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The question is as follow:

Here is how we can think of N-firm Cournot competition. Assume all the firms have the same marginal cost C > 0. Firm 1 chooses Q1, Firm 2 chooses Q2, and so on. The market price P = A – (Q1 + Q2 + … + QN). Assume A > C.

*a) Solve for the Cournot (pure strategy) equilibrium. (Hint: the firms are all the same, so you should expect the equilibrium to be symmetric, that is, Q*1 = Q*2 = …= Q*N.)*

b) Based on your answer to a), show whether the equilibrium profit of a firm increases or decreases in the number of firms, N.

My answer to a is: Let Q be the market output, qi be output firm i,

P = A - Q for Q = ∑qi i=1

Marginal cost of firm i = C which C> 0 and A>C

πi = (A - Q - C)qi

take the derivative,

π'i = A - Q - qi - C since Q = qi + qj + ... + qn, Q' = 1

π'i = A - [(Nqi) + qi] - C since they are identical firms, therefore Q = Nqi

qi* = (A-C)/(N+1)

Since the equilibrium is symmetric, thus

P = A - Nqi*

P = A - N(A-C)/(N+1)

P = (A + NC)/(N+1)

For the profit of individual firm is

π = (A + NC)/(N+1) * (A-C)/(N+1) - C(A-C)/(N+1)

= (A-C)/(N+1) * [ (A + NC)/(N+1) - C ]

= (A-C)/(N+1) * (A-C)/(N+1)

= [(A-C)/(N+1)] ^ 2

For this reason, when N increase, the economic profit of a firm would decrease and vice versa.

Am i doing the right thing so far?

  • 1
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I think you've misunderstood the basic idea of the Cournot equilibrium (which is basically a Nash equilibrium). I suggest to take a look at my answer to your previous question and see how my treatment there differs from your treatment here.

The error lies in that you varied $Q_1$ for all firms simultaneously, whereas a Cournot/Nash equilibrium is defined by each firm varying only its own quantity, keeping the other firms' quantities fixed. So you need to first differentiate with respect to $Q_1$ and then use the symmetry.

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    ... I have edited the answer above, could you help checking if i am almost there... Thanks!2012-11-30
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Determine market price and quantities produced; non-cooperative cournot game

Check the comment section. I do know the answer now, and i'm going to post it later. Probably after the weekend when i've got time.

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    This is not an answer. Supplying links to material is fine, but please elucidate a bit in the answer either what the material says or at least how it relates to the question.2012-11-29