The supply function for a commodity is \( p=3\sqrt{16+x} \). If 9 units of goods are sold, then producer\'s surplus is :
Answer & explanation
Correct answer: option 1
The correct answer is Option (1) → 13
At \( x_0=9 \) the price is
\( \displaystyle p_0=3\sqrt{16+9}=3\times 5=15 \)
Producer’s surplus is the revenue \( p_0x_0 \) minus the area under the supply curve from 0 to \( x_0 \):
\( \displaystyle PS=p_0x_0-\int_{0}^{9}3\sqrt{16+x}\,dx \)
\( \displaystyle \int_{0}^{9}3(16+x)^{1/2}\,dx \)
\( \displaystyle =3\cdot\frac{2}{3}\Big[(16+x)^{3/2}\Big]_{0}^{9} \)
\( \displaystyle =2\left(25^{3/2}-16^{3/2}\right) \)
\( \displaystyle =2(125-64)=122 \)
\( \displaystyle PS=15\times 9-122=135-122=13 \)
Tip: Producer’s surplus uses the area under the supply curve (PS = p0x0 − area), while consumer’s surplus uses the area under the demand curve (CS = area − p0x0).