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Question
The demand function for a commodity is p = e–x .Find the consumer’s surplus when p = 0.5
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Solution
The demand function p = e–x
When p = 0.5
⇒ 0.5 = e–x
`1/2 = 1/"e"^x`
⇒ ex = 2
∴ x = log 2
∴ Consumer’s surplus
C.S = `int_0^x` (demand function) dx – (Price × quantity demanded)
= `int_0^log2 "e"^-x "d"x - (0.5) log 2`
= `(("e"^-x)/(-1))_0^log2 - 1/2 log 2`
= `((-1)/"e"^x)_0^log2 - 1/2 log 2`
= `((-1)/"e"^log2) - ((-1)/"e"^0) - 1/2 log 2`
= `(-1)/2 + ((-1)/"e"^0) - 1/2 log 2`
= `(-1)/2 + 1 - 1/2 log 2`
= `1/2 - 1/2 log 2`
C.S = `1/2 [1 - log 2]` units
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