PSL Course Packet MATH 110 - Pennsylvania State University

PSL Course Packet

MATH 110

Marginal Analysis & Elasticity

Covering Material from Section 3.4

Produced in collaboration with the Penn State Department of Mathematics

Printed copies of this packet are available for FREE in 220 Boucke

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Cost, Revenue, & Profit

Total Cost & Average Cost Functions

Definition 1. The total cost function, C(x) measures the costs incurred from operating a business and is defined by

C(x) = F (x) + V (x) where F (x) denotes the fixed costs (i.e., costs that remain the same regardless of the level of production x) and V (x) denotes the variable costs (i.e., costs that vary depending on the level of production x) of operating a business. Definition 2. The average cost function, C(x), measures the average cost per unit produced and is defined by

C (x) C(x) = .

x

Total Revenue & Average Revenue Functions

Definition 3. The total revenue function, R(x), measures the amount of money received from the sale of x units and is defined by

R(x) = x ? p(x) where x is the number of units demanded and p(x) is the unit price. Definition 4. The average revenue function, R(x), measures the average amount of money received per unit sold and is defined by

R(x) R(x) = .

x

Total Profit & Average Profit Functions

Definition 5. The total profit function, P (x) measures the difference between the total revenue and total cost functions and is defined by

P (x) = R(x) - C(x) where R(x) is the total revenue function and C(x) is the total cost function. Definition 6. The average profit function, P (x), measures the average profit earned per unit produced and sold and is defined by

P (x) P (x) = .

x

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Example 1. A manufacturer of Robot Tutors has a fixed monthly cost of $500 and a processing cost of $9 for each robot tutor produced. Assuming each robot sells for $19, compute the total profit and the average profit per robot when 250 robots are produced and sold.

Step 1. Compute the total cost function, C(x), where x denotes the number of robots produced.

C(x) = F (x) + V (x) = 500 + 9x

total cost equals fixed costs plus variable costs $500 of fixed costs and $9 for each robot

Step 2. Compute the total revenue function, R(x).

R(x) = x ? p(x) = 19x

revenue equals number of units times price per unit since each robot sells for $19

Step 3. Compute the total profit function, P (x).

P (x) = R(x) - C(x) = (19x) - (500 + 9x) = 19x - 500 - 9x = 10x - 500

profit equals revenue minus cost using Steps 1 and 2

Step 4. Plug in x = 250 into the profit function to find the profit associated with the production and sale of 250 robots.

P (250) = 10(250) - 500 = 2500 - 500 = 2000

Step 5. Plug in x = 250 into the average profit function to find the average profit associated with each robot when 250 robots are produced and sold.

P (250) P (250) =

250 2000 = 250

=8

since

P (x)

=

P (x) x

using Step 4

Therefore, each of the 250 robots produced and sold earns an average profit of $8.

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Marginal Analysis

Marginal Cost & Marginal Average Cost Functions

Definition 7. If C(x) denotes the total cost function, then C (x) denotes the marginal cost function, which approximates the extra cost incurred in producing one additional unit

C (x) C(x + 1) - C(x) and C (x) denotes the marginal average cost function.

Marginal Revenue & Marginal Average Revenue Functions

Definition 8. If R(x) denotes the total revenue function, then R (x) denotes the marginal revenue function, which approximates the revenue realized from the sale of one additional unit

R (x) R(x + 1) - R(x) and R (x) denotes the marginal average revenue function.

Marginal Profit & Marginal Average Profit Functions

Definition 9. If P (x) denotes the total profit function, then P (x) denotes the marginal profit function, which approximates the profit generated from the production and sale of one additional unit

P (x) P (x + 1) - P (x)

and P (x) denotes the marginal average profit function.

Marginal Analysis Notation

Name Function Average Marginal Function Marginal Average

Cost

C (x)

C (x)

=

C (x) x

C (x)

C

(x)

=

d dx

C

(x)

Revenue

R(x)

R(x)

=

R(x) x

R (x)

R

(x)

=

d dx

R(x)

Profit

P (x)

P (x)

=

P (x) x

P (x)

P

(x)

=

d dx

P

(x)

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Example 2. The daily demand for the new PBox5 Game Dr. Mathematica-Exam Day of Reckoning is given by

125 1

p(x) =

-x

x+2 2

where x is the number of video games sold each day and p is in dollars. Using the marginal revenue function, R (x), approximate the marginal revenue when 3 video games are sold each day and interpret the result.

Step 1. Compute the total revenue function, R(x)

R(x) = x ? p(x)

125 1

=x

-x

x+2 2

= 125x - 1 x2 x+2 2

Step 2. Compute the marginal revenue function, R (x).

d R (x) = R(x)

dx

d =

125x - 1 x2

dx x + 2 2

125(x + 2) - 125x 1

=

- ? 2x

(x + 2)2

2

125x + 250 - 125x

=

-x

(x + 2)2

250 = (x + 2)2 - x

Step 3. Plug in x = 3 into the marginal revenue function.

250 R (3) = (3 + 2)2 - 3

250 = -3

52 = 10 - 3 =7

Therefore, the total daily revenue would increase by approximately $7 if sales increased from 3 to 4 units each day.

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