The economy of ShortLife has two kinds of jobs, which are the only sources of income for the people. One kind of job pays $200, the other pays $100. Individuals in this economy live for two years. In each year, only half the population can manage to get the high-paying job. The other half has to be content with the low-paying one. At the end of each year, everybody is fired from existing positions, and those people assigned to the high-paying job next year are chosen randomly. This means that at any date, each person, irrespective of past earnings, has probability 1/2 of being selected for the high-paying job.

(a) Calculate the Gini coefficient based on people’s incomes in any one particular period and show that it suggests a good deal of inequality Now calculate each person’s average per period lifetime income and compute the Gini coefficient based on these incomes. Does the latter measure suggest more or less inequality? Explain why.

(b) Now change the scenario somewhat. Suppose that a person holding a job of one type has probability 3/4 of having the same kind of job next year. Calculate the expected lifetime income (per year average) of a person who currently has a high-paying job, and do the same for a person with a low-paying job. Compute the Gini coefficient based on these expected per period incomes and compare it with the measure obtained in case (a). Explain the difference you observe.

(c) Generalize this idea by assuming that with probability p you hold your current job, and with probability 1 − p you change it. Find a formula for inequality as measured by the Gini coefficient for each p, examine how it changes with p, and explain your answer intuitively.