2 b c b 2 w n 15 in the decentralized case individual

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2 b ( c - b ) 2 = W N (15) In the decentralized case, individual city does not value the positive spill-over effect it brings to the other city. However, in the centralized provision, the social benefits from spill-over effects are internalized, and hence more public goods are provided. d) Consider the following subsidies, s 1 = s 2 = b (16) Taking city 2’s contribution as given, city 1 chooses its public good provision level g 1 to maximize the following welfare function: W 1 ( g 1 , g 2 ) = 2 [ a g 1 + b g 1 g 2 ] - cg 1 + bg 1 (17) The first-order condition resulting from this maximization problem is given below: a 1 g 1 + b g 2 g 1 = c - b (18)
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Solving for g 1 gives us the reaction function: g 1 ( g 2 ) = a + b g 2 c - b ! 2 (19) City 2’s reaction function is similar: g 2 ( g 1 ) = a + b g 1 c - b ! 2 (20) From symmetry of the problem we can calculate the equilibrium provision level with subsidy g NS by solving the following equation: g NS = a + b q g NS c - b 2 (21) which is the same as the provision level in (b): g NS = a c - 2 b 2 (22)
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Problem 3: Tiebout Sorting and Segregation a) Roy’s identity implies that the demand for housing can be expressed as: h ( p, y ) = - ∂V ∂p ∂V ∂y = - Bpy ρ p - - 1 py ρ - 1 p - = By p (23) The price and income elasticity of housing demand is given by: e p = ∂h ( p, y ) ∂p × y h ( p, y ) = - By p 2 × p By p = - 1 (24) e y = ∂h ( p, y ) ∂y × y h ( p, y ) = B p × y By p = 1 (25) b) There will exist a household with income y that is exactly indifferent between living in j or j + 1 . This household is characterized by the following indifference condition: V ( g j , p j , y ) = V ( g j +1 , p j +1 , y ) (26) Specifically, - n αg ρ j +1 + h yp - B j +1 i ρ o = - n αg ρ j + h yp - B j i ρ o (27) Collect terms containing y on the left-hand side: y ρ h p - j +1 - p - j i = α h g ρ j - g ρ j +1 i (28) Solve for y y = α h g ρ j - g ρ j +1 i h p - j +1 - p - j i 1 ρ (29) c) Households that indifferent between both communities are still characterized by the indifference condition as above. Since households also differ in
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