Slide 7- 1 Copyright © 2012 Pearson Education, Inc.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3.8 Business and Economics Applications ■ Break-Even Analysis ■ Supply and Demand

Slide 3- 3 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.

Slide 3- 4 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Break-Even Analysis 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). (continued)

Slide 3- 5 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

Slide 3- 6 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

Slide 3- 7 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution a) Total cost is given by C(x) = (Fixed costs) plus (Variable costs) C(x) = 2, x. 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, x) = 8x – 2,400.

Slide 3- 8 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

Slide 3- 9 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley e) ,500 1,000 2,500 2,000 3,000 4,000 3,500 R(x) = 10x C(x) = x P(x) = 8x – Break-even point Loss Gain Wallets sold

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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: 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.

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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.

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Supply and Demand Supply Demand Equilibrium point Price Quantity 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.

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Solution Example Find the equilibrium point for the demand and supply functions given. Since both demand and supply are quantities and they are equal at the equilibrium point, we rewrite the system as (1) (2) (1) (2)

Slide Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 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): Thus, the equilibrium quantity is 440 units, and the equilibrium price is $32. Solution (continued)