If under pure competition demand and supply functions are given by \(p = \sqrt{10-x}\) and \(p =\displaystyle\frac{1}{2} (x-2)\) respectively, where p is price per unit and x is quantity, then the consumer surplus is:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \(\displaystyle\frac{2}{3} [10\sqrt{10}-26]\)
Under pure competition the market settles at the equilibrium, where demand price = supply price:
\( \displaystyle \sqrt{10 - x} = \frac{x - 2}{2} \)
Squaring:
\(4(10 - x) = (x - 2)^2\)
\(40 - 4x = x^2 - 4x + 4\)
\(x^2 = 36\)
The quantity cannot be negative, so \(x_0 = 6\), and \(p_0 = \frac{6 - 2}{2} = 2\) (check: \(\sqrt{10 - 6} = 2\)).
The consumer surplus is the area under the demand curve up to \(x_0\), minus the amount actually spent, \(p_0x_0\):
\( \displaystyle CS = \int_{0}^{6} \sqrt{10 - x}\,dx - 2 \times 6 \)
\( \displaystyle \int_{0}^{6} \sqrt{10 - x}\,dx = \left[-\frac{2}{3}(10 - x)^{3/2}\right]_{0}^{6} \)
\( \displaystyle = -\frac{2}{3}(4)^{3/2} + \frac{2}{3}(10)^{3/2} \)
\( \displaystyle = \frac{2}{3}\left(10\sqrt{10} - 8\right) \)
Writing \(12 = \frac{2}{3} \times 18\):
\( \displaystyle CS = \frac{2}{3}\left(10\sqrt{10} - 8\right) - \frac{2}{3}(18) \)
\( \displaystyle = \frac{2}{3}\left[10\sqrt{10} - 26\right] \)