The supply function of a commodity is \(P = x^3+2x+18\). When 4 units of commodity are sold, then producer surplus is:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 208
The producer surplus at \(x_0\) units is the revenue received minus the area under the supply curve:
\( \displaystyle PS = p_0 x_0 - \int_0^{x_0} S(x)\,dx \)
At \(x_0 = 4\):
\(p_0 = 4^3 + 2(4) + 18 = 64 + 8 + 18 = 90\)
\(p_0 x_0 = 90 \times 4 = 360\)
\( \displaystyle \int_0^4 (x^3 + 2x + 18)\,dx = \left[\frac{x^4}{4} + x^2 + 18x\right]_0^4 \)
\(= 64 + 16 + 72 = 152\)
\(PS = 360 - 152 = 208\)