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Unformatted text preview: 5 Other Provision Mechanisms for Public Goods 5.1 Lindahl Equilibria We have already seen that the &competitive¡voluntary provision model in general implies that the resulting allocation is ine¢ cient. There is however a much studied &market insti- tution¡called the Lindahl equilibrium model that in principle could achieve e¢ ciency. The idea is to think of the amount purchased by each agent as a distinct commodity and have each agent to face a personalized price p i and to have these prices chosen so that all agents agrees on the level of the public good : A Lindahl Equilibrium is then de£ned in the spirit of a standard competitive equilibrium as a vector of prices and an expanded allocation so that 1) each consumer is willing to buy &his share¡of the public good at his personalized price, 2) £rms are willing to supply the level of public good if paid the sum of all personalized prices & the quantity supplied, 3) markets clear. For ease of exposition we continue to consider the example with two agents, A and B: De&nition 1 A Lindahl Equilibrium is a vector p & = & p A & ; p B & ¡ and an allocation & x & A ; x B & ; y & ¡ such that a. p A & + p B & = 1 : The interpretation of p J & is that this is the personalized price (tax) that agent i pays for the public good in terms of of the private good. b. Each consumer maximizes utility, that is & x J & ; y & ¡ solves max x;y u J ( x; y ) s.t. e J ¡ x ¡ p J & y ¢ Notice that each consumer must optimally purchase the same level of the the public good. c. Market clears x A & + x B & + y & £ e A + e B 48 Consider the utility maximization problem for agent i; which after substituting the con- straint becomes max y u J & e J & p J & y; y ¡ : The &rst order condition for an interior solution is thus @u J & x J & ; y & ¡ @x p J & = @u J & x J & ; y & ¡ @y for each agent J; (a¡) p J & = @u J ( x J & ;y & ) @y @u J ( x J & ;y & ) @x Summing over J = A; B we get @u A ( x A & ;y & ) @y @u A ( x A & ;y & ) @x + @u B ( x B & ;y & ) @y @u B ( x B & ;y & ) @x = p A & + p B & = 1 ; which is the condition for a Pareto optimal allocation.which is the condition for a Pareto optimal allocation....
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