## Existence of Marginal Cost Pricing Equilibria with Public GoodsWe examine the existence of marginal cost pricing equilibria under non-convex technology. We formalize marginal cost prices in terms of Clarke's normal cone. First, we are concerned with the formalization of marginal cost prices and the existence problem. Khan (1987) introduced Ioffe's normal cone as a formalization of marginal cost prices because, without free disposal, Clarke's normal cone may be too "big." Although it may not be convex, Khan showed that its lack of convexity is of no consequence for the second welfare theorem, but left open the question as to whether the convexity property is essential for the existence of a marginal cost pricing equilibrium. We answer this question in the negative. Next, we study the existence of marginal cost pricing equilibria, and the related notion of equilibria for the economy with public goods. We provide an alternative proof of the existence of marginal cost pricing equilibria for the private goods economy. Our proof does not depend on the assumption of the presence of one convex producer, which is commonly used in the literature. We also present existence result without free disposal under rather restrictive conditions. Moreover, we provide an alternative proof of the existence of Lindahl-Hotelling equilibria (Khan-Vohra (1987)). Our result can discard several assumptions including the one that public goods are not "bads." Finally, we further extend the result to an economy with public inputs as well as public goods. We propose an equilibrium notion, named Lindahl-Hotelling-Kaizuka Equilibrium, in which producers with non-convex technology are regulated to follow marginal cost pricing with deficits financed through given rules for lump sum taxation; producers with convex technology maximize profits, and consumers maximizes utilities; all producers and consumers are charged Lindahl prices according to their marginal evaluation of public good; and the sum of Lindahl prices over all producers and consumers is equal to its marginal cost (output) price. We provide a set of sufficient condition for the existence of such equilibria. |

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### Contents

ON THE NONEXISTENCE OF A MARGINAL COST PRICING EQUILIBRIUM | 10 |

ON THE EXISTENCE OF MARGINAL COST PRICING EQUILIBRIA | 21 |

ON THE EXISTENCE OF LINDAHLHOTELLING EQUILIBRIA | 51 |

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### Common terms and phrases

1-llpll Appendix to Chapter attainable set Bonnisseau boundary Claim Clarke normal cone Clarke tangent cone common vector consider an economy consumption plan consumption set continuous function contradiction convex producer convex-valued and upper-hemicontinuous Cornet cost pricing equilibrium denote Economic Theory Economies with Increasing equilibrium concept Existence of Equilibria existence of marginal existence proof Existence Theorem follows from Lemma free disposal assumption homeomorphism Hurwicz implies increasing returns interior Ioffe normal cone Journal of Economic Journal of Mathematical Khan Khan-Vohra Lemma 9 Lindahl equilibrium Lindahl prices Lindahl-Hotelling equilibrium llxll sk lump sum taxation mapping marginal cost pricing Mathematical Economics modified demand correspondence monotonic preference Moreover non-convex technology output Pareto optimal point y€Y price vector pricing rule producer h producer with non-convex producers with convex production plan public inputs returns to scale Rn+J Second Welfare Theorem technology set vector of marginal Vohra Welfare Economics