Marginal Analysis - MATH 151 Calculus for Management
Marginal Analysis
MATH 151 Calculus for Management
J. Robert Buchanan
Department of Mathematics
Spring 2014
J. Robert Buchanan
Objectives
After completing this lesson we will be able to calculate: marginal cost, marginal revenue, marginal profit, and marginal average cost.
J. Robert Buchanan
Background
C ( x ) : total cost of producing x items.
R ( x ) : total revenue when x items are sold.
P ( x ) : profit from producing and selling x items.
P ( x ) = R ( x ) − C ( x )
Remark: in business and economics the word marginal indicates a rate of change.
J. Robert Buchanan
C
0
( x ) : marginal cost.
R
0
( x ) : marginal revenue.
P
0
( x ) : marginal profit.
C
0
( x ) can be thought of as an estimate of the change in C ( x ) when x is increased by 1.
Background
C ( x ) : total cost of producing x items.
R ( x ) : total revenue when x items are sold.
P ( x ) : profit from producing and selling x items.
P ( x ) = R ( x ) − C ( x )
Remark: in business and economics the word marginal indicates a rate of change.
C
0
( x ) : marginal cost.
R
0
( x ) : marginal revenue.
P
0
( x ) : marginal profit.
C
0
( x ) can be thought of as an estimate of the change in C ( x ) when x is increased by 1.
J. Robert Buchanan
Marginal Cost
Definition
Marginal cost is the rate of change of total cost per unit change in quantity x .
C
H x
L
500
400
300
200
100
x
2 4
J. Robert Buchanan
6 8
Example
Suppose C ( x ) = 225 + 2 x
2
.
The cost of producing 4 items is C ( 4 ) = 257
The cost of producing 5 items is C ( 5 ) = 275
The cost of producing the 5th item is C ( 5 ) − C ( 4 ) = 18
The marginal cost function is C
0
( x ) = 4 x .
C
0
( 4 ) = 16 which is close to the cost of the 5th item.
J. Robert Buchanan
Marginal Revenue
Definition
Marginal revenue is the rate of change of the total revenue per unit change in sales when the level of sales is x items.
Recall: if the price of an item is p = D ( x ) where D is the demand function, then revenue can be calculated as
R ( x ) = x · p = x D ( x ) .
J. Robert Buchanan
Example
The revenue from producing x items is given by the formula
R ( x ) = − 0 .
3 x
2
+ 48 x .
R ( 20 ) = 840
R ( 21 ) = 875 .
7
R
0
( x ) = − 0 .
6 x + 48
R
0
( 20 ) = 36
J. Robert Buchanan
Marginal Profit
Definition
Marginal profit is the rate of change of the profit per unit change in sales when x items are produced and sold.
P ( x ) = R ( x ) − C ( x )
P
0
( x ) = R
0
( x ) − C
0
( x )
J. Robert Buchanan
Example (1 of 2)
A company can produce x items for a cost of
C ( x ) = 2750 + 30 x + 0 .
1 x
2
. The revenue from the sale of x items is R ( x ) = 100 x − 0 .
05 x
2
.
40 000
30 000
20 000
10 000
500 x
100 200 300
J. Robert Buchanan
400
Example (2 of 2)
The profit function is
P ( x ) = 100 x − 0 .
05 x
2
− ( 2750 + 30 x + 0 .
1 x
2
) = 70 x − 0 .
15 x
2
− 2750 .
For example, P ( 250 ) = 5375 and P ( 251 ) = 5369 .
85.
The break even points are x = 43 .
30 and x = 423 .
36.
The marginal profit is P
0
( x ) = 70 − 0 .
3 x .
Note that P
0
( 250 ) = − 5.
J. Robert Buchanan
Marginal Average Cost
Definition
If C ( x ) is the cost of producing x items, the cost of production per item is called the average cost per item and is written
C ( x ) =
C ( x )
.
x
Definition
Marginal average cost C
0
( x ) is the rate of change of the average cost per unit change in production. This is an approximation of the change in average cost when one additional item is produced.
J. Robert Buchanan
Example (1 of 2)
Suppose the total cost of producing x items is
C ( x ) = 45 + 3 x + 0 .
005 x
2
.
The average cost is
C ( x ) =
45 + 3 x + 0 .
005 x
2 x
=
45 x
+ 3 + 0 .
005 x .
The marginal average cost is
C
0
( x ) = −
45 x 2
+ 0 .
005 .
J. Robert Buchanan
Example (2 of 2)
C
H x
L
10
8
6
4
2
100 200 300
The average cost is minimized at x = 94 .
87.
400
J. Robert Buchanan
500 x
Consumption and Savings
Suppose the total income of a nation is x .
The proportion of the income which is saved is S ( x ) and the proportion which is spent on consumer goods is C ( x ) .
x = C ( x ) + S ( x )
The rate of change in C ( x ) per unit change in x is called the marginal propensity to consume .
The rate of change in S ( x ) per unit change in x is called the marginal propensity to save .
1 = C
0
( x ) + S
0
( x )
C
0
( x ) = 1 − S
0
( x )
J. Robert Buchanan
Example
Suppose the total national consumption is given by the function
C ( x ) = 250 − 0 .
75 x − 0 .
025 x
0 .
8
.
The marginal propensity to consume is
C
0
( x ) = − 0 .
75 − 0 .
02 x
− 0 .
2
= − 0 .
75 −
0 .
02
.
x 0 .
2
The marginal propensity to save is
S
0
( x ) = 1 − C
0
( x ) = 1 − − 0 .
75 −
0 .
02 x 0 .
2
= 1 .
75 −
0 .
02
.
x 0 .
2
J. Robert Buchanan
