Question

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 shovw 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

Answer 1

The Gini coefficient measures income inequality, which is high in the ShortLife economy for single-period earnings but nonexistent when averaged over lifetimes. With increased job security (scenario b), inequality is reduced compared to scenario a. Generally, as the probability of retaining one's current job increases, the Gini coefficient decreases, indicating less income inequality.

Step-by-step explanation:

The question pertains to the Gini coefficient, a measure of income inequality within a society. For scenario (a), with a 50% chance of getting either job each year, the incomes for the individuals for two years could be $200+$200, $200+$100, $100+$200, or $100+$100, with equal probabilities for each combination. To calculate the Gini coefficient for one period, we only consider incomes of $200 and $100, which would result in a high level of inequality for that period. For average per period lifetime income, each person expects to earn ($200+$100)/2 = $150 per year, indicating identical incomes for everyone and a Gini coefficient of 0, which suggests no inequality.

In scenario (b), if a person has a high-paying job there is a 3/4 probability they will retain it next year, giving them an expected income of $200 + 0.75*$200 + 0.25*$100. Conversely, someone with a low-paying job would have an expected income of $100 + 0.75*$100 + 0.25*$200. Calculating the Gini coefficient based on these expected incomes would reveal less inequality than in part (a), as earnings are more stable over time.

For scenario (c), we can generalize the formula for expected lifetime income with probability p for retaining current jobs. As p increases, the stability of income increases, decreasing the Gini coefficient, hence reducing income inequality. The formula for the Gini coefficient would thus depend on the variances in the expected incomes, becoming smaller as p approaches 1.