The effect of exogenous product familiarity on endogenous consumer search

Abstract

When a consumer is familiar with one product but not its competitor, she is faced with a decision: either buy what she knows, or engage in search to learn more. When search is costly, competing firms may attempt to encourage or discourage search by adjusting prices. In this paper we consider how competitive dynamics between two quality differentiated firms are affected if one product enjoys a familiarity advantage. Familiarity is defined as a consumer’s ex-ante knowledge of fit for a particular product. An increase in the level of familiarity for one product allows a firm to charge higher prices since there are more consumers with information on that product relative to the competition. We call this the direct effect of familiarity. However, an increase in familiarity also has an indirect effect, since it gives the rival firm a stronger incentive to decrease price in order to encourage searching, in turn increasing overall competition. The effect of familiarity on profits depends on the magnitudes of these effects, and it is moderated by the level of quality differentiation between products. For very high or very low levels of differentiation, the results are relatively straightforward. However, when the level of differentiation is moderate, the results are more nuanced, with the higher-quality firm realizing higher profits from more familiarity, even if it must lower prices due to the indirect effect. We also find that, contrary to conventional wisdom, overall competition may be higher when firms are more quality differentiated. This is driven by the fact that higher quality differences bolster the indirect effect, with a lower quality firm providing deeper price cuts to counter increased familiarity of a high quality rival. We conclude by examining how changes in the cost of searching impact equilibrium outcomes.

Appendix

1.1 Derivation of demand function given in Eq. 9 of the paper

We provide the technical details of the derivation of the demand function following the threshold for search, $\widehat {v}$. The analysis requires us to also outline which consumers will engage in costly search.

The demand that firm 1 faces from the segment of consumers who are ex-ante familiar with product 1 and not product 2 depends on $\widehat {v}$. Given $\widehat {v}$, some of these consumers will buy product 1 without search, some may buy product 2 without search, and some will engage in search get fully informed before making a purchase decision. First, we remark that since $\overline {v}>p_{1},$ there cannot exist a segment of consumers who are ex ante familiar with product 1 and buy product 2 without engaging in costly search. For this segment of consumers to exist, it must be the case that even with the lowest possible match realization of product 2, post purchase, the consumer is better off buying product 2 (without search) rather than product 1. However, the lowest possible valuation of product 2 is 0 and the lowest possible surplus that consumer may get from buying product 2 is − p₂, which is strictly less than $\overline {v}-$p₁, i.e. the lowest possible surplus the consumers could get from buying product 1. In other words, if a consumer buys product 2 without search and post purchase realizes that its match is arbitrarily close to 0, the initial purchase decision of buying product 2 without search becomes sequentially irrational.

Next we consider consumers who buy product 1 without engaging in search. Given the threshold, firm 1 expects that consumers in this segment who have a match v₁ such that $v_{1}+{\Delta } >\widehat {v}$ will buy product 1 without searching. Note that, since v₁ > 0, if $\widehat {v}-{\Delta } <0$ then all these consumers will buy product 1 without engaging in search.

The rest of the consumers will pay the search cost c, engage in search, and become informed of their match for product 2. Note that, if $\widehat {v} -{\Delta } >1$, then all consumers will engage in search. The threshold $\underline {v_{1}}$ above which no consumers engage in search is given by

$$ \underline{v_{1}}\equiv \max [0,\min \{1,\widehat{v}-{\Delta} \}] $$

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The above observations imply that all consumers with $v_{1}>\underline {v_{1}}$ or v₁ < −Δ will make a purchase without searching. On the other hand, all consumers with −Δ < v₁ < HCode $\underline {v_{1}}$ will engage in search to become fully informed before choosing a product. Firms 1 and 2 compete for this segment of consumers. There are no other segments of consumers to consider given that, from the above expression, $\underline { v_{1}}>-{\Delta }$.

Firm 1 expects to derive demand from both segments of consumers – those who are familiar with product 1 but not product 2 (a segment of size κ), and those who are familiar with product 2 but not product 1 (size 1 − κ ). As mentioned above, all consumers in the first segment with match value $v_{1}>\underline {v_{1}}$ will never search for information on product 2, and thus this entire segment purchases product 1. Since v₁ is drawn from U[0, 1], the proportion of consumers who buy product 1 without engaging in search is $(1-\underline {v_{1}})$. The rest of the consumers, with −Δ < v₁ < HCode $\underline {v_{1}}$, do engage in search and thus become fully informed. Firm 1 competes with firm 2 for these consumers and gets the demand from those whose match for product 1 is such that $v_{1}-p_{1}+\overline {v}\geq v_{2}-p_{2}^{\ast }$, or v₂ ≤ v₁ + Δ for all −Δ < v₁ < HCode $\underline {v_{1}}$. In other words, firm 1 gets F(v₁ + Δ) proportion of consumers for all v₁ < HCode $\underline {v_{1}}$ only if v₁ + Δ > 0 where the F(.) is the cumulative density function of v₂. Finally, as mentioned above, if v₁ < −Δ, firm 1 does not derive any demand from this segment.

Summarizing the above information, therefore, firm 1 expects to get a proportion ${\int \limits }_{{\max \limits } [-{\Delta } ,0]}^{\underline {v_{1}}}F(v_{1}+{\Delta } )dG(v_{1})$ of these consumers, where G(.) is the cumulative density function of v₁ (the domain of integration reflects the fact that, if v₁ < −Δ, then firm 1 gets no demand from this segment of consumers). In other words, the total demand, D_1/1, from this segment of κ consumers is the sum of the above two parts and is given by

$$ D_{1/1}=(1-\underline{v_{1}})+{\int}_{\max [-{\Delta} ,0]}^{\underline{v_{1}} }F(v_{1}+{\Delta} )dG(v_{1}) $$

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Finally, given the assumption of complete market coverage Δ > 0 and that v_i (i = 1, 2) are drawn from U[0, 1], the demand that firm 1 expects to derive from the segment of consumers who are ex-ante familiar with its product and is not familiar with product 2 therefore can be given by

$$ D_{1/1}=(1-\underline{v_{1}})+{\int}_{0}^{\underline{v_{1}}}(v_{1}+{\Delta})dG(v_{1}) $$

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Next consider the set of consumers who are ex-ante familiar with product 2 and not product 1. In the following we outline the demand that firm 1 expects to derive from this segment of consumers given by D_1/2. The segment is (1 − κ) of the market. As in the case of D_1/1, given $\widehat {v},$ some consumers may buy product 1 without search, some may buy product 2 without search, and some will engage in search become fully informed and make purchase decision. First, consider a consumer who buys product 1 without engaging in search. Those consumers who realized v₂ sufficiently low such that $\inf (v_{1})-p_{1}+\overline {v}>$$v_{2}-p_{2}^{\ast }$, or for whom v₂ < Δ holds, are better off buying product 1 without engaging in search since the worst-case utility from the (higher quality) product 1 is higher than their known utility from product 2.

Next consider the consumers who buy product 2 without engaging in search. Those consumers whose realized v₂ is such that $v_{2}>\widehat {v} +{\Delta }$ holds will be better off buying product 2 without engaging in search provided $\widehat {v}+{\Delta } \in (0,1)$. Firm 1 does not expect to derive any demand from these consumers. The remaining consumers will engage in search and become fully informed, and the two firms will compete for them. Firm 1 therefore expects demand from those consumers who, after search, find $v_{1}-p_{1}+\overline {v}\geq v_{2}-p_{2}^{\ast }$ or v₁ > v₂ −Δ for all v₂ < $\underline {v_{2}}$ , where $\underline {v_{2}}$ is given by

$$ \underline{v_{2}}\equiv \max [0,\min \{1,\widehat{v}+{\Delta} \}] $$

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In other words, the expected demand from this group is (1 − G(v₂ −Δ)) given that v₁ is drawn from U[0,1].

Summarizing, the demand that firm 1 expects to face from this segment of consumers is

$$ D_{1/2}={\int}_{\max [{\Delta} ,0]}^{\underline{v_{2}}}(1-G(v_{2}-{\Delta} ))dF(v_{2})+\max [{\Delta} ,0] $$

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Finally, again given the assumption of complete market coverage Δ > 0. Therefore the demand that firm 1 expects to derive from the segment of consumers who are ex-ante familiar with product 2 and is not familiar with its product is

$$ D_{1/2}={\int}_{\Delta }^{\underline{v_{2}}}(1-v_{2}+{\Delta} )dF(v_{2})+{\Delta} $$

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The above demands for each of the segments from which firm 1 expects to derive demand are reported in the main text of the paper.

1.2 Derivation of equilibrium

In order to derive the equilibrium above we first ought to outline the demand function as a function of Δ. Since the demand is also a function of $\underline {v_{i}}$i = 1, 2, there are multiple scenarios which can evolve given Δ. Specifically, four different cases can evolve: (i) $\widehat {v}-{\Delta } >0$ and $\widehat {v}+{\Delta } <1$ (ii) $\widehat {v} -{\Delta } <0$ and $\widehat {v}+{\Delta } <1$ (iii) $\widehat {v}-{\Delta } >0$ and $\widehat {v}+{\Delta } >1$ and (iv) $\widehat {v}-{\Delta } <0$ and $\widehat {v} +{\Delta } >1$. These cases evolve primarily because the functional firm of v_i changes depending on which of the four conditions hold. In other words, v_i introduces discontinuities in the demand function. Given the above information, solving for the above integrals we can write the expected demand function by firm as follows:

$$ \begin{array}{@{}rcl@{}} D_{1}({\Delta} )&=&\left\{ \begin{array}{l} \left( \kappa (\widehat{v}-1)^{2}+\widehat{v}-\frac{\widehat{v}^{2}}{2} +{\Delta} -\kappa \frac{{\Delta}^{2}}{2}\text{ }\right) \text{ if }0<{\Delta} <\min [\widehat{v},1-\widehat{v}] \\ \left( \kappa +(1-\kappa )(\widehat{v}-\frac{\widehat{v}^{2}}{2}+{\Delta} )\right) \text{ if }{\Delta} >\widehat{v}\text{ and }{\Delta} <1-\widehat{v} \\ \left( \frac{1+\kappa (\widehat{v}-1)^{2}+(2-{\Delta} ){\Delta} }{2}\right) \text{ if }{\Delta} <\widehat{v}\text{ and }{\Delta} >1-\widehat{v} \\ \kappa +\frac{(\kappa -1)(({\Delta} -2){\Delta} -1)}{2}\text{ if }{\Delta} > \widehat{v}\text{ and }{\Delta} >1-\widehat{v}\text{ } \end{array} \right.\\ D_{2}({\Delta} )& =&1-D_{1}({\Delta} ) \end{array} $$

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Differentiating the above demand function with respect to Δ gives $D_{1}^{^{\prime }}({\Delta } )$ which again needs to be evaluated for all possible scenarios. Using the fact that in equilibrium Δ = Δ^∗ yields $D_{i}({\Delta }^{\ast })$ and $D_{i}^{^{\prime }}({\Delta }^{\ast })$ for i = 1, 2. There will be four possible values of D_i(Δ^∗) and $D_{i}^{^{\prime }}({\Delta }^{\ast })$ depending on which of the four conditions mentioned above holds. To derive the equilibrium prices we use Eq. 12 above which is $p_{1}^{\ast }({\Delta }^{\ast })=\frac { D_{1}({\Delta }^{\ast })}{D_{1}^{^{\prime }}({\Delta }^{\ast })}$; $p_{2}^{\ast }({\Delta }^{\ast })=-\frac {D_{2}({\Delta }^{\ast })}{ D_{2}^{^{\prime }}({\Delta }^{\ast })}$. Notice since there are four possible $D_{i}({\Delta }^{\ast })$ and $D_{i}^{^{\prime }}({\Delta }^{\ast })$ we will have four candidate prices depending on the value of Δ^∗.

Next, we solve for the equilibrium Δ^∗ using the equality ${\Delta }^{\ast }=p_{2}^{\ast }({\Delta }^{\ast })-p_{1}^{\ast }({\Delta }^{\ast })+\overline {v}$ in equation (13). There will be four sets of Δ^∗ corresponding to four sets of prices. We check that in each of the cases the corresponding bound (given in conditions (i) -(iv) above) on Δ^∗ holds. For instance, Δ^∗ corresponding to the first case ought to satisfy ${\Delta }^{\ast }<1-\widehat {v}$ (which automatically means that Δ^∗ < $\widehat {v})$. Only three of the candidates satisfy the boundary conditions outlined above, and they correspond to cases (i), (iii) and (iv). The conditions (i) stated in terms of $\overline {v}$ is $Max[0,(2\kappa -1)(1-\widehat {v})^{2}]<\overline {v}<\frac {2-\widehat {v}- \widehat {v}^{2}}{1-\kappa -\kappa \widehat {v}}$. This level of $\overline {v}$ is termed as low $\overline {v}$. Condition (iii) above stated in terms of $\overline {v}$ turns out to be $\frac {(1+\kappa (1-\widehat {v})+2\widehat {v} )(1-\widehat {v})}{\widehat {v}}$$<\overline {v}<\frac {\kappa (\widehat {v} -1)^{2}-\widehat {v}(3-2\widehat {v})}{1-\widehat {v}}$. This level of $\overline {v}$ is termed as intermediate $\overline {v}$. Finally, condition (iv) ${\Delta }^{\ast }>\widehat {v}$ holds for $\overline {v}>\frac {(3-2 \widehat {v})\widehat {v}+\kappa (1-3\widehat {v}+2\widehat {v}^{2})}{(1-\kappa )(1-\widehat {v})}$. This level of $\overline {v}$ is termed as high $\overline {v}$. The Δ^∗ corresponding to low, medium and high $\overline {v}$ are given in Table 1 in the paper.

Next, to ensure that the above candidate equilibrium actually exists, we need to make sure that the equilibrium in each of the regions is deviation proof. In the following we simply outline the deviation check for the case when $0<{\Delta }^{\ast }<1-\widehat {v}$ which as mentioned above corresponds to $Max[0,(2\kappa -1)(1-\widehat {v})^{2}]<\overline {v}<\frac {2-\widehat {v}- \widehat {v}^{2}}{1-\kappa -\kappa \widehat {v}}$. The deviation check for equilibrium in any of the other cases can be done in a similar fashion. While testing for unilateral deviations off the region $0<{\Delta }^{\ast }<1- \widehat {v}$, each of the firms can change its price unilaterally so that the resulting Δ, say Δ^d, is an element of one of the other regions mentioned in the demand schedule above, or it can deviate to corner the whole market. In particular, firm 1 can unilaterally change prices post deviation if one of the following holds (a) ${\Delta }^{d}\in (1- \widehat {v},\widehat {v})$; (b) ${\Delta }^{d}\in (\widehat {v},1)$; (c) ${\Delta }^{d}\in (\widehat {v}-1,0)$; (d) ${\Delta }^{d}\in (-\widehat {v},\widehat {v} -1);$ (e) ${\Delta }^{d}\in (-1,-\widehat {v})$ and (f) Δ^d > 1, in which case it corners the whole market, foreclosing the market for firm 2. ${\Delta }^{d}=p_{2}^{\ast }-{p_{1}^{d}}+\overline {v}$ and ${p_{1}^{d}}$ is firm 1’s deviating price.

To prove non-deviation, first notice that to have a deviation in the interior of the other regions it must be that ${\Delta }^{d}=p_{2}^{\ast }-{p_{1}^{d}}+\overline {v}$ or $\overline {v}$$={\Delta }^{d}-p_{2}^{\ast }+{p_{1}^{d}}$ after deviation for all $Max[0,(2\kappa -1)(1-\widehat {v})^{2}]< \overline {v}<\frac {2-\widehat {v}-\widehat {v}^{2}}{1-\kappa -\kappa \widehat {v }}$. Now, consider the deviation to (a) above by firm 1. In such a case, the deviating price by firm 1 will be of the form ${p_{1}^{d}}({\Delta }^{d})=\frac { 1+\kappa (\widehat {v}-1)^{2}-({\Delta }^{d}-2){\Delta }^{d}}{2({\Delta }^{d}-1)} \dot {.}$ The above expression for ${p_{1}^{d}}({\Delta }^{d})$ follows from ${p_{1}^{d}}({\Delta }^{d})=$$\frac {D_{1}({\Delta }^{d})}{D_{1}^{^{\prime }}({\Delta }^{d})}$ for ${\Delta }^{d}\in (1-\widehat {v},\widehat {v})$. $\frac { D_{1}({\Delta }^{d})}{D_{1}^{^{\prime }}({\Delta }^{d})}$ is taken from the demand schedule outlined above. Substituting ${p_{1}^{d}}({\Delta }^{d})$ in ${\Delta }^{d}-p_{2}^{\ast }+{p_{1}^{d}}$ and then differentiating the resulting expression with respect to Δ^d shows that ${\Delta }^{d}-p_{2}^{\ast }+{p_{1}^{d}}$ is a strictly increasing function of Δ^d. In other words, ${\Delta }^{d}-p_{2}^{\ast }+{p_{1}^{d}}$ strictly increases over all ${\Delta }^{d}\in (1-\widehat {v},\widehat {v})$. Notice in the deviating region $\overline {v}$ the appropriate condition is $\frac {(1+\kappa (1-\widehat {v})+2 \widehat {v})(1-\widehat {v})}{\widehat {v}}$$<\overline {v}<\frac {\kappa (\widehat {v}-1)^{2}-\widehat {v}(3-2\widehat {v})}{1-\widehat {v}}$. Define $\overline {v}^{d}$ as an element in $\frac {(1+\kappa (1-\widehat {v})+2 \widehat {v})(1-\widehat {v})}{\widehat {v}}$$<\overline {v}<\frac {\kappa (\widehat {v}-1)^{2}-\widehat {v}(3-2\widehat {v})}{1-\widehat {v}}$ and $\overline {v}^{\ast }$ as a element in $Max[0,(2\kappa -1)(1-\widehat {v} )^{2}]<\overline {v}<\frac {2-\widehat {v}-\widehat {v}^{2}}{1-\kappa -\kappa \widehat {v}}$ which is the region for the equilibrium under consideration. Since $\overline {v}^{d}>$$\overline {v}^{\ast }$, it automatically follows that after deviation $\overline {v}^{d}$$>{\Delta }^{d}-p_{2}^{\ast }+{p_{1}^{d}}$ for all $\overline {v}^{d}\in \left (\frac {(1+\kappa (1- \widehat {v})+2\widehat {v})(1-\widehat {v})}{\widehat {v}},\frac {\kappa (\widehat {v}-1)^{2}-\widehat {v}(3-2\widehat {v})}{1-\widehat {v}}\right )$. This also means that there cannot exist a feasible deviation in the interior of $(1-\widehat {v},\widehat {v})$ including the end point $\widehat {v}$ because at the end point ${\Delta }^{d}-p_{2}^{\ast }+{p_{1}^{d}}$ = $\overline {v }^{\ast }$ for $\overline {v}^{\ast }\in \left (Max[0,(2\kappa -1)(1-\widehat { v})^{2}],\frac {2-\widehat {v}-\widehat {v}^{2}}{1-\kappa -\kappa \widehat {v}} \right )$. This is because for possible ${\Delta }^{d}\in (1-\widehat {v}, \widehat {v}),$ firm 1 has an incentive to monotonically reduce its deviating price, in effect invalidating the fact that any pure deviation to this region is feasible. Following precisely the same approach, one can show that neither firm has a feasible deviation to any of the other regions. In each case, one can establish that with any deviation, ${\Delta }^{d}-p_{2}^{\ast }+{p_{1}^{d}}$ or ${\Delta }^{d}-{p_{2}^{d}}+p_{1}^{\ast }$ is strictly monotonic. This also means that the deviating firm will either monotonically increase or decrease its equilibrium price. A similar approach establishes the non-deviation to other regions outlined as well. This establishes that the equilibrium is deviation proof.

Having outlined the equilibrium result, we now provide the proofs of Propositions.

1.3 Proof of Proposition 1

To derive the equilibrium prices, we use Eq. 12, which is $p_{1}^{\ast }({\Delta }^{\ast })=\frac {D_{1}({\Delta }^{\ast })}{D_{1}^{^{\prime }}({\Delta }^{\ast })}$ ; HCode $p_{2}^{\ast }({\Delta }^{\ast })=-\frac { D_{2}({\Delta }^{\ast })}{D_{2}^{^{\prime }}({\Delta }^{\ast })}$. Such implicit prices are given in Table 3 below. Notice that $p_{i}^{\ast }({\Delta }^{\ast })$ are stated for permissible Δ^∗ which corresponds to low, intermediate and high $\overline {v}$.

Table 3 Equilibrium Prices Implicitly defined as a function of Δ^∗

Substituting Δ^∗ which is given in Table 1 of the paper, we get equilibrium prices $p_{i}^{\ast }$ as a function of Δ^∗. Below we outline the price expressions

$$ p_{1}^{\ast }=\left\{ \begin{array}{l} \frac{\left( \begin{array}{c} (-3+\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v}+\kappa \overline{v}^{2})}- \\ \kappa (4+8\kappa (\widehat{v}-1)^{2}-4(\widehat{v}-2)\widehat{v}-\kappa \overline{v}^{2}+ \\ \overline{v}(2+\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v} +\kappa \overline{v}^{2})}))) \end{array} \right) }{\left( 4\kappa (-1+\kappa \overline{v}-\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v}+\kappa \overline{v}^{2})}))\right) } \text{ for low }\overline{v} \\ \frac{\left( \begin{array}{c} (11+4\kappa (\widehat{v}-1)^{2}-\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}} \\ +\overline{v}(2-\overline{v}+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v }-2)\overline{v}})) \end{array} \right) }{(4(1-\overline{v}+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v} -2)\overline{v}}))}\text{ for Intermediate }\overline{v} \\ \frac{1}{8}\left( -5+5\overline{v}-\frac{3\sqrt{(\kappa -1)(-9+\kappa (\overline{v}-1)^{2}+(\overline{v}-2)\overline{v})}}{(\kappa -1)}\right) \text{ for High }\overline{v} \end{array} \right. $$

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$$ p_{2}^{\ast }=\left\{ \begin{array}{l} \frac{\left( \begin{array}{c} (-3-\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v}+\kappa \overline{v}^{2})} \\ -\kappa (-12+8\kappa (\widehat{v}-1)^{2}-+4(\widehat{v}-2)\widehat{v}-\kappa \overline{v}^{2}+ \\ \overline{v}(2+\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v} +\kappa \overline{v}^{2})}))) \end{array} \right) }{\left( 4\kappa (-1+\kappa \overline{v}-\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v}+\kappa \overline{v}^{2})}))\right) } \text{ for low }\overline{v} \\ \\ \frac{2(\kappa (\widehat{v}-1)^{2}-\frac{1}{16}\left( 1-\overline{v}+\sqrt{ -9+8\kappa (\overline{v}-1)^{2}+(\overline{v}-2)\overline{v}})\right)^{2}}{ \overline{v}-1+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{ v}}}\text{ for Intermediate }\overline{v} \\ \\ \frac{1}{8}-\frac{\overline{v}}{8}-\frac{\sqrt{(\kappa -1)(9+\kappa (\overline{v}-1)^{2}+(\overline{v}-2)\overline{v)}}}{8(\kappa -1)}\text{ for High }\overline{v} \end{array} \right. $$

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Differentiating the above expressions with respect to κ yields $\frac {dp_{i}^{\ast }}{d\kappa }$. The expressions are quite cumbersome to report for case (i). For brevity we do not report the corresponding expressions on the comparative statics. First, consider the case when $Max[0,(2\kappa -1)(1-\widehat {v})^{2}]<\overline {v}<\frac {2-\widehat {v}- \widehat {v}^{2}}{1-\kappa -\kappa \widehat {v}}$, one can evaluate that $\frac {dp_{1}^{\ast }}{d\kappa }>0$. Moreover, $\frac {dp_{2}^{\ast }}{d\kappa }<0$ for $\overline {v}$ arbitrarily close to the lower limit $Max[0,(2\kappa -1)(1-\widehat {v})^{2}]$ and $\frac {dp_{2}^{\ast }}{d\kappa }>0$ for $\overline {v}\rightarrow \frac {2-\widehat {v}-\widehat {v}^{2}}{1-\kappa -\kappa \widehat {v}}$. Since $p_{2}^{\ast }$ is a continuous function of κ it must be that there is exist a $\overline {v}$ in the interior of the interval $\left [ Max[0,(2\kappa -1)(1-\widehat {v})^{2}],\frac {2-\widehat { v}-\widehat {v}^{2}}{1-\kappa -\kappa \widehat {v}}\right ]$ above which $p_{2}^{\ast }$ is an increasing function of κ and below which it is a decreasing function of κ. This establishes the first part of the proposition which corresponds to low $\overline {v}$.

For intermediate $\overline {v},$$\frac {(1+\kappa (1-\widehat {v})+2\widehat {v })(1-\widehat {v})}{\widehat {v}}<\overline {v}<\frac {\kappa (\widehat {v} -1)^{2}-\widehat {v}(3-2\widehat {v})}{1-\widehat {v}},$

$$ \frac{dp_{1}^{\ast }}{d\kappa }=\frac{\left( \begin{array}{c} (\widehat{v}-1)^{2}(-3+4\kappa (1-\widehat{v})^{2}+\widehat{v}^{2}+ \\ \sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}- \\ \overline{v}(2+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{ v}}) \end{array} \right) }{ \begin{array}{c} \sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}} \\ (1-\overline{v}+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2) \overline{v}})^{2} \end{array} }<0 $$

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$$ \frac{dp_{2}^{\ast }}{d\kappa }=-\frac{\left( \begin{array}{c} (\widehat{v}-1)^{2}(13+4\kappa (\widehat{v}-1)^{2}+\overline{v}^{2}+ \\ \sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}- \\ v(2+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}) \end{array} \right) }{ \begin{array}{c} \sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}} \\ (1-\overline{v}+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2) \overline{v}})^{2} \end{array} }<0 $$

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Finally, for high $\overline {v},$$\overline {v}>\frac {(3-2\widehat {v}) \widehat {v}+\kappa (1-3\widehat {v}+2\widehat {v}^{2})}{(1-\kappa )(1-\widehat { v})},$

$$ \frac{dp_{1}^{\ast }}{d\kappa }=\frac{3}{2(1-\kappa )(\sqrt{(\kappa -1)(-9+\kappa (\overline{v}-1)^{2}+(\overline{v}-2)\overline{v})}}>0 $$

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$$ \frac{dp_{2}^{\ast }}{d\kappa }=\frac{1}{2(1-\kappa )(\sqrt{(\kappa -1)(-9+\kappa (\overline{v}-1)^{2}+(\overline{v}-2)\overline{v})}}>0 $$

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This establishes all parts of Proposition 1.

1.4 Proof of Proposition 2

${\Pi }_{i}({\Delta }^{\ast })=p_{i}^{\ast }({\Delta }^{\ast })D_{i}({\Delta }^{\ast })$ for $i=1,2.D_{i}({\Delta }^{\ast })$ is the equilibrium demand that each firm faces given that in equilibrium Δ = Δ^∗. $D_{i}({\Delta }^{\ast })$ is given in the following table:

	$D_{1}({\Delta }^{\ast })$	$D_{2}({\Delta }^{\ast })$
Low $\overline {v}$	$\kappa (\widehat {v}-1)^{2}+\widehat {v}-\frac {\widehat {v }^{2}}{2}+{\Delta }^{\ast }-\frac {\kappa {\Delta }^{\ast ^{2}}}{2}$	$\frac {1}{2 }\left (2+(\widehat {v}-2\right ) \widehat {v}-2{\Delta }^{\ast }+\kappa$
		$(-2(\widehat {v}-1)^{2}+{\Delta }^{\ast 2}))$
Intermediate $\overline {v}$	$\frac {1}{2}\left (1+\kappa (\widehat {v} -1\right )^{2}-({\Delta }^{\ast }-2){\Delta }^{\ast })$	$\frac {1}{2}\left (-\kappa (\widehat {v}-1\right )^{2}+({\Delta }^{\ast }-1)^{2})$
High $\overline {v}$	$\kappa +\frac {1}{2}(\kappa -1)(-1+({\Delta }^{\ast }-2){\Delta }^{\ast })$	$\frac {1}{2}(1-\kappa )({\Delta }^{\ast }-1)^{2}$

Given $p_{i}^{\ast }({\Delta }^{\ast })$ in Table 3 and $D_{i}({\Delta }^{\ast })$ one can get ${\Pi }_{i}({\Delta }^{\ast })$. Substituting Δ^∗ given in Table 1 yields ${\Pi }_{i}^{\ast }$ as a function of market primitives (κ and c).

Next, we differentiate ${\Pi }_{i}^{\ast }$ with respect to κ to get $\frac {d{\Pi }_{i}^{\ast }}{d\kappa }$ which again has three parts corresponding to low, intermediate, and high values of $\overline {v}$.

For low $\overline {v}$, or when $Max[0,(2\kappa -1)(1-\widehat {v})^{2}]< \overline {v}<\frac {2-\widehat {v}-\widehat {v}^{2}}{1-\kappa -\kappa \widehat {v }}$, the comparative static expressions are again quite cumbersome. However, one can compute and show that $\frac {d{\Pi }_{1}^{\ast }}{d\kappa }>0$. Moreover $\frac {d{\Pi }_{2}^{\ast }}{d\kappa }|_{v\rightarrow Max[0,(2\kappa -1)(1-\widehat {v})^{2}]}<0$ and $\frac {d{\Pi }_{2}^{\ast }}{d\kappa } |_{v\rightarrow \frac {2-\widehat {v}-\widehat {v}^{2}}{1-\kappa -\kappa \widehat {v}}}$ > 0. Since ${\Pi }_{2}^{\ast }$ is a continuous function of κ it must be that there is exist a $\overline {v}$ in the interior of the interval $\left [ Max[0,(2\kappa -1)(1-\widehat {v})^{2}],\frac {2-\widehat { v}-\widehat {v}^{2}}{1-\kappa -\kappa \widehat {v}}\right ]$ above which ${\Pi }_{2}^{\ast }$ is an increasing function of κ and below which it is a decreasing function of κ.

For intermediate $\overline {v},$$\frac {(1+\kappa (1-\widehat {v})+2\widehat {v})(1-\widehat {v})}{\widehat {v}}<\overline {v}<\frac {\kappa (\widehat {v} -1)^{2}-\widehat {v}(3-2\widehat {v})}{1-\widehat {v}},$

$$ \frac{d{\Pi}_{1}^{\ast }}{d\kappa }=\frac{\left( \begin{array}{c} (\widehat{v}-1)^{2}(5+ 12\kappa (\widehat{v}-1)^{2}+\overline{v}^{2}+ \\ \sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}- \\ \overline{v}(2+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{ v}})(11+4\kappa (\widehat{v}-1)^{2} \\ -\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}+ \\ \overline{v}(2+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{ v}}) \end{array} \right) }{ \begin{array}{c} 16\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}} \\ (1-\overline{v}+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2) \overline{v}})^{2} \end{array} }>0 $$

(31)

$$ \frac{d{\Pi}_{2}^{\ast }}{d\kappa }=\frac{\left( \begin{array}{c} (\widehat{v}-1)^{2}(21 + 12\kappa (\widehat{v}-1)^{2}+\overline{v}^{2}+ \\ \sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}- \\ \overline{v}(2+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{ v}})(-5+4\kappa (\widehat{v}-1)^{2} \\ -\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}+ \\ \overline{v}(2+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{ v}}) \end{array} \right) }{ \begin{array}{c} 16\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}} \\ (1-\overline{v}+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2) \overline{v}})^{2} \end{array} }<0 $$

(32)

Finally, for high $\overline {v},$$\overline {v}>\frac {(3-2\widehat {v}) \widehat {v}+\kappa (1-3\widehat {v}+2\widehat {v}^{2})}{(1-\kappa )(1-\widehat { v})},$

$$ \frac{d{\Pi}_{1}^{\ast }}{d\kappa }=\frac{\left( \begin{array}{c} (-67+\kappa^{2}(\overline{v}-1)^{4}+\sqrt{(\kappa -1)(-9+\kappa (\overline{v }-1)^{2}+(\overline{v}-2)\overline{v})} \\ -\kappa (\overline{v}-1)^{2}(6+\sqrt{(\kappa -1)(-9+\kappa (\overline{v} -1)^{2}+(\overline{v}-2)\overline{v})} \\ +\overline{v}(-4+2\overline{v}-\sqrt{(\kappa -1)(-9+\kappa (\overline{v} -1)^{2}+(\overline{v}-2)\overline{v})})) \\ +\overline{v}(-3(4+\sqrt{(\kappa -1)(-9+\kappa (\overline{v}-1)^{2}+(\overline{v}-2)\overline{v})})+ \\ \overline{v}(-4+\overline{v}-\sqrt{(\kappa -1)(-9+\kappa (\overline{v} -1)^{2}+(\overline{v}-2)\overline{v})})))) \end{array} \right) }{64(\kappa -1)(\sqrt{(\kappa -1)(-9+\kappa (\overline{v}-1)^{2}+(\overline{v}-2)\overline{v})}}>0 $$

(33)

$$ \frac{d{\Pi}_{2}^{\ast }}{d\kappa }=\frac{\left( \begin{array}{c} (1+\kappa (\overline{v}-1)-\overline{v}+ \\ \sqrt{(\kappa -1)(-9+\kappa (\overline{v}-1)^{2}+(\overline{v}-2)\overline{v} )})(3+\kappa (\overline{v}-1)^{2}- \\ \sqrt{(\kappa -1)(-9+\kappa (\overline{v}-1)^{2}+(\overline{v}-2)\overline{v} )} \\ +\overline{v}(2-\overline{v}+\sqrt{(\kappa -1)(-9+\kappa (\overline{v} -1)^{2}+(\overline{v}-2)\overline{v})})) \end{array} \right) }{256(\kappa -1)^{2}(\sqrt{(\kappa -1)(-9+\kappa (\overline{v} -1)^{2}+(\overline{v}-2)\overline{v})}}>0 $$

(34)

This establishes all parts of Proposition 2.

1.5 Proof of Proposition 3

First, notice that when $\overline {v}$ is high, $\overline {v}>\frac {(3-2 \widehat {v})\widehat {v}+\kappa (1-3\widehat {v}+2\widehat {v}^{2})}{(1-\kappa )(1-\widehat {v})},$ equilibrium price $p_{i}^{\ast }$ is not a function of c, establishing the last part of the proposition.

For $\overline {v}$ sufficiently low or when $\overline {v}\in \left [ Max[0,(2\kappa -1)(1-\widehat {v})^{2}],\frac {2-\widehat {v}-\widehat {v}^{2}}{ 1-\kappa -\kappa \widehat {v}}\right ]$,

$$ \begin{array}{@{}rcl@{}} \frac{dp_{1}^{\ast }}{d\widehat{v}} &=&\frac{ \begin{array}{c} 2(2\kappa -1)(\widehat{v}-1)(5+\kappa^{2}(8(\widehat{v}-1)^{2}+\overline{v} ^{2}) \\ (\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v}+\kappa \overline{v}^{2})}- \\ \kappa (12+4(\widehat{v}-2)\widehat{v}+ \\ \overline{v}(2+\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v} +\kappa \overline{v}^{2})}))) \end{array} }{ \begin{array}{c} \sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v}+\kappa \overline{v}^{2})} \\ (1-\kappa \overline{v}+\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2 \overline{v}+\kappa \overline{v}^{2})} \end{array} }<0\text{ for }\kappa >1/2\text{ }\\ &\Longrightarrow &\text{ }\frac{dp_{1}^{\ast }}{dc}>0\text{ for }\kappa >1/2 \end{array} $$

(35)

The opposite holds true for $\kappa <\frac {1}{2}$

$$ \begin{array}{@{}rcl@{}} \frac{dp_{2}^{\ast }}{d\widehat{v}} &=&-\frac{ \begin{array}{c} 2(2\kappa -1)(\widehat{v}-1(5+\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v} -1)^{2}-2\overline{v}+\kappa \overline{v}^{2})} \\ +\kappa (4-4(\widehat{v}-2)\widehat{v}+\kappa (8(\widehat{v}-1)^{2}+ \overline{v}^{2})- \\ \overline{v}(2+\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v} +\kappa \overline{v}^{2})}))) \end{array} }{ \begin{array}{c} \sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v}+\kappa \overline{v}^{2})} \\ (1-\kappa \overline{v}+\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2 \overline{v}+\kappa \overline{v}^{2})} \end{array} }\\ &&>0\text{ for }\kappa >1/2\text{ }\\ &\Longrightarrow &\text{ }\frac{dp_{2}^{\ast }}{dc}<0\text{ for }\kappa >1/2 \end{array} $$

(36)

The opposite holds true for $\kappa <\frac {1}{2}$

Finally, for intermediate $\overline {v}$ or $\frac {(1+\kappa (1-\widehat {v})+2\widehat {v})(1-\widehat {v})}{\widehat {v}}<\overline {v}<\frac {\kappa (\widehat {v}-1)^{2}-\widehat {v}(3-2\widehat {v})}{1-\widehat {v}}$

$$ \begin{array}{@{}rcl@{}} \frac{dp_{1}^{\ast }}{d\widehat{v}} \!\!&=&\!\!\frac{\left( \begin{array}{c} 2\kappa (\widehat{v}-1)(-3+4\kappa (1-\widehat{v})^{2}+\overline{v}^{2}+ \\ \sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}-\overline{v }(2+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}) \end{array} \right) }{\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}} (1-\overline{v}+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2) \overline{v}})^{2}}\\ &&>0\text{ }\\ &\Longrightarrow &\text{ }\frac{dp_{1}^{\ast }}{dc}<0\text{ } \end{array} $$

(37)

$$ \begin{array}{@{}rcl@{}} \frac{dp_{2}^{\ast }}{d\widehat{v}} \!&=&\!\!-\frac{\left( \begin{array}{c} 2\kappa (\widehat{v}-1)(13+4\kappa (\widehat{v}-1)^{2}+\overline{v}^{2}+ \\ \sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}-v(2+\sqrt{ 9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}) \end{array} \right) }{\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}} (1-\overline{v}+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2) \overline{v}})^{2}}\\ &&>0\text{ }\\ &\Rightarrow &\text{ }\frac{dp_{2}^{\ast }}{dc}<0 \end{array} $$

(38)

This establishes all parts of Proposition 3.

1.6 Proof of Proposition 4

Akin to equilibrium prices, for high $\overline {v},$ equilibrium profit is not a function of c. Hence equilibrium profit is invariant to changes in search cost c.

For $\overline {v}$ sufficiently low or when $\overline {v}\in$$\left [ Max[0,(2\kappa -1)(1-\widehat {v})^{2}],\frac {2-\widehat {v}-\widehat {v}^{2}}{ 1-\kappa -\kappa \widehat {v}}\right ]$,

$$ \begin{array}{@{}rcl@{}} \frac{d{\Pi}_{1}^{\ast }}{d\widehat{v}} \!\!&=&\!\!\frac{\left( \begin{array}{c} ((2\kappa -1)(\widehat{v}-1)(13+ \\ \kappa^{2}(24(\widehat{v}-1)^{2}+\overline{v}^{2})+\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v}+\kappa \overline{v}^{2})}- \\ \kappa (20 + 12(\widehat{v}-2)\widehat{v}+\overline{v}(2+ \\ \sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v}+\kappa \overline{v}^{2})}))) \\ (3-\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v}+\kappa \overline{v}^{2})}+\kappa (4+8\kappa (\widehat{v}-1)^{2}- \\ 4(\widehat{v}-2)\widehat{v}-\kappa \overline{v}^{2}+ \\ \overline{v}(2+\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v} +\kappa \overline{v}^{2})})))) \end{array} \right) }{ \begin{array}{c} (8\kappa \sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v} +\kappa \overline{v}^{2})} \\ (1-\kappa \overline{v}+\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2 \overline{v}+\kappa \overline{v}^{2})}^{2} \end{array} }\\ &&<0\text{ for }\kappa >1/2\text{ }\\ &\Longrightarrow &\text{ }\frac{d{\Pi}_{1}^{\ast }}{dc}>0\text{ for }\kappa >1/2 \end{array} $$

(39)

The opposite holds true for $\kappa <\frac {1}{2}$

$$ \begin{array}{@{}rcl@{}} \frac{d{\Pi}_{2}^{\ast }}{d\widehat{v}} &=&-=\frac{\left( \begin{array}{c} ((2\kappa -1)(\widehat{v}-1)(13+\kappa^{2}(24(\widehat{v}-1)^{2}+\overline{v }^{2})+ \\ \sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v}+\kappa \overline{v}^{2})}- \\ \kappa (4+ 12(\widehat{v}-2)\widehat{v}+ \\ \overline{v}(2+\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v} +\kappa \overline{v}^{2})}))) \\ (3-\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v}+\kappa \overline{v}^{2})}+ \\ \kappa (-12+8\kappa (\widehat{v}-1)^{2}- \\ 4(\widehat{v}-2)\widehat{v}-\kappa \overline{v}^{2}+ \\ \overline{v}(2+\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v} +\kappa \overline{v}^{2})})))) \end{array} \right) }{ \begin{array}{c} (8\kappa \sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2\overline{v} +\kappa \overline{v}^{2})} \\ (1-\kappa \overline{v}+\sqrt{9+\kappa (8(2\kappa -1)(\widehat{v}-1)^{2}-2 \overline{v}+\kappa \overline{v}^{2})}^{2} \end{array} }\\ &&>0\text{ for }\kappa >1/2\text{ }\\ &\Longrightarrow &\text{ }\frac{d{\Pi}_{2}^{\ast }}{dc}<0\text{ for }\kappa >1/2 \end{array} $$

(40)

Finally, for intermediate $\overline {v}$ or $\frac {(1+\kappa (1-\widehat {v})+2\widehat {v})(1-\widehat {v})}{\widehat {v}}<\overline {v}<\frac {\kappa (\widehat {v}-1)^{2}-\widehat {v}(3-2\widehat {v})}{1-\widehat {v}}$

$$ \begin{array}{@{}rcl@{}} \frac{d{\Pi}_{1}^{\ast }}{d\widehat{v}} &=&\frac{\left( \begin{array}{c} \kappa ((\widehat{v}-1)(5+ 12\kappa (\widehat{v}-1)^{2}-2\overline{v}+ \overline{v}^{2}+ \\ \sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}-\overline{v }\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}) \\ +(11+4\kappa (\widehat{v}-1)^{2}+ \\ 2\overline{v}-\overline{v}^{2}-\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}+ \\ \overline{v}\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}) \end{array} \right) }{ \begin{array}{c} 8\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}} \\ (1-\overline{v}+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2) \overline{v}})^{2} \end{array}}\\ &&<0\text{ or} \end{array} $$

(41)

$$ \begin{array}{@{}rcl@{}} \text{ }\frac{d{\Pi}_{1}^{\ast }}{dc} &>&0\text{ } \end{array} $$

(42)

$$ \begin{array}{@{}rcl@{}} \frac{d{\Pi}_{2}^{\ast }}{d\widehat{v}} &=&\frac{\left( \begin{array}{c} \kappa ((\widehat{v}-1)(21 + 12\kappa (\widehat{v}-1)^{2}-2\overline{v}+ \overline{v}^{2}+ \\ \sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}- \\ \overline{v}\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}} )+(-5+4\kappa (\widehat{v}-1)^{2}+ \\ 2\overline{v}-\overline{v}^{2}-\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}}+ \\ \overline{v}\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}} ) \end{array} \right) }{8\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2)\overline{v}} (1-\overline{v}+\sqrt{9+8\kappa (\widehat{v}-1)^{2}+(\overline{v}-2) \overline{v}})^{2}}\\ &&>0\text{ or} \end{array} $$

(43)

$$ \begin{array}{@{}rcl@{}} \text{ }\frac{d{\Pi}_{2}^{\ast }}{dc} &<&0 \end{array} $$

(44)

This establishes all parts of Proposition 4.

1.7 Proof of Proposition 5

For brevity, we provide a shorter version of the proof of Proposition 5. Specifically, we do not report all the expressions that are included in evaluating the comparative statics, but instead offer those complete expressions as available from the authors on request. For low $\overline {v}, \overline {v}\in \left [ Max[0,(2\kappa -1)(1-\widehat {v})^{2}],\frac {2- \widehat {v}-\widehat {v}^{2}}{1-\kappa -\kappa \widehat {v}}\right ] ,$ we first compute the consumer surplus by substituting the equilibrium, $p_{i}^{\ast }(i=1,2)$ and Δ^∗. The expressions are provided in the tables above. Next, we differentiate the resulting expression with respect to κ and evaluating the resulting expression yields the result stated in the proposition. For intermediate $\overline {v}$ or $\frac { (1+\kappa (1-\widehat {v})+2\widehat {v})(1-\widehat {v})}{\widehat {v}}< \overline {v}<\frac {\kappa (\widehat {v}-1)^{2}-\widehat {v}(3-2\widehat {v})}{1- \widehat {v}}$, we use the appropriate $p_{i}^{\ast }(i=1,2)$ and Δ^∗ that is provided in the tables above to compute the equilibrium consumer surplus. For $\overline {v}$ we check the corners. In other words, we evaluate it for c = 0 and $c=\frac {1}{8}$. Next, we verify that $\frac {dCS}{d\kappa }<0$ for c = 0 and $\overline {v}$ in the intermediate region. Furthermore, $\frac {dCS}{d\kappa }>$ for $c=\frac {1}{8}$ and $\overline {v}$ in the intermediate region. Since CS is a continuous function of c and $\overline {v}$ the result in Proposition 5 (part ii) follows. Finally, for high $\overline {v}$ or $\overline {v}>\frac {(3-2 \widehat {v})\widehat {v}+\kappa (1-3\widehat {v}+2\widehat {v}^{2})}{(1-\kappa )(1-\widehat {v})},$ again substitution of $p_{i}^{\ast }(i=1,2)$ and Δ^∗ from table 1 and 2 yields the appropriate expression for the consumer surplus. Differentiating the resulting expression with respect to κ and evaluating the sign of $\frac {dCS}{d\kappa }$ shows that the equilibrium consumer surplus monotonically declines with respect to κ

This establishes all parts of Proposition 5.