Economic Growth 3rd Edition Weil Solutions Manual
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Chapter 4 Population and Economic Growth requiri ng a computer or calculator Note: Special icons in the margin identify problems requiring requiring calculus
and those
.
Solutions to Problems 1.
To find the average growth rate of the population, we use the following equation: L (1 g) g )n L , t
t n
where Lt is the population the population at time time t , g is the growth rate, and n is the duration of growth. Substituting L L 2; 2 ; in n 100,000; t and t n 7 billion, billion, we we can can rewrite rewrite and solve solve the equation equation for for the growth growth rate. rate.
2(1 g) g )100,000 7, 000, 000, 000, 1
7,000, 000,000 100,000 7,000, 1 0.000219. 2
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 inco me per capita shifts outward. At this point, we are at Point B. Here, population growth growth will be positiv positivee because because of the high level of income income per capita. capita. As population population grows grows and income 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.
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b. We begin begin from Point A, where where the population population size is stable with with no growth. 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 have half the population population with a higher higher income per capita level allowing allowing population 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 with no growth. With 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.
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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. In a randomized controlled trial, one would have to randomly vary either the quantity or the quality of children in a treatment group and co mpare the children in this treatment group to childr en in a control group. For example, providing enhanced education to the treatment group represents an exogenou s downward shock to the cost of having higher qualit y children. Providing family planning to a treatment group would r epresent an exogenous downward shock to the quantity of children. Using twins would be a good natural experiment. Since
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twin births are basically random, they provide an identif ying exogenous variation of quantity. One can compare the quality of children who wer e born as twins to the qualit y of children who were born alone. 5. 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
y
1
X
A 1 n X X Y A
X , ss ss
yY ,
ss ss
X
Y
nY
Y
0, and X 5% for Country X, and for Countr y Y, we use X 20%, n X 0, the values Y 5%, nY 4%, and 5%. Also, set 1 / 3 and A X AY . This yields, We now substitute in the values
Y
1 (1 / 3) 1 (1 / 3
0.2
y X , ss ss
y ss Y , ss
0 0.05
0.05
0.04 0.05
1 2
1 2 36 4 2.68.
5 9
5
Therefore, we conclude the ratio of Country X to Y in their steady-state levels of inco me per capita to be near 2.68.
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6. Because the equation for capital capital accumulation suggests that in the steady state,
f (k ) ( n )k , 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. equilibrium. It is is important to note that a country cannot transition from A to B or vice versa.
7.
Country A and B are identical in every every respect but for their population growth growth rates. That is, n A n B . However, this implies that their respective steady states are not equal. Writing the reduced ratio equation, as on page 98, we get:
y A, A, ss ss
y
, ss B ss
n 1
A n A B
B
.
y , ss . Utilizing the fact that n A n B and A B , we can conclude that y A, A, ss ss ss 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. B
8.
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) b.
[(2/3) (2/3) (1/3) (1/3)] 1 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
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We graph the equation, Lˆ y 100, in the figure below.
First, we divide divide both sides of the the production production function function by L by L and rearrange to get: 1
1
1
Y L2 X 2 X 2 , y L L L Therefore, L
X y 2
.
For X For X 1,000,000 the figure is shown below.
c.
In the steady state, the growth rate of population is zero, zero, Lˆ 0. Using this value and rearranging the first equation, we solve for the steady-state value of income per capita: Lˆ 0 y 100, y ss 100
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Substituting in this value into the production function, we back out the value of L ss as follows: L ss
X 2 ss
1, 000, 000 100. (100)2
The steady-state population is 100. a. The steady-state level of income per worker is characterized by yˆ 0. 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:
10.
AX ln( y ln( y)) ln( Ax ln( Ax)) ln ln( A)) ln( X ln( X ) ln( L), L), ln( A L d d ln( y ln( y)) ln( A ln( A)) ln( X ln( X ) ln( L), L), dt dt ˆ ˆ yˆ A X . L
Therefore, the growth rate of income per worker must equal the growth rate of productivity plus the growt h rate of lan d minus the population growth rate. Since land X land X , and productivity productivity A A,, are ˆ ˆ L , y ˆ constant, X These values tell us that and because we are interested in the steady A 0.
state value of income per capita, yˆ 0 Lˆ . Now, we plug plug the growth growth rate of of population population into the equation that relates population growth to income-per-capita to arrive at our solution. y 100 , 100 y ss 100. 100. Lˆ 0
b.
Referring back to our our equation relating the growth rate of income income per capita, productivity, productivity, land, ˆ ˆ and population, we can set X 0 and A 0.1.
ˆ ˆ 0.1 Lˆ . yˆ A X L In the steady-state level of income per worker, yˆ must equal zero. Thus, we have that L that Lˆ must
equal 0.1. Using this value to solve for y ss in the equation relating population growth and income per capita, capita, Lˆ 0.1
y 100 , 100
y ss 110. 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.
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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: 80
30
0.5dx 30 25 55. 55. 1dx 0.5dx 30
0
Equivalently, one could find the area using geometry. B H (40 20)(1) 20.
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: 79
29
(i) 1 0.5 0 30 25 0 55. i
0
80
30
Therefore, the life expectancy at birth is 55 years. b. To calculate calculate the total fertility fertility rate, we must must find the area under under the age-spe age-specific cific fertility fertility rate function. This amounts to solving equation: 40
1dx 20.
20
Equivalently, one could find the area using geometry. B H (40 20)(1) 20.
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: 39
19
F (i ) 0 1 0 0 20 0 20. F( i
0
20
40
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 be expected expected to give birth to. First First noting that the probability of being alive 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 alive from age 30 to 39 is 0.5 with with the age-spec age-specific ific fertility rate at one one child per woman, woman, we solve the following equation: 39
29
F (i) (1 1) (0.5 1) 10 5 15. (i) F( i
20
30
That is, the rate of reproduction 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 . ,
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A.2.
Chapter 4
Population and Economic Growth
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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, 39
29
F (i ) (1 0.2) (0.5 (i) F( i
20
30
0) 2 0 2.
Adjusting for the net rate of reproduction for Country X is 2. For Country Y, ,
29
39
20
30
F (i) (1 0) (0.5 2.0) 0 1 1. (i) F( i
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 t he 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 probability of being being alive alive changes changes in those those 10 years, years, we have a difference difference in in the net net rate of reproduction. ,
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