Since households also differ in α it is useful to

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the indifference condition as above. Since households also differ in α , it is useful to rewrite this indifference condition as y ρ = α h g ρ j - g ρ j +1 i h p - j +1 - p - j i (30)
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Rearranging and taking logs yields ln( α ) - ρ ln( y ) = K j where K j = ln p - j +1 - p - j g ρ j - g ρ j +1 The boundary indifference condition is thus a line in the (ln y, ln α )-space. The slope is determined by ρ , the intercept is given by K j . Note that ρ < 0 so the slope is negative. Higher income households must have lower preferences for public goods to be indifferent. d) Consider an equilibrium in which two communities have different housing price, say p 1 < p 2 (31) We are interested in an equilibrium in which both cities have positive population. The level of public good in community 2 should be higher than that of community 1, g 1 < g 2 (32) Otherwise all people will choose community 1 because it is cheaper and better. The set of individuals that are indifferent between community 1 and 2 are given by the equation derived in problem (c). Plotting this line, the equilibrium sorting will then look as follows:
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Note that holding tastes for public goods fixed at some level α we have perfect sorting of households by income. But since households differ in tastes, there will not be perfect sorting by income in equilibrium.
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Problem 4: Externalities and Cap & Trade a) The social optimum requires that Marginal Benefit = Marginal Cost for each firm. Firm A: 3 x 2 A = 300. x A = 10 Firm B: 2 x B = 300. x B = 150 b) Consider the case of equal reduction by 80 units. The benefits are given by 300 * 160 = 48 , 000. The costs are 80 2 + 80 3 = 518 , 400. This policy is not optimal since the costs far outweigh the benefits. c) Consider the case of a $ 300 subsidy per unit of abatement. Consider the decision problem for firm A: max 300 x A - x 3 A (33) The FOC is 3 x 2 A = 300. Hence x A = 10. Firm B faces the following problem: max 300 x B - x 2 B (34) The FOC is 2 x B = 300 and hence x B = 150.
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