Constrained Consumptions, Lipschitzian Demands, and Regular Economies

We consider an exchange economy where the consumers face linear inequality constraints on consumption. We parametrize the economy with the initial endowments and constraints. We exhibit sufficient conditions on the constraints implying that the demand is locally Lipschitzian and continuously differentiable on an open dense subset of full Lebesgue measure. Using this property, we show that the equilibrium manifold is lipeomorphic to an open, connected subset of an Euclidean space and that the lipeomorphism is almost everywhere continuously differentiable. We prove that regular economies are generic and that they have a finite odd number of equilibrium prices and local differentiable selections of the equilibrium prices.