Business and Business and Economic Economic ApplicationsApplications

Summary of Business Summary of Business Terms and FormulasTerms and Formulas

x is the number of units produced (or sold)x is the number of units produced (or sold) p is the price per unitp is the price per unit R is the total revenue from selling x unitsR is the total revenue from selling x units

R = xpR = xp C is the total cost of producing x unitsC is the total cost of producing x units

is the average cost per unitC

CC

x

Summary of Business Summary of Business Terms and FormulasTerms and Formulas

P is the total profit from selling x unitsP is the total profit from selling x units

P = R – C P = R – C

The break-even point is the number of The break-even point is the number of units for which R = C.units for which R = C.

MarginalsMarginals

(Marginal revenue) (extra revenue

from selling one additional unit)

(Marginal cost) (extra cost of producing

one additional unit)

(Marginal profit) (extra profit from selling

one additio

dR

dx

dC

dx

dP

dx

nal unit)

Using Marginals as Using Marginals as ApproximationsApproximations

A manufacturer determines that the profit A manufacturer determines that the profit derived from selling derived from selling xx units of a certain item is units of a certain item is given by given by P = 0.0002xP = 0.0002x33 + 10x. + 10x.

a. Find the marginal profit for a production level a. Find the marginal profit for a production level of 50 unitsof 50 units

b. Compare this with the actual gain in profit b. Compare this with the actual gain in profit obtained by increasing the production from 50 obtained by increasing the production from 50 to 51 units.to 51 units.

Demand FunctionDemand Function

The number of units The number of units xx that consumers that consumers are willing to purchase at a given price are willing to purchase at a given price pp is defined as the demand functionis defined as the demand function

p = f(x)p = f(x)

Finding the Demand Finding the Demand FunctionFunction

A business sells 2000 items per month at A business sells 2000 items per month at a price of $10 each. It is predicted that a price of $10 each. It is predicted that monthly sales will increase by 250 items monthly sales will increase by 250 items for each $0.25 reduction in price. Find for each $0.25 reduction in price. Find the demand function corresponding to the demand function corresponding to this prediction.this prediction.

StepsSteps

First find the number of units produced:First find the number of units produced:

102000 250

0.25

2000 1000(10 )

2000 10000 1000

12,000 1000

px

p

p

p

StepsSteps

Now solve this equation for Now solve this equation for pp

12,000 1000

12 12 , 20001000 1000

x p

x xp or x

Finding Marginal Finding Marginal RevenueRevenue

A fast-food restaurant has determine that A fast-food restaurant has determine that the monthly demand for its hamburgers is the monthly demand for its hamburgers is

Find the increase in revenue per hamburger Find the increase in revenue per hamburger (marginal revenue) for monthly sales of (marginal revenue) for monthly sales of 20,000 hamburgers.20,000 hamburgers.

60,000

20,000

xp

Finding Marginal Finding Marginal RevenueRevenue

Because the total revenue is given by Because the total revenue is given by R = xp,R = xp, you have you have

and the marginal revenue isand the marginal revenue is

260,000 1(60,000 )

20,000 20,000

xR xp x x x

1(60,000 2 )

20,000

dRx

dx

Finding Marginal Finding Marginal RevenueRevenue

When When xx = 20,000 the marginal revenue = 20,000 the marginal revenue

Now lets look at the graph. Notice that Now lets look at the graph. Notice that as the price decreases, more as the price decreases, more hamburgers are sold. (Make sense?)hamburgers are sold. (Make sense?)

1(20,000) 60,000 2(20,000)

20,000

20,000$1/

20,000

dR

dx

unit

Finding Marginal ProfitFinding Marginal Profit

Suppose that the cost of producing those Suppose that the cost of producing those same same xx hamburgers is hamburgers is

C = 5000 + 0.56C = 5000 + 0.56xx

(Fixed costs = $5000; variable costs are (Fixed costs = $5000; variable costs are 5656¢ per hamburger)¢ per hamburger)

Finding Marginal ProfitFinding Marginal Profit

Find the total profit and the marginal Find the total profit and the marginal profit for 20,000, for 24,400, and for profit for 20,000, for 24,400, and for 30,000 units.30,000 units.

Solution: Because Solution: Because P = R – C, P = R – C, you can you can use the revenue function to obtainuse the revenue function to obtain

Finding Marginal ProfitFinding Marginal Profit

2

2

2

2

1(60,000 ) (5000 0.56 )

20,000

60,0005000 0.56

20,000 20,000

3 0.56 500020,000

2.44 500020,000

P x x x

x xx

xx x

xx

Finding Marginal ProfitFinding Marginal Profit

Now do the derivative to find a marginalNow do the derivative to find a marginal

2.4410,000

dP x

dx

DemandDemand 20,00020,000 24,40024,400 30,00030,000

ProfitProfit $23,800$23,800 $24,768$24,768 $23,200$23,200

Marginal Marginal profitprofit

$0.44$0.44 $0.00$0.00 - $0.56- $0.56

Finding Maximum ProfitFinding Maximum Profit

In marketing a certain item, a business In marketing a certain item, a business has discovered that the demand for the has discovered that the demand for the item is item is

The cost of producing The cost of producing xx items is given by items is given by C = 0.5x + 500. Find the price per unit C = 0.5x + 500. Find the price per unit that yields a maximum profit.that yields a maximum profit.

50p

x

Finding Maximum ProfitFinding Maximum Profit

From the given cost function, you obtainFrom the given cost function, you obtain P = R – C = xp – (0.5x + 500).P = R – C = xp – (0.5x + 500).

Substituting for Substituting for pp (from the demand function) (from the demand function) producesproduces

50(0.5 500) 50 0.5 500P x x x x

x

Finding Maximum ProfitFinding Maximum Profit

To find maximums using calculus, you now To find maximums using calculus, you now do a derivative of the profit equation and do a derivative of the profit equation and then set it equal to zero.then set it equal to zero.

250.5

250 0.5

250.5 0.5 25

50 2500

dP

dx x

x

xx

x x

Finding Maximum ProfitFinding Maximum Profit

This gives us the number of units needed to This gives us the number of units needed to be produced in order to get the maximum be produced in order to get the maximum profit. How do we find the price?profit. How do we find the price?

50 50$1.00

502500p

Minimizing the Average Minimizing the Average CostCost

A company estimates that the cost (in A company estimates that the cost (in dollars) of producing dollars) of producing xx units of a certain units of a certain product is given by product is given by

C = 800 + 0.4x + 0.0002xC = 800 + 0.4x + 0.0002x22.. Find the production level that minimizes Find the production level that minimizes

the average cost per unit.the average cost per unit.

Minimizing Average CostMinimizing Average Cost Substituting from the given equation for Substituting from the given equation for CC

produces produces 2

2

22

2 2

800 0.04 0.0002

8000.04 0.0002

Finding the derivative and setting it equal to 0 yields

8000.0002 0

8000.0002 0.0002 800

8004,000,000 2000

0.0002

C x xC

x x

xx

dC

dx x

xx

x x x units

PracticePractice

Your turnYour turn

Homework (From CD Appendix G)Homework (From CD Appendix G) p. G5 problems 1 – 33 oddp. G5 problems 1 – 33 odd