Page 360 - DMTH201_Basic Mathematics-1

Unit 14: Business Applications of Maxima and Minima

In order that MC curve is U-shaped, the MC function should represent a parabola with axis Notes
pointing vertically upward. Further, in order that total cost function makes economic sense, the
vertex of the parabola must lie in positive quadrant.

d MC b
For minima of MC, we have 6ax 2b 0 x
dx 3a
d MC
2
a
Further, 2 6 , which should be positive for minima.
dx
This implies that a > 0. Also, since x, the output level, should be positive, therefore
b < 0.
b 2 b 3ac b 2
Now min. MC = 3a 2b c
3a 3a 3a
2
This will be positive only if b < 3ac. Since a > 0, this condition also implies that c > 0. Further, the
constant term d, which represents the total fixed cost, is always positive.

14.4 Economic Applications (Continued)

14.4.1 Maximisation of Profits

Profit is the difference between total revenue and total cost of a producer or firm. We know that
total revenue as well as total costs are often expressed as functions of level of output, x. If we
write TR = R(x) and TC = C(x), then the profit p can be written as p(x) = R(x) – C(x).

We want to find that value of x so that p(x) becomes maximum. The conditions for maxima of
p(x) are:
First order condition
x = R x C x = 0, or R x C x or MR(x) = MC(x)

Let x satisfy this equation. Then, we can write R x C x
e e e
Here x is termed as the profit maximising or equilibrium output. Note that the first order condition
e
is also termed as the equilibrium condition.
Second order condition
In order that profit (x) is maximum at x , we should have x < 0.
e e
This condition implies that R x e C x or R x e C x , i.e. the slope of marginal revenue
e
e
curve must be less than slope of the marginal cost curve at equilibrium point.
Alternatively, we can express total revenue and total cost as functions of price, where price and
quantity are related by the demand function x = (p). Thus, we can also express profit of the firm
as a function of price. The first and second order condition maximum profits, in this case, can be
written as p e 0 and p < 0 respectively.
e
14.4.2 Profit Maximisation by a Firm under Perfect Competition

A firm under perfect competition is a price taker i.e. price is constant. Therefore, the only option
before it is to choose that level of output at which its profits are maximised.

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