The Optimal Deterrence of Tax Evasion: The Trade-off Between Information Reporting and Audits

Transcription

1 The Optimal Deterrene of Tax Evasion: The Trade-off Between Information Reporting and Audits Yulia Paramonova Department of Eonomis, University of Mihigan Otober 30, 2014 Abstrat Despite the widespread reognition of the effetiveness of information reporting to inrease tax ompliane, existing tax theory onsiders tax audits to be the main tool to prevent evasion. The paper extends tax theory by modeling information reporting as an additional enforement instrument that allows aquiring information about taxpayers inome. The paper solves the problem of the IRS ommissioner who, having limited resoures to deter tax evasion, has to maximize tax revenue by alloating resoures between audits and information reporting. I derive the optimal resoure alloation whih is governed by the value of information reporting - how muh the aquired information helps to failitate tax audits. I find that the value of information reporting, beause of the strategi behavior of taxpayers, depends on the number of audits as an inverse U-shaped funtion. As a result, the optimal level of information depends on the budget suh that it first inreases, but then dereases with the budget, while the optimal number of audits always inreases with the budget. This implies that there is a budget at whih it is no longer optimal to expand the information reporting system. Applying the terminology of utility theory, we would say that information reporting is not a normal good. I am grateful for omments and advie from Joel Slemrod, Tilman Börgers, James R. Hines Jr., Christopher House, Mike Gideon, Gareth Myles, Wojieh Kopzuk and two anonymous referees. I wish to thank partiipants at the Mihigan PFFL and Regular Seminars, PET 2014, Nuremberg Taxation, Soial Norms and Compliane 2014 onferenes. Any errors or omissions are my own. 1

2 1 Introdution Information reporting is onsidered a dependable tool for improving tax ompliane. 1 When mathed with tax reords, it provides information that enables the tax authority to verify the amount of inome reported by taxpayers in their returns. Consequently, many ountries use information reporting. Most OECD ountries have information reporting for dividend and interest inomes some for inomes from independent personal servies, royalties, and patents. 2 The United States has a substantial information reporting system and has expanded it reently. It now requires banks to report the gross amount of merhant payment ard transations and brokers to report the adjusted ost basis for ertain seurities. But, information reporting has its osts that are usually overlooked. Those osts inlude expenses needed to publiize rules, to eduate taxpayers, and to perform and improve mathing of information returns with tax returns. For example, during the period 2009 through 2012 the IRS spent about $110 million just for developing the mathing program needed to implement the new information reporting requirements. 3 employ new resoures, it has to divert resoures from tax audits. Unless the tax ageny is able to These observations show that information reporting is highly valuable for improving ompliane. But, at the same time, information reporting imposes some tangible osts involving trading off resoures with tax audits. This raises some serious questions. How many resoures should the tax authority devote to information reporting expansion? More generally, what is the optimal tax enforement poliy? The existing tax enforement literature does not answer these questions, beause it has not modeled information reporting it onsiders tax audits as the main tool to prevent tax evasion. To address these limitations, this paper extends optimal tax enforement theory by onsidering information reporting as an additional tax enforement instrument, and introduing it into the existing tax evasion framework. The paper fouses on the problem of the tax authority that has to maximize tax revenue olletion by optimally deterring tax evasion. 4 The tax authority has a limited budget, and therefore it must trade-off its resoures between auditing people and improving information reporting. This paper models information reporting as an instrument that allows aquiring informa- 1 Aording to the existing empirial literature, inome soures that are subjet to some information reporting are haraterized by a higher tax ompliane than inome soures that are not (Kleven et al. (2011), Gordon and Li (2009), Alm and MKee (2009)). 2 In some of these ountries some of inome soures are also subjet to withholding. See OECD (2004). 3 For more details see GAO (2011). 4 This paper leaves aside equity issues of taxation and enforement. 2

3 tion about taxpayers inome. Speifially, in the model, information reporting provides the tax authority with signals. Based on a signal, the tax authority an onstrut preditions about the taxpayer inome distribution. Through improving information reporting and its expansion, a more aurate signal an be obtained. This allows onstruting a more aurate predition about the taxpayer inome distribution and failitates onduting tax audits, beause it helps to divide taxpayers into more distint audit lasses. Based on this model, I find that optimal tax enforement poliy involves a ombination of auditing and information reporting. Moreover, the optimal level of information reporting depends on the tax ageny s budget suh that it first inreases, but then dereases with the budget, while the optimal number of audits always inreases with the budget. This implies that there is a level of budget at whih it is no longer optimal to improve information reporting. It is beause of a strategi behavior of taxpayers that leads to a partiular substitution pattern between signal auray and the number of audits. Speifially, the number of audits that an be substituted by an inrease in signal auray, provided that the tax revenue stays the same, first rises but then delines with the number of people audited. In the ore of the model lies the differene between audits and information reporting, whih I oneptualize in the amount of information eah of these tools reveals. Both audits and information reporting help the tax authority to aess information about taxpayers inome, but they do this differently. An audit helps to reveal full information about one taxpayer, while information reporting helps to reveal partial information about many, or all, taxpayers. For example, information reporting on payments to independent ontrators helps the IRS to disover ases when the reipients of these payments underreport their gross profits. 5 However, it does not help to disover information about independent ontrators expenses, and thus their taxable inome is still unertain. 6 In general, information reporting helps to redue only some unertainty about true taxable inome. This justifies modeling information reporting as signals that help to make a more aurate predition of the inome distribution. As a first step in understanding the relationship between tax audits and information reporting, this paper analyzes how the audit poliy depends on the auray of information about taxpayers. This paper extends the model of Sanhez and Sobel (1993) and onsiders audit probability as a funtion of reported inome, as well as the signal. Based on the signals, taxpayers an be divided into audit lasses, whih allows audit funtion to be hosen 5 In the US, information about payments to independent ontrators is reported on 1099-MISC form. 6 Of ourse, information reporting on different inome soures is helpful to disover taxable inome to different extents. 3

4 separately for eah audit lass. Within an audit lass, the optimal audit strategy is to examine those taxpayers whose reported inome is below the utoff inome speifi to this audit lass. I find that optimal audit poliy requires the marginal revenue of an audit to be the same for all signals. This result has valuable pratial impliations. Speifially, it implies that the tax authority should fous audits on those taxpayers about whom it has less aurate signals; in other words, on those who have more opportunity to evade taxes. For example, if we onsider two groups of taxpayers wage earners and self-employed individuals then audits should be foused on the self-employed, i.e., those who haraterized by less aurate information. The seond step of the analysis fouses diretly on the trade-off between how muh of the resoures to spend on improving signal auray and how muh to spend on onduting tax audits. The trade-off arises beause the tax authority has limited resoures. The deision to alloate resoures depends on how effetively an inrease in signal auray, as ompared to an inrease in the number of audits, helps to redue tax evasion and raises tax revenue. When a ertain amount of tax revenue has to be olleted, an inrease in signal auray allows saving on tax audits. Based on this observation, the value of signal auray an be defined in terms of the number of saved audits through the inrease in signal auray. An analysis of the marginal rate of substitution between signal auray and the number of audits shows that the value of signal auray first inreases but then dereases as the number of audits inreases. This substitution pattern arises beause optimal auditing has to aount for taxpayers strategi behavior. The intuition for the delining value of signal auray is the following. Reall that without a signal only low inome taxpayers are audited. Signals help to divide taxpayers into audit lasses and therefore help to rediret tax audits toward some high-inome taxpayers. As signal auray inreases, taxpayers an be divided into more distint audit lasses and more high-inome taxpayers an be examined. However, the value of dividing taxpayers into audit lasses delines as more audits an be onduted, beause more taxpayers with higher inome have been audited. Moreover, when the number of audits inreases, the marginal tax revenue of inreasing number of lasses delines faster than the additional tax revenue of inreasing number of audits. To examine the property of the optimal solution, this paper studies omparative statis with respet to the budget and osts. When a hange in the tax authority s budget is onsidered, I find that the optimal signal auray depends on the tax ageny s budget suh that it first rises with the budget, but delines later on. This implies that there is a level of budget at whih it is no longer optimal to expand information reporting. At the same time, 4

5 the optimal number of audits always inreases with the budget. This paper disusses several important impliations of the results. The first impliation is that investments in information reporting are espeially ritial for a tax authority with sare resoures. This suggests that developing ountries ould benefit by using information reporting. The seond impliation onerns tax authorities that have substantial resoures for tax enforement. These tax authorities might be lose to the point at whih it is no longer optimal to expand information reporting. Therefore, for them it is espeially important to ondut a areful estimation of potential benefits and osts of further expanding information reporting. The rest of the paper is organized as follows. Setion 2 reviews the related literature to show how this paper builds on existing theory and extends it. Setion 3 presents the model that inorporates information reporting into the tax evasion framework by introduing the onept of signal auray. Setion 4 first haraterizes the optimal tax audit rule and how it depends on signal auray. Then, taking into aount the dependene of the optimal audit rule on signal auray, the paper determines the optimal level of signal auray and the optimal number of audits. Finally, the analysis of the omparative statis with respet to the tax authority s budget and osts reveals poliy relevant findings. Setion 5 disusses poliy impliations of the results. Setion 6 onludes. 2 Related Literature This setion overviews the existing optimal tax enforement literature in order to lay the groundwork for the model in this paper. Moreover, this setion helps to asertain a onept of the auray of information, whih is a new notion to the literature. While existing literature apprehends that tax authorities an ondition audit strategy on information beyond reported inome, it does not rekon that the auray of information an be hanged. It is this endogenizing of the auray of information that is implemented in the rest of this paper. Sanhez and Sobel (1993) solve for the optimal audit rule in a model where the tax authority observes only reported inome and hooses the probability of audit based on that reported inome. The optimal audit strategy in this ase is a utoff rule: to audit with high enough probability the taxpayers with reported inome below a ertain utoff and to not audit those with a reported inome above that utoff. Sothmer (1987) notes that, in pratie, the tax authority an observe both reported inome and orrelates like profession, age, and gross inome as reported independently by 5

6 the employer. Given that suh information allows taxpayers to be segregated into audit lasses, the enforement ageny will find it lurative to ondition the probability of an audit on audit lass, as well as on reported inome, partiularly if the audit lass is a good signal of inome (p. 229). Her model examines audit rules within an audit lass. The other paper that allows onditioning of tax audits on information about taxpayers is Maho-Stadler and Perez- Castrillo (2002). In ontrat to Sothmer (1987), they assume only three disrete inome levels and that signals are unobserved by the taxpayers. Nevertheless, the main findings in those papers are similar. However, both papers treat signal as free and exogenous, whih preludes the possibility of addressing the optimal improvement in the auray of observed information. Similar to the model in this paper, Menihini and Simmons (2014) onsider a ostly state verifiation model where a prinipal entering a ontrat with an ex-post informed agent, as well as auditing, has the option of aquiring ostly imperfet information about future revenues. In ontrast to the urrent paper, their model has only two inome realizations and two signal values. 7 The model in this paper has a ontinuous distribution of inomes and signals. Also, their optimization problem of maximizing of an agent expeted inome subjet to the prinipal s partiipation onstraint differs from the urrent paper optimization problem that solves for the maximum tax revenue olletion given that agents maximize their expeted inome. Lastly, they do not allow the prinipal to hoose the size of the orrelation between the signal and state realization, while this paper allows the prinipal (tax authority) to hoose signal auray. To desribe the auray with whih the tax authority an observe true information (relevant for tax administration purposes), a new term, observability of the tax base, has been reently introdued by Slemrod and Traxler (2010). They define observability as the auray of the tax base measurement that the tax authority an obtain. In their model, the tax authority an influene observability by hanging the amount of investment spent on tax enforement. They determine the optimal level of observability that is defined by the trade-off between the soial osts from allowing more inauray and the net revenue gains from having a heaper-to-administer, but more apriious, tax system. Their model is similar to the model in this paper in its idea to endogenize information observed by the tax authority; however, their model does not allow for tax evasion and, thus, tax enforement is beyond its sope. Kleven and Kopzuk (2011) note that the auray of information onerning appliants 7 The extension of the model has three inome realizations. 6

7 to soial programs is an important fator affeting program design. They show that the government an influene auray of this information by setting a omplexity of sreening tests. Their model haraterizes the optimal soial program in whih poliy makers an hoose the rigor of sreening, along with a benefit level and eligibility riterion. While their settings are suffiiently different from those onsidered in this paper, the idea that government an influene and, hene, optimally hoose the auray of information is parallel. Besley and Persson (2011) argue that tax bases like inome taxes and value added taxes an only beome effetive through extensive government investments in tax ompliane. This supports the importane to improve information reporting from a high level of generalization. Aording to them, in the proess of administrative infrastruture development the ountries are able to move from olleting around 10 perent of national inome towards olleting around 40 perent, and to shift from trade taxes and exise taxes towards labor inome and other broad bases. In their framework a forward-looking government will deide to invest in fisal apaity in order to build a more effetive tax system. However, the ways to improve administrative infrastruture are beyond their sope. Boserup and Pinje (2013) introdue information reporting into a model of tax evasion in a way that is different than this paper. They assume that one part of taxpayer inome inome reported by third parties is ompletely observed and that other part of inome is ompletely unobserved by the tax authority. This is a reasonable assumption, but it preludes the general ase when information reporting provides only partial information about taxable inome. Also, in their model information reporting is exogenous and the tax authority annot affet its extent. As has been disussed so far, the importane of auray of observed information for the state poliy and, in partiular, for tax poliy has been aknowledged in many studies. The next natural move is to onsider information auray as one tax enforement instrument and to determine its optimal level. I present this move in what follows. 3 Model This setion presents a model that desribes taxpayers who evade their taxes and the tax authority that fights tax evasion. The key innovative part of the model is signals; they represent information olleted by the tax authority through information reporting. This model presents a way to inorporate information reporting into the existing tax evasion framework. 7

8 Consider an eonomy onsisting of risk-neutral individuals haraterized by their inome, i. Inome is exogenous and distributed on R aording to.d.f. F 0 ( ), whih is assumed to be ontinuously differentiable with density f 0 (i) = F 0(i). Eah individual is subjet to an inome tax at rate t and is required to file a tax report. The tax authority ollets taxes and performs audits to enfore tax ompliane. The tax authority does not observe taxpayers true inome, and a priori believes that taxpayers true inome is distributed aording to F 0 ( ). However, the tax authority reeives a signal about eah taxpayer s inome, s R. 8 Given a signal, s, about a taxpayer s inome, the tax authority forms an updated belief about the taxpayer s inome distribution, whih is the onditional distribution funtion F (i s). The assoiated onditional density funtion is denoted by f(i s). The tax authority uses a signal when it hooses the probability of an audit, p(r, s). So, the probability of an audit depends not only on taxpayer s reported inome, r, but also on the signal, s. In pratie, the tax authority has different types of information that might be helpful to predit a taxpayer s true inome, inluding a taxpayer s age, oupation, marital status, information from previous periods, as well as from third-party reporting of different inome types. Third-party information reporting is espeially valuable in prediting a taxpayer s true inome. I assume that the tax authority is able to aggregate this information to onstrut a univariate predition of true inome. This predition is implied by the signal, s. To haraterize the dispersion of the onditional distribution of inome, I introdue the signal auray, a. Correspondingly, I denote the onditional distribution of inome by F a (i s), indiating that it depends on the signal auray, a. I define signal auray, a, as the inverse of the standard deviation of the onditional distribution of inome, whih requires imposing additional assumptions. Speifially, assume that for any signal, s, the onditional distribution of inome is F a (i s) = G(a(i s)), (1) where G is a symmetri distribution funtion with zero expetation and unit variane, and the expetation of the onditional distribution of inome is equal to s. The onditional p.d.f., f a (i s), is orrespondingly equal to ag(a(i s)). As an be seen from the definition of G(a(i s)), signal auray, a, is equal to the inverse of the standard deviation of the onditional inome. Importantly, ondition (1) assumes that all signals are haraterized by the same auray, 8 Signal s is one-dimensional. 8

9 a. This impliitly assumes that all taxpayers have idential inome soures. For example, half may be wage and salary inome and half rental inome. However, this assumption is needed only for the analysis in Setion 4.2. As we go along, I will point out when this assumption is imposed and when not. The distribution of signals also depends on auray, a. Let us denote the.d.f. of signals by H a (s) and p.d.f. of signals by h a (s). An integration of the onditional distribution of inome over all signals should give the p.d.f. of inome, that is, ˆ f 0 (i) = ag(a(i s))h a (s)ds. (2) I assume that, when signal auray hanges, the distribution of inome, f 0 (i), does not hange. For this to be true, the distribution of signals, h a (s), has to hange with signal auray. Indeed, when signal auray hanges, the onditional distribution of inome - beliefs based on signals - hanges. To satisfy ondition (2), the distribution of signals, h a (s), should orrespondingly adjust. Moreover, in order for the assumption on the onditional distribution of inome (1) and the assumption that the distribution of inome, f 0 (i), does not depend on signal auray to be satisfied, the support of the distribution of inome should be unbounded. 9 she is entitled to reeive a tax refund of t i. 10 This permits a taxpayer to have a negative inome, in whih ase An example that satisfies all the assumptions imposed above obtains when the onditional distribution of inome is normal with expetation equaled to s and variane equaled to 1 a 2. Letting Φ( ) and ϕ( ) to denote the.d.f. and the p.d.f. of the standard normal distribution, the onditional.d.f. of inome is F a (i s) = Φ(a(i s)) and the onditional p.d.f. is f a (i s) = aϕ(a(i s)). Additionally, the distribution of true inome is also normal with expetation µ and variane Ω 2. The distribution of signals that onforms to these assumptions is a normal distribution with expetation µ and variane Σ 2 = Ω 2 1 (i.e., the.d.f. is H(s) = Φ( s µ ), a 2 Σ the p.d.f. is h(s) = 1 s µ 11 ϕ( )). Σ Σ The tax authority may inrease signal auray at some resoure osts, for example, by 9 You an request a proof of this assertion from the author. 10 Having negative inome does not hange the inentives of a taxpayer. A taxpayer with negative inome has inentives to understate the true inome beause understatement of true inome allows to laim a higher tax refund. 11 In the ase of the standard normal distribution it an be shown that the notion of signal auray agrees with the notion of informativeness introdued by Blakwell: more aurate signals are also more informative. Aording to one of the Blakwell s definitions of the more informative experiment, if the posteriors onstruted after observing the outome of experiment P are mean-preserving spread of the posteriors onstruted after observing the outome of experiment Q then experiment P is alled to be more informative than experiment Q. More information about Blakwell s informativeness an be found in Borgers (2009). 9

10 improving information reporting. Speifially, the tax authority an ahieve signal auray, a, by investing K(a). The ost funtion, K( ) > 0, is assumed to be inreasing (i.e., K ( ) > 0). Later, I have an in-depth disuss of what these osts inlude. A taxpayer is assumed to know the signal and its auray. The taxpayer knows that the probability of audit is based on her reported inome, r, and on the known signal, s. If a taxpayer s report is not audited, then the taxpayer owes the tax based on the reported inome, tr. If her report is audited, then the taxpayer owes taxes based on her true inome and also a penalty proportional to the onealed taxes at rate π, ti + (1 + π)t(i r). Eah taxpayer minimizes her expeted tax payment: min r {tr + p(r, s)(1 + π)t(i r)} (3) Define this minimum as T (i, s, p( )) and the optimal reported inome that minimizes the above taxpayer s problem as r(i, s). though the audit probability. Note that the signal affets the optimal report only The tax authority hooses the probability of an audit, p(r, s), and signal auray, a, to maximize the tax revenue given limited resoures to ondut tax audits and invest in signal auray. 12 While the tax authority does not observe the true inome of eah individual, it knows the onditional distribution of inome based on signals, F a (i s), and the distribution of signals over the entire population, H a (s). Thus, the tax authority problem is { max T (i, s, p( ))df a (i s)dh a (s) } p(r,s),a s.t. p(r(i, s), s)df a (i s)dh a (s) + K(a) B, where is the ost of an audit, and B is a budget of the tax authority. In what follows the tax authority is assumed to have a limited budget (i.e., 0 B < ) meaning that the tax 1 authority is unable to audit every taxpayer with probability at least, in whih ase all evasion would be deteted. 12 First, formulating the tax authority problem in this way rather than maximizing total welfare allows my analysis to fous on tax enforement problem. Seond, I onsider onstraint tax revenue maximization rather than net tax revenue maximization beause, as the model of hierarhial poliy desribed by Sanhez and Sobel (1993) shows, soiety benefits when the government ontrols the budget of the tax authority rather than allows the tax authority to maximize net tax revenue. In their model, the government first selets a tax funtion, a level of publi good expenditure, and the tax authority s budget and then delegates the responsibility to ollet taxes to the tax authority. The model shows that the budget provided by the government is always less than the net revenue-maximizing budget. An extra dollar in the tax authority s budget not just raises extra tax revenue, but also imposes heavier tax burden on soiety and hene dereases soial welfare. Sine I onsider only tax authority s enforement problem, I assume that the tax authority is subjet to the budget onstraint. (4) 10

11 The osts of improving signal auray, K(a), inludes several omponents. First, if improving signal auray is ahieved through expanding information reporting, then K(a) inludes administrative osts of information reporting expansion. Those osts arise beause the tax authority has to publiize new rules in soial media, reate and publish new tax forms, et. Seond, the ost of improving signal auray inludes the ost of proessing the information returns olleted from third parties. This is an important omponent of information reporting ost that is usually left in the shadow, but atually deserves a areful examination. Eah year the IRS reeives an enormous number of information returns. For example, in 2013 the number of information returns reeived was 2 billion. 13 To math all these information returns with tax returns, a sophistiated mathing program had to be developed and used. In 2009 the IRS initiated the so-alled Information Reporting and Doument Mathing (IRDM) program in order to implement the two new information reporting requirements K and 1099-B information returns. To develop and support this program, IRS spent approximately $110 million during the period 2009 through Even without expanding information reporting, there are osts assoiated with proessing and improving information reporting. An improvement in information reporting that inreases signal auray an be ahieving through a better examination of information reports, but this requires investments in omputer, software, and programs. Additionally, the IRS has been modernizing its system in order to failitate the information returns filing and submitting proess. 15 Moreover, during the last deade the IRS has been onstantly working to improve the mathing proess of existing programs and refining the methodology of seleting ases for taxpayer ontat. However, the GAO (2009) report pointed out that the Automated Underreporter (AUR) program a program responsible for mathing 1099-MISC information returns urrently has a narrow reah and pursues less than half of 1099-MISCrelated ases in the AUR inventory (p. 5). Thus, a sope for further improvement of information reporting remains. The third important omponent of information reporting osts is the ompliane osts imposed on third parties (firms/taxpayers). Compliane osts inlude the osts of getting taxpayer identifiation numbers, buying software, traking reportable payments, filing returns, and mailing opies to taxpayers or paying to tax preparer. Contrary to ommon belief, the GAO (2007) report found that the size of existing ompliane osts was relatively 13 See SOI Tax Stats - Information Reporting Program - IRS Data Book Table 14, available at 14 For more details see GAO (2011). 15 For further details see IRS IT Modernization Vision & Strategy (Otober 2007). 11

12 small. Speifially, [o]ne small business employing under five people told GAO of possibly spending 3 to 5 hours per year filing Form 1099 information returns manually, using an aounting pakage to gather the information. An organization with more than 10,000 employees estimated spending less than.005 perent of its yearly staff time on preparing and filing Forms 1099, inluding reord-keeping. Two external parties reported pries for preparing and filing Forms 1099 with IRS of about $10 per form for 5 forms to about $2 per form for 100 forms, with one of them harging about $.80 per form for 100,000 forms (p. 3). Even though the size of the ompliane osts seems to be moderate, it should not be exluded from onsideration. The ost of improving signal auray, K(a), may inlude a ompensation for imposed ompliane osts to third parties. 4 Analysis This setion desribes the solution to problem (4). To failitate the analysis, I subdivide it into two steps. 16 First, by holding signal auray fixed, I determine how the optimal audit poliy an depends on signal auray. Seond, taking into aount the dependene the optimal audit poliy on signal auray, I derive the optimal level of signal auray and the optimal number of tax audits. The main fous in that subsetion is on the trade-off between the number of tax audits and level of signal auray. The analysis is onluded by omparative statis whih examines how hange in the budget and osts affet the solution. Finding from this subsetion brings poliy relevant impliations. 4.1 The Optimal Audit Poliy: How Does It Depend on Signal Auray? In this subsetion signal auray is taken as fixed and the optimal audit poliy is determined. For notational simpliity, the supersript a is omitted. So, the tax authority problem is { } max T (i, s, p( ))df (i s)dh(s) p(r,s) s.t. p(r(i, s), s)df (i s)dh(s) + K(a) B, (5) A simple solution strategy to takle problem (5) an be disovered by examining the nature of the problem. Reall that signals are observed by the tax authority and by taxpayers as well. Consequently, signals annot be manipulated by taxpayers. Moreover, signals provide 16 This two-step approah allows relying on the previous results for the optimal audit rule. 12

13 information about inome distribution, whih allows the tax authority to segregate taxpayers into audit lasses. Thus, the tax authority an treat all taxpayers with signal s as belonging to one audit lass. For eah audit lass, i.e., for eah signal, the tax authority an determine the optimal audit funtion given the amount of resoures assigned to ondut audits within this audit lass. Let s denote this amount of resoures by B(s) = p(r(i, s), s)df (i s). The sub-problem of hoosing optimal audit funtion within an audit lass given resoures B(s) an be analyzed using arguments employed by Sanhez and Sobel (1993). Then, the alloation of resoures among audit lasses, i.e., the size of B(s) for any s, an be hosen to maximize the total tax revenue subjet to the onstraint B(s)dH(s) B K(a). In pursuane of this strategy, a solution of problem (5) whih is desribed in Proposition 1 an be found. Note that while Proposition 1 relies on arguments employed by Sanhez and Sobel (1993), it extends them to the ase when the support of the inome distribution is unbounded. 17 Additionally, Proposition 1 does not require any assumptions about onditional distribution of inome F (i s) exept that for eah signal s the onditional inverse hazard rate γ(i s) = 1 F (i s) f(i s) is stritly dereasing in i. So, the onditional distribution funtion ould have bounded or unbounded support, whih w.l.o.g. is denoted by [l(s), h(s)]. Also, signals need not to have the same auray, hene the variane of the onditional distribution of inome ould be different for different signals. Proposition 1. Assume that for eah signal s the onditional inverse hazard 1 F (i s) rate γ(i s) = is stritly dereasing in i. The optimal audit funtion f(i s) whih solves (5) satisfies: where the optimal audit trigger β(s) satisfies 1, if r < β(s) p (r, s) =, (6) 0, if r β(s) γ(β(s) s) = = λ t() < λ t() if l(s) β(s) h(s), (7) if β(s) = l(s) and λ - Lagrange multiplier - is determined so that the budget onstraint is satisfied: 17 Sanhez and Sobel (1993) onsider only the ase when the support of the inome distribution is bounded. 13

14 ˆ Proof. See proof in Appendix. F (β(s) s)dh(s) = (1 + π)[b K(a)]. (8) Proposition 1 shows that the optimal audit rule depends both on reported inome and signal. For a given signal s, the optimal audit poliy is a utoff rule: audit with probability 1 if reported inome is below the utoff β(s), and do not audit if reported inome is above the utoff β(s). 18 Note that with suh an audit poliy in plae, a taxpayer optimal strategy is to hoose reported inome equal to true inome, r(i, s) = i, if her true inome i is less than the utoff β(s) and to hoose reported inome equal to the utoff β(s), r(i, s) = β(s), if her true inome i is greater or equal to the utoff β(s). 19 The dependene of the audit trigger β(s) on the signal is ruial. Beause of this dependene, it is not low reports, but low reports onditional on signals that are audited by the tax authority. This onditioning on signals makes the derived audit rule more realisti than the simple utoff rule like one in Sanhez and Sobel (1993) whih was ritiized for mismathing the reality. 20 Indeed, the derived audit rule whih onditions audit probability on signal/audit lass easily agrees with the experiene that audit probability seems to inrease with reported inome. Even when the tax authority audits taxpayers with low inome reports within an audit lass with higher probability than it audits high-report taxpayers, the observed probability of audit whih pools audit lasses ould rise with reported inome. Taxpayers with high reported inome might have high probabilities of audit beause they are in audit lasses that signal high inome. Conditions (7) and (8) whih determine the audit trigger β(s) have intuitive interpretations. Condition (8) guarantees that resoures spent on audits are equal to the available resoures for audits. Condition (7) basially requires the marginal revenue of an audit to be the same among all signals, i.e., among all audit lasses. This an be seen by onsidering a small hypothetial inrease of the utoff from β(s) to β(s) + ɛ, for a given signal s. In the result of suh an inrease, those who have inome greater than β(s), whose mass is 18 I all β(s) the audit trigger or utoff inome, and use these terms interhangeably. The term audit trigger is more self-explanatory, the term utoff inome is used in the literature. 19 The utoff rule results in the solution of the model beause this paper following mehanism design literature adopts a ommitment priniple assuming that the prinipal announes an audit strategy suffiient to indue truthful reports by the agent and then stiks to it despite it being ostly. 20 Melumad and Mookherjee (1989) point out that this type of solution has redibility problem. This rule is not redible beause the tax authority has to audit those who do not evade. However, if we interpret this stati model from a dynami prospetive - the tax authority has to audit those who not evade today in order that they do not evade tomorrow - this rule looks more reasonable. 14

15 equal to (1 F (β(s) s)), will inrease the paid taxes on tɛ. But, this will require the tax authority to ondut f(β(s) s)ɛ additional audits at ost. Condition (7) requires that the ratio of marginal revenue to marginal ost of the hypothetial inrease of the utoff inome, tɛ(1 F (β(s) s)) = t() γ(β(s) s), equals λ (the Lagrange multiplier) for signals that require f(β(s) s)ɛ() 1 auditing at the optimum. Beause marginal ost of an audit,, is onstant, ondition (7) implies that the marginal revenue of an audit is the same for all signals. Condition (7) also overs the ase when taxpayers with a ertain signal are not audited. It is the ase when β(s) = l(s). If for some signals the funtion γ(i s) is lower than λ, t() even for the lowest inome l(s), then the audit trigger β(s) is equal to l(s). 21 This situation an arise if l(s) is finite and some signals are relatively more aurate than others, so that audits of the taxpayers with suh signals annot generate high enough tax revenue. In turn, this an happen when some taxpayers have high degree of voluntary ompliane beause of higher information reporting, and thus less opportunity to evade. For instane, those taxpayers might be wage earners, and it might be optimal to onentrate audits only on selfemployed people, leaving wage earners unaudited. A rigorous example of a situation when some taxpayers are not audited is onsidered in the Appendix A. The result that optimal audit poliy requires the marginal revenue of an audit to be the same for all signals has valuable pratial impliations. Speifially, it implies that the tax authority should fous audits on those taxpayers about whom it has less aurate signals whih means on those who have more opportunity to evade taxes. To see this, onsider two groups of taxpayers suh that the first group is haraterized by signal auray a 1 and seond group is haraterized by signal auray a 2 whih is greater than the first signal auray (i.e., a 1 < a 2 ). Assume that the onditional distributions of inome are F a 1 (i s 1 ) = G(a 1 (i s 1 )) and F a 2 (i s) = G(a 2 (i s 1 )) for the first and seond groups orrespondingly. Then, the following orollary shows that the number of people audited in the group with signal auray a 1 is greater than the number of people audited in the group with signal auray a 2 when a 1 < a 2. Corollary. If a 1 < a 2 then G(a 1 (β(s 1 ) s 1 )) > G(a 2 (β(s 2 ) s 2 )). Proof. See proof in Appendix. This orollary shows that optimal audit poliy presribes the tax authority to ondut more audits among taxpayers with less aurate signals than among taxpayers with more aurate signals. Intuitively, prior to audits the expeted audit revenue from taxpayers with 21 γ(i s) shows the proportion of people with inome greater than i to people with inome exatly equal to i for a given signal s. 15

16 a low aurate signal is higher then from taxpayers with a high aurate signal. Therefore, the tax authority should ondut more audits among the former ones in order for the marginal revenue of an audit to be the same for both groups. This result that the tax authority should fous audits on those taxpayers about whom it has less aurate signals losely mathes reality. However, prior to this paper there has been no theory whih would provide a sound ground for it. 4.2 The Optimal Signal Auray and the Optimal Number of Audits In this subsetion I determine the optimal signal auray as well as the optimal number of audits. To do this, I return to the ase when signal auray is not fixed and an be hanged by the tax authority by investing resoures. subsetion of how the optimal audit rule depends on signal auray. Also I rely on the result from the previous The optimal poliy involves a trade-off of how muh to invest in signal auray and how muh of resoures to spend on audits. To be able to fous on this trade-off, we need to use assumption (1) imposed on the onditional distribution of inome, whih assumes that all signals are haraterized by the same auray. This assumption is that F (i s) = G(a(i s)). Given this assumption, the utoff funtion that haraterizes the audit poliy an be derived in expliit form. To make inferene about the utoff funtion that is defined by (7) and (8), it is helpful to analyze the onditional inverse hazard rate funtion, γ(i s) = 1 G(a(i s)). ag(a(i s)) As the distribution funtion on its own, the onditional inverse hazard rate funtion, γ(i s) = 1 G(a(i s)) ag(a(i s)), depends on i and s additively. Therefore, the FOC (7), whih is 1 G(a(β(s) s)) = λ, implies that β(s) s is a onstant that does not depend ag(a(β(s) s)) t() on s and depends only on a. By defining δ β(s) s, the other FOC (8) the budget onstraint an be simplified to G(aδ) = P (i.e., P probability 1 (B K(a)). Defining the right-hand side by (B K(a))) and notiing that P is the number of taxpayers audited (with ), helps to derive the utoff inome whih is β(s) = s + δ = s + 1 a G 1 (P ). (9) For sake of brevity, I will refer to P as the number of audits. Note that 0 P < 1, beause 0 B <. As (9) shows, the utoff inome inreases in the magnitude of signal, s, and in the number of audits, P. Intuitively, the higher is the magnitude of signal, s, the higher is the expeted 16

17 true inome, and therefore the higher is the utoff inome. Also, the greater is the number of audits, P, that are onduted, the higher is the utoff inome. Simply, beause to audit more people, the tax authority needs to have a high utoff inome. The utoff inome (9) is also affeted by signal auray, a. When signal auray inreases, the onditional density of inome given signal, G(a(i s)), shrinks toward its expeted value, s, and the utoff inome does the same. Speifially, if P 1 2 (i.e., G 1 (P ) 0), then an inrease in a leads to an inrease in the utoff inome. If P > 1 2 (i.e., G 1 (P ) > 0), then an inrease in a leads to a derease in the utoff inome. 22 Intuitively, as signal auray inreases, a signal provides more aurate predition of inome whih is refleted by the onditional density of inome beoming more onentrated (lower variane) around that signal. This implies that onditional on the signal the probability of inome being in lose proximity of the signal is higher, therefore the utoff inome moves toward the signal. Importantly, when signal auray inreases, shrinking of the onditional density of inome ours along with widening of the density of signals. The density of signals beome more dispersed around the mean. For instane, when all distributions are normal, the distribution s µ of signals is H(s) = Φ( ), with variane Ω 2 1 inreasing in signal auray, a. Thus, Ω 2 1 a 2 a 2 as signal auray inreases, signals beome more distint and better predit inome. Using expression (9) for the utoff inome, we an alulate the tax revenue. We start by alulating the tax revenue olleted from a taxpayer whose signal is s and then we alulate the total tax revenue by integrating over all signals. The tax revenue whih is olleted [ β(s) from a taxpayer with signal s is T (s) =t iag(a(i s))di +β(s) ]= ag(a(i s))di β(s) t[s + 1P E a G[z z G 1 (P )] + 1(1 P a )G 1 (P )], where E G [z z G 1 (P )] is an expetation of a random variable with the.d.f. G( ), denoted by z, onditional on z G 1 (P ), i.e., E G [z z G 1 (P )] = G 1 (P ) z g(z) dz. The total tax revenue is obtained by integrating T (s) P over all signals and is equal to T R = T (s)h(s)ds = = t [ µ + 1 a (P E G[z z G 1 (P )] + (1 P )G 1 (P )) ]. (10) Let us define R(P ) as R(P ) P E G [z z G 1 (P )] + (1 P )G 1 (P ). The expression t R(P ) has a simple interpretation and is equal to the expeted tax olletion from a taxpayer with signal s = 0 if a = 1 and orrespondingly the utoff inome is equal to G 1 (P ). Speifially, with probability P suh a taxpayer has a true inome lower than G 1 (P ), in whih ase she honestly reports her true inome equal in expetation to E G [z z G 1 (P )]. With 22 Here, I onsider an exogenous hange in a meaning that the number of audits, P, fixed. 17

18 probability (1 P ) her true inome is greater than G 1 (P ), in whih ase she reports the utoff inome, G 1 (P ). Beause R(P ) E G [z] = 0, the tax revenue, T R = t(µ + 1 R(P )), inreases a with a. Beause R (P ) = > 0, the tax revenue inreases with P. Interestingly, 1 P g(g 1 (P )) R (P ) is equal to inverse hazard rate funtion evaluated at i = G 1 (P ) given s = 0 and a = 1, i.e., R (P ) = γ(g 1 (P ) s = 0, a = 1). Finally, the tax revenue is T R(a, P ) = t(µ + 1 R(P )) (11) a As (11) shows, to raise higher tax revenue, the tax authority an use two tools: to inrease signal auray and to inrease the number of audits. First, an inrease in signal auray, a, raises the tax revenue by proportionally shrinking R(P ) funtion. It is beause an inrease in signal auray inreases the average inome of those who are audited and, in ase when P 1, inreases reported inome of those who are not audited, whih is equal to the utoff 2 inome. When P > 1, an inrease in signal auray, on the opposite, dereases the utoff 2 inome, but it is still inreases the average inome of those who are audited. Seond, an inrease in the number of audits raises the tax revenue by inreasing R(P ) funtion. It is beause an inrease in the number of audits inreases the utoff inome. Reall that P = (B K(a)) and notie that this expression is exatly equivalent to the budget onstraint P + K(a) = B. Given this, we an state the tax authority problem of finding the optimal signal auray, a, and the optimal number of audits, P, whih maximize the tax revenue, as max t(µ + 1 R(P )) a L a, 0 P a (12) s.t. P + K(a) = B, where a L = 1. To insure that auray a is not smaller than 1 the inverse of the standard Ω Ω deviation of unonditional inome distribution, I impose a a L and assume K(a) = 0 for a a L. Problem (12) onsiders maximization of tax revenue, T R(a, P ) = t(µ+ 1 R(P )) subjet to a the budget onstraint P + K(a) = B. As Figure 1 shows, this problem an be presented graphially in (P, a) spae in terms of finding the highest iso-revenue urve T R(a, P ) = onst given the budget onstraint P + K(a) = B. An iso-revenue urve an be expressed as a T R=onst = R(P ) From this expression it is lear that iso-revenue urves are stritly µ T R/t. onvex if R(P ) is stritly onave. 23 If K(a) is onvex, then the budget onstraint urve 23 Note that, when G( ) is equal to the.d.f. of the standard normal distribution, Φ( ), as in the example 18

19 Figure 1: Iso-revenue urve and the budget onstraint is onave. The strit onvexity of the iso-revenue urves and the onavity of the budget onstraint urve ensure that there exists a unique optimal auray, a. If there is a point where an iso-revenue urve is tangent to the budget onstraint urve, then it is the interior solution; otherwise, the solution is a orner solution. Formally, the ratio the FOCs for the maximization problem (12) an be written as follows: 24 T R a T R P 1 + π K (a ) = R(P ) a R (P ) 1 + π = 0 if a > a L, P > 0 K (a ) = > 0 if a > a L, P = 0 < 0 if a = a L, P > 0. (13) The solution of (12) is determined by equation (13) and the budget onstraint: Equation (13) has a simple interpretation. 1 + π P + K(a ) = B. (14) At an interior optimum, the ratio of the marginal tax revenue from inreasing signal auray to the marginal tax revenue from inreasing the number of audits is equal to the ratio of the marginal ost of inreasing signal auray to the marginal ost of inreasing the number of audits. above, R(P ) is indeed onave. It an be shown that R R(P ) (P ) = ϕ 2 (Φ 1 (P )) < 0 24 There are two FOCs for problem (12) w.r.t. a and w.r.t. P whih inlude a Lagrange multiplier. We an divide one by the other to eliminate the Lagrange multiplier. 19

20 Figure 2: The optimal solution (P, a ) as intersetion of the FOC ondition and the budget onstraint The following proposition formalizes the onditions defining the optimal solution. Proposition 2. i) If γ(z) = 1 G(z) is stritly dereasing and K(a) is onvex g(z) then the point the auray a and the number of audits P whih satisfies to the ondition (13) and (14) is the unique solution of (12); ii) If additionally K (a L ) = 0 then the optimal solution is an interior solution haraterized by Proof. In the appendix. R(P ) a R (P ) = 1 + π K (a ). (15) Proposition 2 provides onditions whih ensure the existene of a unique interior optimum. Relying on this result and imposing these assumptions hereafter we an examine the properties of the optimum. Figure 2 illustrates how the interior solution an be determined as the intersetion of FOC (15) and budget onstraint (14). As Figure (2) shows, FOC (15) has an inverse U-shape. This indiates a speifi relationship between signal auray and the number of audits whih an be desribed as the following. By the analogy to utility theory, let us define the marginal rate of substitution of signal auray, a, for the number of audits, P, as the ratio of the marginal tax revenue from inreasing signal auray to the marginal tax revenue from inreasing the 20

21 number of audits, that is, MRS a, P = T R a T R P = R(P ) ar (P ). (16) The marginal rate of substitution of signal auray, a, for the number of audits, P, shows how many audits ould be saved through an inrease in signal auray keeping tax revenue the same. Based on analogy with utility theory for two normal goods, one might expet that the marginal rate of substitution would derease with signal auray and inrease with the number of audits. MRS a, P is indeed dereases with signal auray, a. However, MRS a, P first inreases with the number of audits, P, but then it dereases the number of audits. This auses a partiular substitution pattern between signal auray and the number of audits. This substitution pattern an be desribed by onsidering how many audits the tax authority an save through an inrease in signal auray provided that the tax revenue is the same. This number of saved audits initially inreases with the number of audits, but then dereases with the number of audits. Sine, the left hand side of FOC (15) is exatly equal to MRS a, P, the shape of the FOC urve in Figure 2 reflets this substitution pattern. Intuition To build intuition behind this substitution part, reall that in this model signals failitates tax audits, beause they allows the tax authority to divide taxpayers into audit lasses and to ondut audits within eah audit lass separately. To illustrate how the tax revenue is inreased by dividing people into audits lasses, I onsider a simple example. It also illustrates how this additional revenue depends on the number of audits onduted. The example onsiders two ases when the tax authority onduts audits without obtaining any signals about taxpayers inome, and when the tax authority does obtain signals allowing to divide taxpayers into audit lasses and ondut audits within eah audit lass. To ompare these two ases, I desribe how the audits are onduted and what the reported inome is. For simpliity, the example assumes that taxpayer s true inome is distributed uniformly on [ 1, 1]. The rest of the assumptions about taxpayer are the same. First, onsider a ase when the tax authority does not have any signals about taxpayers inome and therefore all taxpayers belong to one audit lass. Aording to Proposition 1, the audit rule in this ase is to audit those taxpayers whose reported inome is less than the audit trigger β, with probability 1, and not audit those whose reported inome is above the audit trigger β. If we denote the total number of audits, whih the tax authority an ondut by P, then the audit trigger β is equal to 2P 1. This is indiated in Figure 3 (a). The next panel of the figure Figure 3 (b) shows how reported inome depends on true inome. When true inome is less than the audit trigger β, taxpayers report inome equaled 21

22 Figure 3: The example illustrating the value of dividing taxpayer into audit lasses (a) (b) () (d) to true inome. When true inome is above the audit trigger β, reported inome is equal to the audit trigger β. Given reported inome, we an alulate the tax revenue in this ase. It is equal to T R 1 = t(1 P ) 2. This ase provides a baseline to whih the following ase is ompared. Consider the seond ase when the tax authority is able to obtain two signals suh that onditional on the first signal, s 1, inome is uniformly distributed on [ 1, 0] and onditional on the seond signal, s 2, inome is uniformly distributed on [0, 1]. The probability of eah signal is 1. These signals allow the tax authority to divide taxpayers into two audit lasses 2 and ondut audits within eah lass separately. Figure 3 () illustrates how the audits are alloated within eah audit lass. For an easy omparison with Figure 3 (a), Figure 3 () shows the joint distribution of inome and signal (i.e., f(i, s) = f(i s) 1 2 ). The optimal audit strategy in eah audit lass is alulated aording to Proposition 1. If a taxpayer s signal is s 1 then she is audited if her reported inome is less than the audit trigger β 1. If taxpayer s signals is s 2 then she is audited if her reported inome is less than 22

23 Figure 4: The example illustrating the value of dividing taxpayer into audit lasses: the ase of large number of audits (a) (b) () (d) the audit trigger β 2. If the total number of audits is P, then the audit trigger β 1 is equal to P 1 and the audit trigger β 2 is equal to P. This implies that P audits are onduted 2 in the first audit lass and P audits are onduted in the seond audit lass. The number 2 of audits onduted in eah lass is the same beause the marginal revenue of an additional audit should be the same in both audit lasses. 25 This audit strategy indues reported inome as shown in Figure 3 (d). Reported inome is equal to true inome when true inome i belongs to either [ 1, β 1 ) or [0, β 2 ), reported inome is onstant and equal to β 1 when i [β 1, 0], and reported inome is onstant and equal to β 2 when i [β 2, 1]. 26 As an be seen, reported inome in this ase differs from reported inome in the baseline ase shown in Figure 3 (b). When audits are based on signals, reported inome is substantially higher for i [0, 1] and lower for i [β 1, 0]. So, dividing taxpayer into audits 25 Reall that the ost of an audit,, is onstant for all taxpayers. 26 When true inome is equal to 0, reported inome ould be equal to β 1 or 0 depending on whether the signal is s 1 or s 2. 23

24 lasses helps to deter evasion by some high-inome taxpayers. The tax revenue in this ase is higher than tax revenue in baseline ase and is equal to T R 2 = t 2 (1 P )2. Interestingly, in the formula T R 2 = t 2 (1 P )2 the division of t(1 P ) 2 by 2 resembles the division of R(P ) by auray a in the formula for tax revenue T R(P ) = t(µ + 1 R(P )) in general model. Overall, the omparison of the ase when audits a are based on signals with the baseline ase shows that ability to divide taxpayers into audit lasses helps to inrease tax revenue beause audits an be redireted toward some highinome taxpayers. The additional tax revenue obtained by dividing taxpayers into audit lasses dereases with the number of audits P. To show this, Figure 4 presents exatly the same example as Figure 3 but when the number of audits P is bigger. As an be seen, there are more high-inome taxpayers who are audited in the baseline ase when the number of audits P is bigger. As a result, the additional revenue from dividing taxpayers into audit lasses is smaller. This also an be seen diret from the formula T R 2 T R 1 = t 2 (1 P )2. Indeed, T R 2 T R 1 dereases with P. What this example annot illustrate is how the revenue from inreasing the number of audit lasses hanges relative to the revenue from inreasing the number of audits. relative hange is important for understanding the substitution pattern, beause the substitution between signal auray and the number of audits is determined by the relative hange in revenue. By definition, the substitution pattern depends on how marginal revenue from inreasing signal auray hanges relative to the marginal tax revenue from inreasing the number of audits. For understanding the behavior of the relative hange in revenue, we have to rely on the general model. This The onsequenes of this substitution pattern are revealed in the next subsetion where I analyze how the optimal solution depends on the budget of the tax authority. 4.3 Comparative statis In this setion I analyze how the budget and osts affet the optimal signal auray and the optimal number of audits. I examine first an effet of a hange in the budget and seond an effet of a hange in the osts of audits and osts of improving signal auray. The finding of inverse U-shaped dependene of the optimal signal auray on the budget leads to important poliy impliations. 24

25 Figure 5: The effet of budget, B, hange on the optimal solution (P, a ) Budget Change The effet of the tax authority s budget hange an be analyzed using Figure 2. An inrease in the budget orresponds to a horizontal shift of the budget onstraint urve to the right. As the budget onstraint urve shifts to the right, the optimal solution moves along the FOC urve to the right. As a result, the optimal auray first inreases and then dereases with the budget, B, while the number of audits always inreases with the budget. The following proposition formally states this result. Proposition 3. Assume that assumptions i) and ii) of Proposition 2 are satisfied. There exist a threshold budget, B, suh that when the budget is small, B < B, the optimal signal auray, a, inreases with the budget, B. However, when the budget is large, B > B, the optimal signal auray, a, dereases with the budget. The optimal number of audits, P = P (a ), always inreases with the budget, B. 27 Proof. In the appendix. 27 To address a potential uriosity regarding whether the solution to the net revenue maximization problem lie to the left or to the right of the threshold budget, B, I provide the answer. It ould lie either to the left or to the right of the threshold budget, B. The reason for this is the following. The solution to the net revenue maximization problem the point on the FOC urve depited in Figure 2 for whih the Lagrange multiplier from Proposition 1 is equal to one. This point an be determined as an intersetion the FOC urve depited in Figure 5 with a = t() R (P ) whih is dereasing and onvex urve. Note that the slope of a = t() R (P ) depends on the value of t, π, and. Therefore, the intersetion an our to the left or to right of the pik of the ar. 25