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Math Help - Please help I have exam ? Optimization problem

  1. #1
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    Please help I have exam ? Optimization problem

    A firm produces good A and good B, and faces demand functions for these two goods:
    qA = 1000 * (20 − pA)

    qB = 1000 * (20 − pB)
    Its total cost function is: TC (qA, qB) = 4(qA + qB)+ (qA+qB)2/4000

    What are the profit-maximising quantities of the two goods to make and sell?


    ANSWER: qA = qB = 5333.33. this is the answers but I want to know how the answer was obtained.
    with many thanks.
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    Re: Please help I have exam ? Optimization problem

    Please any help????
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  3. #3
    MHF Contributor MarkFL's Avatar
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    Re: Please help I have exam ? Optimization problem

    Total profit is total revenue minus total cost. Revenue is units demanded times price per unit. Hence we have:

    Total revenue:

    R\left(q_A,q_B \right)=20\left(q_A+q_B \right)-\frac{q_A^2+q_B^2}{1000}

    and so the total profit is:

    P\left(q_A,q_B \right)=R\left(q_A,q_B \right)-C\left(q_A,q_B \right)

    P\left(q_A,q_B \right)=\left(20\left(q_A+q_B \right)-\frac{q_A^2+q_B^2}{1000} \right)-\left(4\left(q_A+q_B \right)+\frac{\left(q_A+q_B \right)^2}{4000} \right)

    P\left(q_A,q_B \right)=16\left(q_A+q_B \right)-\frac{q_Aq_B}{2000}-\frac{q_A^2+q_B^2}{800}

    Next, find the partial derivatives and equate to zero to determine the critical points:

    P_{q_A}\left(q_A,q_B \right)=16-\frac{q_B}{2000}-\frac{q_A}{400}=0

    This implies: 5q_A+q_B=32000

    P_{q_B}\left(q_A,q_B \right)=16-\frac{q_A}{2000}-\frac{q_B}{400}=0

    This implies: 5q_B+q_A=32000

    Together, these imply: q_A=q_B=\frac{16000}{3}
    Thanks from SAM123
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