Notes on Linear Taxation

14.471 Notes on Linear Taxation 1 Overview • Two models – single agent (Ramsey), no lump sum tax – agent heterogeneity and lump sum tax • Two approaches – primal – dual • Mixed Taxation 2 Single Agent Ramsey • consumers: max u( x ) x ∑ qi xi ≤ 0 i e.g u(c1 , c2 , . . . , cn , l ) and ∑ pi (1 + τi )ci = (1 − τ l )wl • CRS technology (inputs are supressed) F (y) ≤ 0 e.g. ∑ p̄i yi − l ≤ 0 • Remark: Production efficiency holds so that F (y) = 0 at optimum (implies intermediate inputs go untaxed) without CRS this result requires profit taxes (see Diamond-Mirrlees) 1 • First Best : – MRSijh = MRSijh – MRSijh = MRTij – F = 0 (efficient production; with inputs this requires a marginal condition equating the relative marginal products across goods) • Firms max py y F (y) ≤ 0 • government ∑ p i gi ≤ ∑ t i x i • market clearing: x i + gi = y i ∀i • note: we could have u(c, g), but in what follows g is fixed, so we supress the depen­ dence. • Definition: A Competitive Equilibrium (CE) with taxes is p, q, c 1. x solves the consumer’s maximization max u( x ) x ∑ qi xi ≤ 0 i 2. y solves the profit maximization max py y F (y) ≤ 0 3. x, g, t, p satisfy the government budget constraint ∑ p i gi ≤ ∑ t i x i 4. markets clear x i + gi = y i ∀i • Result: CE ⇐⇒ F ( x + g) = 0 and agent optimization (1) • note: second condition involves x and q only 2 • First Best max u( x ) x,q F ( x + g) = 0 • Second Best max u( x ) x,q F ( x + g) = 0 x ∈ arg max u( x ) x q·x ≤ 0 • we have two variables x, q but they are related through the last condition • At this point, from consumer maximization we can approach things from... – primal: solve q as a function of x – dual: solve x as a function of q • both approaches are useful 2.1 Dual • define V (q, I ) = max u( x ) x q·x ≤ I and let xi (q, I ) denote the solution (Marshallian/uncompensated demand) e(q, v) ≡ min q · x x u( x ) = v and let xic (q, v) = eqi (q, v) denote the solution (Hicks/compensated demand) • we abuse notation: V (q) = V (q, 0) • Second Best: max V (q) q s.t. F ( x (q, 0) + g) = 0 • property: x c (q, V (q)) = x (q, 0) 3 • equivalently max V (q) 2.2 F ( x c (q, V (q) + g) = 0 s.t. q Optimality condition • We have the first order condition ∂V ∂F (q, 0) − κ ∑ ∂q j ∂yi i • By Roy’s identity ∂V ∂q j ∂ xic ∂V ∂v ∂q j =0 ∂x c ∂V ∂xic − xj i ∂q j ∂v ∂I =0 = − x j ∂V ∂I : 1 ∂V ∂F − xj −∑ κ ∂I ∂yi i • Now use that ∂xic ∂xic ∂V + ∂q j ∂v ∂q j = ∂xi ∂I and pi = ∂F ∂yi to get ∂x c ∂x 1 ∂V − xj − ∑ pi i + x j ∑ pi i = 0 ∂I κ ∂I ∂q j i i ∂x c • Now we know that ∑i qi ∂qj = 0 and that i − ∑ pi i ∂x cj ∂qi ∂xic = ∂q j = ∂xic ∂q j ∑ ti i by symmetry so that ∂xic ∂q j i • Also, we know that ∑i qi ∂x ∂I = 1 so that ∂x ∑ pi ∂Ii = 1 − ∑ ti ∑ ti ∂xic = −xj θ ∂q j i • Thus, we obtain i i 4 ∂xi ∂I where θ≡− 1 ∂V ∂x + 1 − ∑ ti i κ ∂I ∂I i • or equivalently (using symmetry) ∂x cj ∑ ti ∂qi = − x j θ. i • interpretation: – each good is “discouraged” by a common percentage θ, i.e. interpret (falsely) as an estimate of how much good x j fell due to taxation. – DWL= e(q, V (q)) − ∑ ti xic ( p, V (q)) 1 ∂DWL = constant xi pi ∂τi intuitive: marginal DWL is proportional to revenue base (mg cost = mg benefit) 2.3 Primal • Primal solves q from x • consumer optimization x ∈ arg max u( x ) x q·x ≤ 0 • necessary and sufficient conditions: ∃λ > 0 s.t. (assuming local non-satiation) qi = λui ( x ) q·x = 0 thus (imlementability condition) ∑ ui ( x ) x = 0 5 • Result: reverse is also true: if ∑ ui ( x ) x = 0 then ∃q such that x ∈ arg maxx u( x ) s.t. q · x ≤ 0. • Second best max u( x ) F ( x + g) = 0 ∑ ui ( x ) x = 0 • Lagrangian: L = u( x ) + μ ∑ ui ( x ) xi − γF ( x + g) • FOC (1 + μ)ui ( x ) + μ ∑ uij ( x ) x j = γFi ( x + g) j • implication u (x) ij u i ( x ) 1 + μ + μ ∑ j ui ( x ) x j Fi ( x + g) = ukj ( x ) Fk ( x + g) uk ( x ) 1 + μ + μ ∑ j uk ( x ) x j • since p Fi ( x + g) = i Fk ( x + g) pk ui ( x ) q = i uk ( x ) qk • tax rate (where qi = τi pi ) u (x) 1 + μ + μ ∑ j uij( x) x j τi i = u (x) τk 1 + μ + μ ∑ j ukj( x) x j k • exercise: show that if U ( G ( x1 , x2 , . . . , xn ), x0 ) and G is homogeneous of degree 1 then τ1 = τ2 = · · · = τn . 6 2.4 Many Agents Dual • Second Best (dual) max ∑ λh V h (q, I )π h F (∑ x c,h (q, V h (q, I ))π h + g) = 0 s.t. q,I h • note about I: – we can impose I = 0; – typically we do not want to: captures a lump sum transfer/tax – if we allow I free then productive efficiency is obvious • more generally – Pareto problem not convex – cannot maximize weighted utility – but pareto weights for local optimality condition • Define Lagrangian L= ∑ λh V h (q, I )π h − γF(∑ xc,h (q, V h (q, I ))π h + g) h h • FOCs: (using same identities as before) − ∑ λh x hj h ∂xic,h ∂xic,h h ∂V h h π − γ ∑ Fi xi π h = 0 − ∂I ∂q ∂I j h,i ∑ λh h ∂x h ∂V h h π − γ ∑ Fi i π h = 0 ∂I ∂I h,i • notation: – population average: E h [·] = ∑h [·]π h – adjusted pareto weight: βh ≡ λh ∂V h γ ∂I 7 • we arrive at the condition  Eh  ∑ t l l ∂ x cj ,h ∂ql  x hj "  = X j Eh Xj ∂xlh h −1 + β + ∑ t l ∂I l !# •• Note that if we have homothetic and separable preferences then x hj Xj is independent of j. So from here we can see a uniform tax result. • if we have a lump sum then: " Eh so we can write ∂ xlh h −1 + β + ∑ t l ∂I l  Eh  ∑ t l l where β̂h = βh + ∑l tl ∂ x c,h j ∂ql   = X j Covh # =0 " x hj Xj # , β̂h ∂xlh ∂I . • We get two intuitive cases: – βˆh is constant; – x hj Xj is independent of j. Then back to regular case. • Q: Pareto inefficiency? • A: If #agents < #goods maybe cannot find βh that solve these equations • Suppose utility is U i ( G ( x1 , . . . , x N1 ), H ( x N1 +1 , . . . , x N )) and G, H are h.o.d. 1 • Result: tax uniformly within each group. • Proof: treat goods ( x1 , x2 , . . . , x N1 ) and ( x N1 , x2 , . . . , x N ) as inputs into production of G and H. 8 3 Mixed Taxation: Atkinson-Stiglitz • Notation: consumption goods x ∈ Rm labor (in efficiency units) Y∈R B budget set • Given B consumers solve: ( xi , Y i ) ∈ arg max U i ( x, Y ) ( x,Y )∈ B • Technology (linear) ∑ p j xij πi ≤ ∑ Yi i,j i • Feasibility. previous 2 conditions hold. • if Bi allowed to be dependent on i then we can get the first best (Welfare theorem) • ...but here B is independent of i so we are in the second best • Assume: ui ( x, Y ) = U i ( G ( x ), Y ) • Result: uniform taxation is efficient (Atkinson-Stiglitz). B AS ≡ {( x, Y )| p · x ≤ Y − T (Y )} Indeed, anything else is Pareto inefficient! • Exercise to get to result... 1. start from B0 that uses commodity taxes 2. create new B that is “better” Here “better”: save resources and same utility. (Why better?) • really can start from any arbitrary B0 • Note: “two stage” budgeting (given any B)... Define: b = {( g, Y )|∃ x s.t. g = G ( x ) and ( x, Y ) ∈ B} 9 then agents solve (outer stage): arg max U i ( g, Y ) g,Y ∈b • Idea: given B0 we have some b0 . We change B1 but keep implied b1 = b0 . Then we get the same choices of Y i and the same utility for each agent. Good choice: B1 = B AS ≡ {( x, Y )|∃ g s.t. p · x ≤ eG ( g, p) and ( g, Y ) ∈ b0 } where eG ( g, p) ≡ minx p · x s.t. g = G ( x ), is the expenditure function for G. • Equivalently if we define bˆ ≡ {(y, Y )|∃ g s.t. y = eG ( g, p) and ( g, Y ) ∈ b} then B AS ≡ {( x, Y )| p · x ≤ y and (y, Y ) ∈ b̂} which has an obvious income tax interpretation. • This will save resources as long as x choices change. Why? 4 Pigouvian Taxation • now assume – single agent – lump sum taxation – but externalities • utility u( x, x̄ ) concave in both x and x̄ • technology F ( x + g) = 0 • in equilibrium x̄ = x 10 • agent solves (takes x̄ as given) max u( x, x̄ ) q·x = I x ⇒ u x ( x e , x e ) = λq u xi ( x e , x e ) qi ⇒ = qj uxj ( xe , xe ) • Social optimum max u( x, x ) F ( x + g) = 0 x ⇒ u x ( x ∗ , x ∗ ) + u x̄ ( x ∗ , x ∗ ) = γFx ( x ∗ + g) ⇒ Fx u x ( x ∗ , x ∗ ) + u x̄i ( x ∗ , x ∗ ) pi = i = i ∗ ∗ pj Fx j u x j ( x , x ) + u x̄ j ( x ∗ , x ∗ ) • To make xe = x∗ a necessary condition is that both conditions hold, implying pi /qi = p j /q j 1+ 1+ u x̄i ( x ∗ ,x ∗ ) u xi ( x ∗ ,x ∗ ) u x̄ j ( x ∗ ,x ∗ ) u x j ( x ∗ ,x ∗ ) • Theorem: if p and q to satisfy this equation, then there exists an income I (i.e. lump sum tax/transfer) so that the agent chooses x = x ∗ . Proof: (sketch) Use Lagrangian sufficiency theorem. 5 Application to Intertemporal Taxation • neoclassical growth model • simplifying assumptions – single agent first – no uncertainty 11 • technology c t + gt + k t +1 ≤ F ( k t , L t ) + (1 − δ ) k t where F is CRS • preferences ∞ ∑ βt u(ct , Lt ) t =0 • budget constraints – agents ct + k t+1 + qt,t+1 Bt+1 ≤ (1 − τt )wt Lt + Rt k t + (1 − κtB ) Bt where R t = 1 + κ t (r t − δ ) we also need some no-ponzi conditions q0,t = q0,1 q1,2 · · · qt−1,t lim q0,T BT ≥ 0 T →∞ – government: gt + Bt ≤ τt wt Lt + κt rt k t + qt,t+1 Bt+1 • without loss of generality: κtB = 0 t = 1, 2, . . . • Firms: max{ F (Kt , Lt ) − wt Lt − rt Kt } Kt ,Lt necessary and sufficient conditions FL (Kt , Lt ) = wt FK (Kt , Lt ) = rt • Definition of an equilibrium: – agents maximize given prices and taxes 12 – firms maximize – government budget constraint satisfied – market clears: goods, capital and bonds • adding up both budget constraints gives gt + c t + k t +1 ≤ w t L t + (1 + r t − δ ) k t = F ( k t , L t ) + (1 − δ ) k t which is just the resource constraint • solving Bt forward ∞ ∑ q0,t (ct − (1 − τt )wt Lt − Rt kt + kt+1 ) ≤ (1 − κ0B ) B0 t =0 unless q t +1 R t +1 = q0,t+1 R t +1 = 1 q0,t t = 0, 1, . . . there is an arbitrage • cancelling: ∞ ∑ q0,t (ct − (1 − τt )wt Lt ) ≤ R0 k0 + (1 − κ0B ) B0 t =0 • now we can just apply the primal approach • implementability condition: ∞ ∑ βt (uc,t ct + u L,t Lt ) = uc,0 ( R0 k0 + (1 − κ0B ) B0 ) t =0 • Lagrangian L≡ ∞ ∑ βt W (ct , Lt ; μ) − μuc,0 ( R0 k0 + (1 − κ0B ) B0 ) t =0 where W (c, L; μ) ≡ u(c, L) + μ (uc (c, L)c + u L (c, L) L) • optimality conditions obtained from max L s.t. resource constraint 13 • first order conditions: − WL (ct , Lt ; μ) = FL (Kt , Lt ) Wc (ct , Lt ; μ) Wc (ct , Lt ; μ) = βR∗t+1 Wc (ct+1 , Lt+1 ; μ) where R∗t+1 ≡ Fk (k t+1 , Lt+1 ) + 1 − δ is the social rate of return • for agent wt (1 − τt ) = − u L (ct , Lt ) uc (ct , Lt ) uc (ct , Lt ) = βRt uc (ct+1 , Lt+1 ) • implications 1 − τt = u L (ct , Lt ) Wc (ct , Lt ; μ) uc (ct , Lt ) WL (ct , Lt ; μ) R t +1 uc (ct , Lt ) Wc (ct+1 , Lt+1 ; μ) = ∗ R t +1 uc (ct+1 , Lt+1 ) Wc (ct , Lt ; μ) • results: – a form of labor tax smoothing: ∗ the entire sequence of gt has an impact on the tax through μ ∗ no special role for current gt , conditional on current allocation ∗ clearer in special cases: if u(c, L) = c 1− σ Lγ −α γ 1−σ with σ > 0 and γ > 1 then τt = τ̄ – at a steady state the tax on capital is zero (Chamley-Judd): ct → c̄ ⇒ – initial tax on capital and bonds: ∗ equivalent to a lump sum tax ∗ optimal to expropriate 14 Lt → L̄ R t +1 →1 R∗t+1 ∗ if upper bound on tax rates, then they will binding • last result leads to time inconsistency: – plan to... ∗ tax initial capital highly ∗ tax future capital at zero – will plan be carried out? can we commit to it? ∗ if not, and reoptimize once and for all then raise capital again ∗ if reoptimize all the time (discretion): expect high taxes, which lowers wel­ fare • with heterogeneous agents – allow a lump sum (poll) tax – first two results hold: tax smoothing and Chamley-Judd – the last conclusion less clear: ∗ Pareto analysis ∗ depends on distribution of assets and redistributive intent – even if capital levy is optimal, it may be bounded, and correct intuition is not based on a lump sum tax – time inconsistency also more subtle: in general not time consistent, but de­ pends on evolution of wealth 15