Cournot Model - Atlas of Economic Models
 

Cournot Model

Contents

  1. Cournot Model
    1. Standard Imperfect Competition Framework
      1. Setup and Assumptions
    2. Cournot Model
      1. Solving
      2. Example: CRS Production and Linear Demand
    3. Todo

Standard Imperfect Competition Framework

Setup and Assumptions

  1. A single homogeneous good G, price denoted by $$p$$
  2. Aggregate demand: $$Q = Q(p)$$. Unless otherwise stated this is common knowledge of all firms, and is continuously differentiable and invertible to give a well-defined function p(Q).
  3. Firms $$F_{i}, i = 1, ... N$$ (allow $$N = \infty$$). May also have $$i \in [0,1]$$. Firms produce outputs $$q_{i}$$. Let $$Q = \sum q_{i}$$. We will often abuse notation by using $$q$$ for the vector of outputs $$(q_{1},...,q_{i},..., q_{N})$$.
  4. Firms have known cost functions: $$C_{i}(q)$$. Marginal cost is $$C'(q) = c(q)$$. A simple case is that of constant returns to scale (CRS): $$C(q) = cq$$ where c is unit cost.

Cournot Model

Several firms competing on quantity, not price. Firm i's profits are given by:

$$ \[ \Pi_{i} = q_{i} \cdot p(Q) - C(q_{i}) \] $$

The Cournot approach is to find a 'Nash' equilibrium, that is an output vector $$q$$, such that for each i, firm i's output choice, $$q_{i}$$, maximizes profits given other firm's output choices, $$q_{-i}$$. That is $$q_{i}$$ is a solution of:

$$ \[ max_{q_{i}} \Pi_{i} = max_{q_{i}} ( q_{i} \cdot p(q_{i} + Q_{-i}) - C(q_{i}) ) \] $$

Solving

The first order condition (FOC) is:

$$ \[ p' q_{i} + p - C' = 0 \Rightarrow \frac{p'}{p - C'} q_{i} = -1 \] $$

Solving explicitly obviously requires knowledge of $$C$$ and $$p(Q)$$ which here are left general. However it is possible to show the existence of equilibrium in the 2-firm case using Strategic Complement Methods. [TODO: explain how].

Example: CRS Production and Linear Demand

  1. CRS production: $$C(q_{i}) = c q_{i}$$ for constant unit cost c, hence $$C' = c$$.
  2. Linear demand: $$p(Q) = A - B \cdot Q$$

Thus we have:

$$ \[ \frac{B q_{i}}{A - B (q_{i} + Q_{-i}) - c} = 1 \Rightarrow q_{i} = \frac{A - c - B Q_{-i}}{2B} \] $$

Imposing symmetry in equilibrium requires $$q_{i} = q_{j}, \forall i,j$$. Thus

$$ \[ q_{i} = \frac{A - c}{(N+1)B} \] $$

$$ \[ p = A - BN(\frac{A - c}{(N+1)B}) = \frac{N \cdot c + A}{N+1} = c + \frac{A-c}{N+1} \] $$

$$ \[ \Pi_{i} = (p-c) \cdot q_{i} = \frac{ (A - c)^{2} }{ B \cdot (N + 1)^{2} } \] $$

Rmk: In the limit we obtain the competitive outcome: $$\lim_{N \rightarrow \infty} p = c, \lim_{N \rightarrow \infty} \Pi_{i} = 0$$.

Todo

  • Discuss Cournot as reduced form of a 2-stage capacity choice + Bertrand game
  • Discuss existence of Cournot equilibria for general demand + cost functions (see Vives 1999)

TodoMoreInfo