On optimal public good provision with tax evasion

On optimal public good provision with tax evasion

Journal of Public Economics On optimal tax evasion 45 (1991) 127-133. public North-Holland good provision with Josef Falkinger* Unioersity o...

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of Public


On optimal tax evasion

45 (1991)




good provision


Josef Falkinger* Unioersity of Linz, Altenbergerstrajk, Received


4040 Linz, Austria

1990, revised version




It is proved that for public goods with zero income effects (also called Ziff public goods) tax evasion does not affect the optimal level of public expenditure. Further examples show that, in general, tax evasion may lead to less but also to more public expenditure.

1. Introduction Cowell and Gordon (1988) analyzed the influence of public good provision on tax evasion behavior [see also Falkinger (1988)]. This paper addresses the reverse question of how tax evasion behavior influences the decision on the optimal level of public good provision. Kolm (1973) has already emphasized the importance of integrating the analysis of tax evasion behavior and the determination of optimal expenditure on public goods. However, he does not answer the question of how the optimal level of public expenditure should react to tax evasion. Sandmo (1981, p. 286) pointed out that it would be desirable ‘to analyze the interesting question of whether evasion should lead to less expenditure on public goods’. At the same time, referring to the work of Atkinson and Stern (1974), he warns us that no simple answer to this question is to be expected in general. The present paper provides a definite answer for the case that public goods have zero income effects, which plays an important role in the analysis of public good provision [cf. Green and Laffont (1979); Cowell and Gordon (1988) speak of Ziff public goods]. It is proved that in this case the degree of tax evasion prevailing in an economy has no influence at all on the optimal size of public good provision. Evasion should not lead to less expenditure on public goods nor should it lead to more. In general, however, neither of these two alternatives can be excluded (under reasonable conditions), as further examples (presented in section 4 of the paper) show. *I wish to thank helpful comments. 0047-2727/91/$03.50



K. Brunner,




Science Publishers

and an anonymous

B.V. (North-Holland)

referee for

.I. Falkinger, Optimal public good provision


2. The general problem The taxpayer’s utility is a function U(x,g) of private consumption, x, and the level of public good provision, g. We assume pure non-rivalness so that each taxpayer enjoys the full amount g. In the tradition of Allingham and Sandmo (1972), tax evasion is modeled as expected-utility maximizing behavior. Let y denote true income, t the uniform tax rate, and e the amount of tax evaded. The taxpayer chooses e so as to maximize




where x,=y(l-t)-se,



p is the probability of being caught, and s is the penalty rate. It is assumed that the individual takes the supply of public goods as given and does not take account of any impact of his decision to evade on this supply. This seems to be the most plausible assumption in a large economy. The first-order condition for a maximum of (1) is then’

F(e,4 8) = ( 1- 14Ux(xu,d - PsU,(xd~ d = 0.


Together with (2), (3) defines individual behavior, e, in dependence of t and g. (The dependence on p and s plays no role in the following analysis.) Let n be the number of (identical) individuals. The government is supposed to maximize a utilitarian welfare function, i.e. EU(x(e(t,g), t),g), subject to the restriction that public expenditure, g, equals expected tax yields. In view of (2), the budget restriction reads: G(e(t,g),t,g)-g-n[ty-(l-p-pps)e]=O.



L(r,g) =EU(x(&g), conditions:’





r),g) -nG(e(r,g),



and we

forming obtain

Lt= - C(l-P)U,(x,,g) +PU,(xd,g)ly + Wy--_(

By eliminating

1 this can be reduced

‘Notice that e>O implies 1 -p-ps>O. 2Notice that EU,= F(e, t.g) =0, according


to (3).

the the


Lagrangian first-order

= 0,


J. Falkinger,


Optimal public good provision

(1-PP,b”?d +Pu,(xd,g) _ (1-P)u,(x,,g)+pu,(x,,g)

In contrast, without tax evasion g,, would be determined by

1 +n(l 1 -(l





the optimal

level of public



which corresponds to Samuelson’s (1954) criterion for optimal expenditure on public goods. Comparing condition (7) with condition (8), we see that tax evasion affects both sides of the Samuelsonion rule, according to which the sum of the marginal rates of substitution (CMRS) between the public and the private good has to be equal to the marginal rate of transformation (MRT). The left-hand side of condition (7) is a modification to c MRS. Since tax evasion involves uncertainty, MRS= V,/U, is substituted by EU,/EU,. What is the effect of this modification? Define Ex=( 1 -p)x,+px,. Use (4) to obtain from (2): x,=y-g/n+p(l +s)e, x,=y-g/n+p(l +s)e-(1 +s)e, and Ex=y-g/n. Then Jensen’s inequality tells us that ECT,P U,( y-g/n,g) and EU,2 U,( y -g/n,g) if U,, U, are convex/concave functions in x, respectively. Thus, tax evasion may increase or decrease the relevant marginal rates of substitution. The right-hand side of condition (7) is a modification to the marginal rate of transformation. Since tax evasion changes the cost of financing public consumption, MRT= 1 is substituted by [l +n(l -P--ps)e,]/[l -(l -pps)(e,/y)]. This value may be higher or lower than unity depending on the signs of e, and e,, respectively. For instance, e,> 0 means that an increase in g requires a more than proportionate change in the tax rate since tax evasion increases with g. e,>O means that a higher tax rate leads to a less than proportionate increase in expected tax yields since tax evasion increases with t. Thus, the marginal rate of transformation of private income into public goods is comparatively higher than in the no-evasion case. If e,, e, ~0, we have a distortion in the opposite direction. These general reflections show that the impact of tax evasion on optimal public good provision depends on the curvatures of U, and U, and on the signs of eg and e,. No unambiguous conclusion is to be expected in general. The interesting question is how the various effects look like if reasonable restrictions are made. The further analysis deals with important cases.

3. Public goods with zero income effects Consider

the case that g has zero income

effects, which





J. Falkinger, Optimal public good provision

marginal rate of substitution, U,/U,, the individuals can be represented concave utility function and f is transformation. Note that we have takes the form:

is independent of x. The preferences of by U(x,g) =f(x+v(g)), where v is a a monotonously increasing concave U,/U,= v’(g). Therefore, condition (7)

nu’k) = Cl+n(l -P-P4e,llCl -Cl -p--p4 and condition


(8) takes the form:

nu’(g,) = 1. Condition



(3) reduces to

F(e,4 g) = ( I-


+ v(g)) - psf’(xd


v(g)) = 0.


Hence, e, = - F,/F, = yS/K,

e, = - F,/F, = - u’(g)S/K,

with S = (1 - p)f”(x” + u(g)) - psf"(xd + v(g)) and Ps2f”(x, + u(g)). By substituting (12) condition (9) reduces to

K = (1 - p)f”(x,


(12) + u(g)) +


which is fulfilled if and only if nv’(g) = 1, i.e. if g =g,. This proves that under the assumption that public goods have zero income effects tax evasion has no impact on the optimal level of public good provision.

4. Additive utility functions The following examples of additive utility functions show that, under reasonable assumptions, tax evasion may lead to less public expenditure but may also imply a higher optimal level of public good provision. For an additive utility function U(x,g)= U(X)+v(g), with u’>O, U” ~0, v’ > 0, and v” ~0, condition (3) reduces to F(e, t,g) =( 1 -p)u’(x,)


Since es= - F,/F, =O, the conditions

=O. (7) and (8) take the form:



J. Falkinger, Optimal public good provision

nu’k) = Eu’/Cl- ( 1-


~4 WY)1



nu’k,) = U'(Y -_g,m


respectively, with Eu’-(1 -p)u’(x,) +pu’(x,). In view of condition (14), we have e, = - F,/F, = y[( 1 - p)u”(x,) - psu”(x,)]/[(

1 - p)u”(x,)

+ ps2u”(xJ].

Since ps=( 1 -p)u’(x,)/u’(x,), according to (14), e, may also be written in the form e, = y[R(x,) - R(x,)]/[R(x,) + sR(xJ], where R(x) = - u”(x)/u’(x) is the index of absolute risk aversion. Substituting this into condition (15), we obtain nu’(g)z Eu’ if R(x,) 2 R(xJ. From Jensen’s inequality we know that Eu’Iu’(y-g/n) if ~“‘20. (Remember x,=y-g/n+p(l+s)e, xd= y-g/ n +p( 1+ s)e - (1 + s)e, and Ex = y-g/n.) Thus, we can conclude that nu’(g) > u’(y-g/n) and therefore [in view of (16) and the concavity of u and u] g< g,, if the utility function for private consumption is such that ~“‘20 and R’ZO with one inequality holding strictly. However, this implies nothing for utility functions with u”’ ~0 or R’ ~0. Since there is a general presumption that absolute risk aversion is decreasing with income, we have to ask whether the conclusion that the optimal level of public goods should be lower with tax evasion (which makes appeal to intuition) holds if R’ < 0. The following examples show that it does not necessarily. Since R’ <0 implies ti” >O, we cannot take advantage of Jensen’s inequality and the argument employed above. We show by examples that in the case R’g,, or g=g, can be excluded. Take, for r>O, ~(x)=(l-r)-~x~-~, with u(x) =ln x if r = 1. Since R = r/x, we have R’ < 0. Define tl = (1 -p)/(ps). Then condition (14) reduces to x ” =x d &‘, which gives us


et= -YP


and Eu’=x,‘p(l


where /I-(c?“l)/(l +scc”‘). Use (4) and ( 17) to calculate


y - ty = y -g/n



- (1 - p - ps) ( y - ty)P. Thus, (19)

.I. Falkinger, Optimal public good provision


Using (17), (18) and (19), condition

(15) can be rewritten




with ~~(1+(1-p-ps)fi)‘~‘(1-s~))‘p(l+s). Comparing (20) with condition (16) and taking into account the concavity of u and u, we conclude that gg,, and g=g, if /AL), p 1. Hence, gg,. Take 1 +(l -p-ps)fi=(l+s)[(l


i.e. u(x)=~x”~.



Then 1 -sfi=(l

- l)/( 1 +~a”~), +s)/(l +scP)

~=(~r~-l)/(l+sc1~), +s)/(l +sa2) and



shows that ,LLL< 1. Hence, g>g,.

Example 3 g=g,. Take r= 1, i.e. u(x) =lnx. Then /?=(al)/(l +s.x) and p = (1 - sfi)- ‘p( 1 + s) = (1 + m)p. By definition of a, we have pm = 1- p. Hence, p= 1 and g=g,. This example shows that the result goods have zero income effects.


is not



if public

jBy definition, a=(l-p)/ps. Thus p~a”~=(l-p)r~“* and p=[(l-p)aLi2+p][p+ ~“~)+(1-p)~=pZ+2p(l-p)(a+1)/(2a”~)+(1-p)Z. Now, (1-p)a-“2]=pz+(l-p)p(~“*+~ a+ 1_2&z=(a”z -1)2>0, therefore cc+1>2~~“~ and p>(p+l-pp)‘=l. 4psa2=(1-p)a. Thus, p=[(l-p)~~+p]~~~~[p+(l-p)a] and p
References Allingham, Michael G. and Agnar Sandmo, 1972, Income tax evasion: A theoretical analysis, Journal of Public Economics 1, 323-338. Atkinson, Anthony B. and Nicholas Stern, 1974, Pigou, taxation and public goods, Review of Economic Studies 41, 119-128. Cowell, Frank A. and James P.F. Gordon, 1988, Unwillingness to pay: Tax evasion and public good provision, Journal of Public Economics 36, 3055321. Falkinger, Josef, 1988, Tax evasion and equity: A theoretical analysis, Public Finance/Finances Publiques 43, 388-395.

J. Falkinger, Optimal public good provision


Green, Jerry R. and Jean-Jacques Laffont, 1979, Incentives in public decision-making (NorthHolland, Amsterdam). Kolm, Serge-Christophe, 1973, A note on optimum tax evasion, Journal of Public Economics 2, 265-270. Samuelson, Paul A., 1954, The pure theory of public expenditure, Review of Economics and Statistics 36, 3877389. Sandmo, Agnar, 1981, Income tax evasion, labour supply, and the equity - efficiency tradeoff, Journal of Public Economics 16, 265-288.