The demand function \(P\) for maximising a profit monopolist is given by \(P=274-x^2\), while the marginal cost is \(4+3x\) for \(x\) units of commodity. The consumer surplus is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → 486
A monopolist maximises profit where marginal revenue = marginal cost.
\(R = Px = 274x - x^3\)
\( \displaystyle MR = \frac{dR}{dx} = 274 - 3x^2 \)
Setting MR = MC:
\(274 - 3x^2 = 4 + 3x\)
\(x^2 + x - 90 = 0\)
\((x + 10)(x - 9) = 0\)
\(x_0 = 9, \quad P_0 = 274 - 81 = 193\)
The consumer surplus is the area under the demand curve minus the amount actually spent, \(P_0x_0\):
\( \displaystyle CS = \int_{0}^{9} (274 - x^2)\,dx - P_0x_0 \)
\( \displaystyle \int_{0}^{9} (274 - x^2)\,dx = \left[274x - \frac{x^3}{3}\right]_{0}^{9} \)
\(= 2466 - 243 = 2223\)
\(P_0x_0 = 193 \times 9 = 1737\)
\(CS = 2223 - 1737 = 486\)
Why the other options are incorrect
Option (1) 274 — Incorrect. This is the price at x = 0, where the demand curve meets the price axis.
Option (2) 1737 — Incorrect. This is the amount spent, \(P_0x_0\), which is subtracted, not the surplus.
Option (3) 193 — Incorrect. This is the equilibrium price \(P_0\).