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

Answer 1

The elasticity at the demand

η_d = `((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