Econ3101 - Equilibrium Function - Economics Assessment Answer

Econ 3101 - Equilibrium Function - Economics Assessment Answer

Task:

Questions: 

1. Find the optimal bidding function in a first-price auction with n bidders, if the distribution of bidders’ values is F(x) = x, and x is distributed in the interval [0,1].
What happens to the optimal bidding function as n grows large? Intuitively interpret this result (i.e., explain why this should be so).
2. Consider a third-price auction with n>3 bidders (the highest bidder wins but pays a price equal to the third-highest bid).
a. Assume that the equilibrium function is increasing in the bidder’s value. Intuitively try to reason out how the equilibrium bid β(v) should relate to v. Is it equal to v, greater, or less?
b. Can you use the Revenue Equivalence Principle to find the equilibrium bidding function?

3. Consider a sealed-bid second price auction with two bidders. The seller has a value 0 for the object. Each bidder has a value drawn from the uniform distribution F(v)=v on [0, 1], i.e., the probability that his value is less than x is just x.
a. Given that the highest bidder has a value vH, you know how to calculate the expected value vL of the lower bidder, conditional on this value being less than vH. Calculate the expected revenue of the seller.
b. Now suppose the seller can also submit a bid r, which is his reservation price. If the highest bid is lower than r, then the object is not sold. If the highest bid is greater than r, then the highest bidder gets the object at a price equal to the larger of the other two bids. For an arbitrary r, what is the probability that:
i. Both buyers bids exceed r,
ii. Exactly one buyer bids more than r (be careful, you need to count the probability that it is bidder 1 or bidder 2),
iii. Both buyers bid less than r ?
c. From part b., calculate the sellers expected revenue for an arbitrary reserve value (you will have to think carefully about the revenue in case b i.).

d. Find the optimal reserve.
e. If you can’t solve the whole problem, provide some intuitive reasoning about whether the optimal reserve should be 0 or greater than 0. Suppose the seller sets a very small reserve, e.g., 1/100. Think of when he would lose and when he would gain, how much he would lose or gain, and the probabilities of these events.

4. Two executives a and b in a firm are competing for promotion to a higher position. The promotion has utility 1 to each contestant. Remaining in the previous position without promotion has utility that is normalised to zero.
Each contestant i puts in effort xi. Given the effort choices, i wins the contest with

probability

pi(xi
, xj ) = xi
xi + xj
.
If i wins then his payoff is pi(xi

, xj ).1 − ci(xi).

Effort has a different cost to each contestant. Specifically,

ca(x) = 1
2
x
2
, cb(x) = 1
4
x
2
.a. Suppose the two contestants choose effort simultaneously in Cournot fashion. Find the equilibrium choices and the probability that each contestant wins.
b. Suppose a has been with the firm for longer, and hence is considered the ‘senior’ guy. If he contests and loses to the more junior b then he will have some ‘loss of face’ which causes disutility −d. Assume Cournot competition. How large must
d be for a to not compete for the promotion?
c. (Extra-credit] Suppose a chooses first (Stackelberg leader) and then b chooses having observed a’s effort. Find the equilibrium choices and the probability that each contestant wins.

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