Past exam of the mathematics course of the University of Cambridge / 2026 / ii / Paper 1 / 30L / d / Solution

The marginal utility price isThe value function is concave; its tangent at zero lies above its graph. Evaluating that supporting inequality at the utility-indifference displacement $Y-(1+r)\pi (Y)$ gives
$E[U^{\prime }(X_0)(Y-(1+r)\pi (Y))]\ge 0$, hence ${\pi }_0(Y)\ge \pi (Y)$.