Online Aendix for The Timing and Method of Payment in Mergers when Acquirers Are Financially Constrained Alexander S. Gorbenko USC Marshall School of Business Andrey Malenko MIT Sloan School of Management October 017 0, Detailed roof of stes in Proosition 1. Ste 1: Consider i v l. If b i, v l, α i, v l b i, v h, α i, v h, then b i, v l, α i, v l =. By contradiction, suose otherwise. Since b i, v l, α i, v l b i, v h, α i, v h, offer V v l b i, v l, α i, v l reveals that the bidder s tye is v l. Therefore, it must satisfy: α i, v l V v l + b i, v l. Consider a deviation by tye v l from offer b i, v l, α i, v l to offer 0, V v l. The value of this offer is, if erceived as coming from tye v l, and above, if erceived as coming from tye v h with ositive robability. Thus, it satisfies the no default condition that its value, evaluated according to the beliefs of the seller, is at least. However, the ayoff from this offer to tye v l is strictly higher: since λ i > 1. Therefore, b i, v l, α i, v l = V v l V v l α i, v l V v l b i, v l > 1 α i, v l V v l λ i b i, v l. 0, V v l. Assumtion 1 CKIC. According to Assumtion 1 CKIC, if bidder i submits offer b, α satisfying 1 α V v h λ i b max {1 α i, v h V v h λ i b i, v h, V o }, A1 1 α V v l λ i b < max {1 α i, v l V v l λ i b i, v l, V o }, A then the seller must believe that bidder i s synergy is v h. The intuition is as follows. The left-hand sides 1
of A1 and A are the ayoffs of bidder i, if it acquires the target for b, α, if its synergy is v h and v l, resectively. The right-hand sides of A1 and A are the ayoffs of of bidder i with synergy v h and v l, resectively, if it follows the equilibrium strategy. Thus, conditions A A1 mean that the low-synergy bidder is strictly worse off deviating to offer b, α, while the high-synergy bidder is otentially better off. According to CKIC, it is unreasonable for the seller to believe that such an offer comes from tye v l, so the seller must believe that it comes from tye v h. Ste : Consider i v l. If b i, v l, α i, v l b i, v h, α i, v h, then b i, v h, α i, v h = 1. 1 γ i, V v h γ i, where γ i = 1 + 1 λ i 1 1 V v l V v h Since b i, v l, α i, v l b i, v h, α i, v h, offer b i, v h, α i, v h must satisfy: α i, v h V v h + b i, v h A3 1 α i, v h V v l λ i b i, v h V v l A4 The first inequality is the condition that the value of offer b i, v h, α i, v h is at least. The second inequality is the condition that tye v l is not better off deviating from 0, to b i, v h, α i, v h. Let V v l us show that b i, v h, α i, v h must be such that both A3 and A4 bind. First, by contradiction suose that A3 is slack. If α i, v h = 0, then tye v h is better off deviating to offer b i, v h ε, α i, v h for an infinitesimal ε > 0. If α i, v h > 0, consider a deviation by tye v h to b i, v h + ε 1, α i, v h ε for infinitesimal ε 1 > 0 and ε > 0, satisfying 1 α i, v h + ε V v h λ i b i, v h ε 1 > 1 α i, v h V v h λ i b i, v h, A5 1 α i, v h + ε V v l λ i b i, v h ε 1 < V v l. A6 For examle, for an arbitrary infinitesimal ε > 0, let ε 1 = ε λ i V v h + V v l. Then, according to CKIC, the target believes that offer b i, v h + ε 1, α i, v h ε is submitted by tye v h. Since A3 is slack and ε 1 and ε are infinitesimal, the value of offer b i, v h + ε 1, α i, v h ε, as erceived by the target, exceeds. Furthermore, since A5 holds, tye v h is better off buying the target for b i, v h + ε 1, α i, v h ε than for b i, v h, α i, v h, which is a contradiction with the statement that b i, v h, α i, v h is the otimal offer for tye v h. Hence, A3 binds. Second, by contradiction suose that A4 is slack. Since b i, v l, α i, v l b i, v h, α i, v h and b i, v l = 0, it must be that b i, v h > 0. Consider a deviation to b i, v h εv v h, α i, v h + ε for an infinitesimal ε > 0. Since A4 is slack and ε is infinitesimal, b, α = b i, v h εv v h, α i, v h + ε satisfies A. Therefore, it is erceived as coming from tye v h. Hence, the seller values it the same as offer b i, v h, α i, v h. Hence, since b i, v h, α i, v h satisfies A3, so does b i, v h εv v h, α i, v h + ε. However, tye
v h is better off buying the target for b i, v h εv v h, α i, v h + ε than for b i, v h, α i, v h: 1 α i, v h ε V v h λ i b i, v h εv v h = 1 α i, v h V v h λ i b i, v h + λ i 1 εv v h > 1 α i, v h V v h λ i b i, v h, since λ i > 1 and ε > 0. Therefore, both A3 and A4 bind. Solving this system of two equations yields b i, v h, α i, v h = 1 γ i,, where γ i = 1 + 1 v h v l V v h γ i 1. λ i 1 Π T +Π B +v h Ste 3: Consider i v l. It cannot be that b i, v l, α i, v l = b i, v h, α i, v h. By contradiction, suose there is such offer b i, α i. If α i > 0, consider a deviation by tye v h to b i + ε 1, α i ε for infinitesimal ε 1 > 0 and ε > 0 satisfying ε V v h λ i ε 1 > 0 > ε V v l λ i ε 1 For examle, for an arbitrary infinitesimal ε > 0, let ε 1 ε λ i V v l + V v h. Then, according to CKIC, the seller must believe that offer b i + ε 1, α i ε comes from tye v h. Since b i, α i is valued by the seller at least at, ε and ε 1 are infinitesimal, and v h exceeds the average of v h and v l, offer b i + ε 1, α i ε is valued by the seller at more than. Furthermore, since ε V v h λ i ε 1 > 0, tye v h is strictly better off acquiring the target for b i + ε 1, α i ε than for b i, α i. Hence, there is a rofitable deviation for tye v h, which is a contradiction. Ste 4: i v l = Π T +v l r µ X t. Since b i, v l, α i, v l = 0, V v l, the bidder with synergy v l bids u to the rice i v l at which it is indifferent between acquiring the target for 0, and losing the auction: 1 i v l V v l = V o, V v l which yields i v l = V v l V o = Π T +v l r µ X t. V v l 0, Ste 5: Consider i v h ε, i v h] for an infinitesimal ε > 0. It must be that b i, v h, α i, v h =. By the argument in ste, A3 binds. By contradiction, suose that b i, v h > 0. V v h Since i v h ε, i v h], b i, v h, α i, v h is such that the ayoff of the bidder with synergy v h, 1 α i, v h V v h λ i b i, v h, is in the neighborhood of V o. Consider a deviation to b, α = b i, v h εv v h, α i, v h + ε for an infinitesimal ε > 0. Since ε is infinitesimal, 1 α V v h λ i b is in the neighborhood of V o. Since v l < v h, we conclude that 1 α V v l λ i b < V o. 3
Therefore, b, α satisfies A. In addition, b, α satisfies A1: 1 α V v h λ i b = 1 α i, v h V v h λ i b i, v h + λ i 1 εv v h > 1 α i, v h V v h λ i b i, v h, since λ i > 1. Therefore, offer b, α is erceived as coming from tye v h, and tye v h is better off deviating to b, α from b i, v h, α i, v h, which is a contradiction. Hence, b i, v h = 0. Since A3 binds, α i, v h = V v h 0,. Finally, notice that V v h satisfies CKIC. Indeed, there exists no deviation that benefits tye v h and satisfies A3, since A3 binds and b i, v h = 0. Ste 6: i v h = Π T +v h r µ X t. Since b i, v h, α i, v h = 0, V v h for close to i v h, tye v h bids u to the rice i v h at which it is indifferent between acquiring the target for 0, and losing the auction: 1 i v h V v h = V o, V v h which yields i v h = V v h V o = Π T +v h r µ X t. Ste 7: b i i v, v, α i i 0, v, v = 1 γ i i v Π l, T +v l Π T +Π B +v h γ i, where γ i = yields 0, i v V v Π T +v Π T +Π B +v 1 + 1 V v h, v {v l, v h }, and b i i v l, v h, α i i v l, v h = v h v l 1. λ i 1 Π T +Π B +v h Plugging i v = Π T +v r µ X t into ΠT +v Π T +Π B +v, 0. Plugging i v l = Π T +v l r µ X t into the exression for b i i v l, v h, α i i v l, v h from ste yields the last exression. Proof of Lemma 1. By contradiction, suose that X i s is not decreasing for some i {1, }. Without loss of generality, suose this is for i = 1. Since X i s is monotone, there can be two cases: 1 X i s is increasing in s for both i {1, }; X 1 s is increasing in s, but X s is decreasing in s. For each case, consider bidder 1 with signal s 1 laying the strategy of initiating the auction at threshold, if it has not been initiated yet. First, consider case 1. If s is low enough so that X s <, the auction is initiated by bidder at threshold X s. The exected surlus of bidder 1 from the auction in this case is s 1 1 s ψ 1 v h, v l X s r µ. Otherwise, the auction is initiated by bidder 1 at threshold. Its exected surlus from the auction in this case is s 1 1 s ψ 1 v h, v l r µ. Thus, the exected value to bidder 1 at any time t rior to initiation 4
of the auction is Π B X t Xt β s + X 1 + X 1 max u [0,t] X u X 1 Xt X s s 1 1 s ψ 1 v h, v l I df s 1 F X 1 max u [0,t] X u β s 1 1 s ψ 1 v h, v l X s I df s 1 F X 1 max u [0,t] X u. The otimal threshold maximizes this exected value. Differentiating in and s 1 yields β 1 X β t s β 1 X 1 1 s ψ 1v h, v l df s 1 F X 1 max u [0,t] X u < 0. Second, consider case. Since X is decreasing, the argument of Section 3. alies and the ayoff to bidder 1 is given by 1. Differentiating in and s 1 yields β 1 X β t β 1 X 1 s 1 z ψ 1 v h, v l df s F X 1 max u [0,t] X u < 0. By Tokis s theorem Tokis 1978, the otimal initiation threshold of bidder 1 is decreasing in s 1 in both cases, which is a contradiction. Proof of Lemma. Take the full derivative of equations δ i x, ζi x, ζ i x = 0 in x: δ i x + δ i ζ i ζ i x + δ i ζ j x = 0, i {1, }. Multily the equation for δ i by δj, the equation for δ j by δi : δ i δ j + δ i δ j ζ i x + δ i δ j ζ j x = 0; x ζ i δ j x δ i + δ j δ i ζ j x + δ j ζ i δ i ζ i x = 0. Subtract the latter equation from the former one, observing that the third term in the first equation and the second term in the second equation cancel out: In our case, δi x δi δ j δ j δ i ζ i x = δ j δ i δ i δ j. ζ i ζ i x x = ζ i x mζ j x > 0; similarly, δj x > 0. Also, δ i = xζ i x mζ j x A7 0 and δj = xmζ i x > 0. Therefore, the right-hand side of A7 is strictly negative. Because X i s is strictly 5
decreasing in s [s, s], ζ i x < 0 for any x [ Xi s, X s ]. Therefore, δ i ζ i δ j δj ζ i δ i > 0 for any x [ Xi s, X i s ]. Because the derivations aly for any i {1, }, we conclude that δ1 ζ 1 δ ζ δ1 ζ δ ζ 1 > 0 for all x { min i {1,} Xi s, max i {1,} Xi s }, that is, for all x, at which the auction could occur in equilibrium. Details of the solution of the model in Section 3. For brevity, denote ξ i β r µi β 1 ψ i v h,v l. Then, 13 becomes X i s = sm ξ i X 1 j X. i s For transarency, in what follows we unack m X 1 j X [ i s as 1 E z z 1 X j Xi s ]. Without loss of generality, suose that bidder 1 faces higher financial constraints: λ 1 > λ. By Proosition 5, X 1 s > X s for any s. Let ŝ 1 be the signal at which X 1 ŝ 1 = X s. Let ŝ be the signal at which X 1 s = X ŝ. First, consider bidder with signal s ŝ. When X t is about to hit its initiation threshold X s, bidder did not learn anything about the signal of bidder 1, since no threshold X 1 s, s [s, s] has been assed yet. Therefore, for any s ŝ, X s = s 1 E [z]. Second, consider bidder 1 with signal s. When X t is about to hit its initiation threshold X 1 s, bidder 1 believes that the signal of bidder cannot exceed ŝ, since otherwise bidder would have initiated the auction earlier. Therefore, X 1 s = s 1 E [z z ŝ ]. Since ŝ is defined by X 1 s = X ŝ, it satisfies: ŝ 1 E [z] = s 1 E [z z ŝ ]. A8 Third, consider bidder 1 with signal s ŝ 1. When X t is about to hit its initiation threshold X 1 s, bidder 1 believes that bidder with any signal would have initiated the auction before. By our belief udating rule, we assume that bidder 1 believes that bidder s signal is lowest ossible, s. Therefore, for any s ŝ 1, X 1 s = s 1 s. 6
Fourth, consider bidder with signal s. When X t is about to hit its initiation threshold X s, bidder believes that the signal of bidder 1 cannot exceed ŝ 1, since otherwise bidder 1 would have initiated the auction earlier. Thus, X s = s 1 E [z z ŝ 1 ]. Since ŝ 1 is defined by X 1 ŝ 1 = X s, it satisfies: ŝ 1 1 s = s 1 E [z z ŝ 1 ]. A9 Finally, consider bidder 1 with signal s 1 [ŝ 1, s] or bidder with signal s [s, ŝ ]. Bidders with such valuations initiate the auction at thresholds in interval [ X1 ŝ 1 = X s, X 1 s = X ŝ ]. Take any x in this interval. It is the initiation threshold of bidder 1 with some signal in [ŝ 1, s], denoted s 1 x = 1 X 1 x. It is also the initiation threshold of bidder with some signal in [s, ŝ ], denoted s 1 x = X x. For each arameter x, equation 13 in the aer yields a system of two equations with two unknowns, s 1 and s : x = x = s 1 1 E[z z s ] s 1 E[z z s 1 ] A10 A11 For each x, the solution to this system is s 1 x and s x. Equililbrium initiation thresholds X 1 s and X s are the inverses of s 1 x and s x, resectively. In what follows, we secialize this solution to the case of uniform distribution of signals over [s, s] with 0 < s < s 1. Examle: uniform distribution of signals In this case, equations for ŝ 1 and ŝ, A8 and A9, simlify to ŝ 1 s+ s =, s 1 s+ŝ ŝ 1 1 s =. s 1 s+ŝ 1 7
The solutions are: ŝ 1 = ŝ = ξ s 1 s 1 s + ξ, A1 1 s s 1 s ξ 1 s. A13 ξ1 ξ 1 s For examle, in the limit ξ 1, we have ŝ 1 s and ŝ s. That is, if both bidders have aroximately identical financial constraints, there is aroximately no range of thresholds X at which only one of the two bidders initiates the auction with ositive robability. Solutions A1-A13 immediately give X 1 s for s ŝ 1 and X s for s ŝ : X 1 s = X s = s 1 s for s ŝ 1, s 1 s+ s for s ŝ. Finally, the system of equations A10-A11 simlifies to: x = x = s 1 1 s+s s 1 s+s 1 ξ Substituting s = into the first equation and simlifying, we obtain: x 1 s+s 1 xs 1 x s s 1 sx 1 s + s 1 ξ = x s s 1. This yields a quadratic equation for s 1, one of which roots the smaller one belongs to interval [s, ŝ 1 ]. Hence, the case of uniform distribution of signals has a closed form solution for inverse initiation functions. Direct initiation thresholds are also recovered from them by inversion. Details of the equilibrium construction for the model with entry deterrence Section 5.1. Ste 1: no deterrence region, s [s, ŝ 1. Consider a bidder with signal s [s, ŝ 1 and time t satisfying max u t X u X d ŝ 1. If such bidder initiates the auction at threshold, its exected 8
value at time t is X β X 1 t Π B + Xt s1 zψv h, v l s I df z F X 1 max u t X u X 1 max u t X u β Xt + X 1 s1 zψv h, v l Xz X z I df z F X 1 max u t X u, which coincides with 8 in the basic model. Therefore, by the same argument, the equilibrium initiation threshold in this range is given by X s = X s = β I β 1 sm s ψ v h, v l. A14 Ste : artial deterrence region, s ŝ 1, ŝ. Consider a bidder with signal s ŝ 1, ŝ and any time t satisfying max u t X u X fd ŝ and max u t X u < X d ŝ 1. If such bidder initiates the auction at threshold, its exected value at time t is Π B X t + Xt β X 1 s1 zψv h, v l s I df z F X 1 max u t X u β r µi Xt ψv + sψv h, v l h,v l 1 X 1 df z z A15 s F X 1 max u t X u X 1 max u t X u { β Xt + X 1 max s1 zψv h, v l Xz } X z I, 0 df z F X 1 max u t X u. In A15, the first term reflects the stand-alone value of the bidder; the second term reflects the exected value from the auction initiated by the bidder, assuming that the rival articiates; the third term reflects additional value from the auction given that the bidder does not articiate if its signal is low enough; and the fourth term reflects the exected value from the auction initiated by the rival. Differentiating A15 with resect to and alying the equilibrium condition that the maximum must be reached at 9
= X s, we obtain the following differential equation on X s in this region, denoted X d s: β 1 sψv h, v l X d s f = r µi ψv h,v l 1 s X d s F s 1 s s I ψ v h, v l r µi ψv h,v l 1 s X d s z df z βi F s 1 s X d s X d s 1 s 3 Xd s. A16 Ste 3: full deterrence region, s ŝ, s]. Consider a bidder with signal s ŝ, s] and any time t satisfying max u t X u < X fd ŝ. If such bidder follows the strategy of initiating the auction at threshold, rovided that the rival has not initiated the auction yet, and not articiating in the auction initiated by the bidder, its exected value at time t is Π B X t + Xt β X 1 sψv h, v l s I df z A17 F X 1 max u t X u. In A17, the first and second terms reflect the stand-alone value of the bidder and the exected value from the auction, resectively. If the signal of the rival exceeds X 1, the rival initiates the auction before X t reaches threshold, and the bidder gets zero ayoff, because it does not enter the auction. If the signal of the rival is below X 1, the bidder initiates the auction first at threshold and acquires the target for V v l V o, because the rival does not enter the auction. Thus, the ayoff to the bidder from the auction in this case is sψv h, v l r µ I. Differentiating A17 with resect to and alying the equilibrium condition that the maximum must be reached at = X s, we obtain the following differential equation on X s in this region, denoted X fd s: β 1 sψv h, v l X fd s βi = sψv h, v l X fd s I f s F s X fd s X fd s. A18 Ste 4: equations for ŝ 1 and X d ŝ 1. Cutoff tye ŝ 1 and the initial value condition in differential 10
equation A16, Xd ŝ 1, must satisfy: = β Xd ŝ 1 ŝ1 ŝ1 1 zψ v h, v l X ŝ 1 X ŝ 1 s ŝ 1 1 z1 ŝ1 s { z > r µi ψv h,v l 1 ŝ 1 X d ŝ 1 I df z } ψv h, v l X d ŝ 1 I df z ; A19 β 1 ŝ 1 ψv h, v l X d ŝ 1 1 f = r µi ψv h,v l 1 ŝ 1 X d ŝ 1 F ŝ 1 ŝ1 r µi ψv h,v l 1 ŝ 1 X d ŝ 1 ŝ 1 I ψ v h, v l 1 ŝ 1 Xd ŝ 1. df z z βi F ŝ 1 A0 where 1 { } is an indicator function. A19 is the indifference condition stating that tye ŝ 1 must be indifferent between initiating the auction at threshold X ŝ 1 and facing entry of the rival with robability one and initiating the auction at threshold X d ŝ 1 < X ŝ 1 and facing entry of the rival only if its signal z is suffi ciently high. If A19 did not hold, then either tye s just above ŝ 1 would be better off deviating from initiating the auction at threshold X d s to threshold X ŝ 1 if the left-hand side of A19 exceeded the right-hand side or tye s just below ŝ 1 would be better off deviating from initiating the auction at threshold X s to threshold Xd ŝ 1. Hence, A19 must hold in equilibrium. A0 states that the action of the lowest tye in the signaling region, ŝ 1, coincides with its action in the game without signaling incentives, that is, in the modified game in which signal ŝ 1 is truthfully revealed to the rival bidder when tye ŝ 1 initiates the auction. This is the threshold that maximizes ŝ1 β s1 zψv h, v l I df z F ŝ 1 + sψv 1 β h, v l s which is analogous to A15 but sets X 1 s r µi ψv h,v l 1 ŝ 1 df z z F ŝ 1, A1 = ŝ 1. We refer the reader to Mailath 1987 for the formal argument, 1 but the intuition is as follows. Suose a bidder is the lowest tye in region [ŝ 1, ŝ ], and suose that its initiation threshold X d ŝ 1 violates A0. If it deviates to threshold that maximizes A1, its exected ayoff A15 increases for two reasons: first, the ayoff, assuming same entry of the rival z >, increases; second, the entry of the rival cannot increase, because the current r µi ψv h,v l 1 ŝ 1 1 See Grenadier and Malenko 011 for an adatation of this argument to signaling games in the real otions context. 11
equilibrium belief of the rival ŝ 1 is already the lowest. Therefore, ŝ 1 and X d ŝ 1 satisfy A19-A0. Ste 5: equations for ŝ and X fd ŝ. Cutoff tye ŝ and X fd ŝ must satisfy: = ŝ s { ŝ 1 z1 z > } r µi ψv h,v l 1 ŝ X ψv d ŝ h, v l X d ŝ I df z β Xd ŝ ŝ ψ v h, v l X fd ŝ X fd ŝ I, A X fd ŝ = I ψ v h, v l 1 ŝ ŝ. A3 A is the indifference condition saying that tye ŝ must be indifferent between initiating the auction at threshold X ŝ and facing entry of the rival if its signal is high enough and initiating the auction at threshold X fd ŝ < X d ŝ and not facing entry of the rival. The roof of A is similar to the roof of A19. A3 must hold for the following reason. First, Xfd ŝ cannot exceed A3, because otherwise entry of tyes of the rival that are close enough to ŝ is not deterred, which contradicts the assertion that tyes just above ŝ deter entry of the rival with robability one. Second, if Xfd ŝ were below A3, tye ŝ would be better off deviating from threshold X fd ŝ to threshold A3, because the entry of the rival is deterred in both cases and the exected ayoff of the bidder is strictly increasing in the initiation threshold in this range. Therefore, ŝ and X fd ŝ satisfy A-A3. References Grenadier, S. R., and A. Malenko. 011. Real otions signaling games with alications to cororate finance. Review of Financial Studies 4:3993 4036. Mailath, G. J. 1987. Incentive comatibility in signaling games with a continuum of tyes. Econometrica 55:1349 65. Tokis, D. M. 1978. Minimizing a submodular function on a lattice. Oerations Research 6:305 1. 1