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Microeconomic Questions: Optimization and Decision-Making

## Determining the Proportion of the Population Working in Different Occupations

1. In a population of individuals, there is a distribution of ìoccupation 1îability, which is uniformly distributed on the interval [0; 1]: If a type individual works in occupation 1, their wage income is w1 = r1:
All individuals are equally productive in occupation 2, and the wage paid there is w2 = r2:

The objective of each individual is to maximize their income.
1. If r1 = 8 and r2 = 4; Önd the proportion of the population working in occupation 1.
2. If r1 = 8 and r2 = 4; Önd the average wage in occupation 1.
3. Answer both (a) and (b) when r1 = 8 and r2 = 6:

2. An individual lives two periods, and their objective is to maximize their lifetime (sum of two period) income. The productivity of an individual at a given Örm is unknown until she works there, and can only be inferred from the productivity realizations she has at the Örm). There are two types of worker-Örm job matches, ìgoodî ones and ìbadî ones, and they are equally represented in the population (that is, the probability of Önding a good match is 0:5 which is the probability of Önding a bad match). At a bad match, the individual always produces two units of output, yt = 2 for all t: At a good match, the individual produces three units of output with probability 0:9 in any period, and produces 1 unit of output in any period with probability 0:1: Based on y1; she will attempt to determine whether the match is good or bad.

At the end of the Örst period, the individual observes and is paid her productivity in the period, y1: Based on that observation, she can then decide to try a di§erent Örm in the second period or remain at her Örst period Örm. Whatever she chooses, at the end of the second period her productivity y2 will be realized she will be paid that amount. To summarize, the output distributions in any period t are given by Probability distribution of output by type of job Output Level ìBadî job ìGoodî job
yt = 1 0 0:1
yt = 2 1 0
yt = 3 0 0:9

Note that the output realizations are independent over time. If someone in a ìgoodî job in period 1 drew an output level of 3; they still have a probability of a draw of 1 next period of 0.1.
1. Describe the optimal turnover decision, that is, determine who would leave their employer after period 1 and who would stay based on their output realization.
2. Determine the average wage earned in the population in both periods, y1 and y2:
3. An individual lives 3 periods, and seeks to maximize w1  c1 + w2  c2 + w3  c3; where wt is the wage in period t and ct is the cost paid to Önd a new job in period t: Whenever the individual wants to Önd a new job (including period 1), she must pay a cost of C = 3 (so that c1 = 2 for all individuals; if they kept that same job in periods 2 and 3; then c2 = c3 = 0). Whenever an individual Önds a job, their wage w at the job is determined by a draw from the Uniform distribution on the interval [0; 10]: Then the expected value of the wage at a job is given by Ew = 5:

1. Let the wage that the individual had in period 2 be given by w2: For what values of w2 would the individual decide to Önd a new job in period 3?
2. Let the wage that the individual had in period 1 be given by w1: For what values of w1 would the individual decide to Önd a new job in period 2?
3. Determine the average wage in period t; wt ; t = 1; 2; 3:
4. Determine the proportion of people Önding new jobs in period t; t ; where 1 = 1:
5. Does length of time at a job, known as job tenure, ìcauseîthe wage increases observed in periods 2 and 3? Why or why n