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Name: Email ID: @psu.edu ECON402 HW 6

Please scan your answers to this homework assignment and upload to Canvas. You must show your

work in order to receive credit.

These homework assignments can be difficult. Feel free to work together by asking questions on the

discussion board. Donâ€™t ask for an answer; let us know where you are stuck and we can help you get

un-stuck. Donâ€™t just â€œgive awayâ€ the answers, but help your fellow students think through where they

are getting stuck. 1 Marcia, Marcia, Marcia (15 points) From textbook question Ch. 10, #1

Greg is deciding whether to ask Marcia out on a date. However, Greg isnt sure whether Marcia likes

him, and he would rather not ask if he expects to be rejected. Whether Marcia likes Greg is private

information to her. Thus, her preferences regarding Greg constitute her type. Greg does not have any

private information. Assume that there is a 25% chance that Marcia likes Greg. The Bayesian game is

shown here. Should Greg ask Marcia? Find a BNE that answers this question. ECON 402 HW 6 2 Page 2 Auctions: First Price vs. Second Price (10 points) Fill in the blanks in the table below.

Bidder Value A 92 B 95 C 31 D 75 E 88 1st price bid 2nd price bid a) Who wins the first price auction and what price do they pay?

b) Who wins the second price auction and what price do they pay? 3 NE in Auctions (20 points) Consider a first-price sealed bid auction with n risk-neutral bidders (weâ€™ve been examining risk neutral

bidders all along). Each bidder has a private value independently drawn from a uniform distribution on

[0,1]. That is, for each bidder, all values between 0 and 1 are equally likely. The complete strategy of

each bidder is a bid function that will tell us, for any value v, what amount b(v) that bidder will choose

to bid.

It is proposed that the equilibrium bid function for n = 2 is b(v) = v/2 for each of the two bidders.

That is, if we have two bidders, each should bid half her value.

a) (4 points) Suppose youâ€™re bidding against just one opponent whose value is univormly distributed on

[0,1] and always bids half her value. What is the probability that you will win if you bid b=.1? If you

bid b=.4? If you bid b=.6?

b) (4 points) Put together the answers to part a). What is the correct mathematical expression for p(win),

the probability that you win, as a function of your bid b?

c) (4 points) Find an expression for the expected profit you make when your value is v and you bid is

b, given that your opponent is bidding half her value. Remember that there are two cases: either you

win the auction or you lose the auction. You need to average the profit between these cases.

d) (4 points) What is the value of b that maximizes your expected profit? This should be a function of

your value v.

e) (4 points) Use your results to argue that it is a NE for both biddres to follow the same bid function

b(v) = v/2. 4 Lemons (20 points) In the country of Autolandia, used cars all sell for the same price. In Autolandia, the quality of cars is

uniformly distributed with sellers valuing the cars from $0 to $1,000 and buyers valuing the cars from

$100 to $1,100 (buyers value each individual car at $100 more than the sellers). For example, consider

a market with 1,001 cars. There will be one worth $0 to a seller/$100 to a buyer, one worth $1 to a

seller/$101 to a buyer, etc. up to one worth $1,000 to a seller/$1,100 to a buyer. ECON 402 HW 6 Page 3 a) (5 points) If neither the buyer nor the seller knows the quality of a car, will any cars sell? If so, at

what price?

b) (5 points) If only the seller knows the quality of a car, what will a buyer think is the true value of a

car with price p to the seller of that car? How much would a buyer value the average car for sale at

price p? Give a numerical example.

c) (5 points) In the version of the market where only the seller knows the true quality of the car, what

will be the equilibrium price?

d) (5 points) What percentage of the total number of cars get sold? 5 No such thing as a free lunch (10 points) A local charity has been given a grant to serve free meals to the homeless in its community, but it is

worried that its program might be exploited by nearby college students who are always interested in a

free meal. Both a homeless person and a college student receive a payoff of 10 for a free meal. The cost

m2

m2

of standing in line for the meal is 400

for a homeless person and 250

for a college student, where m is the

number of minutes spent waiting in line. Assume that the staff of the charity cannot observe the true

type of people coming for free meals.

a) (5 points) What is the minimum wait time, m, that will achieve separation of types?

b) (5 points) After a while, the charity finds that it can successfully identify and turn away college

students half fo the time. College students who are turned away receive no free meal and also incur

a cost of 5 for their time and embarrassment. Will the ability to partially identify college students

reduce or increase the answer in part a)? Explain. 6 To Stand or Not to Stand, that is the Question (25 points) In a scene at the end of The Princess Bride (same movie as the poison scene from lesson L01-1), the

hero, Wesley, confronts the evil Prince Humperdinck. This interaction can be modeled as the following

game. Wesley can either be strong or weak, as randomly picked by nature with equal probability. Wesley,

knowing if he is strong or weak, can then choose to stand or continue lying on the bed. The prince

observes this decision but does not know Wesleyâ€™s type and chooses to fight or surrender. The prince can

beat a weak Wesley (Wesley was â€mostlyâ€ dead only a few hours earlier) and can really beat up a weak

Wesley who stays in bed. If a weak Wesley stands and the prince fights, then the base payoffs are 1 for

the prince and -1 for Wesley. If Wesley stays in bed, then the prince gets 2 and Wesley still gets -1. If

Wesley is weak and the prince surrenders, regardless of whether Wesley has stood or not the base payoffs

are 1 for Wesley and 0 for the prince. A strong Wesley, however, will destroy the prince and enjoy doing

it should the prince try to fight. It is easier for Wesley to beat the Prince if he stands, though, so in

that case he gets a payoff of 3 while the prince gets -2. If Wesley stays in bed and the Prince chooses

to fight, the payoffs are 1 for Wesley and -1 for the prince. In the event that the prince chooses not to

fight, Wesley gets 2 if he stands and 1 if he stays in bed while in either case the prince gets 0. The final

complication to the payoffs is that if Wesley is weak, standing costs him some extra amount c, while it

costs a strong Wesley nothing.

a) (5 points) derive the extensive form of the game

b) (10 points) derive a pooling BNE where q = 0.5 represents the princeâ€™s initial belief that Wesley is

weak. Be sure to include what are the princeâ€™s beliefs about Wesleyâ€™s type if he observes Wesley in

bed or observes him standing. Find the range of values for c that makes the belief valid.

c) (10 points) derive a separating BNE for this game as well as the range of c over which it is valid.

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