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32 BIAIS AND FAUGERON-CROUZET The investor with a bad signal is better off bidding for S shares than for ¯Q if c > ¯Q N−1 S πl[v(l) − p] . S(1 − k + N − (l + 1)) + (l + 1) ¯Q l=0 Denote K1 the right-hand side of this inequality. Now consider the informed investor with a good signal. She is better off bidding for ¯Q than for S shares if That is, if N−1 ¯Q S πl[vl+1 − p] − c S(1 − k + N − (l + 1)) + (l + 1) ¯Q l=0 N−1 S > S πl[vl+1 − p] . S(1 − k + N − l) + (l) ¯Q l=0 N−1 πl[vl+1 − p]S l=0 ¯Q × S(1 − k + N − l) + l ¯Q + ¯Q − S > c. S(1 − k + N − l) + l ¯Q + S Denote K2 the left-hand side of the inequality. Note that it is indeed positive, since ¯Q S(1 − k + N − l) + l ¯Q + ¯Q > S S(1 − k + N − l) + l ¯Q + S . Proof of Proposition 3. To establish that the strategies stated in the proposition form an equilibrium, we need to prove that the investor with a good signal, anticipating that the other investors follow the equilibrium strategy, prefers to purchase S/(N + 1) shares at price p ¯ , rather than bidding more aggressively, to increase her market share. Indeed, if she does not want to do so, then a fortiori, the uninformed investor or the informed investor with bad news do not want to undercut, while they are still willing to buy a constant amount at p ¯ . Anticipating that the other investors follow their equilibrium strategies, she faces the residual supply curve: S S − N N + 1 − σ (p − p ¯ ) .
IPO AUCTIONS 33 Her task is to choose the price p which maximizes her expected profit, (p), where S (p) = + Nσ (p − p) (E(v | g) − p), N + 1 which is equal to the product of the residual supply curve by the unit profit. To prove that she finds it optimal to purchase S/(N + 1) shares at price p ¯ , we need to show that this is decreasing for all prices above p ¯ .Now ∂(p) p S = Nσ (E(v | g) − p) − + Nσ (p − p) N + 1 is decreasing in p. Hence, to show that it is negative for all prices larger than or equal to p ¯ , we only need to show that it is negative for p ¯ , i.e., that is S Nσ (E(v | g) − p) − < 0, ¯ N + 1 σ< 1 N(N + 1) S E(v | g) − p ¯ Proof of Proposition 4. First, consider the case of the informed investor with a good signal. Her equilibrium expected gain is π N−1 (vn − p) S − c. N Could she obtain more by deviating from her equilibrium strategy? First, suppose she demands a lower amount q, still at price pn. Ifq > S, her expected gain is strictly lower than in equilibrium, since she just obtains less shares. If q ≤ S, then her expected gain is π N−1 S (vn − p)S (N − 1) ¯Q + S ′ which is lower than her equilibrium gain if π N−1 N − 1 ( ¯Q − S) (vn − p)S N (N − 1) ¯Q + S . > c. Second, suppose she places a bid at a lower price than pn, say P.IfP < pn−1, the IPO price is pn−1, while if P ≥ pn−1, then the IPO price is above P. In both cases she obtains 0 shares.
Page 1 and 2: Journal of Financial Intermediation
Page 3 and 4: IPO AUCTIONS 11 TABLE 1 IPO Selling
Page 5 and 6: IPO AUCTIONS 13 with the empirical
Page 7 and 8: IPO AUCTIONS 15 that the others wil
Page 9 and 10: IPO AUCTIONS 17 investor has the po
Page 11 and 12: IPO AUCTIONS 19 First note that con
Page 13 and 14: IPO AUCTIONS 21 FIG. 1. Buy orders
Page 15 and 16: IPO AUCTIONS 23 Consistent with our
Page 17 and 18: IPO AUCTIONS 25 TABLE 3 IPO price R
Page 19 and 20: IPO AUCTIONS 27 FIG. 3. OLS regress
Page 21 and 22: IPO AUCTIONS 29 investor’s anonym
Page 23: p(0) = v0, this can be rewritten as
Page 27 and 28: IPO AUCTIONS 35 Cornelli, F., and G

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