Lecture Notes on Corrective Taxation

Lecture Notes on Corrective Taxation • simplest departure from welfare theorems • very relevant: environmental policy, congestion, rent seeking • today: avoid redistribution issues 1 Pigouvian Taxation with a Single Agent • First assume – single agent – lump sum taxation – but externalities • Utility u( x, x̄ ) concave in both x and x̄ • technology is constant returns, resource constraint: F ( x + g) = 0 • by Euler’s theorem this is equivalent to Fx ( x + g) · ( x + g) = 0 letting p j = Fx j this is simply • Social optimum p · ( x + g) = 0 max u( x, x ) x F ( x + g) = 0 ⇒ u x ( x ∗ , x ∗ ) + u x̄ ( x ∗ , x ∗ ) = γFx ( x ∗ + g) Fx u x ( x ∗ , x ∗ ) + u x¯i ( x ∗ , x ∗ ) p ⇒ i = i = i ∗ ∗ pj Fx j u x j ( x , x ) + u x̄ j ( x ∗ , x ∗ ) 1 • in equilibrium – government budget constraint (q − p) · x + T = p · g implies consumer budget constraint q·x+T = 0 – consistency x̄ = x but do not impose this on agent optimization! – agent solves, takes x̄ and T as given: max u( x, x̄ ) q·x+T = 0 x ⇒ u x ( x e , x e ) = λq u xi ( x e , x e ) qi ⇒ = qj uxj ( xe , xe ) • To make xe = x∗ a necessary condition is that both conditions hold, implying pi /qi = p j /q j u x̄i ( x ∗ ,x ∗ ) u xi ( x ∗ ,x ∗ ) 1+ 1+ u x¯ 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: Compute p from optimum, compute q from conditions above, then just com­ pute T from budget constraint. Apply Lagrangian sufficiency theorem. • Remarks: – no explicit role for elasticity of demand – correct damage per unit consumed, not target reduction • reinterpretation: “internality” or Paternalism 2 – suppose no externality, but agent maximizes wrong utility û( x ) paternistic planner cares about u∗ ( x ) then we can think of above model as one where agent has utility u( x ) + u∗ ( x̄ ) − u( x̄ ) – behavioral biases can sometimes lead us to something similar – internalities and naivety ∗ you ignore (or downplay) some effects from your consumption, such as health from smoking ∗ you are optimistic about certain probabilities that affect your saving deci­ sions 2 Pigouvian Taxation with Many Agents • Now: – suppose many agent types i ∈ I with weights π i – agent specific lump sum taxes T i – without externalities: first best – with externalties: need Pigouvian taxes • Utility ui ( xi , x̄ ) concave in both xi and x̄ = ∑ xi π i assume agent type i takes xi as given (atomostic within each group) • implicit assumption: everyone contributes the same to externalities; relax later. • technology F ( x + g) = 0 • Social optimum max ∑ λh uh ( x h , x̄ )π h F ( x̄ + g) = 0 x first order conditions are x̄ = λh uhx j ( x h , x̄ ) = η j ∑ λh uhx̄j (xh , x̄)π h + ηj = γFj (x̄ + g) h 3 ∑ xh π h • substituting gives ∑ λh u x¯ j ( x h , x̄ )π h λh u x j ( x h , x̄ ) h canceling ηj ∑ h uhx̄ j ( x h , x̄ ) uhx j ( x h , x̄ ) η j + η j = γFj ( x̄ + g) π h + η j = γFj ( x̄ + g) • dividing Fx p ⇒ i = i = pj Fx j h h h u x̄i ( x ,x̄ ) 1 + π ∑ h h h h u xi ( x , x̄ ) u xi ( x h ,x̄ ) uhx̄ ( x h ,x̄ ) uhx j ( x h , x̄ ) h 1 + ∑h π uh j ( xh ,x̄) η 1+ = i ηj 1 + xj • in equilibrium – government budget constraint (q − p) · x + ∑ T i π i = p · g i – implied by consumer budget constraints q · xi + T i = 0 – consistency x̄ = x but do not impose this on agent optimization! – agent solves, takes x̄ and T i as given max ui ( xi , x̄ ) x q · xi + T i = 0 ⇒ uix ( xi , x̄ ) = λq uixi ( xi , x̄ ) qi ⇒ = i i qj u x j ( x , x̄ ) • To make x j = x j∗ a necessary condition is that both conditions hold, implying η qi = i qj ηj 4 γFj −η j ηj γFi −ηi ηi uhx̄ ( x h ,x̄ ) pi /qi = p j /q j 1 + ∑h π h uhi ( xh ,x̄) xi 1 + ∑h πh uhx¯ ( x h ,x̄ ) = j uhx j ( x h ,x̄ ) 1+ 1+ γFj −η j ηj γFi −ηi ηi • Theorem: set 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: Same as before. • Remarks – role of λi is very implicit, not in formula; lump sum taxes help here; – in general T i = T j but this has nothing to do with externalities; – there are optima with T i = T j = T for the right λi ; (e.g. solve equilibrium fixed point with T i = T and setting p/q as in above formula then compute λi from social planner optimum); • Conclusion: corrective tax on goods restores efficiency 3 More General • Utility ui ( xi , x̄ ) x̄ = • Technology ∑ xi π i F ( x + g, x̄ + g) = 0 assume firms maximize profits using first argument • Define F̄ ( x̄ ) = F ( x̄, x̄ ) • Social optimum max ∑ λh uh ( x h , x̄ )π h F̄ ( x̄ + g) = 0 x first order conditions are x̄ = λh uhx j ( x h , x̄ ) = η j ∑ λh uhx̄j (xh , x̄)π h + ηj = γF̄j (x̄ + g) h 5 ∑ xh π h • substituting gives ηj ∑ h ⇒ uhx̄ j ( x h , x̄ ) uhx j ( x h , x̄ ) π h + η j = γF̄j ( x̄ + g) h h h u x¯i ( x ,x̄ ) 1 + π ∑ h h h u xi ( x , x̄ ) uhxi ( x h ,x̄ ) uhx̄ ( x h ,x̄ ) uhx j ( x h , x̄ ) 1 + ∑h π h uh j ( xh ,x̄) F̄xi = F̄x j xj and F̄xi = F̄x j F¯xi F Fxi xi Fx j F¯x j Fx j = F¯xi pi Fxi p j F¯x j Fx j • agent solves, takes x̄ and T i as given max ui ( xi , x̄ ) q · xi + T i = 0 x ⇒ uix ( xi , x̄ ) = λq uixi ( xi , x̄ ) qi ⇒ = i i qj u x j ( x , x̄ ) • To make x j = x j∗ a necessary condition is that both conditions hold, implying η qi = i qj ηj F¯x j pi /qi = p j /q j 4 uhx̄ ( x h ,x̄ ) Fx j 1 + ∑h π h uhi ( xh ,x̄) F̄xi Fxi 1 + ∑ h π h uh xi h u x¯ ( x h ,x̄ ) j xj (x Optimal Public Goods • Samuelson (1969) • we omitted g in utility function; let us add it ui ( x i , g ) we ignore externalities now 6 h ,x̄ ) • optimality condition ∑ λi uig (xi , g)π h = γFg (x + g) i • rearranging uhg ( x h , g) ∑ uh ( x h , g) π h = i xi Fg ( x + g) pg = Fxi ( x + g) pxj avoids λi weights 5 Non Pigouvian Correction • Diamond (1973) Bell journal paper • relax assumption that externality is just aggregate consumption – congestion: some slow and some fast drivers – environment: clean and dirty cars • simplifying assumptions: – ignore redistribution issues with quasilinear utility – focus on single good – constant producer price p • Utility for agent h U h ( α1 , α2 , · · · , α H ) + μ h • note: initially not assumed atomostic (can approach that by making H very large) 5.1 Additively Separable • suppose externality is additive U h = u h ( α h ) + v h ( α1 , α 2 , · · · , α h −1 , α h +1 , · · · , α n ) + μ h • agent has some demand � � max uh (αh ) + vh (α1 , α2 , · · · , αh−1 , αh+1 , · · · , αn ) + μh αh ( p + t)αh + μh = mh ⇒ u hl ( α h ) = p + t ⇒ α h∗ = α h ( p + t ) 7 • Welfare criterion W= ∑ U h (α) + ∑ μh h subject to p ∑ αh + ∑ μh = ∑ mh • indirect welfare W (t) = ∑ U h (α1 ( p + t), α2 ( p + t), . . . , α H ( p + t)) − p ∑ αh ( p + t) + ∑ mh h h W l (t) = ∂U h il ∑ ∑ ∂αi α ( p + t) − p ∑ αhl ( p + t) h i ∂U h il α − t ∑ α hl ∂αi i6=h =∑∑ h setting W l (t) = 0 gives t= ∑h weighted average of externalities ∑i6=h tives 5.2 ∂U h i l ∂αi α α hl − ∑ h ∑i 6= h ∂U h i l ∂αi α , weighted by consumer demand deriva­ Demand Interactions • without separability ∂U h = p+t ∂αh ⇒ αh∗ = αh ( p + t, α1 , . . . , αh−1 , αh+1 , . . . , α H ) demands affect each other: – fewer drivers on highway increase individual demand for driving • for given price we need to think of the fixed point αh∗ = αh ( p + t, α1∗ , . . . , αh−1∗ , αh+1∗ , . . . , α H ∗ ) ∀h a system of H equations in H unknowns • same first order condition then applies for this fixed point relation – beware! not pure price elasiticies – some could be positive, so not weighted sum 8 • example – with p = 1 maximum is at t = 0! – changing parameters, could be negative t... – Not weighted sum of externalities 5.3 An Aggregator • imagine externality affects everyone equally γ = Γ ( α1 , . . . , α H ) and utility U h (αh , γ) + μh • demand is then • fixed point again αh ( p + t, γ) γ = Γ(α1 ( p + t, γ), . . . , α H ( p + t, γ)) one equation in one unknown • implicit function theorem dγ ∑ Γh ∂α ∂t = h dt 1 − ∑ Γh ∂α ∂γ h denominator is feedback effect: – tax increases cost of driving... – ... but less drivers makes driving more attractive • if agents take into account their own effect on γ then this is just a special case of what we did before • assume agents ignore their effect on γ (justification: atomistic within agent h types as before) ∂U h = p+t ∂αh 9 • welfare W (t) = ∑ U h (αh ( p + t), γ) − p ∑ αh ( p + t, γ) + ∑ mh h W l (t) = h ∂ h U −p ∂αh ∑ h h ∂ h dγ ∂ h α α + ∂t dt ∂γ + dγ ∂ h U ∑ dt h ∂γ using first order condition W l (t) = t ∂ ∑ ∂t αh + h • hence at t = 0 we have W l (0) > 0 if dγ dt dγ ∂ h α ∑ dt h ∂γ ∑h ∂ h ∂γ U • solving gives ∗ t =− + >0 ∂αh ∂t ∂αh ∂t ∑ Γh ∂U h ∑ ∂γ ∑ dγ ∂ h U ∑ dt h ∂γ ∑ ∑ ∂αh ∂t ∂αh ∂t −Δ and Δ≡ ∑ Γh = ∑ ∂αh ∂t ∂αh ∂γ ∑ ∂αh ∂α ∂α − ∑ Γh h ∑ h ∂γ ∂γ ∂t ⎛ ⎞ ∂αh ∂αh ∂α ∂γ ∑ ∂th ⎝∑ Γh ∂t∂αh − ∑ Γh ∂αh ⎠ ∑ ∑ ∂t so Δ is the difference of two covariances of Γh with either ∂γ ∂αh ∂t ∂α ∑ ∂th or ∂αh ∂γ ∂α ∑ ∂γh . • since Δ is typically not zero then tax is not weighted sum of Pigouvian tax 6 Rent Seeking • we modeled externalities as affecting utility directly • sometimes can think of affecting income • example: rent seeking – output in some sector is μ( E) where E is total labor effort – payment (wage) for unit of effort is μ( E) E 10 – actual marginal benefit is μl ( E) = 6 μ( E) E unless μ( E) = μ̄E • application: fishing in a particular bay, overfishing without tax • here, formally, the distortion is that without taxes prices agents face are not equal to Fx j – but similar conclusions to utility case – sometimes can substiute income into utility 7 Imperfect Instruments • In Diamond we have imperfect instruments in that taxes are not agent specific – some agents may not pollute as much – but we charge everone the same tax – this leads to a 2nd best problem, unlike before where we attained the first best • Another source of imperfect instruments: taxes are coarse – can only tax gasoline, or miles driven, not use of a particular road – can only tax total fish caught, not fishing in particular locations • Rotschild-Scheuer (2011): application to rent seeking • important: take into account “general equilibrium” effects from fixed point • example: it may be that dγ/dt = 0 • Suppose constant output in rent-seeking sector μ( E) = μ̄ = 1 • Suppose two types of agents – producers: θ = 1 and ϕ = 1 (they can do both things) – rent-seekers: θ = 0 and ϕ = ϕ R ∈ (0, 1 (they can only rent seek) • total rent-seeking effort is E = ϕR eR + λP eP where λ P is the fraction of producers that rent-seek and e P is their effort (rent seekers only work in rent-seeking sector) 11 • productive agents may be... – indifferent, if E = 1, since then μ( E)/E = 1/E = 1 – not working in rent seeking sector if E > 1 – working only in rent seeking sector if E < 1 • I’ll focus on the interior equilibrium (one can show that this is where the optimum lies) E = ϕR eR + μP eP = 1 ⇒ λ P (eR , eP ) = 1 − ϕR eR eP • output is the sum of rent seeking output production μ̄ + (1 − λ P (e R , e P ))e P • a Utilitarian will maximize output net of effort W = μ̄ + (1 − λ P (e R , e P ))e P − = e p + ϕR eR − 1 γ 1 γ e − e γ P γ R 1 γ 1 γ e − e γ P γ R • Imagine it can control efforts, one motivation for this is that it controls a linear tax on each agent type (see below). Note, it cannot control the occupational choice. • The first order conditions are then γ −1 =1 γ −1 = ϕR ep eR 1 (this gives e p = 1 and e R = ϕ Rγ−1 < 1 so the equilibrium is indeed interior with λ P ∈ (0, 1)) • Since the wage is w = 1 for producers and w = ϕ R for rent seekers, these first order conditions coincide with that of the agent facing no distortionary taxes τp = 0 τR = 0 12 8 A Decomposition • Kopczuk (2003) provides a two step “as if” result: – first, correct the Pigouvian tax – then, solve the remaining tax problem • A useful way to think or decompose things, even though it doesn’t necessarily lead to any clear results without more structure • Considers the following problem max v(t, T; P, p; x̄ ) t,T,x̄ X (t, T; P, p) = x̄ R = I (t, T; P, p) + tx̄ G (t, T ) = 0 • The last constraint captures some constraints on taxes. For example, this could in­ corporate that taxes on some other goods have to equal t, so that the tax on X is not perfectly targeted as in Diamond (1973, RAND). • For any baseline tax t p we can change variables t = t p + s and write max v(t p + s, T; P, p; x̄ ) s,T,x̄ X (t p + s, T; P, p) = x̄ R − t p x̄ = I (t p + s, T; P, p) + sX (t p + s, T; P, p) G (t p + s, T ) = 0 • Of course only t∗ = t p + s∗ is determined i.e. s∗ varies one for one with t p • The first order condition for x̄ is v x (t∗ , T ∗ ; P, p; x̄ ) + λ∗ + μ∗ t p = 0 • Now define the tax t p to ensure that λ∗ = 0, so that tp = − v x (t∗ , T ∗ ; P, p; x̄ ) μ∗ • With this value of t p we get the same first order conditions if we look at the system max v(t p + s, T; P, p; x̄ ) s,T,x̄ R − t p x̄ = I (t p + s, T; P, p) + sX (t p + s, T; P, p) G (t p + s, T ) = 0 13 • We can drop x̄ from the maximization and we can rewrite this as max v(s, T; P, p + t p ; x̄ ) s,T R − t p x̄ = I (t p , T; P, p + t p ) + sX (s, T; P, p + t p ) G (t p + s, T ) = 0 14