w06_ass2_a - GS/ECON 5300 Answers to Assignment 2 March...

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GS/ECON 5300 Answers to Assignment 2 March 2006 Q 1. What should the tax rate τ on the return to saving be, if the government must raise some given revenue R by taxation of labour income at the rate t , and of the return to saving at the rate τ , if the economy consists of identical people with the following characteristics? People live for two periods, and work only in the first period. Their preferences can be represented by the utility function U ( L, c 1 , c 2 ) = α ln (1 - L ) + β ln c 1 + γ ln c 2 where L is work done in period 1, c i is consumption in period i , and α + β + γ = 1. The wage rate w is exogenous, as is the return r to saving. Since savings tax revenue is collected in the second period, the government revenue requirement is that the present value of revenue from the two taxes (wage tax revenue collected in period 1 and savings tax revenue collected in period 2) equal R . The government can borrow and lend at the same rate r as individuals. A 1. First, what is the indirect utility for a consumer whose direct untility function is U x ) = n i =1 a i ln x i , with n i =1 a i = 1? With these Cobb–Douglas preferences, the person’s Marshallian demand for good i would be x M i p, Y ) = a i Y p i (1 - 1) where Y is her (exogenous) income, and p i the price of good i . That means that her indirect utility function, V p, Y ) U x M p, Y )) is V p, Y ) = ln Y + n i =1 a i ln a i - n i =1 a i ln p i (1 - 2) where I have used the fact that ln AB = ln A + ln B . In the question, there are 3 goods : present leisure, present consumption, and future consump- tion. If ω w (1 - t ) ρ r (1 - τ ) denote the net–of–tax wage rate and rate of return to saving respectively, then, using present consumption as the num´ eraire, her “exogenous” income is Y ω (1 - 3) since ω is what she would earn, net of tax, if she worked full time. Here the price of leisure is also ω . The price of present consumption is 1, since it is the num´ eraire, and the price of future consumption is 1 / (1 + ρ ). Therefore, using (1 - 2), her indirect utility is V ( ω, 1 , 1 1 + ρ ; ω ) = ln ω + α ln α + β ln β + γ ln γ - α ln ω + γ ln (1 + ρ ) (1 - 4) 1
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where I have used the facts that ln 1 = 0, and that ln A/B = ln A - ln B . From (1 - 4), ∂V ∂ω = 1 - α ω (1 - 5) ∂V ∂ρ = γ 1 + ρ (1 - 6) Now the government’s budget constraint is twL + 1 1 + r τrS = R (1 - 7) where S is the person’s saving : this constraint says that the present value of taxes raised must equal R , and that the government uses the market interest r to discount second–period tax payments. From the person’s Marshallian demand function (1 - 1), 1 - L = α (1 - 8) c 1 = βω (1 - 9) which then imply that L = 1 - α (1 - 10) and S = ωL - c 1 = ω (1 - α - β ) = γω (1 - 11) Hence the government’s problem is to pick t and τ so as to maximize the indirect utility of a representative agent, subject to the budget constraint (1 - 7), which now can be written tw (1 - α ) + w (1 - t ) τ γr 1 + r (1 - 12) = R From the definition of ω and ρ , ∂ω ∂t = - w (1 - 13) ∂ρ ∂τ = - r (1 - 14) Using (1 - 5) and (1 - 6), maximizing utility subject to the government budget constraint (1 - 12) implies first–order conditions L ∂t = - 1 - α 1
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