# Establishment Of Cost Function Analysis

Illustration 53

Presume an industry faces a cost function of F = 16 + 8a + a^2.

1. What is the industry’s fixed cost?

2. Obtain an expression for the industry’s average variable cost and marginal cost.

Solution

1. As fixed cost of the industry does not change with productivity the term in the provided cost function which has no productivity (a) term will be the fixed cost. From the provided cost function it is apparent that fixed cost is 16.

2. Total variable cost TVC

=          TC – TFC

=          (16 + 8a + a^2) – 16

=          8a + a^2

And                             AVC   =          TVC    =          8a + a^2
a                         a

=          8 + a

Marginal cost is the first derived of aggregate or total cost function or total variable cost function.

Therefore,

MC      =          ΔTVC  =          8 + 2a
Δa

Illustration 54

A consumer manufacturing company has the following variable cost function:

TVC    =          400O + 18O^2 + 0.5O^3

If the company’s fixed cost are equal to \$100 millions dollars then determine:

1. Total Cost Function

2. Marginal Cost Function

3. Average Variable Cost Function

4. Average Total cost function

5. At what productivity levels average variable cost and marginal cost will be minimum?

Solution

As total cost is the sum of total fixed cost and total variable cost,

TC = TFC + TVC, we get the total cost function as below:

TC       =          100 + 400O + 18O^2 + 0.5O^3

To ascertain the marginal cost we take the first derivative of the total variable cost function with respect to productivity O. Therefore,

MC      =          dTC     =          100 – 36O + 1.5O^2
dO

To determine the average total cost and average variable cost, we divide the respective total costs by the output level.

AC      =          TC       =          100 + 400 O18 O^2 + 0.5 O^3
O                      O       O              O             O

TC       =          100 + 400 – 18 O + 0.5 O^2
O                      O

and                  AVC   =          TVC    =          400 – 18O + 0.5 O^2
O

It is also supportive to know at what level of productivity average variable cost takes on its minimum value. To ascertain the level of output at which average variable cost is minimum we have to consider first derivative of the average variable cost AVC function and set this derivative equal to null.

Therefore, considering the first derivative of AVC function,

AVC   =          400 – 18 O + 0.5 O^2, we have:

d (AVC)          =          - 18 + 1 O
dO

setting it equal to null we have

=        - 18 + 1 O        =          0

1 O      =          18

O         =          18

Therefore, at output level ‘O’ equal to 18, average variable cost will stay minimum.

Output or productivity at which MC Function is Minimum

MC      =          400 – 36 O + 1.5 O^2

To find the productivity level at which MC is minimum, we have to set the first derivative of MC function equal to null. The first derivative of MC function is

d MC   =          - 36 + 3 O
d O

Setting d MC  equal to null, we have:
dO

=          - 36 + 3 O        =          0

3 O      =          36

O         =          36 / 3

=          12

Therefore, at output level 12, marginal cost stays minimum.

It is apparent that marginal cost takes on the minimum value at an output level smaller than that at which AVC is minimum.

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