The optimal grouping of commodities for indirect taxation

Transcription

1 The optimal grouping of commodities for indirect taxation Pascal Belan, Stéphane Gauthier and Guy Laroque

2 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.

3 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)...

4 Consumer problem Consumer c maximizes subject to or equivalently G G G u g (x g, c)µ g dg + m (1 + t g )x g µ g dg + m w c, (u g (x g, c) (1 + t g )x g ) µ g dg. The solution is x g = ξ g (t g, c) for each g G and the corresponding indirect utility (up to the constant w c ) writes v g (t g, c)µ g dg. G

5 Unconstrained Ramsey Problem The social planner chooses the tax rates (t g ) g G maximizing ( ) α(c) v g (t g, c)µ g dgdν(c) λ t g ξ g (t g, c)µ g dgdν(c) R C G C G or equivalently with L g (t g ) = C G L g (t g ) µ g dg, (α(c)v g (t g, c) dν(c) λt g ξ g (t g, c)) dν(c) being the social contribution of good g.

6 A single peaked assumption We impose that L g (t g ) is single peaked: 1. if it is increasing with t g, then good g should be made infinitely expensive ; if it is decreasing with t g, it should be made free ; 2. if it is first increasing and then decreasing, the optimal taxe rate satisfies the Ramsey rule: where L g (t g ) = 0 a g (tg R ) = 1 tr g t g λ 1 + tg R ε g (tg R ) a g (tg R ) = C α(c) ξ g (t g, c) X g (t g ) dν(c) is the social weight (Feldstein characteristic) of good g.

7 A graphical representation L g ( t g ) 0 R t g t g

8 A more useful one ε ε λ R g R g t t a + = λ a / 1 ε λ R g R g t t a ' ' = λ / a g ' λ / a g g ε ' ε g g g'

9 Tax rule with a finite number of rates If there is a finite number of available tax rates t k (k = 1,..., K). A tax rule is defined by: 1. a collection of groups (G k ) of goods taxed at a common rate, with k G k = G 2. the level of each rate (t k ) An optimal tax rule maximizes L(t, (G k )) = (V g (t k ) λt k X g (t k )) µ g dg. k G k 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(, (G k )).

10 Diamond tax rates The groups are treated as aggregate commodities to which the Ramsey rule is applied: a Gk (t k ) λ = 1 t k 1 + t k ε Gk (t k ) where and a Gk (t k ) = a g (t k ) X g (t k ) G k X Gk (t k ) µ g dg, ε Gk (t k ) = C ε g (t k ) X g (t k ) X Gk (t k ) µ g dg.

11 Optimal groups Theorem 1. A necessary condition for optimality is that any good g be attached to a group k such that L g (t k ) L g (t h ) for every h = 1,..., K. Thus, under the single peakedness assumption, 1. if L g is increasing, g should be taxed at t K ; if it is decreasing, it should be taxed at t Otherwise, if t R g t K, it should be taxed at t K, if t R g t 1, it should be taxed at t 1, if tk t R g t k+1, it should be taxed at either t k or t k+1.

12 An illustration L g ( t g ) 0 t1 2 R t t g t3 t4 t g

13 One more a / λ 1 a t2 = 1 1+ t λ 2 ε R a tg = 1 λ 1+ t R g ε 0 a = 1 t 3 λ 1+ t3 ε ε

14 The case of constant elasticities Let 1/εg 1/εg A g (c) (c) x (c) u g (x, c) = 1 1/ε g (c) x A g (c) ln A g (c) for ε g (c) > 0, and ε g (c) 1 for ε g (c) = 1. We impose that ε g (c) = ε g for single-peakedness to be preserved under aggregation. Then, for t < t, L g (t) < L g (t ) a g λ < φ ( ε g, t, t ) where φ is convex in ε g (but not necessarily monotonic) and asymptote to (1 ε g )t/(1 + t) when ε g goes to infinity.

15 A complete characterization a / λ Taxed at rate t 2 Taxed at rate t ε

16 Ramsey s insights: with a zero tax rate a / λ 1 t 1 = 0 0 t 2 < t 3 t 2 > 0 ε

17 Ramsey s insights: without a zero tax rate a / λ t 1 < t 2 < t 3 t 2 > 0 ε

18 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

19 A more general framework Preferences of consumer c are represented by u c (x, l) and her budget constraint writes (1 + t g )x g µ g dg w(c)l T (w(c)l). G

20 Diamond first-order condition ( a g /λ + X g )µ g dg g G k X g + t k + g G k t g h + C t h X Gh \{g} t g T (Y (c)) Y c t g dν(c)µ g dg = 0, where a g = C α(c)ρ(c)(ξ g /X g )dν(c) and ρ(c) is the marginal utility of income of consumer c. This can be rewritten as and, for an individual good a Gk λ b G k = 1 t k 1 + t k ε Gk a g λ b g = 1 t k 1 + t k ε g.

21 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.

22 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.

23 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 α c ρ c and the marginal cost of public funds λ. The problem has many solutions!

24 Moreover, since Diamond first-order conditions are homogenous of degree 1 in (α c ρ c, λ): we can only expect to recover the ten ratios α c ρ c /λ. Let choose the normalization α c ρ c 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 (α c ρ c, λ) and using the normalization condition leads to: ˆλ = 1.11, α 1 ρ 1 = 0.03, α 4 ρ 4 = 0.54 and α 5 ρ 5 = 0.43.

25 Is the grouping optimal?

26 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?