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 highquality firm induces consumer deliberation by setting a high price whereas the lowquality firm prevents deliberation by charging a low price. Compared to the case of complete information, the price of the highquality firm can be distorted upward to facilitate consumer deliberation, or distorted downward to avoid the lowquality firm’s imitation. In an extension we show that dissipative advertising can facilitate quality signaling. The highquality firm can utilize advertising spending to avert imitation from the lowquality 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.
In few exceptions the term “search” is used where purchase is possible without search (e.g., Kuksov and VillasBoas 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.
See also Kuksov and Lin (2010) on firms’ optimal provision of information in competitive markets.
 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.
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 redefine V _{ H } and V _{ L } as the expected product valuations integrating over all residual uncertainty that cannot be resolved by deliberation.
 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.
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 outofequilibrium belief for such p.
 7.
 8.
Note that \(\phantom {\dot {i}\!}\kappa _{L} < \hat {\kappa } < q_{L} (1  q_{L})\).
 9.
 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.
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.
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.
The reason why p _{ H }>p is the following. Since c ≥ κ _{ 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 outofequilibrium price and so p≠p _{ H }) or p _{ H }>1−c/q _{ L } ≥ p.
 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.
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.
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.
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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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 }. TypeL 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 typeH 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, typeL firm should always behave in the same way as she does under complete information. According to Lemma 1, we must have c ≥ κ _{ L }. We characterize the deliberationon 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— c ≥ q _{ 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 c≤q _{ H }−q _{ L }. (B) The equilibrium prices are unique.

(A)
[“Only if” part]: Suppose c>q _{ H }−q _{ L } and the equilibrium exists. TypeH firm wants to deviate to p _{ L } because π _{ H } = q _{ H }−c<q _{ L } = p _{ L }, a contradiction. [“If” part]: TypeH firm would not deviate to p _{ L } because π _{ H } ≥ p _{ L }. TypeL 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 outofequilibrium price. For p<p _{ L }, typeL firm has no incentive to deviate; neither does typeH 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, typeL firm would not deviate because π _{ L } = p _{ L }>p q _{ L }; typeH 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 outofequilibrium belief μ(q _{ L }p)=1 because the consumer always opts out. The separating equilibrium with such an outofequilibrium belief survives the intuitive criterion because typeL firm wants to deviate under μ(q _{ H }p)=1.

(B)
We prove this part by contradiction. Suppose the equilibrium price of typeH 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 \(\phantom {\dot {i}\!}\pi _{L} = p_{L} > p_{H}^{\prime \prime } q_{L}\), typeL 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, typeH 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 }]^{−1}≤c<q _{ L }(1−q _{ L })
This is very similar to that of Lemma 4. The main difference is that here typeL 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 typeH 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 c ≥ κ _{ L }. (B) The equilibrium prices are unique.

(A)
[“Only if” part]: TypeL firm’s equilibrium strategy requires that c ≥ κ _{ 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 })). TypeH 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 }. TypeL firm is indifferent between p _{ L } and p _{ H }. We now examine the outofequilibrium 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 }}], typeH firm is not willing to deviate under the belief μ(q _{ L }p)=1 because p _{ H }>p and the consumer deliberates;^{Footnote 13} typeL 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 outofequilibrium 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 typeL firm always prefers a deviation under a belief μ(q _{ H }p)=1.

(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 typeH firm. Suppose it is. If \(\phantom {\dot {i}\!}p_{H}^{\prime } \in (c / q_{L} (1  q_{L}), 1  c / q_{H}]\), then typeL 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, typeL 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, typeH firm would deviate, a contradiction.
Proof of Proposition 1
Define
It can be verified that
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 }≤c≤κ _{ 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) TypeH 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, typeH firm in a separating equilibrium follows her completeinformation optimal strategy: p _{ H } = P _{ H }=1−c/q _{ H }. (iii) TypeH 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
TypeL firm adopts her completeinformation optimal strategy. We discuss the existence conditions in four cases separately. The separating equilibria refer to the nodeliberation 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 typeH firm. Since both types of firms are indifferent between (p _{ H },a _{ H }) and (p _{ L },a _{ L }), it is sufficient to examine the outofequilibrium decisions. For any (p,a) such that p−a≤q _{ L }, no firm prefers a deviation under any belief. For any (p,a) such that p−a>q _{ L } and p>q _{ H }, the consumer opts out. For any (p,a) such that p−a>q _{ L } and p≤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 any a ^{′}≠p _{ H }−p _{ L } cannot be the equilibrium advertising spending of typeH 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 typeH firm iff c ≥ q _{ 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 }, typeL firm does not want to deviate to p because p q _{ L }<q _{ L }. However, since q _{ H }−c>q _{ L }, typeH 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 outofequilibrium decisions. For any (p,a) such that p−a≤q _{ L }, no firm prefers to deviate. For any (p,a) such that p−a>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 }−c≤q _{ L }; so, no deviation takes place. For any (p,a) such that p−a>q _{ L } and p≤c/(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 typeH 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 typeH 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 typeH 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 outofequilibrium decisions. For any (p,a) such that p−a≤c/(1−q _{ L }), no firm prefers a deviation. For any (p,a) such that p−a>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 p−a>c/(1−q _{ L }) and p≤q _{ 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 typeH 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 typeH 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 typeH 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 outofequilibrium 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, typeL 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, typeH 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 outofequilibrium decisions. For any (p,a) such that p−a≤c/(1−q _{ L }), neither type of firm prefers a deviation. For any (p,a) such that p−a>c/(1−q _{ L }) and p>1−c/q _{ H }, the consumer opts out. For any (p,a) such that p−a>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 p−a>c/(1−q _{ L }) and p≤c/(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 outofequilibrium belief survives the intuitive criterion. It can be verified that a ^{′}≠p _{ H }−p _{ L } cannot be typeH 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
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 }]^{−1} ≥ q _{ 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 typeL 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.

(A)
TypeL firm is indifferent between (p _{ H },a _{ H }) and (p _{ L },a _{ L }). TypeH 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 outofequilibrium decisions. For any (p,a) such that p≤c/(1−q _{ L }), no firm is willing to deviate because even typeL 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 }−a≤q _{ L }−c, then typeL firm cannot make a profitable deviation; typeH 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 typeL 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 })<p≤c/(1−q _{ H }), if p−a≤q _{ L }−c, then no firm has any incentive to deviate. Otherwise (i.e., p−a>q _{ L }−c), the consumer deliberates or opts out under a belief μ(q _{ L }p,a)=1. TypeL firm can never make a profitable deviation under that belief because she has obtained her completeinformation maximal profit. TypeH firm does not want to deviate either under that belief because she is unwilling to deviate to the highest deliberationinducing 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 typeL firm can make a profitable deviation (\(\phantom {\dot {i}\!}\because p  a > q_{L}  c\)).

(B)
We prove this part by contradiction. Suppose the equilibrium decisions of typeH 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 typeL 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 typeL 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, typeH 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): typeL 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, typeH 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 deliberationonhigh 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 typeL 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 c≤q _{ H }−q _{ L }. (B) The equilibrium decisions of typeH firm are unique. (A): If c>q _{ H }−q _{ L }, then the equilibrium does not exist (see the proof of Lemma 4). Now suppose c≤q _{ H }−q _{ L }. Neither type of firm would deviate to the other type of firm’s equilibrium decisions because c≤q _{ H }−q _{ L }. We next examine the outofequilibrium decisions. For any (p,a) such that p≤q _{ 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 }<p≤c/(1−q _{ H }) and p−a≤q _{ L }, typeL firm would not deviate; so, typeH 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 }<p≤c/(1−q _{ H }) and p−a>q _{ L }, the consumer opts out under μ(q _{ L }p,a)=1. Such an equilibrium survives the intuitive criterion because typeL 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 typeH 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 outofequilibrium decisions (1−c/q _{ H },0), the consumer deliberates or opts out under any belief. Hence typeL firm would never deviate. Yet under the only reasonable belief μ(q _{ H }p _{ H },a _{ H })=1 in light of the intuitive criterion, typeH 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 typeL firm are p _{ L } = c/(1−q _{ L }) and a _{ L }=0. Since the consumer does not deliberate on p _{ L }, c ≥ κ _{ 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 typeH firm are unique.

(A)
If c>[1/q _{ H }(1−q _{ L })+1/q _{ H }]^{−1}, typeH firm can make a profitable deviation to (p _{ L },a _{ L }) because p _{ H } q _{ H }−a _{ H } = q _{ H }−c−a _{ 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. TypeL firm does not want to deviate to (p _{ H },a _{ H }) because p _{ H } q _{ L }≤p _{ L }; typeH 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 }. TypeL 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 }; typeH 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 outofequilibrium decisions. For any (p,a) such that p≤c/(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 })<p≤c/(1−q _{ H }), if p−a≤c/(1−q _{ L }), then neither type of firm prefers a deviation. Otherwise (i.e., p−a>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 }), typeL firm does not want to deviate because c ≥ κ _{ L }; typeH 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 typeL 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 }−a≤c/(1−q _{ L }), then typeL firm does not want to deviate under any belief; typeH 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 typeL firm can make a profitable deviation (because p q _{ L }−a>c/(1−q _{ L })).

(B)
We prove this part by contradiction. Suppose the equilibrium decisions of typeH 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 typeL 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 outofequilibrium 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, typeL 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, typeH 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): TypeL firm is not willing to deviate to any outofequilibrium 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, typeH 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 deliberationonhigh type of separating equilibrium exists iff κ _{ L }≤c≤κ _{ 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 c ≥ q _{ 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, typeH 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, typeH firm’s profit is c q _{ H }/q _{ L }(1−q _{ L }). Comparing the profits, we have
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 typeH 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 poolingnondeliberation 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 poolingnondeliberation 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 poolingnondeliberation 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 poolingnondeliberation 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 poolingnondeliberation 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 poolingnondeliberation 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 poolingdeliberation 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 poolingnondeliberation 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 poolingdeliberation 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 typeL firm always prefers a deviation under an outofequilibrium 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) Poolingnondeliberation: 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 outofequilibrium 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) Poolingdeliberation: 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 typeL 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 ^{∗}, typeL firm has no incentive to deviate to p under any belief. Yet under the only reasonable belief μ(q _{ H }p)=1, typeH 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 }−c≤p ^{∗}. 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 })], typeL 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 typeH 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 outofequilibrium 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 typeL firm can make a profitable deviation under μ(q _{ H }p)=1.
Note that p ^{∗} ≥ (1−c/q _{ L })q _{ H } is equivalent to c ≥ q _{ 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 c ≥ q _{ 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 \(\phantom {\dot {i}\!}q_{L} (1  \bar {q}_{0} / q_{H}) \ge \bar {q}_{0} (1  \bar {q}_{0})\):
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) Poolingnondeliberation: 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) Poolingdeliberation: 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) Poolingnondeliberation: 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 } ⇔ c ≥ q _{ 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) Poolingdeliberation: 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 p≤c/(1−q _{ L }), no firm strictly prefers a deviation under any belief because p≤p ^{∗} 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 typeL firm does not want to deviate under any belief (iff p<p ^{∗} q _{ L }), then even under μ(q _{ H }p)=1 typeH 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 typeL 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) Poolingnondeliberation: 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 typeH 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, typeH 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 })], typeL 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 }], typeH firm would not deviate even under μ(q _{ H }p)=1. Condition (b) implies that, even when the consumer deliberates, if typeL 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, typeH 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 c ≥ p ^{∗}(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 quasiconvex 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) Poolingdeliberation: 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 p≤c/(1−q _{ L }), no firm strictly prefers a deviation under any belief because p≤p ^{∗} 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
To summarize the result, we will show that (i) there exist poolingnondeliberation equilibria iff c ≥ γ _{ N }, and (ii) there exist poolingdeliberation equilibria iff c≤γ _{ D }. We first prove part (ii). There exists a poolingdeliberation 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 poolingdeliberation 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 poolingnondeliberation 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 poolingnondeliberation 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 c ≥ q _{ 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 poolingnondeliberation 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 c ≥ q _{ 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 poolingnondeliberation 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/s1112901691745
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Keywords
 Deliberation
 Signaling
 Dissipative advertising
JEL Classification
 D82
 D83
 L15
 M3