Public Economics - Lecture 4: Taxation of commoditiesmarcsangnier.free.fr/.../ecopub_m1_2014-2015_lec4.pdf · 2014-2015,Springsemester AixMarseilleSchoolofEconomics. PublicEconomics-Lecture4:

Public EconomicsLecture 4: Taxation of commodities

Marc [email protected]

2014-2015, Spring semesterAix Marseille School of Economics

Public Economics - Lecture 4: Taxation of commodities

1 Introduction

2 Tax incidence

3 Optimal commodity taxation

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Public Economics - Lecture 4: Taxation of commoditiesIntroduction

1 Introduction

2 Tax incidence


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Public Economics - Lecture 4: Taxation of commoditiesIntroduction

• What happens when a tax is introduced or changed?• Knowing about it, what tax system should we design?

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Public Economics - Lecture 4: Taxation of commoditiesTax incidence

1 Introduction

2 Tax incidencePartial equilibrium incidenceTax salienceGeneral equilibrium incidence


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What is tax incidence?

• Tax incidence is the study of the effects of tax policies on pricesand the distribution of welfare.

• Effects on prices, quantities, profits, utilities, inputs prices andquantities, capital returns, etc. . .

• Positive analysis as first step in policy evaluation before lookingfor the social welfare maximizing policy.

• Empirical analysis is important as theory is frequently inconclu-sive.

• Ideally, we want to know the effect of a tax change on utilitylevels of all agents.

• Realistically, we usually look at impact on prices or income ofsome agents.

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Partial equilibrium incidence


Kotlikoff, Laurence J. & Summers, Lawrence H., 1987. “Tax incidence,” Handbook ofPublic Economics, in: A. J. Auerbach & M. Feldstein (ed.), Handbook of PublicEconomics, edition 1, volume 2, chapter 16, pages 1043-1092, Elsevier.Key assumptions:

• Economy with only two goods: approximation of incidence ina multi-good model if:

• The market being taxed is “small”,• There are no close substitutes or complements in the utility

function.• Tax revenue is not spent on the taxed good: used to buy theuntaxed one, or simple thrown away.

• Perfect competition.

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No tax

Quantity

e

0

D(p)

S(p)

Q∗

p∗

Firms’ surplus

Consumers’ surplus

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Unit tax levied on consumers

Quantity

e

0

D(p)

S(p)

Q∗

p∗

D(p + t)

Q′

p′

Consumers’ lossFirms’ loss

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Unit tax levied on producers

Quantity

e

0

D(p)

S(p)

Q∗

p∗

S ′(p + t)

Q′

p′

Consumers’ lossFirms’ loss

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Setup of the model

• Two goods: x and y .• The government levies an excise tax on good x :

• Excise tax: levied on a quantity, fixed in nominal terms;• Ad-valorem tax: fraction of price.

• Let p denote the pretax price of x and q = p + t denote thetax inclusive price of good x .

• Good y is the untaxed numeraire.

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• Representative consumer has wealth Z and utility u (x , y).• Facing price q, the consumer demands quantity D(p) of good

x as q = p + t.• Let εD denote the price elasticity of demand:

εD =∂D(p)/D(p)

∂p/q .

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• Firms are price-takers.• Use c(S) units of y to produce S units of good x , with c ′(S) >0 and c ′′(S) ≥ 0.

• Firms choose supply S in order to maximize profit at pretaxprice p:

max pS − c(S)⇒ p = C ′ (S(p)) .

• Let εS denote the price elasticity of supply:

εS =∂S(p)/S(p)

∂p/p .

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Equilibrium

• Equilibrium condition:

S(p∗) = D(p∗ + t),

implicitly defines p∗(t).• The objective is to characterize ∂p/∂t and ∂q/∂t, the effectsof a tax increase on unit revenue from firms and unit cost forconsumers.

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• Implicitly differentiate the equilibrium condition with respect tot and p:

∂S∂p dp =

∂D∂p dp +

∂D∂t dt.

• This yields:dpdt =

∂D∂p

1∂S∂p −

∂D∂p,

or:dpdt =

εDεS − εD

,

with −1 < dpdt < 0.

• The change in unit cost for consumers is :

dqdt = 1+ dp

dt =εS

εS − εD.

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Who’s bearing the burden of the tax ?• Consumers bear the entire burden when:

• εD = 0, i.e. the demand is inelastic (day-to-day demand forgas);

• εS = +∞, i.e. the supply is perfectly elastic (perfectly compet-itive industry).

• Producers bear the entire burden when:• εS = 0, i.e. the supply is inelastic (short term fixed supply, e.g.

housing);• εD = −∞, the demand is perfectly elastic (there exists a close

substitute and demand shifts to it when price increases).

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Perfectly inelastic demand

Quantity

e

0

D(p)

S(p)

Q∗

p∗

S ′(p + t)

p′

Consumers’ loss

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Perfectly elastic supply (1)

Quantity

e

0

S(p)

D(p)

Q∗

p∗

S ′(p + t)p′

Q′

Consumers’ loss

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Perfectly elastic supply (2)

Quantity

e

0

S(p)

D(p)

Q∗

p∗

D′(p + t)

Q′

Consumers’ loss

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Key findings

• Equilibrium is independent of who nominally pays the tax.• Less elastic side of the market bears relatively more the taxationburden.

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Empirical evidence (1)

Joseph J. Doyle Jr. & Krislert Samphantharak, 2008. “$2.00 Gas! Studying theeffects of a gas tax moratorium,” Journal of Public Economics, Elsevier, vol. 92(3-4),pages 869-884, April.

• Question: Who bears the burden of the gas tax?• Context: Gas prices spike above $2 in 2000, near election.• Political desire to provide tax relief.• Temporary reform:

• 5% sales tax on gas was suspended between July 1st and October30th in Indiana and Illinois.

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• Identification relies on difference in differences: compares treatedstates to neighboring states.

• Observations at the station level (s).• Baseline econometric specification:

Pricest = α1 (Illinois or Indiana)s + α2 (Reform period)t+α3 (Illinois or Indiana)s × (Reform period)t + . . .

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Difference in (log) prices against time.Source: Doyle and Samphantharak (2008)

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Main findings:• 70% of tax reductions is passed to consumers in the form oflower prices.

• 80% to 100% of tax reinstatements is passed to consumers onthe from of higher prices.

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Empirical evidence (2)

Gabrielle Fack, 2005. “Pourquoi les ménages pauvres paient-ils des loyers de plus enplus élevés ?,” Économie et Statistique, Programme National Persée, vol. 381(1),pages 17-40.

• Question: Who are the beneficiaries of housing subsidies?• Context: Housing subsidies in France from 1973 to 2002.• Political desire to help less privileged households.• Gradual extension of the covered population and sharp exten-sion from 1991 to 1993:

• Difference-in-differences: Comparing new beneficiaries to thosethat still don’t meet eligibility criteria.

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Share of beneficiaries by income quartile.Source: Fack (2005)

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1st to 2d quartile differences in subsidies per square meter (black squares) and in rentper square meter (grey diamonds).Source: Fack (2005)

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Main findings:• About 80% of housing subsidies translate into higher prices.• So, only about 20% really contributed to lowering housing costsface by households.

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Tax salience

Tax salience

• Central assumption of many models (including the previousanalysis): Taxes are equivalent to prices, that is:

dDdt =

dDdp .

• In practice, taxes may have different effects on demand whetherpeople are aware or not.

• Tax salience:Tax a is more salient than tax b if calculating the

gross-of-tax price under a requires less computationthan calculating gross-of-tax price under b.

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Tax salience

Raj Chetty & Adam Looney & Kory Kroft, 2009. “Salience and Taxation: Theory andEvidence,” American Economic Review, American Economic Association, vol. 99(4),pages 1145-77, September.

• Test whether salience (visibility of tax-inclusive price) affectsbehavioral responses to commodity taxation. In other terms,does the effect of a tax on demand depend on whether it isincluded in the posted price?

• Field experiment: change salience of tax implemented at a su-permarket belonging to a major grocery chain.

• Data: weekly price and quantity sold by product.

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Tax salience

Source: Chetty, Looney and Kroft (2009)

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Tax salience

Experimental difference in differences:• Treatment group:Cosmetics, deodorants and hair care accessories in one largestore in northern California during three weeks in 2006.

• Control groups:• Other products from the same store and in same category (e.g.

toothpaste, skin care, shave);• Same products in two nearby stores.

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Tax salience

Source: Chetty, Looney and Kroft (2009)33 / 59


Tax salience

• Key finding: salience matters.• Additional findings (not shown here): price changes and taxchanges have different effects.

• All in all, the change in demand is larger the more salient thetax.

• Conclusive findings: taxes on producers have greater incidenceon producers than non-salient taxes levied on consumers.

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General equilibrium incidence

General equilibrium incidence

• Trace incidence back to original owners of production factors.• Benchmark static analysis by Harberger (1962).

Arnold C. Harberger, 1962. “The Incidence of the Corporation Income Tax,” Journalof Political Economy, University of Chicago Press, vol. 70, pages 215.

• Short-run analysis of a closed economy: fixed total supply oflabor L and capital K .

• Full employment of L and K .• Perfect competition.• Only two production sectors.• Yet, heavy analytical derivation.• Hard to predict anything precisely: results depend on elasticitiesof demand (preferences), of substitution, etc.

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Public Economics - Lecture 4: Taxation of commoditiesOptimal commodity taxation

1 Introduction

2 Tax incidence

3 Optimal commodity taxationInverse elasticity ruleRamsey ruleProduction efficiency

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• Goal is to maximize social welfare subject to revenue constraint.• First best:

• Suppose we have perfect information, complete markets, perfectcompetition, and lump sum taxes feasible (at no cost).

• Second welfare theorem implies that any Pareto-efficient alloca-tion can be achieved as a competitive equilibrium with appro-priate lump-sum transfers (or taxes).

• Economic policy problem reduces to the computation of thelump-sum taxes necessary to reach the desired equilibrium. Noequity-efficiency trade-off.

• Problems:• No way to make people reveal their characteristics at no cost:

to avoid paying a high lump-sum, a skilled person would pretendto be unskilled.

• Government has to set (distortionary) taxes as a function ofeconomic outcomes: income, property, consumption of goods.

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• End up with second best inefficient taxation:Its impossible to redistribute or raise revenue for public goodprovision without generating efficiency costs.

• Here, discuss optimal commodity taxation.• Main (qualitative) results in optimal commodity taxation the-ory:

• Inverse elasticity rule;• Ramsey rule;• Diamond and Mirrlees production efficiency.

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Inverse elasticity rule


• Simplifying assumption:Goods’ demands are independent from each other, i.e. thereare no cross-price effects between goods.

• Representative consumer utility function:

U(l , x1, x2, . . . , xN),

where l is labor supply and xi is the demand for good i .• Budget constraint:

N∑i=1

qixi = l ,

where qi = ti + pi is the price face by consumers (ti is the taxand pi is producers’ unit revenue).

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• Consumer’s optimal choice:

∀i , ∂U∂xi

= αqi , and∂U∂l = −α,

where α is marginal utility of income.

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• Government revenue is:

R =N∑

i=1tixi .

Which can be rewritten as:N∑

i=1qixi = R +

N∑i=1

pixi ,

where R is the total amount of money needed by the govern-ment.

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• Government’s program:

maxxi

L = U(l , xi) + λ

[ N∑i=1

qixi − R −N∑

i=1pixi

],

with:

l =N∑

i=1qixi .

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• First order condition with respect to xi :

∂U∂xi

+∂U∂l

[qi + xi

∂qi∂xi

]+ λ

[qi + xi

∂qi∂xi− pi

]= 0.

• Using solutions from the consumer’s optimal choice, we get:

(λ− α)xi∂qi∂xi

+ λ(qi − pi) = 0

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• Since εi =∂xi∂qi

qixi, we get the inverse elasticity rule:

∀i , tipi + ti

= −λ− αλ

1εi,

where α is marginal utility of income for consumers and λ isthe marginal utility cost of government revenue. Since taxesare distortionary, λ > α and ti > 0.

• Rewriting:∀i , pi + ti

ti

1εi

= − λ

λ− α.

• This rule means that the tax on good i should be inverselyrelated to the price elasticity of this good.

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Ramsey rule

Ramsey rule

• Ramsey (1927) tax problem:The government set taxes on uses of income in order to raiserevenue and minimize social loss; which goods should be taxed?

• Key assumptions:• Lump-sum taxation is not available.• Leisure cannot be taxed.• Production prices are fixed and normalized to one:

pi = 1 and qi = 1+ τi .

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Ramsey rule

Setup of the model

• N goods indexed by i = 1, . . . ,N.• Mass one of individuals who maximize utility

u (x1, . . . , xN , l) ,

subject to budget constraint

N∑i=1

qixi ≤ wl + Z ,

where qi is consumption price of good i , wl is labor income andZ is exogenous wealth.

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Ramsey rule

Consumer’s problem

• Using Lagrangian multiplier α, first order conditions can bewritten as:

∂u∂xi

= αqi ∀i , and∂u∂l = αw .

• These conditions yield implicit demand functions

xi (q,Z ) ,

and indirect utility function

V (q,Z ) ,

where q = (w , q1, . . . , qN).

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Ramsey rule

Government’s problem

• Given consumer’s behavior, the government chooses taxes inorder to maximize V (q,Z ), subject to revenue requirement

N∑i=1

τixi (q,Z ) ≥ E

• Lagrange expression for the government can be written as:

L = V (q,Z ) + λ

( N∑i=1

τixi (q,Z )− E).

• Remember that for all i , qi = 1+ τi , such that to choose τi isidentical to choose qi .

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Ramsey rule

• First order condition for all i :

∂L∂qi

=∂V∂qi︸︷︷︸

Private welfare loss

+λ xi︸︷︷︸Mechanical effect

+λN∑

j=1τj∂xj∂qi︸︷︷︸

Behavioral effect

= 0.

• Using Roy’s identity(

∂V∂qi

= −αxi), we get:

(λ− α) xi + λN∑

j=1τj∂xj∂qi

= 0.

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Ramsey rule

• Optimal tax rates satisfy a system of N equations and N un-knowns of the form:

N∑j=1

τj∂xj∂qi

= −xiλ(λ− α) .

• Hard to interpret.

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Ramsey rule

• Let us define

θ = λ− α− λ∂∑N

j=1 τjxj

∂Z .

• θ measures the value for the government of introducing 1 elump-sum tax:

• Direct value for the government is λ;• Private loss for individuals is α;• Loss in tax revenue due to the behavioral response is the re-

maining term.

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Ramsey rule

• Remember Slutsky equation :∂xj∂qi

=∂hj∂qi− xi

∂xj∂Z ,

where hj(.) is Hicksian demand function for good j .• After substituting and rearranging terms, we get a nice Ramseyrule:

∀i , 1xi

N∑j=1

τj∂hi∂qj

= − θλ.

• The optimal tax system should be such that the compensateddemand for each good is reduced in the same proportion relativeto the no-tax situation.

• That is, one should limit the distortion in terms of quantitiesrather than prices.

• Should tax heavier goods whose demand is less responsive toprice changes.

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Ramsey rule

Limitations of Ramsey rule

• Redistributive motives are not taken into account.• Necessities are more inelastic than luxuries.• Optimal Ramsey tax system is likely to be regressive.• Diamond (1975) extends Ramsey model to take redistributivemotives into account.

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Ramsey rule

Diamond, P. A., 1975. “A many-person Ramsey tax rule,” Journal of PublicEconomics, Elsevier, vol. 4(4), pages 335-342, November.

• Ramsey model where individuals differ in endowments and wherethe government seeks to maximize the sum of individual utili-ties.

• Optimal tax is still inversely proportional to the elasticity.• But tax rate for goods consumed by the poorest is shifted down,whereas tax rate for goods consumed by the rich is shifted up.

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Production efficiency


• Previous analysis ignored production by assuming that producerprices are fixed.

Diamond, Peter A & Mirrlees, James A, 1971. “Optimal Taxation and PublicProduction I: Production Efficiency” American Economic Review, American EconomicAssociation, vol. 61(1), pages 8-27, March.Diamond, Peter A & Mirrlees, James A, 1971. “Optimal Taxation and PublicProduction II: Tax Rules,” American Economic Review, American EconomicAssociation, vol. 61(3), pages 261-78, June.

• Relax this assumption and model production.• Derive optimal tax policy to achieve production efficiency.

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Setup of the model

• Many consumers, indexed by h = 1, . . . ,H.• Many goods, indexed by i = 1, . . . ,N.• Production prices are not constant. The production possibilitiesof the economy are represented by a production set.

• Key assumption: firms’ profits do not enter the social welfarefunction (fully taxed profits or production functions with con-stant return to scale).

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• The government chooses the vector q = p + τ in order tomaximize

W(V 1(q), . . . ,V H(q)

)s.t.

N∑i=1

τiXi(q) ≥ E ,

where Xi is the sum of individual demands for good i givenafter tax prices q.

• The constraint can be replaced by

X (q) =H∑

h=1xh(q) ∈ Y ,

where Y is the production set which takes into account therevenue requirement E of the government.

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Results and consequences

Results:• Production efficiency result: at the optimum level of taxes q∗,the allocation X (q∗) is on the boundary of Y .

• Optimal tax system à la Diamond (1975).Consequences:

• Public sector should be efficient:• Should face the same prices as the private sector;• Should choose production with the unique goal of maximizing

profits, not generating government revenues.• Intermediate goods (neither direct inputs, nor outputs con-sumed by individuals) should not been taxed:

• Taxing transactions between firms would distort (aggregate)production and prevent production efficiency.

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Comments:• Results rely on two key assumptions:

• Government needs to be able to set a differentiated tax rate foreach input or output;

• Government needs to be able to tax away fully pure profits.• These two assumptions effectively separate the production’sproblem from the the consumption’s one.

• These assumptions may be challenged.

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End of lecture.

Lectures of this course are inspired from those taught by R. Chetty,G. Fields, N. Gravel, H. Hoynes, and E. Saez.

Documents

Public Economics - Lecture 4: Taxation of commoditiesmarcsangnier.free.fr/.../ecopub_m1_2014-2015_lec4.pdf · 2014-2015,Springsemester AixMarseilleSchoolofEconomics. PublicEconomics-Lecture4: