’s index of ethnic fractionalization is,
	where is the fraction of the population belonging to group i. Given that there are three ethnic populations with respective fractions of 0.5, 0.25, and 0.25, we substitute and solve.
	Thatis, the index of ethnic fractionalization is 0.625.
	5. In the figure below, we begin at Point A and move to Point B after a rightward shift in the Y(M) curve. From Point B, the final equilibrium position will be either Point C or Point D. If the economy is described by a relatively flatter MC(Y) curve,...
	6. The starting point of the economy is in an equilibrium position, Point A in the figure below. With an exogenous increase in the modernization of culture, the M(Y) curve shifts upward. For any given level of income, the level of modernization of cul...
	7. This is an exercise to be done by the student. Results will vary.
	8. Again, this is an exercise to be done by the student. Results will vary.
	Chapter 14 Culture
	( Solutions to Problems
	1. Many countries, regions, ethnic groups, organizations, and other existing groups can be analyzed to provide an acceptable answer to this question. As such, this solution is only an example. The following is taken from the culture of a professional ...
	a. Teamwork and trust are cultural attributes that are necessary to the success of a professional sports team. In professional American football, a play of the game is only successful if each member of the team is successful in accomplishing their res...
	b. Unhealthy competition and “superstar” dynamics (not being a team player) are cultural attributes that prevent a team from being successful. Professional sports are characterized by a highly competitive environment, and often times, the level of com...
	c. General discipline is an example of a cultural attitude that has been intentionally instilled in the members of a sports team. For instance, in baseball, it is discipline at the plate, discipline in attending practice, discipline in maintaining one...
	2. A residual for social capability is the difference between the actual value of social capability and the fitted value. The fitted value is obtained from a fitted line that gives a value of social capability for every value of GDP per capita. In Fig...
	3. In Figure 14.3, the two countries have roughly the same level of social capability. However, Thailand is significantly poorer. Since we would expect countries with the same levels of social capability to end up with similar incomes in the long run,...
	4. A country’s index of ethnic fractionalization is,
	where is the fraction of the population belonging to group i. Given that there are three ethnic populations with respective fractions of 0.5, 0.25, and 0.25, we substitute and solve.
	That is, the index of ethnic fractionalization is 0.625.
	5. In the figure below, we begin at Point A and move to Point B after a rightward shift in the Y(M) curve. From Point B, the final equilibrium position will be either Point C or Point D. If the economy is described by a relatively flatter MC(Y) curve,...
	6. The starting point of the economy is in an equilibrium position, Point A in the figure below. With an exogenous increase in the modernization of culture, the M(Y) curve shifts upward. For any given level of income, the level of modernization of cul...
	7. This is an exercise to be done by the student. Results will vary.
	8. Again, this is an exercise to be done by the student. Results will vary.
	Chapter 16 Resources and the Environment at the Global Level
	( Solutions to Problems
	1. To find the annual growth rate of energy intensity of output over this period, we utilize the following equation, where a hat over the variable denotes its growth rate.
	where I is the energy intensity of output, R is energy consumption, y is GDP per capita, and L is the population. Thus, we must derive the right-hand side values from the table. Upon inspection of the table over the 35 year period, each variable has ...
	Therefore,
	The energy intensity fell by two percent per year over this period.
	2. a. Because the stock of fish at time t is 20, we use the equation for the growth in the quantity of fish to determine
	Additionally, the stock of fish and the size of the harvest have been constant for a long period of time. Mathematically, this implies that,
	Thus, must equal implying that
	b. To find the optimal stock of fish in the lake, we differentiate with respect to
	Setting the above equation to 0 and solving for we get The optimal stock of fish that maximizes the growth in the stock of fish will be 50.
	At this stock level of fish, we repeat the analysis in Part (a) to arrive at the maximum sustainable yield.
	Thus, must equal implying that
	3. Current total energy use is calculated by summing over all country groups, the product of the population and energy use per capita. The results are shown in the “Current Energy Use” column in millions. The total energy use calculated with a countr...
	4. For a given rate of use, the number of years a given supply will last is given by the following equation (on page 475):
	where g is the growth rate of use, U(0) is the annual usage of the resource at time 0, S(0) is the available stock of the resource at time 0, and T is the time at which the resource runs out. We are given g to be two percent. U(0) is not given, but w...
	Therefore, the resource will run out in approximately 152.23 years at the current growth rate of usage.
	5. In the first year after the price increase of gasoline by 100 percent to two dollars per gallon, the quantity of gasoline consumed will decrease in both the short-term increase and long-term increase scenarios. The first year is a short-run analysi...
	6. a. Since individuals know that there will be extra oil coming online in five years, they know the price will be lower then it otherwise would have been in the future. Therefore, they will try to sell more oil now while the price is still relativel...
	b. Since people know that there will be more oil in the future, they know the price will be lower in the future. Therefore, when they choose capital goods (such as cars) they will figure in a low average price for oil and buy goods that are not energy...
	c. Since both supply and demand rise, the effect on price is ambiguous.
	7. Other types of pollution discussed in the text are examples of negative externalities that arise as a result of poorly defined property rights. In the case of air pollution, a firm will not internalize the cost of pollution and will not produce the...
	8. In the short run, a firm is unable to substantially change the mix of factors of production. Thus a pollution tax will not lower pollution much, but will raise a lot of revenue. However, in the long run, the firm is able to adjust the factors of pr...
	( Solutions to Appendix Problems
	A.1. The rate of extraction, is one percent per year and the rate of total output growth for 36 years, is two percent per year (using the rule of 72, growth  72/doubling time). Utilizing the following equation relating the rate of extraction and t...
	As a result, the rate of technological progress is three percent per year.
	To find the number of years it will take for the stock of the resource to be reduced to one-eighth of its initial level, we first define an arbitrary initial level as X(0) and utilize the equation for the stock of the resource at any time t.
	The stock of the resource must be one-eighth of its initial level, X(0). We can represent this by writing X(t)  (1/8)X(0). Substituting in this value, along with our given rate of extraction, we can solve for t.
	Taking the log of both sides and rearranging,
	Therefore, in approximately 208 years, the stock of the resource will be reduced to one-eighth of its initial level. Alternatively, we could just use the rule of 72, which says that if something shrinks at one percent per year, it will halve after 7...