Signaling firm value to active investors

Abstract

Active investors provide risk-sharing and value-adding effort in form of advising, networking, monitoring, etc. This paper demonstrates a conflict between two key objectives for high-quality entrepreneurs: to elicit such investor effort and to signal the firm’s type by retaining shares. This conflict may give rise to stable (and economically meaningful) pooling equilibria for startup firms. More established firms, with access to multiple signals, can always realize both of these objectives but may still decide to forego investor effort if eliciting it would require them to deviate substantially from the cost-minimizing signal mix. In comparison with otherwise identical pure-exchange settings (with passive investors), we find that the potential for investors to be active always increases the signaling cost in case of noncontractible investor effort, whereas the effect is ambiguous if investor effort is contractible. At the same time, we identify conditions under which signaling is welfare-enhancing as it helps guide investors’ effort towards more promising ventures.

Appendix

Proof of Lemma 1

Using the pricing function in (2), the founder’s objective U = (1 − α)P(α, k) + α(θ + ϕk) − R(α) can be restated as

$$ U=(1- \alpha )\,\hat{\theta }+\alpha\theta +[(1-\alpha)\,\hat{\phi}+\alpha\phi-1]k-R(\alpha). $$

In a separating equilibrium $\hat{\theta }=\theta$ and $\hat{\phi}=\phi$ and hence for a type-i founder:

$$ U_i =\theta _i +( \phi_i - 1) k - R(\alpha), \hbox{ for } i=\{ L, H \}. $$

Therefore L will choose α = 0 and k = 0 because ϕ _L  < 1.

For the high-type founder we proceed by solving the Langrangean associated with Program ${\mathcal{P}}_1.$ With λ and μ as, respectively, the multipliers for the nonmimicry constraint [rewritten as in (7)] and the non-negativity constraint, and

$$ Y(\alpha)\equiv (1 - \alpha )\phi_H+ \alpha\phi_L - 1, $$

as defined in the main text, we arrive at the Langrangean:

$$ {\mathcal{L}}= \theta _H+(\phi_H-1)k-R(\alpha) -\lambda \left\{ (1 - \alpha )\Updelta\theta +Y(\alpha ) k - R(\alpha) \right \} +\mu\alpha. $$

The solution is characterized by the following conditions:

$$ \frac{\partial {\mathcal{L}}} {\partial \alpha}=-(1-\lambda )\rho \sigma^2 \alpha+\lambda [\Updelta \theta +(\Updelta\phi) k] + \mu =0, $$

(21)

$$ \lambda \left\{ (1 - \alpha )\Updelta\theta +Y(\alpha) k - R(\alpha)\right\}=0, $$

(22)

$$ \mu \alpha=0 , $$

(23)

where (22)–(23) hold with complementary slackness.

We first show that α > 0. Suppose not, and instead α = 0. Then,

$$ (1 - \alpha )\Updelta\theta +Y(\alpha) k - R(\alpha)=\Updelta\theta +(\phi_H-1)k>0, $$

which violates the nonmimicry constraint (7). Thus, α > 0. By complementary slackness of (23), μ = 0. Second, we show that λ > 0. Suppose instead that λ = 0. By (21), then, μ = ρσ²α. As a result, μ > 0, since we have established in the first step that α > 0. However, this contradicts the complementary slackness of (23). Thus, λ > 0. Third, we show that λ < 1. By μ = 0, this follows immediately from (21). Finally, consider the choice of investor effort by the high-type firm. For convenience, and with slight abuse of notation, we will prove the result as if k were continuous; then the partial derivative

$$ \frac{\partial {\mathcal{L}}} {\partial k}=(1-\lambda )(\phi_H-1)+\lambda \alpha (\phi_H-\phi_L) $$

(24)

is strictly positive everywhere. Thus, H chooses k = K in the separating equilibrium.

Proof of Proposition 1

By Lemma 1, we know that the nonmimicry constraint (7) is binding. Hence we apply the Implicit Function Theorem to get

$$ \frac{d\alpha^{**}}{d K}=\frac{Y(\alpha^{**})}{\Updelta\theta +(\Updelta\phi)K+R^{\prime }(\alpha^{**})} $$

and, therefore,

$$ \frac{d^2\alpha^{**}} {d K^2}=\frac{-Y(\alpha^{**})\Updelta\phi-\frac{d\alpha^{**}} {d K}[\Updelta\phi(\Updelta\theta +(\Updelta\phi) K)+(\phi_H-1)\rho \sigma^2]} {[\Updelta\theta +(\Updelta\phi)K+R^{\prime }(\alpha^{**})]^2}. $$

Thus, $\alpha ^{{**}} \left( K \right)$ is a decreasing convex function if $Y\left( {\alpha ^{{**}} \left( K \right)} \right) < 0$ and increasing concave otherwise. This characterization of ${{d\alpha ^{{**}} } \mathord{\left/ {\vphantom {{d\alpha ^{{**}} } {dK}}} \right. \kern-\nulldelimiterspace} {dK}}$ is incomplete however as it depends on the endogenous variable $\alpha ^{{**}} \left( \cdot \right).$ To close the argument, we use the fact $\alpha ^{{**}} \left( K \right) \equiv \alpha _{N} \left( K \right)$ and the expression in footnote 14 to arrive at the closed-form term:

$$ \frac{d\alpha^{**}} {d K}=\frac{1}{\rho \sigma^2} \left(\frac{(\Updelta\phi) Z(K)+\rho \sigma^2 (\phi_H-1)} {\sqrt{(Z(K))^2+2 \rho \sigma^2 [\Updelta\theta +(\phi_H-1)K]}}-\Updelta\phi \right), $$

for $Z(K)\equiv \Updelta\theta +(\Updelta\phi) K$ (see footnote 14). To determine the sign of ${{d\alpha ^{{**}} } \mathord{\left/ {\vphantom {{d\alpha ^{{**}} } {dK}}} \right. \kern-\nulldelimiterspace} {dK}},$ we square the individual terms inside the bracket:

$$ \begin{aligned} & \left( \frac{(\Updelta\phi) Z(K)+\rho \sigma^2 (\phi_H-1)} {\sqrt{(Z(K))^2+2 \rho \sigma^2 [\Updelta\theta +(\phi_H-1)K]}} \right)^2-(\Updelta\phi)^2 \\ =& \frac{\big((\Updelta\phi) Z(K)+\rho \sigma^2 (\phi_H-1)\big)^2-(\Updelta\phi)^2\big((Z(K))^2+2 \rho \sigma^2 [\Updelta\theta +(\phi_H-1)K]\big)} {(Z(K))^2+2 \rho \sigma^2 [\Updelta\theta +(\phi_H-1)K]} \\ =& \frac{\rho \sigma^2 [\rho \sigma^2(\phi_H-1)^2-2\Updelta\theta \Updelta\phi(1-\phi_L)]} {(Z(K))^2+2 \rho \sigma^2 [\Updelta\theta +(\phi_H-1)K]}. \end{aligned} $$

Clearly, when

$$ \sigma^2> \frac{2(\Updelta\theta )(\Updelta\phi)(1-\phi_L)} {\rho (\phi_H-1)^2}\equiv \sigma^2_0, $$

the above expression is positive, so that ${{d\alpha ^{{**}} } \mathord{\left/ {\vphantom {{d\alpha ^{{**}} } {dK > 0,}}} \right. \kern-\nulldelimiterspace} {dK > 0,}}$ and vice versa. Simple algebra then shows that α _LP (σ ²₀ ) = α₀.

Lastly, rewriting the expression in footnote 14 yields

$$ \alpha_N(K)=\frac{2[\Updelta\theta +(\phi_H-1)K]}{\sqrt{(Z(K))^2+2 \rho \sigma^2 [\Updelta\theta +(\phi_H-1)K]}+ Z(K)}. $$

Taking limits, we find that $\lim_{K\to \infty} \alpha^{**}(K) = \lim_{K\to \infty} \alpha_N(K)=\frac{\phi_H-1} {\Updelta\phi}=\alpha_0.$ This completes the proof of Proposition 1.

Proof of Lemma 3

We prove existence of a pooling equilibrium with α = 0. Given $\Upphi>1$, this would result in the investor always exerting high effort (k = K). Following standard procedures for applying the Intuitive Criterion, consider the most favorable beliefs on the part of investors: if a founder deviates from the pooling equilibrium by retaining α > 0, the investor will believe he is H and price the shares accordingly. As in the main text, denote the pooling payoff to each type of founder by $U^{PE}(\alpha=0,k=K)=\Uptheta+(\Upphi-1)K$ and type i’s payoff from deviating by choosing α > 0 by

$$ \begin{aligned} U_{H}^{{dev}} \left( \alpha \right) = & \left\{ {\begin{array}{*{20}c} {\theta _{H} + \left( {\phi _{H} - 1} \right)K - R\left( \alpha \right),} & {{\text{if}}\,\alpha \le \tilde{\alpha }} \\ {\theta _{H} - R(\alpha ),} & {{\text{if}}\,\alpha > \tilde{\alpha }} \\ \end{array} } \right. \\ U_{L}^{{dev}} \left( \alpha \right) = & \left\{ {\begin{array}{*{20}c} {\left( {1 - \alpha } \right)\theta _{H} + \alpha \theta _{L} + Y\left( \alpha \right)K - R\left( \alpha \right)} & {{\text{if}}\,\alpha \le \tilde{\alpha }} \\ {\left( {1 - \alpha } \right)\theta _{H} + \alpha \theta _{L} - R\left( \alpha \right),} & {{\text{if}}\,\alpha > \tilde{\alpha }} \\ \end{array} } \right.. \\ \end{aligned} $$

As illustrated in Fig. 3, the pooling equilibrium satisfies the Intuitive Criterion if and only if:

$$ \begin{aligned} \mathop {\lim }\limits_{{\delta \to 0}} U^{{dev}} _{{\text{L}}} \left( {\tilde{\alpha } - \delta } \right) = & \left( {1 - \tilde{\alpha }} \right)\theta _{H} + \tilde{\alpha }\theta _{L} + Y\left( {\tilde{\alpha }} \right)K - R\left( {\tilde{\alpha }} \right) \\ > U^{{PE}} \left( {\alpha = 0,K} \right) = & \Uptheta + \left( {\Upphi - 1} \right)K. \\ \end{aligned} $$

(25)

$$ > \mathop {\lim }\limits_{{\delta \to 0}} U^{{dev}} _{H} \left( {\tilde{\alpha } + \delta } \right) = \theta _{H} - R\left( {\tilde{\alpha }} \right). $$

(26)

Inequality (25) can be rewritten as follows:

$$ (1-p-\tilde \alpha)[\Updelta\theta +(\Updelta\phi)K] > R(\tilde \alpha). $$

Given $\tilde{\alpha}=(\phi_H-1)/\phi_H$, this inequality holds if and only if

$$ \phi_H<\frac{1}{p} \ \ \ \hbox{\ and\ } \ \ K>\frac{R(\tilde \alpha)}{(1-p-\tilde \alpha)\Updelta\phi}-\frac{\Updelta\theta}{\Updelta\phi} \equiv K_1. $$

Meanwhile, inequality (26) holds if and only if

$$ K>\frac{(1-p)\Updelta\theta -R(\tilde \alpha)}{\Upphi-1}\equiv K_2. $$

Thus, there exists a (stable) pooling equilibrium with k = K and α = 0, if and only if ϕ _H  < 1/p and $K> \underline {K} \equiv \max\{K_1,K_2\}.$ For future reference, denote by σ ² _K the level of operating risk at which K ₁(·) = K ₂(·). Using the above expressions for K ₁(·) and K ₂(·) gives

$$ \sigma_K^2=\frac{2 \Updelta\theta (\phi_H)^2(1-p \phi_H)}{\rho (1-\phi_H)^2 \phi_L}>0. $$

Proof of Proposition 2

We proceed in two steps. First, we show that $\tilde K < \underline K$, which implies that, given $\sigma^2>\tilde{\sigma}^2$ and ϕ _H  < 1/p, a separating equilibrium with efficient effort choice exists for small K, while a pooling equilibrium exists for large K—but those two equilibria are mutually exclusive. This implies that for high K, there will be two types of equilibria, the pooling one and a separating one with k = 0. In that case we show in a second step that H’s payoff will always be greater under pooling.

Step 1. Suppose investor effort is noncontractible, $\sigma^2>\tilde{\sigma}^2$, and ϕ _H  < 1/p. By Lemma 2, $\sigma^2>\tilde{\sigma}^2$ implies that there exists a separating equilibrium with investors exerting effort k = K in high-type firms if and only if $K{\leq}\tilde{K}$, where $\tilde{K}$ is defined by $\alpha_N(\tilde{K} |\sigma^2) \equiv \tilde{\alpha}$. It is a matter of straightforward algebra to show that

$$ \tilde K=\frac{\rho \sigma^2(\phi_H-1)^2-2(\Updelta\theta ) \phi_H} {2(\phi_H-1)\phi_H\phi_L}. $$

(27)

Meanwhile, by Lemma 3, ϕ _H  < 1/p implies that there exists a pooling equilibrium with k = K and α = 0 for all firms if and only if $K> \underline {K}$, where $\underline {K}=\max\{K_1,K_2\}\geq K_1.$ Straightforward but tedious algebra yields

$$ K_1-\tilde K=\frac{2(\Updelta\theta ) \phi_H(1-p\phi_H)(1-\phi_L)+\rho\sigma^2 (\phi_H-1)^2(\Upphi-1)} {2(\phi_H-1)(1-p\phi_H)\phi_L(\Updelta\phi) }. $$

Both denominator and numerator are positive, and therefore $\tilde K<K_1\leq \underline {K}.$

As a result, for $K{\leq}\tilde{K}$ (Case 1 of Proposition 2), there only exists the separating equilibrium with k = K for H and k = 0 for L. For $\tilde{K}<K<\underline{K}$ (Case 2), there only exists the separating equilibrium with zero effort (k = 0) for any firm. Also since $\alpha_{LP}<\tilde \alpha$ for this region, retaining α _LP share would induce the investor to exert effort and thereby upset the postulated equilibrium; the equilibrium α thus has to be $\tilde{\alpha}.$ For $K \geq \underline K$ (Case 3), there exist both the separating equilibrium without effort and the pooling equilibrium with α = 0 and k = K.

Step 2. Next, we compare the high-type founder’s payoffs under the two existing equilibria in Case 3 where $K \geq \underline K$. H’s payoff under pooling is $U_H^{PE}(\alpha=0,k=K)=\Uptheta+(\Upphi-1)K,$ whereas under the separating equilibrium without effort it equals $U_{H}^{SE}(\alpha=\tilde{\alpha},k=0) ={\theta} _H-R(\tilde{\alpha}).$ As shown in the proof of Lemma 3, a necessary condition for the existence of a pooling equilibrium is inequality (26), which is precisely $U_H^{PE}(\alpha=0,k=K)>U_H^{SE}(\alpha =\tilde \alpha,k=0)$. Hence, the existence of the pooling equilibrium with α = 0 ensures that the payoff to the high-type founder under the pooling equilibrium exceeds that under the separating equilibrium.

Proof of Corollary 2

We only need to show that there exist values of K such that inequality (15) and the investor’s incentive constraint (11) are simultaneously satisfied. Note that (15) is satisfied whenever

$$ K \geq \frac{p R(\alpha_N(K))}{(1-p)(1-\phi_L)}. $$

A sufficient condition for this to hold is that

$$ K \geq \frac{p \rho \sigma^2}{2(1-p)(1-\phi_L)}\equiv K_w. $$

Comparing K _w with $\tilde{K}$, as stated in (27), yields that $K_w<\tilde{K}$ if and only if

$$ \Updelta\theta \leq \Updelta\theta _w\equiv \frac{\rho \sigma^2(\phi_H-1)}{2 \phi_H (1-p)(1-\phi_L)}\times \eta , $$

for ${\eta}{\equiv}(1-\phi_L) ({\phi_H}-1)-p ({\phi_H}+\phi_L-1).$ For a feasible solution to obtain, $\Updelta\theta _w$ must be positive, which holds if and only if η ≥ 0, or equivalently:

$$ p\leq p_w \equiv \frac{(\phi_H-1)(1-\phi_L)}{\phi_H + \phi_L -1}. $$

It is straightforward to see that p _w  ∈ [0, 1], hence it is well defined. Lastly, note that even for small p our maintained assumption that $\Upphi=p\phi_H+(1-p)\phi_L \geq 1$ will be satisfied provided ϕ _H is sufficiently high.

Proof of Lemma 4

The founder’s objective can be restated as

$$ U=(1 - \alpha)\hat{\theta }+\alpha\theta +[(1-\alpha)\hat{\phi}+\alpha\phi-1]k-R(\alpha)-V(m-\theta ). $$

In a separating equilibrium $\hat{\theta }=\theta$ and $\hat{\phi}=\phi.$ Hence, for a type-i founder:

$$ U_i =\theta _i +( \phi_i - 1) k - R(\alpha) - V( m -\theta _i ), \ \hbox{\ for\ } i=\{L,H\}. $$

Therefore, L will choose α = 0, m = θ _L , and k = 0 because ϕ _L  < 1.

For H, we again solve the Langrangean for the separating signaling equilibrium. Denoting the multipliers for the nonmimicry and non-negativity constraints by λ, and μ, we have:

$$ \begin{aligned} {\mathcal{L}}=& \theta _H+(\phi_H-1)k-R(\alpha) -V(m-\theta _H) \\ & -\lambda \{ (1 - \alpha )\Updelta \theta +Y(\alpha) k - R(\alpha) - V( m -\theta _L)\} +\mu\alpha. \end{aligned} $$

The solution is characterized by the following conditions:

$$ \frac{\partial {\mathcal{L}}}{\partial \alpha}=-(1-\lambda )\rho \sigma^2 \alpha+\lambda [\Updelta \theta + (\Updelta\phi) k] + \mu =0 , $$

(28)

$$ \frac{\partial {\mathcal{L}}}{\partial m}= -V^{\prime }(m-\theta _H)+\lambda V^{\prime }(m-\theta _L)=0, $$

(29)

$$ \lambda \{ (1 - \alpha )\Updelta\theta +Y(\alpha) k - R(\alpha) - V( m -\theta _L)\}=0 , $$

(30)

$$ \mu \alpha=0, $$

(31)

where (30)–(31) hold with complementary slackness.

We first show that μ = 0. Suppose not, and instead μ > 0. Then, by complementary slackness of (31), α = 0. Hence, (28) becomes $\lambda \left[ {\Updelta \theta + \left( {\Updelta \phi } \right)k} \right] + \mu = 0,$ which cannot hold for μ > 0 (since λ ≥ 0)—a contradiction. Thus, μ = 0.

Second, we show that α > 0. Suppose not, and instead α = 0. Then, together with μ = 0, (28) becomes $\lambda \left[ {\Updelta \theta + \left( {\Updelta \phi } \right)k} \right] = 0.$ Hence λ = 0, which in turn leads to m = θ _H by (29). Also, λ = 0 implies k = K, in equilibrium, because from H’s point of view the partial derivative in (24) (again treating k as a continuous variable) will be strictly positive if λ = 0. However, by the maintained assumption that $V(\Updelta\theta )<(\phi_H-1)K+\Updelta\theta$, we know that (α = 0, m = θ _H , k = K) violates the nonmimicry constraint (17). Therefore, α > 0.

The steps showing that 0 < λ < 1 are identical to those in the proof of Lemma 1 and therefore omitted. It remains to show that m > θ _H . By (29) and 0 < λ < 1, either V′(m − θ _H ) = V′(m − θ _L ) = 0 or V′(m − θ _L ) > V′(m − θ _H ) > 0. Clearly, V′(m − θ _H ) = V′(m − θ _L ) = 0 means m ≤ θ _L , which cannot be H’s optimal choice. So (29) and 0 < λ < 1 together imply that V′(m − θ _L ) > V′(m − θ _H ) > 0, which in turn implies m > θ _H . Finally, since the derivative in (24) is strictly positive everywhere, high-type founders will choose k = K in the separating equilibrium.

Proof of Lemma 5

Throughout the proof, α and m take their optimal values, $\alpha = \alpha ^{{**}} \,{\text{and}}\,m = m^{{**}}$ respectively (that is, we suppress the “**”). In the efficient separating equilibrium, the nonmimicry constraint is binding, and therefore:

$$ (1- \alpha )\Updelta\theta +[(1 - \alpha )\phi_H+ \alpha\phi_L - 1] K - R(\alpha) - V( m -\theta _L)=0 $$

(32)

By (28) and (29),

$$ - \rho \sigma ^{2} \alpha + \lambda \left( {\Updelta \theta + \left( {\Updelta \phi } \right)K + \rho \sigma ^{2} \alpha } \right) = 0 $$

(33)

for λ = V′(m − θ _H )/V′(m − θ _L ). To save on notation, we introduce the following shorthand: let F ¹ denote the left-hand side of (33) and F ² the left-hand side of (32).

Differentiating (32) and (33) with respect to σ² and v, we have (with subscripts denoting partial derivatives): for i = 1, 2, 

$$ \begin{aligned} F^i_{\alpha}\frac{d \alpha}{d \sigma^2}+F^i_{m} \frac{d m}{d \sigma^2}+F^i_{ \sigma^2}&=0\\ F^i_{\alpha}\frac{d \alpha}{d v}+F^i_{m} \frac{d m}{d v}+F^i_{v}&=0 , \end{aligned} $$

where, for

$$ A(\alpha) \equiv \Updelta\theta +(\Updelta\phi) K+\rho\sigma^2\alpha $$

we have:

$$ \begin{aligned} F^1_{\alpha}&=-(1-\lambda )\rho\sigma^2<0, \\ F^2_{\alpha}&=-(\Updelta\phi)K-\Updelta\theta -\rho\sigma^2\alpha=-A(\alpha)<0, \\ F^1_{m}&=A(\alpha)\frac{\partial \lambda }{\partial m}=A(\alpha) \frac{\Updelta\theta } {(m-\theta _L)^2}>0, \qquad \hbox{invoking\ Assumption\ 1}, \\ F^2_{m}&=-v(m-\theta _L)<0, \\ F^1_{ \sigma^2}&=-(1-\lambda )\alpha \rho<0, \\ F^2_{\sigma^2}&=-\frac{\alpha^2}{2}\rho <0, \\ F^1_{v}&=0, \\ F^2_{v}&=-\frac{(m-\theta _L)^2}{2}<0. \end{aligned} $$

By Cramer’s rule:

$$ \begin{aligned} \frac{d \alpha}{d \sigma^2} &= \frac{F^1_{m} F^2_{\sigma^2}-F^2_{m} F^1_{\sigma^2}}{F^1_{\alpha} F^2_{m}-F^1_{m} F^2_{\alpha}}, \qquad \frac{d m}{d \sigma^2} = \frac{F^2_{\alpha} F^1_{\sigma^2}-F^1_{\alpha} F^2_{\sigma^2}}{F^1_{\alpha} F^2_{m}-F^1_{m} F^2_{\alpha}},\\ \frac{d \alpha}{d v} &= \frac{F^1_{m} F^2_{v}-F^2_{m} F^1_{v}} {F^1_{\alpha} F^2_{m}-F^1_{m} F^2_{\alpha}}, \qquad \frac{d m}{d v} = \frac{F^2_{\alpha} F^1_{v}-F^1_{\alpha} F^2_{v}}{F^1_{\alpha} F^2_{m}-F^1_{m} F^2_{\alpha}}. \end{aligned} $$

It is immediate from the signs of the individual components that dα/dσ² < 0, dα/d v < 0, and d m/d v < 0. As for d m/dσ², the denominator (F ¹_α F ² _m  − F ¹ _m F ²_α ) is positive, and the numerator equals:

$$ \begin{aligned} F^2_{\alpha} F^1_{\sigma^2}-F^1_{\alpha} F^2_{ \sigma^2}&=(1-\lambda )\alpha \rho \left[A(\alpha)-\frac{\rho \sigma^2 \alpha}{2}\right]\\ &=(1-\lambda )\alpha\rho\left[\Updelta\theta +(\Updelta \phi) K+\frac{\rho \sigma^2 \alpha}{2}\right]>0 \end{aligned} $$

Thus, d m/dσ² > 0. This completes the proof of Lemma 5.

Proof of Proposition 3

We proceed in two steps. First, we argue that if σ² or v are sufficiently high, $Y\left( {\alpha ^{{**}} } \right) > 0$ will hold. On the other hand, $Y\left( {\alpha ^{{**}} } \right) < 0$ will hold for σ² and v sufficiently low. Secondly, we show that the respective sign of $Y\left( {\alpha ^{{**}} } \right)$ implies the properties stated in the proposition.

Step 1. Denote by α^o(m) the value of α which satisfies the binding nonmimicry constraint (19) for any m. Clearly, dα^o/dm < 0, that is, α and m are substitutes in preventing mimicking. Since $m^{{**}} > \theta _{H}$ (by Lemma 4), $\alpha ^{{**}} = \alpha ^{o} \left( {m = m^{{**}} } \right) < \alpha ^{o} \left( {m = \theta _{{\text{H}}} } \right).$ At the same time, α^o(m = θ _H ) < α _N , because V _L (m = θ _H ) > 0, where α _N is the level of share retention necessary to deter mimicking by L in case α is the sole available signal. By Proposition 1, α _N  < α₀ for any σ² > σ ²₀ . Hence, σ² > σ ²₀ implies that $\alpha ^{{**}} < \alpha _{0} ,\,{\text{so}}\,{\text{that}}\,Y\left( {\alpha ^{{**}} } \right) > 0.$

When $v \to 2\frac{(\phi_H-1)K+\Updelta\theta }{(\Updelta\theta )^2}$ (the upper bound given in footnote 32), misreporting becomes so costly that $\alpha ^{{**}} \to 0,\,{\text{and}}\,{\text{hence}}\,Y\left( {\alpha ^{{**}} } \right) > 0.$ By continuity, there must exist a threshold for v such that for any v greater than this threshold, $Y\left( {\alpha ^{{**}} } \right) > 0.$

For σ² and v sufficiently low, $Y\left( {\alpha ^{{**}} } \right) < 0.$ Identifying these thresholds involves a polynomial of higher order and cannot be solved analytically. A numerical example parameterizing the setting in Assumption 1 as {ρ = 0.1, ϕ _H  = 2, ϕ _L = 0.5, θ _H  = 2, θ _L  = 1} shows that for any σ² ≤ 5 and v ≤ 0.2, $Y\left( {\alpha ^{{**}} } \right) < 0$ will hold.

Step 2. We prove this step by employing the same method as in the proof of Lemma 5. Throughout the proof, α and m take their optimal values, i.e. $\alpha ^{{**}} \,{\text{and}}\,m^{{**}}$ respectively (that is, we again suppress the “^**”). Also, by Lemma 4, high-type founders elicit investor effort k = K, so in the following we replace k with K.

In the efficient separating equilibrium, as argued in the proof of Lemma 5 (see there for definitions of F ¹ and F ²),

$$ F^1 = F^2 =0. $$

Differentiating with respect to K yields

$$ F^i_{\alpha} \frac{d \alpha} {d K}+F^i_{m} \frac{d m}{d K}+F^i_{K}=0 ,\quad i=1, 2. $$

While F ¹_α , F ²_α , F ¹ _m , and F ² _m have been derived in the proof of Lemma 5, note that:

$$ \begin{aligned} F^1_{K}&=\lambda (\Updelta\phi)>0,\\ F^2_{K}&=Y(\alpha) . \end{aligned} $$

By Cramer’s rule:

$$ \frac{d \alpha}{d K} = \frac{F^1_{m} F^2_{K}-F^2_{m} F^1_{K}} {F^1_{\alpha} F^2_{m}-F^1_{m} F^2_{\alpha}} \quad \hbox{\ and\ } \quad \frac{d m} {d K} = \frac{F^2_{\alpha} F^1_{K}-F^1_{\alpha} F^2_{K}}{F^1_{\alpha} F^2_{m}-F^1_{m} F^2_{\alpha}}. $$

If Y(α) ≥ 0, it follows that dα/dK > 0, given the signs of the individual components derived above. Similarly, dm/dK < 0 if Y(α) < 0.

To see how K affects the high type’s total signaling cost, it is helpful to observe that Program ${\mathcal{P}}_3$ is equivalent to minimizing H’s total signaling cost C(α, m) subject to the nonmimicry constraint (19). Hence by revealed preference, if Y(α) > 0, then an increase in K will raise C(α, m), and vice versa.

Proof of Proposition 4

We first prove that there exists no (stable) pooling equilibrium under the stated conditions. Suppose such a pooling equilibrium with α and m did exist. The pooling payoffs to H and L, respectively, would then be

$$ \begin{aligned} U_H^{PE}(\alpha,m)&=(1-\alpha)\Uptheta+\alpha \theta _H +[(1-\alpha)\Upphi+\alpha\phi_H-1]k(\alpha,m,\Upphi)-R(\alpha)-V(m-\theta _H), \\ U_L^{PE}(\alpha,m)&=(1-\alpha)\Uptheta+\alpha \theta _L +[(1-\alpha)\Upphi+\alpha\phi_L-1]k(\alpha,m,\Upphi)-R(\alpha)-V(m-\theta _L). \end{aligned} $$

Therefore,

$$ U_H^{PE}(\alpha,m)-U_L^{PE}(\alpha,m)=\alpha(\Updelta \theta )+\alpha(\Updelta \phi) k(\alpha,m,\Upphi)+ \int\limits_{{m - \theta _{H} }}^{{m - \theta _{H} + \Updelta \theta }} {V^{\prime } \left( t \right)dt} , $$

where

$$ k(\alpha,m,\Upphi)\in\arg\max_{k\in\{0,K\}} \ \left[(1-\alpha)\Upphi-1\right]k = \left \{\begin{array}{ll}K & \hbox {if } (1-\alpha)\Upphi\geq 1,\\ 0 & \hbox {if } (1-\alpha)\Upphi<1 . \end{array}\right. $$

The off-equilibrium payoffs for a deviating founder who retains the same α and chooses a higher m ^# would equal

$$ \begin{aligned} U_H^{dev}(\alpha,m^\#)&=\theta _H+(\phi_H-1)k(\alpha,m^\#,\phi_H)-R(\alpha)-V(m^\#-\theta _H), \\ U_L^{dev}(\alpha,m^\#)&=(1-\alpha)\theta _H+\alpha \theta _L+Y(\alpha)k(\alpha,m^\#,\phi_H)-R(\alpha)-V(m^\#-\theta _L). \end{aligned} $$

Therefore,

$$ U_H^{dev}(\alpha,m^\#)-U_L^{dev}(\alpha,m^\#)=\alpha(\Updelta \theta )+\alpha(\Updelta \phi) k(\alpha,m^\#,\phi_H)+ \int\limits_{{m^{\# } - \theta _{H} }}^{{m^{\# } - \theta _{H} + \Updelta \theta }} {V^{\prime } \left( t \right)dt,} $$

where

$$ k(\alpha,m^\#,\phi_H)\in\arg\max_{k\in\{0,K\}} \ \left[(1-\alpha)\phi_H-1\right]k = \left \{ \begin{array} {ll} K & \hbox {if } (1-\alpha)\phi_H\geq 1, \\ 0 & \hbox {if } (1-\alpha)\phi_H<1. \end{array} \right. $$

A sufficient condition for such a pooling equilibrium to be eliminated by the Intuitive Criterion is that there exists some (α, m ^#) such that U ^dev _H (α, m ^#) > U ^PE _H (α, m) and U ^dev _L (α, m ^#) < U ^PE _L (α, m). A necessary condition for both these inequalities to hold is that

$$ \begin{aligned} &U_H^{dev}(\alpha,m^\#)-U_H^{PE}(\alpha,m)+U_L^{PE}(\alpha,m)-U_L^{dev}(\alpha,m^\#) \\ = &\alpha(\Updelta\phi) [k(\alpha,m^\#,\phi_H)-k(\alpha,m,\Upphi)]+\int_{m^\#-\theta _H}^{m^\#-\theta _H+\Updelta\theta }V^{\prime }(t)dt -\int_{m-\theta _H}^{m-\theta _H+\Updelta\theta }V^{\prime }(t)dt \\ \geq & \int_{m^\#-\theta _H}^{m^\#-\theta _H+\Updelta\theta }V^{\prime }(t)dt-\int_{m- \theta _H}^{m-\theta _H+\Updelta\theta }V^{\prime }(t)dt\\ > &0 . \end{aligned} $$

The last inequality holds by convexity of V(·) and m ^# > m. To prove sufficiency, simply choose m ^# such that U ^dev _H (α, m ^#) > U ^PE _H (α, m) while U ^dev _L (α, m ^#) < U ^PE _L (α, m). Hence, there cannot exist any stable pooling equilibrium.

The proof for the existence of separating equilibria follows similar lines and is only sketched here. The investor’s incentive constraint (20) does not directly depend on m, except through the investor’s belief function $\hat{\phi}.$ Holding fixed $\alpha<\tilde{\alpha}$, we therefore arrive at a standard signaling game in which H chooses m so that the nonmimicry constraint (17) is satisfied as an equality for any $\alpha<\tilde{\alpha}.$

Proof of Proposition 5

In any separating equilibrium, by (20), the investor will choose k = K if and only if (1 − α)ϕ _H  − 1 ≥ 0. The low-type founder will choose α = 0, m = θ _L , and induce no effort because ϕ _L  < 1. The high-type founder has the choice between eliciting high investor effort or not, by way of choosing α. These two options are expressed by the following two subprograms, which are mutually exclusive and commonly exhaustive versions of Program ${\mathcal{P}}_4$ for the alternative levels of induced investor effort:

Subprogram I—Induce high investor effort:k = K

$$ \begin{aligned} U_H^*(k=K) \ \equiv \ &\max_{\alpha,m} \ \{\theta _H+(\phi_H-1)K-R(\alpha) - V( m -\theta _H )\}\\ {{\text{s.t.:}}}\ & (1-\alpha)\phi_H-1\geq 0\\ & (1 - \alpha )\Updelta\theta +Y(\alpha)K- R(\alpha) - V( m -\theta _L )\leq 0\\ & \alpha\geq 0 \end{aligned} $$

(IC-I)

Subprogram II—Induce no investor effort:k = 0

$$ \begin{aligned} U_H^*(k=0) \ \equiv \ & \max_{\alpha,m}\ \{\theta _H-R(\alpha) - V( m -\theta _H )\} \\ {{\text{s.t.:}}}\ &(1-\alpha)\phi_H-1<0 \\ & (1 - \alpha )\Updelta\theta - R(\alpha) - V( m -\theta _L )\leq0 \\ &\alpha\geq0 \end{aligned} $$

(IC-II)

Slightly abusing notation, we denote by $\left( {\alpha _{{\text{I}}}^{*} \left( K \right),m_{{\text{I}}}^{*} \left( K \right)} \right)$ the solution to Subprogram I, by $\left( {\alpha _{{{\text{II}}}}^{*} \left( 0 \right),m_{{{\text{II}}}}^{*} \left( 0 \right)} \right)$ the solution to Subprogram II, and by $U_{H}^{*} \left( K \right) \equiv U_{H} \left( {m_{{\text{I}}}^{*} \left( K \right),\alpha _{{\text{I}}}^{*} \left( K \right),K} \right)$ and $U_{{\text{H}}}^{*} \left( 0 \right) \equiv U_{{\text{H}}} \left( {m_{{{\text{II}}}}^{*} \left( 0 \right),\alpha _{{{\text{II}}}}^{*} \left( 0 \right),0} \right),$ respectively, the corresponding payoffs. Finally, let $U_{{\text{H}}}^{{**}} \left( K \right) \equiv U_{{\text{H}}} \left( {m^{{**}} \left( K \right),\alpha ^{{**}} \left( K \right),k^{{**}} \left( K \right) = K} \right)$ be H’s payoff under the relaxed Program ${\mathcal{P}}_3.$

Given Assumption 2 ($\alpha^{**}(K)>\tilde{\alpha}$ for any K), Subprogram I corresponds to ${\mathcal{P}}_3$ with the additional incentive constraint (IC-I) binding for any K, so $U_{{\text{H}}}^{{**}} \left( K \right) > U_{{\text{H}}}^{*} \left( K \right)$ for all K. Note that $\alpha^{**}(0)=\lim_{K {\to}0}\alpha^{**}(K).$ Since Assumption 2 implies $\alpha^{**}(0)>\tilde{\alpha}$, Subprogram II coincides with the unconstrained ${\mathcal{P}}_3$ with K = 0. Therefore,

$$ U_H^*(0)=U_H^{**}(K=0)=\lim_{K\to 0}U_H^{**}(K)>\lim_{K\to 0} U_H^*(K), $$

(34)

where we use the fact that $U_{{\text{H}}}^{{**}} \left( K \right)$ is continuous. Define:

$$ \begin{aligned} \Uplambda \equiv \ & U_H^*(K)-U_H^*(0) \\ = \ &(\phi_H-1)K-R(\tilde \alpha) - V( m^*_I(K) -\theta _H )+R(\alpha^*_{{{II}}}(0)) +V(m^*_{{{II}}}(0) -\theta _H ) \\ = \ &(\phi_H-1)K-(1-\tilde \alpha)\Updelta\theta -\tilde \alpha\phi_LK- V( m^*_I(K)-\theta _H )+ V( m^*_I(K) -\theta _L ) \\ &+[1-\alpha^*_{{{II}}}(0)]\Updelta\theta + V( m^*_{{{II}}}(0) -\theta _H )-V( m^*_{{{II}}}(0) -\theta _L ) \\ =\ &\tilde \alpha(\Updelta\phi) K+[\tilde \alpha-\alpha^*_{{{II}}}(0)]\Updelta\theta - V(m^*_I(K) -\theta _H )+ V( m^*_I(K) -\theta _L ) \\ & +V( m^*_{{{II}}}(0) -\theta _H )-V(m^*_{{{II}}}(0) -\theta _L ) \end{aligned} $$

Here, we have used the fact that nonmimicry constraints are binding and $\alpha _{{\text{I}}}^{*} \left( K \right) = \tilde{\alpha } = 1 - {1 \mathord{\left/ {\vphantom {1 {\phi _{H} }}} \right. \kern-\nulldelimiterspace} {\phi _{H} }}$ when the incentive constraint (IC-I) is binding. Then:

$$ \begin{aligned} \frac{d \Uplambda}{d K}=\ & \tilde \alpha(\Updelta\phi)-V'( m^*_I(K) -\theta _H )\frac{dm^*_I(K)}{dK} +V'( m^*_I(K) -\theta _L )\frac{d m^*_I(K)}{dK} \\ =\ & \tilde \alpha (\Updelta\phi)+\left(1-\frac{V'( m^*_I(K) -\theta _H )}{V'( m^*_I(K) -\theta _L )}\right)\tilde \alpha\phi_L\\ >&0. \end{aligned} $$

Here we have used that $d m^*_I(K)/dK=\tilde{\alpha}\phi_L/V'( m^*_I(K) -{\theta}_L )>0,$ which follows by applying the Implicit Function Theorem to the binding nonmimicry constraint, together with $\alpha^*_I(K)\equiv \tilde{\alpha}$ being a constant (due to the binding incentive constraint (IC-I)).

Lastly, (34) implies that $\lim_{K\to 0}\Uplambda<0.$ Combining $d \Uplambda / d K>0$ with the continuity property of $\Uplambda$ proves there exists some $\hat{K}>0$ such that the high-type firm elicits investor effort k = K for $K{\geq}\hat{K}$, and k = 0 for $K < \hat{K}.$

Proof of Corollary 3

Case 1: If $\sigma^2>\tilde{\sigma}^2$, then, by Lemma 2, ${\alpha_N}(K)<\tilde{\alpha}$ for small K. Using similar arguments as in the proof of Proposition 3, $\alpha^{**}(K)<\alpha^o(m=\theta _H|K)<\alpha_N (K)<\tilde{\alpha}$ for small K. Second, since $\tilde{\sigma}^2>\sigma_0^2$, $\sigma^2>\tilde{\sigma}^2$ implies σ² > σ ²₀ . Then, by Proposition 3, an increase in K increases $\alpha ^{{**}} \left( K \right).$ Third, $\lim_{K\to \infty} \alpha^{**}(K)=\alpha_0 >\tilde{\alpha}.$ Therefore, by continuity of $\alpha ^{{**}} \left( K \right),$ if $\sigma^2>\tilde{\sigma}^2$, there must exist a threshold K _o such that when K < K _o , $\alpha^{**}(K)<\tilde \alpha$, and vice versa. That is, if K < K _o , then the investor’s incentive constraint (20) will be slack and the solution to the constrained program ${\mathcal{P}}_4$ will coincide with that to the unconstrained program ${\mathcal{P}}_3.$ In that case, $\alpha ^{*} \left( K \right) = \alpha ^{{**}} \left( K \right)\,{\text{and}}\,m^{*} \left( K \right) = m^{{**}} \left( K \right).$ At the same time, $\alpha^{**}(K)<\tilde \alpha<\alpha_0$ implies $Y\left( {\alpha ^{{**}} } \right) > 0.$

As we have shown in the proof of Proposition 3, Cramer’s rule yields

$$ \begin{aligned} \frac{d \alpha}{d K}=&\frac{F^1_{m} F^2_{K}-F^2_{m} F^1_{K}}{F^1_{\alpha} F^2_{m}-F^1_{m} F^2_{\alpha}}>0, \quad {{\text{for}}}\ \ Y(\alpha)>0, \\ \frac{d m}{d K}=&\frac{F^2_{\alpha} F^1_{K}-F^1_{\alpha} F^2_{K}}{F^1_{\alpha} F^2_{m}-F^1_{m} F^2_{\alpha}}, \end{aligned} $$

and therefore

$$ \frac{d \alpha / d K }{\alpha}-\frac{d m / d K }{m} = \frac{(F^1_{m}F^2_{K}-F^2_{m} F^1_{K})/ \alpha -(F^2_{\alpha} F^1_{K}-F^1_{\alpha} F^2_{K})/ m} {F^1_{\alpha} F^2_{m}-F^1_{m} F^2_{\alpha}}. $$

(35)

The denominator of this ratio is positive, while the numerator can be rewritten as:

$$ \begin{aligned} & \frac{F^1_{m} F^2_{K}-F^2_{m} F^1_{K}}{\alpha} - \frac{F^2_{\alpha} F^1_{K}-F^1_{\alpha} F^2_{K}}{m} \\ =& \left(\frac{F^1_{m}}{\alpha}+\frac{F^1_{\alpha}}{m}\right)F^2_{K} -\left( \frac{F^2_{m}}{\alpha}+\frac{F^2_{\alpha}}{m}\right)F^1_{K} \\ =& \left[\frac{A(\alpha)\Updelta\theta }{\alpha (m-\theta _L)^2}- \frac{(1-\lambda )\rho\sigma^2}{m}\right]Y(\alpha) - \left[-\frac{V^{\prime }(m-\theta _L)}{\alpha}- \frac{A(\alpha)} {m}\right] \lambda (\Updelta\phi) \\ = & \left[\frac{(\Updelta\theta +(\Updelta\phi) K)(\Updelta\theta )}{\alpha (m-\theta _L)^2} +\frac{\rho \sigma^2 (\Updelta\theta )} {(m-\theta _L)^2}-\frac{\rho \sigma^2 (\Updelta\theta )} {m(m-\theta _L)} \right]Y(\alpha) +\left[\frac{V^{\prime }(m-\theta _L)} {\alpha}+\frac{A(\alpha)}{m}\right]\lambda (\Updelta\phi) \\ > &0, \quad {{\text{for}}}\ \ Y(\alpha)>0. \end{aligned} $$

We conclude that the expression in (35) is strictly positive, and thus:

$$ \frac{d (\frac{\alpha}{m})} {d K}=\frac{\frac{d \alpha} {d K}m-\frac{d m}{d K}\alpha} {m^{2}}>0. $$

If K > K _o , then $\alpha^{**}(K)>\tilde \alpha$, and the investor’s incentive constraint (20) will be binding. Then, to elicit high effort from the investor, $\alpha^*(K)=\tilde \alpha$, a constant. Also, the nonmimicry constraint will be binding at the optimal solution—thus

$$ (1-\tilde \alpha)\Updelta\theta +Y(\tilde \alpha)K-R(\tilde \alpha) - V(m^*(K)-\theta _L) \equiv 0 $$

holds in an open neighborhood around K. Applying the Implicit Function Theorem:

$$ \frac{d m^*}{d K}=\frac{Y(\tilde \alpha)}{V'( m^*_I(K) -\theta _L )}. $$

Both numerator and denominator are positive, as $Y(\tilde {\alpha})=\tilde{\alpha}\phi_L>0.$ Thus, $m^{*}$ is increasing in K, and the ratio $\left( {{{a^{*} } \mathord{\left/ {\vphantom {{a^{*} } {m^{*} }}} \right. \kern-\nulldelimiterspace} {m^{*} }}} \right)$ is decreasing in K, because $\alpha^*(\cdot)\equiv \tilde{\alpha}$, a constant.

Case 2: If σ² and v are sufficiently low, that is, Assumption 2 holds, then by Proposition 5, the high-type founder elicits k = K for $K{\geq}\hat{K}$, and k = 0 for $K<\hat{K}.$ Therefore, if $K < \hat{K},\alpha ^{*} \left( K \right) \equiv \alpha ^{{**}} \left( 0 \right)$ and $m^{*} \left( K \right) \equiv m^{{**}} \left( 0 \right),$ which are each independent of K. If $K\geq \hat{K}$, to ensure high investor effort, $\alpha^*(K)\equiv\tilde {\alpha}$ would have to hold. Then, the same argument as in Case 1 shows that $m^{*}$ will be increasing in K, and hence the ratio $\left( {{{a^{*} } \mathord{\left/ {\vphantom {{a^{*} } {m^{*} }}} \right. \kern-\nulldelimiterspace} {m^{*} }}} \right)$ will be decreasing in K. This completes the proof of Corollary 3.