Note on a profit-raising entry effect in a differentiated Cournot oligopoly market with network compatibility

Abstract

Based on a differentiated Cournot oligopoly model with network externalities, we consider how an increase in the number of firms affects an individual firm’s output and profit, i.e., a profit-raising entry effect. We demonstrate that if the degree of network compatibilities is larger (smaller) than that of product substitutability, an increase in the number of firms increases (decreases) the individual firm’s output and profit. That is, a profit-raising (-lowering) entry effect arises. In the case of a profit-raising entry, if the number of firms is sufficiently large, consumer surplus declines although social welfare increases overall because this decline is offset by a sufficient producer surplus increase.

Appendix

We consider the case of responsive expectations, in which consumers form expectations for network sizes after firms make their output decisions. This implies that firms can commit to their output levels, so that consumers trust in the output levels and then form expectations for the network sizes (active beliefs), i.e., $q_{i}^{E} = q_{i} ,$ where $i = 1, \ldots ,n$. Thus, it holds that $S_{i}^{E} = S_{i} = q_{i} + \alpha Q_{ - i} ,$ $i, - i = 1, \ldots ,n,i \ne - i,$ where $S_{i}$ denotes the actual network size of firm i’s product. Hereafter we assume that $i, - i = 1, \ldots ,n,i \ne - i.$ Taking Eqs. (2) and (3), we derive the following inverse demand function of product i:

$$p_{i} = A - (1 - \phi )q_{i} - (\gamma - \phi \alpha )Q_{ - i} ,$$

(18)

where we assume that $1 - \phi > \left| {\gamma - \phi \alpha } \right|(n - 1),$ which implies that the own-price effect exceeds the summation of the cross-price effect, i.e., $\left| {\frac{{\partial p_{i} }}{{\partial q_{i} }}} \right| > \sum\nolimits_{ - i = 1}^{n} {\left| {\frac{{\partial p_{i} }}{{\partial q_{ - i} }}} \right|} .$ In particular, there is an upper limit on the number of firms as follows: $\overline{n} \equiv \frac{1 - \phi }{{\left| {\gamma - \phi \alpha } \right|}} + 1 > n.$

Equation (4) can be rewritten as:

$$\pi_{i} = p_{i} q_{i} = \left\{ {A - (1 - \phi )q_{i} - (\gamma - \phi \alpha )Q_{ - i} } \right\}q_{i} .$$

(19)

The first-order condition for profit maximization by firm i is given by:

$$\frac{{\partial \pi_{i} }}{{\partial q_{i} }} = p_{i} - (1 - \phi )q_{i} = A - 2(1 - \phi )q_{i} - (\gamma - \phi \alpha )Q_{ - i} = 0.$$

(20)

Using equation (20), we derive the following reaction function for firm i, which differs from the reaction function in the case of passive expectations:

$$q_{i} = \frac{A}{2(1 - \phi )} - \frac{\gamma - \phi \alpha }{{2(1 - \phi )}}Q_{ - i} .$$

(21)

Equation (21) shows that the strategic relationship between firms depends on the degree of network compatibilities and product substitutability, i.e., $\frac{{\partial q_{i} }}{{\partial q_{ - i} }} > ( < )0 \Leftrightarrow \phi \alpha > ( < )\gamma .$ Thus, a strategic complement (substitute) emerges if the degree of network compatibilities is larger (smaller) than that of product substitutability.

Using equation (21), we obtain a symmetric Cournot equilibrium under responsive expectations, i.e., $q_{i} = q_{ - i} = q^{r} .$ as follows:

$$q^{r} = \frac{A}{2(1 - \phi ) + (\gamma - \phi \alpha )(n - 1)},$$

(22)

where $q^{r} > 0 \Leftrightarrow \tilde{n} \equiv \frac{2(1 - \phi )}{{\phi \alpha - \gamma }} + 1 > n,$ if $\phi \alpha > \gamma .$¹⁰ Superscript r denote the case of responsive expectations.

We obtain the following entry effect on the individual output:

$$\frac{{dq^{r} }}{dn} = \left( {q^{r} } \right)^{2} \frac{\phi \alpha - \gamma }{A} > ( < )0 \Leftrightarrow \phi \alpha > ( < )\gamma .$$

(23)

Equation (23) demonstrates that if the degree of network compatibilities is larger (smaller) than that of product substitutability, an increase in the number of firms increases (decreases) the output level. This is because strategic complements (substitutes) arise under larger (smaller) network compatibilities.

Using the first-order condition of Eq. (20), the individual firm’s profit is represented by $\pi^{r} = (1 - \phi )(q^{r} )^{2} .$ Thus, it follows that

$$\frac{{d\pi^{r} }}{dn} = 2(1 - \phi )q^{r} \frac{{dq^{r} }}{dn} > ( < )0 \Leftrightarrow \phi \alpha > ( < )\gamma .$$

(24)

Equations (23) and (24) show that Proposition 1 holds in the case of responsive expectations.

Similarly, with respect to the entry effect on total output, $Q^{r} = nq^{r} ,$ it holds that:

$$\frac{{dQ^{r} }}{dn} = q^{r} + n\frac{{dq^{r} }}{dn} = \frac{2(1 - \phi ) - (\gamma - \phi \alpha )}{{2(1 - \phi ) + (\gamma - \phi \alpha )(n - 1)}}q^{r} > 0.$$

(25)

Thus, an increase in the number of firms increases total output. However, because of $p^{r} = (1 - \phi )q^{r} ,$ we have $\frac{{dp^{r} }}{dn} = (1 - \phi )\frac{{dq^{r} }}{dn}.$ Based on (A.6), if $\phi \alpha > \gamma ,$ the price increases.

Therefore, by the same procedure as in Sect. 2.4, we can derive the same results as in the case of passive expectations, i.e., Proposition 2.