Optimal Tax Progressivity: An Analytical Framework

Transcription

1 Optimal Tax Progressivity: An Analytical Framework Jonathan Heathcote Federal Reserve Bank of Minneapolis Kjetil Storesletten Oslo University and Federal Reserve Bank of Minneapolis Gianluca Violante New York University Federal Reserve Bank of Atlanta January 14th, 2013 Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 1 /41

2 Question How progressive should labor income taxation be? Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 2 /41

3 Question How progressive should labor income taxation be? Argument in favor of progressivity: missing markets 1. Social insurance of privately-uninsurable lifecycle shocks 2. Redistribution with respect to initial conditions Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 2 /41

4 Question How progressive should labor income taxation be? Argument in favor of progressivity: missing markets 1. Social insurance of privately-uninsurable lifecycle shocks 2. Redistribution with respect to initial conditions Arguments against progressivity: distortions 1. Labor supply 2. Human capital investment 3. Public provision of good and services 4. Redistribution from high to low diligence types Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 2 /41

5 Overview of the framework Huggett (1993) economy: -lived agents, idiosyncratic productivity risk, and a risk-free bond in zero net-supply, plus: 1. additional private insurance (other assets, family, etc) Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 3 /41

6 Overview of the framework Huggett (1993) economy: -lived agents, idiosyncratic productivity risk, and a risk-free bond in zero net-supply, plus: 1. additional private insurance (other assets, family, etc) 2. differential innate diligence & (learning) ability 3. endogenous skill investment + multiple-skill technology 4. endogenous labor supply 5. government expenditures valued by households Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 3 /41

7 Overview of the framework Huggett (1993) economy: -lived agents, idiosyncratic productivity risk, and a risk-free bond in zero net-supply, plus: 1. additional private insurance (other assets, family, etc) 2. differential innate diligence & (learning) ability 3. endogenous skill investment + multiple-skill technology 4. endogenous labor supply 5. government expenditures valued by households Tractable equilibrium framework closed-form SWF Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 3 /41

8 Overview of the framework Huggett (1993) economy: -lived agents, idiosyncratic productivity risk, and a risk-free bond in zero net-supply, plus: 1. additional private insurance (other assets, family, etc) 2. differential innate diligence & (learning) ability 3. endogenous skill investment + multiple-skill technology 4. endogenous labor supply 5. government expenditures valued by households Tractable equilibrium framework closed-form SWF Ramsey approach: tax instruments & mkt structure taken as given Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 3 /41

9 The tax/transfer function T(y i ) = y i λy 1 τ i The parameter τ measures the rate of progressivity: τ = 1 : 0 < τ < 1: τ = 0 : τ < 0 : full redistribution ỹ i = λ progressivity T (y) T(y)/y > 1 no redistribution flat tax 1 λ regressivity T (y) T(y)/y < 1 Break-even income level: y 0 = λ 1 τ Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 4 /41

10 The tax/transfer function T(y i ) = y i λy 1 τ i The parameter τ measures the rate of progressivity: τ = 1 : 0 < τ < 1: τ = 0 : τ < 0 : full redistribution ỹ i = λ progressivity T (y) T(y)/y > 1 no redistribution flat tax 1 λ regressivity T (y) T(y)/y < 1 Break-even income level: y 0 = λ 1 τ Restrictions: (i) no lump-sum transfer & (ii) T (y) monotone Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 4 /41

11 Empirical fit to US micro data PSID , age 20-60, N = 13,721: R 2 = 0.96: τ US = Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 5 /41

12 Empirical fit to US micro data PSID , age 20-60, N = 13,721: R 2 = 0.96: τ US = Log of post government income Log of pre government income Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 5 /41

13 Marginal and average tax rates Tax Rate Average Tax Rate Marginal Tax Rate Household Income x 10 5 At the estimated value of τ US = Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 6 /41

14 Demographics and preferences Perpetual youth demographics with constant survival probability δ Preferences over consumption (c), hours (h), publicly-provided goods (G), and skill-investment (s) effort: U i = v i (s i )+E 0 (βδ) t u i (c it,h it,g) t=0 v i (s i ) = 1 κ i s 2 i 2µ u i (c it,h it,g) = logc it exp(ϕ i ) h1+σ it 1+σ +χlogg κ i Exp(η) ϕ i N ( vϕ 2,v ϕ ) Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 7 /41

15 Technology Output is CES aggregator over continuum of skill types: Y = [ 0 N (s) θ 1 θ ] θ θ 1 ds, θ (1, ) Aggregate effective hours by skill type: N(s) = 1 0 I {si =s}z i h i di Aggregate resource constraint: Y = 1 0 c i di+g Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 8 /41

16 Individual efficiency units of labor logz it = α it +ε it α it = α i,t 1 +ω it with ω it N ( v ω 2,v ω ) α i0 = 0 i ε it i.i.d. over time with ε it N ( v ε 2,v ε ) ϕ κ ω ε cross-sectionally and longitudinally Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 9 /41

17 Individual efficiency units of labor logz it = α it +ε it α it = α i,t 1 +ω it with ω it N ( v ω 2,v ω ) α i0 = 0 i ε it i.i.d. over time with ε it N ( v ε 2,v ε ) ϕ κ ω ε cross-sectionally and longitudinally Pre-government earnings: y it = p(s i ) }{{} skill price exp(α it +ε it ) }{{} efficiency determined by skill, fortune, and diligence h it }{{} hours Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 9 /41

18 Government Disposable (post-government) earnings: ỹ i = λy 1 τ i Government budget constraint (no government debt): G = 1 0 [ yi λy 1 τ i ] di Government chooses (G,τ), and λ balances the budget residually WLOG: G g Y Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 10 /41

19 Markets Competitive good and labor markets Perfect annuity markets against survival risk Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 11 /41

20 Markets Competitive good and labor markets Perfect annuity markets against survival risk Competitive asset markets (all assets in zero net supply) Non state-contingent bond Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 11 /41

21 Markets Competitive good and labor markets Perfect annuity markets against survival risk Competitive asset markets (all assets in zero net supply) Non state-contingent bond Full set of insurance claims against ε shocks If v ε = 0, it is a bond economy If v ω = 0, it is a full insurance economy If v ω = v ε = v ϕ = 0 & θ =, it is a RA economy Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 11 /41

22 A special case: the representative agent max C,H U = logc H1+σ 1+σ +χloggy s.t. C = λy 1 τ and Y = H Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 12 /41

23 A special case: the representative agent max C,H U = logc H1+σ 1+σ +χloggy s.t. C = λy 1 τ and Y = H Market clearing: C +G = Y Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 12 /41

24 A special case: the representative agent max C,H U = logc H1+σ 1+σ +χloggy s.t. C = λy 1 τ and Y = H Market clearing: C +G = Y Equilibrium allocations: logh RA (τ) = 1 1+σ log(1 τ) logc RA (g,τ) = log(1 g)+logh RA (τ) Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 12 /41

25 Optimal policy in the RA economy Welfare function: W RA (g,τ) = log(1 g)+χlogg+(1+χ) log(1 τ) 1+σ 1 τ 1+σ Welfare maximizing (g,τ) pair: g = χ 1+χ τ = χ Solution for g will extend to heterogeneous-agent setup Allocations are first best (same as with lump-sum taxes) Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 13 /41

26 Recursive stationary equilibrium Given (G,τ), a stationary RCE is a value λ, asset prices {Q( ),q}, skill prices p(s), decision rules s(ϕ,κ,0), c(α,ε,ϕ,s,b), h(α,ε,ϕ,s,b), and aggregate quantities N(s) such that: households optimize markets clear the government budget constraint is balanced Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 14 /41

27 Recursive stationary equilibrium Given (G,τ), a stationary RCE is a value λ, asset prices {Q( ),q}, skill prices p(s), decision rules s(ϕ,κ,0), c(α,ε,ϕ,s,b), h(α,ε,ϕ,s,b), and aggregate quantities N(s) such that: households optimize markets clear the government budget constraint is balanced In equilibrium, bonds are not traded b = 0 allocations depend only on exogenous states α shocks remain uninsured, ε shocks fully insured Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 14 /41

28 Equilibrium skill choice and skill price

29 Equilibrium skill choice and skill price Skill price has Mincerian shape: logp(s;τ) = π 0 (τ)+π 1 (τ)s(κ;τ) s(κ;τ) = π 1 (τ) = ηµ(1 τ) θ η θµ(1 τ) κ (skill choice) (marginal return to skill)

30 Equilibrium skill choice and skill price Skill price has Mincerian shape: logp(s;τ) = π 0 (τ)+π 1 (τ)s(κ;τ) s(κ;τ) = π 1 (τ) = ηµ(1 τ) θ η θµ(1 τ) κ (skill choice) (marginal return to skill) Distribution of skill prices (in levels) is Pareto with parameter θ

31 Equilibrium skill choice and skill price Skill price has Mincerian shape: logp(s;τ) = π 0 (τ)+π 1 (τ)s(κ;τ) s(κ;τ) = π 1 (τ) = ηµ(1 τ) θ η θµ(1 τ) κ (skill choice) (marginal return to skill) Distribution of skill prices (in levels) is Pareto with parameter θ Offsetting effects of τ on s and p on pre-tax wage inequality var(logp(s;τ)) = 1 θ 2 Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 15 /41

32 Equilibrium consumption allocation logc (α,ϕ,s;g,τ) = logc RA (g,τ)+(1 τ)logp(s;τ) }{{} skill price +(1 τ)α }{{} unins. shock (1 τ)ϕ }{{} pref. het. + M(v ε ) }{{} level effect from ins. variation Response to variation in (p, ϕ, α) mediated by progressivity Invariant to insurable shock ε Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 16 /41

33 Equilibrium hours allocation logh (ε,ϕ;τ) = logh RA (τ) ϕ }{{} pref. het. 1 σ(1 τ) M(v ε) }{{} level effect from ins. variation + 1 σ ε }{{} ins. shock Response to ε mediated by tax-modified Frisch elasticity 1ˆσ = 1 τ σ+τ Invariant to skill price p, uninsurable shock α, and size of g Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 17 /41

34 Social Welfare Function Planner chooses constant (g, τ) Make two assumptions: 1. Planner puts equal weight on period utility of all currently alive agents, discounts future at rate β 2. Skill investments are fully reversible Then, SWF becomes average period utility in the cross-section Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 18 /41

35 Exact expression for SWF log(1 τ) W(g,τ) = log(1+g)+χlogg +(1+χ) (1+ ˆσ)(1 τ) 1 (1+ ˆσ) [ ( ) 1 +(1+χ) θ 1 log ηθ + θ ( ) ] θ µ(1 τ) θ 1 log θ 1 1 ( ( )) ( )] 1 τ 1 τ [ log 2θ (1 τ) 1 θ θ (1 τ) 2 v ϕ 2 ( ) δ v ω (1 τ) 1 δ 2 log 1 δexp τ(1 τ) 2 v ω 1 δ (1+χ)σ 1ˆσ 2 v ε 2 +(1+χ)1ˆσ v ε Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 19 /41

36 Representative Agent component log(1 τ) W(g,τ) = log(1+g)+χlogg +(1+χ) (1+ ˆσ)(1 τ) 1 (1+ ˆσ) }{{} Representative Agent Welfare = W RA (g,τ) [ ( ) 1 +(1+χ) θ 1 log ηθ + θ ( ) ] θ µ(1 τ) θ 1 log θ 1 1 ( ( )) ( )] 1 τ 1 τ [ log 2θ (1 τ) 1 θ θ (1 τ) 2 v ϕ 2 ( ) δ v ω (1 τ) 1 δ 2 log 1 δexp τ(1 τ) 2 v ω 1 δ (1+χ)σ 1ˆσ 2 v ε 2 +(1+χ)1ˆσ v ε Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 20 /41

37 Exact expression for SWF W(τ) = χlogχ (1+χ)log(1+χ)+(1+χ) +(1+χ) [ 1 θ 1 log 1 2θ (1 τ) [ log ( ( 1 log(1 τ) (1+ ˆσ)(1 τ) 1 (1+ ˆσ) ) ηθ + θ ( ) ] θ θ 1 log θ 1 ( )) ( )] 1 τ 1 τ θ θ µ(1 τ) (1 τ) 2 v ϕ 2 ( ) δ v ω (1 τ) 1 δ 2 log 1 δexp τ(1 τ) 2 v ω 1 δ (1+χ)σ 1ˆσ 2 v ε 2 +(1+χ)1ˆσ v ε Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 21 /41

38 Skill investment component log(1 τ) W(τ) = χlogχ (1+χ)log(1+χ)+(1+χ) (1+ ˆσ)(1 τ) 1 (1+ ˆσ) [ ( ) 1 +(1+χ) θ 1 log ηθ + θ ( ) ] θ µ(1 τ) θ 1 log θ 1 }{{} 1 2θ (1 τ) }{{} avg. education cost productivity gain = loge [(p(s))] = log(y/n) [ ( ( )) ( )] 1 τ 1 τ log 1 θ θ }{{} consumption dispersion across skills (1 τ) 2 v ϕ 2 ( ) δ v ω (1 τ) 1 δ 2 log 1 δexp τ(1 τ) 2 v ω 1 δ (1+χ)σ 1ˆσ 2 v ε 2 +(1+χ)1ˆσ v ε Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 22 /41

39 Skill investment component Optimal degree of progressivity (τ) Elasticity of substitution in production (θ) Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 23 /41

40 Optimal marginal tax rate at the top Diamond-Saez formula (Mirlees approach): t = 1 1+ζ u +ζ c (θ 1) = 1+σ θ +σ Decreasing in θ (increasing in thickness of Pareto tail) Our model: there is a range where τ is increasing in θ When complementarity is high, cost of distorting skill investment is large Key difference: exogenous vs endogenous skill distribution Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 24 /41

41 Uninsurable component W(τ) = χlogχ (1+χ)log(1+χ)+(1+χ) +(1+χ) [ 1 θ 1 log 1 2θ (1 τ) [ log ( ( 1 log(1 τ) (1+ ˆσ)(1 τ) 1 (1+ ˆσ) ) ηθ + θ ( ) ] θ θ 1 log θ 1 ( )) ( )] 1 τ 1 τ θ θ µ(1 τ) (1 τ) 2 v ϕ }{{ 2 } cons. disp. due to prefs ( ) δ v ω (1 τ) 1 δ 2 log 1 δexp τ(1 τ) 2 v ω 1 δ }{{} consumption dispersion due to uninsurable shocks (1 τ) 2 v α 2 (1+χ)σ 1ˆσ 2 v ε 2 +(1+χ)1ˆσ v ε Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 25 /41

42 Insurable component W(τ) = χlogχ (1+χ)log(1+χ)+(1+χ) +(1+χ) [ 1 θ 1 log 1 2θ (1 τ) [ log ( ( 1 log(1 τ) (1+ ˆσ)(1 τ) 1 (1+ ˆσ) ) ηθ + θ ( ) ] θ θ 1 log θ 1 ( )) ( )] 1 τ 1 τ θ θ µ(1 τ) (1 τ) 2 v ϕ 2 ( ) δ v ω (1 τ) 1 δ 2 log 1 δexp τ(1 τ) 2 v ω 1 δ 1 v ε 1 (1+χ)σ +(1+χ) } ˆσ 2 {{ 2 } ˆσ v ε }{{} hours dispersion prod. gain from ins. shock=log(n/h) Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 26 /41

43 Parameterization Parameter vector {χ,σ,θ,v ϕ,v ω,v ε } Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 27 /41

44 Parameterization Parameter vector {χ,σ,θ,v ϕ,v ω,v ε } To match G/Y = 0.19: χ = 0.23 Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 27 /41

45 Parameterization Parameter vector {χ,σ,θ,v ϕ,v ω,v ε } To match G/Y = 0.19: χ = 0.23 Frisch elasticity (micro-evidence): σ = 2 Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 27 /41

46 Parameterization Parameter vector {χ,σ,θ,v ϕ,v ω,v ε } To match G/Y = 0.19: χ = 0.23 Frisch elasticity (micro-evidence): σ = 2 cov(logh,logw) = 1ˆσ v ε var(logh) = v ϕ + 1ˆσ 2v ε var 0 (logc) = (1 τ) 2 ( v ϕ + 1 θ 2 ) var(logw) = 1 θ 2 + δ 1 δ v ω Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 27 /41

47 Parameterization Parameter vector {χ,σ,θ,v ϕ,v ω,v ε } To match G/Y = 0.19: χ = 0.23 Frisch elasticity (micro-evidence): σ = 2 cov(logh,logw) = 1ˆσ v ε v ε = 0.17 var(logh) = v ϕ + 1ˆσ 2v ε v ϕ = var 0 (logc) = (1 τ) 2 ( v ϕ + 1 θ 2 ) θ = 2.16 var(logw) = 1 θ 2 + δ 1 δ v ω v ω = Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 27 /41

48 Optimal progressivity 1 Social Welfare Function welfare change rel. to optimum (% of cons.) Welfare Gain = 0.5% τ * = τ US = Baseline Model Progressivity rate (τ) Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 28 /41

49 Optimal progressivity: decomposition welfare change rel. to optimum (% of cons.) Social Welfare Function (1) Rep. Agent τ = Progressivity rate (τ) Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 29 /41

50 Optimal progressivity: decomposition welfare change rel. to optimum (% of cons.) (2) + Skill Inv. τ = Social Welfare Function (1) Rep. Agent τ = Progressivity rate (τ) Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 29 /41

51 Optimal progressivity: decomposition welfare change rel. to optimum (% of cons.) (2) + Skill Inv. τ = Social Welfare Function (1) Rep. Agent τ = Progressivity rate (τ) (3) + Pref. Het. τ = Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 29 /41

52 Optimal progressivity: decomposition welfare change rel. to optimum (% of cons.) (2) + Skill Inv. τ = Social Welfare Function (1) Rep. Agent τ = Progressivity rate (τ) (3) + Pref. Het. τ = (4) + Unins. Shocks τ = Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 29 /41

53 Optimal progressivity: decomposition welfare change rel. to optimum (% of cons.) (2) + Skill Inv. τ = Social Welfare Function (1) Rep. Agent τ = (5) + Ins. Shocks τ * = Progressivity rate (τ) (3) + Pref. Het. τ = (4) + Unins. Shocks τ = Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 29 /41

54 Actual and optimal progressivity Average tax rate Actual US τ US = Utilitarian τ = Income (1 = average income) Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 30 /41

55 Alternative SWF Utilitarian SWF embeds desire to insure and to redistribute wrt (κ, ϕ) Switch off desire to redistribute Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 31 /41

56 Alternative SWF Utilitarian SWF embeds desire to insure and to redistribute wrt (κ, ϕ) Switch off desire to redistribute Economy with heterogeneity in (κ,ϕ), and χ = v ω = τ = 0 Compute CE allocations Compute Negishi weights s.t. planner s allocation = CE Use these weights in the SWF Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 31 /41

57 Alternative SWF Utilitarian κ-neutral ϕ-neutral Insurance-only Redist. wrt κ Y N Y N Redist. wrt ϕ Y Y N N Insurance wrt ω Y Y Y Y τ Welf. gain (pct of c) Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 32 /41

58 Optimal progressivity: alternative SWF Average tax rate Actual US τ US = Utilitarian τ = Ins. Only τ = Income (1 = average income) Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 33 /41

59 Alternative assumptions on G Case Utilitarian Insurance-only G Y τ Welf. gain τ Welf. gain g endogenous χ = % % ḡ exogenous χ = % % Ḡ exogenous χ = % % With χ = 0, no externality and no push towards regressivity If G fixed in level, lower progressivity translates 1-1 into higher c Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 34 /41

60 Irreversible skill investment Assume skills are fixed after being chosen Planner has an additional incentive to increase progressivity: Can compress consumption inequality, while discouraging skill investment only for new generations (not for current ones) Needed to preserve tractability: production is segregated by age Planner chooses τ = Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 35 /41

61 Irreversible skill investment Assume skills are fixed after being chosen Planner has an additional incentive to increase progressivity: Can compress consumption inequality, while discouraging skill investment only for new generations (not for current ones) Needed to preserve tractability: production is segregated by age Planner chooses τ = Hybrid model: optimal progressivity is, again, τ = Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 35 /41

62 Political Economy Think of g and τ as outcomes of a majority rule voting process Median voter theorem applies in our model! Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 36 /41

63 Political Economy Think of g and τ as outcomes of a majority rule voting process Median voter theorem applies in our model! Agents agree over spending and choose g = χ/(1+χ)... but they disagree over τ In spite of multi-dimensional heterogeneity over (ϕ, κ, α), preferences are single-peaked in τ Median voter chooses τ med = More progressivity because median voter poorer than average Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 36 /41

64 Progressive consumption taxation c = λ c 1 τ where c are expenditures and c are units of final good Implement as a tax on total (labor plus asset) income less saving Consumption depends on α but not on ε Can redistribute wrt. uninsurable shocks without distorting the efficient response of hours to insurable shocks Higher progressivity and higher welfare Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 37 /41

65 Budget constraints 1. Beginning of period: innovation ω to α shock is realized 2. Middle of period: buy insurance against ε: b = Q(ε)B(ε)dε, where Q( ) is the price of insurance and B( ) is the quantity E 3. End of period: ε realized, consumption and hours chosen: c+δqb = λ[p(s)exp(α+ε)h] 1 τ +B(ε) Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 38 /41

66 No bond-trade equilibrium Micro-foundations for Constantinides and Duffie (1996) CRRA, unit root shocks to log disposable income In equilibrium, no bond-trade c t = ỹ t Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 39 /41

67 No bond-trade equilibrium Micro-foundations for Constantinides and Duffie (1996) CRRA, unit root shocks to log disposable income In equilibrium, no bond-trade c t = ỹ t Unit root disposable income micro-founded in our model: 1. Skill investment+shocks: wages 2. Labor supply choice: wages pre-tax earnings 3. Non-linear taxation: pre-tax earnings after-tax earnings 4. Private risk sharing: after-tax earnings disp. income 5. No bond trade: disposable income = consumption Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 39 /41

68 Equilibrium risk-free rater ρ r = (1 τ)((1 τ)+1) v ω 2 Intertemporal dis-saving motive = precautionary saving motive Key: precautionary saving motive common across all agents r τ > 0: more progressivity less precautionary saving higher risk-free rate Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 40 /41

69 Upper tail of wage distribution x 10 4 Top 1pct of the Wage Distribution 2 Model Wage Distribution Lognormal Wage Distribution Density Wage (average=1) Heathcote-Storesletten-Violante, Optimal Tax Progressivity p. 41 /41