# cournot competition with N-firms

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?

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This site supports MathJax to allow mathematical formulas and equations to be written using $\LaTeX$. Please see the MathJax Tutorial or drop by chat for help. – robjohn Nov 29 '12 at 15:53

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 am sorry that i do not really understand cournot equilibrium.. could you please explain it to me in more detail? my understanding to it is that firms are deciding their own output level with respect to their expectation about how others will set their output level. So to solve this question for N-firms with hint that the equilibrium is symmetric, i set Q1 to be any one of the firms output and (n-1)Q to be the output level for the rest of the firms that this firm expect them to have. Am i right with is statement? – Steve Nov 30 '12 at 6:48
@Steve: No; as I suspected, you've fundamentally misunderstood the idea of the Cournot equilibrium. It's hard for me to explain because I don't know what you know or don't know; e.g. do you know game theory and Nash equilibria? A Cournot equilibrium is essentially just a Nash equilibrium. If you don't know what that means, I suggest to read up on Nash equilibria at Wikipedia. The basic idea is not that each firm decides based on their expectation about others, but that in equilibrium each firm's choice is optimal with respect to the other firms' choices. – joriki Nov 30 '12 at 9:45
yes, i do, i have learnt pure strategic NE, Mixed strategic NE, Subgame perfect NE. Maybe the exercises that i have done for these topics usually have payoff matrix or allow me to construct one for it...but for cournot equilibrium...i have no idea what is going on. from your point of view, do you mean that i could also construct a matrix for cournot comprising output and price of the two firms? – Steve Nov 30 '12 at 15:00
sorry...i am talking about duopoly example... – Steve Nov 30 '12 at 15:01
@Steve: You can't literally construct a matrix because the options form a continuum, but other than that the principle is the same: A set of strategies for the players is in equilibrium if each player's strategy is the best response to the other player's strategy; so the one player's strategy has to be varied (in this case by differentiation; in the matrix case by comparing rows and columns) while keeping the other player's strategy fixed. – joriki Nov 30 '12 at 15:15

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. – robjohn Nov 29 '12 at 15:55