# Tax evasion: A model

## Tax evasion: A model

1. rntroctuction %kkM&~~M of ~~CCQIW ti &&Wual income tax returns is a widespread ph~rttm~~o~:No doubt there wit1be some income eamerS in ail XOI&~~W...
1. rntroctuction

%kkM&~~M of ~~CCQIW ti &&Wual income tax returns is a widespread ph~rttm~~o~:No doubt there wit1be some income eamerS in ail XOI&~~Wwho w3E &deed &ate their true income in their income tax [email protected]‘.I#s a matter M principb. There will abo be others for whiom an h9neSt imccMnetax Mum is &y one 0f the set of possible returns. Such a peisoxr vrritisubmit an hane.st return only if it is ‘uptimaV (til some suitably defined sease) for him trodc, so. The purpoz~ af this paper is to provide a formal analysis of this problem. i First, the optimum (in a Sense to be made precise in the next section) proportion of income to be understated wil; be derived as a functk [email protected][email protected], prtsbabiiity sf detection af understatement ‘ad &e prq+qtks of the &x function. Second, it will be sh\$lNn that ghn the income disttibution, a proportionate tax function yielding the s;urtreOotat rwwe as a progressive tax function in the absence of undentaterkmt of incume, ,tifl ykld kzver ex.pected revenue in the ird, the problem of psence sf optimsll undePs;[email protected] income. ~ptimd allocation of resoletrees towards detection of tax av~ida~n~ewill be casnsi&mL

2. The model Let us consider an individual whose true income is y. Suppose he adds that the probability that he wilt be detected if &und~tates ixt his in~
DifferentWing [email protected]) with respect to X we get (denoting ikrkatives by ptimes):

ft c3.nbe easily evaluated that:

+yw-z)(l

-li)TH((l--?t)y)

.

(6)

TN. &Mum,

whidh together with T(0) T’(y) > 0 will correspond assume that for all X > 0, that the pnalty muftiplier

ofx,

Tax evu&w: u moded

341

= 0 will yield a proporeionate rate of tax. to a progressive tax structure. Let us also P(X) 2 0: U(X) > 0, P”(A) 2 8. his mecans is a positive, increasitrg and convex function

Under these assum;ptians it is clear that &\$/ax < 0, and a#/ar < 0. A&ICI 3#/i)y 2 0 when + = 0. With the reasanabk additionakaswmptions tiat p(O) = 0 i.e. penalty multipher when there is no understatement -af i~~ame and T’(0) = 0 i.e. tht the maqinai rate of talxis zero at ZIG ~KXM~~,it can be seen from (3) that @(O,y, S) > 0 and cpC1,r,R)UandO
p&bus, the optimal proportion A*’by which decreases as the probability of detection nr that ahrl;lry = --(a#/ay)/(a+/aX) > 0 if T’ is positive. Hence: 2: Given a progressive tax function, and a probability of irtdependent of income y, the richer a person, the large]-is proportion b> which he til understate his income. It ShouM b noted that while Proposition f and Corollal;y I h,\$Ad bus even if A is a function of income, Corollary 2 need not hok. in the GISTwhew 1cis an increasing function of income. For in such a case d’rP/dy = dA*fay + (iO.*/dn)/(dni/\$y). While ah*;ay> O!,a?P/a?r<:0, since dn/dy > 0 the sign of d?P/dz is indeterminate withrout additional assumptions. For instance, if we assume that the margin4 rate of t;.lxis ~;~nstzmt,then A* diecreases as y increases if 71is an incre:asing functir, of income, leading to:

income increases.

where kt is the [email protected]~p&k3n uf [email protected] income. -It is ts -be remetilbeti& that Xg will in genwal depi=ndeony, [email protected])etc. G&w h&sinieoara y before taxes, an individual chocws his X so as to m&&e his expected iwome. after taxes a.rkdpenalties. This is equiv&at to MS mM&ihlg the expected taxes and penalties since pre-tax \$rw~rne is &ven. Hence, #ven T,Q), the expected value of taxes and peu4.&iesthat results f?am his choice of X other than Xz wil! be kger. In pariculw if the indMdua1 sets X = h? (the optimal value for his income JJ and the probability ?p,had the tax futwtion lxm T, fy)) rather th8n iset X = I\$ his expeclted value of taxes and penalties will be higher. Thus ?trx;[email protected]>y

+

Tyy)l

+(l - n)T, ((1 - X\$)jv}

Hence integrating both sides of (11) after ~~lt~~~~~;ati~~~ by get:

NOW,if we assume s is independent of JJ then, ?tr ‘is in&:pen&nt of y and

turn to the question OFthe detemtirzati~n sf x, the pr&&biof detecting the qnderstateme& of &c~~e in an income tax return.. simpik5ty fet us assume that af is a15increasirrg,concave function af the a~~~l~~ spent on the scrutiny of a return.. Let w trtrther s;’ aunt x to be spent on a return is the samt;=4%~ali furtoc%ionT(y) the govem%nent is interested in izing the tifferenee betwelen the cost of scrutiny and the exm n;rities. Thus the ~a~~~~d is now

TYV.Srinivaspn, Tax evasion: a model

+

{

345

1 - nr(x)}\$T(y(I- A”)) l(y) dy - x

where h* is, as earlier, the optimal proportion of understatment of income. Taking the derivative of 2 with respect to x we get (usirfg the fagt that X* is optimal imp&s the partial of Z with respect to X* is ZW0): dz z = n’(~),f[X*[email protected]*)y+T(y)

-T{(l-X*:)y}II(y)dy-=l

and \$2

= n”(x)~[x*~(x*)y + T(y) - T{y(l. - A*))1hj-4 dy

ax2 + (3r’(x)}2J[{?wqh”) +m*)YI +T’{y(l-XS)}]

s

?(y)dy

l

From the concavity of n(x) and the fact that ala*/br< 0, we get d2Z/&? 9 0 This impties that if a solution exists for dZ,kb = 0 it is unique. A se; of sufficient conditions for the existence of 2~sol&ion to dZ/d~ = 0 are lirn,++r’(~) = QOand Km,, Q)R’(X)= 0. We :;haUassume these conditions to holds.Thus the optimum x is drzterminedby

It is easy to interpret tlhb quation. The left-hand side represents the niargbal product iti terms of expected revenue and penalties (per return) per unit increase of the expenditure per return on (Scrutiny.This is the product af the maq\$nal increase a’(x) in the probability of detection of understatemes~t of income and the increalseand ave age revenue and penalties per return Fberunit increase in the probabifityrn(x).

remarks Xtis vev obvious that the: model of this pa

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