Core and Equilibrium

Debreu and Scarf analyze the core of a large economy, inspired by Edgeworth (Debreu, G., & Scarf, H. (1963, September). A limit theorem on the core of an economy. International Economic Review, 4 (3), 235–246; Edgeworth, F. Y. (2003 (originally 1881)). Mathematical psychics: An essay on application of mathematics to the moral sciences. In P. Newman (Ed.), F. Y. Edgeworth: Mathematical Psychics and further papers on political economy (pp. 1–174). Oxford: Oxford University Press). Consider an exchange economy in which each consumer has a utility function and an endowment of goods. An allocation of goods across the consumers is in the core if no coalition of consumers can trade their endowments among themselves so at least one member of the coalition gains and no member loses. For an economy with many consumers, the intuitive idea is that an allocation of goods lies in the core if and only if it is attainable as a general market equilibrium. For an allocation in the core, there exists a price vector such that it is a general market equilibrium. Conversely, for a general market equilibrium, the allocation is in the core. We develop their analysis via primal/dual convex optimization modelling whether an allocation is in the core of a large economy. The primal is a maximization choosing simultaneously across both allocations and coalitions, to see whether any coalition can do better. For a core allocation, no coalition can do better and the optimum value is zero. If the allocation is not in the core, then some coalition can do better, and the optimum value is positive. The dual is a general-equilibrium minimization, looking for an equilibrium price vector. For a core allocation, the dual solution solves for a price vector for which the allocation is a general equilibrium. Conversely, if the allocation is not in the core, then the dual has no solution.