Mathematical Social Sciences 10 (1985) 81-89

North-Holland

81

TAX E V A S I O N A N D T H E P R I S O N E R ' S D I L E M M A

Daniel GOTTLIEB* Research Department, Bank of Israel, Jerusalem, Israel

Communicated by F.W. Roush

Received 3 December 1984

This analysis expands the model of tax evasion suggested by Allingham and Sandmo (1972) to include public goods, financed by revenues from taxation and penalties. We argue that this leads to a Pareto inferior equilibrium outcome of individual declarations both in models of competitive and interdependent behaviour, thus linking the paradox to the Prisoner's Dilemma, well known from game theory. It is further claimed that a government led by utilitarian welfare standards will perpetuate tax evasion in the case of positive variable costs of detection.

Key words." Tax evasion; prisoner's dilemma; public goods.

1. Introduction

Tax evasion has been analysed by Allingham and Sandmo (1972), Srinivasan (1973) and others. Although this subject has been widely discussed, no attempt has yet been undertaken to introduce the supply of a public good into these models.

The present analysis expands the basic model suggested by Allingham and Sand- mo to include a public good financed by revenues from taxation and penalties.

In Section 2 it is argued that certain weak conditions on the parameters of the model imply that the equilibrium outcome of income declarations to the tax autho- rities, based on individual maximisation of expected utility, lacks the feature of Pareto optimality.

In Section 3 the problem is analysed in a game-theoretic framework, linking the paradox to the well known Prisoner's Dilemma.

Section 4 discusses the problem of overcoming the Prisoner's Dilemma applied to taxation in a utilitarian framework.

2. The basic model

Consider an economy in which n individuals (i= 1,. . . , n) have to pay a propor- tional tax (0) on their exogenously given income (IV,.) to the government which then uses the collected tax funds to supply the society with a public good (G).

*I am indebted to Dr. A. Rubinstein and Professor E. Shesinski for helpful comments.

0165-4896/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)

82 D. Gottlieb / Tax evasion

Individuals derive utility from the consumption of private and public goods. Private consumption is represented by after-tax income in the individual's utility function.

The individual can take advantage of the informational dichotomy between him and the authorities concerning his true income. He can choose to declare only part (Xi) of his income for tax purposes, thereby risking detection with probability (p) and penalty (F) per dollar on the evaded tax. 1

Let Yi denote the level of income after successful evasion and zi the level when the tax payer is audited:

Yi= Wi- OSi,

Zi = Wi- OXi- FO(Wi- Xi). (1)

All individuals are assumed to have identical expected utility functions, which are assumed to be additive over income and the public good, strictly increasing, strictly concave and twice differentiable. Each individual is supposed to maximise his ex- pected utility with respect to his income declaration subject to his budget constraint, i.e. solve the problem

max E Ui = (1 -p) U(yi) +pU(zi) + V(G) for all i. (2) x,

The government maximises some welfare function (see Section 4) subject to the maximised private expected utilities and a balance budget, by choosing optimal values for the tax rate (0) and the audit rate (p). Penalty (F) is fixed by the legal system and therefore exogenous to the present model. Detection involves a certain cost. This cost is assumed to be twice differentiable with a positive marginal cost, i.e.

C'(p)>0.

The government operates within a balanced budget:

(3)

n

ET=(1-p)O ~ X~+p ~ [OXi+FO(W~-X~)]-C(p)=G. i = 1 i

(4)

The element of uncertainty with respect to public funds is assumed to be eliminated in each period by an equalising fund (p) for which

E(/~) =0.

Simplifying (4) the balanced budget becomes:

G=O ~ Xi+pFO ~ (Wi Xi)-C(p)" i i

(5)

1We adopt Yitzhaki's (1974) suggestion to relate the penalty to the amount of tax evaded rather than to income evaded. Thus, the substitution effect in the comparative static individual reaction to changes in the tax rate is eliminated (see below).

D. Gottlieb / Tax evasion 83

We analyse two cases of individual decision. (1) The competitive case. The individual takes the level of public goods supplied

as given and neglects his own and other people's direct influence on tax funds (and therefore on the supply of the public good). This assumption seems reasonable for

a large society. (2) The interdependent case. It is assumed that the tax payer takes into considera-

tion the direct effect of his and other people's declarations on the supply of public goods. Game-theoretic tools are necessary for this analysis.

3. The competitive case

By maximising expected utility, the individual regards the supply of the public good as given, G = (~. Let X* denote optimum declaration. The first- and second- order conditions for maximum expected utility are

E U x = - ( 1 - p ) U ' ( y * ) + p ( F - 1 ) U ' ( z * ) = O for all i, (6)

o r

U'(y*) p ( F - 1)

U'(z*) 1 - p

D-EUx , x,=O2[(1-p)U"(y*)+p(F- 1)2U"(z*)]<O for all i. (7)

An interior solution for X; requires that

p + (1 -p ) U'(Wi) <pF< 1 for all i. 2 (8) U'(W~(1 -FO))

The comparative static results arising from changes in the values of the para- meters p, 0 and F correspond to those of Allingham and Sandmo (1972) despite the introduction of a public good into the model. We obtain:

O0 0(1 - p) U' (y*) [X* { R A (z*) - R A (YT)} + F( Wi- XT) R A (z/*)],

D

ox? op

ox? OF

0 [U'(y*) + ( F - 1) U'(z*)] > 0, (9)

D

PO[u,(z , )_O(F_l)( * " * - - - - - - W i - S } ) U ( z i )1>0. D

2The left-hand side can be rewritten:

p ( F - 1) U'(Wi)

(1 - p) U'( Wi(1 -FO))

In this form this condition may easily be compared with the first-order condition (6).

84 D. Gottlieb / Tax evasion

The tax payer increases his optimal declaration in response to an increase in the audit and penalty rates with respect to the tax rate only under the assumption of decreasing absolute risk aversion.

Equilibrium versus Pareto optimum - a dilemma

Let the outcome * * (Xl ,X~, ..., X/*, ..., X*) be denoted by X*. This outcome repre- sents a competitive equilibrium since all tax payers are at their atomistic equilibria. Intuitively one would expect such an outcome to be optimal. When isolated, each individual reaches maximum expected utility. However, the social outcome is Pareto inferior. This can be seen by letting each individual add an increment h to his

declaration X*. Let Ri(h ) denote the ith person's expected utility after all members of the society have increased their own declarations by an increment h:

Ri(h) = (1 - p ) U ( W i - O(X* + h)) + p U ( W i - O(X* + h) - FO(W i - X * - h))

+ V(O ~i (X*+h)+pFO ~ ( W ~ - X * - h ) - C ( p ) ) f o r a l l i . ; (10)

We wish to show that R~(h) is positive for all individuals; in other words, there exists a small enough h such that everyone's welfare can be improved by each person ad- ding an increment to his declaration:

R~(h) = 0 [ - (1 - p ) U ' ( y * ) + p U ' ( z * ) ( F - 1) + n(1 - p F ) V'(G*)]

for all i. (11)

By substituting (6) into (11) we obtain:

R~(h) = On(1 - p F ) V'(G*) for all i. (12)

Assuming an interior solution (see (8)), the derivative R~(h) will be positive for all individuals. Therefore the equilibrium outcome X* cannot be Pareto optimal. In an economy with tax evasion all tax payers prefer everybody to increase their tax pay- ment marginally. However, such a decision violates each tax payer's own atomistic equilibrium. Consequently, each person will join an agreement of declaring an addi- tional increment mei~ely to induce others to increase their declarations while he will remain at his initial optimum declaration.

We have demonstrated that the equilibrium outcome is Pareto inferior. Com- pared to the outcomes of voluntary full compliance (with declaration H), 3 a Pareto improvement can only be achieved if the following statement holds:

U(Hi)+ V ( G ( H ) ) > ( 1 - p ) U ( y * ) + p U ( z * ) + V(G(X*)) for all i. (13)

3 Voluntary full compliance refers to the situation in which tax payers agree to raise their declarations to I4I/at the given audit rate.

D. Got t l i eb / Tax evas ion 85

Since

G(H) -G(X*)=O(1-pF) ~ (Wi-X*) , i

y*- O( X*),

Hi-

it can be shown by the use of the mean value theorem that

O(1-pF) ~ (Wi-X*)V ' (G(H))>(1-p)O(Wi-X*)U' (H~) i

- pO(F- 1)( W~ - X~') U' (Hi)

for all i, (14)

is a sufficient condition for (13). Rearranging (14) we get:

2 i (Wi -X*) U'(Hi) > for all i. (15)

Wi-X* V'(G(H))

Notice that in the global sense, Pareto inferiority of the equilibrium outcome depends on a considerably more restrictive assumption than at the margin.

4. The interdependent case

The competitive assumption appears to be acceptable in dealing with tax behaviour in large societies. Casual observation suggests, however, that in small societies tax payers feel much more concerned about other people's tax compliance since anybody's evasion incurs an increasing percentage loss of total tax funds. In- dividual decisions become interdependent and may be more adequately analysed by the use of game-theoretic tools.

The situation can be accurately described by a non-cooperative non-zero-sum game with n players (i--1, ..., n). Each participant seeks to maximise his expected utility or payoff by choosing an optimal strategy from the convex, closed and bounded set R i, R i = {Xi[0 ~ X i ~_~ W/}. The individual's (expected) payoff function is EUi(X1,X 2,. . . ,Xi, . . . ,Xn). The product set, which is also convex, closed and bounded is denoted by R, where

R = R 1 x R 2 x . . . × R i x . . . x R n .

Similarly to Rosen (1965), we assume that EUi(X) is continuous in X and concave in Xi for each fixed value of (Xl, . . . ,Xi_1,Xi+l, . . . ,Xn). 4 Rosen proved that an equilibrium point (in the sense of Nash, 1951) X ° = (X °, ... , X °) exists such that

4See Rosen (1965, p. 522).

86 D. Gottlieb / Tax evasion

EUi(X °) max {EUi(X °, . . . ,Si, o o o :

for all i.

Rosen proceeds by showing that the assumption of strictly diagonally concave payoff functions ascertains uniqueness of the equilibrium outcome. In the Appendix it is demonstrated that our model satisfies the condition for strict diagonal concavi-

ty. At such an outcome no player can improve his payoff by a unilateral deviation

from his strategy X °. In other words, the game reaches equilibrium when each player holds on to his 'best' strategy. In maximising his expected utility, the tax payer takes into account his own marginal impact on the supply of public goods. The first- and second-order conditions are modified. )?i denotes the optimum strategy. Namely

EU~,= - (1 - p ) U ' ( f ' i ) + P ( F - 1 ) U ' f f i ) + ( 1 - p F ) V ' ( G ) = 0 for all i, (6')

(7 ') EU.~,i = / ) i = (1 - p ) O"(.yi) + p ( F - 1)2U"(Zi) + (1 - p F ) z V"(G.) < O,

and so are the conditions for an inferior solution:

P + ( 1 - p ) U ' ( W / ) - ( 1 - p F ) V ' ( G ) < p F < I for all i. (8') U'(W~(1 -FO))

Adding condition (15) yields a situation that bears strong similarity to the well- known Prisoner's Dilemma. 5 Both are characterised by a complete separation of the Pareto inferior equilibrium outcome and Pareto optimal disequilibria. Pareto improvements can only be achieved by a joint increase in declarations. The players agree to a contract but each of them breaches it, hoping to gain from the others' compliance. In other words, the ith person's preference ordering is as follows, in descending order:

(1) E U(Si)+ V(G(H -i, S i ) ) , where )?i denotes the optimal declaration of person i, given all others declare their true income.

(2) u ( n i ) + V(G(H)). (3) E U(X i + V(G(f~)). (4) u ( n i ) + v (Go( - i , Hi)).

The standard solution to the Prisoner's Dilemma is social enforcement. In our case this would imply that the government raise the audit rate p to that level which renders cheating unprofitable, that is to p = 1 IF. Let us scrutinise this suggestion in the next section.

5See Luce and Raiffa (1957, chapter 5, sections 4 and 5). Sen (1967) discussed the case of more than two persons, and refers to this situation as the 'Isolation Paradox ' .

D. Gottlieb / Tax evasion 87

5. The optimal audit rate

By considering tax evasion a negative phenomenon, it seems intuitively reasonable to set the audit rate at p - 1/F, as can be seen from conditions (8) and (8'). We may ask ourselves if a utilitarian oriented government would opt for such a policy. Let the government maximise welfare (S), represented by the sum of individual expected utilities and which is twice differentiable, subject to a balanced budget constraint (5) and the tax payer's first-order condition (6); 6 i.e.

/ , /

m a x S = ~ [(1-p)U(y*)+pU(z*)+ V(G*)] (16) p i = 1

s.t. (5) G*=O ~ X*+pFO ~ (Wi-X*)-Ctp), i = 1 i

(6) -(1-p)U'(y*)+p(F-1)U'(z*)=0 for all i.

The first-order condition is:

2 i -uO, t)+U(z*)+ v'(o*)

[ \{OX*']-C'(P)I]017 J =0 (17) × OF ~i ( W , - ' X * ) + 0 ( 1 - p F ) ~i At .p>_ 1/F the first derivative of the social welfare function yields:

~[p>_,/F: ~ [V'(G(H))O(1-pF) ~ \--~-pj-C'(1)]. (17

From equations (3) and (9) it follows that (17') is negative; this result is illustrated in Fig. 1.

OS

............ X

) P

Fig. 1.

6For simplicity we assume competitive behaviour, although the qualitative result remains unchanged in the case of interdependent behaviour.

88 D. Gottlieb / Tax evasion

6. Conclusions

It has been demonstrated that the introduction of a public good into the model of tax evasion creates a paradox which bears strong similarity to the Prisoner's Dilemma.

Though the paradox occurs permanently in the neighbourhood of the equilibrium outcome, the Pareto superiority of the full-compliance outcome to the equilibrium outcome demands a stronger sufficient condition than merely an interior solution.

It was shown that a government led by utilitarian welfare standards will bring about parameter conditions which perpetuate tax evasion in the case of positive variable costs of detection. This result suggests that our intuitive quest for the elimination of tax evasion cannot be rationalized on utilitarian grounds.

The utilitarian welfare function bears yet another shortcoming. It seems paradox- ical to use up public funds in the attempt to reduce tax evasion while at the same time it is favoured by assigning a positive weight to the individual's gain from fraudulent behaviour.

Appendix

According to Rosen (1965), 7 the function

n

a(x , r )= ~., r iEUi(X) , i = I

is called diagonally strictly concave (d.s.c) for x e R and fixed r > 0 if for every X °, X 1 ~ R, we have:

( X 1 - X ° ) ~ g ( X O, r) + ( X ° - X 1 )~g(X 1, r) > O, (AI)

where g is defined as:

g ( X °, r) =

O E U1 (X °) r l

OXl

O E Un (X °) rn

(A2)

However, according to his Theorem 2 it has to be shown that 9 P E E n, g > 0 ,

such that tr(X, r') is d.s.c. Then the equilibrium solution of the game is unique. Condition (A1) can then be rewritten as follows:

(OEUi(Xo) r i (X] - X °) \ OXi

i = 1

7See Rosen (1965, pp. 523-524).

D. Gottlieb / Tax evasion 89

Without loss of generality ~ 1 <_l<n, x t > X °. The concavity assumption (see Sec- tion 4 above) implies that:

aEUAX °) OEUt(X 1) >

ax~ ox~ Then setting rj, j = 1, .. . , n, j :# l , arbitrarily at rj= 1, it is sufficient to choose rt such that

r I >

0 OEU./(X°) 1~ (xJ - x) ) ( axj -

OEUi(X~)) axj

(XI_xO)(OEUI(X °) OEU,(X')~ axt axt /

References

M.G. Allingham and A. Sandmo, Income tax evasion: A theoretical analysis, J. Publ. Econ. 1 (1972)

323-338. R.D. Luce and H. Raiffa, Games and Decisions (Wiley, New York, 1957) Chapter 5. J.F. Nash, Noncooperative games, Ann. Math. 54 2 (1951) 286-295. J.B. Rosen, Existence and uniqueness of equilibrium points for concave N-person games, Econometrica

33 (3) (1965) 520-534. A.K. Sen, Isolation, assurance and the social rate of discount, Quart. J. Econ. 81 (1967) 112-124. B. Singh, Making honesty the best policy. J. Publ. Econ. 2 (1973) 257-263. T.N. Srinivasan, Tax evasion. A model, J. Publ. Econ. 2 (1973) 339-346. L. Weiss, The desirability of cheating incentives and randomness in the optimal income tax, J. Polit.

Econ. 84 (6) (1976) 1343-1351. S. Yitzhaki, A note on 'Income tax evasion: A theoretical analysis', J. Publ. Econ. 3 (1974) 201-202.