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Question
The demand and supply functions under perfect competition are pd = 1600 – x2 and ps = 2x2 + 400 respectively. Find the producer’s surplus
Solution
pd = 1600 – x2 and ps = 2x2 + 400
Under the perfect competition pd = ps
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
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