Diversity and Quantity Choice in a Horizontally Differentiated Duopoly

Abstract

The Hotelling model is extended where not only can consumers choose to buy from both sellers (multi-brand purchase), buying multiple units from one seller is also considered (within-brand multi-unit purchase). When an increase of the demand for within-brand multi-purchase exceeds a threshold: (i) firms’ strategies are switched from single-brand equilibrium with higher prices to multi-brand equilibrium with lower prices, if the incremental value from consuming an additional unit of the same product is independent of the preference for diversity; (ii) the direction of such equilibrium-switch is reversed if the two types of multi-purchase are substitutes.

Appendix: Proofs

Supplemental Proof for Proposition 1

In this section, we provide more details regarding the inequalities that shall be compared through case (a-1) to case (c-3), assuming that r_A ≥ r_B.

First, we rule out the possibility that one firm charges ${p_{i}^{M}}$ whereas the other firm charges $p_{-i}^{S}$. Suppose firm A charges ${{p}_{A}^{M}}$ evaluated at ${{p}_{B}^{S}}({{p}_{A}^{M}}) < \hat {p}_{B}\Leftrightarrow$

$$ \beta+r_{A}\mu > \frac{2}{4(\sqrt{2}-1)}(r_{A}-r_{B})\mu+\frac{6}{4\sqrt{2}-1}t. $$

(A.1)

When Eq. A.1 holds, the condition ${{p}_{A}^{M}} > \hat {p}_{A} \Leftrightarrow$

$$ \beta+r_{A}\mu < \frac{2(3-\sqrt{2})}{7}(r_{A}-r_{B})\mu+\frac{2(3-\sqrt{2})(1+\sqrt{2})}{7}t $$

cannot hold evaluated at μ < 3t. Therefore, firm B will not charge ${{p}_{B}^{S}}$. Similarly, suppose firm A charges ${{p}_{A}^{S}}({{p}_{B}^{M}})$ evaluated at ${{p}_{B}^{M}} > \hat {p}_{B}\Leftrightarrow \beta +r_{A} \mu < \frac {1}{2\sqrt {2}-1}(r_{A}-r_{B}) \mu + \frac {2}{2\sqrt {2}-1}t$, the condition ${{p}_{A}^{S}}({{p}_{B}^{M}}) < \hat {p}_{A}\Leftrightarrow \beta +r_{A} \mu >\frac {23(29-8\sqrt {2})}{713}(r_{A}-r_{B})\mu +\frac {(29-8\sqrt {2})(18 + 24\sqrt {2})}{713}t$ cannot hold, i.e., firm B will not charge ${{p}_{B}^{M}}$.

• (a) When $\hat {p}_{A} < {{p}_{A}^{M}}$, the following condition holds:

  $$ \beta+r_{A}\mu < \frac{2(\sqrt{2}-1)}{2\sqrt{2}-1}(r_{A}-r_{B})\mu+\frac{2}{2\sqrt{2}-1}t. $$

  (A.2)

  Suppose $\hat {p}_{B} > {{p}_{B}^{M}}$, then

  $$ \beta+r_{A}\mu>\frac{1}{2\sqrt{2}-1}(r_{A}-r_{B})\mu+\frac{2}{2\sqrt{2}-1}t, $$

  which contradicts Eq. A.2. Therefore, for case (a), Eq. A.2 holds such that $\hat {p}_{B} < {{p}_{B}^{M}}$, i.e., $\left ({{{p}_{A}^{S}}}^{*},{{{p}_{B}^{S}}}^{*}\right )$ is a unique equilibrium.

• (b) When ${{p}_{A}^{M}} < \hat {p}_{A} < \sqrt {2}{{p}_{A}^{M}}$, it follows that

  $$ \frac{2(\sqrt{2}-1)}{2\sqrt{2}-1}(r_{A}-r_{B})\mu+\frac{2}{2\sqrt{2}-1}t < \beta+r_{A}\mu < (2-\sqrt{2})(r_{A}-r_{B})\mu+\sqrt{2}t. $$

  (A.3)

  • (1) The intersection of Eq. A.3 and $\hat {p}_{B} < {{p}_{B}^{M}}$ is

    $$ \frac{2(\sqrt{2} - 1)}{2\sqrt{2} - 1}(r_{A}-r_{B})\mu+\frac{2}{2\sqrt{2}-1}t \!<\! \beta+r_{A}\mu \!<\!\frac{1}{2\sqrt{2}-1}(r_{A}-r_{B})\mu+\frac{2}{2\sqrt{2}-1}t. $$

    (A.4)

    Therefore, when Eq. A.4 holds, the unique equilibrium is $\left ({{{p}_{A}^{S}}}^{*},{{{p}_{B}^{S}}}^{*}\right )$.

  • (2) The intersection of Eq. A.3 and ${{p}_{B}^{M}} < \hat {p}_{B} < \sqrt {2} {{p}_{B}^{M}}$ is

    $$ \frac{1}{2\sqrt{2}-1}(r_{A}-r_{B})\mu+\frac{2}{2\sqrt{2}-1}t < \beta+r_{A}\mu < (\sqrt{2}-1)(r_{A}-r_{B})\mu + \sqrt{2}t. $$

    (A.5)

    However, evaluated at Eq. A.5, there are two equilibria. For firm A,

    $$ {\pi_{A}^{M}}\left( {{p}_{A}^{M}},{{p}_{B}^{M}}\right)<{\pi_{A}^{S}} \left( {{{p}_{A}^{S}}}^{*},{{{p}_{B}^{S}}}^{*}\right)\Leftrightarrow \beta+r_{A}\mu < \frac{\sqrt{2}}{3}(r_{A}-r_{B})\mu+\sqrt{2}t, $$

    (A.6)

    which is a sufficient condition for

    $$ {\pi_{B}^{M}}\left( {{p}_{A}^{M}},{{p}_{B}^{M}}\right)<{\pi_{B}^{S}} \left( {{{p}_{A}^{S}}}^{*},{{{p}_{B}^{S}}}^{*}\right)\Leftrightarrow\beta+r_{A}\mu < \left( 1-\frac{\sqrt{2}}{3}\right)(r_{A}-r_{B})\mu+\sqrt{2}t. $$

    (A.7)

    Evaluated at Eq. A.5, both Eq. A.6 and Eq. A.7 hold. That is, $\left ({{p}_{A}^{M}},{{p}_{B}^{M}}\right )$ is not a stable equilibrium because both firms have incentives to deviate to $\left ({{{p}_{A}^{S}}}^{*},{{{p}_{B}^{S}}}^{*}\right )$.

  • (3) The intersection of Eq. A.3 and $\sqrt {2}{{p}_{B}^{M}} < \hat {p}_{B}$ is

    $$ (\sqrt{2}-1)(r_{A}-r_{B})\mu+\sqrt{2}t<\beta+r_{A}\mu < (2-\sqrt{2})(r_{A}-r_{B})\mu+\sqrt{2}t. $$

    (A.8)

    Evaluated at Eq. A.8, the intersection of ${{p}_{A}^{M}}$ and ${{p}_{B}^{M}}$ exists. But the existence of the intersection of ${{p}_{A}^{S}}$ and ${{p}_{B}^{S}}$ depends crucially on the relative size of β + r_Aμ, in particular:

    • (i) If ${{p}_{A}^{S}}$ and ${{p}_{B}^{S}}$ intersects, we have ${{p}_{B}^{S}}|_{p_{A}= \sqrt {2}{{p}_{A}^{M}}} > \hat {p}_{B} \Leftrightarrow$

      $$ (\sqrt{2}-1)(r_{A}-r_{B})\mu+\sqrt{2}t<\beta+r_{A}\mu < \frac{\sqrt{2}}{3}(r_{A}-r_{B})\mu+\sqrt{2}t. $$

      (A.9)

      If Eq. A.9 holds, there are two equilibria. However, the second inequality of Eq. A.9 is equivalent to Eq. A.6, which is also a sufficient condition for Eq. A.7. Therefore, evaluated at Eq. A.9, the stable equilibrium is $\left ({{{p}_{A}^{S}}}^{*},{{{p}_{B}^{S}}}^{*}\right )$.

    • (ii) Evaluated at Eq. A.8, if ${{p}_{A}^{S}}$ and ${{p}_{B}^{S}}$ do not intersect, the parametric values correspond to

      $$ \frac{\sqrt{2}}{3}(r_{A}-r_{B})\mu+\sqrt{2}t<\beta+r_{A}\mu < (2-\sqrt{2})(r_{A}-r_{B})\mu +\sqrt{2}t. $$

      (A.10)

      Then, the unique equilibrium is $\left ({{{p}_{A}^{M}}}^{*},{{{p}_{B}^{M}}}^{*}\right )$.

• (c) When $\sqrt {2}{{p}_{A}^{M}} <\hat {p}_{A}$,

  $$ \beta+r_{A}\mu > (2-\sqrt{2})(r_{A}-r_{B})\mu +\sqrt{2}t. $$

  (A.11)

  Suppose $\hat {p}_{B} < \sqrt {2}{{p}_{B}^{M}}$, then

  $$ \beta+r_{A}\mu < (\sqrt{2}-1)(r_{A}-r_{B})\mu +\sqrt{2}t, $$

  which contradicts Eq. A.11. Therefore, when Eq. A.11 holds, the unique equilibrium is $\left ({{{p}_{A}^{M}}}^{*},{{{p}_{B}^{M}}}^{*}\right )$. The cutoff conditions derived above are summarized in Fig. 2.

□

Supplemental Proof for Proposition 3

The intersection points of the best responses given by Eq. 9 are determined by the relative positions of ${p_{i}^{M}}(p_{-i})$, ${p_{i}^{S}}(p_{-i})$ and $\tilde {p}_{-i}$ (see Fig. 3). Evaluated at $\tilde {p}_{-i}$, ${p_{i}^{M}}(\tilde {p}_{-i})=\frac {\left ((1+r)(\beta -r\mu )-rt \right )\left (2r\sqrt {2(1+r^{2})} + (1+r)(1+r^{2}) \right ) }{2(1-r)(1+r^{2})(1+4r+r^{2})}$, ${p_{i}^{S}}(\tilde {p}_{-i})= \frac {\left ((1+r)(\beta -r\mu )-rt\right ) \left ((1+r)\sqrt {2(1+r^{2})}+4r\right ) }{2(1-r)(1+r)(1+4r+r^{2})}$, and when 1 ≥ r ≥ 0,

$$ 1\leq\frac{{p_{i}^{S}}(\tilde{p}_{-i})}{{p_{i}^{M}}(\tilde{p}_{-i})} = \frac{(1+r^{2})\left( (1+r)\sqrt{2(1+r^{2})}+4r\right) }{(1+r) \left( 2r\sqrt{2(1+r^{2})}+(1+r)(1+r^{2})\right)} \leq \sqrt{2}. $$

In the proof of Proposition 1, we have shown the relative slopes of ${{p}_{A}^{S}}$ and ${{p}_{B}^{S}}$. Here, ${{p}_{A}^{M}}$ and ${{p}_{B}^{M}}$ are also upward sloping, where $\frac {\partial {{p}_{A}^{M}}}{\partial p_{B}} = \frac {r}{1+r^{2}} < \frac {1}{2} = \frac {\partial {{p}_{A}^{S}}}{\partial p_{B}}$ and $\left (\frac {\partial {{p}_{B}^{S}}}{\partial p_{A}}\right )^{-1} = 2 > \left (\frac {\partial {{p}_{B}^{M}}}{\partial p_{A}}\right )^{-1} = \frac {1+r^{2}}{r}>\frac {1}{2}$.

• (i) When $\tilde {p}_{-i}<{p_{i}^{M}}(\tilde {p}_{-i}) < {p_{i}^{S}}(\tilde {p}_{-i}) \Leftrightarrow$

  $$ \beta-r\mu < K^{S}:= \frac{4(1-r+r^{2})\sqrt{2(1+r^{2})} +2-7r+2r^{2}-5r^{3} }{(1-r)(7-2r+7r^{2})}t, $$

  (A.12)

  the unique intersection point is $\left ({{{p}_{A}^{S}}}^{*},{{{p}_{B}^{S}}}^{*}\right )$.

• (ii) When ${p_{i}^{M}}(\tilde {p}_{-i}) < {p_{i}^{S}}(\tilde {p}_{-i})< \tilde {p}_{-i} \Leftrightarrow$

  $$ \beta-r\mu>K^{M}:= \frac{(1+r)\sqrt{2(1+r^{2})}-3r-r^{2}}{(1-r)(1+r)}t, $$

  (A.13)

  the unique intersection point is $\left ({{{p}_{A}^{M}}}^{*},{{{p}_{B}^{M}}}^{*}\right )$.

• (iii) When ${p_{i}^{M}}(\tilde {p}_{-i})<\tilde {p}_{-i}<{p_{i}^{S}}(\tilde {p}_{-i})\Leftrightarrow K^{S}<\beta -r\mu <K^{M}$, there are two points of intersection: $\left ({{{p}_{A}^{M}}}^{*},{{{p}_{B}^{M}}}^{*}\right )$ and $\left ({{{p}_{A}^{S}}}^{*},{{{p}_{B}^{S}}}^{*}\right )$. However, evaluated at K^S < β − rμ < K^M, the condition

  $$ \beta-r\mu < \frac{\sqrt{2}(1-r+r^{2})+r\sqrt{1+r^{2}}}{(1-r)\sqrt{1+r^{2}}}t $$

  (A.14)

  holds such that ∀i and 0 < r < 1, ${\pi _{i}^{S}}\left ({{{p}_{A}^{S}}}^{*},{{{p}_{B}^{S}}}^{*}\right ) > {\pi _{i}^{M}}\left ({{{p}_{A}^{M}}}^{*},{{{p}_{B}^{M}}}^{*}\right )$. That is, K^S < β − rμ < K^M is a sufficient condition of (i.e., a subset of) Eq. A.14.

Finally, we validate the above arguments by checking whether the equilibrium prices are offered consistently with consumers’ decisions. Evaluated at $\left ({{{p}_{A}^{M}}}^{*},{{{p}_{B}^{M}}}^{*}\right )$, the demand for multi-brand purchase is positive, i.e., $\tilde {x}_{B}\left ({{{p}_{A}^{M}}}^{*},{{{p}_{B}^{M}}}^{*}\right ) < \tilde {x}_{A}\left ({{{p}_{A}^{M}}}^{*},{{{p}_{B}^{M}}}^{*}\right )\Leftrightarrow$

$$ \beta-r\mu > \frac{t}{1+r^{2}}, $$

(A.15)

where Eq. A.15 is a necessary condition of Eq. A.13. Evaluated at $\left ({{{p}_{A}^{S}}}^{*},{{{p}_{B}^{S}}}^{*}\right )$, the demand for multi-brand purchase is eliminated, i.e., $\tilde {x}_{B}\left ({{{p}_{A}^{S}}}^{*},{{{p}_{B}^{S}}}^{*}\right )>\tilde {x}_{A}\left ({{{p}_{A}^{S}}}^{*},{{{p}_{B}^{S}}}^{*}\right )\Leftrightarrow$

$$ \beta-r\mu < \frac{3-r}{2(1+r)}t, $$

(A.16)

where Eq. A.16 is implied by β − rμ < K^M. □

To summarize (Fig. 4), when β − rμ is greater than the threshold given by Eq. A.13, the equilibrium prices are $\left ({{{p}_{A}^{M}}}^{*},{{{p}_{B}^{M}}}^{*}\right )$; when β − rμ is lower than the threshold given by Eq. A.13, the equilibrium prices are $\left ({{{p}_{A}^{S}}}^{*},{{{p}_{B}^{S}}}^{*}\right )$.

Equilibria with Quadratic Costs and Fixed Locations

When prices are higher relative to β and μ, the demand for multi-brand purchase is zero. A consumer who is indifferent between choosing brand A and B locates at $\hat {x}^{S}$. When prices are lower relative to β and μ, the demand for multi-brand purchase is positive. A consumer who is indifferent between buying brand A only and both brands, and the one who is indifferent between buying brand B only and both brands, locate at

$$ \hat{x}_{B}=1-\frac{\sqrt{\left( \beta-p_{B}+r(\mu-p_{B})\right)t}}{t},~\hat{x}_{A}=\frac{\sqrt{\left( \beta-p_{A}+r(\mu-p_{A})\right)t}}{t}, $$

respectively.

The remaining procedure follows the similar manner as shown by Proposition 1. When $\hat {x}_{B}>\hat {x}_{A}$, given p_B, firm A solves ${{p}_{A}^{S}}=\arg \max_{p_{A}}(1+r)p_{A}\hat {x}^{S}$; Given p_A, firm B solves ${{p}_{B}^{S}}=\arg \max_{p_{B}}(1+r)p_{B}(1-\hat {x}^{S})$. When $\hat {x}_{B}<\hat {x}_{A}$, firm A solves ${{p}_{A}^{M}}=\arg \max_{p_{A}}(1+r)p_{A}\hat {x}_{A}$ and firm B solves ${{p}_{B}^{M}}=\arg \max_{p_{B}}(1+r)p_{B}(1-\hat {x}_{B})$. Evaluated at $\hat {p}_{-i}$, charging ${p_{i}^{S}}$ according to $\hat {x}_{B}>\hat {x}_{A}$ and charging ${p_{i}^{M}}$ according to $\hat {x}_{B}<\hat {x}_{A}$ are equally profitable, i.e., ${\pi _{i}^{M}}({p_{i}^{M}})={\pi _{i}^{S}}(\hat {p}_{-i})$. Hence, the conditional reactions are given by

$$ p_{i}^{BR}=\left\{\begin{array}{ll} \frac{(1+r)p_{-i}+t}{2(1+r)},~~~~\hfill p_{-i}> \hat{p}_{-i} = \frac{4}{3}\frac{(3t)^{\frac{1}{4}}(\beta+r\mu)^{\frac{3}{4}}}{1+r}-\frac{t}{1+r}\\ \frac{2}{3}\frac{\beta+r\mu}{1+r},~~~~\hfill p_{-i} < \hat{p}_{-i} = \frac{4}{3}\frac{(3t)^{\frac{1}{4}}(\beta+r\mu)^{\frac{3}{4}}}{1+r}-\frac{t}{1+r} \end{array}\right.. $$

Next, by checking the relative positions of ${p_{i}^{M}}$, ${p_{i}^{S}}(\hat {p}_{-i})$ and $\hat {p}_{i}$ and the resulting points of intersection of ${p}_{A}^{BR}$ and ${p}_{B}^{BR}$, the symmetric equilibrium is given by

$$ \left\{\begin{array}{ll} {{p_{i}^{S}}}^{*} = \frac{t}{1+r},~{\pi_{i}^{S}}=\frac{t}{2},~~~~\hfill \beta+r\mu < \frac{6^{\frac{2}{3}}3^{\frac{1}{3}}t}{4}\\ {{p_{i}^{M}}}^{*}=\frac{2}{3}\frac{\beta+r\mu}{1+r},~{\pi_{i}^{M}}=\frac{2\sqrt{3}(\beta+r\mu)^{\frac{3}{2}}}{9\sqrt{t}}~~~~\hfill \beta+r\mu > \frac{6^{\frac{2}{3}}3^{\frac{1}{3}}t}{4} \end{array}\right.. $$

□

Supplemental Proof for Proposition 4

In Proposition 4, the threshold for regime switch, denoted $\widehat {\beta +r\mu }:=-\frac {t}{12} + \frac {t}{12}\left [(28+3\sqrt {87})^{\frac {1}{3}} +\frac {1}{(28+3\sqrt {87})^{\frac {1}{3}}}-1\right ]^{2}$, is found by ${\pi _{i}^{S}}|_{a=0,b=1} = {\pi _{i}^{M}}|_{a=b=\frac {1}{2}}\left (\widehat {\beta +r\mu }\right )$. For the remaining, we complete the proof by ruling out possible deviations.

Suppose that the initial equilibrium corresponds to the multi-brand equilibrium, i.e., when $\beta +r\mu >\widehat {\beta +r\mu }$ holds with minimum differentiation. By Eq. 12, price deviation in stage 2 is unprofitable. Consider the deviations in stage 1 by relocating at a = 0, given $b=\frac {1}{2}$, and then search for the optimal prices in stage 2.

Given a = 0 and $b=\frac {1}{2}$ chosen in stage 1, and the demand for multi-brand satisfies the condition $\hat {x}_{b}>\hat {x}_{a}$ (single-brand equilibrium), then the equilibrium prices in stage 2 are ${{p}_{A}^{S}}|_{a=0,b=\frac {1}{2}} = \frac {5}{12}\frac {t}{1+r}$ and ${{p}_{B}^{S}}|_{a=0,b=\frac {1}{2}} = \frac {7}{12}\frac {t}{1+r}$ (by plugging a = 0 and $b=\frac {1}{2}$ into Eq. 11). The equilibrium profits are ${\pi _{A}^{S}}|_{a=0,b=\frac {1}{2}}=\frac {25}{144}t$ and ${\pi _{B}^{S}}|_{a=0,b=\frac {1}{2}} = \frac {49}{144}t$. If firm A deviates its price by setting ${{p}_{A}^{M}}$ such that $\hat {x}_{b}<\hat {x}_{a}$ (multi-brand equilibrium), then the equilibrium price and profit are ${{p}_{A}^{M}}|_{a=0,b=\frac {1}{2}} = \frac {2}{3}\frac {\beta +r\mu }{1+r}$ and ${\pi _{A}^{M}}|_{a=0,b=\frac {1}{2}} = \frac {2\sqrt {3}(\beta +r\mu )^{\frac {3}{2}}}{9\sqrt {t}}$. Charging ${{p}_{A}^{M}}|_{a=0,b=\frac {1}{2}}$ is less profitable than charging ${{p}_{A}^{S}}|_{a=0,b=\frac {1}{2}}$ provided that $\beta +r\mu < \frac {450^{\frac {2}{3}}3^{\frac {1}{3}}}{144}t$. Similarly, if firm B deviates by setting ${{p}_{B}^{M}}$ such that $\hat {x}_{b} < \hat {x}_{a}$, such deviation is not profitable provided that ${\pi _{B}^{M}}|_{a=0,b=\frac {1}{2}} < {\pi _{B}^{S}}|_{a=0,b=\frac {1}{2}} \Leftrightarrow \beta +r\mu < -\frac {t}{12}+\frac {t}{12}\left (\frac {1}{2}(155+7\sqrt {489})^{\frac {1}{3}}+\frac {2}{(155+7\sqrt {489})^{\frac {1}{3}}}-1\right )^{2}$, under which, the condition $\beta +r\mu < \frac {450^{\frac {2}{3}}3^{\frac {1}{3}}}{144}t$ holds such that neither firm will deviate.

Next, if the demand for multi-brand satisfies the condition $\hat {x}_{b}<\hat {x}_{a}$ and given $a=0 < b=\frac {1}{2}$ chosen in stage 1, consider the possible price deviations in stage 2. Before deviation, prices and profits are given by Eq. 12 where location parameters are replaced by a = 0 and $b = \frac {1}{2}$. Given ${{p}_{B}^{M}}|_{a=0,b=\frac {1}{2}}$, if firm A deviates by setting ${{p}_{A}^{S}} = \frac {24(\beta +r\mu )+7t+2\sqrt {\left (12(\beta +r\mu )+t\right )t} }{72(1+r)}$ such that $\hat {x}_{b}>\hat {x}_{a}$, the profit is ${\pi _{A}^{S}}|_{{{p}_{B}^{M}}\left (a=0,b=\frac {1}{2}\right )} = \frac {\left (24(\beta +r\mu )+7t+2\sqrt {\left (12(\beta +r\mu )+t\right )t } \right )^{2}}{5184t}$. Deviating is unprofitable provided that ${\pi _{A}^{S}}|_{{{p}_{B}^{M}}\left (a=0,b=\frac {1}{2}\right )} < {\pi _{A}^{M}}|_{a=0,b=\frac {1}{2}}\Leftrightarrow \beta +r\mu >0.32t$.⁵ Similarly, given ${{p}_{A}^{M}}|_{a=0,b=\frac {1}{2}}$, if firm B deviates by setting ${{p}_{B}^{S}} = \frac {8(\beta +r\mu )+9t}{24(1+r)}$, it gives ${\pi _{B}^{S}}|_{{{p}_{A}^{M}}\left (a=0,b=\frac {1}{2}\right )} = \frac {\left [8(\beta +r\mu )+9t\right ]^{2}}{576t}$. Such deviation is unprofitable provided that ${\pi _{B}^{S}}|_{{{p}_{A}^{M}}\left (a=0,b=\frac {1}{2}\right )}<{\pi _{B}^{M}}|_{a=0,b=\frac {1}{2}}\Leftrightarrow \beta +r\mu > 0.41t$.

Therefore, when $\beta +r\mu > -\frac {t}{12}+\frac {t}{12}\left (\frac {1}{2}(155+7\sqrt {489})^{\frac {1}{3}}+\frac {2}{(155+7\sqrt {489})^{\frac {1}{3}}}-1\right )^{2}$, given a = 0 and $b=\frac {1}{2}$, both firms will deviate to offer prices such that the demand for multi-brand is positive. However, when the demand for multi-brand is positive, firm A has no incentive to relocate at a = 0 in stage 1, as shown by Eq. 12, where $\frac {\partial {\pi _{A}^{M}}}{\partial a} > 0$. When β + rμ < 0.41t, both firms will charge ${p_{i}^{S}}$ to eliminate multi-brand purchase. In such a case, the location $(a,b)=\left (0,\frac {1}{2}\right )$ is not optimal, because moving b to 1 makes firm B strictly better off. Similarly when $0.41t < \beta +r\mu < -\frac {t}{12}+\frac {t}{12}\left (\frac {1}{2}(155+7\sqrt {489})^{\frac {1}{3}}+\frac {2}{(155+7\sqrt {489})^{\frac {1}{3}}}-1\right )^{2}$, deviating to $(a,b)=\left (0,\frac {1}{2}\right )$ in stage 1 is not profitable, because $-\frac {t}{12}+\frac {t}{12}\left (\frac {1}{2}(155+7\sqrt {489})^{\frac {1}{3}}+\frac {2}{(155+7\sqrt {489})^{\frac {1}{3}}}-1\right )^{2} < \widehat {\beta +r\mu }$ and ${\pi _{i}^{M}}|_{\beta +r\mu <\widehat {\beta +r\mu }} < {\pi _{i}^{S}}|_{\beta +r\mu <\widehat {\beta +r\mu }}$.

Now suppose the initial condition satisfies $\beta +r\mu < \widehat {\beta +r\mu }$ with maximum differentiation given by Eq. 13. Clearly, when fixing (a,b) = (0, 1), price deviation in stage 2 is unprofitable. Consider firm B deviates by relocate at $b=\frac {1}{2}$, given a = 0. Evaluated at $(a,b)=\left (0,\frac {1}{2}\right )$, charging ${{p}_{B}^{S}}$ in stage 2 must generate an incentive to move b to 1 in stage 1. Alternatively, charging ${{p}_{B}^{M}} = \frac {12(\beta +r\mu )-t+\sqrt {\left (12(\beta +r\mu )+t\right )t}}{18(1+r)}$ according to $\hat {x}_{b} < \hat {x}_{a}$ gives ${\pi _{B}^{M}}|_{a=0,b=\frac {1}{2}}$, which is less profitable than that before deviating in stage 1, i.e., ${\pi _{B}^{M}}|_{a=0,b=\frac {1}{2}} < {\pi _{B}^{S}}|_{a=0,b=1}=\frac {t}{2}$ provided that $\beta +r\mu < \widehat {\beta +r\mu }$. □