normative-tax-analysis-III-second slides

Intertemporal Optimal Taxation

Intertemporal Optimal Taxation: OutlineCapital Income Taxation with Linear and Nonlinear Taxation 1. Representative Household Basic two-period model Capital income taxation? Time-consistent taxation Time-inconsistent preferences Bequests 2. Heterogenous Households: Nonlinear Taxation Basic two-period model Di erences in discount rates Varying ability over time Uncertainty

Intertemporal Optimal Taxation

Two-Period Optimal Commodity Taxation

Present and future consumption: c1, c2 Variable labor (= h c0 ) in present period Utility: u (c1, c2, ) Consumer prices: 1+θ1, 1+θ2, w (p1= p2= 1) Intertemporal budget constraint: (1+θ1 )c1+ (1+θ2 )c2=w 1+r or, q1 c1+ q2 c2= w

where q1, q2 are present value consumer prices

Intertemporal Optimal Taxation

Tax EquivalencesProportional commodity taxθ1=θ2=θ equivalent to labor income tax tw in present value terms (though time pro les of revenues di er) Budget constraint with an income tax at the rateθm: c1+ c2= (1 θm )w 1+ (1 θm )r

= Equivalent toθ1,θ2 withθ2>θ1 Anyθ1,θ2 can be replicated by Wage and capital income tax: tw, tr (dual income tax) Income and wage tax: tm, tw Income and value-added tax: tm,θ

Intertemporal Optimal Taxation

Optimal Two-Period Tax StructureGiven present value of government revenue Three-commodity Ramsey tax applies:τ1θ1/(1+θ1 )ε11+ε22+ε10==τ2θ2/(1+θ2 )ε11+ε22+ε20= τ1=τ2 orθ1=θ2 ifε10=ε20= tw optimal ifε10=ε20= tw and tr> 0 ifε20<ε10 (c2 more complementary with leisure than c1 ) Generally, tr= tw Case for schedular taxation (dual income tax)

Intertemporal Optimal Taxation

OLG Extension: Atkinson-Sandmo

Young supply labor, consume and save; old consume Population grows at rate n If r> n, increase in K increases steady state welfare In absence of intergenerational transfers, if r> n, may be preferable to augment Corlett-Hague tax with further tax on capital to increase, given form of utility function (Atkinson-Sandmo, King) Mitigated by use of consumption vs wage tax (Summers), or debt policy/intergenerational transfers

Intertemporal Optimal Taxation

Four-Commodity CaseHousehold utility: u (c1, c2, 1, 2 ) Two-period budget constraint with all taxes: (1+θ1 )c1+ (1+θ2 )c2= (1 θw 1 )w1 1+ (1 θr )r1

+

(1 θw 2 )w2 2 1+ (1 θr )r

(Tax rates on labor and consumption can vary over time) Only 3 tax rates needed to control 3 relative prices Example 1: Commodity taxes zero (θ1=θ2= 0): c1+ c2= (1 θw 1 )w1 1+ (1 θr )r1

+

(1 θw 2 )w2 2 1+ (1 θr )r

Example 2: Wage taxes zero (tw 1= tw 2= 0): (1+θ1 )c1+ (1+θ2 )c2= w1 1+ (1 θr )r1

+

w2 2 1+ (1 θr )r

Generally, need eitherθ1=θ2 or tw 1= tw 2

Intertemporal Optimal Taxation

Zero Capital Income Tax?Case 1:θ1=θ2= 0. No need for tr to tax c1 relative to c2 if: Expenditure function implicitly separable: e (A(q1, q2, u ), (1 θw 1 )w1, (1 θw 1 )w2, u ) Example: u (f (c1, c2 ), u (f (c1, c2 ),1, 2)

with f (·) homothetic

1, 2)

=

1 σ x1 x 1 σ

+ v ( 1)+β 2+ v ( 2) 1 σ 1 σ

Case 2: tw 1= tw 2= 0. No need for tr to tax w1 Expenditure function implicitly separable: e (q1, q2, u, B (w1, w2 ), u )) Generally, eitherθi or twi must be time-varying Suppose not=

1

versus w2

2

if:

Intertemporal Optimal Taxation

Chamley-Judd Zero-Capital Tax Case

Suppose Preferences are u (c1,1)

+β u (c2,

2)

Wage rate is identical in both periodsβ= 1/(1+ r ) (steady state)= Optimal tr= 0,θ1=θ2, tw 1= tw 2= c1= c2 and1

=

2

(Steady state)

Optimal for capital taxes to be zero in the long run in a representative-agent dynamic model

Intertemporal Optimal Taxation

Proof of Zero trAll prices and taxes are in present value terms Consumer prices: q1= 1, q2= p2+ tc 2,ω1= w1+ tw 1,ω2= w2+ tw 2 Household: Max u (c1, 1 )+β u (c2, 2 ) s.t. c1+ q2 c2=ω1 FOCs c1, c2: FOCs1, 2 1 uc

1+ω2 2

=α,

2β uc

=αq2

:

u 1= αω12)

β u 2= αω21 1

Government Lagrangian: L= u (c1, 1 )+β u (c2,1+γ[uc c1

+λ[w1

+ w21

2

c1 p2 c2 R]2 2]

+

2β uc c2

+u

+βu

The rst-order conditions are:1 1 1 uc λ+γ[uc+ ucc c1+ u 1c 1]= 0 2β uc 1 2

(c1 ) (c2 ) ( 1) ( 2)

λp2+

u+λw1+

2γβ[uc 1

β u+λw2+

2+ ucc c2+ u 2c 2]= 0 1γ[u+ uc c1+ u 1 1]= 0 2γβ[u 2+ uc c2+ u 2 2]= 0

Intertemporal Optimal Taxation

Proof of tr= 0, continuedSince p2=β (= 1/(1+ r ) and w2=β w1, conditions (c2 ) and ( 2 ) become: 2 2 2 uc λ+γ[uc+ ucc c2+ u 2c 2]= 0 (c2 )2 u 2+λw1+γ[u 2+ uc c2+ u 2 2]

=01

( 2)2

(c1 ), (c2 ), ( 1 ) and ( 2 ) satis ed if c1= c2 and 1= u 2 and u 1= u 2 So, uc c Using household FOCs:2 uc q2 p2==1=, 1 ucββ

=

u2ω2 w2==1= 1βω1β w1 u

= q2= p2, so no tax on capital income= q2/q1= w2/w1, so labor taxes are same over time

Intertemporal Optimal Taxation

In nite-Horizon (Ramsey) CaseNote: In multi-period context, constant tax on capital equivalent to increasing tax on consumption over time (Bernheim): suggests a low capital tax rate, or a capital tax rate that varies over time Utility; u (x0,0)

+

∞ t t=1β u (xt, t )

Taxes allowed on wages and capital Capital income tax → 0 in long run (Chamley-Judd) If u (x, )= x 1 σ/(1 σ )+ v ( ), capital tax zero for t> 0 Assumes representative agent model: but Ricardian equivalence violates biology/anthropology (Bernheim-Bagwell) Assumes full commitment

Intertemporal Optimal Taxation

Multi-Period OLG ModelTwo-period life-cycle Zero-capital tax no longer generally applies unlessSteady state with no saving, or Utility u (x, )= x 1 σ/(1 σ )+ v ( )

Liquidity constraints favor capital taxes (Hubbard-Judd)Reallocate tax liabilities to future periods

Especially with wage uncertainty (Aiyagari)Excessive precautionary saving

Simulations suggest high capital income tax (Conesa-Kitao-Krueger)

Intertemporal Optimal Taxation

Time-Consistent TaxationThe Problem Taxpayers take long-run and short-run decisions Long-run decisions, like saving, create asset income that is xed in the future Shor

t-run decisions, like labor supply, create income in the same period Second-best optimal tax policy is determined before long-run decisions are taken Second-best tax policies are generally time-inconsistent: even benevolent governments will choose to change tax policies after long-run decisions are undertaken If households anticipate such re-optimizing, the outcome will be inferior to the second-best Governments may implement policies up front to mitigate that problem

Intertemporal Optimal Taxation

General Consequences of Inability to CommitExcessive capital taxation (Fischer) Excessive taxation of quasi-rents of natural resources Samaritan’s dilemma (Bruce-Waldman, Coate): Government unable to help those who have chosen not to help themselves Soft budget constraint between national and sub-national governments Mitigated by various measuresRestriction to consumption taxation Incentives for asset accumulation Mandatory saving Under-investment in tax enforcement Social insurance Training

Intertemporal Optimal Taxation

Commodity Tax Case: An Illustrative ModelBased on Fischer 1981 Rev Econ Dyn& Control and Persson and Tabellini survey in Handbook of Public Economics Two periods, two goods (c1, c2 ) and labor in period 2 ( ) Quasilinear utility: u (c1 )+ c2+ h(1 ) Time endowment 1, wealth endowment 1 Wage rate= 1, interest rate= 0 Second-period taxes: tk, t on k, Fixed government revenue R Consumer problem Max{c1,= }

u (c1 )+ (1 t )+ (1 tk )(1 c1 )+ h(1 )

= c1 (1 tk ), c1 (1 tk )< 0, k (1 tk )= 1 c1 (1 tk ) (1 t ), (1 t )> 0

Indirect utility: v (tk, t ), with vtk= (1 c1 ), vt= Note: Marginal utility of incomeα= 1

Intertemporal Optimal Taxation

Government Policy (Second Period)Max{tk,t}

v (tk, t ) s.t. t (1 t )+ tk k (1 tk )= Rλ 1 1 tkλ 1 1 t=> 0,=>0 1 tλη 1 tkληk

Second-best tax:

whereη= (1 t )/ andηk= (1 tk )k/k Ex post, government will reoptimize by treating k as xed and set tk as high as possible (e.g. tk= 1) Households anticipate this and reduce saving Time-consistent equilibrium is inferior to second-best Government may react by providing ex ante saving incentives Inability to commit may be responsible for high capital income and wealth tax rates in practice Widespread use of investment and savings incentives Same phenomenon applies to human capital investment, investment by rms and housing If R endogenous, perceived MCPF too high: over-spending

Intertemporal Optimal Taxation

Time-Inconsistent PreferencesThe Case of Sin Taxes (O’Donoghue and Rabin) Assumptions Households consume xt, zt in period t∈[0, T] Utility: ut= v (xt,ρ) c (xt 1,γ )+ zt, cx, vxρ, cxγ> 0 Income m, producer prices unity Government imposes taxθ on x, returns lump-sum revenue a Per period decision utility: u (x, z )= v (x,ρ) β c (x,γ )+ z Experienced utility: u (x, z )= v (x,ρ) c (x,γ )+ z Ideal Behaviour Max u (x, z ) s.t. x+ z= m= vx (x ,

ρ) cx (x ,γ ) 1= 0, z = m x Actual Behaviour Max u (x, z ) s.t. (1+θ)x+ z= m+ a vx (x (θ),ρ) β cx (x (θ),γ )= 1+θ z (θ, a)= m+ a (1+θ)x (θ)=

Intertemporal Optimal Taxation

Optimal Sin TaxesWhen t= 0: x (0)≥ x (0) asβ≤ 1 Identical households Optimal tax:θ = (1 β )cx (x )= Pigouvian tax on externality imposed on one’s self Heterogeneous households 1. Ifβ= 1 for all households,θ = 0 2. Ifβ< 1 for all,θ > 0, but rst best not achieved due to heterogeneity inγ,ρ,β 3. Ifβ< 1 for some,β= 1 for others,θ > 0: second-order e ect of small tax ifβ= 1, rst-order e ect ifβ< 1 Note: Should the government interfere with consumer behaviour in the rst place? (Paternalism or not)

Intertemporal Optimal Taxation

BequestsMotives Voluntary I: Altruism Voluntary II: Joy of giving Involuntary: Unintended Strategic: Requited transfer E cient Taxation Externality of voluntary transfers (bene ts to donors and donees): Pigouvian subsidy on bequests Taxation of involuntary transfers fully e cient Equitable Taxation Voluntary& strategic transfers: tax donors and donees Double counting? Ricardian equivalence? Equality of opportunity arguments

Intertemporal Optimal Taxation

Dynamic Optimal Nonlinear TaxationNew Dynamic Public Finance n heterogeneous dynasties that live inde nitely Wages evolve in each stochastically i.i.d. Labor supply and saving chosen each period given uncertainly about future wages Intertemporal (lifetime) optimal nonlinear income taxation: government observes income and saving Focus on tax wedge on savings and labour tax smoothing Key Issues Commitment by government Implementation of second-best optimum (tax wedges) Market failures: no wage insurance; liquidity constraints Main insights learned from two-period models=

Intertemporal Optimal Taxation

Basic Two-Period, Two-Type Case (Diamond)Setting cij= consumption in period j by type i (i, j= 1, 2)1 i

= yi1/wi labour supply by type i in period 1 only

2 Utility: u (ci1 ) h( 1 i )+β u (ci )

Lifetime tax schedule (gov. observes ci1, ci2, or s ) Government problem (full commitment assumed)1 max n1 u (c1 ) h 1 y1 y1 2 1 2+β u (c1 )+n2 u (c2 ) h 2+β u (c2 ) w1 w2 2 c1 1+r 2 c2 1+r

s.t.1 1 n1 y1 c1 1 u (c2 ) h

1 1+ n2 y2 c2

=R

(λ) (γ )

1 y2 y1 2 1 2+β u (c2 )≥ u (c1 ) h 1+β u (c1 ) w2 w2

本文来源：https://www.bwwdw.com/article/oq2j.html