Assume a worker (W) can produce output (X) for an employer by increasing his effort (E) according to the following equation: X = (1.5)E. Assume that the compensation paid by the employer (Y) increases the utility of the worker, while the amount of effort decreases his utility as such: U(E,Y) = Y - (E2)/2. Assume that price of output is 1 and that labor costs are the only costs of production, such that total profit, p, equals p = Y - X. Assume that the next-best option for the worker provides them a utility of zero.
(a) Given only the information provided above, what is the maximum amt of profit the firm can earn?
(b) If the employer did not want to stipulate the output (and cannot verify the effort level), then, using a pay-schedule, what level of base-pay will render the same level of profit (as in (a)) when the piece-rate pay is 1?
(c) Assume that the worker's risk-aversion level is measured by R = 2. Also, assume that output is stochastic, such that the expected output from a given level of effort is E[X] = (1.5)E - E[Z], where the average value of Z is E[Z] = 0 and its variance is Var[Z] = 2. What is the loss of profits for the employer if they choose a new optimal B, while keeping the piece-rate pay at 1?
(d) Can a different piece-rate pay be chosen by the employer such that their profits are maximized under the conditions of question (c)? If so, what is this optimal level of P?