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3.8
Business and
Economics
Applications
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Break-Even Analysis
Supply and Demand
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
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.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 3
Break-Even Analysis
(continued)
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).
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 4
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.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 5
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) Graph the total-cost, total-revenue, and total-profit
functions. Determine the break-even point.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 6
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 by
P(x) = R(x) – C(x)
= 10x – (2,400 + 2x)
= 8x – 2,400.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 7
Solution
d) Total profit will be
P(500) = 8(500) – 2,400
= 4,000 – 2,400
= $1,600.
e) The graphs of each of the three functions are
shown on the next slide. R(x), C(x), and P(x)
are all in dollars.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 8
e)
R(x) = 10x
Break-even point
4,000
3,500
3,000
2,500
2,000
1,500
1,000
500
C(x) = 2400 + 2x
P(x) = 8x – 2400
0 50
100 150 200 250 300 350 400 450 500 550
Wallets sold
-2500
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Slide 3- 9
Gains occur where the revenue is greater than the
cost. Losses occur where the revenue is less than the
cost. The break-even point occurs where the graphs
of R and C cross. Thus to find the break-even point ,
we solve the system:
R( x)  10 x,
C ( x)  2,400  2 x.
Using substitution we find that x = 300. The
company will break even if it produces and sells 300
wallets and takes in a total of R(300) = $3,000 in
revenue. Note that the break-even point can also be
found by solving P(x) = 0.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 10
Supply and Demand
As the price of a product varies, the amount
sold varies. Consumers will demand less as
price goes up. Sellers will supply more as the
price goes up.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 11
Supply and Demand
Supply
Quantity
Equilibrium point
Demand
Price
The point of intersection is called the equilibrium
point. At that price, the amount that the seller will
supply is the same amount that the consumer will buy.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 12
Example
Find the equilibrium point for the demand and
supply functions given.
D( p)  3000  80 p, (1)
S ( p)  120  10 p.
(2)
Solution
Since both demand and supply are quantities and
they are equal at the equilibrium point, we rewrite
the system as
q  3000  80 p, (1)
q  120  10 p.
(2)
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 13
Solution (continued)
Solving using substitution we find the
equilibrium price is $32. To find the quantity,
we substitute $32 into either equation D(p) or
S(p). We use S(p):
S (32)  120  10(32)  440.
Thus, the equilibrium quantity is 440 units,
and the equilibrium price is $32.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Slide 3- 14