15. A worker with access to a competitive wage W-500, can choose between a high effort level e=1or a low effort level e=0 with respective costs c(e-1)=50, c(e=0)=0 and operates a stochastic technology yielding a high level product Q (as opposed to a low level product q) with probability P=1/3 if e=1, or with probability p=1/4 if e-0 (under circumstances making it socially efficient a high effort level). Then, the efficiency wage of optimal contract of a risk-neutral principal is: a) 100 b) 500 c) 1000 d) 1100. e) none of the previous statements a-d is correct.
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- Natasha has utility function u(I) = (10*I)0.5, where I is her annual income (in thousands). (a) Is she a risk loving, risk averse or risk neutral individual? She is [risk loving, risk adverse, risk neutral] , as her utility function is [concave, convex, linear] (b) Suppose that she is currently earning an income of $40,000 (I = 40) and can earn that income next year with certainty. She is offered a chance to take a new job that offers a 0.6 probability of earning $44,000 and a 0.4 probability of earning $33,000. She should [take, not take] the new job because her expected utility of (approximately) [18.27,19.82,20,20.95,21.14] is [greater than, less than, equal to] her current utility of [18.27,19.85,20,20.95,21.14] ., and you are considering a self-employment opportunity that may pay $10,000 per year or $40,000 per year with equal probabilities. What certain income would provide the same satisfaction as the expected utility from the self-employed position? a) $22,500 b) $15,000 c) $27,500 d) $25,00Suppose that Mira has a utility function given by U=2I+10√I. She is considering two job opportunities. The first job pays a salary of $40,000 for sure. The second job pays a base salary of $20,000 but offers the possibility of a $40,000 bonus on top of your base salary. She believes that there is a probability of p=0.50 that she will earn the bonus. What is the expected salary of the second job? Which offer gives Mira a higher expected utility? Based on this information, is Mira risk adverse, risk neutral, or risk-loving?
- Let W0 represents an individual’s current wealth and U(W) is this individual’s von Neumann-Morgenstern utility index (or utility function) that reflects how s/he feels about various levels of wealth. Assume this individual marginal utility of wealth decreases a wealth increases. Which of the following statements is true? a. This individual will prefer to keep his or her current wealth rather than taking a fair gamble. b. For this individual, a 50-50 chance of winning or losing c dollars yields less expected utility than does refusing the bet. c. This individual is said to be risk averse. d. All of the above.Consider the constant relative risk aversion utility of wealth function from Chapter 3 for an investor with gamma parameter equal to 0.25: U(W) = W^(0.25)/(0.25) = 4W^(0.25). Suppose this investor is faced with a 50-50 bet to receive nothing or to receive 1000 dollars. What's a fair price for this bet to the investor? I.e., what is the certainty equivalent wealth (CEW) associated with this bet, for this investorthere are two states of the world next period. State 1 occurs with probability π. The individual has a utility function U(C)=ln(C) and a subjective discount factor of 0.8. The utility maximising individual chooses consumption in the first period of 1.8 units. In the second period, consumption in state 1 is 2.5, and consumption in state 2 is 2. if the price of the contingent claim in state 1 divided by the price of the contingent price in state 2 equals 0.2, the calculate what π equals
- 1. Consider an insurance contract with the premium r=$200 and payout q=$800 a.) John has healthy-state income IH = $900 and sick-state income IS = $100. He has a probability of illness p = 0.2. Is the contract fair and/or full for John? b.) What is John’s expected income without this insurance contract? What is John’s expected income with this insurance contract?Abigail is a consumer whose utility is a function of her total wealth W. u(W ) = log W. Suppose that Abigail begins with initial wealth of A = 100 but will suffer a serious illness with probability π = 0.15 which will require extensive treatment costing L = 80. To hedge against this risk, Abigail considers buying a health insurance policy. She may buy as much insurance I as she wishes at a cost of p per dollar of coverage, so her payoffs in each state are Healthy Ill Probability 0.85 0.15 No Insurance 100 20 Claim 0 I Premium −pI −pI a) Show that Abigail is risk averse. b) Suppose that the insurance premiums are actuarially fair so that p = 0. Find Abigail’s expected wealth E[W ] and expected utility E[u(W )] as functions of how much insurance she buys I. c) How much insurance should Abigail buy?Abigail is a consumer whose utility is a function of her total wealth W. u(W ) = log W. Suppose that Abigail begins with initial wealth of A = 100 but will suffer a serious illness with probability π = 0.15 which will require extensive treatment costing L = 80. To hedge against this risk, Abigail considers buying a health insurance policy. She may buy as much insurance I as she wishes at a cost of p per dollar of coverage, so her payoffs in each state are Healthy Ill Probability 0.85 0.15 No Insurance 100 20 Claim 0 I Premium −pI −pI a) Now suppose that the insurance company raises premiums to p = 0.2 so that they are no longer actuarially fair. Find Abigail’s expected wealth E[W ] and expected utility E[u(W )]. b) How much insurance should Abigail buy now?
- V7 Consider a owner-manager problem in which πgross = 2e + ε [manager has control over e, ε are factors outside of manager’s control, ε~N(0,σ2 )] The owner pays the manager a salary of s out of the gross profits. Manager’s cost of effort = e2 /2. Manager has constant risk aversion utility function. σ 2 = 4 A = 1 a) What is the first-best outcome for manager utility, manager effort, and net profits of the owner? b) Now consider that the owner cannot observe manager effort and offers a salary tied to gross profits: s(πgross) = a + b πgross What is the second-best outcome for manager utility, manager effort, and net profits of the owner?An investor has a power utility function with a coefficient of relative risk aversion of 3. Compare the utility that the investor would receive from a certain income of £2 with that generated by a lottery having equally likely outcomes of £1 and £3. Calculate the certain level of income which, for an investor with preferences as above, would generate identical expected utility to the lottery described. How much of the original certain income of £2 the investor would be willing to pay to avoid the lottery? Detail the calculations and carefully explain your answer.Suppose an individual earns income $600 when they are sick, and $1000 when they are healthy. Suppose this individual is sick with probability p = 0.5 and has a utility function over income, I, of U(I) = ln(I). Is this individual risk-averse, risk neutral or risk-loving? Suppose she is able to purchase insurance at any amount from at an actuarially fair price. Fully describe the amount she would purchase (payout, premium and final outcomes). Verify that she is better off with the contract in part b, as opposed to being uninsured. Suppose insurance company A offers a payout q = $400 (when she is sick) at a premium of r = $220 and insurance company B offers a payout of $200 at a premium of $100. Company A's contract is: Actuarially fair or unfair? Is it full or partial insurance? Company B's contract is: Actuarially fair or unfair? Is it full or partial insurance? Which contract does this individual prefer? Suppose contract A is unfair, but offers full coverage at price . Contract B…