The Welfare Theorems

The economist's notion of social desirability is the Pareto order.

A consumption plan $x$ is Pareto-better than consumption plan $x'$, written $x\succ_P x'$, iff for all $n$, $x_n\succeq_nx'_n$, and for some consumer $k$, $x_k\succ x'_k$. An allocation $z=(x,y)$ is Pareto optimal iff it is feasible, and if for no other feasible consumption plan $z'=(x',y')$ is it true that $x'\succ_P x$.

How do we know an optimum exists? In exchange economies this is not hard. The set of feasible allocations is obviously compact, so suitable continuity assumptions on preferences should do the trick. When production is possible, compactness of the set of feasible allocations is not so obvious. Debreu (Theory of Value, Ch. 6.2.) gives us an answer.