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Final Exam Microeconomic Theory II (8030)
发布时间:2022-04-27
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Final Exam Microeconomic Theory II (8030)
2022
1. (10) Consider the following game:
1u
@
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(0,1) (3,2) (-1,3) (1,5)
a) (5) Write down the normal form representation of the game and find its Nash Equilibria.
b) (5) Find a sequential equilibrium of the game (a profile of strategies and beliefs).
2. (10) Veronica has decided to sell her comic book collection in order to help pay for a new car. The only people interested in bidding for the collection are her friends, Archie and Betty. In order to get a good price and yet be fair to her friends, she has decided to have a first-price, sealed-bid auction with no reserve. Archie considers the collection worth VA and Betty considers it worth VB . For Archie, Betty’s valuation is an independent random variable uniformly distributed between 0 and 1. Similarly for Betty, Archie’s valua- tion is an independent random variable uniformly distributed between 0 and
1. Show that this static game has the following symmetric, pure-strategy
Bayesian Nash equilibrium: bA(∗)(VA ) = VA and bB(∗)(VB ) =
VB . (Hint: In
order to prove that this is an equilibrium strategy for both players, it is suffi-
cient to show that bA(∗) is Archie’s best response to the belief that bB(∗) is Betty’s
strategy. Note also that Archie’s payoff function is (VA − bA ))prob(bA > bB ),
3. (20) Assume that a consumer’s von Neumann-Morgenstern utility of wealth is u(w) = √w , and her initial wealth is w0 = 100. Suppose that there are but two loss levels, l1 = 0 and l2 = 51. There are two effort levels, e = 0 and e = 1. The consumer’s disutility of effort is given by the function d(e), where d(0) = 0 and d(1) = 1/3. Suppose that the loss probabilities, πl (e), are given below:
πl1 (0) = 1/3, πl2 (0) = 2/3; πl1 (1) = 2/3, πl2 (1) = 1/3. Assume that there is only one insurance company that is risk neutral.
(a) (2) Verify that probabilities given above satisfy the monotone likelihood
ratio property.
(b) (2) Find the consumer’s reservation utility.
(c) (2) What effort level will th consumer exert if no insurance is available?
(d) (2) Show that if information is symmetric, then it is optimal for the insurance company to offer a policy that induces high effort.
(e) (2) Show that the policy in part (d) will not induce high effort if infor-
mation is asymmetric.
(f) (10) Compute the insurance policy (premium and coverage) that would
induce high effort.
4. (10) Consider the two-player version of the all-pay auction and vi , vj be the private valuations independent and identically distributed on a uniform distribution from [0,1].
(a) Find a monotone increasing bidding function, b(v), that forms a sym-
metric Nash Equilibrium.
(b) Compute the expected revenue from the two-player all-pay auction.
(c) Show that the two-player all-pay auction and the two-player first-price auction raise the same expected revenue.