Note on Excess Capacity in a Monopoly Market with Network Externalities

Abstract

Using a capacity-then-production choice model, we consider whether excess capacity results hold, when there is one provider, i.e., a monopoly or a public firm, in a network product and service market. In the case of a monopoly, if consumers form expectations of network sizes after (before) a capacity-scale decision, the capacity scale is larger than (equal to) the production quantity. Thus, excess capacity results hold (do not hold). Furthermore, in the case of a public firm, excess capacity results do not hold, irrespective of the timing of consumer expectations. However, this result depends on the specification of the utility function.

Appendices

Appendix 1

The Case of Responsive Expectations

In the case of responsive expectations, in which consumers believe the announced production quantity of the monopoly (active beliefs, i.e., y = q), the monopoly can commit to the production quantity before consumers form expectations of network size. Accordingly, the inverse demand function, i.e., Eq. (2), is revised as $p\left(q,y=q;n\right)=A-\left(1-n\right)q=\overset{\frown }{p}\left(q;n\right).$ Thus, the profit function, i.e., Eq. (4), is expressed as:

$$ \Pi \left(q,x;n\right)=\overset{\frown }{p}\left(q;n\right)q-C\left(q,x\right)=\left\{A-\left(1-n\right)q\right\}q-\frac{c}{2}{\left(x-q\right)}^2. $$

In this case, we obtain the same results as those in a pure monopoly market where excess capacity results do not hold, i.e., ${q}^{\ast \ast \ast }={x}^{\ast \ast \ast }=\frac{A}{2\left(1-n\right)}.$

Furthermore, we can demonstrate that excess capacity results do not hold in the first-best policy.

Appendix 2

The First-best Policy

1.1 Appendix 2.1

The Case of Ex Post Expectations

Given x and y, the government decides the production quantity to maximize social welfare. Based on Eq. (12), because the FOC is given by $\frac{\partial W\left(q,x,y;n\right)}{\partial q}=A-q+ ny+c\left(x-q\right)=0,$ it holds that ${q}^F={q}^F\left(x,y;n\right)=\frac{A+ cx+ ny}{1+c}.$ Under the fulfilled expectations, i.e., y = q, we obtain ${\tilde{q}}^F={\tilde{q}}^F\left(x;n\right)=\frac{A+ cx}{1+c-n}.$ Thus, Eq. (12), is rewritten as:

$$ W\left({\tilde{q}}^F\left(x;n\right),x;n\right)=A{\tilde{q}}^F-\frac{1}{2}\left(1-n\right){\left({\tilde{q}}^F\right)}^2-\frac{c}{2}{\left(x-{\tilde{q}}^F\right)}^2. $$

Using the above equation, the FOC of maximization of social welfare with respect to the capacity scale is given by $\frac{\partial W\left({\tilde{q}}^F\left(x;n\right),x;n\right)}{\partial x}=\frac{c}{1+c-n}\left\{A-\left(1-n\right)x\right\}.$ Thus, we obtain the following capacity scale and production quantity: ${x}^{F\ast }=\frac{A}{1-n}$ and ${q}^{F\ast }=\frac{A+{cx}^{F\ast }}{1+c-n}=\frac{A}{1-n}={y}^{F\ast }.$

1.2 Appendix 2.2

The Case of Ex Ante Expectations

In this case, the game of the sequential decisions of production quantity and then capacity scale is identical to the game of simultaneous decisions of these levels, given the expected network sizes. In particular, because of Eq. (12), we derive the following FOCs: $\frac{\partial W\left(q,x;y,n\right)}{\partial q}=A-q+ ny+c\left(x-q\right)=0$ and $\frac{\partial W\left(q,x;y,n\right)}{\partial x}=-c\left(x-q\right)=0.$ At the first stage, the fulfilled expectations, i.e., y = q, hold, and we obtain the following capacity scale and production quantity in the equilibrium: ${y}^{F\ast \ast }={q}^{F\ast \ast }={x}^{F\ast \ast }=\frac{A}{1-n}.$

Based on Appendixes 2.1 and 2.2, excess capacity results do not hold in the first-best policy.

Appendix 3

The Case of Ex Ante Expectations Based on a General Formulation

1.1 Appendix 3.1

Monopoly

The profit function of a monopoly is generally expressed as:

$$ \Pi \left(q,x,y;n\right)=\frac{\partial U\left(q,y;n\right)}{\partial q}q-C\left(q,x\right)=p\left(q,y;n\right)q-C\left(q,x\right). $$

In the case of ex ante expectations, given the expected network sizes, the FOCs of profit maximization with respect to production quantity and then capacity scale are given by:

$$ \frac{\mathrm{\partial \Pi}\left(q,x,y;n\right)}{\partial q}=p\left(q,y;n\right)+\frac{\partial p\left(q,y;n\right)}{\partial q}q-\frac{\partial C\left(q,x\right)}{\partial q}=0, $$

$$ \frac{\mathrm{\partial \Pi}\left(q,x,y;n\right)}{\partial x}=-\frac{\partial C\left(q,x\right)}{\partial x}=-c\left(x-q\right)=0. $$

Furthermore, in the first stage, the fulfilled expectations hold, i.e., y = q. Because it holds in the equilibrium that y^∗∗ = q^∗∗ = x^∗∗, excess capacity results do not arise.

1.2 Appendix 3.2

The First-best Policy

The social welfare function is generally expressed as:

$$ W\left(q,y,x;n\right)=U\left(q,y;n\right)-C\left(q,x\right). $$

Following Appendix 3.1, we have the following equations:

$$ \frac{\partial W\left(q,x,y;n\right)}{\partial q}=\frac{\partial U\left(q,y;n\right)}{\partial q}-\frac{\partial C\left(q,x\right)}{\partial q}=p\left(q,y;n\right)-\frac{\partial C\left(q,x\right)}{\partial q}=0, $$

$$ \frac{\partial W\left(q,x,y;n\right)}{\partial x}=-\frac{\partial C\left(q,x\right)}{\partial x}=-c\left(x-q\right)=0, $$

$$ y=q. $$

Thus, because it holds in the equilibrium that y^{F ∗ ∗} = q^{F ∗ ∗} = x^{F ∗ ∗}, the excess capacity results do not hold.