M.Sc Thesis
M.Sc Student Shami Labib Efficient Price Guided Allocations and Trading Allocations in Economies with Finite Number of Private and Public Goods Department of Industrial Engineering and Management Professor Benyamin Shitovitz

Abstract

In the theory of public goods economies two equilibrium concepts are commonly used, the Nash equilibria and the Lindahl equilibria. In 1954 Samuelson showed that in a public good economy a necessary and sufficient condition for efficient production of a public good is that the summation of the subjective marginal rates of substitution will be equal to the technical rate of substitution in production of the public good. One of the main results in the economic literature is that if there are at least two players it follows that the Nash allocation is not Pareto-optimal. In contrast, since there is a personalized price vector for a Lindahl allocation which satisfies the Samuelson condition, it follows that the Lindahl equilibria is Pareto optimal; however, it is not necessarily Pareto dominating the Nash equilibria.

In 2012 Perets, Shitovitz and Spiegel presented a new equilibrium concept called "Trading Equilibrium", which is based on an personalized price mechanism that yields a core allocation that strictly Pareto dominates the unique Nash allocation in that market.

In this thesis we generalize the discussion to economies with one private good, L public goods and finite number of households H={1,?,H} such that every household (consumer) has a strictly positive initial endowment of the private good. In addition, we assumed that the individuals in the economy have achieved an "initial" allocation, and we proved by "abstract economies" techniques the existence of equilibrium prices that define an efficient price guided allocation with respect to the "initial" allocation that has been achieved above. The result we have received is interesting in economics theory since it yields an allocation that is efficient and Pareto dominates the initial allocation from which we emerge, which is equivalent to the efficiency of competitive allocations in economies without public goods.

In the third section of our research, we concentrated on finite economies with n consumers, K private goods and L public goods, where all consumers have the same strict convex preferences. We also assumed the strict ordinal normality of both private and public goods and ordinal seperability of the utility functions. In that economy we assumed that the production technology of the public goods is linearly additive and each consumer contributes a positive amount from each private good to each public good at the Nash equilibrium. Under these assumptions we have shown that the Nash allocation is unique and symmetric. We also proved the existence of Trading allocation which based on the personalized prices for each player and each  public good, and we have also shown that the Trading allocation is a Foley core allocation that Pareto dominates the unique Nash allocation in that economy.

The main result in this research shows that in economies with more than one public good, and households with different utility functions, meaning that every consumer has different preferences over the public goods, there exists an efficient price guided allocation that dominates any other feasible allocation, however, it is not necessarily unique or in the Foley's core.

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