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Question
The supply function of certain goods is given by x = a`sqrt("p" - "b")` where p is unit price, a and b are constants with p > b. Find elasticity of supply at p = 2b.
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Solution
Given that x = a`sqrt("p" - "b")`
Elasticity of supply: ηs = `"p"/x * "dx"/"dp"`
x = a`sqrt("p" - "b")`
`"dx"/"dp" = "a"(1/(2sqrt "p - b"))`
ηs = `"p"/x * "dx"/"dp"`
`= "p"/("a"sqrt("p" - "b")) xx "a" xx 1/(2sqrt("p" - "b"))`
`= "p"/(2("p - b"))`
Hint for differentiation
Use y = `sqrtx`
`"dy"/"dx" = 1/(2sqrtx)`
(or) x = `"a"sqrt("p - b")`
x = `"a"("p - b")^(1/2)`
`"dx"/"dp" = "a" * 1/2 ("p - b")^(1/2 - 1)`
`= "a"/2 ("p - b")^(- 1/2)`
`= "a"/2 1/(sqrt ("p - b"))`
When p = 2b, Elasticity of supply: ηs = `"2b"/(2("2b" - "b")) = "2b"/"2b" = 1`
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