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66. Sell a Lot and Make No Profits:

Bertrand’s Economics Paradox

The Question:

Two companies, Toastee and Presso, produce and sell coffee makers that also toast bagels. They are what is know in economics as a duopoly because they are the only two corporations in the world that produce that contraption.

The cost of producing and distributing each unit is $100. Toastee is the first to go to market. Pricing its contraption at $115, the corporation makes a nice $15 profit on each unit. Presso comes next, and offers their product for $110. True, they make only $10 profit per unit, but wow, do they rip into Toastee market share. Not to be outdone, Toastee adjusts its unit price to $105. Now Presso is faced with a dilemma: it can lower its price again, to $100, at which price it would capture the entire market. But profitability would be zero. And Toastee would have no choice but to draw level.

The logical end to this price war is that Toastee and Presso realize that they are no longer earning any money. So they might as well stop producing and go out of business.



The Paradox:

Well, that depends on whom you want to listen to. According to Joseph Bertrand (1822-1900), a French mathematician who was also interested in economics, the answer is Yes. According to his compatriot, the mathematician and philosopher Antoine Augustine Cournot (1801-1877) however, the answer is No.

Which model is correct? We know, of course, that in most industries that are dominated by only two competitors (duopolies), or by just a few competitors (oligopolies), the firms remain in business and are profitable. So, paradoxically, Bertrand cannot have been quite correct, even though his analysis does not seem incorrect. Cournot’s model, which will be explained below, corresponds more closely to reality.



In the early days of economic inquiry, some thinkers thought deeply about models of the economy. However, apart from arithmetic illustrations and examples, their work consisted mainly of words: they described their observations, recounted anecdotes and explained their conclusions.

Thus, compared to, say, physics, medicine or chemistry, economics was not considered a serious science, until mathematics entered the scene. The discipline turned into a serious science only after mathematical models were developed that suggest how to optimize something, be it wealth, profits or the utility for money. This occurred only in the late 19th century when neoclassical economists, Cournot and Bertrand among them, began to make use of mathematical methodology and tools.[1]



Cournot had a different idea than the one Bertrand had about how duopolies and oligopolies operate. In Cournot’s analysis, the companies compete on quantity, not on price.

Both firms know the demand curve for a product (the lower the price, the higher the demand). Each firm contemplates what quantity the other firm might produce in order to maximize its profits, and then decides for itself what quantity it should produce in order to maximize its own profit.

In our example, Presso knows that Toastee must make a decision about the quantity that it will produce. But there is a whole range of quantitites among which Toastee could choose. So Presso determines -- given each of Toastee’s possible output levels -- what quantities it should produce. Toastee, for its part, does the same. After some to and fro, the quantities of the two firms converge to a combined output level that determines the price that they can. (Actually, according to Cournot, all this occurs simultaneously.) The resulting price will be somewhat lower than the exorbitant monopoly price, but higher than the punishing perfect market price. In this manner, both Toastee and Presso can be profitable and remain in business.

For a market dominated by a duopoly, the Cournot model is more realistic than Bertrand’s. But there are other reasons why Bertrand’s model does not describe reality. For example, both Toastee and Presso have only limited capacities for production. Toastee may be located on the West Coast and delivery to the East Coast is prohibitively expensive. Presso may launch a branding campaign which claims that its product is superior to Toastee’s.


Technical supplement:

The most advantageous market for consumers is one in which there are many producers, and perfect competition drives the price down nearly to the level of marginal cost. In general, the producers make a small but reasonable profit and consumers are guaranteed nearly the lowest possible price.

On the other hand, in a monopoly, the sole producer is able to charge any price he wants. His profits are enormous and consumers pay through their noses.

A duopoly is somewhere in between: the quantity produced is greater than in a monopoly but lower than under perfect competition. And the price that is charged is lower than in a monopoly, but higher than under perfect competition.

* * *

What if Toastee and Presso decide to play nice with each other, thus becoming a monopoly?

In order to protect consumers from corporations that wield and abuse monopy powers, monopolies and cartels are usually outlawed. An exception are utility companies, like power companies or, in times long gone, telephone companies, who are sometimes allowed exceptionally high monopoly prices in lucrative locations, in order to entice them to operate also in unprofitable geographic regions without forcing them to go bankrupt



© George Szpiro, 2019



[1] This changed again in the late 20th century with the emergence of behavioral economics.

Correctiosns, comments, observations:   

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