Vertical differentiation beyond the uniform distribution

Abstract

The assessment of the way distributive shocks, such as increased polarization or higher inequality, affect vertically differentiated markets has been severely hampered by the standard reference to uniform distributions. In this paper we offer the first proof of existence of a subgame perfect Nash equilibrium in a vertically differentiated duopoly with uncovered market, for a large set of symmetric and asymmetric distributions of consumers, including, among others, all logconcave distributions. The proof relies on the ‘income share elasticity’ representation of the consumers’ density function. Some illustrative examples are also provided to assess the impact of distributive shocks on market equilibrium.

Appendices

Appendix A. The second order conditions at the price stage

In this and in the following appendices we shall make use of the following simplifying notation: $f_{i}\equiv f\left( \theta _{i}\right)$, $F_{i}\equiv F\left( \theta _{i}\right)$, $\eta _{i}\equiv \eta \left( \theta _{i}\right)$, $\pi _{i}\equiv \pi \left( \theta _{i}\right)$, $i=H,L$.

The SOC of firm H requires:

$$\begin{aligned} \frac{\partial ^{2}\Pi _{H}}{\partial p_{H}^{2}}=-\frac{1}{\Delta }\left( 2f_{H}+\frac{p_{H}}{\Delta }f_{H}^{\prime }\right) <0. \end{aligned}$$

By using (1) and recalling the definitions of $\eta \left( \theta \right)$ and $\pi \left( \theta \right)$ it can be rewritten as

$$\begin{aligned} -\frac{1}{\Delta }\frac{\left( 1-F_{H}\right) }{\theta _{H}}\left( 2\eta _{H}+\pi _{H}-1\right) <0, \end{aligned}$$

which is indeed the case for all $\theta _{H}>0$, if Property 3 holds.

The SOC of firm L is satisfied if

$$\begin{aligned} \frac{\partial ^{2}\Pi _{L}}{\partial p_{L}^{2}}=-2\left( \frac{f_{H}}{ \Delta }+\frac{f_{L}}{s_{L}}\right) +p_{L}\left( \frac{f_{H}^{\prime }}{ \Delta ^{2}}-\frac{f_{L}^{\prime }}{s_{L}^{2}}\right) <0, \end{aligned}$$

i.e.

$$\begin{aligned} f_{H}+\frac{\Delta }{s_{L}}f_{L}+\frac{1}{2}\frac{p_{L}}{\Delta }\left( f_{H}^{\prime }-\left( \frac{\Delta }{s_{L}}\right) ^{2}f_{L}^{\prime }\right) >0. \end{aligned}$$

(A1)

Equation (3) allows to reformulate $\Delta /s_{L}$ and $p_{L}/\Delta$ in terms of $\theta _{H}$ and $\theta _{L}$ only:

$$\begin{aligned} \frac{\Delta }{s_{L}}&=\frac{\theta _{L}}{\theta _{H}}\frac{\eta _{H}}{ 1-\eta _{H}}, \end{aligned}$$

(A2)

$$\begin{aligned} \frac{p_{L}}{\Delta }&=\frac{\theta _{H}}{\eta _{H}}\left( 1-\eta _{H}\right) , \end{aligned}$$

(A3)

so that using (5$'$) and the definitions of $\eta \left( \theta \right)$ and $\pi \left( \theta \right)$, the inequality (A1) can be rewritten as

$$\begin{aligned} \eta _{H}+\frac{1}{2}\left( 1-\eta _{H}\right) \left( \pi _{H}-1\right) + \frac{1}{2}\dfrac{2-\eta _{H}}{1-\eta _{L}}\eta _{L}\dfrac{\eta _{H}}{1-\eta _{H}}\left( 1+\pi _{L}\right) >0. \end{aligned}$$

At equilibrium, the first two terms are positive since Eq. (3) ensures $\eta _{H}<1$ and Property 3 ensures that $\left( \pi _{H}-1\right) >-2\eta _{H}$. The last term is positive since Property 2 and Eq. (3) imply (a) $\eta _{L}<\eta _{H}<1$, and (b) $\pi _{L}>-\eta _{L}>-1$

Appendix B. The comparative statics of the price stage

By totally differentiating (1) and (2), we obtain the following linear system:

$$\begin{aligned} \begin{pmatrix} \dfrac{\partial ^{2}\Pi _{H}}{\partial p_{H}^{2}} &{} \dfrac{\partial ^{2}\Pi _{H}}{\partial p_{H}\partial p_{L}} \\ \dfrac{\partial ^{2}\Pi _{L}}{\partial p_{L}\partial p_{H}} &{} \dfrac{ \partial ^{2}\Pi _{L}}{\partial p_{L}^{2}} \end{pmatrix} \begin{pmatrix} dp_{H} \\ dp_{L} \end{pmatrix} =- \begin{pmatrix} \dfrac{\partial ^{2}\Pi _{H}}{\partial p_{H}\partial \Delta } \\ \dfrac{\partial ^{2}\Pi _{L}}{\partial p_{L}\partial \Delta } \end{pmatrix} d\Delta _{s}- \begin{pmatrix} \dfrac{\partial ^{2}\Pi _{H}}{\partial p_{H}\partial s_{L}} \\ \dfrac{\partial ^{2}\Pi _{L}}{\partial p_{L}\partial s_{L}} \end{pmatrix} ds_{L} \end{aligned}$$

where the above derivatives can be written as:

• $\dfrac{\partial ^{2}\Pi _{H}}{\partial p_{H}^{2}}=-2f_{H}\dfrac{1}{ \Delta }-\dfrac{p_{H}}{\Delta ^{2}}f_{H}^{\prime }$; $\dfrac{\partial ^{2}\Pi _{H}}{\partial p_{H}\partial p_{L}}=f_{H}\dfrac{1}{\Delta }+\dfrac{ p_{H}}{\Delta ^{2}}f_{H}^{\prime }$;

• $\dfrac{\partial ^{2}\Pi _{L}}{\partial p_{L}\partial p_{H}}=f_{H} \dfrac{1}{\Delta }-\dfrac{p_{L}}{\Delta ^{2}}f_{H}^{\prime }$;

• $\dfrac{\partial ^{2}\Pi _{L}}{\partial p_{L}^{2}}=-\dfrac{2}{\Delta } f_{H}-\dfrac{2}{s_{L}}f_{L}+\dfrac{p_{L}}{\Delta ^{2}}f_{H}^{\prime }-f_{L}^{\prime }\dfrac{\theta _{L}}{s_{L}}$;

• $\dfrac{\partial ^{2}\Pi _{H}}{\partial p_{H}\partial \Delta } =f_{H}\left( \dfrac{\theta _{H}}{\Delta }+\dfrac{p_{H}}{\Delta ^{2}}\right) + \dfrac{p_{H}}{\Delta }f_{H}^{\prime }\dfrac{\theta _{H}}{\Delta }$;

• $\dfrac{\partial ^{2}\Pi _{H}}{\partial p_{H}\partial s_{L}} =-f_{H}\left( \dfrac{\theta _{H}}{\Delta }+\dfrac{p_{H}}{\Delta ^{2}}\right) -\dfrac{p_{H}}{\Delta }f_{H}^{\prime }\dfrac{\theta _{H}}{\Delta }$;

• $\dfrac{\partial ^{2}\Pi _{L}}{\partial p_{L}\partial \Delta } =f_{H}\left( \dfrac{p_{L}}{\Delta ^{2}}-\dfrac{\theta _{H}}{\Delta }\right) + \dfrac{p_{L}}{\Delta }f_{H}^{\prime }\dfrac{\theta _{H}}{\Delta }$;

• $\dfrac{\partial ^{2}\Pi _{L}}{\partial p_{L}\partial s_{L}} =f_{H}\left( \dfrac{\theta _{H}}{\Delta }-\dfrac{p_{L}}{\Delta ^{2}}\right) - \dfrac{p_{L}}{\Delta }f_{H}^{\prime }\dfrac{\theta _{H}}{\Delta }+2\dfrac{ \theta _{L}}{s_{L}}f_{L}+\dfrac{\theta _{L}^{2}}{s_{L}}f_{L}^{\prime }$,

and are evaluated at equilibrium.

Denoting with A the coefficients matrix, its determinant can be written as:

$$\begin{aligned} \left| A\right|= & {} \frac{1}{\Delta ^{2}}f_{H}^{2}\left( 2+\pi _{H}\right) \\&+\,\frac{1}{\Delta ^{2}}\left\{ \frac{\Delta }{s_{L}}\left( \left( 1+\pi _{L}\right) f_{L}\right) \left( \left( 1+\pi _{H}\right) f_{H}+\frac{p_{L}}{ \Delta }f_{H}^{\prime }\right) \right\} . \end{aligned}$$

Property 2 ensures that $\left( 1+\pi _{H}\right) >0$, provided that $\eta _{H}<1$, which is indeed the case at the price stage equilibrium. Since $\theta _{L}<\theta _{H}$, Property 2 also ensures that at equilibrium $\eta _{L}<1$ so that $\left( 1+\pi _{L}\right) >0$. Therefore $\left| A\right| >0$ if:

$$\begin{aligned} \left( 1+\pi _{H}\right) f_{H}+\frac{p_{L}}{\Delta }f_{H}^{\prime }>0. \end{aligned}$$

Recalling Eq. (A3), at equilibrium we can reformulate this inequality as

$$\begin{aligned} \frac{f_{H}}{\eta _{H}}\left( 2\eta _{H}+\pi _{H}-1\right) >0, \end{aligned}$$

which is indeed the case under Property 3.

We can now perform the following comparative statics exercises.

The effect on $p_{L}^{*}$ of a change in $s_{H}$

First notice that for given $s_{L}$, $dp_{L}^{*}/ds_{H}=dp_{L}^{*}/d\Delta$. By applying Cramer’s rule:

$$\begin{aligned} \frac{dp_{L}^{*}}{d\Delta }=\frac{1}{\left| A\right| }\frac{p_{L} }{\Delta ^{3}}f_{H}^{2}\left( 2+\pi _{H}\right) , \end{aligned}$$

where the RHS is evaluated at equilibrium, and is positive under Property 2.

The effect on $p_{H}^{*}$ of a change in $s_{L}$

By Cramer’s rule and using the definition of $\pi \left( \theta \right)$, we have:

$$\begin{aligned} \dfrac{dp_{H}^{*}}{ds_{L}}=-\tfrac{\frac{p_{H}}{\Delta } f_{H}^{2}\left( 2+\pi _{H}\right) +\frac{\Delta }{s_{L}}\left( 1+\pi _{L}\right) f_{L}\left( \frac{p_{H}}{\Delta }\left( \theta _{H}-\theta _{L}\right) f_{H}^{\prime }+\left( \theta _{H}-\theta _{L}\right) f_{H}+ \frac{p_{H}}{\Delta }f_{H}\right) }{f_{H}^{2}\left( 2+\pi _{H}\right) +\frac{ \Delta }{s_{L}}\left( 1+\pi _{L}\right) f_{L}\left( \left( 1+\pi _{H}\right) f_{H}+\frac{p_{L}}{\Delta }f_{H}^{\prime }\right) }, \end{aligned}$$

(B1)

where the RHS is evaluated at equilibrium. By using (A2) for $\Delta /s_{L}$, (A3) for $p_{L}/\Delta$, $p_{H}/\Delta =p_{L}/\Delta +\theta _{H}$, the definitions of $\eta$ for f, the definition of $\pi$ for $f^{\prime }$, and then Eq. (5$'$) for $\left( 1-F_{L}\right)$, we obtain that at the price equilibrium equation (B1) collapses to:

$$\begin{aligned} \dfrac{dp_{H}^{*}}{ds_{L}}=-\frac{\theta _{H}}{\eta _{H}} \tfrac{\left( 2+\pi _{H}\right) \left( 1-\eta _{H}\right) +\left( 1+\pi _{L}\right) \eta _{L}\frac{2-\eta _{H}}{1-\eta _{L}}\left( \pi _{H}+\eta _{H}-\left( \pi _{H}+\eta _{H}-1\right) \frac{\theta _{L}}{\theta _{H}}\right) }{\left( 2+\pi _{H}\right) \left( 1-\eta _{H}\right) +\left( 1+\pi _{L}\right) \eta _{L}\frac{2-\eta _{H}}{1-\eta _{L}}\left( \frac{2\eta _{H}+\pi _{H}-1}{\eta _{H}}\right) }, \end{aligned}$$

(B2)

which under Properties 2 and 3 is unambiguously negative.

The sign of $\left( \frac{dp_{L}^{*}}{ds_{L}}-\theta _{L}^{*}\right)$

Consider now $dp_{L}^{*}/ds_{L}$. By applying Cramer’s rule and using the definition of $\pi \left( \theta \right)$, we obtain

$$\begin{aligned} \dfrac{dp_{L}^{*}}{ds_{L}}=\theta _{L}\dfrac{\frac{\Delta }{ s_{L}}\left( 1+\pi _{L}\right) f_{L}\left[ \left( 1+\pi _{H}\right) f_{H}+ \frac{p_{L}}{\Delta _{s}}f_{H}^{\prime }\right] -\frac{s_{L}}{\Delta } f_{H}^{2}\left( 2+\pi _{H}\right) }{\frac{\Delta }{s_{L}}\left[ \left( 1+\pi _{L}\right) f_{L}\right] \left[ \left( 1+\pi _{H}\right) f_{H}+\frac{p_{L}}{ \Delta _{s}}f_{H}^{\prime }\right] +f_{H}^{2}\left( 2+\pi _{H}\right) } . \end{aligned}$$

(B3)

By using (A2) for $\Delta /s_{L}$, (A3) for $p_{L}/\Delta$, the definitions of $\eta$ for f, the definition of $\pi$ for $f^{\prime }$, and then Eq. (5$'$) for $\left( 1-F_{L}\right)$, we obtain that at the price equilibrium equation (B3) collapses to:

$$\begin{aligned} \dfrac{dp_{L}^{*}}{ds_{L}}=\theta _{L}\dfrac{\left( 1+\pi _{L}\right) \eta _{L}\frac{2-\eta _{H}}{1-\eta _{L}}\left( 2\eta _{H}+\pi _{H}-1\right) -\frac{\theta _{H}\left( 1-\eta _{H}\right) ^{2}}{\theta _{L}} \left( 2+\pi _{H}\right) }{\left( 1+\pi _{L}\right) \eta _{L}\frac{2-\eta _{H}}{1-\eta _{L}}\left( 2\eta _{H}+\pi _{H}-1\right) +\eta _{H}\left( 2+\pi _{H}\right) \left( 1-\eta _{H}\right) }. \end{aligned}$$

(B4)

Since under Property 3 the ratio in (B4) is surely less than one, we get:

$$\begin{aligned} \dfrac{dp_{L}^{*}}{ds_{L}}-\theta _{L}^{*}=\theta _{L}\left( \tfrac{\left( 1+\pi _{L}\right) \eta _{L}\frac{2-\eta _{H}}{ 1-\eta _{L}}\left( 2\eta _{H}+\pi _{H}-1\right) -\frac{\theta _{H}\left( 1-\eta _{H}\right) ^{2}}{\theta _{L}}\left( 2+\pi _{H}\right) }{\left( 1+\pi _{L}\right) \eta _{L}\frac{2-\eta _{H}}{1-\eta _{L}}\left( 2\eta _{H}+\pi _{H}-1\right) +\eta _{H}\left( 2+\pi _{H}\right) \left( 1-\eta _{H}\right) } -1\right) <0, \end{aligned}$$

(B5)

where the RHS is evaluated at equilibrium.

The sign of $\left( \frac{dp_{H}^{*}}{ds_{H}}-\theta _{H}^{*}\right)$

Consider now $dp_{H}^{*}/ds_{H}=dp_{H}^{*}/d\Delta$. By applying Cramer’s rule and using the definition of $\pi \left( \theta \right)$, we obtain:

$$\begin{aligned}&\frac{dp_{H}^{*}}{d\Delta }\nonumber \\&=\frac{\left( f_{H}\left( \tfrac{ p_{L}}{\Delta }\pi _{H}-\theta _{H}\right) f_{H}\left( 1+\tfrac{p_{H}}{ \Delta }\frac{\left( \pi _{H}-1\right) }{\theta _{H}}\right) -f_{H}\left( \theta _{H}+\tfrac{p_{H}}{\Delta }\pi _{H}\right) \left( f_{H}\left( \tfrac{ p_{L}}{\Delta }\frac{\left( \pi _{H}-1\right) }{\theta _{H}}-2\right) - \tfrac{f_{L}\Delta }{s_{L}}\left( \pi _{L}+1\right) \right) \right) }{ f_{H}^{2}\left( 2+\pi _{H}\right) +\frac{\Delta }{s_{L}}\left[ \left( 1+\pi _{L}\right) f_{L}\right] \left[ \left( 1+\pi _{H}\right) f_{H}+\frac{p_{L}}{ \Delta }f_{H}^{\prime }\right] } \end{aligned}$$

(B6)

By using (A2) for $\Delta /s_{L}$, (A3) for $p_{L}/\Delta$, the definitions of $\eta$ for f, the definition of $\pi$ for $f^{\prime }$, Eq. (5$'$) for $\left( 1-F_{L}\right)$, and noting that $p_{H}/\Delta =p_{L}/\Delta +\theta _{H}=\theta _{H}/\eta _{H}$, we obtain that at the price equilibrium equation (B6) collapses to:

$$\begin{aligned} \frac{dp_{H}^{*}}{d\Delta }{=}\theta _{H}\tfrac{\left( \left( \frac{1-\eta _{H}}{\eta _{H}}\right) \pi _{H}-1\right) \left( \eta _{H}+\pi _{H}-1\right) \left( 1-\eta _{H}\right) -\left( \eta _{H}+\pi _{H}\right) \left( \left( \left( 1-\eta _{H}\right) \frac{\left( \pi _{H}-1\right) }{ \eta _{H}}-2\right) \left( 1-\eta _{H}\right) -\frac{\eta _{L}\left( 2-\eta _{H}\right) }{\left( 1-\eta _{L}\right) }\left( \pi _{L}+1\right) \right) }{ \eta _{H}\left( 1-\eta _{H}\right) \left( 2+\pi _{H}\right) +\left( 1+\pi _{L}\right) \frac{\eta _{L}\left( 2-\eta _{H}\right) }{\left( 1-\eta _{L}\right) }\left( \pi _{H}+2\eta _{H}-1\right) } \end{aligned}$$

(B7)

Since under Property 3 the ratio in (B7) is surely greater than one, we get:

$$\begin{aligned} \frac{dp_{H}^{*}}{d\Delta }-\theta _{H}>0. \end{aligned}$$

(B8)

Appendix C. The second order condition at the quality stage

The proof that the Second Order Condition at the quality stage for firm L is satisfied, relies on three preliminary steps.

Step I It is useful to calculate $\Psi ^{\prime }\left( \theta _{H}\right)$ at $\Psi \left( \theta _{H}\right) =0$. By deriving Eq. (11) and recalling that $\theta _{L}=g\left( \theta _{H}\right)$ according to (5$'$), we get that at $\Psi \left( \theta _{H}\right) =0$:

$$\begin{aligned} \Psi ^{\prime }\left( \theta _{H}\right)= & {} \frac{\eta _{H}^{\prime }\eta _{L}\theta _{L}}{1-\eta _{L}}\frac{2-\eta _{H}}{1-\eta _{H}}+\left( 1-\eta _{H}\right) \left( \frac{\partial p_{HL}}{\delta \theta _{H}}+\frac{ \partial p_{HL}}{\delta \theta _{L}}g^{\prime }\left( \theta _{H}\right) +1\right) \\&-\frac{\eta _{H}^{\prime }}{1-\eta _{L}}\eta _{L}\theta _{L}+ \frac{2-\eta _{H}}{1-\eta _{L}}\left( \frac{\eta _{L}\theta _{L}\eta _{L}^{\prime }}{1-\eta _{L}}+\eta _{L}^{\prime }\theta _{L}+\eta _{L}\right) g^{\prime }\left( \theta _{H}\right) . \end{aligned}$$

Collecting terms the above expression can be simplified into:

$$\begin{aligned} \Psi ^{\prime }= & {} \frac{\eta _{H}^{\prime }\eta _{L}\theta _{L}}{ \left( 1-\eta _{L}\right) \left( 1-\eta _{H}\right) }+\left( 1-\eta _{H}\right) \left( \frac{\partial p_{HL}}{\delta \theta _{H}}+\frac{\partial p_{HL}}{\delta \theta _{L}}g^{\prime }\left( \theta _{H}\right) +1\right) \\&+\,\frac{2-\eta _{H}}{1-\eta _{L}}\left( \frac{\theta _{L}\eta _{L}^{\prime }}{1-\eta _{L}}+\eta _{L}\right) g^{\prime }\left( \theta _{H}\right) . \end{aligned}$$

Using now the definitions of $\eta$ and $\pi$, we can write:

$$\begin{aligned} \eta _{L}^{\prime }=\frac{\eta _{L}}{\theta _{L}}\left( \pi _{L}+\eta _{L}\right) ;\quad \eta _{H}^{\prime }=\frac{\eta _{H}}{\theta _{H}}\left( \pi _{H}+\eta _{H}\right) , \end{aligned}$$

which used into $\Psi ^{\prime }\left( \theta _{H}\right)$ leads to:

$$\begin{aligned} \Psi ^{\prime }\left( \theta _{H}\right) =&\,\frac{\theta _{L}}{ \theta _{H}}\frac{\eta _{H}\left( \pi _{H}+\eta _{H}\right) \eta _{L}}{ \left( 1-\eta _{L}\right) \left( 1-\eta _{H}\right) }+\left( 1-\eta _{H}\right) \left( \frac{\partial p_{HL}}{\partial \theta _{H}}+\frac{ \partial p_{HL}}{\partial \theta _{L}}g^{\prime }\left( \theta _{H}\right) +1\right) \nonumber \\&+\frac{2-\eta _{H}}{1-\eta _{L}}\eta _{L}\frac{\pi _{L}+1}{1-\eta _{L}}g^{\prime }\left( \theta _{H}\right) . \end{aligned}$$

(C1)

Step II. Notice that at the price equilibrium the effects of changes in $s_{L}$ on the value of $\theta _{H}$ and $\theta _{L}$ can be written as follows:

$$\begin{aligned} \theta _{LL}&\equiv \frac{\partial \theta _{L}^{*}}{\partial s_{L}}= \frac{1}{s_{L}}\left( p_{LL}-\theta _{L}\right) <0, \end{aligned}$$

(C2)

$$\begin{aligned} \theta _{HL}&\equiv \frac{\partial \theta _{H}^{*}}{\partial s_{L}}= \frac{1}{\Delta }\left( p_{HL}+\theta _{H}-p_{LL}\right) . \end{aligned}$$

(C3)

The sign of $\theta _{LL}$ can be easily established from (B5). As far as $\theta _{HL}$ is concerned, we recall from Lemma 2 that for (5$'$) to be satisfied:

$$\begin{aligned} \theta _{LL}=g^{\prime }\theta _{HL} \end{aligned}$$

(C4)

so that, given $g^{\prime }>0$, $\theta _{HL}<0$.

Step III For future reference, it is useful to solve the FOC of firm L at the quality stage in terms of $p_{LL}$. From Eq. (8), this FOC can be written as:

$$\begin{aligned} p_{LL}\left( F_{H}-F_{L}\right) +p_{L}f_{H}\theta _{HL}=p_{L}f_{L}\theta _{LL}. \end{aligned}$$

By using (C4) for $\theta _{LL}$ and (C3) for $\theta _{HL}$, the above can be reformulated as:

$$\begin{aligned} p_{LL}=\frac{\left( \frac{p_{L}}{\Delta }f_{L}g^{\prime }-\frac{p_{L}}{ \Delta }f_{H}\right) }{\left( F_{H}-F_{L}+\frac{p_{L}}{\Delta } f_{L}g^{\prime }-\frac{p_{L}}{\Delta }f_{H}\right) }\left( p_{HL}+\theta _{H}\right) . \end{aligned}$$

Consider now the ratio at the RHS of the above expression. Recalling Eqs. (5$'$) and (A3), the expression for $g^{\prime }$ and the definitions of $\eta$, we obtain:

$$\begin{aligned} \tfrac{\left( \frac{p_{L}}{\Delta }f_{L}g^{\prime }-\frac{p_{L}}{\Delta } f_{H}\right) }{\left( F_{H}-F_{L}+\frac{p_{L}}{\Delta }f_{L}g^{\prime }- \frac{p_{L}}{\Delta }f_{H}\right) }=\tfrac{\left( 1-\eta _{H}\right) \left( 1+\pi _{H}-\pi _{L}\right) \left( 1-\eta _{L}\right) }{\left( 1-\eta _{H}+\eta _{L}\right) \left( 1+\pi _{L}\right) +\left( 1-\eta _{H}\right) \left( 1-\eta _{L}\right) \left( 1+\pi _{H}-\pi _{L}\right) }, \end{aligned}$$

so that, using Eq. (10) for $\left( p_{HL}+\theta _{H}\right)$ we get:

$$\begin{aligned} p_{LL}=-\theta _{L}\tfrac{\eta _{L}\left( 2-\eta _{H}\right) \left( 1+\pi _{H}-\pi _{L}\right) }{\left( 1-\eta _{H}+\eta _{L}\right) \left( 1+\pi _{L}\right) +\left( 1-\eta _{H}\right) \left( 1-\eta _{L}\right) \left( 1+\pi _{H}-\pi _{L}\right) }. \end{aligned}$$

(C5)

Now we proceed to evaluate the SOC for firm L at the quality stage. The first order derivative of the profits of firm L with respect to quality (Eq. (8) in text) can be written as:

$$\begin{aligned} \frac{\partial \Pi _{L}}{\partial s_{L}}=\frac{p_{L}}{\Delta }f_{H}\left( p_{HL}+\theta _{H}\right) +f_{L}\theta _{L}^{2}+\left( F_{H}-F_{L}-\frac{ p_{L}}{\Delta }f_{H}-\theta _{L}f_{L}\right) p_{LL}. \end{aligned}$$

Therefore, given Eq. (2), the second order derivative is:

$$\begin{aligned} \frac{\partial ^{2}\Pi _{L}}{\partial s_{L}^{2}} =&\,\frac{\partial }{ \partial s_{L}}\left( \frac{p_{L}}{\Delta }f_{H}\left( p_{HL}+\theta _{H}\right) \right) -p_{LL}\frac{\partial }{\partial s_{L}}\left( \frac{p_{L} }{\Delta }f_{H}\right) \nonumber \\&+\frac{\partial }{\partial s_{L}}\left( f_{L}\theta _{L}^{2}\right) +p_{LL} \frac{\partial }{\partial s_{L}}\left( F_{H}-F_{L}-\theta _{L}f_{L}\right) . \end{aligned}$$

(C6)

Using (C2) in the first two terms, and the definition of $\pi _{L}$ in the last two terms of (C6), we obtain:

$$\begin{aligned} \dfrac{\partial ^{2}\Pi _{L}}{\partial s_{L}^{2}}= & {} \left( \left( 2p_{LL}+\dfrac{p_{L}}{\Delta }\right) f_{H}+\dfrac{p_{L}}{\Delta } f_{H}^{\prime }\left( p_{HL}+\theta _{H}-p_{LL}\right) \right) \theta _{HL}\\&+\,f_{L}\left( \theta _{L}-p_{LL}\right) \left( 1+\pi _{L}\right) \theta _{LL}+\tfrac{p_{L}}{\Delta }f_{H}\left( \dfrac{\partial p_{HL}}{ \partial \theta _{H}}+1\right) \theta _{HL}\\&+\,\dfrac{p_{L}}{\Delta }f_{H} \dfrac{\partial p_{HL}}{\partial \theta _{L}}\theta _{LL}. \end{aligned}$$

which, using (A3), (C4), and the definition of $\eta$, results into:

$$\begin{aligned} \dfrac{\partial ^{2}\Pi _{L}}{\partial s_{L}^{2}}=&\left( \left( 2p_{LL}+\tfrac{p_{L}}{\Delta }\right) f_{H}+\tfrac{p_{L}}{\Delta } f_{H}^{\prime }\left( p_{HL}+\theta _{H}-p_{LL}\right) \right) \theta _{HL}\nonumber \\&+f_{L}\left( \theta _{L}-p_{LL}\right) \left( 1+\pi _{L}\right) g^{\prime }\theta _{HL}+\left( 1-F_{H}\right) \left( 1-\eta _{H}\right) \left( \dfrac{\partial p_{HL}}{\partial \theta _{H}}+1\right) \theta _{HL}\nonumber \\&+\dfrac{\partial p_{HL}}{\partial \theta _{L}}g^{\prime }\theta _{HL}. \end{aligned}$$

(C7)

Consider now (C1). By multiplying both sides by $\theta _{HL}$ and rearranging terms, it can be rewritten as:

$$\begin{aligned}&\left( 1-\eta _{H}\right) \left( \left( \dfrac{\partial p_{HL}}{ \partial \theta _{H}}+1\right) \theta _{HL}+\dfrac{\partial p_{HL}}{\partial \theta _{L}}g^{\prime }\theta _{HL}\right) \\&\quad =\theta _{HL}\Psi ^{\prime }\left( \theta _{H}\right) -\left( \dfrac{\theta _{L}}{\theta _{H}}\dfrac{\eta _{H}\left( \pi _{H}+\eta _{H}\right) \eta _{L}}{\left( 1-\eta _{L}\right) \left( 1-\eta _{H}\right) } \theta _{HL}+\dfrac{2-\eta _{H}}{1-\eta _{L}}\eta _{L}\dfrac{\pi _{L}+1}{ 1-\eta _{L}}g^{\prime }\theta _{HL}\right) . \end{aligned}$$

(C1′)

Using (C1$'$) into (C7) we obtain:

$$\begin{aligned}&\dfrac{\partial ^{2}\Pi _{L}}{\partial s_{L}^{2}}=\left( 1-F_{H}\right) \theta _{HL}\Psi ^{\prime }\left( \theta _{H}\right) +f_{L}\left( \theta _{L}-p_{LL}\right) \left( 1+\pi _{L}\right) g^{\prime }\theta _{HL}\nonumber \\&\quad +\left( \left( 2p_{LL}+\tfrac{p_{L}}{\Delta }\right) f_{H}+\tfrac{ p_{L}}{\Delta }f_{H}^{\prime }\left( p_{HL}+\theta _{H}-p_{LL}\right) \right) \theta _{HL}\nonumber \\&\quad -\left( 1-F_{H}\right) \left( \dfrac{\theta _{L}\eta _{H}\left( \pi _{H}+\eta _{H}\right) \eta _{L}}{\theta _{H}\left( 1-\eta _{L}\right) \left( 1-\eta _{H}\right) }\theta _{HL}+\dfrac{2-\eta _{H}}{1-\eta _{L}}\eta _{L}\dfrac{\pi _{L}+1}{1-\eta _{L}}g^{\prime }\theta _{HL}\right) . \end{aligned}$$

(C8)

The first term in (C8) is negative, since $\theta _{HL}<0$ (see Step II of this appendix) and $\Psi ^{\prime }>0$ at equilibrium. In the sequel we prove that the sum of the remaining terms is equal to zero at equilibrium. By using repeatedly Eqs. (5$'$), (10), (A3) as well as the expression for $g^{\prime }$ and the definitions of $\eta$ and $\pi$, tedious algebra shows that this amounts to proving that:

$$\begin{aligned}&2p_{LL}\dfrac{\eta _{H}}{\theta _{H}}+\left( 1-\eta _{H}\right) +\left( 1-\eta _{H}\right) \dfrac{1}{\theta _{H}}\left( \pi _{H}-1\right) \left( p_{HL}+\theta _{H}\right) \\&\quad -\left( 1-\eta _{H}\right) \dfrac{1}{\theta _{H}}\left( \pi _{H}-1\right) p_{LL}+\dfrac{\eta _{H}}{\theta _{H}}\left( 2+\pi _{H}\right) \theta _{L}-p_{LL}\dfrac{\eta _{H}}{\theta _{H}}\left( 2+\pi _{H}\right) \\&\quad -\left( \dfrac{\theta _{L}}{\theta _{H}}\dfrac{\eta _{H}\left( \pi _{H}+\eta _{H}\right) \eta _{L}}{\left( 1-\eta _{L}\right) \left( 1-\eta _{H}\right) }+\eta _{H}\dfrac{\left( 2+\pi _{H}\right) }{1-\eta _{L}}\dfrac{ \theta _{L}}{\theta _{H}}\right) =0. \end{aligned}$$

By substituting $\ p_{HL}+\theta _{H}$ from Eq. (10), collecting terms, and substituting for $p_{LL}$ from Eq. (C5), this expression can be transformed into:

$$\begin{aligned}&\left( 1-\eta _{H}\right) \\&\quad -\tfrac{\theta _{L}\eta _{L}\left( 2-\eta _{H}\right) }{\theta _{H}\left( 1-\eta _{L}\right) \left( 1-\eta _{H}\right) }\left( \tfrac{ \left( 1-\eta _{H}+\eta _{L}\right) \left( 1+\pi _{L}\right) \left( 2\eta _{H}+\pi _{H}-1\right) +\eta _{H}\left( 1-\eta _{L}\right) \left( 1-\eta _{H}\right) \left( 1+\pi _{H}-\pi _{L}\right) }{\left( 1-\eta _{H}+\eta _{L}\right) \left( 1+\pi _{L}\right) +\left( 1-\eta _{L}\right) \left( 1-\eta _{H}\right) \left( 1+\pi _{H}-\pi _{L}\right) }\right) =0. \end{aligned}$$

The term $\theta _{L}\eta _{L}\left( 2-\eta _{H}\right) /\theta _{H}\left( 1-\eta _{L}\right) \left( 1-\eta _{H}\right)$ can again be substituted from Eq. (10), so that we get:

$$\begin{aligned}&\left( 1-\eta _{H}\right) \nonumber \\&\quad +\left( \tfrac{p_{HL}}{\theta _{H}}+1\right) \left( \tfrac{\left( 1-\eta _{H}+\eta _{L}\right) \left( 1+\pi _{L}\right) \left( 2\eta _{H}+\pi _{H}-1\right) +\eta _{H}\left( 1-\eta _{L}\right) \left( 1-\eta _{H}\right) \left( 1+\pi _{H}-\pi _{L}\right) }{\left( 1-\eta _{H}+\eta _{L}\right) \left( 1+\pi _{L}\right) +\left( 1-\eta _{L}\right) \left( 1-\eta _{H}\right) \left( 1+\pi _{H}-\pi _{L}\right) }\right) =0. \end{aligned}$$

(C9)

Now, we deal with the term $\left( p_{HL}/\theta _{H}\right) +1.$ At equilibrium, given (B2) and (10):

$$\begin{aligned}&\left( \tfrac{p_{HL}}{\theta _{H}}+1\right) =\\&\quad -\left( 1-\eta _{H}\right) \tfrac{\left( 1+\pi _{L}\right) \eta _{L}\frac{2-\eta _{H}}{1-\eta _{L}}+\left( 1-\eta _{H}\right) \left( 2+\pi _{H}\right) }{\left( 1+\pi _{L}\right) \eta _{L}\frac{2-\eta _{H}}{1-\eta _{L}}\left( 2\eta _{H}+\pi _{H}-1\right) +\left( 1-\eta _{H}\right) \left( 2+\pi _{H}\right) \eta _{H}+\left( 1+\pi _{L}\right) \left( \pi _{H}+\eta _{H}-1\right) \left( 1-\eta _{H}\right) } \end{aligned}$$

which, substituted into (C9) yields:

$$\begin{aligned} 1-\tfrac{\left( 1-\eta _{H}+\eta _{L}\right) \left( 1+\pi _{L}\right) \left( 2\eta _{H}+\pi _{H}-1\right) +\eta _{H}\left( 1-\eta _{L}\right) \left( 1-\eta _{H}\right) \left( 1+\pi _{H}-\pi _{L}\right) }{\left( 1+\pi _{L}\right) \eta _{L}\left( 2-\eta _{H}\right) \left( 2\eta _{H}+\pi _{H}-1\right) +\left( 1-\eta _{H}\right) \left[ 2+\pi _{H}\right] \eta _{H}\left( 1-\eta _{L}\right) +\left( 1+\pi _{L}\right) \left( \pi _{H}+\eta _{H}-1\right) \left( 1-\eta _{H}\right) \left( 1-\eta _{L}\right) }=0 \end{aligned}$$

which is actually true. Therefore Eq. (C8) collapses to

$$\begin{aligned} \frac{\partial ^{2}\Pi _{L}}{\partial s_{L}^{2}}=\left( 1-F_{H}\right) \theta _{HL}\Psi ^{\prime }\left( \theta _{H}\right) <0. \end{aligned}$$