PROBLEM (3) (Moral hazard) A firm owns expensive machinery to be operated by a worker. If the worker operates very carefully (works hard, e=1) the value of the machinery (net of depreciation) at the end of the year would be $100k with probability %90, and $40k with the remaining probability %10. If the worker works carelessly (shirks, e=0) the probabilities are %50 and %50 respectively. The worker experiences a disutility from having to work carefully that is worth $10k. Working carelessly has no disutility cost for him. The worker also has an outside option of quitting and working for another company that gives him $50k, without any disutility cost of carefulness whatsoever. The worker is risk neutral, and cares about the expected income she will make minus the disutility cost $10k if she is working carefully. The firm is also risk neutral, and cares about the expected value of the machinery minus the worker's compensation payments. (a) If the firm can observe whether the worker is working carefully or not and contract upon it, how much should the firm offer the worker so that she stays with the firm (and works carelessly)? Stay with the firm and work hard? (b) What is the expected utility of the firm in each case in (a), when the firm pays the worker those minimum wages you calculated in (a)? (c) Suppose the firm cannot observe the care level of the worker but can observe the machinery's worth at the end of the year, and can pay the worker accordingly; say firm pays worker ws if the machinery is worth $100k, w if it is worth $40k. Write down the inequalities the compensation scheme wu , wi should satisfy so that the worker would prefer to stay with the firm and work carefully (rather than carelessly)? (d) Naturally, to maximize their expected utility, the firm will choose the wu, wi pair (or pairs; there may be many) from among the ones you have identified in (c) to minimize the expected compensation. Writing down this problem (minimize expected compensation subject to contraints you found in (c)) solve for the optimal compensation scheme(s). (e) Compare the minimal expected compensation the worker receives on average in (d) with the compensation in (a) (when firm makes the worker work carefully). What is the information rent (agency cost) in this case