Marginal profit, and maximizing profit & average profit

Marginal profit, and maximizing profit & average profit

Lessons

Marginal Profit (MP) is the additional profit that is gained when you increase the unit by one. It is also the derivative of the profit function. In other words,

MP=P(q)=R(q)C(q)MP = P'(q) = R'(q) - C'(q)

Average Profit (AP) is the amount of profit generated per unit. In other words,

AP(q)=P(q)q=R(q)C(q)qA P(q) = \frac{P(q)}{q} = \frac{R(q) - C(q)}{q}

In this section, we would want to find the quantity qq, which maximizes profit and average profit. To maximize profit, we would want to solve for:

P(q)=0P'(q) = 0

To maximize average profit, we would want to solve for:

AP(q)=0A P'(q) = 0

  • Introduction
    Marginal Profit, and Maximizing Profit & Average Profit Overview:
    a)
    Understanding and Maximizing Marginal Profit

    b)
    Understanding and Maximizing Average Profit


  • 1.
    Marginal Profit

    Given the following information, find the marginal profit and the value of qq which maximizes the profit. Lastly, calculate the maximum profit.

    a)
    R(q)=2q2+50q+6,C(q)=200+10qR(q) = -2q^{2} + 50q + 6, C(q) = 200 + 10q

    b)
    R(q)=10q2+10,C(q)=2qR(q) = -\frac{10}{q^{2}} + 10, C(q) = 2q

    c)
    p(q)=2q+400p(q) = -2q + 400, fixed cost is $200200, costs 4040$ per unit to make

    d)
    q(p)=(300p)3q(p) = \frac{(300 - p)}{3}, fixed cost is $100100, variable cost is $2q22q^{2}


  • 2.
    Average Profit

    Given the following information, find the marginal average profit and the value of qq which maximizes the average profit:

    a)
    R(q)=q2+35q,C(q)=100+5qR(q) = -q^{2} + 35q, C(q) = 100 + 5q

    b)
    R(q)=100q+400,C(q)=5qR(q) = -\frac{100}{q} + 400, C(q) = 5q

    c)
    p(q)=2q+50 p(q) = -2q + 50, fixed cost is $5050, costs $1010 per unit to make

    d)
    q(p)=400p2q(p) = \frac{400 - p}{2}, fixed cost is $288288, variable cost is $20q20q