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Prof. Dr. W. Emons Economics of Information Spring 2012 Problem Set 1 The house of a representative consumer will burn down with a probability of p ∈ (0, 1) in the next period; this would cause a monetary damage of D > 0. Therefore, the consumer has an available income of Y −D in case of a damage. There will be no ﬁre with probability 1 − p and the consumer has the income Y . The consumer is risk averse, i.e. she has a von-Neumann-Morgenstern utility function U (·) with U > 0 and U < 0. The consumer can buy insurance contracts (π, E), where π is the insurance premium and E the gross indemnity. a) Draw the indiﬀerence curve in (π, E)-space, i.e. the premium on the ordinate and the gross indemnity on the abscissa. Calculate the slope and the curvature of the indiﬀerence curves. b) Assume the consumer lives in a large country with consumer protection. Competition leads to a large number of insurance companies. All of them oﬀer the premium rate q = p. Consumers choose indemnity. Draw the insurance contracts in the graph above and show that the consumer chooses full insurance. c) Assume that the consumer lives in a small country with producer pro- tection. There is only one insurance company which is free to determine its contracts. Which contract will the monopolist oﬀer in order to in- crease its expected return? Prof. Dr. W. Emons Economics of Information Spring 2012 Problem Set 2 Consider a consumer who can inﬂuence the probability of damage with his eﬀort level e. The utility of the consumer is U (y, e) = ln(y) − e. Further, the following information is given: y - income of the consumer e - the consumer’s eﬀort level with e ∈ 0, ln 2 5 w - the consumer’s wealth s - monetary damage with w > s > (3/4)w π - insurance premium E - gross indemnity 1/10, if e = ln 25 p(e) - probability of damage where p(e) = 3/10, if e = 0 a) Show that the consumer will always make an eﬀort if there is no insur- ance. b) Determine the insurance contracts (π, E) which set the consumer in- diﬀerent with respect to the eﬀort level. Draw his indiﬀerence curves in (π, E)-space. c) Calculate ﬁrst-best. What is the problem with ﬁrst-best? d) Which contracts (π, E) allow an insurance company not to make losses if the company cannot observe eﬀort levels? e) Determine second- and third-best. What is the problem here? Draw a graph depicting ﬁrst-, second- and third-best. f) What equilibrium results from a competitive market without commu- nication between insurance companies? g) Discuss the welfare properties of ﬁrst-, second-, and third-best. Prof. Dr. W. Emons Economics of Information Spring 2012 Problem Set 3 A restaurant owner (principal) wants to hire someone to run his restaurant (agent). The agent can choose between exercising an eﬀort e = 2 or not ¯ exercising an eﬀort e = 0. His reservation utility is u = 10 if he doesn’t accept the job. Let w ≥ 0 be the agent’s wage. His utility is hence w − e if he accepts the job, u= 10 if he rejects the job oﬀer. The principal’s revenues R(e) depend on the eﬀort level of the agent: H = 15 for e = 2, R(e) = L=9 for e = 0. The principal oﬀers the agent a wage function which maximizes the princi- pal’s proﬁts. a) Give a formal description of the agent’s participation constraint. Calcu- late the incentive compatibility constraint of the agent if the principal wants him to take eﬀort level e = 2. b) Calculate the wage function (or the contract) which maximizes the principal’s proﬁts. How high are proﬁts? Now assume that revenues are inﬂuenced by the mood of the guests: ˜ H = 16 with probability 4/5, R(2) = L = 11 with probability 1/5, ˜ H = 16 with probability 1/5, R(0) = L = 11 with probability 4/5. Assume further that the utility function of the agent contains the expected wage E[w] instead of w. c) Give a formal description of the participation and the incentive com- patibility constraints of the agent. d) Calculate the wage function (or the optimal contract) that the principal oﬀers to the agents. e) Compare results from ?? and ??. When are proﬁts higher? And wages? How could the model be changed to get a more realistic description of the principal-agent problem? Prof. Dr. W. Emons Economics of Information Spring 2012 Problem Set 4 Consider the principal-agent model we have looked at in the lecture. Let B(an ) = M πnm Sm , n = 1, . . . , N . Assume m=1 L := max {B(a) − C o (a)} − max {B(a) − C(a)} . a∈A a∈A This means that L is a measure for the costs arising for the principal because the agent’s action is not observable. Prove the following statements: a) L ≥ 0. b) If the agent is risk neutral then L = 0. c) Let an be a ﬁrst-best action, i.e. an ∈ arg max an ∈A B(an ) − C o (an ). If πnm > 0 ⇒ πn m = 0 ∀an = an then L = 0. d) Let A be ﬁnite and an be a ﬁrst-best action. If at least for one Sm it is true that πnm = 0 and πn m > 0 ∀an = an then L = 0. o e) Assume an is a ﬁrst-best action. If an ∈ arg min an ∈A C (an ) then L = 0. Prof. Dr. W. Emons Economics of Information Spring 2012 Problem Set 5 Consider a market for second-hand cars of a certain type. Car owners have a utility parameter αj = 1 and know the quality of their cars. Non-owners have a utility parameter αi = 4/3 and cannot judge the quality of an individual car. However, they know the distribution of the qualities x ∈ [0, 1] of cars of this type, which has been published recently in a renowned automobile magazine. The distribution has the density 1/51, for 0 ≤ x ≤ 1/2, f (x) = 101/51, for 1/2 < x ≤ 1. a) Consider the whole market where there are more non-owners than own- ers. Determine the supply function of the owners. Also determine the demand function of the non-owners if they take into account the behav- ior of the sellers. Find all possible equilibria and rank them according to the Pareto criterion. b) On the Bernese local market there are 15 non-owners. Further there are 5 owners in each quality class 0, 1/2, and 1. Non-owners still assume the above quality distribution. Determine supply, demand, and all possible equilibria. Prof. Dr. W. Emons Economics of Information Spring 2012 Problem Set 6 Consider an economy where – in contrast to the model we looked at in the lecture – education increases productivity. An individual of group 1 achieving education level y incurs the costs c1 (y) = y and his productivity becomes y θ1 (y) = 1 + . 4 An individual of group 2 achieving education level y has the costs y c2 (y) = 2 resulting in productivity y θ2 (y) = 2 + . 4 Fraction λ ∈ (0, 1) of the employees belong to group 1. a) Consider wage functions of the form w1 if y ∈ [0, y ), ¯ w(y) = w2 if y ≥ y . ¯ Find all Spence equilibria. b) Compare the equilibria according to the Pareto criterion. c) Discuss the hypothesis that too much is invested in education in the framework of the model. Prof. Dr. W. Emons Economics of Information Spring 2012 Problem Set 7 Individuals earn a ﬁxed income y per year. Assume each individual has a probability of accident p only known to him. The damage caused by an accident is 1. Let us denote the individual’s net income with the random ˜ variable y . An individual’s utility is the expected income minus a risk term ˜ given by the variance of y . Individuals maximize their utilities. Assume they have the possibility to insure themselves. An insurance contract is described by a yearly premium π and an indemnity 1 − s in case of a damage where s ∈ [0, 1] is the deductible. a) Calculate the utility of a consumer with and without insurance. Assume that all individuals have the same probability of accident p. b) For which π(s) does the contract (π(s), 1 − s) result in zero proﬁts if all individuals insure themselves? c) For which values of s is the contract more attractive for insurants than no insurance? What level of deductible maximizes utility? Assume there are two groups of insurants, those with ph = 1/4 and those with pl = 1/8. d) What level does s have to be at, so that a contract leading to zero proﬁts with 1/8-insurants isn’t more attractive for 1/4-insurants than the (1/4, 1)-contract? e) Show that the contract (π(s), 1 − s) found is better for 1/8-insurants than the (1/4, 1)-contract. f) What “costs” of lost utility do 1/8-individuals bare in order to distin- guish themselves from 1/4-individuals? Prof. Dr. W. Emons Economics of Information Spring 2012 Problem Set 8 Consider the following credit market with asymmetric information. An en- trepreneur of type x can invest amount L and achieves the random revenue ˜ 2L/x with probability x, Z= 0 with probability 1 − x. The investment level L is exogenously given and independent of x. The entrepreneur doesn’t have any own means to invest. He considers borrowing L with the duty of paying back (1 + r)L. He is only liable with revenues from the project itself. a) Each entrepreneur knows his own type (i.e. his x) and is willing to borrow if, and only if, in case of success the revenue is at least as high as credit costs. Which entrepreneurs will borrow if the interest rate is r? b) Banks do not know the type of an entrepreneur. However, they do know that x is distributed with density f (x) = 1 for x < 0 ≤ 1 in the populace. How does the interest rate inﬂuence the probability of repayment expected by the bank? How high is the expected average repayment of the entrepreneurs who are willing to borrow at an interest rate r? c) Banks reﬁnance themselves at the interest rate i = 10%. They are risk neutral and compete in a market with free entry. Outline the conditions for interest rate r to be an equilibrium interest rate. d) Show that there is no equilibrium interest rate r < ∞. Explain this result. Prof. Dr. W. Emons Economics of Information Spring 2012 Problem Set 9 A consumer has the utility function u(c1 , c2 ) = ln c1 + ln c2 , resulting from consumption c1 in period 1 and c2 in period 2. His period 1 income is y1 = 0; his period 2 income is y2 = 1000. He can borrow L in the ﬁrst period with interest rate r. He has to repay (1 + r)L in period 2. a) Determine the classical demand. b) The debtor can ﬁle for bankruptcy and avoid repayment. Then his income will be conﬁscated up to a remainder Z. When is ﬁling for bankruptcy advantageous? c) Determine the classical credit market equilibrium if the remainder Z is distributed with probability 1 f (Z) = , 0 < Z < 1000, 1000 in the population and the banks reﬁnance themselves with the interest rate i = 10%. d) Assume no one considers ﬁling for bankruptcy. Then i = r in a credit market equilibrium. Which people (i.e. which Z-values) would beneﬁt if everyone repaid their credits? Are there people who would beneﬁt from living in a world without bankruptcy (in the sense used in this problem set)? What are the consequences from your answer for the actual behavior of people? Prof. Dr. W. Emons Economics of Information Spring 2012 Problem Set 10 Joe’s garage has a monopoly for ﬁxing muﬄers in Zappaville, a small town with 64 inhabitants, each owning one car. The probability that a car’s muﬄer is in bad shape is p = 1/2, the probability that it is in good shape is 1 − p = 1/2. A muﬄer in bad shape will work with probability ql = 1/4, good muﬄers with probability qh = 3/4. If the muﬄer works, the utility of driving a car is 1, otherwise 0. Joe needs d = 1/16 hours to check whether the muﬄer works well. He needs r = 1/8 to repair it. Repairing means turning a muﬄer from bad into good shape. Joe can, however, “repair” a good muﬄer: in this case he unnecessarily works r units of time on the muﬄer leaving it in good shape. Customers wait at Joe’s garage while he works on their vehicles and can therefore observe whether he checks or repairs their car’s muﬄer. When he is not in his garage, he works at McDonald’s for 1 dollar per hour. a) Customers observe his capacity L (i.e. the hours per day he is in his garage). Show that it is an equilibrium that Joe keeps his garage open L = 8 hours a day and charges D = 1/8 dollars for checking the muﬄers and R = 1/4 for repairing them. b) Now assume that customers cannot observe the opening hours L of Joe’s garage (Joe could go to work at McDonald’s during shop hours or work in his garage until midnight). Show that the unique equilibrium is Joe working L = 8 hours in his garage and charging D = 3/16 for checking and R = 1/8 for repairing the muﬄer.

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posted: | 10/31/2012 |

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