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Question
If the supply function for a product is p = 3x + 5x2. Find the producer’s surplus when x = 4
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Solution
The supply function p = 3x + 5x²
When x = 4
⇒ p = 3(4) + 5(4)²
p = 12 + 5(16)
= 12 + 80
p = 92
∴ x0 = 4 and p0 = 92
Producer’s Surplus
P.S = `x_0"p"_0 - int_0^(x_0) "g"(x) "d"x`
= `(4)(92) - int_0^4 (3x + 5x^2) "d"x`
= `368 - [(3x^2)/2 + (5x^3)/3]_0^4`
= `368 - {(3/2 (4)^2 + 5/3 (4)^3) - [0]}`
= `368 - {3/2 (16) + 5/3 (64)}`
= `368 - [24 + 320/3]`
= `368 - 24 - 320/3`
= `344 - 320/3`
= `(1032 - 320)/3`
= `712/3`
= `237.3`
∴PS = 237.3 units
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