Page 364 - DMTH201_Basic Mathematics-1

Unit 14: Business Applications of Maxima and Minima

Profits are maximised when x = 10. Notes
When a tax of 3 per unit is imposed, the total cost is written as
1 1
C x = x 3 6x 2 30x 20 3x x 3 6x 2 33x 20
t 3 3

1 3 2 1 3 2
Profit t x = 10x 3 x 6x 33x 20 3 x 6x 23x 20

Now t x = x 2 12x 23 0, for max. x 2 12x 23 0

12 144 92 12 7.21
x = . Thus, x = 9.6 or 2.4
2 2
Further it can be shown that t x 0 , when x = 9.6. Therefore, post-tax equilibrium occurs at
lower level of output.

14.4.3 Profit Maximisation by a Monopoly Firm

Let the firm faces an inverse demand function p = f(x). Then we can write the total revenue of the
firm as R(x) = p.x = x.f(x). Assuming the cost function as C(x), we can write the profit function as
p(x) = x.f(x) – C(x). As before, the profit maximising conditions are x = 0 and x < 0.

1
1. The equilibrium condition can be written as MR(x) = MC(x) or p 1 =
Notes
MC(x). Thus p > MC(x) when > 1 (note that a profit maximising monopolist
always operates on the elastic portion of the demand curve). Since p = MC(x)
for a perfectly competitive firm, this implies that price charged by a monopolist
will be higher for producing the same level of output.

2. Like a perfectly competitive firm, there is no supply curve of a monopoly
1
firm. To show this, we solve the equilibrium condition p 1 = MC(x) for x.
The solution for x will be a function of p and . This function can be regarded
as a supply function only if is constant. However, we know that often h is
different at different points of the demand curve.

1
Example: The demand and cost functions of a monopolist are given to be x 500 p
2
and C x 3 59x 2 1315x 2000 respectively. Find his profit maximising level of output and price.
Solution:
1
We can write the demand function as p 500 x or p 1000 2x
2
Therefore, the profit function of the monopolist is
x = 1000 2x x x 3 59x 2 1315x 2,000
= x 3 57x 2 315x 2000
We have, x = 3x 2 114x 315 0 or x 2 38x 105 0 for max. p
x 35 x 3 = 0 x = 35 or 3

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