Protecting Sticky Consumers in Essential Markets

Abstract

This paper studies regulatory policy interventions that are aimed at protecting sticky consumers who are exposed to the risk of being taken advantage of. We model heterogeneous consumer switching costs alongside asymmetric market shares. This setting encompasses many markets in which established firms are challenged by new entrants. We identify circumstances under which such interventions can be counterproductive: with regard to the stated consumer protection objective and also with regard to the complementary aim to promote competition.

Appendices

Proofs

Proof of Lemma 5.1

The result follows immediately from equations (7) and (8) and assumptions 1 and 2.

An alternative argument proceeds as follows. Assumption 1 implies H and RH. Suppose that the opposite were true: $\sigma _A < \sigma _B$. Then, by RH and Assumption 2, for $\sigma ^D_A<s<\sigma ^D_B$,

$$\begin{aligned} \frac{F_A(\sigma ^D_A)}{f_A(\sigma ^D_A)}\le \frac{F_A(s)}{f_A(s)} \le \frac{F_B(s)}{f_B(s)}\le \frac{F_B(\sigma ^D_B)}{f_B(\sigma ^D_B)}, \end{aligned}$$

and so the last two first-order conditions imply $p^D_B-m^D_B\le p_A^D-m^D_A$. H and Assumption 2 imply,

$$\begin{aligned} \frac{1-F_B(\sigma ^D_B)}{f_B(\sigma ^D_B)}\le \frac{1-F_B(s)}{f_B(s)}\le \frac{1-F_A(s)}{f_A(s)}\le \frac{1-F_A(\sigma ^D_A)}{f_A(\sigma ^D_A)}, \end{aligned}$$

and the first two first-order conditions in turn imply $p^D_B\le p^D_A$. Therefore, the two inequalities imply $\sigma ^D_B=p^D_B-p^D_A+m^D_A \le p^D_A-p^D_B+m^D_B=\sigma ^D_A$, which is a contradiction. $\square$

Proof of Proposition 5.1

From $\sigma ^D_A\ge \sigma ^D_B$ and the first-order conditions, it follows that $2(p^D_A- p^D_B)\ge m_A^D-m_B^D$. So it is sufficient to prove that $m^D_A\ge m^D_B$.

From the definitions of $\sigma _A$ and $\sigma _B$,

$$\begin{aligned} \sigma ^D_A+\sigma ^D_B=m^D_A+m^D_B, \end{aligned}$$

and from the first-order conditions,

$$\begin{aligned} \sigma ^D_A= & {} p_A^D-p^D_B+m^D_B=m^D_A+\frac{F_B(\sigma ^D_B)}{f_B(\sigma ^D_B)} -\frac{F_A(\sigma ^D_A)}{f_A(\sigma ^D_A)}\\ \sigma ^D_B= & {} p_B^D-p^D_A+m^D_A=m^D_B+\frac{F_A(\sigma ^D_A)}{f_A(\sigma ^D_A)} -\frac{F_B(\sigma ^D_B)}{f_B(\sigma ^D_B)}. \end{aligned}$$

Lemma 5.1, together with Assumption 2, implies that $\frac{F_B(\sigma ^D_B)}{f_B(\sigma ^D_B)}-\frac{F_A(\sigma ^D_A)}{f_A(\sigma ^D_A)}\le 0$. Therefore, $0\le m^D_A-\sigma ^D_A=\sigma ^D_B-m^D_B$, and hence

$$\begin{aligned} m^D_B\le \sigma ^D_B\le \sigma ^D_A\le m^D_A. \end{aligned}$$

$\square$

Proof of Proposition 5.2

It is sufficient to prove that $\sigma ^{u}_A \le \sigma ^{D}_A$, which implies, by the first-order conditions for firm A and Assumption 2, that $p^{u}_A\ge p^D_A$. This, together with $\sigma ^D_A>\sigma ^D_B>0$ under HBPD, as shown in Lemma 5.1, by Assumption 2 implies also $p^{u}_B\ge p^D_B-m^D_B$.⁵⁷

Suppose to the contrary that $\sigma ^{u}_A > \sigma ^{D}_A$. Then, $p^{u}_A\le p^D_A$ by the first-order conditions and Assumption 2. This ranking of prices of firm A, together with the supposition, implies also that $p^{u}_B\le p^D_B-m^D_B$.⁵⁸ Therefore, $p^{u}_B-c\le p^D_B-c-m^D_B$, and hence

$$\begin{aligned} \frac{1-x_0+x_0F_A(\sigma ^{u}_A)}{x_0f_A(\sigma ^{u}_A)}\le \frac{F_A(\sigma ^D_A)}{f_A(\sigma ^D_A)}. \end{aligned}$$

Notice that $x_0=1$ implies $\sigma ^{u}_A=\sigma ^D_A$. Since the lefthand side of the inequality is decreasing in $x_0$, Assumption 2 implies that $\sigma ^{u}_A<\sigma ^D_A$ for $\frac{1}{2}<x_0<1$, which is a contradiction to the supposition. $\square$

Proof of Lemma 5.2

Suppose, to the contrary, that $\sigma ^i_j<\sigma ^e_j$. Then, a customer of firm j with s such that $\sigma ^i_j<s<\sigma ^e_j$, switches externally, but not internally, iff

$$\begin{aligned} p_k^L-m_k^L+s<pj^L-m_j^L+\alpha s, \end{aligned}$$

or iff

$$\begin{aligned} (1-\alpha )s < p_j^L-p_k^L-(m_j^L-m_k^L). \end{aligned}$$

A customer of firm j with $s'<\sigma ^i_j$ switches internally, but not externally, iff

$$\begin{aligned} p_k^L-m_k^L+s'>pj^L-m_j^L+\alpha s', \end{aligned}$$

or iff

$$\begin{aligned} (1-\alpha )s' > p_j^L-p_k^L-(m_j^L-m_k^L). \end{aligned}$$

But then, $s'>s$, which is a contradiction. $\square$

Proof of Proposition 5.3

Suppose external switching is from A to B and $\sigma ^e_A>0$. Then,

$$\begin{aligned} \pi ^L_A= & {} x_0p^L_A\left( 1-F_A(\sigma ^e_A)\right) -m_A^L x_0\left( F_A(\sigma ^i_A) -F_A(\sigma ^e_A)\right) \\ \pi ^L_B= & {} (1-x_0)p_B^L-m_B^L(1-x_0)F_B\left( \sigma _B^i\right) +x_0(p^L_B-m_B^L)F_A\left( \sigma ^e_A\right) . \end{aligned}$$

The first-order conditions for the firms’ profit maximization problem yield

$$\begin{aligned} \sigma _A^{*i}= & {} \frac{m_A^{*L}}{\alpha }=\frac{1-F_A(\sigma ^{*i}_A)}{f_A(\sigma ^{*i}_A)}\\ \sigma _B^{*i}= & {} \frac{m_B^{*L}}{\alpha }=\frac{1-F_B(\sigma ^{*i}_B)}{f_B(\sigma ^{*i}_B)}\\ \frac{p^{*L}_A-m^{*L}_A}{1-\alpha }= & {} \frac{1-F_A(\sigma _A^{*e})}{f_A(\sigma ^{*e}_A)}\\ \frac{p^{*L}_B-m^{*L}_B}{1-\alpha }= & {} \frac{1-x_0+x_0F_A(\sigma ^{*e}_A)}{x_0f_A(\sigma ^{*e}_A)}. \end{aligned}$$

Therefore,

$$\begin{aligned} \sigma ^{*e}_A= & {} \frac{p^{*L}_A-m^{*L}_A}{1-\alpha }-\frac{p^{*L}_B-m^{*L}_B}{1-\alpha }\\= & {} \frac{2x_0\left( 1-F_A(\sigma ^{*e}_A)\right) -1}{x_0f_a(\sigma ^{*e}_A)}, \end{aligned}$$

which shows that $\sigma ^e_A >0$ if, and only if, $x_0\ge \frac{1}{2}$. $\square$

Proof of Proposition 5.4

Since $\sigma ^{L_e}_B=0$, $\sigma ^D_B>0$ implies that $p^D_A-m^D_A< p^L_A-m^L_A$; $\sigma ^D_A>\sigma ^{L_e}_A$ implies $p^D_B-m^D_B<p^L_B-m^L_B$; and $\sigma ^{L_i}_j>\sigma ^D_j$ implies $p^L_j>p^D_j$, $j\in \{A,B\}$. $\square$

Proof of Proposition 5.5

This result is another corollary to Proposition 5.3. The Corollary 5.3.1 shows that $\sigma ^e_A =\sigma ^u_A$ does not depend on $\alpha$. So the challenger’s profit depends only on $p^{L}_B$ - with $m^L_B=0$ -, and (18) shows that this price is increasing in $\alpha$. If “leakage” is only imposed on A, then the challenger competes as in the case of UP—less vigorously – and its reaction function shifts according to (12), rather than (21). At the same time, A also competes with itself, and this is greater when the cost of internal switching is lower. Hence A’s means to discriminate are reduced relative to conventional HBPD, and that benefits the challenger. In the limit, as $\alpha$ tends to zero, this yields the UP outcome which is the most profitable for both firms. $\square$

Proof of Proposition 5.6

The firms’ profit maximization problem, subject to the ratio constraints—$m_A-\beta p_A\le 0, m_B-\beta p_B\le 0$—is to maximise

$$\begin{aligned} \pi _A= & {} p_Ax_0(1-F_A(\sigma _A))+(p_A-m_A)(1-x_0)F_B(\sigma _B)+\lambda _A(m_a-\beta p_A)\\ \pi _B= & {} p_B(1-x_0)(1-F_B(\sigma _B))+(p_B-m_B)x_0F_A(\sigma _A)+\lambda _B(m_B-\beta p_B), \end{aligned}$$

and the first-order conditions yield

$$\begin{aligned} p^{R}_A= & {} \frac{1-F_A(\sigma ^{R}_A)}{f_A(\sigma ^{R}_A)}+\lambda ^R_A\frac{\beta -1}{x_0f_A(\sigma ^R_A)}\\ p^{R}_B= & {} \frac{1-F_B(\sigma ^{R}_B)}{f_B(\sigma ^{R}_B)}+\lambda ^R_B\frac{\beta -1}{(1-x_0)f_B(\sigma ^R_B)}\\ p^{R}_A-m^{R}_A= & {} \frac{F_B(\sigma ^{R}_B)}{f_B(\sigma ^{R}_B)}+\lambda ^R_A\frac{1}{(1-x_0)f_B(\sigma ^R_B)}\\ p^{R}_B-m^{R}_B= & {} \frac{F_A(\sigma ^{R}_A)}{f_A(\sigma ^{R}_A)}+\lambda ^R_B\frac{1}{x_0f_A(\sigma ^R_A)}, \end{aligned}$$

where $\sigma ^{R}_j$ are defined analogously to $\sigma ^D_j$ and the Kuhn-Tucker Theorem yields $\lambda ^R_j\ge 0$, $j\in \{A,B\}$. Also,

$$\begin{aligned} \sigma ^R_A= & {} \frac{1-2F_A(\sigma ^R_A)}{f_A(\sigma ^R_A)}+\frac{1}{x_0f_A(\sigma ^R_A)}(\lambda ^R_A(\beta -1)-\lambda ^R_B) \end{aligned}$$

(22)

$$\begin{aligned} \sigma ^R_B= & {} \frac{1-2F_B(\sigma ^R_B)}{f_B(\sigma ^R_B)}+\frac{1}{(1-x_0)f_B(\sigma ^R_B)}(\lambda ^R_B(\beta -1)-\lambda ^R_A). \end{aligned}$$

(23)

Since $\lambda ^R_A(\beta -1)-\lambda ^R_B$ and $\lambda ^R_B(\beta -1)-\lambda ^R_A$ are non-positive, Assumption 2 implies that $\sigma ^R_j\le \sigma ^D_j$ and so $p^R_j\ge p^D_j$, $j\in \{A,B\}$. When $\beta$ is large enough so that the ratio constraints do not bind, $\lambda ^R_A=\lambda ^R_B=0$ and (22) and (23) imply the HBPD result. When $\beta =0$, then, from Proposition (5.2) $\sigma ^R_B=0$ and so (23) implies $\lambda ^R_A+\lambda ^R_B=1-x_0$, so that (22) implies that $\sigma ^R_A=\sigma ^u_A$. $\square$

Price Discrimination as Entry Deterrence

In order to investigate the incumbent’s HBPD as an entry deterrence strategy in our setting, suppose that the market is dynamic, in the sense that over time some customers leave the market while others join the market. Specifically, for a market size $\tau$, suppose that the fraction of ’new’ customers that the challenger can capture—$(1-x_0)\tau$—to ’old’ customers that are served by the incumbent—$x_0\tau$—is a constant $\kappa =\frac{1-x_0}{x_0}$; and suppose that $\tau$ itself is ex ante uncertain. Then the challenger’s profits - when pricing uniformly and facing a price discriminating incumbent and conditional on $x_0$ - are

$$\begin{aligned} \pi ^*_B(\tau )=\tau \frac{\left( \kappa \left( 1-F_B(\sigma ^*_B)\right) +F_A(\sigma ^*_A)\right) ^2}{\kappa f_B(\sigma ^*_B)+f_A(\sigma ^*_A)}. \end{aligned}$$

Notice that

$$\begin{aligned} \sigma ^*_A= & {} p^*_A-p^*_B=\frac{1-F_A(\sigma ^*_A)}{f_A(\sigma ^*_A)} -\frac{\kappa \left( 1-F_B(\sigma ^*_B)\right) +F_A(\sigma ^*_A)}{\kappa f_B(\sigma ^*_B)+f_A(\sigma ^*A)}\\ \sigma ^*_B= & {} p^*_B-p^*_A+m^*_A=\frac{\kappa \left( 1-F_B(\sigma ^*_B)\right) +F_A(\sigma ^*_A)}{\kappa f_B(\sigma ^*_B)+f_A(\sigma ^*A)} -\frac{F_B(\sigma ^*_b)}{f_B(\sigma ^*_B)}: \end{aligned}$$

The switching costs of the marginal customer depend on $\kappa$, but not on $\tau$.

Then, if the challenger firm has not entered the market yet, its expected profit from entering is given by

$$\begin{aligned} {\mathbb {E}}\left[ \pi ^*_B(\tau )\right] ={\mathbb {E}}\left[ \tau \right] \frac{\left( \kappa \left( 1-F_B(\sigma ^*_B)\right) +F_A(\sigma ^*_A)\right) ^2}{\kappa f_B(\sigma ^*_B)+f_A(\sigma ^*_A)}. \end{aligned}$$

When the incumbent prices uniformly,

$$\begin{aligned} {\mathbb {E}}[\pi ^{*u}_B(\tau )]={\mathbb {E}}\left[ \tau \right] \frac{\left( \kappa +F_A(\sigma ^{*u}_A)\right) ^2}{f_A(\sigma ^{*u}_A)}, \end{aligned}$$

and this is seen to exceed ${\mathbb {E}}[\pi ^*_B(x_0)]$.⁵⁹ So, as the challenger compares expected profit from entry with any sunk cost that is associated with entry, price discrimination on the part of the incumbent, as opposed to uniform pricing, erects a higher entry barrier for a uniform-pricing challenger.