13.10: Exercises for Chapter 13

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EXERCISE 13.1

Georgina is contemplating entering the job market after graduating from high school. Her future lifespan is divided into two phases: An initial one during which she may go to university, and a second when she will work. Since dollars today are worth more than dollars in the future she discounts the future by 20%, that is the value today of that future income is the income divided by 1.2. By going to university and then working she will earn (i) -$60,000; (ii) $600,000. The negative value implies that she will incur costs in educating herself in the first period. In contrast, if she decides to work for both periods she will earn $30,000 in the first period and $480,000 in the second.

If her objective is to maximize her lifetime earnings, should she go to university or enter the job market immediately?
If instead of discounting the future at the rate of 20%, she discounts it at the rate of 50%, what should she do?

EXERCISE 13.2

Imagine that you have the following data on the income distribution for two economies.

	Quintile share of total income
First quintile	4.1	3.0
Second quintile	9.6	9.0
Third quintile	15.3	17.0
Fourth quintile	23.8	29.0
Fifth quintile	47.2	42.0
Total	100	100

On graph paper, or in a spreadsheet program, plot the Lorenz curves corresponding to the two sets of quintile shares. You must first compute the cumulative shares as we did for Figure 13.3.
Can you say, from a visual analysis, which distribution is more equal?

EXERCISE 13.3

The distribution of income in the economy is given in the table below. The first numerical column represents the dollars earned by each quintile. Since the numbers add to 100 you can equally think of the dollar values as shares of the total pie. In this economy the government changes the distribution by levying taxes and distributing benefits.

Quintile	Gross income $m	Taxes $m	Benefits $m
First	4	0	9
Second	11	1	6
Third	19	3	5
Fourth	26	7	3
Fifth	40	15	3
Total	100	26	26

Plot the Lorenz curve for gross income to scale.
Now subtract the taxes paid and add the benefits received by each quintile. Check that the total income is still $100. Calculate the cumulative income shares and plot the resulting Lorenz curve. Can you see that taxes and benefits reduce inequality?

EXERCISE 13.4

Consider two individuals, each facing a 45 year horizon at the age of 20. Ivan decides to work immediately and his earnings path takes the following form: Earnings =20,000+1,000t–10t², where the t is time, and it takes on values from 1 to 25, reflecting the working lifespan.

In a spreadsheet enter values 1... 25 in the first column and then compute the value of earnings in each of the 25 years in the second column using the earnings equation.
John decides to study some more and only earns a part-time salary in his first few years. He hopes that the additional earnings in future years will compensate for that. His function is given by 10,000+2,000t–12t². In the same spreadsheet compute his annual earnings for 25 years.
Plot the two earnings functions you have computed using the 'charts' feature of Excel. Does your graph indicate that John passes Ivan between year 10 and year 11?

EXERCISE 13.5

In the short run one half of the labour force has high skills and one half low skills (in terms of Figure 13.2 this means that the short-run supply curve is vertical at 0.5). The relative demand for the high-skill workers is given by , where W is the wage premium and f is the fraction that is skilled. The premium is measured in percent and f has a maximum value of 1. The W function thus has vertical and horizontal intercepts of .

Illustrate the supply and demand curves graphically, and illustrate the skill premium going to the high-skill workers in the short run by determining the value of W when f=0.5.
If demand increases to what is the new premium? Illustrate your answer graphically.

EXERCISE 13.6

Consider the foregoing problem in a long-run context, when the fraction of the labour force that is high-skilled is more elastic with respect to the premium. Let this long-run relative supply function be .

Graph this long-run supply function and verify that it goes through the same initial equilibrium as in Exercise 13.5.
Illustrate the long run and short run on the same diagram.
What is the numerical value of the premium in the long run after the increase in demand? Illustrate graphically.