Pareto optima, non-convexities and regulated market equilibria

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In a differentiable setting without convexity assumptions, a typical local Pareto optimum can be sustained by a combination of a price system and a system of market regulations. Some consumers are required to consume fixed amounts of some basic commodity bundles, and some producers are required to produce fixed amounts of other basic commodity bundles. The agents are otherwise free to maximize utility and profit (locally). No more than l–1 agents are subject to regulation, where l is the number of commodities.