Paper 4: Fundamentals of Business Mathematics & Statistic

FUNDAMENTALS OF BUSINESS MATHEMATICS AND STATISTICS I 3.49

Application :
Few terms : The term marginal cost indicates the changes in the total cost for each additional unit of
production.

If total cost = c, output = q, then c = f(q) and
dc
dq= marginal cost.

Now, average cost

c f q( )

q q

= =

For example, let c = q^3 – 2q^2 + 4q + 15, to find average cost and marginal cost. Here c is a function
of q.
Total cost c = q^3 – 2q^2 + 4q + 15

Average cost =
c q 2q 4q 15^32215
q= − q+ + =q 2q 4− + +q

Marginal cost = dc ddq dq= (q 2q 4q 15 3q 4q 4^3 −^2 + + )=^2 − +

Note : The total cost is represented by the constant 15, for even if the quantity produced is zero., the cost
equal to 15 will have to be incurred by the firm.
Again since this constant 15 drops out during the process of deriving marginal cost, obviously the magnitude
of fixed cost does not affect the marginal cost.
Minimum Average Cost : The minimum average cost can be determined by applying first and second
derivatives, which will be clear from the following example.
Example 106 :If the total cost function is c = 3q^3 – 4q^2 + 2q, find at what level of output, average cost be
minimum and what level will it be?
Solution:
Total cost ( = TC) = 3q^3 – 4q^2 + 2q.

Average cost ( )^2
AC c 3q 4q 2,
= =q= − +

Marginal cost (MC)

d(TC) 9q 8q 2 2

= dq = − +

Now

d(AC) 6q 4 ;

dq = − making

d AC( ) 0

dq = we get 6q – 4 = 0,

q.^2

= 3

At this level average cost function will be minimum if ( )

2

2

d AC 0.

dq > Now ( )

2

2

d AC 6 0

dq = > which shows

average cost is minimum. Hence average cost will be minimum at an output level of^23 and its value will

be

2 2 2 2

3 3 − 4 3 +2 .= 3