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16 BIAIS AND FAUGERON-CROUZET shares: E(p(n)) = E(vn) − kπ(1 − π) N−1 (v1 − v0). • The optimal mechanism can be implemented with the following price schedule: p(n) = vn, ∀n < N, 1 − π p(N) = vN − k π N (v1 − p(0)). The proposition has a simple intuition. The mechanism designer must ensure that investors with a good signal announce their information truthfully rather than pretending they have a bad signal. To makes misrepresentation unattractive, the optimal mechanism minimizes the amount allocated to investors who announce bad information. This implies setting this amount equal to 0, except when it is not possible, because all informed investors announce bad signals. In that situation it is not possible to exclude all the informed agents, because the retail investors cannot absorb the whole issue. Hence underpricing must be used, in addition to quantity discrimination, to induce truthful revelation. This gives rise to informational rents for the large investors. The corresponding wedge between the expected proceeds and the expected value of the shares is proportional to the amount which the uninformed retail investors cannot buy (kS). Indeed, if the uninformed retail investors could buy all the shares, allocating them the whole issue would be a simple way to avoid adverse selection and sell all the shares at their a priori expected value, thus eliminating underpricing. Note that, as in Rock (1986), a winner’s curse problem, arises for the uninformed investors in the optimal mechanism, as the amount they are allocated is maximízed when the the value of the stock is lowest. Yet, they are still willing to participate to the auction, because the mechanism is designed to satisfy their individual rationality constraint. Next we discuss the extent to which our results are robust to the informational requirements of our approach: • In contrast with Allen and Faulhaber (1989), Welch (1989) or Chemmanur (1993), we do not assume that corporate insiders selling their shares in the IPO have private information about the market value of the stock. While clearly important and interesting, the case where there is two-way information asymmetry, i.e., both the sellers and the buyers have private signals, is beyond the scope of the present paper. Note that Chemmanur (1993) analyzes two-way information asymmetry, in the context of fixed-price offers. • In our framework, it is possible to exclude investors who report bad signals, except when all of them do so. This directly reflects our assumption that each large
IPO AUCTIONS 17 investor has the potential to absorb the whole issue. If instead we assumed, like Benveniste and Spindt (1989) that the bidding capacity of the informed investors was below S, the threat to be excluded from the allocation would be less strong. Correspondingly, quantity discrimination would be a less powerful tool, and it would be necessary to rely on underpricing to a larger extent. In that context, the optimal price schedule would entail underpricing in several states, as in Benveniste and Spindt (1989). • In our model, underpricing is decreasing in the number of bidders, and the outcome of the mechanism goes to the competitive outcome (no underpricing) as the number of bidders goes to infinity. Note, however, that it would not be very reasonable to assume infinitely many bidders in our framework. In practice, there is only a limited number of professional investors who, like the large traders we consider in the model, have private information about the value of the shares and have the capacity to absorb the whole issue. With the pricing schedule presented in Proposition 1, underpricing arises when the IPO price is high. This suggests that underpricing and IPO prices are likely to be positively correlated. Now, with this pricing schedule, the covariance between underpricing and IPO prices cov(v − p, p) simplifies to This is positive if π N (1 − π N )(vN − p(N))(p(N) − E(v | b)). E(v) − E(v | b)) > kπ(1 − π) N (v1 − v0), which is likely to hold, especially when k is relatively low. This positive covariance can be interpreted in terms of underreaction of prices to demand. Note that, if no large investor could absorb the whole issue, and correspondingly there was underpricing in more states of the world, as in Benveniste and Spindt (1989), underreaction would also arise. Benveniste and Spindt (1989) and Benveniste and Wilhelm (1990) provide an interpretation of the Book Building method along the lines of this type of optimal mechanism. The indications of interest transmitted by investors to the investment bank are similar to messages about signals, and their reflection in the price adustment and discretionary allocation set by the intermediary are similar to the direct mechanism. Consistent with this theory, Hanley (1993) found positive correlation between price adjustment and underpricing in Book Building data. 3. THEORETICAL ANALYSIS OF FIXED PRICE OFFERS, MARKET CLEARING AUCTIONS, AND MISES EN VENTE 3.1. Fixed Price Auction Our analysis in this section is similar to Rock (1986). The differences are the following. First, unlike Rock (1986), we do not impose the condition that informed
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Page 3 and 4: IPO AUCTIONS 11 TABLE 1 IPO Selling
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Page 13 and 14: IPO AUCTIONS 21 FIG. 1. Buy orders
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Page 19 and 20: IPO AUCTIONS 27 FIG. 3. OLS regress
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Page 27 and 28: IPO AUCTIONS 35 Cornelli, F., and G

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