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Math Help - Economics

  1. #1
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    Economics

    Hi Everyone,

    I recently got this question as a homework exercise, and I tried to do it but am having a lot of trouble and would love any help you can give.

    Thanks guys, here's the question:


    The market for luxury female shoes consists of two firms: Milono Shoes and Blanc Shoes. Milono Shoes has a patented technology that permits it to produce its stiletto shoes in a manner that is more efficient than its rival, Blanc Shoes. As a result, Milono Shoes is the first to choose its profit-maximising output level in the market. Milono Shoes is considering whether or not it should undertake a merger with Blanc Shoes as an external consultant has suggested that a merger would be profitable. The inverse demand function for luxury female shoes is given by P = 800 4Q, Milono Shoes costs are CM(QM) = 40QM and the costs for Blanc Shoes are given by CB(QB) = 40QB.
    Would it be profitable for Milono Shoes to merge with Blanc Shoes? If not, explain why not.


    Again, thanks for your help in advance guys.

    Kind regards,

    Kelly
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  2. #2
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    Simple answer: Merges are (almost) always good for the producers in IO-models. In this simple model you face a Stackelberg game to be solved by backward induction. For BS the output of MS is given. Thus:

    \pi_{BS} = (800 - 4(Q_{MS}+Q_{BS}) Q_{BS} - 40Q_{BS}

    If you optimize this profitfunction for the output of BS you have this company's optimal outputlevel ( Q^*_{BS}) with respect to what MS has chosen. Using this outputlevel (MS will antizipate this level, too.) you can now optimize the MS profits:

    \pi_{MS} = (800 - 4(Q_{MS}+Q^*_{BS}) Q_{MS} - 40Q_{MS}

    Now you know what both Q will be and therefore what the total Q will be. This will yield both companies profits. You will see that the sum of both profits is lower then the profit in a monopoly and therefore a merge would be profitable.
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  3. #3
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    Hi,
    Just in reply to that answer.
    If the sum of both profits are lower than the monopoly. Would it be more profitable for the MS to not merge?
    Sorry, Im just curious
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  4. #4
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    No, because if they merged they could split the profits in a way that both sides earn more.
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  5. #5
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    Correct me if im wrong,

    But I decided to work it out and found out that total profits for MS was $18,050 whilst total profits for BS was $7,125. I also worked out that if MS wasnt to merge their profit would be $39,900.

    Could someone give me a working out of how to get to the correct answer?
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  6. #6
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    \frac{\partial \pi_{BS}}{\partial Q_{BS}} = 800 - 4Q_{MS} - 8 Q_{BS} - 40 = 0 \Rightarrow Q_{BS} = 95 - \frac{1}{2}Q_{MS}

    Then:

    \frac{\partial\pi_{MS}}{\partial Q_{MS}} = 800 - 8Q_{MS} - 95 + \frac{2}{2}Q_{MS} - 40 = 0 \Rightarrow Q_{MS} = 95

    Thus: Q_{BS} = 47.5

    If I now fill in, the market price will be 230 so BS earns: 9025 and MS: 18050, Total: 27075

    In a monopoly, the maximization problem will be:

    \frac{\partial\pi}{\partial Q} = 800 - 8 Q_{BS} - 40 = 0 \Rightarrow Q_{BS} = 95

    Thus \pi = 39900 > 27075
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