Price competition and quality differentiation with multiproduct firms

Abstract

This paper examines a two-stage competition where firms simultaneously choose the number of products and qualities in the first stage, and then compete in prices. It is shown that a monopolist must sell a single product. In addition, in any equilibrium of multiproduct duopoly, there are segmented patterns of quality differentiation. Entangled configurations never emerge because each firm has an incentive to reduce the number of products facing direct competition with its rival. This result contrasts sharply with the equilibrium of non-segmented quality differentiation when firms compete in quantities. Furthermore, we find that the high-quality firm never offers more products than the low-quality firm, and quality differentiation between firms is greater than that within a firm.

Appendices

Appendix 1: Proof of Proposition 1

Proof

To ensure that it is not profitable for firm $A$ to deviate to configuration $(a)$, we must have $f\le 0.003\frac{\overline{\theta }^{3}}{\alpha }$. However, given $(q_{a1}^{*},q_{a2}^{*})=$ $(0.4\frac{\overline{ \theta }}{\alpha },0.2\frac{\overline{\theta }}{\alpha })$, we solve the first-order condition whereby firm $B$ maximizes its profit $\pi _{B}(aab)$ and then obtain $q_{b1}^{**}=0.105\frac{\overline{\theta }}{\alpha }$ and $\pi _{B}(0.4\frac{\overline{\theta }}{\alpha },0.2\frac{\overline{ \theta }}{\alpha },0.105\frac{\overline{\theta }}{\alpha })=0.005\frac{ \overline{\theta }^{3}}{\alpha }-f$. To ensure that firm $B$ has no incentive to enter the market, we require that $f\ge 0.005\frac{\overline{ \theta }^{3}}{\alpha }$. Thus, configuration $(aa)$ cannot be an SPNE. $\square$

Appendix 2: Proof of Remark 1

• (i) The following proof shows that the candidate $(aba)$ in Table 1, where $(q_{a1}^{*},q_{a2}^{*},q_{b1}^{*})= (0.421\frac{\overline{ \theta }}{\alpha },0.033\frac{\overline{\theta }}{\alpha },0.214\frac{ \overline{\theta }}{\alpha })$, cannot be selected as an SPNE.

Proof

Given $q_{b1}=0.214\frac{\overline{\theta }}{\alpha }$, by solving the first-order conditions whereby firm $A$ maximizes its profit $\pi _{A}(aab)$ by selling two products with qualities $(q_{a1},q_{a2})$ such that $q_{a1}>q_{a2}>0.214\frac{\overline{\theta }}{\alpha }$, we derive $q_{a1}=0.466\frac{\overline{\theta }}{\alpha }$ and $q_{a2}=0.398\frac{ \overline{\theta }}{\alpha }$. Substituting them into the profit of firm $A$ , we obtain $\pi _{A}(0.466\frac{\overline{\theta }}{\alpha },0.398\frac{ \overline{\theta }}{\alpha },0.214\frac{\overline{\theta }}{\alpha })=0.01533 \frac{\overline{\theta }^{3}}{\alpha }-2f$, which is always higher than $\pi _{A}(0.421\frac{\overline{\theta }}{\alpha },0.033\frac{\overline{\theta }}{ \alpha },0.214\frac{\overline{\theta }}{\alpha })=0.01526\frac{\overline{ \theta }^{3}}{\alpha }-2f$. $\square$

• (ii) The following proves that the candidate $(aab)$ in Table 1, where $(q_{a1}^{*},q_{a2}^{*},q_{b1}^{*})=(0.457\frac{\overline{\theta } }{\alpha },0.371\frac{\overline{\theta }}{\alpha },0.183\frac{\overline{ \theta }}{\alpha })$, cannot be selected as an SPNE.

Proof

Given that firm $B$ sells a single product with quality $0.183\frac{ \overline{\theta }}{\alpha }$, we solve the first-order condition whereby firm $A$ maximizes its profit $\pi _{A}(ab)$ and derive $q_{a1}=0.401\frac{ \overline{\theta }}{\alpha }$ and $\pi _{A}(0.401\frac{\overline{\theta }}{ \alpha },0.183\frac{\overline{\theta }}{\alpha })=0.0178773\frac{\overline{ \theta }^{3}}{\alpha }-f$. Accordingly, to ensure that firm $A$ will offer two products rather than one, it must be the case that $0.0182949\frac{ \overline{\theta }^{3}}{\alpha }-2f\ge 0.0178773\frac{\overline{\theta }^{3} }{\alpha }-f$, i.e., $f\le 0.0004176\frac{\overline{\theta }^{3}}{\alpha }$. Moreover, we solve the first-order condition that firm $B$ maximizes its profit $\pi _{B}(aabb)$ as firm $A$ sells two products with qualities $(0.457\frac{\overline{\theta }}{\alpha },0.371\frac{\overline{\theta }}{ \alpha })$, and obtain $q_{b1}=0.195\frac{\overline{\theta }}{\alpha }$, $q_{b1}=0.097\frac{\overline{\theta }}{\alpha }$ and $\pi _{B}(0.457\frac{ \overline{\theta }}{\alpha },0.371\frac{\overline{\theta }}{\alpha },0.195 \frac{\overline{\theta }}{\alpha },0.097\frac{\overline{\theta }}{\alpha } )=0.011071\frac{\overline{\theta }^{3}}{\alpha }-2f$. Therefore, to ensure that firm $B$ will not deviate to selling two products, we require that $0.0106517\frac{\overline{\theta }^{3}}{\alpha }-f\ge 0.0110711\frac{ \overline{\theta }^{3}}{\alpha }-2f$, i.e., $f\ge 0.0004194\frac{\overline{ \theta }^{3}}{\alpha }$. Thus, the candidate $(aab)$ cannot be an SPNE since, if it is profitable for firm $A$ to sell two products, firm $B$ must have an incentive to deviate to selling two products. $\square$

Appendix 3: No deviation conditions of configurations $(a), (abb)$ and $(aabb)$

$$\begin{aligned}&{\mathbf{Monopoly\,\, with\, \,a\,\, single\,\, product\,\, (a)}}. \\&\pi _{A}^{*}(a)\ge \max \{\pi _{A}(aa)\}\ \text { and} \\&0\ge \max \{\pi _{B}(ab),\pi _{B}(ba)\}\text { ; }0\ge \max \{\pi _{B}(abb),\pi _{B}(bab),\pi _{B}(bba)\}. \\&\pi _{A}^{*}(a)>0. \\&{\mathbf{Sandwich\,\, (abb).}} \\&\pi _{A}^{*}(abb)\ge \max \{\pi _{A}(bab),\pi _{A}(bba)\}\ \text { and } \pi _{B}^{*}(abb)\ge \max \{\pi _{B}(bab),\pi _{B}(bba)\}\text {.} \\&\pi _{A}^{*}(abb)\ge \max \{\pi _{A}(ab),\pi _{A}(ba)\} \ \text { and } \\&\pi _{A}^{*}(abb)\!\ge \! \max \{\pi _{A}(aabb),\pi _{A}(abba),\pi _{A}(abab),\pi _{A}(bbaa),\pi _{A}(baab),\pi _{A}(baba)\}\text {.} \\&\pi _{A}^{*}(abb)>0 \ \text { and }\ \pi _{B}^{*}(abb)>0. \\&{\mathbf{Enclosure\,\, (aabb).}} \\&\pi _{A}^{*}(aabb)\ge \max \{\pi _{A}(abba),\pi _{A}(abab),\pi _{A}(bbaa),\pi _{A}(baab),\pi _{A}(baba)\}\ \text { and } \\&\pi _{B}^{*}(aabb)\ge \max \{\pi _{B}(abba),\pi _{B}(abab),\pi _{B}(bbaa),\pi _{B}(baab),\pi _{B}(baba)\}. \\&\pi _{A}^{*}(aabb)\ge \max \{\pi _{A}(abb),\pi _{A}(bab),\pi _{A}(bba)\}\ \text { and} \\&\pi _{B}^{*}(aabb)\ge \max \{\pi _{B}(aab),\pi _{B}(aba),\pi _{B}(aab)\}. \\&\pi _{A}^{*}(aabb)>0\ \text{ and } \pi _{B}^{*}(aabb)>0. \end{aligned}$$