1 – Introduction
1On the other hand, when explaining how competitive equilibrium is reached, one is used to refer to an auctioneer who would propose and revise the price that buyers and sellers then have to accept. This again is somewhat inadequate, since auctioneers actually exist only on special markets, such as those concerning stock exchange or commodity future exchange. There is one aspect of economic theory which has been, and remains, particularly problematic: the study of pricemaking behaviour and the determination of market prices. Given the centrality of prices in economic theory, this may seem somewhat surprising. However, economic theory has tended to circumvent the direct problem of price determination by assuming the existence of an agent, a deus ex machina, whose sole job is to aggregate together agents’ decisions, taken in various contexts, and to establish the required market prices. This agent is usually referred to as the auctioneer. So, for example, in the canonical model of general equilibrium exchange it is the auctioneer who prevents trade from taking place at disequilibrium prices where some agent must find their economic choices are rationed. Alternatively, in the model of strategic market behaviour based upon quantitysetting decisions, the Cournot model, it is the auctioneer who sends back the required marketclearing price after traders have decided what quantity to send to the market. Although the auctioneer is a convenient fiction which has permitted much elegant modeling of economic decisions, the invocation of the auctioneer is a serious problem for economic theory and deserves deeper understanding.
2Despite these problems regarding how market prices are determined, there is a line of thought, dating back almost to the establishment of the auctioneer, which takes a quite different approach. In Bertrand [1883] it was considered what would happen if sellers were to directly compete by setting prices. In this case there is no need for an auctioneer as trading prices are chosen by sellers. At this early stage, it was erroneously concluded that there could be no market equilibrium if sellers were to set prices. Instead, market prices would display an inexorable spiral downwards. Since this important insight, there has been substantial work which has formalized Bertrand’s insight using game theory, and which has shown the assumption of the nonexistence of a price equilibrium to be incorrect under certain circumstances. [2] Moreover, even if markets are far from competitive pricemaking behaviour may result in the perfectly competitive market outcome. This is the striking result which is usually referred to as the ‘Bertrand paradox.’ When there are at least two sellers in the market, which have identical and constant marginal costs, the market demand possesses a finite chokeoff price, then the unique symmetric equilibrium is for all sellers to quote the marginal cost to buyers: the perfectly competitive market outcome prevails. [3]
3In the last two decades there has been renewed interest amongst economic theorists to examine under which conditions we should expect pricemaking behaviour by strategic sellers to result in the perfectly competitive market outcome. This literature has tended to take two different approaches to the types of market contacts which sellers offer to buyers. In Bertrand competition it is assumed that sellers post prices with a commitment to supply all the demand forthcoming from buyers. In Edgeworth competition it is assumed that sellers post prices in the market with no commitment to supply any particular quantity so that sellers would never meet more demand than their competitive supplies. [4] In the next section of the paper we review, in some detail, the findings of this literature and find that it does not provide a deep foundation for the perfectly competitive market outcome to emerge from sellers acting as price makers. Moreover, the assumptions regarding the types of contracts which sellers can form with buyers represent polar cases.
4Although models of direct pricemaking behaviour by economic agents dispenses with the fiction of the auctioneer we should note that there is an alternative approach to contract formation which gives a foundation for perfect competition without the auctioneer. This can be found in Edgeworth [1881]. The early ideas of Edgeworth regarding contract formation, and the later formalization of the core, provides a foundation for pricetaking behaviour at least in large markets. Debreu and Scarf [1963] showed that in a pure exchange economy with standard preferences the only allocations which remain in the core as the economy is replicated are Walrasian allocations. [5] Therefore, if traders can form coalitions and trade amongst themselves, the only outcomes we should expect in large economies are allocations close to Walrasian equilibria. [6]
5Despite the elegance and deep insight of the DebreuScarf result it does have one drawback: when coalitions form and trade they do so without resorting to the price mechanism. It does not, therefore, provide an explanation of how trade takes place via the price mechanism. Is it not possible to provide a theory of pricemaking behaviour in which sellers directly set prices but traders are able to form coalitions and trade amongst themselves? That is, can we provide a theory of price determination which combines the richness of contract formation embodied in the notion of the core with agents acting as price makers. This is the aim of this paper. An informal outline of contract formation based upon agents acting as pricemakers but being able to form coalitions will be set out and a new core concept, the ‘Bertrand core,’ will be introduced. It should be noted that the relationship between Edgeworth and Bertrand’s ideas has been commented upon before. MasColell et al. [1995: 655] draws attention to the close relationship between Bertrand price competition and the form of competition which takes place in the Edgeworth core. We find that there are some deep similarities between the Edgeworth core and the Bertrand core. Whereas Walrasian allocations are contained in the Edgeworth core we find that pricetaking equilibria are contained in the Bertrand core. Moreover, we find that there is a pricemaking analogue of the DebreuScarf result: as the set of market traders becomes large the only prices which remain in the Bertrand core are competitive equilibria. Therefore a new pricemaking foundation for competitive equilibrium emerges from the Bertrand core.
6In the next section of the paper we summarize the most recent results regarding contract formation in pricesetting games. Then, in section three, we consider the new approach to contract formation which is at the centre of the Bertrand core. In the final section we draw some conclusions regarding future research in this area.
2 – Pricemaking Behaviour and Market Contracts
7The benchmark result in the literature on pricesetting games is the Bertrand paradox. As noted above, if there are at least two sellers in the market which have identical and constant marginal costs, the market demand is continuous and possesses a finite chokeoff price then the unique symmetric pure strategy Nash equilibrium of the pricesetting game is for all sellers to quote the marginal cost. In any other pure strategy equilibrium all trade takes place at marginal cost. If one considers the mixed extension of the game then it can be shown that all equilibria are in pure strategies [Harrington, 1989]. However, if the market demand does not possess a finite chokeoff price there may be mixed equilibria with sellers earning positive profits [Baye and Morgan, 1996]. The assumption that sellers have constant marginal costs obscures somewhat the types of market contracts which sellers offer in the marketplace because at any price above marginal cost sellers will be willing to supply all the demand forthcoming to them. This assumption, that sellers explicitly offer contracts which guarantee to supply all the demand forthcoming to them at the price they quote, has subsequently become the defining feature of Bertrand competition. [7]
8In an in influential paper, Dastidar [1995] considered a classical Bertrand price game, where sellers commit to serving all the market demand forthcoming to them, with the sole difference that sellers had strictly convex cost functions. In this game there is a continuum of pure strategy Bertrand equilibria. If sellers are asymmetric then assuming that the market demand is split between sellers in proportion to their competitive supplies in the event of price ties it can be shown that the competitive equilibrium always belongs to the Bertrand equilibrium set. [8] Dastidar [2001] showed that if sellers’ cost functions are sufficiently convex then the collusive price may belong to the Bertrand equilibrium set. Therefore the collusive outcome may be achievable in a oneshot game. Hoernig [2002] considered the mixed extension of this pricing game and found that there exist different types of mixed strategies. Sellers may play mixed strategies placing probability mass upon any finite number of pure equilibrium prices which give positive profits or play mixed strategies with continuous supports which place an atom at the upper bound. Moreover, Dastidar [1997] showed that even in a market with convex costs Bertrand competition is more competitive than Cournot competition in the sense that there always exists a Bertrand equilibrium price which is lower than the Cournot price. [9]
9Chowdhury and Sengupta [2004] considered when the refinement of coalitionproofness reduces the equilibrium set in Bertrand games. They found that if sellers have symmetric costs then the game admits a unique coalitionproof Bertrand equilibrium. It was established that if the price space is discrete and sufficiently fine then this unique coalitionproof equilibrium can be implemented in a game in which sellers sequentially announce prices. They also showed that if one considers sequences of economies then as the number of sellers in the market becomes large the set of coalitionproof equilibria coincides with the competitive equilibrium of the market provided all sellers are active in the limit. This is especially interesting because in a related paper Novshek and Chowdhury [2003] showed that if sellers have Ushaped average costs then as sellers become small compared to the market demand the set of Bertrand equilibria may be empty or constitutes an interval including the competitive equilibrium price. Therefore it appears that admitting coalitional deviations is what is decisive in refining the equilibrium set.
10When sellers have increasing returns to scale costs most of the literature indicates that equilibrium existence is problematic. Shapiro [1989] was the first to note that if one adds an avoidable fixed cost to the standard Bertrand game with constant returns to scale variable costs then there exists no pure strategy price equilibrium. [10] This result has been strengthened by Dastidar [2006] which showed that if sellers have symmetric concave costs, and price ties are broken by the equal sharing rule, then a pure strategy equilibrium fails to exist. It is wellknown, though, that if the sharing rule is changed so that one firm is selected randomly from the set of sellers tieing at the minimum price to serve all the market demand, this sharing rule is usually referred to as “winnertakesall,” then this reestablishes the existence of pure strategy equilibrium. [11] Up until recently the existence of a mixed strategy equilibrium when sellers have concave costs has been an open question. However, Baye and Kovenock [2008] provided a simple and elegant proof, by considering the upper and lower bounds of any mixed strategy, that in the case of an avoidable fixed cost no mixed strategy equilibrium exists. Moreover, their proof can be applied to the general case where sellers have symmetric concave costs. Despite these negative existence results, it seems clear that the nonexistence of equilibrium is intimately related to assumption that sellers have symmetric costs and the price space is the real line. [12] Chowdhury [2002] considered a market with increasing returns to scale costs where the price space was discrete a showed that there exist multiple tremblinghand perfect Bertrand equilibria. As the price space becomes increasingly fine both these equilibria converge to the limit price outcome.
11The other common cost function, that of initial increasing returns to scale and later decreasing returns to scale, has only recently received attention in Bertrand games. Saporiti and Coloma [2010] considered a market with symmetric sellers which have a possibly avoidable fixed cost and a strictly convex variable cost function which gives rise to a Ushaped average cost function. They established that if the set of competitive equilibria is nonempty then the set of Bertrand equilibria is nonempty. However, the market may possess a Bertrand equilibrium but fail to possess a competitive, or pricetaking, equilibrium. [13] Furthermore, they uncovered a previously unexplored relationship between the existence of Bertrand equilibrium and the nonsubadditivity of the cost function by showing that the nonsubadditivity of the cost function at the oligopoly breakeven price is a necessary and sufficient condition for the existence of a Bertrand equilibrium. [14] Yano [2006a] considered a price game with free entry by sellers with ushaped average costs where in addition to posting prices sellers also specified which quantities they are willing to sell at that price. Therefore this game had some similarities with Edgeworth games as sellers could ration the market demand. It was shown that the longrun competitive equilibrium could be sustained under certain conditions. Remarkably, even though sellers are symmetric, this game also admits an equilibrium where the law of one price may fail. That is, there may exist a pure strategy equilibrium in which a homogeneous good is traded at different prices. In a related paper Yano [2006b] showed that the standard Bertrand equilibrium can be sustained as an equilibrium of the game and the Edgeworth criticism is valid if one abandons the assumption of free entry.
12The recent research on price games has focussed upon two specific areas: sharing rules and uncertainty. Hoernig [2007] considered arbitrary profit functions, not necessarily derived from demand and costs, and looked for the properties of sharing rules which guarantee the existence of equilibrium. A sharing rule is weakly tiedecreasing if payoffs at ties are weakly below nontied payoffs and a sharing rule is coalition monotone if the sum of players payoffs do not decrease as more sellers are added to a price tie. It was established that if profit functions satisfy a continuity condition, are weakly tiedecreasing, coalition monotone and the sum of payoffs is upper semicontinuous then there exists a mixed strategy Bertrand equilibrium. Sufficient conditions for the existence of pure strategy equilibrium were also provided. Bagh [2010] also considered Bertrand games with deterministic sharing rules which decrease in price and showed that if sellers have symmetric convex costs a pure strategy equilibrium exists and this holds even if one permits certain types of discontinuous demands.
13There are only a small number of papers which have studied the impact of uncertainty and incomplete information in Bertrand games. Spulber [1995] considered the case where there is a continuum of cost types and sellers had weakly convex costs. It was shown that the game possesses a pure strategy equilibrium which converges to marginal cost pricing as the market became large. Janssen and Rasmusen [2002] analyzed a price game where there was a fixed set of sellers, with constant marginal cost and an exogenous probability that a firm is inactive. They established that the game has a mixed strategy equilibrium and showed that under certain specifications the expected Bertrand price may be higher than the Cournot price a reversal of the usual outcome. Routledge [2010b] considered the classical Bertrand game, with a fixed set of sellers with constant returns to scale costs, where the cost type of each firm was drawn from a binary distribution, and showed that there exists a mixed strategy Bayesian equilibrium.
14As noted at the beginning, there is an alternative approach to contract formation which assumes that sellers post prices in the market with no commitment to supply, or an inability to supply, any particular quantity— so no seller would ever supply more than their competitive supply. This is the assumption used in the literature on Edgeworth games. [15] In this case, Bertrand’s original speculation about the nonexistence of a price equilibrium often turns out to be correct. If a pure strategy price equilibrium exists then this equilibrium is the competitive equilibrium [Baye and Kovenock, 2008]. However, if the competitive equilibrium is not a pure strategy price equilibrium it is a nontrivial exercise to show that the game possesses a mixed strategy price equilibrium. This is because the game is one with discontinuous payoffs. Nevertheless, Dixon [1984] showed that in a market with sellers which have identical convex costs, the existence results in Dasgupta and Maskin [1986] can be applied, to prove that a mixed strategy equilibrium exists under both the efficient and proportional rationing rules.
15There are a number of problems with the mixed strategy equilibria in Edgeworth games. First, it is not easy to explain why sellers should randomize across prices. Second, the equilibria are such that sellers never place probability mass upon the competitive equilibrium prices. Therefore the results do not provide any pricemaking foundation for competitive equilibrium.
16The analysis of Edgeworth equilibria in mixed strategies has been extended to consider what happens as the number of sellers in the market becomes large. In this case a pricemaking foundation for competitive equilibrium can be provided but this foundation depends upon the rationing rule. Two rationing rules have featured prominently in the literature: efficient and proportional rationing. Under efficient rationing the buyers with the highest valuations buy from the lowest price seller whereas under proportional ration the lower priced sellers receives an equal fraction of the demand from buyers regardless of their valuations. If the rationing rule is efficient rationing then as the number of sellers in the market increases the distribution, and the support, of the mixed strategy equilibrium converges to an atom at the competitive equilibrium price. However, if the market demand is rationed according to the proportional rationing rule then as the market becomes large the distribution, but not the support, of the mixed strategy equilibrium converges to an atom at the competitive equilibrium price [Allen and Hellwig, 1986a,b]. The difference between these two convergence results means that under the proportional rule even if there is a large, but finite, number of sellers in the market there is always nonzero probability of prices close to the monopoly price of the market being played. Vives [1986] analyzed an Edgeworth game where sellers had capacity constraints and considered what happens as the size of each seller reduces with respect to the market demand. A closed form expression for the mixed strategy equilibrium under the efficient rationing rule was provided and it was shown that this equilibrium strategy converged, both support and distribution, to an atom at the competitive price as each seller’s capacity tends to zero.
17In summary, the message which has come out of this research regarding pricesetting behaviour by sellers is that the competitive equilibrium of the market may be sustained as an equilibrium but this foundation for competitive behaviour is weak and subject to the form of the market contract. In Bertrand competition, where sellers commit to supplying any quantity forthcoming from buyers, a pure strategy equilibrium exists under quite general conditions and the competitive equilibrium belongs to the set. However, this provides only a weak foundation for competitive equilibrium, as other prices, including possibly the collusive outcome, may also be equilibrium outcomes. On the other hand, when sellers do not provide a commitment to supply any quantity forthcoming, the existence of a pure strategy equilibrium is problematic, and, even as markets become large, prices higher than the competitive price may still be quoted by sellers.
18This inability of the standard models to provide a convincing pricemaking foundation for competitive equilibrium leads us to question whether it is possible to provide a richer approach to contract formation which can provide a more credible foundation for perfect competition which does not rely upon sellers posting random prices in the market or arbitrary rationing rules. In the next section of the paper we present the outline of such an approach. This alternative theory of price formation combines Bertrand’s original insight regarding price competition with Edgeworth’s insight regarding recontracting.
3 – Rethinking Bertrand and Edgeworth: Pricemaking Behaviour and Coalition Markets
19In this section we present the outline of a new approach to contract formation amongst pricemaking sellers which admits the possibility that a coalition of traders may form a market and trade amongst themselves. This section draws upon the model presented in Routledge [2010a] and readers wanting to find greater mathematical substance behind the ideas expressed in this section are referred to that paper. Following the tradition of much of the literature described in the previous section we consider the market for a perfectly homogeneous good. In the market there is a finite set of buyers B = {1,…,b}, b ≥ 2, and a finite set of sellers S = {1,…, s}, s ≥ 2. It will be assumed that each seller has a cost function which is strictly convex and each buyer in the market has a wellbehaved demand function.^{�} We shall want to consider a Bertrand price competition game between possible subsets of buyers and sellers so let χ^{B} denote all the nonempty subsets of buyers and χ^{S} denote all the nonempty subsets of sellers. We shall let G = (B, S) denote the grand coalition of all buyers and sellers. For any set of traders T′ = (B′, S′)×χ^{B} ×χ^{S} consider a classical Bertrand price game between these buyers and sellers. In the game, each seller simultaneously and independently quotes a price to the buyers with a commitment to supply all demand forthcoming from the buyers. A pure strategy Bertrand equilibrium is a vector of prices such that no seller can increase their profit by unilaterally changing their price. We shall let E(T′) denote the pure strategy Bertrand equilibria of the market formed by the T′ traders.
20Now we shall want to consider the possibility that a coalition of traders, buyers and sellers, can form a coalition and trade amongst themselves. In any coalition market sellers act noncooperatively in setting prices. Therefore, in any coalition market trade must take place at a pure strategy Bertrand equilibrium price for the coalition market. Let π_{i}(p, T′) denote the profit of seller i in the market with T′ traders when i the vector of prices quoted to buyers is p. We shall say that a coalition of traders A ⊂ T′ has an improvement upon price vector p if there exists a p′ ∈E(A) such that min p′ < min p and π_{i}(p′, A) > π_{i}(p, T′). That is to say, a subset of traders has an improvement upon a price vector if they can form a coalition and trade at a new equilibrium price vector which results in buyers obtaining the good at a lower price, min p′ < min p, and sellers in the coalition market earn higher profits, π_{i}(p′, A) > π_{i}(p, T′). Given this notion of an improvement upon a price vector we can now define the Bertrand core. A price vector p is in the Bertrand core if p∈E(G) and no coalition of traders, A ⊂ G, has an improvement upon p. Let C(G) denote the price vectors in the Bertrand core.
21There are a number of points to note about the Bertrand core. First, the wellknown relationship between the Edgeworth core and Walrasian equilibria is in a sense reversed: the prices in the Bertrand core are a subset of the Bertrand equilibrium prices. Therefore the Bertrand core is a re nement of the Bertrand equilibrium set. Second, the types of contracts which sellers can form with buyers is richer than the standard contracts considered in the existing literature. In this model sellers may form contracts with a strict subset of the buyers and may still supply more than their competitive supply but less than the whole market demand. Finally, the Bertrand core is not a wholly cooperative solution concept. It admits the possibility that a coalition of traders may recognize that it is in their common interest to form a market and trade amongst themselves. However, once the market is formed the sellers act noncooperatively in quoting prices to the buyers. Later in the paper we shall discuss a related concept in which sellers exhibit greater cooperation and a related solution concept.
22A question which naturally occurs is whether the Bertrand core, C(G), is a nonempty set? Although we shall not dwell on the details of the result it is straightforward to show that the Bertrand core is nonempty by establishing that, under the conditions described here, the market possesses a competitive equilibrium and all sellers quoting the competitive equilibrium price is a vector which belongs to the Bertrand core. We now illustrate some of the ideas in the following simple example.
23The previous example illustrates that in a small market there may be prices in the Bertrand core which are significantly different from the competitive equilibrium price. We now consider whether this remains the case when the set of traders in the market increase. The result we shall discover is that as the set of traders becomes large the only contracts which remain in the Bertrand core are those which are ‘close’ to the competitive equilibrium price. Therefore the Bertrand core is able to provide a pricemaking foundation for competitive equilibrium without resorting to problematic assumptions such as sellers playing mixed strategies or arbitrary rationing rules in the market. To understand which contracts remain in the Bertrand core as the set of traders becomes large we consider the standard replication process. The r = 1, 2,… replication of a given grand coalition, G, is the market in which there is r number of each type of buyer and seller in the market. Following the notation used earlier, we shall let C_{r}(G) denote the price vectors in the rreplicated Bertrand core and P_{r}(G) will denote the set of competitive equilibrium prices of the rreplicated market. We now present the limit result regarding pricemaking contracts and the Bertrand core.
24Theorem (limit result on the Bertrand core). For any ε > 0 there exists an ∈ N such that min p − p^{C} < ε for all p∈C_{r}(G), p^{C}∈P_{r}(G) whenever
25The result contained in the theorem tells us that however close we require market prices to be to the competitive equilibrium of the market there exists some finite replication of the market such that all trading prices in the Bertrand core take place sufficiently close to the competitive equilibrium provided the market is replicated sufficiently many times. Remarkably, the result in Theorem 1 remains valid even when the equilibrium set of the market remains unchanged under the replication procedure [see Example 2 in Routledge, 2010a]. The result is also similar to the DebreuScarf result with the advantage that all trade takes place at prices determined directly by the traders.
26It was noted above that when a coalition forms and trades amongst themselves it is assumed that sellers act noncooperatively in offering price contracts to the buyers. This is the limited cooperation which is at the heart of the Bertrand core. However, one might consider the possibility that if traders are willing to cooperate in forming a coalition then they might also cooperate in setting prices within the coalition. In Chowdhury and Sengupta [2004] limited cooperation between sellers was considered by studying the set of coalitionproof Bertrand equilibria. Loosely speaking, an equilibrium price vector is a coalitionproof Bertrand equilibrium if no subset of sellers can enact a joint change in their prices and improve their profits. Then, after the set has changed their prices, no further subset of the deviators can undermine this change by also enacting a joint change in their prices. The notion of coalitionproofness was introduced by Bernheim et al. [1987] and captures the idea that players in a game may be able to communicate to improve their outcomes but cannot form binding agreements, so any outcome must be at least a Nash equilibrium of the game under consideration. In Routledge [2010a] the possibility of coalition markets forming and greater cooperation by sellers was considered. To study this possibility the concept of the coalitionproof Bertrand core was introduced. A price vector is said to be in the coalitionproof Bertrand core if it is a coalitionproof Bertrand equilibrium for the grand coalition and no subset of traders can form a market and trade at a coalitionproof equilibrium vector which is an improvement for both buyers and sellers. An example was presented which demonstrated that the coalitionproof Bertrand core may be empty.
27Generally, it is not easy to compare the coalitionproof Bertrand equilibria with the Bertrand core. This is because the Bertrand core is in some respects a stronger concept than coalitionproofness and in other respects it is a weaker concept. It is stronger in the sense that it allows both buyers and sellers to form coalitions and trade amongst themselves whereas in coalitionproof Bertrand equilibria it is only the sellers in the market with may cooperate to change their prices. However, the Bertrand core is weaker in the sense that it allows trade to take place at price vectors which may not be coalitionproof. Therefore if we consider cooperation amongst sellers in setting prices to be likely, and coalitions of buyers and sellers can form, then the appropriate solution concept is the coalitionproof Bertrand core.
28In this section the aim has been to provide some rather informal insights into a new approach to the formation of pricemaking contracts. The notion of the Bertrand core is an original combination of the early insights of Bertrand and Edgeworth regarding contract formation and economic exchange. The most powerful insight which this new core concept provides is to show that in markets with a large number of traders pricemaking behaviour results in contracts which are close to the competitive equilibrium and this result remains valid even when the equilibrium set of the market may remain unchanged. However, when traders can form coalitions, and sellers are willing to cooperate in setting prices, but cannot form binding contracts, the coalitionproof Bertrand core may be empty. In this final case it is problematic for us to give predictions regarding which contracts are likely to emerge in the market.
Conclusion
29The aim of this paper has been to outline a novel approach to priceformation without resorting to the fiction of the auctioneer or extreme assumptions regarding contract types. The trading game shows that even when sellers act as pricemakers we can provide a foundation for perfect competition in large markets. Moreover, the model is tractable enough to be extended in a number of different directions to provide additional insights regarding pricemaking contracts.
30First, it may be possible to analyze the Bertrand core with asymmetric information. As the Bertrand core shares a number of similarities with the Edgeworth core analyzed in general equilibrium exchange it would be interesting to see whether it is possible to analyze the Bertrand core with differential information [Glycopantis and Yannelis, 2005]. For example, if sellers have asymmetric information about each other’s costs and the set of buyers in the market is the Bertrand core nonempty? A Bertrand game with incomplete information was studied in Routledge [2010b]. Moreover, what happens as the set of traders in the market becomes large? As the Bertrand core is a new concept these types of questions have previously not been possible to analyze in the context of pricemaking contracts.
31Second, what happens when both buyers and sellers exhibit strategic behaviour. In the trading game analyzed here the novelty is that the demand side of the market is explicitly modelled and this permits a rich set of trading possibilities. However, if the number of buyers in the market is small it is possible that buyers as well as sellers may behave strategically. The Bertrand core assumes that strategic behaviour is only exhibited by the sellers; buyers simply accept the best contract offered to them. Therefore a more realistic model might admit the possibility of bilateral strategic behaviour and would involve defining a new core concept.
32Finally, we have assumed throughout that sellers cannot write binding contracts so that any solution must be at least a Nash equilibrium of the game under consideration. It would be interesting to analyze completely cooperative solution concepts such as the core and the bargaining set in the game and compare them with the Bertrand core and the coalitionproof Bertrand core. In Kaneko [1977] this was done for a trading game with constant average costs but could be extended to cover the trading game presented here.
Notes

[1]
School of Social Sciences, University of Manchester. Email: R.R.Routledge@liverpool.ac.uk I should like to thank all the attendees of the Workshop on Perfect Competition, University of Nanterre, January 2011, for their helpful comments.

[2]
For an authoritative survey of the literature on Bertrand and Edgeworth games the reader is referred to Vives [1999, Ch.5].

[3]
The assumption that the market demand possesses a finite chokeoff price is not a trivial technicality. If no finite chokeoff price exists, and monopoly profit is unbounded above, then there may be symmetric mixed strategy equilibria in which sellers price above marginal cost and earn high profits [Baye and Morgan, 1996].

[4]
Papers using the Bertrand assumption include Dastidar [1995], Novshek and Chowdhury [2003] and Chowdhury and Sengupta [2004]; papers using the Edgeworth assumption include Allen and Hellwig [1986a,b] and Dixon [1984].

[5]
This result has been widely generalized. If the set of traders increases arbitrarily, provided all traders do not vanish as a fraction of the limit economy, and the limit economy is regular, then the core equivalence result holds [Hildenbrand and Kirman, 1988:190–9].

[6]
Of course, given that economies possess, generically, a finite set of equilibria, the theory does not provide us with an explanation of which Walrasian allocations we should expect to emerge in large economies.

[7]
An assumption that is satisfied in markets such as those for gas and electricity.

[8]
A simple proof of the existence of a Bertrand equilibrium in a market with convex costs, and capacity sharing at ties, is to show that all sellers quoting the competitive equilibrium price is a Bertrand equilibrium.

[9]
Klaus and Brandts [2008] reported results of experimental Bertrand games with convex costs and found that players tended to price well above the competitive equilibrium price and in a significant number of cases players priced above the Cournot price.

[10]
Sharkey and Sibley [1993] show that if the fixed cost is sunk, and sellers simultaneously choose entry and price, then there exists a mixed strategy equilibrium where sellers mix between entering and over a range of prices with the upper bound being the monopoly price. Moreover, as the market becomes large the equilibrium displays convergence away from the competitive outcome.

[11]
Vives [1999: 119] shows the under winnertakesall competition the zero profit monopoly price is the unique pure strategy equilibrium. Baye and Morgan [2002] found that a sufficient condition for the existence of a zero profit pure strategy equilibrium in a winnertakesall price game is the existence of a monopoly breakeven price.

[12]
If sellers have asymmetric costs with different monopoly breakeven prices then there always exist mixed strategy equilibria where the more efficient firm serves the market and less efficient sellers mix on an interval above this price. This type of equilibrium was first discovered by Deneckere and Kovenock [1996] and was later restated by Blume [2003].

[13]
It is an open question of whether the nonexistence of a competitive equilibrium implies the nonexistence of a Bertrand equilibrium in markets where cost functions are continuous. The reason for this not being true in Saporiti and Coloma [2010] is that in their framework the cost function was discontinuous at the origin.

[14]
A function is said to be strictly subadditive if for every we have with z = x + y

[15]
Sometimes referred to as ‘BertrandEdgeworth’ games.