Johannes Spinnewijn - darp.lse.ac.uk

Commodity Taxes Ramsey Model Applications

Lecture 8: Optimal Commodity Taxation

Johannes Spinnewijn

London School of Economics

Lecture Notes for Ec426

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Agenda

1 Potential reasons for imposing commodity taxes

2 Effi cient revenue collection:

1 First-best taxation2 Second best taxation: The Ramsey optimal tax model

3 Applications

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Reasons for Imposing Commodity Taxes

1 To satisfy a revenue requirement for financing public goods:Ramsey taxes (single-agent)

2 To redistribute from rich to poor:Ramsey taxes (many-agent)

3 To correct externalities and other market failures:Pigouvian taxes

4 To correct internalities associated with bounded rationality:Paternalistic taxes

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Excess Burden

Excess Burden (or Deadweight Loss)

to generate £ 1 of revenue, welfare of those taxed is reduced bymore than £ 1 because the tax distorts incentives and behavior

Optimal size of a policy: Marginal Value = Marginal EB

Key intuition:MEB ≈ τ × ∆x

MEB increases with the size of the taxMEB is smaller the less responsive the agents are

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Illustration: Excess Burden

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Basic Model

Utility u(x1, x2, L) with market goods x1 and x2, and leisure L

Agents’budget constraint:

q1x1 + q2x2 = wh

or q1x1 + q2x2 + wL = wT

Define:

endowment of time, T

hours worked, h = T − Lprices q1 and q2

wage rate, w

wT is the market value of time endowment (full income).

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First-best Revenue Collection

Consider a uniform tax on all consumption goods, includingleisure, at rate τ

(1+ τ)q1x1 + (1+ τ)q2x2 + (1+ τ)wL = wT

or q1x1 + q2x2 + wL =wT1+ τ

A uniform tax on all consumption goods is equivalent to a taxon full income (which is exogenous)

This is a lump-sum tax which is non-distortionary ⇒First-best solution!

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Is First-Best Feasible?The government does not observe the value of leisure, itobserves only market transactions ⇒ labor is taxable, leisureis not

Consider a uniform rate τ on goods x1, x2 and labor

(1+ τ)q1x1 + (1+ τ)q2x2 = (1+ τ)wh

(1+ τ) cancels out:

tax system does not collect any revenue

tax on goods is offset exactly by a subsidy on labor

If leisure cannot be taxed, the tax system has to distortrelative prices in order to collect any revenue ⇒ Second-bestsolution

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Ramsey Model: Representative HouseholdConsider a single representative consumer - ignoring equityconcerns

The consumer solves the following problem:

max u(x) subject to px = y

x = (x0, ..., xN ) is a vector of goods, and p = (p0, ..., pN ) avector of consumer prices

This gives the FOC for good i : ∂u∂xi= λpi

Consider the following functions:Uncompensated demand function xi (p, y)

Indirect utility function v(p, y) = u(x(p, y)) with ∂v∂y = λ

Roy’s identity ∂v∂pi= −xiλ (follows from the envelope theorem)

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Dual Representation

The dual problem solves

min px = y subject to u(x) ≥ u

Define the following functions and identities

Compensated demand function xci (p,u) . It is homogenous ofdegree zero in p

Expenditure function e(p,u) = p · xc (p,u)

Slutsky equation ∂xi∂pj=

∂x ci∂pj− xj ∂xi

∂y

Slutsky symmetry implies ∂x ci∂pj

=∂x cj∂pi

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Taxes & NormalizationAssumption: Government can tax all market goods and labor atdifferent rates, but it cannot tax leisure directly

Let good 0 be (minus) labor and goods 1, ...,N be marketgoods

exogenous income y is zero

budget is px = 0

Tax rates are t = (t0, .., tN ).Budget is homogenous of degree zero in p ⇒ one tax rate canbe normalized

Convention: normalize t0 = 0 so that labor is untaxed

Key distinction: Untaxability of leisure is a restriction; no taxon labor is normalization (confusion about this in earlyliterature)

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Government ProblemThe govt solves the optimal tax problem where

maxp1,...,pN

v(p, y) s.t.N

∑j=1(pj − qj ) · xj (p, y) ≥ R

R is an exogenous revenue requirement

Fixed producer prices q = (q0, ..., q1) (linear productiontechnology)

Consumer prices pi = qi + ti∀i :By setting t1, .., tN , the govt controls p1, .., pN

Define the Lagrangian with µ = the marginal welfare loss ofincreasing R

L = v(p, y) + µ

(N

∑j=1(pj − qj ) · xj (p, y)− R

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Deriving the Ramsey Rule

The FOC for pi is given by

∂v∂pi

+ µ[xi +N

∑j=1tj

∂xj∂pi] = 0

Three terms: (i) direct welfare efect, (ii) mechanical revenueeffect, and (iii) behavioral revenue effect

Insert Roy’s identity, Slutsky equation, and Slutsky symmetry

Define the social marginal utility of income

α ≡ λ+ µN

∑j=1tj

∂xj∂y

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Ramsey Rule

Classic formulation of the Ramsey rule

1xi

N

∑j=1tj

∂xci∂pj

= −µ− α

µ(1)

Common proportional output reduction interpretation of (1)

Using Slutsky symmetry, we may write the Ramsey rule as

1xi

N

∑j=1tj

∂xcj∂pi

= −µ− α

µ(2)

I find formulation (2) better for intuition

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Interpretation (1): Proportional Output Reduction

Assume either that taxes are small or that compensateddemands are linear in prices

In either case, the change in compensated demand for good idue to the imposition of all taxes is ∆xci = ∑N

j=1 tj∂x ci∂pj

The Ramsey rule then states that ∆x cix cishould be constant

across goods

∆x cix ciis sometimes labeled the index of discouragement

Proportional output reduction interpretation is really not veryuseful

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Interpretation (2)

Numerator ∑Nj=1 tj

∂x cj∂pi

is the behavioral revenue effect (due tocompensated responses) of a marginal increase in ti

Denominator xi is the mechanical revenue effect of a marginalincrease in ti

General insight: Behavioral revenue loss from a marginal taxincrease = marginal deadweight loss

Ramsey rule in words: The marginal DWL per dollar ofcollected revenue must be the same for all tax instruments

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The Ramsey Rule and Tax Policy

The Ramsey rule provides a very general characterization andis pretty intuitive

But the rule is not operational for policy: It is a statementabout compensated quantity changes, not about tax policy

The literature has considered various special cases to obtaindirect statements about tax policy

Some classic results:1 The inverse elasticity rule

2 The Corlett-Hague rule

3 Uniform commodity tax theorem

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The Inverse Elasticity Rule (1)

Assume that compensated cross-effects between taxed goodsare zero, ∂xci /∂pj = 0 for i 6= j and i , j = 1, ..,N.

Ramsey rule (1) becomes

tipi= −(µ− α

µ)1εii> 0

where εii =∂x ci∂pi

pixiis the compensated own-price elasticity

In words: "The tax rate on each good is inversely proportional

to its own-price elasticity."

A famous result that relies on very strong assumptions aboutpreferences

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Inverse Elasticity Rule (2)Homogeneity of degree zero of compensated demands implies(using Euler’s Theorem)

N

∑j=1

εij = 0 where εij =∂xci∂pj

pjxi

Zero cross-effects between taxed goods implies εi0 + εii = 0

Inverse elasticity rule can be written as

tipi=

(µ− α

µ

)1

εi0> 0

Cross-elasticity εi0 reflects substitutability btw goods i and 0

In words: "The tax rate on each good is inversely proportionalto its substitutability with leisure (i.e., high taxes on leisurecomplements)." 21 / 24


A Uniform Commodity Tax Theorem

Consider a uniform tax structure on taxed goods: tj = θpj forj = 1, ..,N

Insert this into the Ramsey rule and use Euler’s Theorem

εi0 =1θ(

µ− α

µ), i = 1, ..,N

Uniform taxation requires that all taxed goods are equallycomplementary to leisure

Deaton (1976, 1981): suffi cient condition for uniform taxation

u(x0, x1, ..., xn) = u(x0, f (x1, ..., xn) and f homothetic

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Uniformity vs Selectivity?

Question: Should we impose uniform commodity taxes ⇐⇒uniform taxes on all taxable goods?

Given that uniform commodity taxation is identical to anincome tax, if uniformity is optimal, we could abolishcommodity taxes altogether

The Ramsey model shows that selective taxation is optimalunless preferences have a very special structure

But are there other arguments for uniformity outside themodel?

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Problems with Selectivity

1 Ignorance: We lack knowledge of the elasticities on whichRamsey taxation should be based

2 Administration and complexity

3 Creation of new goods: New goods may be put on the marketonly for tax avoidance reasons

4 Political economy: lobbying, bribery to get lenient taxtreatment

5 Fairness: Individuals will be treated differently just becausethey have different tastes

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Johannes Spinnewijn - darp.lse.ac.uk

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