public economics taxation ii: optimal taxation · 2016-04-26 · i. iturbe-ormaetxe (u. of...

Public EconomicsTaxation II: Optimal Taxation

Iñigo Iturbe-Ormaetxe

U. of Alicante

Winter 2012

I. Iturbe-Ormaetxe (U. of Alicante) Public Economics Taxation II: Optimal Taxation Winter 2012 1 / 98

Outline

1 Commodity Taxation I: Ramsey Rule

2 Commodity Taxation II: Production E¢ ciency

3 Income Taxation I: Mirrlees Model


Optimal Commodity Taxation: Introduction

We use what we know about incidence and e¢ ciency costs to analyzeoptimal design of commodity taxes

What is the best way to design taxes given equity and e¢ ciencyconcerns?

The literature on optimal commodity taxation focuses on linear taxsystems

Non-linear tax systems are studied with income taxation


Second Welfare Theorem

The starting point is the Second Theorem of Welfare Economics: anyPareto optimal outcome can be achieved as a competitive equilibriumwith appropriate lump-sum transfers of wealth

This result requires the same assumptions as the First Theorem plusone more (and some convexity requirements):

1 Complete markets (no externalities)2 Perfect information3 Perfect competition4 Lump-sum taxes/transfers across individuals feasible

If all assumptions hold, there is no equity-e¢ ciency trade-o¤ and theoptimal tax problem is trivial. Implement LSTs that meetdistributional goals given revenue requirement

Problem: informationI. Iturbe-Ormaetxe (U. of Alicante) Public Economics Taxation II: Optimal Taxation Winter 2012 4 / 98

2nd Welfare Theorem: Information Constraints

To set optimal LSTs, government needs to know individualcharacteristics (ability or earning capacity)

Impossible to make people reveal ability at no cost: If asked, theyhave incentives to misreport skill levels

Tax instruments must be a function of observable economic outcomes(income, property, consumption of goods)

This distorts prices, a¤ecting behavior and generating DWL

Information constraints force us to move from 1st best world of thesecond welfare theorem to 2nd best world with ine¢ cient taxation: Itis impossible to redistribute or raise revenue for public goods withoutgenerating e¢ ciency costs


Central Results in Optimal Tax Theory

1 Ramsey (1927): inverse elasticity rule

2 Mirrlees (1971): optimal income taxation

3 Diamond and Mirrlees (1971): production e¢ ciency

4 Atkinson and Stiglitz (1976): no consumption taxation with optimalnon-linear (including lump sum) income taxation

5 Chamley, Judd (1983): no capital taxation in innite horizon models


Ramsey (1927) Tax Problem

Government sets taxes on uses of income in order to fulll two goals:

1 Raise total revenue R

2 Minimize utility loss for agents in economy. Or to minimize excessburden of taxation


Ramsey Model: Key Assumptions

1 Lump sum taxes are prohibited

2 Cannot tax all commodities (leisure untaxed)

3 Production prices p are xed. Consumer prices are q = p + t


Set-up

A population of identical individuals (equivalent to a singlerepresentative consumer) =) No equity concerns

The consumer maximizes u(x), where x = (x0, .., xN ), subject toq1x1 + ..+ qNxN Z , where Z is non-wage incomeFrom the point of view of government is the same to set (t1, .., tN )than to set (q1, ..qN ), given (p1, ..pN )

The FOCs are: ∂u∂xi= αqi , where α = ∂V

∂Z is the marginal utility ofincome for the individual

We get (Marshallian) demands xi (q,Z ) and indirect utility functionV (q,Z ), where q = (q1, ..qN )


Excess burden (using Equivalent Variation)

Government solves:

m«axqfV (q,Z )g s.t. tx =

N

∑i=1tixi (q,Z ) R.

Equivalent to minimize the excess burden of taxation:

m«¬nqEB(q) = E (q,V (q,Z )) E (p,V (q,Z )) R,

subject to the same revenue requirement

Why? Because E (p,V (q,Z )) is monotonically increasing in V (q,Z )and R and Z E (q,V (q,Z )) are constant. Maximizing V (q,Z ) isthe same as minimizing E (p,V (q,Z )). The excess burden oftaxation is evaluated at the utility level V (q,Z ) that holds in thepresence of taxation


Uniform taxation

If we could tax all commodities, the problem is trivial. Consideruniform taxation, where q = φp and φ > 1 is chosen such that(φ 1)px = RExcess burden is Z E (p,V (q,Z )) R. Substituting:

E (φp,V (φp,Z )) E (p,V (φp,Z )) (φ 1)px(φp,Z )

Since E (., .) is homogeneous of degree 1 on prices:

φE (p,V (φp,Z )) E (p,V (φp,Z )) (φ 1)px(φp,Z )= (φ 1)E (p,V (φp,Z )) (φ 1)px(φp,Z )

Finally, using the fact that:E (p,V (φp,Z )) pxC (φp,V (φp,Z )) px(φp,Z ) :

(φ 1)px(φp,Z ) (φ 1)px(φp,Z ) = 0


Uniform taxation 2

With uniform taxation, excess burden is zero. This is equivalent toimposing a LST

The constraint for the household is φpx = Z or:

px = Z (φ 1)Zφ

If we could tax all commodities, uniform taxation would be optimal

Next we see what can we do if we cannot tax all commodities


Ramsey Model: Governments Problem

The Lagrangian of government is:

LG = V (q,Z ) + λ[∑itixi (q,Z ) R ]

) ∂LG∂qi

=∂V∂qi|z

Priv. WelfareLoss to Indiv.

+ λ[ xi|zMechanicalE¤ect

+∑jtj∂xj/∂qi| z BehavioralResponse

] = 0

Using Roys identity ( ∂V∂qi= αxi ):

(λ α)xi + λ ∑jtj∂xj/∂qi = 0


Ramsey Optimal Tax Formula

Optimal tax rates have to satisfy a system of N equations and Nunknowns:

∑jtj

∂xj∂qi

= xiλ(λ α) for i = 1, ..,N

Here note that one commodity cannot be taxed (think of commodity0)

Also note that ∂xj/∂qi = ∂xj/∂tiUsing a perturbation argument we can arrive at the same formula


Ramsey Formula: Perturbation Argument

Suppose government increases ti by dti

The e¤ect of this tax increase on social welfare is the sum of e¤ect ongovernment revenue and on private surplus

The marginal e¤ect on government revenue is:

dR = xidti +∑jtjdxj

The marginal e¤ect on private surplus is:

dU =∂V∂qidti = αxidti

At an optimum these two e¤ects cancel each other:

dU + λdR = 0


Ramsey Formula: Using Slutsky

Recall the Slutsky equation:

∂xj/∂qi = ∂hj/∂qi xi∂xj/∂Z

We substitute into formula above:

(λ α)xi + λ ∑jtj [∂hj/∂qi xi∂xj/∂Z ] = 0

) 1xi

∑jtj

∂hi∂qj

= θ

λ

where θ = λ α λ ∂∂Z (∑j tjxj )


Ramsey Formula: Compensated Elasticity Representation

The term θ is independent of i and measures the value for thegovernment of introducing a 1 euro LST

θ = λ α λ∂(∑jtjxj )/∂Z

Three e¤ects:1 Direct value for the government is λ (recall this is the Lagrangemultiplier of the government budget constraint)

2 Loss in welfare for the individual is α (this is the Lagrange multiplier ofthe consumers problem)

3 Behavioral e¤ect ! loss in tax revenue of ∂(∑j tjxj )/∂Z


Intuition for Ramsey Formula

1xi

∑jtj

∂hi∂qj

= θ

λ

Suppose that E is small so that all taxes are also small. Then the taxtj on good j reduces the consumption of good i (holding utilityconstant) by approximately:

dhi = tj∂hi∂qj

The summatory on the left represents the total reduction inconsumption of good i (because of all taxes)

Dividing by xi yields the percentage reduction in consumption of eachgood i = index of discouragementof the tax system on good i

The Ramsey formula says that these indexes must be equal acrossgoods at the optimum


Inverse Elasticity Rule

Using elasticities, we can write Ramsey formula as:

N

∑j=1

tj1+ tj

εcij =θ

λ,

where εcij is the compensated elasticity (from Hicksian demands). Wehave normalized pj = 1

In the special case in which all cross-elasticities are zero we obtain theclassic inverse elasticity rule:

ti1+ ti

=θ

λ

1εii

More elastic goods should have lower taxes


Uniform Taxation?

Consider a case with only three goods 0, 1, 2 where 0 is untaxed. Wehave two equations (there are 2 taxes):

t11+ t1

εc11 +t2

1+ t2εc12 =

θ

λt1

1+ t1εc21 +

t21+ t2

εc22 =θ

λ

Rearranging, we obtain:

t1/(1+ t1)t2/(1+ t2)

=εc22 εc12εc11 εc21


Uniform Taxation?

Using the property that εc11 + εc12 + εc10 = 0 :

t1/(1+ t1)t2/(1+ t2)

=εc20 + εc21 + εc12εc10 + εc12 + εc21

We have t1 = t2 only when εc10 = εc20. This happens if both goods areequally complementary with respect to the untaxed good (thenumeraire)

If εc10 < εc20, good 1 should be taxed at a higher rate. This result iscalled Corlett-Hague theorem: With preferences that are notseparable between taxable goods and the untaxed good, governmentshould deviate from uniform taxation by taxing more heavily goodsthat are more complementary to the untaxed good


Corlett-Hague Theorem

When the untaxed good is leisure, this means higher taxes for goodsthat are complementary to leisure (lower taxes for goods that arecomplementary to labor)

Classical example: swimsuits and pizza. Swimsuits should be taxedmore than pizza

Deaton nds a necessary and su¢ cient condition under which we getuniform taxation as a prescription of the Ramsey rule. It is calledQuasi-separabilityof the utility function: marginal rates ofsubstitution between goods on a given indi¤erence curve must beindependent of the consumption of the untaxed good (for example,Cobb-Douglas)


Limitations of Ramsey Formula

According to Ramsey Formula government should tax inelastic goodsto minimize e¢ ciency costs

We do not take into account redistributional issues (representativeconsumer)

Problem: necessities tend to be more inelastic than luxury goods

Optimal Ramsey tax system is likely to be regressive

Diamond (1975) extends Ramsey model to take redistributiveconcerns into account


Diamond 1975: Many-Person Model

H individuals with utilities u1, .., uh, .., uH

Aggregate consumption of good i is:

Xi (q) = ∑h

xhi

Government chooses tax rates ti and a lump sum transfer T 0 tomaximize social welfare:

m«axW (V 1, ..,V H ) s.t.N

∑i=1tiXi R + T

Consider the e¤ect of increasing the tax on good i by dti


Diamond: Perturbation Argument

E¤ect of perturbation on revenue:

dR = Xidti +∑jtjdXj = dti [Xi +∑

jtj

∂Xj∂qi]

E¤ect on individual hs welfare (recall ∂V h∂qi

= αhxhi ):

duh =∂V h

∂qidti = αhxhi dti

E¤ect on total private welfare:

dW = ∑h

(∂W/∂V h)αhxhi dti = dti [∑h

βhxhi ]

where βh = (∂W/∂V h)αh is hs social marginal utility of wealth

At the optimum:dW + λdR = 0


Diamond: Many-Person Tax Formula

Solving yields formula for optimal tax rates:

∑jtj

∂Xj∂qi

= Xiλ[λ ∑h βhxhi

Xi]

If government does not care for redistribution: βh = α constant: Backto Ramsey Formula

With redistributive tastes, βh will be lower for higher incomeindividuals

The new term

∑h βhx hiXi

is the average social marginal utility,

weighted by consumption of good i


Diamond Formula: Special Case

When all uncompensated cross price elasticities are zero, optimal taxrates satises:

ti1+ ti

=1εuii

1 ∑h βhxhi

λXi

!Again ti must be inversely proportional to the elasticity, but the termin brackets is not constant across goods

For goods that are consumed by people with a high weight in theSWF, the term (∑h βhxhi )/(λXi ) is large

The optimal tax rate for these goods is lower (elasticities being thesame)

The opposite happens for goods consumed by people with low weightin the SWF (the rich)


Optimal Transfer

In this model, it is optimal for the government to pay a uniformtransfer T on top of tax rates

With redistributive tastes, this uniform transfer is strictly positive(T > 0)

Without redistributional considerations, T = R

This is ruled out by the constraint T 0 : the poor cannot a¤ord topay a LST


Diamond and Mirrlees (1971)

Diamond and Mirrlees (1971) relax the assumption that producerprices are xed. They model production

Two main results:

1 Production e¢ ciency: even in an economy where rst-best isunattainable, optimal policy maintains production e¢ ciency(production e¢ ciency: cannot produce more output with same inputs,cannot produce same output with fewer inputs)

2 Characterize optimal tax rates with endogenous prices and show thatRamsey rule can be applied


Model

Robinson Crusoe economy: one consumer, one rm, two goods (labor,consumption). Labor is used as input to produce consumption that issold by the rm to the consumer (x = f (l))

Government cannot use a LST (we have in mind a case with manyindividuals)

Government directly chooses allocations and production subject torequirement that allocation must be supported by an equilibrium pricevector

Government levies tax t on consumption to fund revenue requirementR

Individual budget constraint: (1+ t)c l (pre-tax prices are 1)

By changing t we get demand as a function of tax rates: the O¤ercurve


O¤er Curve


Social Planners problem

Governments problem is:

m«axτV (q) = u(c(q), l(q))

subject to 2 constraints:

1 Revenue constraint: tc R

2 Production constraint: x = f (l)

Replace these constraints by (l , c) 2 H where H is the production set(taking into account the revenue constraint)


Production Set with Revenue Requirement


1st Best: Optimal LST


2nd Best: Optimal Distortionary Tax


Production E¢ ciency Result in One-Consumer Model

Key insight: allocation resulting from optimal distortionary tax is stillon the boundary of the production set (production e¢ ciency): Despitethe need to distort consumption choices to raise revenue (orredistribute), the production side should not be distorted

The equilibrium price vector q places the consumer on the boundary,subject to revenue requirement

With a LST, there is tangency between the boundary of theproduction set and consumers indi¤erence curve, yielding higherwelfare


Diamond and Mirrlees: General Model

With many rms, the condition of production e¢ ciency is that themarginal rate of technical substitution (MRTS) between any twoinputs is the same for all rms (recall that MRTSKL = ∂f /∂K

∂f /∂L . Here fis the production function)

With two rms, x and y and two inputs, K and L, with an optimaltax schedule:

MRTSxKL = MRTSyKL

Without taxation, prot maximizing rms in competitive marketsalways reach production e¢ ciency

The same result is obtained with taxation, provided all rms face thesame after-tax prices for inputs, so input taxes are not di¤erentiatedbetween rms


No Taxation of Intermediate Goods

Intermediate goods: goods that are neither direct inputs or outputsto individual consumption

Taxes on transactions between rms would distort production

Consider two industries, with labor as the primary input: intermediategood A, nal good B

Industry A uses labor (lA) to produce good A with a one-to-onetechnology: xA = lA

Industry B uses good A and labor lB to produce nal good B :xB = F (lB , xA). CRS technology

With wage rate w , producer price of good A is pA = w

Suppose that good A is taxed at rate τ. The cost for rm B of goodA is w + τ



Firm B chooses lB and xA to maximize F (lB , xA) wl (w + τ)xA

We get Fl = w and FA = w + τ > Fl

Aggregate production is ine¢ cient. Reduce lB and increase lA a smallamount

Then xA increases. Total production of good B increases. Tax revenuerises (government budget constraint satised)



To sum up, the Diamond-Mirrlees result gives a rationale for

Non-taxation of intermediate goods: VAT as correct form of indirecttaxationNon-di¤erentiation of input taxes between rmsNo tari¤s should be imposed on goods and inputs imported or exportedby the production sector

A key assumptions is that consumption side is independent of theproduction side


Notation and Basic Concepts

Let T (z) denote tax liability as a function of earnings z

Transfer benet with zero earnings is T (0) 0 (demogrant orlumpsum grant)

EXAMPLE (Linear Tax): T (z) = T (0) + τz , with T (0) 0 and0 < τ < 1. After tax or net income is:

z T (z) = T (0) + (1 τ)z

Here the marginal tax rate is constant (τ)

Generically, marginal tax rate is T 0(z): individual keeps 1 T 0(z) foran additional euro of earnings


Notation and Basic Concepts

When moving from zero to z earnings, net income rises from T (0)to z T (z). Individual keeps a fraction:

z T (z) + T (0)z

= 1 T (z) T (0)z

We call τp = [T (z) T (0)]/z , the participation tax rate. Individualkeeps fraction 1 τp of earnings when moving from 0 to z :

z T (z) = T (0) + z [T (z) T (0)] = T (0) + z(1 τp)

If T (0) = 5000, z = 10000, T (10000) = 2000, τp = 0,7(1 τp = 0,3)

Break-even earnings point z: point at which T (z) = 0


No Behavioral Responses

All individuals have same utility u(c), strictly increasing and concave,where c is after-tax income

We take income z as xed for each individual: c = z T (z). Incomez has density h(z)

Utilitarian government maximizes the sum of utilities:Z ∞

0u(z T (z))h(z)dz

Subject to the budget constraintRT (z)h(z)dz R (multiplier λ)


No Behavioral Responses

The Lagrangian is:

L = [u(z T (z)) + λT (z)]h(z)

The FOC is:[u0(z T (z)) + λ]h(z) = 0

We have u0(z T (z)) = λ ) z T (z) = c constant for all z

Then c = z R, where z =Rzh(z)dz average income

Government should conscate all income and equalize all after-taxincomes


Issues with Simple Model

A marginal tax rate of 100% would destroy incentives to work =)The assumption that z is exogenous is not realistic

Optimal income tax theory incorporates behavioral responses sincethe work of Mirrlees (1971)

Even without behavioral responses people could object a tax of100%. Can be seen as conscatory

There are political constraints that put a bound on what governmentscan do


Second Welfare Theorem

People have di¤erent abilities to earn income

According to the 2nd Welfare Theorem, any Pareto e¢ cient allocationcan be reached as a competitive equilibrium, provided that a suitableredistribution of initial endowments through individualized lump-sumtaxes can be done

Because of informational problems, government cannot do this. Needto use distortionary taxes and transfers based on observables (incomeand consumption) to redistribute

There is a conict between e¢ ciency and equity


E¢ ciency-Equity Trade-O¤

Tax revenue can be used to transfer programs aimed to reduceinequality. This is desirable if society is concerned by inequality andpoverty

Taxes and transfers reduce incentives to work. High tax rates createine¢ ciency if individuals react to taxes

This generates an equity-e¢ ciency trade-o¤


Mirrlees: Behavioral Responses

The crucial point in Mirrlees (1971) is that we have to take intoaccount distortions on labor supply

There is a continuum of individuals who di¤er in productivity (wage)w distributed with density f (w)

Individuals maximize utility u(c , l), subject to the constraintc = wl T (wl), where c is consumption (or after-tax income), l islabor supply, and T (wl) is a tax-transfer schedule. Solution yieldsindirect utility V (w)

Government cannot observe w . Only observes wl


Mirrlees: Behavioral Responses

The problem of the government is to nd a tax schedule T () tomaximize:

SWF =ZG (u(c , l))f (w)dw)

s.t. resource constraint:ZT (wl)f (w)dw E

and individual FOC: w(1 T 0)uc + ul = 0

where G (.) is increasing and concave

The general solution to this problem is very complex and provideslittle intuition. Due to this, Mirrlees carried out several simulationsusing a Cobb-Douglas utility function, a log-normal wage distributionand a classical utilitarian SWF


Mirrlees 1971: Results

Optimal tax schedule is approximately linear above a certainthreshold. Other authors as Tuomala (1984) found this result is notrobust

Get T () < 0 at the bottom (transfer) and T () > 0 further up (tax)

Due to complexity, a few general results:

1 0 T 0(.) 1

2 The marginal tax rate T 0(.) should be zero at the top if skilldistribution bounded


After Mirrlees

Deep impact on information economics

Little impact on practical tax policy until recently

Piketty (1997), Diamond (1998) and Saez (2001) connect Mirrleesapproach to empirical literature

Obtain formulas in terms of labor supply elasticities

Three lines: 1) Revenue-maximizing linear tax (La¤er curve); 2) Topincome tax rates; 3) Full income tax schedules


La¤er Curve

With a constant tax rate τ, reported income z depends on 1 τ(net-of-tax rate)

Total tax revenue is R(τ) = τ z(1 τ). This has an inverse-Ushape: R(τ = 0) = R(τ = 1) = 0

Tax rate τ that maximizes R:

0 = R0(τ) = z τdz/d(1 τ)

) τMAX = 1/(1+ ε)

where ε = [(1 τ)/z ]dz/d(1 τ) is the elasticity of reportedincome

Strictly ine¢ cient to have τ > τ (if ε = 0, τMAX = 1; ifε = 1, τMAX = 1/2; etc.)


Optimal Top Income Tax Rate

Now assume that the marginal tax rate is constant (τ) above somexed income threshold z

We derive the optimal value of τ using a perturbation argument

Assume no income e¤ects εc = εu = ε

Total number of individuals above z is N

We denote by zm(1 τ) their average income (depends on net-of-taxrate 1 τ)


Optimal Top Rate

A small change dτ > 0 above z has three e¤ects:

1. A mechanical increase in tax revenue of dM = N [zm z ]dτ

2. A behavioral response:

dB = Nτdzm = Nτdzm

d(1 τ)dτ

= N τ

1 τ

1 τ

zmdzm

d(1 τ)zmdτ = N τ

1 τεzmdτ


Optimal Top Rate

3. A welfare e¤ect: If government values marginal consumption ofrich at g 2 (0, 1), dW = gdM. Here g depends on the curvature ofu(c) and on the SWF

Summing the three terms:

dM + dW + dB = dM(1 g) + dB =

= Ndτ

(1 g)[zm z ] ε

τ

1 τzm

Optimal τ is such that dM + dW + dB = 0 )

τTOP1 τTOP

=(1 g)(zm/z 1)

ε(zm/z)


Optimal Top Rate

We can solve:

τTOP =(1 g)( zmz 1)

(1 g)( zmz 1) + ε zmz

τTOP decreases with g and with ε. It increases with zm/z (thethickness of top tail)

Not an explicit formula since zm/z also a function of τ


Optimal Tax Rate

The top tail of the income distribution is closely approximated by aPareto distribution. Density function is f (z) = a ka/z1+a wherea > 1 is the Pareto parameter (also F (z) = 1 ka/za)

With a Pareto distribution the ratio zm/z is the same for all z in thetop tail and it is equal to a/(a 1)

For the US, reasonable values are z = $400, 000 and zm = $1,2million, so that zm/z = 3 and a = 1,5. For France, a slightly higherbetween 2.2 and 2.3


Optimal Tax Rate

Since zm/z = a/(a 1), with a Pareto distribution the formula canbe simplied into:

) τTOP =1 g

1 g + a ε

Example: ε = 0,5, g = 0,5, a = 1,5) τTOP = 40%

In France, taking ε = 0,5, g = 0,5, a = 2,2, we get τTOP = 31,25%


Bounded Distributions

Top earner gets zT , and second earner gets zS

Then zm = zT when z > zS ) zm/z ! 1 when z ! zT )

dM = Ndτ[zm z ]! 0 < dB = Ndτετ

1 τzm

Optimal τ is zero for z close to zT

This result has little practical relevance since it applies to the topearner only


Revenue Maximizing Tax Rate

To nd the revenue maximizing top tax rate we assign 0 weight towelfare of top incomes

With an utilitarian SWF ) g = uc (zm)! 0 when z ! ∞

With a Rawlsian SWF ) g = 0 for any z > m«¬n(z)

If g = 0, τTOP = τMAX = 1/(1+ aε)

Example: a = 2 and ε = 0,5 ) τ = 50%. Result extremely sensibleelasticity value (a = 2, ε = 2 ) τ = 20%)

The La¤er rate can be obtained as a special case with z = 0

) zm/z = ∞ = a/(a 1)) a = 1) τMAX = 1/(1+ ε)


General Case: Optimal Non-Linear Income Tax

Now consider general problem of setting optimal T (z)

Let H(z) = CDF of income [population normalized to 1] and h(z) itsdensity [endogenous to T (.)]

Let g(z) = social marginal value of consumption for taxpayers withincome z in terms of public funds

Let G (z) be the average social marginal value of consumption fortaxpayers with income above z [G (z) =

R ∞z g(s)h(s)ds/(1H(z))]


Small Tax Reform Perturbation


General Non-Linear Income Tax

Consider small reform: increase T 0 by dτ in small interval (z , z + dz)

Mechanical revenue e¤ect

dM = dzdτ(1H(z))

Before the change, an individual with income z + dz pays taxesΩ+ τdz , where Ω represents taxes paid by an individual with incomez

After the change, she will pay taxes Ω+ (τ + dτ)dz . The di¤erenceis dτdz

Mechanical welfare e¤ect

dW = dzdτ(1H(z))G (z)



Behavioral e¤ect: substitution e¤ect δz inside small band [z , z + dz ]:

dB = h(z)dz T 0 δz = h(z)dz T 0 dτ ε(z) z/(1 T 0)

This is because we can write:

δz =δz

d(1 T 0)1 T 0z

d(1 T 0)1 T 0 z = ε(z)

dτ

1 T 0 z

At the optimum dM + dW + dB = 0



Optimal tax schedule satises:

T 0(z)1 T 0(z) =

1ε(z)

1H(z)zh(z)

[1 G (z)]

T 0(z) decreasing in g(z 0) for z 0 > z [redistributive tastes]

T 0(z) decreasing in ε(z ) [e¢ ciency]

T 0(z) decreasing in h(z)/(1H(z)) [density]

To see connection with top tax rate: consider z ! ∞

G (z)! g , (1H(z))/(zh(z))! 1/a

ε(z ) ! ε ) T 0(z) = (1 g)/(1 g + a ε) = τTOP


Negative Marginal Tax Rates Never Optimal

Suppose T 0 < 0 in band [z , z + dz ]

Increase T 0 by dτ > 0 in band [z , z + dz ]

dM + dW > 0 because G (z) < 1 for any z > 0 (with declining g(z)and G (0) = 1)dB > 0 because T 0(z) < 0 [smaller e¢ ciency cost]

Therefore T 0(z) < 0 cannot be optimal

Marginal subsidies also distort local incentives to work

Better to redistribute using lump sum grant


Numerical Simulations of Optimal Tax Schedule

Formula above is a condition for optimality but not an explicitformula for optimal tax schedule. Distribution of incomes H(z)endogenous to T (.)

Therefore need to use structural approach (specication of primitives)to calculate optimal T (.)

Saez (2001) uses two utility functions:

u(c , l) = log(c l1+ k

1+k

)

u(c , l) = log(c) log( l1+ k

1+k

)

With the rst, no income e¤ects and ε = 1/k. With the second,εC = 1/k, and εU goes to zero as z goes to innity.Calibrate the exogenous skill distribution F (w) such that actual T (.)yields empirical H(z)


Results

Dashed line: optimal linear tax rate

Optimal tax rates are clearly U-shaped:

Decreasing from $0 to $75,000 and then increase until $200,000

Above $200,000, optimal rates close to asymptotic level


Diamond (1988)

No income e¤ects: u(c , l) = c + V (1 l)

Constant labor supply elasticity

SWF G (u) increasing and strictly concave

Ability distribution is Pareto above the mode (wm)

At least above wm , marginal tax rates should be increasing in w

Not robust if utility function is u(c , l) = U(c) + V (1 l), with Uconcave: Dahan and Strawczynski (2000)


Combining Income and Commodity Taxes

Most governments tax both income and goods

In Spain, di¤erential commodity taxes with non-linear income tax

1 VAT has three rates: 18% (general), 8% and 4%. 8% applies to food,health, transportation. 4% to necessities, books,..

2 There are additional excise taxes (impuestos especiales) on somegoods (alcohol, gasoline, cigarettes)

3 Imposes capital income taxes

What is the best combination of taxes?


Commodity and Income Taxation

K goods c = (c1, .., cK ) with pre-tax price p = (p1, .., pK )

Individual h has utility uh(c1, .., cK , z)

Suppose there is a nonlinear income tax on earnings z . Cangovernment raise welfare using commodity taxes t = (t1, .., tK )?

More instruments cannot hurt:

m«axt ,T (.)

SWF m«axt=0,T (.)

SWF


Atkinson and Stiglitz: Commodity Taxation is Superuous

Atkinson and Stiglitz (1976) show that:

m«axt ,T (.)

SWF = m«axt=0,T (.)

SWF

Commodity taxes not useful if uh(c1, .., cK , z) satisfy 2 conditions:

1 Weak separability between (c1, .., cK ) and z in utility

2 All individuals have the same sub-utility function of consumption:

uh(c1, .., cK , z) = Uh(v(c1, .., cK ), z)


Atkinson-Stiglitz: Intuition

If tax system is too progressive (relative to the optimum), a highability individual will choose to work less to mimic income level(z = wl) of a lower ability individual

Both pay same income tax and have the same after-tax income, butthe high ability individual has more leisure. Government cannot useincome tax to redistribute

What about di¤erentiated good taxes? Is it possible to reduce theincentive to mimic?

No. With separability and homogeneity, conditional on earnings z ,consumption choices c = (c1, .., cK ) do not provide information onability. Two individuals with the same z , but di¤erent abilities, choosethe same consumption bundle


Atkinson-Stiglitz: Intuition

Di¤erentiated commodity taxes t1, .., tK create a tax distortion withno benet

It is optimal to have uniform taxation. In particular, zero taxation ofgoods

It is better to do all the redistribution with the individual income tax

With only linear income taxation (Diamond-Mirrlees 1971, Diamond1975), di¤erential commodity taxation can be useful tonon-linearize the tax system


Atkinson-Stiglitz: Implications

A consequence of uniform taxation is that we should not subsidizeparticular goods (housing, health) for the purpose of redistribution

Another implication is that even a government who would like toredistribute should not do so by taxing luxury goods more than basicgoods

A striking result, although it has limited practical relevance


Atkinson-Stiglitz: Criticisms

First, the result depends on a technical condition, weak separability.Without it, better to tax more heavily goods which arecomplementary to leisure

Second, the theorem requires that all individuals must have the samepreferences. With di¤erences in tastes, the result is not longer true(Saez (2002) in the Journal of Public Economics)

Now the optimal non-linear income tax should be supplemented notonly by taxes on commodities that are complementary to leisure, butalso by taxes on commodities for which high-income people have arelatively strong taste


The case for uniform taxation

In spite of the above, there are still some arguments in favor ofuniform taxation:

Government can lack information on compensated cross-priceelasticities needed to compute optimal taxes

Changes in tastes and technologies may require frequent adjustmentsin taxes

A uniform VAT is easier to administer and less susceptible to fraud

Government would eliminate incentives for special interest groups tolobby for low tax rates on particular goods


Should we tax capital income?

The Atkinson-Stiglitz theorem has the implication that capital shouldnot be taxed: since a capital tax is a tax on future consumption butnot on current consumption, it violates the prescription of uniformtaxation

Not only this, since a capital tax imposes an ever-increasing tax onconsumption further in the future, its violation of the principle ofuniform taxation is extreme

Consider a simple life-cycle model. Consumer supplies L labor whenyoung and enjoys consumption C1 (when young) and C2 (when old)

Taxes on labor are T (wL), where w is pre-tax wage and the return tocapital (r) is taxed at rate trFinally, we denote savings when young by S


Model

The constraints are:

S = wL T (wL) C1 and C2 = [1+ r(1 tr )]S

We can combine them into:

C1 +C2

1+ r(1 tr )= wL T (wL)

The term on the left can be written as:

C1 +1

1+ r+

rtr(1+ r)(1+ r(1 tr ))

C2


Model 2

A tax on capital income is like an excise tax on future consumption,because it raises the relative price of future consumption above1/(1+ r), which is the price without taxes on capital incomeIf total time to 1, leisure when young is 1 LConsider that lifetime utility U(1 L,C1,C2) can be written as:

U(1 L,C1,C2) = U(1 L, v(C1,C2))

According to Atkinson-Stiglitz, if we interpret C1 and C2 as twodi¤erent commodities, they should be taxed uniformly, provided thatthe government is using the income tax for redistributional purposes


Interpretation

Without precise information about the degree of complementarity orsustitutability between leisure and C1 and C2, a possibility is toconsider that both of them are equally complements with leisure

This implies that the marginal e¤ective tax rate on capital incomeshould be zero

However, if leisure and future consumption are complements, apositive capital income tax would be optimal. Empirically, leisureseems to be increasing with age. Then, tax rates on capital incomeshould be positive


Innite horizon models

In the standard neoclassical growth model with innitely-livedindividuals, an optimal income tax policy entails setting capitalincome taxes equal to zero in the long run. This is the classicalChamley-Judd result

Intuition: assume that pre-tax return of capital in the steady state isr , and there is a tax rate on capital τ

Capital taxation changes the relative price of consumption at date tand at date t + T by a factor:

1+ r1+ r(1 τ)

TWithout taxes, the relative price of Ct+T in terms of Ct is

11+r

T.

With taxes it is

11+r (1τ)

TI. Iturbe-Ormaetxe (U. of Alicante) Public Economics Taxation II: Optimal Taxation Winter 2012 83 / 98

Innite horizon models 2

If τ = 0, that factor is 1

If τ 6= 0, when T goes to innity the factor goes either to zero (whenτ < 0) or to +∞ (when τ > 0)

Taxing (or subsidizing) capital would distort in an explosive way thechoice between current and future consumption

This result does not require any separability assumption

However, the result does not hold in OLG models: See Erosa andGervais (2002) and Conesa, Kitao and Krueger (2009). The latter usea OLG models where households face borrowing constrains and aresubject to idiosyncratic income shocks. They calibrate their model tothe US economy and obtain an optimal tax rate on capital income of36% (and a at tax of 23% on labor income, together with a $7,200deduction)


Taxation and Savings: Evidence

Key assumption in Chamley-Judd and Atkinson-Stiglitz results: peopleoptimize their savings decisions

Recent evidence challenges this assumption

Madrian and Shea (2001) study employee 401(k) enrollment decisionsand contribution rates at a U.S. corporation:

Most people adhere to company defaults and do not make activesavings choices

Suggests that defaults may have much bigger impacts on savingsdecision than net-of-tax returns


Elasticity of Taxable Income

Labor supply elasticity is estimated to be close to zero for males(higher for married women). This implies that e¢ ciency cost of taxinglabor income is expected to be low

However, taxes trigger many other behavioral responses: changes inthe form of compensation (e.g, employer-paid health insurance);moving away from taxed goods or activities to those that are taxedmore lightly or not taxed at all; trying to avoid taxes by shiftingincome intertemporally (for instance, diverting income into atax-deferred account); tax evasion

The elasticity of taxable income (ETI) tries to capture all these: itmeasures how reported taxable income (TI) responds to tax changes.The bottom line is that the e¢ ciency cost of taxation can be muchhigher than is implied if labor supply is the only margin of behavioralresponse


Tax Elasticity of Top Incomes

Most empirical studies have found that the behavioral response to taxchanges concentrates on top earners. We focus on them

As above, we assume that all incomes above an income threshold zface a constant marginal tax rate τ and we assume the number of topearners is N

The optimal tax on top earners (assuming the weight thatgovernment attaches to them is zero) is:

τTOP =1

1+ aε,

where a is the Pareto parameter and ε is the elasticity of top earners



The key ingredient is the elasticity ε of top earners. If, for instance,ε = ,25 and a = 1,5, τTOP = ,73, substantially higher than thecurrent 42.5 percent top US marginal tax rate (all taxes combined)

The optimal value would be 42.5% if ε is extremely large, around .9(or if the marginal consumption of very high income earners is highlyvalued, g = ,72)

A large literature has focused on the response of reported income,either adjusted gross incomeor taxable incometo net-of-tax rates

Found large and quick responses of reported incomes along the taxavoidance margin at the top of the distribution, smaller e¤ects onreal economic responses



For example, in the US exercises of options surged in 1992 before the1993 top rate increase took place

Tax Reform Act of 1986 led to a shift from corporate to individualincome because it became more advantageous to be organized as abusiness taxed solely at the individual level rather than as acorporation taxed rst at the corporate level

With tax avoidance or evasion opportunities, the tax base in a givenyear is very sensitive to tax rates. The optimal tax rate is low. But..

If avoidance channels is re-timing or income shifting, there are taxexternalities: changes in tax revenue in other periods or other taxbases, optimal taxes must not decrease

Also the tax avoidance or evasion component of ε can be reducedthrough base broadening and tax enforcement



Gruber and Saez (2002) nd a substantial taxable income elasticityestimate (ε = ,57) at the top. However, they also nd a smallelasticity (ε = ,17) for income before any deductions

With the rst estimate, τTOP = ,54 but with the second τTOP = ,80

Taking as xed state and payroll taxes, such rates correspond to topfederal rates of .48 and .76, respectively

In any case, there is some agreement that ε = ,57 is a conservativeupper bound


Example: Income shifting

Taxable income can be also reduced because individuals shift fromtaxable compensation toward untaxed fringe benets. Or fromindividual income toward corporate income; deferred compensation,etc.

Assume that a fraction s < 1 of incomes that disappear from TI afteran increase in taxes dτ are shifted to corporate income that is taxedon average at rate t < τ

The tax that maximizes revenue is:

τs =1+ aεts1+ aε

> τTOP =1

1+ aε

If a = 1,5, ε = 0,5, τ = 0,35, s = 1/2 and t = 0,3,τs = ,636 > τTOP = ,57


Chetty, Looney and Kroft (2009)

Until now, we assume agents respond to tax changes in the same waythey react to price changes

Suppose that for some commodity the producer price is 10 euros andthere is a tax of 1 euro per unit, so nal consumer price is 11 euros.An increase in producer price of 50 cents has the same e¤ect onpurchases than an increase in 50 cents in the tax

Chetty-Looney-Kroft challenge this. They nd that salience (orvisibility) of taxes a¤ect consumerspurchase decisions, becauseconsumers are inattentive to taxes

For example, sometimes taxes on commodities are not displayed inposted prices

They nd that commodity taxes that are included in posted pricesthat consumers see when shopping have larger e¤ects on demand


Theory

Two goods, called x and y , are supplied perfectly elastically. Price ofy is 1 and we call p the pre-tax price of x

Good y is untaxed and x pays sales tax at rate τS so total price isq = (1+ τS )p

Consumers see p when shopping because sales tax is not included inthe posted price. They have to calculate q, making q less salientthan p

Call x(p, τS ) the demand as a function of posted price and sales tax.With rational shoppers, demand depends only on total price:x(p, τS ) = x((1+ τS )p, 0)

With rational consumers, a 1% increase in p and a 1% increase in(1+ τS ) reduce demand by exactly the same amount. If εx ,p andεx ,1+τS are the elasticities with respect to p and (1+ τS ) :εx ,p = εx ,1+τS


Degree of underreaction

In contrast, ChLK propose consumers underreact to the tax τS

because it is less salient: εx ,p > εx ,1+τS

Log-linearizing demand x(p, τS ) :

log x(p, τS ) = α+ β log p + θr β log(1+ τS )

The parameter θr captures the degree to which consumers underreactto the tax. In particular:

θr =∂ log x/∂ log(1+ τS )

∂ log x/∂ log p=

εx ,1+τS

εx ,p

The null hypothesis is θr = 1. They propose two strategies toestimate θr


Manipulate tax salience

They conducted an experiment in a supermarket chain for three weeks

Stores do not include the sales tax in the price tag on the shelves fortaxable goods like cosmetics. Taxes only appear on the sales slip afterpaying at the cash register

ChLK adjusted the price tags to display prices including the 7.375%sales tax. The result was a decline in sales of those items by about8% relative to the control groups

Making visible the sales tax reduce purchases, suggesting that mostconsumers do not normally take into account the sales tax on suchproducts

They estimate a value of bθr 0,35. That is, there is substantialunderreaction


Alternative approach

In a second test they used data over a long period of time of alcoholpurchases to compare the e¤ect of price changes with tax changes

Alcohol is subject to an excise tax (salient) and the less-salient salestax that appears only at the cash register

They found a much larger e¤ect of increases in the excise tax than toincreases in the sales tax. They obtain an estimate of bθr 0,06The explanation by ChLK is not that consumers are not aware of theexistence of taxes, but that they simply do not bother to computetax-inclusive prices


Policy implications

Standard theory predicts that taxes create an e¢ ciency cost only tothe extent that it reduces demand for the taxed good

Here, even if the demand for the taxed good does not change, taxesmay have a substantial e¢ ciency cost. The reason is that the tax candistort consumption of other goods. A consumer that is not attentiveto taxes on some good may end up spending too much on it and,thus, will have less money to spend on other goods

For instance, an individual who does not take into account taxes oncars, would over-spend on the car and end up with less money leftthan he would like for food, reducing economic welfare


Policy implications

Another implication is that now statutory incidence matters: supposea phone has $39.99 in the price tag, but adding taxes and costumerfees, actual price is $47.00. If taxes were levied by the rm, the onlyway the rm can pass taxes to costumers is by raising prices in theprice tags, reducing demand

Firms will bear a higher burden if the tax is levied on them ratherthan on consumers


public economics taxation ii: optimal taxation · 2016-04-26 · i. iturbe-ormaetxe (u. of...

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