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 market