The supply function for a commodity is \( p=x^2+x+3 \). The producer’s surplus when \( x=3 \) is.
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → \( \displaystyle\frac{45}{2} \)
Producer’s surplus is the revenue actually received minus the least revenue producers would accept:
\( \displaystyle \text{PS}=x_0p_0-\int_{0}^{x_0}p\,dx \)
At \( x_0=3 \):
\( \displaystyle p_0=9+3+3=15 \)
\( \displaystyle x_0p_0=3\times 15=45 \)
\( \displaystyle \int_{0}^{3}(x^2+x+3)\,dx=\left[\frac{x^3}{3}+\frac{x^2}{2}+3x\right]_{0}^{3} \)
\( \displaystyle =9+\frac{9}{2}+9 \)
\( \displaystyle =\frac{45}{2} \)
\( \displaystyle \text{PS}=45-\frac{45}{2}=\frac{45}{2} \)
Why the other options are incorrect
Option (2) \( \displaystyle\frac{32}{3} \) — Incorrect. The surplus is \( 45-\frac{45}{2}=\frac{45}{2} \), not \( \frac{32}{3} \).
Option (3) 45 — Incorrect. 45 is only the total revenue \( x_0p_0 \); the area under the supply curve must be subtracted.
Option (4) \( \displaystyle\frac{62}{3} \) — Incorrect. The surplus is \( 45-\frac{45}{2}=\frac{45}{2} \), not \( \frac{62}{3} \).
Tip: Producer’s surplus: rectangle minus area under the supply curve. Consumer’s surplus: area under the demand curve minus rectangle.