Econ 805 Fall 2009 Problem Set 1-

1. Discrete Auctions with Continuous Types

Consider an auction for a single good, with two bidders, each with a private value drawn independently from the uniform distribution on [0, 100]. The seller decides to hold a first price sealed-bid auction, but bids are constrained to be taken from the set {0, 25, 50, 75}. If both bid 0, nobody will be awarded the object; if they bid the same, the winner will be chosen at random (with equal probabilities).

(a) Does revenue equivalence hold in this setting? Why or why not?

(b) Show that in any equilibrium, neither bidder bids higher than his or her value. (Recall that weakly dominated strategies may be played in equilibrium, but that strategies which do not survive iterated elimination of strictly dominated strategies may not.)

(c) Show that in any equilibrium, nobody bids 75.

(d) Show that in any equilibrium, nobody with value above 25 bids 0.

(e) Show that in any equilibrium, bids are weakly increasing in types.

(f) Calculate the unique symmetric equilibrium of this auction. (It may help to note that parts (b)-(e) have reduced potential equilibrium strategies to the value at which bidders stop bidding 25 and start bidding 50.)

Now suppose that bids are instead constrained to be taken from the set {0, 50}, with the same rules.

(g) What are equilibrium strategies?

(h) What is the seller's expected revenue? Is this more or less than he would earn in an ordinary (unconstrained) first-price auction?

(i) What is the expected payoff to each type of bidder? Is there any type that prefers this (constrained) auction to an ordinary first-price auction?

2. K-Unit Auctions with Unit Demand

Consider an N-bidder auction for K identical objects, with 1 < K < N. Each bidder only wants one object. Private valuations are independent, and drawn from a common probability distribution F.

(a) Use the envelope theorem to show that revenue equivalence holds. (You do not need to explicitly calculate V (t), but you do need to express it as a function of auction primitives. Feel free to define shorthand notation for messy expressions.)

(b) The natural analog to the one-item second-price auction is an auction where the K highest bidders win the items, and they all pay a price equal to the K + 1st-highest bid, or the highest losing bid. Show that in this auction, it is a dominant strategy to bid your type.

(c) The natural analog to the first-price auction is an auction where the K highest bidders win the items, and they each pay their own bid. Calculate the expected revenue in this auction if F is the uniform distribution on [0, 100].

3. Auctions and Price-Discriminating Monopolists

(Based on Jeremy Bulow and John Roberts (1989), "The Simple Economics of Optimal Auctions," Journal of Political Economy 97, and problem 7 from Klemperer ATP)

Suppose a monopolist can produce an arbitrary amount of a good for free. There is a continuum of customers with mass 1, who have valuations for the good distributed on the interval [v, v] with distribution F, so price P would lead to demand of 1 - F(P); or put another way, demand Q can be achieved at price F^-1(1 - Q).

(a) Show that if P is the price at which demand is Q, then the derivative of revenue with respect to Q is equal to

P - (1 - F(P)/f(P))

At price P, the marginal customer values the good at v = P. So the marginal revenue from (lowering price just enough to) selling to the incremental customer with value v is v - (1-F(v)/f(v)), which is why this expression is referred to as marginal revenue.

(b) Show that when F is regular, the monopolist maximizes revenue by selling to all customers with v - (1-F(v)/f(v)) > 0 and not the others.

Now suppose the monopolist faces N different markets i of equal size, and can set a different price P_i in each market. Customers in market i have valuations with distribution F_i on [v_i, v­_i^-]. Selecting a price P_i for market i can be thought of as choosing an allocation rule

p_i : [v_i, v­_i^-] → [0, 1]

where a customer with value v_i in market i gets the good with probability pi(vi); but subject to the constraint that pi must be nondecreasing and equal to either 0 or 1 almost everywhere. (By setting a price P_i, the monopolist implicitly sets p_i(v_i) = 1 for v_i > P_i and 0 for v_i < P_i.)

(c) Show that a customer with value v_i in market i gets surplus

_{v_i}∫^v_i p_i(x)dx

(d) Since total surplus = consumer surplus + producer surplus, we can write the seller's revenue as the value of all goods sold, minus the surplus left to consumers. Show that this is

revenue = _i=1∑^N_{v_i}∫^v_ivp_i(v)f_i(v)dv - ∑_{v_i}∫^v_if_i(v)(_{v_i}∫^v_ip_i(x)dx)dv

(e) Show that this is equal to

∑_{v_i}∫^v_i p_i(v)(v -(1 - F_i(v)/f_i(v)))f_i(v)dv

the sum of the marginal revenue of all the customers he sells to.

(f) If each F_i is regular, describe the solution to the monopolist's problem when he has a limited supply of Q divisible units of the good to sell, and when he is unconstrained, and relate the problem to an auctioneer's choice of an optimal auction.

(Recall that when choosing an allocation rule p, the monopolist is subject to the constraint that each p_i must be nondecreasing and either 0 or 1 almost everywhere; but that if the unconstrained maximizer satisfies this constraint, it is also the constrained maximizer.)

(g) To see the limitations of this analogy, let Q = 1 and N = 2.

i. Solve the monopolist's problem when customers in market 1 have valuations for the good which are distributed uniformly on [0, 1], and customers in market 2 have valuations which are distributed uniformly on [1, 2].

ii. Calculate the optimal auction with two bidders, with the first bidder's valuation distributed uniformly on [0, 1] and the second bidder's valuation distributed uniformly on [1, 2].

iii. Calculate the expected revenue in each, and explain the difference.