Page 351 - DMTH201_Basic Mathematics-1

Basic Mathematics – I

Notes d TR dp p dx
condition implies that = p x 0 or 1 i.e. h = 1. Thus maxima of total revenue
dx dx x dp
occurs at a level of output where elasticity of demand is unity.

Example: The inverse demand function facing a monopolist is p x ( , > 0). Find
the price charged and quantity sold for maximum monopoly revenue. Show that the elasticity at
this point is unity.
Solution:

TR = p.x = x .x x x 2
d TR
= 2 x 0, for maxima, x .
dx 2

The monopoly price p = .
2 2
Second order condition
2
d TR
dx 2 = –2 < 0.
2
2
d TR d TR
Note that 2 = 2 . Thus the second order condition implies that the marginal revenue
dx dx
should be falling at the point , .
2 2
dp
For elasticity of demand at the point , on the demand function p x , we have =
2 2 dx
dx 1
– or dp .

dx p 1 2
= dp x 2 1. Hence elasticity of demand is unity.

Example: A wholesaler of pencils charges 24 per dozen on orders of 50 dozens or less.
For orders in excess of 50 dozens, the price is reduced by 20 paise per dozen in excess of 50
dozens. Find the size of the order that maximises his total revenue.

Solution:
Let x be the number of dozens in an order.
When x 50, TR = 24x

When x > 50, the price charged per dozen is given by
p = 24 0.20 x 50 34 0.20x
This is the equaton of a straigh line passing through the point (50, 24) with slope =
– 0.20.
Thus, TR = p.x = (34 – 0.2x).x = 34x – 0.2x 2
We note here that TR will have maxima only when x > 50 because, when x £ 50, TR is a straight
line and hence has no maxima.

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