Yet another Sheep

There is a very nice way of understanding the second welfare theorem that is related to the constuction explained in my last post. The argument is very easy, especially in an exchange economy. Take any Pareto efficient distribution and use it as the endowment. Now if a equilibrium exists from this endowment it is by the first welfare theorem Pareto efficient. Let´s look at this equilibrium. We know that nobody can be better off without someone being worse off since the distribution of the endowment was Pareto efficient. But since everyone can be as good off as before by keeping their endowment, we know that everyone is indifferent between their endowment and the equilibrium allocation. So everybody could simply keep their endowment- and we still have a equilibrium allocation. So we can support the efficient distribution by prices, which "proves" the second welfare theorem.

The problem is of course that we had to assume that a equilibrium exists, starting with a efficient distribution as the endowment. All the additional assumptions we need for the second welfare theorem in contrast to the first welfare theorem are only needed for establishing existence. If we assume existence, we don't need convexity, continuity or anything alike.

The complete argument, also treating the case of production, can be found in a nice paper by Maskin and Roberts: On the Fundamental Theorems of General Equilibrium.