Simon Fraser University Department of Economics Economics ECON

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					                                      Simon Fraser University
                                      Department of Economics


     Economics ECON - 302                                                                  Daniel Monte
     Microeconomic Theory II: Strategic Behavior                                           Spring 2008


                                               Problem Set 6


    1. Insurance and Risk
    Consider the following scenario. A person has a house worth 1,000,000. The person has to
decide how much insurance to buy. There is a probability of 0.02 that the house will be destroyed
in an earthquake. If this terrible event happens, and the person is uninsured, his wealth drops to
zero. If there is no earthquake, the house remains with the same value of 1,000,000. Insurance
costs $1 dollar for every $50 dollars of coverage (Example it would cost $20,000 to fully insure the
house).

   a) What is the expected wealth if the person decides not buy insurance?
   EV (W ) = 0:02 0 + 0:98 1; 000; 000 = 980; 000
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   b) Suppose that the person’ utility function is U = W 2 . What is his expected utility if he
                               s
      t
doesn’ buy any insurance?
   EU (W ) = 0:02 (0)0:5 + 0:98 (1; 000; 000)0:5 = 0; 98 1; 000 = 980

    c) What is the certain wealth level that would make this person equally happy as not buying
insurance and running the risk?
    EU (CE) = 980
         1
    (CE) 2 = 980
    CE = 9802 = 960; 400:

    d) Given the price of the insurance, what is the optimal amount of insurance?
                                                                             1
    price of insurance is $1 dollar for every $50 dollars of coverage, thus 50 dollars for every dollar
of coverage.

                                          r                    r
                                                1                                 1
                               max 0:02    I      I + 0:98         1; 000; 000      I
                                 I             50                                50
   FOC                1                                             1
   0:02   49
    2     50   I 49
                 50
                      2
                          + 0:98 1
                                 2
                                      1
                                     50   1; 000; 000        1
                                                            50 I
                                                                    2
                                                                        =0
                      1                                 1
                 49                              1
   0:02 49     I 50  = 0:98 1; 000; 000
                      2
                                                50 I
                                                        2

     49                1
   I 50 = 1; 000; 000 50 I

   I = 1; 000; 000

   e) Is the price of the insurance actuarially fair?
                            1                                                        1
   Yes, since price is = 50 and the probability of loss is               = 0:02 =   50




                                                        1
   f ) Did the person fully insure?
   Yes, since she bought 1,000,000 of insurance, which is actually the total value of the house.


              s
   2. Jensen’ inequality
   A concave function f : R ! R is characterized by the condition that:
                                     Z           Z
                                  f     xdF         f (x) dF;                                      (1)

                                                                           s
    for any distribution F : R ! [0; 1]. Condition (1) is known as Jensen’ inequality.
    Consider some lottery F (x). The certainty equivalent for this lottery is denoted CE, while the
expected value of this lottery is denoted EV: Suppose that a decision maker has a concave utility
u (x).

                  s
   a) Use Jensen’ inequality to show that u (CE) u (EV ) ; for lottery F (x).
                                                                       R
   Note that the expected utility of u (x) given lottery F is given by: u (x) dF:
         R
   Also: xdF is the expected value of lottery F .
                s
   From Jensen’ inequality we have that:
      R        R
   u xdF          u (x) dF                                  R
   But we know from the de…nition of CE that: u (CE) = u (x) dF:
   Thus:
                                        Z
                                    u      xdF         u (CE)

                                           u (EV )        u (CE)


   b) Show that CE EV
   Since u (x) is strictly increasing, u (EV )   u (CE) imples that EV    CE.

   c) Show that the risk premium is rp 0.
   rp EV CE
   From part b we know that EV      CE , thus rp         0.

    3. Bayesian Updating (Based on Sobel (1985))
    There are two agents in the economy: a policy maker and an adviser. The adviser can be a
                                               t
friend or an enemy. The policy maker doesn’ know if the adviser is a friend or an enemy, but he
has prior beliefs that the adviser is a friend with probability (thus, with probability 1 , the
policy maker thinks that the adviser is an enemy).

    a) Assume that a ‘ friend’always tells the truth. Whereas an ‘  enemy’always lies. If the policy
maker observes that the adviser has told the truth, what is his posterior belief that the adviser is
a friend?
    Lets denote his posterior by pT rue , using Bayesian updating, we have that:


                                            Pr (truthjF riend) Pr (F riend)
       Pr (F jtruth) =
                           Pr (truthjF riend) Pr (F riend) + Pr (truthjEnemy) Pr (Enemy)
                                 1
                       =                  =1
                           1 + (1     )0

                                                     2
   b) Now assume that a ‘                                         enemy’tells the truth with some
                            friend’always tells the truth, but an ‘
probability q > 0. If the policy maker observes that the adviser has told the truth, what is his
posterior belief that the adviser is a friend?


                                             Pr (truthjF riend) Pr (F riend)
        Pr (F jtruth) =
                            Pr (truthjF riend) Pr (F riend) + Pr (truthjEnemy) Pr (Enemy)
                                  1
                       =                   =
                            1 + (1     )q      + (1     )q



    c) Now assume that a ‘    friend’always tells the truth and an ‘  enemy’always lies. However, the
adviser can only observe the true message with probability > 1 . I.e. if the adviser tells the truth,
                                                                    2
the policy maker will observe the truth with probability , but will observe a lie with probability
(1     ). Similarily, if the adviser lies, the policy maker observes a lie with probability ; but will
observe a truth with probability (1        ). If the policy maker observes a truth, what is his posterior
belief that the adviser is a friend?


                                             Pr (truthjF riend) Pr (F riend)
        Pr (F jtruth) =
                            Pr (truthjF riend) Pr (F riend) + Pr (truthjEnemy) Pr (Enemy)
                       =
                               + (1     ) (1    )




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