In microeconomics, the Bertrand–Edgeworth model of price-setting oligopoly looks at what happens when there is a homogeneous product (i.e. consumers want to buy from the cheapest seller) where there is a limit to the output of firms which they are willing and able to sell at a particular price. This differs from the Bertrand competition model where it is assumed that firms are willing and able to meet all demand. The limit to output can be considered as a physical capacity constraint which is the same at all prices (as in Edgeworth’s work), or to vary with price under other assumptions.
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History
Joseph Louis François Bertrand (1822–1900) developed the model of Bertrand competition in oligopoly. This approach was based on the assumption that there are at least two firms producing a homogenous product with constant marginal cost (this could be constant at some positive value, or with zero marginal cost as in Cournot). Consumers buy from the cheapest seller. The Bertrand–Nash equilibrium of this model is to have all (or at least two) firms setting the price equal to marginal cost. The argument is simple: if one firm sets a price above marginal cost then another firm can undercut it by a small amount (often called epsilon undercutting, where epsilon represents an arbitrarily small amount) so that the equilibrium is zero (this is sometimes called the Bertrand paradox).
The Bertrand approach assumes that firms are willing and able to supply all demand: there is no limit to the amount that they can produce or sell. Francis Ysidro Edgeworth considered the case where there is a limit to what firms can sell (a capacity constraint): he showed that if there is a fixed limit to what firms can sell, then there may exist no pure-strategy Nash equilibrium (this is sometimes called the Edgeworth paradox).
Martin Shubik developed the Bertrand–Edgeworth model to allow for the firm to be willing to supply only up to its profit maximizing output at the price which it set (under profit maximization this occurs when marginal cost equals price). He considered the case of strictly convex costs, where marginal cost is increasing in output. Shubik showed that if a Nash equilibrium exists, it must be the perfectly competitive price (where demand equals supply, and all firms set price equal to marginal cost). However, this can only happen if market demand is infinitely elastic (horizontal) at the competitive price. In general, as in the Edgeworth paradox, no pure-strategy Nash equilibrium will exist. Huw Dixon showed that in general a mixed strategy Nash equilibrium will exist when there are convex costs. Dixon’s proof used the Existence Theorem of Partha Dasgupta and Eric Maskin. Under Dixon's assumption of (weakly) convex costs, marginal cost will be non-decreasing. This is consistent with a cost function where marginal cost is flat for a range of outputs, marginal cost is smoothly increasing, or indeed where there is a kink in total cost so that marginal cost makes a discontinuous jump upwards.
Later developments and related models
There have been several responses to the non-existence of pure-strategy equilibrium identified by Francis Ysidro Edgeworth and Martin Shubik. Whilst the existence of mixed-strategy equilibrium was demonstrated by Huw Dixon, it has not proven easy to characterize what the equilibrium actually looks like. However, Allen and Hellwig were able to show that in a large market with many firms, the average price set would tend to the competitive price.
It has been argued that non-pure strategies are not plausible in the context of the Bertrand–Edgworth model. Alternative approaches have included: