## Abstract

Once oligopoly theory became an interesting target for researchers looking for application of nonlinear dynamics in economics, the supply of Cournot models almost exploded. Not so for the Bertrand case. It is important to realize that the difference does not lie in whether price or quantity is used for optimizing—both work equally well in all monopolistic markets. The difference lies in whether the competing commodities are considered as identical or as slightly different substitutes by the consumers. They cannot be both at the same time! Bertrand modelling was hampered because economists never supplied a convincing demand theory for close substitutes, at least not any precise model that might work in a global setting such as a dynamic process wandering around in an extended phase space. There, however, is one possibility to model substitutes, once proposed by Kelvin Lancaster. In his “new theory of demand” it is not the marketed commodities, but their “properties” as measurable performance scores, that enter the utility function. A marketed commodity is thus regarded as a bundle of such property scores, so the consumer can even combine different marketed commodities to obtain a desired property mix. The property score vector of each commodity is designed by the producer. For close substitutes the vectors are similar but not identical. This makes it possible to model Bertrand price competition using precise demand functions, which, of course, change whenever a design is changed. Unfortunately, the model setup does not permit solving for explicit reaction functions, which we need for cooking up a computer experiment program. Fortunately, it is possible to define close approximations to the true reaction functions—an unusual strategy, but it works. We find interesting dynamics scenarios even with this model. A particular advantage of using the Lancaster approach is that we can also consider choice of design as a means of competition along with pricing. As the Lancasterian properties get well defined shadow prices within the model, the value of each design can be quantified, and the competitors can choose a change whenever the value of the best possible design exceeds that of the current design by a sufficient amount. In this way we can see whether the system converges to identical designs or if the competitors rather seek very different characteristics of their products. Other solutions, such as periodic, may be possible, which may explain cyclic recurrence of designs. This study is not carried out in the book, but left as a stub to the reader. There is also another related stub. Recently the present author saw a few contributions trying to combine Cournot and Bertrand by letting the competitors use both price and quantity adjustments. These simply do not work as **a commodity either is homogenous, or it is not**. There is one possibility though to put sense in this wish to combine, and that is to assume that there are different groups of consumers, some who regard the commodities as identical, some who do not. In this case both competitors might use price discrimination and sell in the different markets. A model, again based on Lancaster is devised as a stub, but analysis is left to the reader.

## References

- Agliari A, Gardini L, Puu T (2005) Global bifurcations in duopoly when the Cournot point is destabilized through a subcritical Neimark bifurcation. Int Game Theory Rev 8:1–20CrossRefGoogle Scholar
- Bertrand J (1883) Théorie mathématique de la richesse sociale. Journal des Savants 48:499–508Google Scholar
- Chamberlin EH (1933) The theory of monopolistic competition. A reorientation of the theory of value. Harvard University Press, Cambridge, MAGoogle Scholar
- Cournot A (1838) Récherces sur les principes mathématiques de la théorie des richesses. Dunod, ParisGoogle Scholar
- Edgeworth FY (1897) La teoria pura del monopolio. Giornale degli Economisti 15:13–31Google Scholar
- Frisch R (1965) Theory of production. D. Reidel Publishing Company, Dordrecht, Holland (Translation from Norwegian of “Innledning til Produksjonsteorien” 1927, various editions.)Google Scholar
- Hicks JR (1947) Value and capital. Oxford University Press, OxfordGoogle Scholar
- Johansen L (1972) Production functions. North-Holland, AmsterdamGoogle Scholar
- Lancaster KJ (1966) A new approach to consumer demand. J Polit Econ 74:132–157CrossRefGoogle Scholar
- Puu T (2017) A new approach to modeling Bertrand duopoly. Rev Behav Econ 4:51–67CrossRefGoogle Scholar
- Tramontana F, Gardini L, Puu T (2010) Global bifurcations in a piecewise smooth Cournot duopoly. Chaos Solitons Fractals 43:15–23CrossRefGoogle Scholar
- von Stackelberg H (1934) Marktform und Gleichgewicht. Julius Springer, BerlinGoogle Scholar