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Question
The elasticity of demand with respect to price for a commodity is given by `((4 - x))/x`, where p is the price when demand is x. Find the demand function when the price is 4 and the demand is 2. Also, find the revenue function
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Solution
The elasticity at the demand
ηd = `((4 - x))/x`
`- "p"/x ("d"x)/"dp" = ((4 - x)/x)`
`"p"/x ("d"x)/"dp" = ((x - 4)/x)`
`1/x[x/(x -4)] "d"x = 1/"p" "dp"`
Integrating on both sides
`int 1/((x - 4)) = int 1/"p" "dp"`
log |x – 4| = log |p| + log k
log |x – 4| = log |pk|
⇒ (x – 4) = pk ........(1)
When p = 4 and x = 2
(2 – 4) = 4k
⇒ – 2 = 4k
k = `(-1)/2`
Equation (1)
⇒ (x – 4) = `"p"((-1)/2)`
– 2(x – 4) = p
⇒ p = 8 – 2x
Revenue function R = px = (8 – 2x)x
R = 8x – 2x2
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