Lecture 2 [0.3cm] Optimal Indirect...

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Lecture 2 Optimal Indirect Taxation March 2014

Transcript of Lecture 2 [0.3cm] Optimal Indirect...

Lecture 2

Optimal Indirect Taxation

March 2014

Optimal taxation: a general setup

Individual choice criterion, for i = 1, . . . , I :

U(ci , `i , θi )

Individual anonymous budget constraint

B(ci , `i , θi , prices) ≥ 0.

Social objective

W (U(c1, `1, θ1), . . . ,U(cI , `I , θI ))

Feasibility, for instance

F (I∑

i=1

ci + G ,I∑

i=1

`iθi ) ≤ 0.

Problem: choose the B that maximizes the social objective (aiming topay for the war and/or to redistribute), given the agents’ behaviors withmarket clearing prices.

Ramsey vs. Mirrlees

Ramsey: constraints are put directly on the B function, i.e. taxes ontraded goods are linear in the amounts traded. The constraints aresupposed to stem from institutional features of the economy which arenot spelled out.

Mirrlees: all constraints come from informational asymmetries. Thestructure of the economy is common knowledge, but the governmentdoes not know who is who. In the most studied case, it can only maketaxes conditional on income (or efficient labor) `θ, and not on ` and θseparately. This leads to a formulation with incentive constraints, akin toa mechanism design problem.

An attempt to define direct/indirect taxes

Atkinson, Canadian J. of Economics, 1977.

‘Historically the distinction no doubt arose from the method ofadministration, in that the taxpayer handed over income tax directly tothe revenue authorities, but only paid sales taxes indirectly via thepurchase of goods. (· · · ) the phrase ‘assise directement’ was apparentlyin use for personal taxes in France in the sixteenth century, and thesewere contrasted with excise taxes.’

‘According to Buchanan (1970), ‘direct taxation is defined as taxationimposed upon the person who is intended to be the final bearer of theburden of payment.’

‘Direct taxes may be adjusted to the individual characteristics of thetaxpayer, whereas indirect taxes are levied on transactions irrespective ofthe circumstances of buyer or seller. (· · · ) direct taxes can bepersonalized or tailored to the particular economic and socialcharacteristics of the household being taxed.’

Indirect taxes in France

Recettes nettes du budget général

2007(en milliards d'euros)

2008(en milliards d'euros)

Impôt sur le revenu 56,8 60,5Impôt sur les sociétés 51,5 53,9Taxe intérieure sur les produits pétroliers 17,6 16,9Taxe sur la valeur ajoutée 131,1 135,0Autres recettes fiscales 10,9 5,8Recettes fiscales nettes 267,9 272,1

Source : ministère du Budget, des Comptes publics et de la Fonction publique.

Plan

1. Tax incidence

2. Ramsey: optimal indirect taxes

3. Direct taxes: the intensive and extensive models

4. Separability, production efficiency: how to best combine direct andindirect taxes

5. Capital taxation

6. Application: labor supply, Prescott

Plan of the talk

1. The main ideas in a single consumer, no substitution and no incomeeffects setup.

2. The general formula

3. Back to a representative agent, three goods case: Corlett-Hague.

I. A simplified frameworkThere is only one household (efficiency) who maximizes

U(X, L) =n∑

i=1

Ui (Xi )− L

subject to her budget constraintn∑

i=1

qiXi ≤ L,

with respect to the variables (X = (X1, . . . ,Xn), L). The qi = pi + ti areconsumer prices which differ from the production prices pi by the ‘excise’tax ti . Labor is the numeraire and is not taxed.

Preferences are additive, increasing concave in the consumption goods,quasilinear in labor: the demand for good ξi (qi ) only depends on its ownprice (neither cross price nor income effects) and decreases with respectto qi .

The indirect utility of the consumer is

V (q) =n∑

i=1

[Ui (ξi (qi ))− qiξi (qi )] .

Production and second best program

Production takes place at constant returns to scale, with exogenouslyfixed production prices p.

The authority chooses the tax vector t = q− p which maximizes V (q)subject to a revenue requirement R

n∑i=1

tiξi (qi ) ≥ R.

The Lagrangian is

L(q,λ) = V (q) + λ

[n∑

i=1

tiξi (qi )− R

].

First order conditions

The n FOCs with respect to ti are

∂V

∂qi(q) + λ

[ξi (qi ) + ti

∂ξi∂qi

(qi )

]= 0, i = 1, . . . , n

to which one must add the budget constraint

n∑i=1

tiξi (qi )− R = 0.

Using the expression of the indirect utility or appealing to Roy’s identity,the FOCs can be rewritten as

−ξi (qi ) + λ

[ξi (qi ) + ti

∂ξi∂qi

(qi )

]= 0.

The marginal cost of public funds

The FOCs imply that λ is larger than 1: one unit of public expenditurecosts more than one unit of private good (marginal cost of public funds).One notes

θ =λ− 1

λ> 0.

Second order conditions

Important remark: The Lagrangian is not necessarily concave.

L′′qi = (2λ− 1)ξ′i + λtiξ

′′i .

Introducing elasticities

The absolute value of the price elasticity of demand is

εi (qi ) = − qiξi (qi )

∂ξi∂qi

(qi ) > 0.

The FOC

−ξi (qi ) + λ

[ξi (qi ) + ti

∂ξi∂qi

(qi )

]= 0

is equivalent to (after division by ξi )

λ− 1 + λtiqi

qiξi (qi )

∂ξi∂qi

(qi ) = 0.

The inverse elasticity rule

I All the goods should be taxed.

I The optimal tax rate is inversely proportional to the price elasticityof (compensated) demand.

tipi + ti

=

(1− 1

λ

)1

εi (qi )=

θ

εi (qi ).

The discouragement index

Another reading of the inverse elasticity rule:

tiξi (qi )

∂ξi∂qi

(qi ) = −θ.

The LHS gives the rate of change in the demand of good i which followsthe introduction of a small tax on this good:

dXi =∂ξi∂ti

dti '∂ξi∂ti

ti ⇒dXi

Xi' tiξi (qi )

∂ξi∂qi

.

Taxation should discourage the demand for every good in the sameproportion θ (θ > 0): The LHS is the ‘discouragement index’ (Mirrlees,1976).

Other instruments: lump sum taxes

Assume that the authority can collect T in a lump-sum fashion. TheLagrangian rewrites:

L(t,T , λ) = V (t)− T + λ

[∑i

tiξi (ti ) + T − R

].

Substitute a lump-sum tax dT to indirect taxes, maintaining theaggregate collected tax constant. The resulting change in the socialobjective is

dL =∑i

∂L∂ti

dti +∂L∂T

dT = dV .

dV =∂L∂T

dT = (λ− 1)dT = λθdT .

Other instruments: wage tax

A normalization issue: Should one tax good k when tk > 0 at theoptimum ?

We have assumed up to now that labor income is not taxed.

Take two different tax structures:

1. In the first one goods are taxed at rates t while labor is not taxed;

2. In the second one goods are taxed at rates t′ and labor at rate τ ′.

With the second tax structure the household budget constraint is

n∑i=1

(pi + t ′i )Xi = (1− τ ′)L.

Let ti be such that pi + ti = (pi + t ′i )/(1− τ ′), i.e.,

ti = t ′i +τ ′

1− τ ′(pi + t ′i ) .

Then both tax structures are equivalent for the household.

The aggregate tax revenues coincide in both cases:

n∑i=1

tiXi =n∑

i=1

t ′iXi +τ ′

1− τ ′n∑

i=1

(pi + t ′i )Xi =n∑

i=1

t ′iXi + τ ′L.

Therefore ti = 0 does not mean that good i should be tax-free, but thatit should be taxed (resp. subsidized) as much as labor is subsidized (resp.taxed).

II. The general Ramsey rule

A more general setup, with cross price and income effects and a numberof consumers, to assess efficient indirect tax structures. We keep labouras the untaxed numeraire.

The typical h household chooses (X, L) which maximizes U(X, L) s.t.

n∑i=1

qiXi ≤ L + M,

where M represents nonlabor income.

Consumer demand is ξhi (q,M) for all i , while labor supply is ξhL(q,M).

Indirect utility is

V h(q,M) = U(ξh1 (q,M), . . . , ξhn(q,M), ξhL(q,M)

).

The marginal utility of income is V h′M ≡ αh.

The government program

The authority chooses q which maximize

W [V 1(q,M1), . . . ,V H(q,MH)]

subject to (multiplier λ)

H∑h=1

n∑j=1

(qj − pj)ξhj (q,Mh) ≥ R.

The derivation of the Ramsey formula

The first order condition in qi gives

H∑h=1

∂W

∂V h

∂V h

∂qi= −λ

H∑h=1

ξhi +n∑

j=1

tj∂ξhj∂qi

.

By Roy’s identity, αh denoting the marginal utility of income ofconsumer h:

∂V h

∂qi= −αhξhi .

Let βh = (∂W /∂V h)αh be the social marginal utility of income ofconsumer h. Then

H∑h=1

βhξhi = λ

H∑h=1

ξhi +n∑

j=1

tj∂ξhj∂qi

.

Ramsey formula: the right hand side

The Slutsky equation gives

∂ξhj∂qi

= Shji − ξhi

∂ξhj∂Mh

.

Substituting

H∑h=1

βh

λξhi =

H∑h=1

ξhi +H∑

h=1

n∑j=1

tj

(Shji − ξhi

∂ξhj∂Mh

).

Let bh = βh/λ+∑n

j=1 tj∂ξhj /∂M

h be the net social marginal utility ofincome of consumer h. Then

n∑j=1

tj

H∑h=1

Shij = −ξi

(1−

H∑h=1

bhξhiξi

).

The final result

The empirical covariance between the net social marginal valuation ofincome and consumption of good i is

ri =1

H

H∑h=1

(bh

b− 1

)(ξhiξi− 1

)=

1

H

H∑h=1

bh

b

ξhiξi− 1

∑nj=1 tj

∑Hh=1 S

hij /H

ξi= −

(1−

H∑h=1

bh

H

ξhiξi

)= −1 + b + rib.

The (absolute value of the) left hand side is the discouragement index, ameasure of the change in compensated demand. It varies with the good icontrary to the initial example, depending on the correlation between thesocial weight and the share in the use of the good. The governmentshould discourage less the consumption of goods bought by agents with ahigh net social marginal utility of income.

III The representative consumer

All the income effects are the same, so that bh is independent of h.∑nj=1 tjSij

ξi= −(1− b).

Summing up the equalities over the goods i after multiplication with thetax rate ti gives

n∑j,i=1

tjSij ti = −(1− b)I∑

i=1

tiξi .

The left hand side is negative by the negative definiteness of the Slutzkyterms: b must be smaller than one is some tax revenue is collected.Therefore the discouragement indexes are all nonnegative: the optimaltax system does not encourage the consumption of any good.

Corlett-Hague (1953): the two-good case

The FOC aret1S11 + t2S12 = −(1− b)ξ1

t1S21 + t2S22 = −(1− b)ξ2

which give

t1 =1− b

D(S12ξ2 − S22ξ1)

t2 =1− b

D(S21ξ1 − S11ξ2)

Playing with Slutzky and elasticitiesThe compensated demand for good i is homogeneous of degree zero inprices, so that noting 0 the leisure good which is also the numeraire:

S10 + q1S11 + q2S12 = 0

S20 + q1S21 + q2S22 = 0.

Define the compensated elasticities

εij =qjSijξi

.

Thenε10 + ε11 + ε12 = 0,

and similarly for good 2.

Factorizing ξ1ξ2/q2 in the t1 formula yields

t1 =1− b

D

ξ1ξ2q2

(ε12 − ε22),

or

t1 = −1− b

D

ξ1ξ2q2

(ε10 + ε11 + ε22).

Corlett-Hague concluded

t1q2 − t2q1 = −1− b

Dξ1ξ2(ε10 − ε20)

Near zero tax rates, this says that the tax rate on good 1 should be largerthan that on good 2, t1/q1 > t2/q2, if and only if ε10 < ε20, i.e. good 1is more complementary to leisure than good 2.