Page 326 - DMTH201_Basic Mathematics-1

Unit 12: Successive Differentiation

dp 1 p Notes
(iii) We are given p = a – bx . b Thus d
dx b x
Since elasticity of demand depends upon p (or x) and thus, will be different at different
points on the demand curve.
1 p p
When = 1, we can write 1 or b
d
b x x
p b 1 b
or b  , from demand equation
a p x a p
a
p = a – p or 2p = a or p =
2
a
Thus, the elasticity of demand is unity when p = .
2
(iv) Let x f(x) = c where c is a constant. Differentiating both sides w.r.t. x, we have f(x) + xf (x) =
f ( ) x
0 or 1. We note that expression on the left hand side is elasticity of demand of
x
xf ( )
the function p = f(x). Thus = 1. Since is independent of x (or p), hence elasticity of
d d
demand is unity at every point on the demand curve p = f(x).
We note that x f(x) is the total outlay (or expenditure) of the consumer. Thus when total
outlay of the consumer is constant the demand is unitary elastic at every point. It can also be
shown that p = f(x), in this case, will represent a rectangular hyperbola with centre at (0, 0)
and asymptotes as the axes of the coordinate system.
dx p
(v) The elasticity of demand, d , can also be written as
dp x
dx /dp marginal demand function
=
d
x /p average demand function
Hence the result.

Example

(i) The price elasticity of demand of a commodity when price = Rs 10 and quantity demanded
= 25 units, is given to be 1.5. Find the demand equation of the commodity on the assumption
that it is linear.
2
(ii) Find the elasticity of demand of the inverse demand function p = 3x – 100x + 800 when x
= 10. Find, approximately, the percentage change in demand if price rises by 4%. Also find
the elasticity at new price, quantity combination.
Solution.

dx p dx 10
(i) Given = 1.5, 1.5 or 1.5
d
dp x dp 25
dx 1.5 25
Thus, 3.75
dp 10
The required demand equation will be a straight line passing through the point (25, 10)
1
with slope =
3.75

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