Cournot competition

Cournot competition is an economic model used to describe industry structure. It so called after Antoine Augustin Cournot (1801-1877) after he observed competition in a spring water duopoly. It has the following features:

• There is more than one firm and all firms produce a homogeneous product;
• Firms do not cooperate;
• Firms have market power;
• The number of firms is fixed;
• Firms compete in quantities, and choose quantities simultaneously;
• There is strategic behaviour by the firms.

An essential assumption of this model is that each firm aims to maximize profits, based on the expectation that its own output decision will not have an effect on the decisions of its rivals. Price is a commonly known decreasing function of total output. All firms know Failed to parse (Missing texvc executable; please see math/README to configure.): N , the total number of firms in the market, and take the output of the others as given. Each firm has a cost function Failed to parse (Missing texvc executable; please see math/README to configure.): c_i(q_i) . Normally the cost functions are treated as common knowledge. The cost functions may be the same or different among firms. The market price is set at a level such that demand equals the total quantity produced by all firms. Each firm takes the quantity set by its competitors as a given, evaluates its residual demand, and then behaves as a monopoly.

Graphically finding the Cournot duopoly equilibrium

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

Failed to parse (Missing texvc executable; please see math/README to configure.): p_1

= firm 1 price, Failed to parse (Missing texvc executable; please see math/README to configure.): p_2
= firm 2 price

Failed to parse (Missing texvc executable; please see math/README to configure.): q_1

= firm 1 quantity, Failed to parse (Missing texvc executable; please see math/README to configure.): q_2
= firm 2 quantity

Failed to parse (Missing texvc executable; please see math/README to configure.): c

= marginal cost, identical for both firms

Equilibrium prices will be:

Failed to parse (Missing texvc executable; please see math/README to configure.): p_1 = p_2 = P(q_1+q_2)

This implies that firm 1’s profit is given by Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): P(0+q_2)=P(q_2) . If firm 1 produces Failed to parse (Missing texvc executable; please see math/README to configure.): q_1'

then price is given by Failed to parse (Missing texvc executable; please see math/README to configure.): P(q_1'+q_2)

. More generally, for each quantity that firm 1 might decide to set, price is given by the curve Failed to parse (Missing texvc executable; please see math/README to configure.): d_1(q_2) . The curve Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): 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 - Failed to parse (Missing texvc executable; please see math/README to configure.): r_1(q_2)

- with twice the slope of Failed to parse (Missing texvc executable; please see math/README to configure.): d_1(q_2)
and with the same vertical intercept. The point at which the two curves (Failed to parse (Missing texvc executable; please see math/README to configure.): c
and Failed to parse (Missing texvc executable; please see math/README to configure.): r_1(q_2)

) intersect corresponds to quantity Failed to parse (Missing texvc executable; please see math/README to configure.): q_1''(q_2) . Firm 1’s optimum Failed to parse (Missing texvc executable; please see math/README to configure.): 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 Failed to parse (Missing texvc executable; please see math/README to configure.): q_2 . Diagram 2 considers two possible values of Failed to parse (Missing texvc executable; please see math/README to configure.): q_2 . If Failed to parse (Missing texvc executable; please see math/README to configure.): q_2=0 , then the first firm's residual demand is effectively the market demand, Failed to parse (Missing texvc executable; please see math/README to configure.): d_1(0)=D . The optimal solution is for firm 1 to choose the monopoly quantity; Failed to parse (Missing texvc executable; please see math/README to configure.): q_1''(0)=q^m

(Failed to parse (Missing texvc executable; please see math/README to configure.): q^m
is monopoly quantity). If firm 2 were to choose the quantity corresponding to perfect competition, Failed to parse (Missing texvc executable; please see math/README to configure.): q_2=q^c
such that Failed to parse (Missing texvc executable; please see math/README to configure.): P(q^c)=c

, then firm 1’s optimum would be to produce nil: Failed to parse (Missing texvc executable; please see math/README to configure.): q_1''(q^c)=0 . This is the point at which marginal cost intercepts the marginal revenue corresponding to Failed to parse (Missing texvc executable; please see math/README to configure.): d_1(q^c) .

• It can be shown that, given the linear demand and constant marginal cost, the function Failed to parse (Missing texvc executable; please see math/README to configure.): q_1''(q_2)

is also linear. Because we have two points, we can draw the entire function Failed to parse (Missing texvc executable; please see math/README to configure.): q_1''(q_2)

, see diagram 3. Note the axis of the graphs has changed, The function Failed to parse (Missing texvc executable; please see math/README to configure.): 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 interception point of the reaction curves. See diagram 4.

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

Calculating the equilibrium

In very general terms, let the price function for the (duopoly) industry be Failed to parse (Missing texvc executable; please see math/README to configure.): P(q_1+q_2)

and firm i have the cost structure Failed to parse (Missing texvc executable; please see math/README to configure.): 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): Failed to parse (Missing texvc executable; please see math/README to configure.): \Pi\ i = P(q_1+q_2).q_i - C_i(q_i) . The best response is to find the value of Failed to parse (Missing texvc executable; please see math/README to configure.): q_i

that maximises Failed to parse (Missing texvc executable; please see math/README to configure.): \Pi\ i
given Failed to parse (Missing texvc executable; please see math/README to configure.): q_j

, with Failed to parse (Missing texvc executable; please see math/README to configure.): i \ne \ j , i.e. given some output of the opponent firm, the output that maximises profit is found. Hence, the maximum of Failed to parse (Missing texvc executable; please see math/README to configure.): \Pi\ i

with respect to Failed to parse (Missing texvc executable; please see math/README to configure.): q_i
is to be found. First derive Failed to parse (Missing texvc executable; please see math/README to configure.): \Pi\ i
with respect to Failed to parse (Missing texvc executable; please see math/README to configure.): q_i

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial \Pi\ i }{\partial q_i} = \frac{\partial P(q_1+q_2) }{\partial q_i}.qi + P(q1+q2) - \frac{\partial C_i (q_i)}{\partial q_i}

Setting this to zero for maximisation:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial \Pi\ i }{\partial q_i} = \frac{\partial P(q_1+q_2) }{\partial q_i}.qi + P(q1+q2) - \frac{\partial C_i (q_i)}{\partial q_i}=0

The values of Failed to parse (Missing texvc executable; please see math/README to configure.): q_i

that satisfy this equation are the best responses. The Nash equilibria are where both Failed to parse (Missing texvc executable; please see math/README to configure.): q_1
and Failed to parse (Missing texvc executable; please see math/README to configure.): q_2
are best responses given those values of Failed to parse (Missing texvc executable; please see math/README to configure.): q_1
and Failed to parse (Missing texvc executable; please see math/README to configure.): q_2

.

An example

Suppose the industry has the following price structure: Failed to parse (Missing texvc executable; please see math/README to configure.): P(q_1+q_2)= a - (q_1+q_2)

The profit of firm i (with cost structure Failed to parse (Missing texvc executable; please see math/README to configure.): C_i(q_i)
such that Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial ^2C_i (q_i)}{\partial q_i^2}=0
and Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial C_i (q_i)}{\partial q_j}=0, j \ne \ i
for ease of computation) is:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Pi\ i = \bigg(a - (q_1+q_2)\bigg).q_i - C_i(q_i)

The maximisation problem resolves to (from the general case):

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial \bigg(a - (q_1+q_2)\bigg) }{\partial q_i}.qi + a - (q_1+q_2) - \frac{\partial C_i (q_i)}{\partial q_i}=0

Without loss of generality, consider firm 1's problem:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial \bigg(a - (q_1+q_2)\bigg) }{\partial q_1}.q1 + a - (q_1+q_2) - \frac{\partial C_1 (q_1)}{\partial q_1}=0

Failed to parse (Missing texvc executable; please see math/README to configure.): \Rightarrow \ - q_1 + a - (q_1+q_2) - \frac{\partial C_1 (q_1)}{\partial q_1}=0

Failed to parse (Missing texvc executable; please see math/README to configure.): \Rightarrow \ q_1 = \frac{a - q_2 - \frac{\partial C_1 (q_1)}{\partial q_1}}{2}

By symmetry:

Failed to parse (Missing texvc executable; please see math/README to configure.): \Rightarrow \ q_2 = \frac{a - q_1 - \frac{\partial C_2 (q_2)}{\partial q_2}}{2}

These are the firms' best response functions. For any value of Failed to parse (Missing texvc executable; please see math/README to configure.): q_2 , firm 1 responds best with any value of Failed to parse (Missing texvc executable; please see math/README to configure.): q_1

that satisfies the above. In Nash equilibria, both firms will be playing best responses so solving the above equations simultaneously. Substituting for Failed to parse (Missing texvc executable; please see math/README to configure.): q_2
in firm 1's best response:

Failed to parse (Missing texvc executable; please see math/README to configure.): \ q_1 = \frac{a - (\frac{a - q_1 - \frac{\partial C_2 (q_2)}{\partial q_2}}{2}) - \frac{\partial C_1 (q_1)}{\partial q_1}}{2}

Failed to parse (Missing texvc executable; please see math/README to configure.): \Rightarrow \ q_1* = \frac{a + \frac{\partial C_2 (q_2)}{\partial q_2} - 2*\frac{\partial C_1 (q_1)}{\partial q_1}}{3}

Failed to parse (Missing texvc executable; please see math/README to configure.): \Rightarrow \ q_2* = \frac{a + \frac{\partial C_1 (q_1)}{\partial q_1} - 2*\frac{\partial C_2 (q_2)}{\partial q_2}}{3}

The Nash equilibrium is at Failed to parse (Missing texvc executable; please see math/README to configure.): (q_1*,q_2*) . Making suitable assumptions for the partial derivatives (for example, assuming each firm's cost is a linear function of quantity and thus using the slope of that function in the calculation), the equilibrium quantities can be substituted in the assumed industry price structure Failed to parse (Missing texvc executable; please see math/README to configure.): P(q_1+q_2)= a - (q_1+q_2)

to obtain the equilibrium market price.

Cournot competition with many firms and the Cournot Theorem

For an arbitrary number of firms, 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:

Failed to parse (Missing texvc executable; please see math/README to configure.): \ q_i = q = \frac{a-c} {(N+1)}

which is each individual firm's output

Failed to parse (Missing texvc executable; please see math/README to configure.): \sum q_i = Nq = \frac{N(a-c)} {(N+1)}

which is total industry output

and

Failed to parse (Missing texvc executable; please see math/README to configure.): \ p = \frac{a} {N+1} + \frac{Nc} {N+1}

is the market clearing price.

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.

Failed to parse (Missing texvc executable; please see math/README to configure.): \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 Failed to parse (Missing texvc executable; please see math/README to configure.): N

firms, when fixed costs for each firm are Failed to parse (Missing texvc executable; please see math/README to configure.): F

, we have the endogenous number of firms:

N=(a-c)/√F-1

and a production for each firm equal to:

q=√F

This equilibrium is usually known as Cournot equilibrium with endogenous entry (or Marshall equilibrium).

Implications

• 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. However cartels are usually illegal, so firms have some motive to tacitly collude using self-imposing strategies to reduce output, which (ceteras paribus) raises price and thus increases profits.

Bertrand versus Cournot

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

• Bertrand predicts 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.

Stackelberg versus Cournot

• The Stackelberg and Cournot models are similar because in both, competition is on quantity.
• The first move gives the leader in Stackelberg a crucial advantage.
• There is also the important assumption of perfect information in the Stackelberg game: the follower must observe the quantity chosen by the leader, otherwise the game reduces to Cournot.
• Above is an example of too much information hurting a player.
• In Cournot competition, it is the simultaneity of the game (the imperfection of knowledge) that results in neither player (ceteris paribus) being at a disadvantage.

See also

• Nash equilibrium
• Game theory
• Bertrand competition
• Stackelberg competition
• Tacit collusion

References

ar:نقطة كورنو

da:Cournot-konkurrence de:Cournot-Oligopol hu:Cournot-duopólium it:Oligopolio di Cournot