Welfare Economics of Incomplete Markets

When markets are complete, Arrow's Theorem implies that equilibrium is optimal. Not so, however, when markets are incomplete.

Here is the direct derivation of the optimality of interior equilibria for complete markets from the equilibrium FOC. \begin{align*} D_{sl}U^i(x^i)-\lambda^i_sp_{sl}&=0,\\ -\lambda^i_0q_s+\lambda^i_s&=0,\\ \text{etc.}& \end{align*}