N-player Cournot

1
N-player Cournot

• Econ 414

2

• Demand function P a Q
• Cost function Ci(qi) cqi
• n firms, i 1,2,,n. So Q q1 q2 qn.
• Solve using a representative firm i.
• pi qi(a Q c)
• qi(a ?q-i c)
• FOC (a 2qi ?q-i c) 0
• Solve for firm is best response function
• qi (a ?q-i c)/2
• This gives linear equations which we could solve
  simultaneously (for i 1,2,.,n)

3

• Instead, we will impose symmetry.
• It should be clear from the symmetric nature of
  the problem and the best response functions that
  the solution will be symmetric ie q1 q2
  qn.We could see for example that simultaneously
  solving the best response functions for q1 and q2
  will imply that q1 q2, and we could repeat this
  for all other pairs of equation.
• Thus, we can impose qi q for all i on our
  representative firm best response function.
• This implies
• q (a (n-1)q c)/22q a (n-1)q
  c(n1)q a cq (a c)/(n1). So this is
  our unique Nash equilibrium.

4

• To find prices and profits, we can substitute
  this solution for q into our original demand
  function and profit function.
• Industry output Q nq n(a c)/(n1)
• Market price P a n(a c)/(n1) (a
  nc)/n1
• Firm profitpi (a c)/(n1)a c n(a
  c)/(n1) (a c)/(n1) (a c)/(n1) (a
  c)/(n1)2

5

• This then is a generalization of our duopoly
  case, where n 2. Substitute n 2 into the
  solutions before, and see that we get our duopoly
  outcomes.q1 q2 (a c)/3pi (a c)/32
• Look also at how the model converges to perfectly
  competitive outcomes as n ?8.qi ? 0Q ? a cP
  ? cpi ? 0.