Non-redistributive second welfare theorems
Résumé
The second welfare theorem tells us that social welfare in an economy can be
maximized at an equilibrium given a suitable redistribution of the endowments. We examine
welfare maximization without redistribution. Specifically, we examine whether the clustering
of traders into k submarkets can improve welfare in a linear exchange economy. Such an
economy always has a market clearing ε-approximate equilibrium. As ε→ 0, the limit of these
approximate equilibria need not be an equilibrium but we show, using a more general price mechanism than the
reals, that it is a “generalized equilibrium”. Exploiting this fact, we give a polynomial
time algorithm that clusters the market to produce ε-approximate equilibria
in these markets of near optimal social welfare, provided the number of goods and
markets are constants. On the other hand, we show that it is NP-hard to find an
optimal clustering in a linear exchange economy with a bounded number of goods
and markets. The restriction to a bounded number of goods is necessary to obtain
any reasonable approximation guarantee; with an unbounded number of goods, the
problem is as hard as approximating the maximum independent set problem, even
for the case of just two markets.