Quality and Price Competition in Spatial Markets

Abstract

We analyse the effects of competition on quality provision in spatial markets where providers compete on both price and quality and where travel distance is a key factor for consumers’ choice of provider. Examples of such markets include health care, long-term care, child care and education. While policy makers in many countries have introduced measures to stimulate consumer choice, and thus competition, in such markets, the scientific literature has not produced a consensual view about the resulting effects on quality provision. In this chapter we present a unified theoretical framework that synthesises a number of different insights from the existing theoretical literature. We show that the relationship between competition and quality provision is complex and highly ambiguous and that it is likely to depend on several factors related to both the demand and supply side of the market, such as providers’ degree of motivation or risk aversion, the cost relationship between quality and output and the presence of income effects in demand.

Appendices

Appendix 1

In this appendix we provide existence conditions for the Nash equilibrium in the static duopoly game and underlying details of the comparative statics results.

The second-order conditions for Provider i’s optimal choices of price and quality are given by

$$\displaystyle \begin{aligned} \frac{\partial ^{2}\Omega _{i}}{\partial p_{i}^{2}}&=v^{\prime }\left( \pi _{i}\right) \frac{1}{2\tau }\left[ \left( p_{i}-\frac{\partial c\left( D_{i},q_{i}\right) }{\partial D_{i}}\right) u^{\prime \prime }\left( y\left( p_{i}\right) \right) \right.\\&\quad \left.-\left( 2+\frac{\partial ^{2}c\left( D_{i},q_{i}\right) }{\partial D_{i}^{2}}\frac{u^{\prime }\left( y\left( p_{i}\right) \right) }{ 2\tau }\right) u^{\prime }\left( y\left( p_{i}\right) \right) \right] <0 \end{aligned} $$

(12.39)

and

$$\displaystyle \begin{aligned} \frac{\partial ^{2}\Omega _{i}}{\partial q_{i}^{2}} &=v^{\prime \prime }\left( \pi _{i}\right) \left[ \left( p_{i}-\frac{\partial c\left( D_{i},q_{i}\right) }{\partial D_{i}}\right) \frac{\beta }{2\tau }-\frac{ \partial c\left( D_{i},q_{i}\right) }{\partial q_{i}}\right] ^{2}\\ &\quad -v^{\prime }\left( \pi _{i}\right) \left[ \left( \frac{\beta }{2\tau } \frac{\partial ^{2}c\left( D_{i},q_{i}\right) }{\partial D_{i}^{2}}+\frac{ \partial ^{2}c\left( D_{i},q_{i}\right) }{\partial D_{i}\partial q_{i}} \right) +\frac{\partial ^{2}c\left( D_{i},q_{i}\right) }{\partial q_{i}^{2}} \right]\\ &\quad +b^{\prime \prime }\left( q_{i}\right)< 0. \end{aligned} $$

(12.40)

Since the first-order condition for the optimal price implies $p_{i}>\partial c\left ( D_{i},q_{i}\right ) /\partial D_{i}$, then (12.39) always holds, while (12.40) holds if output and quality are cost substitutes (i.e., $\partial ^{2}c\left ( D_{i},q_{i}\right ) /\partial D_{i}\partial q_{i}>0$) or, in case of cost complementarities, if the magnitude of $\left \vert \partial ^{2}c\left ( D_{i},q_{i}\right ) /\partial D_{i}\partial q_{i}\right \vert$ is not too large.

From (12.7)–(12.8), which implicitly define the symmetric Nash equilibrium, we derive

$$\displaystyle \begin{aligned} \frac{\partial F_{p}}{\partial p^{\ast }}=-\frac{u^{\prime }\left( y\left( p^{\ast }\right) \right) }{2\tau }+\left( p^{\ast }-\frac{\partial c\left( \frac{1}{2},q^{\ast }\right) }{\partial D_{i}}\right) \frac{u^{\prime \prime }\left( y\left( p^{\ast }\right) \right) }{2\tau }<0, \end{aligned} $$

(12.41)

$$\displaystyle \begin{aligned} \frac{\partial F_{q}}{\partial q^{\ast }}&=-v^{\prime \prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) \frac{\partial ^{2}c\left( \frac{1 }{2},q^{\ast }\right) }{\partial q_{i}^{2}}\\&\quad -v^{\prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) \left[ \frac{\partial ^{2}c\left( \frac{1 }{2},q^{\ast }\right) }{\partial D_{i}\partial q_{i}}\frac{\beta }{2\tau }+ \frac{\partial ^{2}c\left( \frac{1}{2},q^{\ast }\right) }{\partial q_{i}^{2}} \right] +b^{\prime \prime }\left( q^{\ast }\right) <0, \end{aligned} $$

(12.42)

$$\displaystyle \begin{aligned} \frac{\partial F_{p}}{\partial q^{\ast }}=\frac{\partial ^{2}c\left( \frac{1 }{2},q^{\ast }\right) }{\partial D_{i}\partial q_{i}}\frac{u^{\prime }\left( y\left( p^{\ast }\right) \right) }{2\tau }>\left( <\right) 0\mbox{ if }\frac{ \partial ^{2}c\left( \frac{1}{2},q^{\ast }\right) }{\partial D_{i}\partial q_{i}}>\left( <\right) 0, \end{aligned} $$

(12.43)

$$\displaystyle \begin{aligned} \frac{\partial F_{q}}{\partial p^{\ast }}&=\frac{v^{\prime \prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) }{2}\left[ \left( p^{\ast }-\frac{ \partial c\left( \frac{1}{2},q^{\ast }\right) }{\partial D_{i}}\right) \frac{ \beta }{2\tau }-\frac{\partial c\left( \frac{1}{2},q^{\ast }\right) }{ \partial q_{i}}\right] \\&\quad +v^{\prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) \frac{\beta }{2\tau }>0, \end{aligned} $$

(12.44)

$$\displaystyle \begin{aligned} \frac{\partial F_{p}}{\partial \tau }=\left( p^{\ast }-\frac{\partial c\left( \frac{1}{2},q^{\ast }\right) }{\partial D_{i}}\right) \frac{ u^{\prime }\left( y\left( p^{\ast }\right) \right) }{2\tau ^{2}}>0, \end{aligned} $$

(12.45)

$$\displaystyle \begin{aligned} \frac{\partial F_{q}}{\partial \tau }=-v^{\prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) \left( p^{\ast }-\frac{\partial c\left( \frac{1}{2 },q^{\ast }\right) }{\partial D_{i}}\right) \frac{\beta }{2\tau ^{2}}<0. \end{aligned} $$

(12.46)

Existence and stability of the Nash equilibrium require that ∂F _p∕∂p ^∗ < 0, ∂F _q∕∂q ^∗ < 0 and $\left ( \partial F_{p}/\partial p^{\ast }\right ) \left ( \partial F_{q}/\partial q^{\ast }\right ) -\left ( \partial F_{q}/\partial p^{\ast }\right ) \left ( \partial F_{p}/\partial q^{\ast }\right ) > 0$. Using (12.41)–(12.44), the latter condition is equivalent to the condition

$$\displaystyle \begin{aligned} &4\tau \left( \begin{array}{c} u^{\prime }\left( y\left( p^{\ast }\right) \right) \\ - u^{\prime \prime }\left( y\left( p^{\ast }\right) \right) \left( p^{\ast }-\frac{\partial c\left( \frac{1}{2},q^{\ast }\right) }{ \partial D_{i}}\right) \end{array} \right)\\ &\times\left( \begin{array}{c} \frac{\partial ^{2}c\left( \frac{1}{2},q^{\ast }\right) }{\partial q_{i}^{2}} \left( v^{\prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) +v^{\prime \prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) \right) \\ -b^{\prime \prime }\left( q^{\ast }\right) \end{array} \right)\\ &+\frac{\partial ^{2}c\left( \frac{1}{2},q^{\ast }\right) }{\partial D_{i}\partial q_{i}}\left( \begin{array}{c} 2\tau u^{\prime }\left( y\left( p^{\ast }\right) \right) v^{\prime \prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) \frac{\partial c\left( \frac{1}{2},q^{\ast }\right) }{\partial q_{i}} \\ -\beta \left[ \begin{array}{c} u^{\prime }\left( y\left( p^{\ast }\right) \right) v^{\prime \prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) \\ +2 u^{\prime \prime }\left( y\left( p^{\ast }\right) \right) v^{\prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) \end{array} \right] \left( p^{\ast }-\frac{\partial c\left( \frac{1}{2},q^{\ast }\right) }{\partial D_{i}}\right) \end{array} \right) \\ &\quad >0. \end{aligned} $$

(12.47)

If output and quality are cost substitutes, this condition is satisfied if the degree of provider risk-aversion (measured by $-v^{\prime \prime }\left ( \pi \right ) /v^{\prime }\left ( \pi \right )$) is not too large. In case of cost complementarity, we need to impose the additional condition that the magnitude of $\left \vert \partial ^{2}c\left ( D_{i},q_{i}\right ) /\partial D_{i}\partial q_{i}\right \vert$ is not too large.

Applying Cramer’s Rule, the effects of a marginal reduction in τ on equilibrium prices and qualities are given by, respectively,

$$\displaystyle \begin{aligned} \frac{\partial p^{\ast }}{\partial \tau }=\frac{\frac{\partial F_{q}}{ \partial \tau }\frac{\partial F_{p}}{\partial q^{\ast }}-\frac{\partial F_{p} }{\partial \tau }\frac{\partial F_{q}}{\partial q^{\ast }}}{\frac{\partial F_{p}}{\partial p^{\ast }}\frac{\partial F_{q}}{\partial q^{\ast }}-\frac{ \partial F_{q}}{\partial p^{\ast }}\frac{\partial F_{p}}{\partial q^{\ast }}} \end{aligned} $$

(12.48)

and

$$\displaystyle \begin{aligned} \frac{\partial q^{\ast }}{\partial \tau }=\frac{\frac{\partial F_{q}}{ \partial p^{\ast }}\frac{\partial F_{p}}{\partial \tau }-\frac{\partial F_{p} }{\partial p^{\ast }}\frac{\partial F_{q}}{\partial \tau }}{\frac{\partial F_{p}}{\partial p^{\ast }}\frac{\partial F_{q}}{\partial q^{\ast }}-\frac{ \partial F_{q}}{\partial p^{\ast }}\frac{\partial F_{p}}{\partial q^{\ast }}} . \end{aligned} $$

(12.49)

A sufficient (but not necessary) condition for the numerator in (12.48) to be positive (and thus for ∂p ^∗∕∂τ to be positive) is ∂F _p∕∂q ^∗≤ 0, which requires $\partial ^{2}c\left ( 1/2,q^{\ast }\right ) /\partial D_{i}\partial q_{i}\leq 0$. By applying the relevant expressions in (12.41)–(12.46), (12.48) can be expressed as (12.9) in Sect. 12.2. Using that expression, it is straightforward to show that $\partial p^{\ast }/\partial \tau >\left ( <\right ) 0$ if

$$\displaystyle \begin{aligned} \sigma _{v}:=-\frac{v^{\prime \prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) }{v^{\prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) }<\left( >\right) 1-\frac{b^{\prime \prime }\left( q^{\ast }\right) }{v^{\prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) \frac{ \partial ^{2}c\left( \frac{1}{2},q^{\ast }\right) }{\partial q_{i}^{2}}}. \end{aligned} $$

(12.50)

What is the scope for a price increase as a result of more competition, that is, ∂p ^∗∕∂τ < 0? Using again (12.48), ∂p ^∗∕∂τ < 0 requires

$$\displaystyle \begin{aligned} \frac{\partial F_{p}}{\partial q^{\ast }}>\frac{\frac{\partial F_{p}}{ \partial \tau }\frac{\partial F_{q}}{\partial q^{\ast }}}{\frac{\partial F_{q}}{\partial \tau }}. \end{aligned} $$

(12.51)

On the other hand, equilibrium existence requires

$$\displaystyle \begin{aligned} \frac{\partial F_{p}}{\partial q^{\ast }}<\frac{\frac{\partial F_{p}}{ \partial p^{\ast }}\frac{\partial F_{q}}{\partial q^{\ast }}}{\frac{\partial F_{q}}{\partial p^{\ast }}}. \end{aligned} $$

(12.52)

Thus, a price increase as a result of more competition is possible only if

$$\displaystyle \begin{aligned} \frac{\frac{\partial F_{p}}{\partial p^{\ast }}\frac{\partial F_{q}}{ \partial q^{\ast }}}{\frac{\partial F_{q}}{\partial p^{\ast }}}>\frac{\frac{ \partial F_{p}}{\partial \tau }\frac{\partial F_{q}}{\partial q^{\ast }}}{ \frac{\partial F_{q}}{\partial \tau }}. \end{aligned} $$

(12.53)

Using (12.41)–(12.46), this condition is equivalent to the condition

$$\displaystyle \begin{aligned} \frac{\sigma _{u}}{\sigma _{v}}>\frac{-\left( \left( p^{\ast }-\frac{ \partial c\left( \frac{1}{2},q^{\ast }\right) }{\partial D_{i}}\right) \beta -2\tau \frac{\partial c\left( \frac{1}{2},q^{\ast }\right) }{\partial q_{i}} \right) }{2\beta \left( p^{\ast }-\frac{\partial c\left( \frac{1}{2},q^{\ast }\right) }{\partial D_{i}}\right) }, \end{aligned} $$

(12.54)

which is identical to the condition for ∂q ^∗∕∂τ < 0 , given by (12.13) in Sect. 12.2. Thus, higher prices as a result of more competition is only possible if quality also increases.

Appendix 2

In this appendix we provide a relatively brief and general overview of the solution procedures for deriving the open-loop and feedback closed-loop equilibria presented in Sect. 12.2.3. For further details, we refer to Cellini et al. (2018).

1.1 I Open-Loop Solution

Given the optimisation problem of Provider i defined by (12.17), the current-value Hamiltonian is given by¹⁹

$$\displaystyle \begin{aligned} H_{i}&=\left( p_{i}-c\right) \left( \frac{1}{2}+\frac{\beta \left( q_{i}-q_{j}\right) }{2\tau }-\frac{p_{i}-p_{j}}{2\tau }\right) \\&\quad -\frac{\gamma }{2}I_{i}^{2}-\frac{\theta }{2}q_{i}{}^{2}+\mu _{i}\left( I_{i}-\delta q_{i}\right) +\mu _{j}\left( I_{j}-\delta q_{j}\right), \end{aligned} $$

(12.55)

where μ _i and μ _j are the current-value co-state variables associated with the two state equations. The solution is a set of linear differential equations that satisfy the following conditions: (i) ∂H _i∕∂I _i = 0, (ii) ∂H _i∕∂p _i = 0, (iii) $\dot {\mu }_{i}=\rho \mu _{i}-\partial H_{i}/\partial q_{i}$, (iv) $\dot {q} _{i}=\partial H_{i}/\partial \mu _{i}$ and (v) $\dot {\mu }_{j}=\rho \mu _{j}-\partial H_{i}/\partial q_{j}$. In addition, the solution has to satisfy the transversality condition lim_t→+∞e ^−ρtμ _i(t)q _i(t) = 0. The second-order conditions are satisfied if the Hamiltonian is concave in the control and state variables (Léonard and van Long 1992), which requires $H_{I_{i}I_{i}}\ =-\ \gamma <0$, $H_{p_{i}p_{i}}=-1/\tau <0$, $H_{q_{i}q_{i}}=-\theta <0$ and $H_{p_{i}p_{i}}H_{q_{i}q_{i}}-\left ( H_{q_{i}p_{i}}\right ) ^{2}=\left ( 4\tau \theta -\beta ^{2}\right ) /4\tau ^{2}>0$. Thus, the solution exists if the cost parameters τ and/or θ are sufficiently large relative to the marginal willingness to pay for quality, β.

From conditions (i)–(v) we can derive

$$\displaystyle \begin{aligned} p_{i}=w+\tau +\frac{\beta (q_{i}-q_{j})}{3}, \end{aligned} $$

(12.56)

$$\displaystyle \begin{aligned} \dot{I}_{i}=\left( \delta +\rho \right) I_{i}+\frac{\theta }{\gamma }q_{i}- \frac{\beta \left( p_{i}-c\right) }{2\tau \gamma }, \end{aligned} $$

(12.57)

which characterise the optimal prices set in each period and the optimal dynamic investment path. In the steady state, $\dot {q}_{i}=\dot {I}_{i}=0$, q _i = q _j = q ^OL and p _i = p _j = p ^OL. Using (12.56)–(12.57) and the dynamic constraint (12.14), the steady-state prices and qualities are given by (12.18)–(12.19) in Sect. 12.2.

1.2 I Feedback Closed-Loop Solution

To solve for the feedback closed-loop solution, we restrict attention to stationary linear Markovian strategies, which we derive using the Hamilton-Jacobi-Bellman (HJB) equation. Since the instantaneous objective function and the linear dynamic constraint of Provider i constitute a linear-quadratic problem, we define the value function as

$$\displaystyle \begin{aligned} V^{i}(q_{i},q_{j})=\alpha _{0}+\alpha _{1}q_{i}+\alpha _{2}q_{j}+(\alpha _{3}/2)q_{i}^{2}+(\alpha _{4}/2)q_{j}^{2}+\alpha _{5}q_{i}q_{j}. \end{aligned} $$

(12.58)

If we define the investment strategies as functions I _i = ϕ _i(q _i, q _j) and I _j = ϕ _j(q _i, q _j), the above-defined value must satisfy the HJB equation, given by

$$\displaystyle \begin{aligned} \rho V^{i}(q_{i},q_{j})=\max \left \{ \begin{array}{c} \left( p_{i}-c\right) \left( \frac{1}{2}+\frac{\beta \left( q_{i}-q_{j}\right) }{2\tau }-\frac{p_{i}-p_{j}}{2\tau }\right) -\frac{\gamma }{2}I_{i}^{2}-\frac{\theta }{2}q_{i}{}^{2} \\ +V_{q_{i}}^{i}(q_{i},q_{j})\left( I_{i}-\delta q_{i}\right) +V_{q_{j}}^{i}(q_{i},q_{j})\left( I_{j}-\delta q_{j}\right) \end{array} \right \} . \end{aligned} $$

(12.59)

Maximisation of the right-hand side of (12.59) with respect to I _i yields $V_{q_{i}}^{i}=\gamma I_{i}$, which after substitution yields

$$\displaystyle \begin{aligned} I_{i}=\phi _{i}(q_{i},q_{j})=\frac{\alpha _{1}+\alpha _{3}q_{i}+\alpha _{5}q_{j}}{\gamma },\quad i,j=1,2;\quad i\neq j. \end{aligned} $$

(12.60)

Maximisation of the right-hand side of (12.59) with respect to p _i yields

$$\displaystyle \begin{aligned} \frac{1}{2}+\frac{\beta \left( q_{i}-q_{j}\right) }{2\tau }-\frac{p_{i}-p_{j} }{2\tau }-\left( p_{i}-w\right) \frac{1}{2\tau }=0,\quad i,j=1,2;\quad i\neq j. \end{aligned} $$

(12.61)

Solving the two-equation system defined by (12.61) yields

$$\displaystyle \begin{aligned} p_{i}=w+\tau +\frac{\beta \left( q_{i}-q_{j}\right) }{3},\quad i,j=1,2;\quad i\neq j. \end{aligned} $$

(12.62)

By substituting I _i = ϕ _i(q _i, q _j), I _j = ϕ _j(q _i, q _j), $V_{q_{i}}^{i}(q_{i},q_{j})=\alpha _{1}+\alpha _{3}q_{i}+\alpha _{5}q_{j}$, $V_{q_{j}}^{i}=\alpha _{2}+\alpha _{4}q_{j}+\alpha _{5}q_{i}$ and V ⁱ, as defined by (12.58), into the HJB equation, we obtain

$$\displaystyle \begin{aligned} &\left( \rho \alpha _{0}-\frac{1}{2}\tau -\frac{1}{2\gamma }\alpha _{1}^{2}- \frac{1}{\gamma } \alpha _{1}\alpha _{2}\right)\\ &\quad +q_{i}\left(\alpha _{1}\left( \delta +\rho \right) -\frac{1}{3 }\beta -\frac{1}{\gamma }\alpha _{1}\alpha _{3}-\frac{1}{\gamma }\alpha _{2}\alpha _{5}-\frac{1}{\gamma }\alpha _{1}\alpha _{5}\right)\\ &\quad +q_{j}\left( \alpha _{2}\left( \delta +\rho \right) +\frac{1}{3}\beta - \frac{1}{\gamma }\alpha _{2}\alpha _{3}-\frac{1}{\gamma }\alpha _{1}\alpha _{4}-\frac{1}{\gamma }\alpha _{1}\alpha _{5}\right)\\ &\quad +q_{i}^{2}\left( \alpha _{3} \left( \delta +\frac{1}{2}\rho \right) -\frac{1}{2\gamma }\alpha _{3}^{2}-\frac{1}{\gamma }\alpha _{5}^{2}+ \frac{1}{2}\theta -\frac{1}{18}\frac{\beta ^{2}}{\tau }\right) \\ &\quad +q_{j}^{2}\left( \alpha _{4}\left( \delta +\frac{1}{2}\rho \right) - \frac{1}{\gamma }\alpha _{3}\alpha _{4}-\frac{1}{2\gamma } \alpha _{5}^{2}- \frac{1}{18}\frac{\beta ^{2}}{\tau }\right)\\ &\quad +q_{i}q_{j}\left( \left( 2\delta +\rho \right) \alpha _{5}+\frac{1}{9} \frac{\beta ^{2}}{\tau }-\frac{2}{\gamma }\alpha _{3}\alpha _{5} - \frac{1}{\gamma }\alpha _{4}\alpha _{5}\right)\\ &\quad =0 \end{aligned} $$

(12.63)

There are six possible solutions. By imposing the (sufficient but not necessary) condition

$$\displaystyle \begin{aligned} \frac{9}{4}\tau \left( \left( \delta +\frac{1}{2}\rho \right) ^{2}\gamma +\theta \right) \geq \beta ^{2}, \end{aligned} $$

(12.64)

it can be shown that only one of these solutions satisfy the criteria for equilibrium existence.²⁰ In this unique solution, the coefficients of the equilibrium dynamic investment strategy, (12.21), are given by

$$\displaystyle \begin{aligned} \alpha _{1}=\frac{\beta \gamma }{3\left( \gamma \left( \delta +\rho \right) -\alpha _{3}\right) }>0, \end{aligned} $$

(12.65)

$$\displaystyle \begin{aligned} \alpha _{3}=\left( \delta +\frac{1}{2}\rho \right) \gamma -\left( \frac{ 36\tau \left( \left( \delta +\frac{1}{2}\rho \right) ^{2}\gamma +\theta \right) -5\beta ^{2}}{4\beta ^{2}}-\frac{81\tau \psi }{16\beta ^{2}\gamma } \right) \sqrt{\psi }<0, \end{aligned} $$

(12.66)

$$\displaystyle \begin{aligned} \alpha _{5}=-\frac{1}{2}\sqrt{\psi }<0, \end{aligned} $$

(12.67)

where

$$\displaystyle \begin{aligned} \psi :=\frac{4\gamma }{27}\left( 6\left( \left( \delta +\frac{1}{2}\rho \right) ^{2}\gamma +\theta \right) -\frac{\beta ^{2}}{\tau }-2\sqrt{3\varphi }\right) >0 \end{aligned} $$

(12.68)

and

$$\displaystyle \begin{aligned} \varphi :=\left( 3\left( \left( \delta +\frac{1}{2}\rho \right) ^{2}\gamma +\theta \right) -\frac{\beta ^{2}}{\tau }\right) \left( \left( \delta +\frac{ 1}{2}\rho \right) ^{2}\gamma +\theta \right) >0. \end{aligned} $$

(12.69)

The steady-state solution, (12.22)–(12.23), is then derived by setting q _i = q _j = I _i∕δ, which implies $\dot {I}_{i}=0$.

Appendix 3

In this appendix we provide the underlying details of the comparative statics results in the n-firm Salop model presented in Sect. 12.3.

From (12.27)–(12.28), which implicitly define the symmetric Nash equilibrium, we derive

$$\displaystyle \begin{aligned} \frac{\partial G_{p}}{\partial p^{\ast }}=-\frac{u^{\prime }\left( y\left( p^{\ast }\right) \right) }{\tau }+\left( p^{\ast }-\frac{\partial c\left( \frac{1}{n},q^{\ast }\right) }{\partial D_{i}}\right) \frac{u^{\prime \prime }\left( y\left( p^{\ast }\right) \right) }{\tau }<0, \end{aligned} $$

(12.70)

$$\displaystyle \begin{aligned} \frac{\partial G_{q}}{\partial q^{\ast }}&=-v^{\prime \prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) \frac{\partial ^{2}c\left( \frac{1 }{n},q^{\ast }\right) }{\partial q_{i}^{2}}\\&\quad -v^{\prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) \left[ \frac{\partial ^{2}c\left( \frac{1 }{n},q^{\ast }\right) }{\partial D_{i}\partial q_{i}}\frac{\beta }{\tau }+ \frac{\partial ^{2}c\left( \frac{1}{n},q^{\ast }\right) }{\partial q_{i}^{2}} \right] +b^{\prime \prime }\left( q^{\ast }\right) <0, \end{aligned} $$

(12.71)

$$\displaystyle \begin{aligned} \frac{\partial G_{p}}{\partial q^{\ast }}=\frac{\partial ^{2}c\left( \frac{1 }{n},q^{\ast }\right) }{\partial D_{i}\partial q_{i}}\frac{u^{\prime }\left( y\left( p^{\ast }\right) \right) }{\tau }>\left( <\right) 0\mbox{ if }\frac{ \partial ^{2}c\left( \frac{1}{2},q^{\ast }\right) }{\partial D_{i}\partial q_{i}}>\left( <\right) 0, \end{aligned} $$

(12.72)

$$\displaystyle \begin{aligned} \frac{\partial G_{q}}{\partial p^{\ast }}&=v^{\prime \prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) \frac{1}{n}\left[ \left( p^{\ast }- \frac{\partial c\left( \frac{1}{n},q^{\ast }\right) }{\partial D_{i}}\right) \frac{\beta }{\tau }-\frac{\partial c\left( \frac{1}{n},q^{\ast }\right) }{ \partial q_{i}}\right] \\&\quad +v^{\prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) \frac{\beta }{\tau }>0, \end{aligned} $$

(12.73)

$$\displaystyle \begin{aligned} \frac{\partial G_{p}}{\partial n}=-\frac{1}{n^{2}}\left( 1+\frac{\partial ^{2}c\left( \frac{1}{n},q^{\ast }\right) }{\partial D_{i}^{2}}\frac{ u^{\prime }\left( y\left( p^{\ast }\right) \right) }{\tau }\right) <0, \end{aligned} $$

(12.74)

$$\displaystyle \begin{aligned} \frac{\partial G_{q}}{\partial n}=\frac{v^{\prime }\left( \pi \left( p^{\ast },q^{\ast }\right) \right) }{n^{2}}\left[ \frac{\partial ^{2}c\left( \frac{1 }{n},q^{\ast }\right) }{\partial D_{i}^{2}}\frac{\beta }{\tau }+\frac{ \partial ^{2}c\left( \frac{1}{n},q^{\ast }\right) }{\partial D_{i}\partial q_{i}}\right] \lessgtr 0. \end{aligned} $$

(12.75)

Existence and stability of the Nash equilibrium require that ∂G _p∕∂p ^∗ < 0, ∂G _q∕∂q ^∗ < 0 and $\left ( \partial G_{p}/\partial p^{\ast }\right ) \left ( \partial G_{q}/\partial q^{\ast }\right ) -\left ( \partial G_{q}/\partial p^{\ast }\right ) \left ( \partial G_{p}/\partial q^{\ast }\right ) >0$. In qualitative terms, these conditions are similar to the equivalent conditions in the duopoly version of the model.

Applying Cramer’s Rule, the effects of a marginal increase in n on equilibrium prices and qualities are given by, respectively,

$$\displaystyle \begin{aligned} \frac{\partial p^{\ast }}{\partial n}=\frac{\frac{\partial G_{q}}{\partial n} \frac{\partial G_{p}}{\partial q^{\ast }}-\frac{\partial G_{p}}{\partial n} \frac{\partial G_{q}}{\partial q^{\ast }}}{\frac{\partial G_{p}}{\partial p^{\ast }}\frac{\partial G_{q}}{\partial q^{\ast }}-\frac{\partial G_{q}}{ \partial p^{\ast }}\frac{\partial G_{p}}{\partial q^{\ast }}} \end{aligned} $$

(12.76)

and

$$\displaystyle \begin{aligned} \frac{\partial q^{\ast }}{\partial n}=\frac{\frac{\partial G_{q}}{\partial p^{\ast }}\frac{\partial G_{p}}{\partial n}-\frac{\partial G_{p}}{\partial p^{\ast }}\frac{\partial G_{q}}{\partial n}}{\frac{\partial G_{p}}{\partial p^{\ast }}\frac{\partial G_{q}}{\partial q^{\ast }}-\frac{\partial G_{q}}{ \partial p^{\ast }}\frac{\partial G_{p}}{\partial q^{\ast }}}. \end{aligned} $$

(12.77)