COMMUNICATION BETWEEN SHAREHOLDERS 1

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1 COMMUNICATION BTWN SHARHOLDRS 1 A B. O A : A D Lemma B.1. U to µ Z r 2 σ2 Z + σ2 X 2r ω 2 an additive constant that does not deend on a or θ, the agents ayoffs can be written as: 2r rθa ω2 + θ µ Y rcov Y, Z r 2 θ2 σ 2 Y. Proof. We have that V ar θ ax + Y + Z = θ 2 a 2 V ar X + θ 2 V ar Y + V ar Z + 2Cov θy, Z, since by assumtion Cov X, Y = Cov X, Z = 0. Thus θ ax + Y + Z r V ar θ ax + Y + Z 2 = θaµ X + θµ Y + µ Z r 2 θ2 a 2 r 2 θ2 σ 2 Y r 2 σ2 Z rθcov Y, Z = θa µ X + r 2 θ2 a 2 + θ µ Y rcov Y, Z r 2 θ2 σ 2 Y + µ Z r 2 σ2 Z = 2r r 2 θ 2 a 2 2rθa µ + µ2 σ 4 X + θ µ Y rcov Y, Z r 2 θ2 σ 2 Y + µ Z r 2 σ2 Z + 2rσ 4 X µ 2 = 2r rθa ω2 + θ µ Y rcov Y, Z r 2 θ2 σ 2 Y + µ Z r 2 σ2 Z + σ2 X 2r ω2. Lemma B.2 Controlling Investor s New-Venture Related Demand. Assume the action a C Ω is otimally chosen by controlling investor C under the advice of a exert investor with, θ. Let D C r C, a denote the controlling investor C s new-venture related demand function. Then D C r C, a C = σ2 X ρ2ρ1 θ for θ 4ρ1 2 C 3 4 r θ, θ. For levels of θc smaller than 3 4 θ demand equals zero, and it is negative for levels of larger than θ. Proof. From Proosition A.2, the investor s equilibrium highest ayoff V C ρ equals aroximately: V C ρ Differentiating the above, we have that: V C ρ 24 if 0 ρ 3 4 4ρ1 if 3 4 ρ 1 ρ13ρ1 6 1ρ 6 4ρ1 if 1 ρ 0 if ρ 3 4 σ 2 ρ2ρ1 X 4ρ1 2 if 3 4 ρ ρ1 2 if 1 ρ

2 2 N. ANTIĆ AND N. PRSICO Since D C r C, a C = V Cρ, demand of the controlling investor is given by: 0 if θ C 3 4 θ D C r C, a σ C 2 X ρ2ρ1 θ if 3 4ρ1 2 4 θ θ 2 θ 1 4ρ1 2 if θ Corollary B.3 Features of Controlling Investor s New-Venture Related Demand. The controlling investor s new-venture related demand is zero on 0, 3 4 θ, it is increasing on 3 4 θ, θ, and it is negative on θ,. The controlling investor urchases a ositive amount of shares [ ] 5σ only if rices are in 0, 2 X 192 θ. Proof. D C r C, a C is constant on 0, 3 4 θ. DC r C, a C is increasing on 3 4 θ, θ since: D C r C, a ρ C 2 4ρ ρ 1 2ρ 1 2 8ρ 4ρ 1 4ρ 1 3 [ 1 = 4ρ 1 2 2ρ + 2ρ 1 8ρ 2ρ 1 ] 4ρ 1 1 [ ] = 4ρ 1 3 4ρ 1 2 8ρ 2ρ 1 = 1 4ρ 1 3 > 0. Lemma B.4 xert Investor s New-Venture Related Demand. Assume the action a C ω n is otimally chosen by controlling investor C under the advice of an exert investor with risk aversion arameter and holdings θ. Let D θ, a C denote the exert investor s demand function. For θ 0, / we have D θ, a C = σ2 X 8ρ 2 8ρ+1 2 > 0. Demand is negative for 16ρ 2 8ρ+1 θ /, 4 /3. Demand is also negative for θ 4 /3, 1. Proof. From Proosition A.2, the exert investor s equilibrium highest ayoff V ρ equals aroximately: where ρ = / θ. V ρ ρ ω if 0 ρ ρ 4ρ1 if 3 4 ρ 1 ρ1 ρ4ρ1 4ρ 3 if 1 ρ

3 COMMUNICATION BTWN SHARHOLDRS 3 Case ρ < 3/4. This case corresonds to θ > 4 /3. D θ, a C = V ρ θ = 2 = 2 = 2 θ θ 2 θ ω 2 1 2ρ 1 < ω Case 3 4 ρ 1. Since σ 2 X 3 1ρ 4ρ1 = D θ, a C = V ρ 4ρ1 2, we have that: = V ρ = = = 4ρ 1 2 ρ θ 4ρ 1 2 ρ2 4ρ 1 2. θ θ C θ 2 ρ 1 θ Case 1 ρ. This case corresonds to θ /. On the range 1 ρ we have: V ρ = ρ 1 4ρ 3 6 ρ 4ρ 1 3 8ρ 2 8ρ + 1 = 6 ρ 2 4ρ 1 2 Now: So D θ, a C = V ρ D θ, a C = = V ρ 3 6ρ θ θ ρ 1 θ 8ρ 2 8ρ + 1 4ρ 1 2

4 4 N. ANTIĆ AND N. PRSICO = σ2 X 8ρ 2 8ρ ρ 2. 8ρ + 1 Which means that: D θ, a C 2 1 2ρ 1 2 if ρ 3 4 ρ2 4ρ1 2 if 3 4 ρ 1 2 8ρ 2 8ρ+1 4ρ1 2 > 0 if 1 ρ. Corollary B.5 Features of xert Investor s New-Venture Related Demand. The exert investor s new-venture related demand is ositive and decreasing in θ over the interval 0, /. D 0, a C = σ2 X 4 and D θc, a C = σ2 X 18. Since exert investor s demand is negative or zero elsewhere, if the exert investor urchases any ositive amount of shares in equilibrium he urchases within the interval 0, /. The exert investor urchases / only if rices are in [ ] 0, σ2 X 18 θc. Proof. The function 8ρ2 8ρ+1 is ositive if ρ > 1; it has value 1/9 at ρ = 1; and it aroaches 16ρ 2 8ρ+1 1/2 as ρ. The function is increasing in ρ since d 8ρ 2 8ρ+1 dρ = 16ρ > 0 since ρ > 1/4. 4ρ1 2 4ρ1 3 Therefore, since ρ is a decreasing function of θ, we have that D θ, a C is decreasing in θ on 0, /.

5 COMMUNICATION BTWN SHARHOLDRS 5 A C. O A : T W P I There is an entrenched, controlling investor who owns share of the comany. There is a continuum of non-controlling investors, who are essentially noise traders that have erfectly elastic demand at some rice. An exert investor starts out with a share endowment denoted by t, he observes ω and then he trades. The exert investor can sell all but ε of his endowment, or buy more shares u to a maximum holding of 1, at the rice set by the non-controlling investors,. After observing the exert investor s trade, the two engage in chea talk communication and the controlling investor chooses a. C.1. M In this model trading takes lace under asymmetric information; therefore, trading serves as a signal of the exert s information. The assumtion of constant rice i.e., noise traders with erfectly elastic demand simlifies the analysis, but the analysis would not break down if we assumed a rice function that is increasing in the exert s net trade. Finally, this model is most comarable to the entrenched shareholder setting discussed in Section 5.2, in that the controlling shareholder s holdings are fixed. C.2. The equilibrium is characterized by two regions. A low-ω region where the exert investor acquires less than 1 ; in this region the exert investor s trade fully reveals ω to the controlling investor. And, for some arameter values, a high-ω region where the exert investor acquires exactly 1 shares and then relies on chea talk to convey information. The analysis is insired by the signaling model in Section 4 of Kartik 2007, where in our case, the exert investor signals through trading shares. However, in the resent aer the cost/benefit of signaling is endogenous because the share value is a function of the receiver s action. So his results cannot be alied directly.

6 6 N. ANTIĆ AND N. PRSICO C.3. F In the fully-searating region the controlling investor learns the exact state ω based on the exert investor s net trade after he learns the information. Denote the exert s ex ost osition after trading by t ω. Suose the exert investor retrades into t ω. If t ω is fully searating, uon observing it the controlling investor correctly infers that the state is ω and she takes action a = ω/ recall that r C = 1. The exert investor s utility is then: t ω ω 2 ω t ω t, C.1 2 where denotes the rice of shares. This is the utility after the exert and now informed investor has re-traded his osition using a searating trading strategy t. This utility has been derived based on Lemma B.1 where Z stands for the non-random variable t ω t, the amount of money aid to achieve the new osition t ω. We want to show that the trading function described by the following differential equation: t t ω 1 ω ω = t ω for ω > 0. C.2 t ω 1 ω 2 + reresents the searating region of an equilibrium. Lemma C.1. Features of the solution to the diff erential equation Let t solve the differential equation C.2 with initial condition t 0 = ε, where 0 < ε < /. Then: t 0 = 0; t ω > 0 for all ω > 0; and the function t ω achieves / asymtotically, but not for finite ω. Proof. quation C.2 indicates that t 0 = 0; and furthermore that t ω > 0 as long as, simultaneously: t ω 1 < 0 t ω 1 ω 2 + > 0. C.3 Rearranging, we get the following suffi cient condition for t ω > 0: ω < t ω 1 < 0. 2

7 COMMUNICATION BTWN SHARHOLDRS 7 This condition certainly holds in the neiborhood of ω, t ω = 0, ε. As ω increases, the left-most inequality is imlied by cannot fail before the right-most. To see this, suose by contradiction that the left-most inequality fails first. Then there must be an ω > 0 such that the denominator of equation C.2 equals zero. But then since θc t ω1 < 0, equation C.2 imlies lim t ω =, ω ω contradicting the assumtion that the left-most inequality fails. Thus, t ω > 0 for all ω as long as t ω is below /. For ω large enough t ω must achieve / at least asymtotically, for otherwise: lim t ω 1 = ξ < 0, ω and then taking limits in equation C.2 we would have: t ω = ξ t ω < 0, ξω contradicting the roerty that t ω > 0 for all t ω < /. Let us show that t ω achieves / asymtotically only. Suose by contradiction that there is a finite ω such that θc t ω 1 = 0. Then from C.2: lim ω ω lim ω ω ω log ω log log t ω t ω 1 t ω 1 t ω 1 t ω 1 = ω = A t ω 1 = ex t ω = ω = A for ω ω = Aω + B for ω ω [ ] Aω + B def = A for ω ω Hence the function θc t ω 1 must aroach zero as a negative exonential recall that the constant A is negative. But a negative exonential never achieves zero, hence we have contradicted the assumtion that ω is finite. Corollary C.2. Let t solve the differential equation C.2 with initial condition t 0 = ε. Then t 1 /.

8 8 N. ANTIĆ AND N. PRSICO As t ω the function t ω goes to zero hence t ω becomes very flat. This means that slight variations in shares convey a lot of information about the exert s signal. < 1 this effi cient signaling takes lace at ownershi levels below 1. In this case the signaling equilibrium can be erfectly searating and revealing for all ω. For the equilibrium to involve some ooling it must be that > 1. Corollary C.3. Since the t that solves the differential equation C.2 with initial condition t 0 = ε is ositive and nondecreasing in ω, the roduct t ω ω is strictly increasing for all ω. If Lemma C.4. Best resonse roerty Let t solve the differential equation C.2 with initial condition t 0 = ε. Suose the controlling investor exects tye ω to lay t ω. Then any exert tye ω refers t ω to t ω for any ω > 0. Proof. Suose the exert investor who knows the state ω acquires t ω ; then his utility is: t ω ω 2 ω t ω t. C.4 2 Denote this function by u ω, ω. Differentiating this utility function with resect to ω yields the following first-order condition: [ σ 2 0 = X t ω ω ] ω t ω ω t ω. ω C.5 For ω = ω to be a maximum, this first-order condition must hold at ω = ω. We now show that the condition holds if t solves the differential equation C.2. To see this, set ω = ω and rewrite the first-order condition as follows: [ σ 2 0 = X t ω ω ] ω ω t ω ω t ω = t ω ω ω [t ω ω + t ω] t ω [ = t ω ω ω ω + θ ] C t ω t ω ω ω t ω which holds if t solves the differential equation C.2. C.6

9 COMMUNICATION BTWN SHARHOLDRS 9 The second-order conditions for a maximum require that the exert s utility function C.4 be single-eaked. Denoting this function by u ω, ω, the first order conditions C.5 can be written as: u 1 ω, ω ω=ω = 0. Using C.5 we can write: u 1 ω, ω = u 1 ω, ω + ω ω = ω ω t ω ω ω, t ω ω ω where the second equality reflects the first order conditions. Since by Lemma C.3 t ω ω > ω 0, this exression shows that the first derivative is is ositive for ω < ω and negative otherwise. Hence u ω, ω is indeed single-eaked as a function of ω and it attains a maximum at ω = ω. Lemma C.5. Initial condition of equilibrium trading strategy There are otentially many signalling equilibria, each associated with a different value of t 0. The controling investor is indifferent among them all. The one that is most referred by the exert investor is the one with the smallest t 0 = ε. Proof. The family of strategies identified by differential equation C.2 indexed by its initial conditions t 0, give rise to a family of signaling equilibria. Higher initial conditions result in a ointwise-higher strategies t. Irresective of the initial condition t 0, all strategies identified by differential equation C.2 induce the same fully informed controlling agent s action. Therefore the exert investor s referred equilibrium within this family of signaling equilibria is the one where his holdings t ω are closest to the exert investor s referred holdings conditional on the controlling agent being fully informed. We now show that the exert investor s referred holdings conditional on the controlling agent being fully informed are lower than any equilibrium signaling strategy. To see this, notice that if the controlling investor knows ω the exert s roblem is: The first order conditions read: max t 2 t ω ω 2 t t. t F I 1 ω 2 = 0,

10 10 N. ANTIĆ AND N. PRSICO where t F I denotes the full-information otimal holdings for the exert investor. A slight rearrangement yields: t F I 1 ω 2 + = 0. From C.3, which must hold in equilibrium, we have: t ω 1 ω 2 + > 0, which in comarison to the revious equation verifies that for any ω, t F I, is smaller than the equilibrium trading level t ω. Therefore the equilibrium that is most referred by the exert investor is the one with the smallest t 0 = ε. Proosition C.6. Characterization of fully searating equilibrium Let t solve the differential equation C.2 with initial condition t Suose t 1 1. Then there is a fully searating equilibrium where all exert tyes in [0, 1] trade according to t. The amount of shares acquired after retrading cannot exceed /. C.4. C Denote by t ω;,,, solve the differential equation C.2 with initial condition t 0 = ε. We want to see whether, as the arameters,,, change, whether the effect on t is monotonic. To do this, we will comare two solutions t ω; and t ω; with >. If whenever t ω; = t ω; we have t ω; > t ω; then the two solutions never cross and we have our monotonicity result. Lemma C.7. > imlies t ω; > t ω;. Proof. Suose t ω; = t ω; = t. Then from differential equation C.2 we have: r t ω; = t > t = t ω;. 2 + r 2 + σ 2 X It follows that t ω; > t ω;. Lemma C.8. r > imlies t ω; > t ω; r.

11 COMMUNICATION BTWN SHARHOLDRS 11 Proof. Since >, the following inequalities hold for any t > 0 : [ r t θ ] [ C r < t θ ] C θ } C θ {{} C θ }{{}} C θ {{} C }{{} ω 2 + t 1 r < r 2 + r t > r t 1 If t ω; = t ω; = t then the above inequality reads: t ω; > t ω; r. ω 2 + So > imlies that if t ω; = t ω; = t then t ω; > t ω;. It follows that t ω; > t ω; r. t + Lemma C.9. θ C > imlies t ω; > t ω; θ C if and only if t ω < 2. Proof. Suose t ω; = t ω; = t. Then in light of differential equation C.2 the following inequalities are equivalent: t ω; > t ω; θ C r θ t > C t 2 + r θ 2 + θ C C r t 1 t 1 θ < C 2 + r θ 2 + θ C C ] [ ω 2 + θ C < θ t 1 t 1 C t 1 θ C < θ } C θ t 1 C {{}}{{} [ t 1 θ t 1 C 1 t 1 θ } C {{} < 1 θ C θ t 1 C }{{} ω 2 + θ ] C

12 12 N. ANTIĆ AND N. PRSICO The inequality holds if: 0 < θ θ 2 t 1 θ = 2 θ 3 t + 1 θ 2, which is equivalent to t < θ 2. C.5. P The ooling region has the form [ω, 1]. On that region all tyes urchase the maximum amount of available shares in our case, 1 and so no signal is conveyed by share osition. Instead, communication takes lace via chea talk. Tye ω has to be indifferent between buying t ω and being erfectly revealed, or buying 1 and ooling with the lowest interval in the chea talk equilibrium artition. Given an equilibrium characterized by a threshold ω, a full-revelation trading function t, and a artition {ω, ω 1, ω 2,...} of the ooling region, tye ω s ayoff from buying t x for any x < ω is, by C.4: U x; ω = t x x 2 ω t x t + σ2 X ω 2 for any x < ω, 2 2 whereas tye ω s ayoff from ooling at 1 and inducing any action y > ω is: U y; ω = 1 y 2 ω 1 t + σ2 X ω 2 for any y > ω. 2 2 Lemma C.10. Higher tyes are more inclined to ool Fix x < ω < y. Then U y; ω U x; ω is increasing in ω. Proof. U y; ω U x; ω = 2 = 2 1 y 1 y ω The derivative with resect to ω is: 2 ω 1 t + σ2 X σ2 X 2 1 y ω t x x ω t x x 2 ω + t x t 2 + t x 1 + t x x ω

13 = σ2 X COMMUNICATION BTWN SHARHOLDRS 13 1 y t x x, which is ositive because 1 > t x and y > x. From this, we have that the equilibrium is characterized by a cutoff tye, which is indifferent between searating and ooling, i.e., a tye ω which solves: 0 = t ω ω 2 ω t x t + σ2 X 1 y 2 1 ω ω + 1 t 2 2 [ = t ω ω 2 ω 1 y ] 2 1 ω ω + 1 t ω. C.7 2 We turn to a numerical examle to illustrate our equilibrium. In this examle, we assume that the controlling investor is entrenched and owns share > 0.5 of the comany. In the grahs that follow, we assume that = 5, = 0.6, = 1, = 5 and t 0 = ε = Mathematica code can be rovided. We tried different initial conditions and the results are identical for ε between 10 3 and 10 9 beyond this we exhaust machine recision for this roblem.

14 14 N. ANTIĆ AND N. PRSICO Utility from ooling - searating ω Panel A: Difference in utility between ooling and searating for candidate cutoff tyes quity holding in equilibrium ω Panel B: xert investor s equity choice of equity as a function of his rivate information F 4. quilibrium construction in the model with retrading Panel A of Figure 4 shows the utilities from ooling vs searating for ossible cutoff-tyes. That is, given a otential cutoff-tye on the x-axis, we comute that tye s exected utility from the chea talk equilibrium if it and tyes above engaged in chea talk and subtract that tye s exected utility from searating trading according to the t function, which solves

15 COMMUNICATION BTWN SHARHOLDRS 15 differential equation C.2, and the controlling investor taking her referred action. We see that there exists a unique cutoff tye, ω 0.42 and that tyes below ω refer to searate, while tyes above refer to ool. Panel B of Figure 4 shows the level of equity held by the exert investor after trading on rivate information in the equilibrium constructed. xert investors who learn the state is above ω urchase 1 = 0.4 shares and engage in chea talk. xert investors who learn the state is below ω, erfectly signal ω by uchasing a number of shares which is strictly increasing in the state ω. We further exlore our model by considering some comarative statics. In articular, we ask what haens when the holdings of the controlling investor,, change. As shown in Figure 5, the cutoff tye which is indifferent between searating and ooling, ω, is decreasing in. This means that the set of tyes which is engaging in chea talk, [ω, 1], is increasing in, which is quite intuitive: if is larger, tyes need to urchase fewer shares in order to ool to a holding of 1 and thus the gain of engaging in chea talk as oosed to revealing your tye needs to be smaller to make this tradeoff aealing. Cutoff tye, ω F 5. Comarative static with resect to the shareholdings of the controlling investor: the larger the holding the larger the set of tyes which ool and engage in chea talk.