should the good and the selfish be taxed differently?

Scand. J. of Economics, 2012DOI: 10.1111/j.1467-9442.2012.01705.x

Should the Good and the Selfish be TaxedDifferently?∗

Ngo Van LongMcGill University, Montreal QC H3A 2T7, [email protected]

Frank StahlerUniversity of Tubingen, Tubingen, DE-72074, [email protected]

Abstract

In this paper, we consider an environment where individual actions have externalities and twotypes of agents exist: agents with social preferences (the good) and selfish agents. Selfishagents have pay-off functions that do not take into account social welfare. The pay-off ofan agent is a linear combination of social welfare and the pay-off of a selfish agent. Wedemonstrate that the corrective tax rates that maximize social welfare do not depend on thedegree of social preferences. Hence, the good and the selfish should not be taxed differently.

Keywords: Efficiency-inducing taxation; externalities; social preferences

JEL classification: H21

I. Introduction

How should taxation treat individuals who have preferences that incorporateboth a selfish component and a component that reflects social preferences?To our knowledge, this question has not been formally discussed in thebody of literature on optimal taxation. This lack of interest is somewhatsurprising, given the frequent appeal that good citizens should have a senseof social responsibility. Of course, an appeal for social responsibility is allthat can be done if individual actions cannot be influenced by measuressuch as taxes and subsidies. How should the government respond to the ex-istence of agents who have social preferences if it can influence individualactions? For example, should the government – if possible – make envi-ronmental taxes dependent on the degree of social preferences? Is thereanother dimension to be taken into account when designing an optimaltax/subsidy scheme? In this paper, we demonstrate an invariance theo-rem: social preferences do not play any role in policy design as long as all

∗We are grateful to Ted Bergstrom and two anonymous referees for very useful comments.Any remaining errors are our own.

C© The editors of The Scandinavian Journal of Economics 2012. Published by Blackwell Publishing, 9600 GarsingtonRoad, Oxford, OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.

2 N. V. Long and F. Stahler

relevant actions can be taxed/subsidized and agents agree on common sense(i.e., they agree on what should be done for the group as a whole).

The issue of taxing activities that give rise to externalities has a long-standing tradition in economics. Since the early work of Pigou (1920), it hasbeen well known that the outcome of individual actions is typically Paretoinefficient if individuals, in the pursuit of private interests, take actionsthat generate external effects, directly affecting the wellbeing of others.Without coordination, in a world ridden with external effects, selfish utilitymaximization normally results in a suboptimal outcome.

In order to internalize externalities and to increase social welfare, Pigou(1920) proposed a corrective tax (subsidy) for a negative (positive) ex-ternality.1 In this paper, we ask whether the optimal tax/subsidy schemeshould be changed if a subgroup of individuals is not entirely selfish butalso has social preferences. These good agents have common sense – theyknow what would be best for the whole group and they take this intoaccount when making decisions. We model their behavior such that theiractions are no longer directed at maximizing private utility; they seek tomaximize a linear combination of private utility and social welfare. Shouldthese agents be taxed differently? In this paper, we show that the good andthe selfish should not be taxed differently.

The taxation literature does not seem to have touched this issue fromthis perspective. However, the focus has been on issues of fairness and re-distribution. Fairness has been considered in the context of optimal taxation(Alesina and Angeletos, 2005; Fleurbaey and Maniquet, 2006, 2007), andexperimental evidence suggests that agents are inequality averse (Ackertet al., 2007).2 This body of literature is more concerned with redistri-bution, whereas we focus on allocational effects of taxation. We assumethat issues of fairness and redistribution could be solved by another set ofinstruments.

The public economics literature does not lack invariance results, so ourcontribution is not completely novel in this respect. Bergstrom et al. (1986)have shown that redistribution does not change the private provision of apublic good, as long as the set of contributors does not change.3 Atkinson

1 Alternative solutions include negotiation (Coase, 1960), and setting up a market for exter-nalities (Arrow, 1970). When the regulator does not have perfect information, implementationof efficient allocations as subgame perfect equilibria can be achieved by mechanism designs(see Moore and Repullo, 1988; Varian, 1994). For a review of the literature on implemen-tation under complete information, see Moore (1992). See Palfrey (2002) for a review ofimplementation theory for both complete and incomplete information environments.2 As for fair allocations, previous studies have dealt with the existence and the properties ofallocations, which are considered fair. For example, Nishimura (2004) discusses this issuefor allocations that are considered envy-free.3 See Groves and Ledyard (1977) for the pioneering paper on a mechanism to solve thefree-rider problem.

C© The editors of The Scandinavian Journal of Economics 2012.

Should the good and the selfish be taxed differently? 3

and Stiglitz (1976) have demonstrated that differential commodity taxesare unnecessary if an optimal non-linear income tax has been set.4 Ourinvariance theorem is closest to the rotten kid theorem of Becker (1974),who has considered a set-up in which a benevolent principal transfersresources to (selfish) agents. He has shown that redistribution does notchange consumption, as long as all of these agents are still supported,because selfish agents behave such that they maximize the group’s income(see also Bergstrom, 1989). In our model, however, there is no benevolentprincipal on whom other (selfish) agents are dependent. Agents in ourmodel do not differ in terms of their social status in a hierarchy, but theyare instead different in terms of their social preferences.

The remainder of the paper is organized as follows. In Section II, wedevelop the basic model and the invariance theorem. In Section III, weshow that our results extend to non-individualized taxes. In Section IV,we show that our model is not restricted to the functional specificationintroduced in Section II. We conclude in Section V.

II. The Model

There are n different agents. We label those agents who maximize theirown pay-off as selfish agents. Those agents who also take social welfareinto account are labeled agents with social preferences; they are the goodin our model. An individual agent will be denoted by i . Let ai be agent i’saction where ai ∈ R+, and let a−i be the vector of actions of his rivals.5

Let us first define the selfish component of each pay-off. The gross pay-offof an agent i , before tax and ignoring any social preferences, is

u(i) = u(i)(ai , a−i ).

The function u is assumed to be twice differentiable and strictly concavein ai . We assume that the net pay-off is the sum of the gross pay-off uand the tax bill ti ai , so that the pay-off arising from the actions (ai , a−i ) –again ignoring any social preferences – is equal to

u(i)(ai , a−i ) = u(i)(ai , a−i ) − ti ai ,

where ti is the agent-specific tax rate, imposed by the government, andtaken as given by each agent. We assume that the tax bill depends linearly

4 More recently, Kaplow (2006) has shown that Pigouvian taxes can be best, even in the caseof labor supply distortions because of income taxation.5 Our results would not change if ai is the vector of actions of agent i such that ai ∈ R

n+,

and a−i denotes the matrix of actions of all other agents.



on the level of actions of agent i .6 As for social welfare, we assume thatit can be represented by

W = W (a1, a2, . . . , an).

This specification assumes that all agents agree on its functional form;therefore, W represents what is called common sense.7

However, agents differ with respect to the degree by which they careabout social welfare, and therefore we distinguish two different types ofagents. The selfish type does not take social welfare into account, whereasthe agent with social preferences does so, to a certain extent. We supposethat any agent i seeks to maximize the objective function

V (i)(θi , ai , a−i ) = (1 − θi )u(i)(ai , a−i ) + θi W (ai , a−i ), 0 ≤ θi < 1,

where θi is i’s degree of social preferences. A selfish agent is characterizedby θi = 0, and an agent with social preferences is characterized by θi >

0. Note that agents differ in two dimensions. First, their gross pay-offu(i) might differ, even if the same actions are carried out by all agents,including themselves. Second, agents might have different degrees of socialpreferences, even if their gross pay-off functions coincide.

We begin our analysis with the case where all n agents are selfish:θi = 0 for all i . This case will serve as a benchmark for the taxation ofall agents. If the social planner had direct control over all ai , she wouldchoose the vector a = (a1, a2, . . . , an) to maximize welfare. The first-orderconditions for social welfare maximization consist of n equations

∂W (a∗)

∂ai= 0, for i = 1, 2, . . . , n. (1)

We assume that the second-order conditions are satisfied, and that thefirst-order conditions uniquely identify the socially optimal vector a∗ =(a∗

1 , a∗2 , . . . , a∗

n ). We now show that the social optimum can be achieved byimposing an optimally chosen tax vector t∗ = (t∗

1 , t∗2 , . . . , t∗

n ). In particular,we prove the invariance result that the optimal tax vector is independent ofthe degree of social preferences of any agent.

It is convenient to use the following notation:

Wi (a∗i , a∗

−i ) ≡∂W (a∗i , a∗

−i )

∂ai≡ ∂W (a∗)

∂ai.

6 We confine ourselves to linear taxes. It should be clear that non-linear, individualized taxes,which severely punish agent i if he deviates from some a∗

i , would also do the job. We dealwith non-individualized taxes in Section III.7 Note that W could be based on individual pay-offs such that W =W [u(1)(a1, a−1), u(2)(a2, a−2), . . . , u(n)(an, a−n)], but this is not required. See also ourexample at the end of this section and in Section III.



A selfish agent i takes as given the tax rate ti and the actions of all otheragents, denoted by a−i . Maximizing V (i)(0, ai , a−i ) = u(i)(ai , a−i ) with re-spect to ai yields the first-order condition

∂ u(i)(ai , a−i )

∂ai≡ u(i)

i (ai , a−i ) ≡ u(i)i (ai , a−i ) − ti = 0. (2)

We arrive at the following.

Lemma 1. The tax vector t∗ = (t∗1 , t∗

2 , . . . , t∗n ) defined by

t∗i = u(i)

i (a∗i , a∗

−i ) − Wi (a∗i , a∗

−i ) (3)

implements the socially optimal action vector as a Nash equilibrium of thegame played by n selfish agents.

Proof : Suppose that the government sets ti according to equation (3). Thefirst-order condition (2) of agent i , evaluated at t∗

i , is

u(i)i (ai , a−i ) − [u(i)

i (a∗i , a∗

−i ) − Wi (a∗i , a∗

−i )] = 0. (4)

Clearly, if agent i thinks that all other agents choose a−i = a∗−i , then the

first-order condition (4) is satisfied by choosing ai = a∗i . �

We now show that the same tax vector is optimal and that it achieves thesame action vector, even if one, some, or all agents have social preferences.Suppose that ti is set at t∗

i , as given by equation (3), and that an agent ibelieves that all other agents will choose a∗

−i . Then, the objective functionof agent i with social preferences is

V (i) = (1 − θi )[u(i)(ai , a∗

−i ) − t∗i ai

]+ θi W (ai , a∗−i ),

and the corresponding first-order condition is

(1 − θi )[u(i)

i (ai , a∗−i ) − t∗

i

]+ θi Wi (ai , a∗−i ) = 0. (5)

Clearly, this first-order condition is satisfied by choosing ai = a∗i , which

is the socially optimal action defined by equation (1), because the terminside the square brackets is zero, and so is Wi (a∗

i , a∗−i ). We have proven

the following.

Proposition 1. The social optimum can be achieved as a Nash equilib-rium by applying a vector of individual taxes. The optimal tax vector isindependent of the vector of parameters θi , which represents the degree ofsocial preferences. At the optimal tax, the Nash equilibrium action of eachagent is independent of θi .



As an example, let us suppose that the social planner’s social welfarefunction is the sum of individual gross pay-offs:

W (a1, a2, . . . , an) =n∑

k=1

u(k)(ak, a−k) (6)

Then, the social optimum (a∗1 , a∗

2 , . . . , a∗n ) satisfies

∂W

∂ai=

n∑k=1

u(k)i (a∗

k , a∗−k) = 0,

and the optimal tax is equal to the sum of the external effects on otherindividuals:

ti = −∑k �=1

u(k)i (ak, a−k). (7)

A modified form of specification (6) can be proposed. Let W denotethe sum of net pay-offs, that is,

W =n∑

k=1

[u(k)(ak, a−k) − tkak + Tk], (8)

where Tk is a lump-sum payment to agent k, such that the budget isbalanced:

n∑k=1

(tkak − Tk) = 0.

In Section IV, we show that including a budget constraint but not al-lowing lump-sum taxes or other distributional instruments will invalidateour invariance result.8 However, with lump-sum taxes, it can be shownthat tax scheme (VII) is also invariant with respect to the degree of socialpreferences, as long as each individual i , when maximizing

V (i) ≡ (1 − θi )[u(i)(ai , a∗

−i ) − t∗i ai

]+ θi W (ai , a∗−i ),

anticipates that∑

Tk depends on his actions such that9

n∑k=1

Tk = t∗i ai +

∑j �=i

t∗j a∗

j .

8 We thank a referee for pointing this out.9 We are indebted to a referee for this insightful observation.



Thus, agent i maximizes

V (i) = (1 − θi )[u(i)(ai , a∗

−i ) − t∗i ai

]+ θi

×⎧⎨⎩t∗

i ai +∑j �=i

t∗j a∗

j +n∑

k=1

[u(k)(ak, a−k) − tkak

]⎫⎬⎭ .

The first-order condition

(1 − θi )[u(i)

i (ai , a∗−i ) − t∗

i

]+ θi

{t∗i +

[n∑

k=1

u(k)i (ak, a−k)

]− t∗

i

}= 0

demonstrates that our invariance result holds again. Therefore, our analysiscan be easily extended to the case in which welfare is the sum of netpay-offs.

What is the intuition for our invariance theorem? Social preferences im-ply that an agent balances the social and the selfish parts of his preferences.The selfish part needs the tax in order to behave well; the other part doesnot need it, but the tax also does no harm. Proposition 1 shows that thetax rate does not depend on the relative strengths of the two parts; it issufficient to deal with the bad part, the good part will behave well anyway.In the following sections, we generalize this result.

III. Non-Individualized Tax Schemes

The model of Section II and Proposition 1 have both relied on the feasi-bility of individual tax rates. In particular, the government is supposed toknow each individual’s gross pay-off function. Usually, taxes are not indi-vidualized, but they depend on an observable level of certain actions. Wenow explore how our results generalize if tax rates must not be individual-specific, but the government might use a tax scheme that depends onobservable actions only.10

Let T (a) denote the (non-individualized) tax scheme that depends on theobservable level of actions, which is denoted by a.11 Because the tax isnot individualized, the net pay-off of agent i is now equal to

u(i)(ai , a−i ) = u(i)(ai , a−i ) − T (ai )ai ,

10 In this model, the government knows that there are n agents with n (generally) distinct andknown objective functions, but it does not know which agent has which objective function.This also holds for each individual agent. This assumption is in line with the optimal taxationliterature – agents can pretend to be of a certain type but they cannot pretend to be of atype that does not exist.11 Note that T has no subscript because taxes no longer depend on individuals but they arethe same for all individuals who take the same level of actions.



where there is no subscript for the tax. Each selfish agent maximizes hernet-payoff over his action and takes into account his effect on his tax bill.The first-order condition for a selfish agent i is

u(i)i (ai , a−i ) = u(i)

i (ai , a−i ) − T (ai ) − dT (ai )

daiai = 0. (9)

Note that we now observe a tax avoidance effect [dT (ai )/dai ]ai becausethe actions will vary the tax rate.

We now show how an efficiency-inducing tax schedule T ∗(.) can befound. Again, let a∗ ≡ (a∗

1 , a∗2 , . . . , a∗

n ) be the socially optimal vector ofactions. We assume that all a∗

i are strictly positive. Without loss of gener-ality, we relabel individuals so that

0 < a < a∗1 ≤ a∗

2 ≤ · · · ≤ a∗n . (10)

After this relabeling, we define, for any individual i , his “divergence ofprivate marginal benefit from social marginal benefit” by

φi ≡ u(i)i (a = a∗

i , a∗−i ) − Wi (a = a∗

i , a∗−i ).

Let us introduce the following assumption.

Assumption 1. (monotonicity)

0 < φ1 ≤ φ2 ≤ · · · ≤ φn. (11)

This assumption implies that the more an individual’s private marginalbenefit (evaluated at the social optimum) exceeds the social marginal ben-efit, the higher is his activity level. Note that φi are (known) real numbers,not functions.

If condition (10) holds with strict inequality (i.e., 0 < a < a∗1 < a∗

2< · · · < a∗

n ), then Assumption 1 is all we need. If we want to treat themore general case, where equality holds for some pair (i, j), then we needan additional assumption, as follows.

Assumption 2.

a∗i = a∗

j ⇒ φi = φ j . (12)

This assumption states that at the socially optimal allocation, if twoagents are assigned the same activity level, their marginal utilities withrespect to their own activity (evaluated at the optimal level) are equal, eventhough their utility functions u(i)(·) and u( j)(·) can be different.

In view of inequalities (10) and (11) as well as condition (12), we cannow construct a monotone non-decreasing function ω(a), which has thefollowing properties:12

12 There are infinitely many functions that satisfy these properties. We can take any of them.



(i) ω(a∗i ) = φi for i = 1, 2, . . . n;

(ii) ω(a) is continuously differentiable, with derivative ω′(a) ≥ 0.

We define the function

�(a) ≡∫ a

0ω (a) da.

Notice that �(0) = 0, so that, as will become clear below, an agent whochooses a = 0 will pay no tax. Our construction implies that

�′(a) = ω(a).

Hence, by property (i) above,

�′(a∗i ) = u(i)

i (a = a∗i , a∗

−i ) − Wi (a = a∗i , a∗

−i ).

Our tax function is now constructed as follows:

aT (a) ≡ �(a).

The selfish individual i’s first-order condition (9) is then

u(i)i (ai , a−i ) − �′(ai ) = 0.

It follows that by choosing ai = a∗i , the individual achieves his private

optimum (given the tax schedule), which coincides with the social optimum.The second-order condition

u(i)i i (ai , a−i ) − �′

i (ai ) = u(i)i i (ai , a−i ) − ω′(ai ) < 0

is satisfied because u(i)i i < 0 and ω′ ≥ 0. We are now able to prove the

following.

Proposition 2. The social optimum can be achieved as a Nash equilibriumby applying a non-individualized tax scheme if Assumptions 1 and 2 hold.The optimal tax scheme is independent of the vector of parameters θi ,which represents the degree of social preferences.

Proof : The invariance follows easily from the optimal behavior of an agentwith social preferences. His net pay-off is

V (i) = (1 − θi )[u(i)(ai , a∗

−i ) − T (ai )]+ θi W (ai , a∗

−i ).

This yields the first-order condition

(1 − θi )

[u(i)

i (ai , a∗−i ) − T (ai ) − dT (ai )

daai

]+ θi Wi (ai , a∗

−i ) = 0.



0.45 0.5 0.55 0.6 0.65 0.7ai

0.5

1

1.5

2

2.5

uii

Fig. 1. Optimal activity levels

This condition is fulfilled for ai = a∗i if tax scheme aT (a) ≡ �(a) is

applied. �

Let us illustrate our results using an environmental economics examplewith heterogeneous agents. Let ai be the amount of resources used byindividual i , and let Z = ∑

a2i be the resulting pollution. Assume that

individual i’s utility function is

u(i) ≡ δi ln ai − 1

4Z2,

where 0 < δ1 < δ2 < · · · < δn . Let social welfare be the sum of individualutility levels:

W =n∑

i=1

u(i) = −n

4Z2 +

n∑i=1

δi ln ai .

Define � ≡ ∑ni=1 δi . In Appendix A, we prove that the optimal tax sched-

ule can be taken to be

T (a) = (n − 1)a

2

√�

n. (13)

This tax schedule will also lead to an optimal allocation if one, some, orall agents are “green”, in the sense that they partially take into account theeffect of their resource use on others. Figure 1 demonstrates the results forthree different agents.13 It shows the marginal gross pay-off u(i)

i for eachagent when the other agents choose the socially optimal activity levels. The

13 The simulation assumes δ1 = 0.8, δ2 = 1, and δ3 = 1.2, and it implies T (ai ) = ai anddT (ai )/dai = 1. The marginal tax effect is thus equal to 2ai . The optimal activity levels areequal to a∗

1 = 0.5164, a∗2 = 0.5773, and a∗

3 = 0.6325, respectively.



thick line is the marginal tax effect T (ai ) + ai dT (ai )/dai , which intersectsthe u(i)

i lines at the optimal activity levels.Our example also offers some guidance on the shape of the tax scheme.

In our example, T (a) is linear in activity levels because the divergencebetween private marginal benefit and social marginal benefit is linear ina,14 and thus �(a) is quadratic. For other specifications, the divergencemight be increasing in a at an increasing (decreasing) rate. Then, wewould expect the optimal tax scheme to be progressive (regressive).

IV. Discussion

We now discuss how our results can be extended to alternative specifi-cations, and we explore the conditions under which our results might nothold true. Would the invariance property in Proposition 1 still hold if theutility functions were not quasi-linear? We suppose that there are F fac-tors of production, G final consumption goods, and I individuals. Let αi

f

(respectively, cig) be the amount of factor f supplied by (respectively, final

good g consumed by) individual i . We suppose that the utility function ofa selfish individual i is

U (i) = U (i)(αi

1, αi2, . . . , α

iF , ci

1, ci2, . . . , ci

G, α−i , c−i),

where α−i and c−i are the vector of factor supplies and consumptions ofother individuals, respectively. The social welfare function is

W(α1

1, α12, . . . , α

1F , c1

1, c12, . . . , c1

G, α21, α

22, . . . , α

2F , c2

1, c22, . . . , c2

G, . . .)

≡ W (α, c) ,

where all technological constraints have been taken into account.Appendix B shows that our invariance results also hold in this environ-

ment, and thus they are not confined to quasi-linear environments.15 Whatis the basic foundation of our invariance theorem, and why does it hold ingeneral? Suppose that the allocation that maximizes social welfare has beenfound. We consider the action of an entirely selfish agent, given that allactivity levels by other agents are first-best because of taxation. Taxationof his activity will induce him to set his activity at the right level. Thetax is such that the selfish agent’s after-tax pay-off is maximized when hechooses the level that is implied by the welfare-maximizing allocation. So,the selfish agent cannot do better. Now, suppose that the agent ceases tobe entirely selfish and gives some weight to social preferences. He stillcares about the after-tax pay-off, but he also cares about social welfare. As

14 See equation (A3).15 Note that we can also consider the different activity levels as activity levels in time, andthus our analysis suggests that our basic results can also be extended to intertemporal models.



the existing tax already maximizes social welfare and the after-tax pay-off,this agent will maintain the same activity level as before. So, there is noneed to change the tax rates.

The invariance theorem relies on the feasibility of taxing all relevantactivities. If at least one activity that gives rise to an externality problemcannot be taxed (or subsidized), Propositions 1 and 2 will not hold. Inthis case, the tax scheme can only be second-best, because it can onlyindirectly influence untaxed activities. This includes the case where socialwelfare is the sum of net pay-offs, as in equation (8), and there is a bindingbudget constraint, but only activities can be taxed (lump-sum taxes are notavailable). This is shown in Appendix C for an economy with two agents.Then, the requirement of a binding budget constraint makes any policydependent on the degree of social preferences. It is the lack of instrumentsthat invalidates the invariance results because the corrective taxes also haveto play a fiscal role.

We should also mention that our invariance theorem will not hold underasymmetric information (e.g., in a Mirrlees-type optimal taxation model),because unobservable actions remain untaxed by definition. In this case,social preferences will definitely improve social welfare. Finally, our invari-ance theorem fails if taxation deals with the effects of envy in a group atthe same time, unless this trade-off can be dealt with by the specificationof the welfare function W .

V. Concluding Remarks

In this paper, we have developed a model in which individual agents aredifferent in two dimensions. First, the pay-offs they derive for themselvesmight differ because of different pay-off functions. Second, the extent towhich they take welfare aspects into account (their degree of social pref-erences) can vary between individuals. We have shown that the degreesof social preferences felt by individual agents have no bearing on the op-timal tax rate and on the optimal allocation, which is implemented as aNash equilibrium. Hence, efficiency-inducing tax policy, which might ap-ply to any activity that creates an externality problem, should take intoaccount heterogeneity in terms of individual pay-offs functions only, andnot heterogeneity in terms of social preferences.

We have also demonstrated that, under a relatively mild assumption,non-individualized tax schemes are insensitive to the degree of social pref-erences. This is an important result because it demonstrates clearly that asocial planner only needs to know the distribution of types in terms of thepay-off functions. Unless certain activities cannot be taxed and unless theseactivities give rise to externalities, the social planner does not need to knowthe degree of social preferences of individuals in order to be able to design



optimal policies. From the point of view of Pigouvian taxation, our resultsdemonstrate that, in quasi-linear environments, taxes should be equal to theexternality cost, and they should not depend on social preferences.

Appendix A: Proof of Equation (13)

In the absence of any regulation, each selfish individual will maximize u(i)

by choosing ai , taking Z−i as given. This leads to n first-order conditions:

δi = a2i Z . (A1)

Adding up over all n first-order conditions leads to � = Z2. The unregu-lated Nash equilibrium in this game implies the aggregate pollution levelof Z = √

�. Each individual’s resource use amounts to ai = √δi/Z =√

δi/√

�.The socially optimal pollution level can be derived from maximization

of W , which gives n first-order conditions:

Wi = δi

ai− nai Z = 0.

Adding up all n first-order conditions yields � = nZ2. Thus, the sociallyoptimal level of pollution is given by

Z∗ =√

�/n, (A2)

and the resulting optimal resource use, a∗i , is

a∗i =

√δi

nZ∗ =√

δi√n�

.

The marginal utility and the marginal welfare, computed at the optimallevel of resource use, are equal to

u(i)i (a = a∗

i , a∗−i ) = δi

a∗i

− a∗i

⎛⎝a∗2i +

∑j �=i

a∗2j

⎞⎠ = δi

a∗i

− a∗i Z∗

and

Wi (a = a∗i , a∗

−i ) = δi

a∗i

− na∗i

⎛⎝a∗2i +

∑j �=i

a∗2j

⎞⎠ = δi

a∗i

− na∗i Z∗,

respectively. Thus,

φi ≡ u(i)i (a = a∗

i , a∗−i ) − Wi (a = a∗

i , a∗−i ) = (n − 1)a∗

i Z∗. (A3)



Expression (A3) shows that Assumption 1 (monotonicity) is satisfied. Wecan then construct the function ω(a) = (n − 1)aZ∗. Integration leads to

�(a) = (n − 1)Z∗ a2

2,

and the tax function is thus given by

T (a) = �(a)

a= (n − 1)Z∗a

2. (A4)

(Here, each individual treats Z∗ as given.) Inserting Z∗ from equation (A2)yields the tax function (A4). Taking as given the tax function (13) andassuming that j chooses a∗

j , individual i finds that his first-order conditionis

δi

ai− ai

⎡⎣a2i +

∑j �=i

(a∗j )

2

⎤⎦− (n − 1)ai

√�/n = 0.

The choice of ai = a∗i will satisfy this first-order condition. The second-

order condition is also satisfied:

− δi

a2i

− Z∗ − 2a2i − (n − 1)Z∗ < 0. �

Appendix B: Generalization of the Invariance Property

The socially optimal allocation, denoted by a hat, is assumed to be theunique solution of the (F + G)I first-order conditions, for i = 1, 2, . . . , I ,f = 1, 2, . . . , F , and g = 1, 2, . . . , G:

∂W

∂αif

(α1

1, α12, . . . , α

1F , c1

1, c12, . . . , c1

G, α21, α

22, . . . , α

2F , c2

1, c22, . . . , c2

G, . . .) = 0,

∂W

∂cig

(α1

1, α12, . . . , α

1F , c1

1, c12, . . . , c1

G, α21, α

22, . . . , α

2F , c2

1, c22, . . . , c2

G, . . .) = 0.

Corresponding to the optimal allocation, we can determine the (pre-tax)price of factor f (denoted by w f ) and the (pre-tax) price of good g (de-noted by pg), where f = 1, 2, . . . , F and g = 1, 2, . . . , G. Let τ i

f and t ig be

the individualized tax per unit of αif and ci

g, respectively. The consumptionof good 1 does not generate externalities, so t i

1 = 0 for all i . Let Li be thelump-sum transfer to individual i . The budget constraint of individual i isthen

p1ci1 = Li +

∑f

(w f − τ i

f

)αi

f −∑

g

(pg + t i

g

)ci

g. (B1)



Consider, for the moment, the case where all individuals are self-ish. The optimal taxes must ensure that the optimal allocation can beimplemented as a Nash equilibrium, that is, for each i , the vector(αi

1, αi2, . . . , α

iF , ci

1, ci2, . . . , ci

G) maximizes individual i’s utility:

U (i)(αi

1, αi2, . . . , α

iF ,

Li +∑(

w f − τ if

)αi

f −∑(

pg + t ig

)ci

g

p1, ci

2, . . . , ciG , α−i , c−i

),

Thus, for f = 1, 2, . . . , F

∂U (i)

∂αif

⎡⎣αi1, α

i2, . . . , α

iF ,

Li +∑(

w f − τ if

)αi

f −∑(

pg + t ig

)ci

g

p1, ci

2, . . . , ciG , α−i , c−i

⎤⎦= 0 (B2)

and for g = 2, . . . , G

∂U (i)

∂aig

⎡⎣αi1, α

i2, . . . , α

iF ,

Li +∑(

w f − τ if

)αi

f −∑(

pg + t ig

)ci

g

p1, ci

2, . . . , ciG , α−i , c−i

⎤⎦= 0. (B3)

Setting the hat values αif and ci

g into the system of (F + G)I equations(B1)–(B3), we can determine the (F + G − 1)I tax rates (recall that t i

1 = 0)and I lump-sum transfer amounts Li for i = 1, 2, . . . , I .

Now, suppose that some agents have social preferences. Very generally,the objective function of an agent with social preferences can be written asa function that depends on the utility of a selfish agent and social welfare:

V i [U i (·), W (·)], V iU > 0, V i

W ≥ 0.

The utility of a selfish agent is a special case of V i for which V iW = 0.

The first-order conditions for an agent with social preferences are

V iU

∂U (i)

∂αif

⎡⎢⎢⎢⎣αi1, . . . α

iF ,

∑f

(w f − τ i

f

)αi

f −∑g �=1

(pg + tg

)ci

g

p1, ci

2, . . . , ciG , α−i , c−i

⎤⎥⎥⎥⎦+ V i

W

∂W

∂αif

(α, c) = 0,



V iU

∂U (i)

∂cig

⎡⎢⎢⎢⎣αi1, . . . α

iF ,

∑i

(wi − τi )αi −∑j �=0

(p j + t j )a j

p0, ci

2, . . . , ciG, α−i , c−i

⎤⎥⎥⎥⎦+ V i

W

∂W

∂cig

(α, c) = 0,

for f = 1, 2, . . . , F and g = 2, 3, . . . , G, respectively. Clearly, the opti-mal tax vector when all agents are selfish is also optimal when someor all agents have social preferences, because ∂W/∂ci

g (α, c) = 0 and∂W/∂αi

f (α, c) = 0 .

Appendix C: Model with a Budget Constraint in the Absence ofLump-Sum Taxes

We assume that there are two agents, 1 and 2, each maximizingV (i)(θi , ai , a j ). We assume that individual maximization yields reactionfunctions ai (t1, t2). The social planner maximizes the sum of net pay-offs

W = u(1) [a1(t1, t2), a2(t1, t2)] − t1a1(t1, t2)

+ u(2) [a1(t1, t2), a2(t1, t2)] − t2a2(t1, t2),

subject to t1a1(t1, t2) + t2a2(t1, t2) ≥ B, assuming that B is not too large.Furthermore, we assume that the constraint is binding. Maximizing theLagrangian L = W + λ [t1a1(t1, t2) − t2a2(t1, t2) − B] yields the first-ordercondition for the optimal tax t1

∂L

∂t1= ∂ a1

∂t1

[u(1)

1 − t1 + u(2)1

]+ ∂ a2

∂t1

[u(2)

2 − t2 + u(1)2

]− a1

+ λ

(a1 + t1

∂ a1

∂t1+ t2

∂ a2

∂t1

)= 0,

and a similar condition for the optimal tax t2.

If both agents are selfish (i.e., θi = 0), u(i)i − ti = 0, and the first-order

condition is

∂L

∂t1= ∂ a1

∂t1u(2)

1 + ∂ a2

∂t1u(1)

2 − a1 + λ

(a1 + t1

∂ a1

∂t1+ t2

∂ a2

∂t1

)= 0.

If both agents care for social welfare only (i.e., θi = 1), u(i)i + u( j)

i − ti =0, and the first-order condition is

∂L

∂t1= −a1 + λ

(a1 + t1

∂ a1

∂t1+ t2

∂ a2

∂t1

)= 0.



This condition is clearly different from the one for θi = 0, not only becausetwo terms have disappeared, but also because ∂ ai/∂t1 changes with θi .Thus, the invariance theorem does not hold. (This argument also showsthat the budget constraint will always be binding when θi = 1, because ithas no solution for λ = 0.)

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First version submitted September 2009;final version received October 2010.


should the good and the selfish be taxed differently?

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