Redistributive Taxation in a Partial-Insurance Economy

Redistributive Taxation in a Partial-Insurance Economy Jonathan Heathcote Federal Reserve Bank of Minneapolis and CEPR Kjetil Storesletten Federal Reserve Bank of Minneapolis and CEPR Gianluca Violante New York University, CEPR and NBER NYU Macro Lunch, April 1st, 2010 Heathcote-Storesletten-Violante, Redistributive Taxation p. 1/34

Taxation Two classic roles for government taxation: 1. Provision of public goods 2. Redistribution / insurance At the same time, the design of taxes and transfers must be sensitive to private incentives In light of these objectives, how progressive should the tax system be? Heathcote-Storesletten-Violante, Redistributive Taxation p. 2/34

More specifically How does the optimal rate of progressivity depend upon... 1. the elasticity of labor supply 2. the level of inequality in the economy 3. the amount of risk and privately-provided insurance 4. the desire for consuming public goods Heathcote-Storesletten-Violante, Redistributive Taxation p. 3/34

More specifically How does the optimal rate of progressivity depend upon... 1. the elasticity of labor supply 2. the level of inequality in the economy 3. the amount of risk and privately-provided insurance 4. the desire for consuming public goods Our contribution: tractable framework that delivers insights on the trade-offs Heathcote-Storesletten-Violante, Redistributive Taxation p. 3/34

Approach: Ramsey problem with incomplete markets Take tax instruments and market structure as given Generalization: 1. nonlinear tax/transfer system (no lump-sum taxes) 2. endogenous choice of G 3. heterogeneous agents and incomplete markets redistributive motive HSV vs. Conesa-Krueger and Bohacek-Kejak: more restrictive tax function, but analytical solution Heathcote-Storesletten-Violante, Redistributive Taxation p. 4/34

The Model (H-S-V, 2009) Equilibrium heterogeneous-agents model featuring: 1. differential labor productivity + idiosyncratic productivity risk 2. flexible labor supply and risk-free bond (self-insurance) 3. additional risk-sharing (financial markets, family, etc.) Heathcote-Storesletten-Violante, Redistributive Taxation p. 5/34

The Model (H-S-V, 2009) Equilibrium heterogeneous-agents model featuring: 1. differential labor productivity + idiosyncratic productivity risk 2. flexible labor supply and risk-free bond (self-insurance) 3. additional risk-sharing (financial markets, family, etc.) 4. nonlinear tax/transfer system 5. government expenditures valued by households Heathcote-Storesletten-Violante, Redistributive Taxation p. 5/34

Technology and resource constraint Aggregate technology linear in effective labor: Y = w i h i di y i di Resource constraint: Y = c i di + G Heathcote-Storesletten-Violante, Redistributive Taxation p. 6/34

Demographics and preferences Perpetual youth demographics with constant survival probability δ Heathcote-Storesletten-Violante, Redistributive Taxation p. 7/34

Demographics and preferences Perpetual youth demographics with constant survival probability δ Preferences over sequences of private consumption, hours worked, and public good: with period-utility: U(c i, h i, G) = E 0 (βδ) t u(c it, h it, G t ) t=0 u (c it, h it, G t ) = c1 γ it 1 1 γ ϕ h1+σ χg1 γ it 1 + σ + t 1 1 γ Heathcote-Storesletten-Violante, Redistributive Taxation p. 7/34

Individual labor productivity Individual endowments of efficiency units of labor: lnw it = α it + ε it Heathcote-Storesletten-Violante, Redistributive Taxation p. 8/34

Individual labor productivity Individual endowments of efficiency units of labor: lnw it = α it + ε it α it component follows unit root process α it = α i,t 1 + ω it with ω it F ω and α i0 F α0 Heathcote-Storesletten-Violante, Redistributive Taxation p. 8/34

Individual labor productivity Individual endowments of efficiency units of labor: lnw it = α it + ε it α it component follows unit root process α it = α i,t 1 + ω it with ω it F ω and α i0 F α0 ε it component is transitory ε it i.i.d. with ε it F ε Heathcote-Storesletten-Violante, Redistributive Taxation p. 8/34

Financial and insurance markets Assets traded competitively (all in zero net supply) Perfect annuity against survival risk Non-contingent bond Complete markets for ε shocks Heathcote-Storesletten-Violante, Redistributive Taxation p. 9/34

Financial and insurance markets Assets traded competitively (all in zero net supply) Perfect annuity against survival risk Non-contingent bond Complete markets for ε shocks Market structure v ε = 0 bond economy v α = 0 full insurance In between: partial insurance Heathcote-Storesletten-Violante, Redistributive Taxation p. 9/34

Government Two-parameter tax/transfer function to redistribute/insure and finance public good G Heathcote-Storesletten-Violante, Redistributive Taxation p. 10/34

Government Two-parameter tax/transfer function to redistribute/insure and finance public good G Disposable post-government earnings: ỹ i = λy 1 τ i Government budget constraint (no government debt): G = [yi λy 1 τ i ] di Given (G, τ), λ balances the budget in equilibrium Heathcote-Storesletten-Violante, Redistributive Taxation p. 10/34

Our model of fiscal redistribution The parameter τ measures the rate of progressivity: ln(ỹ i ) = const + (1 τ) ln(y i ) 1. τ = 0 ỹ i = λy i : no redistribution, i.e. flat tax (1 λ) 2. τ = 1 ỹ i = λ: full redistribution Then, if τ > 0: 1. The system is progressive: T (y) T(y)/y = 1 λ(1 τ)y τ 1 λy τ > 1 y 2. The system generates a transfer (ỹ i > y i ) for low earnings Heathcote-Storesletten-Violante, Redistributive Taxation p. 11/34

Empirical relevance of our model for taxes / transfers CPS 1980-2005, positive labor income: 1,026,829 obs. Estimated slope of model line (R 2 = 0.88) yields τ = 0.26 Heathcote-Storesletten-Violante, Redistributive Taxation p. 12/34

Empirical relevance of our model for taxes / transfers CPS 1980-2005, positive labor income: 1,026,829 obs. Estimated slope of model line (R2 = 0.88) yields τ = 0.26 6 Log Household Disposable Income 8 10 12 14 Fiscal Progressivity in Year 2000 6 8 10 12 Log Household Pre Government Income Data 14 Model Heathcote-Storesletten-Violante, Redistributive Taxation p. 12/34

Household s Problem V (α, b) = max c,h,b,b( ) E [ u (c, h, G) + δβ Ω V (α + ω, b )df ω ] df ε subject to Q ( )B ( ) dε = b E c + qδb = B (ε) + λ (exp(α + ε) h) 1 τ ε c 0, h 0, b B b 0 = 0 A stationary equilibrium is a set of prices (q, Q ( )) and a policy (G, τ, λ) s.t. (i) agents take these as given and optimize, (ii) markets clear, and (iii) the government budget is balanced Heathcote-Storesletten-Violante, Redistributive Taxation p. 13/34

No bond trading equilibrium In equilibrium, agents choose not to save/borrow, and individual allocations only depend on (α, ε) Heathcote-Storesletten-Violante, Redistributive Taxation p. 14/34

No bond trading equilibrium In equilibrium, agents choose not to save/borrow, and individual allocations only depend on (α, ε) Generalization of Constantinides and Duffie (JPE, 1996) who assume unit root shocks to log-disposable income Our environment micro-founds I(1) disposable income: 1. Individual wage process richer than unit root 2. Flexible labor supply earnings endogenous 3. Private risk sharing 4. Non-linear taxation and transfers Heathcote-Storesletten-Violante, Redistributive Taxation p. 14/34

Equilibrium risk-free rate r Under log-normality of the shocks, closed form for r With inelastic labor (σ = ) and linear taxes (τ = 0): ρ r γ = (γ + 1) v ω 2 where (γ + 1) is the coefficient of relative prudence With inelastic labor and non-linear taxes (τ 0): ρ r γ = (1 τ) (γ (1 τ) + 1) v ω 2 r τ > 0: more progressivity more government insurance less precautionary saving higher risk-free rate Heathcote-Storesletten-Violante, Redistributive Taxation p. 15/34

Equilibrium allocations: hours worked lnh (α, ε) = 1 (1 τ) ( σ + γ) [(1 γ) lnλ + ln(1 τ) ϕ] Representative agent M h (v ε ) Wealth effect + 1 γ σ + γ α Unins. shock + 1 σ ε Insurable shock Heathcote-Storesletten-Violante, Redistributive Taxation p. 16/34

Equilibrium allocations: hours worked lnh (α, ε) = 1 (1 τ) ( σ + γ) [(1 γ) lnλ + ln(1 τ) ϕ] Representative agent M h (v ε ) Wealth effect + 1 γ σ + γ α Unins. shock + 1 σ ε Insurable shock Tax-modified Frisch elasticity (decreasing in τ): 1 σ 1 τ σ + τ γ measures the relative strength of income vs. substitution effect for uninsurable shocks Heathcote-Storesletten-Violante, Redistributive Taxation p. 16/34

Equilibrium allocations: consumption ln c (α) = 1 σ + γ [(1 + σ) lnλ + ln(1 τ) ϕ] Representative agent + M c (v ε ) Wealth effect + π(γ, σ, τ)α Uninsurable shocks Heathcote-Storesletten-Violante, Redistributive Taxation p. 17/34

Equilibrium allocations: consumption ln c (α) = 1 σ + γ [(1 + σ) lnλ + ln(1 τ) ϕ] Representative agent + M c (v ε ) Wealth effect + π(γ, σ, τ)α Uninsurable shocks The transmission coefficient of a permanent uninsured shock: [ ] σ + γ π(γ, σ, τ) = (1 τ) σ + γ + τ (1 γ) TAX PROGRESSIVITY σ + 1 σ + γ LABOR SUPPLY Heathcote-Storesletten-Violante, Redistributive Taxation p. 17/34

Equilibrium allocations: consumption ln c (α) = 1 σ + γ [(1 + σ) lnλ + ln(1 τ) ϕ] Representative agent + M c (v ε ) Wealth effect + π(γ, σ, τ)α Uninsurable shocks The transmission coefficient of a permanent uninsured shock: [ ] σ + γ π(γ, σ, τ) = (1 τ) σ + γ + τ (1 γ) TAX PROGRESSIVITY Quantitatively (γ = σ = 2, τ = 0.26): σ + 1 σ + γ LABOR SUPPLY 0.60 = 0.74 1.07 = 0.79 0.75 Heathcote-Storesletten-Violante, Redistributive Taxation p. 17/34

Government s problem (F α, F ε ) are the aggregate states, exogenous and time-invariant: given (τ, G), economy is in a stationary no bond-trade equilibrium Government puts weight β t on the welfare of all agents born at dates t =,...,, and chooses a pair (τ, G) The Social Welfare Function becomes: W(τ, G) 1 u (c (α; τ, G), h (α, ε; τ, G), G)dF ε df α 1 β Heathcote-Storesletten-Violante, Redistributive Taxation p. 18/34

Government s problem (F α, F ε ) are the aggregate states, exogenous and time-invariant: given (τ, G), economy is in a stationary no bond-trade equilibrium Government puts weight β t on the welfare of all agents born at dates t =,...,, and chooses a pair (τ, G) The Social Welfare Function becomes: W(τ, G) 1 u (c (α; τ, G), h (α, ε; τ, G), G)dF ε df α 1 β The government chooses the pair (τ, G) to maximize W(τ, G) subject to: (i) (c, h ) are competitive equilibrium allocations; and (ii) the government budget constraint is satisfied Heathcote-Storesletten-Violante, Redistributive Taxation p. 18/34

Roadmap for welfare analysis Assumption: log-normal shocks Assumption: log-utility over private and public goods (γ = 1) Heathcote-Storesletten-Violante, Redistributive Taxation p. 19/34

Roadmap for welfare analysis Assumption: log-normal shocks Assumption: log-utility over private and public goods (γ = 1) 1. No utility from public goods (χ = 0) Instrument chosen: τ 2. Valued G (χ > 0) Instruments chosen: (τ, G) Heathcote-Storesletten-Violante, Redistributive Taxation p. 19/34

Social welfare function (χ = 0) Representative agent (v α = 0, v ε = 0): W RA (τ) = ϕ + ln(1 τ) (1 τ) 1 + σ Welfare maximizing progressivity: τ = 0 Heathcote-Storesletten-Violante, Redistributive Taxation p. 20/34

Social welfare function (χ = 0) Representative agent (v α = 0, v ε = 0): W RA (τ) = ϕ + ln(1 τ) (1 τ) 1 + σ Welfare maximizing progressivity: τ = 0 Heterogeneous agents (v α > 0, v ε > 0): W (τ) = W RA (τ) + 1ˆσ v ε ln(y/h) ( ) 1 vε σ ˆσ 2 2 var(ln h) (1 τ) 2 v α 2 var(ln c) Heathcote-Storesletten-Violante, Redistributive Taxation p. 20/34

Comparative statics on the optimal progressivity rate Heathcote-Storesletten-Violante, Redistributive Taxation p. 21/34

Comparative statics on the optimal progressivity rate W (τ) is globally concave in τ if σ 2 Heathcote-Storesletten-Violante, Redistributive Taxation p. 21/34

Comparative statics on the optimal progressivity rate W (τ) is globally concave in τ if σ 2 τ v α > 0: more uninsurable risk more public insurance W(τ) τ τ=0 > 0 iff v α > 0 strictly positive solution for τ τ σ > 0: less elastic labor supply less severe distortions Heathcote-Storesletten-Violante, Redistributive Taxation p. 21/34

Comparative statics on the optimal progressivity rate W (τ) is globally concave in τ if σ 2 τ v α > 0: more uninsurable risk more public insurance W(τ) τ τ=0 > 0 iff v α > 0 strictly positive solution for τ τ σ > 0: less elastic labor supply less severe distortions τ v ε < 0: more insurable risk more severe distortions Heathcote-Storesletten-Violante, Redistributive Taxation p. 21/34

Comparative statics on the optimal progressivity rate W (τ) is globally concave in τ if σ 2 τ v α > 0: more uninsurable risk more public insurance W(τ) τ τ=0 > 0 iff v α > 0 strictly positive solution for τ τ σ > 0: less elastic labor supply less severe distortions τ v ε < 0: more insurable risk more severe distortions τ ϕ = 0: independent of the disutility of work Heathcote-Storesletten-Violante, Redistributive Taxation p. 21/34

Optimal progressivity with χ = 0 0.55 0.56 0.57 v α Utility of RA (G nonvalued) Optimal Social Welfare 0.58 0.59 0.6 v ε 0.61 0.62 Actual US 0.63 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 Progressivity rate (τ) Parameterization: σ = 2, v α = v ε = 0.14 τ (σ, v α, v ε ) = 0.21 Heathcote-Storesletten-Violante, Redistributive Taxation p. 22/34

Valued government consumption: χ > 0 Define g G/Y Welfare of the representative agent (v α = v ε = 0): W RA (1 + χ) ln(1 τ) (1 τ) (τ, g) = (1 + χ)ϕ + 1 + σ + ln(1 g) + χ lng Welfare-maximizing fiscal policy is given by the pair: g = χ 1 + χ Samuelson s condition τ = χ Regressive taxation Heathcote-Storesletten-Violante, Redistributive Taxation p. 23/34

Intuition Allocations (C, H, G ) induced by (g, τ ) are first best Heathcote-Storesletten-Violante, Redistributive Taxation p. 24/34

Intuition Allocations (C, H, G ) induced by (g, τ ) are first best Optimal regressivity (τ = χ) achieves both: desired average tax rate (to finance G) zero marginal tax rate at H (as with a lump-sum tax) Public good externality: H is larger than it would be absent taxes: taxation is used to increase hours worked to socially efficient level Heathcote-Storesletten-Violante, Redistributive Taxation p. 24/34

Valued govt. consumption and heterogeneity W (τ, g) = W RA (τ, g) + (1 + χ) [ 1 ˆσ v ε ln(y/h) ( ) 1 vε σ ˆσ 2 2 var(ln h) ] (1 τ) 2 v α 2 var(ln c) Heathcote-Storesletten-Violante, Redistributive Taxation p. 25/34

Valued govt. consumption and heterogeneity W (τ, g) = W RA (τ, g) + (1 + χ) [ 1 ˆσ v ε ln(y/h) ( ) 1 vε σ ˆσ 2 2 var(ln h) ] (1 τ) 2 v α 2 var(ln c) Optimal public good provision is unchanged: g = χ/(1 χ) Trade-off in determining optimal rate of progressivity: More rigid labor supply (higher σ) more progressive taxation More uninsurable risk (higher v α ) more progressive taxation More insurable risk (higher v ε ) flatter taxation Stronger taste for G (higher χ) more regressive taxation Heathcote-Storesletten-Violante, Redistributive Taxation p. 25/34

Progressive or regressive taxation? Parameter space can be divided into two regions: χ > v α (1 + σ) τ < 0 χ = v α (1 + σ) τ = 0 χ < v α (1 + σ) τ > 0 Heathcote-Storesletten-Violante, Redistributive Taxation p. 26/34

Progressive or regressive taxation? Parameter space can be divided into two regions: χ > v α (1 + σ) τ < 0 χ = v α (1 + σ) τ = 0 χ < v α (1 + σ) τ > 0 Insurable risk v ε irrelevant because at τ = 0 labor supply response to insurable shocks is undistorted Heathcote-Storesletten-Violante, Redistributive Taxation p. 26/34

Progressive or regressive taxation? Parameter space can be divided into two regions: χ > v α (1 + σ) τ < 0 χ = v α (1 + σ) τ = 0 χ < v α (1 + σ) τ > 0 Insurable risk v ε irrelevant because at τ = 0 labor supply response to insurable shocks is undistorted With v α = v ε = 0.14, and χ = 0.25 (g = 0.2) σ = 0.8 τ = 0.00 (flat) σ = 2.0 τ = 0.07 (optimal) σ = 6.3 τ = 0.26 (actual US) Heathcote-Storesletten-Violante, Redistributive Taxation p. 26/34

Optimal progressivity with χ = 0.25 1.14 1.16 1.18 Utility of RA (G valued) Optimal Social Welfare 1.2 1.22 v α v ε 1.24 1.26 1.28 Actual US 0.3 0.2 0.1 0 0.1 0.2 0.3 Progressivity rate (τ) Heathcote-Storesletten-Violante, Redistributive Taxation p. 27/34

Average tax rate: actual US vs optimal Average tax rate 0.5 0.4 0.3 0.2 0.1 0 0.1 0.2 τ = 0.26 τ = 0.21 τ = 0.07 Actual US Optimal when G nonvalued Optimal when G valued 0.5 1 1.5 2 2.5 3 3.5 4 Labor Income Heathcote-Storesletten-Violante, Redistributive Taxation p. 28/34

Progressive consumption taxation President s Advisory Panel on Tax Reform (2005) lists a progressive consumed income tax among its proposals Implementation: progressive income tax with full deduction for savings Argument: avoids distortions to capital accumulation, while retaining scope for redistribution Additional argument: consumption taxes redistribute wrt. uninsurable shocks without distorting the efficient response of hours to insurable shocks Heathcote-Storesletten-Violante, Redistributive Taxation p. 29/34

Progressive consumption taxation Individual budget constraint: y = c = λc 1 1 τ where c are expenditures and c physical units Welfare can be expressed as: W (τ, g) = W RA (τ, g) + ( 1 + χ 2 ) 1 σ v ε ln(y/h) (1 τ) 2 v α 2 var(ln c) No misallocation of labor under this tax scheme one can achieve higher welfare than with income taxation Heathcote-Storesletten-Violante, Redistributive Taxation p. 30/34

Concluding remarks Tractable incomplete-markets model to study the two key roles of fiscal policy: redistribution/insurance and public good provision Characterization of optimal fiscal policy ( Ramsey-style ) What s next? Solve model for general CRRA preferences (γ 1) Quantify how much of G is pure public good (e.g., defense) and how much is a transfer (e.g., public education) Politico-economic analysis Relationship with static Mirlees problem Heathcote-Storesletten-Violante, Redistributive Taxation p. 31/34

Political economics Policy is determined in a repeated voting game Each period the agent with median α picks (τ, G) Heathcote-Storesletten-Violante, Redistributive Taxation p. 32/34

Political economics Policy is determined in a repeated voting game Each period the agent with median α picks (τ, G) No strategic interaction between successive median voters each median voter solves a static maximization problem For γ = 1 and α N ( v α 2, v α ) the outcome of the voting game is equal to the Ramsey policy... same redistribution, but for different motives: Ramsey planner values equality given utilitarian SWF Median voter benefits since α median = v α 2 < α mean = 0 Heathcote-Storesletten-Violante, Redistributive Taxation p. 32/34

Solving for competitive equilibrium 1. Conjecture no bond-trade: α uninsured and ε fully insured 2. Economy as continuum of groups indexed by α: within group, the Welfare Theorems apply Use group-planner problem to derive allocations, taking tax function (λ ) as given 3. Use agents FOC to back out shadow" bond price for each group, i.e., E t [MRS t,t+1 ] 4. Verify no-bond-trading equilibrium: check that shadow bond price is independent of island-specific characteristics 5. Given eq. allocations, solve for λ via aggregate government budget constraint Heathcote-Storesletten-Violante, Redistributive Taxation p. 33/34

Government s problem (F α, F ε ) are the aggregate states, exogenous and time-invariant Government puts weight β t on the welfare of all living agents born at dates t =,...,, and chooses a sequence {τ t, G t } t=0 The SWF becomes: W({τ t, G t } t=0) = t=0 βt W(τ t, G t ) where W(τ, G) u (c (α; τ, G), h (α, ε; τ, G), G)dF ε df α The dynamic Ramsey problem reduces to a static problem The government chooses the pair (τ, G) to maximize W(τ, G) subject to: (i) (c, h ) are competitive equilibrium allocations; and (ii) the government budget constraint is satisfied Heathcote-Storesletten-Violante, Redistributive Taxation p. 34/34