Leading indicator variables and managerial incentives in a dynamic agency setting

Abstract

This paper studies, in a dynamic agency setting, how incentives and contractual efficiency are affected by leading indicators of firms’ future financial performance. In our two-period model, a leading indicator variable provides a noisy forecast of the uncertain return from the manager’s long-term effort, and both contracting parties cannot refrain from renegotiating contract terms based on updated information. We find that the leading indicator can reduce the manager’s long-term effort incentive, as it allows the firm owner to capture more of the resulting return through renegotiated wages (i.e., the manager is held up). By reducing the uncertainty about future aggregate cash flows, the leading indicator also exacerbates the “ratchet” effect and discourages the manager’s short-term effort. In equilibrium, as the leading indicator becomes more accurate in forecasting future cash flows, the first-period contract attaches higher explicit weights to both the forward-looking leading indicator and backward-looking cash flow, and yet the manager may find it optimal to reduce both the short- and long-term efforts. We further show that with a more accurate leading indicator variable, the explicit incentive on the lagging cash flow may increase more than that on the leading indicator, and the equilibrium firm profit may decrease and diverge from the manager’s equilibrium efforts.

Appendix

Proof of Lemma 1

Substituting the optimal second period effort $e_{2}^{\ast }=v\cdot \beta _{2}$ into the expression of π₂ (β₂), the optimal $\beta _{2}^{\ast }$ can then be solved from the first-order condition for the maximization of π₂ (β₂):

$$v^{2}-\left( v^{2}+\rho\cdot{Var}[c_{2}|c_{1},f]\right) \cdot\beta_{2}^{\ast}= 0. $$

The firm profit for the second period then equals

$$\begin{array}{@{}rcl@{}} \pi_{2}\left( \beta_{2}^{\ast}\right) & =&v^{2}\cdot\beta_{2}^{\ast} -\frac{1}{2}[\left( v^{2}+\rho\cdot{Var}[c_{2}|c_{1},f]\right) \cdot\beta_{2}^{\ast}]\cdot\beta_{2}^{\ast}\\ & =&v^{2}\cdot\beta_{2}^{\ast}-\frac{1}{2}v^{2}\cdot\beta_{2}^{\ast} =\frac{v^{2}}{2}\cdot\beta_{2}^{\ast}. \end{array} $$

□

Proof of Lemma 2

Because

$${Var}[c_{2}|c_{1},f]=h_{\varepsilon}^{-1}+\left( h_{\varepsilon }+h_{0}\right)^{-1}+\left( h+h_{\eta}\right)^{-1}\text{,} $$

$\beta _{2}^{\ast }$ strictly increases in h. Given that κ_ε is independent of h, $\beta ^{I}\equiv -\gamma \cdot \beta _{2}^{\ast } \cdot \kappa _{\varepsilon }$ must strictly decrease in h as well, i.e., $\frac {\partial \beta ^{I}}{\partial h}<0$.

It can be easily verified that

$$\begin{array}{@{}rcl@{}} \lambda^{I} & =&\gamma\cdot\beta_{2}^{\ast}\cdot\left( 1-\kappa_{f}\right) \\ & =&\frac{\gamma h_{\eta}v^{2}}{(h_{\eta}+h)(v^{2}+\rho{Var} [\varepsilon_{2}|c_{1}])+\rho}, \end{array} $$

which strictly decreases in h. Therefore, $\frac {\partial \lambda ^{I} }{\partial h}<0$. □

Proof of Lemma 3

From Eq. 20, the first-order condition yields

$$\beta_{1}^{\ast}=\frac{v^{2}(1+\kappa_{\varepsilon}\gamma\beta_{2}^{\ast} )}{v^{2}+\rho\left( h_{0}^{-1}+h_{\varepsilon}^{-1}\right) }\text{.} $$

Since $B_{1}^{\ast }=\beta _{1}^{\ast }+\beta ^{I}$, we have

$$B_{1}^{\ast}=\frac{v^{2}-\rho\gamma\beta_{2}^{\ast}h_{0}^{-1}}{v^{2} +\rho\left( h_{0}^{-1}+h_{\varepsilon}^{-1}\right) }. $$

Because $\beta _{2}^{\ast }$ strictly increases in h, $\beta _{1}^{\ast }$ strictly increases in h and $B_{1}^{\ast }$ strictly decreases in h. It follows that $e_{1}^{\ast }=v\cdot B_{1}^{\ast }$ also strictly decreases in h. □

Proof of Lemma 4

From Eq. 21, the first-order condition yields

$$\lambda^{\ast}=\frac{w^{2}(\gamma-\gamma\beta_{2}^{\ast}\left( 1-\kappa_{f}\right) )}{w^{2}+\rho\left( h_{\eta}^{-1}+h^{-1}\right) }. $$

Since Λ^∗ = λ^∗ + λ^I, we have

$${\Lambda}^{\ast}=\frac{\gamma w^{2}+\rho\gamma\beta_{2}^{\ast}h^{-1}}{w^{2} +\rho\left( h_{\eta}^{-1}+h^{-1}\right) }. $$

It is obvious that the numerator of λ^∗ increases in h (from the proof of Lemma 2) and the denominator decreases in h. Thus, λ^∗ increases in h. To simplify derivation, let Ω ≡ v² + ρ Var[ε₂|c₁]. Then

$$\lambda^{I}=\frac{\gamma h_{\eta}v^{2}}{(h_{\eta}+h){\Omega}+\rho}\text{.} $$

By the Implicit Function Theorem (applied to the FOC of Eq. 19), $\frac {d{\Lambda }^{\ast }}{dh}$ has the opposite sign as the derivative of the marginal risk premium with respect to h, i.e.,

$$\frac{d{\Lambda}^{\ast}}{dh}\overset{sign}{=}-\left[ \lambda^{\ast}\cdot \frac{\partial{Var}\left[ f\right] }{\partial h} -{Var}\left[ f\right] \cdot\frac{\partial\lambda^{I}}{\partial h}\right] . $$

Because both $\frac {\partial {Var}\left [ f\right ] }{\partial h}$ and $\frac {\partial \lambda ^{I}}{\partial h}$ are negative, it can be verified with the expression of λ^∗ and λ^I that

$$\frac{d{\Lambda}^{\ast}}{dh}\overset{sign}{=}-\left[ \frac{\lambda^{\ast} }{\frac{\partial\lambda^{I}}{\partial h}}-\frac{{Var}\left[ f\right] }{\frac{\partial{Var}\left[ f\right] }{\partial h} }\right] \overset{sign}{=}\psi_{0}\left( w\right) \cdot h^{2}+\psi_{1}\left( w\right) \cdot h+\psi_{2}\left( w\right) ,$$

where ψ_i (⋅)^′s (for i = 0, 1, 2) are all linear functions of w² that satisfy:

$$\begin{array}{@{}rcl@{}} && \psi_{0}\left( w\right) \overset{sign}{=}h_{\eta}\left( {\Omega} -v^{2}\right) \cdot w^{2}-v^{2}\rho,\\ && \psi_{1}\left( w\right) \overset{sign}{=}\left( \rho+h_{\eta}\left( {\Omega}-v^{2}\right) \right) \cdot w^{2}-v^{2}\rho,{~and}\\ && \psi_{2}\left( w\right) \overset{sign}{=}\left( \rho+h_{\eta}\left( {\Omega}-v^{2}\right) \right) \cdot w^{2}-v^{2}\rho\cdot\frac{h_{\eta}{\Omega} }{\rho+h_{\eta}{\Omega}}. \end{array} $$

It is obvious that Ω − v² = ρV ar[ε₂|c₁] > 0.

Let w⁰ denote the positive w at which ψ₂ (w) = 0, i.e., $w^{0}=\left (\frac {h_{\eta }{\Omega }}{\rho +h_{\eta }{\Omega }}\cdot \frac {v^{2}\rho }{\left (\rho +h_{\eta }\left ({\Omega }-v^{2}\right ) \right ) }\right )^{\frac {1}{2}}$. Therefore, when w ≤ w⁰, ψ_i (w) ≤ 0 for all i (i = 0, 1, 2) and hence $\frac {d{\Lambda }^{\ast }} {dh}<0$. Let w¹ denote the positive w at which ψ₀ (w) = 0, i.e., $w^{1}=\left (\frac {v^{2}\rho }{h_{\eta }\left ({\Omega }-v^{2}\right ) }\right )^{\frac {1}{2}}$. Therefore, when w ≥ w¹, ψ_i (w) ≥ 0 for all i and $\frac {d{\Lambda }^{\ast }} {dh}>0$. For w ∈ (w⁰,w¹), ψ₀ (w) < 0 and ψ₂ (w) > 0, which indicates that $\frac {d{\Lambda }^{\ast }}{dh}$ intersects the positive half of the h-axis only once, from positive to negative.

Given that w⁰ < w¹, we have the following:

• (i) when w ≤ w⁰, Λ^∗ decreases in h;

• (ii) when w ≥ w¹, Λ^∗ increases in h; and

• (iii) when w⁰ < w < w¹, Λ^∗ first increases and then decreases in h.

Since b^∗ = Λ^∗w, the same results apply to b^∗.

Now we show that the optimal bonus rates remain unchanged in the presence of the “take-the-money-and-run” constraint. If the manager were to adopt the “off-equilibrium” strategy of quitting after the first period, he would choose

$$e_{1}^{off}=v\beta_{1}^{\ast},{~and~}b^{off}=w\lambda^{\ast}. $$

Comparing the expected bonuses and effort costs from {e1off, b^off} with those from the equilibrium effort choices $\left \{ e_{1}^{\ast },\text { }b^{\ast }\right \}$, it is easy to see that the manager’s first-period certainty equivalent increases by

$$\begin{array}{@{}rcl@{}} \delta & =&\frac{1}{2}\left( e_{1}^{off}\right)^{2}-\left( v\beta_{1}^{\ast}e_{1}^{\ast}-\frac{1}{2}\left( e_{1}^{\ast}\right)^{2}\right) +\frac{1}{2}\left( b^{off}\right) ^{2}-\left( w\lambda^{\ast}b^{\ast} -\frac{1}{2}\left( b^{\ast}\right)^{2}\right) \\ & =&\frac{1}{2}v^{2}\left( \gamma\beta_{2}^{\ast}\right)^{2}\kappa_{\varepsilon}^{2}+\frac{1}{2}w^{2}\left( \gamma\beta_{2}^{\ast}\right)^{2}\left( 1-\kappa_{f}\right)^{2}. \end{array} $$

To lock in the manager, a sufficiently large amount of the fixed pay must be deferred to the second period so that $CE_{1}^{off}\equiv -{\Delta }+\delta \leq CE_{1}^{\ast }= 0$. In other words, ${\Delta }\geq \frac {1}{2}v^{2}\left (\gamma \beta _{2}^{\ast }\right )^{2}\kappa _{\varepsilon }^{2}+\frac {1}{2} w^{2}\left (\gamma \beta _{2}^{\ast }\right )^{2}\left (1-\kappa _{f}\right )^{2}$. Such deferred compensation has no impact on the expected firm profit or the manager’s efforts. □

Proof of Proposition 1

Using the expressions for λ^∗ and $\beta _{1}^{\ast }$, tedious but straightforward algebra shows that for all h_η > 0, $\frac {\partial \left (\lambda ^{\ast }/\beta _{1}^{\ast }\right ) }{\partial h}$ has the same sign as the polynomial below:

$${\Psi}(h_{\eta})={\Psi}_{3}\cdot h_{\eta}^{3}+{\Psi}_{2}\cdot h_{\eta}^{2}+{\Psi} _{1}\cdot h_{\eta}-\gamma h^{2}v^{2}\rho^{2}\kappa_{\varepsilon}, $$

where ${\Psi }_{j}^{\prime }s$ (for j = 1, 2, 3) are polynomial functions of other parameters of the model. Because $\lim \limits _{h_{\eta }\rightarrow 0} {\Psi }(h_{\eta })=-\gamma h^{2}v^{2}\rho ^{2}\kappa _{\varepsilon }<0$, $\frac {\partial \left (\lambda ^{\ast }/\beta _{1}^{\ast }\right ) }{\partial h}<0$ for sufficiently small but positive h_η. □

Proof of Proposition 2

Given the expression for π_s and that $\frac {\partial \beta _{2}^{\ast } }{\partial h}=\frac {\rho \left (\beta _{2}^{\ast }\right )^{2}}{v^{2}\left (h+h_{\eta }\right )^{2}}$, we have

$$\begin{array}{@{}rcl@{}} \frac{d\pi_{s}^{\ast}}{dh} & =&\frac{\partial\pi_{s}^{\ast}\left( \beta^{I},h\right) }{\partial h}+\frac{\partial\pi_{s}^{\ast}\left( \beta^{I},h\right) }{\partial\beta^{I}}\frac{\partial\beta^{I}}{\partial h}\\ & =&\frac{\gamma}{2}\rho\left( \beta_{2}^{\ast}\right)^{2}\left( h+h_{\eta}\right)^{-2}-\rho\beta_{1}^{\ast}\kappa_{\varepsilon}\left( h_{0}^{-1}+h_{\varepsilon}^{-1}\right) \gamma\frac{\partial\beta_{2}^{\ast} }{\partial h}\\ & =&\left[ \frac{v^{2}}{2}-\frac{\rho\beta_{1}^{\ast}}{h_{0}}\right] \gamma\frac{\partial\beta_{2}^{\ast}}{\partial h}. \end{array} $$

Because $\frac {\partial \beta _{2}^{\ast }}{\partial h}>0$, it can be verified that

$$\frac{d\pi_{s}^{\ast}}{dh}\overset{sign}{=}\frac{v^{2}}{2}-\frac{\rho\beta_{1}^{\ast}}{h_{0}}\overset{sign}{=}\varphi_{1}\left( h_{0}\right) \cdot h+\varphi_{0}\left( h_{0}\right) , $$

where φ₁ (h₀) and φ₀ (h₀) are given by:

$$\begin{array}{@{}rcl@{}} \varphi_{1}\left( h_{0}\right) & =&\left( v^{4}h_{\varepsilon}^{2} + 2v^{2}\rho h_{\varepsilon}+\rho^{2}\right) {h_{0}^{2}}\\ && +\left( v^{4}h_{\varepsilon}^{3}+ 2v^{2}\rho h_{\varepsilon}^{2}+\rho^{2}h_{\varepsilon}\right) h_{0}\\ && -\left( 2\rho^{2}h_{\varepsilon}^{2}+v^{2}\rho h_{\varepsilon}^{3} + 2v^{2}\gamma\rho h_{\varepsilon}^{3}\right) \text{, and}\\ \varphi_{0}\left( h_{0}\right) & =&h_{\eta}\varphi_{1}\left( h_{0}\right) +\rho h_{\varepsilon}k(h_{0}), \end{array} $$

with k(h₀) ≡ (h₀ + h_ε) (h₀h_εv² + ρh₀ − ρh_ε).

Notice that φ₀ (h₀), φ₁ (h₀), and k(h₀) are quadratic functions of h₀ and intersect with the positive half of the h₀-axis only once, from negative to positive.

Let $h_{0}^{-}$, $h_{0}^{+}$, and ${h_{0}^{k}}\equiv \frac {h_{\varepsilon }\rho }{v^{2}h_{\varepsilon }+\rho }$ denote the unique positive roots of φ₀ (h₀), φ₁ (h₀), and k(h₀), respectively. Because $\varphi _{1}\left ({h_{0}^{k}}\right ) =-2v^{2}\gamma \rho h_{\varepsilon }^{3}<0$, we have $h_{0}^{+}>{h_{0}^{k}}$. Therefore

$$\varphi_{0}\left( h_{0}^{+}\right) >0\text{, }\varphi_{1}\left( h_{0}^{-}\right) <0{,~and~}h_{0}^{-}<h_{0}^{+}. $$

Let

$$\bar{h}\equiv-\frac{\varphi_{0}\left( h_{0}\right) }{\varphi_{1}\left( h_{0}\right) }=-h_{\eta}-\frac{\rho h_{\varepsilon}k(h_{0})}{\varphi_{1}\left( h_{0}\right) }. $$

We then have the following:

• When $h_{0}\leq h_{0}^{-}$, φ₁ < 0, φ₀ ≤ 0, and $\pi _{s}^{\ast }$ decreases in h for all h > 0.

• When $h_{0}^{-}<h_{0}<h_{0}^{+}$, φ₁ < 0, φ₀ > 0,and $\pi _{s}^{\ast }$ increases (decreases) in h for $h<(>)\bar {h}$.

• When $h_{0}\geq h_{0}^{+}$, φ₁ ≥ 0, φ₀ > 0, and $\pi _{s}^{\ast }$ increases in h for all h > 0.

□

Proof of Proposition 3

By the Envelope Theorem, $\frac {d\pi ^{l}}{dh}$ has the opposite sign as the derivative of the total risk premium with respect to h, i.e.,

$$\frac{d\pi_{l}^{\ast}}{dh}\overset{sign}{=}-\left[ \frac{\lambda^{\ast}} {2}\cdot\frac{\partial\text{Var}\left[ f\right] }{\partial h}-\text{Var}\left[ f\right] \cdot\frac{\partial\lambda^{I} }{\partial h}\right] . $$

Because both $\frac {\partial \text {Var}\left [ f\right ] }{\partial h}$ and $\frac {\partial \lambda ^{I}}{\partial h}$ are negative, it can be verified that

$$\frac{d\pi_{l}^{\ast}}{dh}\overset{sign}{=}-\left[ \frac{1}{2}\cdot \frac{\lambda^{\ast}}{\frac{\partial\lambda^{I}}{\partial h}}-\frac {{Var}\left[ f\right] }{\frac{\partial{Var} \left[ f\right] }{\partial h}}\right] \overset{sign}{=}\phi_{0}\left( w\right) \cdot h^{2}+\phi_{1}\left( w\right) \cdot h+\phi_{2}\left( w\right) , $$

where ϕ_i (⋅)^′s (for i = 0, 1, 2) are all linear functions of w² that satisfy:

$$\begin{array}{@{}rcl@{}} && \phi_{0}\left( w\right) \overset{sign}{=}h_{\eta}\left( {\Omega} -2v^{2}\right) \cdot w^{2}-2v^{2}\rho,\\ && \phi_{1}\left( w\right) \overset{sign}{=}\left( \rho+h_{\eta}\left( {\Omega}-\frac{3}{2}v^{2}\right) \right) \cdot w^{2}-2v^{2}\rho,{~and}\\ && \phi_{2}\left( w\right) \overset{sign}{=}\left( \rho+h_{\eta}\left( {\Omega}-v^{2}\right) \right) \cdot w^{2}-2v^{2}\rho\cdot\frac{{\Omega} h_{\eta }}{\rho+{\Omega} h_{\eta}}. \end{array} $$

Let w⁻ denote the positive w at which ϕ₂ (w) = 0. It can be verified that ϕ₀ (w⁻) < 0 and ϕ₁ (w⁻) < 0. Therefore, when w ≤ w⁻, ϕ_i(w) ≤ 0 for all i and hence $\pi _{l}^{\ast }$ decreases in h. For w > w⁻, we consider the following two cases:

• (i) If Ω = v² + ρV ar[ε₂|c₁] ≤ 2v², then ϕ₀ (w) < 0 for any w. Since ϕ₂ (w) > 0, ϕ₀ ⋅ h² + ϕ₁ ⋅ h + ϕ₂ intersects the positive half of the h axis only once, from positive to negative.

• (ii) If Ω = v² + ρV ar[ε₂|c₁] > 2v², then there exists a $w^{+}\equiv \left (\frac {2v^{2}\rho h_{\eta }^{-1}}{{\Omega }-2v^{2}}\right )^{\frac {1}{2}}$ such that ϕ₀ (w) > (<)0 for w > (<)w⁺. It can be easily verified that ϕ₁ (w⁺) > 0. Therefore

  • when w⁻ < w < w⁺, ϕ₀ < 0, ϕ₂ > 0, and ϕ₀ ⋅ h² + ϕ₁ ⋅ h + ϕ₂ intersects the positive half of the h axis only once, from positive to negative.

  • when w ≥ w⁺, ϕ_i (w) ≥ 0 for all i, and hence π_l increases in h.

Taken together, define

$$w^{+}\equiv \genfrac{\{}{.}{0pt}{}{\left( \frac{2v^{2}\rho h_{\eta}^{-1}}{{\Omega}-2v^{2} }\right)^{\frac{1}{2}}\equiv\left( \frac{2v^{2}\rho h_{\eta}^{-1}} {\rho{Var}[\varepsilon_{2}|c_{1}]-v^{2}}\right)^{\frac{1}{2} }{, when~}\rho~{Var}[\varepsilon_{2}|c_{1}]>v^{2}} {{\kern1pc}+\infty, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ when ~\rho~{Var}[\varepsilon_{2}|c_{1}]\leq v^{2}} . $$

Then for w ∈ (w⁻,w⁺), there always exists an $\tilde {h}$ such that $\pi _{l}^{\ast }$ increases (decreases) in h for $h<(>)\tilde {h}$. It is obvious that when w⁺ is finite, it decreases in V ar[ε₂|c₁].

Substituting the expressions for w⁰ and w¹ from Lemma 4 into ϕ₀ (w) and ϕ₂ (w), we have

$$\phi_{0}\left( w^{1}\right) <0{~and~}\phi_{2}\left( w^{0}\right) <0. $$

Therefore, w⁰ < w⁻ and w¹ < w⁺. □

Proof of Proposition 4

To prove the first part of (i), notice that combining the proof of Propositions 2 and 3, it is obvious that when $h_{0}<h_{0}^{+}$ and w < w⁺ (h₀), total firm profit π (h) must decrease in h for $h>h^{\ast }\equiv max\{\bar {h},\tilde {h}\}$. Since π (h) is continuous in h, it must attain a maximum over the compact set h ∈ [0,h^∗]. Note that the maximum cannot be achieved at h = h^∗ unless $\bar {h}=\tilde {h}$ because π^′(h^∗) < 0. Also, the maximum must be a global maximum since π (h) decreases in h for h > h^∗. The second part of (i) and (ii) are obvious from the previous propositions. □

Proof of Proposition 5

Given that Λ = λ + γβ₂, the firm owner’s maximization problem is equivalent to choosing {β₁,β₂,Λ} to maximize the expected net firm profit given by the following expression:

$$\begin{array}{@{}rcl@{}} {\Pi} & =&\underset{\{\beta_{1},\text{ }\beta_{2},\text{ }{\Lambda}\}}{max}\{ve_{1}\left( \beta_{1}\right) -m\left( e_{1}\left( \beta_{1}\right) \right) -m\left( b\left( {\Lambda}\right) \right) +\gamma(ve_{2}\left( \beta_{2}\right) +wb\left( {\Lambda}\right) -m\left( e_{2}\left( \beta_{2}\right) \right) )\\ && -\frac{1}{2}\rho((\beta_{1}+\gamma\beta_{2}\kappa_{\varepsilon} )^{2}{Var}[c_{1}]+({\Lambda}-\gamma\left( 1-\kappa_{f}\right) \cdot\beta_{2})^{2}{Var}[f]+{\gamma\beta_{2}^{2}} {Var}[c_{2}|c_{1},f])\}, \end{array} $$

where e₁ (β₁) = vβ₁, e₂ (β₂) = vβ₂, and b (Λ) = w(λ + γβ₂) ≡ wΛ denote the induced effort choices.

The FOCs for the optimal $\left \{ {\beta _{1}^{0}},{\beta _{2}^{0}}\text {, } {\Lambda }^{0}\right \}$ are as follows:

$$\begin{array}{@{}rcl@{}} 0 & =&v^{2}-{\beta_{1}^{0}}v^{2}-\rho\left( {\beta_{1}^{0}}+\gamma \kappa_{\varepsilon}{\beta_{2}^{0}}\right) {Var}[c_{1}],\\ 0 & =&\gamma w^{2}-{\Lambda}^{0}w^{2}-\rho\left( {\Lambda}^{0}-\gamma\left( 1-\kappa_{f}\right) {\cdot\beta_{2}^{0}}\right) {Var}[f],{~and}\\ 0 & =&\gamma^{2}w^{2}+\gamma v^{2}-\gamma{\Lambda}^{0}w^{2}-{\beta_{2}^{0}}\gamma v^{2}-\rho\gamma\kappa_{\varepsilon}\left( {\beta_{1}^{0}}+\gamma \kappa_{\varepsilon}{\beta_{2}^{0}}\right) {Var}[c_{1}]\\ && -\rho\gamma\kappa_{f}\left( {\Lambda}^{0}-\gamma\left( 1-\kappa_{f}\right) {\beta_{2}^{0}}\right) {Var}[f]-\rho\gamma\beta_{2}^{0}{Var}[c_{2}|c_{1},f]. \end{array} $$

The above system of equations entails the following unique solution:

$$\begin{array}{@{}rcl@{}} {\beta_{2}^{0}} & =&\frac{v^{2}+\rho(\frac{w^{2}\gamma}{(w^{2}+\rho {Var}[f])h}-\frac{v^{2}}{(v^{2}+\rho{Var} [c_{1}])h_{0}})}{v^{2}+\rho{Var}[c_{2}|c_{1},f]+\rho\gamma (\frac{h_{\eta}}{\left( h+h_{\eta}\right) }\frac{w^{2}}{\left( w^{2} +\rho{Var}[f]\right) h}+\frac{\kappa_{\varepsilon}v^{2} }{\left( v^{2}+\rho{Var}[c_{1}]\right) h_{0}})}\\ & =&\frac{v^{2}+\rho\cdot\left[ \tilde{\lambda}\left( 1-\kappa_{f}\right) {Var}[f]-\tilde{\beta}_{1}\kappa_{\varepsilon} {Var}[c_{1}]\right] }{v^{2}+\rho\cdot{Var} [c_{2}|c_{1},f]+\rho\cdot\left[ \tilde{\lambda}\left( 1-\kappa_{f}\right)^{2}{Var}[f]+\gamma\tilde{\beta}_{1}\kappa_{\varepsilon} ^{2}{Var}[c_{1}]\right] },\\ {\beta_{1}^{0}} & =&\frac{v^{2}-{\rho\gamma\beta_{2}^{0}}h_{0}^{-1}}{v^{2} +\rho{Var}[c_{1}]},{~and}\\ {\Lambda}^{0} & =&\frac{\gamma\left( w^{2}+{\rho\beta_{2}^{0}}h^{-1}\right) }{w^{2}+\rho{Var}[f]}, \end{array} $$

where $\tilde {\lambda }\equiv \frac {\gamma \cdot w^{2}}{w^{2}+\rho {Var}[f]}$ and $\tilde {\beta }_{1}\equiv \frac {v^{2}}{v^{2} +\rho {Var}[c_{1}]}$.

It can be verified with straightforward math that the second-order condition is satisfied (i.e., the Hessian matrix is negative definite), and $\beta _{2}^{0}\in \left (0,1\right )$ such that ${\beta _{1}^{0}}\in (0,\tilde {\beta }_{1})$ and ${\Lambda }^{0}\in (\tilde {\lambda },\gamma )$.

By the Envelope Theorem and given that γ < 1, we have

$$\begin{array}{@{}rcl@{}} && \frac{d{\Pi}}{dh}\overset{sign}{=}-\frac{\partial(({\Lambda}^{0}-\gamma\left( 1-\kappa_{f}\right) {\beta_{2}^{0}})^{2}{Var}[f])}{\partial h}-\gamma({\beta_{2}^{0}})^{2}\frac{\partial{Var}[c_{2}|c_{1} ,f]}{\partial h}\\ && \overset{sign}{=}({\Lambda}^{0}-\gamma\left( 1-\kappa_{f}\right) \beta_{2}^{0})^{2}h^{-2}-2({\Lambda}^{0}-\gamma\left( 1-\kappa_{f}\right) \beta _{2}^{0}){\gamma\beta_{2}^{0}}h^{-1}(h_{\eta}+h)^{-1}+\gamma({\beta_{2}^{0}} )^{2}(h_{\eta}+h)^{-2}\\ && >\left[ \left( {\Lambda}^{0}-\gamma\left( 1-\kappa_{f}\right) \beta_{2}^{0}\right) h^{-1}-{\gamma\beta_{2}^{0}}\left( h+h_{\eta}\right)^{-1}\right]^{2}\\ && \geq0. \end{array} $$

Therefore, the firm profit π strictly increases in h. □