University of California, Davis Department of Economics Giacomo Bonanno. Economics 103: Economics of uncertainty and information PRACTICE PROBLEMS
|
|
- Gavin Morgan
- 5 years ago
- Views:
Transcription
1 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? Page of 0
2 **** 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
3 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, the lottery will be written more simply as 0 Ben is offered a choice between A = ] and B = 3000 He chooses B. Which of the following will Ben choose if he satisfies the axioms of expected utility? C = and D = **** PRACTICE PROBLEM 7 (Expected Utility) **** Consider the following lotteries: $3000 $000 $000 $500 L = 0 0 L $3000 $000 $000 $500 = L 3 = $3000 $000 $000 $500 L = $3000 $000 $000 $ 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
4 **** 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 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 and z z z3 L **** 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 (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
5 **** 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 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? L =, L = (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
6 *** 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 (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
7 **** 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
8 **** 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 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 = 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
9 **** 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 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
10 (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 = x = 6 Page 0 of 0
11 (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 $ (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
12 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
13 **** 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
14 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 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
15 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, 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
16 **** 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, 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
17 (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
18 *** 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 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 (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
19 **** 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 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
20 **** 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
ANSWERS TO PRACTICE PROBLEMS oooooooooooooooo
University of California, Davis Department of Economics Giacomo Bonanno Economics 03: Economics of uncertainty and information TO PRACTICE PROBLEMS oooooooooooooooo PROBLEM # : The expected value of the
More informationPh.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2017
Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2017 The time limit for this exam is four hours. The exam has four sections. Each section includes two questions.
More informationEconomics Honors Exam Review (Micro) Mar Based on Zhaoning Wang s final review packet for Ec 1010a, Fall 2013
Economics Honors Exam Review (Micro) Mar. 2017 Based on Zhaoning Wang s final review packet for Ec 1010a, Fall 201 1. The inverse demand function for apples is defined by the equation p = 214 5q, where
More informationChoice under risk and uncertainty
Choice under risk and uncertainty Introduction Up until now, we have thought of the objects that our decision makers are choosing as being physical items However, we can also think of cases where the outcomes
More informationThe trade-offs associated with getting an education
Department of Economics, University of California, Davis Professor Giacomo Bonanno Ecn 103 Economics of Uncertainty and Information The trade-offs associated with getting an education Usually higher education
More informationMicroeconomics of Banking: Lecture 2
Microeconomics of Banking: Lecture 2 Prof. Ronaldo CARPIO September 25, 2015 A Brief Look at General Equilibrium Asset Pricing Last week, we saw a general equilibrium model in which banks were irrelevant.
More informationManagerial Economics
Managerial Economics Unit 9: Risk Analysis Rudolf Winter-Ebmer Johannes Kepler University Linz Winter Term 2015 Managerial Economics: Unit 9 - Risk Analysis 1 / 49 Objectives Explain how managers should
More informationPh.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program August 2017
Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program August 2017 The time limit for this exam is four hours. The exam has four sections. Each section includes two questions.
More informationHow do we cope with uncertainty?
Topic 3: Choice under uncertainty (K&R Ch. 6) In 1965, a Frenchman named Raffray thought that he had found a great deal: He would pay a 90-year-old woman $500 a month until she died, then move into her
More informationECONOMICS OF UNCERTAINTY AND INFORMATION
ECONOMICS OF UNCERTAINTY AND INFORMATION http://greenplanet.eolss.net/eolsslogn/searchdt_advanced/searchdt_cate... 1 of 7 11/19/2011 5:15 PM Search Print this chapter Cite this chapter ECONOMICS OF UNCERTAINTY
More informationInsurance, Adverse Selection and Moral Hazard
University of California, Berkeley Spring 2007 ECON 100A Section 115, 116 Insurance, Adverse Selection and Moral Hazard I. Risk Premium Risk Premium is the amount of money an individual is willing to pay
More informationPAULI MURTO, ANDREY ZHUKOV
GAME THEORY SOLUTION SET 1 WINTER 018 PAULI MURTO, ANDREY ZHUKOV Introduction For suggested solution to problem 4, last year s suggested solutions by Tsz-Ning Wong were used who I think used suggested
More informationUNCERTAINTY AND INFORMATION
UNCERTAINTY AND INFORMATION M. En C. Eduardo Bustos Farías 1 Objectives After studying this chapter, you will be able to: Explain how people make decisions when they are uncertain about the consequences
More informationMICROECONOMICS COMPREHENSIVE EXAM
MICROECONOMICS COMPREHENSIVE EXAM JUNE 2012 Instructions: (1) Please answer each of the four questions on separate pieces of paper. (2) Please write only on one side of a sheet of paper (3) When finished,
More informationEXTRA PROBLEMS. and. a b c d
EXTRA PROBLEMS (1) In the following matching problem, each college has the capacity for only a single student (each college will admit only one student). The colleges are denoted by A, B, C, D, while the
More informationName. Final Exam, Economics 210A, December 2014 Answer any 7 of these 8 questions Good luck!
Name Final Exam, Economics 210A, December 2014 Answer any 7 of these 8 questions Good luck! 1) For each of the following statements, state whether it is true or false. If it is true, prove that it is true.
More informationMidterm 2 (Group A) U (x 1 ;x 2 )=3lnx 1 +3 ln x 2
Econ 301 Midterm 2 (Group A) You have 70 minutes to complete the exam. The midterm consists of 4 questions (25,30,25 and 20 points). Problem 1 (25p). (Uncertainty and insurance) You are an owner of a luxurious
More informationTotal /20 /30 /30 /20 /100. Economics 142 Midterm Exam NAME Vincent Crawford Winter 2008
1 2 3 4 Total /20 /30 /30 /20 /100 Economics 142 Midterm Exam NAME Vincent Crawford Winter 2008 Your grade from this exam is one third of your course grade. The exam ends promptly at 1:50, so you have
More informationExercises for Chapter 8
Exercises for Chapter 8 Exercise 8. Consider the following functions: f (x)= e x, (8.) g(x)=ln(x+), (8.2) h(x)= x 2, (8.3) u(x)= x 2, (8.4) v(x)= x, (8.5) w(x)=sin(x). (8.6) In all cases take x>0. (a)
More informationModels and Decision with Financial Applications UNIT 1: Elements of Decision under Uncertainty
Models and Decision with Financial Applications UNIT 1: Elements of Decision under Uncertainty We always need to make a decision (or select from among actions, options or moves) even when there exists
More informationExercises - Moral hazard
Exercises - Moral hazard 1. (from Rasmusen) If a salesman exerts high e ort, he will sell a supercomputer this year with probability 0:9. If he exerts low e ort, he will succeed with probability 0:5. The
More informationFirst Welfare Theorem in Production Economies
First Welfare Theorem in Production Economies Michael Peters December 27, 2013 1 Profit Maximization Firms transform goods from one thing into another. If there are two goods, x and y, then a firm can
More informationEC202. Microeconomic Principles II. Summer 2009 examination. 2008/2009 syllabus
Summer 2009 examination EC202 Microeconomic Principles II 2008/2009 syllabus Instructions to candidates Time allowed: 3 hours. This paper contains nine questions in three sections. Answer question one
More informationIf Tom's utility function is given by U(F, S) = FS, graph the indifference curves that correspond to 1, 2, 3, and 4 utils, respectively.
CHAPTER 3 APPENDIX THE UTILITY FUNCTION APPROACH TO THE CONSUMER BUDGETING PROBLEM The Utility-Function Approach to Consumer Choice Finding the highest attainable indifference curve on a budget constraint
More informationMicroeconomics Qualifying Exam
Summer 2018 Microeconomics Qualifying Exam There are 100 points possible on this exam, 50 points each for Prof. Lozada s questions and Prof. Dugar s questions. Each professor asks you to do two long questions
More informationEcon 101A Final exam Mo 18 May, 2009.
Econ 101A Final exam Mo 18 May, 2009. Do not turn the page until instructed to. Do not forget to write Problems 1 and 2 in the first Blue Book and Problems 3 and 4 in the second Blue Book. 1 Econ 101A
More informationAdvanced Microeconomic Theory
Advanced Microeconomic Theory Lecture Notes Sérgio O. Parreiras Fall, 2016 Outline Mathematical Toolbox Decision Theory Partial Equilibrium Search Intertemporal Consumption General Equilibrium Financial
More informationMicroeconomic Theory August 2013 Applied Economics. Ph.D. PRELIMINARY EXAMINATION MICROECONOMIC THEORY. Applied Economics Graduate Program
Ph.D. PRELIMINARY EXAMINATION MICROECONOMIC THEORY Applied Economics Graduate Program August 2013 The time limit for this exam is four hours. The exam has four sections. Each section includes two questions.
More informationFinal Examination: Economics 210A December, 2015
Name Final Examination: Economics 20A December, 205 ) The island nation of Santa Felicidad has N skilled workers and N unskilled workers. A skilled worker can earn $w S per day if she works all the time
More informationSection 9, Chapter 2 Moral Hazard and Insurance
September 24 additional problems due Tuesday, Sept. 29: p. 194: 1, 2, 3 0.0.12 Section 9, Chapter 2 Moral Hazard and Insurance Section 9.1 is a lengthy and fact-filled discussion of issues of information
More informationPhD Qualifier Examination
PhD Qualifier Examination Department of Agricultural Economics May 29, 2015 Instructions This exam consists of six questions. You must answer all questions. If you need an assumption to complete a question,
More informationAssignment 5 Advanced Microeconomics
LONDON SCHOOL OF ECONOMICS Department of Economics Leonardo Felli S.478; x7525 Assignment 5 Advanced Microeconomics 1. Consider a two consumers exchange economy where the two people (A and B) act as price
More informationBest Reply Behavior. Michael Peters. December 27, 2013
Best Reply Behavior Michael Peters December 27, 2013 1 Introduction So far, we have concentrated on individual optimization. This unified way of thinking about individual behavior makes it possible to
More informationMicroeconomics of Banking: Lecture 3
Microeconomics of Banking: Lecture 3 Prof. Ronaldo CARPIO Oct. 9, 2015 Review of Last Week Consumer choice problem General equilibrium Contingent claims Risk aversion The optimal choice, x = (X, Y ), is
More informationExpected Utility And Risk Aversion
Expected Utility And Risk Aversion Econ 2100 Fall 2017 Lecture 12, October 4 Outline 1 Risk Aversion 2 Certainty Equivalent 3 Risk Premium 4 Relative Risk Aversion 5 Stochastic Dominance Notation From
More informationMock Examination 2010
[EC7086] Mock Examination 2010 No. of Pages: [7] No. of Questions: [6] Subject [Economics] Title of Paper [EC7086: Microeconomic Theory] Time Allowed [Two (2) hours] Instructions to candidates Please answer
More informationMaster in Industrial Organization and Markets. Spring 2012 Microeconomics III Assignment 1: Uncertainty
Master in Industrial Organization and Markets. Spring Microeconomics III Assignment : Uncertainty Problem Determine which of the following assertions hold or not. Justify your answers with either an example
More informationMicroeconomic Theory May 2013 Applied Economics. Ph.D. PRELIMINARY EXAMINATION MICROECONOMIC THEORY. Applied Economics Graduate Program.
Ph.D. PRELIMINARY EXAMINATION MICROECONOMIC THEORY Applied Economics Graduate Program May 2013 *********************************************** COVER SHEET ***********************************************
More informationECON Microeconomics II IRYNA DUDNYK. Auctions.
Auctions. What is an auction? When and whhy do we need auctions? Auction is a mechanism of allocating a particular object at a certain price. Allocating part concerns who will get the object and the price
More informationPhD Qualifier Examination
PhD Qualifier Examination Department of Agricultural Economics May 29, 2014 Instructions This exam consists of six questions. You must answer all questions. If you need an assumption to complete a question,
More informationDepartment of Economics The Ohio State University Final Exam Questions and Answers Econ 8712
Prof. Peck Fall 016 Department of Economics The Ohio State University Final Exam Questions and Answers Econ 871 1. (35 points) The following economy has one consumer, two firms, and four goods. Goods 1
More informationChoice under Uncertainty
Chapter 7 Choice under Uncertainty 1. Expected Utility Theory. 2. Risk Aversion. 3. Applications: demand for insurance, portfolio choice 4. Violations of Expected Utility Theory. 7.1 Expected Utility Theory
More informationPAULI MURTO, ANDREY ZHUKOV. If any mistakes or typos are spotted, kindly communicate them to
GAME THEORY PROBLEM SET 1 WINTER 2018 PAULI MURTO, ANDREY ZHUKOV Introduction If any mistakes or typos are spotted, kindly communicate them to andrey.zhukov@aalto.fi. Materials from Osborne and Rubinstein
More informationGeneral Examination in Microeconomic Theory SPRING 2014
HARVARD UNIVERSITY DEPARTMENT OF ECONOMICS General Examination in Microeconomic Theory SPRING 2014 You have FOUR hours. Answer all questions Those taking the FINAL have THREE hours Part A (Glaeser): 55
More informationCONSUMPTION THEORY - first part (Varian, chapters 2-7)
QUESTIONS for written exam in microeconomics. Only one answer is correct. CONSUMPTION THEORY - first part (Varian, chapters 2-7) 1. Antonio buys only two goods, cigarettes and bananas. The cost of 1 packet
More informationModels & Decision with Financial Applications Unit 3: Utility Function and Risk Attitude
Models & Decision with Financial Applications Unit 3: Utility Function and Risk Attitude Duan LI Department of Systems Engineering & Engineering Management The Chinese University of Hong Kong http://www.se.cuhk.edu.hk/
More information3. Other things being equal, a lump sum tax is at least as good for a consumer as a sales tax that collects the same revenue from him.
Section I: or This section is worth a total of 10 marks. There are 10 questions worth 1 mark each; answer all of them. Simply indicate if you think the statement is true or false. 1. A consumer with convex
More informationFINAL EXAMINATION VERSION B
William M. Boal Signature: Printed name: FINAL EXAMINATION VERSION B INSTRUCTIONS: This exam is closed-book, closed-notes. Simple calculators are permitted, but graphing calculators, calculators with alphabetical
More informationPAPER NO.1 : MICROECONOMICS ANALYSIS MODULE NO.6 : INDIFFERENCE CURVES
Subject Paper No and Title Module No and Title Module Tag 1: Microeconomics Analysis 6: Indifference Curves BSE_P1_M6 PAPER NO.1 : MICRO ANALYSIS TABLE OF CONTENTS 1. Learning Outcomes 2. Introduction
More informationChapter Four. Utility Functions. Utility Functions. Utility Functions. Utility
Functions Chapter Four A preference relation that is complete, reflexive, transitive and continuous can be represented by a continuous utility function. Continuity means that small changes to a consumption
More informationCHOICE THEORY, UTILITY FUNCTIONS AND RISK AVERSION
CHOICE THEORY, UTILITY FUNCTIONS AND RISK AVERSION Szabolcs Sebestyén szabolcs.sebestyen@iscte.pt Master in Finance INVESTMENTS Sebestyén (ISCTE-IUL) Choice Theory Investments 1 / 65 Outline 1 An Introduction
More informationUTILITY ANALYSIS HANDOUTS
UTILITY ANALYSIS HANDOUTS 1 2 UTILITY ANALYSIS Motivating Example: Your total net worth = $400K = W 0. You own a home worth $250K. Probability of a fire each yr = 0.001. Insurance cost = $1K. Question:
More information12.2 Utility Functions and Probabilities
220 UNCERTAINTY (Ch. 12) only a small part of the risk. The money backing up the insurance is paid in advance, so there is no default risk to the insured. From the economist's point of view, "cat bonds"
More informationANASH EQUILIBRIUM of a strategic game is an action profile in which every. Strategy Equilibrium
Draft chapter from An introduction to game theory by Martin J. Osborne. Version: 2002/7/23. Martin.Osborne@utoronto.ca http://www.economics.utoronto.ca/osborne Copyright 1995 2002 by Martin J. Osborne.
More informationEcon 101A Final exam Mo 19 May, 2008.
Econ 101 Final exam Mo 19 May, 2008. Stefano apologizes for not being at the exam today. His reason is called Thomas. From Stefano: Good luck to you all, you are a great class! Do not turn the page until
More informationUncertainty. Contingent consumption Subjective probability. Utility functions. BEE2017 Microeconomics
Uncertainty BEE217 Microeconomics Uncertainty: The share prices of Amazon and the difficulty of investment decisions Contingent consumption 1. What consumption or wealth will you get in each possible outcome
More informationChapter 23: Choice under Risk
Chapter 23: Choice under Risk 23.1: Introduction We consider in this chapter optimal behaviour in conditions of risk. By this we mean that, when the individual takes a decision, he or she does not know
More informationDARTMOUTH COLLEGE, DEPARTMENT OF ECONOMICS ECONOMICS 21. Dartmouth College, Department of Economics: Economics 21, Summer 02. Topic 5: Information
Dartmouth College, Department of Economics: Economics 21, Summer 02 Topic 5: Information Economics 21, Summer 2002 Andreas Bentz Dartmouth College, Department of Economics: Economics 21, Summer 02 Introduction
More informationECON Micro Foundations
ECON 302 - Micro Foundations Michael Bar September 13, 2016 Contents 1 Consumer s Choice 2 1.1 Preferences.................................... 2 1.2 Budget Constraint................................ 3
More information(0, 1) (1, 0) (3, 5) (4, 2) (3, 10) (4, 8) (8, 3) (16, 6)
1. Consider a person whose preferences are represented by the utility function u(x, y) = xy. a. For each pair of bundles A and B, indicate whether A is preferred to B, B is preferred to A, or A is indifferent
More informationEcon 101A Final Exam We May 9, 2012.
Econ 101A Final Exam We May 9, 2012. You have 3 hours to answer the questions in the final exam. We will collect the exams at 2.30 sharp. Show your work, and good luck! Problem 1. Utility Maximization.
More informationName ECO361: LABOR ECONOMICS SECOND MIDTERM EXAMINATION. November 5, Prof. Bill Even DIRECTIONS
Name ECO361: LABOR ECONOMICS SECOND MIDTERM EXAMINATION November 5, 2015 Prof. Bill Even DIRECTIONS The exam contains a mix of short answer and essay questions. Your answers to the 20 short answer portion
More informationMassachusetts Institute of Technology Department of Economics Principles of Microeconomics Final Exam Wednesday, October 10th, 2007
Page 1 of 7 Massachusetts Institute of Technology Department of Economics 14.01 Principles of Microeconomics Final Exam Wednesday, October 10th, 2007 Last Name (Please print): First Name: MIT ID Number:
More informationA. Introduction to choice under uncertainty 2. B. Risk aversion 11. C. Favorable gambles 15. D. Measures of risk aversion 20. E.
Microeconomic Theory -1- Uncertainty Choice under uncertainty A Introduction to choice under uncertainty B Risk aversion 11 C Favorable gambles 15 D Measures of risk aversion 0 E Insurance 6 F Small favorable
More informationFoundational Preliminaries: Answers to Within-Chapter-Exercises
C H A P T E R 0 Foundational Preliminaries: Answers to Within-Chapter-Exercises 0A Answers for Section A: Graphical Preliminaries Exercise 0A.1 Consider the set [0,1) which includes the point 0, all the
More informationPreferences - A Reminder
Chapter 4 Utility Preferences - A Reminder x y: x is preferred strictly to y. p x ~ y: x and y are equally preferred. f ~ x y: x is preferred at least as much as is y. Preferences - A Reminder Completeness:
More informationEcon 101A Final exam May 14, 2013.
Econ 101A Final exam May 14, 2013. Do not turn the page until instructed to. Do not forget to write Problems 1 in the first Blue Book and Problems 2, 3 and 4 in the second Blue Book. 1 Econ 101A Final
More informationProbability. An intro for calculus students P= Figure 1: A normal integral
Probability An intro for calculus students.8.6.4.2 P=.87 2 3 4 Figure : A normal integral Suppose we flip a coin 2 times; what is the probability that we get more than 2 heads? Suppose we roll a six-sided
More informationLecture 6 Introduction to Utility Theory under Certainty and Uncertainty
Lecture 6 Introduction to Utility Theory under Certainty and Uncertainty Prof. Massimo Guidolin Prep Course in Quant Methods for Finance August-September 2017 Outline and objectives Axioms of choice under
More informationEcn Intermediate Microeconomic Theory University of California - Davis October 16, 2008 Professor John Parman. Midterm 1
Ecn 100 - Intermediate Microeconomic Theory University of California - Davis October 16, 2008 Professor John Parman Midterm 1 You have until 6pm to complete the exam, be certain to use your time wisely.
More informationChapter 1 Microeconomics of Consumer Theory
Chapter Microeconomics of Consumer Theory The two broad categories of decision-makers in an economy are consumers and firms. Each individual in each of these groups makes its decisions in order to achieve
More informationmarket opportunity line fair odds line Example 6.6, p. 120.
September 5 The market opportunity line depicts in the plane the different combinations of outcomes and that are available to the individual at the prevailing market prices, depending on how much of an
More informationLecture 18 - Information, Adverse Selection, and Insurance Markets
Lecture 18 - Information, Adverse Selection, and Insurance Markets 14.03 Spring 2003 1 Lecture 18 - Information, Adverse Selection, and Insurance Markets 1.1 Introduction Risk is costly to bear (in utility
More informationEconomics 171: Final Exam
Question 1: Basic Concepts (20 points) Economics 171: Final Exam 1. Is it true that every strategy is either strictly dominated or is a dominant strategy? Explain. (5) No, some strategies are neither dominated
More informationEconomics 102 Summer 2014 Answers to Homework #5 Due June 21, 2017
Economics 102 Summer 2014 Answers to Homework #5 Due June 21, 2017 Directions: The homework will be collected in a box before the lecture. Please place your name, TA name and section number on top of the
More informationLesson Exponential Models & Logarithms
SACWAY STUDENT HANDOUT SACWAY BRAINSTORMING ALGEBRA & STATISTICS STUDENT NAME DATE INTRODUCTION Compound Interest When you invest money in a fixed- rate interest earning account, you receive interest at
More informationUNIT 1 THEORY OF COSUMER BEHAVIOUR: BASIC THEMES
UNIT 1 THEORY OF COSUMER BEHAVIOUR: BASIC THEMES Structure 1.0 Objectives 1.1 Introduction 1.2 The Basic Themes 1.3 Consumer Choice Concerning Utility 1.3.1 Cardinal Theory 1.3.2 Ordinal Theory 1.3.2.1
More informationTime, Uncertainty, and Incomplete Markets
Time, Uncertainty, and Incomplete Markets 9.1 Suppose half the people in the economy choose according to the utility function u A (x 0, x H, x L ) = x 0 + 5x H.3x 2 H + 5x L.2x 2 L and the other half according
More informationLECTURE NOTES ON MICROECONOMICS
LECTURE NOTES ON MICROECONOMICS ANALYZING MARKETS WITH BASIC CALCULUS William M. Boal Part 4: General equilibrium and market power Chapter 13: General equilibrium Problems (13.1) [Efficiency versus fairness]
More informationd. Find a competitive equilibrium for this economy. Is the allocation Pareto efficient? Are there any other competitive equilibrium allocations?
Answers to Microeconomics Prelim of August 7, 0. Consider an individual faced with two job choices: she can either accept a position with a fixed annual salary of x > 0 which requires L x units of labor
More informationPart 4: Market Failure II - Asymmetric Information - Uncertainty
Part 4: Market Failure II - Asymmetric Information - Uncertainty Expected Utility, Risk Aversion, Risk Neutrality, Risk Pooling, Insurance July 2016 - Asymmetric Information - Uncertainty July 2016 1 /
More informationComparison of Payoff Distributions in Terms of Return and Risk
Comparison of Payoff Distributions in Terms of Return and Risk Preliminaries We treat, for convenience, money as a continuous variable when dealing with monetary outcomes. Strictly speaking, the derivation
More informationPh.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2015
Ph.D. Preliminary Examination MICROECONOMIC THEORY Applied Economics Graduate Program June 2015 The time limit for this exam is four hours. The exam has four sections. Each section includes two questions.
More informationNotes on a Basic Business Problem MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W
Notes on a Basic Business Problem MATH 104 and MATH 184 Mark Mac Lean (with assistance from Patrick Chan) 2011W This simple problem will introduce you to the basic ideas of revenue, cost, profit, and demand.
More informationUtility and Choice Under Uncertainty
Introduction to Microeconomics Utility and Choice Under Uncertainty The Five Axioms of Choice Under Uncertainty We can use the axioms of preference to show how preferences can be mapped into measurable
More informationApril 28, Decision Analysis 2. Utility Theory The Value of Information
15.053 April 28, 2005 Decision Analysis 2 Utility Theory The Value of Information 1 Lotteries and Utility L1 $50,000 $ 0 Lottery 1: a 50% chance at $50,000 and a 50% chance of nothing. L2 $20,000 Lottery
More information3. Other things being equal, a lump sum tax is at least as good for a consumer as a sales tax that collects the same revenue from him.
Section I: or This section is worth a total of 10 marks. There are 10 questions worth 1 mark each; answer all of them. Simply indicate if you think the statement is true or false. 1. A consumer with convex
More informationCharacterization of the Optimum
ECO 317 Economics of Uncertainty Fall Term 2009 Notes for lectures 5. Portfolio Allocation with One Riskless, One Risky Asset Characterization of the Optimum Consider a risk-averse, expected-utility-maximizing
More informationChapter 6: Supply and Demand with Income in the Form of Endowments
Chapter 6: Supply and Demand with Income in the Form of Endowments 6.1: Introduction This chapter and the next contain almost identical analyses concerning the supply and demand implied by different kinds
More informationProblem Set 2. Theory of Banking - Academic Year Maria Bachelet March 2, 2017
Problem Set Theory of Banking - Academic Year 06-7 Maria Bachelet maria.jua.bachelet@gmai.com March, 07 Exercise Consider an agency relationship in which the principal contracts the agent, whose effort
More informationDepartment of Economics The Ohio State University Midterm Questions and Answers Econ 8712
Prof. James Peck Fall 06 Department of Economics The Ohio State University Midterm Questions and Answers Econ 87. (30 points) A decision maker (DM) is a von Neumann-Morgenstern expected utility maximizer.
More informationPortfolio Management
MCF 17 Advanced Courses Portfolio Management Final Exam Time Allowed: 60 minutes Family Name (Surname) First Name Student Number (Matr.) Please answer all questions by choosing the most appropriate alternative
More informationExpected Utility and Risk Aversion
Expected Utility and Risk Aversion Expected utility and risk aversion 1/ 58 Introduction Expected utility is the standard framework for modeling investor choices. The following topics will be covered:
More informationMEASURES OF RISK-AVERSION
Department of Economics, Universit of California, Davis Professor Giacomo Bonanno Ecn 03 Economics of Uncertaint and Information Lecture MESURES OF RISK-VERSION Can we answer the question: Which of two
More information2 Maximizing pro ts when marginal costs are increasing
BEE14 { Basic Mathematics for Economists BEE15 { Introduction to Mathematical Economics Week 1, Lecture 1, Notes: Optimization II 3/12/21 Dieter Balkenborg Department of Economics University of Exeter
More informationAS/ECON 2350 S2 N Answers to Mid term Exam July time : 1 hour. Do all 4 questions. All count equally.
AS/ECON 2350 S2 N Answers to Mid term Exam July 2017 time : 1 hour Do all 4 questions. All count equally. Q1. Monopoly is inefficient because the monopoly s owner makes high profits, and the monopoly s
More informationBEEM109 Experimental Economics and Finance
University of Exeter Recap Last class we looked at the axioms of expected utility, which defined a rational agent as proposed by von Neumann and Morgenstern. We then proceeded to look at empirical evidence
More informationExercise List 2: Market Failure
Universidad Carlos III de Madrid Microeconomics II ME&MEIM Exercise List 2: Market Failure Exercise 1. A good of two qualities, high (H) and low (L), is traded in competitive markets in which each seller
More informationASHORTCOURSEIN INTERMEDIATE MICROECONOMICS WITH CALCULUS. allan
ASHORTCOURSEIN INTERMEDIATE MICROECONOMICS WITH CALCULUS Roberto Serrano 1 and Allan M. Feldman 2 email: allan feldman@brown.edu c 2010, 2011 Roberto Serrano and Allan M. Feldman All rights reserved 1
More informationPrice Theory Lecture 9: Choice Under Uncertainty
I. Probability and Expected Value Price Theory Lecture 9: Choice Under Uncertainty In all that we have done so far, we've assumed that choices are being made under conditions of certainty -- prices are
More information