Behavior-Based Price Discrimination and Product Choice

Abstract

We study a two-period model of behavior-based price discrimination, as in Fudenberg and Tirole (RAND J Econ 31(4):634–657, 2000), but we allow firms to make their product choices in the first period. We show that the only possible equilibrium involves maximal differentiation. This is in contrast to Choe et al. (Manag Sci 64(12):5669–5687, 2018), where equilibrium features less-than-maximal differentiation when competition is in personalized pricing. Thus, our result highlights an important interplay between the type of price competition and product choice.

Appendix

Proof of Proposition 1 (Sketch)

Recall that there are three possible outcomes in $\tau = 2$: (1) $z = z_A$; (2) $z_A< z < z_B$; and (3) $z_B = z$. We use subscript $k = 1, 2, 3$ to denote each of these outcomes. For each k, denote the location of the $\tau = 1$ marginal consumer by $z_k$ and each firm’s total discounted profit by $\varPi _{ik}$, $i = A, B$.

First, consider the $\tau = 1$ pricing game given (a, b). For each k, firms simultaneously choose $p_A, \ p_B$ to maximize $\varPi _{Ak}, \ \varPi _{Bk}$, which leads to reaction functions $p_{Ak}(p_B; a, b)$ and $p_{Bk}(p_A; a, b)$. These reaction functions represent locally optimal prices given k. For firm i, the ‘true’ reaction function is derived from comparing $\varPi _{ik}, k = 1, 2, 3$ to find the $p_{ik}$ that leads to a global optimum. We then solve the two true reaction functions simultaneously for equilibrium prices $p_A^*(a, b)$, $p_B^*(a, b)$, and the marginal consumer’s location $z^*(a, b)$. In the technical appendix, we show that the solution exists only when $k = 2$ with sufficient conditions as given in Proposition 1; in other cases $k = 1, 3$, the two reaction functions do not intersect.

Next, consider the $\tau = 1$ location game: Substitute $p_A^*(a, b)$, $p_B^*(a, b)$, and $z^*(a, b)$ into each firm’s profit function and denote them by $\varPi ^*_{A2}(a, b), \varPi ^*_{B2}(a, b)$, where the second subscript indicates that the pricing equilibrium is possible only when $k = 2$. Differentiating these profit functions, one can show ${\partial \varPi _{A2}^*}/{\partial a} < 0$ for all $a, b \in [0, 1]$ and ${\partial \varPi _{B2}^*}/{\partial b} > 0$ for all $a, b \in [0, 1]$. Thus $a = 0, \ b = 1$ is a candidate equilibrium, which leads to the two-way poaching outcome in $\tau = 2$. Substituting $a = 0, b = 1$ into $p_A^{*}(a, b)$, $p_B^{*}(a, b)$, (1), (2), (3), and (4) gives us the equilibrium that is stated in the proposition. $\square$