Review of Revenue, Cost and Profit

We define the revenue R to be the total amount of money coming into the company, the cost C the total amount of money coming out of the business, and the profit P is the revenue minus the cost.  When we say marginal, we mean the derivative with respect to x the number of items sold.

For example the marginal cost is

dC
Marginal Cost  =
dx

If we let p be the price per unit, then we have

 Revenue  =  Price*Units Sold or           R = px

Example:

Suppose you own a snow board rental shop and have determined that the demand equation for your snow boards is

x
p  =  \$20 -
10

(At \$20 per rental you wont sell any)  and the cost equation is

C = 50 + 3x

(\$50 fixed costs and \$3 per snow board rental)  What is the marginal profit in terms of x?  What price should you charge to maximize profits?

Solution:

The revenue is

So that profit is

The marginal profit is

dP                   x
=  17 -
dx                    5

To find the maximum profit we set the marginal profit equal to zero and solve:

x
17 -         = 0
5

x = 105

Thus the price we should set is

105
p  =  20 -             =  \$9.95 per rental
10

Exercise:

Suppose that the cost for a truck driver is \$7.50 per hour and that the cost to operate the truck is 0.002v2 where v is the average speed of the truck.  How fast should you recommend your driver to drive in order to minimize the total costs?

Average

Recall that the average is the total divided by the number of items.  Hence, the average cost is the total cost C divided by the average cost x.

Example:

Find the minimum average cost if

C  =  2x2 + 5x + 18

Solution:

The average cost is

C                        18
A  =           =  2x + 5 +
x                          x

18
A'  =  2 -
x2
We set

18
2 -           =  0
x2

to get

x = 3

Since

36
A''  =
x3

plugging in 3 gives a positive value.  By the second derivative test, we see that 3 is a minimum.  The minimum average cost is

Cmin  =  2(9) + 5(3) + 18 = 51

Exercise:

Find the maximum average revenue if the demand equation is

p = 500 + 10x - x2

Elasticity

We define the price elasticity of demand by

 Definition                                     rate of change of demand         elasticity =                                                                                 rate of change of price                               or                    p/x         h =                                  dp/dx We say that a product is      elastic  if  |h| > 1      inelastic  if  |h| < 1

The idea is that a product is elastic if a drop in price results in a significant rise in demand.  A product is inelastic if a drop in price does not result in a significant rise in demand.

Example

The demand function for a product is

p = 50 + x - x2

determine the elasticity when x = 4.

Solution

We see that

p  =  38,     dp/dx  =  1 - 2x  =  -7

hence

38 / 4              38
=                 =  -
-7                 28

so that the product is elastic.

Exercise

Determine the elasticity for

x2
p =
at      x = 10
100x - 1