Principles of Optimal Taxation

This lecture Principles of optimal taxes Focus on linear taxes (VAT, sales, corporate, labor in some countries) (Almost) no heterogeneity across consu...

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Principles of Optimal Taxation Mikhail Golosov

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This lecture

Principles of optimal taxes Focus on linear taxes (VAT, sales, corporate, labor in some countries) (Almost) no heterogeneity across consumers highlight the key driving forces behind taxes and distortions associate with them sidestep questions of the optimal taxation of redistribution: large topic in itself many insights do not depend on it

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Plan

1 2 3 4

Optimal commodity taxation Optimal intermediate goods taxation Taxation of capital income Tax smoothing

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Optimal commodity taxation

Static economy General equilibrium 4 main elements: consumers …rms government market clearing

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Consumers

A representative consumer supplies labor l and consumes n di¤erent consumption goods. Normalizing his wage rate to 1, the representative consumer solves consumer’s problem (1) max U (c1 , ..., cn , l ) c ,l

s.t.

∑ pi (1 + τi )ci

=l

i

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Firms

A large number of …rms operate identical, constant returns to scale technology to produce consumption goods. The …rm solves …rm’s problem (2) max ∑ pi xi x ,l

s.t

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l

i

F (x1 , ..., xn , l ) = 0

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Government

The government has to rely on commodity taxes to …nance exogenous expenditures fgi g. Government’s budget constraint (3) is

∑ pi gi i

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= ∑ pi τ i ci i

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Market clearing

Market clearing condition (4) is ci + gi = xi

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8 i

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De…nition of Competitive Equilibrium With taxes fτ i g and government purchases fgi g, allocations fci , l g and prices fpi g are CE if and only if the following conditions are satis…ed. consumers take fpi g as given and solve consumer’s problem (1). …rms take fpi g as given, solve …rm’s problem (2) and make 0 pro…t in equilibrium. government’s budget constraint (3) is satis…ed. market clearing condition (4) is satis…ed.

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Question: How to …nd fτ i g to …nance government expenditure fgi g so that welfare is maximized? 2 approaches: 1 2

Express everything as a function of τ and maximize w.r.t. τ directly; Use "primal"/"Ramsey" approach.

We will take the second approach. Idea: …nd necessary and su¢ cient conditions on fci , l g that should be true in any CE, and then …nd the fci , l g that satisfy these conditions and maximize the welfare.

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Consumer’s FOCs: Uci = λpi (1 + τ i ) Ul =

λ

which implies that pi (1 + τ i ) =

Uci Ul

Substitute back into consumer’s budget constraint to get

∑ Uci ci + Ul l = 0

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Theorem For any exogenous stream (g1 , ..., gn ) consider (c1 , ..., cn , l ) that satisfy

∑ Uci ci + Ul l = 0 F (c1 + g1 , ..., cn + gn , l ) = 0 Then there exists a competitive equilibrium with taxes for which (c1 , ..., cn , l ) are equilibrium allocations This may seem a little surprising since we have n + 1 variables and only two constraints. This means that there exist many solutions to this system of equations. For any solution that satis…es these conditions, we can …nd some taxes that would implement them.

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Re-constructing equilibrium from allocations Pick any (c1 , ..., cn , l ) that satis…es the conditions above. Construct prices: from …rm’s problem we have pi = λFi 1 = λFl Therefore, pi =

Fi (c1 + g1 , ..., cn + gn , l ) Fl (c1 + g1 , ..., cn + gn , l )

Construct taxes: from consumer’s FOCs 1 + τi =

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U ci (c ,l ) U l (c ,l )

pi

=

Uci (c , l ) Fl (c1 + g1 , ..., cn + gn , l ) Ul (c , l ) Fi (c1 + g1 , ..., cn + gn , l )

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Remaining su¢ ciency conditions

Are …rms making zero pro…t? Yes, since F is CRS.

∑ Fi c i

+ Fl l = 0

i

Does it raise enough money to …nance the government?

∑ pi gi

= ∑ pi τ i ci

substitute de…nition of prices, taxes and consumer budget constraint to verify that it holds also follows from Walras law

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How to …nd something that maximizes social surplus? max U (c1 , ..., cn , l ) c ,l

s.t.

∑ Uci ci + Ul l = 0

F (c1 + g1 , ..., cn + gn , l ) = 0

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For simplicity that U (c1 , ..., cn , l ) = u1 (c1 ) + ... + un (cn ) + v (l ) Consider the FOCs

(1 + λ)ui0 (ci ) + λui00 (ci )ci = γFi (1 + λ)v 0 (l ) + λv 00 (l )l = γFl Let Hi =

ui00 ci /ui0 , and Hl =

v 00 l /v 0 . Then

(1 + λ ) (1 + λ )

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λHi Ui F = i λHl Ul Fl

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We know that 1 + τi =

Ui Fl Ul Fi

Therefore, 1 + τi =

(1 + λ ) (1 + λ )

λHl λHi

τi λ ( Hi Hl ) = 1 + τi (1 + λ) λHl Combining with the same condition for good j, we get τi 1 +τ i τj 1 +τ j

=

Hi Hj

Hl Hl

Hi > Hj implies that τ i > τ j .

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What is Hi?

Consumer theory: consume solves max ∑ ui (ci ) s.t.

∑ pi ci

m

The FOC of the consumer’s maximization problem becomes Ui (ci (p, m )) = λ(p, m )pi Di¤erentiate this with respect to non-labor income Uii

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∂ci ∂λ U ∂λ = pi = i ∂m ∂m λ ∂m

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This implies that Hi

Uii ci = Ui

ci λ

∂λ ∂m ∂c i ∂m

Income elasticity of demand: ηi =

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∂ci m ∂m ci

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We have Hi = Thus,

∂λ ∂m ∂c i ∂m

m λ m ci

=

∂λ m ∂m λ

ηi

.

ηj Hi = Hj ηi

where η i is income elasticity of demand. From τi 1 +τ i τj 1 +τ j

=

Hi Hj

Hl Hl

this implies that if a good has a higher income elasticity, it should be taxes at a lower rate. So it is optimal to tax necessities at a higher rate than luxury goods.

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Lesson 1

Spread out tax distortions across all goods Tax more heavily the goods for which demand is inelastic Higher taxes distort inelastic goods less ! deadweight burden is smaller Remark: the result that necessities should be taxed at a higher rate than luxuries is not very robust derived under assumption that all agents are identical if we allow for heterogeneity and income taxation, often obtain a uniform commodity taxation result: if consumption is weakly separable from labor, tax all goods at the same rate, do all the redistribution through labor income taxation.

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Intermediate goods

How would we tax goods that consumers do not consume directly such as intermediate goods? A general result (Diamond and Mirrlees (1971)) is that economy should always be on the production possibility frontier with optimal taxes. This implies that intermediate goods should not be taxed.

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Two sectors: Final goods sector has technology f (x, z, l1 ) = 0 where z is an intermediate good. Intermediate goods sector has technology h(z, l2 ) = 0

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Consumers maximize their utility subject to budget constraint. max U (c, l1 + l2 ) p (1 + τ )c

s.t.

w (l1 + l2 )

Final goods sector maximizes its pro…t subject to feasibility constraint. max px s.t.

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wl1

q (1 + τ z )z

f (x, z, l1 ) = 0

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FOCs w = γfl q (1 + τ z ) = γfz so that

fl w = fz q (1 + τ z )

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Intermediate goods sector maximizes its pro…t subject to feasibility constraint max qz

wl2

h(z, l2 ) = 0

s.t. FOCs

q = γhz w = γhl so that

hl = hz hl = hz

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w q

(1 + τ z )

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fl fz

(2.1)

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Government budget constraint τpc + τ z qz = pg Market clearing c +g = x Following steps similar to those we did before, we can derive the implementability constraint Uc c + Ul (l1 + l2 ) = 0

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The social planner’s problem is max U (c, l1 + l2 ) s.t.

Uc c + Ul (l1 + l2 ) = 0 f (c + g , z, l1 ) = 0 h(z, l2 ) = 0

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FOC w.r.t z: fz γf + hz γh = 0 or

fz = hz

γh γf

FOC w.r.t l1

[l1 ] : Ul (1 + λ) + λ(Ull (l1 + l2 ) + Ucl c ) = fl γf FOC w.r.t l2

[l2 ] : Ul (1 + λ) + λ(Ull (l1 + l2 ) + Ucl c ) = hl γh which implies that fl γ = h hl γf or

fl hl = fz hz This suggests that when taxes are set optimally, the marginal rate of transformation should be undistorted across goods. Comparing with the condition for CE (2.1) we see that in the optimum τ z = 0 Golosov ()

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Lesson 2

Tax consumption goods but not intermediate goods The same …nal bundle of consumption can be achieved with either consumption or intermidate taxes ... but intermediate taxes distort more by misallocating intermediate inputs

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Limitations

externalities (obvious) intermidiate goods are not used as a consumption good if can, tax …nal consumption but not intermidiate consumption, but that may not be feasible. if cannot, the results need not apply, similar to what we show below.

perfect competitition may need to tax them if cannot tax monopoly’s pure pro…ts

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Optimal capital taxation

Dynamic economy Government: …nances a stream of government purchases gt . Assume that government can use only linear taxes. No lump sum taxes. No taxation of capital in the …rst period (equivalent to lump sum tax)

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Environment

Representative in…nitely lived agent with utility ∑t∞=0 βt u (ct , lt ). Government: Needs to …nance gt . Chooses taxes to …nance gt government debt bt to smooth out the distortions

Representative agent with taxes.

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Consumer’s problem (1) max ∑ βt u (ct , lt ) c ,l ,k

s.t.

(1 + τ ct )ct + kt +1 + bt +1

(1

τ kt )(1 + (rt

δ))kt + (1

τ lt )wt lt + Rt bt

k0 = k¯ 0

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Firm’s problem (2) max F (kt , lt ) k ,l

wt lt

rt kt

Government budget constraint (3) gt + Rt bt

τ lt wt lt + τ kt (1 + (rt

δ)) kt + τ ct ct + bt +1

Market clearing (4) ct + gt + kt +1

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F (kt , lt ) + (1

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δ)kt

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De…nition: CE with taxes fτ lt , τ kt g and government purchases fgt g is allocations fct , lt , kk , bt ) and prices fwt , rt g s.t. consumers take fwt , rt g as given and solve consumer’s problem (1) …rms take fwt , rt g as given, solve producer’s problem and make 0 pro…t in equilibrium (2) government’s budget constraint is satis…ed (3) markets clear (4)

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Some observations Observation 1: Irrelevance of some taxes FOCs: ul (t ) = uc (t )

(1 τ lt )wt 1 + τ ct

p βuc (t + 1) (1 + τ ct +1 ) = t +1 = uc (t ) pt (1 + τ ct ) (1 τ kt +1 )(1 + rt +1

(1 + rt +1

δ)

δ) = Rt +1

Too many taxes, can get rid of some. We will assume that τ ct = 0 for al t Equivalently, we could assume that τ kt = 0 and have

(1 + τˆ ct ) = (1 (1 + τˆ ct +1 )

τ kt +1 )

Positive tax on capital and constant tax on consumption is equivalent to zero tax on capital and increasing tax on consumption. Golosov ()

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Observation 2: Nothing fancy about dynamics Instead of thinking about period t consumption, think about period 0 consumption of a good with label "t": equivalent to the static commodity taxation problem with in…nitely many goods.

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Observation 3: Non-distortionary taxation of capital in period 0 Note that taxes on capital in period 0 does not distort any decisions: equivalent to a lump sum tax. If government could use this tax, it would set it at a very high level to get enought revenues to …nance all future gt . Assume (without any justi…cation) that this tax is unavailable to make the problem interesting τk 0 = 0

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Observation 4: (as before) Many ways to ensure that distortions hold Here: tax gross return on capital 1 + r δ Could instead (as usually done in practice) tax net return r δ : nothing changes in the analysis.

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Finding necessary conditions Proceed as before: substitute FOCs into budget constraint: uc (t )ct + ul (t )lt + uc (t ) [kt +1 + bt +1 ]

1

β

uc (t

1) [kt + bt ]

Feasibility ct + kt +1 + gt

F (kt , lt ) + (1

δ)kt

These conditions necessary. Depends on 4 variables: c, l, k, b. Equivalently can re-write in terms of c, l, k, a where a (t + 1)

uc (t ) [kt +1 + bt +1 ]

so that we get uc (t )ct + ul (t )lt + a(t + 1)

β

1

a (t )

Sum over all the periods to get

∑ βt [uc (t )ct + ul (t )lt ] = uc (0)k0 Golosov ()

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(ImC)

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Optimal taxes

Solve for the optimal allocations max ∑ βt u (ct , lt ) s.t. (F), (ImC).

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FOCs:

βt uc (t ) + η [ βt ucc (t )ct + βt uc (t ) λ t = [ Fk ( t + 1 ) + ( 1

βt ucl (t )lt ] = λt δ)]λt +1

Therefore

=

uc (t ) + η [ucc (t )ct + uc (t ) ucl (t )lt ] uc (t + 1) + η [ucc (t + 1)ct +1 + uc (t + 1) ucl (t + 1)lt +1 ] β(Fk (t + 1) + (1 δ))

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(*)

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Theorem No capital taxes in the steady state Proof. Suppose ct ! c, kt ! k, lt ! l. Then the equation above says β ( Fk ( t + 1 ) + ( 1

δ)) = 1

Consumer (Euler) in the steady state

(1

τ k ) β (1 + (r

δ)) = 1

and r = Fk . These equations give

(1

τk ) = 1

so that τ k = 0

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Theorem 1 c1 1 σ Then τ t = 0 for all t > 1.

Suppose u (c, l ) =

σ

+ v (l ) (actually need much weaker conditions).

Proof. In this case ucl = 0 and ucc c =

= =

σuc and

uc + η [ucc c + uc ucl l ] σuc +1 uc 1 + η uc uc (1 + η [1 σ])

so that (*) becomes uc (t ) = β ( Fk ( t + 1 ) + ( 1 uc (t + 1)

δ))

De…nition of taxes on capital immidiately implies that τ kt = 0 for all t > 1 Golosov ()

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Discussion

General results: high tax on capital in the beginning, goes to zero. labor taxes typically positive government revenues high in the beginning, decrease over time: budget surplus in the beginning, de…cit later on Judd (JPubE 1987) Add heterogeneity. Two types of agents, capitalists (who do not work and own capital) and workers (who work but cannot save). Capital and labor taxes not only create distortions but also redistributed from capitalists to workers. Showed a start result that even if the planner cares only about workers, still taxes are zero on capitalists in the long run.

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Lesson 3

Tax labor/consumption, not capital Capital distortions quickly accumulate due to compounding Contrast with a naive view that want to distribute distortions across all sources of income

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Time consistency

Compute the optimal policy from t = 0 perspective high capital taxes early, go to zero (say, by t = 100)

Compute the optimal policy from t = 100 perspective The two will not be the same government has strong incentives to deviate from the optimal policy and choose di¤erent taxes later on

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Time consistency II

If government cannot commit, agents will take that into account when taxes are announced Invest less, even though they are promised low taxes tomorrow know that tomorrow government cannot keep its promise and will revert to high taxes

Welfare losses can be large without commitment Kydland and Prescott’s 2004 Nobel prize Lesson 4: Optimal policy is not time consistent. It is important to be able to commit and avoid temptation ex-post

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Taxation over business cycle

Big topic, sketch one general idea Suppose preferences are u (ct , lt ) = ct ul (ct , lt ) = (1

1 1 +γ 1 +γ lt

τ lt )wt

Suppose gt follows some stochastic process Ignore capital (or set taxes on capital to zero)

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Implementability constraint

As before uc (t )ct + ul (t )lt + a(t + 1) But now uc (t ) = 1 Cannot sum, due to uncertainty SP solves max E ∑ βt [ct

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β

1

a (t )

v (lt )]

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Taxes as random walk

FOCs imply that ul (t ) = Et ul (t + 1) if v (l ) is quadratic τ t = Et f τ t + 1 g

(*)

Taxes follow random walk independent of the stochastic process for gt

Implication: consider a positive shock for gt government revenues must go up for (*) to be satis…ed, they must go up in all future periods by the same (small) amount from government b.c., debt must go up a lot, be repaid over time

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Lesson 5

Tax smoothing: smooth tax distortions in response to shocks, use debt to help doing that

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Summary

Lesson Lesson Lesson Lesson Lesson

1: 2: 3: 4: 5:

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tax elastic goods less than less elastic do not tax intermediate goods do not tax capital commitment is important smooth taxes in response to shocks

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