In oligopoly theory, conjectural variation is the belief that one firm has an idea about the way its competitors may react if it varies its output or price.
Suppose you have two firms producing the same good, so that the industry price is determined by the combined output of the two firms (think of the water duopoly in Cournot's original 1838 account).
Now suppose that each firm has what is called the "Bertrand Conjecture" of −1.
In this case, each firm believes that the other will imitate exactly any change in output it makes, which leads (with constant marginal cost) to the firms behaving like a single monopoly supplier.
The notion of conjectures has maintained a long history in the Industrial Organization theory ever since the introduction of Conjectural Variations Equilibria by Arthur Bowley in 1924[1] and Ragnar Frisch (1933)[2] (a useful summary of the history is provided by Giocoli[3]).
Not only are conjectural variations (henceforth CV) models able to capture a range of behavioral outcomes – from competitive to cooperative, but also they have one parameter which has a simple economic interpretation.
CV models have also been found quite useful in the empirical analysis of firm behavior in the sense that they provide a more general description of firms behavior than the standard Nash equilibrium.
As Stephen Martin has argued:There is every reason to believe that oligopolists in different markets interact in different ways, and it is useful to have models that can capture a wide range of such interactions.
Conjectural oligopoly models, in any event, have been more useful than game-theoretic oligopoly models in guiding the specification of empirical research in industrial economics.
[4]The CVs of firms determine the slopes of their reaction functions.
What happens if we require the actual slope of the reaction function to be equal to the conjecture?
Some economists argued that we could pin down the conjectures by a consistency condition, most notably Timothy Bresnahan in 1981.
[5] Bresnahan's consistency was a local condition that required the actual slope of the reaction function to be equal to the conjecture at the equilibrium outputs.
, and as a get bigger, the consistent conjecture increases (becomes less negative) but is always less than zero for finite a.
The concept of consistent conjectures was criticized by several leading economists.
[6][7] Essentially, the concept of consistent conjectures was seen as not compatible with the standard models of rationality employed in Game theory.
It was realized that this approach could provide a foundation for the evolution of consistent conjectures.
Huw Dixon and Ernesto Somma[8] showed that we could treat the conjecture of a firm as a meme (the cultural equivalent of a gene).
They showed that in the standard Cournot model, the consistent conjecture was the Evolutionarily stable strategy or ESS.
From an evolutionary perspective, those types of behavior that lead to higher payoffs become more common."
In the long run, firms with consistent conjectures would tend to earn bigger profits and come to predominate.
For simplicity, let us follow Cournot's 1838 model and assume that there are no production costs, so that profits equal revenue
With conjectural variations, the first order condition for the firm becomes:
This first order optimization condition defines the reaction function for the firm, which states, for a given CV, the optimal choice of output given the other firm's output.
The CV term serves to shift the reaction function and most importantly later its slope.
To solve for a symmetric equilibrium, where both firms have the same CV, we simply note that the reaction function will pass through the x=y line so that:
, then we obtain the perfectly competitive outcome with price equal to marginal cost (which is zero here).
is a simple behavioral parameter which enables us to represent a whole range of possible market outcomes from the competitive to the monopoly outcome, including the standard Cournot model.
In this case the slope of the reaction functions is −1/2 which is "inconsistent" with the conjecture.
As we can see from this example, when a=0 (marginal cost is horizontal), the Bertrand conjecture is consistent