Seminario de Investigación. Life-Cycle Models. Jorge Mondragón Minero ITAM. March 25, 2015

Transcription

1 Life-Cycle Models Jorge Mondragón Minero ITAM March 25, 2015

2 Introduction Evidence Wages across USA and Europe have been increasing since 1970 Differences in TFP between USA and Europe affect the return of high productive workers Difference in education (higher in Europe because it is free) Stock of capital is lower in Europe than in the USA (differences in taxation) Europe taxes are higher than in the USA. Europe has a bigger welfare government

3 Introduction Progressive Wedge PW [0.5, K0.5] = 1 1 τ(k0.5) 1 τ(0.5) (τ(y s ))Marginal Tax Average Tax( τ(y s ))

4 Assumptions Individuals have one unit of time that can be allocated: Leisure Working Investing in Human-Capital

5 Assumptions Assume β(1 + r) = 1 Human-Capital is produced using the following technology: Q s = A j (h s n s i s ) α A j : Individual Ability h s : Individual stock of Human-Capital n s i s : Fraction of individual working time that is spent training

6 Assumptions The cost of training is borne by workers: w s = P h h s (1 i s ) P h : Price of Human-Capital Law of Motion of Human-Capital: h s+1 = h s + Q s

7 Assumptions Labor Income: y s = w s n s = P h h s (1 i s )n s = P h h s n s P h C(Q s ) P h h s n s : Potential Wage

8 Maximization Problem The problem of individual type j is: max {Q s,a s+1,h s+1,c s,n s} S s=1 s.t. S β s 1 u(c s, 1 n s ) s=1 c s + a s+1 = (1 τ(y s ))y s + (1 + r)a s h s+1 = h s + Q s y s = P h h s n s P h C(Q s ) where C(Q s ) = ( Qs A j s ) 1 2

9 State Variables There are three state variables: a s : Assets h s : Human-Capital s: Age (it matters because individuals retire)

10 Value Function V (h s, a s, s) = = max {u((1 + r)a s + (1 τ(y s ))y s a s+1, 1 n s )+ {Q s,a s+1,h s+1,n s} S s=1 + βv (h s+1, a s+1, s + 1)} s.t. y s = P h h s n s P h C(Q s ) h s+1 = h s + Q s

11 Note Tax Liability: τ(y)y Marginal Income Tax: τ(y) + τ (y)y = τ(y)

12 First Order Conditions a s+1 : u c (c s, 1 n s ) βv a (h s+1, a s+1, s + 1) = 0 n s : u n (c s, 1 n s ) + u c (c s, 1 n s )(P h h s (1 τ(y s )) τ (y s )P h h s y s ) = = u n (c s, 1 n s ) + u c (c s, 1 n s )P h h s (1 τ(y s ) τ (y s )y s ) = = u n (c s, 1 n s ) + u c (c s, 1 n s )P h h s (1 τ(y s )) = 0 Q s : u c (c s, 1 n s )[(1 τ(y s ))( P h C (Q s )) τ (y s )y s ( P h C (Q s )) + βv h (h s+1, a s+1, s + 1)] = 0

13 First Order Conditions Therefore... u c (c s, 1 n s ) = βv a (h s+1, a s+1, s + 1) (1) u n (c s, 1 n s ) = u c (c s, 1 n s )P h h s (1 τ(y s )) (2) u c (c s, 1 n s )P h C (Q s )(1 τ(y s )) = βv h (h s+1, a s+1, s + 1) (3)

14 Envelope Conditions V a (h s, a s, s) = u c (c s, 1 n s )(1 + r) V h (h s, a s, s) = u c (c s, 1 n s )[(1 τ(y s ))P h n s τ(y s )P h n s y s ] βv h (h s+1, a s+1, s + 1) Then... V a (h s, a s, s) = u c (c s, 1 n s )(1 + r) (4) V h (h s, a s, s) = u c (c s, 1 n s )P h n s (1 τ(y s )) + V h (h s+1, a s+1, s + 1) (5)

15 Equations If we iterate in (5)... V h (h s, a s, s) = u c (c s, 1 n s )P h n s (1 τ(y s ))+ + βu c (c s+1, 1 n s+1 )P h n s+1 (1 τ(y s+1 ))+ + β 2 u c (c s+2, 1 n s+2 )P h n s+2 (1 τ(y s+2 )) β S s u c (c S, 1 n S )P h n S (1 τ(y S ))

16 Equations From (3)... u c (c s 1, 1 n s 1 )P h C (Q s 1 )(1 τ(y s 1 )) = = βu c (c s, 1 n s )P h n s (1 τ(y s ))+ + β 2 u c (c s+1, 1 n s+1 )P h n s+1 (1 τ(y s+1 ))+ + β 3 u c (c s+2, 1 n s+2 )P h n s+2 (1 τ(y s+2 )) β S s+1 u c (c S, 1 n S )P h n S (1 τ(y S ))

17 Equations From (1) and (4)... Hence... u c (c s 1, 1 n s 1 ) = β(1 + r)u c (c s, 1 n s ) u c (c s 1, 1 n s 1 ) u c (c s, 1 n s ) = 1

18 Solution Finally... P h C (Q s 1 ) Marginal Cost of Investing in Human-Capital = [ P h β 1 τ(y s) 1 τ(y s 1 ) n s + β 2 1 τ(y s+1) 1 τ(y s 1 ) n s β S s+1 1 τ(y ] S) 1 τ(y s 1 ) n S Marginal Benefit of Investing in Human-Capital

19 Special Cases When the labor supply is inelastic and the taxes are constant throughout time (n s = 1 and τ(y s ) = τ): C (Q s 1 ) = β + β β S s+1 The marginal benefit DOES NOT depend on the tax rate. More progressive tax systems reduce the benefit of investing in Human-Capital: 1 τ(y s ) 1 τ(y s 1 ) < 1

20 Elastic Labor Supply Marginal Income Labor Supply Marginal Benefit of Investing in Human-Capital Optimal Level of Human-Capital Endogenous Labor Supply Amplify the Effect of Tax on Human-Capital Progressive Wedge: PW (y s, y s+k ) = 1 1 τ(y s+k) 1 τ(y s ) n i n AVG