Cournot competition

The state of equilibrium... is therefore stable; i.e. if either of the producers, misled as to his true interest, leaves it temporarily, he will be brought back to it.

— Antoine Augustin Cournot, Recherches sur les Principes Mathematiques de la Theorie des Richesses (1838), translated by Bacon (1897).

Antoine Augustin Cournot (1801-1877) first outlined his theory of competition in his 1838 volume Recherches sur les Principes Mathematiques de la Theorie des Richesses as a way of describing the competition with a market for spring water dominated by two suppliers (a duopoly).[2] The model was one of a number that Cournot set out "explicitly and with mathematical precision" in the volume.[3] Specifically, Cournot constructed profit functions for each firm, and then used partial differentiation to construct a function representing a firm's best response for given (exogenous) output levels of the other firm(s) in the market.[3] He then showed that a stable equilibrium occurs where these functions intersect (i.e. the simultaneous solution of the best response functions of each firm).[3]

The consequence of this is that in equilibrium, each firm's expectations of how other firms will act are shown to be correct; when all is revealed, no firm wants to change its output decision.[1] This idea of stability was later taken up and built upon as a description of Nash equilibria, of which Cournot equilibria are a subset.[3]

This section presents an analysis of the model with 2 firms and constant marginal cost.

${\displaystyle p_{1}}$ = firm 1 price, ${\displaystyle p_{2}}$ = firm 2 price

${\displaystyle q_{1}}$ = firm 1 quantity, ${\displaystyle q_{2}}$ = firm 2 quantity

${\displaystyle c}$ = marginal cost, identical for both firms

Equilibrium prices will be:

${\displaystyle p_{1}=p_{2}=P(q_{1}+q_{2})}$

This implies that firm 1's profit is given by ${\displaystyle \Pi _{1}=q_{1}(P(q_{1}+q_{2})-c)}$

• Calculate firm 1's residual demand: Suppose firm 1 believes firm 2 is producing quantity ${\displaystyle q_{2}}$. What is firm 1's optimal quantity? Consider the diagram 1. If firm 1 decides not to produce anything, then price is given by ${\displaystyle P(0+q_{2})=P(q_{2})}$. If firm 1 produces ${\displaystyle q_{1}'}$ then price is given by ${\displaystyle P(q_{1}'+q_{2})}$. More generally, for each quantity that firm 1 might decide to set, price is given by the curve ${\displaystyle d_{1}(q_{2})}$. The curve ${\displaystyle d_{1}(q_{2})}$ is called firm 1's residual demand; it gives all possible combinations of firm 1's quantity and price for a given value of ${\displaystyle q_{2}}$.

• Determine firm 1's optimum output: To do this we must find where marginal revenue equals marginal cost. Marginal cost (c) is assumed to be constant. Marginal revenue is a curve - ${\displaystyle r_{1}(q_{2})}$ - with twice the slope of ${\displaystyle d_{1}(q_{2})}$ and with the same vertical intercept. The point at which the two curves (${\displaystyle c}$ and ${\displaystyle r_{1}(q_{2})}$) intersect corresponds to quantity ${\displaystyle q_{1}''(q_{2})}$. Firm 1's optimum ${\displaystyle q_{1}''(q_{2})}$, depends on what it believes firm 2 is doing. To find an equilibrium, we derive firm 1's optimum for other possible values of ${\displaystyle q_{2}}$. Diagram 2 considers two possible values of ${\displaystyle q_{2}}$. If ${\displaystyle q_{2}=0}$, then the first firm's residual demand is effectively the market demand, ${\displaystyle d_{1}(0)=D}$. The optimal solution is for firm 1 to choose the monopoly quantity; ${\displaystyle q_{1}''(0)=q^{m}}$ (${\displaystyle q^{m}}$ is monopoly quantity). If firm 2 were to choose the quantity corresponding to perfect competition, ${\displaystyle q_{2}=q^{c}}$ such that ${\displaystyle P(q^{c})=c}$, then firm 1's optimum would be to produce nil: ${\displaystyle q_{1}''(q^{c})=0}$. This is the point at which marginal cost intercepts the marginal revenue corresponding to ${\displaystyle d_{1}(q^{c})}$.

• It can be shown that, given the linear demand and constant marginal cost, the function ${\displaystyle q_{1}''(q_{2})}$ is also linear. Because we have two points, we can draw the entire function ${\displaystyle q_{1}''(q_{2})}$, see diagram 3. Note the axis of the graphs has changed, The function ${\displaystyle q_{1}''(q_{2})}$ is firm 1's reaction function, it gives firm 1's optimal choice for each possible choice by firm 2. In other words, it gives firm 1's choice given what it believes firm 2 is doing.

• The last stage in finding the Cournot equilibrium is to find firm 2's reaction function. In this case it is symmetrical to firm 1's as they have the same cost function. The equilibrium is the intersection point of the reaction curves. See diagram 4.

• The prediction of the model is that the firms will choose Nash equilibrium output levels.

In very general terms, let the price function for the (duopoly) industry be ${\displaystyle P(q_{1}+q_{2})}$ and firm ${\displaystyle i}$ have the cost structure ${\displaystyle C_{i}(q_{i})}$. To calculate the Nash equilibrium, the best response functions of the firms must first be calculated.

The profit of firm i is revenue minus cost. Revenue is the product of price and quantity and cost is given by the firm's cost function, so profit is (as described above): ${\displaystyle \Pi _{i}=P(q_{1}+q_{2})\cdot q_{i}-C_{i}(q_{i})}$. The best response is to find the value of ${\displaystyle q_{i}}$ that maximises ${\displaystyle \Pi _{i}}$ given ${\displaystyle q_{j}}$, with ${\displaystyle i\neq j}$, i.e. given some output of the opponent firm, the output that maximises profit is found. Hence, the maximum of ${\displaystyle \Pi _{i}}$ with respect to ${\displaystyle q_{i}}$ is to be found. First take the derivative of ${\displaystyle \Pi _{i}}$ with respect to ${\displaystyle q_{i}}$:

${\displaystyle {\frac {\partial \Pi _{i}}{\partial q_{i}}}={\frac {\partial P(q_{1}+q_{2})}{\partial q_{i}}}\cdot q_{i}+P(q_{1}+q_{2})-{\frac {\partial C_{i}(q_{i})}{\partial q_{i}}}}$

Setting this to zero for maximization:

${\displaystyle {\frac {\partial \Pi _{i}}{\partial q_{i}}}={\frac {\partial P(q_{1}+q_{2})}{\partial q_{i}}}\cdot q_{i}+P(q_{1}+q_{2})-{\frac {\partial C_{i}(q_{i})}{\partial q_{i}}}=0}$

The values of ${\displaystyle q_{i}}$ that satisfy this equation are the best responses. The Nash equilibria are where both ${\displaystyle q_{1}}$ and ${\displaystyle q_{2}}$ are best responses given those values of ${\displaystyle q_{1}}$ and ${\displaystyle q_{2}}$.

For an arbitrary number of firms, ${\displaystyle N>1}$, the quantities and price can be derived in a manner analogous to that given above. With linear demand and identical, constant marginal cost the equilibrium values are as follows:

Market demand; ${\displaystyle \ p(q)=a-bq=a-bQ=p(Q)}$

Cost function; ${\displaystyle \ c_{i}(q_{i})=cq_{i}}$, for all i

${\displaystyle \ q_{i}=Q/N={\frac {a-c}{b(N+1)}},}$

which is each individual firm's output

${\displaystyle \sum q_{i}=Nq={\frac {N(a-c)}{b(N+1)}},}$

which is total industry output

${\displaystyle \ p=a-b(Nq)={\frac {a+Nc}{N+1}},}$

which is the market clearing price, and

${\displaystyle \Pi _{i}=\left({\frac {a-c}{N+1}}\right)^{2}\left({\frac {1}{b}}\right)}$ , which is each individual firm's profit.

The Cournot Theorem then states that, in absence of fixed costs of production, as the number of firms in the market, N, goes to infinity, market output, Nq, goes to the competitive level and the price converges to marginal cost.

${\displaystyle \lim _{N\rightarrow \infty }p=c}$

Hence with many firms a Cournot market approximates a perfectly competitive market. This result can be generalized to the case of firms with different cost structures (under appropriate restrictions) and non-linear demand.

When the market is characterized by fixed costs of production, however, we can endogenize the number of competitors imagining that firms enter in the market until their profits are zero. In our linear example with ${\displaystyle N}$ firms, when fixed costs for each firm are ${\displaystyle F}$, we have the endogenous number of firms:

${\displaystyle N={\frac {a-c}{\sqrt {Fb}}}-1}$

and a production for each firm equal to:

${\displaystyle q={\frac {\sqrt {Fb}}{b}}}$

This equilibrium is usually known as Cournot equilibrium with endogenous entry, or Marshall equilibrium.[4]

• Output is greater with Cournot duopoly than monopoly, but lower than perfect competition.
• Price is lower with Cournot duopoly than monopoly, but not as low as with perfect competition.
• According to this model the firms have an incentive to form a cartel, effectively turning the Cournot model into a Monopoly. Cartels are usually illegal, so firms might instead tacitly collude using self-imposing strategies to reduce output which, ceteris paribus will raise the price and thus increase profits for all firms involved.

Although both models have similar assumptions, they have very different implications:

• Since the Bertrand model assumes that firms compete on price and not output quantity, it predicts that a duopoly is enough to push prices down to marginal cost level, meaning that a duopoly will result in perfect competition.
• Neither model is necessarily "better." The accuracy of the predictions of each model will vary from industry to industry, depending on the closeness of each model to the industry situation.
• If capacity and output can be easily changed, Bertrand is a better model of duopoly competition. If output and capacity are difficult to adjust, then Cournot is generally a better model.
• Under some conditions the Cournot model can be recast as a two-stage model, where in the first stage firms choose capacities, and in the second they compete in Bertrand fashion.

However, as the number of firms increases towards infinity, the Cournot model gives the same result as in Bertrand model: The market price is pushed to marginal cost level.

• Aggregative game
• Bertrand competition
• Bertrand–Edgeworth model
• Conjectural variation
• Game theory
• Nash equilibrium
• Stackelberg competition
• Tacit collusion

• Holt, Charles. Games and Strategic Behavior (PDF version), PDF
• Tirole, Jean. The Theory of Industrial Organization, MIT Press, 1988.
• Oligoply Theory made Simple, Chapter 6 of Surfing Economics by Huw Dixon.