Editing Oligopoly (section)

==== ''Cournot-Nash model'' ====
{{Main|Cournot competition}}

The [[Antoine Augustin Cournot|Cournot]]–[[John Nash (mathematician)|Nash]] model is the simplest oligopoly model. The model assumes that there are two equally positioned firms; the firms compete on the basis of quantity rather than price, and each firm makes decisions on the assumption that the other firm's behaviour is unchanging.<ref>This statement is the Cournot conjectures. Kreps, D.: A Course in Microeconomic Theory page 326. Princeton 1990.</ref> The market demand curve is assumed to be linear, and marginal costs constant.

In this model, the [[Nash equilibrium]] can be found by determining how each firm reacts to a change in the output of the other firm, and repeating this analysis until a point is reached where neither firm desires to act any differently, given their predictions of the other firm's responsive behaviour.<ref>Kreps, D. ''A Course in Microeconomic Theory''. page 326.  Princeton 1990.</ref>

The equilibrium is the intersection of the two firm's ''reaction functions'', which show how one firm reacts to the quantity choice of the other firm.<ref>Kreps, D. ''A Course in Microeconomic Theory''. Princeton 1990.{{page needed|date=September 2010}}</ref> The reaction function can be derived by calculating the first-order condition (FOC) of the firms' optimal profits. The FOC can be calculated by setting the first derivative of the objective function to zero. For example, assume that the firm <math>1</math>'s demand function is <math>P = (M - Q_2) - Q_1</math>, where <math>Q_2</math> is the quantity produced by the other firm , <math>Q_1</math> is the amount produced by firm <math>1</math>,<ref>Samuelson, W & Marks, S. ''Managerial Economics''. 4th ed. Wiley 2003{{page needed|date=September 2010}}</ref> and <math>M=60</math> is the market. Assume that marginal cost is <math>C_M=12</math>. By following the profit maximisation rule of equating marginal revenue to marginal costs,{{Clarify|date=July 2023|reason=unsure what this means; wikilink or explain}} firm <math>1</math> can obtain a total revenue function of <math>R_T = Q_1 P = Q_1 (M - Q_2 - Q_1) = MQ_1 - Q_1 Q_2 - Q_1^2</math>. The marginal revenue function is <math>R_M = \frac{\partial R_T}{\partial Q_1} = M - Q_2 - 2 Q_1</math>.<ref group="note"><math>R_M = M - Q_2 - 2Q_1</math>. can be restated as <math>R_M = (M - Q_2) - 2Q_1</math>.</ref>
:<math>R_M = C_M</math>
:<math>M - Q_2 - 2Q_1 = C_M</math>
:<math>2Q_1 = (M - C_M) - Q_2</math>
:<math>Q_1 = \frac{M - C_M}{2} - \frac{Q_2}{2} = 24 - 0.5 Q_2</math> [1.1]
:<math>Q_2 = 2(M - C_M) - 2Q_1 = 96 - 2Q_1</math> [1.2]

Equation 1.1 is the reaction function for firm <math>1</math>. Equation 1.2 is the reaction function for firm <math>2</math>. The Nash equilibrium can thus be obtained by solving the equations simultaneously or graphically.<ref>Pindyck, R & Rubinfeld, D: Microeconomics 5th ed. Prentice-Hall 2001{{page needed|date=September 2010}}</ref>

Reaction functions are not necessarily symmetric.<ref>Pindyck, R & Rubinfeld, D: Microeconomics 5th ed. Prentice-Hall 2001</ref> Firms may face differing cost functions, in which case the reaction functions and equilibrium quantities would not be identical.