Cournot competition

Cournot competition is an economic model used to describe an industry structure in which companies compete on the amount of output they will produce, which they decide on independently of each other and at the same time. It is named after Antoine Augustin Cournot (1801–1877) who was inspired by observing competition in a spring water duopoly.[1] It has the following features:

An essential assumption of this model is the "not conjecture" 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 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.

History

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]

Graphically finding the Cournot duopoly equilibrium

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

p_1 = firm 1 price, p_2 = firm 2 price
q_1 = firm 1 quantity, q_2 = firm 2 quantity
c = marginal cost, identical for both firms

Equilibrium prices will be:

p_1 = p_2 = P(q_1+q_2)

This implies that firm 1’s profit is given by \Pi_1 = q_1(P(q_1+q_2)-c)

Calculating the equilibrium

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): \Pi_i = P(q_1+q_2) \cdot q_i - C_i(q_i). The best response is to find the value of q_i that maximises \Pi_i given q_j, with i \ne j, i.e. given some output of the opponent firm, the output that maximises profit is found. Hence, the maximum of \Pi_i with respect to q_i is to be found. First take the derivative of \Pi_i with respect to q_i:

\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:

\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 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 - (q_1+q_2) The profit of firm i (with cost structure C_i(q_i) such that \frac{\partial ^2C_i (q_i)}{\partial q_i^2}=0 and \frac{\partial C_i (q_i)}{\partial q_j}=0, j \ne i for ease of computation) is:

\Pi_i = \bigg(a - (q_1+q_2)\bigg) \cdot q_i - C_i(q_i)

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

\frac{\partial \bigg(a - (q_1+q_2)\bigg) }{\partial q_i} \cdot q_i + a - (q_1+q_2) - \frac{\partial C_i (q_i)}{\partial q_i}=0

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

\frac{\partial \bigg(a - (q_1+q_2)\bigg) }{\partial q_1} \cdot q_1 + a - (q_1+q_2) - \frac{\partial C_1 (q_1)}{\partial q_1}=0
\Rightarrow \ - q_1 + a - (q_1+q_2) - \frac{\partial C_1 (q_1)}{\partial q_1}=0
\Rightarrow \ q_1 = \frac{a - q_2 - \frac{\partial C_1 (q_1)}{\partial q_1}}{2}

By symmetry:

\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 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:

\ q_1 = \frac{a - \left(\frac{a - q_1 - \frac{\partial C_2 (q_2)}{\partial q_2}}{2}\right) - \frac{\partial C_1 (q_1)}{\partial q_1}}{2}
\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}
\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 symmetric Nash equilibrium is at (q_1^*,q_2^*). (See Holt (2005, Chapter 13) for asymmetric examples.) 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 - (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:

Market demand; \ p(q)=a-bq=a-bQ=p(Q)

Cost function; \ c_i(q_i)=cq_i , for all i

\ q_i = Q/N = \frac{a-c} {b(N+1)},

which is each individual firm's output

\sum q_i = Nq = \frac{N(a-c)} {b(N+1)},

which is total industry output

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

which is the market clearing price, and

\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.

\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 N firms, when fixed costs for each firm are F, we have the endogenous number of firms:

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

and a production for each firm equal to:

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

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

Implications

Bertrand versus Cournot

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

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.

See also

References

  1. 1 2 Varian, Hal R. (2006). Intermediate microeconomics: a modern approach (7th ed.). W. W. Norton & Company. p. 490. ISBN 0-393-92702-4.
  2. Van den Berg et al. 2011, p. 1
  3. 1 2 3 4 Morrison 1998
  4. Etro, Federico. Simple models of competition, page 6, Dept. Political Economics -- Università di Milano-Bicocca, November 2006
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