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Math Help - Cournot-Nash Equilibrium Problem, wrong reaction curve?

  1. #1
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    Cournot-Nash Equilibrium Problem, wrong reaction curve?

    Hey all, i posted this over at reddit math help but they didn't respond and the one responder said i made a mistake but i'm not sure what it is? I haven't integrated in forever and he said i messed up my integration, so i'm turning to you guys with hopes you'll explain it

    The relevant parts of the question are:
    Two companies produce a product at $3 per. There are no other costs, and the two companies are trying to maximize profits (Y).
    Q1=12-P1+.5P2
    Q2=12-P2+.5P1
    Y1=Q2(P1-3)
    Y2=Q1(P2-3)

    A) Find the equilibrium price and profit for each firm.
    Here is what i have:
    π1=TR-TC
    π1=P1Q1-3Q1
    π1=P1(12-P1+.5P2)-3(12-P1+.5P2)
    π1=15P1-P12 +.5P1P2-1.5P2-36
    dπ1/dP1=15-2P1-P2
    15-2P1-P2=0
    -2P1=P2-15
    P1=7.5-.5P2 (Reaction curve for firm 1 and 2)
    Substitute P2 into P1 to find equilibrium price
    P1=7.5-.5(7.5-.5P1)
    P1=3.75+.25P
    *P1=5 (equilibrium price)
    Plug P into demand function
    Q1=Q2=12-5+.5(5)=9.5
    Put P and Q into profit equation
    Y1=Y2=Q(P-3)
    *Y=9.5x2=19 (in thousands)
    So i think there's a mistake with my work because i'm having an issue with another section later.

    B) If the firms collude and choose a joint best price P, the profit for each becomes Y1-Y2=(P-3)(12-P+.5P). What price maximizes their joint profit?
    π=PQ-3Q
    π=P(12-P+.5P)-3(12-P+.5P)
    π=12P-.5P2 -36+3P-1.5P
    π=13.5P-.5P2 -36
    dπ/dP=13.5-P
    *P=13.5 (profit maximizing price)

    C) Suppose one firm chooses to defect while the other maintains the agreed price, what is the best defecting price? What is the resulting profit? (here is where i run into issues)
    Plug 13.5 into original reaction curve
    P2=7.5-.5(13.5)
    P2=.75

    This is clearly wrong, because cost per product is $3, and this induces massively negative profit. Furthermore, i know that the right answer is 13.49, and my professor told me that there must be an error in one of my calculations (i emailed him). I knew the answer would be 13.49 but i still can't get it with the math (a detail of the problem is that both companies have loyal clients and floaters who will go to the firm with the lower price, so undercutting by 1 cent is the maximizing strategy).
    Does anyone see where i am going wrong?
    Last edited by sAfuRos; April 24th 2012 at 04:04 PM.
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  2. #2
    Junior Member ignite's Avatar
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    Re: Cournot-Nash Equilibrium Problem, wrong reaction curve?

    I think your approach is incorrect.You are given Q1 and Q2 as functions of P1,P2.You will first need to find P1 as a function of Q1 and Q2 and P2 as a function of Q1 and Q2.
    Then maximise firm profits wrt quantity of that firm not wrt price as you had wrongly done.
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  3. #3
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    Re: Cournot-Nash Equilibrium Problem, wrong reaction curve?

    I think your approach is incorrect.You are given Q1 and Q2 as functions of P1,P2.You will first need to find P1 as a function of Q1
    i think you can do the optimisation on either variable .

    The OP has made an algebra mistake here:
    π1=15P1-P12 +.5P1P2-1.5P2-36
    dπ1/dP1=15-2P1-P2
    it should be
    \frac{dn_1}{dP_1}=15-2p_1 + 0.5p_2 =0
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  4. #4
    Junior Member ignite's Avatar
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    Re: Cournot-Nash Equilibrium Problem, wrong reaction curve?

    Here we are talking about Cournot-Nash Equilbrium where both firms set Quantity and not price.
    Please have a look at : Nash equilibrium - Wikipedia, the free encyclopedia
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