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

Marginal Cost = dC/dx

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

 Revenue = Price*Units Sold R = px

Example:

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

p = \$20 - x/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

R = px = (20 - x/10)(x) = 20x - x2/10

So that profit is

P = R - C =  (20x - x2/10) - (50 + 3x) = 17x - x2/10 - 50

The marginal profit is

dP/dx  = 17 - x/5

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

17 - x/5 = 0

x = 105

Thus the price we should set is

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

Exercise:

Suppose that the cost for a truck driver is \$7.50 per hour and that the cost to operate the truck is v2/50000 per mile 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

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

A' = 2 - 18/x2

We set

2 - 18/x2  = 0

to get

x = 3

Since

A'' = 36/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         elasticity = (rate of change of demand)/(rate of change of price) or         h = (p/x)/(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

h = (38/4)/(-7) = -38/28

so that the product is elastic since its absolute value is larger than 1.

Exercise:

Determine the elasticity for

p = x2 /(100x - 1) at x = 10