On Tax Fraud Actiyity"

Transcription

1 OsakaPrefecture On Tax Fraud Actiyity" 37 Shigeru Watanabe' I. Introduction The rdlationship between tax evasion and monopoly has seldom been analyzed since the pioneering work by Allingham and Sandmo (1972)i) except Kreutzerand Lee (1986),Kreutzer and Lee have shown that the tax on the monopoly profit increasesthe productionleve12),which means the increase of social surplus. However, the model developedby Kreutzer and Lee is too specific due to following points ;(1> it neglects the possibilitythat the tax evasion might be detected,(2)the rate of costs overstatement isassumed to be given, (3) other way to evade tax than overstating the costs isnot considered. We will generalize the model considering the above points. The purposes of thispaper are to show that (1)profittax rate is not useful as a tool fordecreasing the tateof proceeds understatement and that of costs overstatement,though effective forthe increaseof production,(2)penalty tax rate3} iseffective iordecreasing above two rates though itdecreasesthe production level, (3)operating both the profit tax rate and the penalty tax rate under some conditions, we can decreasethe rates of proceeds understatement and costs overstatement with the increaseof the production level, In the next section, we will generalize the model developedby Kreutzer and Lee considering the points discussedabove. The results of thispaper are summarized inthe concluding section. II. Possibility of Deteetionand Understatement of Proceeds Expected Profit Err will be shown as * Associate Professor of Economics, School of Economics, of Osaka Prefecture. * * I would like to thank to professors G. Suhama, S.Ito and Y. Tomita for their helpful comments. 1) Tax Evasion has been analyzed in many papers. See for example Yitzhaki(1974)and Watanabe (1986, 1987). ' 2) Needlessto say, in the.casewhere the tax evasion dgesn'texist, the tax on the monopoly profit will have no effect on the level of production. 3) The penalty tax rate is neglected in Kreutzer and Lee (1986), as the possibilityto be detected isneglected,

2 OsakaPrefecture 38 B, Uitiv.OsahaP,of D. Vbl.es EzP (Q)Q-C(Q)-t {P (Q)(1-e)Q-(1+6)C (Q)} -(1 -le (E)) q(d) Ft,SC(Q) -k (E)(1-q (a))ft ep (Q)Q -k (E)q (6){cSC(Q)+eP<Q)Q} F4 (1) where Q}P (Q),C (Q)ț and 6 denotequantity supplied, price, cost function, profit tax rate and the rate of cost overstatement respectively and e, Z} (e)(k'(e)>o, le" (e)>o), q (d)(q'(6)>o,q" (cr)>e)and F denote the rate of proceeds uinderstatement, the possibility forthe understatement to be detected,the possibility forthe overstatement to be detectedand the penalty tax rate respectively. The model developedby Kreutzerand Lee corresponds to the case where the possibility of detectionand the understatement of proceeds fe (E) O, q(6) O and E O. In addition, the rate of costs overstatement a is F assumed to be given inkreutzerand Lee (1986)Ḥowever, inour generalized 6 isan endogenous variable as well as e and Q. Firstorder conditions with respect to Q, e and 6 are,shown as are not considered, then OEz aq - {1 -t+te + {-1 (1-leF)} +t+ ld MR (1-qF)}MC o. (2) eezti) Oe (Q)Q (1-kf(e) Ee-le(E)F) o, (3) aentc e6 o, (Q)(1-q'(cr)RS-q(") F) (4)

3 OsakaPrefecture 1989 On 7laxFlaud Activily 39 Second order conditions are satisfied.`} o2ez E,,- <O, (5) OQ2 E3, El, ' >o, 4, G, (6) El, E,, E,, D Ei, Ele, a, <O. (7) a, a, Eig, The effect of the change in the tax rate on Q, E and 6 will be investigatedin the following. Differentiating the firstorder ' conditions with respect to t,we get aq 1 at t-- E,,4, 4,> O, (8) D a2ez where E14E F (e2k'mr+62q'mc )/( 1 -t)> o,5} aqat G2<e. Eb3<O andd<o,6) ae at O, (9) a6 at O. <10 )7) 4)5)6)7) See MathematicalAppendix [1). See Mathematical Appendix (2]. See MathematicalAppendix [1). E12 O from MathematicalAppendix (1) and G3--a3a,Eb4 derivedfrom the model. O will be straightforward

4 OsakaPrefecture 4o B. Uitiv.OsakaRrqt[D. Vbl.33 Hence from (8) we can see that the profit tax rate is an effective tool for increasingthe production level,raisingthe profit tax rate would increasethe production level. However, from (9) and (10), the profittax rate couldn't be a useful instrumentferdecreasingthe rate of proceeds understatement and that of costs overstatement, Next, we will examine the effect of the penalty tax rate on a E and cr. Differentiating the first'order conditions with respect to F, we ebtain OQ 1 OF -- Ei, Eh, Ei,< O, (11) D where E,,-t (cfemr+6qmc)< O, {12) OE 1 - EliE>sEb3<O, (13) af D where Eh,-tP (Q)Q(k' (e)e+k (e))<o, ( 14 ) a6 1 OF -- ll, a, &,<O, (15) D where Elis-tC (Q)(q'(6)6+q (6))<O. ' Therefore,frpm (11),(13)and <15), raising penalty rate would decreasethe rate of proceeds understatement and that of costs overstatement, though itwould also decrease the production level. Hence, the penalty rate is a useful tool for decreasing the inequityby the costs overstatement and proceeds understatement. However, raising the penalty rate would have a negative effect on the levelof the preduction. On the other hand, the profit tax rate is not useful as an instrument for decreasingthe inequityby the costs overstatement and the proceeds understatement, though itisa useful tool for the increaseof the production level. In the following,we will investigate the way to attain two targets at the same time;(1)increasing the production level,{2)decreasing the inequityby tax evasion. t/t t

5 OsakaPrefecture 1989, On 71zxF)u"d Activity 41 We consider the case where the profittax rate israised with the increaseof the penalty rate. The effect of raising the penalty rate in thiscase on the output level will be shown as dq aq dt + aq. (16) clf at czf af In addition, ifthe profit tax rate and the penalty marginal increment, the equation {16)will be rewritten as rate are raised by same dq oq aq +. (17) `tr7 ot OF Inthe same way, the effects on e and6 will be shown as ' de OE ae ae + <O, (18) cm at af af d" oo acr a6 + af ot af af <O, (19) from (9),(10),(13)and (15). From (8),(11) and (17), we obtain dq 1 cif' --4,G, (&+G,), D 1 t t --a2 a3 {MC-MR+eleFMR G---)+6qFMCG--)}, ( 2o )s) D F F Eler where iii,i2- (sqr are elasticities with respect toeand 6. le q ' 8) See MathematicalAppendix(3).'

6 OsakaPrefecture 42 B. Uheiv.OsahaP,ef D, Vbl.33 SinceE>2<O.Eig,<o,D<eg) conditions dqfor >O. ne and MC>MRiO), j,>- (ī - 1, t F N 2) are sufficient Therefore,the higherthe penalty rate relative to the profit tax rate, or the largerthe elasticities with respect to e and 6, the higher the possibility that the sufficient conditions are satisfied, On the other hand, the rate of proceeds understatement and that of costs overstatement are decreasedcrom(18)and (19). Hence, operating both the prqfit tax rate and the penalty rate under the above conditions, we can decreasethe rates of proceeds understatement and costs overstatement with the increase of the production level, III. ConcludingRemarks We have examined the way to attain two targets at the same time ;(1) increasingthe production levelof the monopoly, which means the increaseof social surplus and (2) decreasingthe inequityof the tax evasion by the understatement of proceeds and the overstatement of costs. Raisingthe profittax rate increases the production level, which isthe same result as Kreutzerand Lee (1986)Ḥewever, the profit tax rate isnet a useful instrumentfordecreasingthe rate of proceeds understatement and that of costs overstatement. On the other hand, the penalty rate is a useful teel for decreasingthe inequity by the costs overstatement and proceedsunderstatement. Hewever, raising the penalty rate would have a hegative effect en the production level.ithas been derivedthat raising both the profit tax rate and the penalty rate by same marginal incrementunder sorne conditions, we can attain the above two targets at the same time. 9)10)SeeMathematical Appendix (1). See Mathematical Appendix (2].

7 OsakaPrefectureUniverFitY 1989 On 7lax Fraud Activdy 43 Mathematical Appendix (1] Eii {1-t+te (1-leF)}MR' + {-1+t+te(1-qF)} MC'. From (3},1 - kf le'fe.frem (2),'weget {-1+t+tcr(I-qF)}- {1-t+te (1-leF)}MR/MC. Hence, Eii(1 -t+ te2fe'f)(mrr-mc'mr/mc)< o, where t<1, k'>o, MR'<O and MC'>O, Ell E,2 G, S, E,, ]El,,> O, a2en where Ei2' aqoe {t(1-k (E)F)-tEk,F} MRO 02Epu from (3 ) and E>2aE2 -tp (Q)Q (k"e+2k') F< o. Eil El, E,, D E,,,E}, E,, E,,4,Ei,<O, a, a, Eig, a2ez where Ei3' a6, -tc (Q)(q"O+24)F<O, and Ei3E23O as well as Ei2 O are derivedstraightforward from the model (2] Ei4 {-1+E(1mleF)} MR+ {1+0(1-qF)} MC. From(2),Ei4(MC-MR)/t. Weget 1 -lefk'fe and 1 -qf4as from (3) and (4).

8 OsakaPrefecture 44 B. Univ.OstthaPwf D. Vbl,33 Then, Ei4 (- 1 +le'fe2) MR+(1 +q'f"2) MC. Hence, Ei4 (MC-A(R)/t(-1 +k'fe2) MR+( 1 +q'rs2) MC. Therefore,MC - MR t 1-t (E2ktIilnfR + 02 qt FMC)> O, Hence,Ei4 <MC - MR)/t C3) F (e2k'mr+62q'mc)>o. 1-t ' Ei4+Eis {"1+e(1-kF)} nlr+ {1+e(1-qF)} MC - t (ekmr+ 6k7MC) (- I+e2k'F)MR+(1+62q'F) MC -t (EkMR + eqmc) (from<3) and (4)) MC - MR +,kanr (-E. kt m!). k F +OqnvC (-qt--). q F e t References M. G. Allinghamand Sandmo, "`Income Tax Evasion: A TheoreticalAnalysis,"lournal ofibublic Economics 1,1972. D. Kreutzerand R. L. Dwight, "On Taxation & UnderstatedMonopoly Profits,']IVdetional7}zx fournat 2,1986, S. Watanabe, "Income Tax Evasion,"PublicChoiceStudies,8, S.Watanabe, "Tax Evasion and Asymmetric Laffer Curve, 'Porbtic Choice Sttcdies10, 1987, S.Yitzhaki, "A Note on Income Tax Evasion, ';loumal of PbeblicEconomics 3, 1974.