ps4_sol - EC314-Fall 2010 Solutions Problem Set 4 Matt...

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EC314-Fall 2010 Solutions – Problem Set 4 Matt Turner (updated: 18 August 2010) . 1. 2. (a.) 0 n l f = 1 f = 2 k = wn + lk (isocost) 2 a a 2 b b 0 n l f = 1 f = 2 k = w n + l k (isocost) 2 a a 2 b b (b.) To minimize costs, we always want to be at the corner of the isoquant, along the dashed line in the figures. Along this locus, we have n a = l b . 1
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3. 4. 5. (a.) We have u ( x ) = w - r ( x ) - cx . Let u = 0 and let x denote the edge of the city. Then x satisfies u ( x ) = w - c x = 0, so that x = w c . In equilibrium, the rent gradient adjusts so that utility is equal throughout the city, u ( x ) = w - cx - r ( x ) = 0, which means that r ( x ) = w - cx for all x x and 0 for more distant locations. (b.) Let δ be the market discount factor and let α = δ 10 . The land transfer tax requires payments tr ( x ) in years 9,19,29,... Thus, V ( x ) = ( δ 9 + δ 19 + δ 29 + ... ) tr ( x ) = δ 9 tr ( x )( 1 + δ 1 0 + δ 20 + ... ) = δ 9 tr ( x )( 1 + α + α 2 + ... ) = δ 9 tr ( x ) Σ t = 0 α t 2
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= δ 9 tr ( x ) 1 1 - α = δ 9 tr ( x ) 1 - δ 10 We need Tr ( x ) such that V ( x ) = Σ t = 0 δ t Tr ( x ) = V ( x ) = 1 1 - δ Tr ( x ) = ( 1 - δ ) V ( x ) = Tr ( x ) (c.) The after tax rent gradient R ( x ) = ( 1 + T ) r ( x ) , where r is what landlords keep.
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