Can coupons counteract strategic consumer behavior?

Abstract

In multi-period environments, the presence of strategic consumers induces monopolist retailers to inter-temporally compete with themselves. Targeting consumers with price-discount coupons is a proposed mechanism to overcome this inter-temporal competition. Targeting consumers with coupons can counteract strategic consumer behavior, but this mechanism cannot completely eradicate the negative implications imposed by the presence of such consumers. Additionally, the quality of information available (regarding the consumers’ valuations) may play an important role in the targeting decisions. Specifically, we illustrate how the retailer may absent completely from targeting efforts when the quality of information and the proportion of strategic consumers are sufficiently low. Lastly, we consider the optimal investment in information solicitation, demonstrating the trade-off between a low investment, which result in low quality of targeting capability, and a high investment which improves the effectiveness of the targeting efforts.

Appendix: Proofs and additional statements

In this Section we sketch the a proofs for Lemma 1 and Corollary 1. Proof of Proposition 1 can be found in Sect. I.B. (page 16) of Lazear (1986) and Proposition 2 can be trivially demonstrated. Proposition 3, and Proposition 4 are direct consequences of first order conditions and hence omitted.

Proof of Lemma 1

Recall that the Lemma states: In the Perfect Information case, $\beta =1$, where consumers are myopic, $\alpha =0$, targeting only the set of consumers with valuation in [$p^*_1-c_1, p^*_2$], i.e., $\gamma _{1}=[p^*_1-c_1, p^*_2]$, achieves the optimal revenues. Where $p^*_1 \text{ and } p^*_2$ are the the sub-game perfect equilibrium prices resulting in case all consumers behave rationally. In order to prove this Lemma we need to show that there is no benefit of offering a coupon to consumers outside of $\gamma _{1}=[p^*_1-c_1, p^*_2]$. We use arguments regarding marginal contributions to do so We look at adding a small proportion of the population, of size $\epsilon \ge 0$, to the set of targeted consumers. On the one hand, targeting consumers with valuation $\gamma _{1}=[p^*_1-c_1-\epsilon , p^*_2]$ does not provide any added benefit, ceteris paribus, since the consumers added to the targeted population do not purchase the product as their valuation is less than $p_1-c$. Using the same logic, targeting $\gamma _{1}=[p^*_1-c_1, p^*_2+\epsilon ]$, means we provide a coupon to the population $[p^*_2, p^*_2+\epsilon ]$ to purchase the product at $p^*_1-c_1$ instead of at $p^*_2$, which in turns means we could have raised the second period price to $p^*_2+\epsilon$ to the rest of the consumers, i.e., this can’t be optimal.

Proof of Corollary 1

This result is archived by construction of a pricing policy, which provides the retailer with the same optimal revenue she receives when facing myopic consumers. The construction is such that it is resistant to strategic consumer behavior. Given that the optimal pricing policy facing myopic consumers and N periods is: $p_1\ge p_2 \ge ...\ge p_N$. As targeting is perfect, target, with coupon $c_t$ the population with valuation: $\gamma _t$ = [$p_{N+1-t},p_{N-t}$], and the coupon is such that the effective price in period $c_t$ is $p_1-p_{N+1-t}$, and this is for $t \in {1,2,...,N}$. The constructed price is such that no consumer can rationally expect that he will face a lower price in the future, and as targeting is perfect, no consumers are offered a coupon that is not intended for their use. Essentially, as the desired the “skimming” policy is such that it refers to increasing valuations, countering the effects of strategic consumers.