Question

Under perfect competition for a commodity the demand and supply laws are P_d = `8/(x + 1) - 2` and P_s = `(x + 3)/2` respectively. Find the consumer’s and producer’s surplus

Answer 1

P_d = `8/(x + 1) - 2` and P_s = `(x + 3)/2`

Under Perfect Competition,

P_d = P_s 

`8/(x + 1) - 2 = (x + 3)/2`

`8/((x + 1)) - ((x + 3))/2` = 2

`(8(2) - (x + 3)(x + 1))/(2(x + 1))` = 2

16 – (x² + 3x + x + 3) = 2[2(x + 1)]

16 – (x² + 4x + 3) = 4(x + 1)

16 – x² – 4x – 3 = 4x + 4

x² + 4x + 4x + 4 + 3 – 16 = 0

x² + 8x – 9 = 0

(x + 9)(x – 1) = 0

⇒ x = – 9 or x = 1

The value of x cannot be negative x = 1 when x₀ = 1

p₀ = `8/(1 + 1) - 2`

⇒ p₀ = `8/2 - 2`

p₀ = 4 – 2

⇒ p₀ = 2

C.S = `int_0^x` f(x) dx – x₀p₀

= `int_0^1 (8/(x + 1) - 2) "d"x - (1)(2)`

= `{8{[log(x + 1)] - 2x} int_0^1 - 2`

= 8 {[log (1 + 1) – 2(1)] – 8 [log (0 + 1) – 2(0)]} – 2

= [8 log (2) – 2 – 8 log1] – 2

= `8 log(8/2) - 2 - 2`

C.S = (8 log 2 – 4) units

P.S = `x_0"p"_0 - int_0^(x_0) "g"(x)  "d"x`

= `(1)(2) - int_0^1 ((x + 3)/2)  "d"x^(1/2)`

= `2 - 1/2 [(x + 3)^2/2]_0^1`

= `2 - 1/4 {[x + 3]^2}_0^1`

= `2 - 1/4 [(1 + 3)^2 - (0 + 3)^2]`

= `2 - 14 [4^2 - 3^2]`

⇒ `2 - 1/4 [16 - 9]`

= `(8 - 7)/4`

= `1/4`

∴ P.S = `1/4` units