Marginal Functions in Economics
BY
DR. JULIA ARNOLD AND MS. KAREN
OVERMAN
USING TAN’S 5TH EDITION APPLIED
CALCULUS FOR THE MANAGERIAL ,
LIFE, AND SOCIAL SCIENCES TEXT
Cost Function
Suppose the total cost of manufacturing refrigerators per
week was
C(x)  8000 200x  0.2x2.....(0  x  400)
Now imagine you wanted to know the actual cost of
manufacturing the 251st refrigerator.
Since the equation above gives you the cost of making x
refrigerators, to do this you would need to find the total
cost of making 251 refrigerators and subtract from it the
cost of making 250 refrigerators, or C(251)-C(250).
Find C(251)-C(250).
C (251)  C (250)
8000 200(251)  0.2(251) 2  8000 200(250)  0.2(250) 2
200(251)  0.2(251) 2  200(250)  0.2(250) 2
50200 12600.20  50000 12500
99.80
So we have found the actual cost to manufacture the 251st
refrigerator was $99.80
The actual cost of making the 251st refrigerator is $99.80.
Notice however, that
C (251 )  C (250 )
 C (251 )  C (250 )
251  250
is the slope formula of a secant line between the x values of 250
and 251 or the average cost over the interval [250, 251].
Let’s compare the actual cost with C’(250)
C(x)  8000 200x  0.2x2 .....(0  x  400)
C' (x)  200  .4x
C' (250)  200  .4(250)  $100
The difference between the actual cost, $99.80 and C’(250)
is only 20 cents. Which was easier to compute?
I hope you are saying the C’ way.
You may be asking yourself: What is C’(250)?
We can say C’(250) represents the approximate cost of making one
more unit.
The actual cost incurred in producing an additional unit of a
certain commodity given that a plant is already at a certain level
of operation is called the marginal cost.
Economists have defined the marginal cost function to be the
derivative of the corresponding total cost function. In other
words, if C is a total cost function, then the marginal cost
function is defined to be its derivative C’. Thus, the
adjective marginal is synonymous with derivative of.
Average Cost Functions
Suppose C(x) is a total cost function. Then the average cost
function, denoted by C(x) (read C bar of x) is denoted by:
C (x )
C x  
x
This function represents the average cost of producing x
units of the commodity.
The derivative of C(x) is called the marginal average cost
function and measures the rate of change of the average cost
function with respect to the number of units produced.
d  C (x ) 
C' x  
dx  x 
Revenue Function
The Revenue Function is R(x) = px where x is the number of
units produced and p is the selling price. The selling price p
is related to the quantity x of the commodity demanded.
This relationship p= f(x) is called a demand equation.
Thus R(x) = xf(x)
The derivative R’(x) is called the marginal revenue
function, and measures the rate of change of the revenue
function with respect to the number of units produced.
Profit Function
The Profit Function is P(x) = R(x) - C(x) , where x is the
number of units produced, C(x) is the cost function and R(x)
is the revenue function.
The derivative P’(x) is called the marginal profit function,
and measures the rate of change of the profit function
with respect to the number of units produced and
provides us with a good approximation of the actual profit
or loss realized from the sale of the (x + 1)st unit,
assuming the xth unit has been sold.