Profit raising entry under mixed behavior

Abstract

This paper provides a new theoretical justification for entry to raise incumbent’s profit. In an industry with product differentiation and when firms use different strategic variables, we show that entry by an additional price setter may end up benefitting another price setting incumbent. This is due to the response of existing quantity setters, who contract their output after entry. Such reduction in output, which creates demand for the price setters, may more than compensate the reduction in price that is brought about by entry, provided that the products are not sufficiently differentiated.

Appendices

Appendix I

In this appendix we present the proofs of the results in the paper.

Proof of Lemma 1

Let there be n firms, k of which set quantities. In order to have different types of firms we assume that $k\in \left[ 1,n-1\right]$. Without loss of generality let these firms be firm 1, ..., k. Firms $k+1,...,n$ set prices. There are such $n-k$ firms. The demand for the product of any firm i is given by

$$\begin{aligned} p_{i}=v-\frac{1}{1+u}(nq_{i}+uQ) \end{aligned}$$

Aggregating the demands for the two types of firms we obtain, respectively for the firms that set quantities and for the firms that set prices

$$\begin{aligned} P_{k}&=kv-\frac{1}{1+u}(nQ_{k}+kuQ)\\ P_{n-k}&=(n-k)v-\frac{1}{1+u}(nQ_{n-k}+(n-k)uQ) \end{aligned}$$

with $P_{k}=\sum _{j=1}^{k}p_{j}$ and $P_{n-k}=\sum _{j=k+1}^{n}p_{j}$ and likewise for quantities. As $Q=Q_{k}+Q_{n-k}$, we have

$$\begin{aligned} P_{k}&=kv-\frac{1}{1+u}(nQ_{k}+ku\left( Q_{k}+Q_{n-k}\right) )\\ P_{n-k}&=(n-k)v-\frac{1}{1+u}(nQ_{n-k}+(n-k)u\left( Q_{k}+Q_{n-k}\right) ) \end{aligned}$$

or

$$\begin{aligned} Q_{k}&=\frac{1}{n+ku}\left( 1+u\right) \left( kv-P_{k}-ku\frac{Q_{n-k} }{1+u}\right) \\ Q_{n-k}&=\frac{1}{n+u\left( n-k\right) }\left( 1+u\right) \left( v\left( n-k\right) -P_{n-k}-uQ_{k}\frac{n-k}{1+u}\right) \end{aligned}$$

Consider one of the quantity setting firms. The demand for its product is

$$\begin{aligned} p_{i}=v-\frac{1}{1+u}(nq_{i}+uQ)=v-\frac{1}{1+u}(nq_{i}+u\left( Q_{k} +Q_{n-k}\right) ) \end{aligned}$$

We want to write this as a function of the quantities of the other quantity setting firms, $Q_{k-i}$, and of the prices of the price setting firms, $P_{n-k}$. So,

$$\begin{aligned} p_{i}&=v-\frac{1}{1+u}\left[ nq_{i}+u\left( Q_{k}+\frac{1}{n+u\left( n-k\right) }\left( 1+u\right) \left( v\left( n-k\right) -P_{n-k} -uQ_{k}\frac{n-k}{1+u}\right) \right) \right] \\ \Leftrightarrow p_{i}&=\frac{P_{n-k}u\left( 1+u\right) +nv\left( 1+u\right) -nu\left( Q_{k-i}\right) -nq_{i}\left( n+u\left( n-k+1\right) \right) }{\left( 1+u\right) \left( n+u\left( n-k\right) \right) } \end{aligned}$$

where $Q_{k-i}=Q_{k}-q_{i}$.

Consider now one of the price setting firms. The demand for its product is

$$\begin{aligned} p_{i}&=v-\frac{1}{1+u}(nq_{i}+uQ)\\ q_{i}&=(1+u)\frac{v}{n}-\frac{Q_{k}+Q_{n-k}}{n}u-p_{i}\frac{1+u}{n} \end{aligned}$$

We want to write this as a function of the quantities of the other quantity setting firms, $Q_{k}$, and the prices of the other price setting firms, $P_{n-k-i}$. So,

$$\begin{aligned} q_{i}&=(1+u)\frac{v}{n}-\frac{Q_{k}+\frac{1}{n+u\left( n-k\right) }\left( 1+u\right) \left( v\left( n-k\right) -P_{n-k}-uQ_{k}\frac{n-k}{1+u}\right) }{n}u-p_{i}\frac{1+u}{n}\\ q_{i}&=\frac{P_{n-k-i}u\left( 1+u\right) -p_{i}\left( 1+u\right) \left( n+u\left( n-k-1\right) \right) +nv\left( 1+u\right) -nuQ_{k} }{n\left( n+u\left( n-k\right) \right) } \end{aligned}$$

where $P_{n-k-i}=P_{n-k}-p_{i}$.

Hence, we have

$$\begin{aligned} p_{i}&=\frac{P_{n-k}u\left( 1+u\right) +nv\left( 1+u\right) -nu\left( Q_{k-i}\right) -nq_{i}\left( n+u\left( n-k+1\right) \right) }{\left( 1+u\right) \left( n+u\left( n-k\right) \right) }\\ q_{i}&=\frac{P_{n-k-i}u\left( 1+u\right) -p_{i}\left( 1+u\right) \left( n+u\left( n-k-1\right) \right) +nv\left( 1+u\right) -nuQ_{k} }{n\left( n+u\left( n-k\right) \right) } \end{aligned}$$

or

$$\begin{aligned} p_{i}&=\frac{\sum _{j=k+1}^{n}p_{j}u\left( 1+u\right) +nv\left( 1+u\right) -nu\sum _{\begin{subarray}{c} j=1\\ j\ne i \end{subarray}}^{k}q_{j}-nq_{i}\left( n+u\left( n-k+1\right) \right) }{\left( 1+u\right) \left( n+u\left( n-k\right) \right) },i=1,...,k\end{aligned}$$

(1)

$$\begin{aligned} q_{i}&=\frac{\sum _{\begin{subarray}{c} j=k+1\\ j\ne i \end{subarray}}^{n}p_{j}u\left( 1+u\right) -p_{i}\left( 1+u\right) \left( n+u\left( n-k-1\right) \right) +nv\left( 1+u\right) -nu\sum _{j=1}^{k}q_{j}}{n\left( n+u\left( n-k\right) \right) },i=k+1,...,n \end{aligned}$$

(2)

respectively for the quantity and for the price setting firms .

For the quantity setting firms the profit function is

$$\begin{aligned} \pi _{i}^{Q}&=(p_{i}-c)q_{i}\\&=\left(\frac{\sum _{j=k+1}^{n}p_{j}u\left( 1+u\right) +nv\left( 1+u\right) -nu\sum _{\begin{subarray}{c} j=1\\ j\ne i \end{subarray}}^{k}q_{j}-nq_{i}\left( n+u\left( n-k+1\right) \right) }{\left( 1+u\right) \left( n+u\left( n-k\right) \right) }-c\right)q_{i} \end{aligned}$$

and the first-order conditions for profit maximization are

$$\begin{aligned} \frac{\partial \pi _{i}^{Q}}{\partial q_{i}}=\frac{\sum _{j=k+1}^{n}p_{j}u\left( 1+u\right) +nv\left( 1+u\right) -nu\sum _{\begin{subarray}{c} j=1\\ j\ne i \end{subarray}}^{k} q_{j}-2nq_{i}\left( n+u\left( n-k+1\right) \right) }{\left( 1+u\right) \left( n+u\left( n-k\right) \right) }-c=0 \end{aligned}$$

Due to firm symmetry ($q_{1}=...=q_{k}=q$ and $p_{k+1}=...=p_{n}=p$) this simplifies to

$$\begin{aligned} \frac{\partial \pi _{i}^{Q}}{\partial q_{i}}&=\frac{(n-k)pu\left( 1+u\right) +nv\left( 1+u\right) -nu(k-1)q-2nq\left( n+u\left( n-k+1\right) \right) }{\left( 1+u\right) \left( n+u\left( n-k\right) \right) }-c=0\Leftrightarrow \\ q&=\left( n\left( v-c\right) +u\left( n-k\right) \left( p-c\right) \right) \frac{u+1}{n\left( 2n+u\left( 2n-k+1\right) \right) } \end{aligned}$$

For the price setting firms the profit function is

$$\begin{aligned} \pi _{i}^{P}&=(p_{i}-c)q_{i}\\&=(p_{i}-c)\frac{\sum _{\begin{subarray}{c} j=k+1\\ j\ne i \end{subarray}}^{n}p_{j}u\left( 1+u\right) -p_{i}\left( 1+u\right) \left( n+u\left( n-k-1\right) \right) +nv\left( 1+u\right) -nu\sum _{j=1}^{k}q_{j}}{n\left( n+u\left( n-k\right) \right) } \end{aligned}$$

and the first-order conditions for profit maximization are

$$\begin{aligned} \frac{\partial \pi _{i}^{P}}{\partial p_{i}}=\frac{\sum _{\begin{subarray}{c} j=k+1\\ j\ne i \end{subarray}}^{n}p_{j}u\left( 1+u\right) -p_{i}\left( 1+u\right) \left( n+u\left( n-k-1\right) \right) +nv\left( 1+u\right) -nu\sum _{j=1}^{k}q_{j}}{n\left( n+u\left( n-k\right) \right) }\\ +(p_{i}-c)\frac{-\left( 1+u\right) \left( n+u\left( n-k-1\right) \right) }{n\left( n+u\left( n-k\right) \right) }=0 \end{aligned}$$

or, due to symmetry,

$$\begin{aligned}&\frac{\partial \pi _{i}^{P}}{\partial p_{i}}=\frac{(n-k-1)pu\left( 1+u\right) -p\left( 1+u\right) \left( n+u\left( n-k-1\right) \right) +nv\left( 1+u\right) -nukq}{n\left( n+u\left( n-k\right) \right) }\\&\quad +(p-c)\frac{-\left( 1+u\right) \left( n+u\left( n-k-1\right) \right) }{n\left( n+u\left( n-k\right) \right) }=0\Leftrightarrow \\&\quad p=c+n\frac{\left( 1+u\right) \left( v-c\right) -kqu}{\left( 1+u\right) \left( 2n+u\left( n-k-1\right) \right) } \end{aligned}$$

Solving the system

$$\begin{aligned} q&=\left( n\left( v-c\right) +u\left( n-k\right) \left( p-c\right) \right) \frac{u+1}{n\left( 2n+u\left( 2n-k+1\right) \right) }\\ p&=c+n\frac{\left( 1+u\right) \left( v-c\right) -kqu}{\left( 1+u\right) \left( 2n+u\left( n-k-1\right) \right) } \end{aligned}$$

with respect to p, q we obtain:

$$\begin{aligned} p^{P}&=c+\frac{n\left( 2n+u\left( 2\left( n-k\right) +1\right) \right) \left( v-c\right) }{\left( 2n\left( u+2\right) \left( n+u\left( n-k\right) \right) -u^{2}\left( n+1\right) \right) }\\ q^{Q}&=\frac{\left( 2n+u\left( 2\left( n-k-1\right) +1\right) \right) \left( u+1\right) \left( v-c\right) }{\left( 2n\left( u+2\right) \left( n+u\left( n-k\right) \right) -u^{2}\left( n+1\right) \right) } \end{aligned}$$

Plugging these prices and quantities in (1) and (2) under the symmetry assumption, we obtain the price of a quantity setting firm (say firm 1)

$$\begin{aligned} p_{1}&=\frac{(n-k)pu\left( 1+u\right) +nv\left( 1+u\right) -nu(k-1)q-nq\left( n+u\left( n-k+1\right) \right) }{\left( 1+u\right) \left( n+u\left( n-k\right) \right) }\\&=c+\frac{n\left( 2n+u\left( 2\left( n-k-1\right) +1\right) \right) \left( n+u\left( n-k+1\right) \right) \left( v-c\right) }{\left( 2n\left( 2+u\right) \left( n+u\left( n-k\right) \right) -u^{2}\left( n+1\right) \right) \left( n+u\left( n-k\right) \right) } \end{aligned}$$

and the quantity of a price setting firm (say firm $k+1$)

$$\begin{aligned} q_{k+1}&=\frac{(n-k-1)pu\left( 1+u\right) -p\left( 1+u\right) \left( n+u\left( n-k-1\right) \right) +nv\left( 1+u\right) -nukq}{n\left( n+u\left( n-k\right) \right) }\\&=\frac{\left( 2n+u\left( 2\left( n-k\right) +1\right) \right) }{\left( 2n\left( 2+u\right) \left( n+u\left( n-k\right) \right) -u^{2}\left( n+1\right) \right) }\frac{\left( 1+u\right) \left( n+u\left( n-k-1\right) \right) \left( v-c\right) }{\left( n+u\left( n-k\right) \right) } \end{aligned}$$

It should be noted that if the two types of firms exist, that is, if $1\le n-k\le n-1$, the condition $2n\left( 2+u\right) \left( n+u\left( n-k\right) \right) -u^{2}\left( n+1\right) >0$, which is always true, assures that quantities are positive and that price is above marginal cost.⁴

Finally, profits are

$$\begin{aligned} \pi _{i}^{Q}(n,k)=(p_{1}-c)q^{Q}=\frac{n\left( 2n-u+2u\left( n-k\right) \right) ^{2}\left( n+u\left( n-k+1\right) \right) \left( 1+u\right) \left( v-c\right) ^{2}}{\left( n+u\left( n-k\right) \right) \left( 2n\left( 2+u\right) \left( n+u\left( n-k\right) \right) -u^{2}\left( n+1\right) \right) ^{2}} \end{aligned}$$

and

$$\begin{aligned} \pi _{i}^{P}(n,k)=(p^{P}-c)q_{k+1}=\frac{n\left( 2n+u+2u\left( n-k\right) \right) ^{2}\left( n+u\left( n-k-1\right) \right) \left( 1+u\right) \left( v-c\right) ^{2}}{\left( n+u\left( n-k\right) \right) \left( 2n\left( 2+u\right) \left( n+u\left( n-k\right) \right) -u^{2}\left( n+1\right) \right) ^{2}} \end{aligned}$$

$\square$

Proof of Proposition 1:

Let $k=n-g$. Entry increases the profit of a price setter if and only if

$$\begin{aligned} \pi _{i}^{P}(n+1,n-g)-\pi _{i}^{P}(n,n-g)&>0\Leftrightarrow \\ -\frac{(u+1)}{\left( n+ug\right) \left( n+1+u\left( g+1\right) \right) }\frac{\sum _{i=0}^{8}f_{i}(n,g)u^{i}}{h_{1}(u,n,g)h_{2}(u,n,g)}&>0 \end{aligned}$$

with

$$\begin{aligned} h_{1}(u,n,g)&=\left( u^{2}\left( n\left( 2g-1\right) -1\right) +2nu\left( 2g+n\right) +4n^{2}\right) ^{2}>0\\ h_{2}(u,n,g)&=\left( u^{2}\left( 2g+n+2gn\right) +2u\left( n+1\right) \left( 2g+n+3\right) +4\left( n+1\right) ^{2}\right) ^{2}>0 \end{aligned}$$

and

$$\begin{aligned}&f_{0}(n,g)=64n^{4}\left( n+1\right) ^{4}>0 \\&f_{1}(n,g)=192n^{3}\left( 2n+1\right) \left( n+1\right) ^{3} g+64n^{4}\left( n+4\right) \left( n+1\right) ^{3}>0 \\&f_{2}(n,g)=\left[ \begin{array}{c} 192n^{2}\left( 5n+5n^{2}+1\right) \left( n+1\right) ^{2}g^{2} +384n^{3}\left( 4n+n^{2}+2\right) \left( n+1\right) ^{2}g\\ +16n^{2}\left( 18n^{2}+14n^{3}+n^{4}-7n-3\right) \left( n+1\right) ^{2} \end{array} \right]>0 \\&f_{3}(n,g)=\left[ \begin{array}{c} n^{7}\left( 96g+80\right) +n^{6}\left( 960g^{2}+1264g+240\right) +n^{5}\left( 1280g^{3}+4800g^{2}+3248g-128\right) \\ +n^{4}\left( 3200g^{3}+7296g^{2}+2720g-896\right) +n^{3}\left( 2688g^{3}+4224g^{2}+368g-864\right) \\ +n^{2}\left( 832g^{3}+768g^{2}-320g-272\right) +n\left( 64g^{3} -48g-16\right) \end{array} \right]>0 \\&f_{4}(n,g)=\left[ \begin{array}{c} 16n^{7}+n^{6}\left( 240g^{2}+352g+40\right) +n^{5}\left( 1280g^{3} +2512g^{2}+1024g-392\right) \\ +n^{4}\left( 960g^{4}+5120g^{3}+5440g^{2}+176g-1224\right) +\\ n^{3}\left( 1920g^{4}+6144g^{3}+3920g^{2}-1472g-1288\right) +\\ n^{2}\left( 1152g^{4}+2560g^{3}+640g^{2}-1216g-548\right) +\\ n\left( 192g^{4}+256g^{3}-112g^{2}-240g-68\right) \end{array} \right]>0 \\&f_{5}(n,g)=\left[ \begin{array}{c} 64gn^{6}+n^{5}\left( 320g^{3}+608g^{2}+160g-192\right) +\\ n^{4}\left( 960g^{4}+2528g^{3}+1664g^{2}-744g-648\right) \\ +n^{3}\left( 384g^{5}+2880g^{4}+4352g^{3}+928g^{2}-2048g-832\right) \\ +n^{2}\left( 576g^{5}+2496g^{4}+2432g^{3}-496g^{2}-1632g-468\right) +\\ n\left( 192g^{5}+576g^{4}+288g^{3}-368g^{2}-432g-96\right) -4g \end{array} \right]>0 \\&f_{6}(n,g)=\left[ \begin{array}{c} n^{5}\left( 96g^{2}-32\right) +n^{4}\left( 240g^{4}+512g^{3}+240g^{2} -392g-147\right) \\ +n^{3}\left( 384g^{5}+1312g^{4}+1248g^{3}-288g^{2}-992g-222\right) +\\ n^{2}\left( 64g^{6}+768g^{5}+1616g^{4}+816g^{3}-852g^{2}-916g-152\right) +\\ n\left( 64g^{6}+384g^{5}+544g^{4}+80g^{3}-420g^{2}-316g-45\right) -12g^{2}-12g \end{array} \right] >0 \\&f_{7}(n,g)=\left[ \begin{array}{c} 32g^{5}n^{2}\left( 2g+3n\right) +16g^{3}n\left( g+4n\right) \left( 3gn+4g^{2}+n^{2}\right) \\ +16g^{3}n\left( 25gn+13g^{2}+10n^{2}\right) \\ +8gn\left( -26gn^{2}+11g^{2}n+24g^{3}-8n^{3}\right) \\ -2n\left( 99gn^{2}+180g^{2}n+16g^{3}+4n^{3}\right) \\ -\left( 197gn^{2}+188g^{2}n+12g^{3}+17n^{3}\right) -24g^{2}-69gn-12n^{2}-9g \end{array} \right] \lessgtr 0 \\&f_{8}(n,g)=\left[ \begin{array}{c} 16g^{6}n^{2}+16g^{4}n\left( g+n\right) ^{2}+8g^{4}n\left( 4g+5n\right) \\ -8g^{2}n\left( 4n-3g\right) \left( g+n\right) -g\left( 51gn^{2} +16g^{2}n+4g^{3}+8n^{3}\right) \\ -\left( 4gn^{2}+31g^{2}n+12g^{3}+n^{3}\right) -9g^{2} \end{array} \right] \lessgtr 0 \end{aligned}$$

Note that for $g=1$

$$\begin{aligned} f_{8}(n,1)=-25\left( n+1\right) \left( n-1\right) ^{2}<0 \end{aligned}$$

Regardless of the sign of $f_{7}(n,1)$ there is one variation in polynomial $\sum _{i=0}^{8}f_{i}(n,g)u^{i}$ (the number of variations in a polynomial is defined as the number of times that two of its consecutive terms have different signs) and so, using Descartes’ Rule of Signs, there is one positive root, $u_{r}$. As $f_{0}(n,g)=64n^{4}\left( n+1\right) ^{4}>0$, we have $\pi _{i}^{P}(n+1,n-g)-\pi _{i}^{P}(n,n-g)>0$ for $u>$ $u_{r}$.

When $g=2$ we have that

$$\begin{aligned} f_{7}(n,2)&=12\,678n+19\,082n^{2}+6435n^{3}+376n^{4}-210>0\\ f_{8}(n,2)&=2180n+2412n^{2}+111n^{3}-196>0 \end{aligned}$$

meaning that there are no roots as all coefficients are positive. Finally, given that

$$\begin{aligned} \frac{\partial f_{7}(n,g)}{\partial g}=768g^{3}n-96g^{2}n+1040g^{4} n+384g^{5}n+264g^{2}n^{2}-197n^{2}\\ +480g^{2}n^{3}-198n^{3}+1600g^{3}n^{2}+192g^{2}n^{4}-64n^{4}+832g^{3} n^{3}-416gn^{3}\\ +1520g^{4}n^{2}-720gn^{2}+480g^{4}n^{3}-69n-48g+384g^{5}n^{2}-376gn-36g^{2} -9>0 \end{aligned}$$

and

$$\begin{aligned}&\frac{\partial f_{8}(n,g)}{\partial g}=2(48g^{3}n-24g^{2}n-9g+80g^{4} n+48g^{5}n+80g^{3}n^{2}-51gn^{2}\\&\quad +4n^{3}\left( 8g^{3}-8g-1\right) +80g^{4}n^{2}-12g^{2}n^{2}-18g^{2}\\&\quad +48g^{5}n^{2}-31gn-8g^{3}-2n^{2})>0 \end{aligned}$$

there are also no real roots for $g\ge 2$. $\square$

Appendix II

In this appendix we show that entry by an additional price setter may benefit another price setting incumbent when Singh and Vives (1984)’s demand structure is considered. Assuming the demand for firm i is given by

$$\begin{aligned} p_{i}=a-bq_{i}-b\gamma \sum _{\begin{subarray}{c} j=1\\ {} j\ne i \end{subarray}}^{n}q_{j} \end{aligned}$$

with $a>c$, $b>0$ and $\gamma \in \left[ 0,1\right)$, for the case of duopoly we have that the relevant inverse demand and demand curves are

$$\begin{aligned} p_{1}&=a\left( 1-\gamma \right) -bq_{1}\left( 1-\gamma \right) \left( \gamma +1\right) +\gamma p_{2}\\ q_{2}&=\frac{a}{b}-\gamma q_{1}-\frac{1}{b}p_{2} \end{aligned}$$

and the equilibrium prices and quantities result from

$$\begin{aligned} \frac{\partial \left( \left( p_{1}-c\right) q_{1}\right) }{\partial q_{1}}&=a-c-a\gamma +\gamma p_{2}-2bq_{1}\left( 1-\gamma \right) \left( \gamma +1\right) =0\\ \frac{\partial \left( (p_{2}-c)q_{2}\right) }{\partial p_{2}}&=\frac{a+c-2p_{2}-b\gamma q_{1}}{b}=0 \end{aligned}$$

The best response functions are

$$\begin{aligned} q_{1}&=\frac{a\left( 1-\gamma \right) -c+\gamma p_{2}}{2b\left( 1-\gamma \right) \left( \gamma +1\right) }\\ p_{2}&=\frac{1}{2}\left( a+c-b\gamma q_{1}\right) \end{aligned}$$

that yield equilibrium duopoly prices and quantities:

$$\begin{aligned} p_{2}^{D}&=c+\frac{\left( \gamma +2\right) \left( 1-\gamma \right) }{4-3\gamma ^{2}}\left( a-c\right) \\ q_{1}^{D}&=\frac{2-\gamma }{4-3\gamma ^{2}}\frac{a-c}{b}\\ p_{1}^{D}&=c+\frac{\left( 1-\gamma \right) \left( 2-\gamma \right) \left( \gamma +1\right) }{4-3\gamma ^{2}}\left( a-c\right) \\ q_{2}^{D}&=\frac{\left( \gamma +2\right) \left( 1-\gamma \right) }{4-3\gamma ^{2}}\frac{a-c}{b} \end{aligned}$$

Before entry, profits are given by

$$\begin{aligned} \pi _{1}^{D}&=\frac{\left( \gamma -2\right) ^{2}\left( 1-\gamma \right) \left( \gamma +1\right) }{\left( 3\gamma ^{2}-4\right) ^{2}}\frac{\left( a-c\right) ^{2}}{b}\\ \pi _{2}^{D}&=\frac{\left( \gamma +2\right) ^{2}\left( \gamma -1\right) ^{2}}{\left( 3\gamma ^{2}-4\right) ^{2}}\frac{\left( a-c\right) ^{2}}{b} \end{aligned}$$

After entry, we will have a triopoly with one quantity setter (firm 1) and two price setters (firms 2 and 3). The relevant demand and inverse demand curves are now

$$\begin{aligned} p_{1}&=\frac{1}{\gamma +1}\left( a\left( 1-\gamma \right) +\gamma \left( p_{2}+p_{3}\right) -b\left( 2\gamma +1\right) \left( 1-\gamma \right) q_{1}\right) \\ q_{2}&=\frac{1}{b\left( 1-\gamma \right) \left( \gamma +1\right) }\left( a\left( 1-\gamma \right) -p_{2}+\gamma p_{3}-b\gamma \left( 1-\gamma \right) q_{1}\right) \\ q_{3}&=\frac{1}{b\left( 1-\gamma \right) \left( \gamma +1\right) }\left( a\left( 1-\gamma \right) -p_{3}+\gamma p_{2}-b\gamma \left( 1-\gamma \right) q_{1}\right) \end{aligned}$$

The Nash equilibrium is the solution to

$$\begin{aligned} \frac{\partial \left( \left( p_{1}-c\right) q_{1}\right) }{\partial q_{1}}&=\frac{a-c-a\gamma -c\gamma +\gamma \left( p_{2}+p_{3}\right) -2b\left( 2\gamma +1\right) \left( 1-\gamma \right) q_{1}}{\gamma +1}=0\\ \frac{\partial \left( (p_{2}-c)q_{2}\right) }{\partial p_{2}}&=\frac{a-a\gamma +c-2p_{2}+\gamma p_{3}-b\gamma \left( 1-\gamma \right) q_{1} }{b\left( 1-\gamma \right) \left( \gamma +1\right) }=0\\ \frac{\partial \left( (p_{3}-c)q_{3}\right) }{\partial p_{3}}&=\frac{a-a\gamma +c-2p_{3}+\gamma p_{2}-b\gamma \left( 1-\gamma \right) q_{1} }{b\left( 1-\gamma \right) \left( \gamma +1\right) }=0 \end{aligned}$$

that corresponds to the following best response functions (given the symmetry of the price setters, one can obtain the best response function for the quantity of firm 1 with respect to any common price set by firms 2 and 3, $p_{i}$, as well as firms 2 and 3 equilibrium price for any given level of $q_{1}$):

$$\begin{aligned} q_{1}&=\frac{a-c-a\gamma -c\gamma +\gamma \left( 2p_{i}\right) }{2b\left( 1-\gamma \right) \left( 2\gamma +1\right) }\\ p_{i}&=\frac{1}{2-\gamma }\left( a+c-a\gamma -b\gamma \left( 1-\gamma \right) q_{1}\right) \end{aligned}$$

yielding

$$\begin{aligned} p_{2}^{T}&=p_{3}^{T}=c+\frac{\frac{1}{2}\left( 3\gamma +2\right) \left( 1-\gamma \right) }{2+3\gamma -\gamma ^{2}}\left( a-c\right) \\ q_{1}^{T}&=\frac{1}{2}\frac{\gamma +2}{2+3\gamma -\gamma ^{2}}\frac{a-c}{b}\\ p_{1}^{T}&=c+\frac{\left( 1-\gamma \right) \left( 2\gamma +1\right) \left( \gamma +2\right) }{2\left( \gamma +1\right) \left( 2+3\gamma -\gamma ^{2}\right) }\left( a-c\right) \\ q_{2}^{T}&=q_{3}^{T}=\frac{\left( 3\gamma +2\right) }{2\left( \gamma +1\right) \left( 2+3\gamma -\gamma ^{2}\right) }\frac{a-c}{b} \end{aligned}$$

Equilibrium profits are

$$\begin{aligned} \pi _{1}^{T}&=\frac{\left( 1-\gamma \right) \left( \gamma +2\right) ^{2}\left( 2\gamma +1\right) }{4\left( \gamma +1\right) \left( -3\gamma +\gamma ^{2}-2\right) ^{2}}\frac{\left( a-c\right) ^{2}}{b}\\ \pi _{2}^{T}&=\pi _{3}^{T}=\frac{\left( 1-\gamma \right) \left( 3\gamma +2\right) ^{2}}{4\left( \gamma +1\right) \left( -3\gamma +\gamma ^{2}-2\right) ^{2}}\frac{\left( a-c\right) ^{2}}{b} \end{aligned}$$

The profit of firm 2 increases with entry if and only if

$$\begin{aligned} \pi _{2}^{T}-\pi _{2}^{D}>0\Leftrightarrow h(\gamma ):=-4-8\gamma +\gamma ^{2}+8\gamma ^{3}+4\gamma ^{4}>0 \end{aligned}$$

There is one variation in polynomial $h(\gamma )$ and so, using Descartes’ Rule of Signs, there is one positive root, $\gamma _{r}$. As $h(0)=-4<0$ and $h(1)=1>0$, we have that $\gamma _{r}\in (0,1)$, and for $\gamma >\gamma _{r}$ entry increases the profit of firm 2.

Appendix III

In this appendix we show that entry by an additional price setter may benefit another price setting incumbent when Höffler (2008)’s "consistent formulation" of the demand structure is considered.

Consumers choose quantities and Y (a composite good) to maximize utility

$$\begin{aligned} V=Y+U(.) \end{aligned}$$

subject to the budget constraint $\sum _{i=1}^{n}p_{i}q_{i}+Y=I$. In other words, consumers maximize

$$\begin{aligned} \max _{q_{i},...,q_{n}}V=(I-\sum _{i=1}^{n}p_{i}q_{i})+U(.) \end{aligned}$$

Shubik and Levitan (1980)’s demand follows from the utility function

$$\begin{aligned} U(.)=vQ-\frac{n}{2(1+u)}\left( \sum _{i=1}^{n}q_{i}^{2}+\frac{u}{n} Q^{2}\right) \end{aligned}$$

with $Q=q_{1}+...+q_{n}$. In the main text we have assumed that before entry $n=2$ and after entry $n=3$. An alternative would be to assume that $n=3$ both before and after entry but that before entry there is an infinite price for firm 3’s product (the representative consumer is constrained not to buy the entrant’s product before entry). From utility maximization, demands are given by

$$\begin{aligned} v-\frac{3}{(1+u)}\left( q_{1}+\frac{u}{3}\left( q_{1}+q_{2}+q_{3}\right) \right) -p_{1}&=0\\ v-\frac{3}{(1+u)}\left( q_{2}+\frac{u}{3}\left( q_{1}+q_{2}+q_{3}\right) \right) -p_{2}&=0\\ q_{3}&=0 \end{aligned}$$

yielding

$$\begin{aligned} p_{1}&=\frac{1}{\left( u+3\right) \left( u+1\right) }\left( 3v\left( u+1\right) -3q_{1}\left( 2u+3\right) +up_{2}\left( u+1\right) \right) \\ q_{2}&=\frac{1}{u+3}\left( v\left( u+1\right) -p_{2}\left( u+1\right) -uq_{1}\right) \end{aligned}$$

The equilibrium quantity for firm 1 and price for firm 2 are given by

$$\begin{aligned} p_{2}^{D}&=c+\frac{9\left( u+2\right) }{24u+u^{2}+36}\left( v-c\right) \\ q_{1}^{D}&=\frac{\left( u+6\right) \left( u+1\right) }{24u+u^{2} +36}\left( v-c\right) \\ p_{1}^{D}&=c+\frac{3\left( u+6\right) \left( 2u+3\right) }{\left( u+3\right) \left( 24u+u^{2}+36\right) }\left( v-c\right) \\ q_{2}^{D}&=\frac{9\left( u+2\right) \left( u+1\right) }{\left( u+3\right) \left( 24u+u^{2}+36\right) }\left( v-c\right) \end{aligned}$$

and the corresponding normalized profits are

$$\begin{aligned} \pi _{1}^{D}&=\frac{3\left( 2u+3\right) \left( u+1\right) \left( u+6\right) ^{2}}{\left( u+3\right) \left( 24u+u^{2}+36\right) ^{2}}\\ \pi _{2}^{D}&=\frac{81\left( u+1\right) \left( u+2\right) ^{2}}{\left( u+3\right) \left( 24u+u^{2}+36\right) ^{2}} \end{aligned}$$

with

$$\begin{aligned} \pi _{2}^{T}-\pi _{2}^{D}=\frac{3u\left( u+1\right) \left( 540u+204u^{2} +25u^{3}+432\right) \left( u^{4}-90u^{3}-423u^{2}-648u-324\right) }{4\left( u+3\right) \left( 2u+3\right) \left( 24u+u^{2}+36\right) ^{2}\left( 21u+4u^{2}+18\right) ^{2}} \end{aligned}$$

Once more, there is one variation in polynomial $h(u):=\left( u^{4} -90u^{3}-423u^{2}-648u-324\right)$ and so, using Descartes’ Rule of Signs, there is one positive root, $u_{r}$. As $h(0)=-324<0$, we have that for $u>u_{r}$ entry increases the profit of firm 2.