Download The optimal grouping of commodities for indirect taxation

The optimal grouping of commodities for indirect
taxation
Pascal Belan, Stéphane Gauthier and Guy Laroque
http://www.crest.fr/pageperso/gauthier/vat.pdf
Introduction
A hot public debate about taxation of restaurants in France!
Insights from the theory of indirect taxation: The Ramsey rule.
But this rule does not take into account the (institutional)
constraint on the number of available rates in the UE.
The paper discusses the structure of indirect taxation under such
constraint, both the levels of tax rates and the groups of goods
taxed at each rate.
What the paper highlights
The role of the purported rate, which corresponds to the rate that
would be applied to an individual commodity if it could be taxed
freely, all the other rates being held fixed.
Under a single-peakedness assumption, a good should be taxed to
one of the (typically two) actual rates that are closest to its
purported rate.
An application to the UK: The social welfare function that
rationalizes the current VAT puts most of the weight of the fourth
and the fifth deciles of consumption, about 2/3 of (relevant)
aggregate consumption is correctly taxed, but ’Food out’, which
comprises restaurants and fast foods, is too heavily taxed (some
items in this group should be exempted) ...
Consumer problem
Consumer c maximizes
Z
ug (xg , c)µg dg + m
G
subject to
Z
(1 + tg )xg µg dg + m ≤ wc ,
G
or equivalently
Z
(ug (xg , c) − (1 + tg )xg ) µg dg .
G
The solution is xg = ξg (tg , c) for each g ∈ G and the
corresponding indirect utility (up to the constant wc ) writes
Z
vg (tg , c)µg dg .
G
Unconstrained Ramsey Problem
The social planner chooses the tax rates (tg )g ∈G maximizing
Z Z
Z
Z
α(c)
vg (tg , c)µg dgdν(c)−λ
tg ξg (tg , c)µg dgdν(c) − R
C
G
C
G
or equivalently
Z
Lg (tg ) µg dg ,
G
with
Z
Lg (tg ) =
(α(c)vg (tg , c) dν(c) − λtg ξg (tg , c)) dν(c)
C
being the ’social contribution of good g ’.
A single peaked assumption
We impose that Lg (tg ) is single peaked:
1. if it is increasing with tg , then good g should be made
infinitely expensive ; if it is decreasing with tg , it should be
made free ;
2. if it is first increasing and then decreasing, the optimal taxe
rate satisfies the Ramsey rule:
ag (tgR )
tgR
∂Lg (tg )
=0⇔
=1−
εg (tgR )
∂tg
λ
1 + tgR
where
ag (tgR )
Z
=
α(c)
C
ξg (tg , c)
dν(c)
Xg (tg )
is the social weight (Feldstein characteristic) of good g .
A graphical representation
Lg (t g )
0
t gR
tg
A more useful one
a/λ
a
λ
= 1−
t gR'
1 + t gR'
ε
ag ' / λ
g'
1
ag / λ
g
a
λ
0
εg
ε g'
= 1−
t gR
1 + t gR
ε
ε
Tax rule with a finite number of rates
If there is a finite number of available tax rates tk (k = 1, . . . , K ).
A tax rule is defined by:
1. a collection of groups (Gk ) of goods taxed at a common rate,
with ∪k Gk = G
2. the level of each rate (tk )
An optimal tax rule maximizes
XZ
L(t, (Gk )) =
(Vg (tk ) − λtk Xg (tk )) µg dg .
k
Gk
A strategy to characterize an optimal tax rule:
→ Given the groups, the levels of tax rates maximize L(t, ·)
→ Given the tax rates, the groups maximize L(·, (Gk )).
Diamond tax rates
The groups are treated as aggregate commodities to which the
Ramsey rule is applied:
aGk (tk )
tk
=1−
εG (tk )
λ
1 + tk k
where
Z
aGk (tk ) =
ag (tk )
Gk
and
Z
εGk (tk ) =
εg (tk )
C
Xg (tk )
µg dg ,
XGk (tk )
Xg (tk )
µg dg .
XGk (tk )
Optimal groups
Theorem 1. A necessary condition for optimality is that any good
g be attached to a group k such that Lg (tk ) ≥ Lg (th ) for every
h = 1, . . . , K .
Thus, under the single peakedness assumption,
1. if Lg is increasing, g should be taxed at tK ; if it is
decreasing, it should be taxed at t1 .
2. Otherwise,
I
I
I
if tgR ≥ tK , it should be taxed at tK ,
if tgR ≤ t1 , it should be taxed at t1 ,
if tk ≤ tgR ≤ tk+1 , it should be taxed at either tk or tk+1 .
An illustration
Lg (t g )
0
t1
t2
t gR
t3
t4
tg
One more
a/λ
1
a
λ
= 1−
t2
ε
1 + t2
a
λ
= 1−
t gR
1 + t gR
ε
ε
0
a
λ
= 1−
t3
ε
1 + t3
The case of constant elasticities
Let


x 1/εg (c)

 Ag (c)1/εg (c)
1 − 1/εg (c)
ug (x, c) =
x


 Ag (c) ln
Ag (c)
for εg (c) > 0, and εg (c) 6= 1
for εg (c) = 1.
We impose that εg (c) = εg for single-peakedness to be preserved
under aggregation. Then, for t < t 0 ,
Lg (t) < Lg (t 0 ) ⇔
ag
< φ εg , t, t 0
λ
where φ is convex in εg (but not necessarily monotonic) and
asymptote to (1 − εg )t/(1 + t) when εg goes to infinity.
A complete characterization
a/λ
Taxed at rate t 2
Taxed at rate t3
1
ε
0
Ramsey’s insights: with a zero tax rate
a/λ
t1 = 0
1
t 2 < t3
0
t2 > 0
ε
Ramsey’s insights: without a zero tax rate
a/λ
t1 < 0
1
t 2 < t3
0
t2 > 0
ε
An application
Is the actual tax structure far from the optimal one ?
What kind of reforms, if any, should be implemented ?
→ How restaurants and fast food should be taxed ?
Separability assumptions made so far are very strong in practice:
→ Cross price effects
→ Income effects
A more general framework
Preferences of consumer c are represented by
uc (x, l)
and her budget constraint writes
Z
(1 + tg )xg µg dg ≤ w (c)l − T (w (c)l).
G
Diamond first-order condition
Z
(−ag /λ + Xg )µg dg
g ∈Gk
Z
+
∂Xg X ∂XGh \{g }
+
th
∂tg
∂tg
h
Z
∂Yc
+
T 0 (Y (c))
dν(c)µg dg = 0,
∂tg
C
tk
g ∈Gk
R
where ag = C α(c)ρ(c)(ξg /Xg )dν(c) and ρ(c) is the marginal
utility of income of consumer c. This can be rewritten as
aGk
tk
−bGk = 1 −
εG
λ
1 + tk k
and, for an individual good
ag
tk
−bg = 1 −
εg .
λ
1 + tk
Illustration with data from the U.K.
49% of consumption is taxed at the standard rate (17.5%), 10% at
the reduced rate (5%) and 27% is exempted; the remaining
(tobacco, alcohol, petrol and diesel) is subject to large excise taxes.
We assume that the tax authority takes as given after-tax income
and the current grouping of commodities. If tax rates are optimally
chosen, the Diamond first-order must be satisfied. This gives some
information about social redistributive aims and allows us to draw
a fan corresponding to these weights.
The issue is whether the actual composition of the commodity
groups is optimal.
A digression on uniform commodity taxation
If the Atkinson Stiglitz conditions hold (preferences are separable
between commodities and labor, and the preferences for
commodities are identical across individuals at the microeconomic
level), and if the government can freely tax incomes in a non linear
way, all the goods should be taxed at the same rate.
But:
1. There is no general agreement on the empirical relevance of
the Atkinson Stiglitz conditions. Browning and Meghir (1991)
find some evidence of non separability.
2. We work with fixed after tax incomes, so that our exercise
provides information on the optimal indirect tax rates, given
the current income tax schedule.
To answer this question
We must first recover the implicit social weights consistent with
the current VAT.
We have three Diamond first-order conditions (standard, reduced,
exempted) and a normalization condition on social weights.
We observe (1) the tax rates, (2) own and cross price elasticities
for twenty categories of goods, and (2) the shares of consumption
by (consumption) decile for each category.
The unknown are the ten weights αc ρc and the marginal cost of
public funds λ.
→ The problem has many solutions!
Moreover, since Diamond first-order conditions are homogenous of
degree 1 in (αc ρc , λ): we can only expect to recover the ten ratios
αc ρc /λ. Let choose the normalization
Z
αc ρc dν(c) = 1.
C
Then, an equal lump-sum transfer of 1 unit (of aggregate
consumption) leads to an increase of 1 unit of social welfare: social
welfare is measured in tenths of aggregate consumption.
Minimizing the square of the LHS of Diamond FOC with respect
to (αc ρc , λ) and using the normalization condition leads to:
λ̂ = 1.11,
α1 ρ1 = 0.03, α4 ρ4 = 0.54 and α5 ρ5 = 0.43.
Is the grouping optimal?
Results
If one excludes goods subject to excise taxes (other considerations
than mere redistribution should matter in such cases), it remains
87% of aggregate consumption, and about 2/3 of these 87%
appear to be correctly taxed.
Main exceptions:
Exempted goods that should be taxed:
→ dairy products, fruits and vegetables, other non-VAT foods
Goods taxed at the standard rate that should be exempted:
→ food out and Public transport.
Impact of the redistributive stance of the government:
→ Utilitarist vs. Ralwsian ?