Past exam of the mathematics course of the University of Cambridge / 2004 / iii / Paper 36 / 2 / a / Solution

Use the exact-resource-balance convention for an exchange economy. If $\omega$ is the integrable initial-endowment profile, a feasible consumption allocation is a measurable integrable function $f:A\to {\mathbb{R}}_{+}^L$ withIt is Pareto efficient if there is no other feasible consumption allocation $g$ satisfying $g(a){⪰}_{a}f(a)$ for almost every consumer and $g(a){\succ }_{a}f(a)$ on a set of positive measure. A Pareto improvement makes nobody worse off, up to a null set, and makes a positive-measure group strictly better off. Positive measure is essential: reallocating to a single consumer of zero mass does not constitute an aggregate Pareto improvement.