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# 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 fi 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 , the fi 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 . Equation 6 de fi nes in an implicit optimal labor supply. 1.2 The e ff ects of an increase of τ and t The total di ff erential of a given function f ( x, y ) is given by the formula: df ( x, y ) = ∂f ( x, y ) ∂x dx + ∂f ( x, y ) ∂y dy (7) In particular, if f ( x, y ) = 0 we can use the total di ff erential to derive the following: 0 = ∂f ( x, y ) ∂x dx + ∂f ( x, y ) ∂y dy (8) dy dx = ∂f ( x,y ) ∂x ∂f ( x,y ) ∂y (9) Equation 9 is the formula that we will use to derive the e ff ects of an increase of τ and t on the optimal supply of labor. Let’s start with evaluating the e ff ect of a change in t . In this case we have that f ( h , t ) = 0 is equivalent to equation 6. Hence we need to look at the sign of: dn dt = ∂f ( n ,t ) ∂t ∂f ( n ,t ) ∂n (10) We can easily recover that: ∂f ( n , t ) ∂n = U ll + U lc (1 τ ) w [ U cc (1 τ ) w U cl ](1 τ ) w (11) ∂f ( n , t ) ∂t = U lc (1 τ ) wU cc (12) 2
Since the utility function is nice , we have that 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 ff ect of an increase in 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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