The demand function for a commodity is \( p=45+x-x^{2} \), then the consumer’s surplus at equilibrium price \( p_{0}=25 \) will be:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → \( \displaystyle\frac{425}{6} \)
The equilibrium quantity \( x_{0} \) comes from \( p_{0}=25 \):
\( \displaystyle 45+x-x^{2}=25 \)
\( \displaystyle x^{2}-x-20=0 \)
\( \displaystyle (x-5)(x+4)=0 \)
Quantity cannot be negative, so \( x_{0}=5 \).
Consumer’s surplus is the area under the demand curve minus the amount actually spent, \( p_{0}x_{0}=25\times 5=125 \):
\( \displaystyle CS=\int_{0}^{5}\left(45+x-x^{2}\right)dx-p_{0}x_{0} \)
\( \displaystyle =\left[45x+\frac{x^{2}}{2}-\frac{x^{3}}{3}\right]_{0}^{5}-125 \)
\( \displaystyle =225+\frac{25}{2}-\frac{125}{3}-125 \)
\( \displaystyle =\frac{1350+75-250-750}{6} \)
\( \displaystyle =\frac{425}{6} \)
Tip: Consumer’s surplus = \( \int_{0}^{x_0}D(x)\,dx-p_{0}x_{0} \). Forgetting to subtract \( p_{0}x_{0} \) is the most common error.
CBSE Applied Mathematics (Class XII):
Consumers' Surplus and Producers' Surplus
