Next Tuesday: Amit Gandhi guest lecture on empirical work on auctions Next Wednesday: first problem set due

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1 Econ 805 Advanced Micro Theory I Dan Quint Fall 2007 Lecture 6 Sept Next Tuesday: Amit Gandhi guest lecture on empirical work on auctions Next Wednesday: first problem set due Today: the price-discriminating monopolist problem and marginal revenue Bulow and Roberts, The Simple Economics of Optimal Auctions ) the value of an optimal auction versus the value of an additional bidder Bulow and Klemperer, Auctions Versus Negotiations ) auctions with reserve prices The Price-Discriminating Monopolist Today, we consider the problem of a monopolist who can sell in multiple markets and charge different prices in each; and we show that the problem has a very similar structure to the optimal-auctions problem. We then use this to answer the question of which is more valuable to the seller, running the optimal auction with N bidders or a simple ascending auction with N + 1 bidders. First, consider the following problem, from Bulow and Roberts, The Simple Economics of Optimal Auctions. A monopolist has Q units of a divisible good, which he can sell in N different markets of equal size, which we normalize to be 1. For i {1, 2,..., N}, customers in market i have valuations for the good on the interval [, ], distributed according to the probability distribution function F i. Bulow and Roberts complicate things by assuming the monopolist s supply varies stochastically, but this doesn t end up changing things.) Instead of thinking of the monopolist as setting a price, or a quantity, in each market, Bulow and Roberts think of the monopolist choosing a set of functions p i : [, ] [0, 1] representing the probability that a buyer in market i with value will choose to buy the product. Of course, given a price of P in a given market, this is just 1 when > P and 0 when < P.) The monopolist s capacity constraint then becomes p i v)f i v)dv Q 1

2 Note that if a buyer in market i values the good at v, his surplus is v p i x)dx because p i will generally be either 0 or 1, so this will integrate to v v, where v is the type who is indifferent about buying the good given its price, and therefore values it at exactly the price it is offered at. Given this, we can write the seller s revenue as the total value of all goods sold minus total consumer surplus, which is vp i v)f i v)dv v f i v) p i x)dxdv Changing the order of integration in the second integral, and then evaluating the inner integral, gives = so we can rewrite revenue as v f i v) p i x)dxdv = 1 F i x))p i x)dx = p i v) v 1 F ) iv) f i v)dv f i v) x f i v)p i x)dvdx 1 F i x) p i x)f i x)dx f i x) so this is what the monopolist will maximize, subject to the feasibility of {p i }. Now let s interpret the term in parentheses. Demand in market i is Q i = 1 F i P i ), or P i = Fi 1 1 Q i ), so revenue is Q i Fi 1 1 Q i ). Differentiating with respect to Q i, marginal revenue is Qi Fi 1 1 Q i ) ) = F 1 1 i 1 Q i ) Q i Q i f i 1 Q i ) = P i 1 F ip i ) f i P i ) so going back to our expression for the monopolist s revenue, the term in parentheses is simply marginal revenue in market i at price P i = v, or the marginal revenue of selling to customer n market i. We can rewrite expected revenue as MR i v)p i v)f i v)dv The monopolist s problem, then, becomes very straightforward to solve: sell to every customer with positive marginal revenue if the capacity constraint does not bind, or to the customers in all markets) with the highest marginal revenues if the capacity constraint binds. 2

3 Of course, there is one problem. If F i is regular, then within each market, marginal revenue is increasing in type; so the seller can sell only to the high-marginal-revenue customers simply by setting the correct price, knowing that the high-marginal-revenue types have valuations above this threshold. If F i is not regular, MR i v) is not monotonic in v, in which case it is not possible to cherry-pick only the customers with the highest marginal revenues, since once you announce a price, everyone in that market with values at least that high will want to buy. This is the same as saying that if F i is regular, the constraint that p i be increasing in v does not bind. Like in Myerson, then, some smoothing is required for the irregular case.) Back to Auctions, and Bulow and Klemperer, Auctions Versus Negotiations Like the monopolist s problem, recall from Myerson that we can rewrite the IPV auctioneer s expected revenue as { t 0 + E t p i t) t i 1 F ) } it i ) t 0 U i p, x, a i ) f i t i ) i N i N Consider mechanisms where U i p, x, a i ) = 0. Expected revenue can be rewritten as ) } t 0 E t { i N p i t)mr i t) + 1 i N p i t) So if we think of the seller as being another possible buyer, with marginal revenue of t 0, then the expected revenue is simply the expected value of the marginal revenue of the winner. Jump back to the symmetric case, so F i = F. Continue to assume regularity. In an ordinary second-price or ascending auction, with no reserve price, the object sells to the bidder with the highest type, which is also the bidder with the highest marginal revenue; so the expected revenue in this type of auction what Bulow and Klemperer call an absolute English auction ) is Expected Revenue = E t max{mrt 1 ), MRt 2 ),..., MRt N )} This is Bulow and Klemperer Lemma 1.) The fact that expected revenue = expected marginal revenue of winner also makes it clear why the optimal reserve price is MR 1 t 0 ) this replaces bidders with marginal revenue less than t 0 with t 0. So counting the seller s value from keeping the unsold object) an English auction with an optimal reserve price has expected revenue Expected Revenue = E t max{mrt 1 ), MRt 2 ),..., MRt N ), t 0 } So here s the gist of Bulow and Klemperer, Auctions Versus Negotiations. They compare the simple ascending auction with N + 1 bidders, to the optimal auction with N 3

4 bidders. We discovered last week that with symmetric independent private values, the optimal auction is an ascending auction with a reserve price of MR 1 t 0 ).) The gist of Bulow and Klemperer is that the former is higher, that is, that E max{mrt 1 ), MRt 2 ),..., MRt N ), MRt N+1 )} E max{mrt 1 ), MRt 2 ),..., MRt N ), t 0 } so the seller gains more by attracting one more bidder than by holding the perfect auction. They normalize t 0 to 0, but this doesn t change anything.) Bulow and Klemperer do require both independence and symmetry for the result as I ve stated it, although they relax private values. They allow each bidder s value to depend in pretty much any symmetric way on his own and his opponents signals, provided the game stays symmetric and the signals stay independent. In that case, the optimal auction is not an ascending auction with reserve price, but an ascending auction followed by a take-it-orleave-it offer to the last man standing after everyone else has dropped out. They show that once again, with independent signals and risk-neutral bidders, adding one more bidder and running a straight ascending auction is better in expectation than the optimal mechanism. Last week, we did the example with correlated types where the optimal auction extracted all bidder surplus, and you can t outperform that by finding one more bidder and going back to an ascending auction; but they cite another result that when types are affiliated a particular type of positive correlation that we ll cover in a couple of weeks), the ascendingplus-offer auction is optimal among all mechanisms where losers don t pay anything, the winner when someone wins) has the highest type, and his payment is weakly increasing in his own type; so in this setting affiliated types), adding one more bidder is still better than running the best among all standard-looking mechanisms. So the results of Bulow and Klemperer are basically: Assume symmetry, risk-neutrality, and serious bidders, that is, v t 0. If either i) bidders have private values, or ii) bidders signals are independent or affiliated, expected revenue is higher in an absolute N + 1-bidder English auction than an N-bidder English auction followed by a take-it-or-leave-it offer With independent signals, the latter auction is optimal, so adding an extra bidder and running a simple ascending auction outperforms any feasible sales mechanism With affiliated signals, the latter auction is not optimal, and the optimal auction outperforms the former; but the latter is optimal among auctions where losers don t pay, the winner has the highest signal, and his payment is weakly increasing in his own signal for any realization of his opponents signals; so adding a bidder outperforms any sales mechanism that meets these criteria We ll do the proof in the special case of independent private values. Without private values, the proof is similar, you just have to condition everything on the other bidders 4

5 signals, which get revealed as they drop out in an ascending auction. We ll cover affiliated signals in a few weeks, so we ll ignore that part for now.) The proof is surprisingly simple. All we need to do is to show that E max{mrt 1 ), MRt 2 ),..., MRt N ), MRt N+1 )} E max{mrt 1 ), MRt 2 ),..., MRt N ), t 0 } since the left-hand side is the expected revenue in an ascending auction with N + 1 bidders and no reserve price, and the right-hand side is the expected revenue in the optimal auction with N bidders. First of all, note that E{MRt N+1 )} 0. This is because the revenue from selling to nobody is 0, the revenue from selling to everybody is v 0, and the difference between these is the integral of marginal revenue, integrated over all types, so the expected marginal revenue is v. This is where their serious bidder assumption is crucial. The fact that t 0 = 0 is just a normalization; but the requirement that v t 0 is a real assumption, and their result breaks down without it.) If x is a constant, then the function gy) = max{x, y} is a convex function of y draw it), so by Jensen s inequality, E y max{x, y} max{x, Ey)} If we take an expectation over x, this gives us E x {E y max{x, y}} E x max{x, Ey)} or E max{x, y} E max{x, Ey)} Now let x = max{mrt 1 ), MRt 2 ),..., MRt N )} and y = MRt N+1 ); E max{mrt 1 ), MRt 2 ),..., MRt N ), MRt N+1 )} E max{mrt 1 ), MRt 2 ),..., MRt N ), EMRt N+1 ))} E max{mrt 1 ), MRt 2 ),..., MRt N ), t 0 } Finally and leading to the title of the paper), Bulow and Klemperer point out that negotiations really, any process for allocating the object and determining the price cannot outperform the optimal mechanism, and therefore leads to lower expected revenue than a simple ascending auction with one more bidder. They therefore argue that a seller should never agree to an early take-it-or-leave-it offer from one buyer when the alternative is an ascending auction with at least one more buyer, etc. 5

6 Reserve Prices We haven t really done much with reserve prices yet. In our symmetric IPV world, we ve shown that the optimal reserve price in a second-price auction is MR 1 t 0 ), which does not depend on the number of bidders. A couple of points worth making. In a second-price auction with a reserve price r, bidders with values t > r still have a dominant strategy of bidding their type. Bidders below r won t submit serious bids. Exactly what they do is undetermined, but doesn t affect the outcome.) In a first-price auction with a reserve price r, bidders with values t < r won t submit serious bids. It s also clear that V r) = 0, that is, a bidder with value t equal to the reserve price has expected payoff 0. Expected payoffs, and therefore equilibrium bids, can be calculated for types above r using the envelope theorem, since the equilibrium will be symmetric and bids strictly increasing in types above r: V t) = V r) + t r F N 1 s)ds = t r F N 1 s)ds = F N 1 t) t bt)) By the usual envelope-theorem logic, first- and second-price auctions with the same reserve price will be revenue-equivalent The proofs are all basically identical to the proofs without reserve prices.) By revenue-equivalence, calculating the optimal reserve price in the first-price auction is the same as in the second-price auction. In the second-price auction, the seller s revenue is 0 if v 1 < r revenue = r if v 2 < r < v 1 v 2 if v 2 > r so if we let F 1) and F 2) be the distributions of v 1 and v 2, ERr) = Differentiating with respect to r gives ) ER r) = F 2) r) F 1) r) + ) F 2) r) F 1) r) r v 0 ) + = v r v v 0 )df 2) v) ) f 2) r) f 1) r) r v 0 ) r v 0 )f 2) r) ) F 2) r) F 1) r) f 1) r)r v 0 ) With IPV, and F 1) r) = F N r) F 2) r) = NF N 1 r)1 F r)) + F N r) 6

7 and, differentiating F 1), f 1) r) = NF N 1 r)fr) Plugging these in, ER r) = NF N 1 r)1 F r)) NF N 1 r)fr)r v 0 ) The NF N 1 r) terms drop out, so r does not depend on N; setting this equal to 0 and rearranging gives r 1 F r) fr) = v 0 that is, the optimal reserve price is found by setting marginal revenue equal to v 0, marginal cost OK, we already knew this.) With correlated values, the gain from setting a reserve price is lower: since v 1 and v 2 are more likely to be close together, the events where the reserve price makes you money you gain r v 2 when v 2 < r < v 1 ) are both fewer and, on average, less valuable This is one of my working papers...) 7

8 Finally, if there s time... Last week, we did a bit on second-order stochastic dominance, and mentioned the result that for two random variables X and Y with the same mean, these three conditions are equivalent: E{uX)} E{uY )} for every increasing, concave u SOSD) Y = D X + Z, where EZ X) = 0 for every X x F s)ds x Gs)ds for every x, where F and G are the distributions of X and Y, respectively That answered the question, when does every risk-averse utility maximizer prefer one distribution of outcomes to another? Another question is, when does every utility maximizer risk-averse, risk-neutral, or risk-loving) prefer one distribution of outcomes to another? Now we no longer consider only distributions with the same mean. The following two are equivalent: E{uX)} E{uY )} for every increasing function u FOSD) F s) Gs) for every s Like before, we can express u as a positive sum of basis functions, and use this to show the equivalence between the two conditions. differentiable. Then which we can rewrite as or θ ux) = a + u θ)dθ ux) = a + u θ)1 θ<x dθ ux) = a + u θ)1 x>θ dθ We ll do the special case where u is Since u is increasing, u θ) 0, and so we ve expressed u as a positive sum of basis functions h θ x) = 1 x>θ, that is, the jump functions. By the same logic as before, then, EuX) EuY ) if and only if Eh θ x) Eh θ y) for every θ. If this holds for every θ, we can multiply the inequality by u θ), integrate over θ, and we re done; if it fails for some θ, that s a valid increasing function which doesn t prefer X to Y.) But E {h θ x)} = 1 x>θ df x) = θ df x) = F ) F θ) = 1 F θ) and similarly, E {h θ y)} = 1 Gθ), so X first-order stochastically dominates Y if and only if F θ) Gθ) for every θ. 8

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