Optimal Tax Progressivity: An Analytical Framework · Federal Reserve Bank of Atlanta – January...

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

Question

• How progressive should labor income taxation be?


Question


• Argument in favor of progressivity: missing markets

1. Social insurance of privately-uninsurable lifecycle shocks

2. Redistribution with respect to initial conditions


Question


• 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


Overview of the framework

• Huggett (1993) economy: ∞-lived agents, idiosyncraticproductivity 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









• Tractable equilibrium framework → closed-form SWF

• Ramsey approach: tax instruments & mkt structure taken as given


The tax/transfer function

T (yi) = yi − λy1−τi

• The parameter τ measures the rate of progressivity:

◮ τ = 1 : full redistribution → yi = λ

◮ 0 < τ < 1: progressivity → T ′(y)T (y)/y > 1

◮ τ = 0 : no redistribution → flat tax 1− λ

◮ τ < 0 : regressivity → T ′(y)T (y)/y < 1

• Break-even income level: y0 = λ1

τ


The tax/transfer function

T (yi) = yi − λy1−τi

• The parameter τ measures the rate of progressivity:

◮ τ = 1 : full redistribution → yi = λ

◮ 0 < τ < 1: progressivity → T ′(y)T (y)/y > 1

◮ τ = 0 : no redistribution → flat tax 1− λ

◮ τ < 0 : regressivity → T ′(y)T (y)/y < 1

• Break-even income level: y0 = λ1

τ

Restrictions: (i) no lump-sum transfer & (ii) T ′(y) monotone


Empirical fit to US micro data

• PSID 2000-06, age 20-60, N = 13, 721: R2 = 0.96: → τUS = 0.151


Empirical fit to US micro data

• PSID 2000-06, age 20-60, N = 13, 721: R2 = 0.96: → τUS = 0.151

8.5

99.

510

10.5

1111

.512

12.5

13Lo

g of

pos

t−go

vern

men

t inc

ome

8.5 9 9.5 10 10.5 11 11.5 12 12.5 13Log of pre−government income


Marginal and average tax rates

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5

x 105

−0.4

−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

0.5

Household Income

Tax

Rat

e

Average Tax RateMarginal Tax Rate

• At the estimated value of τUS = 0.151


Demographics and preferences

• Perpetual youth demographics with constant survival probability δ

• Preferences over consumption (c), hours (h), publicly-providedgoods (G), and skill-investment (s) effort:

Ui = vi(si) + E0

∞∑

t=0

(βδ)tui(cit, hit, G)

vi(si) = −1

κi

s2i2µ

ui (cit, hit, G) = log cit − exp(ϕi)h1+σit

1 + σ+ χ logG

κi ∼ Exp (η)

ϕi ∼ N(vϕ2, vϕ

)


Technology

• Output is CES aggregator over continuum of skill types:

Y =

[∫ ∞

0

N (s)θ−1

θ ds

] θ

θ−1

, θ ∈ (1,∞)

• Aggregate effective hours by skill type:

N(s) =

∫ 1

0

I{si=s} zihi di

• Aggregate resource constraint:

Y =

∫ 1

0

ci di+G


Individual efficiency units of labor

log zit = αit + εit

• αit = αi,t−1 + ωit with ωit ∼ N(− vω

2 , vω)

αi0 = 0 ∀i

• εit i.i.d. over time with εit ∼ N(− vε

2 , vε)

• ϕ ⊥ κ ⊥ ω ⊥ ε cross-sectionally and longitudinally


Individual efficiency units of labor

log zit = αit + εit

• αit = αi,t−1 + ωit with ωit ∼ N(− vω

2 , vω)

αi0 = 0 ∀i

• εit i.i.d. over time with εit ∼ N(− vε

2 , vε)

• ϕ ⊥ κ ⊥ ω ⊥ ε cross-sectionally and longitudinally

• Pre-government earnings:

yit = p(si)︸︷︷︸skill price

× exp(αit + εit)︸︷︷︸efficiency

× hit︸︷︷︸hours

determined by skill, fortune, and diligence


Government

• Disposable (post-government) earnings:

yi = λy1−τi

• Government budget constraint (no government debt):

G =

∫ 1

0

[yi − λy1−τ

i

]di

• Government chooses (G, τ), and λ balances the budget residually

• WLOG: G ≡ g · Y


Markets

• Competitive good and labor markets

• Perfect annuity markets against survival risk


Markets



• Competitive asset markets (all assets in zero net supply)

◮ Non state-contingent bond


Markets



• Competitive asset markets (all assets in zero net supply)

◮ Non state-contingent bond

◮ Full set of insurance claims against ε shocks

� If vε = 0, it is a bond economy

� If vω = 0, it is a full insurance economy

� If vω = vε = vϕ = 0 & θ = ∞, it is a RA economy


A special case: the representative agent

maxC,H

U = logC −H1+σ

1 + σ+ χ log gY

s.t.

C = λY 1−τ and Y = H



maxC,H

U = logC −H1+σ

1 + σ+ χ log gY

s.t.


• Market clearing: C +G = Y



maxC,H

U = logC −H1+σ

1 + σ+ χ log gY

s.t.


• Market clearing: C +G = Y

• Equilibrium allocations:

logHRA(τ) =1

1 + σlog(1− τ)

logCRA(g, τ) = log(1− g) + logHRA(τ)


Optimal policy in the RA economy

• Welfare function:

WRA(g, τ) = log(1− g) + χ log g + (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)


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


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


Equilibrium skill choice and skill price


• Skill price has Mincerian shape: log p(s; τ) = π0(τ) + π1(τ)s(κ; τ)

s(κ; τ) =

√ηµ (1− τ)

θ· κ (skill choice)

π1(τ) =

√η

θµ (1− τ)(marginal return to skill)



s(κ; τ) =

√ηµ (1− τ)


π1(τ) =

√η


• Distribution of skill prices (in levels) is Pareto with parameter θ



s(κ; τ) =

√ηµ (1− τ)


π1(τ) =

√η


• 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(log p(s; τ)) =1

θ2


Equilibrium consumption allocation

log c∗(α, ϕ, s; g, τ) = logCRA(g, τ) + (1− τ) log p(s; τ)︸︷︷︸skill price

+(1− τ)α︸︷︷︸unins. shock

− (1− τ)ϕ︸︷︷︸pref. het.

+ M(vε)︸︷︷︸level effect from ins. variation

• Response to variation in (p, ϕ, α) mediated by progressivity

• Invariant to insurable shock ε


Equilibrium hours allocation

log h∗(ε, ϕ; τ) = logHRA(τ)− ϕ︸︷︷︸pref. het.

+1

σε

︸︷︷︸ins. shock

−1

σ(1− τ)M(vε)

︸︷︷︸level effect from ins. variation

• Response to ε mediated by tax-modified Frisch elasticity 1σ = 1−τ

σ+τ

• Invariant to skill price p, uninsurable shock α, and size of g


Social Welfare Function

Planner chooses constant (g, τ)

Make two assumptions:

1. Planner puts equal weight on period utility of all currently aliveagents, discounts future at rate β

2. Skill investments are fully reversible

Then, SWF becomes average period utility in the cross-section


Exact expression for SWF

W(g, τ) = log(1 + g) + χ log g + (1 + χ)log(1− τ)

(1 + σ)(1− τ)−

1

(1 + σ)

+(1 + χ)

[−1

θ − 1log

(√ηθ

µ (1− τ)

)+

θ

θ − 1log

(θ

θ − 1

)]

−1

2θ(1− τ)−

[− log

(1−

(1− τ

θ

))−

(1− τ

θ

)]

− (1− τ)2vϕ2

−

(1− τ)

δ

1− δ

vω2

− log

1− δ exp

(−τ(1−τ)

2 vω

)

1− δ

−(1 + χ)σ1

σ2

vε2

+ (1 + χ)1

σvε


Representative Agent component

W(g, τ) = log(1 + g) + χ log g + (1 + χ)log(1− τ)

(1 + σ)(1− τ)−

1

(1 + σ)︸︷︷︸Representative Agent Welfare = WRA(g, τ)

+(1 + χ)

[−1

θ − 1log

(√ηθ

µ (1− τ)

)+

θ

θ − 1log

(θ

θ − 1

)]

−1

2θ(1− τ)−

[− log

(1−

(1− τ

θ

))−

(1− τ

θ

)]

− (1− τ)2vϕ2

−

(1− τ)

δ

1− δ

vω2

− log

1− δ exp

(−τ(1−τ)

2 vω

)

1− δ

−(1 + χ)σ1

σ2

vε2

+ (1 + χ)1

σvε


Exact expression for SWF

W(τ) = χ logχ− (1 + χ) log(1 + χ) + (1 + χ)log(1− τ)

(1 + σ)(1− τ)−

1

(1 + σ)

+(1 + χ)

[−1

θ − 1log

(√ηθ

µ (1− τ)

)+

θ

θ − 1log

(θ

θ − 1

)]

−1

2θ(1− τ)−

[− log

(1−

(1− τ

θ

))−

(1− τ

θ

)]

− (1− τ)2vϕ2

−

(1− τ)

δ

1− δ

vω2

− log

1− δ exp

(−τ(1−τ)

2 vω

)

1− δ

−(1 + χ)σ1

σ2

vε2

+ (1 + χ)1

σvε


Skill investment component


(1 + σ)(1− τ)−

1

(1 + σ)

+(1 + χ)

[−1

θ − 1log

(√ηθ

µ (1− τ)

)+

θ

θ − 1log

(θ

θ − 1

)]

︸︷︷︸productivity gain = logE [(p(s))] = log (Y/N)

−1

2θ(1− τ)

︸︷︷︸avg. education cost

−

[− log

(1−

(1− τ

θ

))−

(1− τ

θ

)]

︸︷︷︸consumption dispersion across skills

− (1− τ)2 vϕ

2

−

(1− τ)

δ

1− δ

vω2

− log

1− δ exp

(−τ(1−τ)

2 vω

)

1− δ

−(1 + χ)σ1

σ2

vε2

+ (1 + χ)1

σvε


Skill investment component

5 10 15 20 250

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Elasticity of substitution in production (θ)

Opt

imal

deg

ree

of p

rogr

essi

vity

(τ)


Optimal marginal tax rate at the top

• Diamond-Saez formula (Mirlees approach):

t =1

1 + ζu + ζc(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 skillinvestment is large

◮ Key difference: exogenous vs endogenous skill distribution


Uninsurable component


(1 + σ)(1− τ)−

1

(1 + σ)

+(1 + χ)

[−1

θ − 1log

(√ηθ

µ (1− τ)

)+

θ

θ − 1log

(θ

θ − 1

)]

−1

2θ(1− τ)−

[− log

(1−

(1− τ

θ

))−

(1− τ

θ

)]

− (1− τ)2vϕ2︸︷︷︸

cons. disp. due to prefs

−

(1− τ)

δ

1− δ

vω2

− log

1− δ exp

(−τ(1−τ)

2 vω

)

1− δ

︸︷︷︸consumption dispersion due to uninsurable shocks ≈ (1 − τ)2 vα

2

−(1 + χ)σ1

σ2

vε2

+ (1 + χ)1

σvε


Insurable component


(1 + σ)(1− τ)−

1

(1 + σ)

+(1 + χ)

[−1

θ − 1log

(√ηθ

µ (1− τ)

)+

θ

θ − 1log

(θ

θ − 1

)]

−1

2θ(1− τ)−

[− log

(1−

(1− τ

θ

))−

(1− τ

θ

)]

− (1− τ)2 vϕ

2

−

(1− τ)

δ

1− δ

vω2

− log

1− δ exp

(−τ(1−τ)

2 vω

)

1− δ

−(1 + χ)σ1

σ2

vε2︸︷︷︸

hours dispersion

+ (1 + χ)1

σvε

︸︷︷︸prod. gain from ins. shock=log(N/H)


Parameterization

• Parameter vector {χ, σ, θ, vϕ, vω, vε}


Parameterization


• To match G/Y = 0.19: → χ = 0.23


Parameterization


• To match G/Y = 0.19: → χ = 0.23

• Frisch elasticity (micro-evidence): → σ = 2


Parameterization


• To match G/Y = 0.19: → χ = 0.23


cov(log h, logw) =1

σvε

var(log h) = vϕ +1

σ2vε

var0(log c) = (1− τ)2(vϕ +

1

θ2

)

var(logw) =1

θ2+

δ

1− δvω


Parameterization


• To match G/Y = 0.19: → χ = 0.23


cov(log h, logw) =1

σvε → vε = 0.17

var(log h) = vϕ +1

σ2vε → vϕ = 0.035

var0(log c) = (1− τ)2(vϕ +

1

θ2

)→ θ = 2.16

var(logw) =1

θ2+

δ

1− δvω → vω = 0.003


Optimal progressivity

−0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−8

−7

−6

−5

−4

−3

−2

−1

0

1

Progressivity rate (τ)

wel

fare

cha

nge

rel.

to o

ptim

um (

% o

f con

s.)


Baseline Model

τUS= 0.151

τ*= 0.065

Welfare Gain = 0.5%


Optimal progressivity: decomposition

−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−16

−14

−12

−10

−8

−6

−4

−2

0

2

4


wel

fare

cha

nge

rel.

to o

ptim

um (

% o

f con

s.)


(1) Rep. Agentτ = −0.233



−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−16

−14

−12

−10

−8

−6

−4

−2

0

2

4


wel

fare

cha

nge

rel.

to o

ptim

um (

% o

f con

s.)


(2) + Skill Inv.τ = −0.062

(1) Rep. Agentτ = −0.233



−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−16

−14

−12

−10

−8

−6

−4

−2

0

2

4


wel

fare

cha

nge

rel.

to o

ptim

um (

% o

f con

s.)


(2) + Skill Inv.τ = −0.062

(1) Rep. Agentτ = −0.233

(3) + Pref. Het.τ = −0.020



−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−16

−14

−12

−10

−8

−6

−4

−2

0

2

4


wel

fare

cha

nge

rel.

to o

ptim

um (

% o

f con

s.)


(2) + Skill Inv.τ = −0.062

(1) Rep. Agentτ = −0.233

(3) + Pref. Het.τ = −0.020

(4) + Unins. Shocksτ = 0.076



−0.4 −0.3 −0.2 −0.1 0 0.1 0.2 0.3 0.4−16

−14

−12

−10

−8

−6

−4

−2

0

2

4


wel

fare

cha

nge

rel.

to o

ptim

um (

% o

f con

s.)


(2) + Skill Inv.τ = −0.062

(1) Rep. Agentτ = −0.233

(3) + Pref. Het.τ = −0.020

(5) + Ins. Shocksτ* = 0.065

(4) + Unins. Shocksτ = 0.076


Actual and optimal progressivity

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2

−0.1

0

0.1

0.2

0.3

0.4

Income (1 = average income)

Ave

rage

tax

rate

Actual US τUS = 0.151

Utilitarian τ∗ = 0.065


Alternative SWF

Utilitarian SWF embeds desire to insure and to redistribute wrt (κ, ϕ)

Switch off desire to redistribute


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 = vω = τ = 0

• Compute CE allocations

• Compute Negishi weights s.t. planner’s allocation = CE

• Use these weights in the SWF


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∗ 0.065 0.006 0.037 -0.022

Welf. gain (pct of c) 0.49 1.35 0.84 1.92


Optimal progressivity: alternative SWF

0.5 1 1.5 2 2.5 3 3.5 4 4.5 5−0.2

−0.1

0

0.1

0.2

0.3

0.4

Income (1 = average income)

Ave

rage

tax

rate

Actual US τUS = 0.151

Utilitarian τ∗ = 0.065

Ins. Only τ∗ = −0.022


Alternative assumptions on G

Case Utilitarian Insurance-onlyG

Yτ∗ Welf. gain τ∗ Welf. gain

g∗ endogenous χ = 0.233 0.189 0.065 0.49% −0.022 1.92%

g exogenous χ = 0 0.189 0.184 0.07% 0.092 0.20%

G exogenous χ = 0 0.189 0.071 0.45% −0.011 1.78%

• With χ = 0, no externality and no push towards regressivity

• If G fixed in level, lower progressivity translates 1-1 into higher c


Irreversible skill investment

• Assume skills are fixed after being chosen

• Planner has an additional incentive to increase progressivity:

◮ Can compress consumption inequality, while discouraging skillinvestment only for new generations (not for current ones)

• Needed to preserve tractability: production is segregated by age

• Planner chooses τ∗ = 0.137


Irreversible skill investment

• Assume skills are fixed after being chosen

• Planner has an additional incentive to increase progressivity:

◮ Can compress consumption inequality, while discouraging skillinvestment only for new generations (not for current ones)

• Needed to preserve tractability: production is segregated by age

• Planner chooses τ∗ = 0.137

• Hybrid model: optimal progressivity is, again, τ∗ = 0.065


Political Economy

• Think of g and τ as outcomes of a majority rule voting process

• Median voter theorem applies in our model!


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 τmed = 0.083

• More progressivity because median voter poorer than average


Progressive consumption taxation

c = λc1−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 theefficient response of hours to insurable shocks

• Higher progressivity and higher welfare


Budget constraints

1. Beginning of period: innovation ω to α shock is realized

2. Middle of period: buy insurance against ε:

b =

∫

EQ(ε)B(ε)dε,

where Q(·) is the price of insurance and B(·) is the quantity

3. End of period: ε realized, consumption and hours chosen:

c+ δqb′ = λ [p(s) exp(α+ ε)h]1−τ +B(ε)


No bond-trade equilibrium

• Micro-foundations for Constantinides and Duffie (1996)

◮ CRRA, unit root shocks to log disposable income

◮ In equilibrium, no bond-trade ⇒ ct = yt


No bond-trade equilibrium

• Micro-foundations for Constantinides and Duffie (1996)

◮ CRRA, unit root shocks to log disposable income

◮ In equilibrium, no bond-trade ⇒ ct = yt

• 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


Equilibrium risk-free rate r∗

ρ− r∗ = (1− τ) ((1− τ) + 1)vω2

• 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


Upper tail of wage distribution

4 5 6 7 8 9 100

1

2

x 10−4

Wage (average=1)

Den

sity

Top 1pct of the Wage Distribution

Model Wage DistributionLognormal Wage Distribution


Optimal Tax Progressivity: An Analytical Framework · Federal Reserve Bank of Atlanta – January...

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