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# Suppose there are three types of individuals: high productivity (HP), medium productivity (MP) and low productivity (LP).

1. Suppose there are three types of individuals: high productivity (HP), medium productivity (MP) and low productivity (LP). LP and MP workers amount for a proportion of sizes qL and qM of the entire population respectively. Suppose that individuals can choose any schooling level between 0 and 11 years. Further suppose that education gives no productivity 2 enhancements, it only awards a diploma for each year approved. LP individuals find education to be more costly as they progress through the education system relative to MP workers, and MP, in turn, do so relative to HP individuals. That is, while each year of schooling costs HP individuals $20,000, each year of schooling costs MP individuals $40,000, and each year of schooling costs LP individuals $60,000. Firms pay according to the productivity they guess each individual has. Such inference is based depending on the schooling years workers can demonstrate they have by showing the diplomas they obtained. Hence, the (lifetime) salary payed is given by the following scheme:

Salary = 200, 000 + 5, 000y if y < 5

430, 000 + 5, 000y if 5 <= y < 10

600, 000 + 5, 000y if y >= 10

a). If there is no signaling mechanism (i.e., no diplomas), what is the average productivity the firm infers from the pool of all individuals? Remember, this should come from a weighted average of productivities. (Tip: the firm should assume that the proportion of LP in every year of schooling is qL/12, and the proportion of MP in every year of schooling is qM/12.)

b). What is the offer (that is, the payments offered to HP, MP, LP, and the years of schooling required of each) the firm should make if it wants to reach a separating equilibrium using schooling as a signaling mechanism in which HP, MP and LP types choose different levels of schooling? Show that each type of worker has no incentive to choose the schooling years chosen by any of the other two types.