Objectives for Marginal Analysis
The student will be able to compute:
■ Marginal cost, revenue and profit
■ Marginal average cost, revenue and profit
■ The student will be able to solve
applications
Marginal Cost
Marginal Revenue and Marginal Profit
Definition:
If x is the number of units of a product sold in some time interval, then
Total revenue = R(x)
Marginal revenue = R’(x)
If x is the number of units of a product produced and sold in some time interval, then
Total profit = P(x) = R(x) – C(x)
Marginal Cost and Exact Cost
Assume C(x) is the total cost of producing x items. Then the
Actual cost of producing the (x + 1)st item = C(x + 1) – C(x).
The marginal cost is an approximation of the exact cost C’(x) ≈ C(x + 1) – C(x).
Similar statements are true for revenue and profit.
Example 1
The total cost of producing x electric guitars is C(x) = 1,000 + 100x – 0.25x
2.
1. Find the marginal cost function.
2. Find the marginal cost function when 50
electric guitars are produced.
Example 1
3. Use marginal cost to estimate the cost in producing the 51
stguitar.
4. Find the exact cost of producing the 51
stguitar.
Example 1 (continued)
The total cost of producing x electric guitars is C(x) = 1,000 + 100x – 0.25x
2.
1. Find the marginal cost function.
The marginal cost is C’(x) = 100 – 0.5x
2. Find the marginal cost function when 50 electric guitars
are produced.
Example 1 (continued)
3. Use marginal cost to estimate the cost in producing the 51
stguitar.
The cost of producing the 51st guitar is approximately $75.
4. Find the exact cost of producing the 51
stguitar.
The exact cost is C(x + 1) – C(x).
C(51) – C(50) = 5,449.75 – 5375 = $74.75.
HW Problem 13 p. 163
HW Problem 13 p. 164
HW Problem 13 p. 164
a.Compute the actual cost of manufacturing the 41
stunit.
Actual cost in manufacturing the 41
stunit = C(41)- C(40)
= 5584 – 5340
= 244
HW Problem 11 p. 163
Approximations by Increments
Approximations by Increments
Approximations by Increments
HW Problem 15 p. 164
The price-demand equation and the cost function for the production of television sets are given by
where x is the number of sets that can be sold at a price of $p per set, and C(x) is the total cost of producing x sets.
1. Find the marginal cost.
Example 3
x x
x C x
p and ( ) 150 , 000 30
300 30 )
(
The price-demand equation and the cost function for the production of television sets are given by
where x is the number of sets that can be sold at a price of $p per set, and C(x) is the total cost of producing x sets.
1. Find the marginal cost.
Example 3 (continued)
x x
x C x
p and ( ) 150 , 000 30
300 30 )
(
2. Find the revenue function in terms of x.
Example 3
(continued)
2. Find the revenue function in terms of x.
The revenue function is
3. Find the marginal revenue.
Example 3 (continued)
300 30 )
( )
(
x
2x x
p x x
R
2. Find the revenue function in terms of x.
The revenue function is
3. Find the marginal revenue.
The marginal revenue is
4. Find R’(1500) and interpret the results.
Example 2 (continued)
300 30 )
( )
(
x
2x x
p x x
R
300 15 )
(
' x
x
R
2. Find the revenue function in terms of x.
The revenue function is
3. Find the marginal revenue.
The marginal revenue is
4. Find R’(1500) and interpret the results.
Example 2 (continued)
300 30 )
( )
(
x
2x x
p x x
R
300 15 )
(
' x
x
R
200 1500 $
300 )
1500 (
'
R
Example 2 (continued)
5. Graph the cost function and the revenue function on the
same coordinate. Find the break-even point.
Example 2 (continued)
5. Graph the cost function and the revenue function on the same coordinate. Find the break-even point.
0 < y < 700,000 0 < x < 9,000
Solution: There are two
break-even points. C(x)
R(x)
6. Find the profit function in terms of x.
Example 2
(continued)
6. Find the profit function in terms of x.
The profit is revenue minus cost, so 7. Find the marginal profit.
Example 2 (continued)
150000 30 270
) (
2
x x
x
P
6. Find the profit function in terms of x.
The profit is revenue minus cost, so 7. Find the marginal profit.
8. Find P’(1500) and interpret the results.
Example 2 (continued)
150000 30 270
) (
2
x x
x P
270 15 )
(
' x
x
P
6. Find the profit function in terms of x.
The profit is revenue minus cost, so 7. Find the marginal profit.
8. Find P’(1500) and interpret the results.
Example 2 (continued)
150000 30 270
) (
2
