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PS6Solutions - Economics 1011b Problem Set 6 Professor Aleh...

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Economics 1011b Problem Set 6 Professor Aleh Tsyvinski The problem set is due next Tuesday, April 17, by 5 pm in your TF’s mailbox in Lit- tauer. Late problem sets will not be accepted. You can work in groups (discuss the so- lutions, etc). However, you must write the solutions by yourself. Please write down all derivations/explanations. For clarifications (not answers) please contact Leon Berkelmans ([email protected]). Exercise 1. Optimal Taxation Consider an N period economy where some government expenditures g needs to be financed in the first period. The government can use taxes on labor income in all periods, τ 1 ...τ N and is free to set these taxes with the only constraint that the intertemporal budget constraint must hold, i.e. the DPV of tax revenues must equal g, but the government can choose whether to tax agents more in the first period, more in the second, etc. We also assume that the government is benevolent, in the sense that it maximizes households’ utility. Utility is given by: U = N t =1 β t - 1 ( c t - 1 2 h 2 t ) where c t is consumption in period t , h t is labor supplied in period t , and β < 1 is the discount factor. Assume the real interest rate is equal to r , with β (1 + r ) = 1. Also assume that the wage rate is 1. a. Define a Ramsey equilibrium, defining both the household problem and the government problem. Solution: The household problem is: max c 1 ...c N ,h 1 ,...,h N N t =1 β t - 1 ( c t - 1 2 h 2 t ) subject to: N t =1 c t ( 1 1 + r ) t - 1 = N t =1 (1 - τ t ) h t ( 1 1 + r ) t - 1 Substituting the budget constraint into the utility function gives the problem as: max h 1 ,...,h N N t =1 β t - 1 ((1 - τ t ) h t - 1 2 h 2 t ) The solution to this is: h t = (1 - τ t ) 1
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The governments problem is to maximize households welfare, i.e.
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