Stability of pricing cartels in a bilateral price leadership model.

• Author: Veendorp, Emiel C.H.

1. Introduction

   The standard price leadership model focuses on a dominant seller of a homogeneous product who correctly assumes all other sellers to be pricetakers. The price leadership role doesn't necessarily have to be exercised by a single seller but can be shared by a number of firms forming a pricing cartel. This raises the question how variations in the size of this cartel affect the fortunes of those inside and outside the cartel, and which cartel sizes would be stable in the sense that cartel members have no incentive to leave the cartel while members of the competitive fringe have no incentive to join. This issue has been explored by d'Aspremont, Jacquemin, Gabszewicz and Weymark |1~, Donsimoni |2~, and Donsimoni, Economides and Polemarchakis |3~, leading to the conclusion that stable cartels exist whenever the number of firms is finite.

   This paper presents a bilateral extension of the dominant firm price leadership model. Since there is no conceptual reason why the group of firms making the pricing decision could not consist of both buyers and sellers, the question arises naturally whether such bilateral cartels would be viable and stable. The answer does not seem to be obvious. Why would a seller of some intermediate product such as bauxite, cement or tobacco, want to join and why would he be allowed to join, a pricing cartel made up of buyers of that product or conversely? For the interests of buyers and sellers are typically opposed and the cartel is supposed to be merely a pricing cartel that sets a uniform price for all firms outside and inside the cartel and fixes the level of operation for its members (outputs for its member-sellers, inputs for its member-buyers), but that does not engage in profit pooling for its members.

   To address this question we first construct a simple version of a bilateral price leadership model for a market with an arbitrary number of identical buyers and sellers (section II). The effects of changes in cartel size on the welfare of individual buyers and sellers are analyzed in section III where it is shown that the formation of bilateral cartels is potentially profitable for the relevant subset of buyers and sellers. The issue of cartel stability is taken up in section IV: our analysis suggests that stable cartels consist of a relatively small number of buyers and sellers, and that they typically include some bilateral ones but not those with a perfectly balanced composition. The final section offers some concluding comments.

2. A Bilateral Price Leadership Model

   The typical dominant firm price leadership model refers to a market where one seller or a coalition of a subset of sellers makes the pricing decision, while other sellers and all buyers act as pricetakers. There is conceptually no reason why this dominant group of firms could not be made up of a subset of buyers or a subset of buyers and sellers. This last situation will be referred to as a bilateral price leadership model, and the dominant group as the dominant coalition or pricing cartel. This cartel sets the price for all buyers and sellers in the competitive fringes, fixes the relevant input and output levels of its members, but does not pool their profits.

   In the absence of profit pooling the benefits of bilateral cartel formation are not shared equally among its buying and selling members, just as they would not be shared equally among the members of a one-sided cartel if firms were not identical. In the presence of profit pooling on the other hand, the issue of cartel stability would essentially disappear since overall profits of cartel members and those that consider to join the cartel, would always increase (section III), so that eventually all buyers and sellers could be expected to join. Also, the arrangement supporting profit pooling would obviously have to be a much more elaborate and collusive one than the one required for a pricing and rationalization cartel.

   To analyze the implications of different compositions of such a pricing cartel, the argument focuses on a market where market demand and supply would be given by D(P) with D|prime~(P) |is less than~ 0 and S(P) with S|prime~(P) |is greater than~ 0 as long as buyers and sellers act as pricetakers. Let the number of identical buyers and sellers be n and m respectively. In a competitive market the demand of a single buyer would then be

   |D.sub.i~ = D(P)/n (i = 1,...,n)

   and the corresponding marginal and total product curves (in dollar terms)

   M|P.sub.i~ = |D.sup.-1~ (n|Q.sub.i,b~)

   (i = 1,...,n)

   |TP.sub.i~ = integral of, between limits |Q.sub.i,b~ and 0 |D.sup.-1~(n|Q.sub.i,b~)d|Q.sub.i,b~

   where |Q.sub.i,b~ is the input level of a typical buyer. Similarly, the marginal and total cost curves of individual sellers are given by

   M|C.sub.j~ = |S.sup.-1~(m|Q.sub.j,s~)

   (j = 1,...,m)

   |TC.sub.j~ = integral of, between limits of |Q.sub.j,s~ and 0 |S.sup.-1~(m|Q.sub.j,s~)d|Q.sub.j,s~

   where |Q.sub.j,s~ is the output level of a single seller.

   A dominant coalition of N buyers and M sellers maximizes its (joint) profits

   ||Pi~.sub.d~ = N integral of, between limits of |Q.sub.i,db~ and 0 |D.sup.-1~(n|Q.sub.i,db~)d|Q.sub.i,db~ - NP|Q.sub.i,db~

   + MP|Q.sub.j,ds~ - M integral of, between limits of |Q.sub.j,ds~ and 0 |S.sup.-1~(m|Q.sub.j,ds~)d|Q.sub.j,ds~

   with respect to |Q.sub.i,db~, |Q.sub.j,ds~ and P, and subject to the market clearing condition

   N|Q.sub.i,db~ + (n - N)D(P)/n = M|Q.sub.j,ds~ + (m - M)S(P)/m.

   To preserve the spirit of a price leadership model, we assume N |is less than~ n and/or M |is less than~ m.

   Differentiating the corresponding Lagrangian (L = ||Pi~.sub.d~ + |Lambda~(...)) yields the following first order conditions:

   |Delta~L/|Delta~|Q.sub.i,db~ = |ND.sup.-1~(n|Q.sub.i,db~) - NP - N|Lambda~ = 0

   |Delta~L/|Delta~|Q.sub.j,ds~ = MP - M|S.sup.-1~(m|Q.sub.j,ds~ + M|Lambda~ = 0

   |Delta~L/|Delta~P = -N|Q.sub.i,db~ + M|Q.sub.j,ds~ + |Lambda~{(m - M)S|prime~(P)/m - (n - N)D|prime~(P)/n} = 0

   |Delta~L/|Delta~|Lambda~ = M|Q.sub.j,ds~ - N|Q.sub.i,db~ + (m -...