Reference no: EM133198195
Mrs. Vernon owns a gift shop selling expensive glass objects. If an accident involving glass breakage occurs, she will suffer a loss of $4,000. Her income without accident is $13,000. She could choose either low effort ??1 = 0, or high effort ??2 = 1. The probability of accident is 0.75 when ??1 = 0 and it is 0.25 when ??2 = 1. Mrs. Vernon's utility function is ??(??, ??) = √?? - ??, where ?? denotes net wealth and ?? effort. The insurance company cannot monitor Mrs. Vernon's action.
a. If the insurance company offers Mrs. Vernon full insurance, which effort level can it expect from her? What will be the actuarially fair insurance premium? What will be Mrs. Vernon's expected utility?
b. Suppose the insurance policy stipulates a $2,000 deductible (i.e., Mrs. Vernon needs to pay the first $2,000 if there is an accident) and is priced actuarially fairly assuming high effort on the part of Mrs. Vernon. Show that Mrs. Vernon will indeed choose high effort, and that this will give her higher expected utility than in (a).
2. Ted is an attorney. Let ?? denote his effort and ?? his income. He has two levels of effort: ??1 = 1 and ??2 = 4. There are two levels of income: ??1 = 100 and ??2 = 400. The table below shows the probability of each income being realized conditional on his effort:
??1 = 100 ??2 = 400
??1 = 1 0.75 0.25
??2 = 4 0.25 0.75
Ted's utility function is given by ??(??, ??) = √?? - ??.
a. If Ted works by himself in a one-attorney firm, which effort level will he choose? What will be his expected utility?
b. Suppose Ted forms a partnership with another attorney, Noah. Noah is identical to Ted with respect to everything but his risks are independent of Ted's. The two partners share equally the total income of the firm. If Noah chooses ??2, what will be Ted's expected utility if he also chooses ??2? (Hint: What will be Ted's income prospect?)
c. Show that if Noah chooses ??2, Ted's expected utility will be higher if he himself chooses ??1 instead of ??2.
3. A firm is organized as a 2-person partnership. Each worker has a time endowment T = 24. Time can be spent either as leisure (x) or effort (e). The firm's profit function is ?? = 3(??1 + ??2 ), where ??i
In other words, if one partner exerts the high effort, the other has an incentive to shirk. To use game theory jargon, we can say that it's not a Nash equilibrium for both to choose high effort.
denotes Partner ??'s effort. Partner ??'s utility function is given by ?? = ????????, where ???? denotes his income.
a. What is each partner's leisure, effort, income and utility in the fair and efficient outcome? Draw a graph to illustrate.
b. What is each partner's leisure, effort, income and utility in the partnership equilibrium? Illustrate the equilibrium in the same graph that you drew in (a).
4. Solid Numbers, Inc. is an accounting firm organized as a partnership. There are 3 accountants. Each worker has time endowment T = 30. Time can be spent as leisure (x) or effort (e). Solid Number's profit function is ?? = 12(??1 + ??2 + ??3), where ???? denotes worker ??'s effort. Worker ??'s utility function is given by ?? = ????????, where ???? denotes his income.
a. What is each worker's leisure, effort, income and utility in the fair and efficient outcome?
b. What is each partner's leisure, effort, income and utility in the partnership equilibrium? (You can assume symmetry in equilibrium.)
5. Suppose the firm in Problem 3 is organized as an owner and an employee. Both workers still contribute to the production of the firm, but Worker 1 is the owner and Worker 2 the employee.
The owner offers the following contract to the employee: "You must exert ??^units of effort, and you will be paid ??^. If you choose any other level of effort you will be paid nothing." The owner incurs a cost of 12 for monitoring the employee's effort. What will be the value of ??^and ??^ in the contract?
You can assume that half of the monitoring cost is subtracted from each worker's income. What will be each person's leisure, effort, income and utility?