Taxation, Network Externalities, Consumer Suffering, and Profit-Raising Entry: A Cautionary Note

Abstract

Conventional wisdom is that higher competition, implying more active firms, benefits consumers but hurts producer surplus. Under unit taxation/subsidy, we find that the entry of efficient firms improves consumer welfare, while the entry of inefficient firms hurts consumer surplus with network externalities. The entry of efficient firm raises industry profit if the degree of network externalities is relatively high. The latter result is at odds with the conventional wisdom and showed another channel for the possibility of a profit-raising entry. Our results suggest that the strength of network effects should be considered for designing taxation/subsidy and competition policies concurrently.

Appendix

1.1 Proof for Lemma 1

$$\frac{ds}{dn}=\frac{a\left(m+m\right)+c\left(m\left(-2+\lambda \right)-n\lambda \right)}{\left(1-e\right){\left(m+n\right)}^{3}}>0$$

$$\frac{ds}{dm}=\frac{\left(a-c\right)m+n\left(a-c\left(2\lambda -1\right)\right)}{\left(e-1\right){\left(m+n\right)}^{3}}>0$$

$$\frac{ds}{de}=\frac{a\left(m+n\right)-c\left(m+n\lambda \right)}{{\left(e-1\right)}^{2}{\left(m+n\right)}^{2}}$$

1.2 Proof for Proposition 1

$$\frac{d{q}_{i}}{dm}=\frac{ds}{dm}-(1-e)\frac{dQ}{dm}=\frac{d{q}_{j}}{dm}>0$$

$$\frac{d{q}_{j}}{dn}=\frac{ds}{dn}-(1-e)\frac{dQ}{dn}=\frac{d{q}_{j}}{dn}$$

Based on equations $\frac{ds}{dm}$ and$\frac{dQ}{dm}$, $\frac{d{q}_{i}}{dm}$ implies that an increase in the number of inefficient firms increases the outputs of efficient and inefficient firms. Alternatively, given$\frac{d{q}_{j}}{dn}$, the effects of an increase in the number of efficient firms on the outputs are not unidirectional because $\frac{ds}{dn}>0$ and $\frac{dQ}{dn}$.

$$\frac{dQ}{dm}=-\frac{cn\left(-1+\lambda \right)}{\left(-1+e\right){\left(m+n\right)}^{2}}<0$$

$$\frac{dQ}{dn}=\frac{cm(-1+\lambda )}{(-1+e){(m+n)}^{2}}>0$$

$$\frac{dQ}{de}=-\frac{-a(m+n)+c(m+n\lambda )}{{(-1+e)}^{2}(m+n)}>0$$

1.3 Proof for Proposition 2

Given $\frac{d{q}_{i}}{dm}$, the effect of an increase in the number of inefficient firms on the industry profit is positive.$\frac{d\Pi }{dm}=2n{q}_{i}\frac{d{q}_{i}}{dm}+{{q}_{j}}^{2}+2m{q}_{j}\frac{d{q}_{j}}{dm}>0.$

Similarly, using $\frac{d{q}_{i}}{dn}$, we represent the following effect of an increase in the number of efficient firms:

$$\frac{d\Pi }{dn}=2n{q}_{i}\frac{d{q}_{i}}{dm}+{{q}_{i}}^{2}+2m{q}_{j}\frac{d{q}_{j}}{dn}={{q}_{i}}^{2}+2Q\frac{d{q}_{i}}{dn}$$

We can derive the following relationship:

$$\frac{d{q}_{i}}{dn}>(<)0\leftrightarrow A+c(1-\lambda )m(m+n)e>(<)0$$

where $A=a(m+n)-c((2-\lambda )m+\lambda n+(1-\lambda )m(m+n)$).

Using the assumption in the model, i.e., $a>(1+(1-\lambda )n)c$, it holds that.

$a(m+n)>(1+(1-\lambda )n)c(m+n$).

The following relationship arises:

$(1+(1-\lambda )n)(m+n$) > ( <)$((2-\lambda )\mathrm{m}+\lambda n+(1-\lambda )m(m+n))\leftrightarrow (n-m)(m+n+1)>(<)0$.

Similarly,

$$\frac{d\Pi }{\partial n}=\frac{1}{{(-1+e)}^{2}{(m+n)}^{4}}(-{a}^{2}{(m+n)}^{2}-2ac(m+n)(m(-2+\lambda )-n\lambda )+{c}^{2}({(-1+e)}^{2}{m}^{4}{(-1+\lambda )}^{2}+2{(-1+e)}^{2}{m}^{3}n{(-1+\lambda )}^{2}+2mn(-2+\lambda )\lambda -{n}^{2}{\lambda }^{2}+{m}^{2}(-3+{(-1+e)}^{2}{n}^{2}{(-1+\lambda )}^{2}+2\lambda )))>0$$

$$\frac{d\Pi }{\partial m}=\frac{1}{{(-1+e)}^{2}{(m+n)}^{4}}(-{a}^{2}{(m+n)}^{2}+2ac(m+n)(m+n(-1+2\lambda ))+{c}^{2}({m}^{2}(-1+{(-1+e)}^{2}{n}^{2}{(-1+\lambda )}^{2})+2m(n+{(-1+e)}^{2}{n}^{3}{(-1+\lambda )}^{2}-2n\lambda )+{n}^{2}({(-1+e)}^{2}{n}^{2}{(-1+\lambda )}^{2}+(2-3\lambda )\lambda )))<0$$

1.4 Proof for Proposition 3

$$\frac{dSW}{\partial m}=\frac{cn(-1+\lambda )(-a(m+n)+c(m(1+(-1+e)n(-1+\lambda ))+n((-1+e)n(-1+\lambda )+\lambda )))}{(-1+e){(m+n)}^{3}}>0$$

$$\frac{dSW}{\partial n}=\frac{cm(-1+\lambda )(a(m+n)+c(m(-1+(-1+e)n(-1+\lambda ))+(-1+e){m}^{2}(-1+\lambda )-n\lambda ))}{(-1+e){(m+n)}^{3}}>0$$