Why offer lower prices to past customers? Inducing favorable social price comparisons to enhance customer retention

Abstract

Price discrimination policies vary widely across companies. Some firms offer new customers the lowest price; others give preferential prices to their past customers. We contribute to the literature on price discrimination in behavior-based pricing by exploring how customers’ social price comparisons, i.e., comparing one’s price to that received by similar peers, impact the optimal structure of price discrimination. Social price comparisons have a negative (positive) impact on customers’ transaction utility if the price charged to past customers is higher (lower) than a new customer’s price. Using an analytical model with vertically differentiated firms, we show that a firm with relatively large market share will reward its past customers with relatively low prices when social price comparisons have a sufficiently large impact on utility. Furthermore, we find that social price comparisons lead to a relaxation of the price competition for new customers. Thus, both firms can earn higher profits when such comparisons are made than when they are absent. We also examine how other factors, such as horizontal competition and strategic customers, interact with social price comparison concerns to impact pricing strategies. Finally, we show how pricing behavior differs when price comparisons are based on historic reference prices rather than on peers’ prices.

Appendix

1.1 Proof of Proposition 1

Each firm chooses its old customer and new customer prices to maximize expected profit, given by eqs. (5) and (6), respectively. After using eqs. (2), (3), and (4) to substitute for ${\tilde{\alpha}}_{\mathrm{N}}$, ${\tilde{\alpha}}_{\mathrm{L}}$and ${\tilde{\alpha}}_{\mathrm{H}}$, we set the first-order conditions to zero, i.e., $\frac{\partial {\Pi}^L}{\partial {\mathrm{p}}^{L, N}}=\frac{\partial {\Pi}^L}{\partial {\mathrm{p}}^{L,\mathrm{O}}}=\frac{\partial {\Pi}^H}{\partial {\mathrm{p}}^{H, N}}=\frac{\partial {\Pi}^H}{\partial {\mathrm{p}}^{H, O}}\equiv 0$. Solving these four simultaneous equations, we find that the equilibrium prices are⁸:

$$ \begin{array}{c}\hfill {\mathrm{p}}^{L, O}=\frac{1}{2}\left(\frac{1+\beta}{3+2\beta}+\frac{1+\beta}{\left(3+2\beta \right)\left(1+\theta \right)}+\frac{-2+2\tilde{\upalpha}\left(1-\theta \right)+\theta -\beta \left(2+\theta \right)}{3+\left(3-2\theta \right)\theta +6\beta \left(1+\theta \right)}\right)\hfill \\ {}\hfill {\mathrm{p}}^{L, N}=\frac{1+\beta}{3+2\beta}-\frac{1}{6+4\beta \theta +6\theta +4\beta \theta}+\frac{1-2\beta \left(1+\theta \right)-4\tilde{\upalpha}\left(1+\theta \right)}{2\left(3+\left(3-2\theta \right)\theta +6\beta \left(1+\theta \right)\right)}\kern1.2em \hfill \\ {}\hfill {\mathrm{p}}^{H, O}=\frac{1}{2}\left(\frac{1+\beta}{3+2\beta}+\frac{1+\beta}{\left(3+2\beta \right)\left(1+\theta \right)}+\frac{2-2\tilde{\upalpha}\left(1-\theta \right)-\theta +\beta \left(2+\theta \right)}{3+\left(3-2\theta \right)\theta +6\beta \left(1+\theta \right)}\right)\kern1.92em \hfill \\ {}\hfill {\mathrm{p}}^{H, N}=\frac{1+\beta}{3+2\beta}-\frac{1}{6+4\beta \theta +6\theta +4\beta \theta}+\frac{-1+2\beta \left(1+\theta \right)+4\tilde{\upalpha}\left(1+\theta \right)}{2\left(3+\left(3-2\theta \right)\theta +6\beta \left(1+\theta \right)\right)}\kern1.2em \hfill \end{array} $$

(10)

These prices are only valid if $0\le {\tilde{\alpha}}_L\le \tilde{\alpha}\le {\tilde{\alpha}}_H\le 1$ and $0\le {\tilde{\alpha}}_N\le 1$, i.e., all of the consumer segments identified by Fig. 1 are of non-negative size. Let ${{\tilde{\uptheta}}^H}_1$be the value of θ for which the solution in Eq. (10) results in ${\tilde{\alpha}}_H=\tilde{\alpha}$ and ${{\tilde{\uptheta}}^L}_1$ be the value of θ for which these prices result in ${\tilde{\alpha}}_L=\tilde{\alpha}$. Thus, the prices given in Eq. (10) represent an equilibrium only if $0\le \uptheta \le \min \left\{{{\tilde{\uptheta}}_1}^H,{{\tilde{\uptheta}}_1}^L\right\}$.⁹ If this condition is not met, the equilibrium consists of a corner solution in which ${\tilde{\alpha}}_L=\tilde{\alpha}$ and/or ${\tilde{\alpha}}_H=\tilde{\alpha}$ (as will be derived in the subsequent paragraphs). Over the region in which $0\le \uptheta \le \min \left\{{{\tilde{\uptheta}}_1}^H,{{\tilde{\uptheta}}_1}^L\right\}$, using (A2), we find that it is always the case that p^H , O>p^H , N. However, we find that p^L , O<p^L , N if ${\tilde{\upalpha}}^{\prime}\left(\upbeta, \uptheta \right)<\tilde{\alpha}<{\tilde{\upalpha}}^{"}\left(\upbeta, \uptheta \right)$and p^L , O ≥ p^L , N otherwise, where ${\tilde{\upalpha}}^{\hbox{'}}\left(\upbeta, \uptheta \right)=\frac{3+\uptheta \left[3+\uptheta +\left(1+\upbeta \right)\left(2\upbeta \right)\left(1+\uptheta \right)-{\uptheta}^2\right]}{\left(3+2\upbeta \right)\left(1+\uptheta \right)\left(3+\uptheta \right)}$ and ${\tilde{\upalpha}}^{"}\left(\upbeta, \uptheta \right)=\frac{-1+\theta +{\theta}^2+2{\beta}^2\left(1+\theta \right)+\beta \left(2+\theta \left(4+\theta \right)\right)}{1+8\beta +6{\beta}^2+2\left(1+\beta \right)\left(2+3\beta \right)\theta -\left(1+\beta \right){\theta}^2}$. Specifically, ${\tilde{\upalpha}}^{\prime}\left(\upbeta, \uptheta \right)$ is the value of $\tilde{\alpha}$ for which p^L , O=p^L , N and ${\tilde{\upalpha}}^{"}\left(\upbeta, \uptheta \right)$is the value of $\tilde{\alpha}$ for which the prices result in Firm L retaining all of its past consumers, i.e., ${\tilde{\alpha}}_L=\tilde{\alpha}$. In Fig. 2, ${\tilde{\upalpha}}^{\prime}\left(\upbeta, \uptheta \right)$and ${\tilde{\upalpha}}^{"}\left(\upbeta, \uptheta \right)$are represented by the two curves which delineate the very small, bottom-left region of the first graph for which Firm L rewards past customers.

To derive the corner solutions, note that ${{\tilde{\uptheta}}^H}_1$and ${{\tilde{\uptheta}}^L}_1$ are defined implicitly by the equations: ${\tilde{\alpha}}_H={\mathrm{p}}^{H, O}-{\mathrm{p}}^{L, N}+\uptheta \left({\mathrm{p}}^{H, O}-{\mathrm{p}}^{H, N}\right)=\tilde{\alpha}$ and ${\tilde{\alpha}}_L={\mathrm{p}}^{H, N}-{\mathrm{p}}^{L, N}-\uptheta \left({\mathrm{p}}^{L, O}-{\mathrm{p}}^{L, N}\right)=\tilde{\alpha}$. (The closed form solutions, which are functions of β and ᾶ, are too algebraically complex to include here. Please see the authors for the Mathematica file which contains this expression, as well as all other detailed calculations referred to in the Appendix.)

If ${{\uptheta >\tilde{\uptheta}}_1}^H$, Firm H has an incentive to retain all of its past consumers i.e., ${\tilde{\alpha}}_H=\tilde{\alpha}$. This corner solution implies that Firm H sets ${\mathrm{p}}^{\mathrm{H},\mathrm{N}}=\frac{p^{H, O}-{p}^{L, N}+{p^{H, O}}^{\ast}\theta -\tilde{\upalpha}}{\theta}$. Using the remaining FOC’s, $\frac{\partial {\Pi}^L}{\partial {\mathrm{p}}^{L, N}}=\frac{\partial {\Pi}^L}{\partial {\mathrm{p}}^{L, O}}=\frac{\partial {\Pi}^H}{\partial {\mathrm{p}}^{H, O}}\equiv 0$, we find the following equilibrium prices:

$$ \begin{array}{c}\hfill {\mathrm{p}}^{L, O}=\frac{\left(\beta \left(1+\theta \right)+\tilde{\upalpha}+\theta \right)\left(1+\beta \left(2+\theta \right)\right)}{\left(1+\theta \right)\left(3+\left(3-\theta \right)\theta +6{\beta}^2\left(1+\theta \right)+\beta \left(10+\left(10-\theta \right)\theta \right)\right)}\kern15.11999em \hfill \\ {}\hfill {\mathrm{p}}^{L, N}=\frac{\left(\beta \left(1+\theta \right)+\tilde{\upalpha}+\theta \right)\left(\theta +2\beta \left(1+\theta \right)\right)}{\left(1+\theta \right)\left(3+\left(3-\theta \right)\theta +6{\beta}^2\left(1+\theta \right)+\beta \left(10+\left(10-\theta \right)\theta \right)\right)},{\mathrm{p}}^{H, O}=\frac{\tilde{\upalpha}\left(1+\theta \right)+\left(\beta \left(1+\theta \right)+\tilde{\upalpha}+\theta \right)\left(\theta \left(3-\theta \right)+\beta \left(2+4\theta \right)\right)}{\left(1+\theta \right)\left(3+\left(3-\theta \right)\theta +6{\beta}^2\left(1+\theta \right)+\beta \left(10+\left(10-\theta \right)\theta \right)\right)}\hfill \\ {}\hfill {\mathrm{p}}^{H, N}=\frac{\left(\beta \left(1+\theta \right)+\tilde{\upalpha}+\theta \right)\left(2+\theta \left(2-\theta \right)+4\beta \left(1+\theta \right)\right)}{\left(\left(1+\theta \right)\right)\left(3+\left(3-\theta \right)\theta +6{\beta}^2\left(1+\theta \right)+\beta \left(10+\left(10-\theta \right)\theta \right)\right)}\kern14.63999em \hfill \end{array} $$

(11)

However, these prices are only valid if ${\tilde{\alpha}}_L\le \tilde{\alpha}$. Let ${{\tilde{\uptheta}}^H}_2$be the value of θ for which the prices in Eq. (11) result in ${\tilde{\alpha}}_L=\tilde{\alpha}$. In particular, ${{\tilde{\uptheta}}^H}_2$is found by setting ${\tilde{\alpha}}_L={\mathrm{p}}^{H, N}-{\mathrm{p}}^{L, N}-\uptheta \left({\mathrm{p}}^{L, O}-{\mathrm{p}}^{L, N}\right)=\tilde{\alpha}$. This reduces to: ${{\tilde{\uptheta}}^H}_2=\frac{-1+3\tilde{\alpha}-3\beta \left(1-3\tilde{\alpha}-\beta \left(1-2\tilde{\alpha}\right)\right)+\sqrt{{\left(1+\beta \left(3+\beta \right)\right)}^2-2\left(1+\beta \right)\left(3+\beta \left(13+3\beta \left(5+2\beta \right)\right)\right)\tilde{\upalpha}+{\left(1+\beta \right)}^2\left(17+12\beta \left(5+3\beta \right)\right){\tilde{\upalpha}}^2}}{2\left(1+\beta \right)\left(\beta +\tilde{\alpha}\right)}$. The prices in Eq. (11) describe the equilibrium only if ${{\tilde{\uptheta}}^H}_1<\uptheta \le {{\tilde{\uptheta}}^H}_2$. In this region, using Eq. (11), we find that p^L , O>p^L , N when θ < 1/(1 + β) and p^L , O<p^L , N for ${{\tilde{\uptheta}}^H}_2$> θ > 1/(1 + β).

For the region $\uptheta >{{\tilde{\uptheta}}^H}_2$, both Firms L and H will price at a corner solution in which they each retain all of their past customers, i.e., ${\tilde{\alpha}}_H={\tilde{\alpha}}_L=\tilde{\alpha}$. These two constraints imply ${\mathrm{p}}^{\mathrm{H},\mathrm{N}}=\frac{p^{H, O}-{p}^{L, N}+{p^{H, O}}^{\ast}\theta -\tilde{\upalpha}}{\theta}$ and ${\mathrm{p}}^{L, N}=\frac{p^{L, O}-{p}^{H, N}+{p^{H, O}}^{\ast}\theta +\tilde{\upalpha}}{\theta}$. Using the remaining FOC’s $\frac{\partial {\Pi}^L}{\partial {\mathrm{p}}^{L,\mathrm{O}}}=\frac{\partial {\Pi}^H}{\partial {\mathrm{p}}^{H, O}}\equiv 0$, we find that the equilibrium prices:

$$ \begin{array}{c}\hfill {\mathrm{p}}^{L, O}=\frac{\beta \left(1+\beta \left(2-3\tilde{\upalpha}\right)-4\tilde{\upalpha}\right)-\tilde{\upalpha}+\theta -\left(1+\beta \right)\left(3\beta \left(-1+\tilde{\upalpha}\right)+2\tilde{\upalpha}\right)\theta +\left(1+\beta \right)\left(\beta +\tilde{\upalpha}\right){\theta}^2}{\left(1+\beta \right)\left(1+3\beta \right){\left(1+\theta \right)}^2}\kern5.159997em \hfill \\ {}\hfill {\mathrm{p}}^{L, N}=\frac{\beta^2+\left(1+\beta \right)\left(\beta +\tilde{\upalpha}\right)\theta}{\left(1+\beta \right)\left(1+3\beta \right)\left(1+\theta \right)},{\mathrm{p}}^{H, O}=\frac{\tilde{\upalpha}+\beta \left(\beta +4\tilde{\upalpha}+3\beta \tilde{\upalpha}\right)+\left(2+3\beta \right)\left(\beta +\tilde{\upalpha}+\beta \tilde{\upalpha}\right)\theta +\left(1+\beta \right)\left(1+2\beta -\tilde{\upalpha}\right){\theta}^2}{\left(1+\beta \right)\left(1+3\beta \right){\left(1+\theta \right)}^2}\hfill \\ {}\hfill {\mathrm{p}}^{H, N}=\frac{\beta \left(1+2\beta \right)+\left(1+\beta \right)\left(1+2\beta \hbox{-} \tilde{\upalpha}\right)\theta}{\left(1+\beta \right)\left(1+3\beta \right)\left(1+\theta \right)}\kern14.15999em \hfill \end{array} $$

(12)

Using Eq. (12), we find that in this parameter region $\left(\uptheta >{{\tilde{\uptheta}}^H}_2\ \right)$, one firm will offer lower prices to its past customers. Specifically, p^L , O<p^L , N and p^H , O>p^H , N if ${{\tilde{\uptheta}}^L}_2$< θ < min{max{1/(1 + β), $\left(\tilde{\upalpha}-\beta \left(1-3\tilde{\upalpha}\right)\right)/\left(\left(1-3\tilde{\upalpha}\right)\left(1+\beta \right)\right)$},2}. Otherwise, we have p^L , O>p^L , N & p^H , O<p^H , N.

If ${{\uptheta >\tilde{\uptheta}}_1}^L$, Firm L has an incentive to retain all of its past consumers i.e., ${\tilde{\alpha}}_L=\tilde{\alpha}$. This corner solution implies that Firm L sets ${\mathrm{p}}^{L, N}=\frac{p^{L, O}-{p}^{H, N}+{p^{H, O}}^{\ast}\theta +\tilde{\upalpha}}{\theta}$. Using the remaining FOC’s $\frac{\partial {\Pi}^H}{\partial {\mathrm{p}}^{H, N}}=\frac{\partial {\Pi}^L}{\partial {\mathrm{p}}^{L,\mathrm{O}}}=\frac{\partial {\Pi}^H}{\partial {\mathrm{p}}^{H, O}}\equiv 0$, we find the following equilibrium prices:

$$ \begin{array}{c}\hfill {\mathrm{p}}^{L, O}=\frac{2{\beta}^2\left(2-3\tilde{\upalpha}+\theta \right)\left(1+\theta \right)\hbox{-} \tilde{\upalpha}\left(3+2\left(3-\theta \right)\theta \right)+\theta \left(3+\left(1-\theta \right)\theta \right)+\beta \left(4+\left(4-\theta \right)\theta \left(2+\theta \right)\hbox{-} \tilde{\upalpha}\left(12+\left(14-\theta \right)\theta \right)\right)}{\left(1+\theta \right)\left(3+\left(3-\theta \right)\theta +6{\beta}^2\left(1+\theta \right)+\beta \left(10+\left(10-\theta \right)\theta \right)\right)}\kern6.239996em \hfill \\ {}\hfill {\mathrm{p}}^{L, N}=\frac{1+2\theta \hbox{-} {\theta}^3+2{\beta}^2{\left(1+\theta \right)}^2\hbox{-} \tilde{\upalpha}\left(2+\left(2-\theta \right)\theta \right)+\beta \left(1+\theta \right)\left(2-4\tilde{\upalpha}+\left(2-\theta \right)\theta \right)}{\left(1+\theta \right)\left(3+\left(3-\theta \right)\theta +6{\beta}^2\left(1+\theta \right)+\beta \left(10+\left(10-\theta \right)\theta \right)\right)}\kern13.55999em \hfill \\ {}\hfill {\mathrm{p}}^{H, O}=\frac{\left(\beta \left(2+\theta \right)+1\right)\left(2-\tilde{\upalpha}+2\theta +2\beta \left(1+\theta \right)\right)}{\left(1+\theta \right)\left(3+\left(3-\theta \right)\theta +6{\beta}^2\left(1+\theta \right)+\beta \left(10+\left(10-\theta \right)\theta \right)\right)},{\mathrm{p}}^{H, N}=\frac{\left(2\beta \left(1+\theta \right)+\theta \right)\left(2-\tilde{\upalpha}+2\theta +2\beta \left(1+\theta \right)\right)}{\left(1+\theta \right)\left(3+\left(3-\theta \right)\theta +6{\beta}^2\left(1+\theta \right)+\beta \left(10+\left(10-\theta \right)\theta \right)\right)}\hfill \end{array} $$

(13)

However, these prices are only valid if ${\tilde{\alpha}}_H\ge \tilde{\alpha}$. Let ${{\tilde{\uptheta}}^L}_2$be the value of θ for which the prices in (Eq. 13) result in ${\tilde{\alpha}}_L=\tilde{\alpha}$. In particular, ${{\tilde{\uptheta}}^L}_2$is found by setting ${\tilde{\alpha}}_H={\mathrm{p}}^{H, O}-{\mathrm{p}}^{L, N}+\uptheta \left({\mathrm{p}}^{H, O}-{\mathrm{p}}^{H, N}\right)=\tilde{\alpha}$. Substituting the prices from (Eq. 13), his reduces to: ${{\tilde{\uptheta}}^L}_2=\frac{1-3\tilde{\alpha}+2\beta -9\beta \tilde{\alpha}-6{\beta}^2\tilde{\alpha}+\sqrt{{\left(1+2\beta \right)}^2{\left(1-3\left(1+\beta \right)\delta \right)}^2+4\left(1+\beta \right)\left(1+2\beta -\delta \right)\left(1-2\delta +2\beta \left(2+\beta -4\delta -3\beta \delta \right)\right)}}{2\left(1+\beta \right)\left(1+2\beta \hbox{-} \tilde{\alpha}\right)}$. The prices in (Eq. 11) describe the equilibrium only if ${{\tilde{\uptheta}}^L}_1<\uptheta \le {{\tilde{\uptheta}}^L}_2$. In this region, using (Eq. 13), we find that p^H , O>p^H , N when θ < 1/(1 + β) and p^H , O<p^H , N for ${{\tilde{\uptheta}}^L}_2$> θ > 1/(1 + β).

Next, for $\uptheta >{{\tilde{\uptheta}}^L}_2$, as before, we find the best responses for the prices using the FOCs. Both Firms L and H know that they will be at a corner solution, i.e., ${\tilde{\alpha}}_H={\tilde{\alpha}}_L=\tilde{\alpha}$. Using $\frac{\partial {\Pi}^L}{\partial {\mathrm{p}}^{L,\mathrm{O}}}=\frac{\partial {\Pi}^H}{\partial {\mathrm{p}}^{H, O}}\equiv 0$, we find that, for $\uptheta >{{\tilde{\uptheta}}^L}_2$, the equilibrium prices are same as (Eq. 12).

1.2 Proof of Proposition 2

The proof of parts (a) and (b) are derived by substituting eqs. (10) through (12) for the prices in eqs. (5) and (6). In particular, we calculate the difference between profits when there are social price comparisons to the profits when no comparisons are made: Π^L − Π^L(θ = 0) , Π^H − Π^H(θ = 0). For the parameter range $0\le \uptheta < \min \left\{{{\tilde{\uptheta}}_1}^H,{{\tilde{\uptheta}}_1}^L\right\}$, the equilibrium prices, when there are social price comparisons, are given in (Eq. 10). The resulting profit difference is:

$$ {\Pi}^L-{\Pi}^L\Big|{}_{\theta =0}=\kern0.5em \left[-\left(\left(1+\beta \right){\left(3+4\beta \left(1+\beta -\tilde{\upalpha}\right)-6\tilde{\upalpha}\right)}^2+{\left(3\tilde{\upalpha}+2\beta \left(2+2\beta +\tilde{\upalpha}\right)\right)}^2\right)/{\left(3+6\beta \right)}^2\right.\kern10.07999em +\left({\left(2\beta \left(2+2\beta +\tilde{\upalpha}\right)+2\beta \left(4+3\beta \right)\theta +\left(1+2\beta \left(1+\beta -\tilde{\upalpha}\right)-3\tilde{\upalpha}\right){\theta}^2-\left(1+\beta \right){\theta}^3+3\left(\tilde{\upalpha}+\theta \right)\right)}^2\kern2.879999em \right.+\left(3+4{\beta}^2{\left(1+\theta \right)}^2-6\tilde{\upalpha}{\left(1+\theta \right)}^2+2\theta \left(3+\theta -{\theta}^2\right)-2\beta \left(1+\theta \right)\left(-2-\left(3-\theta \right)\theta +2\tilde{\upalpha}\left(1+\theta \right)\right)\right)\kern2.879999em \left(3+\left(3-\theta \right)\left(1-\theta \right)\theta +4{\beta}^3\left(1+\theta \right)-3\tilde{\upalpha}\left(\left(3-\theta \right)\theta +2\right)+\beta \left(7+\theta \left(5+\left(-7+\theta \right)\theta \right)-2\tilde{\upalpha}\left(5+\left(6-\theta \right)\theta \right)\right)\;\right.\kern0.36em \left.\left.\left.+{\beta}^2\left(8+6\theta -4\left(\tilde{\upalpha}+\tilde{\upalpha}\theta +{\theta}^2\right)\right)\right)\right)/\left(\left(1+\theta \right){\left(3+\left(3-2\theta \right)\theta +6\beta \left(1+\theta \right)\right)}^2\right)\right]/{\left(3+2\beta \right)}^2\kern4.319998em $$

(14)

$$ {\Pi}^H-{\Pi}^H\Big|{}_{\theta =0}=\left[-\left({\left(6+2\beta \left(7+4\beta -\tilde{\upalpha}\right)-3\tilde{\upalpha}\right)}^2+4\left(1+\beta \right){\left(3\tilde{\upalpha}+2\beta \left(2+2\beta +\tilde{\upalpha}\right)\right)}^2\right)/{\left(3+6\beta \right)}^2\right.\kern10.91999em +\left({\left(6+2\beta \left(7+4\beta -\tilde{\upalpha}\right)-3\tilde{\upalpha}+6\theta +\beta \left(19+12\beta \right)\theta +\left(-2+3\tilde{\upalpha}+\beta \left(3+4\beta +2\tilde{\upalpha}\right)\right){\theta}^2-\left(1+\beta \right){\theta}^3\right)}^2\right.+\left(2\left(1+\beta \right)\left(3\tilde{\upalpha}+2\beta \left(2+2\beta +\tilde{\upalpha}\right)\right)+\left(-3+9\tilde{\upalpha}+\beta \left(1+12\tilde{\upalpha}+4\beta \left(3+2\beta +\tilde{\upalpha}\right)\right)\right)\theta -\left(1+3\tilde{\upalpha}+2\beta \left(4+3\beta +\tilde{\upalpha}\right)\right){\theta}^2+\left(1+\beta \right){\theta}^3\right)\left(8{\beta}^2{\left(1+\theta \right)}^2+6\tilde{\upalpha}{\left(1+\theta \right)}^2+\theta \left(3+2\left(1-\theta \right)\theta \right)+2\beta \left(1+\theta \right)\left(4+\left(6-\theta \right)\theta +2\tilde{\upalpha}\left(1+\theta \right)\right)\right)/\kern12.95999em \left.\left(\left(1+\theta \right){\left(3+\left(3-2\theta \right)\theta +6\beta \left(1+\theta \right)\right)}^2\right)\right]/{\left(3+2\beta \right)}^2\kern18.24em $$

(15)

If $\tilde{\alpha}>\left(1+\upbeta \right)/\left(4+3\upbeta \right)$ and ${{\tilde{\uptheta}}_2}^H>\uptheta \ge {{\tilde{\uptheta}}_1}^H$, equilibrium prices are given in (Eq. 11), which leads to a profit difference of:

$$ {\Pi}^L-{\Pi}^L\Big|{}_{\uptheta =0}=-\frac{\left(1+4\beta {\left(1+\beta \right)}^2\right){\left(\beta +\tilde{\upalpha}\right)}^2}{{\left(3+2\beta \left(5+3\beta \right)\right)}^2}+\frac{{\left(\beta +\tilde{\upalpha}+\theta +\beta \theta \right)}^2\left(1+\beta \left(1+\beta \right)\left(4+\left(4-\theta \right)\theta +4\beta \left(1+\theta \right)\right)\right)}{\left(1+\theta \right){\left(3+\left(3-\theta \right)\theta +6{\beta}^2\left(1+\theta \right)+\beta \left(10+\left(10-\theta \right)\theta \right)\right)}^2} $$

(16)

$$ {\Pi}^H-{\Pi}^H\Big|{}_{\theta =0}=-\frac{4\left(1+\beta \right){\left(1+2\beta \right)}^2{\left(\beta +\tilde{\upalpha}\right)}^2}{{\left(3+2\beta \left(5+3\beta \right)\right)}^2}-\left(1-\tilde{\upalpha}\right)\left(\delta +\frac{2\beta \left(\beta +\tilde{\upalpha}\right)}{3+2\beta \left(5+3\beta \right)}\right)+\left[\left(\left(\beta +\tilde{\upalpha}+\theta +\beta \theta \right)\right.\right.\left(2\left(1+\beta \right)\left(1+2\beta \right)\left(\beta +\tilde{\upalpha}\right)+\left(-1+3\tilde{\upalpha}+2\beta \left(-1+5\tilde{\upalpha}+\beta \left(2+2\beta +3\tilde{\upalpha}\right)\right)\right)\theta -\left(1+\beta \right)\left(\beta +\tilde{\upalpha}\right){\theta}^2\right)\left.\left(2+\left(2-\theta \right)\theta +4\beta \left(1+\theta \right)\right)\right)/\left({\left(3+\left(3-\theta \right)\theta +6{\beta}^2\left(1+\theta \right)+\beta \left(10+\left(10-\theta \right)\theta \right)\right)}^2\right)\kern1.44em \left.+\left(1-\tilde{\upalpha}\right)\left(\tilde{\upalpha}+\frac{\left(\beta +\tilde{\upalpha}+\theta +\beta \theta \right)\left(\left(3-\theta \right)\theta +\beta \left(2+4\theta \right)\right)}{3+\left(3-\theta \right)\theta +6{\beta}^2\left(1+\theta \right)+\beta \left(10+\left(10-\theta \right)\theta \right)}\right)\right]/\left(1+\theta \right)\kern3.839998em $$

(17)

For either the cases (a) $\tilde{\alpha}>\left(1+\upbeta \right)/\left(4+3\upbeta \right)$and $\uptheta >{{\tilde{\uptheta}}^H}_2$ or (b) $0<\tilde{\alpha}<\left(1+\beta \right)/\left(4+3\beta \right)$and $\uptheta >{{\tilde{\uptheta}}^L}_2$, equilibrium prices are given in (Eq. 12), thus yielding a profit difference of:

$$ {\Pi}^L-{\Pi}^L\Big|{}_{\theta =0}=\frac{\theta \left(\tilde{\upalpha}+2{\tilde{\upalpha}}^2\theta +{\beta}^3\left(2-\tilde{\upalpha}\left(4-9\tilde{\upalpha}\right)\right)\left(1+\theta \right)+\beta \tilde{\upalpha}\left(4+\theta +\tilde{\upalpha}\left(3+9\theta \right)\right)+{\beta}^2\left(\theta +\tilde{\upalpha}\left(3-\theta +4\tilde{\upalpha}\left(3+4\theta \right)\right)\right)\right)}{\left(1+\beta \right){\left(1+3\beta \right)}^2{\left(1+\theta \right)}^2} $$

(18)

$$ {\Pi}^H-{\Pi}^H\Big|{}_{\uptheta =0}=\frac{\theta \left(\theta \hbox{-} \tilde{\alpha}\left(3-2\tilde{\alpha}\right)\theta +{\beta}^3\left(5-\tilde{\alpha}\left(10-9\tilde{\alpha}\right)\right)\left(1+\theta \right)+3\beta \left(1-\tilde{\alpha}\right)\left(1-\tilde{\alpha}+2\theta \hbox{-} 3\tilde{\alpha}\theta \right)+{\beta}^2\left(10+11\theta \hbox{-} 4\tilde{\alpha}\left(5+6\theta \hbox{-} \tilde{\alpha}\left(3+4\theta \right)\right)\right)\right)}{\left(1+\beta \right){\left(1+3\beta \right)}^2{\left(1+\theta \right)}^2} $$

(19)

For the final case in which $0<\tilde{\alpha}<\left(1+\beta \right)/\left(4+3\beta \right)$ and ${{\tilde{\uptheta}}_2}^L>\uptheta \ge {{\tilde{\uptheta}}_1}^L$, equilibrium prices are given by (Eq. 13), which yields a profit difference of:

$$ {\Pi}^L-{\Pi}^L\Big|{}_{\uptheta =0}=\frac{-\left(1+\beta \right){\left(1+2\beta \left(1+\beta \right)\right)}^2-4\left(1+\beta \right)\left(-1-\beta \left(1-2\beta \left(2+\beta \right)\right)\right)\tilde{\alpha}+\left(5+2\beta \left(23+2\beta \left(31+\beta \left(29+9\beta \right)\right)\right)\right){\tilde{\alpha}}^2}{{\left(3+2\beta \left(5+3\beta \right)\right)}^2}+\left(\left(\left(\left(\left(1+\beta \right)\left(1+2\beta \left(1+\beta -2\tilde{\alpha}\right)-2\tilde{\alpha}\right)+\left(1+\beta \left(3+2\beta \left(2+\beta -3\tilde{\alpha}\right)-10\tilde{\alpha}\right)-3\tilde{\alpha}\right)\theta -\left(1+\beta \right)\left(1+\beta -\tilde{\alpha}\right){\theta}^2\right)\kern1.56em \right.\right.\right.\left.\left(1+2\theta -{\theta}^3+2{\beta}^2{\left(1+\theta \right)}^2-\tilde{\alpha}\left(2+\left(2-\theta \right)\theta \right)-\beta \left(1+\theta \right)\left(-2+4\tilde{\alpha}-\left(2-\theta \right)\theta \right)\right)\right]\kern8.519994em /\left(-3-\left(3-\theta \right)\theta -6{\beta}^2\left(1+\theta \right)\hbox{-} \beta \left(10+\left(10-\theta \right)\theta \right)\right)+\tilde{\alpha}\left.\left(2{\beta}^2\left(-2+3\delta -\theta \right)\left(1+\theta \right)+\tilde{\alpha}\left(3+2\left(3-\theta \right)\theta \right)\hbox{-} \theta \left(3+\left(1-\theta \right)\theta \right)\hbox{-} \beta \left(4+\left(4-\theta \right)\theta \left(2+\theta \right)\hbox{-} \tilde{\alpha}\left(12+\left(14-\theta \right)\theta \right)\right)\right)\right)/\left(\left(1+\theta \right)\left(-3-\left(3-\theta \right)\theta -6{\beta}^2\left(1+\theta \right)-\beta \left(10+\left(10-\theta \right)\theta \right)\right)\right)\kern13.07999em $$

(20)

$$ {\Pi}^H-{\Pi}^H\Big|{}_{\uptheta =0}=-\frac{\left(1+4\beta {\left(1+\beta \right)}^2\right){\left(2+2\beta \hbox{-} \tilde{\alpha}\right)}^2}{{\left(3+2\beta \left(5+3\beta \right)\right)}^2}+\frac{{\left(2-\tilde{\alpha}+2\theta +2\beta \left(1+\theta \right)\right)}^2\left(1+\beta \left(1+\beta \right)\left(4-\left(4-\theta \right)\theta +4\beta \left(1+\theta \right)\right)\right)}{\left(1+\theta \right){\left(3+\left(3-\theta \right)\theta +6{\beta}^2\left(1+\theta \right)+\beta \left(10+\left(10-\theta \right)\theta \right)\right)}^2} $$

(21)

We utilize Mathematica to numerically calculate the profit differences over each region. Figure 5 illustrate the values of ${\overline{\tilde{\alpha}}}^L\left(\uptheta, \upbeta \right)$ and ${\overline{\tilde{\alpha}}}^H\left(\uptheta, \upbeta \right)$that are referred to in Proposition 2.

1.3 Proof of Proposition 3

Proposition 3 is based on our model of horizontal competition with asymmetric initial market share, where Firm A’s initial market share is κ ^A (< 1/2). Similar to our analysis of the base model, we find ᾶ^AB, which is defined as the customer who initially purchased from Firm A and is indifferent between repurchasing from Firm A (which yields a utility of V − p^A , O − θ(p^A , O − p^A , N) − t ⋅ x) and switching to Firm B (which yields a utility of V − p^B , N − t ⋅ (1 − x)). Similarly, ᾶ^BA is the customer who initially purchased from Firm B and is indifferent between repurchasing from Firm B (which yields a utility of V − p^B , O − θ(p^B , O − p^B , N) − t ⋅ (1 − x)) and switching to Firm A (which yields a utility of V − p^A , N − t ⋅ x). We define ᾰ^N as the newly-arriving customer who is indifferent between purchasing from either firm. Assuming the transaction cost is one, ᾶ^AB = 1/2 ⋅ (1 + p^B , N + θ ⋅ p^A , N − p^A , O(1 + θ)), ᾶ^BA = 1/2 ⋅ (1 + p^B , O − p^A , N + θ(p^B , O − p^B , N)), and ᾶ^N = 1/2 ⋅ (1 − p^A , N + p^B , N). The repurchasing customers who are located in the interval (0 < κ < ᾶ^AB) on the Hotelling line will repurchase from Firm A, those located at (ᾶ^AB < κ < κ ^A) will switch from Firm A to Firm B, those located at (κ ^A < κ < ᾶ^BA) will switch from Firm B to Firm A, and those located at (ᾶ^BA < κ < 1) will repurchase from Firm B. The profit function for each firm is as follows.

$$ {\Pi}^A={\mathrm{p}}^{A, O}\cdot \left({\overset{\smile }{\alpha}}^{A B}\right)+{\mathrm{p}}^{A, N}\cdot \left({\overset{\smile }{\alpha}}^{BA}-{\upkappa}^A\right)+\beta \cdot {\mathrm{p}}^{A, N}\cdot \left({\overset{\smile }{\alpha}}^N\right) $$

(22)

$$ {\Pi}^B={\mathrm{p}}^{B, O}\cdot \left(1-{\overset{\smile }{\alpha}}^{B A}\right)+{\mathrm{p}}^{B, N}\cdot \left({\upkappa}^A-{\overset{\smile }{\alpha}}^{B A}\right)+\beta \cdot {\mathrm{p}}^{B, N}\cdot \left(1-{\overset{\smile }{\alpha}}^N\right) $$

(23)

Taking the first derivatives of the profit function with respect to their prices and setting the first-order conditions to be zero $\left(\frac{\partial {\Pi}^A}{\partial {\mathrm{p}}^{A, N}}=\frac{\partial {\Pi}^A}{\partial {\mathrm{p}}^{A, O}}=\frac{\partial {\Pi}^B}{\partial {\mathrm{p}}^{B, N}}=\frac{\partial {\Pi}^B}{\partial {\mathrm{p}}^{B, O}}\equiv 0\right)$, we find:

$$ {\mathrm{p}}^{A, N}=\frac{2\left(1+\beta \right)}{3+2\beta}-\frac{1}{\left(3+2\beta \right)\left(1+\theta \right)}-\frac{2\left(-1+2{\kappa}^A\right)\left(1+\theta \right)}{3+\left(3-2\theta \right)\theta +6\beta \left(1+\theta \right)},{\mathrm{p}}^{A, O}=\frac{1+\beta}{3+2\beta}+\frac{1+\beta}{\left(3+2\beta \right)\left(1+\theta \right)}+\frac{-1-2{\kappa}^A\left(-1+\theta \right)+\theta}{3+\left(3-2\theta \right)\theta +6\beta \left(1+\theta \right)}{\mathrm{p}}^{B, N}=\frac{2\left(1+\beta \right)}{3+2\beta}-\frac{1}{\left(3+2\beta \right)\left(1+\theta \right)}-\frac{2\left(-1+2{\kappa}^A\right)\left(1+\theta \right)}{3+\left(3-2\theta \right)\theta +6\beta \left(1+\theta \right)},{\mathrm{p}}^{B, O}=\frac{1+\beta}{3+2\beta}+\frac{1+\beta}{\left(3+2\beta \right)\left(1+\theta \right)}+\frac{\left(-1+2{\kappa}^A\right)\left(-1+\theta \right)}{3+\left(3-2\theta \right)\theta +6\beta \left(1+\theta \right)} $$

(24)

This solution is only valid for ᾶ^AB < κ ^A. Let ${\overset{\smile }{\uptheta}}_1$ be the value of θ for which the prices in (Eq. 24) result in ᾶ^AB = κ ^A. (The closed form solution for ${\overset{\smile }{\uptheta}}_1$ is a function of β and κ ^A, but takes on a functional form that is too long to include here.) Thus, (Eq. 24) gives the equilibrium prices only if θ < ${\overset{\smile }{\uptheta}}_1$. Note that ${\overset{\smile }{\uptheta}}_1$ >0 only if κ ^A >$\frac{3+16\beta +12{\beta}^2}{12+44\beta +24{\beta}^2}$. Over this entire relevant range (θ < ${\overset{\smile }{\uptheta}}_1$), we find that p^A , O>p^A , Nand p^B , O>p^B , N.

For $\uptheta >{\overset{\smile }{\uptheta}}_1$, Firm A will price at the corner solution such that it retains all of its past customers, i.e., ᾶ^AB = κ ^A. In particular, the constraint that ᾶ^AB = κ ^A implies that Firm A sets ${\mathrm{p}}^{\mathrm{A},\mathrm{N}}=\frac{\left(1+\theta \right){p}^{A, O}-{p}^{B, N}+2{\kappa}^A-1}{\theta}$. Setting $\frac{\partial {\Pi}^A}{\partial {\mathrm{p}}^{A, O}}=\frac{\partial {\Pi}^B}{\partial {\mathrm{p}}^{B, N}}=\frac{\partial {\Pi}^B}{\partial {\mathrm{p}}^{B, O}}\equiv 0$, we find that the equilibrium prices are:

$$ {\mathrm{p}}^{A, N}=\frac{-6{\beta}^2{\left(1+\theta \right)}^2+2\beta \left(1+\theta \right)\left(-4+4{\kappa}^A+\left(-4+\theta \right)\theta \right)+{\kappa}^A\left(4-2\left(-2+\theta \right)\theta \right)+\left(1+\theta \right)\left(-3+\theta \left(-3+2\theta \right)\right)}{\left(1+\theta \right)\left(-3+\left(-3+\theta \right)\theta -6{\beta}^2\left(1+\theta \right)+\beta \left(-10+\left(-10+\theta \right)\theta \right)\right)}{\mathrm{p}}^{A, O}=\frac{-3+6{\beta}^2\left(-2+2{\kappa}^A-\theta \right)\left(1+\theta \right)+{\kappa}^A\left(6-4\left(-3+\theta \right)\theta \right)+\theta \left(-9+2\left(-1+\theta \right)\theta \right)+\beta \left(-16+{\kappa}^A\left(24-2\left(-14+\theta \right)\theta \right)+\theta \left(-27+\theta \left(-7+2\theta \right)\right)\right)}{\left(1+\theta \right)\left(-3+\left(-3+\theta \right)\theta -6{\beta}^2\left(1+\theta \right)+\beta \left(-10+\left(-10+\theta \right)\theta \right)\right)}\;{\mathrm{p}}^{B, N}=\frac{\left(\theta +2\beta \left(1+\theta \right)\right)\left(3-2{\kappa}^A+3\theta +3\beta \left(1+\theta \right)\right)}{\left(1+\theta \right)\left(3+\left(3-\theta \right)\theta +6{\beta}^2\left(1+\theta \right)+\beta \left(10+\left(10-\theta \right)\theta \right)\right)},{\mathrm{p}}^{B, O}=\frac{\left(3-2{\kappa}^A+3\theta +3\beta \left(1+\theta \right)\right)\left(1+\beta \left(2+\theta \right)\right)}{\left(1+\theta \right)\left(3+\left(3-\theta \right)\theta +6{\beta}^2\left(1+\theta \right)+\beta \left(10+\left(10-\theta \right)\theta \right)\right)}\kern3.599998em $$

(25)

These prices form a valid equilibrium only if ᾶ^BA ≥ κ ^A. Let ${\overset{\smile }{\theta}}_2$ be the value of θ for which the prices in (Eq. 25) result in ᾶ^BA = κ ^A, Thus, (A11) gives the equilibrium prices for the parameter region, ${\overset{\smile }{\uptheta}}_1\le \uptheta \le {\overset{\smile }{\uptheta}}_2$ where ${\overset{\smile }{\theta}}_2=\frac{3-6{\kappa}^A+\beta \left(8+3\beta -6\left(3+2\beta \right){\kappa}^A\right)+\sqrt{33+196\beta +386{\beta}^2+300{\beta}^3+81{\beta}^4-4\left(1+\beta \right)\left(23+3\beta \left(36+\beta \left(47+18\beta \right)\right)\right){\kappa}^A+4{\left(1+\beta \right)}^2\left(17+12\beta \left(5+3\beta \right)\right){\kappa}^{A2}}}{2\left(1+\beta \right)\left(2+3\beta -2{\kappa}^A\right)}$.

From (Eq. 25), we find that p^A , O>p^A , N over the entire relevant parameter region. Furthermore, p^B , O<p^B , N when θ > 1/(1 + β) and p^B , O>p^B , N when θ < 1/(1 + β).

Next, for ${\overset{\smile }{\uptheta}}_2<\uptheta$, both Firms A and B know that they will be at a corner solution, i.e., ᾶ^AB= ᾶ^BA = κ ^A. These two constraints imply ${\mathrm{p}}^{A, N}=\frac{\left(1+\theta \right){p}^{A, O}-{p}^{B, N}+2{\kappa}^A-1}{\theta}$ and ${\mathrm{p}}^{\mathrm{B},\mathrm{N}}=\frac{\left(1+\theta \right){p}^{B, O}-{p}^{A, N}-2{\kappa}^A+1}{\theta}$. Using $\frac{\partial {\Pi}^A}{\partial {\mathrm{p}}^{A, O}}=\frac{\partial {\Pi}^B}{\partial {\mathrm{p}}^{B, O}}\equiv 0$, we find that the equilibrium prices are:

$$ {\mathrm{p}}^{A, N}=\frac{\beta \left(1+3\beta \right)+\left(1+\beta \right)\left(3\beta +2{\kappa}^A\right)\theta}{\left(1+\beta \right)\left(1+3\beta \right)\left(1+\theta \right)},{\mathrm{p}}^{A, O}=\frac{1+3\theta -3{\beta}^2\left(-2+2{\kappa}^A-\theta \right)\left(1+\theta \right)+2{\kappa}^A\left(-1+\left(-2+\theta \right)\theta \right)+\beta \left(5+\theta \left(10+3\theta \right)+2{\kappa}^A\left(-4+\left(-5+\theta \right)\theta \right)\right)}{\left(1+\beta \right)\left(1+3\beta \right){\left(1+\theta \right)}^2}\;{\mathrm{p}}^{B, N}=\frac{\beta \left(1+3\beta \right)+\left(1+\beta \right)\left(2+3\beta -2{\kappa}^A\right)\theta}{\left(1+\beta \right)\left(1+3\beta \right)\left(1+\theta \right)},{\mathrm{p}}^{B, O}=\frac{\left(1+3\beta \right)\left(-1+2\left(1+\beta \right){\kappa}^A\right)+\left(-1+3{\beta}^2+2\left(1+\beta \right)\left(2+3\beta \right){\kappa}^A\right)\theta +\left(1+\beta \right)\left(2+3\beta -2{\kappa}^A\right){\theta}^2}{\left(1+\beta \right)\left(1+3\beta \right){\left(1+\theta \right)}^2}\kern1.22em $$

(26)

Using (Eq. 26), we find that p^B , O<p^B , N and p^A , O>p^A , N for all ${\uptheta >\overset{\smile }{\uptheta}}_2$.

1.4 Proof of Proposition 4

In Section 4.2, we add forward-looking customers to the base model. This model extension consists of a two-stage game, which can be solved using backwards induction. First, we examine the firms’ decisions in the second-period. Note that, at this stage, each firm will have observed the first-period’s market share, $\tilde{\upalpha}$. The second-period profit functions for the firms are:

$$ \begin{array}{l}{\Pi}_2^L={{\mathrm{p}}_2}^{L, O}\cdot \left({\tilde{\upalpha}}_{\mathrm{L}}\right)+{{\mathrm{p}}_2}^{L, N}\cdot \left({\tilde{\upalpha}}_{\mathrm{H}}-\tilde{\upalpha}\right)+{{\mathrm{p}}_2}^{L, N}\cdot \left({\tilde{\upalpha}}_{\mathrm{N}}\right)\\ {}{\Pi}_2^H={{\mathrm{p}}_2}^{H, O}\cdot \left(1-{\tilde{\upalpha}}_{\mathrm{H}}\right)+{{\mathrm{p}}_2}^{H, N}\cdot \left(\tilde{\upalpha}-{\tilde{\upalpha}}_{\mathrm{L}}\right)+{{\mathrm{p}}_2}^{H, N}\cdot \left(1-{\tilde{\upalpha}}_{\mathrm{N}}\right)\end{array} $$

(27)

Using the FOC’s and solving the simultaneous equations, we arrive at the following equilibrium prices:

$$ {{\mathrm{p}}_2}^{L, N}=\frac{1}{10}\left(4-\frac{1}{1+\theta}+\frac{5\left(1+2\theta +4\tilde{\alpha}\left(1+\theta \right)\right)}{-9+\theta \left(-9+2\theta \right)}\right),{{\mathrm{p}}_2}^{L, O}=\frac{1}{5}\left(1+\frac{1}{1+\theta}+\frac{5\left(2+\tilde{\alpha}\left(-1+\theta \right)\right)}{-9+\theta \left(-9+2\theta \right)}\right){{\mathrm{p}}_2}^{H, N}=\frac{1}{10}\left(4-\frac{1}{1+\theta}-\frac{5\left(1+2\theta +4\tilde{\alpha}\left(1+\theta \right)\right)}{-9+\theta \left(-9+2\theta \right)}\right),{{\mathrm{p}}_2}^{H, O}=\frac{1}{5}\left(1+\frac{1}{1+\theta}+\frac{5\left(-2+\tilde{\alpha}-\tilde{\alpha}\theta \right)}{-9+\theta \left(-9+2\theta \right)}\right) $$

(28)

Note that these prices are valid only if ${\tilde{\upalpha}}_{\mathrm{H}}\ge \tilde{\upalpha}$. Let ${\overset{\ddddot{}}{\uptheta}}_1\left(\updelta \right)$ be the value of θ for which the pricing solution in (Eq. 28) results in ${\tilde{\upalpha}}_{\mathrm{H}}=\tilde{\upalpha}$. Thus, (Eq. 28) gives the equilibrium prices over the region θ ≤ ${\overset{\ddddot{}}{\uptheta}}_1$ .

Now, we solve for the first-period market share, $\tilde{\upalpha}$. Consumers will choose from which firm to purchase in the first-period, accounting not only for the first-period prices charged by each firm (p ₁ ^L, p ₁ ^H), but also anticipating the second-period prices, which are given by (Eq. 28). Then, taking into account the overall utility of the customers, we try to find$\tilde{\upalpha}$. The utility (across both periods) earned by a customer with α quality taste will be $\mathrm{V}+\upalpha -{\mathrm{p}}_1^H+\updelta \cdot \left(\mathrm{V}-{{\mathrm{p}}_2}^{L, N}\right)$ if she purchases from Firm H and switches to Firm L. Her utility would be $\mathrm{V}-{\mathrm{p}}_1^L+\updelta \cdot \left(\mathrm{V}+\upalpha -{{\mathrm{p}}_2}^{H, N}\right)$ if she instead purchases from Firm L in the first-period and then switches to Firm H in the second-period (δ is the discount factor, which we assume is the same for customers and firms). In the second-period pricing structure given by (Eq. 28), the consumer with α = $\tilde{\upalpha}$ will be indifferent between these two purchasing patterns. Thus, we have

$$ \tilde{\alpha}=\left({\mathrm{p}}_1^H-{\mathrm{p}}_1^L-\updelta \left({{\mathrm{p}}_2}^{H, N}-{{\mathrm{p}}_2}^{L, N}\right)\right)/\left(1-\updelta \right) $$

(29)

Now consider the firms’ pricing decisions. The profit functions for the two firms are:

$$ {\Pi}_{Total}^L={\mathrm{p}}_1^L\cdot \left(\tilde{\upalpha}\right)+\updelta \cdot \left\lfloor {\Pi}_2^L\right\rfloor, {\Pi}_{Total}^H={\mathrm{p}}_1^H\cdot \left(1-\tilde{\upalpha}\right)+\updelta \cdot \left\lfloor {\Pi}_2^H\right\rfloor $$

(30)

Substituting for $\tilde{\upalpha}$ using (Eq. 29) and second-period prices using (Eq. 28), we take the derivative of profit (as given in Eq. 30) with respect to first-period prices, and then use these two first-order conditions to solve for the equilibrium first-period prices. Notice that each firm’s price will not only impact the revenue earned from first-period sales, but will also impact first-period market share, which, in turn, affects second-period revenue. The optimal first-period prices are:

$$ {\mathrm{p}}_1^L=\frac{\delta^2\left(\theta \left(\theta \left(\theta \left(4\theta \left(\theta \left(\theta \left(15-4\theta \right)+63\right)-200\right)-1599\right)+148\right)+1305\right)+522\right)+\delta \left(\theta \left(\theta \left(\theta \left(2\theta \left(14\theta -67\right)-149\right)+632\right)+982\right)+369\right)\left(\theta \left(2\theta -9\right)-9\right)-5\left(\theta +1\right){\left(\theta \left(2\theta -9\right)-9\right)}^3}{5\left(\theta +1\right)\left(\theta \left(2\theta -9\right)-9\right)\left(\delta \left(\theta \left(\theta \left(4\theta \left(3\theta -22\right)+79\right)+338\right)+171\right)-3{\left(\theta \left(2\theta -9\right)-9\right)}^2\right)}{\mathrm{p}}_1^H=\frac{1}{30}\left(20-14\delta -\frac{3\delta}{1+\theta}+\frac{9\delta \left(9+11\theta \right)}{9+\left(9-2\theta \right)\theta}-\frac{5\delta \left(198+\theta \left(497+\left(259-58\theta \right)\theta \right)+\delta \left(-138+\theta \left(-341+\theta \left(-163+58\theta \right)\right)\right)\right)}{-3{\left(-9+\theta \left(-9+2\theta \right)\right)}^2+\delta \left(171+\theta \left(338+\theta \left(79+4\theta \left(-22+3\theta \right)\right)\right)\right)}\right) $$

(31)

Substituting the prices from (Eq. 31) back into (Eq. 28) and (Eq. 29), we find the equilibrium second-period prices and the first-period market share are as follows:

$$ \tilde{\alpha}=\frac{{\left(9+\theta \left(9-2\theta \right)\right)}^2-2\delta \left(27+\theta \left(50+\theta \left(8+\theta \left(-15+2\theta \right)\right)\right)\right)}{3{\left(9+\theta \left(9-2\theta \right)\right)}^2-\delta \left(171+\theta \left(338+\theta \left(79+4\theta \left(-22+3\theta \right)\right)\right)\right)}{{\mathrm{p}}_2}^{L, N}=\frac{\left(9+\theta \left(9-2\theta \right)\right)\left(-23+4\theta \left(-13+\theta \left(-5+3\theta \right)\right)\right)+\delta \left(149+\theta \left(489+2\theta \left(216+\theta \left(-17+6\theta \left(-9+2\theta \right)\right)\right)\right)\right)}{5\left(1+\theta \right)\left(-3{\left(-9+\theta \left(-9+2\theta \right)\right)}^2+\delta \left(171+\theta \left(338+\theta \left(79+4\theta \left(-22+3\theta \right)\right)\right)\right)\right)}{{\mathrm{p}}_2}^{L, O}=\frac{\left(9+\theta \left(9-2\theta \right)\right)\left(-29+\theta \left(-51+2\theta \left(-5+3\theta \right)\right)\right)+\delta \left(182+\theta \left(497+\theta \left(326+\theta \left(-67+6\theta \left(-9+2\theta \right)\right)\right)\right)\right)}{5\left(1+\theta \right)\left(-3{\left(-9+\theta \left(-9+2\theta \right)\right)}^2+\delta \left(171+\theta \left(338+\theta \left(79+4\theta \left(-22+3\theta \right)\right)\right)\right)\right)}{{\mathrm{p}}_2}^{H, N}=\frac{\left(9+\theta \left(9-2\theta \right)\right)\left(-58+\theta \left(-137+2\theta \left(-35+6\theta \right)\right)\right)+\delta \left(364+\theta \left(1209+\theta \left(1157+2\theta \left(43+4\theta \left(-26+3\theta \right)\right)\right)\right)\right)}{5\left(1+\theta \right)\left(-3{\left(-9+\theta \left(-9+2\theta \right)\right)}^2+\delta \left(171+\theta \left(338+\theta \left(79+4\theta \left(-22+3\theta \right)\right)\right)\right)\right)}{{\mathrm{p}}_2}^{H, O}=\frac{1}{5}\left(1+\frac{1}{1+\theta}-\frac{5\left(-\left(5+\theta \right)\left(-9+\theta \left(-9+2\theta \right)\right)+2\delta \left(-16+\theta \left(-19+\theta \left(2+\theta \right)\right)\right)\right)}{-3{\left(-9+\theta \left(-9+2\theta \right)\right)}^2+\delta \left(171+\theta \left(338+\theta \left(79+4\theta \left(-22+3\theta \right)\right)\right)\right)}\right) $$

(32)

For all $0\le \uptheta \le {\overset{\ddddot{}}{\uptheta}}_1\left(\updelta \right)$, using the expressions in (Eq. 32), we find p ₂ ^L,O > p ₂ ^L,N and p ₂ ^H,O > p ₂ ^H,N.

Now consider the parameter region $\uptheta \ge {\overset{\ddddot{}}{\uptheta}}_1\left(\updelta \right)$. In this region, we will be at a corner solution in the second period such that the high-quality firm retains all of its past customers, which occurs at ${{\mathrm{p}}_2}^{L, O}=\frac{\tilde{\alpha}+{p_2}^{L, N}+{{p_2}^{H, N}}^{\ast}\theta\ }{1+\theta}$. Using this equation for p ^L,O, along with the 3 other FOC’s, solving the simultaneous equations results in the following equilibrium prices:

$$ \begin{array}{l}{{\mathrm{p}}_2}^{L, N}=\frac{\left(1+\tilde{\alpha}+2\theta \right)\left(2+3\theta \right)}{\left(1+\theta \right)\left(19+\theta \left(19-2\theta \right)\right)},{{\mathrm{p}}_2}^{L, O}=\frac{\left(3+\theta \right)\left(1+\tilde{\alpha}+2\theta \right)}{\left(1+\theta \right)\left(19+\theta \left(19-2\theta \right)\right)}\\ {}{{\mathrm{p}}_2}^{H, N}=\frac{\left(1+\tilde{\alpha}+2\theta \right)\left(6+\left(6-\theta \right)\theta \right)}{\left(1+\theta \right)\left(19+\theta \left(19-2\theta \right)\right)},{{\mathrm{p}}_2}^{H, O}=1-\frac{3-4\tilde{\alpha}}{1+\theta}+\frac{5\left(8-\tilde{\alpha}\left(11-\theta \right)-2\theta \right)}{19+\theta \left(19-2\theta \right)}\end{array} $$

(33)

Similar to the previous procedure, we turn to the first-period. Taking into account the overall utility of the customers, we try to find$\tilde{\upalpha}$. The utility earned by a customer with α quality taste will be $\mathrm{V}+\upalpha -{\mathrm{p}}_1^H+\updelta \cdot \left(\mathrm{V}+\upalpha -{{\mathrm{p}}_2}^{H, O}-\uptheta \cdot \left({{\mathrm{p}}_2}^{H, O}-{{\mathrm{p}}_2}^{H, N}\right)\right)$ if she purchases from Firm H in both periods. Her utility would be $\mathrm{V}-{\mathrm{p}}_1^L+\updelta \cdot \left(\mathrm{V}+\upalpha -{{\mathrm{p}}_2}^{H, N}\right)$ if she instead purchases from Firm L in the first-period and then switches to Firm H in the second-period. The consumer with α = $\tilde{\upalpha}$ will be indifferent between these two purchasing patterns. Thus, we have

$$ \tilde{\alpha}={\mathrm{p}}_1^H-{\mathrm{p}}_1^L+\updelta \left({{\mathrm{p}}_2}^{H, O}-{{\mathrm{p}}_2}^{H, N}\right)\left(1+\uptheta \right) $$

(34)

Taking the same steps as before, the optimal first-period prices are:

$$ \begin{array}{l}{\mathrm{p}}_1^L=\left\{\begin{array}{l}\left(1+\uptheta \right){\left[19+\uptheta \left(19-2\uptheta \right)\right]}^3-\updelta \left(19+\uptheta \left(19-2\uptheta \right)\right)\left(243+\uptheta \left(1339+\uptheta \left(1955+\uptheta \left(651-4\uptheta \left(52-3\uptheta \right)\right)\right)\right)\right)\\ {}+{\updelta}^2\left(-2568+\uptheta \left(757+\uptheta \Big(20694+\uptheta \left(25513+2\uptheta \left(2175-\uptheta \left(1697-214\uptheta +{\uptheta}^2\right)\right)\right)\right)\right)\end{array}\right\}\\ {}/\left\{\left(-19+\uptheta \left(-19+2\theta \right)\right){\left[3\left(1+\uptheta \right)\left(19+\uptheta \left(19-2\uptheta \right)\right)\right]}^2+\updelta \left(235+\uptheta \left(1468+\uptheta \left(2317+2\uptheta \left(429-\uptheta \left(115-6\uptheta \right)\right)\right)\right)\right)\right\}\\ {}{\mathrm{p}}_1^H=\left\{\begin{array}{l}2\left(1+\uptheta \right){\left(\left(19+\uptheta \left(19-2\uptheta \right)\right)\right)}^3-\updelta \left(19+\uptheta \left(19-2\uptheta \right)\right)\left(122+\uptheta \left(1491+\uptheta \left(2846+\uptheta \left(33-2\uptheta \right)\left(37-6\uptheta \right)\right)\right)\right)\\ {}+{\updelta}^2\left(-6178+\uptheta \left(-9465+\uptheta \left(15146+\uptheta \left(31339+2\uptheta \left(4552-\uptheta \left(1779-4\left(39-\uptheta \right)\uptheta \right)\right)\right)\right)\right)\right)\end{array}\right\}\\ {}/\left\{\left(-19+\uptheta \left(-19+2\theta \right)\right){\left[3\left(1+\uptheta \right)\left(19+\uptheta \left(19-2\uptheta \right)\right)\right]}^2+\updelta \left(235+\uptheta \left(1468+\uptheta \left(2317+2\uptheta \left(429-\uptheta \left(115-6\uptheta \right)\right)\right)\right)\right)\right\}\end{array} $$

(35)

Then we find the equilibrium second-period prices and first-period market share are as follows:

$$ \begin{array}{l}\tilde{\alpha}=\frac{\left(1+\theta \right){\left(19+\theta \left(19-2\theta \right)\right)}^2+\delta \left(178-\theta \left(57+\theta \left(802+\theta \left(-9+2\theta \right)\left(-49+4\theta \right)\right)\right)\right)}{3\left(1+\theta \right){\left(19+\theta \left(19-2\theta \right)\right)}^2-\delta \left(235-\theta \left(1468+\theta \left(2317+2\theta \left(429+\theta \left(-115+6\theta \right)\right)\right)\right)\right)}\\ {}{{\mathrm{p}}_2}^{L, N}=\frac{\left(2+3\theta \right)\left(2\left(38+\theta \left(95+\left(53-6\theta \right)\theta \right)\right)+\delta \left(-3+\theta \left(-99+2\theta \left(-59+6\theta \right)\right)\right)\right)}{3\left(1+\theta \right){\left(19+\theta \left(19-2\theta \right)\right)}^2-\delta \left(235+\theta \left(1468+\theta \left(2317+2\theta \left(429+\theta \left(-115+6\theta \right)\right)\right)\right)\right)}\\ {}{{\mathrm{p}}_2}^{L, O}=\frac{\left(3+\theta \right)\left(2\left(38+\theta \left(95+\left(53-6\theta \right)\theta \right)\right)+\delta \left(-3+\theta \left(-99+2\theta \left(-59+6\theta \right)\right)\right)\right)}{3\left(1+\theta \right){\left(19+\theta \left(19-2\theta \right)\right)}^2-\delta \left(235+\theta \left(1468+\theta \left(2317+2\theta \left(429+\theta \left(-115+6\theta \right)\right)\right)\right)\right)}\\ {}{{\mathrm{p}}_2}^{H, N}=\frac{\left(-6+\left(-6+\theta \right)\theta \right)\left(2\left(38+\theta \left(95+\left(53-6\theta \right)\theta \right)\right)+\delta \left(-3+\theta \left(-99+2\theta \left(-59+6\theta \right)\right)\right)\right)}{-3\left(1+\theta \right){\left(-19+\theta \left(-19+2\theta \right)\right)}^2+\delta \left(235+\theta \left(1468+\theta \left(2317+2\theta \left(429+\theta \left(-115+6\theta \right)\right)\right)\right)\right)}\\ {}{{\mathrm{p}}_2}^{H, O}=\frac{\left(1+\theta \right)\left(19+\theta \left(19-2\theta \right)\right)\left(-27+\theta \left(-59+6\left(-6+\theta \right)\theta \right)\right)+\delta \left(-172+\theta \left(282+\theta \left(1947+\theta \left(2070+\theta \left(367+2\theta \left(-91+6\theta \right)\right)\right)\right)\right)\right)}{\left(1+\theta \right)\left(-3\left(1+\theta \right){\left(-19+\theta \left(-19+2\theta \right)\right)}^2+\delta \left(235+\theta \left(1468+\theta \left(2317+2\theta \left(429+\theta \left(-115+6\theta \right)\right)\right)\right)\right)\right)}\end{array} $$

(36)

From (Eq. 36), we find that p ₂ ^H,O > p ₂ ^H,N for all $\uptheta \ge {\overset{\ddddot{}}{\uptheta}}_1\left(\updelta \right)$. More importantly, it is clear that p ₂ ^L,O < p ₂ ^L,N if 3 + θ < 2 + 3 θ, which reduces to θ > ½. (Notice that (p ₂ ^L,O - p ₂ ^L,N) does not depend on δ.)

To illustrate how first-period market share and second-period prices behave, we graph them as a function of θ (assuming δ = .9):

1.5 Proof of Proposition 5

For the historical reference price effect, we start by noting that, by definition, we must have ${\tilde{\upalpha}}_{\mathrm{R}\mathrm{L}}\le {\tilde{\upalpha}}_{\mathrm{R}}$ and ${\tilde{\upalpha}}_{\mathrm{R}\mathrm{H}}\ge {\tilde{\upalpha}}_{\mathrm{R}}$. We follow the same steps as the forward-looking model analysis. First, we look at the firms’ decisions in the second-period. Note that, at this stage, each firm will have observed the first-period’s market share, ${\tilde{\upalpha}}_{\mathrm{R}}$. The second-period profit functions for the firms are the same as the forward-looking model profit function (Eq. 27). First, consider the region $0\le \uptheta \le {\widehat{\overline{\uptheta}}}_1\left(\updelta \right)$. Using the FOC’s and solving the simultaneous equations, we find the equilibrium prices (which are all a function of ${\tilde{\upalpha}}_{\mathrm{R}}$), where ${\widehat{\overline{\uptheta}}}_1\left(\updelta \right)$ is the value of θ with respect to δ for which this solution results in ${\tilde{\upalpha}}_{\mathrm{R}\mathrm{H}}={\tilde{\upalpha}}_{\mathrm{R}}$. Then, we turn to the first-period and determine the first-period market share. Consumers will choose from which firm to purchase in the first-period, accounting not only for the first-period prices charged by each firm (p _R1 ^L, p _R1 ^H), but also anticipating the second-period prices. Then, taking into account the overall utility of the customers, we try to find $\tilde{\upalpha}$. For the marginal customer located at ${\tilde{\upalpha}}_{\mathrm{R}}$, the utility (across both periods) from buying from Firm H in the first period and then switching to Firm L in the second period will be equal to the utility gained through buying from Firm L in the first period and then switching to Firm H in the second period:

$$ {\tilde{\alpha}}_{\mathrm{R}}=\left({\mathrm{p}}_{R1}^H-{\mathrm{p}}_{R1}^L-\updelta \left(\Big({{\mathrm{p}}_{R2}}^{H, N}-{{\mathrm{p}}_{R2}}^{L, N}\right)\right)/\left(1-\updelta \right) $$

(37)

Now we consider the firms’ pricing decisions from total profits, similar to (Eq. 30). We follow the same steps as the forward-looking extension model. The optimal first-period prices are:

$$ {\mathrm{p}}_{R1}^L=\left(1/2\gamma \right)\left(\left(-1+\frac{5\left(1+\gamma \right)\left(-90+8{\gamma}^2\delta -5\gamma \left(9+\delta \right)\right)}{-225+\gamma \left(-225+\left(-35+\gamma \left(69+8\gamma -40\delta \right)\right)\delta \right)}+\frac{405\left(1+\gamma \right)\left(3+\gamma \right)-\left(855+\gamma \left(1222+463\gamma \right)\right)\delta +10\gamma \left(11+15\gamma \right){\delta}^2}{45\left(-27+19\delta \right)+\gamma \left(-1215+\delta \left(862-55\delta +\gamma \left(287+8\gamma \left(-1+\delta \right)+\delta \left(-263+40\delta \right)\right)\right)\right)}\right)\right){\mathrm{p}}_{R1}^H=\left(1/2\gamma \right)\left(-3+\frac{5\left(1+\gamma \right)\left(-90+8{\gamma}^2\delta -5\gamma \left(9+\delta \right)\right)}{-225+\gamma \left(-225+\left(-35+\gamma \left(69+8\gamma -40\delta \right)\right)\delta \right)}+\frac{-405\left(1+\gamma \right)\left(3+\gamma \right)+\left(855+\gamma \left(1222+463\gamma \right)\right)\delta -10\gamma \left(11+15\gamma \right){\delta}^2}{45\left(-27+19\delta \right)+\gamma \left(-1215+\delta \left(862-55\delta +\gamma \left(287+8\gamma \left(-1+\delta \right)+\delta \left(-263+40\delta \right)\right)\right)\right)}\right) $$

(38)

In order to find the equilibrium second-period prices and the first-period market share, we substitute the prices from (Eq. 38) into (Eq. 37) and the (unreported) expressions for second-period prices:

$$ \begin{array}{l}{\tilde{\alpha}}_R=\frac{-405\left(1+\gamma \right)+\left(270+\gamma \left(131+53\gamma \right)\right)\delta +10\left(5-3\gamma \right){\gamma \delta}^2}{45\left(-27+19\delta \right)+\gamma \left(-1215+\delta \left(862-55\delta +\gamma \left(287+8\gamma \left(-1+\delta \right)+\delta \left(-263+40\delta \right)\right)\right)\right)}\\ {}{{\mathrm{p}}_{\mathrm{R}2}}^{L, O}=\frac{-90+8{\gamma}^2\delta -5\gamma \left(9+\delta \right)}{-225+\gamma \left(-225+\left(-35+\gamma \left(69+8\gamma -40\delta \right)\right)\delta \right)}+\frac{225-160\delta +\gamma \left(90+\delta \left(-97+30\delta \right)\right)}{45\left(-27+19\delta \right)+\gamma \left(-1215+\delta \left(862-55\delta +\gamma \left(287+8\gamma \left(-1+\delta \right)+\delta \left(-263+40\delta \right)\right)\right)\right)}\\ {}{{\mathrm{p}}_{\mathrm{R}2}}^{L, N}=\frac{1}{2}\left(1-\frac{\left(1+\gamma \right)\left(-90+8{\gamma}^2\delta -5\gamma \left(9+\delta \right)\right)}{-225+\gamma \left(-225+\left(-35+\gamma \left(69+8\gamma -40\delta \right)\right)\delta \right)}+\frac{45\left(1+\gamma \right)\left(7+\gamma \right)-\left(215+\gamma \left(194+75\gamma \right)\right)\delta +10\gamma \left(-1+3\gamma \right){\delta}^2}{45\left(-27+19\delta \right)+\gamma \left(-1215+\delta \left(862-55\delta +\gamma \left(287+8\gamma \left(-1+\delta \right)+\delta \left(-263+40\delta \right)\right)\right)\right)}\right)\\ {}{{\mathrm{p}}_{\mathrm{R}2}}^{H, O}=\frac{-90+8{\gamma}^2\delta -5\gamma \left(9+\delta \right)}{-225+\gamma \left(-225+\left(-35+\gamma \left(69+8\gamma -40\delta \right)\right)\delta \right)}+\frac{5\left(-45+32\delta \right)+\gamma \left(-90+\left(97-30\delta \right)\delta \right)}{45\left(-27+19\delta \right)+\gamma \left(-1215+\delta \left(862-55\delta +\gamma \left(287+8\gamma \left(-1+\delta \right)+\delta \left(-263+40\delta \right)\right)\right)\right)}\\ {}{{\mathrm{p}}_{\mathrm{R}2}}^{H, N}=\frac{1}{2}\left(1-\frac{\left(1+\gamma \right)\left(-90+8{\gamma}^2\delta -5\gamma \left(9+\delta \right)\right)}{-225+\gamma \left(-225+\left(-35+\gamma \left(69+8\gamma -40\delta \right)\right)\delta \right)}+\frac{-45\left(1+\gamma \right)\left(7+\gamma \right)+\left(215+\gamma \left(194+75\gamma \right)\right)\delta +10\left(1-3\gamma \right){\gamma \delta}^2}{45\left(-27+19\delta \right)+\gamma \left(-1215+\delta \left(862-55\delta +\gamma \left(287+8\gamma \left(-1+\delta \right)+\delta \left(-263+40\delta \right)\right)\right)\right)}\right)\end{array} $$

(39)

For all $0\le \uptheta \le {\widehat{\overline{\uptheta}}}_1\left(\updelta \right)$, using the expressions in (Eq. 39), we find p _R2 ^LO > p _R2 ^LN and p _R2 ^HO > p _R2 ^HN.

Now consider the parameter region ${\widehat{\overline{\uptheta}}}_1\left(\updelta \right)\le \uptheta \le {\widehat{\overline{\uptheta}}}_2\left(\updelta \right)$, where ${\widehat{\overline{\uptheta}}}_2\left(\updelta \right)$ is the value of θ with respect to δ for which the pricing solution from (Eq. 39) results in ${\tilde{\upalpha}}_{\mathrm{R}\mathrm{L}}={\tilde{\upalpha}}_{\mathrm{R}}$. This occurs when ${{\mathrm{p}}_{R2}}^{H, O}=\frac{\tilde{\alpha}+{p_{R2}}^{L, N}+{p_{R1}}^{H\ast}\gamma\ }{1+\gamma}$. Following the same steps as in the forward-looking model, we derive the equilibrium prices and market share:

$$ \begin{array}{l}{\mathrm{p}}_{R1}^L=\frac{\left(-6859{\left(1+\gamma \right)}^2+57\left(1+\gamma \right)\left(81+\gamma \left(23+8\gamma \right)\right)\delta +\left(2568+\gamma \left(12759+\left(10063-392\gamma \right)\gamma \right)\right){\delta}^2-\gamma \left(6745+\gamma \left(4111+456\gamma \right)\right){\delta}^3+196{\gamma}^2\left(-7+2\gamma \right){\delta}^4\right)}{\left(-20577{\left(1+\gamma \right)}^2+19\left(1+\gamma \right)\left(235+\gamma \left(-127+330\gamma +8{\gamma}^2\right)\right)\delta +2\gamma \left(8833+\gamma \left(7327-4\gamma \left(333+17\gamma \right)\right)\right){\delta}^2-2{\gamma}^2\left(233+\gamma \left(3015+76\gamma \right)\right){\delta}^3+8{\gamma}^3\left(368+17\gamma \right){\delta}^4\right)}\\ {}{\mathrm{p}}_{R1}^H=\frac{-13718{\left(1+\gamma \right)}^2+38\left(1+\gamma \right)\left(61+\gamma \left(21+2\gamma \left(58+\gamma \right)\right)\right)\delta +2\left(3089+\gamma \left(11422+\gamma \left(7929-2\gamma \left(599+34\gamma \right)\right)\right)\right){\delta}^2-2\gamma \left(3874+\gamma \left(4285+8\gamma \left(359+\gamma \right)\right)\right){\delta}^3+4{\gamma}^2\left(48+\gamma \left(1189+34\gamma \right)\right){\delta}^4-60{\gamma}^3\left(13+\gamma \right){\delta}^5}{\left(-20577{\left(1+\gamma \right)}^2+19\left(1+\gamma \right)\left(235+\gamma \left(-127+330\gamma +8{\gamma}^2\right)\right)\delta +2\gamma \left(8833+\gamma \left(7327-4\gamma \left(333+17\gamma \right)\right)\right){\delta}^2-2{\gamma}^2\left(233+\gamma \left(3015+76\gamma \right)\right){\delta}^3+8{\gamma}^3\left(368+17\gamma \right){\delta}^4\right)}\\ {}{\tilde{\alpha}}_R=\frac{-\left(-6859{\left(1+\gamma \right)}^2+38\left(1+\gamma \right)\left(-89+\gamma \left(-61+\gamma \left(85+2\gamma \right)\right)\right)\delta +\gamma \left(5639+\gamma \left(6327-20\gamma \left(28+3\gamma \right)\right)\right){\delta}^2-2{\gamma}^2\left(603+\gamma \left(2055+38\gamma \right)\right){\delta}^3+12{\gamma}^3\left(163+5\gamma \right){\delta}^4\right)}{\left(-20577{\left(1+\gamma \right)}^2+19\left(1+\gamma \right)\left(235+\gamma \left(-127+330\gamma +8{\gamma}^2\right)\right)\delta +2\gamma \left(8833+\gamma \left(7327-4\gamma \left(333+17\gamma \right)\right)\right){\delta}^2-2{\gamma}^2\left(233+\gamma \left(3015+76\gamma \right)\right){\delta}^3+8{\gamma}^3\left(368+17\gamma \right){\delta}^4\right)}\\ {}{\mathrm{p}}_{\mathrm{R}2}^{L, O}=\frac{\left(57\left(-76+3\delta \right)+\gamma \left(-7220+4{\gamma}^2\left(-1+\delta \right)\delta \left(1+\delta \right)\left(-57+49\delta \right)+9\delta \left(133+529\delta \right)-4\gamma \left(722+\delta \left(-513+\delta \left(-981+776\delta \right)\right)\right)\right)\right)}{\left(-20577{\left(1+\gamma \right)}^2+19\left(1+\gamma \right)\left(235+\gamma \left(-127+330\gamma +8{\gamma}^2\right)\right)\delta +2\gamma \left(8833+\gamma \left(7327-4\gamma \left(333+17\gamma \right)\right)\right){\delta}^2-2{\gamma}^2\left(233+\gamma \left(3015+76\gamma \right)\right){\delta}^3+8{\gamma}^3\left(368+17\gamma \right){\delta}^4\right)}\\ {}{\mathrm{p}}_{\mathrm{R}2}^{L, N}=\frac{\left(361\left(-8+\gamma \right){\left(1+\gamma \right)}^2+19\left(1+\gamma \right)\left(6+\gamma \left(-39+49\gamma \right)\right)\delta +\gamma \left(2318+\left(1537-869\gamma \right)\gamma \right){\delta}^2+\left(179-851\gamma \right){\gamma}^2{\delta}^3+588{\gamma}^3{\delta}^4\right)}{\left(-20577{\left(1+\gamma \right)}^2+19\left(1+\gamma \right)\left(235+\gamma \left(-127+330\gamma +8{\gamma}^2\right)\right)\delta +2\gamma \left(8833+\gamma \left(7327-4\gamma \left(333+17\gamma \right)\right)\right){\delta}^2-2{\gamma}^2\left(233+\gamma \left(3015+76\gamma \right)\right){\delta}^3+8{\gamma}^3\left(368+17\gamma \right){\delta}^4\right)}\\ {}{\mathrm{p}}_{\mathrm{R}2}^{H, O}=\frac{-361{\left(1+\gamma \right)}^2\left(27+37\gamma \right)+19\left(1+\gamma \right)\left(-172+\gamma \left(-39+\gamma \left(261+4\gamma \left(59+\gamma \right)\right)\right)\right)\delta +\gamma \left(14135+\gamma \left(30708+\gamma \left(14429-8\gamma \left(307+17\gamma \right)\right)\right)\right){\delta}^2-{\gamma}^2\left(8775+\gamma \left(13531+4\gamma \left(1455+4\gamma \right)\right)\right){\delta}^3+8{\gamma}^3\left(342+\gamma \left(602+17\gamma \right)\right){\delta}^4-60{\gamma}^4\left(13+\gamma \right){\delta}^5}{\left(1+\gamma \right)\left(-20577{\left(1+\gamma \right)}^2+19\left(1+\gamma \right)\left(235+\gamma \left(-127+330\gamma +8{\gamma}^2\right)\right)\delta +2\gamma \left(8833+\gamma \left(7327-4\gamma \left(333+17\gamma \right)\right)\right){\delta}^2-2{\gamma}^2\left(233+\gamma \left(3015+76\gamma \right)\right){\delta}^3+8{\gamma}^3\left(368+17\gamma \right){\delta}^4\right)}\\ {}{\mathrm{p}}_{\mathrm{R}2}^{H, N}=\frac{3\left(361\left(-8+\gamma \right){\left(1+\gamma \right)}^2+19\left(1+\gamma \right)\left(6+\gamma \left(-39+49\gamma \right)\right)\delta +\gamma \left(2318+\left(1537-869\gamma \right)\gamma \right){\delta}^2+\left(179-851\gamma \right){\gamma}^2{\delta}^3+588{\gamma}^3{\delta}^4\right)}{\left(-20577{\left(1+\gamma \right)}^2+19\left(1+\gamma \right)\left(235+\gamma \left(-127+330\gamma +8{\gamma}^2\right)\right)\delta +2\gamma \left(8833+\gamma \left(7327-4\gamma \left(333+17\gamma \right)\right)\right){\delta}^2-2{\gamma}^2\left(233+\gamma \left(3015+76\gamma \right)\right){\delta}^3+8{\gamma}^3\left(368+17\gamma \right){\delta}^4\right)}\end{array} $$

(40)

Using the expression in (Eq. 40), we find that ${\mathrm{p}}_{\mathrm{R}2}^{L, O}>{\mathrm{p}}_{\mathrm{R}2}^{L, N}$ for ${\widehat{\overline{\uptheta}}}_1\left(\updelta \right)\le {\uptheta <\overline{\overline{\uptheta}}}_2\left(\updelta \right)$ and ${\mathrm{p}}_{\mathrm{R}2}^{L, O}<{\mathrm{p}}_{\mathrm{R}2}^{L, N}$ for ${\overline{\overline{\uptheta}}}_2\left(\updelta \right)<\uptheta \le {\widehat{\overline{\uptheta}}}_2\left(\updelta \right)$, where ${\overline{\overline{\uptheta}}}_2\left(\updelta \right)$is the value of θ with respect to δ for which the solution results in ${\mathrm{p}}_{\mathrm{R}2}^{L, O}={\mathrm{p}}_{\mathrm{R}2}^{L, N}$ (which is derived by the aid of Mathematica software and shown in Fig. 7).

Finally consider ${\widehat{\overline{\uptheta}}}_2\left(\updelta \right)<\uptheta$. In this region, both firms are at the corner solution in the second period in which they will each retain all of their past customers. This implies that we have the constraint ${{\mathrm{p}}_{R2}}^{L, O}=\frac{\hbox{-} \tilde{\alpha}+{p_{R2}}^{H, N}+{p_{R1}}^{L\ast}\gamma\ }{1+\gamma}$, in addition to the previous constraint that ${{\mathrm{p}}_{R2}}^{H, O}=\frac{\tilde{\alpha}+{p_{R2}}^{L, N}+{p_{R1}}^{H\ast}\gamma\ }{1+\gamma}$. Following the same steps as above, we derive the equilibrium prices and first-period market share:

$$ \begin{array}{l}{\mathrm{p}}_{R1}^L=\frac{-8+\delta \left(3+\delta \right)+2{\gamma}^2\left(1-\delta \right){\left(1+\delta \right)}^2\left(-4+5\delta \right)+\gamma \left(1+\delta \right)\left(-16+\delta \left(19+\delta \right)\right)}{8\left(1+\gamma \left(1+\delta \right)\right)\left(-3-\delta +3\gamma \left({\delta}^2\hbox{-} 1\right)\right)},{\mathrm{p}}_{R1}^H=\frac{-16+\delta \left(3+\delta \right)+2{\gamma}^2\left(1-\delta \right){\left(1+\delta \right)}^2\left(-8+7\delta \right)+\gamma \left(-32+\delta +28{\delta}^2-5{\delta}^3\right)}{8\left(1+\gamma \left(1+\delta \right)\right)\left(-3-\delta +3\gamma \left({\delta}^2\hbox{-} 1\right)\right)}\\ {}{\tilde{\alpha}}_R=\frac{-4+\gamma \left(-4+\delta +5{\delta}^2\right)}{4\left(-3-\delta +3\gamma \left(-1+{\delta}^2\right)\right)},{\mathrm{p}}_{\mathrm{R}2}^{L, O}=\frac{-1-3\delta -\gamma \left(10+\delta \left(5+\delta \right)-2{\gamma}^2\left(1-\delta \right){\left(1+\delta \right)}^2\left(-4+5\delta \right)-17\gamma \left({\delta}^2\hbox{-} 1\right)\right)}{8\left(1+\gamma \left(1+\delta \right)\right)\left(-3-\delta +3\gamma \left({\delta}^2\hbox{-} 1\right)\right)},{\mathrm{p}}_{\mathrm{R}2}^{L, N}=\frac{1}{8}\\ {}{\mathrm{p}}_{\mathrm{R}2}^{H, O}=\frac{1}{72}\left(\Big(48+\frac{4}{1+\gamma}-\frac{18}{1+\gamma \left(1+\delta \right)}-\left(\Big(42\gamma \delta +\frac{3+3\gamma +25\delta +21\gamma \delta}{3+3\gamma +\delta -3{\gamma \delta}^2}\right)/\left(1+\gamma \right)\right),{\mathrm{p}}_{\mathrm{R}2}^{H, N}=\frac{3}{8}\end{array} $$

(41)

Using (Eq. 41), we calculate the price discrimination across new and old customers that occurs in the second period $\left({\mathrm{p}}_{\mathrm{R}2}^{L, O}-{\mathrm{p}}_{\mathrm{R}2}^{L, N},{\mathrm{p}}_{\mathrm{R}2}^{H, O}-{\mathrm{p}}_{\mathrm{R}2}^{H, N}\right)$. These price expressions can also be used to calculate the price shift (increase/decrease) over time: ${\mathrm{p}}_{\mathrm{R}2}^{L, O}-{\mathrm{p}}_{\mathrm{R}1}^{L,},{\mathrm{p}}_{\mathrm{R}2}^{H, O}-{\mathrm{p}}_{\mathrm{R}1}^{H,}$. In the figures below, we illustrate the magnitude and sign of the price discrimination (δ = 0.9 (left) and δ = 0.5 (right)). We also provide the results from the forward-looking model (with social price comparisons) for comparison. Notice that in the left figure, we have p ^L,O - p ^L,N < 0 for all θ > .5, whereas p _R2 ^L,O - p _R2 ^L,N < 0 only for γ > .7. In the right figure, we again have p ^L,O - p ^L,N < 0 for all θ > .5. But, p _R2 ^L,O - p _R2 ^L,N remains positive. These results indicate that, as discussed in the text, social price comparisons offer a more robust explanation for offering lower prices to one’s past customers (than do reference price effects).