Question

The demand and supply functions under perfect competition are p_d = 1600 – x² and p_s = 2x² + 400 respectively. Find the producer’s surplus

Answer 1

pd = 1600 – x² and p_s = 2x² + 400

Under the perfect competition p_d = p_s

1600 – x² = 2x² + 400

1600 – 400 = 2x² + x²

⇒ 1200 = 3x²

⇒ x² – 400

⇒ x = 20 or – 20

The value of x cannot be negative, x = 20 when x₀ = 20;

p₀ = 1600 – (20)²

= 1600 – 400

P₀ = 1200

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

= `(20)(1200) - int_0^20 (2x^2 + 400)  "d"x`

= `24000 - [2(x^3/3) + 400x]_0^20`

= `24000 - {[2/3 (20)^2 + 400(20)] - [0]}`

= `24000 - [2/3 (8000) + 8000]`

= `24000 - 16000/3 - 8000`

= `16000 - 16000/3`

= `1/3 [48000 - 16000]`

∴ P.S = `1/3 [32000]`units