Lecture 3
Optimal Indirect Taxation
Stéphane Gauthier
November 9, 2010
An attempt to define direct/indirect taxes
Atkinson, Canadian J. of Economics, 1977.
‘Historically the distinction no doubt arose from the method of
administration, in that the taxpayer handed over income tax directly to
the revenue authorities, but only paid sales taxes indirectly via the
purchase of goods. (· · · ) the phrase ‘assise directement’ was apparently
in use for personal taxes in France in the sixteenth century, and these
were contrasted with excise taxes.’
‘According to Buchanan (1970), ‘direct taxation is defined as taxation
imposed upon the person who is intended to the final bearer of the
burden of payment.’
‘Direct taxes may be adjusted to the individual characteristics of the
taxpayer, whereas indirect taxes are levied on transactions irrespective of
the circumstances of buyer or seller. (· · · ) direct taxes can be
personalized or tailored to the particular economic and social
characteristics of the household being taxed.’
Indirect taxes in France
Recettes nettes du budget général
Impôt sur le revenu
Impôt sur les sociétés
Taxe intérieure sur les produits pétroliers
Taxe sur la valeur ajoutée
Autres recettes fiscales
Recettes fiscales nettes
2007
(en milliards d'euros)
56,8
51,5
17,6
131,1
10,9
267,9
2008
(en milliards d'euros)
60,5
53,9
16,9
135,0
5,8
272,1
Source : ministère du Budget, des Comptes publics et de la Fonction publique.
A first benchmark: The inverse elasticity rule
There is only one household (efficiency) who maximizes
U(X, L) =
n
X
Ui (Xi ) − L
i=1
s.t. her budget constraint
n
X
qi Xi ≤ L,
i=1
where qi = pi + ti (ti is an ‘excise’).
Preferences are (1) additive and (2) quasilinear: demand for good i only
depends on its own price (neither cross price nor income effects).
She gets
n
X
V (q) =
Ui (Xi (qi )) − L(q).
i=1
The authority chooses t = q − p which minimize V (p) − V (q) s.t.
n
X
ti Xi (qi ) ≥ R.
i=1
→ Discuss the timing and the assumptions on technology
The Lagrangean is
L(q,λ) = V (q) + λ
" n
X
#
ti Xi (qi ) − R ,
i=1
and the (n + 1) FOCs (be careful with FOCs) are
∂V
∂Xk
(q) + λ Xk (qk ) + tk
(qk ) = 0, k = 1, . . . , n
∂qk
∂qk
and
n
X
i=1
ti Xi (qi ) − R = 0.
Appealing to Roy’s identity, the first n FOCs can be rewritten
∂Xk
−Xk (qk ) + λ Xk (qk ) + tk
(qk ) = 0
∂qk
tk
1
1
θ
⇔
= 1−
≡
,
pk + t k
λ εk (qk )
εk (qk )
with θ > 0 and (by the law of demand)
εk (qk ) = −
qk ∂Xk
(qk ) > 0.
Xk (qk ) ∂qk
This is the ’inverse elasticity rule’ :
I All the goods should be taxed.
I
The optimal tax rate is (apparently) inversely proportional to the
price elasticity of (compensated) demand.
Another reading of the ‘inverse elasticity rule’:
tk
θ
tk ∂Xk
=
⇔
(qk ) = −θ.
pk + tk
εk (qk )
Xk (qk ) ∂qk
The LHS gives the change in the demand of good k which follows the
introduction of a small tax on this good:
dXk =
∂Xk
dXk
tk ∂Xk
∂Xk
dtk '
tk ⇒
'
.
∂tk
∂tk
Xk
Xk (qk ) ∂qk
→ Taxation should discourage the demand for every good in the same
proportion θ (θ > 0): The LHS is the ’discouragement index’ (Mirrlees,
1976).
The higher θ = 1 − 1/λ > 0, the higher discouragement should be.
I
A possible interpretation: one should discourage consumption when
the authority puts a high value on collected taxes, λ is high (θ is
close to 1).
I
Here the household marginal utility of income is α = 1.
Hence 1/λ could be the cost (such as evaluated by the authority,
but supported by the household) of a lump-sum income tax on the
household. Such a tax would yield 1 unit of tax and costs 1/λ units
of tax. The authority does not value the household loss when θ is
high (close to 1), and this urges the authority to ‘discourage’
consumption.
Assume that the authority can collect T in a lump-sum fashion. The
Lagrangean rewrites:
"
#
X
L(t, T , λ) = V (t, −T ) + λ
ti Xi (ti ) + T − R .
i
Now, let us substitute a lump-sum tax dT to indirect taxes, maintaining
the whole collected tax constant. The resulting change in the social
objective is
X ∂L
∂L
dti +
dT = dV .
dL =
∂ti
∂T
i
If the initial situation is a Ramsey optimum, then
∂L
= 0, i = 1, . . . , n.
∂ti
dV
∂L dT
dT
1
⇒
=
= (−α + λ)
= 1−
dT = θdT .
λ
∂T λ
λ
λ
A normalization issue: Should one tax good k when tk > 0 at the
optimum ?
By assumption labor (income) is not taxed: when tk > 0 it could be that
good k should be taxed more heavily than (labor) income.
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 t0 and labor at rate τ .
With the second tax structure the household budget constraint is
n
X
(pi + ti0 )Xi = (1 − τ )L.
i=1
Let ti be such that pi + ti = (pi + ti0 )/(1 − τ ), i.e.,
ti = ti0 +
τ
(pi + ti0 ) .
1−τ
Then both tax structures are equivalent for the household.
The collected tax coincide in both cases:
n
X
i=1
ti Xi =
n
X
i=1
ti0 Xi +
n
n
X
τ X
(pi + ti0 )Xi =
ti0 Xi + τ L.
1−τ
i=1
i=1
Therefore tk = 0 does not mean that k should be tax-free, but that it
should be taxed (resp. subsidized) as much as labor is subsidized (resp.
taxed): tk0 /pk = −τ (for τ < 1).
→ Good k complement or substitute to labor ?
Ramsey rule
→ A more general setup, with cross price and income effects, to assess
efficient indirect tax structures.
The household chooses (X, L) which maximizes U(X, L) s.t.
n
X
qi Xi ≤ L + M,
i=1
where M represents some nonlabor income.
Solution is Xi (q, M) for every i, and L(q, M).
Indirect utility is
V (q, M) = U (X1 (q, M), . . . , Xn (q, M), L(q, M)) ,
and the marginal utility of income VM0 ≡ α.
The authority chooses t which maximize
V (p + t, M)
s.t.
n
X
ti Xi (p + t, M) ≥ R.
i=1
The Ramsey rule is (appeal to the Slutsky decomposition):
!
n
n
X
α X ∂Xi
1 ∂Hk
=− 1 − −
ti
≡ −θ.
ti
Xk ∂qi
λ
∂M
i=1
i=1
The LHS is the discouragement index, i.e., the rate of change in the
(compensated) demand for good k when n small rates dti = ti − 0 = ti
are introduced from the laissez-faire.
To interpret θ, assume again that the authority can substitute a small
lump-sum tax dT > 0 to indirect taxes, maintaining the whole amount of
collected tax constant. The change in the authority’s objective is
"
!#
n
X
dL = d V (q, M − T ) + λ
ti Xi (q, M − T ) + T − R
i=1
= dV
n
n
X
X
∂L
∂L
dti +
dT
=
∂ti
∂T
i=1
i=1
n
X
∂L
=
dT
∂T
i=1
n
=
X ∂Xi
∂V
+λ −
ti
+1
−
∂T
∂M
i=1
!!
dT .
Hence:
dV
=
λ
n
α X ∂Xi
ti
1− −
λ
∂M
!
dT = θdT .
i=1
→ θ represents the value (expressed in units of tax) for the tax authority
of one marginal unit of tax collected in a lump-sum way.
From the authority viewpoint the cost of this ‘reform’ comprises:
1. α/λ, the welfare loss of the household, evaluated by the authority;
2. the reform implies a lower household’s income: under normality,
demand is lower, and thus there is a fall in indirect taxes collected,
equal to
n
X
∂Xi
ti
dT .
∂M
i=1
Two parts in this ‘cost’: how the household suffers and whether this
household can be a suitable ‘vache à lait’.
Should demand be still discouraged at all, i.e.,
n
θ ≡1−
α X ∂Xi
−
ti
λ
∂M
≶ 0?
i=1
At Ramsey optimum,
n
X
∂Hk
ti
= −θXk
∂qi
i=1
⇒
n X
n
X
k=1 i=1
n
ti
X
∂Hk
tk Xk = −θR < 0,
tk = −θ
∂qi
k=1
since the Slutsky matrix is negative definite.
→ θ > 0: the lower the ‘cost’ of taxation, the higher the demand should
be discouraged.
Equity versus efficiency
Efficiency requires to discourage demand of every good in the same
proportion. With different households, indirect taxes should be designed
so that there is less discouragement of demand for goods preferred by
household whose social weight is higher.
Examples:
1. Household h has labor productivity w h . Her demand for good k is
Xk (q, w h , M) and she gets indirect utility V (q, w h , M).
2. Household h has nonlabor income M h . Her demand for good k is
Xk (q, w , M h ) and she gets V (q, w , M h ).
3. Household h has tastes U h (X, L). Her demand for good k is
Xkh (q, w , M) and she gets V h (q, w , M).
→ V h (q).
The authority sets q which maximizes W (V 1 (q), . . . , V H (q)) s.t.
n
H
X
X
ti
Xih (q) ≥ R.
i=1
h=1
The Ramsey rule is now:
n
H
H
n
X
X
X
1 ∂Hkh
β h X ∂Xih
ti
=−
1−
−
ti
Xk ∂qi
λ
∂M h
i=1
h=1
i=1
h=1
where
Xk ≡
H
X
h=1
Xkh
and β h =
∂W h
α
∂V h
is ‘household h marginal social utility of income’.
!
Xkh
,
Xk
When the authority taxes h in a lump-sum fashion,
1. h looses αh utils, the authority evaluates this loss at β h utils, and
thus β h /λ units of taxes.
2. h is poorer, reduces her consumption, pays less indirect taxes. Here
you recognize the ‘vache à lait’ characteristics.
The whole loss for the tax authority is therefore
n
bh =
β h X ∂Xih
+
ti
.
λ
∂M h
i=1
This represents the ’social value of a lump-sum income transfer toward
household h’ (Diamond, 1975).
When b h > 1, the tax authority would like to transfer one unit of income
to household h; otherwise, when b h < 1, it would like to confiscate this
unit. Both operations are forbidden by assumption.
If all households were identical, Xkh = Xk /H, the Ramsey rule would
rewrite
H
n
H
X
X
1 X h
1 ∂Hkh
= −(1 − b̄), b̄ ≡
b .
ti
Xk ∂qi
H
i=1
h=1
h=1
h
(In fact, b̄ = b = b for all h.) Note that b̄ < 1 by the law of demand.
The same result would apply if all households were perceived as identical
by the authority, i.e., b h = b̄.
The term b̄ reflects some efficiency purpose. One goes back to the
previous case: Demand for each good should be discouraged in the same
proportion θ = 1 − b̄ > 0. The lower b̄, the lower the cost of taxation
(from the government viewpoint, i.e., agents neither suffer nor react to
taxes), and the higher the discouragement is.
In the general case, let
φk ≡ cov
b h Xkh
,
b̄ X̄k
=
H
1 X h Xkh
b
− 1.
b̄ h=1 Xk
Therefore φk > 0 when good k is consumed by high social value
households (typically, the less-off), and φk < 0 when it is consumed by
low social value households (typically, the better-off).
The Ramsey rule rewrites:
n
H
H
X
X
X
1 ∂Hkh
Xh
ti
=− 1−
bh k
Xk ∂qi
Xk
i=1
h=1
!
= −(1 − b̄ − b̄φk ).
h=1
The equity consideration comes through b̄φk . When φk > 0, equity can
be viewed as reducing the ‘efficient discouragement’: The authority does
not value the overall income loss due to the taxation of good k, and the
demand for this good does not react to this lower income (b̄ is low), but
good k is consumed by household the authority likes.
Externality: Pigovian taxes
Household h has utility U h (X, L, ξ), where ξ is ‘pollution’.
Good 1 is a ‘dirty’ good:
X
ξ=
X1h .
h
Fiscal tools are q and personalized taxes/transfers (T h ).
Agent h chooses (X, L) which maximizes her utility subject to
X
qi Xi ≤ L − T h .
i
At the optimum,
−
∂U h /∂X1h
= qi ≡ 1 + ti for all i,
∂U h /∂Lh
and
Th = L −
X
i
qi Xi .
In a first-best optimum, (q, T h ) maximizes
W V 1 (q, T 1 , ξ), . . . , V H (q, T H , ξ)
subject to
XX
h
ti Xih (q, T h , ξ) +
i
X
T h ≥ R,
h
ξ=
X
X1h (q, T h , ξ).
h
Note that
XX
h
i
ti Xih (q, T h , ξ) +
X
h
Th ≥ R ⇔
X
Xih (q, T h , ξ) + R ≤ Lh .
i
since the budget constraint is binding for all household.
Consider a profile (Xh , Lh ) which maximizes
W U 1 ((X1 , L1 , ξ)), . . . , U H ((XH , LH , ξ))
subject to
XX
h
Xih + R ≤
i
X
Lh .
h
ξ=
X
X1h
h
From the FOC, for all h,
0
−
X ∂U h /∂ξ
∂U h /∂X1h
=
1
+
,
∂U h /∂Lh
∂U h0 /∂Lh0
0
h
∂U h /∂Xih
=1
∂U h /∂Lh
for all i > 1.
Decentralization through a suitable (T h ) and q such that, for all h,
X ∂U h0 /∂ξ
t1 =
> 0, ti = 0 for all i > 1.
∂U h0 /∂Lh0
0
h
The ‘Pigovian’ tax t1 represents the social harm from the ‘dirty’ good.
Second-best optimum
Fiscal tools: q (and possibly a uniform income tax/transfer T , not taken
into account here).
The government chooses q which maximize
W V 1 (q, ξ), . . . , V H (q, ξ)
subject to
XX
h
ti Xih (q, ξ) ≥ R,
i
and, for all ξ,
ξ=
X
X1h (q, ξ).
h
In general, the ‘Pigovian’ result does not hold any longer: q discourages
(resp. encourages) the consumption of goods complementary (resp.
substitute) to the dirty good.
→ Carbon tax ?
First-best rule in a second-best optimum
With utility U h (u h (X, L), ξ), the optimal behavior is Xih (q) and Lh (q).
The FOC in qi of government’s problem is


X
X X ∂Xjh
X ∂W ∂V h ∂ξ
X ∂W ∂V h
+
+ λ
Xih +
tj
=0
h
∂V ∂qi
∂qi
∂V h ∂ξ ∂qi
j
h
h
h
h
Let (tj∗ ) the optimal tax rates ‘in the absence of externality’ (but
evaluated at the optimum where externality matter):


X ∂W ∂V h
X X ∂Xjh
X
 = 0.
Xih +
tj∗
+ λ
∂V h ∂qi
∂qi
h
j
h
Let
t1P =
1 X ∂W ∂V h
.
λ
∂V h ∂ξ
h
Then t1 =
t1∗
+
t1P
and ti =
ti∗
for all i > 1.
h
Redistribution through indirect taxes
Recall that EV ≡ M h − e(p, V h ) ≥ T h ≡ t0 Xh .
→ An upper bound for h’ welfare gain is −T h /M h .
Assume that there are
1. two classes, poor (l) and rich (r ) agents,
2. two types of goods, necessities (1) and luxuries (2),
3. a purely ’redistributive’ indirect tax system, i.e., t1 X1 − t2 X2 = 0
with Xi = nl Xil + nr Xir .
Simple calculations yield (cf. Sah, JPubE 1983)
Tl
q1 X1l /M l
− l <
−1 .
M
q1 X1 /M
Let q1 X1l /M l = 0.8, q1 X1 /M = 0.4. The RHS is 0.5: With indirect taxes
one can transfer to the poorest at most 50% of their income.