Note on the excess entry theorem in the presence of network externalities

Abstract

Focusing on the type of consumer expectations, i.e., passive and responsive expectations, we reconsider the excess entry theorem in the case of Cournot oligopoly with network externalities. We demonstrate that whether the number of firms under free entry is socially excessive or insufficient depends on the type of consumer expectations and the degree of network externalities. That is, in the case of passive expectations, if the degree of network externalities is sufficiently large (small), the number of firms under free entry is socially insufficient (excessive), based on the second-best criteria. However, in the case of responsive expectations, the number of firms under free entry is socially excessive, based on the second-best criteria.

Appendix: Excess entry theorem in a general function case

We assume the surplus function of a representative consumer in a homogeneous product market with network externalities as follows: $V = U(Q,F(S^{E} )) - PQ$. Thus, an inverse demand function with network externalities is given by:

$$P = \frac{\partial U}{\partial Q}(Q,F(S^{E} )) \equiv P(Q,F(S^{E} )).$$

(24)

In the following sections, we generally consider the excess entry theorem in the cases of passive and responsive expectations.

1.1 The case of passive expectations

Given Eq. (24), the profit function of firm i is expressed as: $\pi_{i} = P(Q,F(S^{E} ))q_{i} ,\quad i = 1, \ldots ,n$. Firm i determines the output to maximize the profit, given the other firms’ outputs and expected network sizes by consumers. The FOC is given by:

$$\frac{{\partial \pi_{i} }}{{\partial q_{i} }} = P(Q,F(S^{E} )) + \frac{\partial P}{\partial Q}(Q,F(S^{E} ))q_{i} = 0,\quad i = 1, \ldots ,n.$$

(25)

In a fulfilled and symmetric equilibrium, i.e.,$S^{E} = Q$ and $q_{i} = q_{ - i} = q^{p} ,\quad i, - i = 1, \ldots ,n,i \ne - i$ Eq. (25) can be rewritten as:

$$P(Q^{p} ,F(Q^{p} )) + \frac{\partial P}{\partial Q}(Q^{p} ,F(Q^{p} ))q^{p} = 0,$$

(26)

where $Q^{p} = nq^{p} (n)$.

From Eq. (24), the welfare function with network externalities based on the second-best criteria can be represented by:

$$W^{p} (n) = \int\limits_{0}^{{Q^{p} }} {P(X,F(Q^{p} ))dX - ng} .$$

(27)

Given Eq. (27), the socially second-best number of firms is given by: $n^{p*} = \arg \hbox{max} W^{p} (n)$. In particular, the FOC is given by:

$$\frac{{\partial W^{p} }}{\partial n} = P^{p} \frac{{\partial Q^{p} }}{\partial n} + \int\limits_{0}^{{Q^{p} }} {\frac{\partial P}{\partial F}\frac{\partial F}{{\partial Q^{p} }}\frac{{\partial Q^{p} }}{\partial n}dX} - g = 0,$$

(28)

where $P^{p} \equiv P\left( {Q^{p} ,F(Q^{p} )} \right)$, $\frac{\partial P}{\partial F} = \frac{\partial P}{\partial F}\left( {X,F(Q^{p} )} \right)$, and $\frac{{\partial Q^{p} }}{\partial n} = q^{p} + \frac{{\partial q^{p} }}{\partial n}n$.

Using Eq. (28) and the zero-profit condition, we obtain the following equation:

$$\left. {\frac{{\partial W^{p} }}{\partial n}} \right|_{{P^{p} q^{p} - g = 0}} = P^{p} \frac{{\partial q^{p} }}{\partial n}n + \frac{\partial F}{{\partial Q^{p} }}\frac{{\partial Q^{p} }}{\partial n}\int\limits_{0}^{{Q^{p} }} {\frac{\partial P}{\partial F}dX} .$$

(29)

Following the terminology of Mankiw and Whinston (1986), the first term in Eq. (29) is a business-stealing effect and the second a business-augmenting effect by network externalities. Thus, in general, we obtain the following relationship:

$$\left. {\frac{{\partial W^{p} }}{\partial n}} \right|_{{P^{p} q^{p} - g = 0}} > ( < )0 \Leftrightarrow \frac{\partial F}{{\partial Q^{p} }}\frac{{\partial Q^{p} }}{\partial n}\int\limits_{0}^{{Q^{p} }} {\frac{\partial P}{\partial F}dX} > ( < ) - P^{p} \frac{{\partial q^{p} }}{\partial n}n.$$

(30)

Equation (30) is a general condition to prove whether the number of firms under free entry is socially excessive or not, based on the second-best criteria. For example, as in the text, assuming a linear inverse demand function with an additively separable and linear network function, i.e., $\frac{\partial P}{\partial F} = 1$ and $\frac{\partial F}{{\partial Q^{p} }} = \phi$, we obtain Eq. (20).

1.2 The case of responsive expectations

In this case, because it holds that $S^{E} = Q$, Eq. (24) can be revised as follows: $P = P(Q,F(Q))$. The profit function of firm i is expressed as: $\pi_{i} = P(Q,F(Q))q_{i} ,\quad i = 1, \ldots ,n$. In this case, firms in the market take expected network sizes by consumers into account in advance, given the other firms’ outputs. Thus, the FOC is given by:

$$\frac{{\partial \pi_{i} }}{{\partial q_{i} }} = P(Q,F(Q)) + \left\{ {\frac{\partial P}{\partial Q}(Q,F(Q)) + \frac{\partial P}{\partial F}(Q,F(Q))\frac{\partial F}{\partial Q}(Q)} \right\}q_{i} = 0.$$

(31)

In a symmetric equilibrium, i.e., $q_{i} = q_{ - i} = q^{r} ,\quad i, - i = 1, \ldots ,n,i \ne - i$, Eq. (31) can be rewritten as:

$$P(Q^{r} ,F(Q^{r} )) + \left\{ {\frac{\partial P}{\partial Q}(Q^{r} ,F(Q^{r} )) + \frac{\partial P}{\partial F}(Q^{r} ,F(Q^{r} ))\frac{\partial F}{\partial Q}(Q^{r} )} \right\}q^{r} = 0,$$

(32)

where $Q^{r} = nq^{r} (n)$.

By a similar procedure in the case of passive expectations, the welfare function with network externalities based on the second-best criteria can be represented by:

$$W^{r} (n) = \int\limits_{0}^{{Q^{r} }} {P(X,F(X))dX - ng} .$$

(33)

Given Eq. (33), the socially second-best number of firms is given by: $n^{r*} = \arg \hbox{max} W^{r} (n)$. In particular, the FOC is given by:

$$\frac{{\partial W^{r} }}{\partial n} = P^{r} \frac{{\partial Q^{r} }}{\partial n} - g = 0,$$

(34)

where $P^{r} \equiv P\left( {Q^{r} ,F(Q^{r} )} \right)$ and $\frac{{\partial Q^{r} }}{\partial n} = q^{r} + \frac{{\partial q^{r} }}{\partial n}n$.

Using Eq. (30) and the zero-profit condition, we obtain the following equation:

$$\left. {\frac{{\partial W^{r} }}{\partial n}} \right|_{{P^{r} q^{r} - g = 0}} = P^{r} \frac{{\partial q^{r} }}{\partial n}n,$$

(35)

where $P^{r} \frac{{\partial q^{r} }}{\partial n}n$ expresses a business-stealing effect. That is, Eq. (35) shows that the excess entry theorem generally holds in the case of responsive expectations, irrespective of the presence of network externalities. As in the text, assuming a linear inverse demand function, i.e., Eq. (5), we obtain Eq. (23).