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Math Help - Oligopoly-Bertrand & Cournot model

  1. #1
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    Lightbulb Oligopoly-Bertrand & Cournot model

    Suppose there are only two producers, A and B, in a market producing the same good with the same total cost function
    TC=0.05 Q^2 +100.
    The market demand is given by
    Q=1000-5P.
    a) Find the equilibrium price and total output in the Bertrand model.
    b) Find the equilibrium price and total output in the Cournot model.
    =======================

    Can someone please teach me how to compute these? If the marginal cost were constant, then I think I know how to answer these; for this problem, it's the cost function that is confusing me.
    For the Bertrand model, we set P=MC, and solve for the output, but what is the MC in this case? There are two firms, so do we have to add them to get the MC? But how to add them? What is the equation that I have to solve?
    I hope someone can be kind enough to help me out! Thank you so much!
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  2. #2
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    Quote Originally Posted by kingwinner View Post
    Suppose there are only two producers, A and B, in a market producing the same good with the same total cost function
    TC=0.05 Q^2 +100.
    The market demand is given by
    Q=1000-5P.
    a) Find the equilibrium price and total output in the Bertrand model.
    b) Find the equilibrium price and total output in the Cournot model.
    =======================

    Can someone please teach me how to compute these? If the marginal cost were constant, then I think I know how to answer these; for this problem, it's the cost function that is confusing me.
    For the Bertrand model, we set P=MC, and solve for the output, but what is the MC in this case? There are two firms, so do we have to add them to get the MC? But how to add them? What is the equation that I have to solve?
    I hope someone can be kind enough to help me out! Thank you so much!
    Profit is always maximized when MR=MC. The MC for each firm is 0.1Q. Find the MR equation in terms of Q and then equate MR=MC.
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  3. #3
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    Quote Originally Posted by econstudent View Post
    Profit is always maximized when MR=MC. The MC for each firm is 0.1Q. Find the MR equation in terms of Q and then equate MR=MC.
    MR for the INDUSTRY is:
    MR=200-(2/5)Q

    The MC for EACH FIRM is MC=0.1q.

    The trouble is that the two equations have different context, the MR is for the whole industry, but the MC is for each firm.

    I am not sure how to bring this to a common footing. How can we add the MC?
    What is the equation that we have to solve to get the Bertrand equilibrium price and total output in this case?

    Thanks!
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  4. #4
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    But I think the Bertrand equilibrium says P=MC (rather than MR=MC). It's a price strategy.
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  5. #5
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    Quote Originally Posted by kingwinner View Post
    MR for the INDUSTRY is:
    MR=200-(2/5)Q

    The MC for EACH FIRM is MC=0.1q.

    The trouble is that the two equations have different context, the MR is for the whole industry, but the MC is for each firm.

    I am not sure how to bring this to a common footing. How can we add the MC?
    What is the equation that we have to solve to get the Bertrand equilibrium price and total output in this case?

    Thanks!
    In the Bertrand equilibrium, where firms compete on price, the price will always equal to the marginal cost. Imagine if firm A and B set both their prices above MC by the same amount. Then, firm A can decrease their price by a small amount to capture the entire market share. Firm B can do the same. Thus, both firms will eventually end up at an equilibrium where P=MC. If a firm tries to sell below that then there will be negative profits.

    Since both firms have the same price, then the demand for each firm will just be half of the total demand. Their products are identical, so we assume that consumers will be indifferent between then two and thus, there's a 50% chance that a consumer will buy the product from firm A.

    Therefore, if Q is the total quantity demanded, then Q/2 is the quantity demanded from each firm.

    This is the true for the Cournot Equilibrium as well. Again, the consumers are indifferent so in both markets each firm has 50% of the market share.

    Quote Originally Posted by kingwinner View Post
    But I think the Bertrand equilibrium says P=MC (rather than MR=MC). It's a price strategy.
    P=MR occurs special cases and this is one of them. Another case is the perfectly competitive situation.
    Last edited by econstudent; March 26th 2010 at 08:23 PM.
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