PSL Course Packet MATH 110
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
c The Pennsylvania State University 2018. This material is not licensed for resale.
2
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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3
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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4
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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5
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.
c The Pennsylvania State University. This material is not licensed for resale.
6
Example 3. If the demand function for math self-help videos is given by p(x) = 35 - 0.1x
and the total cost function to manufacture the videos is given by C(x) = 3x + 21
Evaluate the marginal profit function at x = 20 and interpret the result.
Step 1. Compute the total revenue function, R(x). R(x) = x ? p(x) = x(35 - 0.1x) = 35x - 0.1x2
Step 2. Compute the total profit function, P (x). P (x) = R(x) - C(x) = (35x - 0.1x2) - (3x + 21) = 35x - 0.1x2 - 3x - 21 = -0.1x2 + 32x - 21
Step 3. Compute the marginal profit function, P (x). d
P (x) = P (x) dx
= d (-0.1x2 + 32x - 21) dx
= -0.1(2x) + 32 = -0.2x + 32 Step 4. Plug in x = 20 into the marginal profit function. P (20) = -0.2(20) + 32
2 = - (20) + 32
10 = -4 + 32 = 28 Therefore, the total profit will increase by approximately $28 when the 21st video is produced and sold.
c The Pennsylvania State University. This material is not licensed for resale.
7
Example 4. The daily cost (in dollars) of producing x computer screens is given by
C(x) = 9x3 - 30x2 + 90x + 900
where x denotes the number of thousands of screens produced each day. Calculate the marginal average cost function when 3000 screens are produced each day and interpret the result.
Step 1. Compute and simplify the average cost function, C(x).
9x3 - 30x2 + 90x + 900
C(x) =
since C(x) = C(x)/x
x
9x3 30x2 90x 900
=-
++
separate the terms to make differentiation easier
x x xx
= 9x2 - 30x + 90 + 900x-1 simplify
Step 2. Compute the marginal average cost function, C (x).
d C (x) = C(x)
dx
d =
9x2 - 30x + 90 + 900x-1
dx
= 9(2x) - 30 + 0 + (-1)900x-2
900 = 18x - 30 -
x2
Step 3. Plug in x = 3, since x denotes the number of thousands of screens produced each day.
900 C (3) = 18(3) - 30 -
(3)2 = 54 - 30 - 100 = -76
Therefore, the average cost of producing each screen will decrease by approximately $76 if the level of production is increased from 3000 to 4000 screens.
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8
Elasticity of Demand
Definition 10. The price elasticity of demand measures how sensitive customer demand is to a small percentage change in the price of a good. Intuitively, elasticity is computed by the following ratio:
% Change in Quantity Demanded -
% Change in Price Recall that the Law of Demand implies that a positive percent change in price will result in a negative percent change in demand. Consequently, the negative sign appears in the formula for elasticity only to make sure that the final result will be a positive value.
If the price of the good is p and the corresponding quantity demanded is f (p), then the elasticity of demand at price p, E(p), is defined by
pf (p) E(p) = -
f (p)
Example 5. A store has determined that the demand for used lamps is given by f (p) = 500 - 15p
where p is the price (in dollars) of a lamp. Find the price elasticity of demand, E(p).
Step 1. Compute f (p).
d f (p) = (500 - 15p)
dp
Step 2. Compute E(p).
= -15
pf (p) E(p) = -
f (p)
p(-15) =-
500 - 15p 15p
= 500 - 15p
c The Pennsylvania State University. This material is not licensed for resale.
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