Optimal Taxation of Wealthy Individuals

Transcription

1 y Individuals Ali Shourideh, Wharton Carnegie Mellon University, March 2016

2 Introduction Past 30 years: Rapid increase in the concentration of wealth at the top in the United States

3 Introduction Past 30 years: Rapid increase in the concentration of wealth at the top in the United States Saez and Zucman (2015): share of wealth in the top 0.1%: 7% in 1978; 22% in 2013

4 Introduction Past 30 years: Rapid increase in the concentration of wealth at the top in the United States Saez and Zucman (2015): share of wealth in the top 0.1%: 7% in 1978; 22% in 2013 How should the tax schedule treat wealth/capital at the top?

5 Introduction Past 30 years: Rapid increase in the concentration of wealth at the top in the United States Saez and Zucman (2015): share of wealth in the top 0.1%: 7% in 1978; 22% in 2013 How should the tax schedule treat wealth/capital at the top? Current theories are insufficient Rep. agent models: Chamley- Judd; incapable of answering this question Models with labor income risk: counterfactual wealth distribution This paper: develop a framework to analyze optimal taxation of wealth with capital income risk

6 Wealth Inequality Need: theory of wealth inequality:

7 Wealth Inequality Need: theory of wealth inequality: How did the wealthy households become wealthy?

8 Wealth Inequality Need: theory of wealth inequality: How did the wealthy households become wealthy? How is wealth distributed?

9 Wealth Inequality Need: theory of wealth inequality: How did the wealthy households become wealthy? How is wealth distributed? Basic observations: Wealth is highly concentrated at the top: top 1% hold 34.7% of wealth (much higher than earnings)

10 Wealth Inequality Need: theory of wealth inequality: How did the wealthy households become wealthy? How is wealth distributed? Basic observations: Wealth is highly concentrated at the top: top 1% hold 34.7% of wealth (much higher than earnings) Most of it is in the form of risky business

11 Wealth Inequality Need: theory of wealth inequality: How did the wealthy households become wealthy? How is wealth distributed? Basic observations: Wealth is highly concentrated at the top: top 1% hold 34.7% of wealth (much higher than earnings) Most of it is in the form of risky business Models with labor income risk

12 Wealth Inequality Need: theory of wealth inequality: How did the wealthy households become wealthy? How is wealth distributed? Basic observations: Wealth is highly concentrated at the top: top 1% hold 34.7% of wealth (much higher than earnings) Most of it is in the form of risky business Models with labor income risk vs. models with capital income risk

13 Wealth Inequality Power law for wealth; Pr(w > ŵ) = κŵ ν, ν data = 2.3

14 This Paper This paper: wealth inequality : business/capital income risk + financial frictions (lack of diversification)

15 This Paper This paper: wealth inequality : business/capital income risk + financial frictions (lack of diversification) Economy is comprised of entrepreneurs who have investment opportunities: Heterogeneous ex-ante: various qualities Heterogeneous ex-post: Returns are risky

16 This Paper This paper: wealth inequality : business/capital income risk + financial frictions (lack of diversification) Economy is comprised of entrepreneurs who have investment opportunities: Heterogeneous ex-ante: various qualities Heterogeneous ex-post: Returns are risky How should various types of capital be taxed: Capital income from the business Capital income outside of business

17 This Paper This paper: wealth inequality : business/capital income risk + financial frictions (lack of diversification) Economy is comprised of entrepreneurs who have investment opportunities: Heterogeneous ex-ante: various qualities Heterogeneous ex-post: Returns are risky How should various types of capital be taxed: Capital income from the business Capital income outside of business What does an efficient/ fair distribution of wealth look like?

18 Introduction Results:

19 Introduction Results: Endogenous risk-return trade-off in the cross-section determines optimal taxes

20 Introduction Results: Endogenous risk-return trade-off in the cross-section determines optimal taxes Taxes on capital income: Progressive - Key intuition: progressive taxes provide insurance with respect to shocks to rate of return

21 Introduction Results: Endogenous risk-return trade-off in the cross-section determines optimal taxes Taxes on capital income: Progressive - Key intuition: progressive taxes provide insurance with respect to shocks to rate of return Formula for characterizing the efficient tail of long-run wealth distribution

22 Literature Taxation literature: Rep. Agent: Chamley (1986), Judd (1985) τ k = 0

23 Literature Taxation literature: Rep. Agent: Chamley (1986), Judd (1985) τ k = 0 NDPF: Golosov-Kocherlakota-Tsyvinski (2003), Albanesi-Sleet (2006),...: Saving reduces labor supply τ k > 0

24 Literature Taxation literature: Rep. Agent: Chamley (1986), Judd (1985) τ k = 0 NDPF: Golosov-Kocherlakota-Tsyvinski (2003), Albanesi-Sleet (2006),...: Saving reduces labor supply τ k > 0 Mainly ignores capital income risk

25 Literature Taxation literature: Rep. Agent: Chamley (1986), Judd (1985) τ k = 0 NDPF: Golosov-Kocherlakota-Tsyvinski (2003), Albanesi-Sleet (2006),...: Saving reduces labor supply τ k > 0 Mainly ignores capital income risk Heterogeneity in saving motives (discount factor shocks): Saez (2002), Piketty-Saez (2012), Farhi-Werning (2013) Insurance role of progressive taxes: Vickrey(1947), Friedman (1948), Varian (1980)

26 Literature Taxation literature: Rep. Agent: Chamley (1986), Judd (1985) τ k = 0 NDPF: Golosov-Kocherlakota-Tsyvinski (2003), Albanesi-Sleet (2006),...: Saving reduces labor supply τ k > 0 Mainly ignores capital income risk Heterogeneity in saving motives (discount factor shocks): Saez (2002), Piketty-Saez (2012), Farhi-Werning (2013) Insurance role of progressive taxes: Vickrey(1947), Friedman (1948), Varian (1980) More recent literature on wealth distribution: Benhabib, Bisin, and Zhu (2011), Gabaix, et. al. (2015): generate pareto distribution via rate of return shocks Builds on insights from Champernowne (1953) and Simon (1955)

27 Outline Two Period Model Infinite Horizon Model: Taxes Long-run distribution of wealth

28 Two Period Model Two periods: t = 0, 1

29 Two Period Model Two periods: t = 0, 1 Continuum of households: draw θ F(θ) at t = 0.

30 Two Period Model Two periods: t = 0, 1 Continuum of households: draw θ F(θ) at t = 0. Investment technology associated with θ

31 Two Period Model Two periods: t = 0, 1 Continuum of households: draw θ F(θ) at t = 0. Investment technology associated with θ invest at t = 0 Output at t = 1

32 Two Period Model Two periods: t = 0, 1 Continuum of households: draw θ F(θ) at t = 0. Investment technology associated with θ invest at t = 0 Output at t = 1 k 1 y 1 = ɛθk 1

33 Two Period Model Two periods: t = 0, 1 Continuum of households: draw θ F(θ) at t = 0. Investment technology associated with θ invest at t = 0 Output at t = 1 k 1 y 1 = ɛθk 1 ɛ H(ɛ); Eɛ = 1

34 Two Period Model Households consume, c 0, c 1 and invest, k 1

35 Two Period Model Households consume, c 0, c 1 and invest, k 1 Households: u(c 0 ) + βu(c 1 ) { c 1 σ u(c) = 1 σ σ 1 log(c) σ = 1

36 Two Period Model Households consume, c 0, c 1 and invest, k 1 Households: u(c 0 ) + βu(c 1 ) { c 1 σ u(c) = 1 σ σ 1 log(c) σ = 1 Allocations: {c 0 (θ), k 1 (θ), c 1 (θ, ɛ)}

37 Two Period Model Households consume, c 0, c 1 and invest, k 1 Households: u(c 0 ) + βu(c 1 ) { c 1 σ u(c) = 1 σ σ 1 log(c) σ = 1 Allocations: {c 0 (θ), k 1 (θ), c 1 (θ, ɛ)} Feasibility: Θ [c 0 (θ) + k 1 (θ)] df(θ) e 0 c 1 (θ, ɛ)dh(ɛ)df(θ) θk 1 (θ)df(θ) Θ 0 Θ

38 Taxation Problem Budget constraint: c 0 + k 1 + b 1 = e 0 c 1 (ɛ) = ɛθk 1 + Rb 1 T(ɛθk 1, Rb 1 ) Important assumption: No insurance against ɛ other than self insurance

39 Taxation Problem Budget constraint: c 0 + k 1 + b 1 = e 0 c 1 (ɛ) = ɛθk 1 + Rb 1 T(ɛθk 1, Rb 1 ) Important assumption: No insurance against ɛ other than self insurance Optimal Taxation Problem: g(θ)u(θ)df(θ) max T (, ) Θ U(θ) = max u(c 0 ) + β Feasibility subject to B.C. ɛ u(c 1 (ɛ))dh(ɛ)

40 Primal Approach Focus on allocations and back out taxes Taxation problem is equivalent to a mechanism design problem Can only observe y 1 ; k 1, c 0, θ, ɛ: private information

41 Primal Approach Focus on allocations and back out taxes Taxation problem is equivalent to a mechanism design problem Can only observe y 1 ; k 1, c 0, θ, ɛ: private information Incentive compatibility:

42 Primal Approach Focus on allocations and back out taxes Taxation problem is equivalent to a mechanism design problem Can only observe y 1 ; k 1, c 0, θ, ɛ: private information Incentive compatibility: u(c 0 (θ)) + β 0 u(c 1 (θ, ɛ))dh(ɛ) u(c 0 (ˆθ) + k 1 (ˆθ) ˆk) + β 0 u(c 1 (ˆθ, ɛ θˆk ˆθk 1 (ˆθ) ))dh(ɛ)

43 Primal Approach Focus on allocations and back out taxes Taxation problem is equivalent to a mechanism design problem Can only observe y 1 ; k 1, c 0, θ, ɛ: private information Incentive compatibility: u(c 0 (θ)) + β 0 u(c 1 (θ, ɛ))dh(ɛ) u(c 0 (ˆθ) + k 1 (ˆθ) ˆk) + β 0 u(c 1 (ˆθ, ɛ θˆk ˆθk 1 (ˆθ) ))dh(ɛ) choose allocations to maximize welfare subject to I.C.

44 Wedges/Marginal Tax Rates Ex-ante wedges: Outside saving: capital invested in other s businesses

45 Wedges/Marginal Tax Rates Ex-ante wedges: Outside saving: capital invested in other s businesses c 0 (θ) σ = βr(1 τ b (θ)) 0 c 1 (θ, ɛ) σ dh(ɛ)

46 Wedges/Marginal Tax Rates Ex-ante wedges: Outside saving: capital invested in other s businesses c 0 (θ) σ = βr(1 τ b (θ)) 0 c 1 (θ, ɛ) σ dh(ɛ) Inside saving or investment: capital invested in own business

47 Wedges/Marginal Tax Rates Ex-ante wedges: Outside saving: capital invested in other s businesses c 0 (θ) σ = βr(1 τ b (θ)) 0 c 1 (θ, ɛ) σ dh(ɛ) Inside saving or investment: capital invested in own business c 0 (θ) σ = βθ(1 τ k (θ)) 0 ɛc 1 (θ, ɛ) σ dh(ɛ)

48 Wedges/Marginal Tax Rates Ex-ante wedges: Outside saving: capital invested in other s businesses c 0 (θ) σ = βr(1 τ b (θ)) 0 c 1 (θ, ɛ) σ dh(ɛ) Inside saving or investment: capital invested in own business c 0 (θ) σ = βθ(1 τ k (θ)) 0 ɛc 1 (θ, ɛ) σ dh(ɛ) Ex-post taxes: T y (ɛθk 1 (θ), Rb 1 (θ)), T b (ɛθk 1 (θ), Rb 1 (θ))

49 A Preliminary Analysis Simpler problem to built intuition: Planner knows θ

50 A Preliminary Analysis Simpler problem to built intuition: Planner knows θ Suppose that T(y, Rb) = (1 τ b )Rb + (1 τ k )y + T 1

51 A Preliminary Analysis Simpler problem to built intuition: Planner knows θ Suppose that T(y, Rb) = (1 τ b )Rb + (1 τ k )y + T 1 Increase τ k by dτ k and T 1 by dt 1 to keep the entrepreneur indifferent: k 1 dτ k = 1 τ k R (1 τ b ) dt 1

52 A Preliminary Analysis Simpler problem to built intuition: Planner knows θ Suppose that T(y, Rb) = (1 τ b )Rb + (1 τ k )y + T 1 Increase τ k by dτ k and T 1 by dt 1 to keep the entrepreneur indifferent: k 1 dτ k = 1 τ k R (1 τ b ) dt 1

53 A Preliminary Analysis Change in the government surplus:

54 A Preliminary Analysis Change in the government surplus: Mechanical change θkdτ k dt 1 = k 1 τ k [θ (1 τ k ) R (1 τ b )] dτ k

55 A Preliminary Analysis Change in the government surplus: Mechanical change θkdτ k dt 1 = Behavioral change τ k θdk + τ b Rdb = k 1 τ k [θ (1 τ k ) R (1 τ b )] dτ k [ θ τ kk ξ h k,θ + R τ ] bb ξ h b,θ dτ k 1 τ k 1 τ b ξ h k,θ, ξh b,θ : compensated elasticities Optimal Taxes: Mech. + Beh. = 0 θ (1 τ k ) R (1 τ b ) = θτ k ξ h k,θ + Rτ bb(1 τ k ) k(1 τ b ) ξh b,θ

56 A Preliminary Analysis Similar perturbation for τ b and combine with above τ k = θ (1 τ k) R (1 τ b ) 1 τ k ξ h k,θ ξh k,r ξ h ξ h b,θ b,r Optimal taxes are a function of risk-premium and elasticities ξ: complicated formula; without risk: ξ h = 1 1 σ β + (k 1 /c 0 ) 1 1/σ 1 If σ 1 decreasing with k 1 /c 0. Increasing risk-premium and decreasing elasticities progressive taxes

57 Optimal Provision of Incentives Above: Form of insurance is exogenous; Now: allow optimal form of insurance Still assume that θ is observable; IC: k 1 (θ) arg max u(c 0 (θ)+k 1 (θ) ˆk)+β ˆk 0 u(c 1 (θ, Local version: u (c 0 (θ)) = β u(c 1 (θ, ɛ)) ρ(ɛ) R + k 1 (θ) dh(ɛ) ˆk k 1 (θ) ɛ))dh(ɛ)

58 Incentive Compatbility Assumption 1. The distribution of ε satisfies the following: 1.1 The support of H is the set of all positive real numbers, R +, 1.2 The distribution function H satisfies monotone likehood ratio assumption, i.e., ρ (ε) = εh (ε) h(ε) 1 is increasing in ε. 1.3 The monotone likelihood ratio takes a finite value ε = 0, i.e., lim ε 0 εh (ε) h(ε) = ρ (0) + 1 >. 2. Preferences satisfy: σ 1/2 Guarantees local IC implies IC Examples for H: generalized Gamma distribution (Weibul, Gamma, Exponential, etc.); Log-normal does not work

59 Optimal Provision of Incentives c 1 (θ, ɛ): should increase in ɛ

60 Optimal Provision of Incentives c 1 (θ, ɛ): should increase in ɛ incentive for investment

61 Optimal Provision of Incentives c 1 (θ, ɛ): should increase in ɛ incentive for investment Optimal consumption schedule at t = 1: Proposition 1. Suppose that θ θ. Then, any optimal allocation, c 1 (θ, ɛ) satisfies 1 u = αg(θ) [1 + s(θ)ρ(ɛ)] (c 1 (θ, ɛ))

62 Optimal Provision of Incentives c 1 (θ, ɛ): should increase in ɛ incentive for investment Optimal consumption schedule at t = 1: Proposition 1. Suppose that θ θ. Then, any optimal allocation, c 1 (θ, ɛ) satisfies 1 u = αg(θ) [1 + s(θ)ρ(ɛ)] (c 1 (θ, ɛ)) 2. Sensitivity, s(θ), is strictly increasing in θ

63 Optimal Provision of Incentives First result: well-known since Holmstrom (1979)

64 Optimal Provision of Incentives First result: well-known since Holmstrom (1979) Second result: Marginal benefit (higher investment and higher output) of sensitivity is increasing in θ, Marginal cost (more risk) is independent of θ

65 Optimal Provision of Incentives First result: well-known since Holmstrom (1979) Second result: Marginal benefit (higher investment and higher output) of sensitivity is increasing in θ, Marginal cost (more risk) is independent of θ Technical issue: marginal cost is bounded above; θ must be bounded above

66 Optimal Provision of Incentives First result: well-known since Holmstrom (1979) Second result: Marginal benefit (higher investment and higher output) of sensitivity is increasing in θ, Marginal cost (more risk) is independent of θ Technical issue: marginal cost is bounded above; θ must be bounded above Loosely speaking: Endogenously determined risk-return trade-off

67 Implication for Taxes Progressivity is a natural result: it provide insurance. Next: An Inverse Euler Equation to understand this better

68 Implication for Taxes Progressivity is a natural result: it provide insurance. Next: An Inverse Euler Equation to understand this better A perturbation δ c 0 (θ) = u (c 0 ) c 1 (θ) = βδ u (c 1 ) 1 k k (θ) = σ u δ (c 0 ) c 0 keeps individual utility unchanged k: similar to compensated elasticity

69 Modified Inverse Euler Equation Optimality implies that cost of this perturbation must be zero: σ θ R k 1 u (c 0 ) = 1 R c 0 βre [u (c 1 )] LHS is increasing in θ. RHS is measure of distortions Proposition Optimal wedges τ k (θ), τ b (θ) are increasing in θ.

70 Intuition Outside saving/bonds: more productive entrepreneurs are subject to more risk, want to self insure using outside saving higher taxes

71 Intuition Outside saving/bonds: more productive entrepreneurs are subject to more risk, want to self insure using outside saving higher taxes Inside saving/equity: Two opposing effects: higher taxes give more insurance; less incentive to invest If sensitivity is mildly increasing, then insurance effect dominates In the paper: this is the case! (had to resort to Chebyshev s integral inequality!)

72 Private Productivities Analyze using the first order approach:

73 Private Productivities Analyze using the first order approach: u (c 0 (θ)) = β u(c 1 (θ, ɛ)) ρ(ɛ) R + k 1 (θ) dh(ɛ) U (θ) = 1 θ k 1(θ)u (c 0 (θ))

74 Private Productivities Analyze using the first order approach: u (c 0 (θ)) = β u(c 1 (θ, ɛ)) ρ(ɛ) R + k 1 (θ) dh(ɛ) U (θ) = 1 θ k 1(θ)u (c 0 (θ)) The same Inver Euler Equation holds, therefore Proposition Suppose that k 1 (θ)/c 0 (θ) is increasing, then τ b (θ) is increasing with private θ.

75 Private Productivities No distortions at the limits:

76 Private Productivities No distortions at the limits: Proposition At the optimum, τ PI b (θ; R) = τfi b (θ; R), τpi k (θ; R) = τf ki(θ; R), when θ = R, θ

77 Numerical Exercise A period is around 25 years; β = (0.95) 25 R = 1/β 2.77 preferences: σ = 1 Not a lot of evidence on ɛ and θ. Campbell et. al. (2001): st. dev. publicly traded stocks: 50% DeBacker, Panousi and Ramnath (2015): Tax data on business income - Annual st. dev is around 46% - 23% of variation from individual heterogeneity; 77% from risk

78 Numerical Exercise ɛ Γ: var[ɛ] { (0.2) 2 25, (0.36) 2 25, (0.45) 2 25 }

79 Numerical Exercise ɛ Γ: var[ɛ] { (0.2) 2 25, (0.36) 2 25, (0.45) 2 25 } θ [R, θ max ], f(θ) 1/θ θ max { ( ) 25, ( ) 25, ( ) 25}

80 Optimal Taxes

81 Optimal Taxes Private information does not make much of a difference: τ 10 3

82 Equivalent Income Taxes Income Tax: (1 + r(1 ˆτ)) 25 = R(1 τ)

83 Ex-post Taxes

84 Dynamic Extension Dynamic Extension: Do Results survive in a fully dynamic model?

85 Dynamic Extension Dynamic Extension: Do Results survive in a fully dynamic model? Ignored one source of saving: bequests

86 Dynamic Extension Dynamic Extension: Do Results survive in a fully dynamic model? Ignored one source of saving: bequests (Efficient) Long-run distribution of wealth

87 Dynamic Extension OLG extension:

88 Dynamic Extension OLG extension: Continuum of young born at date t

89 Dynamic Extension OLG extension: Continuum of young born at date t Live for two periods

90 Dynamic Extension OLG extension: Continuum of young born at date t Live for two periods Preferences: ] V t = E [log c 0,t + β log c 1,t + ˆβV t+2

91 Dynamic Extension OLG extension: Continuum of young born at date t Live for two periods Preferences: ] V t = E [log c 0,t + β log c 1,t + ˆβV t+2 Altruistic toward future generations

92 Dynamic Extension Individual Technology for the young:

93 Dynamic Extension Individual Technology for the young: Draw θ t F(θ t ); i.i.d. across generations and households.

94 Dynamic Extension Individual Technology for the young: Draw θ t F(θ t ); i.i.d. across generations and households. Invest k t+1 at t; Hire l t+1 at t + 1

95 Dynamic Extension Individual Technology for the young: Draw θ t F(θ t ); i.i.d. across generations and households. Invest k t+1 at t; Hire l t+1 at t + 1 Production at t + 1, y t+1 = (ɛ t+1 θ t k t+1 ) α l 1 α t+1

96 Dynamic Extension Individual Technology for the young: Draw θ t F(θ t ); i.i.d. across generations and households. Invest k t+1 at t; Hire l t+1 at t + 1 Production at t + 1, y t+1 = (ɛ t+1 θ t k t+1 ) α l 1 α t+1 Young at t endowed with one unit of labor

97 Dynamic Extension Individual Technology for the young: Draw θ t F(θ t ); i.i.d. across generations and households. Invest k t+1 at t; Hire l t+1 at t + 1 Production at t + 1, y t+1 = (ɛ t+1 θ t k t+1 ) α l 1 α t+1 Young at t endowed with one unit of labor Provide labor to the old running the firm

98 Dynamic Extension Feasibility:

99 Dynamic Extension Feasibility: [c 0,t + c 1,t + k t+1] = (ɛ t θ t 1 k t ) α l 1 α t

100 Dynamic Extension Feasibility: [c 0,t + c 1,t + k t+1] = l t = 1 (ɛ t θ t 1 k t ) α l 1 α t

101 Dynamic Extension Feasibility: [c 0,t + c 1,t + k t+1] = l t = 1 (ɛ t θ t 1 k t ) α l 1 α t Information: k t+1, c 0,t, c 1,t, ɛ t+1, θ t are private y t+1, l t+1 as well as bequests are observable.

102 Dynamic Extension Planning problem:

103 Dynamic Extension Planning problem: max V 0 + δv 1 subject to feasibility, incentive constraints

104 Dynamic Extension Planning problem: max V 0 + δv 1 subject to feasibility, incentive constraints Complicated problem Labor demand: equate capital labor ratio across all firms Output: y t+1 = κɛ t+1 θ t k t+1

105 Dynamic Extension Planning problem: max V 0 + δv 1 subject to feasibility, incentive constraints Complicated problem Labor demand: equate capital labor ratio across all firms Output: y t+1 = κɛ t+1 θ t k t+1 Separate each generations problem from each other component planning problem

106 Dynamic Extension Planning problem: max V 0 + δv 1 subject to feasibility, incentive constraints Complicated problem Labor demand: equate capital labor ratio across all firms Output: y t+1 = κɛ t+1 θ t k t+1 Separate each generations problem from each other component planning problem Focus on local incentive constraints

107 Recursive Problem Component planning problem in a stationary economy: P (w) = max [αqκθk (θ) c 0 (θ) k (θ) + p c 0,c 1,k,w,U Θ [ +q c1 (θ, y) + qp ( w (θ, y) )] dg (y θ, k (θ)) df ( 0

108 Recursive Problem Component planning problem in a stationary economy: P (w) = max [αqκθk (θ) c 0 (θ) k (θ) + p c 0,c 1,k,w,U Θ [ +q c1 (θ, y) + qp ( w (θ, y) )] dg (y θ, k (θ)) df ( subject to 0

109 Recursive Problem Component planning problem in a stationary economy: P (w) = max [αqκθk (θ) c 0 (θ) k (θ) + p c 0,c 1,k,w,U Θ [ +q c1 (θ, y) + qp ( w (θ, y) )] dg (y θ, k (θ)) df ( subject to 0 U (θ) df (θ) = w [ u (c 0 (θ)) + βu (c 1 (θ, y)) + ˆβw (θ, y)] dg (y θ, k (θ)) = U (θ) 0 U (θ) = 1 θ k (θ) u (c [ ] βu (c 1 (θ, y)) + ˆβw (θ, y) g k (y k (θ), θ) dy = u (c 0 (θ))

110 Recursive Problem Component planning problem in a stationary economy: P (w) = max [αqκθk (θ) c 0 (θ) k (θ) + p c 0,c 1,k,w,U Θ [ +q c1 (θ, y) + qp ( w (θ, y) )] dg (y θ, k (θ)) df ( subject to 0 U (θ) df (θ) = w [ u (c 0 (θ)) + βu (c 1 (θ, y)) + ˆβw (θ, y)] dg (y θ, k (θ)) = U (θ) 0 U (θ) = 1 θ k (θ) u (c [ ] βu (c 1 (θ, y)) + ˆβw (θ, y) g k (y k (θ), θ) dy = u (c 0 (θ)) p: wage, q: interest rate

111 Recursive Problem Component planning problem in a stationary economy: P (w) = max [αqκθk (θ) c 0 (θ) k (θ) + p c 0,c 1,k,w,U Θ [ +q c1 (θ, y) + qp ( w (θ, y) )] dg (y θ, k (θ)) df ( subject to 0 U (θ) df (θ) = w [ u (c 0 (θ)) + βu (c 1 (θ, y)) + ˆβw (θ, y)] dg (y θ, k (θ)) = U (θ) 0 U (θ) = 1 θ k (θ) u (c [ ] βu (c 1 (θ, y)) + ˆβw (θ, y) g k (y k (θ), θ) dy = u (c 0 (θ)) p: wage, q: interest rate p, q: market clearing prices

112 Homogeneity log prefs policy functions are homogeneous in w

113 Homogeneity log prefs policy functions are homogeneous in w c 0 (θ, w) = ĉ 0 (θ) e 1+β 1 ˆβ w k(θ, w) = ˆk(θ) e 1+β 1 ˆβ w c 1 (θ, y, w) = ĉ 1 (θ, y) e 1+β 1 ˆβ w w (θ, y, w) = ŵ(θ, y) + w

114 Homogeneity log prefs policy functions are homogeneous in w c 0 (θ, w) = ĉ 0 (θ) e 1+β 1 ˆβ w k(θ, w) = ˆk(θ) e 1+β 1 ˆβ w c 1 (θ, y, w) = ĉ 1 (θ, y) e 1+β 1 ˆβ w w (θ, y, w) = ŵ(θ, y) + w Identical to the two period problem; extra instrument: promised utility

115 Homogeneity log prefs policy functions are homogeneous in w c 0 (θ, w) = ĉ 0 (θ) e 1+β 1 ˆβ w k(θ, w) = ˆk(θ) e 1+β 1 ˆβ w c 1 (θ, y, w) = ĉ 1 (θ, y) e 1+β 1 ˆβ w w (θ, y, w) = ŵ(θ, y) + w Identical to the two period problem; extra instrument: promised utility Qualitative result: progressive taxes on (outside) saving; similar for inside saving

116 Taxes on bequest Bequests at t + 1 affects incentives to invest at t + 2

117 Taxes on bequest Bequests at t + 1 affects incentives to invest at t + 2 Higher bequest: higher consumption

118 Taxes on bequest Bequests at t + 1 affects incentives to invest at t + 2 Higher bequest: higher consumption Lower incentive to steal/lie:

119 Taxes on bequest Bequests at t + 1 affects incentives to invest at t + 2 Higher bequest: higher consumption Lower incentive to steal/lie: u (truth) > u (lie)

120 Taxes on bequest Bequests at t + 1 affects incentives to invest at t + 2 Higher bequest: higher consumption Lower incentive to steal/lie: u (truth) > u (lie) Bequest subsidies: τ b < 0

121 Taxes on bequest Bequests at t + 1 affects incentives to invest at t + 2 Higher bequest: higher consumption Lower incentive to steal/lie: u (truth) > u (lie) Bequest subsidies: τ b < 0 Bequests relaxes future incentive constraints: more resources leads to higher investment

122 Taxes on bequest Bequests at t + 1 affects incentives to invest at t + 2 Higher bequest: higher consumption Lower incentive to steal/lie: u (truth) > u (lie) Bequest subsidies: τ b < 0 Bequests relaxes future incentive constraints: more resources leads to higher investment Compare with models with labor income risk

123 Long-Run Dynamics Stationary Market clearing interest rate?

124 Long-Run Dynamics Stationary Market clearing interest rate? Complete Markets: (1 + r) 2 = q 2 = 1ˆβ

125 Long-Run Dynamics Stationary Market clearing interest rate? Complete Markets: (1 + r) 2 = q 2 = 1ˆβ Private information:

126 Long-Run Dynamics Stationary Market clearing interest rate? Complete Markets: (1 + r) 2 = q 2 = 1ˆβ Private information: Risky capital: cut back and save in risk free bond

127 Long-Run Dynamics Stationary Market clearing interest rate? Complete Markets: (1 + r) 2 = q 2 = 1ˆβ Private information: Risky capital: cut back and save in risk free bond lower q

128 Long-Run Dynamics Stationary Market clearing interest rate? Complete Markets: (1 + r) 2 = q 2 = 1ˆβ Private information: Risky capital: cut back and save in risk free bond lower q Lower capital stock

129 Long-Run Dynamics Stationary Market clearing interest rate? Complete Markets: (1 + r) 2 = q 2 = 1ˆβ Private information: Risky capital: cut back and save in risk free bond lower q Lower capital stock higher q

130 Long-Run Dynamics Stationary Market clearing interest rate? Complete Markets: (1 + r) 2 = q 2 = 1ˆβ Private information: Risky capital: cut back and save in risk free bond lower q Lower capital stock higher q Log-preferences: q 2 = ˆβ

131 Long Run Dynamics of Wealth Wealth: P(w t )

132 Long Run Dynamics of Wealth Wealth: P(w t ) Dynamics of wealth: P(w t+2 ) = ˆβr(θ t, y t+1 ) P(w t )

133 Long Run Dynamics of Wealth Wealth: P(w t ) Dynamics of wealth: P(w t+2 ) = ˆβr(θ t, y t+1 ) P(w t ) Key property in creating fat-tailed distribution of wealth

134 Long Run Dynamics of Wealth Wealth: P(w t ) Dynamics of wealth: P(w t+2 ) = ˆβr(θ t, y t+1 ) P(w t ) Key property in creating fat-tailed distribution of wealth Martingale property: P(w t ) = E t P(w t+2 )

135 Long Run Dynamics of Wealth Wealth: P(w t ) Dynamics of wealth: P(w t+2 ) = ˆβr(θ t, y t+1 ) P(w t ) Key property in creating fat-tailed distribution of wealth Martingale property: P(w t ) = E t P(w t+2 ) Wealth converges to zero almost surely; households borrow against the labor/capital income of their children

136 Long Run Dynamics of Wealth Natural fix: w t w

137 Long Run Dynamics of Wealth Natural fix: w t w Interpretations:

138 Long Run Dynamics of Wealth Natural fix: w t w Interpretations: Parents cannot borrow against their children s income Planner puts higher weight on future generations

139 Long Run Dynamics of Wealth Natural fix: w t w Interpretations: Parents cannot borrow against their children s income Planner puts higher weight on future generations Implication: (1 + r) 2 = q 2 > ˆβ

140 Long Run Dynamics of Wealth For high enough wealth levels, i.e., w P(w t+2 ) P(w t ) ˆβr(θ t, y t+1 ) as w t

141 Long Run Dynamics of Wealth For high enough wealth levels, i.e., w P(w t+2 ) P(w t ) ˆβr(θ t, y t+1 ) as w t For low levels of wealth: w c(w, θ, y) > w u(w, θ, y); Higher saving rate at the bottom

142 Long Run Dynamics of Wealth For high enough wealth levels, i.e., w P(w t+2 ) P(w t ) ˆβr(θ t, y t+1 ) as w t For low levels of wealth: w c(w, θ, y) > w u(w, θ, y); Higher saving rate at the bottom Since q 2 > ˆβ, Eˆβr(θ t, y t+1 ) < 1.

143 Long Run Dynamics of Wealth For high enough wealth levels, i.e., w P(w t+2 ) P(w t ) ˆβr(θ t, y t+1 ) as w t For low levels of wealth: w c(w, θ, y) > w u(w, θ, y); Higher saving rate at the bottom Since q 2 > ˆβ, Eˆβr(θ t, y t+1 ) < 1. A result by Mirek (2011): Long-run distribution of P(w t ) is Pareto with tail ν:

144 Long Run Dynamics of Wealth For high enough wealth levels, i.e., w P(w t+2 ) P(w t ) ˆβr(θ t, y t+1 ) as w t For low levels of wealth: w c(w, θ, y) > w u(w, θ, y); Higher saving rate at the bottom Since q 2 > ˆβ, Eˆβr(θ t, y t+1 ) < 1. A result by Mirek (2011): Long-run distribution of P(w t ) is Pareto with tail ν: ) ν (ˆβr(θ, y) dgdf = 1 Θ R

145 Long Run Dynamics of Wealth For high enough wealth levels, i.e., w P(w t+2 ) P(w t ) ˆβr(θ t, y t+1 ) as w t For low levels of wealth: w c(w, θ, y) > w u(w, θ, y); Higher saving rate at the bottom Since q 2 > ˆβ, Eˆβr(θ t, y t+1 ) < 1. A result by Mirek (2011): Long-run distribution of P(w t ) is Pareto with tail ν: ) ν (ˆβr(θ, y) dgdf = 1 Θ R Tail: Pr [P(w t ) > A] A ν

146 Tail of the wealth distribution Efficient Tail:

147 Tail of the wealth distribution Efficient Tail: 6 5 Ν CE Annual InterestRate q 1 30

148 Tail of the wealth distribution Incomplete Market:

149 Tail of the wealth distribution Incomplete Market: q ν IM Non Stationary The tail behavior of stationary distribution in the incomplete market model

150 Conclusion Developed a theory of optimal capital taxes consistent with wealth distribution Method for characterization of efficient wealth distribution Possible application: human capital accumulation and distribution of income