Product quality and endogenous firm objectives

Abstract

This paper examines the endogenous operational objectives of firms in a Cournot duopoly under the consideration of vertical product differentiation. We consider a three-stage game, where the owners of the duopoly simultaneously decide the operational objectives of their managers to be profit maximization or revenue maximization at the first stage, and then, based on the objectives, the managers simultaneously choose the quality of their products at the second stage, and compete in quantities at the final stage. The equilibrium results show that, for high heterogeneity of consumers’ quality tastes or restrictive quality improving technology, both firms choose to maximize revenues and produce goods with identical quality. For low heterogeneity of consumers’ quality tastes or superior quality improving technology, both firms choose to maximize profits and produce different quality. For the intermediate level of the heterogeneity of quality tastes and quality improving technology, both firms choose different objectives, and the firm maximizing revenues provides a high-quality product, while the firm maximizing profits provides a low-quality one.

Appendix 1

The profits of firms in each case can be written as:

$$\varPi_{1}^{\text{PP}} = \frac{{\overline{s} }}{{128\overline{\theta } }}\left[ {4757\overline{s}^{2} + 18\overline{\theta } \left( {3a + 10\overline{\theta } } \right) - 3\overline{s} \left( {71a + 660\overline{\theta } } \right)} \right],$$

$$\varPi_{2}^{\text{PP}} = \frac{{\overline{s} }}{{128\overline{\theta } }}\left[ {1583\overline{s}^{2} + 2\left( {a - 2\overline{\theta } } \right)\overline{\theta } - \overline{s} \left( {71a + 292\overline{\theta } } \right)} \right],$$

$$\varPi_{1}^{\text{RR}} = \frac{1}{18}\overline{\theta } \left( {2\overline{\theta } - 3} \right),$$

$$\varPi_{2}^{\text{RR}} = \frac{1}{18}\overline{\theta } \left( {2\overline{\theta } - 3} \right),$$

$$\varPi_{1}^{\text{RP}} = \frac{{3\overline{s} }}{{32\overline{\theta } }}\left[ {456\overline{s}^{2} + 3\overline{\theta } \left( {3b + 5\overline{\theta } } \right) - 2\overline{s} \left( {19b + 87\overline{\theta } } \right)} \right],$$

$$\varPi_{2}^{\text{RP}} = \frac{{\overline{s} }}{{32\overline{\theta } }}\left[ {432\overline{s}^{2} + \left( {b - \overline{\theta } } \right)\overline{\theta } - 36\overline{s} \left( {b + 2\overline{\theta } } \right)} \right],$$

$$\varPi_{1}^{\text{PR}} = \frac{{\overline{s} }}{{32\overline{\theta } }}\left[ {432\overline{s}^{2} + \left( {b - \overline{\theta } } \right)\overline{\theta } - 36\overline{s} \left( {b + 2\overline{\theta } } \right)} \right],$$

$$\varPi_{2}^{\text{PR}} = \frac{{3\overline{s} }}{{32\overline{\theta } }}\left[ {456\overline{s}^{2} + 3\overline{\theta } \left( {3b + 5\overline{\theta } } \right) - 2\overline{\theta } \left( {19b + 87\overline{\theta } } \right)} \right].$$

Assume $k = \frac{{\overline{\theta } }}{{\overline{s} }}$, where $1.5 < k < 2.2$. Then, substituting the above into (7)–(12), the equilibrium can be derived as follows.

The condition for Case (PP) to be the equilibrium:

$$\begin{aligned} \varPi_{1}^{\text{PP}} - \varPi_{1}^{\text{RP}} & = \frac{{\overline{s}^{2} }}{128k}\left[ { - 715 + 456\sqrt {\left( {36 - k} \right)\left( {4 - k} \right)} - 213\sqrt {\left( {71 - 2k} \right)\left( {7 - 2k} \right)} } \right. \\ & \quad \left. { + 54k\left( {2 - 2\sqrt {\left( {36 - k} \right)\left( {4 - k} \right)} + \sqrt {\left( {71 - 2k} \right)\left( {7 - 2k} \right)} } \right)} \right] > 0, \\ \end{aligned}$$

where $\varPi_{1}^{\text{PP}} - \varPi_{1}^{\text{RP}} > 0$ if $1.5 < k < 1.529$:

$$\begin{aligned} \varPi_{2}^{\text{PP}} - \varPi_{2}^{\text{PR}} & = \frac{{\overline{s}^{2} }}{1152k}\left[ {14311 - 639\sqrt {\left( {71 - 2k} \right)\left( {7 - 2k} \right)} - 2k\left( {1282 + 82k} \right.} \right. \\ & \quad \left. {\left. { - 9\sqrt {\left( {71 - 2k} \right)\left( {7 - 2k} \right)} } \right)} \right] > 0, \\ \end{aligned}$$

where $\varPi_{2}^{\text{PP}} - \varPi_{2}^{\text{PR}} > 0$ if $1.5 < k < 1.531$.

Accordingly, Case (PP) is the equilibrium if $1.5 < k < 1.529$.

The condition for Case (RP) to be the equilibrium.

Because $\varPi_{1}^{\text{RP}} - \varPi_{1}^{\text{PP}} = - \left( {\varPi_{1}^{\text{PP}} - \varPi_{1}^{\text{RP}} } \right)$, we may obtain $\varPi_{1}^{\text{RP}} - \varPi_{1}^{\text{PP}} > 0$ when $1.529 < k < 2.2$. Besides, no deviation condition for Firm 2 is written as:

$$\begin{aligned} \varPi_{2}^{RP} - \varPi_{2}^{RR} & = \frac{{\overline{s}^{2} }}{288k}\left\{ {\left[ {9\sqrt {\left( {36 - k} \right)\left( {4 - k} \right)} - 41k - 600} \right]k} \right. \\ & \quad \left. { - 324\left[ {\sqrt {\left( {36 - k} \right)\left( {4 - k} \right)} - 12} \right]} \right\} > 0, \\ \end{aligned}$$

where $\varPi_{2}^{\text{RP}} - \varPi_{2}^{\text{RR}} > 0$ if $1.5 < k < 2.031$.

Accordingly, Case (RP) is the equilibrium, if $1.529 < k < 2.031$.

The condition for Case (PR) to be the equilibrium.

Because of $\varPi_{1}^{\text{PR}} - \varPi_{1}^{\text{RR}} = \varPi_{2}^{\text{RP}} - \varPi_{2}^{\text{RR}}$ and $\varPi_{2}^{\text{PR}} - \varPi_{2}^{\text{PP}} = \varPi_{1}^{\text{RP}} - \varPi_{1}^{\text{PP}}$, the condition of Case (PR) to be the SPNE is the same with that of Case (RP): $1.529 < k < 2.031$.

The condition for Case (RR) to be the equilibrium.

First, $\varPi_{1}^{\text{RR}} - \varPi_{1}^{\text{PR}} = \frac{{\left( { - 2 + k} \right)\overline{s}^{2} }}{18k} > 0$, where $\varPi_{1}^{\text{RR}} - \varPi_{1}^{\text{PR}} > 0$ if $2 < k < 2.2$.

Besides, since $\varPi_{2}^{\text{RR}} - \varPi_{2}^{\text{RP}} = - \left( {\varPi_{2}^{\text{RP}} - \varPi_{2}^{\text{RR}} } \right)$, we obtain $\varPi_{2}^{\text{RR}} - \varPi_{2}^{\text{RP}} > 0$ if $2.031 < k < 2.2$.

Accordingly, Case (RR) is the equilibrium if $2.031 < k < 2.2$.