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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 ˆ
ˆ,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
2. a. Because the stock of fish
t
S
at time t is 20, we use the equation for the growth in the
quantity of fish to determine
.
t
G
60 Weil • Economic Growth, Second Edition
b. To find the optimal stock of fish in the lake, we differentiate
t
G
with respect to
.
t
S
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 2% 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
764
364
1,825
278,690
1,394,300
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
Chapter 16 Resources and the Environment at the Global Level 61
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.
6. a. Since individuals know that there will be extra oil coming online in five years, they know the
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
8. In the short run, a firm is unable to substantially change the mix of factors of production. Thus a
◼ 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,
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.
62 Weil • Economic Growth, Second Edition
Chapter 16 Resources and the Environment at the Global Level 63
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.
Taking the log of both sides and rearranging,
