# Pricing And Productivity Under Monopoly Rivalry

__Illustration 74__

Let us exemplify the monopoly symmetry with a numerical example. Presume for a monopolistic the following demand and total cost function are provided. Determine how much he will manufacture and what price he will charge

V = 500 – 2R

TC = 8V + 0.5V^2

__Solution__

In order to determine the profit optimising solution we have to derive the marginal revenue and marginal cost from the demand and cost equations given below. In order to find out the marginal revenue we have to first obtain the total revenue function.

V = 500 – 2R

2R = 500 – V

R = __500 – V__

2

R = 250 – 0.5V

Total revenue TR = R.V = 250V – 0.5V^2 ….Equation (1)

Differentiating it with respect to productivity V we can get MR. Therefore,

MR = __ΔTR__ = __Δ R.V__ = 250 – V …..Equation
(2)

ΔV ΔV

Marginal cost can be procured by differentiating the total function TC = 18V + 0.02V^2

Therefore, we have

MC = __ΔTC__ = 8
+ V

ΔV

As the profits of the monopolist will be optimised when he equates marginal revenue with marginal cost, setting MR = MC we have

MR = MC

250 - V = 8 + V

2V = 242

V = 242 / 2

= 121

To determine price we substitute the value of productivity V in the demand function (1) above we have

R = 250 – 1*121 = 129

To procure total profits we compute total revenue and total cost

TR = R.V = 121 * 129

15609

TC = 8V + 0.5V^2 = 8*121 + 0.5(121)^2

= 8289

Profit = TR – TC = 15609 – 8289

= 7320

**Therefore, productivity V = 121, Price or Rate R = 129 and Profits = $7,320.**

__Illustration 75__

A monopolist industry manufacturing and supplying cooking fuel to a district faces the following demand function.

V = 200 – 40R

The industry has the following cost function.

TC = 2V
+ 0.025V^2

Ascertain the volume of the cooking fuel he will manufacture and rate he will charge
to optimise profits. What will be amount made by him?

__Solution__

It is to be noted that cost function does not posses fixed cost module as all terms in it comprise the element of productivity V. thus, this represents the long run cost function as in the long run all aspects are variable and there are no fixed costs.

It must also be carefully noted that for the difficulty of profit optimisation for procuring total revenue function, from which marginal revenue can be ascertained is necessary to be denoted in terms of “Price as a function of Volume”. Therefore,

V = 200 – 40R

40R = 200 – V

R = __200 – V__

40

= 5 – 0.025V …..Equation (1)

Incrementing both sides by V

VR = TR = 5V – 0.025V^2

MR = __dTR__ = 5 – 0.05V …..Equation
(2)

dV

The provided Cost Function is

TC = 2V + 0.025V^2

MC = __dTC__ = 2
+ 0.05V …..Equation (3)

dV

For optimising profit

MR = MC

5 – 0.05V = 10 + 0.05V

3 = 1V

Profit optimising productivity V* = 3

To work out for R, we substitute t4he value of V* in the demand function equation (1) above.

R = 2 + 0.05 (3)

= 2.15

Therefore, profit optimisation price is $2.15

Profits π = TR – TC

TR = V*R = 3 * 2.15 = 6.45

TC = 2V + 0.025V^2 = 2*3 + 0.025 (3)^2

= 6.225

**Profits π = 6.45 – 6.225 = 0.225**

__Illustration 76__

A monopolist has the following total cost function.

TC = 20 + 10V

- If the price elasticity of demand for his product is -4, determine what price he will set for his commodity.

- If the price elasticity of demand for his commodity varies to -8, how will he amend his price?

__Solution__

TC = 20 + 10V

MC = __ΔTC__ = 10 …..
(1)

ΔV

The association among MR, price and price elasticity of demand is as follows:

MR = R (1 + 1/e)

Provided that price elasticity of demand is -4

MR = R (1 + 1/-4) = R (1 – ¼)

= ¾ R ….. (2)

At symmetry, MR = MC

¾ R = 10

R = 10 * 4/3 = $13.33

Provided that e = -8

MR = R (1 + 1/-8) = R (1 - 1/8)

= 7/8 R

At symmetry,

MR = MC

7/8 R = 10

R = 10 * 8/7 = 11.43

Therefore, with the enhancement in absolute value of price elasticity of demand, monopolist will reduce the price of his commodity.

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