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.

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

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.

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.

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.

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.