Customer poaching and coupon trading

Abstract

The price discrimination literature typically makes the assumption of no consumer arbitrage. This assumption is increasingly violated in the digital economy, where coupons are traded with increased frequency online. In this paper, we analyze the welfare impacts of coupon trading using a modified Hotelling model where firms send coupons to poach each other’s loyal customers. The possibility of coupon trading renders this important instrument for price discrimination less effective. Moreover, coupon distribution has unintended consequences when coupon traders sell coupons back to a firm’s loyal customers. Consequently, coupon trading may reduce firms’ incentive to distribute coupons, leading to higher prices and profits. We find that, an increase in coupon distribution cost lowers promotion frequency but raises promotion depth, and an increase in the fraction of coupon traders lowers both promotion frequency and promotion depth.

Appendix

Proof of Proposition 1

We divide this proof into two parts.³³ In part 1, we derive the optimal prices and couponing strategies. This is the equilibrium candidate. Then in part 2, we show that neither firm has an incentive to deviate unilaterally.

Part 1: equilibrium candidate

Firms’ profit functions are

$$\begin{aligned} \pi _{1}= & {} p_{1}(1-\alpha )(1-\lambda _{2})L+p_{1}(1-\alpha )(1-\lambda _{1})(p_{2}-p_{1})+p_{1}(1-\alpha )\lambda _{2}(L\nonumber \\&+p_{2}-r_{2}-p_{1})+(p_{1}-r_{1})(1-\alpha )\lambda _{1}(p_{2}-p_{1}+r_{1})+p_{1}\alpha (L+p_{2}-p_{1})\nonumber \\&-r_{1}\alpha \lambda _{1}L-k(\lambda _{1}L)^{2}, \end{aligned}$$

(5)

$$\begin{aligned} \pi _{2}= & {} p_{2}(1-\alpha )(1-\lambda _{1})(L-p_{2}+p_{1})+p_{2}(1-\alpha )\lambda _{1}(L-p_{2}+p_{1}-r_{1}) \nonumber \\&+(p_{2}-r_{2})(1-\alpha )\lambda _{2}(p_{1}-p_{2}+r_{2})+p_{2}\alpha (L-p_{2}+p_{1})\nonumber \\&-r_{2}\alpha \lambda _{2}L-k(\lambda _{2}L)^{2}. \end{aligned}$$

(6)

Taking derivative of $\pi _{2}$ with respect to $p_{2}$, $r_{2}$ and $\lambda _{2}$ respectively, then imposing the symmetry conditions ($p_{1}=p_{2},r_{1}=r_{2}$ and $\lambda _{1}=\lambda _{2}$), we can obtain

$$\begin{aligned}&\displaystyle \frac{\partial \pi _{2}}{\partial r_{2}}=-\lambda _{2}(\alpha L-p_{2}-2r_{2}\alpha +2r_{2}+\alpha p_{2})=0,\end{aligned}$$

(7)

$$\begin{aligned}&\displaystyle \frac{\partial \pi _{2}}{\partial \lambda _{2}}=-2k\lambda _{2}L^{2}-r_{2}\alpha L-\alpha p_{2}r_{2}+r_{2}^{2}\alpha +p_{2}r_{2}-r_{2}^{2}=0,\end{aligned}$$

(8)

$$\begin{aligned}&\displaystyle \frac{\partial \pi _2}{\partial p_2}=L-p_2-\lambda _{2}r_{2}\alpha +\lambda _{2} \alpha p_2+r_{2} \lambda _{2}-\lambda _{2} p_2=0. \end{aligned}$$

(9)

Since the cost of coupon distribution is quadratic in $\lambda$, and the rest is roughly linear in $\lambda$, it must be that the optimal $\lambda _{2}>0$. Then, Eq. (7) implies,

$$\begin{aligned} r_{2}=\frac{\alpha L-p_2+\alpha p_2}{2(\alpha -1)}=\frac{1}{2}\left( p_2- \frac{\alpha L}{1-\alpha }\right) . \end{aligned}$$

(10)

From this expression, we can see that $p_2>r_{2}$.

Next, we substitute the expression for $r_{2}$ into Eq. (8) and solve for $\lambda _{2}$. We obtain

$$\begin{aligned} \lambda _{2}=\frac{(\alpha L-p_{2}+\alpha p_{2})^{2}}{8(1-\alpha )kL^{2}}. \end{aligned}$$

(11)

Using $r_{2}$ and $\lambda _{2}$ in Eq. (9), we can solve for the equilibrium price³⁴

$$\begin{aligned} p_{2}=\frac{\left( \frac{2}{3}A-\frac{3}{2}\frac{-\frac{4}{9}\alpha ^{2}+ \frac{16}{3}k}{A}-\frac{1}{3}\alpha \right) L}{-1+\alpha }, \end{aligned}$$

where

$$\begin{aligned} A=\left( \alpha ^{3}{+}36k\alpha {-}27k{+}3\sqrt{12\alpha ^{4}k{+}96\alpha ^{2}k^{2}{+}192k^{3}-6k\alpha ^{3}-216k^{2}\alpha +81k^{2}}\right) ^{\frac{1}{3 }}. \end{aligned}$$

We can then substitute $p_{2}$ back into the expressions for $r_{2}$ and $\lambda _{2}$. The final expressions are too lengthy to report.

So far, we have used first-order conditions (FOCs) to solve for the optimal choices of prices and promotion intensities. However, FOCs are necessary but not sufficient. We need to make sure that the solution we obtained indeed constitutes an equilibrium. Instead of checking whether the Hessian matrix is negative semidefinite (which is quite messy), we show that neither firm has an incentive to unilaterally deviate from this pair of strategies (Bester and Petrakis use a similar method). Without loss of generality, we fix firm 1’s price and promotion strategies as given in Proposition 1, and show that firm 2 to has no incentive to deviate.

Part 2: firm 2 has no incentive to deviate

Note that, the demand/profit functions depend on the locations of marginal consumers and there are two cases. In the first case, $p_{2}\ge p_{1}$ still holds and thus $l_{a}\ge 0$. In the second case, $p_{2}<p_{1}$. In both cases, we assume that $l_{b1}<0$ and $l_{b2}>0$.³⁵

Start with case 1 where $p_{2}\ge p_{1}$. Firm 2’s deviation profit is given by equation (6), with $p_{1}=p^{*}$, $r_{1}=r^{*}$ and $\lambda _{1}=\lambda ^{*}$. We normalize $L=1$. The optimal choice requires that

$$\begin{aligned} \frac{\partial \pi _{2}^{dev}}{\partial r_{2}}=\frac{\partial \pi _{2}^{dev} }{\partial \lambda _{2}}=0. \end{aligned}$$

Solving the first order conditions, we obtain

$$\begin{aligned} r_{2}^{dev}=\frac{2(1-\alpha )p_2 - (1-\alpha ) p^* -\alpha }{2(1-\alpha )}, \end{aligned}$$

$$\begin{aligned} \lambda _{2}^{dev}= \frac{\alpha ^2+\alpha ^2 (p^*)^2+4\alpha ^2 p_2-2 p^* \alpha ^2+2\alpha p^*-4\alpha p_2-2\alpha (p^*)^2+(p^*)^2}{(1-\alpha )k}. \end{aligned}$$

The first order conditions are necessary and sufficient. Note that, we do not substitute $p^{*}$ in these expressions as they would be too lengthy to report. Notice that, firm 2’s deviation profit depends only on $p_{2}^{dev}$, $\alpha$ and k. We want to check whether firm 2 can increase its profit by choosing $p_{2}^{dev}\ne p^{*}$, i.e., to have

$$\begin{aligned} \pi _{2}^{dev}(p_{2}^{dev})>\pi ^{*},\quad \forall \alpha ,k. \end{aligned}$$

We tried various combinations of $\alpha$ and k, and we found that firm 2 can never increase its profit by choosing a price different than $p^{*}$ . Therefore, firm 2 has no incentive to deviate. We then proceeded to the case of $l_{a}<0$ (i.e. $p_{2}<p_{1}$). The steps are similar and we found that firm 2 has no incentive to deviate if k is not too small relative to L.³⁶ $\square$