Lecture 9
Marginal Analysis
Applications of Rate of Change to Economics: Marginal Analysis
 Let C( x) be the total cost of producing x units of a certain product and R( x) be
the total revenue derived from the sale of x units of the product.
- The rate of change of C with respect to x is called the marginal cost, C ' ( x) , at
the production level of x units.
- The rate of change of R with respect to x is called the marginal revenue, R ' ( x) ,
at the sales level of x units.
- Note that C'( x)  the additional cost of producing the ( x  1) st unit
- R'( x)  the additional revenue derived from the sales of the ( x  1) st unit.
- Similarly, the marginal profit, P ' ( x) is approximately equal to the additional
profit of producing and selling one more unit.
Example
1 3
x  5x 2  30 x  10
3
dollars, while the revenue received from the sale of x units is 39 x  x 2 dollars.
At what value(s) of x will the marginal revenue equal the marginal cost?
1. Suppose that the cost of producing x units of some product is
 In a free enterprise economy a firm will set production in such a way as to maximize
its profit. The Profit is maximized at a production level for which the marginal
revenue equals the marginal cost.
Examples
2. Refer to Example 1; at what level of production the profit is maximized.
3. It is estimated that the demand for steel approximately satisfies the equation
p  256  50x (p in dollars, and x in million tons), and the total cost of producing x
million tons of steel is C( x)  182  56x million dollars. Using differentiation to
determine the production level for which the profit is maximized. What is the
price at that production level?
 Suppose C( x) and AC (x ) denote the total cost and average cost, respectively, of
C ( x)
. It can be shown
x
that when the average cost is minimal, it must be equal to the marginal cost.
producing x units of a certain product. Note that AC ( x) 
Example
4. Suppose the total cost in dollars of manufacturing x units of a certain commodity is
C( x)  3x 2  x  48 .
a. Determine the average cost function, AC (x ) .
b. Determine the production level for which the marginal cost is equal to the
average cost.
Answer to examples
1. 9
4. a.
3x  1  48 / x
2. 9 units
b. 4 units
3.
2 million tons, 156 dollars