Laffer Curver

Tax Avoidance and the Laer Curve

Judith Panades

March 2003

Abstract

This paper analyzes the impact on tax revenue of a change in the tax ratewhen the individuals have at their disposal two alternative methods for re-ducing their tax liabilities: tax evasion and tax avoidance. We nd thatthis impact takes the shape of a Laer curve when the xed cost associatedwith tax avoidance is small enough. We also examine more accurately thecase of a isoelastic utility function in order to obtain stronger results on theexistence of such a Laer curve. Finally, we explore the role played by thedistribution of income by considering a discrete distribution.

JEL classication code: E62, H26.

Keywords: Tax Evasion, Tax Avoidance, Laer Curve

Financial support from the Spanish Ministry of Education through grant PB96-1160-C02-02 is gratefully acknowledged. I want to thank Jordi Caballe, Ines Macho, and Pau Olivella fortheir valuable suggestions. Of course, all errors that remain are entirely my own.

Unitat de Fonaments de lAna`lisi Econo`mica. Universitat Auto`noma de Barcelona. Cor-respondence: Judith Panades i Mart. Departament dEconomia i Histo`ria Econo`mica. EdiciB. Universitat Auto`noma de Barcelona. 08193 Bellaterra (Barcelona). Spain. Tel: (34) 93 58118 02. Fax: (34) 93 581 20 12. E-mail: [email protected]

1. Introduction

To pay taxes is viewed as an unpleasant obligation and, consequently, many tax-payers try to reduce their tax liabilities. Most of the literature has concentratedon tax evasion as a way of achieving such a reduction. However, this method im-plies that the taxpayers bear some risk since, if they are discovered underreportingtheir true income, they are punished with a penalty. The growing complexity ofthe tax code has given rise to tax avoidance as an alternative method of reducingthe payment of taxes. We mean by tax avoidance the set of actions which allowtaxpayers to enjoy a legal reduction in their tax liabilities. In general these actionsare riskless but costly.1 Usual practices of taking full advantage of the tax codeinclude, for instance, income splitting, postponement of taxes, and tax arbitrageacross income facing dierent tax treatments.2 On the other hand, the emer-gence of certain countries, called tax heavens, where the income is lower-taxed ornon-taxed at all, has encouraged to a great extent tax avoidance activities. Forexample, it is well known that some taxpayers have their scal residence locatedin a tax heaven in order to pay less taxes while their productive activities arelocated in a country exhibiting much higher tax rates.3 In this paper, we willconcentrate on the analysis of tax avoidance associated with the existence of taxheavens. Therefore, we will assume that tax avoidance is linked to the source ofincome so that, when an individual becomes avoider, he shelters all his incomefrom that source.

The behavior of the taxpayer geared towards minimizing their tax bill hasevident implications for the government revenues. Even if the exact measurementof both tax evasion and tax avoidance activities is, by denition, dicult toachieve, some estimations have been performed. For instance, the US InternalRevenue Service has estimated in 1994 an overall compliance rate of 87%, whichmay seem quite high in relative terms but, if we translated this rate into dollarterms, we would see that each 1% increase in compliance translates into USD7-10 billion in terms of government revenues (see Lyons, 1996).

The aim of this paper is to analyze the relationship between the tax rate andthe tax revenue raised by the government when agents choose between evadingand avoiding as two alternatives ways of reducing their tax liabilities. If the taxrate aects the attitude of individuals towards tax compliance, then any changein the tax rate aects the tax revenue in two ways. On the one hand, for a given

1In some cases these actions encompass some uncertainty. This happens when the avoidancestrategies depend on uncertain events or when the hoopholes in the tax law are exploited untilthe margin.

2For more details see Stiglitz (1985).3This is a very common practice for rms in Netherlands, since the Netherlands Antilles is

considered a tax heaven jurisdiction. Other tax heavens, located in Europe, are for exampleAndorra and Monaco, which usually shelter a few rich individuals.

1

level of tax compliance, collected taxes increase with the tax rate. On the otherhand, an increase in the tax rate may increase the set of taxpayers who avoid theirtaxes, and this diminishes the tax revenue obtained by the government. Whenthe former eect is stronger for low tax rates and weaker for high tax rates, wewill be facing the typical Laer curve relating the government revenue with thetax rate, that is, this relationship will exhibit an inverted U shape.

The government can achieve a larger tax revenue by increasing the tax rateuntil its largest feasible value, when the revenue-tax rate relationship is increasing.However, when this relationship has the shape of a Laer curve, as a consequenceof tax evasion and tax avoidance activities, the maximum tax revenue is achievedby selecting a tax rate strictly lower than 100%. Therefore, governments seekingto maximize its revenue will never nd optimal to extract all the income fromthe private sector by setting arbitrarily large tax rates even if the income of eachindividual is unaected by the tax rate.

Some authors have analyzed taxpayer behavior using a representative agentapproach when both avoidance and evasion are jointly chosen. These authorsstudy the implications that this choice has on some traditional results of the taxevasion literature but disregard the analysis of the behavior of the governmentrevenue as a function of the tax rate. In fact, the comparative statics is imme-diate in the previous analysis when the nes are proportional to evaded income:revenues increase with the tax rate as there is less evaded income.4 It shouldbe noticed that these authors emphasize the aspects of complementarity betweenthese two activities. We consider instead a framework where the individuals onlychoose one of the two alternatives. Therefore, we are treating tax evasion and taxavoidance as substitutive strategies and, thus, we are somewhat following the ideaof polarization between evaders and shelters proposed by Cowell (1990). Accord-ing to this author, by choosing to shelter income, an individual draws attentionto himself and therefore becomes a prime target for investigation and, thus, it isinadvisable to try evading and avoiding at the same time. The presence of legaltax shelters induces thus a polarization between evaders and avoiders.

In our work, we dispense with the assumption of a representative agent, sincewe consider that all the individuals have a dierent exogenous income even thoughtheir preferences are represented by a common utility function. This means thatnot all the individuals select the same method for paying less taxes. We will as-sume that the income distribution is uniform. We will also explore the robustnessof our results in an example with a discrete income distribution. Such a discretedistribution will allow us to discuss the implications of income polarization con-cerning the characterization of the revenue-tax rate relationship.5 Notice that

4See, for example Cross and Shaw (1982) and Alm (1988) who analyze both the individualand government behavior when avoidance and evasion are jointly selected, under the assumptionthat avoidance is not a risky activity. See also Alm and Mccallin (1990) who carry out a mean-variance analysis when both evasion and avoidance are considered as risky assets.

5Waud (1988) has studied the existence of a Laer curve for the government revenue ina context where both tax evasion and tax avoidance take place. This author establishes the

2

in our model individuals are indexed by dierent levels of income but both theaggregate income and the distribution of income among individuals will remainunaltered when the tax rate changes. Thus, our analysis focuses exclusively onthe contribution of the tax code and the tax enforcement policy to the revenue-tax rate relationship and then it disregards the eects of tax compliance on laborsupply and other macroeconomic variables.

We show in this paper that the Laer curve could emerge in all the scenarioswe consider. In order to get such an existence result we need to assume that thecost of avoidance is suciently low so as to allow almost all the individuals toavoid their income for high values of the tax rate. The robustness of this resultindicates that in our context a policy of high marginal tax rates might not bethe appropriate strategy when the objective of the government is to maximize itsrevenue.

The paper is organized as follows. Section 2 presents a description of theindividual behavior both when the taxpayer is an evader and when he is anavoider. The tax revenue function of the government is presented in Section 3.Section 4 provides the analysis of the tax rate-tax revenue relationship both whenthe non-avoiders are honest and when they evade. In Section 5 we carry out amore detailed analysis on the existence of the Laer curve when a isoelastic utilityfunction is assumed. Section 6 characterizes the Laer curve when the incomedistribution function is not uniform but discrete. Finally, the main conclusionsand extensions of the paper are presented in Section 7. Most of the formal proofsare relegated to the Appendix.

2. Individual behavior

Let us consider an economy with a continuum of agents who are indexed by theirincome y. Each individual receives an idiosyncratic exogenous income y drawn

from an uniform distribution onh0, Y

i. The preferences of agents are represented

by a common Von Neumann-Morgenstein utility function U() dened on after-tax income I. The rst and second derivatives U 0 and U 00 exist and are continuouswith U 0 > 0 and U 00 < 0. Therefore, agents are risk averse. Moreover, we assumethat the utility function U() satises the Inada condition: limI0 U 0(I) =.

In this economy agents are supposed to pay proportional taxes with a at-ratetax [0, 1]. However, agents have two alternative ways of reducing their taxbill. On the one hand, they can misreport part of their income (tax evasion) and,on the other hand, they can avoid the tax payments using legal methods (taxavoidance). Tax evasion is a risky activity because if the individual is inspected,he has to pay the evaded taxes plus a penalty. Although we will assume that tax

existence a Laer curve for the expected tax revenue in a partial equilibrium context with arepresentative agent. See also Waud (1986) who studied the appearance of the Laer curveof the tax revenue when tax evasion and tax avoidance appear simultaneously in a classicalsupply-side model.

3

avoidance is a riskless action, there are positive costs associated with it. In ourmodel, taxpayers can be classied on one of two categories: evaders or avoiders.The case of honest taxpayers also arises as a corner solution of the evaders problemand will also be studied.

2.1. Tax evaders

Let us consider the standard Allingham and Sadmo (1972) model based on aportfolio approach. In what follows, we will adopt the variation of such a modelpresented by Yitzhaki (1974). The individual declares an amount of income equalto x and he will be audited by the tax authorities with probability p. The trueincome y is always discovered by the inspection. Therefore, agents reduce theirtax payments by (y x) whenever inspection does not occur. If an individualis caught evading, he has to pay a proportional ne s > 0 on evaded taxes.We assume that overreporting is never rewarded. Therefore we can restrict thedeclared income x to be no greater than y.6. On the other hand, x is no lower thanzero since the tax code does not feature a loss-oset, that is, negative declaredincome is viewed by the tax authorities as equivalent to zero declared income.We dene the evaded income as e = yx where e [0, y]. Then, the random netincome of an evader with true income y and evaded income e is equal to7

y y + e, if the individual is not inspected,and

y y se, if the individual is inspected.We assume that the parameters dening the tax inspection policy, p and s, aregiven exogenously throughout the paper.

Each taxpayer chooses the amount e of evaded income in order to maximizehis expected utility

E [U (I)] = (1 p)U(y y + e) + pU(y y se) u(e). (2.1)The rst order condition for an interior solution of the maximization of (2.1) is

(1 p)U 0 (y y + e) pU 0 (y y se) s = 0. (2.2)The corresponding second-order condition is automatically satised because ofthe assumption of concavity of the utility function.

The following lemma shows the solutions for the maximization problem (2.1):

Lemma 2.1. The optimal evaded income e (, y) as a function of the tax rate and the individual income y is:

6When an individual declares more than his true income, he will only receive the excess taxcontribution if he is audited. Therefore, s = 1 whenever x > y.

7Note that y y + e = y x.

4

(a) if (1 + s) < 1,

e (, y) =

0 for s 1pp

be(, y) for 1pp U 0(y)U 0(yysy) < s < 1ppy for s

1pp

U 0(y)

U 0(yysy) ,

(b) if (1 + s) 1,

e (, y) =

0 for s 1pp

be(, y) for s < 1pp ,where be(, y) is the solution to equation (2.2) for the evaded income e.Proof. See the Appendix.

Lemma 2.1 states the intuitive result that an evader taxpayer will declareless than his actual income if the penalty s is small enough, whereas the amountdeclared is positive whenever s is large enough. Note that when s 1pp taxpayersbecome honest.

Finally, substituting e (, y) into the expected utility we have

V (, y) = (1 p)U (y y + e(, y)) + pU (y y se(, y)) , (2.3)

which is the maximum value achieved by the expected utility, i.e., the indirectutility function.

2.2. Tax avoiders

The structure of the tax system may oer some ways of reducing the tax bill of theagents. This implies that they will pay less taxes since the avoided income will betaxed at a lower tax rate. To simplify the analysis, let us assume that the avoidedincome is taxed at a zero tax rate.8 Moreover, we consider that tax avoidancetakes the form of sheltering a source of income in a tax heaven. Since we assumethat individuals have only a source of income (typically associated with a singleeconomic activity), when an individual decides to become an avoider, he sheltershis total income.

There exist positive costs associated with the tax avoidance activity given bythe function C(y). These costs include, for example, the cost of obtaining therelevant information about the tax clauses, or the payment to a tax advisor, orthe necessary amount to create a rm in order to declare the personal income

8This analysis can easily be modied to accommodate a positive tax rate for avoided income.

5

as corporate income, which in some countries is lower-taxed. We assume thatC(y) = cy + k, where k is a positive xed cost and c (0, 1) is a proportionalcost per unit of avoided income. Note that the existence of a xed cost willprevent the poorest individuals from avoiding since they can not aord it. Theutility achieved by an avoider is thus U(y cy k). Observe that if there is anindividual who wants to be an avoider then the following two conditions musthold simultaneously: (i) > c and (ii) k (1 c)Y . Condition (i) eliminates thevalues of the tax rate for which tax avoidance never takes place, since for c anindividual obtains more utility being honest than being an avoider.9 Condition(ii) ensures that at least the richest individual will be an avoider for > c. Inorder to simplify the analysis, we will assume that condition (ii) holds in whatfollows.

3. The Government

The government obtains resources from the taxes paid by the individuals whoare not avoiders and from the nes that individuals must pay if they are caughtevading. Since the government inspects each individual with probability p, afraction p of individuals is inspected in this large economy. For the sake ofsimplicity we assume that there are no costs associated with tax inspection.10

Note that the density function f(y) of a uniform distribution of income onh0, Y

iis

f(y) =

1Y

for 0 y Y

0 elsewhere

.

Therefore, the total government tax revenue per capita is given by the followingLebesgue integral

G() =

ZA()

[(1 p) (y e(, y)) + p (y + se(, y))] 1Ydy, (3.1)

where A () is the Lebesgue measurable subset of incomes inh0, Y

ifor which

the individuals are non-avoiders when the tax rate is . Obviously, G () is acontinuous function on [0, 1] . Note that avoiders do not pay taxes so that evadersare the only individuals who pay taxes, although possibly less than what they

9The utlity achieved by a honest tax payer is U (y y). Since the utility function is mono-tonically increasing and c, we have

U (y y) U (y cy) > U (y cy k) .

10Note that the assumption of a zero inspection cost does not aect the present analysis if thegovernment is unable to identify ex-ante who is an evader and who is an avoider. In this case,under a constant probability p of inspection, the total inspection cost will be unaected by theproportion of evaders and avoiders and, thus, it will be independent of the tax rate.

6

should. Note also that the tax rate aects G () through its eects both one (, y) and on the set A (). Then, our objective will be to analyze the behaviorof the government tax revenue G () when the tax rate changes. The next sectionprovides the corresponding analysis.

4. The Laer curve

In this section we will analyze whether the relationship between the tax rate andthe government revenue can be described by a Laer curve. There exist severaldenitions of the Laer curve. In particular we will consider only two of them.The weakest one involves a non-monotone relationship between G() and . Astronger denition of the Laer curve requires the existence of a single maximumfor G() in the open interval (0, 1). In this case the Laer curve displays thetypical inverted U shape. Laer curves conforming the dierent previous twodenitions will be found throughout the paper.

To carry out the analysis we need to separate the case where non-avoidersare honest from the case where they are strict evaders. In the former case un-certainty vanishes since individuals declare their true income, while in the lattercase individuals hide part of their true income and they face up to a potentialinspection which will reduce their net income.

4.1. Honest behavior

In this case the individuals who are not avoiders report their true income sothat e (, y) = 0. Following Lemma 2.1 this occurs when s 1pp . To studyhow the total tax revenue is modied when increases, we need to know whichindividuals become avoiders. This decision depends on the dierence betweenthe cost of being honest and the cost of being an avoider. For example, if eitherthe costs associated with tax avoidance are high or the tax rate is low, poorindividuals will prefer paying their corresponding taxes. Thus, for an individualwith income y, avoiding is optimal whenever11

U(y y) U(y cy k). (4.1)Solving (4.1) for , we obtain that

(y) c+ ky.

Therefore, for < c + ky the individual with income y will prefer being honestrather than avoider. In particular, we can nd the tax rate that leaves the richestindividual indierent between avoiding and being honest, that is,

(Y ) = c+k

Y. (4.2)

11We adopt the innocuous convention that when an individual is indierent between beinghonest and avoiding, he is honest.

7

Observe that (Y ) is greater than the proportional cost of avoiding c if the xed

cost k is strictly positive. Hence, for < Yall the individuals will pay their

corresponding taxes. Obviously, (Y ) < 1 for k < (1 c)Y .Now we need to know how many individuals will be honest for each possible

value of the tax rate above Y. It turns out that, for

h(Y ), 1

ithere exists a

threshold level of income yH() 0, Y

making an individual indierent between

being honest and avoiding. In particular, yH() is the value of y satisfying (4.1)with equality. Formally,

yH() =k

c. (4.3)Therefore, the individuals having incomes lower than yH() will tell the truth,while the individuals with higher incomes will be avoiders. Note that when k = 0,yH() = 0. Hence, all the individuals become either honest or avoiders. Note alsothat yH() decreases with so that, if the tax rate increases, the lower incomeindividuals will also benet from avoidance.12 Hence, for

Y< 1 only the

individuals who can aord the xed cost will be avoiders. In this case, the taxrevenue is given by the total taxes paid for the individuals who are honest.

The following Lemma gives us the expression for the total tax revenue func-tion:

Lemma 4.1. Let s 1pp . Then, the total tax revenue will be given by thefollowing two-part function:

G() =

Y2 if 0

Y

2Y

k

c2

if Y< 1

Proof. See the Appendix.

The next proposition characterizes the behavior of G() with respect to thetax rate.

Proposition 4.2. Assume that s 1pp . Then, the total tax revenue G() isa continuous function on [0, 1] that achieves its unique maximum value when

= Y.


The intuition of this result is quite obvious. When the tax rate of the economy

is lower than Y, all the individuals pay their corresponding taxes and the

12Observe that yH(Y) = Y and, thus, the threshold level yH() will be lower than Y for

Y, 1, since yH() decreases with .

8

tax revenue is increasing in the tax rate. On the other hand, when the tax rate

is higher than Y, two dierent eects take place. The rst one is the same as

before since higher tax rates have a positive eect on the tax revenue. However,there exists now an additional eect since an increase in the tax rate impliesthat some individuals (the richest ones) prefer now becoming avoiders and, inconsequence, they do not pay taxes any longer. This eect has a clear negative

impact on the tax revenue. As the Proposition 4.2 shows, for > Ythe

negative eect outweighs the positive one and the tax revenue becomes decreasingin the tax rate. Note that the Laer curve obtained in this subsection conformswith the stronger denition of such a curve, that is, the curve has a uniquemaximum on (0, 1) .

As an illustration of Proposition 4.2, Figure 1 displays the total tax revenueas a function of the tax rate when c = 0.1, Y = 100, k = 1. In this case we obtain

a standard Laer curve with a maximum at Y= 0.11

[Insert Figure 1 about here]

4.2. Tax Evasion

When s < 1pp . program (2.1) yields a strictly positive amount of evaded income,e(, y) > 0. The decision of being either an avoider or an evader will depend on theindividual income and the values of the parameters of the model. For c all theindividuals will prefer evading, since the avoidance activity is only attractive fortax rates strictly greater than c. Therefore, for 0 < c the government obtainssome positive revenue at least from the nes collected by the inspection.13

For > c, tax avoidance can appear. Thus, an individual prefers avoiding toevading when V (, y) < U(y cy k), and vice versa. Hence, given p, s, c, andk, we can dene y() as the income that leaves an individual indierent betweenevading and avoiding when the tax rate is , that is,

V (, y()) = U(y() cy() k), (4.4)

where the function V (, y) is dened in (2.3) . The following lemma establishesthe existence of such a threshold income level y() :

Lemma 4.3. Assume that s < 1pp , and (1 + s) > 1 + sc. Then there existsan income y() > 0 such that all the individuals having an income lower thany() are evaders, while all those with an income higher than y() are avoiders.


13The government does not know if the individuals will prefer evading all their income orevading only a part of it. Note that the nes collected include the amount of evaded taxes plusthe penalty paid as a punishment to evasion.

9

As we expected, the previous results tell us that the richest individuals avoidthe payment of their taxes while the poorest ones choose the evasion as a way ofreducing their tax liabilities. This is so because the poorest individuals can notaord the xed cost k of avoiding. Notice that assuming that (1 + s) > 1 + sc,implies that (1 + s) > 1, which ensures that the problem (2.1) has an interiorsolution.14 In other words, Lemma 4.3 only applies for 0 < e(, y) < y. This factdoes not constitute a very restrictive assumption since the available data abouttax evasion shows that tax evasion takes place in a partial way.

The next proposition summarizes the main result concerning the existence ofthe Laer curve in this context:

Proposition 4.4. Assume s < (1p)p . If the xed cost k is suciently small,then the total tax revenue G() is non-monotonic in , for [0, 1] .


Note that, when tax evasion occurs and the xed cost of avoidance is low, weare only able to make a characterization of the Laer curve in the weakest sense.To ensure that almost all individuals are avoiders when the tax rate is convergingto one, we need to assume that the xed cost is small enough. Therefore, thetax revenue will take a low value because only the poorest individuals pay taxes.On the contrary, in the absence of tax evasion (see subsection 4.1), we were ableto make a stronger characterization which embodied the existence of a singlemaximum of the revenue function without any additional assumption about thexed cost.

The importance of Proposition 4.4 is evident since a higher tax rate doesnot mean a higher tax revenue if individuals can avoid their income taxes. Forinstance, if individuals did not have the possibility of avoiding, an increase inthe tax rate implies an increase in the tax revenue under decreasing absolute riskaversion.15 This is so because, when the ne is proportional to the amount ofevaded taxes, an increase in the tax rate translates into a rise in the amountthat the taxpayer has to pay as a penalty. This induces less evasion since thesubstitution eect has been eliminated as a consequence of imposing penalties onevaded taxes and not on evaded income. Therefore, the nal outcome will be anincrease in the tax revenue.16

14See Yitzhaki (1974).15A sucient condition to ensure that nobody wants to avoid is k > (1 c)Y .16If tax avoidance is not allowed, then the tax revenue is given by

G( ) =

Z Y0

[(1 p) (y e(, y)) + p (y + se(, y))] 1Ydy.

Calculating the derivative of G() respect to the tax rate we have

dG( )

d=

Z Y0

(1 p)

y e(, y) e(, y)

+ py + ps

e(, y) +

e(, y)

1

Ydy.

10

It is important to remark the important role that tax avoidance costs playin the analysis that we have just carried out. First, the xed cost is necessaryto guarantee that not everybody benets from avoidance. Second, if the costsassociated with the tax avoidance are too high, nobody wants to avoid and thetax revenue function will become increasing in the tax rate.

In the next section, we examine exhaustively an example with an isoelasticutility function which will allow us a more precise characterization of the revenuefunction G () .

5. An example: the isoelastic case

The isoelastic utility function has been widely used in several instances of eco-nomic analysis ranging from nancial economics to macroeconomics. Let us onlyconsider the interesting case where the evasion takes places (i.e. s < 1pp ) sincethe honest case has been fully characterized in Section 4.1 without using anyspecic functional form.

The optimal evasion when the utility function is U(C) = C11 , with > 0,

becomes the following:

e(, y) =

y if h(A1)(1)(1+As)

iy if > .

(5.1)

where A =

ps

1 p 1

, and (0, 1) is the tax rate which separates full frompartial evasion. Notice that A > 1 since s < 1pp . The value of

is

=1

sp1p

1

1 + s, (5.2)

which is obtained rearranging the condition for e < y given by Lemma 2.1. Wecan observe that the optimal evasion is a constant proportion () [0, 1] of thetrue income y, where () can be expressed as

( ) =

1 if

(A1)(1)(1+As) if >

.(5.3)

The decision to be a full or a partial evader depends on the level of the taxrate. When the tax rate is small enough, individuals decide to evade all theirincome because, if they are inspected, the penalty that they must pay is not

It can be seen that dG()d

> 0 since e(,y)

< 0 under the assumption of decreasing absolute

risk (see Yitzhaki, 1974) and s < (1p)p.

11

very large. Nevertheless, when the tax rate increases the ne paid becomes alsohigher and this discourages individuals from evading all their income. Note that (1) = 0. This means that the optimal evasion is equal to zero when = 1 as aconsequence of the assumption limI0 U 0(I) =.

To see if the tax revenue displays the shape of a Laer curve, we need to knowhow the individuals modify their behavior when the tax rate changes. Obviously,if individuals become avoiders, they do not pay taxes any longer and this facthas a signicative impact on the tax revenue. Basically, we will consider twodierent values for the tax rate which imply qualitative changes in the individualsbehavior. On the one hand, we have the value of the tax rate which separatesfull from partial evasion. This value depends on the policy inspection parameters,s and p, and the parameter of the utility function. On the other hand, wehave the proportional cost c of avoidance which aects the decision of becomingavoider. In this framework, we consider two possible scenarios: c and > c.The main dierence between these two cases is that when c the taxpayersare only full evaders for very low values of the tax rate ( ) , while they arefull evaders for a larger range larger of tax rates whenever > c.

5.1. CASE 1: cThe rst step is to calculate the dierent components of the tax revenue functionG (). If 0 , everybody prefers evading since < c. Furthermore,as all the individuals will choose to be full evaders, that is, e (, y) = y .Hence, the tax revenue will be only composed by the penalties paid by the con-sumers who are caught evading.

When the tax rate is higher than , evaders choose to declare only a part oftheir true income, that is, e(, y) < y.On the other hand, all the individuals preferbeing evaders for all [0, c] . Therefore, we need to know if some individualprefers becoming avoider in the interval (c, 1]. Observe that for an individualwith income y we can nd the value of the tax rate which leaves him indierentbetween evading and avoiding. More precisely, this value is such that satises thefollowing equality:

(1 p)y y + (A 1) (1 )

(1 +As)y

1+

p

y y s (A 1) (1 )

(1 +As)y

1= (y cy k)1 . (5.4)

Solving for we get17

b (y) = 1 (1 c)D

kDy

, (5.5)

17We adopt the innocuous convention that when an individual is indierent between evadingand avoiding, he evades.

12

where

D

1 + s

1 +As

(1 p)A1 + p

11

. (5.6)

Observe that b (y) is decreasing in y and this means that for high income levelsb (y) will be small because the rich individuals can meet the cost of avoidancemore easily. In particular, the rst potential avoider is the richest individual, i.e.,the one with y = Y . Therefore, we can nd which is the tax rate that leaves therichest individual indierent between evading or avoiding by substituting y = Yinto the expression (5.5). Thus, we get

b(Y ) = 1 (1 c)D

kDY

. (5.7)

The following Lemma ensures that b Y (c, 1) :Lemma 5.1. Assume s < 1pp , then

(a) D > 1.

(b) b Y (c, 1) .Proof. See the Appendix.

Note that the value b Y is greater than c because the richest individualprefers being an evader when = c. This implies that, if b Y , all theindividuals prefer evading. Thus, as the evasion takes place only partially, thetax revenue comes both from the taxes voluntarily paid and from the nes paidby the inspected individuals.

Finally, when b Y < 1, it is no longer true that the best option forall individuals is to become evaders. As the tax rate is growing the individualswho enjoy higher incomes will tend to prefer avoiding. Therefore, there exists anincome yP () that makes an individual indierent between being a partial evaderand being avoider for a given tax rate. We can calculate explicitly this thresholdlevel since yP () is such that (5.4) holds. The value of yP () is given by

yP () =k

(1 c) (1 )D. (5.8)

Note that Y, 1iensures that yP () < Y . Thus, in this case the tax

revenue is only composed by the payments collected from the individuals who donot avoid.

The following Lemma gives us the expression of the function G() :

13

Lemma 5.2. Let s < 1pp and c. Then, the total tax revenue will be given

by the following three-part function:

G() =

12p (1 + s)Y if 0

12 (1 () + p (1 + s)( ))Y if < b Y 12Y

(1 () + p (1 + s)()) (yP ())2 if b Y < 1Proof. See the Appendix.

The next proposition describes the Laer curve.

Proposition 5.3. Assume that c and s < 1pp . Then, the total tax revenueG() is strictly increasing on

h0, b Y and strictly decreasing on b Y , 1i .


We can illustrate the previous proposition by evaluating the tax rates b(Y )given in expression (5.7) and given in expression (5.2) at p = 0.1, s = 3,c = 0.15, Y = 100, = 2 and k = 1. We get = 0.10566 and b Y = 0.19376.Observe that in this case s < 1pp ,

< c and b Y > c. Figure 2 shows the taxrevenue for these values.


Note that the function G() is increasing in 0, b Y , since all the indi-

viduals prefer being evaders when < b Y . This means that a higher tax rateimplies a higher tax revenue as a consequence of two positive eects. First, theincrease in the tax rate causes a raise of the tax revenue accruing from declaredincome. Second, the individuals declare a higher part of their true income since

evaded income is decreasing in . When > b Y not all the individuals wantto be evaders. In particular, only the individuals with an income greater thanyP () will evade. Thus, in this case we can distinguish two dierent eects ontax revenue. On the one hand, there is a positive eect due to the fact that ahigher tax rate implies that the individuals who are evaders pay more taxes. Onthe other hand, we have a negative eect, since an increase of the tax rate causesa decrease on yP () which implies that less people pays taxes. As it has beenproved, the second eect osets the rst one and the tax revenue is decreasing in

for b Y , 1i . Therefore, the maximum value of the government revenue

is reached at the tax rate b Y .14

Finally, let us point out that for the case < c we have been able to ob-tain a Laer curve in its strongest sense under isoelastic preferences, while thischaracterization was not available without such a parametrization of the utilityfunction (see Proposition 4.4). Moreover, we have dispensed with the more vagueassumption of suciently low xed avoidance cost required in Proposition 4.4.

5.2. CASE 2: > c

Like in case 1 we will calculate the multi-part tax revenue function. We knowthat if < , the evader individuals are full evaders, that is, they evade all theirtrue income. Moreover, all the individuals prefer evading rather than avoidingif c. Therefore, we need to know which will be the behavior of the richestindividual when = . If he prefers being full evader, all the individuals will befull evaders. This situation would be similar as the one analyzed in the case 1.On the other hand, the richest individual could prefer being avoider at = .The following lemma establishes the condition under which such a circumstanceoccurs:

Lemma 5.4. The richest individual of this economy strictly prefers being anavoider to being a full evader at = if and only if

k < Y

(1 c)

h(1 p) + pA1

i 11

. (5.9)


Observe that the fulllment of this condition requires small values for boththe xed cost k and the proportional cost c since this makes easy for the richestindividual to become avoider at = .

Assume from now that condition (5.9) holds. In consequence, we can en-sure that there exists a tax rate smaller than leaving the richest individualindierent between being full evader and avoiding. In particular, this tax ratesatises:

(1 p)hYi1

+ phY Y sY

i1=Y cY k

1. (5.10)

Solving for we obtain

e Y = 11p

1 c k

Y

1 (1p)p 111 + s

. (5.11)

Note that condition (5.9) guarantees that e Y (c, ) .Then, for 0 e Y we have that all the individuals are full evaders. Therefore, the tax revenue willbe equal to the penalties paid by full evaders which have been inspected.

15

When e Y it is not longer true that everybody prefers evadingto avoiding. We dene yT () as the income for which an individual is indierentbetween evading all his true income and avoiding. The value of yT () is18

yT () =k

(1 c) ((1 p) + p(1 s)1) 11.

Observe that k 0, (1 c)Y

and

e Y , i ensure that yT () 0, Y .Hence, the tax revenue is only composed by the penalties paid by the inspectedindividuals who prefer being full evaders.

Finally, if < 1, the evaded income is lower than the true income sincefor > the full evaders become partial evaders.19 In this case the tax revenuecollected by the tax authorities comes both from the taxes voluntarily paid andfrom the penalties paid by the inspected partial evaders.

The following Lemma gives us the expression of the function G() :

Lemma 5.5. Let s < 1pp and > c. Then, the total tax revenue will be given

by the following three-part function:

G() =

12p (1 + s)Y if 0 e Y 12Yp (1 + s) (yT ())

2 if e Y < 12Y

(1 () + p (1 + s)()) (yP ())2 if < 1.


The following proposition gives us the results obtained concerning the exis-tence of the Laer curve:

Proposition 5.6. Assume > c and s < 1pp . The total tax revenue G() isnon monotonic in , for [0, 1] if k < (1 c)Ypc (1 (c) + p (1 + s)(c)).Proof. See the Appendix.

In order to illustrate this result we evaluate the expressions (5.11) and (5.2) atp = 0.1, s = 1.5, c = 0.1, Y = 100, = 2, k = 1. Thus, we have = 0.10566 andb Y = 0.19376. Figure 3 shows the tax revenue function G () for the previousparameters values:


18The threshold yT ( ) is obtained equating the utility from full evasion to the utility fromavoiding.19Note that yT () = y

P ( ) for =

.

16

When > c we only are able to make a characterization of the Laer curve inthe weakest sense like in the general case (see subsection 4.2). However, we donot invoke the assumption of suciently low xed avoidance cost to obtain theLaer curve since in this particular case we have found an explicit formula forthe threshold xed cost of avoidance below which a Laer curve is obtained.

6. A discrete distribution of income

In the previous setup, we have assumed that the income distribution of the indi-viduals of the economy was uniform. However, we can ask if the previous resultsabout the existence of a Laer curve relating tax revenues with tax rates wouldchange if the income distribution is modied. Since it is impossible to addresssuch a question with enough generality, in this section we consider a simple dis-crete income distribution with only three income classes y1, y2 and y3, wherey1 < y2 < y3. Individuals are distributed into these three classes according to theproportions {1,2,3} respectively, where 1+2+3 = 1. In order to simplifythe exposition we assume, without loss of generality, that y1 = 0, y2 = m andy3 = 1, where 0 < m < 1. The combination of the proportions 1,2, and 3 aswell as the size of m, will allow us to analyze dierent types of discrete incomedistributions. Observe that this distribution corresponds to an economy with ahigh degree of income polarization whereas such a polarization was absent undera uniform distribution. For simplicity we adopt the isoelastic utility functiontaken in Section 5. The present analysis will be divided into two parts: rst weassume that c holds and we then will examine the case where c < .

6.1. CASE 1: cLike in Section 5 we will nd the dierent parts of the tax revenue function.Hence, for [0, ] all the individuals evade all their income, and then the taxrevenue is given by the penalties paid by the inspected evaders. In this contextthe individuals with income y1 = 0 do not contribute to the tax revenue becausealthough they declare all their true income, the taxes paid are zero.20

When the current tax rate is greater than , partial evasion takes place.Moreover, for c all the individuals will want to be evaders. However, whenthe tax rate is greater than c, the advantage of evasion over avoidance only comesfrom the xed cost associated to avoidance. Similarly to the case with an uniformdistribution of income, we can nd which is the tax rate that leaves the richestindividuals indierent between avoiding and evading. Formally, we obtain thisvalue substituting the value of y3 into the expression (5.5). This yields

by3 = 1 (1 c) kD. (6.1)

20Note from (5.1) that the individuals with y1 = 0 do not evade.

17

For (, by3 ] everybody is a partial evader. In this case, the tax revenue iscomposed both by the taxes voluntary paid and by the penalties from inspectedindividuals. Note that there exists a discontinuity at = by3 since the richestagents stop paying taxes and the tax revenue then falls. The magnitude of thisjump depends on the proportion 3, since a proportion very large of rich individ-uals implies that a large fraction of people does not pay their taxes any longerand this fact causes a dramatic decrease of the tax revenue. Hence, for > by3only the individuals who have income y2 continue paying taxes. However, as thetax rate raises the benet obtained from evading becomes smaller since a hightax rate diminishes the expected utility of an evader. Then, from expression (5.5)the value of the tax rate that will leave the individuals with income y2 indierentbetween evading and avoiding is

by2 = 1 (1 c)D kDm. (6.2)

Then, by2 is as the maximum value of the tax rate that has associated a positivevalue of the tax revenue since for > by2 the tax revenue becomes zero. Therefore,for (by3 , by2 ] the tax revenue is equal to the taxes voluntarily paid by theindividuals with income y2 and to the nes paid by the inspected ones.

Observe from expression (6.2) that by2 < 1 if k < (1 c)m. This is so becausewhen the xed cost is large enough, the individuals with income y2 never resortto avoidance and thus, they pay taxes until = 1. Obviously, for > by2 allthe individuals except the ones who have income y1 prefer avoiding and, as aconsequence, the tax revenue becomes zero.

The next Lemma gives us the expression of the function G().

Lemma 6.1. Let s < 1pp and c. Then, the total tax revenue will be given

by the following four-part function:

G() =

p (1 + s) (2m+ 3) if 0

(2m+ 3) [ (1 () + p (1 + s)())] if < by32m [ (1 ( ) + p (1 + s)())] if by3 < by20 if by2 < 1.


The following proposition summarizes the results concerning the existence ofthe Laer curve when dierent assumptions on the proportions i when i = 1, 2, 3are imposed.

Proposition 6.2. Assume that s < 1pp and k (0,m(1 c)).

18

(a) if 3 2, then the tax revenue function G() has a single maximum at = by3 ,

(b) if the proportion 3 of rich individuals is small enough, the tax revenuefunction G() has a single maximum at = by2 .Proof. See the Appendix.

The intuition of result (a) relies on the fact that, if 3 is large enough, whenthe tax rate is equal to by3 many rich individuals stop paying taxes because theyprefer avoiding their income. This constitutes a very signicant reduction of thecollected taxes since the great source of revenues comes from the rich individuals.Although the tax rate rises, the tax revenue can not recover the previous valueachieved when = by3 . This is so because the individuals with income y2, whichare the only ones who pay taxes, have not enough income to oset the loss thatthe avoidance of the rich individuals brings about. Observe that the assumptionabout the size of the xed cost k allows the individuals with income y2 to avoid hisincome for > by2 , since otherwise when k (m(1 c), (1 c)) those individualsnever benet from avoidance for (0, 1).

Figure 4 shows the behavior of the tax revenue described in part (a) of Proposi-tion 6.1. The parameters values considered are: p = 0.1, s = 3, = 0.5, 1 = 0.4,2 = 0.2, 3 = 0.4, m = 0.7, c = 0.25, and k = 0.25.

[Insert Figure 4 about here] .

On the other hand, the part (b) of the previous proposition says us thatwhen 3 is quite small there is almost no dierence between G2() and G3().Although rich individuals are avoiders, their weight on the total income is verysmall, and the tax revenue quickly osets the loss caused by the avoidance of therich individuals. Note that if 3 < 2 but 3 and 2 are close enough, we get thesame results as in part (a) of Proposition 6.2.

Figure 5 illustrates, the behavior of the tax revenue described by part (b) ofProposition 6.1 when p = 0.1, s = 3, = 0.5, 1 = 0.1, 2 = 0.8, 3 = 0.1,m = 0.7, c = 0.25, and k = 0.25.

[Insert Figure 5 about here] .

Observe that, even if the results stated in Proposition 6.2 do not dependexplicitly on the value of m, this value plays an important role in the analysis ofthe behavior of the government revenue under alternative income distributions.An increase of m, for given values of 1, 2, and 3, gives rise to two oppositeeects on the government revenues. On the one hand, if the value of the parameterm increases, then the individuals having an income equal to y2 become richer and,therefore, they pay more taxes and contribute more to the revenues raised by the

19

government. On the other hand, from inspection of (6.2) we can check that thevalue of the tax rate that will leave the individuals with income y2 indierentbetween evading and avoiding decreases as m increases. This implies in turn thatthe government is not going to get revenues for lower values of the tax rate.

In order to get a more precise intuition on the role played by the parameterm,let us consider an income distribution where 1 and 3 are small and 2 is large.In this context, whenm is close enough to zero one obtains an income distributionwhere a vast majority of individuals is poor. In this case, the government couldset very high tax rates since poor individuals will keep paying taxes since theycould not aord the avoidance costs. This kind of scal policy ends up being veryregressive since the individuals that contribute to the increase in governmentrevenues are just the ones having lower income. However, when m is close to 1and the vast majority of individuals is thus quite rich, the government should settax rates not very high so as to prevent individuals from avoiding. Obviously,since now the individuals enjoy a higher income, the value of the governmentrevenue is generically higher than in the previous case displaying high tax rateswith poor individuals.

Finally, observe that when the income distribution function is discrete, theexistence of the Laer curve in the weak sense is always guaranteed by the non-continuity of the function G().

6.2. CASE 2: > c

The results obtained in this case will depend on the behavior that individualsadopt at = . More specically, we will have the following scenarios accordingto the values of the parameters:

Scenario A0 < c < ey3 < ey2 < < 1 when k < m

Scenario B0 < c < ey3 < < by2 < 1 when m < k < ,

where

=

1 c

1 p+ pA1

11

,

ey3 = 1 1p (1ck)1 (1p)p 111+s ,and

ey2 = 1h1p(1c km)

1 (1p)p

i 11

1+s . (6.3)

20

The values of ey3 and ey2 have been obtained substituting y3 = 1 and y2 = m inexpression (5.11) .

Scenario A presents a situation where the xed cost k is low enough to induceboth the richest individuals and the individuals with income m to prefer avoidingat = . Using the same arguments as in Sections 5 it is easy to see that for [0, ey3 ] all individuals are full evaders. For (ey3 , ey2 ] the richest individualsavoid their income while the rest of the individuals still are full evaders. Finally,nobody pays taxes for (ey2 , 1]. Hence, the tax revenue function is given by

G() =

p (1 + s) (2m+ 3) if 0 ey3p (1 + s)2m if ey3 < ey20 if ey2 < 1.

In scenario B, the xed cost is not low enough for allowing the individualswith income m to avoid their income when = , then they evade all their trueincome. Then, using the arguments seen in the previous sections it is easy to seethat for [0, ey3 ] everybody will evade. For (ey3, ] the richest individuals(with income y3) are avoiders and the individuals with income m are full evaders,for (, by2] the individuals with income m still being evaders but now theyare partial evaders. Finally, the tax revenue is zero for (by2 , 1] because theonly individuals who do not avoid their income (individuals with income y1) donot even pay taxes. Summarizing, we have that the tax revenue function is thefollowing four-part function:

G() =

p (1 + s) (2m+ 3) if 0 ey3p (1 + s)2m if ey3 < 2m [ (1 () + p (1 + s)())] if < by20 if by2 < 1.

The main result concerning the existence of the Laer curve both in scenario Aand in scenario B is that the Laer curve always exists, and its specic formdepends on the values taken by the proportions 1, 2 and by the income m.Next proposition summarizes the main results about the Laer curve under thetwo scenarios considered above:

Proposition 6.3. Assume that s < 1pp . Then,(a) if k < m and the income m is large enough, the tax revenue function

achieves its unique maximum at = ey3 ,(b) if k < m and the proportion 3 is small enough, the tax revenue function

achieves its unique maximum at = ey2 ,21

(c) if m < k < and the proportion 3 is large enough, the tax revenuefunction achieves its unique maximum at = ey3 ,

(d) if m < k < and the proportion 3 is small enough, the tax revenuefunction achieves its unique maximum at = by2 .Proof. See the Appendix.

The intuition behind the statement (a) is quite clear because, if m is close toone, it means that the individuals with income y2 are rather rich and in con-sequence they will resort to tax avoidance almost for the same tax rates asthe richest individuals. Thus, the tax revenue becomes zero for > ey2 wheney2 (ey3 , ey3 + (m)) , where is decreasing in m and limm1 (m) = 0. Hence,for m large enough we will have that the tax revenue will be zero before thepenalties paid by the inspected individuals with income y2, can oset the loss inthe tax revenue caused by the richest individuals. The intuition of the statements(b), (c) and (d) is supported by the role that the proportion 3 of richest indi-viduals plays. If 3 is large enough the loss in the tax revenue can not be osetby the resources obtained from the individuals with income m. When 3 is smallenough, the previous argument works in the opposite direction since the loss dueto the tax avoidance by the richest individuals is very small and it can be osetby the payments made by the individuals with income y2.

Finally, we have to point out that the scenarios considered when > c do notconstitute a good approach of the situation in the real world since the availabledata of tax evasion reveals that tax evasion takes place in a partial way, i.e.,e < y.

7. Conclusions and extensions

We have shown that the possibility of choosing between avoiding and evadingbrings about a tax revenue function exhibiting the shape of a Laer curve. Thatis, the relationship between tax rates and government revenue is non-monotonic.We have carried out the analysis in a partial equilibrium context where the in-dividuals have the same utility function but dier in their incomes. In all thescenarios studied we have found that the tax revenue always displays a Laercurve under some conditions. This fact has to be taken into account when thedierent governments design the scal policies because, when the tax avoidancephenomenon is present, raising the tax rate might not result in an increase of thetax revenue. Moreover, when the costs associated with tax avoidance are low, toset high tax rates and to implement a strong anti-evasion policy is not only inef-fective but also regressive because all the rich individuals will avoid their incomesand will pay no taxes. Under this policy only the poorest individuals will paytaxes (and/or penalties associated with tax evasion) since they cannot aord thecost of avoidance. In fact, a government could reduce the negative impact on tax

22

revenues accruing from the possibility of tax avoidance by implementing the kindof policies analyzed by Chu (1990). He proposes a new policy called FATOTAwhich allows a target group of corporations to choose between two options: topay a xed amount of taxes set by the tax authorities and thus to be exemptedfrom tax inspection, or to pay only an amount chosen by the corporation but besubject to a positive probability of being audited. This type of policies wouldbecome eective provided the xed amount to be paid by the potential avoiderswere smaller than the costs associated with avoidance.

Some extensions of the present work are possible. We comment on some ofthem. We could introduce a progressive tax function in our framework and toanalyze the eects on government tax revenue of a modication in the progres-siveness of the tax function. The result one should expect is that a higher degreeof progressiveness will stimulate the avoidance of rich individuals and, therefore,the decreasing part of the Laer curve will appear at lower average tax rates.

We could also consider other continuous income distributions, like the lognor-mal or the Pareto ones, which could better t the empirical income distributions.Nevertheless, such functional form would prevent us from getting explicit resultsand we should rely instead on simulations.

Finally, in our model all the individuals have the same utility function andthey only dier in their income. It could be interesting to consider heteroge-nous preferences. To this end, we could assume a distribution on the relevantparameters characterizing the indexes of risk aversion of the individuals.

23

A. Appendix

Proof of Lemma 2.1. Since the function u(e) dened in (2.1) is strictly concave,we obtain the corner solution e = 0 whenever u0(0) 0. This weak inequality isin fact equivalent to s 1pp . Notice that when (1 + s) 1, it is impossibleto have e = y since, then, the income of an inspected individual y (1 + s)ycannot be non-positive as follows from the Inada condition. For (1 + s) < 1, toobtain the corner solution e = y, we need that u0(y) 0. This inequality becomes

s 1 pp

U 0(y)

U 0(y y sy) .

Proof of Lemma 4.1. For 0 Ythe tax revenue is composed by the

taxes that all the individuals pay. This revenue is

G1() =

Z Y0

y 1Ydy.

For Y< 1 the tax revenue is given by the total taxes paid for the

individuals who are honest. This revenue is,

G2( ) =

Z kc

0y 1

Ydy.

Performing the integrals Gi () where i = 1, 2, we obtain the expression ap-pearing in the statement of the Lemma.

Proof of Proposition 4.2. Dierentiating G1 () respect to we havedG1d =

Y2

which is unambiguously positive. On the other hand, dierentiating G2 () re-

spect to we obtain dG2d = k2(+c)

2Y (c)3 , which is unambiguously negative. Obvi-

ously, the maximum value achieved by G() is just the kink point = c+k

Y.

Proof of Lemma 4.3. The proof adapts some of the arguments in Chu (1990).

Step 1. We will prove that V (, y) and U(y cy k) intersect at least once.As when the true income tends to zero, the optimal evasion also tends to zero,

we have that at least for y < kc , U(y cy k) < V (, y), since k is positiveand by assumption > c. Then, we only have to see that there exists an incomelevel such that U(y cy k) > V (, y) for a given . Consider the level evasione0 (, y) =

(1)ys , which is less than y since we were assuming that (1 + s) > 1

holds. It can be seen that

y y + e0 = (1 + s) (1 ) ys

24

and

y y se0 = 0.Therefore, we get that

u0(e0) = (1 p)U 0(1 + s)(1 )y

s

pU 0 (0) s.

Since, by assumption, limI0U 0(I) = we have

limI0

u0(e0) = ,

which implies that be (, y) < e0 (, y) , where be (, y) is the optimal evasion givenin Lemma 2.1. Obviously, this means that

y y + be (, y) < y y + e0 (, y) . (A.1)Now dene y0 as the income that makes equal the net income of avoiding withthe net income of evading for e0 (, y) when the inspection does not occurs. Bydenition we have

y0 cy0 k = y0 y0 + e0 (, y0) .After rearranging terms we have that the value y0 becomes

y0 =sk

s(1 c) (s+ 1)(1 ) .

Condition (1 + s) > 1 + sc guarantees that y0 > 0. Then, for y > y0 we get

y

(1 c) (s+ 1) (1 )

s

> y0

(1 c) (s+ 1) (1 )

s

= k.

Hence, we have

y (1 c) k > y (s+ 1) (1 )s

= y y + e0 (, y) .

Finally by (A.1) it holds that

y (1 c) k > y y + be (, y) .Therefore for y > y0 the following inequality must also hold:

y (1 c) k > y y + be (, y) > V (, y).Summarizing, we have that the individuals with income y > y0 prefer avoidingto evading while the individuals with an income y small enough prefer evading.This guarantees then, that V (, y) and U(y cy k) intersect at least once.

25

Step 2. We will prove that V (, y) and U(y cy k) intersect only once.Let be y any intersection of V (, y) and U(y cy k). This implies that

V (, y) = U(y cy k) and consequently we have that

y y sbe (, y) < y cy k < y y + be (, y)must hold. This inequality implies that

U 0 (y cy k) > U 0 (y y + be (, y)) , (A.2)since the function U() is concave. By the envelop theorem we get that

V (, y)y

= (1 p)U 0 (y y + be (, y)) +pU 0 (y y sbe (, y)) (1 s) ,

which can be simplied using (2.2) as

V (, y)y

=

(1 p)(1 + s)(1 )

s

U 0 (y y + be (, y)) . (A.3)

The condition (1 + s) > 1+ sc, implies that (1 + s) > 1 also holds, and this is

a sucient condition for ensuring that(1p)(1+s)(1)

s

< 1. Thus, using (A.2)

and (A.3) we get that

U 0 (y cy k) > V 0 (, y) . (A.4)

It is immediate to check that V (, y) is a concave function, then the inequality(A.4) means that U() and V () intersect once at most.

Proof of Proposition 4.4. When the tax rate is zero the tax revenue is also zerobecause the government can not collect neither taxes nor nes. For (0, c] allthe individuals prefer being evaders, therefore the government gets some revenueat least from the nes collected by the inspection. Moreover, for (0, c], G()is increasing in since the optimal evasion is decreasing in (see Yitzhaki, 1974).In particular evaluating the tax revenue function given by (3.1) at = c, we havethat G(c) > 0. Nevertheless, we do not know which is the exact value of G(c)because, although all the individuals are evaders, we ignore if they evade all hisincome or only a part of it. This decision depends on the own income, the taxrate and the values of p and s. Our proof leans on showing that G(c) > G(1),because this implies that the tax revenue function can not be monotonic. Let usrst investigate what happens with the optimal evasion when = 1. From theexpression (2.2) , we have that

U 0 (y y se(, y)) = (1 p)ps

U 0 (y y + e (, y)) .

26

We know that y y + e (, y) > 0 for (0, 1) so that condition limI0 U 0(I) =implies that

y y se (, y) > 0.Rearranging the last inequality we have that

0 e (, y) < (1 ) ys

. (A.5)

Therefore,

0 lim1 e (, y) lim1

y(1 )s

= 0.

We see that when the tax rate tends to one the optimal tax evasion tends to zeroor, in other words, the individuals tell the truth independently of their respectiveincomes. This result allows us to say that only the individuals who can not face upto the xed cost of avoidance will pay taxes, whereas the others will be avoiders.Thus, the tax revenue when = 1 will be

G(1) = lim1

Z y()0

[(1 p) (y e (, y)) + p (y + se (, y))] 1Ydy

=

Z k1c

0y1

Ydy =

k2

2Y (1 c)2 ,

since lim1 y

() =lim1 y

H() =

k

(1 c) when e (, y) = 0. Note that lim1 y()

0, Ywhenever k

0, (1 c)Y

. As G(c) does not depend on k, we can make

G(c) > G(1) for a small enough value of k.

Proof of Lemma 5.1.(a) Fix and then consider D dened in (5.6) as a function of s and p, D(s, p).After tedious dierentiation, it can be proved that Ds = 0 and

Dp = 0 only when

s = 1pp . Moreover, the Hessian of D(s, p) evaluated at s =1pp is

H(s, p) =

p3

(1p)p

(1p)p

(1p)1

p(1p)

!,

which is positive semi-denite. Finally, we get thatD(s, p) = 1 whenever s = 1pp .We conclude thus that D(s, p) > 1 for all s 6= 1pp . In particular, D > 1 forps < 1 p.

(b) To see that c < b Y < 1 we only have to prove that 0 < h (1c)D kDY i < 1.The assumption that k

0, (1 c)Y

ensures that

h(1c)D kDY

i> 0, while the

part (a) of this lemma implies thath(1c)D kDY

i< 1.

27

Proof of Lemma 5.2. For 0 the tax revenue is only composed by thenes paid by the inspected individuals. This revenue is

G1() =

Z Y0p (1 + s) y

1

Ydy

For b Y the tax revenue is given by the taxes and penalties paid forall the individuals. This revenue is

G2() =

Z Y0[(1 p)(1 ()) + p (1 + s())] y 1

Ydy

=Z Y0[(1 ()) + p (1 + s)()] y 1

Ydy.

Finally, for b Y < 1 the tax revenue isG3() =

Z yP ()0

[(1 p)(1 ()) + p (1 + s( ))] y 1YdyZ yP ()

0[(1 ()) + p (1 + s)()] y 1

Ydy,

since the tax authorities can only collect payments from the individuals who donot avoid.

Performing the integrals Gi () where i = 1, 2, 3 we obtain the expressionappearing in the statement of the Lemma.

Proof of Proposition 5.3. We prove the proposition by stating two claims.

Claim 1: dG1()d > 0 for (0, ] and dG2()d > 0 for

, b Y i .

Calculating dG1()d we have

dG1()

d=p(1 + s)Y

2> 0,

which is unambiguously positive. In a similar way, we obtain

dG2()

d=Y

2

(1 () + p (1 + s)()) + (p (1 + s) 1) d()

d

. (A.6)

From expression (5.3) we have d()d = 12 < 0, where > 0 is given by

=(A 1)(1 +As)

. (A.7)

Note that the rst term inside the square brackets of (A.6) is positive sincethe parameters satisfy the interior condition s < 1pp . On the other hand, the

28

second term inside the square brackets is also positive since d()d < 0 and(p (1 + s) 1) < 0. Then, G2() is also increasing in the tax rate.

Claim 2: dG3()d < 0 forb Y , 1i .

We have the following derivative:

dG3()

d=1

Y

( (1 ()) + p (1 + s)()) dy

P ()

dy()

+

1

Y

1

2(y())2

1 () + p (1 + s)() + (p (1 + s) 1) d()

d

. (A.8)

To evaluate the sign of the previous expression we need to calculate dy()d . Using

expression (5.8) , we getdyP ()d

= DEyP (), (A.9)

where E = [(1 c) (1 )D] . Since yP () > 0, D > 0 and E > 0 for b Y , we can conclude that dyP ()d < 0. Then, plugging (A.9) in (A.8) andrearranging terms, we have

dG3()

d=(yP ())

2

Y

[ (1 + (1 p (1 + s))) (1 p (1 + s))]

DE

+

(yP ( ))2

Y

1

2(1 + (1 p (1 + s)))

.

Note that the sign of the previous derivative depends only on the sign of theexpression into the brackets. Dene

F ( (1 + (1 p (1 + s))) (1 p (1 + s)))DE

+1

2(1 + (1 p (1 + s))) ,

which can be rewritten as

F =D [1 (1 ps) + (1 + (1 p (1 + s)))] + (1 c) (1 + (1 p (1 + s)))

2E

The sign of F is the same as that of its numerator since E > 0 for b Y .Therefore if our objective is to prove that dG3()d < 0, we have to prove that thefollowing inequality holds:

D [1 (1 p (1 + s)) + (1 + (1 p (1 + s)))] >

(1 c) (1 + (1 p (1 + s))) . (A.10)

29

Note that the tax revenue G3( ) is obtained when the tax rate moves betweenb Y < 1. Then, ifD [1 (1 p (1 + s)) + b (1 + (1 p (1 + s)))] >

(1 c) (1 + (1 p (1 + s))) , (A.11)we can guarantee that inequality (A.10) also holds for

hb Y , 1i since theleft term of inequality (A.10) is increasing in . Thus, substituting

b Y = 1 + kDY

(1 c)D

into (A.11) we get

D

1 (1 p (1 + s)) +

1 +

k

DY (1 c)

D

(1 + (1 p (1 + s)))

>

(1 c) (1 + (1 p (1 + s))) .Rearranging and simplifying, we obtain

2D +k

Y(1 + (1 p (1 + s))) > 2(1 c) (1 + (1 p (1 + s))) . (A.12)

According to lemma 5.1, D > 1. Hence,

(1 c) (1 + (1 ps) (1 p (1 + s))) < 1, (A.13)

becomes a sucient condition for (A.12) . Rearranging the inequality (A.13) wehave

(1 c) p (1 + s) (1 c) c < 0. (A.14)It is easy to see that inequality (A.14) holds if

c >

1 + . (A.15)

Using (5.2) and (A.7) and rearranging terms, we get that the sucient condition(A.15) becomes simply c > , and that is always true by assumption. Hence, itfollows that dG3()d < 0.

Proof of Lemma 5.4. The richest individual wants to be avoider if the utilityfrom avoidance is greater than the expected indirect utility from full evasion at = . This is

(1 p)hYi1

+ phY Y sY

i1 0,

that can be as small as we want, taking values of k low enough. Similarly,evaluating the tax revenue function G() at = c, we have

G(c) =1

2c (1 (c) + p (1 + s)(c))Y > 0.

Thus, it is easy to check thatG(c) > G(1), for k < (1c)Ypc (1 (c) + p (1 + s)(c)).Proof of Lemma 6.1. For [0, ] all the individuals evade all their income,and then the tax revenue is given by the penalties paid by the evaders inspected.Formally, this revenue is given by

G1() = 1p (1 + s) y1 + 2p (1 + s) y2 + 3p (1 + s) y3.

31

Simplifying the last expression we have

G1() = p (1 + s) (2m+ 3) .

For (, by3 ] everybody is a partial evader. In this case, the tax revenue iscomposed both by the taxes voluntary paid and by the penalties from inspectedindividuals. After rearranging the terms the tax revenue is given by

G2() = (2m+ 3) [ (1 () + p (1 + s)())] .Finally for (by3 , by2 ] the tax revenue is equal to the taxes voluntarily paid bythe individuals with income y2 and to the nes paid by the inspected ones. Thisrevenue is

G3() = 2m [ (1 () + p (1 + s)())] .

Proof of Proposition 6.2(a)We need to prove that G(by3) is the maximum value that the tax revenue

function G() can achieve. Computing dG1()d we have

dG1()

d= p (1 + s) (2m+ 3) > 0.

Similarly dierentiating G2(), we obtain after rearranging the terms

dG2()

d= (2m+ 3)

1 +

(A 1) (1 p (1 + s))1 + sA

,

which is unambiguously positive. Then, for [0, by3), we have that G2(by3) isthe highest value of the tax revenue.

On the other hand, if we compute dG3()d we obtain that

dG3()

d= 2m

1 +

(A 1) (1 p (1 + s))1 + sA

> 0.

Then, we only have to prove thatG2(by3) > G3(by2). Evaluating these expressions,we get

G2(by3) = (2m+ 3) (by3) ,and

G3(by2) = 2m (by2) ,where () = [ (1 ()) + p (1 + s) ()] . Comparing these two expressionsis straightforward to see that, if 3 2 we only have to prove that"

m ( (by2) (by3)) (by3)

#< 1, (A.17)

32

to show that G2(by3) > G3(by2). Then, taking expressions (6.1) and (6.2) andplugging then into () , the inequality (A.17) becomes

D B(1 c) + kBmD

> 0, (A.18)

where D =

1 + s

1 +As

(1 p)A1 + p 11 , and B = 1 + (1 p (1 + s)) .

Lemma 5.1 and condition (A.13) ensure that D B(1 c) > 0 so that theinequality (A.18) holds.

(b) Following the proof of (a), it is straightforward to see that the proof of(b) reduces to check when

3 < 2

"m ( (by2) (by3))

(by3)#, (A.19)

holds. Since inequality (A.17) holds, we need a small enough value of 3 to ensurethe fulllment of (A.19) .

Proof of Proposition 6.3.(a) It is immediate to see that dG1()d > 0 and

dG2()d > 0, where G1() =

p (1 + s) (2m+ 3) and G2() = p (1 + s)2m. Then, we only have to provethat G1 (ey3) > G2 (ey2), which is equivalent to prove that

3 < 2m

"ey2 ey3ey3#, (A.20)

after substituting the corresponding values of the tax rate. The expression (6.3)tells us how ey2 depends on m. Then, we get

limm1

"2m

ey2 ey3ey3!#

= 0.

In consequence we can conclude that for am suciently close to one, the inequal-ity (A.20) will hold.

(b) Following the proof of (a), it is straightforward to see that the proof of (b)reduces to check when

3 < 2m

"ey2 ey3ey3#, (A.21)

holds. Obviously, a value of 3 small enough ensures that (A.21) holds.

(c) It is immediate to see that dG1()d > 0,dG2()d > 0 and

dG3()d > 0, where

G1() = p (1 + s) (2m+ 3), G2() = p (1 + s)2m and G3() = 2m ()

33

with () = [ (1 ()) + p (1 + s) ()] . Then, we only have to prove thatG1 (ey3) > G3 (by2) since G2() and G3() are continuous on their respectivedomains. The last inequality is equivalent to

3 > 2m

"by2 ey3ey3#. (A.22)

Thus, when the proportion 3 is large enough, inequality (A.22) holds.

(d) Following the proof of (c), it is straightforward to see that the proof of (d)reduces to check when the following inequality is satised:

3 < 2m

"by2 ey3ey3#.

Obviously, such an inequality holds for a small enough value of 3.

34

References

[1] Allingham, M. G. and A. Sandmo, (1972). Income Tax Evasion: A Theo-retical Analysis. Journal of Public Economics 1, 323-38.

[2] Alm, J., (1988). Compliance costs and the tax avoidance-tax decision.Public Finance Quartely 16, 31-66.

[3] Alm, J. N. and J. Mccallin, (1990). Tax Avoidance and Tax Evasion as aJoint Portfolio Choice. Public Finance / Finances Publiques 45, 193-200.

[4] Chu, C. Y., (1990). Plea Bargaining with the IRS. Journal of Public Eco-nomics 41, 319-333.

[5] Cowell, F., A., (1990). Tax Sheltering and the Cost of Evasion. Journal ofPublic Economics 41, 231-243.

[6] Cross, R., and Shaw, G. K., (1982). On the Economics of Tax Aversion.Public Finance / Finances Publiques 37, 36-47.

[7] Lyons J. T., (1996). The Struggle Againts International Fiscal Fraud: Taxavoidance and Tax Evasion. International Bureau of Fiscal Documentation,March 1996.

[8] Stiglitz, J., E., (1985). The General Theory of Tax Avoidance. NationalTax Journal 38, 325-337.

[9] Waud, R., N., (1986). Tax Aversion and the Laer Curve. Scottish Journalof Political Economy. 33, 213-227.

[10] Waud, R., N., (1988). Tax Aversion, Optimal Tax Rates, and Indexation.Public Finance / Finances Publiques 43, 310-325.

[11] Yitzhaki, S., (1974). A Note on Income Tax Evasion: A Theoretical Anal-ysis. Journal of Public Economics 3, 201-202.

35

-2

-1

0

G

0 0.2 0.4 0.6 0.8 1tau

Figure 1. The function G( ) when the individuals are honest,using logarithmic scale (c = 0.1, Y = 100, k = 1).

-2

-1

0

G

0.2 0.4 0.6 0.8 1tau

Figure 2. The function G( ) when the evasion takes placeand c, using logarithmic scale (p = 0.1, s = 3, c = 0.15,Y = 100, k = 1, = 2).

01

2

G

0 0.2 0.4 0.6 0.8 1tau

Figure 3. The function G( ) when the evasion takes placeand > c, using logarithmic scale (p = 0.1, s = 1.5, c = 0.1,Y = 100, k = 1, = 2).

00.1

0.2

G

0.2 0.4 0.6 0.8 1tau

Figure 4. The function G( ) when the income distributionis discrete and 3 > 2 (p = 0.1, s = 3, c = 0.25, k = 0.25, = 0.5, 1 = 0.4, 2 = 0.2, 3 = 0.4, m = 0.7).

00.2

0.4

G

0.2 0.4 0.6 0.8 1tau

Figure 5. The function G( ) when the income distributionis discrete and 3 < 2 (p = 0.1, s = 3, c = 0.25, k = 0.25, = 0.5, 1 = 0.1, 2 = 0.8, 3 = 0.1, m = 0.7).

Laffer Curver

Documents

Transcript of Laffer Curver