# Stability of pricing cartels in a bilateral price leadership model.

I. 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.

II. 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 - M)S(P)/m - (n - N)D(P)/n = 0.

From the first two equations we have

|Q.sub.i,db~ = D(P + |Lambda~)/n (1)

|Q.sub.j,ds~ = S(P + |Lambda~)/m (2)

where P + |Lambda~ is the common level of marginal product and marginal cost for cartel members. Substituting (1) and (2) in the last two equations and writing |Mu~ = M/m, |Nu~ = N/n yields after some rearranging

|Mu~S(P + |Lambda~) + (1 - |Mu~)S(P) - |Nu~D(P + |Lambda~) - (1 - |Nu~)D(P) = 0 (3)

(1 - |Mu~)S(P) - |Lambda~(1 - |Mu~S|prime~(P) - (1 - |Nu~)D(P) + |Lambda~(1 - |Nu~)D|prime~(P) = 0 (4)

which determine the profit maximizing levels of P and |Lambda~.

Visualizing the last two equations in |Lambda~, P space, one can verify that the first locus intersects the P-axis when P equals the competitive equilibrium price, |P.sup.c~, and that its slope

dP/d|Lambda~ = -{|Mu~S|prime~(P + |Lambda~) - |Nu~D|prime~(P + |Lambda~)}

/{|Mu~S|prime~(P + |Lambda~) - |Nu~D|prime~(P + |Lambda~) + (1 - |Mu~)S|prime~(P) - (1 - |Nu~)D|prime~(P)}

is negative and smaller than one absolutely. The second locus intersects the P-axis above |P.sup.c~ if the cartel is seller dominated (|Mu~ |is greater than~ |Nu~) and below |P.sup.c~ if the cartel is buyer dominated (|Mu~ |is less than~ |Nu~). Its slope

dP/d|Lambda~ = {(1 - |Nu~)D|prime~(P) - (1 - |Mu~)S|prime~(P)}/{(1 - |Nu~)D|prime~(P)

- (1 - |Mu~)S|prime~(P) + |Lambda~(1 - |Nu~)D|double prime~(P) - |Lambda~(1 - |Mu~)S|double prime~(P)}

is unity at |Lambda~ = 0 and remains positive unless the values of |Lambda~, S|double prime~ and/or - D|double prime~ are sufficiently large and of the same sign in which case the locus may have a vertical asymptote for some level of |Lambda~ sufficiently large absolutely. Figures 1(A) and 1(B) illustrate the configuration for a seller dominated and a buyer dominated cartel respectively.

It is clear from these observations that a seller dominated cartel exercises monopoly power as a net seller in the relevant market. To maximize profits, its members operate at (input and output) levels where the market price exceeds the competitive price level, which in turn exceeds the marginal revenue corresponding to the excess demand of non-members,

P* |is greater than~ |P.sup.c~ |is greater than~ MR* = |MC*.sub.j,ds~ = |MP*.sub.i,db~ = P* + |Lambda~*.

For buyer dominated cartels the opposite conclusion holds with its members operating at levels where the market price is less than the competitive equilibrium price, which in turn is less than the marginal outlay (or factor cost) corresponding to the excess supply of non-members,

P* |is less than~ |P.sup.c~ |is less than~ MO* = |MP*.sub.i,db~ = |MC*.sub.j,ds~ = P* + |Lambda~*.

Only cartels with a perfectly balanced composition (|Mu~ = |Nu~) will set their price at the competitive level, thereby realizing the largest possible total (buyer plus seller) surplus for that market.

The solution of the bilateral price leadership model can be illustrated graphically in the usual way. Let market demand and supply be linear, and let m = 3, M = 2, n = 5, N = 1. Figure 2(A) shows the market demand and supply curve for the competitive case as well as the individual marginal product and marginal cost curves. These building blocks are assembled in Figure 2(B) in the following manner: draw the excess demand curve of buyers and sellers in the competitive fringes, |D.sub.f~ - |S.sub.f~ (line AA|prime~), and add the corresponding marginal (BB|prime~). The intersection of this marginal with the horizontal summation (subtraction) of the marginal cost and product curves of cartel members, M|C.sub.j,ds~ - M|P.sub.i,db~ (line CC|prime~), fixes the net supply of cartel members and net demand of pricetakers, which will be sold and bought at price P*.

III. Changes in Cartel Composition and Profitability

Overall cartel profits are bound to increase when the number of buyers or sellers belonging to the cartel increases. Since firms joining the cartel as well as existing cartel members could continue to operate at the same input, output and price level as before, leaving profits unchanged for all firms, cartel profits could subsequently be increased further by reallocating output and input shares within the cartel so as to equalize marginal costs and marginal products of cartel members.

In the absence of profit pooling the analysis should focus on the effect of changes in cartel composition on the profits of individual firms. Since the effects of changes in |Mu~ and |Nu~ are perfectly symmetric we consider changes in |Mu~ only. Most results in this section apply to the case with linear demand and supply functions with an occasional reference to the more general, non-linear case.

Comparative static analysis of equations (3) and (4) with D|double prime~ = S|double prime~ = 0 yields

dP*/d|Mu~ = {S(P)(|Mu~S|prime~ - |Nu~D|prime~) + |Lambda~S|prime~(D|prime~ - S|prime~)}/|Delta~ (5)

d|Lambda~*/d|Mu~ = {|Lambda~S|prime~(|Mu~S|prime~ - |Nu~D|prime~) - S(P)(S|prime~ - D|prime~)}/|Delta~ (6)

d(P* + |Lambda~*)/d|Mu~ = {((1 - |Mu~)S|prime~ - (1 - |Nu~)D|prime~)(-|Lambda~S|prime~ - S(P))}/|Delta~ (7)

with

|Delta~ = ((1 - |Mu~)S|prime~ - (1 - |Nu~)D|prime~)((1 + |Mu~)S|prime~ - (1+ |Nu~)D|prime~) |is greater than~ 0.

It follows that d(P* + |Lambda~*)/d|Mu~ |is less than~ 0 since S(P) + |Lambda~S|prime~ = S(P + |Lambda~), and that dP*/d|Mu~ |is greater than~ 0 and d|Lambda~*/d|Mu~ |is less than~ 0 unless |Lambda~* is sufficiently large.

The ambiguous sign of dP*/d|Mu~ is noteworthy and can be supported graphically. An increase in |Mu~ rotates the negatively sloped locus in Figure 1 clockwise around the intercept on the vertical axis, and shifts the positively sloped locus upward. If |Mu~ |is greater than~ |Nu~ then both effects lead to an increase in the vertical coordinate of their intersection point, whereas if |Mu~ |is less than~ |Nu~ these forces work in opposite directions. This suggests that a cartel may actually lower the price it sets when a seller joins the cartel if that cartel is sufficiently buyer dominated (or |Lambda~* sufficiently large).

That this somewhat counterintuitive result can actually occur and can do so in the general, non-linear case, follows from differentiating equations (3) and (4) totally with respect to |Mu~ and then evaluating the result at |Mu~ = 0, |Nu~ = 1, which gives

dP*/d|Mu~ = {S|prime~(P)(S(P + |Lambda~) - S(P))}/{D|prime~(P + |Lambda~)(S|prime~(P)

- |Lambda~S|double prime~(P)) - S|prime~(P)(S|prime~(P) - |Nu~D|prime~(P + |Lambda~))}

which is negative (since |Lambda~* |is greater than~ 0) unless S|double prime~ is sufficiently large and positive or, in other words, unless marginal cost increases at a sufficiently decreasing rate. The intuitive explanation is that a change in the number of sellers belonging to the cartel affects the size of the cartel as well as that of the competitive fringe. One seller joining a buyer dominated cartel will leave fewer sellers to be exploited monopsonistically by the cartel, which would by itself lead to higher prices. But the same seller will produce more of the product after joining the cartel, which by itself would lead to lower prices.

The effect of a change in cartel composition on the equilibrium price translates directly into the effect on the profit levels of members of the competitive fringes, and can be formulated as follows.

PROPOSITION 1. The profits of sellers (buyers) in the competitive fringe will not necessarily increase (decrease) with the proportion of sellers (buyers) that belong to the cartel, and will not necessarily decrease (increase) with the proportion of buyers (sellers) that belong to the cartel.

Figure 3 illustrates how unbalanced cartels have to be for this counterintuitive result to apply. The values of |Mu~ and |Nu~ for which |Delta~||Pi~.sub.i,fs~/|Delta~|Mu~ |is less than~ 0 (|Delta~||Pi~.sub.i,fb/|Delta~|Nu~ |is less than~ 0) in a market with linear demand and supply (D = 10 - P, S = P) are indicated by the diagonally shaded regions in the upper left (lower right) portion of the diagram.

Turning to the profit levels of cartel members, we have for sellers

|Delta~||Pi~*.sub.j,ds~/|Delta~|Mu~ = (P - M|C.sub.j,ds~)|Delta~|Q.sub.j,ds~/|Delta~|Mu~ + |Q.sub.j,ds~|Delta~P/|Delta~|Mu~

= (-|Lambda~S|prime~|Delta~(P + |Lambda~)/|Delta~|Mu~ + S(P + |Lambda~)|Delta~P/|Delta~|Mu~/m

since MC = P + |Lambda~. Substitution of (5) and (7) and rearranging yields

|Delta~||Pi~*.sub.j,ds~/|Delta~|Mu~ = S(P + |Lambda~)(|Mu~S|prime~ - |Nu~D|prime~)(S - |Lambda~S|prime~)/(m|Delta~)

which is positive due to equation (4). For buyers belonging to the cartel one obtains in a similar fashion

|Delta~||Pi~*.sub.i,db~/|Delta~|Mu~ = {(||Lambda~.sup.2~S|prime~D|prime~ - S(P)D(P))(|Mu~S|prime~ - |Nu~D|prime~) + (D(P)S|prime~ - S(P)D|prime~)(S|prime~ - D|prime~)}/(n|Delta~)

which is negative unless |Lambda~ is sufficiently large and positive. This can be summarized as follows.

PROPOSITION 2. Profits of sellers (buyers) belonging to the cartel will always increase with the proportion of sellers (buyers) belonging to the cartel, but profits of sellers (buyers) belonging to the cartel do not necessarily decrease with the proportion of buyers (sellers) belonging to the cartel.

The values of |Mu~ and |Nu~ for which |Delta~||Pi~.sub.i,db~/|Delta~|Mu~ |is greater than~ 0 (|Delta~||Pi~.sub.j,ds~/|Delta~|Nu~ |is greater than~ 0) for the case with linear demand and supply are indicated by the horizontally (vertically) shaded portions of Figure 3. The diagram shows that the cartel compositions for which |Delta~||Pi~.sub.i,db~/|Delta~|Mu~ |is greater than~ 0 can be less unbalanced and buyer dominated than the ones for which |Delta~||Pi~.sub.i,fb~/|Delta~|Mu~ |is greater than~ 0 since |Delta~||Pi~.sub.i,db~/|Delta~|Mu~ is still positive when |Delta~P/|Delta~|Mu~ = 0. This is due to the fact that buyers that belong to the cartel not only benefit from a possibly lower price but also from an increased level of operation or input use.

Proposition 2 implies that the profits of all individual members of a bilateral cartel could potentially increase when an outsider would join their cartel. It remains to be seen whether outsiders would be interested in joining the cartel. This question is addressed in the next section.

IV. Stable Coalitions in a Bilateral Price Leadership Model

In their paper "On the Stability of Collusive Price Leadership" d'Aspremont, Jacquemin, Gabszewics and Weymark introduce the notions of internal and external stability of pricing cartels in a traditional price leadership model, where a dominant group of sellers imposes a selling price on buyers and a competitive fringe of sellers. They call a cartel internally stable when individual cartel members have no incentive to leave the cartel, and externally stable if individual pricetakers have no incentive to join the cartel. The incentives refer to individual profit levels and the relevant profit levels are those of the firm that contemplates leaving or joining the cartel.

Focusing on individual profit levels seems sensible for a pricing cartel that doesn't pool profits. Focusing on the profit level of the particular firm that considers a change in membership is appropriate in a setting where only sellers (or only buyers) can belong to the pricing cartel. For the profits of existing cartel members are then bound to increase when outsiders join their cartel, while members of the two competitive fringes can only resent but not prevent that cartel members join their ranks. It also seems plausible that the process of cartel formation will continue until a stable cartel is reached and that the outcome will not be affected by initial conditions or by the assumption of single firm movements: whether a stable cartel composition is reached by a single pricetaker joining an existing cartel, or by two of them joining simultaneously followed by one firm leaving the cartel, does not really matter as long as all sellers are assumed to be identical.

In a bilateral price leadership model some of these observations do not apply. The definition of stability can no longer focus on the change in profits of the firm leaving or joining the cartel: sellers belonging to the cartel will always welcome additional sellers, but not necessarily additional buyers, and conversely. The possible refusal to accept new members widens the class of stable cartels so that the outcome of cartel formation depends on initial conditions. It also implies that stable cartels are not necessarily accessible from below: existing cartel members may refuse to have other buyers or sellers join the cartel, even though the resulting composition would not necessarily create inducements for any member to leave that cartel and could therefore be stable by itself.

To accommodate bilateral coalitions a pricing cartel of M sellers and N buyers is said to be internally stable when existing cartel members have no incentive to leave the cartel and join the ranks of pricetakers. For cartels consisting of at least one buyer and at least one seller, this requires that

||Pi~*.sub.i,db~(M,N) |is greater than~ ||Pi~*.sub.i,fb~(M,N - 1) (8)

and

||Pi~*.sub.j,ds~(M,N) |is greater than~ ||Pi~*.sub.j,fs~(M - 1,N). (9)

For cartels consisting of buyers or sellers only, internal stability requires only (8) with M = 0, or (9) with N = 0.

A pricing cartel is said to be externally stable if pricetakers do not have the incentive and/or the permission to join the cartel. Cartels that do not include all buyers are externally stable with respect to buyers if

||Pi~*.sub.i,fb~(M,N) |is greater than~ ||Pi~*.sub.i,db~(M,N + 1) (10a)

||Pi~*.sub.j,ds~(M,N) |is greater than~ ||Pi~*.sub.j,ds~(M,N + 1). (10b)

Cartels that do not include all sellers are externally stable with respect to sellers if

||Pi~*.sub.j,fs~(M,N) |is greater than~ ||Pi~*.sub.j,ds~(M + 1,N) (11a)

and/or

||Pi~*.sub.i,db~(M,N) |is greater than~ ||Pi~*.sub.i,db~(M + 1,N). (11b)

Only conditions (10) or (11) apply to cartels to which all sellers (M = m) or all buyers (N = n) belong. Finally, a pricing cartel is said to be stable if it is internally as well as externally stable with respect to both buyers and sellers.

Derivation of conditions for stability for our specific bilateral price leadership model is cumbersome, and its results rather uninformative. It seems more instructive to locate instead all the stable cartels for selected values of m and n, the total number of buyers and sellers in the market. Doing so we limit ourselves to those that are accessible from below: those that could be reached by a process of individual firms joining or leaving the cartel, assuming the initial cartel to consist of a single seller or buyer only.

The information is summarized in Table I for a market with linear demand and supply curves (D = 10 - P, S = P). It appears that the subset of stable cartels that is accessible from below, consists of three members if the total number of buyers and sellers in the market is approximately the same: three sellers or two sellers and one buyer if the initial cartel consisted of a single seller; three buyers or two buyers and one seller if the initial cartel consisted of a single buyer. When the number of buyers is about double the number of sellers, the stable cartel will have two, three or four members, including at least one seller (and conversely). Finally, when the number of buyers in the market exceeds the number of sellers by a factor of four or more or conversely, the only stable cartels will be those with three buyers or three sellers.

It should be noted that pricing cartels that are stable in the traditional setting, may become unstable if bilateral coalitions are allowed. In line with the results of d'Aspremont, Jacquemin, Gabszewicz and Weymark |1~ a cartel consisting of three sellers would have been stable for all m, n values listed in Table I if buyer membership had been ruled out. But if the number of sellers in the market is about double the number of buyers, then that same coalition would become unstable without such a membership restriction.

Again, the set of all stable cartels would include many other bilateral ones, including all those with a balanced composition. Using the appropriate expressions for profits of cartel and fringe members, one can easily show that

||Pi~*.sub.i,db~(M,N) |is greater than~ ||Pi~*.sub.i,fb~(M,N - 1)

||Pi~*.sub.j,ds~(M,N) |is greater than~ ||Pi~*.sub.j,fs~(M - 1,N)

||Pi~*.sub.i,db~(M,N) |is greater than~ ||Pi~*.sub.i,db~(M + 1,N)

||Pi~*.sub.j,ds~(M,N) |is greater than~ ||Pi~*.sub.j,ds~(M,N + 1)

when cartels are perfectly balanced (Mn = Nm). Figure 3 supports this result. For all cartel compositions in the unshaded portion of the unit square, we have |Delta~||Pi~*.sub.ds~/|Delta~|Nu~ |is less than~ 0 and |Delta~||Pi~*.sub.db~/|Delta~|Mu~ |is less than~ 0 which means that the profits of existing cartel members would decrease if members belonging to the other side of the market would join. Since |Delta~||Pi~*.sub.fb~/|Delta~|Nu~ |is greater than~ 0 and |Delta~||Pi~*.sub.fs~/|Delta~|Mu~ |is greater than~ 0, and ||Pi~*.sub.fb~ = ||Pi~*.sub.db~ and ||Pi~*.sub.fs~ = ||Pi~*.sub.ds~ along the diagonal (|Mu~ = |Nu~), existing cartel members have no incentive to leave such a balanced pricing cartel. Unfortunately, these balanced cartels would not be formed by single buyers or sellers joining existing (unbalanced) cartels since such moves would negatively affect the profits of sellers or buyers already belonging to the cartel.

V. Conclusions

The main conclusion of this paper is that in spite of the basically opposing interests of individual buyers and sellers, bilateral pricing cartels are potentially viable and stable. Though the specific results were obtained for a model with identical buyers and sellers and linear demand and supply curves, it seems plausible that qualitatively similar results could be derived from different specifications of these building blocks.

Formation of a bilateral pricing cartel could be looked upon as a weak form of vertical integration. In our paper, bilateral cartel formation was made contingent on the increased profit levels of all individual firms involved. Vertical integration is more likely to be guided by overall gains and would be profitable, therefore, for all initial compositions of the integrating firm. Increased pricing power in the market that is partially bypassed constitutes an obvious rationale for buyers or sellers to integrate backward or forward. But our model is clearly not designed to evaluate other considerations in favor of or against such moves.

Another conclusion of our paper is that a more balanced composition of the bilateral pricing cartel enhances total buyer plus seller surplus. Together, our conclusions seem to offer some support for Galbraith's countervailing power hypothesis |4~. Countervailing power refers to market power created on one side of the market to benefit the position of firms at that side by offsetting market power on the other side. One could argue that buyers, for example, collectively exercise market power by having some of its members join a pricing cartel of sellers. But attempts to create such balanced market power will be successful only if existing members of the cartel are willing to accept their entry, and will not necessarily increase profits of buyers that remain in the competitive fringe. And although some bilateral cartels are indeed accessible from below, they do not include those with a balanced composition.

Our model does support the notion that countervailing power in the sense of a balanced composition of the pricing cartel, will improve welfare. This result contrasts with the one obtained by Veendorp |5~ for his oligoempory model, that increased concentration on one side of the market will never improve the allocation of resources, whether it results in more balanced market power or not. Unfortunately, these balanced cartels don't come about by decentralized coalition formation, and policy measures offering a helping hand in this direction may be beneficial. In those cases that pricing cartels require government approval, such approval might be made conditional on balanced representation in the cartel of buyers and sellers of the product concerned.

References

1. d'Aspremont, Claude, Alexis Jacquemin, Jean Jaskold Gabszewicz, and John A. Weymark, "On the stability of collusive price leadership." Canadian Economic Journal, February 1983, 17-25.

2. Donsimoni, Marie-Paule, "Stable Heterogeneous Cartels." International Journal of Industrial Organization, December 1985, 451-67.

3. Donsimoni, M.-P., N. S. Economides, and H. M. Polemarchakis, "Stable Cartels." International Economic Review, June 1986, 317-27.

4. Galbraith, John Kenneth. American Capitalism: The Concept of Countervailing Power. Boston: Houghton-Mifflin, 1952.

5. Veendorp, E. C. H. "Oligoemporistic competition and the countervailing power hypothesis." Canadian Economic Journal, August 1987, 519-26.

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.

II. 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 - M)S(P)/m - (n - N)D(P)/n = 0.

From the first two equations we have

|Q.sub.i,db~ = D(P + |Lambda~)/n (1)

|Q.sub.j,ds~ = S(P + |Lambda~)/m (2)

where P + |Lambda~ is the common level of marginal product and marginal cost for cartel members. Substituting (1) and (2) in the last two equations and writing |Mu~ = M/m, |Nu~ = N/n yields after some rearranging

|Mu~S(P + |Lambda~) + (1 - |Mu~)S(P) - |Nu~D(P + |Lambda~) - (1 - |Nu~)D(P) = 0 (3)

(1 - |Mu~)S(P) - |Lambda~(1 - |Mu~S|prime~(P) - (1 - |Nu~)D(P) + |Lambda~(1 - |Nu~)D|prime~(P) = 0 (4)

which determine the profit maximizing levels of P and |Lambda~.

Visualizing the last two equations in |Lambda~, P space, one can verify that the first locus intersects the P-axis when P equals the competitive equilibrium price, |P.sup.c~, and that its slope

dP/d|Lambda~ = -{|Mu~S|prime~(P + |Lambda~) - |Nu~D|prime~(P + |Lambda~)}

/{|Mu~S|prime~(P + |Lambda~) - |Nu~D|prime~(P + |Lambda~) + (1 - |Mu~)S|prime~(P) - (1 - |Nu~)D|prime~(P)}

is negative and smaller than one absolutely. The second locus intersects the P-axis above |P.sup.c~ if the cartel is seller dominated (|Mu~ |is greater than~ |Nu~) and below |P.sup.c~ if the cartel is buyer dominated (|Mu~ |is less than~ |Nu~). Its slope

dP/d|Lambda~ = {(1 - |Nu~)D|prime~(P) - (1 - |Mu~)S|prime~(P)}/{(1 - |Nu~)D|prime~(P)

- (1 - |Mu~)S|prime~(P) + |Lambda~(1 - |Nu~)D|double prime~(P) - |Lambda~(1 - |Mu~)S|double prime~(P)}

is unity at |Lambda~ = 0 and remains positive unless the values of |Lambda~, S|double prime~ and/or - D|double prime~ are sufficiently large and of the same sign in which case the locus may have a vertical asymptote for some level of |Lambda~ sufficiently large absolutely. Figures 1(A) and 1(B) illustrate the configuration for a seller dominated and a buyer dominated cartel respectively.

It is clear from these observations that a seller dominated cartel exercises monopoly power as a net seller in the relevant market. To maximize profits, its members operate at (input and output) levels where the market price exceeds the competitive price level, which in turn exceeds the marginal revenue corresponding to the excess demand of non-members,

P* |is greater than~ |P.sup.c~ |is greater than~ MR* = |MC*.sub.j,ds~ = |MP*.sub.i,db~ = P* + |Lambda~*.

For buyer dominated cartels the opposite conclusion holds with its members operating at levels where the market price is less than the competitive equilibrium price, which in turn is less than the marginal outlay (or factor cost) corresponding to the excess supply of non-members,

P* |is less than~ |P.sup.c~ |is less than~ MO* = |MP*.sub.i,db~ = |MC*.sub.j,ds~ = P* + |Lambda~*.

Only cartels with a perfectly balanced composition (|Mu~ = |Nu~) will set their price at the competitive level, thereby realizing the largest possible total (buyer plus seller) surplus for that market.

The solution of the bilateral price leadership model can be illustrated graphically in the usual way. Let market demand and supply be linear, and let m = 3, M = 2, n = 5, N = 1. Figure 2(A) shows the market demand and supply curve for the competitive case as well as the individual marginal product and marginal cost curves. These building blocks are assembled in Figure 2(B) in the following manner: draw the excess demand curve of buyers and sellers in the competitive fringes, |D.sub.f~ - |S.sub.f~ (line AA|prime~), and add the corresponding marginal (BB|prime~). The intersection of this marginal with the horizontal summation (subtraction) of the marginal cost and product curves of cartel members, M|C.sub.j,ds~ - M|P.sub.i,db~ (line CC|prime~), fixes the net supply of cartel members and net demand of pricetakers, which will be sold and bought at price P*.

III. Changes in Cartel Composition and Profitability

Overall cartel profits are bound to increase when the number of buyers or sellers belonging to the cartel increases. Since firms joining the cartel as well as existing cartel members could continue to operate at the same input, output and price level as before, leaving profits unchanged for all firms, cartel profits could subsequently be increased further by reallocating output and input shares within the cartel so as to equalize marginal costs and marginal products of cartel members.

In the absence of profit pooling the analysis should focus on the effect of changes in cartel composition on the profits of individual firms. Since the effects of changes in |Mu~ and |Nu~ are perfectly symmetric we consider changes in |Mu~ only. Most results in this section apply to the case with linear demand and supply functions with an occasional reference to the more general, non-linear case.

Comparative static analysis of equations (3) and (4) with D|double prime~ = S|double prime~ = 0 yields

dP*/d|Mu~ = {S(P)(|Mu~S|prime~ - |Nu~D|prime~) + |Lambda~S|prime~(D|prime~ - S|prime~)}/|Delta~ (5)

d|Lambda~*/d|Mu~ = {|Lambda~S|prime~(|Mu~S|prime~ - |Nu~D|prime~) - S(P)(S|prime~ - D|prime~)}/|Delta~ (6)

d(P* + |Lambda~*)/d|Mu~ = {((1 - |Mu~)S|prime~ - (1 - |Nu~)D|prime~)(-|Lambda~S|prime~ - S(P))}/|Delta~ (7)

with

|Delta~ = ((1 - |Mu~)S|prime~ - (1 - |Nu~)D|prime~)((1 + |Mu~)S|prime~ - (1+ |Nu~)D|prime~) |is greater than~ 0.

It follows that d(P* + |Lambda~*)/d|Mu~ |is less than~ 0 since S(P) + |Lambda~S|prime~ = S(P + |Lambda~), and that dP*/d|Mu~ |is greater than~ 0 and d|Lambda~*/d|Mu~ |is less than~ 0 unless |Lambda~* is sufficiently large.

The ambiguous sign of dP*/d|Mu~ is noteworthy and can be supported graphically. An increase in |Mu~ rotates the negatively sloped locus in Figure 1 clockwise around the intercept on the vertical axis, and shifts the positively sloped locus upward. If |Mu~ |is greater than~ |Nu~ then both effects lead to an increase in the vertical coordinate of their intersection point, whereas if |Mu~ |is less than~ |Nu~ these forces work in opposite directions. This suggests that a cartel may actually lower the price it sets when a seller joins the cartel if that cartel is sufficiently buyer dominated (or |Lambda~* sufficiently large).

That this somewhat counterintuitive result can actually occur and can do so in the general, non-linear case, follows from differentiating equations (3) and (4) totally with respect to |Mu~ and then evaluating the result at |Mu~ = 0, |Nu~ = 1, which gives

dP*/d|Mu~ = {S|prime~(P)(S(P + |Lambda~) - S(P))}/{D|prime~(P + |Lambda~)(S|prime~(P)

- |Lambda~S|double prime~(P)) - S|prime~(P)(S|prime~(P) - |Nu~D|prime~(P + |Lambda~))}

which is negative (since |Lambda~* |is greater than~ 0) unless S|double prime~ is sufficiently large and positive or, in other words, unless marginal cost increases at a sufficiently decreasing rate. The intuitive explanation is that a change in the number of sellers belonging to the cartel affects the size of the cartel as well as that of the competitive fringe. One seller joining a buyer dominated cartel will leave fewer sellers to be exploited monopsonistically by the cartel, which would by itself lead to higher prices. But the same seller will produce more of the product after joining the cartel, which by itself would lead to lower prices.

The effect of a change in cartel composition on the equilibrium price translates directly into the effect on the profit levels of members of the competitive fringes, and can be formulated as follows.

PROPOSITION 1. The profits of sellers (buyers) in the competitive fringe will not necessarily increase (decrease) with the proportion of sellers (buyers) that belong to the cartel, and will not necessarily decrease (increase) with the proportion of buyers (sellers) that belong to the cartel.

Figure 3 illustrates how unbalanced cartels have to be for this counterintuitive result to apply. The values of |Mu~ and |Nu~ for which |Delta~||Pi~.sub.i,fs~/|Delta~|Mu~ |is less than~ 0 (|Delta~||Pi~.sub.i,fb/|Delta~|Nu~ |is less than~ 0) in a market with linear demand and supply (D = 10 - P, S = P) are indicated by the diagonally shaded regions in the upper left (lower right) portion of the diagram.

Turning to the profit levels of cartel members, we have for sellers

|Delta~||Pi~*.sub.j,ds~/|Delta~|Mu~ = (P - M|C.sub.j,ds~)|Delta~|Q.sub.j,ds~/|Delta~|Mu~ + |Q.sub.j,ds~|Delta~P/|Delta~|Mu~

= (-|Lambda~S|prime~|Delta~(P + |Lambda~)/|Delta~|Mu~ + S(P + |Lambda~)|Delta~P/|Delta~|Mu~/m

since MC = P + |Lambda~. Substitution of (5) and (7) and rearranging yields

|Delta~||Pi~*.sub.j,ds~/|Delta~|Mu~ = S(P + |Lambda~)(|Mu~S|prime~ - |Nu~D|prime~)(S - |Lambda~S|prime~)/(m|Delta~)

which is positive due to equation (4). For buyers belonging to the cartel one obtains in a similar fashion

|Delta~||Pi~*.sub.i,db~/|Delta~|Mu~ = {(||Lambda~.sup.2~S|prime~D|prime~ - S(P)D(P))(|Mu~S|prime~ - |Nu~D|prime~) + (D(P)S|prime~ - S(P)D|prime~)(S|prime~ - D|prime~)}/(n|Delta~)

which is negative unless |Lambda~ is sufficiently large and positive. This can be summarized as follows.

PROPOSITION 2. Profits of sellers (buyers) belonging to the cartel will always increase with the proportion of sellers (buyers) belonging to the cartel, but profits of sellers (buyers) belonging to the cartel do not necessarily decrease with the proportion of buyers (sellers) belonging to the cartel.

The values of |Mu~ and |Nu~ for which |Delta~||Pi~.sub.i,db~/|Delta~|Mu~ |is greater than~ 0 (|Delta~||Pi~.sub.j,ds~/|Delta~|Nu~ |is greater than~ 0) for the case with linear demand and supply are indicated by the horizontally (vertically) shaded portions of Figure 3. The diagram shows that the cartel compositions for which |Delta~||Pi~.sub.i,db~/|Delta~|Mu~ |is greater than~ 0 can be less unbalanced and buyer dominated than the ones for which |Delta~||Pi~.sub.i,fb~/|Delta~|Mu~ |is greater than~ 0 since |Delta~||Pi~.sub.i,db~/|Delta~|Mu~ is still positive when |Delta~P/|Delta~|Mu~ = 0. This is due to the fact that buyers that belong to the cartel not only benefit from a possibly lower price but also from an increased level of operation or input use.

Proposition 2 implies that the profits of all individual members of a bilateral cartel could potentially increase when an outsider would join their cartel. It remains to be seen whether outsiders would be interested in joining the cartel. This question is addressed in the next section.

IV. Stable Coalitions in a Bilateral Price Leadership Model

In their paper "On the Stability of Collusive Price Leadership" d'Aspremont, Jacquemin, Gabszewics and Weymark introduce the notions of internal and external stability of pricing cartels in a traditional price leadership model, where a dominant group of sellers imposes a selling price on buyers and a competitive fringe of sellers. They call a cartel internally stable when individual cartel members have no incentive to leave the cartel, and externally stable if individual pricetakers have no incentive to join the cartel. The incentives refer to individual profit levels and the relevant profit levels are those of the firm that contemplates leaving or joining the cartel.

Focusing on individual profit levels seems sensible for a pricing cartel that doesn't pool profits. Focusing on the profit level of the particular firm that considers a change in membership is appropriate in a setting where only sellers (or only buyers) can belong to the pricing cartel. For the profits of existing cartel members are then bound to increase when outsiders join their cartel, while members of the two competitive fringes can only resent but not prevent that cartel members join their ranks. It also seems plausible that the process of cartel formation will continue until a stable cartel is reached and that the outcome will not be affected by initial conditions or by the assumption of single firm movements: whether a stable cartel composition is reached by a single pricetaker joining an existing cartel, or by two of them joining simultaneously followed by one firm leaving the cartel, does not really matter as long as all sellers are assumed to be identical.

In a bilateral price leadership model some of these observations do not apply. The definition of stability can no longer focus on the change in profits of the firm leaving or joining the cartel: sellers belonging to the cartel will always welcome additional sellers, but not necessarily additional buyers, and conversely. The possible refusal to accept new members widens the class of stable cartels so that the outcome of cartel formation depends on initial conditions. It also implies that stable cartels are not necessarily accessible from below: existing cartel members may refuse to have other buyers or sellers join the cartel, even though the resulting composition would not necessarily create inducements for any member to leave that cartel and could therefore be stable by itself.

To accommodate bilateral coalitions a pricing cartel of M sellers and N buyers is said to be internally stable when existing cartel members have no incentive to leave the cartel and join the ranks of pricetakers. For cartels consisting of at least one buyer and at least one seller, this requires that

||Pi~*.sub.i,db~(M,N) |is greater than~ ||Pi~*.sub.i,fb~(M,N - 1) (8)

and

||Pi~*.sub.j,ds~(M,N) |is greater than~ ||Pi~*.sub.j,fs~(M - 1,N). (9)

For cartels consisting of buyers or sellers only, internal stability requires only (8) with M = 0, or (9) with N = 0.

A pricing cartel is said to be externally stable if pricetakers do not have the incentive and/or the permission to join the cartel. Cartels that do not include all buyers are externally stable with respect to buyers if

||Pi~*.sub.i,fb~(M,N) |is greater than~ ||Pi~*.sub.i,db~(M,N + 1) (10a)

||Pi~*.sub.j,ds~(M,N) |is greater than~ ||Pi~*.sub.j,ds~(M,N + 1). (10b)

Cartels that do not include all sellers are externally stable with respect to sellers if

||Pi~*.sub.j,fs~(M,N) |is greater than~ ||Pi~*.sub.j,ds~(M + 1,N) (11a)

and/or

||Pi~*.sub.i,db~(M,N) |is greater than~ ||Pi~*.sub.i,db~(M + 1,N). (11b)

Only conditions (10) or (11) apply to cartels to which all sellers (M = m) or all buyers (N = n) belong. Finally, a pricing cartel is said to be stable if it is internally as well as externally stable with respect to both buyers and sellers.

Table I. Pricing Cartels (M,N) That Are Stable and Accessible from Below in a Market with m Sellers and n Buyers. m n 5 10 20 40 5 (3,0) (0,3) (3,1) (0,3) (3,0) (0,3) (3,0) (0,3) (2,1) (1,2) (1,1) 10 (3,0) (1,3) (3,0) (0,3) (3,1) (0,3) (3,0) (0,3) (1,1) (2,1) (1,2) (1,1) 20 (3,0) (0,3) (3,0) (1,3) (3,0) (0,3) (3,1) (0,3) (1,0) (2,1) (1,2) (1,1) 40 (3,0) (0,3) (3,0) (0,3) (3,0) (1,3) (3,0) (0,3) (1,1) (2,1) (1,2)

Derivation of conditions for stability for our specific bilateral price leadership model is cumbersome, and its results rather uninformative. It seems more instructive to locate instead all the stable cartels for selected values of m and n, the total number of buyers and sellers in the market. Doing so we limit ourselves to those that are accessible from below: those that could be reached by a process of individual firms joining or leaving the cartel, assuming the initial cartel to consist of a single seller or buyer only.

The information is summarized in Table I for a market with linear demand and supply curves (D = 10 - P, S = P). It appears that the subset of stable cartels that is accessible from below, consists of three members if the total number of buyers and sellers in the market is approximately the same: three sellers or two sellers and one buyer if the initial cartel consisted of a single seller; three buyers or two buyers and one seller if the initial cartel consisted of a single buyer. When the number of buyers is about double the number of sellers, the stable cartel will have two, three or four members, including at least one seller (and conversely). Finally, when the number of buyers in the market exceeds the number of sellers by a factor of four or more or conversely, the only stable cartels will be those with three buyers or three sellers.

It should be noted that pricing cartels that are stable in the traditional setting, may become unstable if bilateral coalitions are allowed. In line with the results of d'Aspremont, Jacquemin, Gabszewicz and Weymark |1~ a cartel consisting of three sellers would have been stable for all m, n values listed in Table I if buyer membership had been ruled out. But if the number of sellers in the market is about double the number of buyers, then that same coalition would become unstable without such a membership restriction.

Again, the set of all stable cartels would include many other bilateral ones, including all those with a balanced composition. Using the appropriate expressions for profits of cartel and fringe members, one can easily show that

||Pi~*.sub.i,db~(M,N) |is greater than~ ||Pi~*.sub.i,fb~(M,N - 1)

||Pi~*.sub.j,ds~(M,N) |is greater than~ ||Pi~*.sub.j,fs~(M - 1,N)

||Pi~*.sub.i,db~(M,N) |is greater than~ ||Pi~*.sub.i,db~(M + 1,N)

||Pi~*.sub.j,ds~(M,N) |is greater than~ ||Pi~*.sub.j,ds~(M,N + 1)

when cartels are perfectly balanced (Mn = Nm). Figure 3 supports this result. For all cartel compositions in the unshaded portion of the unit square, we have |Delta~||Pi~*.sub.ds~/|Delta~|Nu~ |is less than~ 0 and |Delta~||Pi~*.sub.db~/|Delta~|Mu~ |is less than~ 0 which means that the profits of existing cartel members would decrease if members belonging to the other side of the market would join. Since |Delta~||Pi~*.sub.fb~/|Delta~|Nu~ |is greater than~ 0 and |Delta~||Pi~*.sub.fs~/|Delta~|Mu~ |is greater than~ 0, and ||Pi~*.sub.fb~ = ||Pi~*.sub.db~ and ||Pi~*.sub.fs~ = ||Pi~*.sub.ds~ along the diagonal (|Mu~ = |Nu~), existing cartel members have no incentive to leave such a balanced pricing cartel. Unfortunately, these balanced cartels would not be formed by single buyers or sellers joining existing (unbalanced) cartels since such moves would negatively affect the profits of sellers or buyers already belonging to the cartel.

V. Conclusions

The main conclusion of this paper is that in spite of the basically opposing interests of individual buyers and sellers, bilateral pricing cartels are potentially viable and stable. Though the specific results were obtained for a model with identical buyers and sellers and linear demand and supply curves, it seems plausible that qualitatively similar results could be derived from different specifications of these building blocks.

Formation of a bilateral pricing cartel could be looked upon as a weak form of vertical integration. In our paper, bilateral cartel formation was made contingent on the increased profit levels of all individual firms involved. Vertical integration is more likely to be guided by overall gains and would be profitable, therefore, for all initial compositions of the integrating firm. Increased pricing power in the market that is partially bypassed constitutes an obvious rationale for buyers or sellers to integrate backward or forward. But our model is clearly not designed to evaluate other considerations in favor of or against such moves.

Another conclusion of our paper is that a more balanced composition of the bilateral pricing cartel enhances total buyer plus seller surplus. Together, our conclusions seem to offer some support for Galbraith's countervailing power hypothesis |4~. Countervailing power refers to market power created on one side of the market to benefit the position of firms at that side by offsetting market power on the other side. One could argue that buyers, for example, collectively exercise market power by having some of its members join a pricing cartel of sellers. But attempts to create such balanced market power will be successful only if existing members of the cartel are willing to accept their entry, and will not necessarily increase profits of buyers that remain in the competitive fringe. And although some bilateral cartels are indeed accessible from below, they do not include those with a balanced composition.

Our model does support the notion that countervailing power in the sense of a balanced composition of the pricing cartel, will improve welfare. This result contrasts with the one obtained by Veendorp |5~ for his oligoempory model, that increased concentration on one side of the market will never improve the allocation of resources, whether it results in more balanced market power or not. Unfortunately, these balanced cartels don't come about by decentralized coalition formation, and policy measures offering a helping hand in this direction may be beneficial. In those cases that pricing cartels require government approval, such approval might be made conditional on balanced representation in the cartel of buyers and sellers of the product concerned.

References

1. d'Aspremont, Claude, Alexis Jacquemin, Jean Jaskold Gabszewicz, and John A. Weymark, "On the stability of collusive price leadership." Canadian Economic Journal, February 1983, 17-25.

2. Donsimoni, Marie-Paule, "Stable Heterogeneous Cartels." International Journal of Industrial Organization, December 1985, 451-67.

3. Donsimoni, M.-P., N. S. Economides, and H. M. Polemarchakis, "Stable Cartels." International Economic Review, June 1986, 317-27.

4. Galbraith, John Kenneth. American Capitalism: The Concept of Countervailing Power. Boston: Houghton-Mifflin, 1952.

5. Veendorp, E. C. H. "Oligoemporistic competition and the countervailing power hypothesis." Canadian Economic Journal, August 1987, 519-26.

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Author: | Veendorp, Emiel C.H. |
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Publication: | Southern Economic Journal |

Date: | Apr 1, 1993 |

Words: | 4824 |

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