-
Solution to the practice problems
1. 25%. 1.65/(sqrt(1.44)*sqrt(1.21))=1.65/(1.2*1.1)=1.25.
2. A.
3. B.
4. A.
5. D. 6000/200=30 150/15=10.
6. A, D, E
7. C.
8. C.
9. C.
10. E.
11. 1/4. Sort people from the poorest to the richest. Then, the
first quarter of the
population will earn nothing. So, the Lorenz curve passes
through (1/4, 0). Lorenz
curve also goes through (0,0) and (1,1). Hence, the area between
the Lorenz curve and
the 45 degree line is just the area of a triangle with a base of
1/4 and a height of 1,
which is 1/2*1/4*1=1/8. The Gini coefficient is twice this area,
which is 1/8*2=1/4.
12. D.
13. 0.4. (9/10+8/10+3/10)/5=0.4
14. C.
15. D.
16. A.
17. 20%. -10+12/(1+r)=0. Solving, r=0.2
18. C.
19. C.
20. bch.
21. ej.
22. bfe.
23. D.
24. 5 1/2*1+1/2*9=5
25. 4 Expected utility=1/2*10.5+1/2*90.5=2. So, the certainty
equivalent amount
satisfies M0.5=2. Solving for M, we have M=4.
26. 5/9 if M=1 and 5 if M=9. We must have (M-CV) 0.5 90.5=M 0.5
40.5. If M=1,
(1-CV) 0.5=2/3. So, CV=5/9. If M=9, (9-CV) 0.5=2. So, CV=5
27. compensating, contingent, valuation
28.3/2 if M=1 and 45/4 if M=9. We must have M0.5 90.5=(M+EV) 0.5
40.5. If M=1,
(1+EV) 0.5=3/2. So, EV=5/4. If M=9, (9+EV) 0.5=9/2. So,
EV=45/4.
29. A, C.
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30. 2.5. 0.15*10+0.1*10=2.5
31. B.
32. A.
33. C.
34. 0.6. [(0.06+0)/(0.04+0.06)]=0.6
35. A.
36. Chips 1, French fries 0.5
37. A, E.
38. Chips 2, French fries 0
39. Chips 1, French fries 1
40. A. If both produce potatoes:
Expected Utility = [(4+4)/2]^0.5 = 2
If both produce wheat:
Expected utility = (2/5)(3/5)(2)[(16+0)/2]^0.5 +
(2/5)(2/5)[(16+16)/2]^0.5 +
(3/5)(3/5)[(0+0)/2]^0.5 = 1.998
If 1 produce wheat and 1 produce potato:
Expected utility = (2/5)(1)[(4+16)/2]^0.5 +
(3/5)(1)[(4+0)/2]^0.5 = 2.113
41. C.
42. D. Let p be the probability of good weather. The expected
utility from potatoes is
E[u]=4, and that from wheat is E[u]=p x 5 + (1-p) x 3 = 3+2p.
So, one prefers potato
production if and only if 4>3+2p, or p
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Ch1. 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, Richlands GDP per capita is 46 and Poorlands GDP per
capita is 13.
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
3 Richland dollars to 2 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 Richlands 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 3 Richland dollars equals 2
Poorland dollar, we
multiply 2/3 to Richlands GDP per capita, yielding 30.67
Poorland dollars. Thus,
the ratio of Richland GDP per capita to Poorland GDP per capita
is 2:36:1.
d. A natural basket to use is 6 computers and 1 ice cream. The
cost of this basket in
Richland
is 23 Richland dollars. The cost of this basket in Poorland is
13 Poorland dollars.
Equating
the costs of baskets to be one price, the purchasing power
parity exchange rate
must be
23 Richland dollars:13 Poorland dollars.
e. To find the ratio of GDP per capita between Richland and
Poorland, we must first
convert GDP denominations into the same currency. In the
analysis that follows, I
choose to convert GDP denominations into Poorland dollars, but
converting to
Richland dollars is equally correct, similar, and will yield the
same result. From
Part (a), we convert Richlands 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 23 Richland dollars equals 13
Poorland dollars, we
multiply 13/23 to Richlands GDP per capita, yielding 26 Poorland
dollars. Thus
the ratio of Richland GDP per capita to Poorland GDP per capita
is 2:1.
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Ch 3. 2. To find the steady-state value of the country, we refer
to Equation (3.3) on page 63.
1 11 .ssy A
=
Plugging in values: A = 1, = 0.5, = 0.5, and = 0.1, we get:
Simplifying the above equation, we get yss = 5.To find the
current output per
worker, we substitute in k = 900 into the production function to
get:
.
That is, the current output is 30 whereas the steady-state
output level is 5. Therefore, we conclude that ssy y> so the
country is above its steady-state level of output per worker.
-
4. This 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:
1
= 0.1,
2
= 0.2, and
setting =1/3.
For =1/2, we get,
= .50.
Therefore, when =1/3, the ratio is 0.707 and when =1/2, the
ratio is 0.50.
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,
1 11 1 0.5
2(1)(0.25) 5 25.0.05ss
Ak
= = = =
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 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 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.)
-
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.
Ch 4. 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
compare the children in
this treatment group to children in a control group. For
example, providing enhanced
education to the treatment group represents an exogenous
downward shock to the
cost of having higher quality children. Providing family
planning to a treatment group
would represent an exogenous downward shock to the quantity of
children. Using
twins would be a good natural experiment. Since twin births are
basically random,
they provide an identifying exogenous variation of quantity. One
can compare the
quality of children who were born as twins to the quality 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 one by the
other. The steady-state
level ratio for Country X to Y is given by:
We now substitute in the values
X
= 25%, n
X
= 0, and
X
= 10% for Country X, and for
Country Y, we use the values
Y
= 5%, n
Y
= 5%, and
Y
= 10%. Also, set =1/3 and A
i
=
A. This yields
-
Therefore, we conclude the ratio of Country Xs steady-state
level of income per capita
to Country Ys to be near 2.74.
8. a) TFR = 4.
NRR = (1/2) [(1 child) (Probability of reaching age 22) + (1
child)
(Probability of reaching age 26) + (1 child) (Probability of
reaching age
30) + (1 child) (Probability of reaching age 34)]
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
9. a. We graph the equation, 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 12 2 2
,Y L X X
yL 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,
0.L = Using this value and rearranging the first equation, we solve
for the steady-state value of income per capita:
0 100,
100ss
L y
y
= = =
Substituting in this value into the production
function, we back out the value of ssL as follows:
2 2
1,000,000100.
(100)ss ss
XL
y= = =
The steady-state population is 100.
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
i
= + + = + + =
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 i = + = + =
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.
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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 i = + = + =
Adjusting for , the net rate of reproduction for Country X is 2.
For Country Y,
29 39
20 30
( ) ( ) (1 0) (0.5 2.0) 0 1 1.i
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 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.
-
Ch5. 1. To calculate the population of Fantasia in the year
2001, we must first
find the population of each age group. Given 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); 80
three-year-olds
(100 0.8); 10 four-year-olds (100 0.1); 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 0.6 children.
Similarly, of 100 two-
year-olds, each person gives birth to 0.7 children and of the
100
three-year-olds, each person gives birth to 0.2 children.
Therefore, the
number of children born in 2001 will be 100 [0.6+0.7+0.2] = 150.
There will
be 150 zero-year-olds in 2001. Summing the population over each
age group
we get the total population in 2001.
150 + 100 + 100 + 80 + 10 = Total 2001 Population = 440.
5. In 2025: each of the 90 million 020 year-olds from 2005 will
have had 1 girl
and moved on to the next age bracket; two-thirds of the 60
million 2140
year-olds will have died; and all the 4160 year-olds will have
died. So, the
new female population structure will be 90 million 020
year-olds, 90 million
2140 year olds and 20 million 4160 year-olds. In 2045, this
process
continues, so we have 90 million 020 year-olds, 90 million 2140
year-olds,
and 30 million 4160 year-olds. In 2065, the same structure as
2045 exists.
6. 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.
-
Ch6 5. The payment to persons with no schooling (1%) is W; the
payment to individuals with
partial primary schooling (3.4%) is 1.65W, the payment to
individuals with complete
primary schooling (8.8%) is 2.43W; the payment to individuals
with incomplete
secondary schooling (22.1%) is 2.77W; the payment to individuals
with complete
secondary schooling (43.4%) is 3.16W; the payment to individuals
with incomplete
higher schooling (7.6%) is 3.61W; and the payment to individuals
with complete higher
schooling (13.7%) is 4.11W. Summing over the entire population,
we calculate the total
wages earned by the population:
[(1% 1)+(3.4% 1.65)+(8.8% 2.43)+(22.1% 2.77)+(43.4%
3.16)+(7.6%
3.61)+(13.7% 4.11)] W Labour Force = 3.1W Labour Force.
The average wage per worker in this economy is 3.1W.
As in Problem 4, the payment to human capital is the difference
between the total
wage and the wage for raw labor, which is 3.1W W = 2.1W.
Payment to Human Capital/Total Payment = 2.1/3.1 = 0.6774?
The fraction of wages paid to human capital is 67.74
percent.
6. The relative return to 10 years of schooling is 2.77, and the
relative return to 14 years of schooling is 3.61 (from table 6.2).
Denoting ih = 2.77 and jh = 3.61, we can solve for the steady-state
ratio for two countries identical in every respect expect for
education as follows:
Thus, the ratio of output per worker in the steady state is
0.767.
7. The relative return to 12 years of schooling is 3.16, and the
relative return to 4 years of schooling is 1.65. Writing the
steady-state ratio for one country over time and denoting h1950 =
1.65 and h2000 = 3.16, we get:
-
Thus, the ratio of steady-state output per worker for this
country over time is 1.92. If
over 100 years, the steady-state output has increased by a
factor of 1.92, 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 1.31
percent.
Ch 11. 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 11 .ss
w
y Ar
=
Given that the world rental price of capital doubles to ,wr
which is exactly2 ,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 11 1 1
1 1.
2 2 2ss ssw w wy A A A y
r r r
= = = =
And with a value of 0.3 for ,
-
That is, the new steady-state level of income per capita falls
to just over 74 percent of its
original level when the world rental rate of capital doubles in
an economy open to the
world capital market.
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.
Ch 12.1.
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.
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 of 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 ), as government collects from the supplier. Specifically,
100 and (1 ) .d sQ P Q P= =
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
= = +=
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
= =
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.P
Since the amount of the good sold is eq,Q we know that government
revenue is equal to eq eq.P Q 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
=
Recall that
2
2 2 22
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
=
=
Setting the above expression equal to zero, rearranging, and
dividing out common
terms,
-
2(1 2 )(2 ) ( 2)( ) . =
Therefore, solving the above equation gives us: (2 /3). = At
this tax rate, government revenue will be maximized.
6. 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
= = = =
Using the values,
(1/3), = = we now have two equations with two unknowns.
Working through the algebra and solving for the steady-state
values ofssxand
,ssk we
get:
3 2
3
3 2 2
3
(1 ), and
(1 )
ss
ss
Ax
Ak
=
=
Plugging in our steady-state values into the production function
will now yield the
steady-state level of output per worker.
1 11 1 3 2 2 3 2 33 33 3
3 3 2
(1 ) (1 )( )(1 ).ss
A A Ay Ak x A
= = =
b. The value of that will maximize output per worker is the same
value that will maximize
( )(1 ), as 3 2/A is a constant. Therefore, (1/ 2), = a tax rate
of 50 percent.
Ch 13 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.
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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
Cu
mu
lati
ve p
erce
nta
ge
of
ho
use
ho
ld in
com
e Line of Perfect Equality
A
B
C
D
Lorenz Curve
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 CoefficientA
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= = =
+ + +
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.
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.
Ch 15.3. a. Globalization removes a resource constraint that
otherwise would reduce income per capita in many countries. For
example, a country without good agricultural land can import food.
Globalization also means that some countries are rich based on
resources that can be exported (think of oil in Kuwait) that
otherwise would be poor if they had to rely on what could be
produced domestically.
b. One way in which globalization affects the relationship
between a countrys geographic characteristics and its level of
income per capita is through the access to trade. Countries
advantageously positioned geographically are likely to experience
higher growth due to increased trade. Conversely, countries that
are geographically isolated are likely to experience lower growth,
as access to trade is costly and limited. For example, Chinas
coastal cities, as well as countries with eased access to trade
have grown rapidly. Additionally, globalization may allow access to
better technologies and well-suited technologies that increase
income.
-
c. One way in which globalization may affect the relationship
between a countrys climate and its level of income per capita may
be through the transfer of technology that alleviates the
detrimental effects of climate. The U.S. South experienced high
growth rates in income per capita with the introduction of the air
conditioner. Although globalization is not the source of this
technology transfer, globalization likely is able to allow the
transfer of such technologies that affect the climate in a manner
that improves the income per capita of the country. Furthermore,
technologies may increase the productivity of agriculture, for
instance, given any level of climate.
4. a. Improvements in resource extraction are technological
advances that are responsible for the change in the relationship
between a countrys natural resources and income per capita.
Technological advances that allow efficient extraction of
previously unattainable oil have improved income per capita in
several countries.
b. Technological revolutions that reduce transportation costs
change the relationship
between a countrys geographical characteristics and income per
capita. Reduced transportation costs make possible profitable
access and participation in the world market that ultimately
increases income per capita. Falling costs of shipping, flying, and
rail are specific examples that have led to the integration of
geographically isolated areas, and thusly, increased the levels of
income per capita in these areas.
c. Air conditioning and DDT are instances of technological
revolutions that have
changed the relationship between a countrys climate and income
per capita. Air
conditioning allows for a higher rate of productivity and a
higher rate of human
capital accumulation in extremely warm climates. These factors
increase income per
capita. Also, DDT is a technological advance that eliminates the
prevalence of
malaria, a consequence of climate. Malaria eradication increases
the health and
productivity of a country, and thus raises the income per
capita.
Ch 16 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.
,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,
2%.R y L= = =
Therefore,
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 100t 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 = b. To find the
optimal stock of fish in the lake, we differentiate tG with
respect to .tS
(100 ) 100 21 .
100 100 50t 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 100t t
t
S SG
= = =
1 0.t t t t tS S S G H+ = = =
Thus, tG must equal ,tH implying that 25.tH =
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.