A Note on Quality Choice, Monopoly, and Network Externality

Abstract

Introducing network externalities into a model of vertically differentiated products, Lambertini and Orsini (2001, 2003) analyze the implications of a monopolist’s quality choice for social optimum. Moreover, they examine how the network externality affects quality, quantity, price, and social surplus. In this note, by looking at the nature of cost functions and the degree of network externalities, we reconsider their results, at least some of which depend upon the specificity of the cost functions.

Appendix

First, let us assume C(q, x) = c(q)x, where $\varepsilon = \frac{{c\prime {\left( q \right)}q}} {{c{\left( q \right)}}} > 1$. Given Eqs. 8 and 14, we have: $x_{M} {\left( {q,\alpha } \right)} = \frac{{q - c{\left( q \right)}}} {{2{\left( {q - \alpha } \right)}}}$ and $x_{S} {\left( {q,\alpha } \right)} = \frac{{q - c{\left( q \right)}}} {{q - 2\alpha }}$. Hence, the following properties hold:

• (i) $x_{S} {\left( {q,\alpha } \right)} > x_{M} {\left( {q,\alpha } \right)}$ for $\frac{q} {2} > \frac{{c{\left( q \right)}}} {2} > \alpha \ge 0$.

• (ii) $\frac{{\partial x_{M} {\left( {q,\alpha } \right)}}} {{\partial q}} < 0$ and $\frac{{\partial x_{S} {\left( {q,\alpha } \right)}}} {{\partial q}} < 0$.

  Substituting x _M (q, α) and x _S (q, α) into Eqs. 7 and 13, respectively, we obtain:

  $$ {G_{M} {\left( {q,\alpha } \right)} = {{q - 2\alpha + c{\left( q \right)}} \over {2{\left( {q - \alpha } \right)}}}\,{\rm{and}}\,G_{S} {\left( {q,\alpha } \right)} = {{q - 4\alpha + c{\left( q \right)}} \over {2{\left( {q - 2\alpha } \right)}}}.} $$

  Hence, the properties of these functions are:

• (iii) $G_{M} {\left( {q,\alpha } \right)} > G_{S} {\left( {q,\alpha } \right)}$ for $\frac{q} {2} > \frac{{c{\left( q \right)}}} {2} > \alpha$,

• (iv) $c\prime \prime {\left( q \right)} > \frac{{\partial G_{M} {\left( {q,\alpha } \right)}}}{{\partial q}} > 0$ and $c\prime \prime {\left( q \right)} > \frac{{\partial G_{S} {\left( {q,\alpha } \right)}}}{{\partial q}} > 0$ given Eqs. 12 and 18.

Let us define the monopoly equilibrium quality and the social optimum quality, respectively, as: $q_{M} = {\left\{ {q\left| {c\prime } \right.{\left( q \right)} = G_{M} {\left( {q,\alpha } \right)}} \right\}}$ and $q_{S} = {\left\{ {q\left| {c\prime {\left( q \right)}} \right. = G_{S} {\left( {q,\alpha } \right)},} \right\}}$. Taking into account (iii) and (iv), we have shown q _M > q _S for α > 0. If α = 0, it happens to hold that q _M  =  q _S , because $G_{M} {\left( {q,0} \right)} = G_{S} {\left( {q,0} \right)}$.

Second, based on (ii), it holds that $x_{M} {\left( {q_{S} ,\alpha } \right)} > x_{M} {\left( {q_{M} ,\alpha } \right)}$, given q _M > q _S . Moreover, given (i), we have $x_{S} {\left( {q_{S} ,\alpha } \right)} > x_{M} {\left( {q_{S} ,\alpha } \right)}$. Thus, x _S > x _M for α ≥ 0. Q.E.D.