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An industrial organization field exam from 2009 with two questions. The first question discusses deriving the best-response function, estimating the distribution of bids nonparametrically, and recovering the valuation for each bidder given her bid in independent private value auctions. It also explores the presence of an 'insider trader' and the distribution of bids in such a scenario. The second question focuses on proving the existence of a unique price for the seller to quote, the value of n, and the buyer's utility in different scenarios when the realization of r depends on an ex ante investment by the buyer.
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Instructions Answer both questions. The two ques- tions count equally. There are three (3) pages to the exam.
Question 1 A researcher has data on bidding in a series of auctions. The auctioned items have identical qualities and there is one auctioned object per auction. The auction mechanism is the following. All bidders privately record their bids and submiut them to the auctioneer. None of the participants can observe the bids of the other bidders. The auctioneer determines the winner as the bidder who submitted the highest bid. The winner will pay the price equal to her bid. The dataset of the researcher
{bki }N i=1k
k=1 contains bids^ b
k i for each recorded participant i = 1,... , Nk in each auction k = 1,... , K. The participants in all auctions are different. Assume that these are independent private value auctions and all bidders are risk-neutral. In other words, each bidder i has a valuation of the auctioned object vi and if she wins the auction and pays the price p then her utility is ui = vi − p. Assume that valuations are independenly drawn from distribution Fv (·) with finite support [v, v] which is known.
(a) Derive the best-response function for each bidder that will give the optimal bidding strategy given her valuation.
(b) Express the best-response correspondence in terms of the distribution of bids rather than the distribution of the private values Fv (·).
(c) Suggest a method to estimate the distribution of bids nonparametrically
(d) Using the best-response correspondence and the estimated distribution of bids provide a method for recovering the valuation for each bidder given her bid.
(e) Using the answer to the previous question provide a nonparametric es- timator Fbv (·) for the distribution of private values. What can you say about its pointwise asymptotic distribution (i.e., the distribution of Fbv (x) for each x ∈ [v, v])?
(f) Derive the expected social welfare in an auction given the distribution of private values. Using this formula, provide an estimator for the expected social welfare for the considered set of auctions.
Now consider a slightly changed setup. Suppose that the structure of the auction and the data above is the same, but bidder 1 is the same across all auctions and she is the “insider trader.” In other words, bidder 1 has the same valuation in all auctions and she knows the bids of all competing bidders before submitting her bid. However, the remaining bidders are not aware of the presence of such an “insider trader” and submit their bids assuming that the auction proceeds with a sealed-bid setup.
(g) Derive the best-response correspondences for the bidders in the new setup.
(h) Derive the distribution of bids given the distribution of private values and the valuation of the “insider trader” and compare it to the case where there is no “insider trader.”
(i) Provide a non-parametric estimator for the new distribution of bids. Will the estimator be different from the previous case?
(j) Suggest a method that will allow the researcher to detect the presence of an “insider trader” by comparing the distribution of bids with and without such a special bidder.
(k) Will the distribution of valuations Fv (·) be identified from the distribu- tion of bids? In case of the positive answer, provide an explicit mapping between the distribution of bids and the distribution of valuations. Oth- erwise, provide a counter-example where two different sets of valuations lead to the same set of bids.
Question 2
A buyer wishes to buy at most one unit of good from a seller. Assume the seller’s cost is 0. The buyer has a valuation r for the good, which is his private information. The seller knows that r ∼ F : [r, r] → [0, 1], where [r, r] is the support. Assume F (·) is continuously differentiable on its support. Assume that F (·) has the property that its associated hazard rate is increasing. Let the timing be that (i) r is chosen according to distribution F ; (ii) the buyer learns his type; and (iii) the seller makes him a take-it-or-leave-it (tioli) offer.
(a) Prove that there is a unique price, p∗^ ∈ (r, r), for the seller to quote if she wishes to maximize her expected profit.
(b) Let h(·) be the hazard rate associated with the distribution F. Let n = p∗h(p∗). What is the value of n? (Your answer needs to be an actual real number, not an algebraic expression.)