In
oligopoly theory
An oligopoly () is a market in which pricing control lies in the hands of a few sellers.
As a result of their significant market power, firms in oligopolistic markets can influence prices through manipulating the supply function. Firms in a ...
, 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. The firm forms a conjecture about the variation in the other firm's output that will accompany any change in its own output.
For example, in the classic
Cournot model of oligopoly, it is assumed that each firm treats the output of the other firms as given when it chooses its output. This is sometimes called the "Nash conjecture," as it underlies the standard
Nash equilibrium
In game theory, the Nash equilibrium is the most commonly used solution concept for non-cooperative games. A Nash equilibrium is a situation where no player could gain by changing their own strategy (holding all other players' strategies fixed) ...
concept. However, alternative assumptions can be made. 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. This means that if firm A increases its output, it conjectures that firm B will reduce its output to exactly offset firm A's increase, so that total output and hence price remains unchanged. With the Bertrand Conjecture, the firms act as if they believe that the market price is unaffected by their own output, because each firm believes that the other firm will adjust its output so that total output will be constant. At the other extreme is the Joint-Profit maximizing 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
In economics, the marginal cost is the change in the total cost that arises when the quantity produced is increased, i.e. the cost of producing additional quantity. In some contexts, it refers to an increment of one unit of output, and in others it ...
) to the firms behaving like a single
monopoly
A monopoly (from Greek language, Greek and ) is a market in which one person or company is the only supplier of a particular good or service. A monopoly is characterized by a lack of economic Competition (economics), competition to produce ...
supplier.
History
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
Sir Arthur Lyon Bowley, FBA (6 November 1869 – 21 January 1957) was an English statistician and economist who worked on economic statistics and pioneered the use of sampling techniques in social surveys.
Early life
Bowley's father, James Wi ...
in 1924 and
Ragnar Frisch
Ragnar Anton Kittil Frisch (3 March 1895 – 31 January 1973) was an influential Norwegian economist and econometrician known for being one of the major contributors to establishing economics as a quantitative and statistically informed science ...
(1933) (a useful summary of the history is provided by Giocoli). 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.
Consistent conjectures
The CVs of firms determine the slopes of their reaction functions. For example, in the standard Cournot model, the conjecture is of a zero reaction, yet the actual slope of the Cournot reaction function is negative. 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. 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. With linear industry demand and quadratic costs, this gave rise to the result that the consistent conjecture depended on the slope of the marginal cost function: for example, with quadratic costs of the form (see below) cost = a.x
2, the consistent conjecture is unique and determined by ''a''. If ''a=0'' then the unique consistent conjecture is the Bertrand conjecture
, 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. Essentially, the concept of consistent conjectures was seen as not compatible with the standard models of rationality employed in
Game theory
Game theory is the study of mathematical models of strategic interactions. It has applications in many fields of social science, and is used extensively in economics, logic, systems science and computer science. Initially, game theory addressed ...
.
However, in the 1990s
Evolutionary game theory
Evolutionary game theory (EGT) is the application of game theory to evolving populations in biology. It defines a framework of contests, strategies, and analytics into which Darwinism, Darwinian competition can be modelled. It originated in 1973 wi ...
became fashionable in economics. It was realized that this approach could provide a foundation for the ''evolution'' of consistent conjectures.
Huw Dixon
Huw David Dixon (/hju: devəd dɪksən/; born 1958) is a British economist. He has been a professor at Cardiff Business School since 2006, having previously been Head of Economics at the University of York (2003–2006) after being a professor ...
and Ernesto Somma showed that we could treat the conjecture of a firm as a
meme
A meme (; ) is an idea, behavior, or style that Mimesis, spreads by means of imitation from person to person within a culture and often carries symbolic meaning representing a particular phenomenon or theme. A meme acts as a unit for carrying c ...
(the cultural equivalent of a gene). They showed that in the standard Cournot model, the consistent conjecture was the
Evolutionarily stable strategy
An evolutionarily stable strategy (ESS) is a strategy (or set of strategies) that is ''impermeable'' when adopted by a population in adaptation to a specific environment, that is to say it cannot be displaced by an alternative strategy (or set of ...
or ESS.
[Dixon and Somma (2003), Proposition 1 p. 528, (1995) p. 13.] As the authors argued, "Beliefs determine Behavior. Behavior determines payoff. 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.
Mathematical example 1: Cournot model with CVs
Let there be two firms, X and Y, with outputs x and y. The market price P is given by the linear demand curve
so that the total revenue of firm X is then
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:
where
is the firms conjecture about how the other firm will respond, the conjectural variation or CV term. 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.
Note that the Cournot-Nash Conjecture is
, in which case we have the standard Cournot
Reaction function. 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:
so that in symmetric equilibrium
and the equilibrium price is
.
If we have the Cournot-Nash conjecture,
, then we have the standard Cournot equilibrium with
. However, if we have the Bertrand conjecture
, then we obtain the perfectly competitive outcome with price equal to marginal cost (which is zero here). If we assume the joint-profit maximizing conjecture
then both firms produce half of the monopoly output and the price is the monopoly price
.
Hence the CV term
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.
Mathematical example 2: Consistency
Take the previous example. Now let the cost of production take the form: cost = a.x
2. In this case, the profit function (revenue minus cost) becomes (for firm X and analogously for firm Y):
The first-order condition then becomes:
which defines the reaction function for firm X as:
This has slope (in output space)
and analogously for firm Y which (we assume) has the same conjecture. To see what consistency means, consider the simple Cournot conjecture
with constant marginal cost ''a=0''. In this case the slope of the reaction functions is −1/2 which is "inconsistent" with the conjecture. The Bresnehan consistency condition is that the conjectured slope
equals the actual slope
which means that
This is a quadratic equation which gives us the unique consistent conjecture
This is the positive root of the quadratic: the negative solution would be a conjecture more negative than −1 which would violate the second order conditions. As we can see from this example, when ''a=0'' (marginal cost is horizontal), the Bertrand conjecture is consistent
. As the steepness of marginal cost increases (''a'' goes up), the consistent conjecture increases. Note that the consistent conjecture will always be less than 0 for any finite ''a''.
Notes
External links
Conjectural variations and competition policyOffice of Fair Trading Report, 2011.
Series on Mathematical Economics & Game Theory, Volume 2: ''Theory Of Conjectural Variations''by Charles Figuières, Alain Jean-Marie, Nicolas Quérou, Mabel Tidball.
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