Assume a worker (W) can produce output (X) for an employer by increasing his eff

Question

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?

Explanation / Answer

This is a very strange problem because of how the variables are defined. There must be an error in how the problem was copied here or an error in how it was written. (a) p = Y – X, where p is profit, Y is compensation paid by the employer, and X is the output. The only solution is : Max(p)=0 The profit is always at a maximum of zero. That is, if the producer produces X units, then the most it could afford to pay the worker in compensation is Y=X since the price of the good produced is 1. If Y=X, then profit it zero. (b) All pay schedules will yield a profit of zero. See part (a) (c) No loss in profits. p=0 (d) No. p=0.

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