Cournot competition is an economics 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;
• There are barriers to entry;
• 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 N, the total number of firms in the market, and take the output of the others as given. Each firm has a cost function ci(qi). 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 both 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

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

p1 = firm 1 price, p2 = firm 2 price

q1 = firm 1 quantity, q2 = firm 2 quantity

c = marginal cost, identical for both firms

Equilibrium prices will be:

p1 = p2 = P(q1+q2)

This implies that firm 1’s profit is given by $\backslash Pi\backslash \; i\; =\; qi(P(q1+q2)-c)$

• Calculate firm 1’s residual demand: Suppose firm 1 believes firm 2 is producing quantity q2. What is firm 1s optimal quantity? Consider the diagram 1. If firm 1 decides not to produce anything, then price is given by P(0+q2)=P(q2). If firm 1 sets produces q1’ then price is given by P(q1’+q2). More generally, for each quantity that firm 1 might decide to set, price is given by the curve d1(q2). The curve d1(q2) is called firm 1’s residual demand; it gives all possible combinations of firm 1’s quantity and price for a given value of q2.

• 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 - r1(q2) - with twice the slope of d1(q2) and with the same vertical intercept. The point at which the two curves (c and r1(q2)) intersect corresponds to quantity q1’’(q2). Firm 1’s optimum q1’’(q2), depends on what it believes firm 2 is doing. To find an equilibrium, we derive firm 1’s optimum for other possible values of q2. Diagram 2 considers two possible values of q2. If q2=0, then the first firm's residual demand is effectively the market demand, d1(0)=D. The optimal solution is for firm 1 to choose the monopoly quantity; q1’’(0)=q^m (q^m is monopoly quantity). If firm 2 were to choose the quantity corresponding to perfect competition, q2=qc P(q^c)=c, then firm 1’s optimum would be to produce nil: q1’’(q^c)=0. This is the point at which marginal cost intercepts the marginal revenue corresponding to d1(q^c).

• It can be shown that, given the linear demand and constant marginal cost, the function q1’’(q2) is also linear. Because we have two points, we can draw the entire function q1’’(q2), see diagram 3. Note the axis of the graphs has changed, The function q1’’(q2) 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

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In very general terms, let the price function for the (duopoly) industry be $P(q\_1+q\_2)$ and firm i have the cost structure $C\_i(q\_i)$. 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): $\backslash Pi\backslash \; i\; =\; P(q\_1+q\_2).q\_i\; -\; C\_i(q\_i)$. The best response is to find the value of $q\_i$ that maximises $\backslash Pi\backslash \; i$ given $q\_j$, with $i\; \backslash ne\; \backslash \; j$, i.e. given some output of the opponent firm, the output that maximises profit is found. Hence, the maximum of $\backslash Pi\backslash \; i$ with respect to $q\_i$ is to be found. First derive $\backslash Pi\backslash \; i$ with respect to $q\_i$:

$\backslash frac\{\backslash partial\; \backslash Pi\backslash \; i\; \}\{\backslash partial\; q\_i\}\; =\; \backslash frac\{\backslash partial\; P(q\_1+q\_2)\; \}\{\backslash partial\; q\_i\}.qi\; +\; P(q1+q2)\; -\; \backslash frac\{\backslash partial\; C\_i\; (q\_i)\}\{\backslash partial\; q\_i\}$

Setting this to zero for maximisation:

$\backslash frac\{\backslash partial\; \backslash Pi\backslash \; i\; \}\{\backslash partial\; q\_i\}\; =\; \backslash frac\{\backslash partial\; P(q\_1+q\_2)\; \}\{\backslash partial\; q\_i\}.qi\; +\; P(q1+q2)\; -\; \backslash frac\{\backslash partial\; C\_i\; (q\_i)\}\{\backslash partial\; q\_i\}=0$

The values of $q\_i$ that satisfy this equation are the best responses. The Nash equilibria are where both $q\_1$ and $q\_2$ are best responses given those values of $q\_1$ and $q\_2$.

An example

Suppose the industry has the following price structure: $P(q\_1+q\_2)=\; a\; -\; b(q\_1+q\_2)$ The profit of firm i (with cost structure $C\_i(q\_i)$ such that $\backslash frac\{\backslash partial\; ^2C\_i\; (q\_i)\}\{\backslash partial\; q\_i^2\}=0$ and $\backslash frac\{\backslash partial\; C\_i\; (q\_i)\}\{\backslash partial\; q\_j\}=0,\; j\; \backslash ne\; \backslash \; i$ for ease of computation) is:

$\backslash Pi\backslash \; i\; =\; \backslash bigg(a\; -\; b(q\_1+q\_2)\backslash bigg).q\_i\; -\; C\_i(q\_i)$

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

$\backslash frac\{\backslash partial\; \backslash bigg(a\; -\; b(q\_1+q\_2)\backslash bigg)\; \}\{\backslash partial\; q\_i\}.qi\; +\; a\; -\; b(q\_1+q\_2)\; -\; \backslash frac\{\backslash partial\; C\_i\; (q\_i)\}\{\backslash partial\; q\_i\}=0$

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

$\backslash frac\{\backslash partial\; \backslash bigg(a\; -\; b(q\_1+q\_2)\backslash bigg)\; \}\{\backslash partial\; q\_1\}.q1\; +\; a\; -\; b(q\_1+q\_2)\; -\; \backslash frac\{\backslash partial\; C\_1\; (q\_1)\}\{\backslash partial\; q\_1\}=0$

$\backslash Rightarrow\; \backslash \; -\; bq\_1\; +\; a\; -\; b(q\_1+q\_2)\; -\; \backslash frac\{\backslash partial\; C\_1\; (q\_1)\}\{\backslash partial\; q\_1\}=0$

$\backslash Rightarrow\; \backslash \; q\_1\; =\; \backslash frac\{a\; -\; bq\_2\; -\; \backslash frac\{\backslash partial\; C\_1\; (q\_1)\}\{\backslash partial\; q\_1\}\}\{2b\}$

By symmetry:

$\backslash Rightarrow\; \backslash \; q\_2\; =\; \backslash frac\{a\; -\; bq\_1\; -\; \backslash frac\{\backslash partial\; C\_2\; (q\_2)\}\{\backslash partial\; q\_2\}\}\{2b\}$

These are the firms' best response functions. For any value of $q\_2$, firm 1 responds best with any value of $q\_1$ that satisfies the above. In Nash equilibria, both firms will be playing best responses so solving the above equations simultaneously. Substituting for $q\_2$ in firm 1's best response:

$\backslash \; q\_1\; =\; \backslash frac\{a\; -\; b(\backslash frac\{a\; -\; bq\_1\; -\; \backslash frac\{\backslash partial\; C\_2\; (q\_2)\}\{\backslash partial\; q\_2\}\}\{2b\})\; -\; \backslash frac\{\backslash partial\; C\_1\; (q\_1)\}\{\backslash partial\; q\_1\}\}\{2b\}$

$\backslash Rightarrow\; \backslash \; q\_1*\; =\; \backslash frac\{a\; +\; \backslash frac\{\backslash partial\; C\_2\; (q\_2)\}\{\backslash partial\; q\_2\}\; -\; 2*\backslash frac\{\backslash partial\; C\_1\; (q\_1)\}\{\backslash partial\; q\_1\}\}\{3b\}$

$\backslash Rightarrow\; \backslash \; q\_2*\; =\; \backslash frac\{a\; +\; \backslash frac\{\backslash partial\; C\_1\; (q\_1)\}\{\backslash partial\; q\_1\}\; -\; 2*\backslash frac\{\backslash partial\; C\_2\; (q\_2)\}\{\backslash partial\; q\_2\}\}\{3b\}$

The Nash equilibrium is at $(q\_1*,q\_2*)$. 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 $P(q\_1+q\_2)=\; a\; -\; b(q\_1+q\_2)$ to obtain the equilibrium market price.

Cournot competition with many firms and the Cournot Theorem

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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 fallows:

$\backslash \; q\_i\; =\; q\; =\; \backslash frac\{a-c\}\; \{b(N+1)\}$ which is each individual firm's output

$\backslash sum\; q\_i\; =\; Nq\; =\; \backslash frac\{N(a-c)\}\; \{(N+1)b\}$ which is total industry output

and

$\backslash \; p\; =\; \backslash frac\{a\}\; \{N+1\}\; +\; \backslash frac\{Nc\}\; \{N+1\}$ is the market clearing price.

The Cournot Theorem then states that 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.

$\backslash lim\_\{N\backslash rightarrow\; \backslash infty\}\; qN\; =\; \backslash frac\{a-c\}\; \{b\}$

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

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.

Implications

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• 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.

Bertrand versus Cournot

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• 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

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• 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.
• This 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

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

References

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Economics models | Competition | Game theory

Cournot-Oligopol | Cournot-duopólium | 古诺竞争 | Cournot-konkurrence