Lecture 2 Optimal Indirect Taxation. March 2014

Transcription

1 Lecture 2 Optimal Indirect Taxation March 2014

2 Optimal taxation: a general setup Individual choice criterion, for i = 1,..., I : U(c i, l i, θ i ) Individual anonymous budget constraint Social objective Feasibility, for instance B(c i, l i, θ i, prices) 0. W (U(c 1, l 1, θ 1 ),..., U(c I, l I, θ I )) F ( I c i + G, I l i θ i ) 0. Problem: choose the B that maximizes the social objective (aiming to pay for the war and/or to redistribute), given the agents behaviors with market clearing prices.

3 Ramsey vs. Mirrlees Ramsey: constraints are put directly on the B function, i.e. taxes on traded goods are linear in the amounts traded. The constraints are supposed to stem from institutional features of the economy which are not spelled out. Mirrlees: all constraints come from informational asymmetries. The structure of the economy is common knowledge, but the government does not know who is who. In the most studied case, it can only make taxes conditional on income (or efficient labor) lθ, and not on l and θ separately. This leads to a formulation with incentive constraints, akin to a mechanism design problem.

4 An attempt to define direct/indirect taxes Atkinson, Canadian J. of Economics, 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 be 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.

5 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 60,5 Impôt sur les sociétés 51,5 53,9 Taxe intérieure sur les produits pétroliers 17,6 16,9 Taxe sur la valeur ajoutée 131,1 135,0 Autres recettes fiscales 10,9 5,8 Recettes fiscales nettes 267,9 272,1 Source : ministère du Budget, des Comptes publics et de la Fonction publique.

6 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 and indirect taxes 5. Capital taxation 6. Application: labor supply, Prescott

7 Plan of the talk 1. The main ideas in a single consumer, no substitution and no income effects setup. 2. The general formula 3. Back to a representative agent, three goods case: Corlett-Hague.

8 I. A simplified framework There is only one household (efficiency) who maximizes U(X, L) = U i (X i ) L subject to her budget constraint q i X i L, with respect to the variables (X = (X 1,..., X n ), L). The q i = p i + t i are consumer prices which differ from the production prices p i by the excise tax t i. Labor is the numeraire and is not taxed. Preferences are additive, increasing concave in the consumption goods, quasilinear in labor: the demand for good ξ i (q i ) only depends on its own price (neither cross price nor income effects) and decreases with respect to q i. The indirect utility of the consumer is V (q) = [U i (ξ i (q i )) q i ξ i (q i )].

9 Production and second best program Production takes place at constant returns to scale, with exogenously fixed production prices p. The authority chooses the tax vector t = q p which maximizes V (q) subject to a revenue requirement R t i ξ i (q i ) R. The Lagrangian is [ ] L(q,λ) = V (q) + λ t i ξ i (q i ) R.

10 First order conditions The n FOCs with respect to t i are [ ] V ξ i (q) + λ ξ i (q i ) + t i (q i ) q i q i to which one must add the budget constraint t i ξ i (q i ) R = 0. = 0, i = 1,..., n Using the expression of the indirect utility or appealing to Roy s identity, the FOCs can be rewritten as [ ] ξ i ξ i (q i ) + λ ξ i (q i ) + t i (q i ) = 0. q i

11 The marginal cost of public funds The FOCs imply that λ is larger than 1: one unit of public expenditure costs more than one unit of private good (marginal cost of public funds). One notes θ = λ 1 > 0. λ

12 Second order conditions Important remark: The Lagrangian is not necessarily concave. L q i = (2λ 1)ξ i + λt i ξ i.

13 Introducing elasticities The absolute value of the price elasticity of demand is The FOC is equivalent to (after division by ξ i ) ε i (q i ) = q i ξ i (q i ) > 0. ξ i (q i ) q i [ ] ξ i ξ i (q i ) + λ ξ i (q i ) + t i (q i ) = 0 q i λ 1 + λ t i q i q i ξ i (q i ) = 0. ξ i (q i ) q i

14 The inverse elasticity rule All the goods should be taxed. The optimal tax rate is inversely proportional to the price elasticity of (compensated) demand. ( t i = 1 1 ) 1 p i + t i λ ε i (q i ) = θ ε i (q i ).

15 The discouragement index Another reading of the inverse elasticity rule: t i ξ i (q i ) = θ. ξ i (q i ) q i The LHS gives the rate of change in the demand of good i which follows the introduction of a small tax on this good: dx i = ξ i dt i ξ i t i dx i t i t i t i X i ξ i (q i ) ξ i. q i Taxation should discourage the demand for every good in the same proportion θ (θ > 0): The LHS is the discouragement index (Mirrlees, 1976).

16 Other instruments: lump sum taxes Assume that the authority can collect T in a lump-sum fashion. The Lagrangian rewrites: [ ] L(t, T, λ) = V (t) T + λ t i ξ i (t i ) + T R. Substitute a lump-sum tax dt to indirect taxes, maintaining the aggregate collected tax constant. The resulting change in the social objective is dl = L dt i + L dt = dv. t i T i dv = L dt = (λ 1)dT = λθdt. T i

17 Other instruments: wage tax A normalization issue: Should one tax good k when t k > 0 at the optimum? 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.

18 With the second tax structure the household budget constraint is (p i + t i )X i = (1 τ )L. Let t i be such that p i + t i = (p i + t i )/(1 τ ), i.e., t i = t i + τ 1 τ (p i + t i ). Then both tax structures are equivalent for the household. The aggregate tax revenues coincide in both cases: t i X i = t i X i + τ 1 τ (p i + t i )X i = t i X i + τ L. Therefore t i = 0 does not mean that good i should be tax-free, but that it should be taxed (resp. subsidized) as much as labor is subsidized (resp. taxed).

19 II. The general Ramsey rule A more general setup, with cross price and income effects and a number of consumers, to assess efficient indirect tax structures. We keep labour as the untaxed numeraire. The typical h household chooses (X, L) which maximizes U(X, L) s.t. q i X i L + M, where M represents nonlabor income. Consumer demand is ξi h(q, M) for all i, while labor supply is ξh L (q, M). Indirect utility is V h (q, M) = U ( ξ h 1(q, M),..., ξ h n(q, M), ξ h L(q, M) ). The marginal utility of income is V h M αh.

20 The government program The authority chooses q which maximize W [V 1 (q, M 1 ),..., V H (q, M H )] subject to (multiplier λ) H (q j p j )ξj h (q, M h ) R. h=1 j=1

21 The derivation of the Ramsey formula The first order condition in q i gives H W V h H V h = λ ξ i h + q i h=1 h=1 j=1 t j ξ h j q i By Roy s identity, α h denoting the marginal utility of income of consumer h: V h q i = α h ξ h i. Let β h = ( W / V h )α h be the social marginal utility of income of consumer h. Then H H β h ξi h = λ ξ i h ξj h + t j. q i h=1 h=1 j=1.

22 Ramsey formula: the right hand side The Slutsky equation gives ξ h j q i = S h ji ξ h i ξ h j M h. Substituting H h=1 β h λ ξh i = H ξi h + h=1 H h=1 j=1 t j (S ji h ξi h ξ h j M h Let b h = β h /λ + n j=1 t j ξj h/ Mh be the net social marginal utility of income of consumer h. Then ) j=1 t j H h=1 S h ij = ξ i (1 H h=1 b h ξh i ξ i. ).

23 The final result The empirical covariance between the net social marginal valuation of income and consumption of good i is r i = 1 H H ( ) ( ) b h ξ h b 1 i 1 = 1 ξ i H h=1 n j=1 t j H h=1 S h ij /H ξ i = ( 1 H h=1 H h=1 b h b ξ h i ξ i 1 ) b h ξi h = 1 + b + r i b. H ξ i The (absolute value of the) left hand side is the discouragement index, a measure of the change in compensated demand. It varies with the good i contrary to the initial example, depending on the correlation between the social weight and the share in the use of the good. The government should discourage less the consumption of goods bought by agents with a high net social marginal utility of income.

24 III The representative consumer All the income effects are the same, so that b h is independent of h. n j=1 t js ij = (1 b). ξ i Summing up the equalities over the goods i after multiplication with the tax rate t i gives I t j S ij t i = (1 b) t i ξ i. j, The left hand side is negative by the negative definiteness of the Slutzky terms: b must be smaller than one is some tax revenue is collected. Therefore the discouragement indexes are all nonnegative: the optimal tax system does not encourage the consumption of any good.

25 Corlett-Hague (1953): the two-good case The FOC are which give t 1 S 11 + t 2 S 12 = (1 b)ξ 1 t 1 S 21 + t 2 S 22 = (1 b)ξ 2 t 1 = 1 b D (S 12ξ 2 S 22 ξ 1 ) t 2 = 1 b D (S 21ξ 1 S 11 ξ 2 )

26 Playing with Slutzky and elasticities The compensated demand for good i is homogeneous of degree zero in prices, so that noting 0 the leisure good which is also the numeraire: Define the compensated elasticities Then and similarly for good 2. S 10 + q 1 S 11 + q 2 S 12 = 0 S 20 + q 1 S 21 + q 2 S 22 = 0. ε ij = q js ij ξ i. ε 10 + ε 11 + ε 12 = 0, Factorizing ξ 1 ξ 2 /q 2 in the t 1 formula yields or t 1 = 1 b D t 1 = 1 b D ξ 1 ξ 2 q 2 (ε 12 ε 22 ), ξ 1 ξ 2 q 2 (ε 10 + ε 11 + ε 22 ).

27 Corlett-Hague concluded t 1 q 2 t 2 q 1 = 1 b D ξ 1ξ 2 (ε 10 ε 20 ) Near zero tax rates, this says that the tax rate on good 1 should be larger than that on good 2, t 1 /q 1 > t 2 /q 2, if and only if ε 10 < ε 20, i.e. good 1 is more complementary to leisure than good 2.