A Pedagogical Treatment of Bilateral Monopoly Author(s): Roger D. Blair, David L. Kaserman, Richard E. Romano Source: Southern Economic Journal, Vol. 55, No. 4 (Apr., 1989), pp. 831-841 Published by: Southern Economic Association Stable URL: http://www.jstor.org/stable/1059465 Accessed: 30/09/2010 04:17

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http://www.jstor.org A Pedagogical Treatment of Bilateral Monopoly*

ROGER D. BLAIR University of Florida Gainesville, Florida

DAVID L. KASERMAN Auburn University Auburn, Alabama

RICHARD E. ROMANO University of Florida Gainesville, Florida

I. Introduction

Most economists are familiar with the concept of bilateral monopoly: an upstream monopolist sells its output to a single downstream buyer that may also be a monopolist in its output market. The theory of bilateral monopoly has a rich history that can be traced to the writings of Cournot [10] and Menger [31].1 Over the 150 or so years that the problem has been under consideration, however, economists have offered a variety of "solutions" ranging from a completely determinate intermediate good price and output to a completely indeterminate solution within a specified range. Interestingly, this historical divergence of opinion concerning the correct outcome under bilateral monopoly still persists. A clear consensus has not yet emerged. Surprisingly, this lack of unanimity exists despite the fact that Bowley [5] provided the theoretically correct solution in 1928.2 Even more surprising, however, is the apparent popularity of the incorrect solution. A recent survey of the population of intermediate microeconomic texts on the authors' bookshelves revealed that over 80 percent of the current treatments of this topic are in error.3 Table I presents the results of that survey. Those authors who present a correct analysis of bilateral monopoly recognize that optimality requires joint profit maximization. This leads them to the correct conclusion regarding the deter- minate quantity of the intermediate product exchanged. Of the five treatments we have classified

*The authors express appreciationto Leonard Cheng, Bob Ekelund, John Mayo, and an anonymous referee for helpful comments on a priordraft of this paper. The usual caveat applies. 1. These early writers did not consider bilateralmonopoly explicitly but, rather,treated the closely related topic of isolated exchange between two individuals. For an excellent discussion of the historicaldevelopment of the theory of bilateralmonopoly, see Machlup and Taber[27]. 2. Machlup and Taber's [27] paper served to remindthe professionof the correctsolution. 3. Not all microeconomics texts treat the subject of bilateralmonopoly. Among those texts that do not examine bilateral monopoly are the following: Becker [4], Browning and Browning [6], David Friedman[14], Hirshleifer [19], Lindsay [26], McCloskey [30], Quirk [34], Russell and Wilkinson [36], and Varian[42]. Of course, there may be others since we have merely a sample on our collective shelves.

831 832 RogerBlair, David Kaserman,and RichardRomano

P3 - - CCx Pi - - ^ m

P3^^^1 \DQ-CT

0 Xi X2 X3 X=Q MRP = Dx MRxMRQ Figure 1. as correct, however, two (Friedman[15] and Stigler [40]) drawno conclusionabout the inputprice range that will result, and another (Dewey [12]) draws an incorrectconclusion.4 Thus, only two texts (Hendersonand Quandt[17], and Layardand Walters[24]) are both complete and correct.5 Furthermore,none of these authorsprovides an adequaterepresentation of the contractcurve, and little emphasis is placed on the fact that the price of the intermediategood does not function as a rationingdevice but merely serves to divide the jointly maximizedprofits. As for the far more prevalentincorrect treatments of bilateralmonopoly, the majorsource of confusion appearsto be a failureto recognize the importanceof joint profitmaximization through negotiationon both the price and the quantityof the intermediategood. As we shall see below, it is extremely difficultto derive the correctsolution absentthis recognition. We have divided this "Incorrect"group into three parts. The first set includes those treat- ments that are clearly written and are clearly incorrectin some importantway. The second set includes those treatmentsthat generally fail to recognizethe correctsolution, but whose exposition is so unclear that one cannot be sure whetherthe authorsintended to treat a narrowerproblem.6

4. Dewey [12] also examines a collusive solution to the monopoly - "monopsony"problem. His treatmentis simplified because he assumes constant marginalcost. As a result, the marginalfactor cost equals the input price and there can be no monopsonistic exploitation.Thus, while Dewey is correctgiven the assumedconditions, his treatmentis less interestingthan the other correct treatments. 5. Scherer's treatise [37] also provides a correct treatmentalthough his is not an intermediatemicroeconomics text. 6. Baumol's exposition [3] is included in the "Unclear"group because he treats this problem in an unorthodox way by using utility functions with income and the quantityof the inputas arguments.Such a treatmentassumes that the amountof income to be sharedbetween the partiesis independentof the quantityof the inputexchanged. This assumption is violated in the bilateral monopoly model since total profitdepends upon the amountof the intermediategood traded (and final output produced). Much of what follows appearsto be correct, but the subsequentdiscussion of the joint profit maximizing solution is inappropriatein this model. A PEDAGOGICALTREATMENT OF BILATERALMONOPOLY 833

These typically fail to identify the crucial importanceof joint profitmaximization. They also tend to muddy the issue by using a labor union as the monopolist. This makes it extremelydifficult to tell whether these authors meant to examine a special case where the seller could not guarantee the quantity,presumably because a union's controlover its membersis somewhatincomplete. Finally, our third set is a small group of texts that do not appear to appreciatefully the importanceof joint profit maximization, but suggest or hint that such an outcome might result. None of these treatmentsexamines intermediateproduct price in a generalway. Neither the range of possible prices nor the fact that price serves only to divide profitsis identified. Clearly, then, something is amiss in the theory of bilateralmonopoly. Either the majority of the authors surveyed (and those who reviewed drafts for their publishers)are unawareof the correct solution, or they are unconvinced by the prior analysis. The purpose of this paper is both to serve as a reminder of the correct solution and to present what is, hopefully, a more convincing case for its adoption. For comparisonpurposes, we sketch the more popularanalysis before proceeding to the correct approach.

II. Conventional Analysis

The conventional analysis is summarizedin Figure 1. For convenience, we assume a fixed in- put/outputratio equal to one. This assumptionis not critical and, in fact, we relax it in the next section. If the downstreamindustry were competitive in the final outputmarket, the derived de- mand for the input would equal DQ -CT, where DQ is final product demand and CT is the constant cost of transformingone unit of input x into one unit of output Q. Thus, DQ - CT representsthe average net revenue as a functionof the quantityof x employed. With monopoly in the sale of Q, however, the derived demandfor x will be the curve that is marginalto DQ - CT, which is labelled D, in the graph. Thus, D, representsthe net marginalrevenue product of input x.7 The curve labelled MRx is marginal to Dx and representsthe marginalrevenue associated with selling this intermediategood to a downstreamfirm that has monopoly power in Q but not monopsony power in x. Note, however, that D_ cannot constitutethe downstreamfirm's derived demand in the bilateral monopoly situationbecause a monopsonistis not a price taker and does not have a demand curve. Turningto the cost curves, AC. denotes the upstreammonopolist's average cost of producing input x, and MC. is marginalcost. If the supplierof x were to behave as a perfect competitor, MC. would correspond to its supply curve. Then, if the downstreammonopsonist were hiring this input from such a competitor,MFC. would be the marginalfactor cost of the input. Authors adopting the standardapproach typically arriveat their conclusionthat the bilateral monopoly problem is indeterminateby alternatelyassuming that one tradingpartner and then the other behaves as would a perfect competitor.If the upstreamfirm behaves competitively,then its supply curve will correspondto MCx. In this case, the downstreamfirm will arriveat the standard monopsony solution, buying x2 units of the intermediateproduct at a price of P2 per unit. If, on the other hand, the downstreamfirm behaves as a perfect competitorin its hiring decision, then the upstreamfirm will exercise its monopoly power in supplyingthe input. In this case, we have the input monopoly solution at xl andp .

7. By "net marginalrevenue productof X," we mean the additionalrevenue generated by employing an additional unit of x, given that the other input(s) necessary to transformx into Q is (are) adjustedoptimally. 834 Roger Blair, David Kaserman, and Richard Romano

TABLE I. IntermediateMicroeconomics Texts TreatingBilateral Monopoly CORRECT INCORRECT

Dewey [12] I. CompletelyIncorrect M. Friedman[15] Baird[1] Hendersonand Quandt [17] Barrett[2] Layardand Walters[24] Cohen and Cyert [9] Stigler [40] Gould and Ferguson[16] Hibdon [18] Kogiku [21] Kohler[22] Koutsoyiannis[23] Liebhafsky[25] Mahanty[28] Mansfield[29] Thompson[41] II. Unclear Baumol [3] Call and Holahan[7] Carroll[8] Freeman[13] Hyman[20] Nicholson [33] Sher and Pinola [38] Solberg [39] III. Incomplete Daniel [11] Miller [32]

According to the conventional analysis, these two outcomes set the bounds on the equilibrium price-quantity combination. Those textbooks that we have branded "Completely Incorrect" in Table I typically conclude that the solution to the bilateral monopoly model will fall somewhere within the (pl,xl)- (p2,x2) range in Figure 1. As we shall explain fully in the next section, however, the correct solution involves a determinate quantity equal to x3 and an indeterminate price in the interval between p3 and p . The quantity can be determined because any value other than x3 permits both buyer and seller to increase profits. This also implies that both the price and quantity of the final good is determinate in spite of claims to the contrary by many authors. The intermediate good's price, however, cannot be determined by demand and cost conditions since it only serves to divide the (maximized) rents. To pin down this price requires a more complete specification of the bargaining process.

III. Correct Solution-A Revised Approach

Bowley [5] and others have pointed out that there is a profit incentive for cooperation between the upstream and downstream firms under conditions of bilateral monopoly. This same incentive exists in the successive monopoly model as well. In this latter model, however, it is assumed implicitly that the opportunities for negotiation and cooperation will be sufficiently restricted to preclude A PEDAGOGICALTREATMENT OF BILATERALMONOPOLY 835 joint profit maximization. Consider, for example, the case where goods travel by boat down a river past two cannon placements occupied by separate, non-cooperative,profit-maximizers.8 Each charges a tariff for the service of not firing on passing boats. There is no mechanism for enforcing contracts between the two cannon-placementowners, who may be enemies. In this situation, it is reasonableto assume that the joint profitmaximizing solution will not be obtained. The probable solution to this problem is that providedby the successive monopoly model.9 In the bilateral monopoly model, however, some negotiationbetween buyer and seller is required for exchange to take place. The incentive to pursue joint profit maximization arises because joint profits are not maximized at either of the two boundarysolutions presentedin the conventional analysis. Such cooperationmay take the extreme form of vertical integration[40]; [15] or it may come about throughthe bargainingprocess [24]. For the latter, it is importantto realize that the negotiation that takes place must involve quantityif joint profits are to be maxi- mized. In this market setting, however, it is theoreticallyunlikely that one firm would choose price and allow the other to select quantitywithout negotiation.Rather, both price and quantity will be determinedthrough bilateral bargaining.10 To analyze the outcome of this bargainingprocess, we adoptthe following model and nota- tion:

x = intermediateproduct that is tradedunder bilateral monopoly conditions; C(x) = total cost of producingx; y = some other input that is competitivelysupplied at a constantcost of py; Q = Q (x, y) = final outputquantity, a functionof x and y; px = price of the intermediategood, x; and P = P(Q) = final outputinverse demand. Now, if the two monopolists were to verticallyintegrate, the profitfunction of the integrated firm would be

rI = P[Q(x,y)]Q(x,y) - C(x) - py. (1)

Profit maximizationby the vertically integratedfirm would result in the productionand employ- ment of inputs x and y such that

(P + QdP/dQ)(aQ/lx) = dCldx (2a) and

(P + QdP/dQ)(aQ/ay) = py. (2b)

8. This example was providedby the anonymousreviewer. 9. The marketdemand for boat trips equals their marginalvaluation net of any transportcosts. By subtractingthe upstreamcannon placement owner's price of passage from this demand, one obtains the downstreamcannon placement owner's demand. To maximize his profits, the downstreamcannon placement owner equates his marginalrevenue to his marginal cost, and chooses his price of passage accordingly.It follows that the demand faced by the upstreamcannon placement owner is the curve marginal to market demand minus downstreammarginal cost. In turn, he equates the curve marginalto this to his marginalcost to determinehis optimal price of passage. This is the standardsolution to the successive monopoly problem, even though no outputpasses between the two successive monopolists, at least in the usual sense. 10. Machlup and Taber [27] make this point. They speculate that a failure to recognize this essential difference between bilateral monopoly and all other marketstructures accounts for the lack of unanimityamong the authorswriting on this subject. 836 Roger Blair, David Kaserman,and RichardRomano

That is, integratedprofits are maximized where the marginalrevenue products of the inputs are equated to their marginal costs. For the special productionfunction employed in Figure 1, this correspondsto x3 units of output (and input). It is at this outputonly that joint profits are at a maximum. Suppose, however, that the bilateral monopolists do not integratevertically. Instead, they continue to conduct arms-lengthnegotiations on Px and x. Then, Bowley's point, and the point we wish to elaborate furtherhere, is that such negotiationwill necessarilyresult in precisely the same joint profit maximizing quantityof the intermediategood being exchanged (and the same employmentof inputy). As a result, both the price of the final good and its outputare determinate in this model and are equal to the price and outputthat result with verticalintegration. To show this, we shall first parallel the conventionalanalysis by assuming that one party and then the other dominates the negotiationprocess. We do not, however, assume competitive behavior on the part of either firm. Following this examinationof the two extreme cases, we analyze the negotiationprocess where neitherparty dominates.

The DominationSolutions

In the absence of vertical integration,the upstreammonopolist's profit function is given by

rru = p,x -C(x), (3) and the downstreammonopsonist's profit functionis given by

TD = P[Q(x,y)]Q(x,y) - pXx - pyy. (4)

Initially, suppose that the upstreamfirm dominates in the negotiationprocess. By definition, this means that Truwill be maximized. Such maximization,however, is subjectto the constraint thatthe downstreamfirm participate in the exchange, which, in turn,requires that the downstream firm's profit be no less than zero. Further,if the upstreamfirm dominatesthe outcome, it is clear that the downstreamfirm will be allowed to earn no more than zero profit. Therefore, upstream firm dominationleads to the constrainedoptimization problem

Max 7T x,y s.t. 7rD = 0.

Setting equation (4) equal to zero and solving for px yields

Px = {P[Q (, Yx,)]Q ) - pyy}/x. (5)

Substitutingequation (5) into equation(3) yields the unconstrainedmaximand

Tru= P[Q(x,y)]Q(, y) - C(x) - pyy = rrI. (6)

Obviously, maximizationof equation(6) will result in equations(2a) and (2b), i.e., a maximiza- tion of joint profits. Fromequation (5), dominationby the upstreamfirm in the negotiationprocess will lead to an intermediateproduct price equal to the average net revenue of the intermediate product. This price, which is given by p3uin Figure 1, exhauststhe downstreamfirm's profits. A PEDAGOGICALTREATMENT OF BILATERALMONOPOLY 837

Alternatively,suppose that the downstreamfirm dominatesthe negotiationprocess. In that case, the downstreamfirm faces the analogousconstrained optimization problem

Max rD x,y s.t. ITr =0.

Setting equation (3) equal to zero and solving for px yields

Px = C(x)/x. (7)

Substitutingequation (7) into equation(4) yields the unconstrainedmaximand

- = rrX = P[Q(x,y)]Q(x, ) C(x) -pyy rrT. (8)

Again, maximizationwill lead to equations(2a) and (2b) and a maximizationof joint profits. In this case, however, the intermediateproduct price is set equal to the upstreamfirm's averagecost (p3 in Figure 1), which exhausts this firm's profits. Thus, regardless of which firm dominates the bargaining, we find that the same output is chosen, viz, the output that maximizes joint profits. Given this output for the intermediate good, both the price of the final good and its output are determinate.The only thing that differs between the two alternative domination solutions is the price at which the intermediategood is exchanged. This price determines which firm receives the maximizedjoint profits. The two extremes determinedby solving equations(5) and (7) at x3 (or pU andp3 in Figure 1) bound the negotiatedprice. A naturalquestion to ask at this point is: Why would the downstream(upstream) firm ever be willing to settle for less profit than it could earn as a monopsonist(monopolist)? In other words, in Figure 1, the downstreamfirm is worse off at (X3,p) than it would be at (x:,p2), and the upstreamfirm is worse off at (x3, p3 ) than it would be at (xl,pl). So why would either firm ever be willing to move to x3 if such a movement results in profitsbeing driven down to zero? The answer is that neither the monopoly nor monopsonysolution is generallyrelevant since each entails competitive behavior on one side of the market,which is not a characteristicof bilateral monopoly." Neither firm has these solutions as an option.

The Non-DominationSolutions

Now, suppose that neither party dominates the negotiationprocess. Bilateralnegotiation should lead to settlement at some point on the contractcurve. In the bilateralmonopoly case, the contract curve will be the locus of the points of tangencybetween the isoprofitcurves of the upstreamand the downstream firms in x - px space. Taking the total differentialof equation (3), setting dru = 0, and solving for dp_ldx, we obtain the slope of the upstreamfirm's isoprofitcurves

dp,ldx du =o=(dCldx - p,)/x. (9)

11. In our examination of the bargainingprocess below, we posit one case where these solutions play an indirect role. There, they may serve to bound the contractcurve. 838 Roger Blair, David Kaserman, and Richard Romano

Px MCx U

~D p_ ---/--*B - \ " /- ^MRPx

O X3 X

Figure 2.

Assuming increasing marginalcosts of producingx, equation(9) implies thatthe upstreamfirm's isoprofit curves will be U-shaped in x - px space, with higher curves denoting higher levels of profit. The minimum point on each isoprofitcurve intersectsthe marginalcost curve. At points above marginal cost, we can see in (9) thatpx > dCldx, which makes the numeratornegative and, therefore, makes the slope of the isoprofit curve negative. The reverse is true for points below the marginalcost curve. This is depicted in Figure2. Takingthe total differentialof equation(4), settingdTrrD = 0 and arrD/dy = 0,12 and solving for dpx/ldx, we obtain the slope of the downstreamfirm's isoprofitcurves

dpxldxldD=o, oaD/ay=o=[(P + QdPldQ)aQ/lx -p, ]/x. (10)

With a declining net marginal revenue product curve,'3equation (10) implies that the isoprofit curves of the downstream firm will have an inverted U-shape, with lower curves representing higher levels of profit. The maximum point on each isoprofitcurve intersects the net marginal revenue product curve. At points below the marginalrevenue product, (P + QdPldQ)oQ/lx > px, which makes the numeratorpositive and, therefore,makes the slope of the isoprofitcurve positive. The reverse is true for points above the net marginalrevenue product. This is shown in Figure 2. The contractcurve for this bargainingproblem, then, is found where

dp,/dxld, =o=dpxldxl D=o, a7D/a-=O -

Substituting from equations (9) and (10), we have

dCldx = (P + QdP/dQ)aQ/ax

12.orrDWe must set /y = 0 sincethe downstream firm will optimize over v giventhe negotiated solutions for x and px. 13. As in the less general case depicted in Figure 1, by "net marginal revenue product of x," we mean the additional revenue generated by an additionalunit of x given that y is adjustedoptimally. Mathematically,MRPx (P + QdPldQ)QP/lQis alwaysevaluated where (2b) is satisfied. A PEDAGOGICALTREATMENT OF BILATERALMONOPOLY 839 at all points along the contract curve. Thus, negotiationmust lead to the maximizationof joint profits if the firms are to be on the contractcurve. Since the points of tangency must be at the joint profit maximizing quantity of X3, the contract curve is vertical. The bounds of the contractcurve depend on the outcome should the negotiation process break down. Neither bilateralmonopolist need ever have negative profits in the event of a breakdown, so that the contractcurve will never extend below point B or above point A in Figure 2 (where p3 and pu are defined analogouslyto those in Figure 1). If one of the bilateral monopolists could make a credible commitmentto withdrawfrom the marketin the event of a breakdownand make a single take-it-or-leave-itoffer on the contractcurve, then the contract curve becomes infinitely short. For example, if the supplierwere in such a position, it could offer epsilon in profits to the downstreamfirm, and the latterwould accept the offer since the alternative is zero profits. This conforms to the above case of supplierdomination and the contractcurve approachespoint A. Another possibility is where one of the bilateral monopolists can commit to a price of x (but not a quantity) in the event of a breakdown.Consider the case where the input supplier could commit to a price should negotiationfail. Then, the outcome following a breakdownwould be the same as the monopoly solution since the suppliercan choose any point on the demand of the downstream firm, which becomes, in effect a price taker. The lower bound on the con- tract curve is the price which, given x3 is produced, yields the supplier the same amount of profits as in the monopoly solution. In Figure 2, this price is denoted by p which satisfies pX3 - C(x3) = 7T*, where Tru is the profit of the upstreamfirm under monopoly.'4The upper bound on the contractcurve is the price which, given x3 is purchased,yields the downstreamfirm the same profits it obtains in the monopoly solution. This price is denoted by p which satisfies P[Q((X3,y)]Q(X3,y) - PX3 - P\y = 7TD (and also (2b)), where 7TD* is the profit of the down- stream firm under monopoly. The analogousbounds can be found associated with the monopoly solution should instead the downstreamfirm be able to commit itself to a purchaseprice in the event of a breakdown. In either case, there is still a range of prices over which both agents are better off than if negotiationfails. With other assumptionsabout committedbehavior in the event of a breakdownother bounds on the contract curve can be found. This approachto boundingthe contractcurve, however, is subject to the following criticisms. First, it is difficultto imagine how either party can make a credible commitment in the event of a breakdownsince Paretoimprovements will be possible in the "breakdownequilibrium." With only two agents and the possibility of negotiation, a Pareto efficient outcome seems compelling. Second, boundingthe contractcurve still leaves open the question of the final negotiated price. Recent game theoreticresearch (see, for example. Rubin- stein [35]) has focused on the effects of the bargainingprocess on the profit split. Our simple model is not rich enough to deal with this issue. Such models of the bargainingprocess embrace, at least implicitly, a solution on the contractcurve since they focus only on the split of profits. Regardless of how it is determined, it is importantto note that the price of the intermediate good does not serve as a rationing device. Instead, it is merely a means of dividing the joint profits. Since this profit split has no allocative impact, it is only of distributivesignificance that the relative shares may be indeterminate.The output of the intermediateproduct, however, is determinateas is the price of the final productand its quantity.

14. It can be shown that p is above the price where MRP. = MC, but below p' . 840 Roger Blair, David Kaserman, and Richard Romano

IV. Conclusion

The results we have presented here are not new. They have been known by a portion of the profession at least since Bowley's 1928 paper. Our exposition of these results, however, is new and more complete. We hope that our clarificationof the bilateralmonopoly analysis will improve instructionand that a consensus will emerge.

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