Two-dimensional vertical differentiation with attribute dependence

Abstract

This paper studies how the type and magnitude of the interaction between quality characteristics determine the nature of the differentiation strategies of firms. The vertical differentiation models with multi-characteristic product preferences consider that the characteristics are independent. We incorporate a non-additive component into a two-dimensional model. Sufficiently high attribute dependence gives rise to maximal differentiation in both qualities: if the qualities are complements, one firm offers the highest quality in both attributes, while the other firm the lowest; on the other hand, if the qualities are substitutes, each firm specializes in one of the qualities. The usual outcome with maximal differentiation in one quality dimension and minimal in the other is achieved if there is low or no attribute dependence. In our model, the magnitude of attribute dependence determines whether differentiation relaxes or intensifies subsequent price competition. Some examples from the banking sector are presented to illustrate the model.

Appendix

Demands in each possible product configuration

Indifferent consumers given by (2) are represented by a line that is increasing in the asymmetric case and decreasing in the dominant product case. There are four cases for the dominated characteristic framework, being Product 1 dominated by Product 2 (i.e. $x_{1}<x_{2}$ and $y_{1}\le y_{2}$):

• Case i.1. Consumers buy Product 2 only if they have a high willingness to pay for quality in both dimensions. The demand functions are given by

  $$\begin{aligned} D_{1}=1-\int _{\theta _{x}(\theta _{y}=1)}^{1}\int _{{\hat{\theta }} _{y}(\theta _{x})}^{1}d\theta _{y}d\theta _{x}=1-\frac{1}{2}\left[ 1-\theta _{x}(\theta _{y}=1)\right] \left[ 1-\theta _{y}(\theta _{x}=1)\right] \end{aligned}$$

• Case i.2. Characteristic x dominance: All consumers with $\theta _{x}=0$ buy product 1 and all consumers with $\theta _{x}=1$ buy product 2. The demand function for firm 1 is given by

  $$\begin{aligned} D_{1}= & {} \int _{0}^{\theta _{x}(\theta _{y}=1)}\int _{0}^{1}d\theta _{y}d\theta _{x}+\int _{\theta _{x}(\theta _{y}=1)}^{\theta _{x}(\theta _{y}=0)}\int _{0}^{{\hat{\theta }}_{y}(\theta _{x})}d\theta _{y}d\theta _{x}\\= & {} \frac{1}{2}\left[ \theta _{x}(\theta _{y}=0)+\theta _{x}(\theta _{y}=1) \right] . \end{aligned}$$

• Case i.3. Characteristic y dominance: All consumers with $\theta _{y}=0$ buy product 1 and all consumers with $\theta _{y}=1$ buy product 2. The demand function for firm 1 is given by

  $$\begin{aligned} D_{1}= & {} \int _{0}^{1}\int _{0}^{{\hat{\theta }}_{y}(\theta _{x})}d\theta _{y}d\theta _{x}\\= & {} \frac{1}{2}\left[ \theta _{y}(\theta _{x}=0)+\theta _{y}(\theta _{x}=1)\right] . \end{aligned}$$

• Case i.4. Consumers buy Product 1 only if they have a low willingness to pay for quality in both dimensions. The demand function for firm 1 is given by

  $$\begin{aligned} D_{1}=\int _{0}^{\theta _{x}(\theta _{y}=0)}\int _{0}^{{\hat{\theta }} _{y}(\theta _{x})}d\theta _{y}d\theta _{x}=\frac{1}{2}\left[ \theta _{x}(\theta _{y}=0)\right] \left[ \theta _{y}(\theta _{x}=0)\right] . \end{aligned}$$

There are another four cases for the asymmetric characteristic framework (i.e. $x_{1}>x_{2}$ and $y_{1}<y_{2}$):

• Case ii.1. Consumers buy Product 1 only if they have a low willingness to pay for quality y and a high willingness to pay for quality x. The demand function for firm 1 is given by

  $$\begin{aligned} D_{1}=\int _{0}^{\theta _{x}(\theta _{y}=0)}\int _{0}^{{\hat{\theta }} _{y}(\theta _{x})}d\theta _{y}d\theta _{x}=\frac{1}{2}\left[ \theta _{x}(\theta _{y}=0)\right] \left[ \theta _{y}(\theta _{x}=0)\right] \end{aligned}$$

• Case ii.2. Characteristic x dominance: All consumers with $\theta _{x}=0$ buy product 2 and all consumers with $\theta _{x}=1$ buy product 1. The demand function for firm 1 is given by

  $$\begin{aligned} D_{1}= & {} \int _{\theta _{x}(\theta _{y}=0)}^{\theta _{x}(\theta _{y}=1)}\int _{0}^{{\hat{\theta }}_{y}(\theta _{x})}d\theta _{y}d\theta _{x}+\int _{\theta _{x}(\theta _{y}=1)}^{1}\int _{0}^{1}d\theta _{y}d\theta _{x}\\= & {} \frac{1}{2}\left[ 1-\theta _{x}(\theta _{y}=0)+1-\theta _{x}(\theta _{y}=1) \right] \end{aligned}$$

• Case ii.3. Characteristic y dominance: All consumers with $\theta _{y}=0$ buy product 1 and all consumers with $\theta _{y}=1$ buy product 2. The demand function for firm 1 is given by

  $$\begin{aligned} D_{1}=\int _{0}^{1}\int _{0}^{{\hat{\theta }}_{y}(\theta _{x})}d\theta _{y}d\theta _{x}=\frac{1}{2}\left[ \theta _{y}(\theta _{x}=0)+\theta _{y}(\theta _{x}=1)\right] \end{aligned}$$

• Case ii.4. Consumers buy Product 2 only if they have a low willingness to pay for quality x and a high willingness to pay for quality y. The demand function for firm 2 is given by

  $$\begin{aligned} D_{1}=1-\int _{0}^{\theta _{x}(\theta _{y}=1)}\int _{0}^{{\hat{\theta }} _{y}(\theta _{x})}d\theta _{y}d\theta _{x}=1-\frac{1}{2}\left[ 1-\theta _{x}(\theta _{y}=1)\right] \left[ 1-\theta _{y}(\theta _{x}=0)\right] \end{aligned}$$

Demands, prices and profits in each possible pricing equilibria

Consider first the dominated characteristics competition where, for example, $x_{1}<x_{2}$ and $y_{1}\le y_{2}$ (of course, symmetric results can be obtained for $x_{1}\le x_{2}$ and $y_{1}<y_{2}).$ Given these assumptions regarding the product positions, the area below the separating line given by (2) represents firm 1’s demand and the area above represents firm 2’s demand.

Case i.1 [$\theta _{x}(\theta _{y}=0)\ge 1,0\le \theta _{x}(\theta _{y}=1)\le 1,\theta _{y}(\theta _{x}=0)\ge 1$ and $0\le \theta _{y}(\theta _{x}=1)\le 1$]. The demand functions are given by

$D_{1}=1-D_{2}$ with $D_{2}=\frac{1}{2d_{x}d_{y}}\left( p_{1}-p_{2}+d_{y}+d_{x}+\gamma \varDelta \right) ^{2}.$

This expression is defined over the interval

$$\begin{aligned} p_{2}\in \left[ p_{1}+d_{y}+d_{x}+\gamma \varDelta -\sqrt{ 2d_{x}d_{y}},p_{1}+d_{y}+d_{x}+\gamma \varDelta \right] \end{aligned}$$

The two limits of this interval are obtained when $D_{2}=1$ and $D_{2}=0.$

After solving the FOCs for prices we obtain second stage optimal prices

$$\begin{aligned} {\hat{p}}_{1}= & {} \frac{5}{8}a+\frac{3}{8}A\hbox { and } {\hat{p}}_{2}=-\frac{1}{8}a+\frac{ 1}{8}A\hbox { being } a=-d_{x}-d_{y}-\gamma \varDelta ,\hbox { and }\\ A= & {} \sqrt{\left( -d_{x}-d_{y}-\gamma \varDelta \right) ^{2}+8d_{x}d_{y}}. \end{aligned}$$

Then the condition for an equilibrium to exist is given by

$$\begin{aligned} \gamma \le -\frac{d_{x}+d_{y}}{\varDelta } \end{aligned}$$

Finally, second stage optimal profits are

$$\begin{aligned} {\hat{\varPi }}_{1}={\hat{p}}_{1}(1-\frac{2\left( {\hat{p}}_{2}\right) ^{2}}{ d_{x}d_{y}})\hbox { and }{\hat{\varPi }}_{2}=\frac{2\left( {\hat{p}}_{2}\right) ^{3}}{ d_{x}d_{y}}. \end{aligned}$$

Case i.2 [$0\le \theta _{x}(\theta _{y}=1)\le \theta _{x}(\theta _{y}=0)\le 1,\theta _{y}(\theta _{x}=0)\ge 1$ and $\theta _{y}(\theta _{x}=1)\le 0$]. The demand functions are given by $D_{2}=1-D_{1}$and

$D_{1}=\frac{1}{2}\frac{1}{d_{x}}\left( -2p_{1}+2p_{2}-2\gamma \varDelta -d_{y}\right)$.

This expression is defined over the interval

$$\begin{aligned} p_{1}\in \left[ p_{2}+\frac{1}{2}(-d_{y}-2\gamma \varDelta -2d_{x}),p_{2}+ \frac{1}{2}(d_{y}+2\gamma \varDelta )\right] . \end{aligned}$$

The two limits of this interval are obtained when $D_{2}=1$ and $D_{2}=0.$

After solving the FOCs for prices we obtain second stage optimal prices

$$\begin{aligned} {\hat{p}}_{1}=\frac{1}{6}\left( 2d_{x}-d_{y}-2\gamma \varDelta \right) \hbox { and } {\hat{p}}_{2}=\frac{1}{6}\left( 4d_{x}+d_{y}+2\gamma \varDelta \right) \end{aligned}$$

Then the condition for an equilibrium to exist is given by

$$\begin{aligned} \gamma \in \left[ \frac{-4d_{x}-d_{y}}{2\varDelta },\frac{2d_{x}-d_{y}}{ 2\varDelta }\right] . \end{aligned}$$

Finally, second stage optimal profits are

$$\begin{aligned} {\hat{\varPi }}_{1}=\frac{\left( 2d_{x}-d_{y}-2\gamma \varDelta \right) ^{2}}{36d_{x}}\hbox { and }{\hat{\varPi }}_{2}=\frac{\left( -4d_{x}-d_{y}-2\gamma \varDelta \right) ^{2}}{36d_{x}}. \end{aligned}$$

Case i.3 [$\theta _{x}(\theta _{y}=0)\ge 1,\theta _{x}(\theta _{y}=1)\le 0,0\le \theta _{y}(\theta _{x}=1)\le$$\theta _{y}(\theta _{x}=0)\le 0$]. This case is symmetric to case i.2. The demand functions are given by $D_{2}=1-D_{1}$ with $D_{1}=\frac{1}{d_{y}} \left( -p_{1}+p_{2}-\gamma \varDelta -d_{x}\right)$

This expression is defined over the interval

$$\begin{aligned} p_{1}\in \left[ p_{2}+\frac{1}{2}(-d_{x}-2d_{y}-2\gamma \varDelta ),p_{2}+ \frac{1}{2}(-d_{x}-2\gamma \varDelta )\right] . \end{aligned}$$

The two limits of this interval are obtained when $D_{2}=1$ and $D_{2}=0.$

After solving the FOCs for prices we obtain second stage optimal prices

$$\begin{aligned} {\hat{p}}_{1}=\frac{1}{6}\left( -d_{x}+2d_{y}-2\gamma \varDelta \right) \hbox { and }{\hat{p}}_{2}=\frac{1}{6}\left( d_{x}+4d_{y}+2\gamma \varDelta \right) \end{aligned}$$

Then the condition for an equilibrium to exist is given by

$$\begin{aligned} \gamma \in \left[ \frac{-d_{x}+4d_{y}}{2\varDelta },\frac{-d_{x}+2d_{y}}{ 2\varDelta }\right] \end{aligned}$$

Finally, second stage optimal profits are

$$\begin{aligned} {\hat{\varPi }}_{1}=\frac{\left( -d_{x}+2d_{y}-2\gamma \varDelta \right) ^{2}}{36d_{y}}\hbox { and }{\hat{\varPi }}_{2}=\frac{\left( -d_{x}-4d_{y}-2\gamma \varDelta \right) ^{2}}{36d_{y}} \end{aligned}$$

Case i.4 [$0\le \theta _{x}(\theta _{y}=0)\le 1,\theta _{x}(\theta _{y}=1)\le 0,0\le$$\theta _{y}(\theta _{x}=0)\le 1$ and $\theta _{y}(\theta _{x}=1)\le$ 0]. The demand functions are given by $D_{2}=1-D_{1}$ with $D_{1}=\frac{1}{2}\frac{d_{x}}{d_{y}}(\frac{1}{-d_{x}} \left( p_{1}-p_{2}+\gamma \varDelta \right) )^{2}.$

This expression is defined over the interval

$$\begin{aligned} p_{1}\in \left[ \max \left\{ p_{2}-d_{y}-\gamma \varDelta ,p_{2}-d_{x}-\gamma \varDelta \right\} ,p_{2}-\gamma \varDelta \right] \end{aligned}$$

The two limits of this interval are obtained when $D_{2}=1$ and $D_{2}=0.$

After solving the FOCs for prices we obtain second stage optimal prices

$$\begin{aligned} {\hat{p}}_{1}=\frac{1}{8}a+\frac{1}{8}A\hbox { and }{\hat{p}}_{2}=-\frac{5}{8}a+\frac{ 3}{8}A being a=-\gamma \varDelta \hbox { and }A=\sqrt{\gamma ^{2}\varDelta ^{2}+8d_{x}d_{y}}. \end{aligned}$$

Then the condition for an equilibrium to exist is given by

$$\begin{aligned} \gamma \ge \max \left\{ \frac{d_{y}-2d_{x}}{\varDelta },\frac{d_{x}-2d_{y}}{ \varDelta }\right\} \end{aligned}$$

Finally, second stage optimal profits are

$$\begin{aligned} {\hat{\varPi }}_{1}=\frac{2{\hat{p}}_{1}^{3}}{d_{x}d_{y}}\hbox { and } {\hat{\varPi }}_{2}=\hat{p }_{2}\left( 1-\frac{1}{2}\frac{d_{x}}{d_{y}}\left( \frac{1}{-d_{x}}\left( {\hat{p}}_{1}- {\hat{p}}_{2}+\gamma \varDelta \right) \right) ^{2}\right) . \end{aligned}$$

Consider now the asymmetric characteristics competition where, for example, $x_{1}>x_{2}$ and $y_{1}<y_{2}$ (of course, symmetric results can be obtained for $x_{1}<x_{2}$ and $y_{1}>y_{2}).$ Given these assumptions regarding the product positions, the area on the right to the separating line given by (2) represents firm 1’s demand and the area on the left represents firm 2’s demand.

Case ii.1 [$0\le \theta _{x}(\theta _{y}=0)\le 1,\theta _{x}(\theta _{y}=1)\ge 1,\theta _{y}(\theta _{x}=0)\le 0$ and $0\le \theta _{y}(\theta _{x}=1)\le 1$]. The demand functions are given by $D_{2}=1-D_{1}$ with $D_{1}=\frac{1}{-2d_{x}d_{y}}\left( -p_{1}+p_{2}-\gamma \varDelta -d_{x}\right) ^{2}$

This expression is defined for

$\left( -p_{1}+p_{2}-\gamma \varDelta -d_{x}\right) ^{2}\le -2d_{x}d_{y}$ or alternatively over the interval

$$\begin{aligned} p_{1}\in [\max \left\{ p_{2}-d_{y}-\gamma \varDelta -d_{x},p_{2}-\gamma \varDelta \right\} ,p_{2}-d_{x}-\gamma \varDelta ]. \end{aligned}$$

The two limits of this interval are obtained when $D_{2}=1$ and $D_{2}=0.$

After solving the FOCs for prices we obtain second stage optimal prices

${\hat{p}}_{1}=\frac{1}{8}a+\frac{1}{8}A$ and ${\hat{p}}_{2}=-\frac{5}{8}a+\frac{ 3}{8}A$ being $a=-d_{x}-\gamma \varDelta$ and

$$\begin{aligned} A=\sqrt{(-d_{x}-\gamma \varDelta )^{2}-8d_{x}d_{y}} \end{aligned}$$

Then the condition for an equilibrium to exist is given by

$$\begin{aligned} \frac{1}{16}\left( -d_{x}-\gamma \varDelta +\sqrt{\left( -d_{x}+\gamma \varDelta \right) ^{2}-8d_{x}d_{y}}\right) ^{2}\le -2d_{x}d_{y} \end{aligned}$$

Finally, second stage optimal profits are

$$\begin{aligned} {\hat{\varPi }}_{1}= & {} {\hat{p}}_{1}\frac{1}{-2d_{x}d_{y}}\left( -{\hat{p}}_{1}+{\hat{p}} _{2}-\gamma \varDelta -d_{x}\right) ^{2}\hbox { and }\\ {\hat{\varPi }}_{2}= & {} {\hat{p}}_{2}(1- \frac{1}{-2d_{x}d_{y}}\left( -{\hat{p}}_{1}+{\hat{p}}_{2}-\gamma \varDelta -d_{x}\right) ^{2}) \end{aligned}$$

Case ii.2 [$0\le \theta _{x}(\theta _{y}=0)\le \theta _{x}(\theta _{y}=1)\le 1,$$\theta _{y}(\theta _{x}=0)\le 0$ and $\theta _{y}(\theta _{x}=1)\ge 1$]. The demand functions are given by $D_{2}=1-D_{1}$ with $D_{1}=\frac{1}{2}\frac{1}{-d_{x}}\left( -2p_{1}+2p_{2}-2d_{x}-d_{y}-2\gamma \varDelta \right)$

This expression is defined over the interval

$$\begin{aligned} p_{1}\in \left[ p_{2}-\frac{1}{2}\left( 2d_{x}+d_{y}+2\gamma \varDelta \right) ,p_{2}-\frac{1}{2}\left( d_{y}+2\gamma \varDelta \right) \right] . \end{aligned}$$

The two limits of this interval are obtained when $D_{2}=1$ and $D_{2}=0.$

After solving the FOCs for prices we obtain second stage optimal prices

${\hat{p}}_{1}=\frac{1}{6}\left( -4d_{x}-d_{y}-2\gamma \varDelta \right)$ and ${\hat{p}}_{2}=\frac{1}{6}\left( -2d_{x}+d_{y}+2\gamma \varDelta \right)$

Then the condition for an equilibrium to exist is given by

$$\begin{aligned} \frac{1}{2}\left( 2d_{x}-d_{y}\right) \le \gamma \varDelta \le \frac{1}{2} \left( -4d_{x}-d_{y}\right) \end{aligned}$$

(note that in the asymmetric characteristic cases $x_{2}-x_{1}<0$)

Finally, second stage optimal profits are

$$\begin{aligned} {\hat{\varPi }}_{1}=\frac{\left( -4d_{x}-d_{y}-2\gamma \varDelta \right) ^{2}}{-36d_{x}}\hbox { and }{\hat{\varPi }}_{2}=\frac{\left( 2d_{x}-d_{y}-2\gamma \varDelta \right) ^{2}}{-36d_{x}}\end{aligned}$$

Case ii.3 [$\theta _{x}(\theta _{y}=0)\le 0,\theta _{x}(\theta _{y}=1)\ge 1,0\le \theta _{y}(\theta _{x}=0)\le \theta _{y}(\theta _{x}=1)\le 1$]. This case is symmetric to Case ii.2. The demand functions are given by $D_{2}=1-D_{1}$ with $D_{1}=\frac{1}{2d_{y}}\left( 2p_{2}-2p_{1}-d_{x}-2\gamma \varDelta \right)$

This expression is defined over the interval

$$\begin{aligned} p_{1}\in \left[ p_{2}-\frac{1}{2}\left( d_{x}+2d_{y}+2\gamma \varDelta \right) ,p_{2}-\frac{1}{2}\left( d_{x}+2\gamma \varDelta \right) \right] \end{aligned}$$

The two limits of this interval are obtained when $D_{2}=1$ and $D_{2}=0.$

After solving the FOCs for prices we obtain second stage optimal prices

$$\begin{aligned} {\hat{p}}_{1}=\frac{1}{6}\left( -d_{x}+2d_{y}-2\gamma \varDelta \right) \hbox { and }{\hat{p}}_{2}=\frac{1}{6}\left( d_{x}+4d_{y}+2\gamma \varDelta \right) \end{aligned}$$

Then the condition for an equilibrium to exist is given by

$$\begin{aligned} \frac{1}{2}\left( -d_{x}-4d_{y}\right) \le \gamma \varDelta \le \frac{1}{2}\left( -d_{x}+2d_{y}\right) \end{aligned}$$

Finally, second stage optimal profits are

$$\begin{aligned} {\hat{\varPi }}_{1}=\frac{\left( -d_{x}+2d_{y}-2\gamma \varDelta \right) ^{2}}{36d_{y}}\hbox { and }{\hat{\varPi }}_{2}=\frac{\left( -d_{x}-4d_{y}-2\gamma \varDelta \right) ^{2}}{36d_{y}}\end{aligned}$$

Case ii.4 [$\theta _{x}(\theta _{y}=0)\le 0,0\le \theta _{x}(\theta _{y}=1)\le 1,0\le$$\theta _{y}(\theta _{x}=0)\le 1$ and $\theta _{y}(\theta _{x}=1)\ge 1$]. This case is symmetric to case ii.1. The demand functions are given by $D_{1}=1-\frac{1}{-2d_{x}d_{y}}\left( p_{1}-p_{2}+d_{y}+\gamma \varDelta \right) ^{2}$

This expression is defined for

$$\begin{aligned} \left( p_{1}-p_{2}+d_{y}+\gamma \varDelta \right) ^{2}\le -2d_{x}d_{y} \end{aligned}$$

that is obtained when $D_{2}=1$ and $D_{2}=0.$

After solving the FOCs for prices we obtain second stage optimal prices ${\hat{p}}_{1}=\frac{5}{8}a+\frac{3}{8}A$ and ${\hat{p}}_{2}=-\frac{1}{8}a+\frac{ 1}{8}A$ being $a=$$\left( -d_{y}-\gamma \varDelta \right)$ and $A=\sqrt{ (d_{y}+\gamma \varDelta )^{2}-8d_{x}d_{y}}.$

Then the condition for an equilibrium to exist is given by

$$\begin{aligned} \frac{1}{16}\left( d_{y}+\gamma \varDelta +\sqrt{(-d_{y}+\gamma \varDelta )^{2}-8d_{x}d_{y}}\right) ^{2}\le -2d_{x}d_{y} \end{aligned}$$

Finally, second stage optimal profits are

$$\begin{aligned} {\hat{\varPi }}_{1}= & {} {\hat{p}}_{1}\frac{1}{-2d_{x}d_{y}}\left( -{\hat{p}}_{1}+{\hat{p}} _{2}-\gamma \varDelta -d_{y}\right) ^{2}\hbox { and }\\ {\hat{\varPi }}_{2}= & {} {\hat{p}}_{2}\left[ 1-\frac{1}{-2d_{x}d_{y}}\left( -{\hat{p}}_{1}+ {\hat{p}}_{2}-\gamma \varDelta -d_{y}\right) ^{2}\right] . \end{aligned}$$

Proof of Proposition 1

Case i.1 The first stage is solved by decomposing the effect of a marginal increase in quality on own profits into the direct and strategic effect (see for example Degryse and Irmen 2001a, p. 558). Formally, for quality $x_{1},$we have that

$$\begin{aligned} \frac{d{\hat{\varPi }}_{1}}{dx_{1}}={\hat{p}}_{1}\frac{\partial D_{1}}{\partial x_{1}}+{\hat{p}}_{1}\frac{\partial D_{1}}{\partial p_{2}}\frac{d{\hat{p}}_{2}}{ dx_{1}} \end{aligned}$$

and then, the total effect is given by the sign of

$$\begin{aligned} d_{x}\left( \gamma y_{1}+1\right) \left( 7A+a\right) -\frac{1}{2}\left( A-a\right) \left( 3A+a\right) . \end{aligned}$$

Following the same procedure for the other qualities, we obtain that the sign of the total effects are given by $d_{y}\left( \gamma x_{1}+1\right) \left( 7A+a\right) -\frac{1}{2}\left( A-a\right) \left( 3A+a\right)$ for $y_{1}.$ Symmetrically, for firm 2 we have that

$$\begin{aligned} \frac{d{\hat{\varPi }}_{2}}{dx_{2}}={\hat{p}}_{2}\frac{\partial D_{2}}{\partial x_{2}}+p_{2}\frac{\partial D_{2}}{\partial p_{1}}\frac{d{\hat{p}}_{1}}{dx_{2}} \end{aligned}$$

and then the sign of total effects is determined by

$$\begin{aligned}&3d_{x}\left( \gamma y_{2}+1\right) \left( A-a\right) +\frac{1}{2}\left( A-a\right) \left( 5A+3a\right) \hbox { for }x_{2},\hbox { and}\\&3d_{y}\left( \gamma x_{2}+1\right) \left( A-a\right) +\frac{1}{2}\left( A-a\right) \left( A+3a\right) \hbox { for }y_{2}. \end{aligned}$$

After studying the sign of the total effects as a function of the value of $\gamma$, a candidate to equilibrium is found for $x_{1}=0,y_{1}=0,x_{2}=1,y_{2}=1$ with $\gamma \in (-1,\frac{3}{2}\sqrt{2} -2).$ However this candidate does not correspond to this case because it is not compatible with the characteristics that define this case, namely with $\theta _{x}(\theta _{y}=1)>0$ and $\theta _{y}(\theta _{x}=1)>0.$ Consequently, there is no equilibrium candidate in this case i.1.

Case i.2 Following the same procedure we obtain the sign of the total effects for every characteristic:

$$\begin{aligned}&\frac{1}{6}\frac{2d_{x}+d_{y}+2\gamma x_{1}y_{1}-4\gamma x_{2}y_{1}+2\gamma x_{2}y_{2}}{d_{x}^{2}}\hbox { for }x_{1},\\&\frac{1}{3}\frac{2\gamma x_{1}+1}{d_{x}}\hbox { for }y_{1},\\&\frac{1}{6}\frac{4d_{x}-d_{y}+2\gamma x_{1}y_{1}-4\gamma x_{1}y_{2}+2\gamma x_{2}y_{2}}{d_{x}^{2}}\hbox { for } x_{2},\hbox { and }\\&-\frac{1}{3}\frac{2\gamma x_{2}+1}{d_{x}}\hbox { for }y_{2}. \end{aligned}$$

A candidate to equilibrium is found for $x_{1}=0,y_{1}=1,x_{2}=1$ and $y_{2}=1$ with $\gamma \in (-\frac{1}{2},1).$

Case i.3 Similar calculations to those in case i.2 yield the same solution, that is, $x_{1}=1,y_{1}=0,x_{2}=1$ and $y_{2}=1$ for $\gamma \in (-\frac{1}{2},1).$

Case i.4 Using calculations similar to those in Case i.1, we obtain that the sign of the total effects are determined by

$$\begin{aligned}&-4d_{y}d_{x}+a\left( A+a\right) +3\gamma y_{1}d_{x}\left( A+a\right) \hbox { for }x_{1},\\&-4d_{y}d_{x}+a\left( A+a\right) +3\gamma x_{1}d_{y}\left( A+a\right) \hbox { for }y_{1},\\&4d_{x}d_{y}+A(A+a)+\gamma y_{2}d_{x}\left( 7A-a\right) \hbox { for }x_{2},\\&4d_{x}d_{y}+A(A+a)+\gamma x_{2}d_{y}\left( 7A-a\right) \hbox { for }y_{2}. \end{aligned}$$

After studying the sign of the total effects as a function of the value of $\gamma$, a candidate to equilibrium is found for $x_{1}=0$, $y_{1}=0,x_{2}=1$ and $y_{2}=1$ with $\gamma >-1.$

      Consider now the asymmetric characteristics competition where, for example, $x_{1}>x_{2}$ and $y_{1}<y_{2}$. $\square$

Case ii.1 Using calculations similar to those in Case i.1, we obtain that the sign of the total effects are determined by

$$\begin{aligned}&-12d_{x}d_{y}-Aa-A^{2}-3\left( \gamma y_{1}+1\right) d_{x}\left( A+a\right) \hbox { for }x_{1},\\&12d_{x}d_{y}+Aa+A^{2}+3\gamma x_{1}d_{y}\left( A+a\right) \hbox { for }y_{1},\\&4d_{x}d_{y}-Aa-A^{2}-(\gamma y_{2}+1)d_{x}\left( 7A-a\right) \hbox { for }x_{2},\\&-4d_{x}d_{y}+Aa+A^{2}+\gamma x_{2}d_{y}\left( 7A-a\right) \hbox { for }y_{2}. \end{aligned}$$

A candidate to equilibrium is found for $x_{1}=1,$$y_{1}=0,x_{2}=0$ and $y_{2}=1$ for $\gamma <-1$ and for $\gamma >0.$

Case ii.2 Using similar calculations to those in Case i.1, we obtain that the sign of the total effects are determined by

$$\begin{aligned}&-4d_{x}+d_{y}+2\gamma x_{1}y_{1}-4\gamma x_{2}y_{1}+2\gamma x_{2}y_{2}\hbox { for }x_{1},\\&-\frac{1}{3}\frac{2\gamma x_{1}+1}{d_{x}}\hbox { for }y_{1},\\&-(-2d_{x}-d_{y}+2\gamma x_{1}y_{1}-4\gamma x_{1}y_{2}+2\gamma x_{2}y_{2})\hbox { for }x_{2},\hbox {and}\\&-\frac{1}{3}\frac{2\gamma x_{2}+1}{d_{x}}\hbox { for }y_{2}. \end{aligned}$$

A candidate to equilibrium is found for $x_{1}=1,$$y_{1}=0,$$x_{2}=0$ and $y_{2}=1$ for $\gamma <-1/2.$

Case ii.3 This case is symmetric to Case ii.2 and gives a candidate to equilibrium for $x_{1}=0,$$y_{1}=1,$$x_{2}=1$ and $y_{2}=0$ for $\gamma <-1/2.$

Case ii.4 This case is symmetric to case ii.1. Similar calculations to those in case ii.4 yield a candidate to equilibrium is found for $x_{1}=1,$$y_{1}=0,x_{2}=0$ and $y_{2}=1.$

Equilibrium configurations are determined by comparing each firm’s most profitable product position subject to the competitor’s position. After obviating the symmetric configurations, and taking into account the values of $\gamma$ which determine the candidates to equilibrium, we can confine the equilibrium analysis to the comparison of the following equilibrium profits:

$$\begin{aligned} \varPi _{1}(0,1,1,1)= & {} \frac{1}{9}\left( \gamma -1\right) ^{2}\\ \varPi _{2}(0,1,1,1)= & {} \frac{1}{9}\left( \gamma +2\right) ^{2}\\ \varPi _{1}(0,0,1,1)= & {} \frac{1}{256}\left( \sqrt{\gamma ^{2}+8}-\gamma \right) ^{3}\\ \varPi _{2}(0,0,1,1)= & {} \frac{1}{256}\left( 5\gamma +3\sqrt{\gamma ^{2}+8} \right) \left( 32-(\sqrt{\gamma ^{2}+8}-\gamma )^{2}\right) \\ \varPi _{1}(0,1,1,0)= & {} \frac{1}{4}\\ \varPi _{2}(0,1,1,0)= & {} \frac{1}{4} \end{aligned}$$

We obtain an equilibrium position when neither firm has incentives to unilaterally alter its position.

Proof of Lemma 1

We focus our attention on the cases that generate the product equilibrium configurations. In the dominant characteristic cases we have that:

• for case i.2 and case i.3

  $$\begin{aligned} \frac{\partial {\hat{p}}_{i}}{\partial \gamma }=\frac{1}{3}\left( x_{i}y_{i}-x_{j}y_{j}\right) \end{aligned}$$

  that implies that the effect is positive (negative) for the firm that offers a higher (lower) quality.

• for case i.4

  $$\begin{aligned}&\frac{\partial {\hat{p}}_{1}}{\partial \gamma }=-\frac{1}{8}\varDelta +\frac{1}{ 8A}\gamma \varDelta ^{2}\hbox { and }\\&\quad \frac{\partial {\hat{p}}_{2}}{\partial \gamma }=\frac{5}{8}\varDelta +\frac{3}{8A }\gamma \varDelta ^{2}. \end{aligned}$$

We can easily see that the first expression is negative and the second one is positive.

In the asymmetric characteristics case (cases ii.2 and ii.3) the expressions are identical to those obtained in case i.2,  and then the effects on prices are those described in the Lemma 1. $\square$

Proof of Lemma 2

The proof of the strategic relationship along the degrees of differentiation is obtained by calculating the cross derivative of the second-stage optimal profits.

• Case i.1 has no candidates to equilibrium product configurations.

• Case i.2

  $$\begin{aligned} {\hat{\varPi }}_{1}= & {} \frac{1}{36}\frac{\left( 2d_{x}-d_{y}-2\gamma \varDelta \right) ^{2}}{d_{x}}\hbox { and }\\ {\hat{\varPi }}_{2}= & {} \frac{1}{36}\frac{\left( 4d_{x}+d_{y}+2\gamma \varDelta \right) ^{2}}{d_{x}} \end{aligned}$$

Then

$\frac{\partial _{i}^{2}{\hat{\varPi }}}{\partial d_{x}\partial d_{y}}=-\frac{d_{y}+2\varDelta \gamma }{18d_{x}^{2}}<0$ because $d_{y}+2\varDelta \gamma >0$ requires that $\frac{y_{2}}{y_{1}}>\frac{1+2\gamma x_{1}}{1+2\gamma x_{2} },$that always holds for the dominated characteristic case with $\gamma >- \frac{1}{2}.$

• Case i.3 is symmetric to Case i.2.

• Case i.4

  $$\begin{aligned} {\hat{\varPi }}_{1}= & {} \frac{2\left( -\frac{1}{8}\gamma \varDelta +\frac{1}{8}\sqrt{ \gamma ^{2}\varDelta ^{2}+8d_{x}d_{y}}\right) ^{3}}{d_{x}d_{y}}\hbox { and }\\ {\hat{\varPi }}_{2}= & {} \left( \frac{5}{8}\gamma \varDelta +\frac{3}{8}\sqrt{\gamma ^{2}\varDelta ^{2}+8d_{x}d_{y}}\right) \left( 1-\frac{1}{2d_{x}d_{y}}\left( \frac{1}{4}\gamma \varDelta -\frac{1}{4}\sqrt{\gamma ^{2}\varDelta ^{2}+8d_{x}d_{y} }\right) ^{2}\right) . \end{aligned}$$

Then

$$\begin{aligned}&\frac{\partial _{1}^{2}{\hat{\varPi }}}{\partial d_{x}\partial d_{y}}\\&\quad =\frac{ \varDelta ^{6}\gamma ^{6}+32d_{x}^{3}d_{y}^{3}+24\varDelta ^{2}\gamma ^{2}d_{x}^{2}d_{y}^{2}+12\varDelta ^{4}\gamma ^{4}d_{x}d_{y}-\varDelta ^{3}\gamma ^{3}(\varDelta ^{2}\gamma ^{2}+8d_{x}d_{y})\sqrt{\varDelta ^{2}\gamma ^{2}+8d_{x}d_{y}}}{64d_{x}^{2}d_{y}^{2}\left( \sqrt{\varDelta ^{2}\gamma ^{2}+8d_{x}d_{y}}\right) ^{3}}\\&\frac{\partial _{2}^{2}{\hat{\varPi }}}{\partial d_{x}\partial d_{y}}\\&\quad = \frac{ \varDelta ^{6}\gamma ^{6}+288d_{x}^{3}d_{y}^{3}+88\varDelta ^{2}\gamma ^{2}d_{x}^{2}d_{y}^{2}+12\varDelta ^{4}\gamma ^{4}d_{x}d_{y}-\varDelta ^{3}\gamma ^{3}(\varDelta ^{2}\gamma ^{2}+8d_{x}d_{y})\sqrt{\varDelta ^{2}\gamma ^{2}+8d_{x}d_{y}}}{64d_{x}^{2}d_{y}^{2}\left( \sqrt{\varDelta ^{2}\gamma ^{2}+8d_{x}d_{y}}\right) ^{3}} \end{aligned}$$

The numerator in both expressions is always positive for this case.

• Case ii.2 (Case ii.1 gives the same candidate than Case ii.2)

  $$\begin{aligned} {\hat{\varPi }}_{1}= & {} \frac{1}{36}\frac{\left( 4d_{x}+d_{y}+2\gamma \varDelta \right) ^{2}}{-d_{x}}\hbox { and}\\ {\hat{\varPi }}_{2}= & {} \frac{1}{36}\frac{\left( 2d_{x}-d_{y}-2\gamma \varDelta \right) ^{2}}{-d_{x}}\\ \end{aligned}$$

$\frac{\partial _{i}^{2}{\hat{\varPi }}}{\partial d_{x}\partial d_{y}}=\frac{1}{18}\frac{d_{y}+2\varDelta \gamma }{d_{x}^{2}}>0$ because $d_{y}+2\varDelta \gamma >0$ requires that $\frac{y_{2}}{y_{1}}>\frac{1+2\gamma x_{1}}{1+2\gamma x_{2}},$ that always holds for the asymmetric characteristic cases (i.e. with $\gamma <-\frac{1}{2}).$$\square$

• Case ii.3 is symmetric to Case ii.2.

• Case ii.4 is symmetric to Case ii.1.

Proof of Lemma 3

The proof follows from calculating the corresponding cross derivatives of the second-stage optimal profits.

• Case i.1 has no candidates to equilibrium product configurations.

• Case i.2

  $\frac{\partial _{i}^{2}{\hat{\varPi }}}{\partial \varDelta \partial d_{x}} =-\frac{\gamma }{9d_{x}^{2}}\left( d_{y}+2\varDelta \gamma \right) ,$ that is positive if $-1/2<\gamma <0$ and negative if $\gamma >0$ (as mentioned in the proof of Lemma 2, $d_{y}+2\varDelta \gamma >0$ requires that $\frac{y_{2}}{y_{1}}>\frac{1+2\gamma x_{1}}{1+2\gamma x_{2}},$ that always holds for the dominated characteristic case with $\gamma >-\frac{ 1}{2}).$

  $\frac{\partial _{i}^{2}{\hat{\varPi }}}{\partial \varDelta \partial d_{y}} =\frac{\gamma }{9d_{x}}$ Note that this case is characterized by $d_{x}>0$ and requires that $\gamma >-1/2.$

• Case i.3 is symmetric to Case i.2.

• Case i.4

  $$\begin{aligned}&\frac{\partial ^{2}{\hat{\varPi }}_{1}}{\partial d_{x}\partial \varDelta } =-\frac{3}{256}\gamma \frac{4\varDelta ^{5}\gamma ^{5}+96\varDelta \gamma d_{x}^{2}d_{y}^{2}+48\varDelta ^{3}\gamma ^{3}d_{x}d_{y}-4\left( \sqrt{ \varDelta ^{2}\gamma ^{2}+8d_{x}d_{y}}\right) ^{3}\left( \varDelta ^{2}\gamma ^{2}\right) }{d_{x}^{2}d_{y}\left( \sqrt{\varDelta ^{2}\gamma ^{2}+8d_{x}d_{y}} \right) ^{3}}\\&\frac{\partial _{2}^{2}{\hat{\varPi }}_{2}}{\partial d_{x}\partial \varDelta }=- \frac{1}{128}\gamma \frac{6\varDelta ^{5}\gamma ^{5}+272\varDelta \gamma d_{x}^{2}d_{y}^{2}+72\varDelta ^{3}\gamma ^{3}d_{x}d_{y}-6\left( \sqrt{\varDelta ^{2}\gamma ^{2}+8d_{x}d_{y}}\right) ^{3}\left( \varDelta ^{2}\gamma ^{2}\right) }{d_{x}^{2}d_{y}\left( \sqrt{\varDelta ^{2}\gamma ^{2}+8d_{x}d_{y}}\right) ^{3}} \end{aligned}$$

  and $\frac{\partial _{i}^{2}{\hat{\varPi }}_{i}}{\partial d_{y}\partial \varDelta }$ can be expressed by switching $d_{x}$and $d_{y}$ in $\frac{\partial ^{2}\hat{ \varPi }_{i}}{\partial d_{x}\partial \varDelta }$. It is easy to see that the numerator in both expressions is positive and then $sign\frac{\partial ^{2} {\hat{\varPi }}_{i}}{\partial d_{z}\partial \varDelta }=-sign$$\gamma .$ Note that this case requires that $\gamma >0.$

• Case ii.2 (Case ii.1 gives the same candidate than Case ii.2).

$\frac{\partial ^{2}{\hat{\varPi }}_{i}}{\partial \varDelta \partial d_{x}}=\gamma \frac{d_{y}+2\varDelta \gamma }{9d_{x}^{2}}<0$. Note that $d_{y}+2\varDelta \gamma >0$ requires that $\frac{y_{2}}{y_{1}}>\frac{1+2\gamma x_{1}}{1+2\gamma x_{2} },$ that always holds for the asymmetric characteristic cases (i.e. with $\gamma <-\frac{1}{2}).$

$\frac{\partial ^{2}{\hat{\varPi }}_{i}}{\partial \varDelta \partial d_{y}} =-\frac{\gamma }{9d_{x}}$ Note that this case is characterized by $d_{x}<0$ and requires that $\gamma <0$. $\square$

Both cross derivatives are then negative.

• Case ii.3 is symmetric to Case ii.2.

• Case ii.4 is symmetric to Case ii.1.

Equilibrium market configuration with linear fixed costs

The analysis with linear fixed costs can be carried out by making use of the optimal profits for the case of zero production costs just by adding $-c$ or $-2c$ in those cases when the firm decides maximal quality in one or both qualities, respectively. In those cases when the firm decides minimal quality, the profits remain the same than in the section with zero costs. Moreover, a candidate to equilibrium is obtained that was not present in the previous analysis: a SQL outcome defined by a configuration (0, 0, 1, 0) is obtained in Case i.2 (and its symmetric candidate given by (0, 0, 0, 1) in Case i.3) besides the previously obtained (a SQL outcome given by (0, 1, 1, 1)).

After comparing the profit levels corresponding to each candidate to equilibrium and the incentives for firms to deviate within this set of candidates, the resulting equilibrium product market configuration is the following:

• 1. SQL with (0, 0, 1, 0) under the following conditions: $c>-\frac{1}{576}\left( -378\gamma -162\sqrt{\gamma ^{2}+8}+9\gamma ^{3}-9\gamma ^{2}\sqrt{\gamma ^{2}+8}+256\right)$ and $\frac{5}{36}<c<\frac{ 4}{9}$ with $\gamma <0.13959$. More specifically, the conditions are $-\frac{1}{576}\left( -378\gamma -162\sqrt{\gamma ^{2}+8}+9\gamma ^{3}-9\gamma ^{2}\sqrt{\gamma ^{2}+8}+256\right)<c<\frac{4}{9}$ for $-0.34089<\gamma <0.13959,$ and $\frac{5}{36}<c<\frac{4}{9}$ for $\gamma <-0.34089.$

• 2. DQL (i.e. (0, 0, 1, 1)) under the following conditions:

  • if $-0.429\,36<\gamma <0.139\,59$ and $\frac{1}{2304}\left( -296\gamma -9\left( \gamma ^{2}+8\right) ^{\frac{3}{2} }+256\gamma ^{2}+36\gamma ^{3}-27\gamma ^{2}\sqrt{\gamma ^{2}+8}+256\right)<c<-\frac{1}{576}\left( -378\gamma -162\sqrt{\gamma ^{2}+8}+9\gamma ^{3}-9\gamma ^{2}\sqrt{\gamma ^{2}+8}+256\right)$

  • if $\gamma >0.139\,59$ and $\frac{1}{2304}\left( -296\gamma -9\left( \gamma ^{2}+8\right) ^{\frac{3}{2} }+256\gamma ^{2}+36\gamma ^{3}-27\gamma ^{2}\sqrt{\gamma ^{2}+8}+256\right)<c<\frac{1}{128}[60\gamma +\left( \gamma ^{2}+18\right) (\sqrt{\gamma ^{2}+8} -\gamma )]$.

• 3. SQL with (0, 1, 1, 1) under the following conditions: $c<\frac{1}{36}\left( 2\gamma +7\right) \left( 2\gamma +1\right)$ and $c<\frac{1}{2304}\left( -296\gamma -9\left( \gamma ^{2}+8\right) ^{\frac{3}{2 }}+256\gamma ^{2}+36\gamma ^{3}-27\gamma ^{2}\sqrt{\gamma ^{2}+8}+256\right)$ with $-1/2<\gamma <0.19614$. The first restriction is binding for $-1/2<\gamma <-0.30886$ and the second one for $-0.30886<\gamma <0.19614$).

• 4. CQL (i.e. (0, 1, 1, 0)) under the following conditions: $\frac{1}{36}\left( 2\gamma +7\right) \left( 2\gamma +1\right)<c<\frac{5}{36 }$ that corresponds to $\gamma <-0.12197$ (note that for $\gamma <-1/2$ only the upper limit is binding). It is important to note this area partially overlaps with that defined in the DQL outcome. In particular this happens for $\frac{1}{36}\left( 2\gamma +7\right) \left( 2\gamma +1\right)<c<\frac{5}{36 }$ and $\frac{1}{2304}\left( -296\gamma -9\left( \gamma ^{2}+8\right) ^{\frac{3}{2} }+256\gamma ^{2}+36\gamma ^{3}-27\gamma ^{2}\sqrt{\gamma ^{2}+8}+256\right)<c<-\frac{1}{576}\left( -378\gamma -162\sqrt{\gamma ^{2}+8}+9\gamma ^{3}-9\gamma ^{2}\sqrt{\gamma ^{2}+8}+256\right)$ that corresponds to the interval $-0.429\,36<\gamma <-0.12917.$ This means that there are two equilibria in this area. By comparing firms’s profits we can see that both firms are better off in the DQL outcome than in the CQL outcome for the interval of values of c and $\gamma$ defined for this area.