Optimal Linear Taxation of Positional Goods

Optimal Linear Taxation

of Positional Goods∗

Daniel Samano†

University of Minnesota

First draft: November 8, 2008

This draft: April 24, 2009

Abstract

In this article I extend the Mirrlees (1971) framework to incorporate positional goods,

that is, goods that are valued relative to other agents’ consumption of the same good. Con-

strained efficient allocations can be implemented through a non-linear consumption tax on

the positional good together with a marginal income tax with standard Mirrleesian proper-

ties, namely, no distortions at the extremes. The non-linearity in the former tax arises as

I allow for individual specific positional externalities. Due to arbitrage opportunities that

arise with the introduction of a non-linear consumption tax, I restrict this instrument to be

linear. My numerical calculations indicate that the aggregate welfare losses of preventing

arbitrage are very small, nevertheless, large distributional effects occur. For instance, when

the positional externality is increasing in income, individuals at the high end of the income

distribution experience large gains since for them, a flat tax effectively reduces the after tax

price of positional goods. This generates a positive income effect that cannot be fully offset

by increases in the marginal labor income tax as optimality requires no distortions at the top,

thus higher consumption occurs. Conversely, individuals at the bottom of the distribution

experience losses since they effectively face a higher after tax price of the positional good

∗I thank my advisor Narayana Kocherlakota for his excellent guidance on the elaboration of this paper.I also thank Chris Phelan, Ina Simonovska, Hakki Yazici, and all participants of the Public Economics andPolicy Workshop at the University of Minnesota for helpful comments. All errors and omissions are mine.

†University of Minnesota. Contact: Address: 4-101 Hanson Hall, 1925 Fourth Street South, Minneapolis,MN 55455. Email: [email protected]

1

and consequently, a negative income effect. In this case, a marginal income tax reduction

cannot offset such income effect due to incentive problems. Both effects are reduced when

preferences over positional goods are non-homothetic as the income effect of price changes

can be outweighed more effectively by adjustments in the marginal income tax.

1 Introduction

More that a hundred years ago Veblen (1899) coined the term conspicuous consumption to

refer to the consumption incurred by individuals primarily with the goal of attaining status

or social position. In modern capitalist societies at least some luxury goods posses this

characteristic. Why is it that people are willing to pay fortunes to own a mansion in Beverly

Hills, drive a brand new German convertible, have access to exclusive country clubs? No

one can deny the intrinsic value derived from the consumption of those goods; however those

purchases may be also motivated, at least partially, for positional considerations. In other

words, for the status that such goods confers to the buyers.

By definition, status is a social ranking or standing. To the extend that agents preferences

exhibit utility interdependence, the consumption of positional goods, such as luxuries, might

impose a negative externality to the society. In other words, if agents preferences are sensitive

to a ranking based on the consumption of a particular set of goods, the consumption of the

“Joneses” may be harmful. Government intervention to correct this type of externalities

may be desirable but for sure debatable. Some articles such as Frank (2005) have recently

exposed arguments in favor of policy targeting this type of externalities. Among them, the

author claims “tax cuts for the wealthy are spent largely on positional goods. Dollars that

could have been used to pay for additional non positional goods have been spent instead on

larger houses and more expensive cars”.

The goal of this article is to conduct a formal normative analysis of taxation under the

presence of consumption goods that generate positional externalities. I make my analysis in

a framework similar to the one in Mirrlees (1971). Agents in my model are endowed with

heterogeneous privately-known productivity. As is well known, these models capture a con-

flict between redistribution and incentives which results in an endogenous non-degenerate

consumption distribution. As individuals preferences include relative consumption or posi-

tional considerations, consumption inequality becomes harmful and government intervention

a la Pigou is desirable. In the analysis below, I parametrize the strength of positional con-

siderations and analyze optimal tax policy.

2

Not surprisingly, constrained efficient allocations in this environment exhibit a wegde be-

tween non positional (for instance, necessities) and positional (for instance, luxuries) goods.

The literature consensus however, is that taxing luxuries is not efficient. Using the Ramsey

approach to optimal taxation, Atkinson and Stiglitz (1972) shows that it is optimal to tax

goods with low income elasticities rather than high. Thus, necessities must be taxed higher

than luxuries. A uniform commodity taxation result was obtained in Atkinson and Stiglitz

(1976) under a framework with heterogeneous agents like the one analyzed in Mirrlees (1971).

Remarkably, only the assumption of separability between consumption goods and leisure

is needed to derive the latter result. Why is it that in this economic environment tax-

ing positional goods such as luxuries is optimal? In the model that I present, taxation

of the positional good occurs due to Pigouvian considerations. Thus, this instrument cor-

rects over-consumption of a good that generates positional externalities. As preferences in

this model display no utility interdependence, the uniform commodity taxation holds as in

Atkinson and Stiglitz (1976) and all redistribution should be carried out through the labor

income tax.

Under no restrictions on the class of taxes that can be used to implement optimal alloca-

tions, the presence of positional considerations implies that constrained efficient allocations

can be implemented through a non-linear tax on the positional good in combination with

a non-linear labor income tax with standard Mirrleesian properties, namely no distortions

at the extremes. The non-linearity in the consumption of the positional good or “luxury

tax” is driven by the fact that I allow agents to contribute to the positional externality in

an arbitrary way. This is in line with the findings of Samano (2008) which finds that a pro-

gressive labor income tax may be partially rationalized as a Pigouvian one whose role is to

correct consumption externalities. The estimations of the previous paper suggest that such

externality may be increasing in income. The previous implementation however is subject to

arbitrage opportunities across consumption goods since agents face a non-linear consump-

tion tax on positional goods. Thus, a non-arbitrage constraint is imposed by equalizing

the marginal rate of substitution between the positional and the non positional good across

agents. I show that the resulting double constrained efficient allocations can be implemented

through a linear positional tax together with a non-linear labor income tax. The non-linear

income tax that implements double constrained efficient allocations differs from the one im-

plementing constrained efficient ones as the former must offset income effects produced by

the flattening in the “luxury tax”.

Numerical calculations indicate that the aggregate welfare losses of preventing arbitrage

3

are very small, nevertheless, large distributional effects occur. When the positional external-

ity is increasing in income, individuals at the high end of the income distribution experience

large gains since for them, a flat tax effectively reduces the after tax price of positional goods.

This generates a positive income effect that cannot be fully offset by increases in the labor

income tax as optimality requires no distortions at the top, thus the consumption of highly

skilled individuals increases. Conversely, individuals at the bottom of the skills distribution

experience losses since they effectively face a higher after tax price of the positional good

and consequently, a negative income effect. In this case, a marginal income tax reduction

cannot offset such income effect due to incentive problems. Both effects are reduced when

preferences over positional goods are non-homothetic as price changes can be offset by small

adjustments in the marginal income tax. My results suggest that the effectiveness of a lin-

ear “luxury tax” correcting positional externalities would crucially depend on the degree of

non-homotheticity in preferences over positional and non positional goods.

The rest of the paper proceeds as follows. Section 2 presents the model and shows the

characterization of constrained efficient allocations. Section 3 presents the characterization

and one implementation of double constrained efficient allocations. Section 4 presents calcu-

lations of the endogenous distributions and optimal taxes for a parametrized version of the

model. Finally, section 5 concludes.

2 The Model

Consider a static economy populated by a continuum of agents with heterogeneous produc-

tivity or skill. Let θ ∈ Θ, where Θ ≡ [θ, θ] and 0 < θ < θ <∞, be individual’s productivity

distributed according to the density f : Θ → R++. Productivity is privately known to each

agent. An agent with productivity θ has a utility function of the form

U(cn, cl, y, C; θ) = u(cn, cl) − αC − v(y

θ

)

, α ∈ [0, 1)

where cn is a necessity, cl is a luxury good and y is effective output.1 Moreover, let

C ≡

∫

Θ

[ωcn(θ) + (1 − ω)cl(θ)]ψ(θ)dθ, ω ∈ [0, 1/2) (1)

be society’s endogenous consumption benchmark specified as a weighted average of necessities

and luxuries. As usual, preferences satisfy ucn > 0, ucl > 0, u(·) is jointly strictly concave and

1As standard in this literature, I define effective labor as y = θl where l is the amount of time worked.

4

v(·) is a convex function. Also, observe that according to the previous utility specification,

uC = −α, thus, following the terminology of Dupor and Liu (2003), agents exhibit jealousy.

Notice that with the assumption that ω < 1/2, I capture the notion that luxuries provoke

more jealousy than necessities as claimed by Frank (2008).2 In other words, luxuries are

more positional than necessities. Obviously, when ω = 0, only luxuries are positional.

An allocation in this economy is {cn(θ), cl(θ), y(θ)}θ∈Θ, where cn : Θ → R+, cl : Θ →

R+ and y : Θ → R+. Abstracting from government expenditure, I define an allocation

{cn(θ), cl(θ), y(θ)}θ∈Θ to be feasible if

∫

Θ

cn(θ)f(θ)dθ +

∫

Θ

cl(θ)f(θ)dθ =

∫

Θ

y(θ)f(θ)dθ (2)

Observe that in the previous definition I am assuming that both consumption goods are

substitutes in production. This assumption is made for simplicity. A reporting strategy is

a mapping σ : Θ → Θ, where σ(θ) represents the skill announced by an agents with skill θ

in a direct revelation game. Thus, making use of the Revelation Principle, an allocation is

incentive compatible if

u(cn(θ), cl(θ))− αC − v

(

y(θ)

θ

)

≥ u(cn(σ(θ)), cl(σ(θ)))− αC − v

(

y(σ(θ))

θ

)

∀θ, σ(θ) ∈ Θ

(3)

Observe that since C cannot be affected unilaterally by a single agent, it is not a function

of θ. An allocation that is incentive compatible and feasible is said to be incentive-feasible.

Finally, let g : Θ → R+ be the density according to which individuals are weighted by the

benevolent planner.

Definition 1. A constrained efficient allocation is an allocation {cspn (θ), cspl (θ), ysp(θ)}θ∈Θ

that maximizes the following planner problem

∫

Θ

[

u(cn(θ), cl(θ)) − αC − v

(

y(θ)

θ

)]

g(θ)dθ (4)

subject to {cn(θ), cl(θ), y(θ)}θ∈Θ being incentive-feasible and cn(θ), cl(θ), y(θ) ≥ 0∀θ ∈ Θ.

2Empirical evidence of this fact is also presented in Carlsson, Johansson-Stenman, and Martinsson (2007)and Solnick and Hemenway (2005).

5

2.1 Characterization of Constrained Efficient Allocations

The following proposition states the necessary conditions that any interior constrained effi-

cient allocation must satisfy. Let ǫsp(θ) ≡v′(

ysp(θ)θ

)

v′′(ysp(θ)θ

)ysp(θ)θ

.

Proposition 1. Any interior constrained efficient allocation {cspn (θ), cspl (θ), ysp(θ)}θ∈Θ must

be incentive-feasible and satisfy

ucn(cspn (θ), cspl (θ))

v′(ysp(θ)θ

)1θ

− 1 =α

λ

ωψ(θ)

f(θ)+ucn(c

spn (θ), cspl (θ))

θf(θ)

[

1 +1

ǫsp(θ)

]

Isp(θ) ∀θ ∈ Θ (5)

ucl(cspn (θ), cspl (θ))

ucn(cspn (θ), cspl (θ))

=1 + α(1−ω)

λψ(θ)f(θ)

1 + αωλψ(θ)f(θ)

∀θ ∈ Θ (6)

where

λ =1 − ωα

∫

Θψ(θ)

ucn(cspn (θ),cspl

(θ))dθ

∫

Θf(θ)

ucn (cspn (θ),cspl

(θ))dθ

(7)

Isp(θ) ≡

∫ θ

θ

[

g(t)

λ−

f(t)

ucn(cspn (t), cspl (t))

−α

λ

ωψ(t)

ucn(cspn (t), cspl (t))

]

dt ∀θ ∈ Θ (8)

Proof. See Appendix A.

According to Proposition 1, the marginal rate of substitution (MRS) between the neces-

sity and the luxury varies across agents if ψ(θ) is different to f(θ) as observed in expression

6. For the sake of exposition, consider the case where ω = 0, that is, only the luxury good

generates positional externalities. Moreover, assume that the ratio ψ(θ)f(θ)

is strictly increasing,

that is, the consumption of more affluent individuals becomes more harmful for the society

as a whole. In that case, it is optimal to have an increasing MRS between the luxury and the

necessity as income goes up. The previous argument breaks in two cases: either when ψ(θ)

equals f(θ) and when α = 0. In both cases, the MRS across consumption goods is constant

across agents. In the second case, under no utility interdependence, the MRS across goods

equals one (marginal rate of transformation given the assumed technology) so the uniform

commodity taxation result holds.

From a theoretical point of view, the case where the MRS is non-constant across agents

is more challenging to analyze. The reason is that if the planner cannot observe agents’

consumption, individuals could meet in a re-trading markets after being assigned their con-

sumption bundle and exchange consumption goods at a given price. In this re-trading

6

market, they all would end up equalizing their MRS between the luxury and the necessity

to an equilibrium relative price. To expose this better, suppose that the non-constant wegde

were to be implemented by a non-linear tax on the consumption of the positional good or

“luxury tax”. Then, individuals could enter into “non-exclusive” arrangements to exchange

luxuries for necessities until all agents equalized their MRS.3 In order to take into account

the previous fact, we need to refine the notion of constrained efficiency in this environment.

Before formally stating this, we need a few definitions.

3 Double Constrained Efficient Allocations

I start by posing the agent’s problem in the re-trading market contingent on having an-

nounced being of σ(θ) type.

3.1 Agents’s Problem

Given allocations {cn(θ), cl(θ), y(θ)}θ∈Θ and price q, an agent who decides to report σ(θ)

attains utility

V ({cn(θ), cl(θ), y(θ)}θ∈Θ, q | σ(θ)) = maxxn(σ(θ)),xl(σ(θ))

u(xn(σ(θ)), xl(σ(θ)))−αC− v

(

y(σ(θ))

θ

)

(9)

s.t.

xn(σ(θ)) + qxl(σ(θ)) ≤ cn(σ(θ)) + qcl(σ(θ))

xn(σ(θ)), xl(σ(θ)) ≥ 0

where xn((σ(θ)) and xl((σ(θ)) is the private consumption of the necessity and the luxury

respectively and q is the relative price of the luxury good. Moreover, I define

V ({cn(θ), cl(θ), y(θ)}θ∈Θ, q) ≡ maxσ(θ)∈Θ

V ({cn(θ), cl(θ), y(θ)}θ∈Θ, q | σ(θ)) (10)

which represents the utility level attained by optimally announcing σ(θ), given {cn(θ), cl(θ), y(θ)}θ∈Θ

and q.

3The term “non-exclusivity” is used to emphasize that agents are not constrained to trade with one singlepartner.

7

3.2 Equilibrium in the Re-Trading Market

An equilibrium in the re-trading market is strategies {σ(θ)}θ∈Θ, allocations {xn(θ), xl(θ)}θ∈Θ

and a price q such that

i) Taking as given {cn(θ), cl(θ), y(θ)}θ∈Θ and q, agents solve (9) and (10),

ii) Re-trading market clears

∫

Θ

xn(σ(θ))f(θ)dθ =

∫

Θ

cn(θ)f(θ)dθ

∫

Θ

xl(σ(θ))f(θ)dθ =

∫

Θ

cl(θ)f(θ)dθ (11)

Let V ({cn(θ), cl(θ), y(θ)}θ∈Θ) equals V ({cn(θ), cl(θ), y(θ)}θ∈Θ, q) where q is the equilib-

rium price. Given the previous definitions, we are in a position to define efficiency in this

environment.

Definition 2. A double constrained efficient allocation is an allocation {c∗n(θ), c∗l (θ), y

∗(θ)}θ∈Θ

that maximizes the following planner problem

∫

Θ

[


(

y(θ)

θ

)]

g(θ)dθ (12)

s. t.


(

y(θ)

θ

)

≥ V ({cn(θ), cl(θ), y(θ)}θ∈Θ) (13)

∫

Θ

cn(θ)f(θ)dθ +

∫

Θ

cl(θ)f(θ)dθ =

∫

Θ

y(θ)f(θ)dθ (14)

and cn(θ), cl(θ), y(θ) ≥ 0 ∀θ ∈ Θ.

Notice that the previous notion of constrained efficiency takes explicitly into account the

re-trading market for consumption goods across agents as a constraint. Lemma 1 establishes

an equivalence statement of the problem stated in Definition 2.

Lemma 1. A double constrained efficient allocation {c∗n(θ), c∗l (θ), y

∗(θ)}θ∈Θ together with the

equilibrium relative price of luxuries q is a solution to the planner’s problem

maxcn(·),cl(·),y(·),q

∫

Θ

[


(

y(θ)

θ

)]

g(θ)dθ (15)

s. t.

8

ucl(cn(θ), cl(θ))

ucn(cn(θ), cl(θ))= q ∀θ ∈ Θ (16)


(

y(θ)

θ

)

≥ V ({cn(θ), cl(θ), y(θ)}θ∈Θ, q | σ(θ)) ∀σ(θ) 6= θ (17)

∫

Θ

cn(θ)f(θ)dθ +

∫

Θ

cl(θ)f(θ)dθ =

∫

Θ

y(θ)f(θ)dθ (18)

and cn(θ), cl(θ), y(θ) ≥ 0 ∀θ ∈ Θ.

Proof. The proof follows closely da Costa (2009). Let q be the equilibrium relative price

associated with the allocation {cn(θ), cl(θ), y(θ)}θ∈Θ and assume that (13) is satisfied. If

(16) is violated, there is an alternative consumption choice that increases the agents utility

holding strategy σ(θ) = θ fixed. Because q is an equilibrium price, this violates (13). To see

that (17) holds observe that V ({cn(θ), cl(θ), y(θ)}θ∈Θ) is simply (10) at equilibrium price.

Hence, there is no strategy σ(θ) that yields higher utility that truth-telling.

Now assume (16) and (17) are satisfied. When the price is q, agents find in their best

interest to reveal their types truthfully, according to (17). This means that there is no

strategy σ(θ) combined with optimal re-trading that increases utility for the agents at that

price. Equation (16) implies that no-trade is optimal for the agent at the very same price

q. No-trade trivially satisfies (11) which guarantees that q is indeed an equilibrium price.

Therefore, (17) implies (13).

Lemma 1 states that if individual allocations are such that the MRS between consumption

goods is constant across agents, in equilibrium, re-trading does not occur. Observe that the

optimal MRS between luxuries and necessities is yet to be determined.

Theorem 1. A double constrained efficient allocation {c∗n(θ), c∗l (θ), y

∗(θ)}θ∈Θ together with

the equilibrium relative price of luxuries q solve the following planner problem

maxcn(·),cl(·),y(·),q

∫

Θ

[


(

y(θ)

θ

)]

g(θ)dθ (19)

s. t.

ucl(cn(θ), cl(θ))

ucn(cn(θ), cl(θ))= q ∀θ ∈ Θ, (20)


(

y(θ)

θ

)

≥ u(cn(σ(θ)), cl(σ(θ))) − αC − v

(

y(σ(θ))

θ

)

∀θ, σ(θ) ∈ Θ

(21)

9

∫

Θ

cn(θ)f(θ)dθ +

∫

Θ

cl(θ)f(θ)dθ =

∫

Θ

y(θ)f(θ)dθ (22)

and cn(θ), cl(θ), y(θ) ≥ 0 ∀θ ∈ Θ.

Proof. Suppose that a deviating agent announces σ(θ) = θ′. Hence, she receives cn(θ′), cl(θ

′).

Suppose she re-trades, thus [xn(θ′), xl(θ

′)] 6= [cn(θ′), cl(θ

′)]. However, truthful agent θ′

and the deviating agent impersonating θ′ who reported σ(θ) = θ′ have the same income

in the re-trading market. By strict concavity of preferences it must be the case that

[xn(θ′), xl(θ

′)] = [cn(θ′), cl(θ

′)], otherwise there would exist another bundle strictly preferred

to [cn(θ′), cl(θ

′)], implying that the previous bundle is not optimal for truthful agent θ′, thus

we arrive to a contradiction. The previous implies that agents do not re-trade in equilib-

rium. Having established the previous fact, it follows that V ({cn(θ), cl(θ), y(θ)}, q | σ(θ)) =

u(cn(σ(θ)), cl(σ(θ)))−αC−v(

y(σ(θ))θ

)

which together with (21) implies that (17) is satisfied.

The other side of the proof is trivially satisfied as (17) considers the maximum of σ(θ).

3.3 Characterization of Double Constrained Efficient Allocations

The following proposition states the necessary conditions that any interior double con-

strained efficient allocation must satisfy. Let ǫ∗(θ) ≡v′( y

∗(θ)θ

)

v′′( y∗(θ)θ

) y∗(θ)θ

.

Proposition 2. Any interior double constrained efficient allocation {c∗n(θ), c∗l (θ), y

∗(θ)}θ∈Θ

must be incentive-feasible and satisfy

ucn(c∗n(θ), c

∗l (θ))

v′(y∗(θ)θ

)1θ

− 1 =α

λ

ωψ(θ)

f(θ)+ucn(c

∗n(θ), c

∗l (θ))

θf(θ)

[

1 +1

ǫ∗(θ)

]

I∗(θ) ∀θ ∈ Θ (23)

ucl(c∗n(θ), c

∗l (θ))

ucn(c∗n(θ), c

∗l (θ))

=

∫

Θf(θ)B∗(θ)

dθ + α(1−ω)λ

∫

Θψ(θ)B∗(θ)

dθ∫

Θf(θ)B∗(θ)

dθ + αωλ

∫

Θψ(θ)B∗(θ)

dθ∀θ ∈ Θ (24)

where

λ =1 − ωα

∫

Θψ(θ)

ucn(c∗n(θ),c∗l(θ))

dθ∫

Θf(θ)

ucn (c∗n(θ),c∗l(θ))

dθ(25)

B∗(θ) ≡ucncl(c

∗n(θ), c

∗l (θ))

ucn(c∗n(θ), c

∗l (θ))

−uclcl(c

∗n(θ), c

∗l (θ))

ucl(c∗n(θ), c

∗l (θ))

∀θ ∈ Θ (26)

I∗(θ) ≡

∫ θ

θ

[

g(t)

λ−

f(t)

ucn(c∗n(t), c

∗l (t))

−α

λ

ωψ(t)

ucn(c∗n(t), c

∗l (t))

]

dt ∀θ ∈ Θ (27)

10

Proof. See Appendix A.

3.4 Implementation of Double Constrained Efficient Allocations

Agents in this economy trade effective labor for consumption of the necessity and luxury.

There is a single firm that employs all agents. It produces one unit of output for every unit

of effective labor, y. Necessities and luxuries are perfect substitutes in production. Every

unit of effective labor receives a payment of w. Agent are also subject to an income tax

schedule T (y(θ)), assumed to be twice differentiable and to induce no bunching and a linear

tax on the luxury good τ . Without loss of generality, there are not taxes on the consumption

of the necessity cn. An agent with effective labor y pays T (y(θ)) of taxes.

Thus, taking as given T (y(θ)), τ , C and the wage w, the problem solved by the agent

with productivity θ, ∀θ ∈ Θ is

maxcn(θ),cl(θ),y(θ)


(

y(θ)

θ

)

(28)

s.t.

cn(θ) + (1 + τ)cl(θ) ≤ wy(θ)− T (y(θ))

cn(θ), cl(θ), y(θ) ≥ 0

Definition 3. Given a labor tax T (y(θ)), luxury tax τ and C, an equilibrium in this economy

is an allocation {ceqn (θ), ceql (θ), yeq(θ)}θ∈Θ and wage weq such that

i. (ceqn (θ), ceql (θ), yeq(θ)) solve (28) ∀θ ∈ Θ

ii. C =∫

Θ[ωceqn (θ) + (1 − ω)ceql (θ)]ψ(θ)dθ

iii. weq = 1

iv. Government balances its budget

∫

Θ

[T (yeq(θ)) + τceql (θ)]f(θ)dθ = 0

v.∫

Θceqn (θ)f(θ)dθ +

∫

Θceql (θ)f(θ)dθ =

∫

Θyeq(θ)f(θ)dθ

An allocation {cn(θ), cl(θ), y(θ)}θ∈Θ is implementable by the income tax T (y(θ)) and the

luxury tax τ if {cn(θ), cl(θ), y(θ)}θ∈Θ and w are an equilibrium.

11

3.5 Characterization of Optimal Income Tax and Linear Luxury

Tax

Define the following tax mechanism T : y → R,

T (y(θ)) =

{

y∗(θ) − c∗n(θ) − (1 + τ)c∗l (θ) if y(θ) = y∗(θ)

y(θ) otherwise.(29)

where

τ =

α(1−2ω)λ

∫

Θψ(θ)B∗(θ)

dθ∫

Θf(θ)B∗(θ)

dθ + αωλ

∫

Θψ(θ)B∗(θ)

dθ(30)

together with

T ′(y∗(θ))

1 − T ′(y∗(θ))=α

λ

ωψ(θ)

f(θ)+ucn(c

∗n(θ), c

∗l (θ))

θf(θ)

[

1 +1

ǫ∗(θ)

]

I∗(θ) (31)

if y(θ) = y∗(θ).

Proposition 3. Any optimal allocation {c∗n(θ), c∗l (θ), y

∗(θ)} can be implemented by an in-

come tax schedule T (y(θ)) defined by (29) and (31) and a flat tax τ on the positional good

satisfying (30).

Proof. In Appendix A.

Corollary 1 (Proposition 3). Suppose ω = 0, u(cn, cl) =[

ηc1− 1

σn + (1 − η)c

1− ρσ

l

]

σσ−1

, σ <

η, η ≤ 1, then the optimal marginal income tax and luxury tax satisfy

T ′(y∗(θ))

1 − T ′(y∗(θ))=ucn(c

∗n(θ), c

∗l (θ))

θf(θ)

[

1 +1

ǫ∗(θ)

]∫ θ

θ

[

g(t)

λ−

f(t)

ucn(c∗n(θ), c

∗l (θ))

]

dt (32)

τ =α

λ

∫

Θc∗l (θ)ψ(θ)dθ

∫

Θc∗l (θ)f(θ)dθ

(33)

where λ = 1R

Θf(θ)

ucn (c∗n(θ),c∗l(θ))

dθand ucn(c

∗n(θ), c

∗l (θ)) ≡

[

ηc∗n(θ)1− 1

σ + (1 − η)c∗l (θ)1− ρ

σ

]1

σ−1

ηc∗n(θ)− 1σ .

It is important to highlight that results of Corollary 1 hold with or without homothetic

preferences since the utility function (known as generalized elasticity of substitution and

12

introduced in Pakos (2006)) becomes homothetic when ρ = 1.4 In section 4, I compute

constrained and double constrained efficient allocations and implementing taxes using the

previous functional form. Corollaries 2 and 3 show expressions for optimal taxes under

another classes of preferences standard in the literature.

Corollary 2 (Proposition 3). Suppose ω = 0, u(cn, cl) = 11−σ

cn1−σ+ 1

1−ρcl

1−ρ, where σ, ρ > 0

then the optimal marginal income tax and luxury tax satisfy

T ′(y∗(θ))

1 − T ′(y∗(θ))=c∗n(θ)

−σ

θf(θ)

[

1 +1

ǫ∗(θ)

]∫ θ

θ

[

g(t)

λ−

f(t)

c∗n(t)−σ

]

dt (34)

τ =α

λ

∫

Θc∗l (θ)ψ(θ)dθ

∫

Θc∗l (θ)f(θ)dθ

(35)

where λ = 1R

Θc∗n(θ)σf(θ)dθ

.

Corollary 3 (Proposition 3). Suppose ω = 0, u(cn, cl) = 11−σ

cn1−σ − e−ρcl, where σ, ρ > 0

then the optimal marginal income tax and luxury tax satisfy

T ′(y∗(θ))

1 − T ′(y∗(θ))=c∗n(θ)

−σ

θf(θ)

[

1 +1

ǫ∗(θ)

]∫ θ

θ

[

g(t)

λ−

f(t)

c∗n(t)−σ

]

dt (36)

τ =α

λ(37)

where λ = 1R

Θc∗n(θ)σf(θ)dθ

.

4 Welfare Losses (and Gains!) Due to Linear Taxation

on Positional Goods

In this section I compute the model constrained efficient and double constrained efficient

allocations and evaluate the welfare in both environments. I also compute the taxes that

4To observe this, notice that

∂cl

∂y

y

cl=

[

(σ−ρ)(σ−1)

(1−η)η

]−σ

cρl + qcl

ρ[

(σ−ρ)(σ−1)

(1−η)η

]−σ

cρl + qcl

where q is the relative price of the luxury good in terms of the necessity. Thus, ∂cl

∂yycl

= 1 if ρ = 1 and∂cl

∂yycl

> 1 if ρ < 1. In the latter case, these preferences properly represent the good cl as a luxury.

13

implement those allocations. For this quantitative exercise, I assume that preferences are

represented by U(cn, cl, C, y; θ) ≡ u(cn, cl) − αC − 11+φ

(

yθ

)1+φ, φ > 0, where

u(cn, cl) =[

ηc1− 1

σn + (1 − η)c

1− ρσ

l

]

σσ−1

,

with ρ ≤ 1 and σ < 1 with σ < ρ. For ρ < 1, this specification of preferences exhibit

non-homotheticity. That is, as income of individuals goes up the share of disposable income

spent in the luxury good increases.5 As closely estimated by Pakos (2006), I set σ = 0.5 and

vary the parameter that measures the ratio of income elasticities between necessities and

luxuries, ρ, to take values in {0.80, 0.91, 1}.6 Recall that when ρ = 1, preferences exhibit

homotheticity. I set η = 0.75 and φ ∈ {0.5, 1.5, 3}.7 The latter implies an uncompensated

elasticity of labor supply of 2, 1/2 and 1/3 respectively. I assume that the support of the

distribution is Θ = [1,∞) although for computational purposes, I use the domain [1, 6]. I

consider two cases for the exogenous distributions of the model. In the first one, skills density

f(θ) is distributed according to a truncated exponential distribution with parameter λf = 1

whereas g(θ) and ψ(θ) follow the same distribution with parameters λg = 1 and λψ = 1.25

respectively.8 Notice that I assume that the planner is utilitarian since λg = λf . In the

second case, skills are distributed according to a Pareto distribution with parameter kf , kg

and kψ respectively. I set kf = kg = 1.8 and kψ = 1.08. As before, the planner is utilitarian.

These distributions are plotted in Figure 1.

I present my calculations for ω = 0, that is, in this world, only luxuries generate positional

externalities. I let the parameter α ∈ {0.05, 0.15}.9 Welfare losses are measured as in

Golosov and Tsyvinski (2007). That is, let

UspΘ ≡

∫

Θ

[

u(cspn (θ), cspl (θ)) − αCsp −1

1 + φ

(

ysp(θ)

θ

)1+φ]

g(θ)dθ,

5Ait-Sahalia, Parker, and Yogo (2004) also model consumption as a composite of necessities and luxuriesusing non-homothetic preferences.

6The value of ρ is also in line with the work of Costa (2001) who estimates income elasticities for foodand recreation in the U.S. up to 1994.

7In Appendix C I also present an exercise where φ = 0.2 which implies a uncompensated labor supplyelasticity of 5!

8Recall that the exponential distribution is f(θ) = λe−λθ, θ ∈ [0,∞). Thus, a lower tail truncatedexponential distribution at θ = θ is f t(θ) = 1

1−F (θ)f(θ) where F (θ) = 1 − e−λθ.9Samano (2008) obtains a value of α for aggregate consumption close to 0.15 using U.S. and U.K. income

data. Dynan and Ravina (2007) obtain values close to 0.10.

14

1 2 3 4 5 60

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

θ

dens

ity

f(θ) Paretoψ(θ) Paretof(θ) Exponentialψ(θ) Exponential

Figure 1: Skills and externality weighting distributions distributed Pareto and Exponential

thus I find the value λΘ such that

∫

Θ

[

u((1 + λΘ)c∗n(θ), (1 + λΘ)c∗l (θ)) − αC∗ −1

1 + φ

(

y∗(θ)

θ

)1+φ]

g(θ)dθ = UspΘ .

In other words, I find the percentage increase in aggregate consumption λΘ that would

deliver the same aggregate welfare in the double constrained efficient allocations than in the

constrained efficient ones. The subindex Θ in the parameter λ makes explicit that welfare

comparisons are made for all agents in the economy. To quantify non-aggregate welfare

effects, I also calculate the value of λL ≡ λ{[θ,θ]|F (θ)=0.90} and λH ≡ λ{[θ,θ]|F (θ=0.90)}.10

Tables 1 and 2 show the optimal linear positional or “luxury tax” rate and welfare losses

(and gains, whenever the variable λ is negative) for several parameters when skills are dis-

tributed according to an exponential distribution. Tables 3 and 4 show the same information

when skills are Pareto distributed. As expected, the double constrained environment delivers

lower aggregate welfare given that an extra constrained is imposed, namely, MRS between

10To be precise, λL is calculated as∫ θ

θ

[

u((1 + λL)c∗n(θ), (1 + λL)c∗l (θ)) − αC∗ − 11+φ

(

y∗(θ)θ

)1+φ]

g(θ)dθ =

UspL , where U

spL ≡

∫ θ

θ

[

u(cspn (θ), csp

l (θ)) − αCsp − 11+φ

(

ysp(θ)θ

)1+φ]

g(θ)dθ and F (θ) = 0.90. λH is calculated

similarly.

15

goods is constant across agents.

My calculations indicate that the aggregate welfare losses suffered by taxing the positional

good in a linear fashion are very low, particularly when the labor supply is reasonably

inelastic. Thus a linear “luxury tax” does almost as well as a non-linear one in terms of

aggregate welfare. Nevertheless, the previous policy has considerable distributional effects.

Assuming the positional externality is increasing in income, individuals at the high end of

the income distribution experience large gains under linear taxation of the positional good.

The reason is the following: a flat luxury tax effectively reduces the price of luxuries for rich

individuals. The drop in the price generates a positive income effect that cannot be offset

by an increase in the marginal income tax as optimality requires no distortion at the top.

Consequently, the consumption of agents at the top increases. When preferences are non-

homothetic the gains of highly skilled individuals are reduced, not to negligible levels though,

as the income effect generated by the drop in the price of luxuries is smaller. In this case,

a small adjustment in the labor income tax is necessary to exert output from individuals at

the high end of the income distribution. The opposite is true for individuals at the bottom

of the skills distribution. They experience considerable welfare losses since a flat luxury tax

increases the after tax price of luxuries. The previous generates a negative income effect

that is offset by a reduction in the marginal income tax. Such reduction cannot fully offset

the income effect since that would violate incentives. If preferences are non-homothetic, the

adjustment in the income tax is smaller than in the homothetic case.

Observe that for α = 0.15, the luxury tax is around 35% when the labor supply is

inelastic (φ = 3, 1.5) and reaches levels of around 40% when the labor supply is more

elastic (φ = 0.50). When positional considerations are weaker, α = 0.05, this tax is around

11%. The more elastic the labor supply, the higher the optimal luxury tax. Also, notice

that when the labor supply is inelastic, the aggregate welfare losses, represented by the

parameter λΘ are no higher than 0.16%. When labor supply becomes more elastic, this loss

increases up to 0.24%. Regarding non-aggregate welfare measures, when the labor supply is

inelastic (φ = 3, 1.5), individuals at the top decile of the skills distribution experience gains

between 0.61% and 6.67%. Those gains, however, diminish when preferences are highly non-

homothetic, the labor supply is elastic and positional concerns are weaker. Regarding welfare

changes of individuals below the tenth decile of the skills distribution, the welfare losses due

to linear taxation of positional goods are very high when preferences are homothetic, the

labor supply is inelastic and the positional considerations are high. These losses considerably

decrease when preferences exhibit non-homotheticity.

16

Figures 2 to 4 show comparative statics of the endogenous distributions of both con-

sumption goods, effective output, utility levels and the optimal non-linear income tax and

luxury tax that implement constrained efficient (CE) and double constrained efficient (DCE)

allocations when skills are exponential distributed. Figures 5 to 7 show the same information

when skills are distributed according to a Pareto distribution.11 Figures 2 and 5 show the

aforementioned distributions and taxes for different income elasticities, one of them corre-

sponding to homothetic preferences. Observe that when preferences exhibit homotheticity,

as the positional tax becomes flat, the marginal income tax drops to values very close to zero.

The same is not true under non-homothetic preferences. In this case, the flattening of the

positional tax only reduces the marginal labor income tax, however not to negligible levels.

Figures 3 and 6 show CE and DCE allocations and taxes in both environments assuming

two different labor supply elasticities and homothetic preferences in both cases. Clearly,

allocations are very sensitive to this parameter. Notice, however, that even when the labor

supply elasticity is low, the effective output distribution under the DCE is not too different

to the CE one. The reason is that changes in the after tax price of positional goods are offset

by reductions in the marginal labor income tax to the extend that incentives are not vio-

lated. Finally, Figures 4 and 7 present the same set of results considering a very elastic labor

supply (φ = 0.50) and non-homothetic preferences.12 These figures confirm the direction of

adjustments in the marginal income tax upon imposing linearity in the positional tax. The

labor income tax goes down to offset income effects due to changes in the after tax price of

positional goods. Nevertheless, since preferences are non-homothetic, changes in the labor

tax are much more moderate than in the homothetic case. Appendix C presents additional

comparative statics exercises.

Table 1: Summary of Variables when Distribution is Exponential, α = 0.15

Variable ρ = 1 ρ = 0.91 ρ = 0.80φ = 3 φ = 1.5 φ = 0.50 φ = 3 φ = 1.5 φ = 0.50 φ = 3 φ = 1.5 φ = 0.50

Linear Luxury Tax (τ) 34.58 35.94 41.35 35.77 36.87 41.08 37.24 37.91 40.03Welfare Loss (λΘ) 0.06 0.06 0.13 0.05 0.08 0.17 0.06 0.06 0.10Welfare Loss Bottom (λL) 0.94 0.89 1.49 0.71 0.69 0.92 0.52 0.45 0.43Welfare Loss Top (λH ) -3.03 -2.23 -1.46 -2.62 -1.94 -1.17 -2.17 -1.58 -0.85

λf and λψ imply that at the top of the distribution ψ/f = 2.17. All numbers are reported in percentage terms.

11Distributions were calculated using the Epanechnikov kernel with a bandwidth of 0.4 × std(x) × n−1/5

where x is the smoothed variable. The optimal bandwidth of Silverman (1986) over-smoothed the uppertail.

12For presentation purposes, I left out of the plot the bottom tail of the distributions in order to see clearlychanges in the upper tail.

17

Table 2: Summary of Variables when Distribution is Exponential, α = 0.05



λf and λψ imply that at the top of the distribution ψ/f = 2.17. All numbers are reported in percentage terms.

Table 3: Summary of Variables when Distribution is Pareto, α = 0.15



kf and kψ imply that at the top of the distribution ψ/f = 2.17. All numbers are reported in percentage terms.

Table 4: Summary of Variables when Distribution is Pareto, α = 0.05



kf and kψ imply that at the top of the distribution ψ/f = 2.17. All numbers are reported in percentage terms.

5 Conclusions

In this paper I have introduced positional consumption goods within the Mirrlees (1971)

framework. Positional goods are those whose valuation depends on an endogenous con-

sumption benchmark. This consumption benchmark is a weighted average of all agents

consumption of the positional good. As the contribution of individuals to the endogenous

consumption benchmark differs from their population size, constrained efficient allocations

exhibit a non-linear wegde between positional and no positional goods. Constrained effi-

cient allocations can be implemented through a non-linear positional tax together with a

18

0 2 4 6 80

0.1

0.2

0.3

0.4

Necessity Consumption

de

nsity

ρ=1, φ=3, CE ρ=0.91, φ=3, CE ρ=1, φ=3, DCE ρ=0.91, φ=3, DCE

0 1 2 3 40

0.2

0.4

0.6

0.8

Luxury Consumption

de

nsity

0 5 100

0.05

0.1

0.15

0.2

Effective output

de

nsity

0 1 2 3 40

0.2

0.4

0.6

0.8

Utility level

de

nsity

0 0.5 10

0.05

0.1

0.15

F(θ)

Op

tim

al In

co

me

Ta

x R

ate

0 0.5 10.2

0.3

0.4

0.5

0.6

F(θ)

Op

tim

al L

uxu

ry T

ax R

ate

Figure 2: Endogenous distributions and optimal taxes when α = 0.15 and distribution ofskills is exponential. Changes in the income elasticity.

19

0 5 100

0.2

0.4


de

nsi

ty

φ=3, CE φ=1.5, CE φ=3, DCE φ=1.5, DCE

0 2 40

0.2

0.4

0.6

Luxury Consumption

de

nsi

ty

0 5 10 150

0.1

0.2

Effective output

de

nsi

ty

0 2 40

0.2

0.4

0.6

Utility level

de

nsi

ty

0 0.5 10

0.02

0.04

0.06

F(θ)Op

tima

l In

com

e T

ax

Ra

te

0 0.5 10.2

0.4

0.6

0.8

F(θ)Op

tima

l Lu

xury

Ta

x R

ate

Figure 3: Endogenous distributions and optimal taxes when α = 0.15, preferences arehomothetic ρ = 1 and distribution of skills is exponential. Changes in the labor supplyelasticity.

20

5 10 15 20 250

0.1

0.2


de

nsity

ρ=0.91, φ=0.50, CE ρ=0.80, φ=0.50, CE ρ=0.91, φ=0.50, DCE ρ=0.80, φ=0.50, DCE

2 4 6 8 10 12 140

0.2

0.4

Luxury Consumption

de

nsity

10 20 30 40 500

0.02

0.04

0.06

0.08

Effective output

de

nsity

2 4 60

0.2

0.4

0.6

Utility level

de

nsity

0 0.5 10

0.05

0.1

0.15

F(θ)Op

tim

al In

co

me

Ta

x R

ate

0 0.5 10.2

0.4

0.6

0.8

F(θ)Op

tim

al L

uxu

ry T

ax R

ate

Figure 4: Upper tail of endogenous distributions and optimal taxes when α = 0.15 anddistribution of skills is exponential. Changes in the income elasticity.

21

0 2 4 6 80

0.2

0.4


de

nsity

ρ=1, φ=3, CE ρ=0.91, φ=3, CE ρ=1, φ=3, DCE ρ=0.91, φ=3, DCE

0 1 2 3 40

0.2

0.4

0.6

Luxury Consumption

de

nsity

0 5 100

0.1

0.2

Effective output

de

nsity

0 1 2 3 40

0.2

0.4

0.6

Utility level

de

nsity

0 0.2 0.4 0.6 0.80

0.05

0.1

0.15

0.2

F(θ)Op

tim

al In

co

me

Ta

x R

ate

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

F(θ)

Op

tim

al L

uxu

ry T

ax R

ate

Figure 5: Endogenous distributions and optimal taxes when α = 0.15 and distribution ofskills is Pareto. Changes in the income elasticity.

22

0 5 100

0.2

0.4


de

nsity

φ=3, CE φ=1.5, CE φ=3, DCE φ=1.5, DCE

0 2 40

0.2

0.4

0.6

Luxury Consumption

de

nsity

0 5 10 150

0.1

0.2

Effective output

de

nsity

0 2 40

0.2

0.4

0.6

Utility level

de

nsity

0 0.2 0.4 0.6 0.80

0.05

0.1

F(θ)Op

tim

al In

co

me

Ta

x R

ate

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

F(θ)Op

tim

al L

uxu

ry T

ax R

ate

Figure 6: Endogenous distributions and optimal taxes when α = 0.15, preferences arehomothetic ρ = 1 and distribution of skills is Pareto. Changes in the labor supply elasticity.

23

5 10 15 20 250

0.1

0.2


de

nsity

ρ=0.91, φ=0.50, CE ρ=0.80, φ=0.50, CE ρ=0.91, φ=0.50, DCE ρ=0.80, φ=0.50, DCE

5 10 150

0.2

0.4

Luxury Consumption

de

nsity

10 20 30 40 500

0.02

0.04

0.06

0.08

Effective output

de

nsity

1 2 3 4 5 60

0.2

0.4

0.6

Utility level

de

nsity

0 0.2 0.4 0.6 0.80

0.1

0.2

F(θ)Op

tim

al In

co

me

Ta

x R

ate

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

F(θ)Op

tim

al L

uxu

ry T

ax R

ate

Figure 7: Upper tail of endogenous distributions and optimal taxes when α = 0.15 anddistribution of skills is Pareto. Changes in the income elasticity.

24

non-linear income tax with standard properties, namely, no distortions at the extremes. The

previous implementation however, is subject to arbitrage opportunities across consumption

goods. Thus, an extra constraint is imposed: the marginal rate of substitution between

the positional and the positional good must be the same across agents. I have shown that

the resulting double constrained efficient allocations can be implemented through a linear

positional tax together with a non-linear labor income tax.

Aggregate welfare losses in the double constrained environment with respect to the con-

strained efficient environment are very low and in some cases negligible. Nevertheless, large

distributional effects arise. Assuming that the positional externality is increasing in income,

individuals at the high end of the income distribution experience large gains since for them,

a flat tax effectively reduces the after tax price of positional goods, thus higher consumption

occurs. This is so since the drop in the price generates a positive income effect that cannot

be offset by an increase in the marginal income tax as optimality requires no distortion at

the top. The opposite is true for individuals at the bottom of the skills distribution. They

experience considerable welfare losses since a flat luxury tax increases the after tax price of

luxuries. The previous generates a negative income effect that is offset by a reduction in the

marginal income tax. Such reduction cannot fully offset the income effect since that would

violate incentives. When preferences are non-homothetic, small adjustments in the income

tax are more effective offsetting income effects derived from changes in the “luxury tax”.

My results suggest that the effectiveness of a linear consumption tax correcting positional

externalities would crucially depend on the degree of non-homotheticity in preferences over

positional and non positional goods.

An important extension of the current model is to incorporate a production technology

whose marginal rate of transformation between luxuries and necessities is not constant. Such

specification would allow us to incorporate in our quantitative analysis potential sharper

changes in the output of the economy as a result of a good specific tax. Also, it is important

to remark that the results presented assume that agents cannot buy positional goods in

markets with different tax regimes. This consideration would impose an upper bound on

this tax that is not considered in this model.

References

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premium,” Journal of Finance, pp. 2959–3004.

25

Atkinson, A., and J. Stiglitz (1972): “The Structure of Indirect Taxation and Economic

Efficiency,” Journal of Public Economics, 1(1), 97–119.

(1976): “The Design of Tax Structure: Direct Versus Indirect Taxation,” Journal

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Carlsson, F., O. Johansson-Stenman, and P. Martinsson (2007): “Do You Enjoy

Having More than Others? Survey Evidence of Positional Goods,” Economica, 74(4).

Costa, D. (2001): “Estimating real income in the United States from 1888 to 1994: Cor-

recting CPI bias using Engel curves,” Journal of Political Economy, 109(6), 1288–1310.

da Costa, C. E. (2009): “Yet Another Reason to Tax Goods,” Review of Economic Dy-

namics, pp. 363–376.

Dupor, B., and W. Liu (2003): “Jealousy and Equilibrium Overconsumption,” American

Economic Review, 93(1), 423–428.

Dynan, K., and E. Ravina (2007): “Increasing income inequality, external habits, and

self-reported happiness,” American Economic Review, 97(2), 226–231.

Frank, R. (2005): “Positional Externalities Cause Large and Preventable Welfare Losses,”

American Economic Review, 95(2), 137–141.

(2008): “Should public policy respond to positional externalities?,” Journal of

Public Economics, 92(8), 1777–1786.

Golosov, M., and A. Tsyvinski (2007): “Optimal Taxation With Endogenous Insurance

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Minnesota, Working paper.

Silverman, B. (1986): Density Estimation for Statistics and Data Analysis. Chapman &

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26

Solnick, S., and D. Hemenway (2005): “Are positional concerns stronger in some do-

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Appendix

A Proofs

A.1 Proof of Proposition 1

Proof. Replicate proof of Proposition 2 setting η(θ) = 0 ∀θ ∈ Θ.


Proof. The first step is to transform the continuum of incentive compatibility constraints (3)

into a first order condition. Let σ(θ) = θ′ and

W (θ, θ′) ≡ u(cn(θ′), cl(θ

′)) − αC − v

(

y(θ′)

θ

)

(38)

A necessary condition for truthful revelation of type is ∂W (θ,θ′)∂θ′

|θ′=θ = 0, therefore it

follows that

ucn(cn(θ), cl(θ))∂cn(θ)

∂θ+ ucl(cl(θ), cl(θ))

∂cl(θ)

∂θ= v′

(

y(θ)

θ

)

y′(θ)

θ∀θ ∈ Θ (39)

Moreover, under truthful revelation W (θ) = u(cn(θ), cl(θ)) − αC − v(

y(θ)θ

)

and hence,

W ′(θ) = ucn(cn(θ), cl(θ))∂cn(θ)∂θ

+ ucl(cn(θ), cl(θ))∂cl(θ)∂θ

− v′(

y(θ)θ

)

y′(θ)θ

+ v′(

y(θ)θ

)

y(θ)θ2

, which

together with (39) becomes

W ′(θ) = v′(

y(θ)

θ

)

y(θ)

θ2∀θ ∈ Θ. (40)

Define the expenditure function e(W (θ), cl(θ), y(θ), C; θ) to satisfy W (θ) = u(e, cl) −

αC − v(

y(θ)θ

)

. Thus, the planner problem can be restated as

maxW (·),cl(θ),y(·),C,κ

∫

Θ

W (θ)g(θ)dθ (41)

27

s.t∫

Θ

cl(θ)f(θ)dθ +

∫

Θ

e(W (θ), cl(θ), y(θ), C; θ)f(θ)dθ =

∫

Θ

y(θ)f(θ)dθ (42)

W ′(θ) = v′(

y(θ)

θ

)

y(θ)

θ2∀θ ∈ Θ (43)

ecl(W (θ), cl(θ), y(θ), C; θ) = κ ∀θ ∈ Θ (44)

C ≡

∫

Θ

[ωe(W (θ), cl(θ), y(θ), C; θ) + (1 − ω)cl(θ)]ψ(θ)dθ (45)

The corresponding Lagrangian is

L(W (θ), cl(θ), y(θ), C, κ, λ, µ(θ), γ, η(θ)) =

∫

Θ

W (θ)g(θ)dθ

−λ

∫

Θ

[cl(θ) + e(W (θ), cl(θ), y(θ), C; θ)− y(θ)] f(θ)dθ+

∫

Θ

µ(θ)

[

W ′(θ) − v′(

y(θ)

θ

)

y(θ)

θ2

]

dθ

+γ

[

C −

∫

Θ

[ωe(W (θ), cl(θ), y(θ), C; θ) + (1 − ω)cl(θ)]ψ(θ)dθ

]

+

∫

Θ

η(θ) [ecl(W (θ), cl(θ), y(θ), C; θ) − κ] dθ (46)

Using integration by parts, it follows that

∫

Θ

µ(θ)W ′(θ)dθ = µ(θ)W (θ) − µ(θ)W (θ) −

∫

Θ

µ′(θ)W (θ)dθ (47)

thus, we can reexpress the above Lagrangian as

L(W (θ), cl(θ), y(θ), C, κ, λ, µ(θ), γ, η(θ)) =

∫

Θ

W (θ)g(θ)dθ

−λ

∫

Θ

[cl(θ) + e(W (θ), cl(θ), y(θ), C; θ) − y(θ)]f(θ)dθ + µ(θ)W (θ) − µ(θ)W (θ)

−

∫

Θ

µ′(θ)W (θ)dθ −

∫

Θ

µ(θ)v′(

y(θ)

θ

)

y(θ)

θ2dθ

28

+γ

[

C −

∫

Θ

[ωe(W (θ), cl(θ), y(θ), C; θ) + (1 − ω)cl(θ)]ψ(θ)dθ

]

+

∫

Θ

η(θ) [ecl(W (θ), cl(θ), y(θ), C; θ) − κ] dθ (48)

Assuming interior solution, it follows from first order conditions that

W (θ):

g(θ)−λf(θ)eW (W (θ), c(θ)l, y(θ), C; θ)−µ′(θ)−γωψ(θ)eW (W (θ), cl(θ), y(θ), C; θ) = 0 (49)

y(θ):

−λey(W (θ), cl(θ), y(θ), C; θ)f(θ) + λf(θ) −µ(θ)

θ2v′

(

y(θ)

θ

) [

1 +1

ǫ(θ)

]

− γωey(W (θ), cl(θ), y(θ), C; θ)ψ(θ) = 0 (50)

cl(θ):

−λecl(W (θ), cl(θ), y(θ), C; θ)f(θ)−λf(θ)− γωecl(W (θ), cl(θ), y(θ), C; θ)ψ(θ)− γ(1−ω)ψ(θ)

+ η(θ)eclcl(W (θ), cl(θ), y(θ), C; θ) = 0 (51)

C :

− λ

∫

Θ

eC(W (θ), cl(θ), y(θ), C; θ)f(θ)dθ+ γ − γω

∫

Θ

eC(W (θ), cl(θ), y(θ), C; θ)ψ(θ)dθ = 0

(52)

κ :∫

Θ

η(θ)dθ = 0 (53)

together with the boundary conditions µ(θ) = µ(θ) = 0 and where ǫ(θ) ≡v′( y(θ)

θ)

v′′( y(θ)θ

) y(θ)θ

.

Moreover, by implicit differentiation of W (θ) it follows that eW (W (θ), cl(θ), y(θ), C; θ) =

1ucn(cn(θ),cl(θ))

, ey(W (θ), cl(θ), y(θ), C; θ) =v′( y(θ)θ ) 1

θ

ucn(cn(θ),cl(θ)), ecl(W (θ), cl(θ), y(θ), C; θ) = −

ucl(cn(θ),cl(θ))

ucn(cn(θ),cl(θ))

and eC(W (θ), cl(θ), y(θ), C; θ) = αucn(cn(θ),cl(θ))

. Moreover, observe that eclcl(W (θ), cl(θ), y(θ), C; θ) =[

ucl(θ)

ucn(θ)

ucncl(θ)

ucn(θ)−

uclcl(θ)

ucn(θ)

]

.

29

The result follows after manipulating (49)-(53).


Proof. Taking first order conditions in agent’s problem we have

T ′(y(θ))

1 − T ′(y(θ))=ucn(c

eqn (θ), ceql (θ))

v′(

yeq(θ)θ

)

1θ

− 1 ∀θ ∈ Θ (54)

anducl(c

eqn (θ), ceql (θ))

ucn(ceqn (θ), ceql (θ))

= 1 + τ ∀θ ∈ Θ (55)

hence from (29)-(31) it follows that

ucl(ceqn (θ), ceql (θ))


− 1 =

α(1−2ω)λ

∫

Θψ(θ)B∗(θ)

dθ∫

Θf(θ)B∗(θ)

dθ + αωλ

∫

Θψ(θ)B∗(θ)

dθ∀θ ∈ Θ (56)

and


v′(

yeq(θ)θ

)

1θ

− 1 =α

λ

ωψ(θ)

f(θ)+ucn(c

∗n(θ), c

∗l (θ))

θf(θ)

[

1 +1

ǫ∗(θ)

]

I∗(θ) ∀θ ∈ Θ (57)

Finally notice that the fact that the government balances its budget implies that

∫

Θ

ceqn (θ)f(θ)dθ +

∫

Θ

ceql (θ)f(θ)dθ =

∫

Θ

yeq(θ)f(θ)dθ (58)

Thus, from (56)-(58) we conclude that {ceqn (θ), ceql (θ), yeq(θ)}θ∈Θ = {c∗n(θ), c∗l (θ), y

∗(θ)}θ∈Θ.

B Solving the Model Numerically

I solve the model by casting it into a system of differential-algebraic equations. Let

r(θ) ≡

(

1 +1

ǫ(θ)

)−1[

1

v′(y(θ)θ

)−

ey(θ)

v′(y(θ)θ

)

(

1 +γω

λ

ψ(θ)

f(θ)

)

]

(59)

30

thus equation (50) becomes

r(θ) =µ(θ)

λθ2f(θ)(60)

Differentiating (60) and using (59) we obtain

r′(θ) =µ′(θ)

λf(θ)θ2− r(θ)

[

2

θ+f ′(θ)

f(θ)

]

(61)

Finally, using equation (49) we have

r′(θ) =g(θ)

f(θ)λθ2−eW (θ)

θ2−γω

λ

ψ(θ)

f(θ)

eW (θ)

θ2− r(θ)

[

2

θ+f ′(θ)

f(θ)

]

(62)

From incentive compatibility we directly have

W ′(θ) = v′(

y(θ)

θ

)

y(θ)

θ2. (63)

Thus, equations (62) and (63) are the differential equations of the system. The algebraic

equations of the system are the following: from equation (51) we obtain

− ecl(θ) − 1 −γωψ(θ)ecl(θ)

λf(θ)−γ(1 − ω)ψ(θ)

λf(θ)+ η(θ)

eclcl(θ)

λf(θ)= 0 (64)

from problem’s restriction we have

ecl(θ) − κ = 0. (65)

Thus, equations (59), (64) and (65) are the algebraic equations of the system. The variables

of the system are [W (θ), r(θ), y(θ), cl(θ), η(θ)]′. Once the value of these variables is known,

it is straightforward to calculate the value of cn(θ). The solution of the system has to satisfy

the feasibility constraint and∫

Θη(θ)dθ = 0. This can be done by adjusting the value of

κ and the vector of initial conditions.13 Moreover, observe that an initial guess must be

followed to calculate C,∫

θcn(θ)ψ(θ)dθ and

∫

θcn(θ)f(θ)dθ which are required to obtain the

solution of the system through the Lagrange multipliers of the system. We can iterate the

previously guessed values until convergence.

13From the fact that µ(θ) it must be that r(θ) = 0 whenever ω = 0.

31

C Robusteness

In this section I present a sensitivity analysis of some of the parameters not presented in

the main body of the paper. Figure 8 shows comparative statics of changes in the jealousy

parameter, α when the skills distribution is Pareto. As expected, both taxes are increasing

in this parameter. Also, observe that when α is higher, the luxury good consumption dis-

tribution experience sharper changes at the top upon the introduction of the positional flat

tax. Figure 9 show changes in η, the parameter that measures the share of disposable income

assigned ti each good. Figure 10 the endogenous distributions and taxes for different elastic-

ity of substitution across goods, the parameter σ. Figure 11 display distributions and taxes

for changes in the ratio ψ(θ)/f(θ). Figure 12 also display changes in the ratio ψ(θ)/f(θ).

Notice that in this case, the previous ratio is decreasing in income! Finally, Figure 13 shows

the endogenous distributions and taxes for a very high elasticity of labor supply.

32

0 2 4 6 80

0.2

0.4


de

nsity

α=0.15, ρ=0.91, φ=3, CE α=0.05, ρ=0.91, φ=3, CE α=0.15, ρ=0.91, φ=3, DCE α=0.05, ρ=0.91, φ=3, DCE

0 1 2 3 40

0.2

0.4

0.6

Luxury Consumption

de

nsity

0 5 100

0.1

0.2

Effective output

de

nsity

0 1 2 3 40

0.2

0.4

0.6

Utility level

de

nsity

0 0.2 0.4 0.6 0.80

0.1

0.2

F(θ)Op

tim

al In

co

me

Ta

x R

ate

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

F(θ)Op

tim

al L

uxu

ry T

ax R

ate

Figure 8: Endogenous distributions and optimal taxes when distribution of skills is Paretoand preferences are non-homothetic. Changes in α.

33

0 2 4 6 80

0.2

0.4


de

nsity

η=0.75, ρ=0.91, φ=3, CE η=0.80, ρ=0.91, φ=3, CE η=0.75, ρ=0.91, φ=3, DCE η=0.80, ρ=0.91, φ=3, DCE

0 1 2 30

0.2

0.4

0.6

Luxury Consumption

de

nsity

0 5 100

0.1

0.2

Effective output

de

nsity

0 1 2 3 40

0.2

0.4

0.6

Utility level

de

nsity

0 0.2 0.4 0.6 0.80

0.1

0.2

F(θ)Op

tim

al In

co

me

Ta

x R

ate

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

F(θ)Op

tim

al L

uxu

ry T

ax R

ate

Figure 9: Endogenous distributions and optimal taxes when distribution of skills is Paretoand preferences are non-homothetic. Changes in η. When η = 0.80 we have λΘ = 0.15%,λL = 1.21% and λH = −5%

34

0 2 4 6 80

0.2

0.4


de

nsity

σ=0.50, ρ=0.91, φ=3, CE σ=0.70, ρ=0.91, φ=3, CE σ=0.50, ρ=0.91, φ=3, DCE σ=0.70, ρ=0.91, φ=3, DCE

0 1 2 30

0.5

1

1.5

Luxury Consumption

de

nsity

0 5 100

0.1

0.2

Effective output

de

nsity

0 1 2 3 40

0.2

0.4

0.6

Utility level

de

nsity

0 0.2 0.4 0.6 0.80

0.1

0.2

F(θ)Op

tim

al In

co

me

Ta

x R

ate

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

F(θ)Op

tim

al L

uxu

ry T

ax R

ate

Figure 10: Endogenous distributions and optimal taxes when distribution of skills is Paretoand preferences are non-homothetic. Changes in σ. When σ = 0.70 we have λΘ = 0.10%,λL = 0.87% and λH = −3.93%

35

0 2 4 60

0.2

0.4


de

nsity

f/ψ=2.17, ρ=0.91, φ=3, CE f/ψ=2.42, ρ=0.91, φ=3, CE f/ψ=2.17, ρ=0.91, φ=3, DCE f/ψ=2.42, ρ=0.91, φ=3, DCE

0 1 2 30

0.2

0.4

0.6

Luxury Consumption

de

nsity

0 5 100

0.1

0.2

Effective output

de

nsity

0 1 2 3 40

0.2

0.4

0.6

Utility level

de

nsity

0 0.2 0.4 0.6 0.80

0.1

0.2

F(θ)Op

tim

al In

co

me

Ta

x R

ate

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

F(θ)Op

tim

al L

uxu

ry T

ax R

ate

Figure 11: Endogenous distributions and optimal taxes when distribution of skills is Paretoand preferences are non-homothetic. Changes in kψ. When at the top, ψ/f = 2.42 we haveλΘ = 0.21%, λL = 1.62% and λH = −6.77%

36

0 2 4 60

0.2

0.4


de

nsity

f/ψ=2.17, ρ=0.91, φ=3, CE f/ψ=0.63, ρ=0.91, φ=3, CE f/ψ=2.17, ρ=0.91, φ=3, DCE f/ψ=0.63, ρ=0.91, φ=3, DCE

0 1 2 3 40

0.2

0.4

0.6

Luxury Consumption

de

nsity

0 5 100

0.1

0.2

Effective output

de

nsity

0 1 2 3 40

0.2

0.4

0.6

Utility level

de

nsity

0 0.2 0.4 0.6 0.80

0.1

0.2

F(θ)Op

tim

al In

co

me

Ta

x R

ate

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

F(θ)Op

tim

al L

uxu

ry T

ax R

ate

Figure 12: Endogenous distributions and optimal taxes when distribution of skills is Paretoand preferences are non-homothetic. Changes in kψ. When at the top, ψ/f = 0.63 and atthe bottom ψ/f = 1.20 we have λΘ = 0.03%, λL = −0.47% and λH = 2.51%

37

20 40 60 80 100 1200

0.01

0.02

0.03


de

nsity

φ=0.20, ρ=0.91, α=0.15, CEφ=0.20, ρ=0.91, α=0.15, DCE

20 40 600

0.02

0.04

Luxury Consumption

de

nsity

50 100 150 200 250 3000

2

4

6

8x 10

−3

Effective output

de

nsity

2 4 6 8 10 12 140

0.2

0.4

Utility level

de

nsity

0 0.2 0.4 0.6 0.80

0.1

0.2

F(θ)Op

tim

al In

co

me

Ta

x R

ate

0 0.2 0.4 0.6 0.80

0.2

0.4

0.6

F(θ)

Op

tim

al L

uxu

ry T

ax R

ate

Figure 13: Upper tail of endogenous distributions and optimal taxes when distribution ofskills is Pareto and preferences are non-homothetic. A very elastic labor supply (φ = 0.20).We obtained λΘ = 0.32%, λL = 1.85% and λH = −0.93%

38

Optimal Linear Taxation of Positional Goods

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