University of California, Davis Department of Economics Giacomo Bonanno Economics 03: Economics of uncertainty and information PRACTICE PROBLEMS oooooooooooooooo Problem :.. Expected value Problem :.. Expected value Problem 3:.. Attitude to risk Problem :.. Attitude to risk Problem 5:.. Insurance contracts Problem 6:.. Expected utility Problem 7:.. Expected utility Problem 8:.. Expected utility Problem 9:.. Expected utility Problem 0:. Attitude to risk Problem : Attitude to risk Problem :. Risk premium Problem 3: Risk premium Problem : Measure of risk aversion Problem 5: Measure of risk aversion Problem 6: Binary lotteries Problem 7:. Binary lotteries Problem 8: Insurance Problem 9: Insurance Problem 0:. Insurance Problem : Insurance Problem : Insurance Problem 3: Insurance Problem : Risk sharing Problem 5: Risk sharing Problem 6: Risk sharing Problem 7: Risk sharing Problem 8: Risk sharing Problem 9: Risk sharing Problem 30: Principal-Agent Problem 3: Principal-Agent Problem 3: Choice of education Problem 33: Signaling Problem 3: Signaling Problem 35: Adverse selection Problem 36: Adverse selection Problem 37: Adverse selection Problem 38: Adverse selection Problem 39: Adverse selection in insurance Problem 0: Moral hazard in insurance Problem : Moral hazard in insurance Problem : Moral hazard in insurance Problem 3: Search Problem : Search Problem 5: Search Problem 6: Search **** PRACTICE PROBLEM (Expected Value) **** Consider the following lottery: 8 6. What is its expected value? 6 6 6 6 Page of 0
**** PRACTICE PROBLEM (Expected Value) **** z z z3 Consider the following lottery:, where z = you get an invitation to dinner at the White House z = you get to ride a motorcycle with Arnold Schwarzenegger z 3 = you get $600. What is the expected value of this lottery? **** PRACTICE PROBLEM 3 (Attitude to risk) **** Shirley owns a house worth $00,000. The value of the building is $75,000 and the value of the land is $5,000. In the area where she lives there is a 0% probability that a fire will completely destroy the building in a given year (on the other hand, the land would not be affected by a fire). An insurance company offers a policy that covers the replacement cost of the building in the event of fire. There is no deductible. The premium for this policy is $7,500 per year. What attitude to risk must Shirley have in order to buy the insurance policy? Explain your answer. **** PRACTICE PROBLEM (Attitude to risk) **** Bill s entire wealth consist of the money in his bank account: $,000. Bill s friend Bob claims to have discovered a great investment opportunity, which would require an investment of $0,000. Bob does not have any money and asks Bill to provide the $0,000. According to Bob the investment could yield a return of $50,000, in which case Bob will return the $0,000 to Bill and then give him 50% of the remaining $0,000. According to Bob the probability that the investment will be successful is %; The probability that the intial investment of $0,000 will be completely lost is 88%. Bill decides to go ahead with the investment and gives $0,000 to Bob. What is Bill s attitude to risk? **** PRACTICE PROBLEM 5 (Insurance contracts) **** Consider the following diagram. The probability of the bad state (or loss) is 0%. NI is the no insurance point. (a) Interpret each point (including NI) as an insurance contract and express it in terms of premium and deductible. (b) Calculate the expected profit from each contract. (c) Find the equation of the isoprofit line that goes through each contract (including the one that goes through point NI). Page of 0
W wealth in the good state,600,556,80 NI A B o 5 line C,0,390 W wealth in the bad state **** PRACTICE PROBLEM 6 (Expected Utility) **** [Notation: in this and the following problem sets we shall simplify the notation as follows: if, in a lottery, a prize is assigned zero probability, we will drop it from the list. Thus, for example, 50 300 60 300 60 the lottery will be written more simply as 0 Ben is offered a choice between A = 000 0 0.8 0. ] and B = 3000 He chooses B. Which of the following will Ben choose if he satisfies the axioms of expected utility? C = 000 0 0. 0.8 and D = 3000 0 0.5 0.75 **** PRACTICE PROBLEM 7 (Expected Utility) **** Consider the following lotteries: $3000 $000 $000 $500 L = 0 0 L $3000 $000 $000 $500 = 0 0 5 6 6 3 3 L 3 = $3000 $000 $000 $500 L = $3000 $000 $000 $500 0 0 Jennifer says that she is indifferent between lottery L and getting $000 for sure. She is also indifferent between lottery L and getting $000 for sure. Finally, she says that between L 3 and L she would choose L 3. Is she rational according to the theory of expected utility? [Assume that she prefers more money to less.] Page 3 of 0
**** PRACTICE PROBLEM 8 (Expected Utility) **** Consider the following basic outcomes: z = dinner at the White House z = free -week vacation in Europe z 3 = $800 z = a grade of A+ in Ecn 03. Rachel says that her ranking of these outcomes is as written above (z better than z, z better than z 3, etc.). She also says that she is indifferent between z and z z 8 0 0 z z z3 and also that she is indifferent between and. If she satisfies the axioms of expected utility theory, which of the two lotteries L and L below will she choose? z z z3 z L 3 8 8 8 8 and z z z3 L 3 5 5 5 **** PRACTICE PROBLEM 9 (Expected Utility) **** Paul s von Neumann-Morgenstern utility-of-money function is U(m) = ln(m), where ln denotes the natural logarithm. Consider the following two lotteries. $30 $8 $ $8 $8 $30 $8 $8 L and L 5 0 0 0 0 0 0 0 0 (a) What is their expected value? (b) Which of the two does Paul prefer? (c) Plot the function ln(m) for m > 0. d ln( m) (d) Calculate dm. (e) Calculate d ln( m). dm **** PRACTICE PROBLEM 0 (attitude to risk) **** What attitude to risk is displayed by the following utility of money functions? (i) x (the square root of x); (ii) ln(x) (x > 0) (the natural logarithm of x); (iii) x ( x 0 ); (iv) 5x+ ( x 0 ). Page of 0
**** PRACTICE PROBLEM (attitude to risk) **** Let x denote the amount of money (measured in millions of dollars) and suppose that it varies in the interval [0,]. John s utility-of-money function is given by: U(x) = - x + x -. (i) Is he risk-neutral, risk-loving or risk-averse? (ii) Jennifer, on the other hand, has the following utility function: What is her attitude to risk? V(x) = 3 (x x). (iii) Do they have the same attitude to risk? Do they have the same preferences for lotteries? (iv) Give an example of two utility functions that display the same attitude to risk but not the same preferences for lotteries. (v) Compare Jennifer s attitude to risk to John's using the Arrow-Pratt measure of absolute risk-aversion. **** PRACTICE PROBLEM (risk premium) **** 8 6 Amy faces the following lottery 3 and tells you that she considers it 0 6 6 6 equivalent to getting $8. (a) What is the risk premium associated with this lottery for Amy? (b) Is Amy risk-loving, risk-neutral or risk-averse? **** PRACTICE PROBLEM 3 (risk premium) **** Bill is risk neutral. How does he rank the following lotteries? 8 6 80 0 90 L =, L = 7 6 6 6 6 0 0 0 (b) What is the risk premium associated with lottery L for Bill? (c) What is the risk premium associated with lottery L for Bill? Page 5 of 0
*** PRACTICE PROBLEM (Measures of risk aversion) *** Jennifer s von Neumann-Morgenstern utility-of-money function is $30 $8 $ $8 $8 U(m) = m. Consider the following lottery: L 0 0 0 0 0 (a) What is the expected value of L? (b) What is the expected utility of L? (c) What is the risk premium associated with L? du ( m) (d) Calculate dm. d U ( m) (e) Calculate. dm (f) Is Jennifer risk-averse, risk-neutral or risk-loving? (g) Calculate the Arrow-Pratt measure of absolute risk aversion R A (m). (h) Evaluate R A (m) at m = $900 and at m = $,600 ****PRACTICE PROBLEM 5 (Measures of risk aversion) **** Suppose that John has the following von Neumann-Morgenstern utility-of-money function: V ( m) 0 m. Consider the same lottery L as in the previous question. Answer questions (a) (h) for John and comment on the results. **** PRACTICE PROBLEM 6 (Binary lotteries) **** $ y $ z Consider all the lotteries of the form with y 0 and z 0. Consider an 3 3 individual with von Neumann-Morgenstern utility-of-money function U ( m) ln( m). $0 $0 (a) Calculate the expected utility of lottery A 3 3 $0 $0 (b) Calculate the expected utility of lottery B 3 3 (c) In the (y,z)-plane draw the indifference curve that goes to point A = (0,0) and calculate the slope of the indifference curve at that point. (d) In the (y,z)-plane draw the indifference curve that goes to point B = (0,0) and calculate the slope of the indifference curve at that point. Page 6 of 0
**** PRACTICE PROBLEM 7 (Binary lotteries) **** Repeat (a)-(d) of the previous question for the case of an individual who is risk neutral. **** PRACTICE PROBLEM 8 (Full insurance) **** Frank has a wealth of $W. With probability p = he faces a loss of $x. The maximum 0 he is willing to pay for full insurance is $800. The risk premium associated with the lottery corresponding to no insurance is $500. What is the value of x? **** PRACTICE PROBLEM 9 (Full insurance) **** Bob owns a house. The value of the land is $85,000 while the value of the building is $0,000. Bob lives in an area where there is a 5% probability that a fire will completely destroy his house during any year. Bob s utility function is given by: U ( m) 800 (0 m) where m denotes money measured in $0,000 (thus, for example, m = means $0,000). An insurance company offers full-insurance policies (i.e. no deductible). (i) (ii) (iii) What is Bob s expected loss if he does not buy insurance? What is Bob s expected wealth if he does not buy insurance? What is Bob s expected utility if he does not buy insurance? (iv) What is Bob s expected utility if he buys full insurance for a premium of $5,500? (v) Would Bob buy insurance if the annual premium were $5,500? (vi) What is the maximum premium that Bob would be willing to pay? **** PRACTICE PROBLEM 0 (Insurance deductible) **** Barbara has a wealth of $80,000 and faces a potential loss of $0,000 with probability 0%. Her utility-of-money function is U ( m) m. An insurance company offers her the following menu of policies: premium deductible $,30 $500 $, 80 $, 000 $, 0 $,500,60 $,000 (a) What is Barbara s expected utility if she does not insure? (b) For each policy calculate the corresponding expected utility and determine which policy Barbara will choose. Page 7 of 0
**** PRACTICE PROBLEM (Insurance) **** An individual s utility-of-money function is given by u( z) a b e z (where z is the amount of money and a and b are positive constants; recall that e = x de x.788... is a mathematical constant (like ) and that e dx ). (i) What is the individual s attitude to risk? (ii) Calculate the individual s index of absolute risk aversion. (iii) Show that if the individual has initial wealth w and is faced with a potential loss x with probability p, the maximum premium he is willing to pay for full insurance is the same whatever his initial wealth, that is, is independent of w. **** PRACTICE PROBLEM (Insurance) **** You have the following vonneumann-morgenstern utility-of-money function (z is money, ln is the natural logarithm): u(z) = ln(z) An insurance company offers the following menu of choices: if you choose deductible D 0 then your premium is h = 0.8 0. D. Determine the amount of deductible you will choose and the premium you will pay if your initial wealth is w = $0, the size of the potential loss is x = $ and the probability of loss is p = /6. **** PRACTICE PROBLEM 3 (Insurance) **** You have the following vonneumann-morgenstern utility-of-money function: (z is money, ln is the natural logarithm). u(z) = ln(z) An insurance company offers the following price schedule: if you choose deductible D 0 then your premium is h = p( + k)(x D), where p is the probability of loss, k is a positive constant and x is the size of the loss. Determine the amount of deductible you will choose and the premium you will pay in the following cases: (a) (b) your initial wealth is w = $0, the size of the potential loss is x = $, the probability of loss is p = /6 and the value of k is /5. your initial wealth is w = $, the size of the potential loss is x = $6, the probability of loss is p = /9 and the value of k is /8. Page 8 of 0
**** PRACTICE PROBLEM (Risk sharing) **** A Principal wants to hire an Agent to run his firm. The Principal s utility-of-money function is U(m) = ln(m), while the Agent s utility-of-money function is V(m) = m +. There are three possible profit levels: x = $,00, x = $,600 and x 3 = $900. The corresponding probabilities are p, p and p3. Let w 5 5 5 be the payment to the Agent if the outcome is x and similarly for x and x 3. Show that the contract A ( w 700, w 700, w3 700) is Pareto dominated by the contract B ( w, 00, w, 000, w 00). 3 **** PRACTICE PROBLEM 5 (Risk sharing) **** John s von Neumann-Morgenstern utility-of-money function is U(m) = m. He owns a firm and is thinking of hiring Joanna to run the firm for him. Joanna s von Neumann-Morgenstern utility-of-money function is V ( m) m. In the past there were good times when the firm s yearly profits were $,000 and bad times when the firm s yearly profits were $,600. About 3 of the time it was a good year and of the 3 time it was a bad year. He offered Joanna a contract, call it A, that pays her a fixed salary of $,000 a year. (a) What is John s attitude to risk? (b) What is Joanna s attitude to risk? (c) Show that the alternative contract, call it B, that pays Joanna $,500 if the profit turns out to be $,000 and pays her nothing if the profit turns out to be $,600, is as good as contract A for Joanna but gives John a higher expected utility. That is, contract B Pareto dominates contract A. (d) Is contract B Pareto efficient? **** PRACTICE PROBLEM 6 (Risk sharing) **** The owner of a firm (the Principal) hires an Agent to manage the firm. The Principal s utility-of-money function is U ( m) ln( m) (where ln is the natural logarithm), while the x 00 Agent s utility-of-money function is V ( m) 00 e (where e is the number.788 ). The profit of the firm is affected by random events and can turn out to be x =,000 (and this is expected to happen with probability ) or x = $600 (with probability 3 ). The Principal offers the following contract to the Agent: if the profit turns out to be x, I ll pay you w = $00, otherwise I will pay you w = $00. (a) What is the attitude to risk of the Principal? (b) What is the attitude to risk of the Agent? (c) Is the proposed contract Pareto efficient? How do you know? Page 9 of 0
(d) If you were to propose a Pareto superior contract (that is, a contract that both Principal and Agent prefer to the one considered above) which of the following suggestions would you make? (a.) Increase both w and w, (a.) decrease both w and w, (a.3) increase w and decrease w, (a.) decrease w and increase w. Prove your claim using an Edgeworth box. **** PRACTICE PROBLEM 7 (Risk sharing) **** Mr. P wants to hire Ms A to run his firm. If Ms. A works for Mr. P, one of two outcomes will occur: the profit of the firm will be $50 this occurs with probability 3 0 or it will be $00 with probability 7 0. Mr. P s vonneumann-morgenstern utility-of-money function is U(y) = y while Ms. A s vonneumann-morgenstern utility-of-money function is V(w) = 7 3w 6 Consider the following contract: P and A agree that A will get $50 if the profit of the firm turns out to be $50, while she will only get $80 if the profit of the firm turns out to be $00. (i) (ii) (iii) contract. What is P s expected utility from this contract? What is A s expected utility from this contract? Is there a different contract that both P and A prefer? Find a better **** PRACTICE PROBLEM 8 (Risk sharing) **** Consider the following Principal-Agent model. The Principal's utility-of-money function is while the Agent's utility-of-money function is U(y) = y V(w) = w. Let x denote the outcome of the job for which the Agent is hired. Assume that x can only take on the following values: $ 5, $ 3 and $. The probabilities are as follows: outcome x = 5 probability 6 x = 3 3 6 x = 6 Page 0 of 0
(i) Suppose that the Principal offers a fixed wage of $ to the Agent. What is the Principal's expected utility? What is the Agent's expected utility? (ii) Show that a fixed-wage contract is not Pareto efficient by giving an example of a Pareto superior form of payment. **** PRACTICE PROBLEM 9 (Risk sharing) **** Consider the following Principal-Agent model. The Principal s utility-of-money function is U($y) = y, while the Agent s utility-of-money function is V($w) = w. Let x denote the outcome of the job for which the Agent is hired. The possible values of x and the corresponding probabilitites are as follows: x probabiltiy $ $6 $6 $65 $79 5 5 5 5 5 (i) Suppose that the Principal offers the following contract to the Agent, call if contract C: the Agent will be paid $0 no matter what the outcome. What is the Principal s expected utility from this contract? What is the Agent s expected utility? (ii) Consider an alternative contract, call it contract D: the Agent will be paid nothing is the outcome is $ or $6 or $6, while she will be paid $60 if the outcome is either $65 or $79. Is contract D Pareto superior to contact C? **** PRACTICE PROBLEM 30 (Principal-Agent) **** A risk-neutral Principal want to hire an Agent to run her firm. The Agent s utility depends on two things: money and effort. Denote effort by e and assume that it can take on only two values: L (for low) or H (for high). Let the Agent s utility function be given as follows (where w denotes money) 90 w 9 if e = L V ( w, e) 90+w 0 if e= H Assume that if they don t sign a contract they both get a utility of zero. The Agent s effort cannot be observed by the Principal and thus cannot be made part of the contract. There are only three possible outcomes (profit levels): 00, 00, 500. If the Agent works hard, better outcomes are more likely than if the Agent does not work hard: outcome (profits) $ 00 $00 $500 probability if e = L / / / probability if e = H / / / Show that of the following contracts, B is Pareto superior to A. Page of 0
CONTRACT A (fixed wage): the Agent will be paid w = 0, no matter what the profits of the firm are. CONTRACT B (contingent wage): the Agent will get nothing (w = 0) if the profit of the firm is either 00 or 00, while he will get w = 00 if the profit of the firm is 500. **** PRACTICE PROBLEM 3 (Principal-Agent) **** Mr. Owny owns a firm. He can either run the firm himself or hire Ms. Managy to run it for him. If he runs the firm himself he gets a utility of 0. If he hires Ms. Managy, he will not be able to check whether she works hard or not. The firm s profit (denoted by x) under the management of Ms. Managy would be as follows: if Ms. Managy is lazy if Ms. Managy works hard x = 0 x = 00 x = 800 with probability with probability with probability with probability with probability with probability Ms. Managy is presently unemployed and her utility from being unemployed is 0. Mr. Owny has the following utility-of-money function (where y denotes money) U(y) = y while Ms. Managy has the following utility function (where w denotes money and denotes the level of effort, with = H meaning that she works hard and = L meaning that she is lazy) Consider the following contracts: w 8 if = L V( w, ) w 0 if = H CONTRACT A: Mr. Owny hires Ms. Managy and agrees to pay her a fixed wage of $0 (i.e. Ms. Owny s wage will be $0 no matter what the profits of the firm). CONTRACT B: Mr. Owny hires Ms. Managy at the following terms: if the profits of the firm are less than $800, Ms. Managy will get nothing; if the profits are 800 Ms. Managy will get $.. Which of the two contracts would Ms. Managy accept?. Which does she prefer? 3. Which is better for Mr. Owny? Page of 0
**** PRACTICE PROBLEM 3 (Choice of Education ) **** Ann s current job pays $0,000 per year. She is considering quitting her job next year and using her savings to finance a Master s degree that is expected to take two years. After she gets her Master s degree, she expects to earn $65,000 per year. Tuition, fees, books and other expenses amount to $0,000 per year. How many years should Ann plan to work after getting the Master s degree, for it to be a worthwhile investment? Answer this question for the following cases: (a) Ann s discount rate is 0, (b) Ann s discount rate is 5%. **** PRACTICE PROBLEM 33 (Signaling) **** Suppose that there are two groups of individuals: Group L Group H Marginal productivity = Marginal productivity = Proportion in population: 3 Proportion in population: 3 Education does not affect productivity. Workers of both types are able to buy education, at a cost. The amount of education y is a continuous variable and that it is fully verifiable (e.g. through a certificate). Type-L workers face a higher cost of acquiring education than type-h workers: Cost of education for Group L individuals: C L = y Cost of education for Group H individuals: C H = y Employers believe that anybody with a level of education less than y * has a productivity of (and thus are offered a wage of ) while everybody with a level of education greater than or equal to y * has a productivity of (and thus are offered a wage of ). What values of y * give rise to a separating signaling equilibrium where H-types indeed choose a level of education not less than y * than y *? while L-types choose a level of education less **** PRACTICE PROBLEM 3 (Signaling) **** Consider the following modification of Spence s model of signaling in the job market. Education does increase productivity. There are two groups in the population. People in Group I have a productivity of + y (where y is the amount of education) and the cost of acquiring y units of education is $y. Page 3 of 0
People in Group II have a productivity of + y units of education is $ y. and the cost of acquiring y Find all the signaling equilibria, when the employer s beliefs are as follows: if a person has y < y o, then he/she comes from Group I, while if a person has y y o, then he/she comes from Group II Assume that the employer offers a wage which is equal to his estimate of the productivity of the applicant. **** PRACTICE PROBLEM 35 (Adverse selection) **** There are two groups of individuals. Group individuals own cars, while Group don t. There are four possible quality levels for cars as shown in the following table, together with the total number of cars of each quality Quality A B C D Number of cars 00 00 00 00 The value that a group individual attaches to a car is lower than the value of the same car to a group individual, as shown in the following table: Quality A B C D Value to Group $6,000 $5,000 $,000 $3,000 Value to Group $5,00 $,500 $3,600 $,700 The quality of a car is known to the owner but not to the prospective buyer. All individuals are risk-neutral. (a) Write down the lottery that corresponds to picking a car at random and calculate the expected value. (b) Suppose that the price of a second-hand car is $3,800. Should a Group individual be willing to buy a car at that price? Explain your answer. (c) Let P be a price at which cars are traded. What are the possible values of P and how may cars will be traded at that price? **** PRACTICE PROBLEM 36 (Adverse selection) **** Suppose that there are two types of workers: high productivity (H) and low productivity (L). An H-worker generates $5,000 in net revenue for the firm, while an L-type generates only $8,000. Each worker knows if he is high productivity or low productivity, while the firm cannot tell workers apart at the time of hiring. Suppose that of all the Page of 0
workers are of type H and 3 are of type L. All workers are currently employed elsewhere at a salary of $0,000. Clearly, if the firm want to attract any workers it has to offer more than $0,000. (a) What is the firm s expected profit per worker if it offers a salary of $0,00? (b) Is there a wage w that the firm can offer such that: () some people will apply, and () the firm can expect to make positive profit? Suppose that an H-worker would produce 5 units of output for the firm, while an L- worker would produce only 8 units of output. The firm can sell each unit of output for $,000 and it has no other costs, besides the labor costs. Suppose that the firm uses a piece-rate compensation scheme, by offering to hire any worker who applies and paying him/her not a fixed salary, but $b per unit of output produced. The firm is risk-neutral. [Continue to assume that (c) Calculate the firm s expected profit per worker if it offers to pay $,000 per unit of output and hires everybody who applies. (d) Who would apply for a job if the firm offered to pay $900 per unit of output?. (e) Calculate the firm s expected profit per worker if it offers to pay $750 per unit of output and hires everybody who applies. **** PRACTICE PROBLEM 37 (Adverse selection) **** Let the quality of a car be denoted by {$,000, $3,000, $,000, $5,000}. The proportion of cars of quality is given as follows: QUALITY PROPORTION 8 =,000 = 3,000 =,000 = 5,000 3 8 There are 00 cars in total. The utility of a seller who sells a car of quality at price P is P and the utility of not selling is 0. Fill in the following table. If price is $,500 $3,00 $,600 $5,50 $6,00 Number of cars offered for sale 8 Average quality of cars offered for sale (Hint: remember to rescale probabilities, i.e. multiply them all by the same number so that they add up to one) 8 Page 5 of 0
**** PRACTICE PROBLEM 38 (Adverse selection) **** Let the quality of a second-hand car be denoted by {,,3}, where is the number of tune-ups that the car received in the past. The value of a car of quality to the seller is $800. Each potential buyer has an initial wealth of $9,05 and the utility of purchasing a car of quality at price P is 9,05 P,000 (while the utility of not bying is 9,05 95. Let the proportion of cars of each quality be as follows (where q is a number strictly between 0 and ): 3 3 proportion q q. Suppose that the price 3 3 of a second-hand car is P = $,700. [In the following assume that, if indifferent between selling and not selling, the owner of a car would sell and, if indifferent between buying and not buying, a potential buyer would buy.] (a) Are there values of q such that ALL cars are traded? (b) Are there values of q such that all cars of quality = and = are traded? (c) Are there values of q such that only cars of quality = are traded? *** PRACTICE PROBLEM 39 (Adverse selection in insurance) *** There are two types of individuals, the U type (of which there are n u ) and the V type (of which there are n v ). They all have the same wealth of $93,600 and face a potential loss of $7,00. The probability s loss is p u = 0 for the U type and pv for the V type. 5 The von Neumann-Morgenstern utility-of-money function of a U type is U ( m) 00 m while the utility-of-money function of a V type is V ( m) 00 ln( m). The insurance industry is a monopoly and the monopolist cannot tell the two types apart, that is, if a consumer applies for insurance, the monopolist is not able to tell whether the consumer is of type U or of type V. (a) What is the maximum premium that a U type is willing to pay for full insurance? (b) Suppose that the monopolist offers only a full-insurance contract with premium $3,50. Calculate the monopolist s expected profits. (c) Suppose now that the monopolist offers two contracts, the one described in part (b) and a contract with premium of $5 and deductible of $5,000. Calculate the monopolist s expected profits. Page 6 of 0
(d) Suppose that n 00 and n, 00. Is the monopolist better off offering only v the contract of part (b) or the two contracts of part (c)? u (e) Suppose that n 00 and n,700. Is the monopolist better off offering only v the contract of part (b) or the two contracts of part (c)? u *** PRACTICE PROBLEM 0 (Moral hazard in insurance) *** Emily has an initial wealth of $80,000 and faces a potential loss of $36,000. The probability of loss depends on the amount of effort she puts into trying to avoid it. If she puts a high level of effort, then the probability is 5%, while if she exerts low effort the probability is 5%. Her utility-of-money function is U ( m) m if low effort m if high effort (a) If Emily remains uninsured, what level of effort will she choose? (b) If Emily is offered a full insurance contract with premium $,50 and she accepts it, what level of effort will she choose? (c) If Emily is offered a full insurance contract with premium $,50 will she indeed accept it? (d) What is the insurance company s expected profit from a full insurance contract with premium $,50? *** PRACTICE PROBLEM (Moral hazard in insurance) *** Bob owns a house near Lake Tahoe. The house is worth $950,000 and constitutes Bob s entire wealth. The probability that there will be a forest fire next year is 0%. If a forest fire occurs then the house will incur damages equal to $00,000. However, by spending $x on protective measures Bob can reduce the probability that the fire will reach the house from 0% to x. Thus the more he spends, the lower the probability. The most he can 0 5, 000 spend is $,000. Bob s utility of money function is U ( m) 0 ln( m) (a) If Bob is not insured, which of the following four options will he choose: () x = 0, () x = $00, (3) x = $750 and () x = $,000? (b) If Bob is offered a full-insurance contract with premium h, what value of x will he choose? (c) Suppose that Bob is offered a full insurance contract at premium h = $0,000. Will he buy it? Page 7 of 0
*** PRACTICE PROBLEM (Moral hazard in insurance) *** Consider an individual whose von Neumann-Morgenstern utility-of-wealth function is U ( m) m m c if she exerts no effort if she exerts effort with c 0. The individual has an initial wealth of W and faces a potential loss of x. The probability of her incurring a loss is p if she exerts effort and p if she chooses no effort, with 0 p p. e n e (a) In a diagram where on the horizontal axis you measure wealth in the bad state ( W ) and on the vertical axis wealth in the good state ( W ) sketch the indifference curves that go through the no-insurance point (NI) (one corresponding to effort and the other to no effort). (b) For the case where W,500, x, 600, pe, pn calculate the slopes of the 0 0 two curves of part (a) at the NI point. 5 (c) Suppose that W,500, x, 600, pe, pn, c and the individual 0 0 6 decides not to insure. Will she exert effort or not? 3 (d) Suppose that W,500, x, 600, pe, pn, c and the individual decides 0 0 not to insure. Will she exert effort or not? In what follows, assume that () effort is observable and verifiable, () if indifferent between not insuring and insuring the individual will choose to insure and 5 (3) W,500, x, 600, pe, pn, c. 0 0 6 (e) Denote by E the contract given by the intersection of the 5 o line and the indifference curve that goes through the no-insurance point (NI) corresponding to effort and F the contract given by the intersection of the 5 o line and the indifference curve that goes through NI corresponding to no effort. (e.) Find the premium and deductible of contract E. (e.) Find the premium and deductible of contract F. (e.3) Suppose that the monopolist offers contract E, provided that the customer can prove that she chose effort (otherwise the contract will not be offered). What will its expected profits be? (e.) Suppose that the monopolist offers contract F, without any restrictions on effort (that is, the contract is offered no matter whether the customer chose effort or no effort). What will its expected profits be? [Think about the customer s options.] (e.5) Let N be the full-insurance contract that makes the consumer indifferent between () signing contract N and choosing no effort and () choosing NI and exerting effort. Show contract N in the wealth diagram and compute its premium and deductible. n Page 8 of 0
**** PRACTICE PROBLEM 3 (Search) **** An unemployed worker with utility-of-money function U(x) = x is looking for a job. The distribution of wages is as follows: WAGE $0 $0 $30 $0 $50 $60 PROPORTION OF FIRMS 3 What is the optimal search strategy for the worker if the cost of each search is c=$0.5? **** PRACTICE PROBLEM (Search) **** [Optional: more difficult than a typical exam question; try it for fun.] An unemployed worker with utility-of-money function U(x) = x is looking for a job. The wage rate is uniformly distributed across firms in the interval [$0,$60] (that is, the density function is f(x)=0 if x<0 or x>60 and f(x)=/50 if 0 x 60 ). What is the optimal search strategy for the worker if the cost of each search is c=$0.5? **** PRACTICE PROBLEM 5 (Search) **** A risk-neutral worker is looking for a job. She has the following information: possible salary $0 $0 $50 $80 proportion of firms offering it 0 3 0 0 0 However, she does not know which firm offers which salary. In order to find out the salary offered by a particular firm, she needs to contact the firm and this costs her $x (that is, each time she gets in touch with a new firm, she has to face a cost of $x). Find the optimal search strategy for every possible value of x. (Assume that if she has searched a number of times, say n times, and is the highest of the salaries offered to her so far, then she can always get at least, by just going back to a firm that offered this salary). Page 9 of 0
**** PRACTICE PROBLEM 6 (Search) **** Emily wants to buy a computer. She knows that the price distribution is as follows Price Fraction of firms $,00 $,900 $,800 Her wealth is $,6. Her utility of money function is U ( m) m if she has $m and no computer m 0 if she has $m and owns a computer The cost of each search is $80 (she has to take time off from work, drive a long way, etc.) (a) How much is a computer worth to Emily? In other words, what is the maximum price that she would be willing to pay for a computer? (b) Suppose that she searches once and is quoted a price of $,00. Should she search a second time? Page 0 of 0