Prices as signals of product quality in a duopoly

Abstract

In a duopoly model of horizontal and vertical differentiation, where consumers are ex-ante unaware of product qualities, we study the firms’ incentives to signal quality via prices. Consumers, after they observe prices, can evaluate a firm’s product quality before purchase if they incur a search cost. We show that a complete information (undistorted) separating equilibrium and a unique pooling equilibrium (in pure strategies) exist. A lower search cost moves the market equilibrium from pooling to separating and induces a mean-preserving spread in the distribution of the equilibrium prices.

A Appendix: Beliefs and proofs of Lemmas and Propositions

1.1 A.1 Unprejudiced beliefs

In the table below we use $p_H^{*}(HL)=p_H^{*}(LH)\equiv p_H^{*}$ and $p_L^{*}(HL)=p_L^{*}(LH)\equiv p_L^{*}$. The ex-ante beliefs $\mu _{a}^{e}(p_{a},p_{b})$ and $\mu _{b}^{e}(p_{a},p_{b})$ of a firm having a high quality product are defined over the equilibrium price pairs and over price pairs in which one price is an equilibrium price for a given product quality configuration, while the other may be an equilibrium price for a different quality configuration, or a non-equilibrium price. However, they are not defined over price pairs where both prices are non-equilibrium prices, as we solve the game by only considering unilateral deviations and consumer beliefs are consistent with this. (We do not present the cases that are symmetric to the ones that are already in the table).

$p_{a}=$	$p_{b}=$	possible quality configurations	$\mu _{a}^{e}(p_{a},p_{b})$	$\mu _{b}^{e}(p_{a},p_{b})$
$p_{H}^{*}$	$p_{H}^{*}$	(H, L)(L, H)	$\frac{1}{2}$	$\frac{1}{2}$
$p_{H}^{*}$	$p_{L}^{*}$	(H, L)	1	0
$p_{H}^{*}$	$p^{*}$	(H, H)(H, L)(L, L)	$\frac{q^{2}+q(1-q)}{q^{2}+q(1-q)+(1-q)^{2}}$	$\frac{q^{2}}{q^{2}+q(1-q)+(1-q)^{2}}$
$p_{H}^{*}$	$\ne \{p_{H}^{*},p_{L}^{*},p^{*}\}$	(H, L)	1	0
$p_{L}^{*}$	$p_{L}^{*}$	(H, L)(L, H)	$\frac{1}{2}$	$\frac{1}{2}$
$p_{L}^{*}$	$p^{*}$	(H, H)(L, H)(L, L)	$\frac{q^{2}}{q^{2}+q(1-q)+(1-q)^{2}}$	$\frac{q^{2}+q(1-q)}{q^{2}+q(1-q)+(1-q)^{2}}$
$p_{L}^{*}$	$\ne \{p_{H}^{*},p_{L}^{*},p^{*}\}$	(L, H)	0	1
$p^{*}$	$p^{*}$	(H, H)(L, L)	$\frac{q^{2}}{q^{2}+(1-q)^{2}}$	$\frac{q^{2}}{q^{2}+(1-q)^{2}}$
$p^{*}$	$\ne \{p_{H}^{*},p_{L}^{*},p^{*}\}$	(H, H)(L, L)	$\frac{q^{2}}{q^{2}+(1-q)^{2}}$	$\frac{q^{2}}{q^{2}+(1-q)^{2}}$

1.2 A.2 Proof of Lemma 1

A candidate symmetric equilibrium when firms have the same quality, (H, H) or (L, L), is (t, t). Any other symmetric pair of prices when firms have the same product quality cannot be an equilibrium. To see this suppose $(p^{*},p^{*})\ne (t,t)$ is a candidate equilibrium in the (H, H) and (L, L) cases. Consumers, upon observing $(p^{*},p^{*})$ in equilibrium, know that firms have the same quality products. If a firm can unilaterally deviate to $p^{dev}$ without affecting consumer beliefs, then it is clearly better off, since $p^{*}$ is not a best response to $p^{*}$. Any $p^{dev}\ne p_H^{*}(HL)$ or $p^{dev}\ne p_L^{*}(HL)$, leaves consumer beliefs unchanged because consumers observe $(p^{dev},p^{*})$ and realize from the price of the non-deviating firm $p^{*}$ that product qualities are the same. If it happens that the best deviating price is equal to $p_H^{*}(HL)$ or $p_L^{*}(HL)$, in which case consumers may not know for sure who is the deviating firm and what the product qualities are, the deviating firm can set a price $\varepsilon$ away from $p_H^{*}(HL)$ or $p_L^{*}(HL)$, so that consumer beliefs are unchanged, and still be better off.

Next, assume that $(p^{*}_H(HL), p^{*}_L(HL))$ is not equal to the complete information prices given by (1). Then, given the beliefs in Appendix A.1, firm a can deviate to its best response $BR_a(p^{*}_L(HL))\ne p^{*}_H(HL)$ and become better off. Consumers believe that the deviating firm is high quality, based on the price of the non-deviating firm. The unique equilibrium under the unprejudiced beliefs is the complete information equilibrium.

1.3 A.3 Proof of Lemma 2

Firm a has high and firm b has low product qualities, (H, L). The equilibrium prices are $(p^{*}_H(HL),p^{*}_L(HL))$. We begin with firm b’s deviation, from $p_b=p_L^{*}(HL)$ to $p_b^{dev}$. Firm b can deviate to $p_H^{*}(HL)$, to $p^{*}=t$ or to any other price.

First, we assume that $p_b^{dev}=p^{*}_H(HL)$. Consumers observe $(p^{*}_H(HL), p^{*}_H(HL))$ and do not know which firm is the low quality firm, although they know that one firm must be of low quality (given the unprejudiced beliefs consumers know that it cannot be (H, H) or (L, L), because if that was the case one price must have been t). Therefore, from the beliefs in Appendix A.1 we have $\mu _a^e=\mu _b^e=\frac{1}{2}$ and hence $\Delta V^{e}=0$ and the marginal consumer $x^{e}$, given by (2), is at $\frac{1}{2}$. Consumers then who visit either firm realize that firm a is the high quality firm and firm b has a low quality product and is the one that deviated, that is $\Delta V_a^{in}=\Delta V_b^{in}=DV$. Some may switch from b to a, if $\kappa <DV$, but no consumer from a will switch to b. The interim marginal consumer is $x_b^{in}$ given by (3). Since $\Delta V=DV>\Delta _b^{in}-\kappa =DV-\kappa$ the final marginal consumer is $x_b^{in}$. This deviation is unprofitable if and only if

$$\begin{aligned} p^{*}_L(HL)(1-x^e)=\, & {} p^{*}_L(HL)\left( 1-\frac{p^{*}_L(HL)-p^{*}_H(HL)+DV+t}{2t}\right) \nonumber \\\ge\, & {} p^{*}_H(HL)\left( 1-x^{in}_b\right) =\,p^{*}_H(HL)\left( 1-\frac{DV-\kappa +t}{2t}\right) . \end{aligned}$$

(7)

Second, we assume that $p_b^{dev}=p^{*}=t$. Consumers observe $(p^{*}_H(HL),t)$ and know that it cannot be (L, H), because the candidate equilibrium prices in this case are $(p^{*}_L(LH),p^{*}_H(LH))$ and both firms would have to have deviated, for which unprejudiced beliefs assign probability zero. But consumers do not know whether it is (H, H) or (L, L) and one firm raised its price, or (H, L) and the low quality firm deviated. From the ex-ante beliefs presented in Appendix A.1 we have $\mu _a^e=\frac{q^2+q(1-q)}{q^2+(1-q)^2+q(1-q)}$ and $\mu _b^e=\frac{q^2}{q^2+(1-q)^2+q(1-q)}$. So, the expected values for the two firms’ products are

$$\begin{aligned} EV_a=\frac{(q^2+q(1-q))V_H+(1-q)^2V_L}{q^2+(1-q)^2+q(1-q)} \text{ and } EV_b=\frac{q^2V_H+(q(1-q)+(1-q)^2)V_L}{q^2+(1-q)^2+q(1-q)}, \end{aligned}$$

(8)

with $EV_a>EV_b$. Consumers initially sort out according to (2) using the above expected qualities for each firm. The marginal consumer is given by

$$\begin{aligned} x^{e}=\frac{(DV+p^{*}_H(HL)-2t)q(1-q)+2t-p^{*}_H(HL)}{2t(1-q+q^2)}. \end{aligned}$$

The consumers who visit firm b they realize that its quality is low and update their beliefs about the quality of firm a by ruling out (H, H). The interim beliefs of the consumers who visited b about the quality of a being high is $\mu ^{in}_{ab}=\frac{q(1-q)}{(1-q)^2+q(1-q)}$. Using these beliefs, the expected quality of firm a for consumers who first visit b is $EV^{in}_a=\frac{q(1-q)V_H+(1-q)^2V_L}{(1-q)^2+q(1-q)}$. The interim marginal consumer, using (3), is given by

$$\begin{aligned} x^{in}_b=\frac{t-p^{*}_H(HL)+EV^{in}_a-V_L-\kappa +t}{2t}=\frac{qDV-p^{*}_H(HL)-\kappa +2t}{2t}. \end{aligned}$$

Some consumers will switch from b to a if and only if $x^{in}_b>x^{e}\Leftrightarrow \kappa <\kappa _2\equiv \frac{q^3}{1-q+q^2}DV.$

Also it is clear that no consumer will switch from a to b, since a is high quality.

To summarize, when $p_b^{dev}=t$, if $\kappa <\kappa _2$, then $x_b^{in}$ is the relevant marginal consumer, while if $\kappa \ge \kappa _2$ the relevant marginal consumer is $x^{e}$.

We first assume that $\kappa <\kappa _2$. Firm b will not find a deviation from $p^{*}_L(HL)$ to t profitable if and only if

$$\begin{aligned} p^{*}_L(HL)(1-x^e)=\, & {} p^{*}_L(HL)\left( 1-\frac{p^{*}_L(HL)-p^{*}_H(HL)+DV+t}{2t}\right) \nonumber \\\ge & {} \,t\left( 1-x_b^{in}\right) =t\left( 1-\frac{qDV-p^{*}_H(HL)-\kappa +2t}{2t}\right) . \end{aligned}$$

(9)

The RHS of (9) is higher than the RHS of (7) if and only if $\kappa <\frac{DV(p^{*}_H(HL)-qt)}{p^{*}_H(HL)-t}$, which holds in the case we are in, since $\kappa _2<\frac{DV(p^{*}_H(HL)-qt)}{p^{*}_H(HL)-t}$. So, the relevant constraint is only (9).

Firm b can also deviate to $p_b^{dev}\ne t$ and $p_b^{dev}\ne p^{*}_H(HL)$. In this case consumers observe $(p^{*}_H(HL), p_b^{dev})$ and immediately realize from $p^{*}_H(HL)$ that firm a has high quality and firm b has low quality. Therefore, only the complete information prices given by (1) ensure that such a deviation is unprofitable.

Using (1), the no deviation constraint (9) is satisfied if and only if $\kappa \le \kappa _1\equiv \frac{DV(DV-9t(1-q))}{9t}$.⁷ Therefore, the low quality firm does not find a deviation profitable if $\kappa \le \min \left\{ \frac{DV(DV-9t(1-q))}{9t}, \frac{q^3}{1-q+q^2}DV \right\}$.

Note that $\frac{DV(DV-9t(1-q))}{9t}<\frac{q^3}{1-q+q^2}DV$, because even when $\frac{(DV-9t(1-q))}{9t}$ attains its maximum, which happens for $DV=3t$ and $q=1$ and $\frac{q^3}{1-q+q^2}$ attains its minimum which happens for $q=\frac{1}{2}+\frac{\sqrt{21}}{14}$, it is still the case that $\frac{DV(DV-9t(1-q))}{9t}<\frac{q^3}{1-q+q^2}DV$. Therefore, $\frac{q^3}{1-q+q^2}DV$ never binds and the constraint is $\kappa \le \frac{DV(DV-9t(1-q))}{9t}$.

In what follows, we show that there is a profitable deviation when $\kappa \ge \kappa _2$. Let assume that $\kappa \ge \kappa _2$. Firm b will not find a deviation from $p^{*}_L(HL)$ to t profitable if and only if

$$\begin{aligned}&p^{*}_L(HL)\left( 1-\frac{p^{*}_L(HL)-p^{*}_H(HL)+DV+t}{2t}\right) \ge \nonumber \\&t\left( 1-x^{e}\right) =t\left( 1-\frac{[DV+p^{*}_H(HL)-2t]q(1-q)+2t-p^{*}_H(HL)}{2t(1-q+q^2)}\right) . \end{aligned}$$

(10)

The RHS of (10) is higher than the RHS of (7) if and only if

$$\begin{aligned} \kappa <\frac{DV}{p^{*}_H(HL)(1-q+q^2)}\left[ p^{*}_H(HL)-(p^{*}_H(HL)+t)q(1-q)\right] . \end{aligned}$$

After we substitute $p^{*}_H(HL)$ from (1) the above inequality becomes

$$\begin{aligned} \kappa <\kappa _3\equiv \frac{DV[DV(1-q+q^2)+3t(1-2q+2q^2)]}{(1-q+q^2)(DV+3t)}. \end{aligned}$$

We have that $\kappa _3>\kappa _2$. As a result, for $\kappa \in \left[ \kappa _2,\kappa _3\right]$ the relevant constraint is (10), whereas for $\kappa >\kappa _3$ the relevant constraint is (7).

When the relevant constraint is (10), using (1), the no deviation constraint is satisfied for $t<\frac{(1-q+q^2)DV}{9(1-2q+2q^2)}$. However, this contradicts assumption 1.

When the relevant constraint is (7), using (1), the no deviation constraint is satisfied for $\kappa <\kappa _4\equiv \frac{4DV^2}{3(3t+DV)}$. Also, $\kappa _4>\kappa _3$ if and only if $t<\frac{(1-q+q^2)DV}{9(1-2q+2q^2)}$. However, this contradicts assumption 1.

We need $q>\frac{2}{3}$ to ensure that $\frac{DV(DV-9t(1-q))}{9t}>0$ and Assumption 1 is satisfied.

Now let’s turn to firm a’s incentive to deviate. Equilibrium profits for the high quality firm are increasing in the quality difference $V_a-V_b$, which is the highest in equilibrium: any deviation on part of firm a, as we have demonstrated above, will decrease the expected quality of a and will increase that of b. Consumers will attach some probability that firm a is of low quality and firm b is of high quality. Thus, firm a who has high quality has no incentive to deviate.

1.4 A.4 Proof of Lemma 3

Both firms have high quality products, (H, H). The candidate equilibrium prices are $(p^{*}, p^{*})=(t,t)$ and the equilibrium profits $\pi _a=\pi _b=\left( \frac{t}{2},\frac{t}{2}\right)$.

First, let’s consider firm a’s deviation to $p_a^{dev}=\frac{DV}{3}+t$. Consumers, upon observing $\left( \frac{DV}{3}+t, t\right)$, realize that one firm has deviated. So, it can be (H, H), or (L, L) and one firm raised its price to $\frac{DV}{3}+t$, or (H, L) and the low quality firm raised its price to t (they can, however, rule out (L, H), given the unprejudiced beliefs). The ex-ante consumer beliefs about expected product qualities are given by (8). Consumers initially sort out according to (2), using (8)

$$\begin{aligned} x^{e}=\frac{3t-DV-q(3t-4DV)+q^2(3t-4DV)}{6t(1-q+q^2)}. \end{aligned}$$

(11)

After consumers visit firm a they realize that its product is of high quality and they update their beliefs about firm b being high quality, by eliminating (L, L), to $\mu _{ba}^{in}=\frac{q^2}{q^2+q(1-q)}$. Hence, firm b’s expected product quality, for the consumers who visited a first, is $EV^{in}_b=\frac{q^2V_H+q(1-q)V_L}{q^2+q(1-q)}$. The interim marginal consumer for a, using (3), must satisfy

$$\begin{aligned} x^{in}_a=\frac{p^{*}-p^{dev}_a+V_H-EV^{in}_b+\kappa +t}{2t}=\frac{(2-3q)DV+3\kappa +3t}{6t}. \end{aligned}$$

Some consumers will switch from a to b if and only if $x^{in}_a<x^{e}\Leftrightarrow \kappa <-\frac{(1-q)^3DV}{1-q+q^2}$. So, no such switching takes place. Also, no consumer will switch from b to a. This is because those who visited b first had a belief that a had a higher expected quality than b and after their visit to b they realize that both have the same quality. Therefore, the relevant marginal consumer for firm a is $x^{e}$.

Hence, a deviation is unprofitable if and only if

$$\begin{aligned} \frac{t}{2}\ge p_a^{dev}x^{e}= \left( \frac{DV}{3}+t\right) \left( \frac{3t-DV-q(3t-4DV)+q^2(3t-4DV)}{6t(1-q+q^2)}\right) , \end{aligned}$$

which holds if and only if

$$\begin{aligned} t\le \frac{(1-2q)^2DV}{9q(1-q)}. \end{aligned}$$

Recall that we need $3t>DV$, assumption 1. From above we have $t\le \frac{(1-2q)^2DV}{9q(1-q)}$. The two conditions hold simultaneously if and only if $q<\frac{1}{2}-\frac{\sqrt{21}}{14}$, or $q>\frac{1}{2}+\frac{\sqrt{21}}{14}$.

Second, firm a can deviate to $p_a^{dev}\ne \frac{DV}{3}+t$. Consumers do not know whether it is (H, H) or (L, L), but they know that firms have the same quality and one firm has deviated. Therefore, such a deviation will not be profitable.

Finally, it is easy to see that if a firm does not want to deviate to $\frac{DV}{3}+t$, then it would not want to deviate to $-\frac{DV}{3}+t$. This is because in this case the initial consumer beliefs about the expected quality difference is tilted in favor of firm b and a firm’s profit is increasing in the quality differential.

1.5 A.5 Proof of Lemma 4

Both firms have low quality products, (L, L). The candidate equilibrium prices are $(p^{*}, p^{*})=(t,t)$ and the equilibrium profits $\pi _a=\pi _b=\left( \frac{t}{2},\frac{t}{2}\right)$.

First, let’s consider firm a’s deviation to $p_a^{dev}=\frac{DV}{3}+t$. Consumers, upon observing $\left( \frac{DV}{3}+t, t\right)$, realize that one firm has deviated. So, it can be (H, H), or (L, L) and one firm raised its price to $\frac{DV}{3}+t$, or (H, L) and the low quality firm raised its price to t (it cannot be (L, H), since we assume unprejudiced beliefs). The initial consumer beliefs about expected product qualities are given by (8), where $EV_a\ge EV_b$. Consumers initially sort out according to (2), using (8), which yields the same $x^{e}$ as in (11).

After consumers visit firm a they realize that its product is of low quality and they update their beliefs, by eliminating (H, H), and (H, L), so they also learn the quality of firm b, that is $EV_b^{in}=V_L$. The interim marginal consumer for a, using (3), must satisfy

$$\begin{aligned} x^{in}_a=\frac{p^{*}-p^{dev}_a+V_L-V_L+\kappa +t}{2t}=\frac{-DV+3\kappa +3t}{6t}. \end{aligned}$$

Some consumers will switch from a to b if and only if $x^{in}_a<x^{e}\Leftrightarrow \kappa <\frac{q(1-q)DV}{1-q+q^2}$ (where $x^{e}$ is given by (11)). Consumers who visit firm b first, eliminate (H, H), but do not know whether it is (L, L) or (H, L). So their updated beliefs about firm a being high quality is $\mu _{ab}^{in}=\frac{q(1-q)}{q(1-q)+(1-q)^2}$. The expected product quality of firm a becomes $EV_a^{in}=\frac{q(1-q)V_H+(1-q)^2V_L}{q(1-q)+(1-q)^2}>V_L$. The interim marginal consumer for a, using (3), must satisfy

$$\begin{aligned} x^{in}_b=\frac{p^{*}-p^{dev}_a+EV_a^{in}-V_L-\kappa +t}{2t}=\frac{(-1+3q)DV-3\kappa +3t}{6t}. \end{aligned}$$

Some consumers will switch from b to a if and only if $x^{in}_b>x^{e}\Leftrightarrow \kappa <\frac{q^3DV}{1-q+q^2}$ (where $x^{e}$ is given by (11)). But if consumers who visited b first switch to a, they realize that a’s product is of low quality. Given that they initially visited b with the expectation that a has higher quality, $EV_a\ge EV_b$, and now, after they have sunk the cost $\kappa$, they realize that both firms have low quality, they, as we have assumed, costlesly return to b. Therefore, no consumer will switch from b to a.

Thus, there are the following two different cases that we should examine.

Case 1: If $\kappa \le \frac{q(1-q)DV}{1-q+q^2}$, then the market share of the deviating firm a is $x^{in}_a$.

Case 2: If $\kappa \ge \frac{q(1-q)DV}{1-q+q^2}$, then the market share of the deviating firm a is $x^{e}$.

We analyze each one of these two cases below.

When $\kappa \le \frac{q(1-q)DV}{1-q+q^2}$, a deviation on part of firm a is unprofitable if and only if

$$\begin{aligned} \frac{t}{2}\ge p_a^{dev}x_a^{in}= \left( \frac{DV}{3}+t\right) \left( \frac{-DV+3\kappa +3t}{6t}\right) , \end{aligned}$$

which holds if and only if

$$\begin{aligned} \kappa \le \frac{(DV)^2}{3(DV+3t)}. \end{aligned}$$

When $\kappa \ge \frac{q(1-q)DV}{1-q+q^2}$, a deviation on part of firm a is unprofitable if and only if

$$\begin{aligned} \frac{t}{2}\ge p_a^{dev}x^{e}= \left( \frac{DV}{3}+t\right) \left( \frac{3t-DV-q(3t-4DV)+q^2(3t-4DV)}{6t(1-q+q^2)}\right) , \end{aligned}$$

which holds if and only if

$$\begin{aligned} t\le \frac{(1-2q)^2DV}{9q(1-q)}. \end{aligned}$$

Notice that we need $3t>DV$, assumption 1. From above we have $t\le \frac{(1-2q)^2DV}{9q(1-q)}$. The two conditions hold simultaneously if and only if $q<\frac{1}{2}-\frac{\sqrt{21}}{14}$, or $q>\frac{1}{2}+\frac{\sqrt{21}}{14}$.

The next two cases are the same as in the (H, H) case.

Firm a can deviate to $p_a^{dev}\ne \frac{DV}{3}+t$. Consumers do not know whether it is (H, H) or (L, L), but they know that firms have the same quality and one firm has deviated. Therefore, such a deviation will not be profitable.

Finally, it is easy to see that if a firm does not want to deviate to $\frac{DV}{3}+t$, then it would not want to deviate to $-\frac{DV}{3}+t$. This is because in this case the initial consumer beliefs about the expected quality difference is tilted in favor of firm b and a firm’s profit is increasing in the quality differential.

1.6 A.6 Proof of Theorem 1

First, from Lemma 3, we need $t\le \frac{(1-2q)^2 DV}{9q(1-q)}$. Also, $q>\frac{1}{2}+\frac{\sqrt{21}}{14}\approx 0.8273$ (recall from Lemma 2 that $q>\frac{2}{3}$, so $q<\frac{1}{2}-\frac{\sqrt{21}}{14}$ is eliminated). Combined with assumption 1, $DV\in \left[ \frac{9q(1-q)t}{(1-2q)^2}, 3t\right)$. When we combine the conditions from Lemmas 2, 3 and 4 , we arrive at the following conditions that must be satisfied

• $\kappa \le \min \left\{ \frac{DV(DV-9t(1-q))}{9t}, \frac{(DV)^2}{3(DV+3t)}, \frac{q(1-q)DV}{1-q+q^2}\right\}$

• or, $\frac{DV(DV-9t(1-q))}{9t}\ge \kappa \ge \frac{q(1-q)DV}{1-q+q^2}$.

Next, note that $\frac{(DV)^2}{3(DV+3t)}\ge \frac{q(1-q)DV}{1-q+q^2}$ if and only if $t\le \frac{(1-2q)^2 DV}{9q(1-q)}$. Therefore, $\frac{(DV)^2}{3(DV+3t)}$ is redundant. Hence, the constraints become

• $\kappa \le \min \left\{ \frac{DV(DV-9t(1-q))}{9t}, \frac{q(1-q)DV}{1-q+q^2}\right\}$

• or, $\frac{DV(DV-9t(1-q))}{9t}\ge \kappa \ge \frac{q(1-q)DV}{1-q+q^2}$,

which suggests that the only relevant constraint is $\kappa \le \frac{DV(DV-9t(1-q))}{9t}$.

1.7 A.7 Proof of Lemma 5

We assume that firm $j=a$ (located at 0) sets the pooling equilibrium price $p^{*}$, while firm $i=b$ (located at 1) deviates by setting its best response, $BR_i(p^{*})$, and the deviator is perceived as low quality.

We are in the (H, L) case and the low quality firm deviates. Consumers who visit b first confirm their beliefs and do not switch, while consumers who visit a first realize that its quality is higher than the average quality and do not switch either. The deviation profit is given by $p^{dev}(1-x^e)$, where $x^e$ is given by (2) with $\Delta V^e=\overline{V}-V_L=qDV$. The maximum deviation profit is $\frac{(p^{*}+t-qDV)^2}{8t}$. On the equilibrium, consumers who visit the low quality firm, firm b, realize that its quality is lower than the average. So, $\Delta V^{in}_b-\Delta V^{e}=\overline{V}-V_L-0=qDV$, but since $\kappa \ge qDV$ no consumer switches from b to a. Furthermore, those who visit a realize its quality is high and they stay. Hence, the pooling equilibrium profits is $\frac{p^{*}}{2}$, which is higher than the deviation profit if and only if

$$\begin{aligned} p^{*}\in \left[ t+qDV-2\sqrt{tqDV}, t+qDV+2\sqrt{tqDV}\right] . \end{aligned}$$

(12)

Now we are in the (L, H) state and we assume that the high quality firm deviates. No consumer switches from b to a, since b is actually high quality. Those who visit a realize its quality is low but they believe b is also low, so $\Delta V^{in}_a=0$. Given that $\Delta V^e=\overline{V}-V_L=qDV$, we have $\Delta V^{e}-\Delta V^{in}_a=qDV$ and since $\kappa \ge qDV$ no consumer switches from a to b. This implies that the maximum deviation profit is the same as when the low quality firm deviated. Hence, the no deviation pooling price range (12) does not change.

Now we assume that we are in the (L, L) case. For the consumers who first visit a we have $\Delta V^e-\Delta V_a^{in}=qDV+0=qDV$. For the consumers who first visit b we have $\Delta V_b^{in}-\Delta V^e=qDV-qDV=0$. Since $\kappa \ge qDV$, no consumer switches firms. The profit of the deviating firm is the same as in the (H, L) case above. The same is true in the (H, H) case. Therefore, the no deviation price range is given by (12).

So, it is possible to support any $p^{*}$ in (12) as a pooling equilibrium with beliefs $\mu _i(p,p^{*})=0$ for any $p\ne p^{*}$ and $i=a,b$.

1.8 A.8 Proof of Lemma 6

We assume that firm $j=a$ (located at 0) sets the pooling equilibrium price $p^{*}$, while firm $i=b$ (located at 1) deviates by setting its best response, $BR_i(p^{*})$, and the deviator is perceived as low quality. This is in the ex-ante sense, as consumers will update their ex-ante beliefs after visiting a firm.⁸

Let’s start with the (L, H) case. We assume that the high quality firm deviates. Since the high quality firm is perceived as low the consumers who first visit b stay. For the consumers who visit a first we have $\Delta V^e-\Delta V_a^{in}=qDV$ and since $\kappa <qDV$ some switch to b. These consumers discover that the quality of b is higher than expected and purchase from b. Hence, the relevant marginal consumer is $x_a^{in}$, given in (3), with $\Delta V^{in}_a=0$. The deviation profit is given by $p^{dev}(1-x_a^{in})$. The maximum deviation profit for the high quality firm is $\frac{(p^{*}+t-\kappa )^2}{8t}$. The equilibrium profit of the high quality firm is given by $p^{*}(1-x_a^{in})=\frac{p^{*}(t-\kappa +qDV)}{2t}$, where $x_a^{in}$ is given in (3) with prices equal to $p^{*}$ and $\Delta V^{in}_a=V_L-\overline{V}=-qDV$, since some consumers who visit firm a, who is low quality, will also visit firm b, on the expectation of average quality, and since b is of high quality they buy from it. Such a deviation is not profitable if and only if

$$\begin{aligned}&p^{*}\in \Omega _{LH}^H\equiv \left[ t-\kappa +2qDV-2\sqrt{qDV(t-\kappa +qDV)}, t-\kappa \right. \nonumber \\&\left. \quad +2qDV+2\sqrt{qDV(t-\kappa +qDV)}\right] , \end{aligned}$$

(13)

where the subscript of $\Omega$ indicates the state and the superscript the firm that deviates.

Now assume we are in the (H, L) state and the low quality firm deviates from $p^{*}$. The consumers who visit the high quality firm a realize that it has a higher quality than what they expected and those who visit b confirm their ex-ante beliefs. The search cost $\kappa$ has no effect on the deviation profits, so the ex-ante and interim marginal consumers coincide. The deviation profit is $p^{dev}(1-x^e)$, where $x^e$ is given by (2) with $\Delta V^e=\overline{V}-V_L=qDV$. The maximum deviation profit of the low quality firm is $\frac{(p^{*}+t-qDV)^2}{8t}$. The equilibrium profit of the low quality firm is $p^{*}(1-x_b^{in})=\frac{p^{*}(t+\kappa -qDV)}{2t}$, where $x_b^{in}$ is the relevant marginal consumer, given by (3), with prices equal to $p^{*}$ and $\Delta V^{in}_b=\overline{V}-V_L=qDV$. Some consumers on the equilibrium path who visit firm b realize that its quality is low and switch to a and stay (since its quality is high). Such a deviation is not profitable if and only if

$$\begin{aligned}&p^{*}\in \Omega _{HL}^L\equiv \left[ t+2\kappa -qDV-2\sqrt{\kappa (\kappa +t-qDV)}, t\nonumber \right. \nonumber \\&\quad \left. +2\kappa -qDV+2\sqrt{\kappa (\kappa +t-qDV)}\right] . \end{aligned}$$

(14)

Now we assume that we are in the (L, L) case. We have $\Delta V^e=\overline{V}-V_L=qDV$. For the consumers who first visit a we have $\Delta V^e-\Delta V_a^{in}=qDV-0=qDV$. For the consumers who first visit b we have $\Delta V_b^{in}-\Delta V^e=qDV-qDV=0$. If $\kappa < q\Delta V$, only some consumers who visit a switch to b. Since $\kappa >\Delta V-\Delta V_a^{in}=0-0=0$ it must be that $x^f=x_a^{in}$. The deviation profit is $p^{dev}(1-x_a^{in})$. The maximum deviation profit is $\frac{(p^{*}+t-\kappa )^2}{8t}$. The equilibrium profit is $\frac{p^{*}}{2}$. Such a deviation is not profitable if and only if

$$\begin{aligned} p^{*}\in \Omega _{LL}\equiv \left[ t+\kappa -2\sqrt{t\kappa }, t+\kappa +2\sqrt{t\kappa }\right] . \end{aligned}$$

(15)

Finally, we assume that we are in the (H, H) case. The high quality firm deviates and is perceived as low quality. No consumer switches firms, since both firms are of high quality. The deviation profit is $p^{dev}(1-x^e)$, where $x^e$ is given by (2) with $\Delta V^e=\overline{V}-V_L=qDV$. The maximum profit of the deviating firm is $\frac{(p^{*}+t-qDV)^2}{8t}$. The equilibrium profit is $\frac{p^{*}}{2}$. Such a deviation is not profitable if and only if

$$\begin{aligned} p^{*}\in \Omega _{HH}\equiv \left[ t+qDV-2\sqrt{tqDV}, t+qDV+2\sqrt{tqDV}\right] . \end{aligned}$$

(16)

Let $\Omega \equiv \Omega _{LH}^H\cap \Omega _{HL}^L\cap \Omega _{LL}\cap \Omega _{HH}$ be the intersection of the four sets, (13)–(16). If $p^{*}\in \Omega$, then no firm finds a deviation from $(p^{*}, p^{*})$ profitable if it is perceived as low quality, regardless of the quality of the rival. The set $\Omega$ is non-empty for high values of $\kappa$. When $\kappa =qDV$, $\Omega =\left[ t+qDV-2\sqrt{tqDV}, t+qDV+2\sqrt{tqDV}\right]$. Note that in this case $\Omega$ coincides with the price range when $\kappa \ge qDV$, see Lemma 5. The higher the tqDV the higher the price range. By continuity, $\Omega$ is non-empty for $\kappa$’s less than (but close to) qDV. But when $\kappa =0$, the $\Omega _{LL}$ collapses to t, while the $\Omega _{HL}^L$ collapses to $t-qDV$. Therefore, $\Omega$ is empty, implying that for $\kappa$’s close to 0 there does not exist a pooling equilibrium.

1.9 A.9 Proof of Proposition 1

We assume that firm $j=a$ (located at 0) sets the pooling equilibrium price $p^{*}$, while firm $i=b$ (located at 1) deviates to $p^{dev}$ and is perceived as high quality. Market shares of firms will be potentially affected, relative to the pooling equilibrium shares, not only because one price has changed but also because of the change in ex-ante beliefs. A firm can benefit if it deviates from a pooling equilibrium $p^{*}$ for two reasons: (i) it sets a price closer to its best-response to $p^{*}$, holding beliefs about product quality differential fixed and (ii) it can affect the beliefs about product quality differential.

Let’s assume that we are in the (L, H) case. The consumers who visit firm b confirm their beliefs and make a purchase. The consumers who visit a have $\Delta V_a^{in}=V_L-V_H=-DV$. Also, $\Delta V^e=\overline{V}-V_H=-(1-q)DV$ and $\Delta V^e-\Delta V_a^{in}=qDV$ and when $\kappa <qDV$ some consumers who visit a switch to b. Because $\Delta V_a^{in}+\kappa =-DV+\kappa >\Delta V=-DV$ the final marginal consumer is $x_a^{in}$. Thus, the deviation profits are given by $p^{dev}(1-x_a^{in})=\frac{p^{dev}(t-\kappa +DV+p^{*}-p^{dev})}{2t}$, where $x_a^{in}$ is given by (3), with $\Delta V_a^{in}=V_L-V_H=-DV$. In equilibrium, $\Delta V_a^{in}=V_L-\overline{V}=-qDV$. Also, $\Delta V^{e}=0$ and so $\Delta V^e-\Delta V_a^{in}=qDV$. When $\kappa <qDV$ some consumers in equilibrium switch from a to b (no consumer switches from b to a since b has a high quality). Because $\Delta V_a^{in}+\kappa =-qDV+\kappa >\Delta V=-DV$ the final marginal consumer is $x_a^{in}$. The equilibrium profits are $p^{*}(1-x_a^{in})=\frac{p^{*}(t-\kappa +qDV)}{2t}$. Such a deviation, when $\kappa <qDV$, is profitable if and only if⁹

$$\begin{aligned} p^{dev}\in \left[ \frac{p^{*}-\kappa +t+DV}{2}\pm \frac{\sqrt{(DV)^2+((2-4q)p^{*}-2\kappa +2t)DV+(\kappa +p^{*}-t)^2}}{2}\right] . \end{aligned}$$

(17)

When $\kappa \ge qDV$, no consumer switches firms both in equilibrium and out-of equilibrium. The deviation profit is $p^{dev}(1-x^e)=\frac{p^{dev}(t+(1-q)DV+p^{*}-p^{dev})}{2t}$, where $x^e$ is given by (2) with $\Delta V^e=-(1-q)DV$. The equilibrium profit is $\frac{p^{*}}{2}$. A deviation is profitable if and only if

$$\begin{aligned} p^{dev}\in \left[ \frac{p^{*}+t+(1-q)DV}{2}\pm \frac{\sqrt{(1-q)^2(DV)^2+2(1-q)DV(p^{*}+t)+(p^{*}-t)^2}}{2}\right] . \end{aligned}$$

(18)

Next, we assume that we are in the (H, L) state and the low quality firm deviates. The consumers who first visit a stay since its quality is high. For the consumers who first visit b we have $\Delta V_b^{in}-\Delta V^e=qDV-(-(1-q)DV)=DV$. So, if $\kappa <DV$, some consumers who first visit b will switch to a. Because $\Delta V_b^{in}-\kappa =qDV-\kappa <\Delta V=DV$, the final marginal consumer is $x_b^{in}$. The deviation profits are given by $p^{dev}(1-x_b^{in})=\frac{p^{dev}(t+\kappa -qDV+p^{*}-p^{dev})}{2t}$, where $x_b^{in}$ is given by (3), with $\Delta V_b^{in}=\overline{V}-V_L=qDV$. In equilibrium, $\Delta V_b^{in}=\overline{V}-V_L=qDV$ and $\Delta V^{e}=0$. So, $\Delta V_b^{in}-\Delta V^e=qDV$. If $\kappa <qDV$, some consumers in equilibrium who first visit b switch to a (no consumer switches from a to b since a has a high quality). Because $\Delta V_b^{in}-\kappa =qDV-\kappa <\Delta V=DV$ the final marginal consumer is $x_b^{in}$. The equilibrium profits are $p^{*}(1-x_b^{in})=\frac{p^{*}(t+\kappa -qDV)}{2t}$. When $\kappa <qDV$, such a deviation is profitable if and only if

$$\begin{aligned} p^{dev}\in \left[ \min \{\kappa +t-qDV, p^{*}\}, \max \{\kappa +t-qDV, p^{*}\}\right] . \end{aligned}$$

(19)

When $\kappa \in [qDV, DV)$, the equilibrium profit is $\frac{p^{*}}{2}$ (since no consumer switches firms), but the deviation profit is still $p^{dev}(1-x_b^{in})$, the same as when $\kappa <qDV$. A deviation is profitable if and only if

$$\begin{aligned} p^{dev}\in \left[ \frac{p^{*}+t+\kappa -qDV}{2}\pm \frac{\sqrt{t^2+2t(\kappa -p^{*}-qDV)+(p^{*}+\kappa -qDV)^2}}{2}\right] . \end{aligned}$$

(20)

We now assume that we are in the (L, L) case. Consumers who initially visit firm a have $\Delta V^e-\Delta V_a^{in}=-(1-q)DV+DV=qDV$, so when $\kappa <qDV$ some switch to b. Consumers who initially visit firm b have $\Delta V_b^{in}-\Delta V^e=qDV+(1-q)DV=DV$, so when $\kappa <DV$ some switch to a. When $\kappa \in [qDV, DV)$ consumers who visit a stay, those who visit b visit also a. We have $\Delta V_b^{in}-\kappa =qDV-\kappa <\Delta V=0$ and $\Delta V=0>\Delta V^e=-(1-q)DV$. So, the final marginal consumer is $x^{in}_b$, given by (3) with $\Delta V^{in}_b=qDV$. The equilibrium profit is $\frac{p^{*}}{2}$. Deviation and equilibrium profits are the same as in the (H, L) case when the low quality firm deviates and $\kappa \in [qDV, DV)$. Hence, a deviation when $\kappa \in [qDV, DV)$ is profitable if and only if (20) is satisfied.

Now we assume that $\kappa <qDV$. There is now two-sided switching. Since $\Delta V_b^{in}-\kappa>\Delta V>\Delta V_a^{in}+\kappa$ holds (i.e., $qDV-\kappa>0>-DV+\kappa$), the final marginal consumer is $x^f$, given by (4), with $\Delta V=0$. The deviation profit is $p^{dev}(1-x^f)=\frac{p^{dev}(t+p^{*}-p^{dev})}{2t}$. The equilibrium profit is $\frac{p^{*}}{2}$. A deviation is profitable if and only if

$$\begin{aligned} p^{dev}\in \left[ \min \{t,p^{*}\}, \max \{t,p^{*}\}\right] . \end{aligned}$$

(21)

It can be verified that when $\kappa =DV$, (18) is the same with (20). In this case no consumer switches firms, and hence high and low type firms have the same incentives to deviate, in the (H, L) and the (L, L) cases, when they are perceived as high types. The same of course is true for $\kappa >DV$. Moreover, it can be verified that the upper bound of (20) is decreasing as $\kappa$ decreases, but (18) is not a function of $\kappa$. Therefore, for $\kappa \in [qDV,DV)$ the high quality firm can always set a price in (18) and above the upper bound of (20), that cannot be mimicked by the low quality firm in the (H, L) and (L, L) cases, to signal its high quality.

Finally, let’s turn to the $\kappa <qDV$ case. The upper bound of (17) is higher than the upper bound of (19), which again suggests that the high quality firm can credibly signal its high quality. This can be seen as follows. Let $\Delta \pi _H^{dev}\equiv \frac{p^{dev}(t-\kappa +DV+p^{*}-p^{dev})}{2t}-\frac{p^{*}(t-\kappa +qDV)}{2t}$, be the high quality firm’s incentive to deviate in the (L, H) case (we used this to derive (17)). Also let $\Delta \pi _L^{dev}\equiv \frac{p^{dev}(t+\kappa -qDV+p^{*}-p^{dev})}{2t}-\frac{p^{*}(t+\kappa -qDV)}{2t}$ be the low quality firm’s incentive to deviate in the (H, L) case (we used this to derive (19)). $\Delta \pi _H^{dev}$ and $\Delta \pi _L^{dev}$ are inverse U-shaped in $p^{dev}$ and intersect once at $p^{dev}=\frac{2p^{*}(qDV-\kappa )}{(1+q)DV-2\kappa }>0$. Moreover, $\Delta \pi _H^{dev}$ is maximized at $p^{dev}=\frac{p^{*}+t+DV-\kappa }{2}$, while $\Delta \pi _L^{dev}$ is maximized at $p^{dev}=\frac{p^{*}+t-qDV+\kappa }{2}$, where $\frac{p^{*}+t+DV-\kappa }{2}>\frac{p^{*}+t-qDV+\kappa }{2}$. Lastly, the difference in the slopes of $\Delta \pi _H^{dev}$ and $\Delta \pi _L^{dev}$ at the point of intersection is $\frac{(1+q)DV-2\kappa }{2t}>0$. These prove that $\Delta \pi _H^{dev}$ intersects $\Delta \pi _L^{dev}$ once from below and hence the upper $p^{dev}$ that makes $\Delta \pi _H^{dev}=0$ is higher than the upper $p^{dev}$ that makes $\Delta \pi _L^{dev}=0$.¹⁰

1.10 A.10 Proof of Theorem 2

Following Assumption 1, $p^{*}=t\in \left[ t+qDV-2\sqrt{tqDV}, t+qDV+2\sqrt{tqDV}\right]$, which was the set of pooling equilibria identified in Lemma 5. Impartial beliefs eliminate all of these pooling equilibria but one. Given that the search cost is high, consumers are captive. Hence, a low quality firm can costlessly mimic the high quality firm, implying that firms are treated symmetrically and a deviation cannot credibly signal superior product quality. Therefore, the game becomes like a standard Hotelling model with firms having symmetric and known product qualities, i.e., the expected quality is $\overline{V}$. As is well-known, the equilibrium in such a model is (t, t).