The Robinson-Friday barter exchange economy we have been dealing with in previous sections was a situation of bilateral monopoly as Robinson owned all of good x and Friday owned all of good y. The conclusion from this exercise was that the "core" would represent the set of possible final settlements. However, this did not exhaust Edgeworth's treatment of monopolistic situations. We should make a few remarks regarding Edgeworth's so-called "exceptional case", the equilibrium achieved in the case of monopoly pricing. This was considered in detail in Edgeworth (1897).

Edgeworth examined the case when two monopolists bargained on price but made no agreement on the quantities traded. In this case, as he demonstrated, the equilibrium will be off the contract curve. However, he also noted that if the monopolists bargained on quantity as well as price, they would return to the contract curve and the core theory we had before applies.

We can see this result intuitively in Figure 1 (we have omitted the autarky indifference curves to avoid cluttering, but the entire core is still represented by the thick black line segment on the contract curve). Let us consider the simplest case, that of a single monopolist (let him be Friday) and various passive consumers (of which we take Robinson to be a representative). Friday is not a perfectly-discriminating monopolist, i.e. one with control over price and quantity, but rather a monopolist with control only over price. Friday would thus like to set a price that maximizes his utility -- subject to the constraint that Robinson "accepts" it, i.e. it must lie on Robinson's offer curve, OC_R. We see this in Figure 1 to be at point M. This is the tangency of Robinson's offer curve and the highest indifference curve of Friday, U^F¢ . The corresponding monopoly price, then, would be the vector p. At these prices, Robinson will choose to demand M and achieve utility U^R¢ -- while the monopolist Friday, of course, will undertake the corresponding trade and achieve his highest utility U^F¢ .

Fig. 1 - Monopoly Pricing

The monopoly pricing position M in Figure 1 resembles a Stackelberg leadership equilibrium (although, in focusing on prices, Edgeworth is decidedly closer to Bertrand). Yet, as we know, Stackelberg leadership equilibria are not efficient -- and this is true in Figure 1: the solution M is off the contract curve; it is not Pareto-optimal.

As such, Edgeworth considered a further twist to this situation. Notice that at M, there is still scope for further trade. This is represented by the "lens" formed by the indifference curves U^F¢ and U^R¢ . Both Robinson and Friday would be better off if, starting from M, they then agreed to move to a point such as D in the "mini-core" formed by the indifference curves U^F¢ and U^R¢ in Figure 1. Where in this "mini-core" they go is, of course, indeterminate. Thus, although monopoly pricing results in a determinate but suboptimal point M, then if further trade is allowed, the final allocation will be an optimal, but indeterminate, point in the mini-core.

What kind of system would achieve a point such as D rather than M? Edgeworth argued that if the monopolist and the consumer bargained not only over price, but also over quantity, then we would move to a point such as D. Or, rather, if they bargained "sequentially" -- first over price, and then over further trade which includes quantities. Notice that D is above the price line, so there is a sense that, in proposing a further move from M to D, the monopolist Friday is actually agreeing to a price for this extra amount that is "worse" for himself than the monopoly price, p -- but in compensation, he would induce Robinson to move off his offer curve (and thus provide Friday with a trade that more than compensates for the decline in monopoly price). Robinson, of course, would agree to such a package deal. So we can think of the further trade beyond M as being a "bulk-buying" discount that Friday will allow Robinson and Robinson will take.

Arthur L. Bowley (1924: Ch. 1) took Edgeworth's (1897) study in a different direction, defining what he called a "bargaining locus" between agents in a price-only contract economy. This is shown in Figure 2. If there is a two-agent economy, where the contracts between Robinson and Friday can only be specified in terms of price, then the locus of resulting allocations will be represented by the thick line in Figure 2 going from M_F to C and then to M_R.

There are three extremes we can see in Figure 2: (1) when Friday is a monopolistic price-maker and Robinson a price-taker (and so the equilibrium is M_F, where Friday's highest indifference curve U^F¢ is tangent to Robinson's offer curve, OC_R); (2) when Robinson is a monopolistic price-maker and Friday is a price-taker (and so the equilibrium is M_R, where Robinson's highest indifference curve U^R¢ is tangent to Friday's offer curve, OC_F) and (3) where both are price-takers, in which case we end up at the "Walrasian" equilibrium allocation C.

What about the case (4), when both are monopolistic price-makers? Bowley's (1924, 1928) argument is that in such a case, then the monopolists would end up somewhere on the "bargaining locus", M_FCM_R, precisely where depending on their relative bargaining strengths. For instance, M_F is just an extreme case of Friday having all the bargaining power and Robinson none at all. A slightly less powerful Friday would not be at M_F, but somewhere on the offer curve OC_R, closer to C. If the bargaining strengths of Friday and Robinson were equal, or rather off-setting, then they would end up at allocation C. The main result of Bowley's analysis is that if bargaining strength is not equal so that we are at C, then we necessarily end up somewhere off the contract curve, i.e. a suboptimal position, where there is still room for improvement which price-only contracts cannot exploit.

There is an interesting implication in this story: in particular, that "price-and-quantity" deals are, in general, superior to mere "price" deals. Thus, one of Edgeworth's objections to the price-mediated exchange we find in Walrasian or Marshallian theory is precisely this: if Friday and Robinson were rational, they would not consider prices alone (which would yield something on the bargaining locus), but would prefer to deal in contracts which specified both quantity and price (and thus end up somewhere on the contract curve). Fully-specified price-and-quantity contracts, Edgeworth argued, are superior to price contracts. Even in the case of the unilateral monopoly we had in Figure 1, both Robinson and Friday would be better off with price-and-quantity contracts. Price-only contracts, Edgeworth suggests, are simply not rational, regardless of whether you are a price-maker or a price-taker.

Quite later on, Wassily Leontief (1946) introduced monopolists which could specify price-and-quantity contracts. This would allow him to be a "perfectly-discriminating" monopolist: he could, via a fully specified contract, drive the other trader back to his autarky indifference curve, and take all the gains-from-trade for himself. Note that, necessarily, the final outcome of such a case would be on the contract curve, and thus Pareto-efficient. The "bargaining locus" that arises from two perfectly-discriminating monopolists would be the Edgeworthian core itself, exactly where they would end up depending upon their relative bargaining strengths.