Non-redistributive second welfare theorems

International audienceThe 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 thereals, that it is a “generalized equilibrium”. Exploiting this fact, we give a polynomialtime algorithm that clusters the market to produce ε-approximate equilibriain these markets of near optimal social welfare, provided the number of goods andmarkets are constants. On the other hand, we show that it is NP-hard to find anoptimal clustering in a linear exchange economy with a bounded number of goodsand markets. The restriction to a bounded number of goods is necessary to obtainany reasonable approximation guarantee; with an unbounded number of goods, theproblem is as hard as approximating the maximum independent set problem, evenfor the case of just two markets