Announcements
Quiz and HW
Test information
1
Lesson 8
Business Applications:
Break-Even Analysis and Equilibrium Quantity/Price
Cost functions, revenue functions, and profit functions are really important in
business applications.
Cost functions are usually modeled at linear ( C ( x)  mx  b ) where b represents
fixed costs of doing business (rent, utilities, insurance, salaries, benefits, etc.) and
variable costs (represented by mx) are the costs of materials needed to produce
each item. For example, fixed monthly costs might be $1,255 for a small home
business, while the cost per item produced would be $15.88. The cost function
would then be C ( x)  15.88 x  1255 . You would then find the cost of running your
business by evaluating the function a x = the number of items produced in a month.
Average costs would be evaluating the Difference Quotient evaluated appropriately
and the rate of change of costs would be the derivative evaluated at the point you
care about.
Revenue functions are modeled by R( x)  xp where x is the number of items sold
and p is the price per item. This is your income. Note that price is not always
fixed. Price might be a function itself! So the unit price is often given as a
“demand function”. You’ll need to see how it works out for the situation you’re
given. A working definition for revenue is “number of items sold times the
demand function, p”. For example if demand is p  100  5 x , then revenue is
R( x)  x(100  5x)  100 x  5 x 2 . The value of the revenue is the function evaluated at a
given x while the rate of change of revenue is the derivative evaluated at a point.
Profit functions are modeled by the difference between revenue and costs.
P( x)  R( x)  C ( x) .
2
Popper 06, Question 1
3
Example
The revenue in dollars from the sale of x variable-speed jigsaws is given by
R( x)  200 x  0.1x 2
What is p, the demand function?
What is the revenue from the sale of 10 jigsaws?
R(10)  200(10)  0.1(100)  2000  1  1999
In business the derivative of revenue is called MARGINAL revenue. Could we get
it? SURE!
C ( x)  50 x  15
The cost function is
What is the cost of producing 10 jigsaws?
What is the profit?
$515
1999 – 515 = 1484 for 10…$148.40 for one. Nice!
What is the profit function? How do we get this?
Is there such a thing as marginal profit?
How would we get the figures?
4
Popper 06
Question 2
5
Break-Even Analysis
The break-even point in business happens when revenues = costs and the business
is neither losing money nor making money. Managers are VERY interested in the
x value at this point and exactly when it is going to happen, of course.
When the y-value of revenue is higher than the y-value for costs a business is
making profits. In the reverse, not so good. Note that dollars go on the y-axis and
quantity goes on the x-axis.
6
We can find the breakeven point algebraically or graphically.
From above:
R( x)  5 x  x 2
C ( x)  2 x
5x  x2  2 x
GGB:
Intersect[f,g, xi,xf]
Popper 06
Question 3
7
Example
Suppose a manufacturer has monthly fixed costs of $100 and production costs of
$12 per item. The item sells for $20. The demand function is p = x.
C ( x)  12 x  100
R( x)  20 x 2
Where is the break-even?
Equate the two and solve for x. GGB:
Intersect[f, g,0,10]
The negative intersection is extraneous. The x value at the intersection is
2.56, this is the break-even quantity. The breakeven revenue is R(2.56)  131.072
which we would report as $131.07.
What does this mean?
8
Popper 06
Question 7
9
Example
Suppose a company can model its costs with
C ( x)  0.000003x 4  0.04 x 2  200 x  70,000
and the demand function is p  0.02 x  300 .
Find the revenue function?
R( x)  .02 x 2  300 x
HOW?
Find the break-even point?
R( x)  C ( x)
.02 x 2  300 x  0.000003 x 4  0.04 x 2  200 x  70, 000
GGB!
Intersect[<object>,<object>]
10
Ignore the negative, use the 2 positives. HOW?
628 is the smallest positive quantity for which all costs are covered.
Profit is Revenue minus Costs. What is the Profit at the break-even: 0.
P(628) = 0.
Profit at 629 is a positive number. Small, but positive!
Popper 06
Question 8
11
Let’s back all the way up. Suppose you have raw data!
Here’s cost data and demand data, or sales data:
(quantity produced, total cost in dollars)
quantity 50
cost
10040
100
140100
200
18200
500
217400
1000
229300
1200
232600
1500
239300
(quantity demanded, price in dollars)
quantity 50
price
295
100
285
200
280
500
270
1000
250
1200
248
1500
249
Find a cubic regression equation that models costs and a quadratic regression
equation that models demand.
Input; create list; fit poly[<list>, #]
12
What is the point x = 923?
The smallest positive quantity for which all costs are covered.
The break-even quantity.
Popper 06
Question 9
13
Market Equilibrium
The price of goods or services usually settles at a price dictated by the demand for
the item will be equal to the supply of the item. If the price is too high, customers
won’t buy it; if the price is too low, manufacturers have no incentive to produce or
supply the item because their profits will be low.
Market equilibrium occurs when the quantity produced equals the quantity
demanded. At market equilibrium you have the point (equilibrium quantity,
equilibrium price). This point is at the intersection of the supply curve and the
demand curve.
Example
Suppose that we have a demand equation with p is the price of the product in
dollars and x is the quantity of the item in thousands. Our supply equation is given
with p is the price of the unit and x is the number of units the company will make.
demand 5 x  3 p  30  0
supply
52x-30p+45=0
Find the equilibrium price and quantity. I’ll use GGB.
14
Check out the initial work on the functions!
What is the equilibrium quantity?
What is the equilibrium price?
15
Example
The quantity demanded of a certain piece of electronics is 8000 unit when the price
is $260. At a unit price of $200, demand increases to 10,000 units. The
manufacturer will not market the item at a price of $100 or less. However for each
$50 increase in price above $100, the manufacturer will market an additional 1000
units. Assume both supply and demand are linear in nature. Find
The supply equation
The demand equation
The equilibrium quantity and price
16
Popper 06
Question 10
17