# 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,

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

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

$A P(q) = \frac{P(q)}{q} = \frac{R(q) - C(q)}{q}$

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

$P'(q) = 0$

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

$A P'(q) = 0$

• 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 $q$ which maximizes the profit. Lastly, calculate the maximum profit.

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

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

c)
$p(q) = -2q + 400$, fixed cost is $$200$, costs $40$$ per unit to make

d)
$q(p) = \frac{(300 - p)}{3}$, fixed cost is $$100$, variable cost is$$2q^{2}$

• 2.
Average Profit

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

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

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

c)
$p(q) = -2q + 50$, fixed cost is $$50$, costs$$10$ per unit to make

d)
$q(p) = \frac{400 - p}{2}$, fixed cost is $$288$, variable cost is$$20q$