Consumer deliberation and quality signaling

Abstract

Consumers are often uncertain about their product valuation before purchase. They may bear the uncertainty and purchase the product without deliberation. Alternatively, consumers can incur a deliberation cost to find out their true valuation and then make their purchase decision. This paper proposes that consumer deliberation about product valuation can be an endogenous mechanism to enable credible quality signaling. We demonstrate this point in a simple setup in which product quality influences the probability that the product has high valuation. We show that with endogenous deliberation there may exist a unique separating equilibrium in which the high-quality firm induces consumer deliberation by setting a high price whereas the low-quality firm prevents deliberation by charging a low price. Compared to the case of complete information, the price of the high-quality firm can be distorted upward to facilitate consumer deliberation, or distorted downward to avoid the low-quality firm’s imitation. In an extension we show that dissipative advertising can facilitate quality signaling. The high-quality firm can utilize advertising spending to avert imitation from the low-quality firm without distorting price downward, earning a higher profit than that without advertising. However, advertising mitigates the distortion at the expense of consumer surplus and social welfare.

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Notes

  1. 1.

    In few exceptions the term “search” is used where purchase is possible without search (e.g., Kuksov and Villas-Boas 2010; Branco et al. 2012; Ke et al. 2015). This is similar to deliberation considered in this paper. Nevertheless, we will still use deliberation to differentiate from the standard meaning of search in the literature.

  2. 2.

    See also Kuksov and Lin (2010) on firms’ optimal provision of information in competitive markets.

  3. 3.

    This setup is adopted for simplicity. The insights here continue to hold in alternative specifications in which quality directly influences consumer utility, e.g., u = v q or u = v + q, where v captures valuation uncertainty and q represents quality.

  4. 4.

    Our model can accommodate cases in which deliberation cannot resolve all valuation uncertainty before purchase (e.g., the utility of goods with experience attributes can be known only after consumption). In these cases we can re-define V H and V L as the expected product valuations integrating over all residual uncertainty that cannot be resolved by deliberation.

  5. 5.

    Note that deliberation, as defined in this paper, does not represent other cognitive processes (e.g., evaluation and calculation of expected payoffs, reasoning about the firm’s incentive for signaling).

  6. 6.

    If a strategy p is dominated for both types of firms under any belief, then neither firm type would like to deviate to p, and so there is no need to specify the out-of-equilibrium belief for such p.

  7. 7.

    Recall that the lower bound is defined as κ L ≡[1/q L (1−q L )+1/q L ] −1 . The expression for the upper bound κ U is given by Eq. 1 in Appendix A.

  8. 8.

    Note that \(\phantom {\dot {i}\!}\kappa _{L} < \hat {\kappa } < q_{L} (1 - q_{L})\).

  9. 9.

    The expression for \(\phantom {\dot {i}\!}\bar {\kappa }_{U}\) is given by Eq. 3 in Appendix A.

  10. 10.

    Since [1/q L (1−q L )+1/q H ] −1 < min{q L (1−q L ),[1/q H (1−q L )+1/q H ] −1 }, there exists c such that the conditions are satisfied iff [1/q L (1−q L )+1/q H ] −1 <q H (1−q H ) ⇔ q H <1−q L (1−q L ).

  11. 11.

    As long as q H <1−q L (1−q L ), we have κ L <[1/q L (1−q L )+1/q H ] −1 <q H (1−q H ). Hence there exists c such that the conditions hold iff q H <1−q L (1−q L ).

  12. 12.

    If \(\phantom {\dot {i}\!}c < \bar {q} (1 - \bar {q})\), the consumer deliberates or opts out; otherwise, since \(\phantom {\dot {i}\!}p > c / (1 - \bar {q}) \ge \bar {q}\), he opts out.

  13. 13.

    The reason why p H >p is the following. Since cκ L is equivalent to p H ≥ 1−c/q L , we have either p H =1−c/q L >p (recall that p is an out-of-equilibrium price and so pp H ) or p H >1−c/q L p.

  14. 14.

    The consumer either deliberates or opts out after observing \(\phantom {\dot {i}\!}p_{H}^{\prime \prime }\) because \(\phantom {\dot {i}\!}p_{H}^{\prime \prime } > p_{H}^{\prime } > c / (1 - q_{H}) \ge c / (1 - \bar {q})\).

  15. 15.

    It is not possible that \(\phantom {\dot {i}\!}p_{H}^{\prime } = p_{H}\) and \(\phantom {\dot {i}\!}a_{H}^{\prime } < a_{H}\) because \(\phantom {\dot {i}\!}a_{H}^{\prime } \ge p_{H} q_{L} - p_{L}\) and \(\phantom {\dot {i}\!}a_{H}^{\prime } \ge 0\) imply \(\phantom {\dot {i}\!}a_{H}^{\prime } \ge a_{H}\).

  16. 16.

    Under condition (b), it is impossible that p p q H for some price p∈(c/(1−q H ),1−c/q H ] while p p q L for some other price p ∈(c/(1−q H ),1−c/q H ].

  17. 17.

    If \(\phantom {\dot {i}\!}[1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{L}]^{-1} \ge \bar {q}_{0} (1 - \bar {q}_{0})\), then we have \(\phantom {\dot {i}\!}\frac {1}{\bar {q}_{0} (1 - \bar {q}_{0})} \ge \frac {1}{q_{H} (1 - \bar {q}_{0})} + \frac {1}{q_{L}} \Longrightarrow 1 \ge \frac {\bar {q}_{0}}{q_{H}} + \frac {\bar {q}_{0} (1 - \bar {q}_{0})}{q_{L}} \Longrightarrow q_{L} \left (1 - \frac {\bar {q}_{0}}{q_{H}}\right ) \ge \bar {q}_{0} (1 - \bar {q}_{0})\). Similarly to the proof of Lemma 16, this leads to a contradiction.

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Acknowledgments

The authors thank Dmitri Kuksov, Paulo Albuquerque, Sameer Hasija, V. Padmanabhan, Kaifu Zhang, and attendees of 2015 Marketing Science Conference for their valuable comments.

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Correspondence to Liang Guo.

Appendices

Appendix A: Technical Details for Separating Equilibria

A.1 Proof of Lemma 1

(i) & (ii) If the consumer’s deliberation decision does not change, then the firm with the lower price can always make a profitable deviation by imitating the firm with the higher price. (iii) Suppose in equilibrium the consumer deliberates on p L but not on p H . Type-L firm should not prefer to deviate her price p L to p H ; so p H p L q L , which implies p H <p L q H . Yet type-H firm is willing to deviate to p L because p L q H >p H , a contradiction.

A.2 Proof of Proposition 1

In any separating equilibrium, type-L firm should always behave in the same way as she does under complete information. According to Lemma 1, we must have cκ L . We characterize the deliberation-on- p H type of equilibrium by showing the following lemmas. It requires the deliberation cost c<q H (1−q H ), which is omitted for clarity.

Lemma 4

For c ≥ q L (1−q L ), there exists a separating equilibrium iff c≤q H −q L . The equilibrium prices are unique: p H =1−c/q H and p L =q L .

Lemma 5

For [1/q L (1−q L )+1/q H ] −1 ≤c<q L (1−q L ), there exists a separating equilibrium iff c≤[1/q H (1−q L )+1/q H ] −1.Footnote 10 The equilibrium prices are unique: p H =1−c/q H and p L =c/(1−q L ).

Lemma 6

For c<[1/q L (1−q L )+1/q H ] −1 , there exists a separating equilibrium iff q H <1−q L (1−q L ) and c ≥ κ L .Footnote 11 The equilibrium prices are unique: p H =c/q L (1−q L ) and p L =c/(1−q L ).

Proof of Lemma 4— cq L (1−q L )

The proof proceeds in two parts: (A) There exists an equilibrium where p H =1−c/q H and p L = q L iff cq H q L . (B) The equilibrium prices are unique.

  1. (A)

    [“Only if” part]: Suppose c>q H q L and the equilibrium exists. Type-H firm wants to deviate to p L because π H = q H c<q L = p L , a contradiction. [“If” part]: Type-H firm would not deviate to p L because π H p L . Type-L firm would not deviate to p H because π L = p L >(1−c/q H )q L = p H q L . We now show that neither type of firm has any incentive to deviate to any out-of-equilibrium price. For p<p L , type-L firm has no incentive to deviate; neither does type-H firm have any incentive because she does not even want to deviate to p L , not to mention a lower price p<p L . For p>c/(1−q H ), given any perceived quality \(\phantom {\dot {i}\!}\bar {q}\), the consumer either deliberates or opts out because \(\phantom {\dot {i}\!}p > c / (1 - q_{H}) \ge c / (1 - \bar {q})\).Footnote 12 When the consumer deliberates, type-L firm would not deviate because π L = p L >p q L ; type-H firm would not deviate either because π H = p H q H >p q H . For p∈(p L ,c/(1−q H )], neither type of firm has any incentive to deviate under an out-of-equilibrium belief μ(q L |p)=1 because the consumer always opts out. The separating equilibrium with such an out-of-equilibrium belief survives the intuitive criterion because type-L firm wants to deviate under μ(q H |p)=1.

  2. (B)

    We prove this part by contradiction. Suppose the equilibrium price of type-H firm is \(\phantom {\dot {i}\!}p_{H}^{\prime } < 1 - c / q_{H}\). According to Lemmas 1, \(\phantom {\dot {i}\!}p_{H}^{\prime }\) has to be greater than c/(1−q H ). The consumer either deliberates or opts out for \(\phantom {\dot {i}\!}p_{H}^{\prime \prime } = 1 - c / q_{H}\) because \(\phantom {\dot {i}\!}p_{H}^{\prime \prime } > p_{H}^{\prime } > c / (1 - q_{H}) \ge c / (1 - \bar {q})\) (where \(\phantom {\dot {i}\!}\bar {q}\) is any perceived quality). Since \(\phantom {\dot {i}\!}\pi _{L} = p_{L} > p_{H}^{\prime \prime } q_{L}\), type-L firm has no incentive to deviate to \(\phantom {\dot {i}\!}p_{H}^{\prime \prime }\) under any belief. Yet under the only reasonable belief \(\phantom {\dot {i}\!}\mu (q_{H} | p_{H}^{\prime \prime }) = 1\) in light of the intuitive criterion, type-H firm can make a profitable deviation to \(\phantom {\dot {i}\!}p_{H}^{\prime \prime }\), a contradiction.

Proof of Lemma 5— [1/q L (1−q L )+1/q H ]−1c<q L (1−q L )

This is very similar to that of Lemma 4. The main difference is that here type-L firm charges c/(1−q L ) to forestall deliberation. We can show there exists a separating equilibrium iff c≤[1/q H (1−q L )+1/q H ]−1. The condition c≤[1/q H (1−q L )+1/q H ]−1 ensures that type-H firm prefers 1−c/q H to other prices.

Proof of Lemma 6— c<[1/q L (1−q L )+1/q H ]−1

We show the following: (A) There exists an equilibrium where p H = c/q L (1−q L ) and p L = c/(1−q L ) iff q H <1−q L (1−q L ) and cκ L . (B) The equilibrium prices are unique.

  1. (A)

    [“Only if” part]: Type-L firm’s equilibrium strategy requires that cκ L . Suppose q H ≥ 1−q L (1−q L ) and the equilibrium exists. Then p H c/(1−q H ), which is not consistent with the desired consumer decisions (deliberation on p H ), a contradiction. [“If” part]: Note that the consumer deliberates on p H because p H <1−c/q H (implied by the premise c<[1/q L (1−q L )+1/q H ]−1) and p H >c/(1−q H ) (implied by q H <1−q L (1−q L )). Type-H firm is not willing to deviate to p L = c/(1−q L ) because π H = p H q H = c q H /q L (1−q L )>c/(1−q L ) = p L . Type-L firm is indifferent between p L and p H . We now examine the out-of-equilibrium prices. Neither type of firm wants to deviate to any price p<c/(1−q L ), p∈(c/(1−q H ),p H ), or p>1−c/q H under any belief. For any p∈(c/(1−q L ), min{c/(1−q H ),1−c/q L }], type-H firm is not willing to deviate under the belief μ(q L |p)=1 because p H >p and the consumer deliberates;Footnote 13 type-L firm does not want to deviate either because she does not even want to deviate to p H >p. For any p∈(1−c/q L ,c/(1−q H )]∪(p H ,1−c/q H ], no firm wants to deviate under μ(q L |p)=1 because p>1−c/q L implies that the consumer never purchases the product. The equilibrium with the out-of-equilibrium belief μ(q L |p)=1 (for any p∈(c/(1−q L ),c/(1−q H )]∪(p H ,1−c/q H ]) survives the intuitive criterion because type-L firm always prefers a deviation under a belief μ(q H |p)=1.

  2. (B)

    We prove by contradiction that any \(\phantom {\dot {i}\!}p_{H}^{\prime } \in (c / (1 - q_{H}), c / q_{L} (1 - q_{L})) \cup (c / q_{L} (1 - q_{L}), 1 - c / q_{H}]\) cannot be the equilibrium price of type-H firm. Suppose it is. If \(\phantom {\dot {i}\!}p_{H}^{\prime } \in (c / q_{L} (1 - q_{L}), 1 - c / q_{H}]\), then type-L firm wants to deviate to \(\phantom {\dot {i}\!}p_{H}^{\prime }\) because \(\phantom {\dot {i}\!}p_{H}^{\prime } q_{L} > [c / q_{L} (1 - q_{L})] \cdot q_{L} = c / (1 - q_{L}) = p_{L}\), a contradiction. Otherwise, type-L firm is not willing to deviate to any price \(\phantom {\dot {i}\!}p_{H}^{\prime \prime } \in (p_{H}^{\prime }, c / q_{L} (1 - q_{L}))\) because \(\phantom {\dot {i}\!}p_{L} = c / (1 - q_{L}) = [c / q_{L} (1 - q_{L})] \cdot q_{L} > p_{H}^{\prime \prime } q_{L}\).Footnote 14 Yet under the only reasonable belief \(\phantom {\dot {i}\!}\mu (q_{H} | p_{H}^{\prime \prime }) = 1\) according to the intuitive criterion, type-H firm would deviate, a contradiction.

Proof of Proposition 1

Define

$$ \kappa_{U} \equiv \min\left\{ q_{H} (1 - q_{H}), \max\left\{ q_{H} - q_{L}, \left[ \frac{1}{q_{H} (1 - q_{L})} + \frac{1}{q_{H}} \right]^{-1} \right\} \right\}. $$
(1)

It can be verified that

$$\begin{array}{@{}rcl@{}} q_{H} - q_{L} &\ge& q_{L} (1 - q_{L}) \iff \left[\frac{1}{q_{H} (1 - q_{L})} + \frac{1}{q_{H}}\right]^{-1} \ge q_{L} (1 - q_{L}) \iff q_{H} \\ &&- q_{L} \ge \left[\frac{1}{q_{H} (1 - q_{L})} + \frac{1}{q_{H}}\right]^{-1}. \end{array} $$
(2)

Combining Lemmas 4 and 5, for c ≥ [1/q L (1−q L )+1/q H ]−1, there exists a separating equilibrium iff c≤ min{q H (1−q H ), max{q H q L ,[1/q H (1−q L )+1/q H ]−1}}≡κ U (except at the boundary q H (1−q H )). The above condition holds for some c iff [1/q L (1−q L )+1/q H ]−1<q H (1−q H ) (because [1/q L (1−q L )+1/q H ]−1<[1/q H (1−q L )+1/q H ]−1≤ max{q H q L ,[1/q H (1−q L )+1/q H ]−1}); this is equivalent to q H <1−q L (1−q L ). Merging the results with Lemma 6, there exists a separating equilibrium iff q H <1−q L (1−q L ) and κ L cκ U (except at the boundary q H (1−q H )). It can be verified that p H = min{1−c/q H , c/q L (1−q L )} and p L = min{q L , c/(1−q L )}.

A.3 Proof of Proposition 2

(i) Type-H firm charges c/(1−q H ) under complete information because κ H c<q H (1−q H ), but she charges 1−c/q H in a separating equilibrium. Therefore, the price is higher in a separating equilibrium but the profit is higher under complete information (except for the boundary c = κ H ). The social welfare is lower in a separating equilibrium than under complete information because of the incurred deliberation cost in a separating equilibrium. (ii) In this region, type-H firm in a separating equilibrium follows her complete-information optimal strategy: p H = P H =1−c/q H . (iii) Type-H firm charges 1−c/q H under complete information but charges c/q L (1−q L )<1−c/q H in a separating equilibrium. So, the price and the profit are higher under complete information. Since the consumer deliberates for both cases, the social welfare is the same.

A.4 Proof of Lemma 2

Type-L firm adopts her complete-information optimal strategy. We discuss the existence conditions in four cases separately. The separating equilibria refer to the no-deliberation type.

Lemma 7

For c ≥ max{q L (1−q L ),q H (1−q H )}, there always exist separating equilibria. Type L: p L =q L and a L =0; type H: p H ∈(q L ,q H ] and a H =p H −p L .

Lemma 8

For q L (1−q L )≤c<q H (1−q H ), there exist separating equilibria iff c ≥ q H −q L . Type L: p L =q L and a L =0; type H: p H ∈(q L ,c/(1−q H )] and a H =p H −p L .

Lemma 9

For q H (1−q H )≤c<q L (1−q L ), there exist separating equilibria iff c ≥ [1/q H (1−q L )+1/q L ] −1 . Type L: p L =c/(1−q L ) and a L =0; type H: p H ∈(c/(1−q L ),q H ] and a H =p H −p L .

Lemma 10

For c< min{q L (1−q L ),q H (1−q H )}, there exist separating equilibria iff c ≥ [1/q H (1−q L )+1/q H ] −1 . Type L: p L =c/(1−q L ) and a L =0; type H: p H ∈(c/(1−q L ),c/(1−q H )] and a H =p H −p L .

Proof of Lemma 7— c ≥ max{q L (1−q L ),q H (1−q H )}

In this case, p L = q L and a L =0. The equilibrium price p H should satisfy p L <p H q H . We prove that any p H ∈(q L ,q H ] and a H = p H p L can be equilibrium decisions of type-H firm. Since both types of firms are indifferent between (p H ,a H ) and (p L ,a L ), it is sufficient to examine the out-of-equilibrium decisions. For any (p,a) such that paq L , no firm prefers a deviation under any belief. For any (p,a) such that pa>q L and p>q H , the consumer opts out. For any (p,a) such that pa>q L and pq H , neither type of firm wants to deviate under a belief μ(q L |p,a)=1, and such an equilibrium survives the intuitive criterion. It can be verified that any a p H p L cannot be the equilibrium advertising spending of type-H firm.

Proof of Lemma 8— q L (1−q L )≤c<q H (1−q H )

In this case, p L = q L and a L =0. The equilibrium price p H should satisfy p L <p H c/(1−q H ). We prove that any p H ∈(q L ,c/(1−q H )] and a H = p H p L can be equilibrium decisions of type-H firm iff cq H q L . [“Only if” part]: For c<q H q L , we prove by contradiction that the above separating equilibrium does not exist. For p=1−c/q H , type-L firm does not want to deviate to p because p q L <q L . However, since q H c>q L , type-H firm prefers to deviate to (p,0) under the only reasonable belief μ(q H |p,0)=1 according to the intuitive criterion, a contradiction. [“If” part]: It is sufficient to check the out-of-equilibrium decisions. For any (p,a) such that paq L , no firm prefers to deviate. For any (p,a) such that pa>q L and p>c/(1−q H ), the consumer deliberates or opts out under any belief. The highest possible profit any firm can make is (1−c/q H )q H = q H cq L ; so, no deviation takes place. For any (p,a) such that pa>q L and pc/(1−q H ), neither type of firm wants to deviate under a belief μ(q L |p,a)=1, and such an equilibrium survives the intuitive criterion. It can be verified that a p H p L cannot be type-H firm’s equilibrium advertising spending.

Proof of Lemma 9— q H (1−q H )≤c<q L (1−q L )

In this case, p L = c/(1−q L ) and a L =0. The equilibrium price p H must satisfy c/(1−q L )<p H q H . We prove that any p H ∈(c/(1−q L ),q H ] and a H = p H p L can be equilibrium decisions of type-H firm iff c ≥ [1/q H (1−q L )+1/q L ]−1. [“Only if” part]: For c<[1/q H (1−q L )+1/q L ]−1, we prove by contradiction that the above separating equilibrium does not exist. The consumer deliberates or purchases without deliberation on p=1−c/q L . Yet type-H firm wants to deviate to (p,0) because p q H >c/(1−q L ) ( ⇔ c<[1/q H (1−q L )+1/q L ]−1), a contradiction. [“If” part]: We just need to examine the out-of-equilibrium decisions. For any (p,a) such that pac/(1−q L ), no firm prefers a deviation. For any (p,a) such that pa>c/(1−q L ) and p>q H , the consumer opts out because \(\phantom {\dot {i}\!}p > q_{H} \ge \bar {q}\) and \(\phantom {\dot {i}\!}p > q_{H} \ge 1 - c / q_{H} \ge 1 - c / \bar {q}\) (where \(\phantom {\dot {i}\!}\bar {q}\) is any perceived quality). For any (p,a) such that pa>c/(1−q L ) and pq H , the consumer deliberates if p≤1−c/q L but opts out otherwise under a belief μ(q L |p,a)=1. Neither type of firm wants to deviate because the highest profit any firm can earn is (1−c/q L )q H c/(1−q L ) ( ⇔ c ≥ [1/q H (1−q L )+1/q L ]−1), and such an equilibrium survives the intuitive criterion. It can be verified that a p H p L cannot be type-H firm’s equilibrium advertising spending.

Proof of Lemma 10— c< min{q L (1−q L ),q H (1−q H )}

In this case, p L = c/(1−q L ) and a L =0. The equilibrium price of type-H firm needs to satisfy c/(1−q L )<p H c/(1−q H ). We prove that any p H ∈(c/(1−q L ),c/(1−q H )] and a H = p H p L can be equilibrium decisions of type-H firm iff c ≥ [1/q H (1−q L )+1/q H ]−1. [“Only if” part]: For c<[1/q H (1−q L )+1/q H ]−1, we prove by contradiction that p H ∈(c/(1−q L ),c/(1−q H )] and a H = p H p L cannot constitute a separating equilibrium that survives the intuitive criterion. The condition c<[1/q H (1−q L )+1/q H ]−1 is equivalent to (1−c/q H )q H >c/(1−q L ). Consider an out-of-equilibrium decision tuple (p,a) such that p=1−c/q H and a = p q H c/(1−q L )−ε, where ε∈(0,p q H c/(1−q L )]; the consumer deliberates or opts out under any belief. Even if the consumer deliberates, type-L firm does not want to deviate to (p,a) because p q L a = c/(1−q L )−p(q H q L ) + ε<c/(1−q L ) (provided ε<p(q H q L )). Under the only reasonable belief μ(q H |p,a)=1 in light of the intuitive criterion, type-H firm would like to deviate because p q H a = c/(1−q L ) + ε>c/(1−q L ), a contradiction. [“If” part]: We just need to check the out-of-equilibrium decisions. For any (p,a) such that pac/(1−q L ), neither type of firm prefers a deviation. For any (p,a) such that pa>c/(1−q L ) and p>1−c/q H , the consumer opts out. For any (p,a) such that pa>c/(1−q L ) and c/(1−q H )<p≤1−c/q H , the consumer deliberates or opts out under any belief. Since the condition c ≥ [1/q H (1−q L )+1/q H ]−1 guarantees p q H a≤(1−c/q H )q H −0≤c/(1−q L ), no firm would like to deviate under any belief. For any (p,a) such that pa>c/(1−q L ) and pc/(1−q H ), similarly to the proof of Lemma 9, no firm would like to deviate under a belief μ(q L |p,a)=1 because c ≥ [1/q H (1−q L )+1/q H ]−1>[1/q H (1−q L )+1/q L ]−1, and the equilibrium with such an out-of-equilibrium belief survives the intuitive criterion. It can be verified that a p H p L cannot be type-H firm’s equilibrium advertising spending.

Proof of Lemma 2

It can be verified that the equilibrium decisions are consistent with Lemmas 7, 8, 9, and 10). We just need to show that the existence condition is \(\phantom {\dot {i}\!}c \ge \bar {\kappa }_{U}\) and \(\phantom {\dot {i}\!}\bar {\kappa }_{U} \ge \kappa _{U}\), where

$$ \bar{\kappa}_{U} \equiv \left\{\begin{array}{ll} \kappa_{U} \equiv \min\left\{ \max\left\{ q_{H} - q_{L}, \left[ \frac{1}{q_{H} (1 - q_{L})} + \frac{1}{q_{H}} \right]^{-1} \right\},\, q_{H} (1 - q_{H}) \right\}\\ \hspace{69mm} \text{if}\,\, q_{L} (1 - q_{L}) \le q_{H} (1 - q_{H}), \\ \min\left\{ \max\left\{ \left[ \frac{1}{q_{H} (1 - q_{L})} + \frac{1}{q_{L}} \right]^{-1},\, q_{H} (1 - q_{H}) \right\},\ \left[ \frac{1}{q_{H} (1 - q_{L})} + \frac{1}{q_{H}} \right]^{-1} \right\} \\ \hspace{69mm}\text{if}\,\, q_{H} (1 - q_{H}) < q_{L} (1 - q_{L}). \end{array}\right. $$
(3)

Similarly to the proof of Proposition 1, if q L (1−q L )≤q H (1−q H ), then the existence conditions in Lemmas 7, 8, and 10 are equivalent to \(\phantom {\dot {i}\!}c \ge \kappa _{U} \equiv \bar {\kappa }_{U}\). Suppose q H (1−q H )<q L (1−q L ). Since [1/q H (1−q L )+1/q H ]−1>[1/q H (1−q L )+1/q L ]−1, the existence conditions in Lemmas 7, 9, and 10 are equivalent to \(\phantom {\dot {i}\!}c \ge \min \left \{ \max \left \{ [ 1 / q_{H} (1 - q_{L}) + 1 / q_{L} ]^{-1},\, q_{H} (1 - q_{H}) \right \},\ [ 1 / q_{H} (1 - q_{L}) + 1 / q_{H} ]^{-1} \right \} \equiv \bar {\kappa }_{U}\).We next show that \(\phantom {\dot {i}\!}\bar {\kappa }_{U} \ge \kappa _{U}\). If [1/q H (1−q L )+1/q H ]−1q H (1−q H ), then \(\phantom {\dot {i}\!}\bar {\kappa }_{U} \ge q_{H} (1 - q_{H}) \ge \kappa _{U}\). Otherwise, we prove by contradiction that q H q L <[1/q H (1−q L )+1/q H ]−1. From the proof of Proposition 1 (see Inequalities (2)), if the above inequality is violated, then q L (1−q L )≤[1/q H (1−q L )+1/q H ]−1<q H (1−q H ), a contradiction. Therefore, \(\phantom {\dot {i}\!}\bar {\kappa }_{U} = [ 1 / q_{H} (1 - q_{L}) + 1 / q_{H} ]^{-1} = \kappa _{U}\).

A.5 Proof of Lemma 3

The equilibrium decisions of type-L firm are p L =1−c/q L and a L =0; in addition, c<κ L . Since the consumer deliberates, c< min{q L (1−q L ),q H (1−q H )}. Since κ L <q L (1−q L ), c< min{κ L ,q H (1−q H )}. We prove the following results: (A) There exists an equilibrium where p H =1−c/q H and a H =(p H p L )q L =(1−q L /q H )c. (B) The equilibrium decisions are unique.

  1. (A)

    Type-L firm is indifferent between (p H ,a H ) and (p L ,a L ). Type-H firm prefers (p H ,a H ) because p H q H a H = q H −(2−q L /q H )c>q H c q H /q L = p L q H a L . We next check the out-of-equilibrium decisions. For any (p,a) such that pc/(1−q L ), no firm is willing to deviate because even type-L firm does not want to deviate to (c/(1−q L ),0). For any (p,a) such that p>1−c/q H , the consumer opts out. For any (p,a) such that c/(1−q H )<p≤1−c/q H , the consumer deliberates or opts out under any belief. If p q L aq L c, then type-L firm cannot make a profitable deviation; type-H firm cannot make a profitable deviation either because p q H a = p q L a + p(q H q L )≤q L c + p(q H q L ) = p H q H a H −(p H p)(q H q L )≤p H q H a H . Otherwise (i.e., p q L a>q L c), the consumer opts out under a belief μ(q L |p,a)=1 because p>1−c/q L + a/q L ≥ 1−c/q L . Such an equilibrium survives the intuitive criterion because the consumer deliberates and type-L firm always wants to deviate under μ(q H |p,a)=1 (\(\phantom {\dot {i}\!}\because p q_{L} - a > q_{L} - c\)). For any (p,a) such that c/(1−q L )<pc/(1−q H ), if paq L c, then no firm has any incentive to deviate. Otherwise (i.e., pa>q L c), the consumer deliberates or opts out under a belief μ(q L |p,a)=1. Type-L firm can never make a profitable deviation under that belief because she has obtained her complete-information maximal profit. Type-H firm does not want to deviate either under that belief because she is unwilling to deviate to the highest deliberation-inducing price 1−c/q L . Such an equilibrium survives the intuitive criterion because, under a belief μ(q H |p,a)=1, the consumer purchases without deliberation and so type-L firm can make a profitable deviation (\(\phantom {\dot {i}\!}\because p - a > q_{L} - c\)).

  2. (B)

    We prove this part by contradiction. Suppose the equilibrium decisions of type-H firm are \(\phantom {\dot {i}\!}(p_{H}^{\prime }, a_{H}^{\prime }) \neq (p_{H}, a_{H})\). Since the consumer deliberates, \(\phantom {\dot {i}\!}p_{H}^{\prime } \in (c / (1 - q_{H}), 1 - c / q_{H}]\). Since type-L firm is unwilling to deviate to \(\phantom {\dot {i}\!}(p_{H}^{\prime }, a_{H}^{\prime })\), we have \(\phantom {\dot {i}\!}p_{H}^{\prime } q_{L} - a_{H}^{\prime } \le q_{L} - c\). There are two possible cases: (B.1) \(\phantom {\dot {i}\!}p_{H}^{\prime } < p_{H}\); (B.2) \(\phantom {\dot {i}\!}p_{H}^{\prime } = p_{H}\) and \(\phantom {\dot {i}\!}a_{H}^{\prime } > a_{H}\). (B.1): Take decisions \(\phantom {\dot {i}\!}(p_{H}^{\prime \prime }, a_{H}^{\prime \prime })\) such that \(\phantom {\dot {i}\!}p_{H}^{\prime \prime } \in (p_{H}^{\prime }, p_{H})\) and \(\phantom {\dot {i}\!}a_{H}^{\prime \prime } = a_{H}^{\prime } + (p_{H}^{\prime \prime } - p_{H}^{\prime }) q_{L} + \varepsilon \), where ε>0. The consumer deliberates or opts out under any belief. So type-L firm does not want to deviate because \(\phantom {\dot {i}\!}p_{H}^{\prime \prime } q_{L} - a_{H}^{\prime \prime } = p_{H}^{\prime } q_{L} - a_{H}^{\prime } - \varepsilon < q_{L} - c\). Yet under the only reasonable belief \(\phantom {\dot {i}\!}\mu (q_{H} | p_{H}^{\prime \prime }, a_{H}^{\prime \prime }) = 1\) in light of the intuitive criterion, type-H firm would deviate to \(\phantom {\dot {i}\!}(p_{H}^{\prime \prime }, a_{H}^{\prime \prime })\) because

    $$ p_{H}^{\prime\prime} q_{H} - a_{H}^{\prime\prime} = p_{H}^{\prime} q_{H} - a_{H}^{\prime} + (p_{H}^{\prime\prime} - p_{H}^{\prime}) (q_{H} - q_{L}) - \varepsilon > p_{H}^{\prime} q_{H} - a_{H}^{\prime} $$
    (4)

    provided \(\phantom {\dot {i}\!}\varepsilon < (p_{H}^{\prime \prime } - p_{H}^{\prime }) (q_{H} - q_{L})\), a contradiction. (B.2): type-L firm would not deviate to any \(\phantom {\dot {i}\!}(p_{H}^{\prime }, a)\) such that \(\phantom {\dot {i}\!}a \in (a_{H}, a_{H}^{\prime })\) under any belief because \(\phantom {\dot {i}\!}p_{H}^{\prime } q_{L} - a < p_{H} q_{L} - a_{H} = q_{L} - c\). Yet under the only reasonable belief \(\phantom {\dot {i}\!}\mu (q_{H} | p_{H}^{\prime }, a) = 1\) according to the intuitive criterion, type-H firm can make a profitable deviation to \(\phantom {\dot {i}\!}(p_{H}^{\prime }, a)\), a contradiction.

A.6 Proof of Proposition 4

We first prove Lemmas 11 and 12, and then prove Proposition 4 by summarizing the results. The separating equilibrium refers to the deliberation-on-high type.

Lemma 11

For q L (1−q L )≤c<q H (1−q H ), there exists a separating equilibrium iff c≤q H −q L . The equilibrium decisions are unique: p H =1−c/q H , p L =q L , and a H =a L =0.

Lemma 12

For c< min{q L (1−q L ),q H (1−q H )}, there exists a separating equilibrium iff κ L ≤c≤[1/q H (1−q L )+1/q H ] −1 . The equilibrium decisions are unique: p H =1−c/q H , p L =c/(1−q L ), a H = max{p H q L −p L ,0}, and a L =0.

Proof of Lemma 11— q L (1−q L )≤c<q H (1−q H )

The equilibrium decisions of type-L firm are p L = q L and a L =0. We prove the following: (A) There exists an equilibrium where p H =1−c/q H and a H =0 iff cq H q L . (B) The equilibrium decisions of type-H firm are unique. (A): If c>q H q L , then the equilibrium does not exist (see the proof of Lemma 4). Now suppose cq H q L . Neither type of firm would deviate to the other type of firm’s equilibrium decisions because cq H q L . We next examine the out-of-equilibrium decisions. For any (p,a) such that pq L or p>c/(1−q H ), no firm is willing to deviate (see the proof of Lemma 4). For any (p,a) such that q L <pc/(1−q H ) and paq L , type-L firm would not deviate; so, type-H firm does not want to deviate either because she does not even want to deviate to a price q L . For any (p,a) such that q L <pc/(1−q H ) and pa>q L , the consumer opts out under μ(q L |p,a)=1. Such an equilibrium survives the intuitive criterion because type-L firm can make a profitable deviation to (p,a) under a belief μ(q H |p,a)=1. (B): We prove this part by contradiction. Suppose the equilibrium decisions of type-H firm are \(\phantom {\dot {i}\!}(p_{H}^{\prime }, a_{H}^{\prime }) \neq (p_{H}, a_{H})\). We must have \(\phantom {\dot {i}\!}c / (1 - q_{H}) < p_{H}^{\prime } \le 1 - c / q_{H}\). For the out-of-equilibrium decisions (1−c/q H ,0), the consumer deliberates or opts out under any belief. Hence type-L firm would never deviate. Yet under the only reasonable belief μ(q H |p H ,a H )=1 in light of the intuitive criterion, type-H firm would deviate, a contradiction.

Proof of Lemma 12— c< min{q L (1−q L ),q H (1−q H )}

The equilibrium decisions of type-L firm are p L = c/(1−q L ) and a L =0. Since the consumer does not deliberate on p L , cκ L . We show the following results: (A) There exists an equilibrium where p H =1−c/q H and a H = max{p H q L p L ,0} iff c≤[1/q H (1−q L )+1/q H ]−1. (B) The equilibrium decisions of type-H firm are unique.

  1. (A)

    If c>[1/q H (1−q L )+1/q H ]−1, type-H firm can make a profitable deviation to (p L ,a L ) because p H q H a H = q H ca H q H c<c/(1−q L ) = p L = p L a L . We now prove that, if c≤[1/q H (1−q L )+1/q H ]−1, then the aforementioned equilibrium exists. If p H q L p L , then a H =0. Type-L firm does not want to deviate to (p H ,a H ) because p H q L p L ; type-H firm would not deviate to (p L ,a L ) because p L a L <p H q H a H ( ⇔ c≤[1/q H (1−q L )+1/q H ]−1). Otherwise (i.e., p H q L >p L ), a H = p H q L p L . Type-L firm is indifferent between (p H ,a H ) and (p L ,a L ) because p L a L = p L = p H q L a H ; type-H firm is not willing to deviate to (p L ,a L ) because p H q H a H = p H (q H q L ) + p L >p L = p L a L . We next check the out-of-equilibrium decisions. For any (p,a) such that pc/(1−q L ) or p>1−c/q H , no firm would deviate under any belief. For any (p,a) such that c/(1−q L )<pc/(1−q H ), if pac/(1−q L ), then neither type of firm prefers a deviation. Otherwise (i.e., pa>c/(1−q L )), the consumer deliberates or opts out under a belief μ(q L |p,a)=1. If the consumer opts out (iff p>1−c/q L ), no firm is willing to deviate. Otherwise (i.e., p≤1−c/q L ), type-L firm does not want to deviate because cκ L ; type-H firm would not deviate either because

    $$\begin{array}{@{}rcl@{}} p q_{H} - a &<& p_{H} q_{H} - 0 \text{and} p q_{H} - a \le p q_{H} = p (q_{H} - q_{L}) + p q_{L}\\ &<& p_{H} (q_{H} - q_{L})+ p_{L} = p_{H} q_{H} - (p_{H} q_{L} - p_{L})\\ \Longrightarrow p q_{H} - a &<& p_{H} q_{H} - \max\{p_{H} q_{L} - p_{L}, 0\} = p_{H} q_{H} - a_{H}. \end{array} $$
    (5)

    The equilibrium with the belief μ(q L |p,a)=1 survives the intuitive criterion because, under a belief μ(q H |p,a)=1, the consumer purchases without deliberation and so type-L firm can make a profitable deviation (\(\phantom {\dot {i}\!}\because p - a > c / (1 - q_{L})\)). For any (p,a) such that c/(1−q H )<p≤1−c/q H , the consumer deliberates or opts out. If p q L ac/(1−q L ), then type-L firm does not want to deviate under any belief; type-H firm does not want to deviate either because

    $$\begin{array}{@{}rcl@{}} p q_{H} - a \le p_{H} q_{H} - 0 \text{ and } p q_{H} - a = p q_{L} - a + p (q_{H} - q_{L}) \\ \le p_{L} + p_{H} (q_{H} - q_{L}) = p_{H} q_{H} - (p_{H} q_{L} - p_{L})\\ \Longrightarrow p q_{H} - a \le p_{H} q_{H} - \max\{p_{H} q_{L} - p_{L}, 0\} = p_{H} q_{H} - a_{H}. \end{array} $$

    Otherwise (i.e., p q L a>c/(1−q L )), neither type of firm wants to deviate under a belief μ(q L |p,a)=1 (see the derivation of Inequality (??)). The equilibrium with such a belief survives the intuitive criterion because, under a belief μ(q H |p,a)=1, the consumer deliberates and so type-L firm can make a profitable deviation (because p q L a>c/(1−q L )).

  2. (B)

    We prove this part by contradiction. Suppose the equilibrium decisions of type-H firm are \(\phantom {\dot {i}\!}(p_{H}^{\prime }, a_{H}^{\prime }) \neq (p_{H}, a_{H})\). Since the consumer deliberates, \(\phantom {\dot {i}\!}p_{H}^{\prime } \in (c / (1 - q_{H}), 1 - c / q_{H}]\). Since type-L firm is unwilling to deviate to \(\phantom {\dot {i}\!}(p_{H}^{\prime }, a_{H}^{\prime })\), we have \(\phantom {\dot {i}\!}p_{H}^{\prime } q_{L} - a_{H}^{\prime } \le p_{L}\). There are two cases: (B.1) \(\phantom {\dot {i}\!}p_{H}^{\prime } < p_{H}\); (B.2) \(\phantom {\dot {i}\!}p_{H}^{\prime } = p_{H}\) and \(\phantom {\dot {i}\!}a_{H}^{\prime } > a_{H}\).Footnote 15 (B.1): For an out-of-equilibrium decision tuple \(\phantom {\dot {i}\!}(p_{H}^{\prime \prime }, a_{H}^{\prime \prime })\) such that \(\phantom {\dot {i}\!}p_{H}^{\prime \prime } \in (p_{H}^{\prime }, p_{H})\) and \(\phantom {\dot {i}\!}a_{H}^{\prime \prime } = a_{H}^{\prime } + (p_{H}^{\prime \prime } - p_{H}^{\prime }) q_{L} + \varepsilon \) (where ε>0 is very small), the consumer deliberates or opts out under any belief. Thus, type-L firm would not deviate because \(\phantom {\dot {i}\!}p_{H}^{\prime \prime } q_{L} - a_{H}^{\prime \prime } = p_{H}^{\prime } q_{L} - a_{H}^{\prime } - \varepsilon < p_{L}\). Yet under the only reasonable belief \(\phantom {\dot {i}\!}\mu (q_{H} | p_{H}^{\prime \prime }, a_{H}^{\prime \prime }) = 1\) in light of the intuitive criterion, type-H firm can make a profitable deviation to \(\phantom {\dot {i}\!}(p_{H}^{\prime \prime }, a_{H}^{\prime \prime })\) (see Inequality (4) in the proof of Lemma 3), a contradiction. (B.2): Type-L firm is not willing to deviate to any out-of-equilibrium decisions \(\phantom {\dot {i}\!}(p_{H}^{\prime }, a)\) where \(\phantom {\dot {i}\!}a \in (a_{H}, a_{H}^{\prime })\) under any belief because \(\phantom {\dot {i}\!}p_{H}^{\prime } q_{L} - a < p_{H} q_{L} - a_{H} = p_{L}\). However, under the only reasonable belief \(\phantom {\dot {i}\!}\mu (q_{H} | p_{H}^{\prime }, a) = 1\) according to the intuitive criterion, type-H firm can profitably deviate to \(\phantom {\dot {i}\!}(p_{H}^{\prime }, a)\), a contradiction.

Proof of Proposition 4

Following the proof of Proposition 1, the deliberation-on-high type of separating equilibrium exists iff κ L cκ U (except at the boundary q H (1−q H )), and p H =1−c/q H , p L = min{q L ,c/(1−q L )}, and a L =0. Note that max{p H q L p L ,0}=0 provided q L (1−q L )≤c<q H (1−q H ) because p L = q L for cq L (1−q L ).

A.7 Proof of Proposition 6

Compared with Proposition 1, Proposition 4 shows that the equilibrium price 1−c/q H is higher than c/q L (1−q L ) in the basic model. In the extended model, type-H firm earns q H c and spends (1−c/q H )q L c/(1−q L ) on advertising (note that p H q L >p L provided \(\phantom {\dot {i}\!}c < \hat {\kappa }\)). In the basic model, type-H firm’s profit is c q H /q L (1−q L ). Comparing the profits, we have

$$\begin{array}{@{}rcl@{}} q_{H} - c - \left[\left( 1 - \frac{c}{q_{H}}\right) q_{L} - \frac{c}{1 - q_{L}}\right] &=& \left[1 - \frac{c}{q_{H}} - \frac{c}{q_{L} (1 - q_{L})}\right] (q_{H} - q_{L})\\ &&+ \frac{c q_{H}}{q_{L} (1 - q_{L})} > \frac{c q_{H}}{q_{L} (1 - q_{L})}. \end{array} $$

Therefore, the profit is higher in the extended model than in the basic model.

In both the basic and the extended models, the consumer deliberates in equilibrium. However, only in the extended model type-H firm “burns money”. As a consequence, the social welfare is lower in the extended model than in the basic model. Since the firm’s profit is higher whereas the social welfare is lower in the extended model, the consumer surplus is lower in the extended model than in the basic model.

Appendix B: Technical Details for Pooling Equilibria

B.1 Characterization of the pooling equilibria

Since \(\phantom {\dot {i}\!}\bar {q}_{0} (1 - \bar {q}_{0}) \ge \min \{q_{L} (1 - q_{L}), q_{H} (1 - q_{H})\}\), we characterize the equilibrium in seven lemmas.

Lemma 13

For \(\phantom {\dot {i}\!}c \ge \max \{q_{L} (1 - q_{L}), q_{H} (1 - q_{H}), \bar {q}_{0} (1 - \bar {q}_{0})\}\) , only pooling-non-deliberation equilibria exist. Any \(\phantom {\dot {i}\!}p^{*} \in [q_{L}, \bar {q}_{0}]\) can be an equilibrium price.

Lemma 14

For \(\phantom {\dot {i}\!}\max \{q_{L} (1 - q_{L}), q_{H} (1 - q_{H})\} \le c < \bar {q}_{0} (1 - \bar {q}_{0})\) , only pooling-non-deliberation equilibria exist. Any \(\phantom {\dot {i}\!}\hspace *{-.3pt}p^{*} \in [q_{L}, c / (1 - \bar {q}_{0})]\) can be an equilibrium price.

Lemma 15

For \(\phantom {\dot {i}\!}\max \{q_{L} (1 - q_{L}), \bar {q}_{0} (1 - \bar {q}_{0})\} \le c < q_{H} (1 - q_{H})\) , only pooling-non-deliberation equilibria exist, and they exist iff \(\phantom {\dot {i}\!}c \ge q_{H} - \bar {q}_{0}\) . Any \(\phantom {\dot {i}\!}p^{*} \in [\max \{q_{L}, q_{H} - c\}, \bar {q}_{0}]\) can be an equilibrium price.

Lemma 16

For \(\phantom {\dot {i}\!}\max \{q_{H} (1 - q_{H}), \bar {q}_{0} (1 - \bar {q}_{0})\} \le c < q_{L} (1 - q_{L})\) , only pooling-non-deliberation equilibria exist. Any \(\phantom {\dot {i}\!}p^{*} \in [\max \{c / (1 - q_{L}), (1 - c / q_{L}) q_{H}\}, \bar {q}_{0}]\) can be an equilibrium price.

Lemma 17

For \(\phantom {\dot {i}\!}q_{L} (1 - q_{L}) \le c < \min \{q_{H} (1 - q_{H}), \bar {q}_{0} (1 - \bar {q}_{0})\}\) , only pooling-non-deliberation equilibria exist, and they exist iff \(\phantom {\dot {i}\!}c \ge [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{H}]^{-1}\) . Any \(\phantom {\dot {i}\!}p^{*} \in [\max \{q_{L}, q_{H} - c\}, c / (1 - \bar {q}_{0})]\) can be an equilibrium price.

Lemma 18

Suppose \(\phantom {\dot {i}\!}q_{H} (1 - q_{H}) \le c < \min \{q_{L} (1 - q_{L}), \bar {q}_{0} (1 - \bar {q}_{0})\}\) . (i) There exist pooling-non-deliberation equilibria iff \(\phantom {\dot {i}\!}c \ge [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{L}]^{-1}\) . Any \(\phantom {\dot {i}\!}p^{*} \in [\max \{c / (1 - q_{L}), (1 - c / q_{L}) q_{H}\}, c / (1 - \bar {q}_{0})]\) can be an equilibrium price. (ii) There exist pooling-deliberation equilibria iff \(\phantom {\dot {i}\!}c \le [1 / q_{L} (1 - q_{L}) + 1 / \bar {q}_{0}]^{-1}\) . Any \(\phantom {\dot {i}\!}p^{*} \in [\max \{c / (1 - \bar {q}_{0}), 1 - c / q_{L}, c / q_{L} (1 - q_{L})\}, 1 - c / \bar {q}_{0}]\) (except at the boundary \(\phantom {\dot {i}\!}c / (1 - \bar {q}_{0})\) ) can be an equilibrium price.

Lemma 19

Suppose \(\phantom {\dot {i}\!}c < \min \{q_{L} (1 - q_{L}), q_{H} (1 - q_{H}), \bar {q}_{0} (1 - \bar {q}_{0})\}\) . (i) There exist pooling-non-deliberation equilibria iff \(\phantom {\dot {i}\!}c \ge [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{H}]^{-1}\) . Any \(\phantom {\dot {i}\!}p^{*} \in [\max \{c / (1 - q_{L}), q_{H} - c\}, c / (1 - \bar {q}_{0})]\) can be an equilibrium price. (ii) There exist pooling-deliberation equilibria where the consumer deliberates iff \(\phantom {\dot {i}\!}c \le [1 / q_{L} (1 - q_{L}) + 1 / \bar {q}_{0}]^{-1}\) . Any \(\phantom {\dot {i}\!}p^{*} \in [\max \{c / (1 - \bar {q}_{0}), 1 - c / q_{L}, c / q_{L} (1 - q_{L})\}, 1 - c / \bar {q}_{0}]\) (except at the boundary \(\phantom {\dot {i}\!}c / (1 - \bar {q}_{0})\) ) can be an equilibrium price.

Proof of Lemma 13—\(c \ge \max \{q_{L} (1 - q_{L}), q_{H} (1 - q_{H}), \bar {q}_{0} (1 - \bar {q}_{0})\}\)

The consumer does not deliberate on any equilibrium price in any pooling equilibrium. As a result, any equilibrium price should not exceed \(\phantom {\dot {i}\!}\bar {q}_{0}\). First, any price less than q L cannot be an equilibrium price. Otherwise, since the consumer always purchases without deliberation on a price q L under any belief, both types of firms can make a profitable deviation by charging q L .

Next, we prove that any \(\phantom {\dot {i}\!}p^{*} \in [q_{L}, \bar {q}_{0}]\) can be an equilibrium price. For any p<p , neither type of firm would like to deviate under any belief. For any p>q H , the consumer always opts out because \(\phantom {\dot {i}\!}p > q_{H} \ge \bar {q}\) and \(\phantom {\dot {i}\!}p > q_{H} \ge 1 - c / q_{H} \ge 1 - c / \bar {q}\) (where \(\phantom {\dot {i}\!}\bar {q}\) is any perceived quality). For any p∈(p ,q H ], neither type of firm wants to deviate under a belief μ(q L |p)=1 because the consumer opts out. An equilibrium with such a belief survives the intuitive criterion because type-L firm always prefers a deviation under an out-of-equilibrium belief is μ(q H |p)=1.

Proof of Lemma 14—\(\max \{q_{L} (1 - q_{L}), q_{H} (1 - q_{H})\} \le c < \bar {q}_{0} (1 - \bar {q}_{0})\)

(i) Pooling-non-deliberation: Any equilibrium price should not exceed \(\phantom {\dot {i}\!}c / (1 - \bar {q}_{0})\) and should not be less than q L either. By examining out-of-equilibrium price in three parts p<p , p>q H , and p∈(p ,q H ], it can be shown that any \(\phantom {\dot {i}\!}p^{*} \in [q_{L}, c / (1 - \bar {q}_{0})]\) can be an equilibrium price. (ii) Pooling-deliberation: Since \(\phantom {\dot {i}\!}q_{L} \le c / (1 - q_{L}) \le c / (1 - \bar {q})\) and \(\phantom {\dot {i}\!}q_{L} \le \bar {q}\) (where \(\phantom {\dot {i}\!}\bar {q}\) is any perceived quality), the consumer purchases without deliberation on q L . Hence type-L firm is unwilling to choose a price that induces deliberation. So, this type of pooling equilibrium does not exist.

Proof of Lemma 15—\(\max \{q_{L} (1 - q_{L}), \bar {q}_{0} (1 - \bar {q}_{0})\} \le c < q_{H} (1 - q_{H})\)

Any equilibrium price p should satisfy \(\phantom {\dot {i}\!}q_{L} \le p^{*} \le \bar {q}_{0}\). We show that any \(\phantom {\dot {i}\!}p^{*} \in [q_{L}, \bar {q}_{0}]\) can be an equilibrium price iff p q H c. [“Only if” part]: Suppose p <q H c. We prove by contradiction that there is no equilibrium that survives the intuitive criterion. Suppose the equilibrium exists. The consumer either deliberates or opts out for a price p=1−c/q H under any belief because \(\phantom {\dot {i}\!}p > c / (1 - q_{H}) \ge c / (1 - \bar {q})\) (where \(\phantom {\dot {i}\!}\bar {q}\) is any perceived quality). Since p q L <q L p , type-L firm has no incentive to deviate to p under any belief. Yet under the only reasonable belief μ(q H |p)=1, type-H firm would deviate to p because p q H = q H c>p , a contradiction. [“If” part]: Suppose p q H c. We prove that any \(\phantom {\dot {i}\!}p^{*} \in [q_{L}, \bar {q}_{0}]\) can be an equilibrium price. For any p<p , neither type of firm would like to deviate under any belief. For any p>c/(1−q H ), the consumer either deliberates or opts out. When the consumer opts out, no firm would like to deviate; when the consumer deliberates (only if p≤1−c/q H ), no firm strictly prefers a deviation because p q H q H cp . For any p∈(p ,c/(1−q H )], neither type of firm prefers a deviation under a belief μ(q L |p)=1 because the consumer always opts out. Such an equilibrium survives the intuitive criterion because, for any price p∈(p ,c/(1−q H )], type-L firm prefers a deviation under a belief μ(q H |p)=1.

On the one hand, as long as \(\phantom {\dot {i}\!}c \ge q_{H} - \bar {q}_{0}\), there exists a pooling equilibrium (with an equilibrium price \(\phantom {\dot {i}\!}p^{*} \in [\max \{q_{L}, q_{H} - c\}, \bar {q}_{0}]\)). On the other hand, if \(\phantom {\dot {i}\!}c < q_{H} - \bar {q}_{0}\), then no price \(\phantom {\dot {i}\!}p^{*} \in [q_{L}, \bar {q}_{0}]\) satisfies p q H c. Therefore, there exists a pooling equilibrium iff \(\phantom {\dot {i}\!}c \ge q_{H} - \bar {q}_{0}\); any \(\phantom {\dot {i}\!}p^{*} \in [\max \{q_{L}, q_{H} - c\}, \bar {q}_{0}]\) can be an equilibrium price provided \(\phantom {\dot {i}\!}c \ge q_{H} - \bar {q}_{0}\).

Proof of Lemma 16—\(\max \{q_{H} (1 - q_{H}), \bar {q}_{0} (1 - \bar {q}_{0})\} \le c < q_{L} (1 - q_{L})\)

Any equilibrium price p should satisfy \(\phantom {\dot {i}\!}c / (1 - q_{L}) \le p^{*} \le \bar {q}_{0}\). We prove that there exists a pooling equilibrium (with an equilibrium price \(\phantom {\dot {i}\!}p^{*} \in [c / (1 - q_{L}), \bar {q}_{0}]\)) iff p ≥ (1−c/q L )q H . [“Only if” part]: We prove by contradiction that no pooling equilibrium exists with p <(1−c/q L )q H . Suppose p is an equilibrium price. the consumer deliberates or purchases without deliberation under any belief for p=1−c/q L . Hence type-H firm can profitably deviate to p under any belief, a contradiction. [“If” part]: Suppose p ≥ (1−c/q L )q H . We prove that any \(\phantom {\dot {i}\!}p^{*} \in [c / (1 - q_{L}), \bar {q}_{0}]\) can be an equilibrium price. For any p<p or p>q H , no firm wants to deviate. For any p∈(p ,q H ], the consumer either deliberates or opts out under an out-of-equilibrium belief μ(q L |p)=1. When the consumer deliberates (only if p≤1−c/q L ), since p ≥ (1−c/q L )q H , neither type of firm strictly prefers a deviation. Such an equilibrium survives the intuitive criterion because type-L firm can make a profitable deviation under μ(q H |p)=1.

Note that p ≥ (1−c/q L )q H is equivalent to cq L (1−p /q H ). As long as \(\phantom {\dot {i}\!}c \ge q_{L} (1 - \bar {q}_{0} / q_{H})\), there exists a pooling equilibrium (with an equilibrium price \(\phantom {\dot {i}\!}p^{*} \in [\max \{c / (1 - q_{L}), (1 - c / q_{L}) q_{H}\}, \bar {q}_{0}]\)). On the other hand, if \(\phantom {\dot {i}\!}c < q_{L} (1 - \bar {q}_{0} / q_{H})\), then no price \(\phantom {\dot {i}\!}p^{*} \in [c / (1 - q_{L}), \bar {q}_{0}]\) satisfies cq L (1−p /q H ). Therefore, there exists a pooling equilibrium iff \(\phantom {\dot {i}\!}c \ge q_{L} (1 - \bar {q}_{0} / q_{H})\). We prove by contradiction that \(\phantom {\dot {i}\!}q_{L} (1 - \bar {q}_{0} / q_{H}) < \bar {q}_{0} (1 - \bar {q}_{0})\). Suppose \(\phantom {\dot {i}\!}q_{L} (1 - \bar {q}_{0} / q_{H}) \ge \bar {q}_{0} (1 - \bar {q}_{0})\):

$$q_{L} \left( 1 - \frac{\bar{q}_{0}}{q_{H}}\right) \ge \bar{q}_{0} (1 - \bar{q}_{0}) \Longrightarrow q_{H} - \bar{q}_{0} \ge \frac{q_{H} \bar{q}_{0}}{q_{L}} \cdot (1 - \bar{q}_{0}) > q_{H} (1 - \bar{q}_{0}) \ge q_{H} - \bar{q}_{0}, $$

a contradiction. The premise of this lemma implies \(\phantom {\dot {i}\!}c \ge \bar {q}_{0} (1 - \bar {q}_{0}) > q_{L} (1 - \bar {q}_{0} / q_{H})\). Thus, pooling equilibria always exist; any \(\phantom {\dot {i}\!}p^{*} \in [\max \{c / (1 - q_{L}), (1 - c / q_{L}) q_{H}\}, \bar {q}_{0}]\) can be an equilibrium price.

Proof of Lemma 17—\(q_{L} (1 - q_{L}) \le c < \min \{q_{H} (1 - q_{H}), \bar {q}_{0} (1 - \bar {q}_{0})\}\)

(i) Pooling-non-deliberation: Any equilibrium price p satisfies \(\phantom {\dot {i}\!}q_{L} \le p^{*} \le c / (1 - \bar {q}_{0})\). Similarly to the proof of Lemma 15, any \(\phantom {\dot {i}\!}p^{*} \in [q_{L}, c / (1 - \bar {q}_{0})]\) can be an equilibrium price iff p q H c. If \(\phantom {\dot {i}\!}c \ge q_{H} - c / (1 - \bar {q}_{0}) \iff c \ge [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{H}]^{-1}\), then there exists a pooling equilibrium (with an equilibrium price \(\phantom {\dot {i}\!}p^{*} \in [\max \{q_{L}, q_{H} - c\}, c / (1 - \bar {q}_{0})]\)). On the other hand, if \(\phantom {\dot {i}\!}c < [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{H}]^{-1}\), then no \(\phantom {\dot {i}\!}p^{*} \in [q_{L}, c / (1 - \bar {q}_{0})]\) satisfies p q H c. So, there exists a pooling equilibrium iff \(\phantom {\dot {i}\!}c \ge [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{H}]^{-1}\); any \(\phantom {\dot {i}\!}p^{*} \in [\max \{q_{L}, q_{H} - c\}, c / (1 - \bar {q}_{0})]\) can be an equilibrium price. (ii) Pooling-deliberation: No such pooling equilibrium exists (see Part (ii) of the proof of Lemma 14).

Proof of Lemma 18—\(q_{H} (1 - q_{H}) \le c < \min \{q_{L} (1 - q_{L}), \bar {q}_{0} (1 - \bar {q}_{0})\}\)

(i) Pooling-non-deliberation: Any equilibrium price p satisfies \(\phantom {\dot {i}\!}c / (1 - q_{L}) \le p^{*} \le c / (1 - \bar {q}_{0})\). The proof of Lemma 16 can be used to show that a pooling equilibrium exists (with \(\phantom {\dot {i}\!}p^{*} \in [c / (1 - q_{L}), c / (1 - \bar {q}_{0})]\)) iff p ≥ (1−c/q L )q H cq L (1−p /q H ). Therefore, a pooling equilibrium exists iff \(\phantom {\dot {i}\!}c \ge q_{L} [1 - c / (1 - \bar {q}_{0}) q_{H}] \iff c \ge [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{L}]^{-1}\); any \(\phantom {\dot {i}\!}p^{*} \in [\max \{c / (1 - q_{L}), (1 - c / q_{L}) q_{H}\}, c / (1 - \bar {q}_{0})]\) can be an equilibrium price.

(ii) Pooling-deliberation: Any equilibrium price p should satisfy \(\phantom {\dot {i}\!}c / (1 - \bar {q}_{0}) < p^{*} \le 1 - c / \bar {q}_{0}\). First, any equilibrium price p ≥ 1−c/q L . Otherwise, for p=1−c/q L , the consumer deliberates or purchases without deliberation under any belief, and so both types of firms prefer to deviate to p. Thus, we focus on \(\phantom {\dot {i}\!}p^{*} \in [\max \{c / (1 - \bar {q}_{0}), 1 - c / q_{L}\}, 1 - c / \bar {q}_{0}]\) (except at the boundary \(\phantom {\dot {i}\!}c / (1 - \bar {q}_{0})\)). Next, we prove that there exists a pooling equilibrium iff p c/q L (1−q L ). [“Only if” part]: Suppose p <c/q L (1−q L ). For p = c/(1−q L ), the consumer purchases without deliberation under any belief. Yet firms can then profitably deviate to p under any belief because p>p q L , and so no pooling equilibrium exists. [“If” part]: Suppose p c/q L (1−q L ). We prove that any \(\phantom {\dot {i}\!}p^{*} \in [\max \{c / (1 - \bar {q}_{0}), 1 - c / q_{L}\}, 1 - c / \bar {q}_{0}]\) (except at the boundary \(\phantom {\dot {i}\!}c / (1 - \bar {q}_{0})\)) can be an equilibrium price. For any pc/(1−q L ), no firm strictly prefers a deviation under any belief because pp q L . For any p>q H , the consumer always opts out under any belief because \(\phantom {\dot {i}\!}p > 1 - c / q_{H} \ge 1 - c / \bar {q}\) and \(\phantom {\dot {i}\!}p > \bar {q}\) (where \(\phantom {\dot {i}\!}\bar {q}\) is any perceived quality). For any p∈(c/(1−q L ),p ), if the consumer deliberates or opts out under any belief, then no firm wants to deviate. Otherwise, the consumer purchases without deliberation under some belief. Note that the consumer deliberates or opts out under a belief μ(q L |p)=1, and so no firm would like to deviate. Such an equilibrium survives the intuitive criterion because, if type-L firm does not want to deviate under any belief (iff p<p q L ), then even under μ(q H |p)=1 type-H firm would not deviate (because p<p q L <p q H ). For any p∈(p ,q H ], no firm wants to deviate under μ(q L |p)=1 because the consumer always opts out. Such an equilibrium survives the intuitive criterion because type-L firm would deviate to p under μ(q H |p)=1. We have just proved that there exists a pooling equilibrium (with an equilibrium price \(\phantom {\dot {i}\!}p^{*} \in [\max \{c / (1 - \bar {q}_{0}), 1 - c / q_{L}\}, 1 - c / \bar {q}_{0}]\) except at the boundary \(\phantom {\dot {i}\!}c / (1 - \bar {q}_{0})\)) iff p c/q L (1−q L ). Hence there exists a pooling equilibrium iff \(\phantom {\dot {i}\!}c \le (1 - c / \bar {q}_{0}) q_{L} (1 - q_{L})\), which is equivalent to \(\phantom {\dot {i}\!}c \le [ 1 / q_{L} (1 - q_{L}) + 1 / \bar {q}_{0} ]^{-1}\); any \(\phantom {\dot {i}\!}p^{*} \in [\max \{c / (1 - \bar {q}_{0}), 1 - c / q_{L}, c / q_{L} (1 - q_{L})\}, 1 - c / \bar {q}_{0}]\) (except at the boundary \(\phantom {\dot {i}\!}c / (1 - \bar {q}_{0})\)) can be an equilibrium price.

Proof of Lemma 19—\(c < \min \{q_{L} (1 - q_{L}), q_{H} (1 - q_{H}), \bar {q}_{0} (1 - \bar {q}_{0})\}\)

(i) Pooling-non-deliberation: Any equilibrium price p should satisfy \(\phantom {\dot {i}\!}c / (1 - q_{L}) \le p^{*} \le c / (1 - \bar {q}_{0})\). We prove that there exists a pooling equilibrium (with an equilibrium price \(\phantom {\dot {i}\!}p^{*} \in [c / (1 - q_{L}), c / (1 - \bar {q}_{0})]\)) that survives the intuitive criterion iff the following two conditions are both satisfied: (a) p ≥ (1−c/q L )q H ; (b) p p q H or p p q L for any p∈(c/(1−q H ),1−c/q H ]. [“Only if” part]: Suppose p <(1−c/q L )q H or there exists p∈(c/(1−q H ),1−c/q H ] such that p <p q H and p >p q L . We prove by contradiction that no pooling equilibrium exists and survives the intuitive criterion. Suppose an equilibrium exists. For p=1−c/q L , the consumer deliberates or purchases without deliberation under any belief because \(\phantom {\dot {i}\!}p \le 1 - c / \bar {q}\) (where \(\phantom {\dot {i}\!}\bar {q}\) is any perceived quality). Hence if p <(1−c/q L )q H , then type-H firm would deviate to p under any belief, a contradiction. For any p∈(c/(1−q H ),1−c/q H ], the consumer either deliberates or opts out because \(\phantom {\dot {i}\!}p > c / (1 - q_{H}) \ge c / (1 - \bar {q})\). If there exists p∈(c/(1−q H ),1−c/q H ] such that p <p q H and p >p q L , then under the only reasonable belief μ(q H |p)=1, type-H firm can profitably deviate to p, a contradiction. [“If” part]: We prove that any price \(\phantom {\dot {i}\!}p^{*} \in [c / (1 - q_{L}), c / (1 - \bar {q}_{0})]\) can be an equilibrium price provided both of the foregoing two conditions (a) and (b) hold. For any p<p or p>1−c/q H , no firm would like to deviate. For any p∈(p ,1−c/q H ], the consumer either deliberates or opts out under μ(q L |p)=1, and so condition (a) guarantees that no firm is willing to deviate. Such an equilibrium survives the intuitive criterion because of the following. For any p∈(p ,c/(1−q H )], type-L firm wants to deviate to p under μ(q H |p)=1 because the consumer purchases without deliberation. For any p∈(c/(1−q H ),1−c/q H ], type-H firm would not deviate even under μ(q H |p)=1. Condition (b) implies that, even when the consumer deliberates, if type-L firm does not want to deviate under any belief (iff p >p q L ), then under the only reasonable belief μ(q H |p)=1, type-H firm would not deviate either (\(\phantom {\dot {i}\!}\because p^{*} \ge p q_{H}\)).

We have just proved that a pooling equilibrium (with an equilibrium price \(\phantom {\dot {i}\!}p^{*} \in [c / (1 - q_{L}), c / (1 - \bar {q}_{0})]\)) exists iff the foregoing conditions (a) and (b) hold. Condition (b) is equivalent to ∀p∈(c/(1−q H ),1−c/q H ], p p q H or ∀p∈(c/(1−q H ),1−c/q H ], p p q L ;Footnote 16 this is subsequently equivalent to p ≥ (1−c/q H )q H = q H c or p q L c/(1−q H ). Since p q H c implies (a), a pooling equilibrium (with an equilibrium price \(\phantom {\dot {i}\!}p^{*} \in [c / (1 - q_{L}), c / (1 - \bar {q}_{0})]\)) exists iff p q H c or (1−c/q L )q H p q L c/(1−q H ). We next prove that the second condition above, (1−c/q L )q H p q L c/(1−q H ), never holds. There are two cases: (1) q L (1−q L )<q H (1−q H ) and (2) q H (1−q H )≤q L (1−q L ). (1): Recall that p c/(1−q L ); we use this inequality to prove by contradiction that p >q L c/(1−q H ). Suppose p q L c/(1−q H ); this is equivalent to cp (1−q H )/q L . We have \(\phantom {\dot {i}\!}c \ge \frac {p^{*} (1 - q_{H})}{q_{L}} \ge \frac {c (1 - q_{H})}{q_{L} (1 - q_{L})} \Longrightarrow q_{L} (1 - q_{L}) \ge (1 - q_{H}) \ge q_{H} (1 - q_{H})\), which contradicts q L (1−q L )<q H (1−q H ). (2): The condition (1−c/q L )q H p q L c/(1−q H ) is equivalent to c ≥ max{q L (1−p /q H ), p (1−q H )/q L }. We prove by contradiction that the above inequality does not hold. Suppose that c ≥ max{q L (1−p /q H ), p (1−q H )/q L }. Note that max{q L (1−p /q H ), p (1−q H )/q L } is quasi-convex in p and is minimized at \(\phantom {\dot {i}\!}p^{*} = q_{H} {q_{L}^{2}} / [q_{H} (1 - q_{H}) + {q_{L}^{2}}]\). Thus, \(\phantom {\dot {i}\!}c \ge \max \{q_{L} (1 - p^{*} / q_{H}),\,p^{*} (1 - q_{H}) / q_{L}\} \ge q_{L} q_{H} (1 - q_{H}) / [q_{H} (1 - q_{H}) + {q_{L}^{2}}]\). Recall that c<q H (1−q H ) and q H (1−q H )≤q L (1−q L ). The second inequality implies \(\phantom {\dot {i}\!}q_{L} \ge q_{H} (1 - q_{H}) + {q_{L}^{2}} \Longrightarrow c \ge q_{L} q_{H} (1 - q_{H}) / [q_{H} (1 - q_{H}) + {q_{L}^{2}}] \ge q_{H} (1 - q_{H})\), a contradiction.

Thus, a pooling equilibrium exists (with an equilibrium price \(\phantom {\dot {i}\!}p^{*} \in [c / (1 - q_{L}), c / (1 - \bar {q}_{0})]\)) iff p q H c. The existence condition is \(\phantom {\dot {i}\!}c \ge q_{H} - c / (1 - \bar {q}_{0}) \iff c \ge [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{H}]^{-1}\); any \(\phantom {\dot {i}\!}p^{*} \in [\max \{c / (1 - q_{L}), q_{H} - c\}, c / (1 - \bar {q}_{0})]\) can be an equilibrium price.

(ii) Pooling-deliberation: Any equilibrium price p should satisfy \(\phantom {\dot {i}\!}c / (1 - \bar {q}_{0}) < p^{*} \le 1 - c / \bar {q}_{0}\) and p ≥ 1−c/q L . We prove that there exists a pooling equilibrium iff the equilibrium price p c/q L (1−q L ). Similarly to the proof of Lemma 18, no pooling equilibrium exists if p <c/q L (1−q L ). Now suppose p c/q L (1−q L ). We prove that any \(\phantom {\dot {i}\!}p^{*} \in [\max \{c / (1 - \bar {q}_{0}), 1 - c / q_{L}\}, 1 - c / \bar {q}_{0}]\) (except at the boundary \(\phantom {\dot {i}\!}c / (1 - \bar {q}_{0})\)) can be an equilibrium price. For any pc/(1−q L ), no firm strictly prefers a deviation under any belief because pp q L . For any p>1−c/q H , the consumer opts out under any belief because \(\phantom {\dot {i}\!}p > 1 - c / \bar {q}\) and \(\phantom {\dot {i}\!}p > q_{H} \ge \bar {q}\) (where \(\phantom {\dot {i}\!}\bar {q}\) is any perceived quality). For any p∈(c/(1−q L ),p ) or p∈(p ,1−c/q H ], it can be proved that neither type of firm would like to deviate to p under μ(q L |p)=1 and such an equilibrium survives the intuitive criterion. Similarly to part (ii) of the proof of Lemma 18, there exists a pooling equilibrium iff \(\phantom {\dot {i}\!}c \le [ 1 / q_{L} (1 - q_{L}) + 1 / \bar {q}_{0} ]^{-1}\); any \(\phantom {\dot {i}\!}p^{*} \in [\max \{c / (1 - \bar {q}_{0}), 1 - c / q_{L}, c / q_{L} (1 - q_{L})\}, 1 - c / \bar {q}_{0}]\) (except at the boundary \(\phantom {\dot {i}\!}c / (1 - \bar {q}_{0})\)) can be an equilibrium price.

Summary

$$\begin{array}{@{}rcl@{}} \text{Define} \,\,\gamma_{N} &\equiv& \left\{ \begin{array}{ll} \min\left\{\max\left\{ q_{H} - \bar{q}_{0}, \left[ \frac{1}{q_{H} (1 - \bar{q}_{0})} + \frac{1}{q_{H}} \right]^{-1} \right\},\, q_{H} (1 - q_{H}) \right\}\\ \hspace{52mm}\text{if}\,\, q_{L} (1 - q_{L}) \le \bar{q}_{0} (1 - \bar{q}_{0}) \le q_{H} (1 - q_{H}), \\ \min\left\{ \left[ \frac{1}{q_{H} (1 - \bar{q}_{0})} + \frac{1}{q_{H}} \right]^{-1},\, q_{H} (1 - q_{H}) \right.\\ \hspace{52mm}\text{if}\,\, q_{L} (1 - q_{L}) \le q_{H} (1 - q_{H}) < \bar{q}_{0} (1 - \bar{q}_{0}), \\ \min\left\{ \max\left\{ \left[ \frac{1}{q_{H} (1 - \bar{q}_{0})} + \frac{1}{q_{L}} \right]^{-1},\, q_{H} (1 - q_{H}) \right\},\ \left[ \frac{1}{q_{H} (1 - \bar{q}_{0})} + \frac{1}{q_{H}} \right]^{-1} \right\} \\ \hspace{52mm}\text{if}\,\, q_{H} (1 - q_{H}) < q_{L} (1 - q_{L}); \end{array}\right. \end{array} $$
(6)
$$\begin{array}{@{}rcl@{}} \gamma_{D} &\equiv& \min\left\{ \left[ \frac{1}{q_{L} (1 - q_{L})} + \frac{1}{\bar{q}_{0}} \right]^{-1},\,\bar{q}_{0} (1 - \bar{q}_{0}) \right\}. \end{array} $$
(7)

To summarize the result, we will show that (i) there exist pooling-non-deliberation equilibria iff cγ N , and (ii) there exist pooling-deliberation equilibria iff cγ D . We first prove part (ii). There exists a pooling-deliberation equilibrium iff \(\phantom {\dot {i}\!}c < \min \{q_{L} (1 - q_{L}), \bar {q}_{0} (1 - \bar {q}_{0})\}\) and \(\phantom {\dot {i}\!}c \le [1 / q_{L} (1 - q_{L}) + 1 / \bar {q}_{0}]^{-1}\) (see Lemmas 18 and 19). Since \(\phantom {\dot {i}\!}[1 / q_{L} (1 - q_{L}) + 1 / \bar {q}_{0}]^{-1} < q_{L} (1 - q_{L})\), there exists a pooling-deliberation equilibrium iff \(\phantom {\dot {i}\!}c \le \min \{[1 / q_{L} (1 - q_{L}) + 1 / \bar {q}_{0}]^{-1}, \bar {q}_{0} (1 - \bar {q}_{0})\} \equiv \gamma _{D}\) (except at the boundary \(\phantom {\dot {i}\!}\bar {q}_{0} (1 - \bar {q}_{0})\)).

Proof for part (i) is shown in three cases separately: (A) \(\phantom {\dot {i}\!}q_{L} (1 - q_{L}) \le \bar {q}_{0} (1 - \bar {q}_{0}) \le q_{H} (1 - q_{H})\), (B) \(\phantom {\dot {i}\!}q_{L} (1 - q_{L}) \le q_{H} (1 - q_{H}) < \bar {q}_{0} (1 - \bar {q}_{0})\), and (C) q H (1−q H )<q L (1−q L ). Case (C) can be divided further into two subcases: (C.1) \(\phantom {\dot {i}\!}q_{H} (1 - q_{H}) \le \bar {q}_{0} (1 - \bar {q}_{0}) \le q_{L} (1 - q_{L})\) and (C.2) \(\phantom {\dot {i}\!}q_{H} (1 - q_{H}) < q_{L} (1 - q_{L}) < \bar {q}_{0} (1 - \bar {q}_{0})\).

(A) This case summarizes results in Lemmas 13, 15, 17, and 19(i). For \(\phantom {\dot {i}\!}c < \bar {q}_{0} (1 - \bar {q}_{0})\), from Lemmas 17 and 19(i), an equilibrium exists iff \(\phantom {\dot {i}\!}c \ge [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{H}]^{-1}\). For \(\phantom {\dot {i}\!}\bar {q}_{0} (1 - \bar {q}_{0}) \le c < q_{H} (1 - q_{H})\), from Lemma 15, an equilibrium exists iff \(\phantom {\dot {i}\!}c \ge q_{H} - \bar {q}_{0}\). It can be verified that \(\phantom {\dot {i}\!}q_{H} - \bar {q}_{0} \ge \bar {q}_{0} (1 - \bar {q}_{0}) \iff \left [\frac {1}{q_{H} (1 - \bar {q}_{0})} + \frac {1}{q_{H}}\right ]^{-1} \ge \bar {q}_{0} (1 - \bar {q}_{0}) \iff q_{H} - \bar {q}_{0} \ge \left [\frac {1}{q_{H} (1 - \bar {q}_{0})} + \frac {1}{q_{H}}\right ]^{-1}\). So, for c<q H (1−q H ), an equilibrium exists iff \(\phantom {\dot {i}\!}c \ge \max \{q_{H} - \bar {q}_{0}, [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{H}]^{-1}\}\). Combined with Lemma 13, there exists a pooling-non-deliberation equilibrium iff \(\phantom {\dot {i}\!}c \ge \min \left \{ \max \{ q_{H} - \bar {q}_{0}, [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{H}]^{-1} \},\, q_{H} (1 - q_{H}) \right \} \equiv \gamma _{N}\).

(B) This case incorporates results in Lemmas 13, 14, 17, and 19(i). For c<q H (1−q H ), from Lemmas 17 and 19(i), an equilibrium exists iff \(\phantom {\dot {i}\!}c \ge [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{H}]^{-1}\). Together with Lemmas 13 and 14, there exists a pooling-non-deliberation equilibrium iff \(\phantom {\dot {i}\!}c \ge \min \{ [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{H}]^{-1},\, q_{H} (1 - q_{H}) \} \equiv \gamma _{N}\).

(C.1) This case summarizes results in Lemmas 13, 16, 18(i), and 19(i). For cq H (1−q H ), from Lemmas 13, 16, and 18(i), an equilibrium exists iff \(\phantom {\dot {i}\!}c \ge [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{L}]^{-1}\) because \(\phantom {\dot {i}\!}[1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{L}]^{-1} < \bar {q}_{0} (1 - \bar {q}_{0})\).Footnote 17 Since \(\phantom {\dot {i}\!}[1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{L}]^{-1} < [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{H}]^{-1}\), together with Lemma19(i), there exists a pooling-non-deliberation equilibrium iff \(\phantom {\dot {i}\!}c \ge \min \left \{ \max \{ [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{L}]^{-1},\, q_{H} (1 - q_{H}) \},\ [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{H}]^{-1} \right \} \equiv \gamma _{N}\).

(C.2) This case sums up results in Lemmas 13, 14, 18(i), and 19(i). For cq H (1−q H ), from Lemmas 13, 14, and 18(i), an equilibrium exists iff \(\phantom {\dot {i}\!}c \ge [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{L}]^{-1}\) because \(\phantom {\dot {i}\!}[1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{L}]^{-1} < [1 / (1 - q_{L}) + 1 / q_{L}]^{-1} = q_{L} (1 - q_{L})\). Since \(\phantom {\dot {i}\!}[1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{L}]^{-1} < [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{H}]^{-1}\), from Lemma 19(i), there exists a pooling-non-deliberation equilibrium iff \(\phantom {\dot {i}\!}c \ge \min \left \{ \max \{ [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{L}]^{-1},\, q_{H} (1 - q_{H}) \},\ [1 / q_{H} (1 - \bar {q}_{0}) + 1 / q_{H}]^{-1} \right \} \equiv \gamma _{N}\).

Proof of Proposition 3

Recall that \(\phantom {\dot {i}\!}\bar {q}_{0} \equiv (1 - \phi ) q_{L} + \phi q_{H} = q_{L} + (q_{H} - q_{L}) \phi \), which is increasing in ϕ. Hence the monotonicity of γ N and γ D in ϕ is equivalent to that of γ N and γ D in \(\phantom {\dot {i}\!}\bar {q}_{0}\). (i): According to the expression for γ N (see Eq. 6), in all three cases γ N is (weakly) decreasing in \(\phantom {\dot {i}\!}\bar {q}_{0}\). Since γ N is continuous in \(\phantom {\dot {i}\!}\bar {q}_{0}\), it is (weakly) decreasing and continuous in \(\phantom {\dot {i}\!}\bar {q}_{0}\); so, γ N is (weakly) decreasing in ϕ. (ii): According to the expression for γ D (see Eq. 6), \(\phantom {\dot {i}\!}\gamma _{D} = [1 / q_{L} (1 - q_{L}) + 1 / \bar {q}_{0}]^{-1} \le \bar {q}_{0} (1 - \bar {q}_{0})\) iff \(\phantom {\dot {i}\!}q_{L} (1 - q_{L}) \le 1 - \bar {q}_{0} \iff (q_{H} - q_{L}) \phi \le (1 - q_{L})^{2}\). If q H ≤1−q L (1−q L ), we have (q H q L )ϕ<q H q L ≤(1−q L )2. Hence γ D is always equal to \(\phantom {\dot {i}\!}[1 / q_{L} (1 - q_{L}) + 1 / \bar {q}_{0}]^{-1}\), which is increasing in \(\phantom {\dot {i}\!}\bar {q}_{0}\). Thus, γ D is always increasing in \(\phantom {\dot {i}\!}\bar {q}_{0}\); so, it is always increasing in ϕ. Otherwise (i.e., q H >1−q L (1−q L )), for ϕ≤(1−q L )2/(q H q L ), \(\phantom {\dot {i}\!}\gamma _{D} = [1 / q_{L} (1 - q_{L}) + 1 / \bar {q}_{0}]^{-1}\) and so it is increasing in ϕ; for ϕ ≥ (1−q L )2/(q H q L ), \(\phantom {\dot {i}\!}\gamma _{D} = \bar {q}_{0} (1 - \bar {q}_{0})\). Note that ϕ ≥ (1−q L )2/(q H q L ) is equivalent to \(\phantom {\dot {i}\!}q_{L} (1 - q_{L}) \ge 1 - \bar {q}_{0}\), which implies that \(\phantom {\dot {i}\!}\bar {q}_{0} \ge 1 - q_{L} (1 - q_{L}) \ge 1 - 0.25 > 0.5\). Hence, for ϕ ≥ (1−q L )2/(q H q L ), \(\phantom {\dot {i}\!}\gamma _{D} = \bar {q}_{0} (1 - \bar {q}_{0})\) is decreasing in \(\phantom {\dot {i}\!}\bar {q}_{0}\) and thus decreasing in ϕ.

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Guo, L., Wu, Y. Consumer deliberation and quality signaling. Quant Mark Econ 14, 233–269 (2016). https://doi.org/10.1007/s11129-016-9174-5

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Keywords

  • Deliberation
  • Signaling
  • Dissipative advertising

JEL Classification

  • D82
  • D83
  • L15
  • M3