Page 328 - DMTH201_Basic Mathematics-1

Unit 12: Successive Differentiation

Notes
Example
Find the elasticity of supply, , for the following functions:
s
2
(i) x = 2p + p at (a) p = 5 and (b) p = 7.
2
(ii) p = x at x = 5.
Solution:
dx
(i) (a) The supply function is x = 2p + p 2 = 2 + 2p = 12 when p = 5
dp

dx p 12 5
When p = 5, we have x = 10 + 25 = 35 = 1.71.
s
dp x 35
dx
(b) When p = 7, we have = 2 + 14 = 16 and x = 14 + 49 = 63
dp
16 7 16
= 1.77
s 63 9
(ii) The supply function is p = x 2

dp 1 25
2x 10 and p = 25 at x = 5 Thus s 0.5
dx 10 5

Example: Show that for the inverse supply function p = a + bx(b > 0), the supply is elastic
if a > 0, inelastic if a < 0 and unitary elastic if a = 0.

Solution:
dp dx p 1 a bx a
Given p = a + bx, we get b s 1.
dx dp x b x bx
a a
Since, b > 0 and x (the quantity) > 0, will be greater than 1 if 1 1 or 0 a 0.
s
bx bx
a a
Similarly, will be less than 1 if 1 1 or 0 a 0.
s
bx bx
a a
Further, = 1, if 1 1, or 0 a 0.
s
bx bx
Example

The supply of a certain good is given by x a p , b where p (> b) is the price and a and b are
positive constants. Find an expression for , the elasticity of supply, as a function of price. Show
s
that decreases as price (or supply) increases and becomes unity when p = 2b.
s
Solution:
dx a 1 a
Given x a p , b we get p b 2
dp 2 2 p b

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