Second welfare theorem

Question

Could you provide the intuition behind the second welfare theorem, and prove in a few rows the theorem itself?

Answer 1

Here is a short argument due to Maskin and Roberts. I'll give it in the simpler context of a pure exchange economy.

Suppose you have a Pareto efficient allocation, and an equilibrium $(x,p)$ exists starting with this allocation as the endowment specification $e$. Then taking these equilibrium prices and everyone just consuming their endowment is an equilibrium; $(e,p)$ is an equilibrium. Indeed, since an agent's endowment is always in the budget set, and everyone chooses optimally, we must have $x_i\succeq_i e_i$ for every agent $i$. But we cannot have $x_i\succ_i e_i$ for some $i$, because then $x$ would be a Pareto improvement over $e$, which is impossible by assumption. So $x_i\sim_i e_i$ for every $i$, and just consuming the endowment is optimal given $p$. Since $e$ is also clearly feasible, we are done.

The argument is so simple because I have just assumed that the equilibrium $(x,p)$ exists. The standard proofs of the second welfare theorem include an existence proof, and that is also why the usual additional assumptions are needed.