Spatial Pricing in Uncontested Procurement Markets: Regulatory Implications

Juan Sesmero

doi:10.1515/jafio-2016-0013

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Spatial Pricing in Uncontested Procurement Markets: Regulatory Implications

Juan Sesmero

Published/Copyright: August 30, 2016

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From the journal Journal of Agricultural & Food Industrial Organization Volume 16 Issue 1

Abstract

This study derives a general characterization of the misalignment between socially and privately optimal spatial pricing in an uncontested procurement market. In doing so it clarifies the link between firm-gate price markdown (pricing the input below its marginal value product at the firm’s location) and spatial price discrimination (varying markdown by distance). We subsequently examine the implications of our results for regulatory prescriptions. Our analysis reveals that, in the absence of regulation, increased firm-gate price markdown is necessarily accompanied by intensified spatial price discrimination, and that discrimination is always conducted against nearby producers. We find that if regulation targets price markdown at the firm gate, then spatial price discrimination is also inhibited resulting in welfare gains. In contrast, directly targeting spatial price discrimination (as generally prescribed by enacted and proposed legislation) cannot attain a first best, and may in fact result in efficiency losses relative to the unregulated equilibrium.

Keywords: spatial monopsony; spatial price theory; regulation; welfare

JEL Classification: L10; L51; Q12; D4

Appendix

Proof of Lemma 1

. We prove lemma 1 by demonstrating that, under the privately optimal pricing strategy [3] and Assumptions 1 and 2, r=1t implies and is implied by wpr=0. Inserting wpr=0 into eq. [3], results in −f0+1−0−trf′0=0. Since f′0>0 by Assumption 1 and f0=0 by Assumption 2, this condition implies that 1−tr=0 or, equivalently, r=1t. Conversely, inserting r=1t into [3], results in −fwp1/t−wp1/tf′wpr|wpr=wp1/t=0. Provided that f′wpr>0 (which is true under the economically relevant range of prices) and that f0=0 (which is true by Assumption 2), the condition −fwpr−wprf′wpr=0 implies that wpr=0. Therefore r=1t⇔wpr=0 as we wanted to demonstrate ■

Proof of Lemma 2

. We prove lemma 2 by demonstrating that, under first order condition [3], the no-arbitrage condition [4], and Lemma 1, r=1t implies and is implied by 1−wr−tr=0. Inserting r=1t into 1−wr−tr yields −w1/t which, by Lemma 1, is equal to zero. Therefore r=1t⇒1−wr−tr=0. Moreover, the reverse implication is true if 1−wr−tr is monotonic in r because that means there is only one distance at which 1−wr−tr=0, and that distance is r=1t.

The slope of the unitary profit 1−wr−tr with respect to distance r is −w′r−t. The no-arbitrage condition generally states that w′r≥−t. But by first order condition [3], 0<wp0<1, and by Lemma 1, wpr=0 only at r=1t. Therefore the no arbitrage condition must hold with strict inequality, w′r>−t for all r. In turn, this implies that −w′r−t<0 for all distance r which amounts to monotonicity of the unitary profit, as we wanted to demonstrate ■

Proof of Result 2

. The boundary of the market area is characterized by eq. [9]. By Lemma 1 R2p=1t. Moreover by Lemma 2, R1p=1t. Therefore, Rp=1t, as we wanted to demonstrate ■

Proof of Result 3

. By Result 1, ws0=1; and by first order condition [3], wp0<1. Moreover by Result 2, Rp=Rs=1t. Therefore the privately optimal spatial pricing function wpr starts from a lower intercept than the socially optimal one wsr, yet it reaches zero at the same point on the x-axis, 1t. Therefore, under monotonicity of wpr (demonstrated as part of proof of Lemma 2), the privately optimal spatial pricing path must have a smaller slope than the socially optimal one, for at least a range of distances; i. e. w′r>−t. This, in turn, implies spatial price discrimination ■

Proof of Result 4

. Since changes in input supply properties change the intercept but not the boundary of the market area, a change in input supply that reduces wp0 amounts to clockwise rotation of function wpr around Rp=1t. This implies that the function wpr must display a smaller slope as a result of this change, for at least a range of distances r ■

Proof of Result 5

. The first part of the result (regulation that reduces firm-gate price markdown, also diminishes spatial price discrimination) is obvious from the fixity of Rp=1t in combination with monotonicity of wpr. If the intercept of wpr is increased without this affecting the radius, the policy causes a clockwise rotation of the price-distance function. This amounts to a reduction in spatial price discrimination.

Regarding the second part of the result (setting firm-gate price equal to marginal value product, eliminates spatial price discrimination and attains a first best), by Result 1 the first best is attained when the firm-gate price equals the marginal value product of the input, and the slope of the price-distance function is equal to −t. The first condition is true by the action of the regulator. The second holds due to the no-arbitrage condition in combination with non-negative unitary profit. To see this, recall that the cost to the firm from purchasing input from distance r is mpr=wpr+rt and the unitary profit is πpr=1−mpr. Since, by regulation πp0=1−1=0, πp′r must be non-negative or it would otherwise (given monotonicity of πpr) violate the non-negative unitary profit condition (and the firm would shut down). And πp′r≥0 implies mp′r≤0. By the no-arbitrage condition wp′r≥−t and, hence, mp′r=wp′r+t≥0. Therefore, simultaneous fulfillment of the non-negative profit condition and the no-arbitrage condition implies that mp′r=0 which amounts to FOB pricing. Therefore, if the regulator prevents markdown and forces the firm to set wp0=1, the firm will respond by following an FOB strategy, which results in a first best solution ■

Acknowledgments

This research has received financial support from the Purdue Research Foundation.

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Published Online: 2016-8-30

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Keywords for this article

spatial monopsony; spatial price theory; regulation; welfare