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Unformatted text preview: l rate of transformation. That is, in a two consumer, two good world the following equation must be satisfied: MRSA = MRSB = MRT Samuelson's theory basically shows that, in the presence of public goods, this usual Pareto optimal allocation rule does not hold anymore. Instead, we must replace it with the Samuelson allocation rule... MRSA + MRSB 242 = MRT which "adds" (not "equates") all individual marginal rates of substitution and then matches them with the common marginal rate of transformation. EXAMPLE Consider the square root economy with unit factor endowments. We have done this example before and found that the equation of the PPF is X+Y=1 with a marginal rate of transformation of MRT = 1 Let's consider the optimal solutions of the Production economy for the usual case of private goods side-by-side with the case of one private and one public good: PRIVATE 222 Model data Good X private good Good Y private good Pareto Optimal Allocation Rules MRSA = MRSB = MRT XA + XB = X YA + YB = Y X+Y=1 Applying to the square root economy YA = YB = 1 XA XB XA + XB = X YA + YB = Y X+Y=1 PUBLIC 222 Model data Good X public good Good Y private good Samuelson Optimal Allocation Rules MRSA + MRSB = MRT XA = XB = X YA + YB = Y X+Y=1 Applying to the square root economy YA + YB = 1 XA XB XA = XB = X YA + YB = Y X+Y=1 243 Solving for the optimal solution YA = XA YB = XB X = XA + XB = Y = YA + YB = Utility Possibility Frontier (UPF) UA + UB = UB Solving for the optimal solution X = XA = XB = Y = YA + YB = Utility Possibility Frontier (UPF) [UA]2 + [UB]2 = UB UA UA LINDAHL'S PRICING OF PUBLIC GOODS Now that Samuelson has figured out how we should allocate public goods, we need a process for dealing with the problem of pricing them. So while Samuelson's theory addresses the issue of Pareto optimal allocations of public goods, Lindahl's theory attempts to answer the following question: Does the usual concept of market equilibrium "Demand = Supply" still apply in the presence of public goods? This is an important question because if we can answer "yes", then we can use our familiar market equilibrium condition D = S to solve the problem of public good pricing. Since there are no differences in the production of public goods relative to private goods (supply side), the main problem is to investigate whether it makes any sense to have the notion of market demand for public goods. 244 PUBLIC GOODS If good X is a public good, then everyBody...
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- Spring '10