OPTIMAL TAXATION: LESSONS FOR TAX POLICY

OPTIMAL TAXATION: LESSONS FOR TAX POLICY Special Lectures at the University of Tokyo International Program in Economics and Center for International Research on the Japanese Economy by Robin Boadway, Queen s University, Canada December 5 6, 2011

Purpose To study the main results from the optimal tax literature that have been relevant for tax policy Topic Outline 1. Optimal Linear Tax Analysis 2. Optimal Nonlinear Tax Analysis 3. Two-Period Analysis and Capital Income Taxation 4. Further Issues

1 Optimal Linear Tax Analysis Representative household setting Efficient taxation: Ramsey Rule Corlett-Hague Theorem Conditions for uniform taxation Externalities Heterogeneous households Social welfare functions First-best outcomes Optimal linear progressive taxation Commodity taxation: Deaton conditions Production Efficiency Theorem

2 Optimal Nonlinear Tax Analysis Optimal nonlinear income taxation Two wage-type case (Stern-Stiglitz) Multiple discrete wage-types Continuous wage types (Mirrlees) Maximin case Extensive-margin labor supply Income vs commodity taxation Atkinson-Stiglitz Theorem Generalization of Atkinson-Stiglitz Theorem Differential taxation of leisure complements

3 Two-Period Analysis and Capital Income Taxation Representative household models The Corlett-Hague analogue Overlapping-generations case Time-consistent taxation Time-inconsistent preferences Bequests Two wage-type nonlinear taxation Identical preferences Different discount factors Age-dependent taxation Uncertain future wage rates Liquidity constraints

4 Further Issues Human capital investment Uncertain earnings Marginal cost of public funds Time-using consumption Non-tax instruments: minimum wage, in-kind transfers, workfare Involuntary unemployment

Methodological Note: The Envelope Theorem Consider the constrained maximization problem: Max x f (x; y) s.t. g(x; y) = 0, where x = vector of choice variables and y = vector of exogenous variables The Lagrange expression is L = f (x; y) + λg(x; y) The first-order conditions are f (x; y) g(x; y) + λ = 0, i x i x i The solution gives x(y), and a value function F (y) f ( x(y); y ) By the envelope theorem: F (y) y j = L f (y) = + λ g(y) y j y j y j We use this frequently for both consumer maximization problems and social welfare maximization problems

Optimal Commodity Taxation: Identical Households The Ramsey Problem n + 1 commodities: x 1,, x n goods, x 0 = h l leisure Representative household utility: u(x 1,, x n, h l) Producer prices: p i, i = 1,, n and p 0 = w = 1 Taxes: t i, i = 1,, n, t 0 = 0 Consumer prices: q i = p i + t i, i = 1,, n and w = 1 Ad valorem taxes: τ i = t i q i, θ i = t i p i = τ i = t i p i + t i = θ i 1 + θ i, θ i = τ i 1 τ i Government is principal, households are agents

The Household Problem max u(x 1,, x n, h l) s.t. {x i,l} n q i x i l = 0 i=1 Lagrangian expression: [ n ] L(x i, l, α) = u(x 1,, x n, h l) α q i x i l i=1 = uncompensated demands x(q, w), l(q, w) = indirect utility: v(q, w) u ( x(q, w), l(q, w) ) Envelope theorem: v q i = αx i (q, w), i = 1,, n, v w = αl(q, w), v m = α = α is marginal utility of household income

Production and Tax Normalizations Production possibilities are linear: n i=1 p ix i = wl R where R = resources used by government Production constraint is homogeneous of degree zero in p and w: normalize producer prices by w = 1. Consumer demands are homogeneous of degree zero in q and w: normalize consumer prices by w = 1 = t 0 = 0 Subtracting the production constraint from the consumer s budget constraint yields government budget constraint: n t i x i = R i=1 = One of household budget constraint, production constraint & government budget constraint are redundant

The Government s Problem Lagrangian: [ n ] L(t, λ) = v(q) + λ t i x i (q) R i=1 = λ is marginal utility of government revenue = Production constraint redundant First-order conditions: [ ] v n x i t k : + λ x k + t i = 0 k = 1,, n q k q k i=1 λ : n t i x i = R i=1 = t k (R) and λ(r) (No guarantee that second-order conditions satisfied)

Interpretation of Optimal Tax Rules Rewrite FOC using envelope theorem, v/ q i = αx i (q, w): n i=1 t i x i q k = ( λ α λ ) x k k = 1,, n (1) Note that λ > α: Marginal value of a yen to the government exceeds the marginal value of a yen to the household (since it is costly to transfer a yen from households to the government) Using (1), various interpretations can be given to the optimal tax structure =

The Inverse Elasticity Rule (Partial Equilibrium) Assume preferences are quasilinear in leisure and additive: u(x 1,, l) = u 1 (x 1 ) + u 2 (x 2 ) + + u n (x n ) + (h l) = Demand functions for goods: x i (q i ), i = 1,, n Condition (1) then becomes: ( ) x k λ α t k = x k q k λ k = 1,, n Interpretation 1: Inverse elasticity rule: τ k = t ( ) k λ α 1 = k = 1,, n q k λ η kk where η kk = ( x k / q k )(q k /x k ) < 0 (elasticity of demand for x k ) Interpretation 2: Proportional reduction approximation: t k x k = q ( ) k x k x k λ α = = k = 1,, n x k q k x k q k x k λ

The Ramsey Rule The Slutsky equation: x i = x i x i q k q k x k u m = s x i ik x k m i, k = 1,, n Substitute Slutsky equation into (1): i t ( ) is ik λ α = + x i t i = (1 b) x k λ m k i = 1,, n where b = α/λ + t i x i / m, net social marginal utility of income By symmetry of the substitution effect (s ik = s ki ): i t is ki x k = (1 b) k = 1,, n (2) where b < 1 if R > 0: Ramsey proportionate reduction rule

Three-Commodity Case (Harberger) The Ramsey rule (2) can be written k = 1 : t 1 s 11 + t 2 s 12 = (1 b)x 1 k = 2 : t 1 s 21 + t 2 s 22 = (1 b)x 2 Eliminating (1 b), we obtain: Multiplying by q 2 /q 1 : t 1 t 2 = s 22x 1 s 12 x 2 s 11 x 2 s 21 x 1 = s 22/x 2 s 12 /x 1 s 11 /x 1 s 21 /x 2 t 1 /q 1 t 2 /q 2 = τ 1 τ 2 = ε 22 ε 12 ε 11 ε 21 Compensated demands are homogeneous of degree zero, i ε ji = 0, so τ 1 = ε 22 + ε 11 + ε 10 (3) τ 2 ε 11 + ε 22 + ε 20

Interpretation: Corlett-Hague Theorem Since ε 11 + ε 22 < 0, (3) says that τ 1 > τ 2 if ε 10 < ε 20 i.e., x 1 is relatively more complementary with leisure than x 2 Corlett-Hague Theorem: Impose a higher tax rate on the good that is more complementary with leisure Intuition: Leisure is taxed indirectly by taxing its complement Corollary: Taxes are proportional if ε 10 = ε 20 (both goods equally complementary with leisure) Proportional tax on goods equivalent to tax on labor income

Uniform Commodity Taxation If t i /q i = τ for all goods, (2) gives: i t is ki x k i = τ s kiq i x k = τ n ε ki = (1 b) i=1 k = 1,, n By homogeneity of compensated demands, i ε ki = 0 = t i /q i = τ if τε k0 = (1 b), k = 1,, n, or ε k0 = ε j0 j, k = 1,, n All goods must be equally substitutable with leisure

Sufficient Condition for Uniform Taxation Uniform taxation if: Expenditure functions are of form e(f (q, u), w, u) Proof: The compensated demand for good i is given by x i (q, w, u) = e f Substitution effect with respect to w is: f q i s i0 = x i(q, w, u) w = 2 e f w f q i so, ε i0 = s i0w x i = w 2 e f w ( ) e 1 = ε k0 f

Example: u(f (x 1,, x n ), x 0 ), with f ( ) homothetic Intuition: Preferences for goods independent of x 0, income elasticities of demand all unity. Slutsky equation yields: x i q k = x k q i i, k = 1,, n (1) = n i=1 t i x k / q i x k ( ) λ α = λ k = 1,, n or, n i=1 q i x k / q i x k = x k x k ( ) λ α = λ k = 1,, n Proportional tax on l reduces income and thus all goods demands proportionately since relative goods prices q i have not changed

Environmental Taxation Good x 1 affects quality of environment e = f (Hx 1 ), f < 0, f 0 Separable utility: u(x 0,, x n, e) = u(h(x 0,, x n ), e) Household demands x i (q, w) and indirect utility v(q, w, e) Government Lagrangian: [ n ] L(t i, λ) = Hv(q, w, f (Hx 1 (q, w))) + λ Ht i x i (q, w) R FOCs: H [ v q k + H v e f x 1 q k ] [ +λ Hx k + n i=1 i=1 Ht i x i q k Using the envelope theorem and rearranging: [ ] n x i αx k = λ x k + t i + H u e q k λ f x 1 q k i=1 ] = 0 k = 1,, n k = 1,, n

Interpretation Define shadow taxes net of Pigouvian tax: t 1 = t 1 + H u e λ f = t 1 t P, t i = t i, i = 2,, n FOCs: n i=1 t i x i q k = ( λ α λ ) x k k = 1,, n (Analogue of (1) in shadow taxes ti ) i = t i s ki = (1 b) k = 1,, n x k or, i t is ki x k = (1 b) + t Ps k1 x k k = 1,, n Goods more complementary with x 1 (s k1 < 0) discouraged more For x 1, s 11 < 0, so unambiguously discouraged

Heterogeneous Households: The Social Welfare Function Assumed properties of Social Welfare Function (SWF) 1. Welfarism: SWF depends only on utilities (consequentialism): W (n 1, u 1,, n h, u h ) 2. Pareto principle: W ( ) increasing in u i 3. Symmetry, anonymity: u i, u j enter W ( ) in same way 4. Quasi-concavity: convex to the origin social indifference curves Simplest SWF satisfying these properties: W (n 1, u 1,, n h, u h ) = i=1,h n i (u i ) 1 ρ 1 ρ = n i w(u i ), 0 ρ where ρ = w u/w : coefficient of aversion to inequality ρ = 0 : utilitarian; ρ = : maximin Note equivalence of concavity of w( ) and u( )

First-Best Utility Possibilities Frontier (UPF) To characterize optimal redistribution between two household-types Three cases Fixed labor supply, identical utility functions Fixed labor supply, different utility functions Variable labor supply, identical utility functions Fixed labour supply, identical preferences n 1, n 2 type-1 s and -2 s with incomes y 1, y 2 Lump-sum taxes T 1, T 2 Identical utilities u(c) = u(y T ), u > 0 > u Government budget n 1 T 1 + n 2 T 2 = 0 Effect of tax changes: du 1 = u 1 c dt 1, du 2 = u 2 c dt 2 Slope of UPF: du 2 du 1 = u2 c dt 2 uc 1 = n 1 uc 2 dt 1 n 2 uc 1 (symmetric) Social welfare maximized at u 1 = u 2

Fixed Labor Supply, Different Utility Functions Assume type-2 s are better utility generators : u 2 (c) > u 1 (c), uc 2 (c) > uc 1 (c) Slope of UPF: du 2 (c 2 ) du 1 (c 1 ) = n 1 uc 2 (c 2 ) n 2 uc 1 (c 1 ) At c 1 = c 2 = c e : u 2 (c e ) > u 1 (c e ), uc 2 (c e ) > uc 1 (c e ), and du 2 (c e ) du 1 (c e ) = n 1 uc 2 (c e ) n 2 uc 1 (c e ) > n 1 = FIGURE 1 n 2 1. Utilitarian outcome (u): u 1 c (c 1 ) = u 2 c (c 2 ) c 2 > c 1 (give more consumption to the better utility generator) 2. Maximin outcome (m): u 1 (c 1 ) = u 2 (c 2 ) c 1 > c 2 (give more consumption to the less efficient utility generator) 3. As aversion to inequality increases, move from Utilitarian to Maximin outcome

u 2.. n 1 n 2 u. c e m.. 45 o. u 1 Figure 1 UPF with Different Utility Functions

Variable Labor Supply, Same Utility Functions This is classical optimal income tax setting (Mirrlees) Common utility function: u(c, l) (strictly concave) Wage rates differ: w 2 > w 1 Linear production (& no government spending): n 1 w 1 l 1 + n 2 w 2 l 2 = n 1 c 1 + n 2 c 2 Lump-sum taxes T 1, T 2 Representative household behavior Max u(c, l) s.t. c = wl T FOC: u l /u c = w l(w, T ) Indirect utility: v(w, T ) with v w = lu c > 0, v T = u c < 0 Two types: v 1 (w 1, T 1 ), v 2 (w 2, T 2 )

Properties of First-Best UPF Move along First-Best UPF using lump-sum taxes T 1, T 2 : Slope: dv 2 dv 1 = v 2 T dt 2 v 1 T dt 1 = n 1 n 2 v 2 T v 1 T = n 1 n 2 u 2 c u 1 c Concave to the origin since v i TT < 0 Points on UPF: FIGURE 2 Laissez Faire (l): T 1 = T 2 = 0 v 2 > v 1 Maximin (m): v 1 = v 2 On 45 o line Utilitarian (u): T 1, T 2 chosen such that u 1 c = u 2 c and u i l /ui c = w i v 1 > v 2 if c normal (Mirrlees) Intuition for utilitarian case: Efficient for high-wage person to supply more labor, while marginal utility of consumption equalized Note: Tax may not be progressive even under Maximin (Sadka)

v 2. s l s m. u. 45 o N 2.... N 1 v 1 Figure 2 First-Best UPF

An Aside: Different Wages, Different Preferences Suppose utility function is u(c) φ i h(l) with c = w i l T i (where T i is based on wage rate) Household characteristics Abilities: w 2 > w 1 Preference for leisure: φ 2 > φ 1 Normative principles Principle of Compensation: Compensate for differences in ability: Persons with different wage and same preferences should achieve same utility (maximin) Principle of Responsibility: Do not compensate for differences in preference: Persons with same wage and different preferences should pay same tax Result: Impossible to satisfy Principle of Compensation and Principle of Responsibility simultaneously = FIGURE 3

c c = w. 2 l.2l.2h 1h..1l c = w 1 l. l Figure 3 Principles of Compensation and Responsibility

Optimal Linear Taxation: Basic Results Suppose households vary in wage rates w j, j = 1,, h, but have common preferences Government levies commodity taxes and equal per person lump-sum subsidy a Household indirect utilities v j (q, w j, a) Government objective is additive social welfare function W (v 1 ) + W (v 2 ),, +W (v j ),, +W (v h ) Deaton (1979): Uniform commodity taxes optimal if utility weakly separable and goods have linear Engel curves with identical slopes across households Satisfied by Gorman polar form utility function, whose expenditure function for person i takes the form e i (q, u i ) = g i (q) + u i f (q), where f (q) is the same for all = Linear progressive income tax

Optimal Linear Taxation: Special Case The setting: Household with wage rate w Assume two goods, x 1, x 2, and labor, l Utility: u(x 1, x 2, l) Linear income tax t, a and commodity tax θ on x 2 Producer prices for goods = unity Consumer budget x 1 + (1 + θ)x 2 = (1 t)wl + a Household Lagrangian L = u( ) α ( x 1 + (1 + θ)x 2 (1 t)wl a ) = Solution: x 1 (θ, t, a), x 2 (θ, t, a), l(θ, t, a) Indirect utility: v(θ, t, a) where envelope theorem implies v θ = αx 2, v t = αwl, v a = α

Optimal Linear Progressive Income Tax Assume additive social welfare function: w w W ( v(θ, t, a) ) df (w) Government problem: Choose t, a given θ (assuming R = 0) L = w w ( W ( v(θ, t, a) ) + λ ( θx 2 (θ, t, a) + twl(θ, t, a) a )) df (w) First-order conditions evaluated at θ = 0: w ( ( a : W v a + λ tw l )) a 1 df (w) = 0 w t : w w ( ( W v t + λ wl + tw l ) ) df (w) = 0 t

Some Definitions Define: β(w) W v a = W α (marginal social utility of income) b(w) β(w)/λ + tw l/ a (social value of government transfer) Slutsky equation, where w n = (1 t)w and s ll is the own substitution effect: l = s ll + l l w n a = l t = l w n w n t = ws ll wl l a And, define the compensated elasticity of labor supply: ɛ ll s ll (1 t)w l = tw l s ll 1 t t

Interpretation of First-Order Conditions Using these definitions and v t = αwl, the FOCs become: a : (b(w) 1 ) df (w) = 0, = b = 1 t : ( wl b(w) 1 + t ) 1 t ɛ ll df (w) = 0 So, using y = wl and E[b] = 1, ( ) t b(w) 1 wldf (w) 1 t = wlɛll df (w) E[by] E[b]E[y] = wlɛll df (w) Cov[b, y] = yɛll df (w) = Equity ( ) Efficiency (+)

Differential Commodity Taxes Let W(θ) = value function from optimal linear income tax problem. Using the envelope theorem and v θ = αx 2 : dw w ( ( dθ = βx 2 + λ x 2 + tw l ) ) df (w) θ=0 θ w The Slutsky equation corresponding to l/ θ is: l θ = s l l2 x 2 a Substituting this into dw/dθ and rearranging, we obtain: 1 dw w ( λ dθ = x 2 β θ=0 λ + 1 + tw s l2 tw l ) df (w) x 2 a w w The FOC on t above can be written: w ( y β λ + 1 + tw s ) ll l tw df (w) = 0 l a

Differential Commodity Taxes, cont d Deaton shows that optimal θ = 0 if a) preferences are weakly separable in goods, and b) Engel curves are linear. Special case: Quasilinear preferences u(x 1, x 2, l) = x 1 + b(x 2 ) h(l) Then, l = l ( (1 t)w ), x 2 = x 2 (1 + θ) = Deaton conditions satisfied Since l θ = l a = 0 and x 2 independent of w FOC on a becomes : w w ( β λ ) df (w) = 0 and, dw dθ = θ=0 w w ( βx2 + λx 2 ) df (w) = 0 = θ = 0 is optimal

Production Efficiency: Diamond and Mirrlees Let v j (q) = max u(x j ) s.t. n i=0 q ix j i = 0 be j s utility Let y k Y k be producer k s vector of production, where Y k is k s feasible production set The government maximizes some social welfare function W (v 1 (q),, v h (q)) subject to k xj + g = y k k Yk where g is net government production Suppose y k lies in the interior of Y k. The government can choose q independently of p. If it reduces q i for some good that has a positive net demand by consumers, social welfare will rise, and the increase of production is feasible. Thus, in an optimum, y k must be on the boundary of Y k. Intuition: With production inefficiency, reduction in consumer price of a good consumed increases utility; increased demand can be satisfied without sacrificing other goods Taxation of profits important, though violation has unclear effects

Implications and Caveats of Production Efficiency Theorem Do not tax producer inputs (case for VAT) Use producer prices for public production (CBA rule) Caution: only applies if taxes are optimal, though implications of non-optimal taxes not obvious Newbery (1986): If only subset of commodities consumption can be taxed optimally, welfare-improving to impose small tax on production of either untaxed or taxed commodity Choice of VAT versus trade taxes in LDCs with large informal sector: VAT preserves production efficiency in formal sector; trade taxes indirectly tax pure profits If skilled and unskilled labor imperfect substitutes, public sector can affect relative wages by increasing demand for unskilled labor inducing production inefficiency (Naito) Production Efficiency violated internationally (Keen-Wildasin) Production Efficiency Theorem still a useful benchmark