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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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