Break-Even Analysis

When a company manufactures x units of a product, it spends money. This is total cost and can be thought of as a function C, where C(x) is the total cost of producing x units. When a company sells x units of the product, it takes in money. This is total revenue and can be thought of as a function R, where R(x) is the total revenue from the sale of x units.

Total profit is the money taken in less the money spent, or total revenue minus total cost. Total profit from the production and sale of x units is a function P given by

Profit = Revenue – Cost, or

P(x) = R(x) – C(x).

There are two types of costs. Costs which must be paid whether a product is produced or not, are called fixed costs. Costs that vary according to the amount being produced are called variable costs. The sum of the fixed cost and variable cost gives the total cost.

Example

A specialty wallet company has fixed costs that are $2,400. Each wallet will cost $2 to produce (variable costs) and will sell for $10.

a) Find the total cost C(x) of producing x wallet.

b) Find the total revenue R(x) from the sale of x wallet.

c) Find the total profit P(x) from the production and sale of x wallet.

d) What profit will the company realize from the production and sale of 500 wallets?

e) Determine the break-even point.

f) Graph the total-cost, total-revenue, and total-profit functions.

Solution

a) Total cost is given by

C(x) = (Fixed costs) plus (Variable costs)

C(x) = 2,400 + 2x.

where x is the number of wallets produced.

b) Total revenue is given by

R(x) = 10x $10 times the number of wallets sold.

c) Total profit is given byP(x) = R(x) – C(x) = 10x – (2,400 + 2x)

= 8x – 2,400.

d) Total profit will be

P(500) = 8(500) – 2,400

= 4,000 – 2,400

= $1,600.

The break-even point occurs when revenue and costs are equal. Thus to find the break-even point, we solve the equation:

e.)

( ) ( )R x C x10 2 2400x x

8 2400x 300x

The company will break even if it produces and sells 300 wallets. Note that the break-even point can also be found by solving P(x) = 0.

R(x) = 10x

C(x) = 2400 + 2x

P(x) = 8x – 2400

-2500

Break-even point

Loss

Gain

f) Graph the total-cost, total-revenue, and total-profit functions.

0 50 100 150 200 250 300 350 400 450 500 550

500

1,5001,000

2,5002,000

3,000

4,0003,500

Wallets sold

$

Supply and Demand

The equilibrium point is when supply equals demand. At that price, the amount that the seller will supply is the same amount that the consumer will buy.

Equilibrium point

Price

Qua

ntity

Supply

( )S p

Demand( )D p

( ) ( )S p D p

Example