The supply function of a producer is given by \( p=\displaystyle\frac{3}{5}e^{3x} \), where \( x \) denotes a thousand units. If sales are 4000 units, then the producers' surplus is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( \displaystyle\frac{11}{5}e^{12}+\displaystyle\frac{1}{5} \)
Since x is in thousands of units, sales of 4000 units mean \( x_{0}=4 \), and the price is
\( \displaystyle p_{0}=\frac{3}{5}e^{12} \)
The producers' surplus is the revenue \( x_{0}p_{0} \) minus the area under the supply curve:
\( \displaystyle \text{PS}=x_{0}p_{0}-\int_{0}^{4}\frac{3}{5}e^{3x}\,dx \)
\( \displaystyle =4\times\frac{3}{5}e^{12}-\frac{1}{5}\left[e^{3x}\right]_{0}^{4} \)
\( \displaystyle =\frac{12}{5}e^{12}-\frac{1}{5}\left(e^{12}-1\right) \)
\( \displaystyle =\frac{11}{5}e^{12}+\frac{1}{5} \)