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Rethinking Profit-Maximization in Second-Degree Price Discriminating Markets

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Abstract

Standard models of indirect price discrimination generate a separating equilibrium in which all consumers choose bundles on their demand curves. Low-demand consumers voluntarily choose to pay a higher unit price for small quantities because the discount is offered by the monopolist at prohibitively high quantity. These models implicitly assume the low-demand consumer would purchase the higher quantity at the discounted price if the quantity requirement were set at a low quantity, but rational monopoly pricing avoids this. We show that it is possible for a monopolist to offer price/quantity bundles that induce each consumer to choose voluntarily the one preferred by the monopolist for her where each bundle is above the respective consumer’s demand curve. The primary theoretical results include greater profits for the firm and greater total efficiency as output more closely approaches the socially optimal level in both market segments. Resolving an ambiguity from earlier research on price discrimination, the model also unequivocally demonstrates that consumer surplus is reduced. An important implication of the monopolist constraining consumption to be above the demand curve for her product is that she has effectively reduced the demand for some subset of other goods and services the consumers purchase.

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Notes

  1. For simplicity, marginal cost is assumed to be zero; this assumption is relaxed later.

  2. The strictly greater than condition here ensures the consumer chooses the high-price bundle. Weak inequality would ensure she is indifferent between the two.

  3. In the treatment that follows, the variables X and Q are used interchangeably; that is, X and Q are the quantity of the good produced or consumed in the market.

  4. In motivating non-linear demand, consumer utility is incorporated directly in the discussion.

  5. For the consumer strictly to prefer her bundle over not consuming at all, lower \( {q}_1^{\mathrm{CS}0} \) by an arbitrarily small amount, δ1, or \( {q}_1^{CS0} \)=2q1- δ1.

  6. It is self-evident that for the interested consumer in Figure 2, consumer surplus is negative at P1, \( {q}_1^{CS0} \).

  7. Alternatively, they can be left with some arbitrarily small positive consumer surplus by reducing q2 by δ2, close to zero.

  8. If consumer 2 wishes to consume more than \( {q}_1^{min} \) voluntarily, then he is on his demand curve at P1, and the consumer surplus he would get here becomes a minimum amount of consumer surplus he would need to be offered at the low-price/high-quantity bundle the monopolist chooses for him. If he does not capture that much surplus here, there is no separating equilibrium.

  9. Due to the non-linear demand function, it can no longer be assumed that the monopolist chooses X2 = 2X to capture the full surplus.

  10. In fact, an additional constraint must be added to the utility maximization problem.

  11. ‘Enough’ is determined by the monopolist. A similar constraint for consumer 2 is added later.

  12. This makes intuitive sense as well since the monopolist is the one choosing Xmin.

  13. In other words, the income and substitution effects are equal and opposite for the second good. This is true in this case because of the assumption of a single income constraint.

  14. The utility premium he achieves here is even larger when compared to a hypothetical bundle at PX=5 that required consumers to purchase a minimum quantity above his demand curve.

  15. The reader can verify that at PX=5, consumer 2 would get less utility at his optimal bundle (12,4) than he does at (20,2) when PX=4. Also, consumer 1 gets less utility at (20,2) than she does at (10,5).

  16. Thus, δ appears as the lower bound of the integrals throughout the paper.

  17. First order conditions are contained in the Online Supplemental Appendix.

  18. Proof available upon request.

  19. If \( {q}_2>{q}_2^{min} \) and \( {q}_2^{min}>{q}_1 \), then the condition must be met as well.

  20. If \( {q}_2>{q}_2^{min} \) and \( {q}_2^{min}>{q}_1 \), then the condition must be met as well.

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Correspondence to Brendan Cushing-Daniels.

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The author would like to thank the editors and an anonymous referee for helpful comments. An earlier version of the paper benefited from suggestions from participants in the International Atlantic Economic Conference held in New York, 11-14 October 2018. Remaining errors are my own.

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Cushing-Daniels, B. Rethinking Profit-Maximization in Second-Degree Price Discriminating Markets. Atl Econ J 48, 223–235 (2020). https://doi.org/10.1007/s11293-020-09670-6

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