    # Perfect Rivalry Under Short Run Illustration 65

For a perfectly rivalled industry the following short run function is provided.

TC       =          4 + 8V + V^2

If price of the commodity prevailing in the market is \$20 at what level of productivity the industry will optimise profits?

Solution

As the Total revenue is price incremented by volume of productivity, total revenue function is

TR       =          P.V      =          20V

TC       =          4 + 8V + V^2

We describe below the profit optimisation with both the TR – TC approach and MC – MR approach.

TR – TC Approach

Profits             π          =          TR – TC

=          20V – 4 – 8V – V^2

=          12V – V^2 – 4                                    …..Equation (1)

Now, profits will be optimum at the productivity level at which first derivative of profit function with respect to the volume of productivity parities zero. Therefore, by considering the first derivative of profit function (1) and fixing it equal to zero, we have:

=          12 – 2V           =          0
dV

2V       =          12

V*       =          12 / 2   =          6

MR - MC Approach

In this approach profits are optimum at the productivity level at which MR parities MC. We thus first derive the marginal revenue and marginal cost from TR and TC functions.

TR       =          20V

MR      =          d TR    =          20
d V

TC       =          4 + 8V + V^2

MC      =          d TR    =          8 + 2V
d V

in order to ascertain profit optimisation productivity we set MR equal to MC. Therefore,

MR      =          MC

20        =          8 + 2V

2V       =          20 – 8

V*       =          12 / 2   =          6

Illustration 66

An industry manufactures a perishable consumable commodity in a perfect rivalled market. the industry's variable cost function is provided by:

TVC    =          300V – 40V^2 + 2V^3

Where V is level of productivity, determine below what price the industry should shut down manufacturing in the short run?

Solution

In the short run an industry will shutdown operations if the price drops below the level of minimum average variable cost. So we first determine the minimum average variable cost.

AVC   =          TVC    =          300V40V^2 + 2V^3
V                       V           V           V

=          300 – 40V + 2V^2

To ascertain the level of productivity at which average variable cost is minimum we take the first derivative of the AVC function and set it equal to zero.

dAVC =          - 40 V + 4V     =          0
dV

40V     =          4V

V         =          40 / 4   =          10

Now substituting the value of V in the AVC function we know the minimum average variable cost.

AVC   =          300 – 40V + 2V^2

=          300 – 40 (10) + 2 (10) ^2

=          300 – 400 + 200

=          100

Therefore, if price drops below \$100 per unit the industry will be liquidated.

Illustration 67

Provided is the following short run cost function of a firm

TC       =          2000 + 20V^2

Derive the expression for industry’s short run supply curve.

Solution

An industry's short run supply curve is industry’s short run marginal cost curve. To procure marginal cost function, we have to procure the first derivative of aggregate cost function.

Therefore,

MC      =          dTC     =          40V
dV

to obtain the short run supply curve of an industry we set price equal to marginal cost. Therefore,

P          =          40V

V         =          P / 40                           ….Equation (1)

Since the supply curve of an industry is that portion of marginal cost curve that lies above the minimum point of the average variable cost AVC curve.

AVC is minimised at the productivity level where its first derivative parities zero.

AVC   =          TVC    =          20V^2
V                       V

=          20V

Setting its derivative equal to zero we have

d (20V)           =          0
dV

V         =          0

Therefore, AVC is minimised when productivity V is equal to zero. It therefore adopts the whole supply function obtained in Equation (1) such as V = P / 40 denotes the short run supply curve of the industry.

Illustration 68

The following table shows the total cost schedule of a rivalled industry. What is total fixed cost of an industry. Using marginal study ascertain the profit optimisation level of productivity if price of the product for the industry is \$16. Compute the profits at this productivity level.

 Volume Sold in Units Total Cost TC in \$ 0 10 10 18 15 26 20 38 25 45 30 58 35 64 40 79

Solution

As the industry incurs a cost of \$10 even at zero level of productivity, the industry’s total fixed cost is \$10. With respect to ascertain the profit optimisation level of productivity i.e. Symmetry level we have to first compute marginal costs at different levels of productivity. We do so in the below tablet.

 Volume Sold in Units Total Cost TC in \$ Marginal Cost in \$ (TCn – TCn-1) 0 10 - 10 18 8 15 26 8 20 34 8 25 45 9 30 57 11 35 68 11 40 80 12

To optimise profits the rivalled industry will equate price with the marginal cost. It will be seen from the tablet that at productivity level of 30 units, the provided price parities marginal cost \$11. Therefore, the firm’s optimising profit level of productivity is 30 units. Profits = TR – TC = P.V – TC = 30 * 11 – 57 = \$273.

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