APPLICATIONS OF BEHAVIOURAL ECONOMICS TO TAX …people.exeter.ac.uk/gdmyles/papers/pdfs/69 JOES.pdfBEHAVIOURAL EXPLANATIONS OF TAX EVASION 943 that have been proposed. The non-expected

doi: 10.1111/j.1467-6419.2012.00733.x

APPLICATIONS OF BEHAVIOURAL ECONOMICSTO TAX EVASION

Nigar Hashimzade

University of Reading

Gareth D. Myles

University of Exeter and Institute for Fiscal Studies

Binh Tran-Nam

University of New South Wales

Abstract. The paper reviews recent models that have applied the techniques of behaviouraleconomics to the analysis of the tax compliance choice of an individual taxpayer. The constructionof these models is motivated by the failure of the Yitzhaki version of the Allingham–Sandmomodel to predict correctly the proportion of taxpayers who will evade and the effect of an increasein the tax rate upon the chosen level of evasion. Recent approaches have applied non-expectedutility theory to the compliance decision and have addressed social interaction. The models wedescribe are able to match the observed extent of evasion and correctly predict the tax effect butdo not have the parsimony or precision of the Yitzhaki model.

Keywords. Tax evasion; Behavioural economics; Social interaction

1. Introduction

Tax evasion is the illegal concealment of a taxable activity. Measuring how much economic activity isconcealed will always be difficult since those who engage in evasion have every motivation to hide theiractivities. Even so, the estimates that are available from official sources and from academic researchersare in agreement that evasion is an economically significant activity. This emphasizes the importanceof the understanding the decision process of a taxpayer when choosing whether to comply with taxlaw or to engage in evasion. A good theory of the compliance decision is essential for designing a taxstructure that deters evasion.

The economic analysis of an individual taxpayer’s compliance decision can be traced back to thepioneering work of Allingham and Sandmo (1972). They modelled the taxpayer as facing a decisionunder risk with the extent of evasion chosen to maximize expected utility. The risk arises fromthe possibility that a random audit will be conducted by the tax authorities that uncovers the evasion.Yitzhaki (1974) modified the model by expressing the punishment for evasion as a fine levied on unpaidtax which is more in line with practice. The Yitzhaki model formed the basis for later developmentsand we refer to it as the ‘standard model’ of the compliance decision throughout the paper. Thestrength of the standard model is the clarity of its comparative statics predictions. The model predictsthat the chosen level of evasion falls when either the penalty rate or the probability of being caught

Journal of Economic Surveys (2013) Vol. 27, No. 5, pp. 941–977C© 2012 John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden,MA 02148, USA.

942 HASHIMZADE ET AL.

evading are increased. It also predicts that evasion increases with income when absolute risk aversionis decreasing. These results are in agreement with intuitive expectations and with empirical evidence.Unfortunately, there are two key dimensions in which the predictions of the model do not accord withdata or intuition. First, when confronted with values of the audit probability and the fine rate close tothose observed in practice the model predicts that all taxpayers should engage in evasion. This is aconsequence of the sufficient condition for evasion that obtains in the expected utility framework. Forexample, the estimated Arrow–Pratt measure of relative risk aversion for the United States is betweenone and two, but only a value of 30 would explain the observed compliance rate (Graetz and Wilde,1985; Alm et al., 1992).1 Secondly, the model predicts that the level of evasion will fall when the taxrate increases. This result is counter-intuitive but is a direct consequence of the fine being a multipleof the amount of tax which the taxpayer attempted to avoid paying. When the tax rate increases sodoes the total fine for a given amount of evasion, and this hits the taxpayer in the state in which theyhave the least income.

These results have caused the validity of the model to be questioned and have led to an extensivesearch for a better model. The aim of this search is to construct a model of the compliance decisionthat remains grounded in economic theory while simultaneously correcting the imperfect predictions.There have been numerous extensions of the standard model, such as making labour supply endogenous,including a choice between employment in a formal and informal sectors and increasing the complexityof the income tax. Details can be found in the surveys of Pyle (1991) and Sandmo (2005). Theseextensions lead to interesting results but do not provide an answer to the failure of the standard modelwithin the context it addresses.

The standard model, and its extensions noted above, are based on two key assumptions. First, the riskinvolved in the evasion decision is addressed through use of the expected utility theory of von Neumannand Morgernstern (1947). Second, the choice problem of the taxpayer is represented as being entirelyindividualistic: the evasion decision is made without reference to what other taxpayers are doing.The identification of a range of ‘anomalies’ (observed choices that do not match the predictions ofstandard theory, see Thaler, 1994) has inspired the development of behavioural economics. Behaviouraleconomics is a broad school that has come to encompass many non-standard approaches to choice.In the context of this paper we take it to mean approaches to choice with risk that depart from theaxioms of expected utility theory and the location of the evasion decision within a social context. Weconsider the branch of behavioural economics that has constructed models for choice with risk, orwith uncertainty, that modify the axioms of expected utility theory and the branch that addresses otheranomalies by incorporating social interaction into the individual decision process. Non-expected utilityand social interaction are not mutually exclusive and applications have frequently combined elementsfrom both.

There is a growing number of papers contributing to this new literature on the compliance decision.The models now available have differences in assumptions and structure that sometimes hide thecommon components of the analysis. It therefore seems an appropriate time to take stock of thisliterature. By reviewing the recent results it is intended that an assessment can be obtained of whathas been achieved, and whether the new approaches provide a significant advance over the standardmodel. In particular, the focus will be placed on isolating the features of the models that can resolvethe incorrect predictions. This survey provides a complement to that of Pyle (1991) which explores thestandard model and its extensions.

The analysis of the current range of non-expected utility model reaches two clear conclusions. Thefirst is that for reasonable parameter values these models can predict the correct level of evasion. Thisoccurs because the models permit the subjective probability of audit (or ‘weighting’ on the payoffwhen audited) to be greater than the objective probability. The second is that they do not generallyreverse the prediction with respect to the direction of the tax effect. Eide (2002) has already madethis observation for one particular model; what we argue is that it is true for all of the alternatives

Journal of Economic Surveys (2013) Vol. 27, No. 5, pp. 941–977C© 2012 John Wiley & Sons Ltd

BEHAVIOURAL EXPLANATIONS OF TAX EVASION 943

that have been proposed. The non-expected utility models can reverse the tax effect when combinedwith other factors, such as a psychic cost of evasion or audit probabilities that depend upon announcedincome, but this is also true of expected utility. A brief explanation is as follows. All the non-expectedutility models are special cases of the Choquet Expected Utility (CEU) model (Chateauneuf, 1994).The value function for CEU is essentially a weighted sum of the payoff when evasion is successfuland the payoff when the evader is caught. The tax rate does not enter into the determination of theweights, so its effect is felt through the effect upon the payoffs. Although there are differences in detail(for example, through the introduction of a reference point in Prospect Theory), the tax rate enters thepayoff function in a manner broadly similar to how it enters the standard model: typically, the payoffis dependent upon the product of the tax rate and the undeclared income. This results in the penaltypaid on undeclared income being proportional to the tax rate, so there is no substitution effect on theallocation of income across states when the tax rate is changed. An increase in the tax rate just has anincome effect which, with decreasing absolute risk aversion, leads to less evasion (Pyle, 1991). As aconsequence, the optimal choice of evasion level will generally be decreasing in the tax rate.

How, then, can the direction of the tax effect be reversed? The answer lies in ensuring that the levelof evasion enters the payoff function separately from the tax rate. The literature has identified severalways in which this can be achieved. One possibility is to change the structure of the punishment forevasion. The limitation of this approach is that model then fails to represent the actual situation in manycountries.2 An alternative is to introduce additional, non-monetary, elements into the decision problemthat capture aspects of social interaction. The idea that the social environment affects the choices madeby individuals has been an important theme of behavioural economics. Models with social interactioncan predict the correct level of evasion and reverse the tax effect. However, these alternative modelsare either not as parsimonious as the standard model or do not make predictions that are as precise.3

Section 2 introduces the individual compliance decision, describes the notation used throughoutthe paper, outlines the standard model and summarizes its predictions. Section 3 reviews a range ofnon-expected utility models of choice and analyses the outcome of applying them to the compliancedecision. The effect of introducing social interaction is analysed in Section 4. Section 5 offers someconclusions.

2. Tax Evasion

This section sets out the basic analytical framework for the compliance decision that is used throughoutthe paper. The standard model of the compliance decision is then analysed. The shortcomings in thepredictions of the model are identified and used to motivate the development of alternative models.

2.1 Compliance Decision

The standard model of the compliance decision considers an individual taxpayer within a single-periodsetting. The taxpayer has a given amount of income that is not directly observed by the tax authority(including an observed component of income is straightforward) and must choose how much of thisincome to declare. The taxpayer is assumed to know the objective probability of an income declarationbeing audited. If the declaration is audited then the true level of income is revealed with certainty.The discovery of undeclared income results in the payment of tax on the undeclared income plus anadditional fine.

The notation that is used throughout the paper is as follows. The actual level of income (unobserved)is Y , the level of income declared to the revenue service is X and the amount of undeclared incomeis E = Y − X . The amount of tax paid on an income level x (x = {X , Y }) is T = T (x) and the fineis F = F(E). The fine for evasion may be dependent upon the tax structure. In the particular case



of a linear tax and fine, T (x) = t x, and F(E) = f t E , where the tax rate, t , and the fine rate, f ,satisfy f > 0, t > 0. The decision problem of the taxpayer is to choose the amount of evasion, E (or,equivalently, the declaration X ).

After the evasion decision is made one out of two potential states of the world is realized. In thestate of the world in which evasion is successful the taxpayer is left with disposable income Y n , where

Y n = Y − T (X ) (1)

The level of disposable income in the state of the world in which evasion is detected is denoted Y c.This is given by

Y c = Y − T (Y ) − F(E) (2)

These two states of the world occur with objective probabilities 1 − p and p, respectively. Theseprobabilities are determined by the audit activities of the revenue service. The non-expected utilitymodels derive the subjective probabilities from the objective probabilities in a variety of ways whichare discussed in detail later. Since the future state of the world is unknown the taxpayer’s disposableincome is a random variable. Let Y denote the random level of disposable income prior to realizationof the state of nature. When evasion level E is chosen a gain t E is obtained if the evasion is undetectedand a loss − f t E is incurred if it is detected. The taxpayer can then be viewed as choosing a quantityt E of an asset which has return r = 1 when evasion is successful and return r = − f when it is not.Disposable income can be defined as the random variable

Y = [1 − t] Y + r t E (3)

where r is the random return.The payoff obtained from an outcome z is given by the payoff function v(z). It is assumed that v(z)

is increasing in z, so that v′(z) > 0 at those points where v(z) is differentiable. The models that followdiffer with respect to assumptions about the payoff function and the nature of the outcome, z, thatenters the payoff function. The literature on choice theory finds it necessary to distinguish between riskand uncertainty. In a decision with risk there are known probabilities of the alternative outcomes. Theseprobabilities may be objective or subjective. In contrast, in a decision with uncertainty there are noknown probabilities. The analysis of choice requires different techniques in these two circumstances.

2.2 The Standard Model

The standard model is introduced as a baseline against which other models can be assessed. The modeland its predictions are now described briefly. Detailed developments can be found in Myles (1995),Pyle (1991) and Sandmo (2005).

It is assumed that the tax and fine are linear and that the taxpayer maximizes the expected value ofutility from disposable income. The expectation is formed using the objective probabilities. Therefore,the outcomes are the income levels Y c and Y n , and the payoff function v(·) is a utility function U (·)satisfying the standard assumptions U > 0 and U ′′ < 0. The decision problem can be written as

max{E}

V = pU (Y [1 − t] − f t E) + [1 − p] U (Y [1 − t] + t E) (4)

The first- and second-order conditions for an interior optimum are

−p f U ′ (Y c) + [1 − p] U ′ (Y n

) = 0 (5)

and

S ≡ p f 2U ′′ (Y c) + [1 − p] U ′′ (Y n

)< 0



A sufficient condition for tax evasion to take place is obtained by evaluating (5) at E = 0. Doing thisshows that it is optimal to choose E > 0 if

f < [1 − p]/p (6)

It is important to observe that this sufficient condition is independent of preferences, so if one taxpayerchooses to evade then all should evade. Moreover, with the value of f = 1 (which is fairly representativeof international tax systems) the model predicts all taxpayers should evade if p < 0.5. Hence, a valueof audit probability far in excess of what is observed is necessary to avoid all taxpayers choosing toconceal at least part of their incomes. The fact that the data on evasion (Andreoni et al., 1998; Slemrodand Yitzhaki, 2002) show that a large proportion actually choose to declare all income presents thefirst challenge to the model.

We now turn to the comparative statics of an interior optimum. The objective function is strictlyconcave so if there is an interior optimum it must be unique. To provide a concrete example beforeproceeding to the general case we now assume that utility is logarithmic, U (z) ≡ ln(z). The decisionproblem is

max{E}

V = p ln (Y [1 − t] − f t E) + [1 − p] ln (Y [1 − t] + t E) (7)

An interior solution must satisfy

− p f

Y [1 − t] − f t E+ 1 − p

Y [1 − t] + t E= 0 (8)

This condition can be solved to give the level of evasion

E = 1

t

Y [1 − t] [1 − p [1 + f ]]

f(9)

The level of evasion, E , enters the objective function in (7) only through terms involving t E so thesolution for E in (9) involves the term 1

t . This is a feature shared by many models. An increase in thetax rate reduces evasion since

d E

dt= − 1

t2

Y [1 − p [1 + f ]]

f< 0 (10)

Returning to the general problem, the effect of an increase in the tax rate on the level of evasion isdetermined by

d E

dt= − 1

S

∂2V

∂E∂t(11)

Using the Arrow–Pratt measure of absolute risk aversion

RA (I ) = −U ′′ (I )

U ′ (I )(12)

we have

∂2V

∂E∂t= p f U ′(Y c)[Y [RA(Y n) − RA(Y c)] − E[RA(Y n) + RA(Y c) f ]] (13)

If absolute risk aversion decreases as income increases then RA (Y n) < RA (Y c), so

∂2V

∂E∂t< 0 (14)

and the level of evasion will fall as the tax rate rises. This result is counter to intuition and contradictsmuch of the empirical evidence (Clotfelter, 1983; Andreoni et al., 1998).



In assessing the result in (14) it should be stressed that the assumption of absolute risk aversiondecreasing with income cannot be regarded as universally acceptable. This leaves a degree of uncertaintyover the conclusion that higher tax rates and higher income lead to greater tax evasion. In addition,the result is also sensitive to the precise form of the punishment for evasion. If the fine is determinedas in Allingham and Sandmo (1972) by f E , f > t , rather than f t E , then the effect of the tax ratecannot be unambiguously signed even with decreasing absolute risk aversion. However, much of theliterature on tax evasion has been developed on the presumption that the tax rate effect is negative.

There have been several attempts to estimate the standard model using data. A survey of early workin this area and some comments on the difficulties can be found in Pyle (1991). One problem forempirical application is to obtain individual-level data on tax compliance. Witte and Woodbury (1985)used data provided by the Internal Revenue Service but these were aggregated to US three-digit zipcode level. Feinstein (1991) does use individual level data but uses a statistical approach without astructural foundation in choice theory. Both of these studies assume enforcement activity can be treatedas an exogenous variable. However, if the revenue service follows a deliberate strategy such as placinggroups with low compliance rates under closer scrutiny then individual characteristics will be linkedto the likelihood of detection and simultaneous equation bias will occur. Dubin et al. (1987) addressthis by estimating a simultaneous equation model, but their data were aggregated to the level of USstates. As pointed out more recently by Frey and Feld (2002), the standard model does not fit well inthe sense that the estimated coefficients generally lack statistical significance or have signs differentfrom what the model predicts. Cowell (2003) notes further difficulties that arise in appropriatelyspecifying an empirical model. First, sample-selection bias can be caused by absence from the sampleof taxpayers who file no tax return. Second, there is a problem in constructing a variable to proxy forprobability of audit. The probability may vary in unknown ways across taxpayers (both in an objectivesense because of tax authority strategy and in a subjective sense because of taxpayers’ beliefs). Third,opportunities for evasion differ across occupations so there may be an unobserved rationing constrainton some individuals. A comprehensive empirical test of the model requires the resolution of all thesedifficulties.

3. Non-Expected Utility Theory

Expected utility theory follows as a consequence of a set of axioms governing preferences over lotteries(see, for example, Mas-Colell et al., 1995). The Allais paradox (Allais, 1953) provided a simple examplefor which violation of the axioms was commonplace. Further anomalies have since been identified,and there is now considerable empirical and experimental evidence of choice behaviour that violatesthe axioms. The accumulation of this evidence has motivated the construction of alternative models forpreferences that aim to provide a better explanation of observed choices. The purpose of this sectionis to review the predictions that emerge when these alternatives, to which we give the general labelof ‘non-expected utility theory’, are applied to the evasion decision. Models of choice under risk arediscussed first, followed by models of choice under uncertainty.

3.1 Risk

The various models of choice with risk can be formulated using the following framework. Let X ={x1, . . . xn} be a set of n outcomes that partitions the set of states of the world, �, into mutuallyexclusive events. Hence, xi ∩ x j = ∅ if i �= j , and ∪n

i=1xi = �. When there is risk the decision makeris able to assign a probability to each outcome. Consequently, a prospect, y, is a pair of vectors{x, p} , x = (x1, . . . xn) , p = (p1, . . . pn) , where each xi ∈ X , pi > 0, i = 1, . . . , n, and

∑ni=1 pi = 1.

ψ denotes the set of prospects. The fineness of the partition (and hence the cardinality n) may differ



Table 1. Models of Non-Expected Utility for Risky Decisions.

Decision weight Payoff

Expected utility pi U (xi )Rank-Dependent Expected Utility wi (pi ) = ω(

∑ij=1 p j ) − ω(

∑i−1j=1 p j ) U (xi )

Karmarkar (1978) wi (pi ) =pαi

pαi +[1−pi ]α∑nj=1

pαj

pαj +[1−p j ]α, 0 < α ≤ 1 U (xi )

Prospect Theory wi (pi ) v(xi − R)

Handa (1977) w(pi ) xi

between prospects. The prospect {(x), (1)} means that x is obtained with certainty. A payoff functionon the outcomes is denoted by v(xi ).

The decision maker is endowed with a set of preferences over the set of prospects, ψ . Denotethe relation ‘at least as good as’ by � and the relation ‘strictly preferred to’ by �. The outcomesare ordered so that xn � xn−1 � · · · � x1. A value function V on ψ represents the decision-maker’spreferences if

V (y) ≥ V (y′) ⇔ y � y′ (15)

A model of preferences can be viewed as a set of axioms that the preference order, �, over prospectsmust satisfy. These axioms determine the class of functions that can represent preferences for thatmodel, and different axioms lead to different classes. The models of preferences that have been appliedto the evasion decision are special cases of the general formulation

V =n∑

i=1

wi (p)v(xi ) (16)

where wi (p) are the decision weights and v(xi ) the payoff function. A utility function, denoted U (xi ),is a special case of a payoff function that satisfies the additional assumptions U ′ (·) > 0 and U ′′ (·) < 0.

Table 1 summarizes how several non-expected utility models relate to the general form in (16); thosein bold have been applied to the evasion decision so are studied in detail in this section. Further detailsof how a range of models of preference representation fit with this general framework can be found inChateauneuf (1994).

We have already mentioned that many choice anomalies have been observed that are inconsistentwith expected utility theory. The alternative models are not without their own problems. Quiggin (1981)notes that whenever the weighting function is non-linear the preferences may fail to satisfy the axiomof dominance which requires a dominant alternative to be valued higher. Kahneman and Tversky (1979)suggested ‘editing out’ the dominated prospects, but Quiggin shows this leads to intransitivity of thepreference order.

3.1.1 Rank-Dependent Expected Utility Theory

Rank-Dependent Expected Utility Theory (and the special case of Anticipated Utility) is characterizedby the particular recursive manner in which the values of the weighting function are generated from theprobabilities. The alternatives must be ranked from best to worst for the recursive process to be applied.



Anticipated Utility, which developed the analysis of Kahneman and Tversky (1979), was introduced byQuiggin (1981) and formalized in Quiggin (1982). Quiggin and Wakker (1994) provided an axiomaticbasis for Anticipated Utility and for the more general case of Rank-Dependent Expected Utility.

Quiggin (1982) observed that the shortcoming of prior models of non-expected utility was theassumption that the decision maker distorted the probability of extreme outcomes but did not dothe same to ‘intermediate’ outcomes with the same probability. To correct this Quiggin proposed thatthe weight for each outcome should be based, in principle, on all the individual probabilities. Thisgives the general form of the objective function

V =n∑

i=1

w (p1, . . . , pn) U (xi ) (17)

It was argued that this alternative to expected utility theory, which Quiggin called Anticipated Utility,is more consistent with the evidence. The reason for making the decision weights depend on set ofprobabilities is shown by observing that if the weights, w(·), are based on individual probabilities thedecision weights must satisfy w

(1n

) = 1n . To see this, assume they did not and that w( 1

n ) < 1n . Then,

for any X and ε a non-negative random variable, it follows that for sufficiently small ε

U (X ) >n∑

i=1

w

(1

n

)U (X + xi + ε) (18)

even though X + xi + ε ≥ X with probability 1. To avoid this inconsistency the decision weightscannot imply w( 1

n ) < 1n .

A formalization of Anticipated Utility theory is undertaken in Quiggin (1982). The paper observesthat the transitivity and dominance axioms of expected utility theory command virtually unanimousassent but in the Allais paradox, and other observed anomalies, it is the independence axiom thatis violated. The Quiggin (1982) proposal is therefore to drop the independence axiom and, hence,introduce more flexible weighting schemes. Anticipated Utility is based on the assumption that wi (0) =0 and wi (1) = 1. Most importantly, it is also assumed that

wi

(1

2,

1

2

)= 1

2. (19)

The key construction then follows. Given a weighting function wi (p, 1 − p) for a two-outcomeprospect, it is shown how this can be extended to a three-outcome prospect, and then extended furtherrecursively. To do this, for p ∈ [0, 1] define the function ω (p) = w1(p, 1 − p). Then the weights canbe found using the recursion

wi (p) = ω

⎛⎝ i∑

j=1

p j

⎞⎠ − ω

⎛⎝ i−1∑

j=1

p j

⎞⎠ (20)

under the convention that∑0

j=1 p j = 0. It should be stressed here that the outcomes have to be rankedfor this construction to apply.

Quiggin and Wakker (1994) introduce the more general case of Rank-Dependent Expected Utilitywhich does not impose the condition in (19). The general representation of preferences is

V =n∑

i=1

⎡⎣ω

⎛⎝ i∑

j=1

p j

⎞⎠ − ω

⎛⎝ i−1∑

j=1

p j

⎞⎠

⎤⎦ U (xi ) (21)



where ω : [0, 1] → [0, 1], is non-decreasing, and ω(0) = 0, ω(1) = 1. Since ω(∑n

j=1 p j

)= 1, the

representation in (21) can be expressed alternatively as

V = U (xn) −n∑

i=1

ω

⎛⎝ i−1∑

j=1

p j

⎞⎠ [U (xi ) − U (xi−1)] (22)

which, in the two-outcome case, can be written as

V = U (x2) − ω (p1) [U (x2) − U (x1)]

= ω (p1) U (x1) + [1 − ω (p1)] U (x2)(23)

Chateauneuf et al. (2005) provide the necessary and sufficient condition in the Rank-DependentExpected Utility framework for a decision maker to be averse to a monotone mean-preserving increasein risk. They adopt a variant of the Quiggin model where the decision function is a product of a utilityfunction and a weighting function. The analysis is based on the ‘greediness’ of the utility function(an index of non-concavity) and the ‘pessimism’ of the weighting function (how far it is below thediagonal). Define the values

GU = supy≤xU ′(x)

U ′(y)(24)

and

Pω = inf0<v<1(1 − ω(v))/(1 − v)

ω(v)/v(25)

For any strictly increasing function U it follows that GU ≥ 1. GU = 1 if and only if U is concave.Chateauneuf et al. show that the decision maker is monotone risk averse if and only if Pω ≥ GU , whichis a joint restriction on utility and weighting functions. This construction provides a representation ofpreferences and a necessary condition for the decision maker to act rationally in the sense of beingaverse to risk. If they do not satisfy this condition then questions can be raised about their behaviourand the implications of any comparative statics results that arise from application of the preferencerepresentation.

Arcand and Graziosi (2005) analyse the tax evasion decision when the taxpayer has preferencesdetermined by Rank-Dependent Expected Utility. Since the preferences of the taxpayer give y + t E �y − t f E, the value function in (23) can be re-written as

max{E}

V (E) = [1 −�2(1 − p)]U (y − t f E) +�2(1 − p)U (y + t E) (26)

where it is assumed that �2(0) = 0 and �2(1) = 1. In addition, Arcand and Graziosi assume �2(1 − p)is inverse s-shaped (concave, then convex). This has the effect of inflating low probabilities and deflatinglarge probabilities. To contrast the level of evasion for this model with the level of evasion with choicesdetermined by expected utility it is further assumed that:

1. �2(1 − p) < 1 − p2. �2( f

1+ f ) < f1+ f .

Assumption (1) implies that 1 −�2(1 − p) > p, so the subjective probability of audit isinflated above the objective probability. Assumption (2) requires that the penalty rate ishigh.



The main result of the paper is to show that, for a given fine rate, the probability of audit at which fullcompliance arises is lower with Rank-Dependent Expected Utility than it is with expected utility. Thisresult can be proved by observing from the objective function (26) that evasion occurs if V ′(E) > 0 atE = 0 which requires �2(1 − p) − [1 −�2(1 − p)] f > 0. This condition can be written as

�2(1 − p) >f

1 + f(27)

Hence, full compliance will occur for any p > p∗RD where

p∗RD = 1 −�−1

2

(f

1 + f

)(28)

Since �2(1 − p) is inverse s-shaped and �2( f1+ f ) < f

1+ f then

�−12

(f

1 + f

)>

f

1 + f(29)

Collecting these facts together

p∗RD = 1 −�−1

2

(f

1 + f

)< 1 − f

1 + f= f

1 + f= p∗

EU (30)

where p∗EU is the value of probability at which evasion is zero for Expected Utility (see (6)). This

finding is a consequence of the overweighting of the probability of audit. The simulation analysisof Arcand and Graziosi considers alternative weighting schemes for the probabilities but all producesimilar results. What is important is that for some parameter values the level of evasion matches USdata.

It should be noted that although this formulation changes the prediction about the extent of evasionit does not change the comparative static effect of a change in the tax rate, t . Given a probability, p, thechange of variables w2(1 − p) ≡ 1 − q , 1 − w2(1 − p) ≡ q can be made, and the optimization becomesthat of the standard model but with p replaced by q . The qualitative properties of the comparativestatics are therefore unchanged.

3.1.2 Prospect Theory

The basic idea of Prospect Theory is that there is a reference point relative to which gains or lossesare assessed. In addition, the theory often assumes that the structure of the payoff function is differentfor gains and losses relative to the reference point, and that transformed probabilities are used by thedecision maker.

In the Kahneman and Tversky (1979) version of Prospect Theory the decision maker ‘edits andevaluates’ each of their prospects and chooses the one with the highest value. This process can beillustrated by considering a decision maker who is choosing between two prospects. One prospect occurswith probability p1 and has outcome, measured relative to the reference point, R, of y = I + − R. Theother prospect, which occurs with probability p2, has outcome, measured relative to the reference point,of x = I − − R, where I − < I +. The overall value of the edited prospect of the taxpayer is

V = w1(p1)v(x) + w2(p2)v(y) (31)

where wi (pi ) is the decision weight. It is possible under the theory that wi (pi ) ≡ pi . The value function,v(z), satisfies the assumptions

v′(z) > 0 (32)



v′′(z) > 0 for z < 0 and v′′(z) < 0 for z > 0 (33)

The assumption in (33) is the statement of convexity in losses but concavity in gains. According toequation (1) in Kahneman and Tversky (1979), under the assumption that wi (pi ) ≡ pi , the value ofthe taxpayer’s prospect is

V = p1v(x) + [1 − p1] v(y)

= v(y) + p1 [v(x) − v(y)](34)

Tversky and Kahneman (1992) provide the basis for a Cumulative Prospect Theory. This model is aresponse to the fact that their form of Prospect Theory does not satisfy stochastic dominance, so it ispossible for riskier prospects to be valued higher than safer ones. Moving to a cumulative formulationbased on Rank-Dependent Expected Utility overcomes this deficiency. In this framework a prospectis a sequence of pairs {xi , Ai } where xi is the outcome if Ai occurs, and xi > x j for i > j . Theoutcomes are labelled so that xi < 0 for i = −m, . . . , 0, and xi > 0 for i = 1, . . . , n. f + is defined asthe positive part, and f − as the negative part. Cumulative Prospect Theory then has the value

V ( f ) = V ( f +) + V ( f −) (35)

where V ( f +) = ∑ni=1w

+i v (xi ) , V ( f −) = ∑0

i=−m w−i v (xi ) , and the decision weights are given by

⎧⎪⎪⎨⎪⎪⎩w+

n = W + (An) , w−−m = W − (A−m)

w+i = W + (Ai ∪ . . . ∪ An) − W + (Ai+1 ∪ . . . ∪ An)

w−i = W − (A−m ∪ . . . ∪ Ai ) − W − (A−m ∪ . . . ∪ Ai−1)

If there is a positive probability, pi , for each outcome xi then the decision weights become

⎧⎪⎪⎨⎪⎪⎩w+

n = W + (pn) , w−−m = W − (p−m)

w+i = W + (pi + · · · + pn) − W + (pi+1 + · · · + pn)

w−i = W − (p−m + · · · + pi ) − W − (p−m + · · · + pi−1)

(36)

This reduces to the standard form of Prospect Theory when there are only two outcomes.Two issues arise in applying Prospect Theory. First, since there is no natural reference point

or functional form for the payoff, there is considerable flexibility within Prospect Theory for therepresentation of the choice problem. This flexibility explains why the applications of Prospect Theoryto the compliance decision by Yaniv (1999), Bernasconi and Zanardi (2004) and al-Nowaihi andDhami (2007) differ markedly in structure. Second, using a payoff function that is convex in lossesbut concave in gains may have support in psychological evidence but when used in models withoptimizing behaviour it can create substantial analytical difficulties. The basic problem is that theobjective function of the taxpayer will not be globally concave. Therefore, there is no guarantee of aunique maximum nor need a solution to the first-order condition for choice be a global maximum (itcan be a local maximum with the global maximum in a corner, or even a minimum). It is this featureof Prospect Theory that gave rise to the ‘bang-bang’ solution described in al-Nowaihi and Dhami: insome cases, a small change in underlying parameters can cause the taxpayer to leap from a cornersolution with perfect compliance to a corner solution with no compliance.



These analytical issues can be illustrated by reviewing the model of Yaniv (1999) who analysedthe effect of an advance tax payment. With Expected Utility an advance will have no effect upon thecompliance decision. If, instead, Prospect Theory is employed the advance tax payment will have aneffect if it determines the reference point from which gains and losses in different states of the worldare measured. Denote the advance tax payment by D. Referring to Elffers and Hessing (1997), theincome level Y − D is suggested by Yaniv (1999) to be a natural reference point. From this referencepoint, D − t X is the gain if evasion is not detected, and D − t X − f t [Y − X ] is the loss if evasion isdetected. This choice of a reference point results in the gain occurring with certainty, but part of theloss is random.

The objective function studied by Yaniv is4

V = v(D − t X ) + pv([D − t X − f t[Y − X ]] − [D − t X ])

= v(D − t X ) + pv(− f t[Y − X ])(37)

This specification is justified by the assumption that when evaluating their prospects, the taxpayersegregates the certain outcome from the risk in the following way

V = v(Y C ) + [1 − p]v(Y G) + pv(Y L ) (38)

where Y C is the outcome received in both prospects with certainty, Y G ≥ 0 is the gain over the certainoutcome in one of the prospects, that occurs with probability 1 − p, and Y L ≤ 0 is the loss from thecertain outcome in the other prospect, that occurs with probability p. In this model, Y C = D − t X ,Y G = 0, and Y L = − f t[Y − X ] ≤ 0 (so Y L = 0 if the income is declared truthfully). To make theanalyses concrete we consider the commonly used power function form for the payoff (Tversky andKahneman, 1992)

v (z) ={

zβ, z ≥ 0

−γ (−z)β, z < 0(39)

We treat the cases of the advance payment D set above, and below, the true tax liability, tY , separately.Consider the case D > tY . Figure 1 illustrates the ‘standard’ parameterization of the value function

(β = 0.88 and γ = 2.25), and Figure 2 a ‘more curved’ parameterization (β = 0.5 and γ = 4). Forboth figures we employ the parameter values Y = 1, t = 0.2, p = 0.1, f = 2 and D = 0.3. The figuresgraph the value function (37) (solid line), as well as its two components, v(D − t X ) (dotted line) andpv(− f t[Y − X ]) (dashed line). It can be seen immediately that because the value function is notglobally concave, the first- and second-order conditions do not necessarily describe the solution ofthe maximization problem. In Figure 1 there is no point in the interior that satisfies the first-ordercondition, whereas in Figures 2 the first-order condition is only satisfied at an interior minimum butthe maximum is at the X

Y = 1 corner. In neither case will an analysis of the first-order condition providea correct description of compliance behaviour.

We now turn to the case D < tY . Figures 3 and 4 correspond to the same two parameters of thevalue function as for Figure 2 but use D = 0.1 to ensure D < tY and higher values of p and f . InFigure 3 the local maximum at X

Y = 0.33 is also the global maximum, whereas in Figure 4 the localmaximum is at X

Y = 0.47, but the global maximum is at the corner with XY = 1. One can also see that

in both cases the first-order condition has another solution which is the interior minimum.The examples in Figures 1– 4 show that the assumptions Prospect Theory makes about the payoff

function – notably convexity in losses and concavity in gains – result in the need to exercise care whenanalysing the compliance decision. In particular, the first-order condition cannot be simply used as abasis for a comparative statics analysis: the model can have corner solutions, local and global interiormaxima, and outcomes where the only solution to the first-order condition is an interior minimum. We



0.2-2.0

-1.5

-1.0

-0.5

0.0

0.5

0.4 0.6 0.8 1.0X/Y

Figure 1. D > tY , β = 0.88, γ = 2.25.

-1.0

-0.5

0.0

-1.50.2 0.4 0.6 0.8 1.0

X/Y

Figure 2. D > tY , β = 0.5, γ = 4.

now proceed to analyse the effect of choice of reference point on the direction of the tax effect, underthe assumption that first-order condition does correctly describe a global maximum.

Tversky and Kahneman (1992) emphasize the role of gains and losses relative to a reference point.Prospect Theory does not detail how this reference point should be chosen which leaves a degree offreedom in modelling. The choice of reference point can be justified in different ways. For example,Yaniv (1999) argued for income after advance tax payment to be the reference point. Here the sequenceof events being modelled is used to fix the reference point. An alternative justification for a differentchoice of reference point is used by al-Nowaihi and Dhami (2007) for the compliance decision withoutan advance. Their argument is that if the taxpayer evades and is not caught they should always be



0.0

-0.2

0.0

0.2

0.4

0.6

0.2 0.4 0.6 0.8 1.0X/Y

Figure 3. p = 0.25, f = 4.

0.0

-0.1

0.0

0.1

0.2

0.3

0.2 0.4 0.6 0.8 1.0X/Y

Figure 4. p = 0.25, f = 20.

in the ‘gain’ region while if they are caught they should always be in the ‘loss’ region. The onlyreference point that guarantees this for any level of evasion is true income after tax, Y [1 − t]. Hence,the reference point is fixed by appealing to a natural interpretation of gain and loss. Rablen (2010) usesthe perceived fairness of public good provision to construct the reference point. Since the referencepoint is not fixed by the theory we consider it worthwhile to analyse the consequences of choosingdifferent reference points in the Prospect Theory model defined by (31).



Denoting the reference point by R(E, t) we specify the objective function for the taxpayer as

V = w1 (p) v (Y c − R (E, t)) + w2 (p) v (Y n − R (E, t))

= w1 (p) vc + w2 (p) vn

(40)

We are interested in the sign of d Edt where E is the interior maximum of V on (0, Y ). The first-order

condition isdV

d E= w2 (p) v′

n [t − RE ] − w1 (p) v′c [ f t + RE ] = 0 (41)

and we assume the second-order condition is satisfied, so

∂2V

∂E2≡ S < 0

The effect of an increase in tax rate upon the level of evasion is derived from the total differential of(41) and is written as

d E

dt= − F

S(42)

where

F = w2 (p) v′n [1 − REt ] − w1 (p) v′

c [ f + REt ]

− w2 (p) v′′n [t − RE ] [Y − E + Rt ]

+ w1 (p) v′′c [ f t + RE ] [Y + f E + Rt ]

(43)

Since −S > 0, the sign of d Edt is determined by the sign of F . We now consider several alternatives

for the specification of the reference point. In every case it should be stressed that we are assumingthat there is an interior global maximum to which we can apply the comparative statics analysis.

Example 1: R = [1 − t]YIn this case F = t E[w2(p)v′′

n + w1(p)v′′c f 2] so

d E

dt= − E

t< 0

This specification of Prospect Theory provides the entirely unambiguous conclusion that an increasein the tax rate reduces evasion. In this case the standard qualitative prediction holds without requiringany restriction on risk aversion.

Example 2: R = R(t) with Rt �= 0This generalizes Example 1 to give

F = t E[w2 (p) v′′

n + w1 (p) v′′c f 2

] + t [Y + Rt ][w1 (p) v′′

c f − w2 (p) v′′n

]F can be positive if:

1. Either w2 (p) v′′n + w1 (p) v′′

c f 2 > 0 and

E > [Y + Rt ]w2 (p) v′′

n − w1 (p) v′′c f

w2 (p) v′′n + w1 (p) v′′

c f 2

2. Or w2 (p) v′′n + w1 (p) v′′

c f 2 < 0 and

E < [Y + Rt ]w2 (p) v′′

n − w1 (p) v′′c f

w2 (p) v′′n + w1 (p) v′′

c f 2



The first condition implies that the tax effect may be reversed at high levels of evasion providedthe payoff from losses is sufficiently convex. The second condition implies that the tax effect maybe reversed at low levels of evasion provided the payoff from gain is sufficiently concave. Both arepossible, but neither is clear-cut.

Example 3: R(E, t) = [1 − t][Y − E]This reference point gives F = [1 + f ]w1(p)v′

c[ v′′cv′

c[t[1 + f ] − 1] − 1]. F will be positive if

v′′c

v′c

[t[1 + f ] − 1] > 1

In the Prospect Theory framework v′′cv′

cis the loss aversion and is positive if the payoff is convex and

being caught evading places the taxpayer in the loss region. It is therefore a possibility that the taxeffect can be reversed for this specification if the penalty rate is sufficiently high.

These examples demonstrate that the choice of reference point can determine the sign of the taxeffect. For some choices it is possible, but not guaranteed, that the direction will be reversed. The choiceof reference point is therefore significant for the application of Prospect Theory. Schepanski and Kelsey(1990) test the idea that shifting the reference should alter the choice that is made. This means thatthe way an experiment is presented or a choice is framed should have a bearing on the outcome if itcan change the reference point. The paper reports on a literature that has found a weak framing effect,or no framing effect. This is partly counter to the claims of Kahneman and Tversky (1984). The basisof the experiment is that individuals find themselves in a loss, gain or neutral position. Loss meansan unexpected tax bill is due but they have the option of claiming a non-allowable deduction whichwill work as long as they are not audited. The gain is based partly on the refund of tax including anon-allowable deduction with the issue of whether the taxpayer should inform the tax authority that ithas made a mistake. Neutral ones were just told the alternative payoffs that could occur (these were thesame as in loss condition). The experiment is claimed to provide strong evidence of a framing effect.King and Sheffrin (2002) also provide a discussion of Prospect Theory and framing and a furtherexperiment that tests the model.

The use of a weighting function in Prospect Theory alters the necessary condition for evasion tooccur and consequently gives the model flexibility to make better predictions about the extent ofevasion. The structure of the payoff function creates non-concavities so that corner solutions can occur.This requires caution to be exercised when using the first-order condition as a basis for comparativestatics. Prospect Theory does not necessarily reverse the direction of the tax effect: the examples showthat certain choices of the reference point can affect the direction of the tax effect in some situations,but none of the examples is compelling.

3.1.3 Regret and Disappointment

The Disappointment Theory of Loomes and Sugden (1986, 1987) asserts that the payoff is determinedby a combination of the outcome resulting from a choice of action and the average outcome thatcould have occurred. Hence, if a bad outcome arises the payoff is reduced even further by additional‘disappointment’ when this poor outcome is contrasted to what might have been obtained. Thedisappointment gives greater motivation than expected utility theory for the decision maker to avoidactions that potentially have particularly bad outcomes.

Consider an action a that leads to monetary payoff y(a, ω) when state ω occurs. In the risk-neutralform of Disappointment Theory the payoff from this action is given by

V (a) = E[y(a, ω)] + E[D(y(a, ω) − E[y(a, ω)])] (44)



where E[·] is the expectation operator. The disappointment function D(·) satisfies:

1. D(0) = 0,2. D is concave on (−∞, 0),3. D is convex on (0,∞),4. 0 < D′(x) < 1,5. −D(−x) ≥ D(x).

Conditions (2) and (3) imply violation of the independence axiom while (4) ensures first-orderstochastic dominance is satisfied. Under these conditions Disappointment Theory shares with ProspectTheory the assumption of concavity with respect to gains and convexity with respect to losses.

Regret Theory (Loomes and Sugden, 1982) has a similar structure. Let a and b be two risky actions.The representation of preferences is then defined by the condition

a � b ⇔ E[y(a, ω) + R(y(a, ω) − y(b, ω))] > E[y(b, ω) + R(y(b, ω) − y(a, ω))] (45)

Here R is the regret (or rejoicing) function which involves the comparison of what results fromchoosing a with what would have resulted from choosing b. The regret function is assumed to be twicedifferentiable and increasing.

The central feature of Disappointment and Regret is the comparison of the outcome achieved withoutcomes that could have been achieved. When the realized outcome is poor the disappointment reducesthe payoff and provides a disincentive to undertake risky actions. In the context of tax evasion thiswill, ceteris paribus, reduce the chosen level of evasion. We now consider whether the inclusion ofdisappointment modifies the sufficient condition for evasion or the tax effect.

Using (44) the payoff of evasion level E can be written

V = E[Y (E)] − E[D(Y (E) − E[Y (E)])] (46)

From the definitions of Y n and Y c it follows that

E[Y (E)] = pY c + [1 − p]Y n = Y [1 − t] + [1 − p − p f ]t E (47)

Employing this, (46) can be written as

V = Y [1 − t] + [1 − p − p f ] t E

−pD (− [1 − p] [1 + f ] t E) − [1 − p] D (p [1 + f ] t E)(48)

From the objective in (48) it can be seen immediately that the sufficient condition for evasionto occur is that 1 − p − p f > 0, which is identical to that for the standard model. In addition, theobjective is dependent upon t E so that the solution for E will involve a term in 1

t . Therefore the modeldoes not provide a route to reversing the direction of the tax effect.

3.2 Uncertainty

A decision is made under uncertainty when there are no known objective probabilities. All probabilitiesare therefore subjective, and need not obey the usual axioms of probability theory. The aim of theliterature on choice with uncertainty is to accommodate these assumptions within a formal framework.

The Ellsberg Paradox provides experimental evidence that decision makers prefer to avoid theunknown: a bet on the drawing from an urn with a known distribution of coloured balls is preferredto betting on a urn with an unknown distribution. Such preferences cannot be incorporated in a choiceframework based on additive probabilities (see Kischka and Puppe, 1992). This observation motivatedthe development of a choice theory for non-additive probabilities and a formalization of ambiguityaversion.



3.2.1 Non-Additive Probabilities

Consider two sets A and B. If A and B are disjoint, so A ∩ B = ∅, then a standard additive probabilitymeasure satisfies the additivity property

p (A ∪ B) = p (A) + p (B) (49)

The basic problem encountered when moving to uncertainty is that a subjective assessment ofprobability need not satisfy additivity. A subjective probability measure that does not satisfy (49)is called non-additive (or a ‘capacity’). For example, a non-additive measure may have

w (A ∪ B) < w (A) + w (B) (50)

The problem of integrating non-additive set functions is analysed in Schmeidler (1986, 1989).Schmeidler assumes that the non-additive measure satisfies two conditions:

1. Normalization: w (∅) = 0, w (�) = 1, where � is the set of states;2. Monotonicity: F ⊂ G implies w (F) ≤ w (G).

Let (�i )ki=1 be a partition of �, �∗

i an indicator function (so �∗i is equal to 1 on the set �i ), and

a = ∑ki=1 αi�

∗i a finite step function with α1 > α2 > · · · > αk . Let αk+1 = 0. The integral of a over

� is then defined by ∫�

adv =k∑

i=1

[αi − αi+1] v(∪i

j=1� j)

(51)

This representation is extended to functions more general than the step function as follows. Let a bea real valued, bounded function on � and v a non-additive probability. Then the Choquet integral isdefined by ∫

�

adv =∫ 0

−∞[v (a ≥ α) − v(�)] dα +

∫ ∞

0v (a ≥ α) dα (52)

with v(�) = 1. An axiomatic basis for this representation, and of some variants which are morerestrictive, is given in Chateauneuf (1994). This process of integration provides for the developmentof a theory of ambiguity based on subjective preferences which can be applied to represent the choiceproblem of a taxpayer who does not know the audit strategy of the revenue service.

3.2.2 Ambiguity

The subjective non-additive probabilities used earlier are known to the decision maker even though theyare not publicly known. Ambiguity refers to a situation in which the probabilities are unknown, even tothe decision maker, so it involves uncertainty about uncertainty. Ambiguity results from the uncertaintyassociated with knowing which distribution from a set of possible distributions is appropriate in agiven situation. Ellsberg (1961) describes this distinction in terms of unambiguous versus ambiguousprobabilities. Einhorn and Hogarth (1986) distinguish between ‘ignorance’, ‘ambiguity’ and ‘risk’,according to the degree to which alternative probability distributions can be ruled out. Hence, ambiguityis an intermediate state between ignorance (no probability distributions are ruled out) and risk (allprobability distributions but one are ruled out). The amount of ambiguity is then an increasing functionof the number of distributions that are not ruled out.

The motivation for considering ambiguity is to take into account the possibility that the informationa decision maker has about some relevant uncertain event is vague or imprecise, and that this affectschoice. Ambiguity aversion means that the decision maker prefers to bet on unambiguous ratherthan on ambiguous events. Camerer and Weber (1992) define ambiguity more generally, referring to



Fellner (1961) and Frisch and Baron (1988): ‘Ambiguity is uncertainty about probability, created bymissing information that is relevant and could be known’. Ambiguity aversion in this case refers to thereluctance of people to take either side of a bet not only when they fear that others might have moreinformation, but also when such fears are unfounded (Frisch and Baron, 1988).

Among the approaches to modelling ambiguity discussed in Camerer and Weber (1992) is theuse of a second-order probability distribution, � (p (xi )), over the probabilities (Marschak, 1975).Ambiguity can be expressed by a second-order probability when the decision maker knows, or canassign, probabilities to all probability distributions in the set of all possible distributions. The second-order probability view can be taken further by presuming subjective second-order probabilities of(first-order) probabilities that might also be subjective. An alternative way of modelling ambiguity isdescribed in Einhorn and Hogarth (1986) as an anchoring-and-adjustment process. The anchor is aninitial guess about the probability, p. The judged probability is S(p) = p + k, where k is the net effectof the mental adjustment process. The value of k can depend on a range of factors. For example, ifk = ξ (1 − p) − ξ pβ then it depends on: (1) p since −p ≤ k ≤ 1 − p; (2) ξ ∈ [0, 1], the amount ofambiguity (ξ = 0 when p is known exactly; if ξ = 1 then all values of p are possible); (3) β ≥ 0, therelative weighting of probabilities > p and < p (the attitude to ambiguity). A value β > 1 means thatprobabilities higher than the anchor are weighted more than probabilities lower than the anchor. Anaxiomatic approach to ambiguity was developed by Fishburn (1993). The fundamental property thatdistinguishes ambiguity from risk is submodularity which is the property that α(A ∪ B) ≤ α(A) + α(B)if the sets A and B are disjoint. The rationale for this condition is that taking the union of A and B mayreduce or cancel ambiguities associated with each separately, and will not introduce new sources ofambiguity in the combination that outweigh such reductions. More recently, an axiomatic approach thatencompasses several generalizations of subjective expected utility theory was laid out by Ghirardatoand Marinacci (2002).

Examples of preference representations for ambiguity include:1. CEU

V ( f ) =∫

Su ( f (s)) v (ds) (53)

where the integral is taken in the sense of Choquet. A subclass of the CEU ordering is subjectiveexpected utility, a particular case in which v is a probability measure.

2. Maximin expected utility (MEU):

V ( f ) = minP∈C

∫S

u ( f (S)) P (ds) (54)

where C is a compact and convex set of probability measures. Subjective expected utility is a particularcase in which C = {P} for some probability measure P . In this case the decision maker evaluates theoutcome with respect to the probability distribution that yields the lowest possible payoff. Moregenerally, α − MEU preferences assign some weight to the worst- and best-case scenarios. For α ∈[0, 1],

V ( f ) = αminP∈C

∫S

u ( f (s)) P (ds) + [1 − α] maxP∈C

∫S

u ( f (s)) P (ds) (55)

MEU (minimax) corresponds to α = 1, and ‘maximax’ has α = 0.Snow and Warren (2005) apply the idea of ambiguity to the tax evasion decision. Ambiguity in their

model means that there is a lack of precise knowledge of the audit probability so the taxpayer forms asecond-order probability distribution over possible audit probabilities. An increase in ambiguity meansan increase in dispersion of this probability distribution. Snow and Warren show that an increasein uncertainty about the probability of being audited (meaning additional ambiguity) increases tax



compliance for ambiguity-averse taxpayers but reduces compliance for ambiguity lovers. They alsoreport that experimental evidence shows people are heterogeneous with respect to preferences overambiguity. This implies that the sufficient condition for evasion to occur will differ across taxpayers.

Assume that the true probability of audit is p but the taxpayer is ambiguous about this. The subjectiveprobability is denoted π and is distributed according to the cumulative distribution function F (π ; a, p),where a is the measure of ambiguity. F (π ; a, p) is the second-order probability distribution. Anincrease in ambiguity is a mean-preserving spread of F (π ; a, p). It is assumed that π is an unbiasedestimate of p so ∫

πd F = p (56)

Snow and Warren also assume that the perception of π is distorted into a weighting ϕ (π, p). Theobjective function is then given by

V = U (Y n) −∫ϕd F

[U (Y n) − U (Y c)

](57)

Observe that if there is no ambiguity or distortion then∫ϕd F = p and the objective collapses to the

standard expected utility form

V = [1 − p] U (Y n) + pU (Y c) (58)

The functioning of the model can be demonstrated by considering the comparative statics withrespect to true audit probability and the level of ambiguity. Assume that (i) an increase in p causes afirst-order stochastic dominance shift in F (π ; a, p), so Fp < 0; (ii) ϕ (π, p) is monotonically increasingin π ; and (iii) the expected value of ϕp is positive. Then the effect on an increase in p upon the payoffof the taxpayer is given by

∂V

∂ p= −

[∫ϕpd F +

∫ϕd Fp

][U (Y n) − U (Y c)]

= −[∫

ϕpd F +∫ϕπ Fpdπ

][U (Y n) − U (Y c)] < 0

(59)

using integration by parts. An increase in ambiguity results in an increase in risk so∫ τ

0Fadπ ≥ 0 (60)

for all τ ∈ [0, 1 ]. The effect of an increase in ambiguity on payoff is

∂V

∂a= −

[∫ϕd Fa

][U (Y n) − U (Y c)]

= −[∫

ϕππ

∫Fadπdτ

][U (Y n) − U (Y c)]

(61)

The second term follows from applying integration by parts twice. Hence, an increase in ambiguityhas no effect on expected utility if ϕππ = 0. The taxpayer can be said to be ambiguity loving ifϕππ < 0 because this ensures that ∂V

∂a > 0. The converse holds if ϕππ > 0, in which case the taxpayeris ambiguity averse.

This development allows the effect of ambiguity upon the declared level of income to be obtained.The necessary condition for an interior optimum is

∂V

∂X= U ′(Y n) −

[∫ 1

0ϕd F][U ′(Y n) − U ′(Y c)[1 − f ]

](62)



The second-order condition must hold so the sign of d Xda is the same as the sign of

[−

∫ 1

0ϕππ

∫ τ

0ϕπ Fadπdτ

] [U ′(Y n) − U ′(Y c) [1 − f ]

](63)

Since f > 1 the sign of (63) is determined by the sign of the first term. Therefore, an increase inambiguity increases compliance (X rises) if the taxpayer is ambiguity averse. It can also be seendirectly from (62) that since

∫ϕd F is independent of t , the tax rate only enters through the standard

channels of Y n and Y c. Consequently, the presence of ambiguity does not reverse the direction of thetax effect.

The idea of higher or lower ambiguity is an interesting one for the tax compliance decision sinceit raises important policy questions about the likely effect of making auditing processes more or lesstransparent. The result derived shows that increasing ambiguity by making policy less transparent willincrease compliance. The model will also yield a different necessary condition for evasion, so canpredict higher levels of compliance than the standard model but it does not change the direction of thetax effect.

3.3 First-Order and Second-Order Risk Aversion

One of the failings of the standard model of compliance is the over-prediction of the proportion oftaxpayers who will engage in non-compliance. The condition that delivers this conclusion is a directconsequence of the properties of the expected utility function: all indifference curves have the samegradient where they cross the certainty line. This gradient is determined entirely by the probability,p, so is also the same for all taxpayers regardless of personal attitudes toward risk. Segal and Spivak(1990) investigate what is required to relax these conclusions by considering the behaviour of the riskpremium for small gambles.

Consider a gamble ε with E[ε] = 0 and variance σ 2ε . The risk premium of the gamble, rε, is denoted

by π(r ). The issue is the behaviour of the derivative of π(r ) around r = 0. Pratt (1964) showed thatin the expected utility framework

π (r ) ≈ −r2

2σ 2ε

U ′′ (x)

U ′ (x)(64)

where x is the initial wealth. Hence, the first derivative with respect to r is zero when evaluated atr = 0 but the second derivative is generally positive (provided U ′′ (x) < 0). This is called ‘risk aversionof order 2’. The implication is that the indifference curve has the same gradient as the fair odds lineon the certainty line, so the certainty point will not be chosen unless the gamble is actuarially fair.

The alternative scenario is ‘risk aversion of order 1’. In this case

∂π (r )

∂r|r=0+ �= 0 (65)

If the risk premium is symmetric around r = 0 then

∂π (r )

∂r|r=0+ = −∂π(r )

∂r|r=0− (66)

so the risk premium has to be kinked at r = 0. It is this condition that marks the difference betweenthe two orders of risk aversion. The kink in the risk premium implies that the indifference curves arealso kinked on the certainty line. This permits a point with certainty to be chosen even if the gambleis not actuarially fair.



The general expression for the gradient of the indifference curve on the certainty line is

limr→0+

−r p/(1 − p) + π (r )

r − π (r )= − p/(1 − p) + ∂π/∂r |r=0+

1 − ∂π/∂r |r=0+(67)

and in the other direction

limr→0−

−r p/(1 − p) + π (r )

r − π (r )= − p/(1 − p) − ∂π/∂r |r=0−

1 + ∂π/∂r |r=0+(68)

Contrasting (67) and (68) confirms that when there is risk aversion of order 1 the indifference curve iskinked on the certainty line.

Segal and Spivak (1990) demonstrate that risk aversion of order 1 arises (a) with expected utilityif the utility function is not differentiable, or (b) if Rank Dependent Expected Utility applies with anAnticipated Utility function and either a strictly concave or a strictly convex distribution transformation.

The ideas are applied to tax evasion by Bernasconi (1998) who adopts the Segal and Spivak (1990)description of orders of risk aversion to argue that a payoff function with first-order risk aversion canresolve the over-prediction of compliance by the expected utility theory.

Consider applying Rank-Dependent Expected Utility theory to the evasion decision. Denote theprospect {(Y n, Y c) , (1 − p, p)}. When Y n > Y c the objective is

V = w (p) U(Y c

) + [1 − w (p)] U(Y n

)(69)

so the gradient of the indifference curve as Y n → Y c from below is

−1 − w (p)

w (p)(70)

When Y n < Y c the objective becomes

V = w(1 − p)U (Y c) + [1 − w(1 − p)]U (Y n) (71)

and the gradient of the indifference curve as from above is

− w (1 − p)

1 − w (1 − p)(72)

The gradients in (70) and (72) will not, in general, be equal, which shows that Rank-DependentExpected Utility has risk aversion of order 1.

The sufficient condition for tax evasion to occur is derived from (70). Since this differs from thevalue of − p

1−p in the standard model, the sufficient condition for evasion to occur will also be changed.Bernasconi derives the probability of audit that triggers evasion for an individual taxpayer by using theCamerer and Ho (1994) probability weighting function,

w (p) = 1 − [1 − p])γ

[pγ + (1 − p)γ ]1/γ (73)

with γ = 0.56. Combining this weighting function with a standard CRRA utility function generatesthe probabilities displayed in Table 2. The table shows that the absolute value of the gradient of theindifference curve for Rank-Dependent Expected Utility can be significantly less than for expectedutility. The expected return from evasion therefore has to be much greater for evasion to occur.

3.4 Summary

This section has considered the analysis of the tax compliance decision in a number of non-expectedutility models. The primary conclusion is that these models are able to provide a sufficient condition



Table 2. Gradient of Indifference Curves.

Audit probability, p − 1−pp − 1−w(p)

w(p)

0.01 99.0 7.40.02 49.0 5.00.03 32.3 3.90.05 19.0 2.90.09 10.1 2.1

for evasion to take place that differs from that of the standard model. Furthermore, under the realisticassumption that taxpayers over-weight the probability of detection these models are able to producea stricter sufficient condition. This overcomes the problem of the standard model predicting that alltaxpayers should evade given observed audit probabilities and fine rates. Furthermore, if the weightingfunction differs across taxpayers then the models can also justify the observation that only a proportionof taxpayers will choose to evade (contrasted to standard model in which all or none evade).

The predictions of the models for the direction of the tax effect are less successful. In any modelin which the payoff depends upon the product of t and E the optimal choice variable of the taxpayeris a value of t E . Hence, as t changes so does E, to keep t E constant. This makes the optimal Einvolve the term 1/t, so as t increases this has the effect of causing E to fall. This conclusion can onlybe changed if the objective function has either t or E entering as a separate argument in a way thatis sufficiently strong to dominate. One way this can be achieved is to have a variable probability ofdetection. This gives an effect that is dependent upon E but not t , so the tax effect can be reversed.5

This separation can also be achieved through the effects of social interaction which are described inthe following section.

It is worth noting that the complexity of these alternative models makes their empirical analysisdifficult, as the results appear to be sensitive to the choice of the functional forms for the payofffunction and the probability weighting functions. Therefore, while these models predict qualitativeresults that are consistent with intuition and the empirical evidence, there have not been systematicattempts in the literature to estimate the parameters of these models empirically.

4. Social Interaction

The previous section has focussed on different models of non-expected utility that vary in the evaluationof the monetary payoffs. It has been shown that these models can improve the prediction on the degreeof compliance, but generally will not reverse the tax effect unless coupled with additional modifications,such as a variable probability of audit. A second branch of behavioural economics considers the socialenvironment in which choices are made. The models in this literature relax the assumption that eachindividual makes a private and isolated decision. Instead, they view individuals as involved in a range ofsocial interactions that affect the payoffs from different choices. There is a wide variety of models andthese incorporate a large number of different forms of interaction. We have attempted a categorizationbut it should be stressed that there are many overlaps between the different models.

4.1 The Cost of Evasion

There are a number of models of the evasion decision that include costs in addition to the fine if caughtevading. These additional costs can be real financial costs such as the payment for avoidance services



0V*

0=E0E

0>dtdE0<

dtdE

>

Figure 5. Separation of Population.

or the loss of return through using a hidden investment instrument. They can also be psychic costs thatarise through the fear of detection or the shame of being exposed. We interpret these psychic costs asbeing a consequence of the social setting in which the taxpayer operates, so are a result of the loss ofsocial prestige or reputation. The experiments of Baldry (1986) suggest that such costs distinguish theevasion decision from a straightforward gamble. Costs can also arise in terms of the loss of a socialcustom utility when tax evasion is chosen. This class of models are discussed in the next subsection.

The approach adopted by Gordon (1989) to incorporate psychic costs into the evasion decision is toassume that utility is determined by the function

U = U (Y ) − χE (74)

In this utility function χE is the additional psychic cost of undertaking an amount of evasion, E . Theparameter χ ≥ 0 measures the extent to which these psychic costs affect the individual taxpayer. Theremay be some taxpayers for whom χ is close to zero; such taxpayers will exhibit behaviour very similarto that predicted by the standard model. Other taxpayers may have a high value of χ : they will bemuch more reluctant to evade tax. It is helpful to think of the value of χ as being determined by acombination of the taxpayer’s attitude towards honest behaviour and the effects of the taxpayer’s socialpeer group. Individual maximization of the utility in (74) leads to the first-order condition for choiceof E

E[U ′(Y )r t] − χ = 0 (75)

where E[·] is the expectations operator and Y is defined by (3).The level of evasion, E , is positive when the marginal utility of evasion is greater than zero at E = 0.

Hence, evasion occurs when V0 − χ > 0, where V0 ≡ [1 − p − p f ]tU ′(Y [1 − t]). If χ is greater thanV0 then all income will be declared. When the model is applied to a population of taxpayers withindividual values of χ the population will be separated into two groups: high-χ taxpayers will declareall income whereas low-χ taxpayers will evade. This is illustrated in Figure 5.It is tempting to labelhigh-χ taxpayers as ‘honest’ but honesty is a relative concept in this model since all taxpayers withfinite χ will evade when the expected gain is sufficiently great. The effect of changes in the probabilityof detection, the fine and income at the individual level are similar to those of the standard modeldescribed by (4). In contrast, the consequence of any change for the aggregate level of evasion (the sumof evasion across individual taxpayers) is composed of the change in the level of evasion by existingevaders (the ‘intensive’ margin) and the change resulting from more or fewer taxpayers evading causedby the change in V0 relative to χ (the ‘extensive’ margin). Since these effects work in the samedirection, there is no change in the predicted effects on aggregate or individual evasion.

The effect of an increase in the tax rate upon the level of evasion is derived in Gordon (1989). Gordonshows that with decreasing absolute risk aversion there exists some χ∗ < V0 such that ∂E/∂t < 0 ifχ < χ∗ and ∂E/∂t > 0 if V0 > χ > χ∗. This is demonstrated by differentiating (75 ), to show that



the effect of the tax change is

∂E

∂t=

E[U ′′(Y )r t [Y − r E]] − χ

tE [

U ′′(Y ) [r t]2] (76)

For χ = 0, (76) reduces to the standard result which has already been shown to be negative. In thelimit as χ tends to V0 the derivative in (76) is positive since E tends to 0 from a positive value. Withdecreasing absolute risk aversion the numerator is monotonic in χ , so the sign of ∂E/∂t must changeonce as claimed. If there are taxpayers characterized by values of χ that fall in the relevant range thenthere will be some evaders who increase the extent of their evasion as the tax rate increases. In additionto this effect, an increase in t also raises V0, so that previously ‘honest’ taxpayers will begin to evade.It is therefore possible, though not unambiguous, that the introduction of utility cost of evasion cangenerate a positive relation between the tax rate and the extent of evasion.

An alternative to the psychic cost is the ‘conscience parameter’ of Eisenhauer (2006, 2008). In thisformulation of the compliance decision an individual recognizes that evading tax results in free-ridingon the taxes paid by compliant taxpayers. This generates a sense of guilt for the tax evader. The guilt isrepresented by discounting the untaxed income by the moral equivalent of a tax rate. Thus, the utilityfunction in the Eisenhauer model is

V = pU (Y [1 − t] − f t E − κE) + [1 − p]U (Y [1 − t] + t E − κE) (77)

where κ is the ‘moral discount rate’ of the taxpayer, or the ‘shadow price of morality’. The guiltinterpretation implies that κ > 0. However, the model can also work with the assumption that κ < 0which represents a taxpayer who derives pleasure from evading tax. If κ > 0 is sufficiently high(κ > t[1 − p − p f ]) then it is optimal not to evade. The model implies that tax evasion increaseswith its expected financial gain but decreases when morality increases. However, the addition of guiltin this form has the effect of reducing the value of d E/dt , so moving the tax effect further in thewrong direction. Eisenhauer calibrated the model using recent empirical data from a transition economy(Moldova) and from several groups of self-employed US taxpayers; both data sets involve aggregatedata. The empirical results are unfavourable to the model: to generate the observed compliance rates κmust be around 39%, that is, guilt from under-reporting taxable income by one dollar is equivalent tothe loss of 39 cents. This is a high value for the guilt parameter since it exceeds the average value ofobserved tax rates.

Bayer (2006) provides an alternative structure for the cost of evasion (which can be interpretedas financial or psychic) that directly addresses the tax effect. Bayer assumes that the taxpayers arerisk-neutral (to simplify the analysis and permit an explicit solution to be derived) and that concealingincome is costly. The marginal cost of evasion is taken to be proportional to the share of income notreported, so C ′(E) = E

Y ξ where ξ represents individual evasion opportunities or attitudes. The totalcost of evasion, C(E), has a constant component that captures the cost of acquiring information aboutevasion opportunities. The tax liability for income Y is T (Y ) = τ (Y )Y , with τ (Y ) = tY + α , and thepenalty is a linear function of the evaded tax.

Combining these assumptions shows that income if audited and caught evading is

Y c = Y − T (Y ) − f [T (Y ) − T (Y − E)] − C(E) (78)

and income if evasion is successful is

Y n = Y − T (Y − E) − C(E) (79)

Under the assumption of risk-neutrality the objective of the taxpayer can be written

max{E}

V = pY c + [1 − p]Y n (80)



The derivative of the objective function for the choice of the level of evasion is

dV

d E= [1 − p] T ′ (Y − E) − p f T ′ (Y − E) − C ′ (E) (81)

which gives the necessary condition for some evasion to be undertaken

1 − p − p f > 0 (82)

Using (78) and (79) the first-order condition can be solved to give the interior solution for the levelof evasion

E = [1 − p − p f ] [2tY + α]

2t [1 − p − p f ] + 1

ξY

(83)

Differentiating with respect to the tax rate t gives the effect of the tax rate upon evasion as

d E

dt=

2 [1 − p − p f ]

[1

ξ− α [1 − p − p f ]

][

2t [1 − p − p f ] + 1

ξY

]2 (84)

If evasion takes place then 1 − p − p f > 0 and Y > E , so (83) implies that 1ξ

− α [1 − p − p f ] > 0.Hence, at any interior solution it must be the case that ∂E/∂t > 0. The model predicts an increase inthe tax rate, t (or in α) must lead to more evasion.

Two comments on this analysis are in order. First, the result can be extended to general functionalforms for the cost and tax function, so the specific functional forms are not necessary. The assumptionof linear expected utility and constant marginal cost are used to allow an explicit solution to be obtainedfor the level of evasion. Bayer (2006) provides necessary conditions on the cost function for the resultto hold more generally. Second, by considering a population of individuals who are characterized byindividual values of ξ it is possible to separate the population into evaders and non-evaders. Thispermits an analysis of the aggregate level of evasion via the intensive and extensive margins.

The results that have been reviewed show that the introduction of additional costs into the compliancedecision can alter the sufficient condition for evasion and reverse the tax effect.6 Furthermore, theintroduction of individual-specific parameters into the extended representation of costs allows differenttaxpayers to make different choices. This permits the models to be in agreement with empirical andexperimental evidence. The drawbacks, and these also apply to all the social interaction models thatfollow, are that there is now at least one unobserved parameter in the model and the predictions of themodels are not as precise as those of the standard model.

4.2 Social Customs

One explanation for the existence of a psychic cost of evasion is the loss of social prestige that followsdetection as a tax evader. The models that include a social custom in the decision framework makeexplicit the link between the psychic cost and the wider social environment. This permits a range ofinteresting phenomena to be modelled.

The psychic cost of Gordon (1989) can be interpreted as the payoff from a social norm which isexogenous to each taxpayer but it is more convincing to assume that the additional cost is generated byexplicit social interaction. This can be done by assuming that the cost is an increasing function of theproportion of taxpayers who do not evade. This formulation captures the fact that more social prestigewill be lost the more out of step the taxpayer is with the remainder of society. This approach has been



developed by Fortin et al. (2007), Kim (2003) and Myles and Naylor (1996) to show that reputationeffects can lead to multiple equilibria and epidemics of evasion.

Let the utility level for a taxpayer of income Y and facing the tax rate t , who chooses not to evade,be given by

V N E = U (Y [1 − t]) + bR(1 − μ) + c (85)

with b ≥ 0 and c ≥ 0. The additional utility from adhering to the social custom of honest tax paymentis bR(1 − μ) + c, where μ is the proportion of population evading tax and R′ > 0. The parameters band c represent the attitude of the individual taxpayer towards the social custom and can be expected tobe different across taxpayers. The effect of the social custom can be given two interpretations. One isthat the utility, bR(1 − μ), of conforming with the group of honest taxpayers and the utility, c, obtainedfrom following the social custom can be considered to be distinct entities. The distinction betweenthe return to conformity and the return from following the social custom is adopted in Akerlof (1980)and has also been stressed in this context by Cowell (1990). Alternatively, the sum bR(1 − μ) + c canrepresent a general function describing the effect of the social custom but with a linear form chosenfor simplicity. It should be noted that the utility from adhering to the social custom is always derivedfrom honest behaviour, even if the honest are in a minority.

When the decision to evade tax is taken, the resulting level of utility is

V E = max{X}

{pU (Y [1 − t] − f t[Y − X ]) + [1 − p]U (Y − t X )} (86)

The assumption used to write (86) is that the utility from following the social custom is lost whenevasion is chosen. The declaration that maximizes (86) is denoted by X∗ and the maximized level ofutility by V E∗. The optimal level of reported income must satisfy the first-order condition

p f U ′(Y [1 − t] − f t[Y − X∗]) − [1 − p]U ′(Y − t) = 0 (87)

From this condition is can be seen that for given Y , the choice of X∗, and hence the value of V E∗,is independent of b and c. Effectively, if evasion is chosen then the level is identical to that for thestandard model. Hence, if an individual evades tax and deviates from the social custom the extent towhich they evade does not depend on the importance attached to the social custom.

Whether an individual taxpayer evades depends upon the value of V N E relative to V E∗. Evasion willonly take place if V E∗ > V N E . The strict inequality captures the implicit assumption that the taxpayerprefers to follow the social custom if there is no utility loss from doing so. Writing this condition indetail, a taxpayer chooses to evade if

pU (Y [1 − t] − f t[Y − X∗]) + [1 − p]U (Y − t X∗) > U (Y [1 − t]) + bR(1 − μ) + c. (88)

The first point to note is that the model will generate no observations of taxpayers just evading a ‘little’tax. Taxpayers either do not evade or jump straight to evasion level E = Y − X∗. The second pointto note is that the model, in common with the psychic cost models, can have perfect compliance evenwhen the expected financial gain from evasion is positive. This occurs when the gain from evasionis not sufficient to offset the loss from not following the social custom. The key point of the modelis that the choice of whether to evade or not depends on the proportion of the population who areevading, μ. The choice is not made by the taxpayer in isolation but is the outcome of a process ofsocial interaction. The determination of which taxpayers evade is made by finding a self-supportingproportion of evaders. That is, the equilibrium value of μ has the property that precisely proportion μof taxpayers will find it optimal to evade in response to that value. Figure 6 depicts such an equilibrium.

A change in the tax rate has an effect at the intensive margin and the extensive margin. The effect atthe intensive margin – a taxpayer declaring X∗ < Y – is the same as for the standard model. The effectat the extensive margin depends on how the change in the tax rates affects the self-supporting value of



Proportionevading

Proportionchoosing toevade

µ

µ

Figure 6. Social Equilibrium.

μ. It is through this process that the model can generate a range of interesting outcomes. For example,a small change in tax rate may cause a low-μ equilibrium to disappear so that the economy ‘jumps’ toa high-μ equilibrium. Hence, an epidemic of evasion can occur in response to a minor change in thetax system

The effect on the extensive margin can be found by taking given values of Y and t and evaluatingthe critical proportion of evaders required for a taxpayer with characteristics {b, c} to evade. If thereexists a value μ∗ of satisfying

pU (Y c) + [1 − p]U (Y n) = U (Y T ) + bR(1 − μ∗) + c (89)

where Y T ≡ Y [1 − t], then, since the right-hand side is monotonically increasing in μ, a taxpayerof parameters {b, c} evades if μ > μ∗ and pays their full amount of tax if μ ≤ μ∗. Now consider ataxpayer for whom there exists a value μ∗ such that (89) is satisfied. The effect of the tax rate on thecritical proportion, μ∗, can be calculated as

dμ∗

dt=

[pU ′ (Y c) + [1 − p] U ′ (Y n) − U ′ (Y T

)]Y

bR′ (90)

where the first-order condition for the choice has been used to simplify the expression. The effect ofan increase in the tax rate on the critical proportion depends on the expected marginal utility fromevasion compared to the marginal utility of correct declaration. This is not restricted by the model, sothe effect may go in either direction. If dμ∗

dt < 0 then an increase in the tax rate reduces the proportionof evaders required to encourage the (previously) marginal taxpayer to evade. The equilibrium numberof evaders must therefore increase.

This model of a social custom emphasizes the importance of distinguishing between individual andaggregate effects of an increase in the tax rate. For each individual who is already evading it remainsthe case that an increase in the tax rate raises the level of income declared. However, an increase inthe tax rate has the effect of increasing the proportion who evade. The net effect is determined by theresolution of these two. Consequently, aggregate data may show a positive relation between the taxrate and evasion even if the opposite is true at the individual level.



Traxler (2010) modifies the analysis by assuming the social custom is a loss of utility when evading.This gives utility

V E = E[U (X )] − Eiθi c (μ) (91)

where θi ≥ 0 is the individual-specific degree of norm internalization and c (μ) is the strength of thenorm for a given fraction of evaders. It is assumed that c′ (μ) ≤ 0 for μ ∈ [0, 1) so that an increase inthe proportion of taxpayer deviating from the norm lowers the social custom cost of evasion. With thisfunctional form the first-order condition for an interior solution is

−p f tU ′ (Y c) + [1 − p] U ′ (Y n

) = θi c (μ) (92)

If θi = 0 the first-order condition collapses to (5) so the standard necessary condition for evasion tooccur will apply. If θi is sufficiently high then there will be no evasion. Evaluating (92) at X = Y givesthe critical value of θi that separates non-evaders from evaders:

θ (μ) ≡ [1 − p [1 + f ]] tU ′ ([1 − t] Y )

c (μ)(93)

Hence, for given μ, the choice of evasion behaviour is determined by

Ei ={

0 for θi ≥ θ (μ)

E∗i for θi < θ (μ)

(94)

where E∗i = Y − X∗, and X∗ is the solution to (87).

The effect of an increase in the tax rate can be found from (92) as

∂E∗i

∂t= −1

E[U ′′][θi c(μ)

t+ [p f tU ′′(Y c)[Y + f E∗

i ] − [1 − p]tU ′′(Y n)[Y − E∗i ]]] (95)

When θi = 0 this collapses to the standard result described in (13), so that ∂E∗i

∂t < 0. When θi satisfies

θi ≥ θ(μ) ≡ −t

c(μ)[p f tU ′′(Y c)[Y + f E∗

i ] − [1 − p]tU ′′(Y n)[Y − E∗i ]] (96)

then an increase in the tax rate raises the level of individual evasion. This is very similar to the resultin Gordon (1989). What is different here is that the tax effect is dependent upon the extent of evasionin the population through μ. A fall in the proportion of tax evaders reduces θ (μ) so a positive taxeffect applies for a larger range of values of θi .

The introduction of psychic costs and of social norms is capable of explaining some of the empiricallyobserved features of tax evasion which are not explained by the standard expected utility maximizationhypothesis. It achieves this by modifying the form of preferences but the basic nature of the approachis unchanged. The obvious difficulty with these changes is that there is little to suggest precisely howsocial norms and utility costs of dishonesty should be formalized.

4.3 Perceptions of Fairness

The tax compliance decision involves two dimensions of fairness. First, there is perceived trade-offbetween what is paid to the government in taxes and what it received from the government in publicservices. If the services received are viewed as limited in quantity or poor in quality then anythingother than a demand for a low tax payment will be perceived as unfair. Second, there is the allocationof the tax payments across taxpayers. If some taxpayers are believed to pay little then a taxpayerasked to make a large payment will perceive the system to be unfair. The issue of horizontal equity as



judged by individual taxpayers is relevant here, and vertical equity also plays a role. It is possible toview fairness as one aspect of the more general concept of ‘tax morale’. Certainly, when we discusstax morale in the next section we will have cause to consider fairness again. A separate treatment offairness is justified on the grounds that fairness in the senses we have described has been successfullyformalized with interesting implications for the compliance decision and the tax effect.

The concept of fairness with respect to the public services obtained relative to tax payment hasbeen analysed by Cowell and Gordon (1988). In their model taxpayers derive utility from income andfrom a public good. An individual taxpayer can choose how much tax to evade but does not directlychoose the quantity of public good which is determined by the tax rate and by the compliance decisionsof the population of taxpayers. The key feature of the model is that a change in the tax rate has anon-negligible effect on public good supply and this indirectly affects the compliance decision. If theeffect of the public good is sufficiently strong, and operates in the correct direction, then it can reversethe standard result for the tax effect.

A simplified version of the Cowell and Gordon (1988) model can be presented as follows. Let thetaxpayer care about income and public good and write the expected utility function as

V = pU(Y c,G

) + [1 − p] U(Y n,G

)(97)

The taxpayer takes the quantity of public good, G, as given when choosing the level of evasion. Underthis assumption the first-order condition for the choice of evasion is

dV

d E= −p f U c

Y + [1 − p] U nY = 0 (98)

When the tax rate is increased public good supply will also change, and this effect must be incorporatedinto the analysis of the tax effect. The tax effect is therefore given by

d E

dt=

[1 − p] U nY Y [Y − E] − p f U c

Y Y [Y + f E] + [[1 − p] U n

Y G − p f U cY G

] ∂G

∂tS

(99)

where S < 0 from the second-order condition. The first two terms in the numerator are equivalent tothe expression for the standard evasion model. The third term captures the public good effect. This isthe product of the effect of additional public good upon the marginal utility of income and the effectof an increase in tax rate on public good supply. If the product of these terms is negative then theywill offset the standard tax effect and will overturn it if strong enough.

The Cowell and Gordon analysis addresses the aspect of fairness concerning the services providedin return for payment of tax. The second aspect of fairness – the perceived fairness of the distributionof tax payments – has been added to the Cowell and Gordon model by Bordignon (1993). To make theextension Bordignon imposes a social norm upon behaviour, based on the concept of Kantian morality,under which each individual assesses their fair tax payment towards the provision of public goods.The calculation of the fair payment takes into account both the quantity of public good provided andthe tax payments made by other taxpayers. This fair payment provides an upper bound on the extentof tax evasion since the taxpayer will not wish to pay less than the fair payment. To calculate theactual degree of tax evasion each taxpayer then determines a level of evasion from the expected utilitymaximization calculation and evades whichever is the smaller out of this quantity and the previouslydetermined upper bound. This formulation is also able to provide a positive relation between the taxrate and evaded tax for some range of taxes since the tax rate plays a role in determining the upperbound on acceptable behaviour.

Two further contributions are worth describing. Falkinger (1988) analyses the decision problem of anexpected utility maximizing taxpayer who assesses the benefit obtained in exchange for the payment oftax. The analysis shows that evasion may decrease as the beneficial expenditure increases, but the result



is not unambiguous. Falkinger (1995) presents arguments to explain a link between risk aversion andthe perceived degree of fairness in the tax system. One aspect is the link between marginal utility andpublic good provision just described. A second aspect is that there is greater guilt from evasion whenthe system is perceived to be fair. Hence, the degree of fairness can be linked back as a determinantof the psychic cost discussed earlier.

4.4 Tax Morale

Torgler and Schaltegger (2006) define tax morale as the intrinsic motivation to pay taxes or, followingSchmolders (1960), as a set of attitudes regarding tax compliance. Tax morale includes the willingnessto pay tax and the wish to contribute a fair share to the financing of public goods. Tax morale affectsthe compliance decision because the incentive to pay tax honestly is reduced if morale is low. Theconcept of tax morale is easy to grasp at an intuitive level, but it has proved harder to represent withina formal model of the compliance decision. There has been much empirical work on tax morale butthe theoretical work in this area is relatively undeveloped. This is a consequence of the difficulties informalizing the concept of morale in a precise way that is amenable to analysis. In some representationstax morale can appear similar to the social custom costs discussed earlier. In others, it can take a formvery similar to the fairness models.

The literature has identified several dimensions of tax morale which are influenced by a rangeof factors. Changes in any of the dimensions of tax morale can lead to reduced compliance. Thecultural dimension includes socio-demographic and socio-economic factors, as well as national prideand religious customs. The institutional dimension refers to the structure within which the governmentoperates, especially the perception of inclusion in the democratic process and the degree of trust in theinstitutions operating the tax system which combine to determine the level of acceptance of tax policy.Another dimension is the structure of the tax system, particularly its perceived degree of fairness, andthe operation of the system. If the system is widely perceived as unfair then tax morale is reduced. Thekey observation of Frey and Feld (2002) is that external intervention in the form of rewards, commands,rules and regulations, and punishments negatively affect intrinsic motivation to pay tax. Consequently,deterrence policies pursued by tax authorities may have increase compliance in the short term but canbe detrimental in the long term it they undermine tax morale.

Dell’Anno (2009) extended the model of Gordon (1989) to investigate how compliance is affectedby tax morale and how tax morale is dependent on a taxpayer’s intrinsic attitudes to honesty andsocial stigma. Tax morale is separated into the effect of a ‘guilty conscience’ and the effect of ‘publicdisgrace’ which are both costs that reduce the utility of the taxpayer. It is assumed that these attitudesare influenced by the taxpayer’s perception of the extent of tax evasion in the society and of thebehaviour of the tax authority. These ideas can be captured by assuming that the payoff function isgiven by

V = pU(Y c

) + [1 − p] U(Y n

) − vE − φρE (100)

where the psychic cost is vE (v is an individual honesty characteristic), and the level of social stigmais determined by the perceived degree of fairness (φ) and quality of relationship between taxpayer andtax authority (ρ). Anything that reduces perceived fairness, such as an increase in evasion, will reduceφ and lower the social stigma cost of evasion. Similarly, the quality of relationship, ρ, can be affectedby the actions of the tax authority. If an increase in tax rate reduces either, or both, of φ and ρ thenthis will provide a mechanism through which the direction of the tax effect can be reversed comparedto the standard model.

Empirical research has analysed the determinants of tax morale and the effects of changes in taxmorale. In this empirical work tax morale is usually measured by the response to attitudinal questions



on tax payment. Torgler (2003) found evidence that trust in government, pride, and religious convictionhave systematic positive influences on tax morale even after controlling for a range of individualcharacteristics. Weck (1983) found a negative correlation between tax morale and the size of theshadow economy. Torgler (2001, 2005), Alm and Torgler (2006) and Alm et al. (2004) also found astrong correlation between tax morale and the size of shadow economy. For example, Alm and Torgler(2006) found that the variable ‘tax morale’ can explain more than 20% of the total variance of thesize of the shadow economy. Torgler (2003) found that increased tax morale reduced evasion. Freyand Torgler (2007) found a high correlation between perceived tax evasion and tax morale, as well asa strong positive correlation between institutional quality and tax morale. Ashby et al. (2009) foundthat occupational taxpaying culture is important in explaining attitudes towards the tax office and taxminimization. Laboratory experiments conducted by Torgler (2004) showed that tax morale was animportant determinant of tax compliance.7

The empirical evidence demonstrates that tax morale is important for the compliance decision. Thetheoretical analysis of the concept has been limited, so little can be drawn from the concept in termsof predictions about behaviour.

4.5 Summary

In contrast with the standard approach and the non-expected utility models of tax evasion, where thestructural models have been developed but their empirical estimation has proved difficult, the socialinteraction and tax morale literature has been empirically successful but generally is lacking systematicstructural modelling. Further theoretical work in this area seems justified by the empirical studies.

It should be clear from the discussion that the literature has analysed a wide range of socialinteractions affecting a taxpayer’s decision. The central point is that all the contributions emphasizethe idea that the compliance decision is not taken in isolation but is made in the context of the socialenvironment of the taxpayer. This extends the set of factors that the taxpayer has to take into accountcompared to the standard model of evasion. As a consequence, models that include social interactionare able to reverse the tax effect for appropriate choice of structure. A secondary point is that socialinteractions models involve both intensive and extensive margins which impact upon the analysis ofindividual and aggregate evasion levels. However, it is clear that these benefits have to be set againstthe loss of precision of the predictions compared to those of the standard model.

5. Conclusions

The Yitzhaki version of the Allingham–Sandmo model of the individual tax compliance has formedthe standard against which all later developments are judged. The general agreement in the literatureis that the model produces two incorrect predictions. First, the sufficient condition for evasion to takeplace over-predicts the empirically observed level of evasion. Second, the model predicts that the levelof evasion will fall when the tax rate rises (a consequence of the fine for evasion being proportionalto tax evaded). This result runs counter to intuition and is also in conflict with some of the empiricalevidence. These shortcomings have led to the development of a significant literature that reconsidersthe compliance decision using ideas drawn from behavioural economics. This paper has surveyedthese new contributions and assessed whether they perform any better than the standard model as anexplanation of the compliance decision.

We have identified two classes of contribution in the behavioural economics literature. The firstclass modifies the standard model by using non-expected utility theory. Moving to non-expectedutility permits a broader range of possibilities for the structure of preferences and removes some ofthe constraints imposed by expected utility theory. The second class of contributions alter the payoff



structure to include a range of factors that broadly reflect social interaction between taxpayers. Together,these contributions provide a wide range of alternative models of the tax compliance decision. Thisnew literature has incorporated many interesting elements into the tax evasion decision that are absentin the standard model. The models are able to make predictions that fit better with the evidence.However, the gain in the potential to match the evidence has yet to offset the loss in precision relativeto the standard model. The standard model is exemplary in its parsimony and clarity of prediction. Togo beyond it some of the parsimony is lost.

Two observations about the literature we have surveyed follow from this review. First, the use ofweighting functions to replace the objective probabilities changes the threshold probability for evasionto take place and, hence, the proportion of evaders in any given situation. Second, non-expected utilitymodels do not reverse the tax effect as long as the payoff depends on the evaded tax (the productof the tax rate and undeclared income). In order to have an impact upon the direction of the taxeffect the structure of payoffs has to be modified in some significant way. One route is to assumethat the probability of detection is dependent upon the declared income level, but we have argued thatthis captures the context of the evasion decision (as a game between taxpayers and revenue service)rather than a representation of a non-expected utility choice theory. A second route is to introduceconsiderations of social interaction. Among such elements we have discussed psychic costs, socialcustoms and social norms and tax morale. The actual predictions of the model with social interactiondepend upon the precise structure that is adopted. Some variants can reverse the tax effect but otherscannot. All are able to alter the sufficient condition for tax evasion to take place and can potentiallymake predictions consistent with practical observation. The concept of tax morale captures a broadrange of ideas but has so far not received a compelling formalization.

We close this review with some final comments on fruitful directions for future research. It istempting to conclude that the current literature has provided a variety of alternative models and thatit is now the role of empirical and experimental research to determine which is the correct one.We see two reasons why this temptation should be avoided. First, before evasion can take placethere needs to be an opportunity. There are many forms of employment that offer no opportunity toevade because employers report directly to the tax authorities (as is the case with PAYE in the UK).In contrast, self-employment require reporting of income and opens the opportunity to evade. Themodels we have reviewed are silent on the source of income so do not address how occupationalchoice is linked to evasion possibilities and individual characteristics. Second, there is the broaderquestion of whether there can be a single encompassing model that provides the explanation in allcircumstances, for all agents. Perhaps there can be, but one must doubt whether it could actually makeany significant predictions. The standard model is extremely parsimonious and precise in it predictions.As the model becomes more general both of these admirable characteristics are lost, so the benefitof not contradicting the data can only be bought at a cost. A more practical question is whether wereally need a comprehensive understanding. For example, a revenue service designing an audit policymay only need to know that the audit probability is subjective, so can be manipulated, even if the truepayoff function is unknown.

The two key insights of the models we have reviewed remain valuable. Non-expected utility theoryhighlights the fact that individual taxpayers do not make choices based on knowledge of objectiveprobabilities. This is a fact exploited by revenue services in the design of their strategies which aredesigned to convey the fact that the probability of audit is significant and that punishment is harsh.Social interaction stresses that taxpayers do not operate in a private isolation but act within a socialenvironment. It is within this environment that they develop their beliefs and attitudes. This suggeststhat the two strands could be usefully combined to model how social interaction leads to the formationof the beliefs and attitudes that enter into individual decision. Agent-based modelling could be the toolto deliver this especially if combined with the literature on social networks to provide a structure oninteraction.



Acknowledgements

Gareth Myles was a Visiting Professorial Fellow at the Australian School of Taxation (ATAX) duringthe preparation of this paper. He wishes to thank ATAX for their hospitality and support. The paperhas benefited from the comments of three referees, and participants at seminars in Bath, Kent and thejoint HMRC/HMT International Conference on Institutional Taxation Analysis.

Notes

1. It is worth observing that the model assumes income is unobserved by the tax authority. If a withholdingor pay-as-you-earn system is in operation then the analysis only applies to income that falls outside thatsystem. This is important when contrasting the predicted level of evasion with the estimated level ofevasion.

2. Although many countries can punish evasion by imprisonment (and some even apply the death penalty),most cases are settled by the payment of a fine.

3. Another possibility we briefly discuss is to make the probability of detection depend upon the level ofincome declared. See Section 3.4.

4. According to (34) the value of the taxpayer’s prospect should be V = v (D − t X ) +p [v (D − t X − f t [Y − X ]) − v (D − t X )]. There is a difference between this formulation and (37)because Yaniv (1999) applies the probability weight to the value of the difference instead of to thedifference in values. We choose to follow the Yaniv formulation to make our points.

5. The variable probability was first discussed in Allingham and Sandmo (1972), and again in Sandmo(2005) and al-Nowaihi and Dhami (2007). Assume the probability of audit is a decreasing function ofdeclared income, p = p (X ), with p′ < 0. The objective of the taxpayer becomes

V = w1 (p (Y − E)) vc (Y c − R) + w2 (1 − p (Y − E)) vn (Y n − R)

At an interior maximum

d E

dt= w′

1 p′vct − w1v

cEt − w′

2 p′vnt − w2v

nEt

S

where S < 0. If w′1 p′vc

t − w′1 p′vc

t is sufficiently negative then the tax effect can be reversed. Aninteresting question is how we interpret the meaning of a probability that depends on the announcementof income. One possibility is that this reflects the operation of the subjective assessment of probabilitiesby the taxpayer. An alternative explanation, which is more objective in nature, is that it reflects areduced form representation of the auditing process. This is the outcome that arises when the revenueservice extracts a signal from the taxpayer’s declared income to determine the probability of audit (seeReinganum and Wilde, 1986).

6. An alternative interpretation of the cost of evasion is provided by Lee (2001). Lee assumes a taxpayercan reduce the assessed income after audit by paying an additional cost (such as a reduction in returnfrom using concealed investments or the cost of professional advice for securing income in non-taxableforms).

7. Torgler (2002) provides a comprehensive survey of experimental work on tax morale.

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APPLICATIONS OF BEHAVIOURAL ECONOMICS TO TAX …people.exeter.ac.uk/gdmyles/papers/pdfs/69 JOES.pdfBEHAVIOURAL EXPLANATIONS OF TAX EVASION 943 that have been proposed. The non-expected