Free in-network pricing as an entry-deterrence strategy

Abstract

This paper analyzes the entry-deterring power of free in-network pricing with multiple incumbents. Free in-network pricing may deter entry since it creates network externality that intensifies competition. One may expect that a particular entry-deterrent strategy adopted by all incumbents would have more entry-deterring power than when it is adopted by some incumbents only. However, we show that when free-in network plan has entry-deterrence power with two incumbent firms, sometimes one incumbent offering free in-network plan may have more entry deterrence power than both firms offering free in-network plans. In other words, we find that an asymmetric adoption of entry-deterrence strategies by the incumbent firms may be the best for entry deterrence. This result highlights the importance of the strategic choice of the pricing plan as a function of not only the likelihood/cost of entry but also of the plan choices of other firms, and may partially explain the asymmetric strategies used by competing firms.

Appendix: Technical derivations

1.1 Market Shares: Duopoly

As shown in the main text of the paper we have the following consumer utility function (i = 1, 2):

$$ {U}_i=V+\frac{{\left(1-{p}_i^I\right)}^2}{4}{M}_i+\frac{{\left(1-{p}_i^o\right)}^2}{4}\left(3-{M}_i\right)-{f}_i-td, $$

(6)

or, more explicitly,

$$ {U}_i=V+\frac{{\left(1-{p}_1^I\right)}^2}{4}M+\frac{{\left(1-{p}_1^o\right)}^2}{4}\left(3-M\right)-{f}_1-td $$

(7)

$$ {U}_2=V+\frac{{\left(1-{p}_2^I\right)}^2}{4}\left(3-M\right)+\frac{{\left(1-{p}_2^o\right)}^2}{4}M-{f}_2-t\left(1-d\right). $$

(8)

Recall that the market is fully-covered. The indifferent consumer’s position is determined by ${U}_1={U}_2$. Therefore the location of the indifferent consumer is

$$ {d}^{\ast }=\frac{1}{2t}\left[t+{f}_2-{f}_1+M\frac{{\left(1-{p}_1^I\right)}^2-{\left(1-{p}_2^o\right)}^2}{4}+\left(3-M\right)\frac{{\left(1-{p}_1^o\right)}^2-{\left(1-{p}_2^I\right)}^2}{4}\right] $$

(9)

Hence by setting M = d ^∗ + 1 we find the market size of firm 1 and firm 2:

$$ M=\frac{4\left(3t+{f}_2-{f}_1\right)+3{\left(1-{p}_1^o\right)}^2-3{\left(1-{p}_2^I\right)}^2}{8t+{\left(1-{p}_1^o\right)}^2+{\left(1-{p}_2^o\right)}^2-{\left(1-{p}_1^I\right)}^2-{\left(1-{p}_2^I\right)}^2} $$

(10)

$$ 3-M=\frac{4\left(3t+{f}_1-{f}_2\right)+3{\left(1-{p}_2^o\right)}^2-3{\left(1-{p}_1^I\right)}^2}{8t+{\left(1-{p}_1^o\right)}^2+{\left(1-{p}_2^o\right)}^2-{\left(1-{p}_1^I\right)}^2-{\left(1-{p}_2^I\right)}^2} $$

(11)

Since every consumer is rational, her expectation of the market shares is the same as the actual market shares, which implies that the firm profit functions are

$$ {\Pi}_1=\left[\left({p}_1^I-c\right)\frac{1-{p}_1^I}{2}M+\left({p}_1^o-c\right)\frac{1-{p}_1^o}{2}\left(3-M\right)+{f}_1\right]M; $$

(12)

$$ {\Pi}_2=\left[\left({p}_2^I-c\right)\frac{1-{p}_2^I}{2}\left(3-M\right)+\left({p}_2^o-c\right)\frac{1-{p}_2^o}{2}M+{f}_2\right]\left(3-M\right). $$

(13)

Here ∏ _i is the profit of firm i, x _i is the in-network usage, y _i is the out-network usage, ${p}_i^I$ is the in-network per-use fee of firm i, ${p}_i^o$ is the out-network per-use fee of firm i, and f _i is the subscription price of firm i (i= 1, 2), and c is the per-minute cost $\left(c\le \frac{1}{2}\right)$.

1.2 Market Shares: 3 Firms in the Market

Similar to the procedure of calculating the market shares in 2-firm case, solving the indifferent consumers’ utility equations, we have the market shares for firm 1, 2 and 3:

$$ {M}_1=3g(1)\frac{1+g(2)\left[4\left({f}_2-{f}_1\right)+3{\left(1-{p}_1^o\right)}^2\right]+g(3)\left[4\left({f}_3-{f}_1\right)+3{\left(1-{p}_1^o\right)}^2\right]}{g(1)+g(2)+g(3)} $$

(14)

$$ {M}_2=3g(2)\frac{1+g(1)\left[4\left({f}_1-{f}_2\right)+3{\left(1-{p}_2^o\right)}^2\right]+g(3)\left[4\left({f}_3-{f}_2\right)+3{\left(1-{p}_2^o\right)}^2\right]}{g(1)+g(2)+g(3)} $$

(15)

$$ {M}_3=3g(3)\frac{1+g(1)\left[4\left({f}_1-{f}_3\right)+3{\left(1-{p}_3^o\right)}^2\right]+g(2)\left[4\left({f}_2-{f}_3\right)+3{\left(1-{p}_3^o\right)}^2\right]}{g(1)+g(2)+g(3)} $$

(16)

That is

$$ {M}_i=3g(i)\frac{1+g(j)\left[4\left({f}_j-{f}_i\right)+3{\left(1-{p}_1^o\right)}^2\right]+g(k)\left[4\left({f}_k-{f}_i\right)+3{\left(1-{p}_i^o\right)}^2\right]}{g(1)+g(2)+g(3)} $$

(17)

For i , j , k = 1 , 2 , 3 and where $g(i)=\frac{1}{8t+3{\left(1-{p}_i^o\right)}^2-3{\left(1-{p}_i^I\right)}^2}\left(i=1,2\kern0.5em \mathrm{or}\ 3\right)$.

Plug ${x}_i^{\ast }$ and ${y}_i^{\ast }$ into the profit functions as in (23) in the main text of the paper, and we will have the profit functions of the 3 firms.

1.3 Proof of Lemma 1

Suppose there are two firms in competition, and they offer subscription prices f ₁ and f ₂, and per-use fees ${p}_1^I$, ${p}_2^I$, ${p}_1^o$ and ${p}_2^o$. Conssider firm 1’s deviation to ${\widehat{p}}_1^I=c$, ${\widehat{p}}_1^o=c$ and subscription price ${\widehat{f}}_1$ such that its new market share would be the same as the old one: $\widehat{M}=M$, that is:

$$ {\widehat{f}}_1=3t+{f}_2+\frac{3{\left(1-c\right)}^2-3{\left(1-{p}_2^I\right)}^2}{4}-M\left[2t+\frac{{\left(1-c\right)}^2+{\left(1-{p}_2^o\right)}^2-{\left(1-c\right)}^2-{\left(1-{p}_2^I\right)}^2}{4}\right] $$

(18)

Then, firm 1’s profit will be no less than the profit it would have when offering ${p}_1^I$, ${p}_1^o$ and f ₁:

$$ \begin{array}{l}\frac{{\widehat{\Pi}}_1-{\Pi}_1}{M}=\widehat{f}-{f}_1-\frac{\left({p}_1^I-c\right){\left(1-{p}_1^I\right)}^2}{2}M-\frac{\left({p}_1^o-c\right)\left(1-{p}_1^o\right)}{2}\left(3-M\right)\\ {}\kern3.5em =M\frac{{\left({p}_1^I-c\right)}^2}{4}+\left(3-M\right)\frac{{\left({p}_1^o-c\right)}^2}{4}\ge 0.\end{array} $$

(19)

(“=” holds when ${p}_1^I=c$ and ${p}_1^o=c$, implying ${\widehat{f}}_1={f}_1$).

Therefore, no matter what prices a firm offers and whether it pre-announces them, it is dominant for the other firm to offer metered plan, that is, to set ${p}_i^I={p}_i^o=c\left(i=1,2\right)$.

Similarly, one proves that if firm 1 offers free in-network pricing plan, that is, ${p}_1^I=0$, then no matter what price and fees firm 2 offers (with or without pre-announcement), firm 1 can make ${\widehat{p}}_1^o=c$ and${\widehat{f}}_1$ such that $\widehat{M}=M$, to generate profit ${\widehat{\Pi}}_1\ge {\Pi}_1$.

In the case of 3 firms in the market, if firm 3 decides to offer metered plan, then it sets${\widehat{p}}_3^I=c$, ${\widehat{p}}_3^o=c$ and ${\widehat{f}}_3$ such that ${\widehat{M}}_3={M}_3$. From ${\widehat{M}}_3={M}_3$we have:

$$ {\widehat{f}}_3-{f}_3=\frac{1}{4}\left[3{\left(1-c\right)}^2-{M}_3{\left(1-{p}_3^I\right)}^2-\left(3-{M}_3\right){\left(1-{p}_3^o\right)}^2\right] $$

(20)

Therefore:

$$ \begin{array}{l}\frac{{\widehat{\Pi}}_3-{\Pi}_3}{M_3}={\widehat{f}}_3-{f}_3-\frac{\left({p}_3^I-c\right)\left(1-{p}_3^I\right)}{2}{M}_3-\frac{\left({p}_3^o-c\right)\left(1-{p}_3^o\right)}{2}\left(3-{M}_3\right)\\ {}\kern2.95em ={M}_3\frac{{\left({p}_3^I-c\right)}^2}{4}+\left(3-{M}_3\right)\frac{{\left({p}_3^o-c\right)}^2}{4}>=0\end{array} $$

(21)

(“=” holds when ${p}_3^I=c$ and ${p}_3^o=c$).

Similarly, one can prove that if firm 3 offers free in-network pricing plan, that is, ${p}_3^I=0$, then no matter what prices and fees the other firms set (with or without pre-announcement), firm 3 sets ${\widehat{p}}_3^0=c$ and ${\widehat{f}}_3$ such that ${\widehat{M}}_3={M}_3$, to obtain profit ${\widehat{\Pi}}_3\ge {\Pi}_3$.

1.4 Proof of Lemma 2

When there are 2 firms in the market without entry threat, as both firm 1 and firm 2 already pre-announced their per-minute fees ${p}_1^I$, ${p}_1^o$, ${p}_2^I$ and ${p}_2^o$, then we can calculate their first-order and second-order conditions subject to f ₁ and f ₂:

$$ {\Pi}_1=\left[\left({p}_1^I-c\right)\frac{1-{p}_1^I}{2}M+\left({p}_1^o-c\right)\frac{1-{p}_1^o}{2}\left(3-M\right)+{f}_1\right]M $$

(22)

$$ {\Pi}_2=\left[\left({p}_2^I-c\right)\frac{1-{p}_2^I}{2}\left(3-M\right)+\left({p}_2^o-c\right)\frac{1-{p}_2^o}{2}M+{f}_2\right]\left(3-M\right) $$

(23)

First-order conditions:

$$ \frac{\partial {\Pi}_i}{\partial {f}_1}=M+\left[\left({p}_1^I-c\right)\frac{1-{p}_1^I}{2}2M+\left({p}_1^o-c\right)\frac{1-{p}_1^o}{2}\left(3-2M\right)+{f}_1\right]\frac{\partial M}{\partial {f}_1}=0 $$

(24)

$$ \frac{\partial {\Pi}_i}{\partial {f}_1}=\left(3-M\right)-\left[\left({p}_2^I-c\right)\frac{1-{p}_2^I}{2}2\left(3-M\right)+\left({p}_2^o-c\right)\frac{1-{p}_2^o}{2}\left(3-2\left(3-M\right)\right)+{f}_2\right]\frac{\partial M}{\partial {f}_2}=0 $$

(25)

Here M is defined by (10), and

$$ \frac{\partial M}{\partial {f}_1}=-\frac{\partial M}{\partial {f}_2}=\frac{-4}{8t+{\left(1-{p}_1^o\right)}^2+{\left(1-{p}_2^o\right)}^2-{\left(1-{p}_1^I\right)}^2-{\left(1-{p}_2^I\right)}^2} $$

(26)

Second-order conditions:

$$ \begin{array}{l}\kern1em \frac{\partial \Pi}{\partial {f}_1^2}=2\left[\left({p}_1^I-c\right)\frac{1-{p}_1^I}{2}\frac{\partial M}{\partial {f}_1}-\left({p}_1^o-c\right)\frac{1-{p}_1^o}{2}\frac{\partial M}{\partial {f}_1}+1\right]\frac{\partial M}{\partial {f}_1}\\ {}+\left[\left({p}_1^I-c\right)\frac{1-{p}_1^I}{2}2M+\left({p}_1^o-c\right)\frac{1-{p}_1^o}{2}\left(3-2M\right)+{f}_1\right]\frac{\partial^2M}{\partial {f}_1^2}\le 0\end{array} $$

(27)

$$ \begin{array}{l}\kern2em \frac{\partial^2{\Pi}_2}{\partial {f}_2^2}=2\left[\left({p}_2^I-c\right)\frac{1-{p}_2^I}{2}\frac{\partial M}{\partial {f}_2}-\left({p}_2^o-c\right)\frac{1-{p}_2^o}{2}\frac{\partial M}{\partial {f}_2}-1\right]\frac{\partial M}{\partial {f}_2}\\ {}-\left[\left({p}_2^I-c\right)\frac{1-{p}_2^I}{2}2\left(3-M\right)+\left({p}_2^o-c\right)\frac{1-{p}_2^o}{2}\left(3-2\left(3-M\right)\right)+{f}_2\right]\frac{\partial^2M}{\partial {f}_2^2}\end{array} $$

(28)

Here $\frac{\partial^2M}{\partial {f}_1^2}=\frac{\partial^2M}{\partial {f}_1^2}=0$. For the different cases below, we derived first-order conditions to find the optimal fees, and verified that second-order conditions are all satisfied.

Both firms offer metered plans: ${f}_1^{\ast }={f}_2^{\ast }=3t$

• (i) Both firms offer free in-network calling plans: ${f}_1^{\ast }={f}_2^{\ast }=3t+3c\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$

• (ii) Both firms offer flat-fee plans: ${f}_1^{\ast }={f}_2^{\ast }=3t+3c{/}_2$

• (iii) Firm 1 offers metered plan, firm 2 offers free in-network calling plan: ${f_1}^{\ast }=\frac{3\left(6t+{c}^2\right)\left(8t+{c}^2-2c\right)}{2\left(24t+3{c}^2-2c\right)},\kern0.5em {f_2}^{\ast }=\frac{3\left(6t+{c}^2\right)\left(8t+{c}^2-2c\right)}{2\left(24t+3{c}^2-2c\right)}$

• (iv) Firm 1 offers metered plan, firm 2 offers flat-fee plan: ${f}_1^{\ast }=3t+3c\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$4$}\right.$, ${f}_2^{\ast }=3t-c\raisebox{1ex}{$2$}\!\left/ \!\raisebox{-1ex}{$4$}\right.+3c{/}_2$

• (v) Firm 1 offers free in-network, while firm 2 offers flat-fee plan: ${f_1}^{\ast }=\frac{3\left(8t+{c}^2+2c\right)\left(6t+{c}^2-c\right)}{2\left(24t+3{c}^2-2c\right)},\kern0.5em {f_2}^{\ast }=\frac{3\left(8t+{c}^2+2c\right)\left(6t+{c}^2-c\right)}{2\left(24t+3{c}^2-2c\right)}$

Therefore, we have profits as in Table 1.

Since firm 3 does not pre-announce any plan before it enters, from Lemma 1 we know that it will always charges usage fee of 2c. Suppose firm 1 and firm 2 pre-announced their usage fees ${p}_1^I$, ${p}_1^o$, ${p}_2^I$ and ${p}_2^o$. Deriving first-order and second-order conditions with respect to f ₁,f ₂ and f ₃:

$$ {\Pi}_1=\left[\left({p}_1^I-c\right)\frac{1-{p}_1^I}{2}M+\left({p}_1^o-c\right)\frac{1-{p}_1^o}{2}\left(3-{M}_1\right)+{f}_1\right]{M}_1 $$

(29)

$$ {\Pi}_2=\left[\left({p}_2^I-c\right)\frac{1-{p}_2^I}{2}{M}_2+\left({p}_2^o-c\right)\frac{1-{p}_2^o}{2}\left(3-M\right)+{f}_2\right]{M}_2 $$

(30)

$$ {\Pi}_3=\left[\left({p}_3^I-c\right)\frac{1-{p}_3^I}{2}{M}_3+\left({p}_3^o-c\right)\frac{1-{p}_3^o}{2}\left(3-M\right)+{f}_3\right]{M}_3-{F}_3 $$

(31)

First-order conditions:

$$ \frac{\partial {\Pi}_i}{\partial {f}_i}={M}_i+\left[\left({p}_1^I-c\right)\frac{1-{p}_1^I}{2}2{M}_1+\left({p}_1^o-c\right)\frac{1-{p}_1^o}{2}\left(3-2{M}_1\right)+{f}_1\right]\frac{\partial {M}_1}{\partial {f}_1}=0\kern0.5em for\ i=1,2,3 $$

(32)

Here M _i is as in (17) shown earlier.

Second-order conditions:

$$ \begin{array}{l}\kern3em \frac{\partial^2{\Pi}_i}{\partial {f}_i^2}=2\left[\left({p}_1^I-c\right)\frac{1-{p}_1^I}{2}\frac{\partial {M}_1}{\partial {f}_1}-\left({p}_1^o-c\right)\frac{1-{p}_1^o}{2}\frac{\partial {M}_1}{\partial {f}_1}+1\right]\frac{\partial {M}_1}{\partial {f}_1}\\ {}+\left[\left({p}_1^I-c\right)\frac{1-{p}_1^I}{2}2{M}_i+\left({p}_i^o-c\right)\frac{1-{p}_i^o}{2}\left(3-2{M}_i\right)+{f}_i\right]\frac{\partial^2{M}_i}{\partial {f}_i^2}\le 0\kern0.5em for\ i=1,2,3\end{array} $$

(33)

Here $\frac{\partial^2{M}_i}{\partial {f}_1^2}=0$ for $i=$1, 2, 3.

From first- and second-order conditions, we find the equilibrium prices f ₁,f ₂ and f ₃ as in the Appendix Table 3. With the profits as in the table, it is easy to check that if the entrant never enters the market, it is dominant for either incumbent to offer metered plan. Furthermore, if the choice of the plan does not affect entry decision, it is always dominant for each incumbent to offer a metered plan regardless of the plan offered by the other incumbent. The proof is straightforward from the comparison of an arbitrary plan and a plan with marginal cost metered pricing which gives the firm the same per-customer surplus (and thus, due to a more efficient consumer choice of use, results in a higher consumer willingness to pay).

1.5 Proof of Proposition 1

To show that the incumbents have more profits by deterring entry if they can:

If both incumbent firms offer free in-network pricing plans and deter entry, either of them has profit $\frac{36t+9\left({c}^2-c\right)}{8}$, if they both offer metered plans and allow the entrant, then each of them has profit t. It is always the case that $\frac{36t+9\left({c}^2-c\right)}{8}>t$, because for this to hold we only need $14t+\frac{18{c}^2}{4}-\frac{9c}{2}>0$, which always holds when t > 2c. If one incumbent offers free in-network pricing plan and the other metered plan, the profits they have are: $\frac{9{\left(12t+{c}^2-2c\right)}^2\left(8t+{c}^2\right)}{4{\left(24t+3{c}^2-2c\right)}^2}$ for the firm that offers free in-network pricing plan, and $\frac{9{\left(6t+{c}^2\right)}^2\left(8t+{c}^2-2c\right)}{{\left(24t+3{c}^2-2c\right)}^2}$ for the firm that offers metered plan. If they both offer metered plans and the entrant enters, the profits equal to t.

Next, we show that $\frac{9{\left(12t+{c}^2-2c\right)}^2\left(8t+{c}^2\right)}{4\left(24t+3{c}^2-2c\right)}>t\kern0.5em \mathrm{and}\ \frac{9{\left(6t+{c}^2\right)}^2\left(8t+{c}^2-2c\right)}{{\left(24t+3{c}^2-2c\right)}^2}>t$ _:

$\frac{9{\left(12t+{c}^2-2c\right)}^2\left(8t+{c}^2\right)}{4\left(24t+3{c}^2-2c\right)}>7t\iff 18t{\left(12t+{c}^2-2c\right)}^2+\frac{9}{4}{c}^2{\left(6t+{c}^2-2c\right)}^2>t{\left(24t+3{c}^2-2c\right)}^2$.

$18t{\left(12t+{c}^2-2c\right)}^2>t{\left(24t+3{c}^2-2c\right)}^2\iff 126{t}^2+18{tc}^2-48tc+\frac{9}{16}{c}^4-\frac{15}{4}{c}^3+\frac{9}{4}{c}^2>0$, which holds since t ≥ c and 0 ≤ c ≤ 1. Moreover,

$\frac{9{\left(12t+{c}^2-2c\right)}^2\left(8t+{c}^2\right)}{4\left(24t+3{c}^2-2c\right)}>t\iff 126{t}^3+\frac{171}{16}{tc}^4+\frac{3}{4}{tc}^3-\frac{tc^2}{4}-\frac{81}{8}{t}^2{c}^3-\frac{27}{2}tc3-\frac{9}{8}{c}^5+\frac{9}{16}{c}^6>0$, which holds for t ≥ c and 0 ≤ c ≤ 1. Thus, the profits the incumbents get by offering free in-network pricing and deterring entry are always larger than the profits they get by offering a metered plan and allowing entry.

To show that by choosing free in-network plan(s) the incumbents may deter entry, we compare the entrant’s profit corresponding to the plans offered by the incumbents, and see what happens for specific numerical values. As Fig. 3 shows, when the per-minute cost is not too high and the incumbents offer free in-network pricing plan(s), the entrant may have profit lower than t (the profit it has when the incumbents accommodate it).

Finally, let us prove that flat-fee plan has no advantage in entry deterrence and is never used in pure-strategy equilibrium. From Appendix Table 3 it is clear that when one or both of the incumbents offer flat-fee plans, the entrant’s profit is larger but the incumbents’ profits are smaller than in the case where the incumbents both offer metered plans. Moreover, to show that the incumbents offering free in-network and flat-fee plans is worse at entry deterrence than the incumbents offering free in-network metered plans, consider the ratio of entrant’s gross-of-fixed-cost profits:

$\frac{\Pi_3\left( Free, Flat\right)}{\Pi_3\left( Free, Measured\right)}=\frac{{\left(320{t}^2+120{tc}^2-144tc+9{c}^4-18{c}^3\right)}^2}{64{t}^2{\left(40t+9{c}^2-18c\right)}^2}=\frac{{\left(320{t}^2+120{tc}^2-144tc+9{c}^4-18{c}^3\right)}^2}{{\left(320{t}^2+72{tc}^2-144tc\right)}^2}$.

We have: (320t ² + 120tc ² − 144tc + 9c ⁴ − 18c ³) − (320t ² + 72tc ² − 144tc) = 48tc ² + 9c ⁴ − 18c ³ > 0 when c < 1, which is true given our assumptions. Therefore, $\frac{\Pi_3\left( Free, Flat\right)}{\Pi_3\left( Free, Measured\right)}>1$. Also, due to the dominance of metered plan, an incumbent has no incentive to offer flat-fee plan when the other incumbent offers free in-network pricing plan, even when it does not consider entry deterrence.

Table 3 Prices in Triopoly