Does price discrimination make collusion less likely? a delivered pricing model

Abstract

This paper compares the stability of collusion under delivered spatial price discrimination and under uniform pricing. Uniquely using a model of elastic demand, we show that collusion under price discrimination can be more stable thus facilitating collusion and making it more likely. This result holds only when the entire market is competitive. Whenever there exist natural monopoly portions of the spatial market, collusion on the remaining market is less stable with spatial price discrimination making such collusion less likely relative to uniform pricing.

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Notes

  1. 1.

    The school milk markets case in Ohio is discussed in detail by Porter and Zona (1999).

  2. 2.

    Thus these studies differ from those which study collusion in Hotelling type models that assume freight on board (Chang 1992; Hackner 1995). They also differs from those which model the influence of location but assume a game in quantity rather than a game in price (Colombo 2012).

  3. 3.

    Thus, in the Hotelling framework one might imagine a form of third-degree price discrimination in which there are separate groupings of consumers along the unit line segment with different willingness to pay (Columbo 2011). In a recent such examination, Zhang et al. (2019) endogenize the decision of competing firms to engage in price discrimination across such groupings.

  4. 4.

    He assumes that regardless of whether the collusion involves uniform or spatially discriminatory prices, the pricing after deviation is rivalrous spatial discrimination. Again, the discrimination is between different groups of customers and not in the form of delivered pricing.

  5. 5.

    We note that any implied location game vanishes in the FOB case as competition causes the firms to collapse on the center (d'Aspremont et al. 1979) and that with quadratic costs the collusive locations could not be solved.

  6. 6.

    No other solution divides the market between a and 1 − a.

  7. 7.

    When product differentiation, t, is zero both price discrimination and uniform pricing collapse to a non-spatial Bertrand game with the same critical discount factor of 0.5.

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Appendix

Appendix

Proof of Proposition 1

Using the profit expressions derived in Sect. 4 and Eq. (3) we derive

$$\bar{\delta }_{UU} = \left\{ \begin{aligned} & 64a^{4} t^{2} - 256a^{3} t^{2} + 592a^{2} t^{2} - 424at^{2} - 32at + 24t + 92t^{2} - 16\quad \quad \quad \quad \quad \quad \quad \quad \quad \;t \le t_{1} \\ & \;/2( - 128a^{3} t^{2} + 16La^{2} t + 168a^{2} t^{2} - 204at^{2} + 14Lt - 32at - 5t^{2} + 20t - 16) \\ & ( - 3520 + 592K + 637Kt^{2} + 17920a^{6} t^{3} - 23808a^{5} t^{3} - 236352a^{4} t^{3} \quad \quad \quad \quad \quad \quad \quad \quad t > t_{1} \\ & \;12876at^{3} + - 392tK - 23808a^{4} t^{2} - 102144a^{3} t^{2} + 3456a^{2} t^{2} - 51072a^{2} t \\ & \; - 35040at^{2} - 10560at + 10496a^{3} t^{3} - 135336a^{2} t^{3} + 832Ka^{4} t^{2} + 448a^{3} t^{2} \\ & \; + 3744Ka^{2} t^{2} - 392Kat^{2} + 448a^{2} t + 1184Kat - 15893t^{3} - 17520t + 12876t^{2} )/ \\ & \;( - 3520 + 592K + 637Kt^{2} - 108864Kt^{2} + 17920a^{6} t^{3} - 23808a^{5} t^{3} - 12480a^{4} t^{3} \\ & \; + 10496a^{3} t^{3} + 859992a^{2} t^{3} - 49332at^{3} - 392Kt - 23808a^{4} t^{2} - 102144a^{3} t^{2} \\ & \; + 3456a^{2} t^{2} - 51072a^{2} t + 89376at^{2} - 10560at + 3744Ka^{2} t^{2} - 392Kat^{2} \\ & \; - 124416La^{2} t^{2} + 448a^{2} t + 1184Kat + 376795t^{3} + 44688t - 49332t^{2} ) \\ \end{aligned} \right.$$
(A1)

From (23) we have

$$\bar{\delta }_{DD} = - \frac{{64a^{3} t^{2} - 60a^{2} t^{2} + 18at^{2} - 24at - 3t^{2} + 18t - 12}}{{8(8a^{3} t^{2} - 3a^{2} t^{2} + 6a^{2} t - 6at^{2} + 6at + 3t^{2} - 6t + 3)}}$$
(A2)

For \(t \in (0,t_{1} ]\), solving \(\bar{\delta }_{UU} = \bar{\delta }_{DD}\) generates a threshold t′. It is easy to verify that when t > t′, \(\bar{\delta }_{UU} - \bar{\delta }_{DD} < 0\) and when t < t′,\(\bar{\delta }_{UU} - \bar{\delta }_{DD} > 0\).

For \(t \in (t_{1} ,\frac{1}{2 - 3a})\), \(\bar{\delta }_{UU} - \bar{\delta }_{DD} < 0\), for any \(t \in (t_{1} ,\frac{1}{2 - 3a})\).

Therefore, in the admissible range of parameters there exists a threshold t’ such that when t > t’,\(\bar{\delta }_{UU} - \bar{\delta }_{DD} < 0\) and when t < t’,\(\bar{\delta }_{UU} - \bar{\delta }_{DD} > 0\).

Details on the Illustrations without Natural Monopoly Regions

(i) From (23) and (A1) and knowing that a = 0 we have that,

$$\bar{\delta }_{UU} = \left\{ {\begin{array}{*{20}l} {\frac{{91t^{2} + 24t - 16}}{{2(14tM - 5t^{2} + 20t - 16)}}} \hfill & {t \le t_{1} } \hfill \\ {\frac{{N^{3} - (23t + 4)(691t^{2} - 680t + 880)}}{{N^{3} - 108864Mt^{2} + 376795t^{3} - 49332t^{2} + 44688t - 3520}}} \hfill & {\text{otherwise}} \hfill \\ \end{array} } \right.$$
(A3)

and

$$\bar{\delta }_{DD} = \frac{{t^{2} - 6t + 4}}{{8(t^{2} - 2t + 1)}}$$
(A4)

For \(t \in (0,t_{1} ]\), solving \(\bar{\delta }_{UU} = \bar{\delta }_{DD}\) generates a threshold t’ = 0.20. It is easy to verify that when t > t’, \(\bar{\delta }_{UU} - \bar{\delta }_{DD} < 0\) and when t < t’, \(\bar{\delta }_{UU} - \bar{\delta }_{DD} > 0\). For \(t \in (t_{1} ,0.5)\),

$$\bar{\delta }_{UU} - \bar{\delta }_{DD} = \frac{{N^{3} - (23t + 4)(691t^{2} - 680t + 880)}}{{N^{3} - 108864Mt^{2} + 376795t^{3} - 49332t^{2} + 44688t - 3520}} - \frac{1}{8}\frac{{t^{2} - 6t + 4}}{{t^{2} - 2t + 1}} < 0$$
(A5)
  1. (ii)

    From (23) and (A1) and knowing that a = 1/4 we have that,

$$\bar{\delta }_{UU} = \left\{ {\begin{array}{*{20}l} {\frac{{73t^{2} + 64t - 64}}{{4(15O - 95t^{2} + 24t - 32)}}} \hfill & {t \le t_{2} } \hfill \\ {\frac{{P^{3} - (55t + 8)(3187t^{2} - 848t + 3520)}}{{P^{3} - 466560Ot^{2} + 3347243t^{3} - 227688t^{2} + 310848t - 28160}}} \hfill & {t > t_{2} } \hfill \\ \end{array} } \right.$$
(A6)

and

$$\bar{\delta }_{DD} = \frac{{5t^{2} - 48t + 48}}{{2\left( {23t^{2} - 66 + 48} \right)}}$$
(A7)

For \(t \in (0,t_{2} ]\), solving \(\bar{\delta }_{UU} = \bar{\delta }_{DD}\) generates a threshold t’’ = 0.52. It is easy to verify that when \(t > t^{{\prime \prime }}\), \(\bar{\delta }_{UU} - \bar{\delta }_{DD} < 0\) and when \(t < t^{{\prime \prime }}\),\(\bar{\delta }_{UU} - \bar{\delta }_{DD} > 0\). For \(t \in (t_{2} , \, 0.8)\),

$$\begin{aligned} \bar{\delta }_{UU} - \bar{\delta }_{DD} = & - (3133Pt^{2} - 175285t^{3} - 272Pt + 21144t^{2} + 2368P - 186816t - 28160)/ \\ & \quad (466560Ot^{2} - 3133Jt^{2} - 3347243t^{3} + 272Pt + 227688t^{2} - 2368P - 310848t + 28160) - \frac{1}{2}\frac{{5t^{2} - 48t + 48}}{{23t^{2} - 66 + 48}} < 0 \\ \end{aligned}$$
(A8)

Proof of Proposition 2

From (23), we have

$$\begin{aligned} \bar{\delta }_{UU} = & ( - 3520 + 592K + 637Kt^{2} + 17920a^{6} t^{3} - 23808a^{5} t^{3} - 236352a^{4} t^{3} \\ & \quad + 12876at^{3} - 392tK - 23808a^{4} t^{2} - 102144a^{3} t^{2} + 3456a^{2} t^{2} - 51072a^{2} t \\ & \quad - 35040at^{2} - 10560at + 10496a^{3} t^{3} - 135336a^{2} t^{3} + 832Ka^{4} t^{2} + 448a^{3} t^{2} \\ & \quad + 3744Ka^{2} t^{2} - 392Kat^{2} + 448a^{2} t + 1184Kat - 15893t^{3} - 17520t + 12876t^{2} )/ \\ & \quad ( - 3520 + 592K + 637Kt^{2} - 108864Kt^{2} + 17920a^{6} t^{3} - 23808a^{5} t^{3} - 12480a^{4} t^{3} \\ & \quad + 10496a^{3} t^{3} + 859992a^{2} t^{3} - 49332at^{3} - 392Kt - 23808a^{4} t^{2} - 102144a^{3} t^{2} \\ & \quad + 3456a^{2} t^{2} - 51072a^{2} t + 89376at^{2} - 10560at + 3744Ka^{2} t^{2} - 392Kat^{2} \\ & \quad - 124416La^{2} t^{2} + 448a^{2} t + 1184Kat + 376795t^{3} + 44688t - 49332t^{2} ) \\ \end{aligned}$$
(A9)

and

$$\bar{\delta }_{DD} = \left\{ {\begin{array}{*{20}l} { - \frac{{280a^{3} t^{3} - 420a^{2} t^{3} + 72a^{2} t^{2} + 210at^{3} - 72at^{2} - 35t^{3} - 24at + 18t^{2} + 12t - 8}}{{8(8a^{3} t^{3} - 12a^{2} t^{3} + 24a^{2} t^{2} + 6at^{3} - 24at^{2} - t^{3} + 12at + 6t^{2} - 6t + 2)}}} \hfill & {\frac{ 1}{2 - 3a} \le t \le \frac{1}{2 - 4a}} \hfill \\ {\frac{5}{8}} \hfill & {t > \frac{1}{2 - 4a}} \hfill \\ \end{array} } \right.$$
(A10)

For \(t \in \left[ {\frac{1}{2 - 3a}, \, \frac{ 1}{ 2- 4a}} \right]\), we have

$$\begin{aligned} \bar{\delta }_{UU} - \bar{\delta }_{DD} = & ( - 3520 + 592K + 637Kt^{2} + 17920a^{6} t^{3} - 23808a^{5} t^{3} - 236352a^{4} t^{3} \\ & \quad + 12876at^{3} - 392tK - 23808a^{4} t^{2} - 102144a^{3} t^{2} + 3456a^{2} t^{2} - 51072a^{2} t \\ & \quad - 35040at^{2} - 10560at + 10496a^{3} t^{3} - 135336a^{2} t^{3} + 832Ka^{4} t^{2} + 448a^{3} t^{2} \\ & \quad + 3744Ka^{2} t^{2} - 392Kat^{2} + 448a^{2} t + 1184Kat - 15893t^{3} - 17520t + 12876t^{2} )/ \\ & \quad ( - 3520 + 592K + 637Kt^{2} - 108864Kt^{2} + 17920a^{6} t^{3} - 23808a^{5} t^{3} - 12480a^{4} t^{3} \\ & \quad + 10496a^{3} t^{3} + 859992a^{2} t^{3} - 49332at^{3} - 392Kt - 23808a^{4} t^{2} - 102144a^{3} t^{2} \\ & \quad + 3456a^{2} t^{2} - 51072a^{2} t + 89376at^{2} - 10560at + 3744Ka^{2} t^{2} - 392Kat^{2} \\ & \quad - 124416La^{2} t^{2} + 448a^{2} t + 1184Kat + 376795t^{3} + 44688t - 49332t^{2} ) \\ & \quad - \frac{{280a^{3} t^{3} - 420a^{2} t^{3} + 72a^{2} t^{2} + 210at^{3} - 72at^{2} - 35t^{3} - 24at + 18t^{2} + 12t - 8}}{{8(8a^{3} t^{3} - 12a^{2} t^{3} + 24a^{2} t^{2} + 6at^{3} - 24at^{2} - t^{3} + 12at + 6t^{2} - 6t + 2)}} < 0 \\ \end{aligned}$$
(A11)

for all admissible \(a\).And for \(t \in \left[ {\frac{ 1}{ 2 4- a},\frac{4}{{3 - 8a^{2} - 4a}}} \right]\), we have

$$\begin{aligned} \bar{\delta }_{UU} - \bar{\delta }_{DD} = & ( - 3520 + 592K + 637Kt^{2} + 17920a^{6} t^{3} - 23808a^{5} t^{3} - 236352a^{4} t^{3} \\ & \quad + 12876at^{3} - 392tK - 23808a^{4} t^{2} - 102144a^{3} t^{2} + 3456a^{2} t^{2} - 51072a^{2} t \\ & \quad - 35040at^{2} - 10560at + 10496a^{3} t^{3} - 135336a^{2} t^{3} + 832Ka^{4} t^{2} + 448a^{3} t^{2} \\ & \quad + 3744Ka^{2} t^{2} - 392Kat^{2} + 448a^{2} t + 1184Kat - 15893t^{3} - 17520t + 12876t^{2} )/ \\ & \quad ( - 3520 + 592K + 637Kt^{2} - 108864Kt^{2} + 17920a^{6} t^{3} - 23808a^{5} t^{3} - 12480a^{4} t^{3} \\ & \quad + 10496a^{3} t^{3} + 859992a^{2} t^{3} - 49332at^{3} - 392Kt - 23808a^{4} t^{2} - 102144a^{3} t^{2} \\ & \quad + 3456a^{2} t^{2} - 51072a^{2} t + 89376at^{2} - 10560at + 3744Ka^{2} t^{2} - 392Kat^{2} \\ & \quad - 124416La^{2} t^{2} + 448a^{2} t + 1184Kat + 376795t^{3} + 44688t - 49332t^{2} ) \\ & \quad - 5/8 < 0 \\ \end{aligned}$$
(A12)

for all admissible \(a\).

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Heywood, J.S., Li, D. & Ye, G. Does price discrimination make collusion less likely? a delivered pricing model. J Econ 131, 39–60 (2020). https://doi.org/10.1007/s00712-020-00699-4

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Keywords

  • Uniform pricing
  • Spatial price discrimination
  • Collusion stability

JEL Classification

  • L13
  • L41
  • R32