Non-redistributive Second Welfare Theorems
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.
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