Stapled Finance Stapledfin 2

form of a specific security, for example in the form of royalties, or debt owed to the seller by the winning bidder. They show that doing so is beneficial for the seller, because it links the value of the payment (or bid) to the bidders’ private information, and since bidders with higher value estimates are more likely to win, a state-contingent security is more attractive to the seller than an up-front cash payment. To understand the linkage effect, it is useful to model auctions as a direct mechanism (see Myerson (1981)). In a direct mechanism, the bidders are asked to report their “type”, their signal 𝑠𝑖 . Denote a bidder’s report by 𝑠′𝑖 . Depending on the vector of reports s′ , a bidder may win the auction, and she may have to pay an amount 𝑝𝑖 (s′ ) to the seller. Notice that the outcome (who wins, and the payment 𝑝𝑖 (s′ )) depends on the reported signals s′ , only, and not on the true signals. In equilibrium, they depend on the true signal, but only indirectly, because usually the highest-signal bidder submits the highest bid. Security auctions allow for a “linkage” of the true type and the payment, if the security pays different amounts to its owner, depending on the value that is eventually realized: the payment 𝑝𝑖 now also depends on the true signal 𝑠𝑖 . That holds for a variety of securities for which 𝑝𝑖 (s′ , 𝑠𝑖 ) is increasing in 𝑠𝑖 for some 𝑠𝑖 , including debt. Security auctions thus create more competition between bidders, because the value of their bid is more closely linked with their true valuation, and competition from lower-valuation bidders forces up bids from higher-valuation bidders, too. DeMarzo et al. (2005) provide the most general analysis of this type of auctions. Comparing bids across different security types, e.g., cash and equity bids, is difficult, but if given the choice, the sellers will always prefer the “steepest” security available: they prefer equity-bid auctions over debt-bid auctions over cash-bid auctions.10 The reverse is true for the bidders: DeMarzo et al. (2005) (see their Proposition 5) show that in “informal auctions,” where the seller does not require bids in the form of a particular security, the bidders prefer participation in a cash-bids auction over a debt-bid auction over an equity-bid auction. Our paper contributes to this literature in two ways. First, the seller in our setup does not require that bids are made in the form of a particular security. In particular, bidders are allowed to submit cash bids, which according to DeMarzo et al. (2005) are their favorite type of bid. Bidders in our setup choose voluntarily to accept the stapled finance that is on offer, if they win, thereby submitting a security bid (cash plus debt). Assuming that cash bids are always permitted seems realistic when modeling takeover negotiations, because rejected bidders can always extend their offer directly to the target firm’s shareholders, and the target firm’s directors may be violating their fiduciary duties if they reject a cash offer that is attractive to the shareholders. Allowing for cash bids makes our model different from the setup in DeMarzo et al. (2005). For example, the seller does not know whether the winner plans to accept the stapled finance. 10 Equity bids refer to shares in the target, not in the combined firm that arises if the winning bidder merges with the target. For asymmetric information problems that arise in the latter case, see Brusco et al. (2007). 15 Therefore, a bidder’s intentions to accept or decline the stapled finance cannot affect their chances of winning (except indirectly, through the amount being bid). Nevertheless, the seller benefits from “linkage”, because the winner’s price is linked to her valuation if she accepts the stapled finance. Similarly, the losing bidder’s plan to accept or reject does not imply that the winner must make the same decision. Finally, our results do not rely on an up-front investment that the winner has to make, unlike DeMarzo et al. (2005) (see Che and Kim (2008) for a discussion). As we suggested in the introduction, a simple way to understand the intuition is to regard the option to accept the stapled finance as a subsidy that the seller offers to the winning bidder. The bidder’s valuation of the subsidy depends on her valuation of the target: higher valuations of the target go along with lower valuations of the subsidy. Thus, low-valuation bidders increase their bids, since they are being subsidized. However, they are less likely to win than high-valuation bidders, so the only effect of the higher bids from low-valuation bidders may be that they are defeated even with a higher bid, and the winner’s price is increased. Our model’s contribution is to show how linkage or security-bid features can be introduced into an auction even if the seller is not in a position to require security bids: the winning bidder may use debt to finance her bid, but this decision is made (if at all) after the auction has ended and a winner has been declared. Thus, the bidders submit cash bids and only indirectly bid using debt.11 A second contribution of our paper is that it analyzes an institutional setup that is frequently observed in practice. Stapled finance has been used in many countries (including the US) for several years (see Section I above), and our analysis explains what forces make this type of M&A financing beneficial for sellers. Corollary 1 describes the main benefit of arranging stapled finance, the reduction in the bidders’ expected net payoffs. This is possible, as we just discussed, because the seller can link the value of a bid with that bidder’s valuation. The drawbacks of stapled finance are that less value is created, because the bidder with the highest valuation is not necessarily the winner. Reduced value creation is bad for the seller, because there is less value that can be extracted from the bidders. Furthermore, if the winning bidder accepts the stapled finance, this causes a loss to the lender, so the seller needs to offer compensation to the lender when asking her to arrange the stapled finance, to offset the expected loss from making it available. Trading off these benefits and costs determines the optimal design of the stapled finance package. Proposition 3 shows that such a package always exist. We now show that under mild conditions on the distribution function 𝑓 , the problem has a unique solution. 11 Notice that the subsidy needs to be designed carefully; a flat subsidy, say a winning award with a fixed value, does not achieve anything for the seller; and as we show, a wrongly designed stapled finance package can reduce the seller’s expected payoff when facing a trade buyer and a financial buyer (see Section ?? of the Internet Appendix for details). 16 Proposition 4 If 𝑓 is log-concave, then there exists a unique optimal value of 𝑠ˆ > 𝑡. It is the value of 𝑠ˆ that solves ∫ 2(1 − 𝐹 (ˆ 𝑠))𝐹 (ˆ 𝑠) = 𝑓 (ˆ 𝑠) 𝑠ˆ 𝐹 (𝑠𝑖 )𝑑𝑠𝑖 . (14) 𝑡 Proof. See the Appendix. The log-concavity assumption is sufficient but not necessary for the uniqueness of an optimum. Log-concavity is satisfied by most commonly used distribution functions (see Bagnoli and Bergstrom (1989) and An (1998)). We can show that even with 𝑁 > 2 bidders, the optimal 𝑠ˆ is still unique and lies in the interior of the support, if 𝑓 is log-concave. Moreover, we can show that the optimal 𝑠ˆ is decreasing in the number of bidders. This result is not only important from a normative point of view, but it also allows us to compare the predictions of our model with those from other informal explanations of stapled finance. (See the next section and Section ?? of the Internet Appendix for details.) V. Alternative Explanations Stapled finance has not been studied in academic work, yet (with the exception of Boone and Mulherin (2008)), so there are no existing, well-developed alternative explanations for the popularity of this type of M&A financing. However, practitioners and journalists have described benefits that sellers enjoy when arranging stapled finance. We now discuss some of them, and how our predictions differ from those that a more formal treatment of those practitioner explanations would deliver. Practitioners emphasize mostly that various transaction costs are reduced, when stapled finance is arranged. For example, the bidders may be required to base their first-round indicative bids on leverage assumptions implied by the stapled finance, thereby saving due diligence costs. This benefits the seller, because reduced bidding costs attract more bidders, thereby enhancing competition. Another benefit for the seller is that if first-round indicative bids are required to base financial assumptions on the stapled finance, then less information may be inadvertently disclosed to competitors.12 Similarly, with a loan commitment available to any bidder, the chances are reduced that a deal will fall through because the financing could not be lined up; this clearly benefits the seller, whose aim is to raise the highest price, but also to complete the deal. We now discuss these (and other) alternative explanations, and how the predictions from these informal explanations can be distinguished from the predictions that our model makes. Note that formalizing the alternative explanations requires the use of models that do not allow for the bidding effects that we analyze (since otherwise the effects cannot be isolated). The bidding effects are eliminated if all bidders value the option to accept stapled finance equally. That can be 12 In fact, worries about information leaks often lead sellers to instruct their advisory investment bank not to share the stapled finance terms with the target’s competitors. 17 achieved either by removing the limited liability that financial buyers enjoy (see our discussion of trade buyers below, in Section VI), or by assuming that the bidders’ valuations are identical and common knowledge. In either type of model, the stapled finance offer provides equal benefits to all bidders. Consider one important implication of our model, that the stapled finance is accepted by lowvaluation bidders, only. In an alternative model based on transaction costs, and absent the bidding effects, all bidders benefit equally from the stapled finance offer. Therefore, the acceptance or rejection decision should not depend on the winner’s valuation of the target. This has some direct empirical implications. One frequently measured variable around takeover contests is the bid premium. For example, if the target is a listed company, one can compare the winner’s bid with the 52-week high price of the target. Our model predicts that the acceptance of stapled finance should be more likely for lower bid premia, and less likely for high bid premia. In contrast, if stapled finance was offered merely to reduce transaction costs, the bid premium should not have any explanatory power. Another variable of interest is the long-term operating performance of the target after the takeover. If the target remains partly publicly traded, this should be easy to measure. If not, then using plant-level data, it may be possible to measure the profitability or productivity of its operations (following the methodology in Maksimovic and Phillips (2002)). If the winning bidder is a private equity fund that does not disclose any information, then post-takeover performance information can only be gained at the time of a re-listing of the target (IPOs are common exit routes for PE funds), or at the time of a secondary (or tertiary, etc.) buyout (a sale to another buyout fund, another common exit route). Our model predicts that the target’s post-takeover performance should be lower if the winning bidder accepted the stapled finance (since only lowvaluation bidders accept it); while it should be higher if stapled finance was available but declined. In contrast, if stapled finance is arranged only to reduce transaction costs, then the acceptance or rejection decision should not have any relation to the subsequent operating performance. A third variable of interest is the abnormal return that a target’s stock experiences around the time of the announcement of a deal (this is a variable used in event studies). If investors understand the bidding effects, then our model predicts that — since investors can predict the future weak or superior performance — the cumulative abnormal return for a bidder should be lower, if investors expect her to accept the stapled finance; and higher if investors expect her to decline the offer. In contrast, if stapled finance is arranged only to reduce transaction costs, then the expected acceptance or rejection decision should not have any relation to the abnormal returns. Empiricists can also study abnormal returns at the time when information about the first round of indicative bids is revealed to the public. If investors hear that stapled finance is being offered, then all bidders should experience abnormal returns that are lower than the abnormal returns 18 they experience in auctions without stapled finance. This is because the availability of stapled finance enhances the competition between the bidders, whose rents are reduced. In contrast, if stapled finance is arranged only to reduce transaction costs, then its availability should not affect the bidders’ share prices. In fact, if the bidders benefit from the reduction in transaction costs, then the abnormal returns may even be positive (compared with announcements of first-round bids without stapled finance). Another variable that is used frequently by empiricists is the number of bidders participating in a takeover contest (see Boone and Mulherin (2007)). As we discuss in the next section, we have analyzed a model with 𝑁 > 2 bidders. As before, it is always optimal for the seller to arrange stapled finance. We also show that the optimal stapled finance package becomes less “aggressive” as 𝑁 increases: the cut-off 𝑠ˆ decreases (see Section ?? of the Internet Appendix for details). Thus, as 𝑁 increases, the probability that the stapled finance is accepted decreases for two reasons: first, it becomes unattractive for some bidders with signal realizations close to 𝑠ˆ; and second, the probability that the two highest signals are below 𝑠ˆ decreases. In contrast, consider a model based on transaction costs. For example, bidders may have to expend resources to perform due diligence on the possible financial structure of an acquisition. By arranging stapled finance, this duplication of effort can be avoided, and more bidders can be induced to participate in the auction. Such a model also predicts that the seller benefits from arranging stapled finance (she benefits from increased competition). However, the probability of acceptance of the stapled finance may not depend on the number of bidders, or it may even increase, contrary to our model’s prediction. (It may increase if the additional bidders are less experienced and may have difficulties with the financing of the acquisition, if they win.) Similarly, an increasing number of bidders increases the chances of information leakages to the target’s competitors, so arranging stapled finance becomes more beneficial, because the seller can limit the information that is made available to the bidders. Practitioners also emphasize how, by arranging stapled finance, the seller can make sure that a deal will not fall apart just because the winning bidder could not arrange the required financing. If this is the reason for arranging stapled finance, then it must be particularly beneficial (and hence more likely to be offered) in industries that experience fast-changing competitive or technological environments, or other sources of uncertainty that may change the winning bidder’s appreciation of a deal before it is completed. Similarly, deals that are complex and therefore take longer to execute are exposed to a higher risk of not being completed. In contrast, while different degrees of uncertainty affect the aggressiveness of the stapled finance that the seller should offer, it does not change the result that offering stapled finance is always optimal. We are not claiming that the alternative explanations are not valid. Our aim is to provide a novel explanation, that goes beyond mere transaction-cost arguments and yields a richer set of cross-sectional predictions. Both our and the existing explanations seem plausible, and we 19 expect them and our explanation to describe economic forces that are complementary. Testing the relative importance of the explanations empirically is left for future research. However, given the many benefits that sellers enjoy when arranging stapled finance, it should not be surprising that it became popular in a very short period of time. VI. Robustness We have used a simple model to derive the optimality of stapled financing. In this section we discuss how the model can be changed and extended in various directions. We discuss alternative auction models (the first-price auction); different valuation models (a pure common values model, and a model with a smoother valuation function); setups with more than two bidders; and a setup in which one bidder is a “trade buyer” (who does not value the option to default inherent in a stapled financing package). Our results extend to all of these setups: the seller always benefits from arranging stapled finance. As we argued earlier, we have chosen the ascending auction setup because of its realism and because it is easy to analyze. Consider a first-price auction, instead: the bidder who submitted the highest bid wins, and she pays her own bid to the seller; ties are resolved with a coin toss. As before, the decision whether to accept or decline the stapled finance offer is made after the winner’s identity and the price to be paid have been declared. At this stage, the decision depends on the bidder’s signal 𝑠𝑖 , the loan amount 𝐿 and the face value of the loan 𝐷, only. In fact, our analysis above remains valid, and we can use the same cut-off signal 𝑠ˆ. It is straightforward to confirm that the following bidding functions form a Nash equilibrium:13 𝑏𝐹 𝑃 𝐴 (𝑠𝑖 ) = 𝜑𝐸 [max {𝑠, 𝑠ˆ} ∣𝑠 < 𝑠𝑖 ] + (1 − 𝜑)𝐸[𝑡]. (15) As before, the bidders compete away their rents if their signal realizations are below 𝑠ˆ. This increased competition feeds competition with higher signals, too, and high-signal bidders (with signals above 𝑠ˆ) shade their bids to a smaller extent. The outcome of the first-price auction is quite similar to that of the ascending auction: the bids are constant for all 𝑠𝑖 < 𝑠ˆ; a coin toss determines the winner if both signals are below 𝑠ˆ; and the allocation is not affected if at least one signal is above 𝑠ˆ. In both auction setups, a bidder with a signal 𝑠𝑖 < 𝑠ˆ expects a zero payoff, and the allocation is identical, so using the arguments in Myerson (1981) it follows that the seller’s expected price is exactly the same as in the ascending auction. (The same holds for other auctions that generate the same allocation and give bidders with a signal 𝑠𝑖 = 𝑡 a zero expected net payoff.) Suppose we use a common values setup instead of a private values setup, i.e., all bidders value 13 For details, see Section ?? of the Internet Appendix. 20 the target firm equally, and each bidder privately receives some independent information about this value. Common values would not allow for the possibility of bidder-specific synergies with the target firm, which may be a reasonable assumption if the target firm is not unique. The seller’s problem is simplified because allocative distortions do not destroy value if all bidders value the target firm equally. But the bidders must worry about the winner’s curse. The rival’s signal affects the expected value of both the target firm and the stapled finance offer, and the winning bidder must estimate the losing bidder’s signal before deciding whether to accept the stapled finance. Just like in the setup with private values, the equilibrium bidding strategy is constant for low signal realizations and increasing for high signal realizations, and the seller benefits from any stapled finance package that is accepted with positive probability.14 We have also analyzed a setup with more than two bidders. The proofs are somewhat more involved, but some mild assumptions on the distribution of the signals are sufficient to derive the same result, that the seller always benefits from arranging stapled finance. The new assumptions are that 𝑓 is log-concave (see An (1998)) and that the density is strictly positive on the entire support [𝑡, 𝑡]. The proofs are provided in Section ?? of the Internet Appendix. We provide both a direct proof and an indirect proof, that relies on an intuitive argument using reserve prices. If the seller’s valuation is equal to the lowest possible valuation by the bidders (i.e., with a signal realization 𝑡), then log-concavity of 𝑓 implies that the seller always benefits from setting a reserve price. The analysis can be based on Figure 2: if 𝑠ˆ describes both the cut-off for accepting stapled finance and the signal that makes the reserve price binding, then in Regions B, C and D the two rent-extraction tools are equivalent (they have no effect in Region D, and in Regions B and C the winner’s price is increased). In Region A, the reserve price is higher than both bids, so the target is not sold, and the seller realizes only her own valuation; with stapled finance, in contrast, the target is sold to one of the bidders, whose valuation is higher than the seller’s, so the seller can realize this valuation because the bidders compete away their rents in Region A. Therefore, if it is beneficial for the seller to post a reserve price, it must be even more beneficial to arrange stapled finance. Another robustness check concerns the information “technology”. Our model in which the true valuation depends either on an unobserved signal or on the winning bidder’s signal simplifies the analysis, but we can derive the same results using a smoother information structure. More precisely, we have analyzed a model in which the realized valuation depends continuously on both the winning bidder’s signal and an unobserved random variable. As before, the seller always benefits from arranging stapled finance. (For details, see Section ?? of the Internet Appendix.) Finally, in practice takeover contests include “trade buyers” (“strategic buyers”), too, not just financial buyers. Trade buyers are interested in synergies that can be realized after incorporating the 14 Again, details are available in Section ?? of the Internet Appendix. 21 target’s operations and assets into their other operations, for example if the bidder is a competitor, or a supplier or customer of the target. This is important in our setup, because after incorporating the target, it is difficult to distinguish the cash flows and assets of the target and the bidder’s other operations. So the existing cash flows and assets would also support the debt if the bidder accepted the stapled finance, thereby reducing or eliminating the strategic benefits of limited liability that a financial buyer enjoys. We analyze a model with one trade buyer and one financial buyer in Section ?? of the Internet Appendix. As before, the seller always benefits from arranging stapled finance, whose terms are optimally set such that a trade buyer always declines the offer. Not surprisingly, the seller’s benefit of arranging stapled finance is smaller than when facing two financial buyers. The optimal stapled financing package is less aggressive, i.e., the cut-off 𝑠ˆ is lower. However, the seller can improve her expected payoff by also setting a reserve price. Finally, the trade buyer’s ex-ante expected net payoff is reduced when stapled finance is offered: her chances of winning are reduced, and if she wins her price may be higher. In contrast, the financial buyer’s ex-ante expected net payoff is increased: her chances of winning increase, and she may benefit from the stapled finance. (These results may explain why trade buyers seem to overpay in takeover contests, when competing with financial buyers; see Bargeron et al. (2008).) VII. Conclusion We have presented a novel explanation for the benefits that accrue to sellers in M&A situations, if they arrange stapled financing for the eventual buyer. Practitioners have described the benefits as savings in transaction costs, but as we show, there is more to it. Stapled finance affects the competition between the bidders in a takeover contest. It strengthens the position of the weaker bidders, who become stronger competitors and thereby force up the price that the eventual winner expects to pay. This can distort the allocation, and the lender of the stapled finance expects a loss on the loan, so she requires compensation from the seller. Nevertheless, the beneficial effect coming from increased competition more than compensates for those drawbacks, and the seller always benefits from arranging stapled finance. We have used a simple auction model to derive our results. It has the benefit of being tractable and of illustrating the intuition in a simple way. However, the results are quite robust. We have shown that the seller benefits from arranging stapled finance even if we relax various assumptions, to generalize the model. Our results are consistent with recent theoretical results on auctions using security bids. Our key theoretical contribution lies in showing how a seller can make bidders compete more aggressively, by merely offering an option that some bidders find valuable and other bidders find worthless. Ex ante, the seller benefits from arranging stapled finance, and the bidders are made worse off, because 22 the bidders find themselves in a situation that resembles a prisoner’s dilemma. Individually, they may benefit from accepting the option; but since they compete for the right to accept it, overall, the bidders are worse off. It is thus not only outside observers who worry about the aggressiveness of stapled finance packages — our model predicts that the bidders, too, should prefer if stapled finance was never offered, even though they sometimes (ex post) accept it and benefit from it. Ours is the first academic analysis of an institutional arrangement that has become popular very quickly in practice. As discussed earlier, investment banks started offering stapled finance only a few years ago. Initially, the intention may well have been to save transaction costs for the seller (the idea may have developed from the practice of frequently offering bridge financing in M&A transactions). However, as we have shown, other forces may have been responsible for the fast growth in popularity, forces that are understood only after a formal analysis of the bidding incentives in the presence of stapled finance. Judging the relative importance of the different explanations must be left for future empirical work. We have discussed what elements are needed for stapled finance to make the bidding more competitive: the stapled finance is optional for the winner; it is non-recourse; and some bidders are financial buyers. We have also discussed what predictions can be drawn from our model, and how these differ from predictions that transaction cost-based models (once formalized) may deliver. Importantly, our model predicts that by design, stapled finance loans must perform poorly during their lifetime — worse than other loans used to finance M&A transactions. It also predicts that if the stapled finance is accepted by the winner, then the target firm should be expected to perform worse (say, in terms of profitability or productivity) than similar targets whose acquisitions were not financed using stapled finance. These predictions imply that it will be interesting to track the performance of acquisitions that used stapled finance, over the next few years; and that of the stapled finance packages themselves. 23 Appendix: Proofs Proof of Proposition 4 The seller’s expected net payoff increase from arranging stapled finance equals the sum of the expected net payoff increases in the four regions of Figure 2. These four net payoff increases are ⎧   ⎨ 𝜑(𝐸[𝑠𝑖 ∣𝑠𝑖 < 𝑠ˆ] − 𝐸[min{𝑠𝑖 , 𝑠∣𝑠𝑖 , 𝑠𝑗 < 𝑠ˆ}]) Region A 𝜑(ˆ 𝑠 − 𝐸[𝑠𝑖 ∣𝑠𝑖 < 𝑠ˆ])   ⎩ 0 Regions B and C (A1) Region D For Region A, the expected net payoff increase is given in (13). For Regions B and C, the expected net payoff increase equals 𝑣(ˆ 𝑠) − 𝐸[𝑣(𝑠𝑖 )∣𝑠𝑖 < 𝑠ˆ]) =𝜑ˆ 𝑠 − 𝜑𝐸[𝑠𝑖 ∣𝑠𝑖 < 𝑠ˆ]. (A2) In Region D, the availability of stapled finance does not change the outcome. Let ΔΠ denote the increase in the seller’s expected net payoff, if she decides to arrange stapled finance. It is equal to the sum of the probability-weighted expected net payoff increases in the four regions that we just calculated, ( ) ΔΠ = Pr{𝐴}𝜑 𝐸[𝑠𝑖 ∣𝑠𝑖 < 𝑠ˆ] − 𝐸[min{𝑠𝑖 , 𝑠𝑗 }∣𝑠𝑖 , 𝑠𝑗 < 𝑠ˆ] + Pr{𝐵 ∪ 𝐶}𝜑(ˆ 𝑠 − 𝐸[𝑠𝑖 ∣𝑠𝑖 < 𝑠ˆ]) (A3) Substitute Pr{𝐴} by (𝐹 (ˆ 𝑠))2 , and Pr{𝐵} and Pr{𝐶} by 𝐹 (ˆ 𝑠)(1 − 𝐹 (ˆ 𝑠)). After partial integration of both summands in (A3), we can rewrite (A3) as ∫ ΔΠ = 𝜑 𝑡 𝑠ˆ ∫ 𝑠𝑖 ∫ 𝐹 (𝑠𝑗 )𝑑𝑠𝑗 𝑓 (𝑠𝑖 )𝑑𝑠𝑖 + 2(1 − 𝐹 (ˆ 𝑠))𝜑 𝑡 𝑠ˆ 𝐹 (𝑠𝑖 )𝑑𝑠𝑖 . (A4) 𝑡 The first order condition is ∂ ΔΠ = 2𝜑(1 − 𝐹 (ˆ 𝑠))𝐹 (ˆ 𝑠) − 𝜑𝑓 (ˆ 𝑠) ∂ˆ 𝑠 ∫ 𝑠ˆ 𝐹 (𝑠𝑖 )𝑑𝑠𝑖 . (A5) 𝑡 Evaluated at 𝑠ˆ = 𝑡, the first order condition is equal to zero. Evaluated at 𝑠ˆ = 𝑡, it is non-positive. Proposition 3 shows that choosing any 𝑠ˆ > 𝑡 is beneficial. Therefore, an interior optimum must exist. Rewrite (A5) as 1 − 𝐹 (ˆ 𝑠) 2 − 𝑓 (ˆ 𝑠) ∫ 𝑠ˆ 𝑡 𝐹 (𝑠𝑖 )𝑑𝑠𝑖 𝐹 (ˆ 𝑠) 24 = 0. (A6) We now show that the first term in (A6) is decreasing in 𝑠ˆ, and the second term increasing, which implies that the FOC can be satisfied (the two terms are equal) for only one value of 𝑠ˆ > 𝑡. The first term is decreasing if and only if ∂ 𝑓 (ˆ 𝑠) >0 ∂ˆ 𝑠 1 − 𝐹 (ˆ 𝑠) ⇐⇒ ∂𝑓 (ˆ 𝑠) ∂ˆ 𝑠 [ ⇐⇒ − (A7) (1 − 𝐹 (ˆ 𝑠)) + 𝑓 (ˆ 𝑠)𝑓 (ˆ 𝑠) ∂2 ∂ˆ 𝑠2 >0 [1 − 𝐹 (ˆ 𝑠)]2 ] ][ ∂ ] [∂ 𝑠)) ∂ˆ 𝑠)) (1 − 𝐹 (ˆ 𝑠)) (1 − 𝐹 (ˆ 𝑠)) − ∂ˆ 𝑠 (1 − 𝐹 (ˆ 𝑠 (1 − 𝐹 (ˆ [1 − 𝐹 (ˆ 𝑠)]2 (A8) > 0, (A9) which is satisfied because the log-concavity of 𝑓 implies that (1 − 𝐹 ) is also log-concave (see An (1998)). 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