Derive the optimal contract, Microeconomics

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Assume that in the market there exist two types of workers where the principle cannot distinguish types. The two types only differ with respect to the disutility of effort. The disutility is either e^2 or 2e^2.

The utility function for worker one is

U^1(w,e) = w - e^2

and for worker two

U^2(w,e) = w - 2e^2

both types of workers have reservation utility zero

The probability that the worker is of type 1 is q = 1/2

An interested firm is risk neutral and has profits Π (w,e) = e - w

a) Which is the good high productive worker and which is the low productive worker

b) Derive the optimal contract for the firm if it had perfect info about the workers type. What effort levels are demanded and what wages paid. Calculate the firm's profits of both types were employed

c) Formulate the problem when an adverse problem is present

d) Derive the second best contract and calculate the firm's profits

e) Compare the cases of symmetric and assymetric info

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