Question

The demand equation for a products is x = `sqrt(100 - "p")` and the supply equation is x = `"P"/2 - 10`. Determine the consumer’s surplus and producer’s surplus, under market equilibrium

Answer 1

p_d = `sqrt(100 - "p")` and p_s = `sqrt(100 - "p")`

Under market equilibrium,

p_d = p_s 

`sqrt(100 - "p") = "p"/2 - 10`

Squaring on both sides

`(100 - "p") = ("p"/2 - 10)^2`

`100 - "p" = ("p"/2)^2 - 2("p"/2) (10) + (10)^2`

`100 - "p" = "p"^2/2 - 10"p" + 100`

`100 - "p" + 10"p" - 100 = "p"^2/4`

⇒ 9p = `"p"^2/4`

36p = p²

⇒ p² – 36

p = 0

p(p – 36) = 0

⇒ p = 0 or p = 36

The value of p cannot be zero,

∴ p₀ = 36 when p₀ = 36

x₀ = `sqrt(100 - 36)`

= `sqrt(64)`

∴ x₀ = 8

C.S = `int_0^(x_0) "f"(x)  "d"x - x_0"p"_0`

= `int_0^8 (100 - x^2)  "d"x - (8)(36)`

= `[100x - x^3/3]_0^8 - 288`

= `{100(8) - (8)^3/3 - [0]} - 288`

= `800 - 512/3 - 288`

= `512 - 512/3`

= `512 [(3 - 1)/3]`

= `512(2/3)`

= `1024/3`

∴ C.S = `1024/3` units

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

= `(8)(36) - int_0^8 (2x + 20)  "d"x`

= `288 - [(2x^2)/2 + 20x]_0^8`

= `288 - [x^2 + 20x]_0^8`

= 288 – {[(8)² + 20(8)] – [0]}

= 288 – [64 + 160]

= 288 – 224 = 64

PS = 64 Units