Question

Q no 4
A monopolist firm has the total cost function and the demand function :
C(x)= ax + bx +c (a,b,c) >0
P(x) = 6 - ax (B, a>0 )
Show that the firm's optimum level are equal when it fixes
a) Output
b) Price
Explain how a monopolist fixes a price on the basis of demand of the product​

Answer 1

The correct question is : A monopolist firm has the total cost function and the demand function :

C(x ) = ax2 + bx +c (a,b,c) >0

P(x) = β - αx (β, α >0 )

Show that the firm’s optimum level are equal when it fixes

a) Output

b)Price

Explain how a monopolist fixes a price on the basis of demand of the product.

Given :

Cost function C(x ) = ax2 + bx +c       (a,b,c) >0

Demand function P(x) = β - αx           (β, α >0 )

Solution :

Since costs are a function of quantity, the formula for profit maximization we be : π = P(x).x - c(x)

where p(x) = price level of quantity x

c(x) = cost to the firm at quantity x

A monopolist firms optimum level means :

Marginal Revenue = Marginal Cost

Revenue Function = P(x).x

R = (β - ax).x = βx - ax²

Now Marginal revenue function (MR) is :

$MR = \frac{dR}{dx}$

$MR = \frac{d(\beta x -ax^{2} )}{dx} \\MR = \beta - 2ax$

Now Marginal cost function means :

$MC = \frac{d(c(x))}{x} \\MC = \frac{d(ax^{2} +bx +c)}{dx}$

MC = 2ax + b

Marginal Revenue = Marginal Cost

P'(π) = (βx - ax²) - (ax² +bx + c)

P'(π) = β - 2ax - (2ax + b)

P'(π) = 0     (for maximizing profit)

β - 2ax - (2ax + b) = 0

i.e, 2ax + b = - 2ax + β

2ax + b = |-(2ax - β)|        

2ax + b = 2ax - β

Both b , β are constant

∴ Firms optimum level is equal at defined Price and Output

and Monopolist firms fixed price by maximize monopoly profit

i.e, 2ax - b = -2ax + β

4ax = β - b

x = (β - b)/4a

P'(π) = β -2ax - (2ax + b)

P'(π) = -4ax + β -b

To confirm, Profit Maximization , 2nd derivative of P(π) must be -ve

i.e, P'(π) = -4ax - β - b

P''(π) = d(-4ax + β - b)/dx

P''(π) = -4a

∴Monopolist firm will get maximum profit at value x [a = (β - b) / 4a]