c Roberto Barrera, Fall 2015
Math 142 1
1.2 Break-Even Analysis and Market Equilibrium
Mathematical models of cost, revenue, and profits
Two types of costs:
1. Fixed costs:
2. Variable costs:
Total cost:
Revenue:
Profit:
Linear Model
Let x denote the number of units of a given product produced by a firm.
Linear cost model:
Linear revenue model:
Profit:
When the number of goods produced, x, is small, the profit P(x) is negative and the
manufacturer is at a LOSS.
Break-even quantity:
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Example: (J.D.Kim) A manufacturer of garbage disposals, has a monthly fixed cost of
150 dollars and a production cost of 30 dollars for each garbage disposal manufactured.
The unit sell for 60 dollars each.
1. Find the cost function.
2. Findthe revenue function.
3. Find the profit function.
4. Find the break-even quantity.
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Supply and Demand
Let x be the number of goods prodced and p be the price of a good.
Demand equation/curve: If a firm wants to sell more goods, this can be accomplished
by decreasing prices.
Supply equation/curve: gives the price necessary for suppliers to make available x units
to the market. Suppliers of any good want to sell more product of the price is higher.
Equilibrium point/quantity/price:
Example: If the demand equation is pd (x) = − 13 x + 10 and the supply equation is
ps (x) = 21 x + 3, find the equilibrium quantity. (We will see how to solve this with a
calculator later.)
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c Roberto Barrera, Fall 2015
Math 142 Quadratic Models
Quadratic polynomial:
Quadratic equation:
Vertex is the point (h,k) where
h=
k=
Where does the quadratic come from?
Demand curve pd (x) = −cx + b
⇒ Revenue r(x) = pd (x) · x = −(cx + b) · x = −cx2 + bx
Break-Even Analysis
Example: The weekly demand for shampoo produced by a small company is 32 bottles
at a price of $5.00 each. If the price is lowered by $2, then the demand increases to
48 bottles. Also, the producers of the shampoo have weekly fixed costs of $150, and it
costs them $0.18 to make each bottle of shampoo.
a) Find the price-demand function, p(x), where x is the number of shampoo bottles
sold.
b) Find the company’s revenue function.
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c) Find the company’s cost function.
d) Find the company’s profit function.
e) What is the company’s revenue if 50 bottles are sold?
Finding Function Values
1) Enter the function into Y1 by pressing Y = .
2) On the home screen ( 2nd , MODE ), call Y1 by pressing VARS , Y-VARS , FUNCTION ,and
1 . Then, open parenthesis, type the input value, and close parenthesis. You should
have something like Y1 (2). Press ENTER .
f) Find the company’s break-even point (find the break-even point using the intersect
function on the calculator, the quadratic formula, and using the zero function on the
calculator). Round to the nearest integer if necessary.
i) Intersect Function:
1) Press Y = and enter the left side of the equation into Y1 and the right side into Y2 .
(Make sure your Stat Plots 1, 2, and 3 are off –unhighlighted.)
2) Try ZOOM , 6 ; ZOOM , 0 ; or WINDOW (adjust) and then GRAPH . (Remember, you must be able to see the intersection point in your window.)
3) Press 2nd , TRACE , and 5 .
4) Move your cursor close to the intersection point, and press ENTER for the first
curve, ENTER for the second curve, and then ENTER again for “Guess?”.
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iii) Zero Function
1) Enter the function into Y1 by pressing Y = . (Make sure your Stat Plots 1, 2, and 3
are off –unhighlighted.)
2) Try ZOOM , 6 ; ZOOM , 0 ; or WINDOW (adjust) and then GRAPH . (Remember, you must be able to see where the function crosses the x-axis, i.e. where the
function equals 0.)
3) Press 2nd , TRACE , and 2 .
4) The calculator will prompt you for the LeftBound. Use your left or right arrow key
to move the cursor to the left of where the function crosses the x-axis and hit ENTER .
The calculator will then prompt you for the RightBound. Move the cursor to the right
of where the function crosses the x-axis and hit ENTER .
5) The zero will appear at the bottom of the screen.
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Example: (courtesy of Heather Ramsey) The market research department for a company that manufactures widgets established the price-demand function to be p(x) =
2000 − 60x, where p(x) is the wholesale price per widget in dollars at which x widgets
can be sold. This company has fixed costs that amount to $4, 000, and the production
cost per widget is $500.
a) Find the company’s break-even point.
b) Find the output that will produce the maximum revenue, and find the maximum
revenue. Round to the nearest integer if necessary.
c) What price should the company charge per widget to achieve maximum revenue?
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Market Equilibrium
Example: At a price of $2.50 per bushel of a certain crop, a company will supply 8.5
million bushels, and the demand is 9.8 million bushels. If the price per bushel increases
by $0.80, then supply increases to 10.5 million bushels, and demand drops to 7.8 million bushels.
a) Find the demand equation (i.e., price-demand function).
b) Find the supply equation.
c) Find the equilibrium point.
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Linear Depreciation
Example: A printer purchased in 2010 for $2,000 depreciates linearly over 10 years.
If the printer has a scrap value of $200, what will it be worth in 2015? What is the rate
of depreciation?