sol_hw1_11 - Suggested Solution for HW1 February 7, 2011 1...

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Suggested Solution for HW1 February 7, 2011 1 Problem 1: tax on labor income 1.1 Household’s problem The household chooses ( c, l ) to maximise the utility function U ( c, l ) subject to the following budget constraint: c (1 τ ) w (1 l )+ d + t (1) The household’s utility function is well behaved and it satis f es the inada conditions. The lagrangian for the household’s problem is: L ( c, l, λ )= U ( c, l ) λ ( c (1 τ ) w (1 l ) d t ) (2) where λ is the lagrange multiplier on the budget constraint. Since U ( c, l ) is nice ,th e f rst order conditions for this problem are both necessary and su cient. ( c ) U c = λ (3) ( l ) U l = λ (1 τ ) w (4) ( λ )0 = λ ( c (1 τ ) w (1 l ) d t ) (5) We have a system of three equations in three unknowns. Given a parametric form for the utility function we could solve directly for the optimal consump- tion/leisure allocation. In this case we are going to derive the optimal amount of leisure in an implicit way. Notice that the budget constraint will hold with 1
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equality (why?) so we have that at the optimum c =(1 τ ) w (1 l )+ d + t , then by combining equations 3 and 4 we get: U l ((1 τ ) w ( n d + t, 1 n ) (1 τ ) wU c ((1 τ ) w ( n d + t, 1 n )=0 (6) Where I have used the fact that l =1 n .Equa t ion6de f nesinanimplicit optimal labor supply. 1.2 The e f ects of an increase of τ and t The total di f erential of a given function f ( x, y ) is given by the formula: df ( x, y )= ∂f ( x, y ) ∂x dx + ( x, y ) ∂y dy (7) In particular, if f ( x, y we can use the total di f erential to derive the following: 0= ( x, y ) dx + ( x, y ) dy (8) dy dx = ( x,y ) ( x,y ) (9) Equation 9 is the formula that we will use to derive the e f ects of an increase of τ and t on the optimal supply of labor. Let’s start with evaluating the e f ect of a change in t . Inth iscasewehave that f ( h ,t is equivalent to equation 6. Hence we need to look at the sign of: dn dt = ( n ,t ) ∂t ( n ,t ) ∂n (10) We can easily recover that: ( n ) = U ll + U lc (1 τ ) w [ U cc (1 τ ) w U cl ](1 τ ) w (11) ( n ) = U lc (1 τ ) cc (12) 2
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Since the utility function is nice ,wehavethat U cc and U ll are both negative. Moreover we assume that the cross derivative leisure/consumption is posi- tive (namely, we assume that the two goods are substitutes). So equations 11 and 12 are both positive and the e f ecto fanincreasein t is to decrease the optimal labor supply.
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sol_hw1_11 - Suggested Solution for HW1 February 7, 2011 1...

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