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Licensed Unlicensed Requires Authentication Published by De Gruyter September 26, 2013

Loyalty Discounts

Uğur Akgün and Ioana Chioveanu

Abstract

This article analyses the use of loyalty inducing discounts in vertical supply chains. An upstream supplier and a competitive fringe sell differentiated products to a retailer who has private information about the stochastic demand. We compare the market outcomes, when the supplier uses two-part tariffs (2PT), all-unit quantity discounts (AU), and market-share discounts (MS). We show that the retailer’s risk attitude affects supplier’s preferences over these pricing schemes. When the retailer is risk neutral, it bears all the risk and the three schemes lead to the same outcome. When the retailer is risk averse, a 2PT performs the worst from the supplier’s perspective, but it leads to the highest welfare. For a wide range of parameter values (but not for all), the supplier prefers MS to AU. By limiting the retailer’s product substitution possibilities, MS makes the demand for the manufacturer’s product more inelastic. This reduces the amount (share of total profits) the supplier needs to leave to the retailer for the latter to participate in the scheme.

JEL: L42; L12; L13

Appendix

Proof of Proposition 3. With demand uncertainty, the risk-neutral retailer makes quantity choices after observing the realized demand. Hence, the second stage optimizations presented in Section 3 still apply. But, since the contracts are agreed upon before the resolution of the uncertainty, a different risk attitude changes the first stage optimization. When the manufacturer faces a risk-neutral retailer, the participation constraint requires the retailer’s expected profit to be at least equal to the retailer’s expected outside option.

Under a 2PT contract, the upstream manufacturer chooses w and F to maximize

The constraint is increasing in the franchise fee, so the supplier chooses the unit price to maximize It follows that the optimal unit price satisfies the first-order condition

[10]
[10]

Using eqs [1] and [2] and envelope theorem, eq. [10] becomes and implies that, in equilibrium,

[11]
[11]

Let us consider an AU contract which induces the retailer to act on the quantity target only under a low demand. Then, the supplier chooses , and F to maximize

The constraint is increasing in the franchise fee, so the supplier chooses w and to maximize It follows that the optimal unit price satisfies the first-order condition By a similar argument as in the case of a 2PT, it follows that the optimal unit price and franchise fee are given by eq. [11]. In addition, Clearly, the manufacturer cannot improve upon this contract. The optimal 2PT and AU contracts result in the same output levels.

Finally, consider an MS contract that induces the retailer to act on the threshold always. Then, the supplier chooses s, and F to maximize

The constraint is increasing in the franchise fee, so the supplier chooses w and s to maximize Then, the unit price satisfies

[12]
[12]

From eq. [9], using the envelope theorem, eq. [12] becomes Then, the optimal unit price and franchise fee are given by eq. [11]. Again, the quantities purchased by the retailer at equilibrium are the same as in the optimal 2PT contract, and the manufacturer’s expected profit is the same. We conclude that the manufacturer is indifferent between these contracts, and the same level of aggregate welfare is obtained under all three contracts. ■

Proof of Proposition 6. Using the closed-form solution in Tables 2 and 3, we compare the upstream profits, consumer surplus, and total welfare in the optimal AU and MS contracts. Recall that with a risk-averse retailer

Upstream profit comparison:

Note that . Then,

Consumer surplus comparison:

Note that , where

Note that

Thus, and

Welfare comparison:

Note that where Note that Thus, and

Finally, note that In effect, whenever the supplier prefers the optimal MS contract (), total welfare and consumer surplus are strictly higher under the optimal AU contract than under the optimal MS contract. ■

Closed-form solutions with linear demand

Using the linear demand specification presented in Section 4, we can derive closed-form solutions for the quantities and prices of the two products ( and for ), retailer’s profit (R), manufacturer’s profit (U), welfare (W), and consumer surplus (CS) for all contracts. Tables 1, 2, and 3 present the equilibrium outcomes under the optimal 2PT, AU, and MS contracts, respectively.

Table 1

Optimal two-part tariff

Table 2

Optimal all-unit discount

Table 3

Market-share discount

Acknowledgements

We would like to thank the Editors Antonio Cabrales and Till Requate and two anonymous referees for valuable comments that have greatly improved our work. We thank Mark Armstrong, Jo Seldeslachts, Jidong Zhou, and various seminar and conference participants for useful comments. Financial support from the Valencian Economic Research Institute (IVIE) and the European Commission is gratefully acknowledged. The usual disclaimer applies.

References

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  1. 1

    Economies of scale can occur at the overall production level or in fulfilling a specific order, but are less likely to relate to the total purchases of a customer over a certain period.

  2. 2

    The EC guidelines require a dominant firm to provide an objective motivation for a discount scheme that can potentially make it harder for its rivals to compete: “It is incumbent upon the dominant undertaking to provide all the evidence necessary to demonstrate that the conduct concerned is objectively justified.” See paragraph 31 in European Commission (2009).

  3. 3

    Lately, European and North American case law have focused on whether loyalty discounts can serve as exclusionary devices that would violate Article 102 of the EC Treaty or Section 2 of the Sherman Act. In addition, firms’ use of loyalty discounts in the distribution of their products has also been attacked as unlawful primary line price discrimination under the Robinson Patman Act and EC law (Art. 102c).

  4. 4

    Mills (2010) and the Office of Fair Trading (2005) present comprehensive overviews of antitrust cases related to loyalty discounts in US and Europe. See also Kobayashi (2005), European Commission (2005, 2009), and Gual et al. (2005).

  5. 5

    The term “all-unit” quantity discount is used to emphasize that we study rollback rebates. However, our setting also informs on the relative private desirability of “incremental-unit” quantity discounts (that do not rollback to inframarginal units once the target is reached), since they cannot improve upon 2PT under our information/risk setting.

  6. 6

    Nocke and Thanassoulis (forthcoming) show that, in the presence of uncertainty, risk aversion may arise endogenously in supply chains when downstream firms face credit constraints in their future investments.

  7. 7

    For example, Kolay et al. (2004) and Kolay, Shaffer, and Ordover (2004) focus on the relative profitability of these pricing schemes (for the upstream supplier) when they are used for optimal screening. In Section 5, we discuss how our work complements the previous work in understanding the supplier’s preferences over the different pricing schemes, when exclusion cannot be a motive.

  8. 8

    In the EEA, when a dominant supplier restricts a downstream customer’s ability to resell its product to other businesses at the same level of the supply chain, but in other geographic markets, this action may be found incompatible with Article 101 and/or Article 102 of the EC Treaty.

  9. 9

    For instance, consider a setting where only authorized dealers can sell the product.

  10. 10

    Even if the implementation of market-share discounts requires costly monitoring of rival sales, there is a non-trivial range of costs for which the supplier might still strictly prefer using a market-share discount to using all-unit quantity discounts.

  11. 11

    In a full information setting where the incumbent faces second period competition by entrants, Feess and Wohlschlegel (2010) show that all-unit discounts shift rents from the entrants. Greenlee, Reitman, and Sibley (2008) consider a monopolist that faces competition in a second market and shows that bundled loyalty discounts (that condition the rebate on the range of products purchased from the monopolist) have ambiguous welfare effects.

  12. 12

    Another stream of literature analyses loyalty discounts as a source of endogenous switching costs in a dynamic, repeat-purchase context. See, for instance, Banerjee and Summers (1987) and Caminal and Matutes (1990).

  13. 13

    In our setting, the retailers could be, for instance, relatively small convenience stores. It is plausible then that the manufacturer of a branded product offers the same contract to all retailers on a take it or leave it basis, rather than hold bilateral negotiations with each local retailer. So, if the supplier needs to deal with many local retailers, the transaction costs of negotiating individual contracts would justify his ability to make a take it or leave it offer. Finally, the assumption that the manufacturer makes a take it or leave it offer is not as restrictive as it may seem. For a relevant discussion, see footnote 9 in Inderst and Valletti (2009).

  14. 14

    Given that the markets are identical apart from the resolution of the uncertainty, the expected outcome under different contracts can be captured by looking at the ex-ante situation for a given local market. Therefore, we suppress the multiplicity of markets for the rest of this article and refer to a single retailer while analysing the contracts.

  15. 15

    We are very grateful to an anonymous referee for suggesting us to use this functional form to capture retailer risk aversion on a continuous scale.

  16. 16

    See, for instance, Vives (2001).

  17. 17

    The opposite is true when the demand is high, as the threshold in the optimal AU contract does not constrain the retailer’s choice in that case.

  18. 18

    When the demand is low, under the AU contract, the retailer optimizes by choosing the quantity of the competitively supplied good corresponding to the threshold quantity of the manufacturer’s good and, although the sales are still distorted in favour of the competitively supplied good, the distortion is lower as

  19. 19

    When the demand is high, for both contracts the retailer’s choices are governed by the first-order conditions and the distortion in the relative sales of the two goods is higher with the optimal AU contract

  20. 20

    Total welfare is calculated as the difference between the gross utility of the consumers and the costs of production. Recall that the costs of production are normalized to zero in our model.

  21. 21

    When acting on the share threshold, the retailer actually chooses the quantity of only one product (i.e. there is only one first-order condition in this case). The quantity of the substitute product is determined by the share requirement. Without loss of generality, we assume that the retailer chooses the quantity of the competitively supplied product ().

  22. 22

    Nocke and Thanassoulis (forthcoming) show that, with an endogenously risk averse downstream firm, the optimal screening contract induces double marginalization and involves a fixed payment from the upstream firm to the downstream one. They argue that such slotting allowances are a risk sharing device. This interpretation is consistent with our results for the MS contract.

  23. 23

    The contract they consider effectively requires the retailer to satisfy both a market-share target and a quantity target for achieving a certain price. This is different from the market-share contract we consider which only stipulates a market-share threshold.

Published Online: 2013-09-26

©2013 by Walter de Gruyter Berlin / Boston

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