Non-redistributive Second Welfare Theorems

Abstract

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.

This research was supported by Natural Sciences and Engineering Research Council of Canada (NSERC) grant no. 288334, 429598 and by Harold H Helm fellowship.