DRD DISCUSSION PAPER Report No. DRD271 TAX EVASION, CORRUPTlON AND ADMINISTRATION: MONITORING THE PEOPLE'S AGENTS UNDER SYXHETRLC DISBOUESTY ,, by Arvind Virmani Hay 1987 I Deveiopment Reserrch Department Economics and ilesearch S t a f f Worla 8ank ' The World Bank does. not accept r e s p o n s i ~ iilt y f o r t h e views expressed herein which a r e those of t h e author(s; and shoold not be a t t r i b u t e d t o t h e World Bank o r t o its a f f i l i a t e d organizations. The findingr; i n t e r p r e t a t i o n s , and conclusions a r e t h e r e s u l t s of research supported by t h e Bank; t h e y do not ' i l y represent o f f i c i a l policy of t h e Bank'. The designations 'employed, necess% entation of material, and any .mps used i n t h i s docu&nt a r e . s o l e l y t h e pr f o r t h g convenience of t h e reader and do not imply the.expression of an>. ,- opinio@whatsoever on the p a r t of t h e World Bank o r its a f f i l i a t e s concern t h e l e a l s t a t u s of any country, t e r r i t o r y , c i t y , a r e a , o r of its a u t h o r i tiur s , g o r corlcerning t h e d e l i m i t a t i o n s of i t s boundaries, o r national a f f i l i a t i o n . Abstract / Previous work on income tax evasion has assumed that taxpayer.dare / dishonest, while tax collectors are honest. Such a clear "moral" asymmetry is /' 1 ' s /" seldom observed in developing countries. The present papeq sho!.rs how I corruption depends on the type of (corruption detection mechanisms'in) society, and the incentives which can be designed :o ensure honesty or increase net revenue. It also sheds light on.th%wzJ_f: of why governents 7 * , ignore the implications of previous models to rai9.e ~~enaltieson tax evaders. In some circumstances this can drastically reduce net revenues. 1. INTRODUCTION There is by now a considerable literature on tax evasion, starting from the partial equilibrium models representing the evasion decision (Allingham and Sandtco, 1972, Srinivasan, 19731, to determining some of the allocation effects of evasion (Penceval,l979, Watson, 1985). The paper of Sandmo (1981) goes a step further in attempting to derive normative impli- cations in the framework of optimal tax theory. Though,these paperQqdiffer from each other in some respects, there is one asshption co-n8to+them. The assumption of aspmetric honesty between taxpayers and tax col- lectors, is implicit in previous work on tax evasion. In the USA this can be --. ... ' I r 2 I 11%, justified on the practical grounds that the IRS appears to be hanestr,.-. I - . - Though this may historically be a reasonable assumption for the tic5 countries, it is highly questionable for the poor ones. 2' A general model must encompass both possibilities, so that conditions under which the tax bureaucracy is honest can be investigated. Host earlier models share the implicit conclusion that highet penal- ties decrease evasion by raising its cost, and are thus a virtually costleas .* 1. Application of the same reasoning to tarpayers would require a model which ' segmented taxpayers, as mosttof them also appear to be honest and should : be treated as such. # r m 0 2. As the recent publicity about the Philippines has brought home to moat people. The increasing concern with tax evasion in rich countries, and reports cf official corruptoin in government contracting and drug enforeen:snt suggest a fresh look at this assumption even for these countries. way of increasing net revenues (Cowell, 1985). 1' This prescription does not seem to be taken seriously in nuny countries. On the contrary developing countries have offered public amnesty for declaring previously evaded taxes. 4' If the bureaucracy is corrupt the price of evasion detected is not nimply the penalty on evaded taxes, but 8186 depends on the rewards and incentives for the tax collector. The role of evasion penalties can therefore be quite different from that in model s with honead'bureaucrats. ,'Thr;current 1 paper shows how an increase in penalties on incoat8 tax evaders can reduce net reveaue collections. ~m&rhr. - The m a t important results which r;.9~lbi the design of an incentive system for tax cd!h&tWbA. The object;i.br-&L incentive design is assumed to be the maximization of net revenues given the tax system and the societal institutions affecting corruption. In this context, the paper classifies societies into three types, Strong, Corruption Deterring and Weak, in terms of the coefficients related to corruption detection and punishment. 2' A strong society is characterized by nonpositive expectations of marginal returns from corruption (taking bribes), In this case the tax bureaucracy is honest independent of the incentive system (the analysis of this Lase is relagated to the 'appendix). - L 3. *See papers by Kolm (19731, Pishburn (1979), Kemp and Ng (1979), Kmskela f(1983). Polinsky and Shave11 (1979), Singh (1973). and Goode (19811. 4 , I recall a recent (past year) report in the ~asbingtonPost that even the US IRS was discretely offering m e a t y for specific offenses. 5. The cost to the government of detecting evasion have been addressed in earlier papera in tha context of detection probabilites (~andmo,1981, Yatson,.1985), m e s a are assumed fixed in tha present paper, The paper focuses on the other two types of societies. In each case the incentive system for tax collectors can be classified into three ranges -- Honest range, Corrupt range and Degenerate range. If the incentive parameters lie in the degenerate range tax collectors act solely in thier own interests. Evasion is also higher than in the corrupt range. What distinguishes the Corruption Deterring Society (CDS for short) from the Weak \ society is that the corrupt range is characterised.by Rent Transfer. Corruption has no effect on evasion, but its existence leads to a transfer of revenues from the government to the tax evaders and tax collectors. The next section(2) discusses the view of corruption and w:x: which underlies the tax evasion model. The presentatmion of the fom&.model is split up into two sections (3 and 5). Section 3 modcls the bargain between the taxpayer and tax collector, which d e t e r n i ~ ~the price of detected r evasion, conditional on the evasion detected. As the conditions for honest bureaucracy depend solely on the bargaining solution of section 3, these are investigated in section 4. The remainder of the model, pertaining to the taxpayers evasion decision conditional on the price of evasion detected, is given in section 5. The analysis of the Weak society with cotrbption detection prob- - ability independent of the amount of bribes taken by the agent is divided into - two sections. Section 6 considere a binear incentive scheae (linear system), .. and section 7 a quadratic incentive sfheme (exposure share system). = The Corruption Deterring society, with exGctations of marginal returns to the tax collector decreasing with bribes, is analysed in the next two sections. Section 8 considers the linear incentive scheme (bribe limit system), and section 9 very briefly considers the quadratic incentive scheme (bribe- exposure limit system.) Section 10 concludes the paper vith an overview of the results, and outlines some directions for future research. 2. MODEL OVERVIEW With tax collectors potentially as dishonest as the tax payers the model must deal with tvo new issues. One concerns collusion between taxpayers and tax collectors in not exposing detbxted evasion in return for a side payment (bribe). 6' The other is concerned Gith incentives 'for and Anitoring of tax collectors, and is addresed in this paper using a principal-agent framework. -7/ In simplest terms, we can think of the government as 'an,'or@aniz~rbn vhich is collectively owned by the people, azd one of whddg-objectives is to collect taxes. 8' The tax bureaucrats in the government can therefore be vieved as the agents of the people (the principals). As in the shareholder- manager dichotomy in a corporation, the reality is more complex. The elected representatives or elected government usually forms an intermediate super- visory layer betveen the principal and the agents (~irole1985). In the present paper hierarchical details are ignored. No distinc- tion is made between the people and their representatives (the principal), or 6. This is analyzed as a cooperative game between the tvo as in Virmani (1383). 7. The principal and agent are different from those in Reinganum and Uilke (1984). 8. The tax schedule will be assumed fixed prior to the tax collection operations considered here. The expenditure side of government is not considered at all in this paper. between differenr layers of the government bureaucracy (the agent). ?I The paper focuses on a single agent or tax collector, who can be seen as a repre- sentative of the tax bureaucracy, - 101 The tax collector is interested in his own returns subject to the incentive and monitoring system created by the principal. g1 This system is assumed to have two components. One component is a general mechanism for ,;c7 I detection and punishment of corrupt officials. In most societies thia $ ,, encompasses the police, special corruption detection departments, the judiciary, the elected representatives, the press and the public/conaumer interest groups. GI The probability of detection d k k - ke 4ss&-l6 gengka~ to depend on the total amount of bribe taken. If cort,uption is dee&t!U'all bribes are forfeited plus a penalty is levied which is assumed measurable in monetary units. For the present purpose we can define societies in terms of the corruption detection probability that characterizes it, The Strong society has a non-concave, the Corruption Deterring Society a convex, and the Weak -*- - 9. Issues connected with corruption of elected representativea will therefore not be addressed. Once the tax schedule is fixed the people are assumed to have identical objectives for the organization, i.e. maximize net revznue from the tax bureaucracy, - a * - 1 If there are N tax collectors/inspectors, each one can be qsumed to cover 1 1 of the entire sample of taxpayers. ~ The paper thetefore ignores any interaction between the inspectors. 11. Tax inspector is assumed to be risk neutral. 12. Thoukh this element is probably most critical in determining cross country variations in official honesty/corruption, it will be taken as given for an individual country, the focus of the present paer. In some poor countries the elected governwnt and other public agents may themselves be corrupt. Consideration of this issue is avoided in the present paper as it w u l d nuke the analysis intractable. society a non-convex, marginal expectation of returns from bribes. For expositional simplicity a Linear function is assumed for both the strong and weak societies. z1The Strong Society is then defined by nonpositive expectation of marginal returns from bribes at the zero bribe level. The Corruption Deterring Society is defined as having positive but decreasing marginal expectation of return from bribes. Though such ,a functiod tould 3 arise from various combinations of detection probability and penalt~llevels, Linear penalties and convex detection probabilities are assumed. The Weak Society is defined as having constant positive marginal expectation of returns ---.. '4 * from bribes. The second component of the system is specific to the tax bureau- cracy, and central to the present analysis. The amount of income declared by any taxpayer is known to both the agent and the principal, The major role of the agent is to detect and expose evasion. The principal has no direct dealing with individual tax payers and therefore no direct infomution on evasion by any taxpayer or sub-set of taxpayers. The only information he receives regularly is the amount of detected evasion which is exposed by the agent. The simplest snd most direct way to monitor the agent is through the amount of evasion exposed by him. 14' The design of the exposure linked reward structure is important in detkmining how much, if any, of detected .9 * evasion will be exposed to the principal. - The taxpayers (Y, say) are tsumed to b~ divided into m sub-sets with (j=l,...m) individuals having identical fixed income in each sub-set. It j 13. As bribes are always zero in the strong society, the assumption has absolutely no effect on the results for this case. 14. If the total taxable income in the economy was known, which is seldom the case, an alternative would be to judge declared plus detected income against thir. is assumed that checking is done on a random basis. 11' .The s u e proportion of evaders from each sub-set are detected, and the distribution of thoae detected is identical to that of the evaders. Any evasion detected ia specific to an individual (and to a given year or years), aa detection has meaning only if the evader can be legally prosecuted. Any bribe for not exposing a certain m u n t of detected evasion must be negotiated by the evader and the tax inspector, and doer ilot directly involve other evaders orithe principal. Determination of the bribe and expoaure level can be viewed aa a bilateral bargain between the two, conditional on the amount of de- tection. 6 1 ~t will be repteaanted as a Naah (19563 'tioopcrati*e ;sp-mena the evader and the detector (agent) againat the p r i n c f p . l " ( V f ~ . 1983). The atartin* baaia for the aeal is the non-cooperative aolution with each acting independently. - 181 An assumption implicit in moat models of income tax evasion ic that evasion is merely miarepreaentation by the taxpayer. Though thia mry be true for small amounts of labor incoma from the infornul market, in general cvaaion 15. In many countries each tar collec~oria.aaaigned a certain number of taxpayers, and he deala on a personal baaia with a11 taxpayera in hia assigned area. tfe has to approve, after checking if necessary, every return filed by his set of taxpayera. Detailed checking may still be random, however. i a 4 16. Examination of colluaiv~dealain developing countriea ruggcata that they are more akin to a bilayral monopoly bargain than a tax inspector 0 monopoly. . @ 0 17. The role of the people as individual tax payers is, however, quite distinct from their role &a collective "ownera" of government. Just aa the role of houaeholda aa ahareholdera is quite distinct from their role as consumers or suppliers. Ic also has some similarity vith the free- rider problem in public gooda. 18. With agent detection effort and investigation probability fixed, threats have no meaning. A atarting point given by a threat ~olutianonly becomes relevant if one of there ia variable. involves the concealment of income and information, and the introduction of false or misleading information. 19' Expenditures and labor costs incurred in evasion constitute a potentially important source af resource costs of evasion and corruption to the economy (Krueger, 1974, Bhagvati, 1982,). It is assumed that costs must be incurred in concealing income. A paraiiel assumption in these papers is that of "a11 or none detection" on investigation. From the information perspective, it ic unlikely that all types of concealment activity is of equal qua'.ity and equally easy to detect. note importantly detection is meaningful only if it can be proved in court (Creenberg (1984)). This vill be representd?.hrtltepresent paper by making incoma tax evasion detected a convex function H of evasion X. -201 To simplify the exposition, the presentation will focus on two special cabb#. The 'linear tax' case in which marginal tax rates are constant, and the 'all- or-none detection' case in vhich tax rates can be non-Linear but 8(.) = X. The formal model of bargaining is given in the next section. Section 4 considers the problem of honest bureaucracy in the context of the bargaining solution, before completing the model in section 5. 3. EVAS~ONPRICE UNDER CORRUPTION Though for simplicity of exposition, the model is frn,;red as a single '2 - period one, it involves two stages. In the first stage. each risk neutral - - - 0 * taxpayer decides on how arch of received income (Zi) to concal (Xi). - 21/ 19. For instance, the creation of false expense vouchers, sale of goods on the grey market and maintenance of illegal investment accounts in a foreign country. 20. Am alternative assumption is to makt the proportiori of evasion detected a random variable, with different probabilities for detectins different proportions of income. These two approaches could also be combined. The latter was tried, but becomes too complicated in sort cases. This is based on the cost of concealment (s(Xi)), and his expectation of evasion cost for each level of evasion. At the second stage, the tax agent randomly selects a proportion ( x ) of taxpayers for investigation. An amount of concealed income Yi(5 Xi) is detected for each of those investigated. Detection depends in general on both the total income and the amount of income concealed (Yi=li(xi,Zi)). 21' The a. tual price of evasion.detected is , determined at this stage. In general a part of the evasion detected is revealed to the principaL or formally exposed on official records (Wilyi). The taxpayer has to pay a tax on the evaded income exposed (W~=T(X~+W~~-T(X~)),01- y penalw, a s s w d Lo be proportional to the taxes evaded (Powi). z'--The taxpayer pay. the tax inspector a bribe (bi) for not exposing some or all oi the detected evasion (Yi-Wi), if this will reduce his total detection costs (ui=(l+~,)vi+bi). Ae noted earlier the bribe and exposure levels are assumed to be determined by a bilateral bargain between the taxpayer and the tax col- lector. The remaining tax and penalties are then assumed to be paid by the taxpayer to the government, and any rewards or incentive by the government to the tax collect~rs. Penalties are then levied on any corrupt tax officials . I detected. In formally modeling thqproblem, I start with the second stage prob- lem of determininb the bribe and exposure levels for given detection. -!- Tnese determine the total cost and pridl per unit of detected evasion. The variable D 21. Risk neutrality ie assumed to keep the collusion stage as simple and tranrparent as porsible. 22. It will alro depend on the effort expended by the inspector. The variable effort care is not considered in this paper. 23. T(.) ir the tax function which is nonlinear in general, but ir arrumed to be linear in one of the two rub-carer considered. Table 1: List of Variables Zi = Incoaa received (fixed) by taxpayer i, i=l...N . zi=T(Zi) = Tax due on income Zi. T'> 0, 1">0. T'=tl (constant) if 1'4. ~ri&(s) represent 1st (2nd) differentials. Di,di=T(Di) = Income and income tax declarr'd by i. xi,xi=T(Zi)-T(Di) = Incom and income tax crncealed by ,i. yi=H(Xi,Zi) = Concealed income detected by tax inapector on investigation of i, H1,Hll " 0, H2,HI2 < 0. Yi= T ( Y ~ + D ~ - T D=~ Sxtra tax due if detected avasion is full^ uqee4!&?. , by inspector agent to prinicipal. Y~,W~=T(Y~+D~)-P(D~) Detected evasion exposed by agent and tax due on it (raapectively) . s(Xi) = Resource cost of evasion to taxpayer i. S'>O, S"ZO = Po Penalty payable by taxpayer per unit of evaded tax. r = Probability of being investigated by the tax inapector. bi Bribe taken by tax inspector from taxpayer i. b= I b = Total bribes taken by tax inspector, b=.#.b . j - J % j u b POW^ = Coat of evasion detetted. e=&+elb = Probability corruptionlbribery being detected. . ~ ~ 2 -0 i A, = Penalty pcr'unit of bribe on dircovery of corruption. '! R = R ~ + R ~ * W + R ~ *=W ~Reward for exposing tax evasion. R1>O , R1+2B2v > 0 for all relivant v, r t vj, _rL v . j j+i j ~=b(l-po-plb)+~ = Expectation of =eturnr-to ageat(p=~(l+~~)). C(Xi)=T(Di)+~(Xi)+rui = Expectation of tax plus evasion coot: to tax payer i. C a t T(D )+r( (~+P,-R~)W+R~W 21-1, = kpectation of net revenues J j to the principal. definitions are listed in table 1. For those investigated, tax evasion of yi detected. Detected taxpayer and agent jointly determine wi and bi. The noncooperative solution representing the bargaining base or threat point is given by, Taxpayer: Min = -ui = -(l+po)~;-bi, bi The tax bureaucrat maximizes expectation of returnr from briber aad~reward from principal. Bureaucrat : Max B=b( 1-po-plb)+Po+R1u+R2w 2 , (2) W. 1 - 1.' ,~ a 7s ' The solution of (1) and (2) ir earily shown to be bi=O, w i w V . whica yields, F=F,=-( ~+P,)w~, B=B0=b(l-po-plb)+Ro+Rl(yi+~)+R2(~i )2 - The Nash Co-operative Bargain is therefore given by the joint maxi- mization of L (eqn. 4) where F and B are the gains to the taxpaysr and tax agent respectively, . Mar L=F B=(F-Fo)(B-Bo), - - subject to B > 0, .-F > 0, where - - - w b:' i' r -P=(F-F~)=(I+P~)(~~-W~)-~~, -B=(B-B~)=~~(1-po-2plb+plbi) - ( R ~ + ~ ~ R ~ ) ( ~ ~ - W ~-vi2) 1-RZ(Y;2 - 8 The necessary conditions for a bargaining solution ace, . Lb dL/dbi (1-po-2plb)P - 0 Lw - - a dl/dwi = ((R1+2wR2)E (l+Po)B) = ( R1+2wR2 - (,l+Po)(l-po-2plb))F- -< 0 (7) The second line of (7) ir obtained from the firat by rubrtitution of ( 6 ) . Lb > 0, lb < 0 agd Lw > 0, can be ruled out because they contradict conditions (4) and (5). If F = 0, the taxpayer is indifferent between the bargaining and - the honest solution a case which occurs for the linear system and is therefore ignored till section 6. Equations ( 4 ) to (7) determine the bribe bi, the exposure wi on which penalty has to be paid by the firm, and the cost per unit of evasion detected ui. These are functions of yi the detected evasion. A graphical depiction of the solution can be obtained by considering the trade off between bribes and exposure for the bureaucrat relative to the trade-off for the taxpayer. Taking the total differential of (2) with B constant and of (1) with F constant, and comparing t r h ~ s ~ o y r c & c h e - ~ & W d i line, we have using the inequality sign in (71, This is depicted in figure 1 as the solution points D (corner) and I (interior). H represents the no-bargain solution. Before setting up and solving the first stage rroblem (section 51, I digress somewhat to consider the conditions for an honest bureaucracy. . I 4. HONEST BUREAUCRACY - i In this pgper, exfectation of marginal returns to the agent from - - taking bribes (l-po-2plb),'and s the marginal reward to him from exposure (to principal) of detected evasion (It1+2wIX2), will both be assumed positive. When the former is negative at the zero bribe level honesty is ensured when the marginal reward is non-negative, as it is in the normal case in which bureau- crats get a fixed wage. This case is considered fully in appendix 1. The Figure 1. Bargaining Solution more interesting cases are ones in which evasion is not ruled out by very high detection probability for corrupt bureaucrats. An honest bureaucracy will be defined as one in which there is no bargaining solution between it and tha taxpayers. 29' All detected evasion is exposed and incurs a penalty, and there are no side payments or bribes. The results hinge on the relationship between the marginal reward parameter R1 and a critical value of this parameter which is termed Re, defined as, ,' Rc (1-po)(l+Po). The results are presented in the following propositions. Proposition 1. Honest Bureaucracy The bureaucracy is honest d e n reward parameter Rl is greater than or equal to the critical value Re, and the revard parameter R2 is non- negative. - 251 Proof. Using (7) 4, = 81-Rc+2wB2+2pl(1+Po)b > 0 for R1>Rc, which contradicta condition (71, so that there is no bargaining solution. 26' If Rl=Rc, 4;2(wR2+pl(1+Po)b)~ 0. The only possible solution is $=O, implying that either rO(if pl=O) or b=O(if R2e) or both are zero. If pl=O we must have from (61, bi = (l+Po + (R1+2R2yi)/(2(1-pol) 2yi > * I (l+Po)yi or Easioa is exposed will be terwd the degenerate solution of the corrupt bureaucracy. 25. The case in which both R2 and pl are zero results in i~differencebetwen corruption and honesty, and is ignored here but considered in section 6. 26. L,BO implies wi-yi and therefore L=-bi20,ie. bi=O. This m a n s -B-0 and Lw=O contradiction. w i ) > 0, a contradiction. A similar contradiction results if both these parameters are non-zero. The reason for this result is simply that the reward to the bureau- crat for exposing all evasion detected is too strong compared to the benefit (net of costs) of evasion. Under the as~uaptionsof the theorem,,: R1 - Rc > -pl(l+Po)b-2uR2 for all u and b, c0ntradiCtin8 equation (7) and (8). No bribes are therefore taken. This outcore can be depicted as point B in figure 1. There is also one case in which an honest bureaucracy cea-rw&& even if the marginal reward parameter R1 is less than the critical value PC. The condition of the R2 parameter is still the su+, but 8n extra condition must be imposed. The corruption detection probability sust be independent of the amount of bribes taken by the tax agent (plrO). This is shovn in the next proposition. Proposition 2. With R1 less than Rc.aqd R2 non-negative, an honest bureaucracy can resclt if plsO. Proof. Ueing ( 7 )%e have R1+2wR2=Rc. Substituting thia in (4) to (6) and a rimplifying we haye for bribe bi and evaaion cost ui, - This means that P < 0, and the taxpayer would ruffer a loss in a - bargaining rolution. An honest bureaucracy epposrs superficially to be much less likely when the marginal reward is declining with total exposure. This impression arises because so far we have focussed on the parameter R1 instead of the equilibrium marginal reward. From (7) it is obvious that if marginal reward ( ~ ~ + 2 w R>Rc, L,>O there cannot be a bargaining solution. ~ ) The bureaucracy must be honest if the equilibrium marginal reward is equ1.to ar'(exceeds the critical value (8,). Define a marginal reward limit Ra am follows. ' Ra = Itc-2yR2 ;PC as R2 :0, where y = & yip if the total evasion detected. i Then the following proposition follows. - 6 - . Proposition 3. In an incentive system with a quadratic marginal reward, the bureaucracy must be honest if R1 is equal to or greater than the reward limit Ra, and R2 is nonpooitive. Proof. Assume to the contrary that a bargaining solution exists (b>O) with R1>Ra. Substituting this in (7) we have for R29, k=2pl( l+Po)b + R1-Rc +2wR2 > 2pl(l+Po)b - 2P2(~-w) > U, as y>u>O. - - This contradicts (7). QED. The rest of the paper focuses on the corrupt b~reaucracy. The honest ? bureaucracy cases do however enter into a consideration,of- the principal's optimizaion problem. . 5. TXE EVASION DECISION AND NET REVEWES I I I In this section I return to the first stage problem of the tax- I payer. The taxpayer must determine how much to evade given the evasion price (function) conditional on evasion detected. The standard assumption of fixed income for each taxpayer is made. Formally his problem in to minimize the expected cost C(.) of income taxation and evasion, given the cost of evasion detected. Rin. C(Xi) f(Zi -x.) + ~(x.)+ n (bi+(l*Po)wi), 1 X. 1 subjectto %SO, LdO(eqns. ( 4 ) to(6)) . I' , < Forming and minimizing the Lagtangian and simplifying the necessary conditions for an interior solution, we have for the 'linear tax' and 'all-or- none detection' cases, tI (1-r Hcd H~(x~,z~))for st^, St(Xi) = { (10) TVD.1 (1-wncd) for H(. )=xi 1 The marginal cost of evasion detected is denoted HCd and given by equations (11) or (12) in the interior and corner solution cases respectively. ncd = L~~ P1 - R2)/(2V), if ~ " ~ 0 , (11) %Y =2((R1+2rP2)+R2(yi~i)), ~ = ( ( p ~ ( l + ~ ~ ) ~ - ~ ~ ) ( l - p ~ - ~ p ~ b ) - p ~ ~ ~ ~ ) , HCd = ( (l+~~)(l-p~-2p~b) (R1+2R2yi) )/U, if L, + < 0, U~2(1~po~2plb+plbi+p1(1+PO~~i ) a (12) U and V are related to the sufficient conditions for'a bargaining equilibrium and must both be non-negative if ( 6 ) and (7) represent a maxima. ~~u'ation(10) shows that the margipal resource coat of evarion must i equal the expectation of tax saving from marginal evasion. Uhat dirtinguisher this condition from the usual analysis of asymmetric dishonesty (with propor- tional penalty) is that the marginal cost of evasion detected is not constant ic general. It will be assumed that the second order conditions for a minima are satirfied. This ran. that either S" or Hll (2nd differential of H with rspect to X) or I"' m a t be poritive, and any decline in marginal cost of detected evarion m a t not be too large. 21' Given thir arrumption the effect of an inereaae in the price of evarion is to decreara evarion. Equtionr (61, (7) and (10) reprerent a solution of the evasion problem for an individual taxpayer. A# this solution depends in general on the total briber taken and the total exposure by tbe tax agent, it represents a "partial aolution". The "complete equilib~iw"must be determined by solving the entire set of equationr riarultaneoorly. In principle this in- volves rolving 2Y'(Y'=U/+) equations for the avarion price, and N equations - for evasion levelr. The chief problem that simultaneity creater ir that iae general there u y k multiple rolutionr. Thouah thir porribility is noted where relevant, the formal analyrir will focus on rituationr in vhich a unique equilibrium exirtr. There are, houcver, only #'+I independent equations for determining the 2 ~ 'variables in the Y' evarion prices, as (7) ir identical for all taxpayerr. With aquation (7) e o m n to all bargain#, U-1 variables can, within certain boundr, be set arbitrarily. The rearon for thir outcome is that the tax collector care,# ,only about the total briber'he receiver and the total esporure that he u k e r to the principal. He ir'indifferent to the precire dirtriblation of the bribes recieved and exporure levels of the . '? - taxpayers ~ o r e ~ e v a r i ohe detects. n \ s In thesenera1 care with a11 paruterr non-zero, equation (7) acts ar a bribe-exporure limit to the bargaining solution. With R2 positive, it 27- m a t is gw+tl MCd+tl dmd/dyi>0 for the linur tax care, and ' + c ~ ' + T 'c t ~ ~ ~ / dfor~a11~or nothing detection. y > also represents the agents trade-off between bribes and exposure. As shown in subsequent aections (7) reduces to a bribe limit or exposure Limit in special cases. The principale# problem can now be formally stated. The Principal's objective is to maximize expectation of returns from tax and penalty payments on tar payer net of cost of papent@ to agent. - 281 This problem is difficult to solve for the g m m d -.&,.the -. paper focuses , on the special cases mentioned in the introducth.- I start the analysis of the weak society, defined by a Linear prob- ability of corruption detection (pl=O), by considering a linear incentive system (R2=0). Tbe necersary condition8 determining the 2nd .Btaga bribe exposure solution (equations 6 and 7) reduce to, Tte sign of the second equation determines the corner solutions for exposure 28. Both the cortr and benefit8 of the general corruption detection system are taken am given, md not conaidered explicitly in thir paper. From equation ( 7 ' )and section 4, it is apparent that there are three sub cases, depending on value of the marginal reward relative to the critical value Rc. la. Honest Bureaucracy This is the case which was considered in proposition 1. lb. Transitional Bureaucracy (exposure indeterminate) In this case the appropriate bribe can-%e 'de!tetnabd 'for -.uti .a6 exposure, but the precise level is i n d e t ~ i n a t e . In el?irite-&edwahe. evader and the agent ate indifferent between different bribe exposure combinations. lc. regenerate Bureaucracy (no exposure) In this case the probability of corruption detection is Low, and the savings from nonexpoaure cf detected evasion high relative to the rewards of exposure. No exposure therefore accurs,.and the agent acts solely in 6 his own interests. = In case la the solution of the taxpayer's first stage problem of how ** much to evade is exactly identical to the honest bureaucracy case (appendix - - ** - 1). The imporFant thing to note an tho present context is that the amount of * evarion by the taxpayer is independent of the reward to the agent. An in- crease in the marginal reward therefore merely increases the reward costs of the principle unless the fixed part is simultaneously adjusted downward. In case lc the taxpayer's evasion costs are all in the form of a bribe, which is still linearly related to detected evasion. It now also de- pends on the marginal reward and corruption detection system effecting the agent (po). Evasion is therefore negatively related to marginal reward R1, penalty for agent corruption A. and its probability of detection P o * Increasing the marginal reward has a positive effect on tax payments, but ' costs nothing since exposure of detected evasion remains zero.* Putting the three cases together the principal's maximization problem (13) can be represented graphicaliy as in figure 2. The optimal incentive scheme is represented by the maxima at R = R = l + ~ ad+--* , in expectation of returns at Rc is equal to I (l+~,)p,y~-. , This simple case illustrates most starkly how penalties on tax payerr interact with the reward, the penalties and the detection probabilities for agents, to determine exposure and bribes. In particular an increase in penal- ties on tax evasion can lead to a sharp deterioration in the government's returns as shown in figure 3. This is because higher penalties make it more profitable for the tax evader and agent to collude, tipping the balance of the system from honesty to corruption. 9' The effect of a deterioration of the social environment which reduces the probability of corruption detection or penaltier on it is similar. - 30' This can bring about a sudden sharp increase in @rruption .. and deteriorsti~nin returns to the principal (figure 4). 29. This provides some justification for schemes vhich allow part evasion to be declared to the government for a temporary reduction in penalties. 30. For instance, if tax agents begin to feel that jail is not inevitable if corruption is detected. Figure 2. ~rincipal's Net Revenue Figure 3. Effect of Penalties on Taxpayers Figure 4. Bffect of Changes in Corruption Detection 7. EXPOSURE SHARE SYSTEM This section considers the quadratic incentive system, in the context of a weak society (pl=O). With the'marginal reward parameter R2 positive the system is honest as shown in propositions 1 and 2. It will therefore be as- sumed negative in this section. he marginal reward parameter R1 is less than(or equal to) the critical reward level Rc, the left hand sidqsof equation (7) (reproduced as (15)) is negative (zero). There is no exposure of de:ected evasion, and we have the degenerate sub-case, with the coat ~f evasion equal to the bribe (equation (14)). u.1 = bi = (1+P0)yi/2 + yi(R1+R2yi)/(2(1-p,)) , (14) R1-Rc +2wR2 C 0 for w > 0 - (15 The marginal cost of evasion detected is therefore, HCd (1+P0)/2 + (R1+2R2yi)/(2(1-po)) (16) By differentiating (161, and recalling tha: evasion is ne~ativelyrelated to the marginal cost of evasion, we see that evasion is negatively related to both incentive parameters. As the principal's net revenue depends only on declared taxes, this will rise as Rl is raised to Rc, and R2 is raised towards zero.. Net revenues will also rise with penalties on tax evasion. I I the marginal --ewerdparameter Rl is greater than the critical reward IL-tel Rc, a corruption solution is obtained with positive but partial exposure of detected evasion. Equations (6) and (7) reduce, on simplification 9 and substitution to equations (17) and (181, giving the Exposure Share sub- - case.Ef In contrast to the linear incentive scyeme, complete honesty does not result from a marginql r e ~ a r dparameter R1 greater than the critical level Rc. There is no gontra.liction, however, as the marginal reward in thisicase is not R1, but Rl*2R2w, and this is equal to Rc (equation (17)). This case should be compardd to the 'ineer one in which the taxpayer and tax agent were indif- ferent between the honest and degenerate solution. Thua,re~_Lacementof a -.) . Linear by a declining marginal ?eward moves the systes_t.owards corruption. The formal analys's of the exposure share sub-case is somewhat more complicmted. For an individual tax payer equation (17) determines the amount of detekted evasion exposed (wi) given the exposure of all other evasion detected (w(i)). - While bargaining with the tax collector, and in determining his evabion, the taxpayer treats the latter as given (i.c. w ~ = w ~ - ~ ( ~ ) )32/ . In ("co@pletel') equilibrium, equations (17) and (18) must be solved simultan- eously for all taxpayers detected. As equation (17) is identical for all tax payers,lN1-1 variables can be arbitrarily set'without affecting the bargaining solutioh (as noted in section 5). - - 9 d 31. I n t h e exposure share system tacal exposure is implicitly limited to w, (i.e. WSwm where w,,, is obtained by solving (17'1. 32. In a formal rational expectation context, this can be thought of as the myapic assumption. 1 Table Corruption Outcomes in the Weak Society (pl = 0) - I EXPOSURE SHARE HONEST SYSTEM LINEAR SYSTEM - SYSTEM RZ > 0 R2 0 R2 < 0 R1 < Rc HONEST DEGENERATE DEGENERATE I " I 8 C t -,7- ," i t i R1 = Rc HONEST INDIF~ERENT/ DEGENERATE i HONEST [ b j I R1 < Rc HONEST HONEST CORRUPT i 1 . -- a ; I One simple plausible rule for setting wi, which ensures consistency with (17) in equilibrium, is as follows. - 331 The cost and marginal cost of evasion detected are therefore, u. = bi+(l+po)wi = (l+po)yi + !42(yi-aiwm/y)2/((2( l - ~ )~ ) 1 ticd = l+Po+ R2 (yi-aiwm/y)/(l-po) ($20)i Differentiation of (20) and (17') shows that the price ok evirion detected, and consequently income declared, must be positively related to the reward parameters R1 and R2. --- .- - , . . Proposition 4. In the exposure limited sub-case, the principal can maximize net revenues by putting, Rl equal to the tax payer's penalty factor (l+Po), and R2 equal to a negative value Rb. Proof. Increasing R1 and R2 will increase declared incomc and associated tax revenues as shown above. Consider the change in net revenues from penalties and rewards ( l + ~ ~ - R ~ - w R ~with ) w R1 and R2* On differentiation and simplifi- cation these reduce respectively to, Thus an increase in R1 or R2 leads to an increase in total net revenue as long - as R1 does not exceed l+Po and R2 is less than zeta(and system is corrupt). 33. Again, ai=yi fulfills these conditions. Define a reward limit Rb as follows, By putting R2=Rb and Rl = (l+Po) into equation ( 6 ) (the source of (I))), it is seen that, L, = po(l+~o)il-wly) > O a s d y - - so that (23) holds cnly if w-y. Complete honesty is ensured if R1=l+Po, and R2zRb. As R1 and R2 have no effect on evasion under an honest system, and revenues from 'ekposurenet of .rewards decline, net revenues are maximized at Rl=l+Po, R2=Rb. QED Let y* be the total evasion detected wd&-azi4i~,es'r Yyp- - , Rb = -po(l+~o)/(2y*), where the right hand side is ~ t i e r r t - &.-*-A- incentive parameters as long the bureaucracy is honest. Therefore we must a1so have, Corqolary 4.1 The incentive system remains optimal despite changes in its concavity, as long as the marginal reward parameters satisfy the constraint, Thus, in a weak society, with its linear corruptioa detection system, the simplest incentive system maximising net revsnues is also linear. The . @ marginal reward must be set equal to the critical rate Rc. The broad pattern can be aumnarized as in table 2. - i -d* 5. BRIBE LIMIT SYVEH- w I turn now"to the Corruption Deterring Society, defined by a convex corruption detection probability (~~'0). This section considers the linear incentive scheme (R2=O). With marginal reward greater than or equal to the critical value, the essential features of the solution are the same as in the - 29 - Linear system. With the marginal reward R1 less than the critical value Re, the bargaining solution given by equations (6) and (7) reduces to, bi (2(1-po-2plb) + plbi) (B1 +(1+Po)(1-po-2plb))(~i~i) R1-(l+Po)(l-po-2plb) ~ ~ - ~ ~ + 2 p ~ ( cO+ ~ ~ ) b l (7") As in the exposure share system both an isterior (corruption) and a degenerate solution are possible, with the former arising when (7") is zero, and the latter when it ir negative. The conditions under which each will prevail are more complicated and interesting. In general rquktion (7 ) repre- sents the "bribe-exporure" limit. With R2 zero, it reducer to a bribe limit b,, where b, is obtained by setting (7") to zero and solving for b. b, (R~-R~)/(z~~(~+P~)) (22) less than or equal to thin limit. When the rystem is not bribe limited, it degenerates into a completely corrupt one. Only when the system is bribe lpmited, will the agent expose to the principal, a part of the evasion that he datecto. Whether orhot the system will be bribe limited dependr on ihe total - bribes received in the degenerate case, bo (say) relative to bm ( eqn. (22)). Putting wi=O in equation (6") and solving forithe bribe, we have, . - 6 + + +/- +/- + +/- - e bi? bi(R1,Po, pop pl; yi, b(i)), where k(i) = I b (23) j+i J fot each taxpayer whose evasion is detected. The signr of the partialr given abdve the equation are obtained by totally differentiating equation - 30 - '(6"). g1 The possibility of a negative effect of b(i) on bi suggests that there may be multiple equilibria. 24' Assuming that there is a unique bquilibrium, the N' simultaneous equations for the N' evaders detected by the &gent, can be solved as functions of the exogenous parameters. Suming these tunctions gives tL- total bribe bo in the degenerate case. The degenerate ... ~olutionwill therefore prevail if thim tot81 bribe, bo=b(R1; ); is less dhan the bribe limit. Prom (22) it is apparent that the bribe limit declines l[inearly with the marginal reward (El) offered by the principal to the agent. 4 degenerate solution can be led out for Sl=Bc because this implies (from q"), -0, b=O a contradiction. Therefore, bo ritbe; c m l 9, 'at least $rice or lie entirely above it. h e two posiflb2c-caees are shown in figurer 58 abd 5b respectively. -37/ Figure 5a identifies a second important marginal reward level Rm, which represent& the minimrun marginal reward necessary for ensuring that the tbx bureaucracy is not totally corrupt. Wher the marginal reward is less than R&, the (fixed) urginal return from exposure is less than fram total bCibes. Nothing is exposed and we have a degenerate solution. If the reward C fdom exposure. It becomes profitable for the agent to limit his total bribes iflied critical level, w have a corrupt'system with partial exposure. 1 . as yi 5 r* ' 4 ~ ~ / ~ ~ ( l * ~as~ ) ~ ) 1 in 'ppnndix 2 * dbi/db(i) :0 Figure 5a. Bribe-Limit Case Figure Sb. Bribe-Limit Care I Above Bc the incentive system ensures that the agent is completely hen-t. In the context of this section the weak economy wieh linear incen- tives, has a minimun reward level R,,, equal to the critical value 8,. The presence of declining marginal returns from bribes opens up an intermediate region by effectively reducing B, below Re. In the care of figure 5b this region expands to fill the vhole solution space, with RE effectively equal to zero The differences between the bribe Limit urd:linear cases,,can be seen in terms of equation (8) and the corresponding figure 1 (reproduced here as figure 6). As shown in figure 6, the taxpayers trade-off function between .. bribes and exposure is Linear (slope=-( (+Po)) in b a k ' c u ~iq'~b.,Ii-r - I case, the agents trade-off function ( ~ )1between brQg+r+a&+wurt. is linear with slope -Bl/(l-po). It intersects the taxpayers trade-off Line at a corner point iS (point D) vhen R1 ir lore (less) than Re. In the bribe limit case the agent's trade-off function (P2) has slope -~~/(l-p~-2p~b),so that as R1 falls belov Rc there is a range of values for which the solution is at an interior pint such as I. It has so~netimcsbeen rpeculated that corruption is purely a transfer - of rents from the government to its agents and the taxpayer. In the degener- ate sub-care this is clearly not so as the price of evasion detected is a non- . t linear function of the evasion detected and various parameters (equatian 23). The bribe-limited sub-case seems to support this speculation. - The i existence of corruption effectively results in m 3nfra-mrginal transfer from the principal to the agec: and the taxpayers. Th~eurginalcost of evuion is -' a identical to the case of the honest bureaucracy. M a t is, 3 The effect of chanies in penalties Po on the marginal price of evasion detection can be negative in the degenerate case. This reinforces the result obtained in the linear system. - 33 - Figure 6. Bribe-Limited Solution Probosition 5. Corruption as Transfers The bribe limited sub-case results in an infra-aurginal fall in the cord of evasion. As the marginal cost of evasion is identical to that under the \honestbureaucracy, so is the amount of evasion. ~ro4f. As in the exposure share sub-case, equation (7") is identica1;for ail taxdayers, and N'-1 variables can be set arbitrarily. A simple rule for setding bi, which ensures consistency with (22) in equilibrium, is, bi 4 ai bm/y ,ai > 0 as yi > 0, Lai = y = Z yi, (24) i i with ai constrained so that (6") yield a non-negatiu erpsura valise leer h evasbon detected (yi). 3' The amount of ei*rp6r;re is then obtained by subsiituting (24) in (6"). That is, The host of detected evasion is in thir case, (26) (26' 1 QED The marginal cost of evasion detected is l+P, (eqn. 26') which ir the I 1 srwias in the chrC of an honest bureaucracy. The second term on the right of - (26) i s independent of detected evarion yi. *01 It represents the infra- uridnal subsidy recieved by all evaders as a consequence cf corruption. - Tho%* taxpayerr benefit from the existence of corruption, the amount of e v a ~ b nis unaffected by the existence of corruption. 39. solution which satisfies these constraints is ai-yi. equilibrium a. must satisfy (17) so that it will tend to be notonically retted to yi. In the final part of this section I consider the principal's problem of maximizing net revenues C (equation (13)). In the bribe limited sub-case, ,changes in R1 and other parameters affectiag the tax agent have no effect on Ithe evasion decision. The principle's net revenue therefore depends on total iexposure w which can be obtained by summing (25) over all evaders detected. Differentiating C (eqn.13) with respect to R1, substituting and simplifying we lind that the slope of C can have either sign, but must be negative at R1=Rc (appendix 2). In the degenerate sub-case the opposite situation previili. .Para- Aeter changes affect only the evasion cost, but have no effect on anpsure and 4onsequently on the principal's reward coats. The "partial equilibriumn dffect (i.e. all other bribes held fixed) of R1 on evasion (bribe price) is (egative (positive) in this case (appendix 2). Asswing that the total effect qs negative, as it is likely to be if 8 unique "complete equilibriuo" exists, the solution of the principal's problca can be shown as in figure 7. The gtimal marginal reward (Il*)will then lie between the two critical reward 11.vels identified above (G 0 and Rl > R ~ ( ~ ( ~ + P ~ ) - ~ W ] - - 4P[p1(l +P )(R1+2vP2) - R2(l-p-2plb+plE)I a Lbb Lw Lbw Lwb -[(l+P )(l-po-2plb)- (Rl+2vP2)12 > 0 0 for a nuxinu. - If Lw-0 Rl+2R,w (l+Po)(l-po-2plb) and the above differentials reduce to, In the dogmerate sub-care of the btih limit solutioa, detected . I m f leealling that b-b(i)+bi, - treating b a# e.quaoou., and totally d e n \b= -2(1-p -2plb+pl?J < 0 (A2.2) 0 I (1+Po)yi . (l+Po)y ' Let b i l 2 + V. Therefore b = + V 1 1 . , I / where y I yi v'Lv.. I 1 i i i1 Substitutirg these in A2.1 and simplifying, we have 1 > * "'i ,, -F m e n Y. < v im~liea - 0 * If yiS y for all i, and bi muat be positively related tob(i). A Thin rill enmun that equllibr!~i m unique. 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