Financial hedging and competitive strategy for value-maximizing firms under quantity competition

Abstract

Inspired by the growing use of financial hedging among competitive firms nowadays, we develop a game-theoretical model to investigate the problem of applying financial hedging to improve a firm’s competitive strategy. A distinctive setting of the model is that the firm value is a concave function of the firm profit, which is consistent with the empirical evidences in finance literature. After proving the unique existence of the Nash equilibrium, we examine the effects of financial hedging on the equilibrium and yield some novel results. In particular, our analysis suggests that in a competitive market, financial hedging is not just to protect a firm’s bottom line; perhaps more importantly, effective financial hedging schemes can help increase the firm value by boosting the firm’s production, raising the market share, and improving its profitability.

Appendix

Proof of Lemma 1

• (i) The concavity of $V_1 \left( {q_1 ;q_2 } \right)$ follows directly from the inequality (3). Similarly we can prove that $V_2 \left( {q_2 ;q_1 } \right)$ is concave in $q_2$.

• (ii) It is straightforward to verify that $V_1 \left( 0 \right) =0$. Then, for $q_2 \in \left[ {0,q_M } \right]$ we have $V_1^{\prime } \left( {0;q_2 } \right) =u_1^{\prime } \left( 0 \right) E\left[ {\left( {{\tilde{Y}} -c_1 -kq_2 } \right) } \right]$. If $q_2 >\frac{E\left( {{\tilde{Y}} } \right) -c_1 }{k}$, then $V_1^{\prime } \left( {0;q_2 } \right) <0$, and from the concavity of the function $V_1 \left( {q_1 } \right)$ the solution of $V_1^{\prime } \left( {q_1 ;q_2 } \right) =0$ will be negative. Thus, we have ${\hat{q}} _1 \left( {q_2 } \right) =0$ when $q_2 >\frac{E\left( {{\tilde{Y}} } \right) -c_1 }{k}$. Otherwise, if $0\le q_2 \le \frac{E\left( {{\tilde{Y}} } \right) -c_1 }{k}$, then $V_1^{\prime } \left( {0;q_2 } \right) \ge 0$. Thus, the solution of $V_1^{\prime } \left( {q_1 ;q_2 } \right) =0$ is non-negative. Combined with the fact that $V_1^{{\prime }{\prime }} \left( q \right) <0$, we know that ${\hat{q}} _1 \left( {q_2 } \right)$ is the unique solution of $V_1^{\prime } \left( {q_1 ;q_2 } \right) =0$ when $0\le q_2 \le \frac{E\left( {{\tilde{Y}} } \right) -c_1 }{k}$.

• (iii) The proof follows directly from (ii) above.

• (iv) It is straightforward to verify that $V_i \left( {q_M } \right) <0=V_i \left( 0 \right)$, which means that for each firm, producing $q_M$ units of product is strictly worse than producing nothing, regardless of the production quantity of the competitor. Thus, ${\hat{q}} _i \left( q \right) <q_M$. $\square$

Proof of Proposition 1

(i) Using Eq. (2) and differentiating $V_1^{\prime } \left( {q_1 ;q_2 } \right)$ with respect to $q_2$, we have

$$\begin{aligned} \frac{\partial V_1{^{\prime }}\left( {q_1 ;q_2 } \right) }{\partial q_2 }= & {} -\,kq_1 E\left[ {\left( {{\tilde{Y}} -c_1 -k\left( {2q_1 +q_2 } \right) } \right) u_1^{{\prime }{\prime }} \left( {\pi _1 \left( {q_1 ,q_2 ;{\tilde{Y}} } \right) } \right) } \right] \\&-\,kE\left[ {u_1^{\prime } \left( {\pi _1 \left( {q_1 ,q_2 ;{\tilde{Y}} } \right) } \right) } \right] \\= & {} \gamma _1 kq_1 E\left[ {\left( {{\tilde{Y}} -c_1 -k\left( {2q_1 +q_2 } \right) } \right) u_1^{\prime } \left( {\pi _1 \left( {q_1 ,q_2 ;{\tilde{Y}} } \right) } \right) } \right] \\&-\,kE\left[ {u_1^{\prime } \left( {\pi _1 \left( {q_1 ,q_2 ;{\tilde{Y}} } \right) } \right) } \right] \end{aligned}$$

where the second equality follows from the fact that $u_1^{\prime } \left( z \right) =e^{-\gamma _1 z}$. We are only interested in analyzing the sign of the above partial derivative at $V_1^{\prime } \left( {q_1 ;q_2 } \right) =0$, in which we know from Eq. (2) that $E\left[ {\left( {{\tilde{Y}} -c_1 -\,k\left( {2q_1 +q_2 } \right) } \right) u_1^{\prime } \left( {\pi _1 \left( {q_1 ,q_2 ;{\tilde{Y}} } \right) } \right) } \right] =0$. Thus, the following inequality holds:

$$\begin{aligned} \frac{\partial V_1{}^{\prime }\left( {q_1 ;q_2 } \right) }{\partial q_2 }=-\,kE\left[ {u_1^{\prime } \left( {\pi _1 \left( {q_1 ,q_2 ;{\tilde{Y}} } \right) } \right) } \right] <0 \end{aligned}$$

which implies that $V_1 \left( {q_1 ;q_2 } \right)$ is (strictly) submodular in $\left( {q_1 ,q_2 } \right)$ around the point $\left( {{\hat{q}} _1 \left( {q_2 } \right) ,q_2 } \right)$. Together with Lemma 1 (ii), we know that the best response ${\hat{q}} _1 \left( {q_2 } \right)$ is a decreasing function with ${\hat{q}} _1^{\prime } \left( {q_2 } \right) <0$ for $0\le q_2 <\frac{E\left( {{\tilde{Y}} } \right) -c_1 }{k}$ (Athey 2002). Moreover, to estimate the lower bound of ${\hat{q}} _1^{\prime } \left( {q_2 } \right)$, one can easily verify the following inequality:

$$\begin{aligned} \frac{-1}{{\hat{q}} _1^{\prime } \left( {q_2 } \right) }=\frac{E\left[ {-u_1^{{\prime }{\prime }} \left( {\pi _1 \left( {q_1 ,q_2 ;{\tilde{Y}} } \right) } \right) \left( {{\tilde{Y}} -c_1 -\,k\left( {2q_1 +q_2 } \right) } \right) ^{2}} \right] }{kE\left[ {u_1^{\prime } \left( {\pi _1 \left( {q_1 ,q_2 ;{\tilde{Y}} } \right) } \right) } \right] }+2>2 \end{aligned}$$

which implies that ${\hat{q}} _1^{\prime } \left( {q_2 } \right) >-0.5$. Notice that the above results are independent of the curvature parameter $\gamma _1$ and the distribution of the risk factor ${\tilde{Y}}$. Thus, one can similarly derive the results for ${\hat{q}} _2 \left( {q_1 } \right)$.

(ii) Firstly, writing out $V_1^{\prime } \left( {q_1 } \right)$ in terms of the density function $f\left( y \right)$ of ${\tilde{Y}}$, we have

$$\begin{aligned} V_1^{\prime } \left( {q_1 ;q_2 } \right)= & {} \mathop \int \nolimits _{y_1 }^{ y_U } u_1^{\prime } \left( {\pi _1 \left( {q_1 ,q_2 ;y} \right) } \right) \left( {y-c_1 -\,k\left( {2q_1 +q_2 } \right) } \right) f\left( y \right) dy\\&-\mathop \int \nolimits _{y_L }^{ y_1 } u_1^{\prime } \left( {\pi _1 \left( {q_1 ,q_2 ;y} \right) } \right) \left( {c_1 +k\left( {2q_1 +q_2 } \right) -y} \right) f\left( y \right) dy \end{aligned}$$

where $y_1 =c_1 +k\left( {2q_1 +q_2 } \right)$. From the fact that $u_1^{\prime } \left( z \right) =e^{-\gamma _1 z}$, we can calculate the following partial derivative:

$$\begin{aligned} \frac{\partial V_1^{\prime }(q_{1};\gamma _1)}{\partial \gamma _1}= & {} -\mathop \int \nolimits _{y_1 }^{ y_U } u_1^{\prime } \left( {\pi _1 \left( {q_1 ,q_2 ;y} \right) } \right) \pi _1 \left( {q_1 ,q_2 ;y} \right) \left( {y-c_1 -\,k\left( {2q_1 +q_2 } \right) } \right) f\left( y \right) dy\\&+\mathop \int \nolimits _{y_L }^{ y_1 } u_1^{\prime } \left( {\pi _1 \left( {q_1 ,q_2 ;y} \right) } \right) \pi _1 \left( {q_1 ,q_2 ;y} \right) \left( {c_1 +k\left( {2q_1 +q_2 } \right) -y} \right) f\left( y \right) dy \end{aligned}$$

We also estimate this derivative at $V_1^{\prime } \left( {q_1 ;q_2 } \right) =0$. Notice that $\pi _1 \left( {q_1 ,q_2 ;y} \right) =\left( {y-c_1 -\,\,k\left( {q_1 +q_2 } \right) } \right) q_1 >kq_1^2$ for $y>y_1$, while $\pi _1 \left( {q_1 ,q_2 ;y} \right) <kq_1^2$ for $y>y_1$. Thus, the following inequality holds:

$$\begin{aligned}&\frac{\partial V_1^{\prime }\left( {q_1 ;\gamma _1 } \right) }{\partial \gamma _1 }<-\,kq_1^2 \left[ {\mathop \int \nolimits _{y_1 }^{ y_U } u_1^{\prime } \left( {\pi _1 \left( {q_1 ,q_2 ;y} \right) } \right) \left( {y-c_1 -\,k\left( {2q_1 +q_2 } \right) } \right) f\left( y \right) dy}\right. \\&\quad \left. {-\mathop \int \nolimits _{y_L }^{ y_1 } u_1^{\prime } \left( {\pi _1 \left( {q_1 ,q_2 ;y} \right) } \right) \left( {y-c_1 -\,k\left( {2q_1 +q_2 } \right) } \right) f\left( y \right) dy} \right] =-\,kq_1^2 V_1^{\prime } \left( {q_1 } \right) =0 \end{aligned}$$

It then follows that that ${\hat{q}} _1 \left( {q_2 ;\gamma _1 } \right)$ will decrease as $\gamma _1$ increases. Obviously, similar result holds for ${\hat{q}} _2 \left( {q_1 ;\gamma _2 } \right)$. $\square$

Proof of Proposition 2

The existence of the solution $q_1^*$ follows directly from the continuity of the function $G_1 \left( {q_1 } \right)$ and the following two inequalities: $G_1 \left( 0 \right) =-{\hat{q}} _1 \left( {{\hat{q}} _2 \left( {q_1 } \right) } \right) <0$ and $G_1 \left( {q_M } \right) =q_M -{\hat{q}} _1 \left( 0 \right) >0$. Then, the uniqueness of $q_1^*$ is due to the monotonicity of the function $G_1 \left( {q_1 } \right)$ as follows:

$$\begin{aligned} G_1^{\prime } \left( {q_1 } \right) =1-{\hat{q}} _1^{\prime } \left( {{\hat{q}} _2 \left( {q_1 } \right) } \right) {\hat{q}} _2^{\prime } \left( {q_1 } \right) >1-\frac{1}{4}=\frac{3}{4} \end{aligned}$$

Now set $q_2^*={\hat{q}} _2 \left( {q_1^*} \right)$. From Proposition 1 (i), we know that ($q_1^*,q_2^*)$ is the unique Nash equilibrium. $\square$

Proof of Lemma 2

This lemma follows directly from the fact that an affine transformation of random variables will not affect the stochastic dominance relationship between these random variables (see, e.g., Shaked and Shanthikumar 2007).

Proof of Proposition 3

The first part of this proposition can be verified directly following the proof of Proposition 1, because our proving process for that proposition does not rely on the distribution of the random variable ${\tilde{Y}}$. Then, the second part of this proposition follows from Proposition 2. $\square$

Proof of Proposition 4

Given an effective financial hedging (${\tilde{Y}} _{H1} \succ {\tilde{Y}} )$, for any fixed pair $\left( {q_1 ,q_2 } \right)$ we know from Lemma 2 that $\pi _1 \left( {q_1 ,q_2 ;{\tilde{Y}} _{H1} } \right) \succ \pi _1 \left( {q_1 ,q_2 ;{\tilde{Y}} _{H1} } \right)$; thus, for the (strictly) increasing and concave function $u_1 \left( \cdot \right)$, we have $V_{1,H1} \left( {q_1 ,q_2 } \right) =E\left[ {u_1 \left( {\pi _1 \left( {q_1 ,q_2 ;{\tilde{Y}} _{H1} } \right) } \right) } \right] >E\left[ {u_1 \left( {\pi _1 \left( {q_1 ,q_2 ;{\tilde{Y}} } \right) } \right) } \right] =V_1 \left( {q_1 ,q_2 } \right)$ (Shaked and Shanthikumar 2007). It then follows that

$$\begin{aligned} V_{1,H1} \left( {q_{1,H1}^*;q_{2,H1}^*} \right)= & {} \mathop {\max }\limits _{q_1 } V_{1,H1} \left( {q_1 ;{\hat{q}} _2 \left( {q_1 } \right) } \right) \ge V_{1,H1} \left( {q_1^*;{\hat{q}} _2 \left( {q_1^*} \right) } \right) \\> & {} V_1 \left( {q_1^*;{\hat{q}} _2 \left( {q_1^*} \right) } \right) =V_1 \left( {q_1^*;q_2^*} \right) \end{aligned}$$

In other words, the firm value is increased with financial hedging. $\square$

Proof of Proposition 5

First of all, define a new function $g\left( y \right) =u_1^{\prime } \left( {\pi _1 \left( {q_1 ,q_2 ;y} \right) } \right) \left( {y-c_1 -\,k\left( {2q_1 +q_2 } \right) } \right)$. We can calculate its first- and second-order derivatives as follows:

$$\begin{aligned} g'\left( y \right) = u_1^{''}\left( {{\pi _1}\left( {{q_1},{q_2};y} \right) } \right) \left( {y - {c_1} - k\left( {2{q_1} + {q_2}} \right) } \right) {q_1} + u_1'\left( {{\pi _1}\left( {{q_1},{q_2};y} \right) } \right) \end{aligned}$$

and

$$\begin{aligned} g''\left( y \right)= & {} u_1^{'''}\left( {{\pi _1}\left( {{q_1},{q_2};y} \right) } \right) \left( {y - {c_1} - k\left( {2{q_1} + {q_2}} \right) } \right) q_1^2 + 2u_1^{''}\left( {{\pi _1}\left( {{q_1},{q_2};y} \right) } \right) {q_1} \\= & {} u_1^{''}\left( {{\pi _1}\left( {{q_1},{q_2};y} \right) } \right) {q_1}\left[ {2 - {\gamma _1}{\pi _1}\left( {{q_1},{q_2}} \right) + {\gamma _1}kq_1^2} \right] \end{aligned}$$

Notice that $u_1^{{\prime }{\prime }} \left( \cdot \right) <0$. Thus, given Assumption 1 we have ${g}^{{\prime }{\prime }}(y)<0$. It follows that for any effective hedge such that ${\tilde{Y}} _{H1} \succ {\tilde{Y}}$, we have

$$\begin{aligned} \frac{\partial }{\partial q_1 }V_{1,H1} \left( {q_1 ;q_2 } \right) =E\left[ {g\left( {{\tilde{Y}} _{H1} ;q_1 } \right) } \right] >E\left[ {g\left( {{\tilde{Y}} ;q_1 } \right) } \right] =\frac{\partial }{\partial q_1 }V_1 \left( {q_1 ;q_2 } \right) \end{aligned}$$

which implies that ${\hat{q}} _{1,H1} \left( {q_2 } \right) >{\hat{q}} _1 \left( {q_2 } \right)$ (Athey 2002). $\square$

Proof of Proposition 6

Let us define $G_{1,H1} \left( {q_1 } \right) =q_1 -{\hat{q}} _{1,H1} \left( {{\hat{q}} _2 \left( {q_1 } \right) } \right)$. Obviously, $G_{1,H1} \left( {q_{1,H1}^*} \right) =0$. Besides, from Proposition 5 we know that under Assumption 1, ${\hat{q}} _{1,H1} \left( {q_2 } \right) >{\hat{q}} _1 \left( {q_2 } \right)$. Combined with the fact that ${\hat{q}} _{1,H1} \left( q \right)$ is a decreasing function, we have

$$\begin{aligned} G_{1,H1} \left( {q_1^*} \right) =q_1^*-{\hat{q}} _{1,H1} \left( {{\hat{q}} _2 \left( {q_1^*} \right) } \right) <q_1^*-{\hat{q}} _1 \left( {{\hat{q}} _2 \left( {q_1^*} \right) } \right) =0 \end{aligned}$$

In addition, from Proposition 3 we have $G_{1,H1}^{\prime } \left( q \right) =1-{\hat{q}} _{1,H1}^{\prime } \left( {{\hat{q}} _2 \left( q \right) } \right) {\hat{q}} _2^{\prime } \left( q \right)>1-\frac{1}{4}>0$. Taken together, we must have $q_{1,H1}^*>q_1^*$.

Similarly, we can define $G_{2,H1} \left( {q_2 } \right) =q_2 -{\hat{q}} _2 \left( {{\hat{q}} _{1,H1} \left( {q_2 } \right) } \right)$. Under Assumption 1, it is easy to verify that

$$\begin{aligned} G_{2,H1} \left( {q_2^*} \right) =q_2^*-{\hat{q}} _2 \left( {{\hat{q}} _{1,H1} \left( {q_2^*} \right) } \right) >q_2^*-{\hat{q}} _2 \left( {q_1^*} \right) =0 \end{aligned}$$

and $G_{2,H1}^{\prime } \left( q \right) >0$. It then follows that $q_{2,H1}^*<q_2^*$.

Further, by definition we have $q_2^*={\hat{q}} _2 \left( {q_1^*} \right)$ and $q_{2,H1}^*={\hat{q}} _2 \left( {q_{1,H1}^*} \right)$. From the Lagrange mean value theorem, there exists a ${\bar{q}} _1 \in \left( {q_1^*,q_{1,H1}^*} \right)$ such that $q_{2,H1}^*-q_2^*={\hat{q}} _2^{\prime } \left( {{\bar{q}} _1 } \right) \left( {q_{1,H1}^*-q_1^*} \right)$. From proposition 1, ${\hat{q}} _2^{\prime } \left( {{\bar{q}} _1 } \right) \in \left( {-0.5,0} \right)$, which means that $q_{2,H1}^*-q_2^*>-0.5\left( {q_{1,H1}^*-q_1^*} \right)$. It then follows that

$$\begin{aligned} q_{1,H1}^*+q_{2,H1}^*-\left( {q_1^*+q_2^*} \right) =\left( {q_{1,H1}^*-q_1^*} \right) +\left( {q_{2,H1}^*-q_2^*} \right) >\frac{1}{2}\left( {q_{1,H1}^*-q_1^*} \right) \end{aligned}$$

Under Assumption 1, we already know that $q_{1,H1}^*>q_1^*$; thus, $q_{1,H1}^*+q_{2,H1}^*-\left( {q_1^*+q_2^*} \right) >0$. $\square$

Proof of Proposition 7

Let us define a function as follows: $G_H \left( q \right) =q-{\hat{q}} _H \left( q \right)$. Obviously, $G_H \left( {q_H^*} \right) =0$. Under Assumption 1, from Proposition 5 we know that ${\hat{q}} _H \left( {q_1^*} \right) >{\hat{q}} \left( {q_1^*} \right) =q_2^*$. Thus,

$$\begin{aligned} G_H \left( {q_1^*} \right) =q_1^*-{\hat{q}} _H \left( {q_1^*} \right) <q_1^*-q_2^*=0 \end{aligned}$$

Together with the inequality that $G_H^{\prime } \left( q \right) =1-{\hat{q}} _H^{\prime } \left( q \right) >0$, we have $q_H^*>q^{*}$.

Then, it is straightforward to verify the following:

$$\begin{aligned} V_H \left( {q_H^*} \right) =\mathop {\max }\limits _q V_H \left( q \right) \ge V_H \left( {q^{*}} \right) >V\left( {q^{*}} \right) \end{aligned}$$

which is just analogous to Proposition 4. $\square$