Nonprofit organizations are a critical part of society as well as a growing sector of the economy. For funders

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1 MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Vol. 13, No. 4, Fall 2011, pp ssn essn INFORMS Effcent Fundng: Audtng n the Nonproft Sector Natale Prvett, Feryal Erhun Department of Management Scence and Engneerng, College of Engneerng, Stanford Unversty, Stanford, Calforna {nprvett@stanfordalumn.org, ferhun@stanford.edu} Nonproft organzatons are a crtcal part of socety as well as a growng sector of the economy. For funders there s an ncreasng and pressng need to ensure that socety reaps the most socal beneft for ther money whle also developng the nonproft sector as a whole. By routnely scrutnzng nonproft reports n an effort to deduce whether a nonproft organzaton s effcent, funders may beleve that they are, n fact, gvng responsbly. However, we fnd that these nonproft reports are unrelable, supportng a myrad of emprcal research and revealng that report-based fundng methods do not facltate effcent allocaton of funds. In response, we develop audt contracts that put funders n a poston to enact change. Audtng, perhaps obvously, supports funders; however, we fnd that t also benefts the populaton of nonprofts. Moreover, audtng results n mproved effcency for the nonproft sector overall. Indeed, our conclusons ndcate that nonprofts may want to work wth funders to ncrease the use of audtng, consequently ncreasng effcency for the sector overall and mpactng socety as a whole. Key words: publc sector; nonproft sector; audt contracts; adverse selecton; prncpal agent framework; resource allocaton; operatonal transparency Hstory: Receved: May 11, 2009; accepted: Aprl 3, Publshed onlne n Artcles n Advance September 2, Introducton At the nterface of operatons management and publc sector applcaton les a wealth of untapped research possbltes. Nonproft operatons n partcular have not been tradtonally consdered wthn the realm of operatons management research; however, recent trends n nonproft organzatons call for more substantal nvestgaton (Brest and Harvey 2008, Bradley et al. 2003). In lght of these dscussons, ths paper seeks to examne the relatonshp between a funder and nonprofts as contractual n nature to address the followng key ssues: 1. Do report-based fundng methods facltate the effcent allocaton of funds? 2. How can audt contracts be mplemented n the sector? What effects do these contracts have on both the funder and nonprofts? 3. Can audtng be used to mprove the performance of the nonproft sector as a whole? Nonproft organzatons are a sgnfcant and growng segment of the economy n addton to makng crtcal socetal contrbutons. Nonprofts provde servces to those who otherwse would not receve them, n many cases partnerng wth the government. They provde the opportunty for ndvduals to volunteer and gve back to socety, whch has become a promnent part of many cultures. Nonprofts also provde a large amount of publc goods that no one drectly pays for but from whch everyone reaps benefts, such as advocatng for cleaner ar. Beyond these socal contrbutons, the nonproft sector s economc mpact should not be underestmated. In 2009, the Internal Revenue Servce (IRS) regstered approxmately 1.5 mllon nonproft organzatons, whch account for 10% of employment n the Unted States (Sherlock and Gravelle 2009). There are many enttes that sustan the nonproft sector fnancally. Foundatons, themselves nonproft organzatons, and governments both act as key grantmakng nsttutons. In 2007, the Bll and Melnda Gates Foundaton alone pad approxmately two bllon dollars n grants, fundng nonprofts workng n global health, poverty, and educaton; and the Natonal Scence Foundaton, a Unted States government entty, dstrbuted bllons of dollars to scentfc research and educatonal actvtes. Worldwde, foundatons have contnued to grow and expand, most dramatcally n the Unted States where more than 72,000 grant-makng foundatons granted a record $42.9 bllon n 2007 (Foundaton Center 2008, Prewtt 2006). Because funders want assurance that ther donatons are used wsely, effcency s a crtcal measure. In the nonproft sector, effcency s about gettng [more] mleage out of the money [nonprofts] spend (Herzlnger 1996, p. 98). The most common cost-centered operatonal defnton of effcency measures the rato of expenses drectly forwardng the msson to total expenses, numbers that can be drawn from the organzaton s publcly avalable IRS Form 990. Ths effcency communcates a relatve 471

2 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector 472 Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS fnancal condton, hghlghtng smlartes n fnancal goals whle controllng for dfferences over tme and across nonproft organzatons (Chabotar 1989), thus enhancng understandng of these organzatons and helpng to create nformed decsons about fnancal support. Fundng methods, partcularly grants, are n essence contracts exchangng money for some socetal return where nonproft organzatons act as sellers of certan socetal actvtes or delverables (Brest and Harvey 2008). To allocate grants, many funders scrutnze fnancal statements and publc reportng forms, such as the IRS 990, n efforts to deduce whether a nonproft organzaton s operatonally effcent (Frumkn and Km 2001). In dong so, funders may beleve that they are, n fact, funnelng funds to the hghest-return organzatons. Ths research descrbes and examnes ths basc fundng stuaton, whch s based merely on the nonproft s reported cost-centered operatonal effcency and the funder s allocaton. However, under these very basc terms, our analyss shows that these nonproft effcency reports are unrelable. Ths result of a lack of effcency relablty and ts causes are well documented (e.g., Qualty , Schwnn and Wllams 2001, Frumkn and Keatng 2003, Trussel 2003, Jones and Roberts 2006, Krshnan et al. 2006, Gordon 2007, Keatng et al. 2008), confrmng that our results accurately descrbe the basc stuaton where funders are unable to effectvely dstngush between effcent and neffcent organzatons. Regardless of the bass for the unrelablty, ths theoretcal result leads to the concluson that a smple contract based solely on funds allocated n response to an effcency report s not effectve. Thus, n response to our frst research queston, we show that report-based fundng methods, n fact, do not facltate effcent allocaton of funds. Although the current reports of effcences are unrelable, a 2008 study on the nonproft marketplace confrms that effcences ndeed are verfable (Wllam and Flora Hewlett Foundaton and McKnsey & Company 2008). Thus, n response to the second queston, we develop audt contracts that can be mplemented wthn the funder nonproft relatonshp. In these contracts, the funder has the opton to audt the nonprofts effcences after awardng them funds. In case there s a dscrepancy between the effcency that the nonproft reports and ts true effcency observed after audtng, the funder may ask the nonproft to pay a penalty back. We fnd that usng audt contracts, the funder can guarantee truthful effcency reports from the nonprofts and thus attan operatonal transparency. The nonproft populaton also prefers these audt contracts. As a result, we may see funders and nonprofts workng together to ncrease audtng wthn the sector and mprove ther respectve stuatons. Ths can be seen at work n both the nonproft sector push to ncrease voluntary self-audtng and legslatve efforts requrng audts, such as the Calforna Nonproft Integrty Act of 2004 (Calforna Government Code 2007). The mplementaton of audt contracts can also mprove the performance of the nonproft sector overall. More specfcally, the funder s choce of penalty postons the sector anywhere between the current, neffcent stuaton and the transparent, effcent stuaton. Ths mples that through mplementaton of audt contracts wth approprately chosen penaltes, benchmark performance can be acheved for the nonproft sector as a whole. Thus, the use of audtng may put funders n a poston to enact change n the nonproft sector and ncrease overall sector effcency. The remander of ths paper s organzed as follows: Secton 2 revews the related lterature. Secton 3 descrbes the general model framework. Sectons 4 and 5 develop the contracts, where ncentve and relablty ssues are explored n theoretcal detal. Secton 6 nvestgates the effects of these dfferent contracts through both theory and numercal examples. In 7, results are extended to the cases of constraned budget, uncertanty n producton, and cost of audtng. Fnally, 8 concludes by revewng ths work n lght of the orgnal research questons presented n the ntroducton. For expostonal smplcty, all proofs are presented n an onlne addendum (avalable at Throughout the paper, the operator E x s used as the expectaton over random varable x and vectors are represented wth boldface. 2. Lterature Revew Wth the nonproft sector expandng n sze and nfluence, t may be surprsng that there s relatvely lttle research, especally analytcal research, nvolvng nonproft operatons Operatons Management n the Nonproft Sector McCardle et al. (2009) use a utlty-based donor model to analyze the behavor of nonproft donatons n the presence of publczed tered fundrasng structures. Ther model can be used by organzatons to make effectve decsons regardng the mplementaton of tered fundrasng structures, whch they show to generate larger donatons. Smlarly, Toyasak and Wakolbnger (2010) study whether an ad agency should earmark ts donatons whle rasng funds for an emergency. The authors conclude that earmarkng leads to a lower overall fundrasng cost percentage and should be used f fundrasng costs and goals are hgh and donors wllngness to donate to a specal

3 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS 473 fund s low. De Vércourt and Lobo (2009) nvestgate the revenue management problem encountered by nonprofts engaged n for-proft ventures as a means to fund ther msson, more specfcally how to allocate funds among nvestment, servng revenue customers, and servng msson customers. Harrson and Lybecker (2005) explore such prce (Bertrand) competton between nonproft and for-proft hosptals, and show that the motvaton of nonprofts has great mportance and mpact on ths competton. In these heterogeneous markets where for-profts and nonprofts compete, contenton can arse over supposed nonproft regulatory and tax advantages. Lu and Wenberg (2004) analyze these supposed advantages and conclude that t s not these advantages, but rather the dfference n the objectve functons of the frms that cause the compettve behavor we observe. Verheyen (1998) examnes the allocaton of nonproft nternal budgets, specfcally unversty and hosptal budgets. The author dscusses these allocaton ssues n the context of several dfferent prncpal agent models, but does not develop or solve such models. Our research enlarges ths body of lterature by applyng the prncpal agent framework to the funder nonproft relatonshp as well as examnng the role of nformaton, ncentves, and audtng, all of whch have been dentfed as crtcal research areas (Zarc and Brandeau 2007, Laffont 1994) yet are unexplored n nonproft operatons Nonproft Evaluaton There s a large, multdscplnary body of lterature on nonproft evaluaton, whch provdes many dfferent technques for measurng outputs (.e., more easly measured, tangble returns) and outcomes (.e., less clearly measured, less tangble returns). For example, Stufflebeam (2001) dentfes, descrbes, and assesses 22 dfferent nonproft evaluaton approaches; and Martn and Kettner (1996) put such performance measurement nto perspectve by delvng nto the nuances of ts practce. Amed toward the nonproft practce audence, Newcomer (1997) gves an overvew of both desgn and use of evaluaton and performance measures. Herzlnger (1996) calls for ncreased accountablty to restore publc trust, partcularly through fnancal evaluaton. She detals four questons to perform fnancal assessment, whch are mplemented n The Boston Foundaton s assessment of and call to acton for the fscal health of Massachusetts nonprofts (Keatng et al. 2008). As a more holstc fnancal evaluaton framework, socal cost-beneft analyss values projects, programs, etc. as the net socal beneft mnus the net socal cost where all mpacts must be monetzed; Boardman et al. (2006) and Levn (1983) both provde revews. Our research draws on these methods n developng the funder s perspectve of nonproft contrbuton. Beyond ths, our modelng of the funder s utlty, partcularly the portons derved from the nonprofts funded, extends the theory and understandng of how these methods can be ncorporated nto strategc decson makng and theoretcal modelng Capacty and Resource Allocaton Allocaton s wdely studed n both operatons management and economcs lterature. More recently, researchers have tackled nformaton problems n capacty and resource allocaton, specfcally the neffcences created because of ncentve msalgnments (e.g., Harrs et al. 1982; Cachon and Larvere 1999a, b; Rajan and Rechelsten 2004; Karabuk and Wu 2005), whch are most n lne wth the goals of ths paper. Cachon and Larvere (1999a, b) study nterfrm capacty allocaton problems where a sngle suppler s allocatng capacty among retalers wth prvate demand nformaton. Karabuk and Wu (2005), Rajan and Rechelsten (2004), and Harrs et al. (1982), on the other hand, examne ntrafrm capacty and resource allocaton problems. Karabuk and Wu (2005) study ncentve algnment for a sngle decson maker allocatng capacty to managers wth prvate demand nformaton usng bonus payments and partcpaton charges. Rajan and Rechelsten (2004) and Harrs et al. (1982) model a common resource beng allocated among dvsons where the output of each dvson s an ncreasng functon of allocaton along wth productvty and manageral effort, both of whch are the dvsonal managers prvate nformaton. Transfer prcng mechansms are found optmal. Our fundng model s certanly n lne wth these streams of lterature; t presents an allocaton of funds among horzontal agents wth prvate cost nformaton n the form of an effcency rato. In addton, our decson maker has outsde opportuntes and we seek to use audtng mechansms to resolve asymmetrc nformaton. Because of the nature of the utltes n our model, our objectve functon ncorporates the funder s as well as part of the nonprofts objectves makng the soluton more socally optmal. Therefore, we are examnng a problem wth both shared and conflctng objectves between the prncpal and agent(s) Incentves In seekng to algn ncentves and resolve problems due to asymmetrc nformaton, many researchers rely on optmal contracts (.e., contracts that nclude contngences for each possble value of asymmetrc nformaton). Such contracts may be arbtrarly complex, makng them dffcult to wrte, mplement, and admnster n practce. Recently, we observed a trend n operatons management lterature where smple (n terms of the number of contngences ncluded)

4 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector 474 Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS yet effcent contracts are studed as an alternatve to complex optmal contracts (e.g., Larvere and Porteus 2001, Kayış et al. 2011, Kalkancı et al. 2011). In a smlar sprt, we also do not work wth complex optmal contracts, but develop smple, mplementable audt contracts that can be appled gven the current state of the nonproft sector Audtng A wealth of economc lterature exsts around audtng, partcularly n relevng ncentve ssues; we focus on a few representatve peces here. Baron and Besanko (1984) explore the use of audtng n the context of regulaton of frms, specfcally n cost reportng, where the regulator can order consumer refunds n the case of overstated costs. The authors explore audtng as a means to structure frm ncentves for truthful cost reportng under asymmetrc nformaton. Usng a related model, Laffont and Trole (1986) analyze ncentve contracts, both regulatory and procurement, wth the addton of a moral hazard component, whch prevents the regulator from rewardng hgh costs. Our scenaro, however, s not regulatory n nature. Though we study audtng n the sprt of ths lterature, we do not dscuss the detals of performng such audts. Tools, methods, and examples of performng audts n the nonproft sector can be found n Brdge et al. (2009) and Carman (2009). Our funder s seekng her own objectves, whch, n fact, are both conflctng and shared wth the nonprofts. These shared objectves between the funder and nonprofts complcate the stuaton and produce unque results. We also do not merely look at optmal contracts that acheve frst-best performance as we examne the funder s perspectve, the nonproft perspectve, and the sector effcency perspectve. Instead, we devse alternatve contracts to mprove effcency, as well as the stuaton for all partes and the sector. 3. Model Defnton The relatonshp between nonprofts and the funders who support them s not trval. The funder supples resources to nonprofts n the form of grant allocatons wth the expectaton that these nonprofts produce returns n the form of socal output or outcome. An example of output mght be the number of shelters dstrbuted followng a natural dsaster by WorldVson, a nonproft humantaran organzaton. An example of outcome mght be ncreased apprecaton and educaton regardng the envronment by the Serra Club, a nonproft envronmental organzaton. However, as dscussed n 2, there s a growng body of lterature on nonproft evaluaton, whch provdes many dfferent technques for measurng program outputs and outcomes. Consderng ths lterature and our stylzed model framework, we treat output and outcome equvalently. Accordngly, n the remander of the paper, we use the term output to nclude both output and outcome. Although both the funder and the nonprofts am to maxmze ths socal output, they each have other, possbly conflctng, components of ther respectve objectves. More specfcally, the funder looks to maxmze the value for her dollar whereas nonprofts look to maxmze the dollars they receve (.e., ther allocaton). Each nonproft has an unobservable effcency type, whch descrbes how well the nonproft converts allocatons nto outputs/outcomes furtherng ther msson. In addton, nonprofts can also exercse effort to further ther msson; ths effort s also unobservable. These unobservables are n tenson wth the funder s objectves of gvng responsbly and fulfllng a duty to the sector. Note that effcency s characterzed as a type as opposed to a decson varable for the nonproft because although nonprofts can ncrease ther effcences, these changes don t just happen (Neuhoff and Searle 2008, p. 33). Instead these changes reflect a decson that entals a process much longer than the fundng process; thus, they are outsde the scope of ths paper. The scenaro that we analyze conssts of one funder and N 1 nonprofts. All partes are expected utlty maxmzers. The funder does not know the true effcency types of the nonprofts wth certanty, but knows that each true effcency type, 0 1, s dstrbuted wth probablty densty functon f and cumulatve dstrbuton functon F, = 1 N. Note that although the support of the effcency dstrbuton s 0 1, an effcency type of one does not necessarly mean that the nonproft s channelng all of ts funds to msson crtcal actvtes. Some funds may be used for actvtes such as fundrasng and overhead, however, any upper bound can be normalzed to one to accommodate other lmts. The effcency defnton adopted here descrbes only part of how effectvely the nonprofts accomplsh ther msson; as such, t s a smplfcaton of a complex concept. Nonproft organzatonal effectveness s multdmensonal and, therefore, cannot be encapsulated n a sngle measure (Herman and Renz 1999). Yet ths rato s the most common costcentered operatonal defnton of effcency n the lterature and s also commonly employed by audtors, accredtors, meda, and charty oversght analysts to compare the operatons of organzatons wth smlar mssons, wth the goal of determnng whch organzatons have the leanest operatons (Frumkn and Km 2001, p. 270); for example, see Chabotar (1989), Herzlnger (1996), Hager and Flack (2004), Krshnan et al. (2006), and Gordon et al. (2007). For many analysts, ths defnton translates nto accountablty

5 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS 475 by provdng nsght nto management and admnstraton spendng (Hager and Flack 2004), producton costs, and translaton of the supply-chan nto fnancal values through recordng and analyzng the costs assocated wth products/actvtes undertaken (Bagnol and Megal 2011, p. 150). The fundng sequence of events for a gven contract s as follows: () The funder announces some allocaton scheme, possbly based on types, A, for each nonproft. () Nonproft announces an effcency, ˆ, perhaps through an applcaton process. These effcency types are also typcally avalable through IRS data submtted by the nonprofts, such as the IRS 990 Form. Ths announcement may or may not be true. () The funder allocates resources wthn the contractual framework to maxmze her utlty consderng the nonprofts announcements and any constrant on her budget. (v) Each nonproft chooses an effort to maxmze hs utlty based on ths allocaton. Nonproft s utlty, u, and producton (output), y, are determned by u = A e2 2 + y (nonproft utlty) (1) y = 2 e A (nonproft output producton) (2) where 0 s a multpler descrbng how much the nonproft values allocaton as compared to output, A s the allocaton made from the funder to nonproft, e s nonproft s effort, and s the true effcency type of nonproft. The nonproft s utlty n Equaton (1) s ncreasng n the allocaton he receves from the funder and n hs output and s decreasng n the effort he exerts. For ther survval, nonprofts depend on funds; therefore, the nonproft s utlty s ncreasng n the allocaton he receves. The nonproft gans utlty n output because output supports the nonproft s msson. Nonprofts may value allocaton more than, less than, or the same as output; both output and allocaton contrbute to the nonproft s utlty. Ths comparatve value s descrbed by the multpler. Although the nonproft would lke to support hs msson, he prefers to do ths wth the least effort. Ths may be so that the effort, such as volunteer tme, can be spread over other programs. The producton functon n Equaton (2) ndcates that the nonproft must have postve effort, effcency, and allocaton to produce output. Furthermore, the nonproft producton functon s concave n effort, e, effcency type,, and allocaton, A, sgnfyng dmnshng returns and boundedness of producton, smlar to related models n the lterature (e.g., Cachon 2003). We ntally assume that output can be measured unambguously; we relax ths assumpton and study producton uncertanty n 7.2. The funder s expected utlty, U f, s U f = N =1 ( E c y + B N =1 ) E A (3) where c > 0 s the funder s monetary valuaton of each unt of nonproft s output, s the return on an outsde opportunty, and B s the funder s budget. The frst term of the funder s utlty, N =1 E c y, s the total expected value from the nonprofts output. The funder would lke to maxmze each nonproft s output, y, and each output s worth some amount c. The c term s ndvdualzed to each nonproft and may ncorporate the extent to whch the work of nonproft supports the msson of the funder. The funder only observes the output, y, and effcency report, ˆ, of each nonproft, but does not observe the nonproft s effort choce, e, nor true effcency,. The second term of the funder s utlty, B N =1 E A, represents an outsde opportunty. Although the funds allocated may ncrease the nonprofts output, these funds are also valued by the funder consderng that they could be used for other opportuntes. In fact, the funder may decde that the outsde opportunty s worth more than the (socal) return of nvestng that money nto a partcular nonproft. For example, de Vércourt and Lobo (2009) nvestgate usng ths money n a forproft venture or nvestng t n order to supplement the ncome stream. Note that the expectaton of A over s because the funder s allocaton scheme may depend on nonproft types as outlned n the sequence of events. Unlke a for-proft corporaton that s presumed to dstll the objectves of ther shareholders, managers, employees, and clents nto a sngle measure of shareholder value, the nonproft has no sngle prmary nterest group that s nvarably and clearly defned, homogenous wth respect to nterests, and whose goals are easly expressble and transferable nto the organzaton for assessment of alternatve courses of acton (Speckbacher 2003, p. 268). Because of ths vared abundance of stakeholders and purposes, quantfyng and modelng the objectves of a nonproft organzaton can be ambguous and contentous. Our formulatons are thus some frst attempts to model the fundng dynamcs between funders and nonprofts and were developed through an understandng of the feld, whch we ganed through personal communcaton (Foundaton Center 2009, Phlanthropy and Cvl Socety Research Workshop 2010, Meredth 2009), feld reports, and academc and trade publcatons. Furthermore, our modelng assumptons, such as funder utlty ncreasng n output and effort (Equaton (3)) and effcency modeled as asymmetrc nformaton, are consstent wth relevant economcs lterature (e.g., Easley and

6 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector 476 Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS O Hara 1983). Smlar to related models n the lterature (e.g., Cachon 2003), we express effort cost as a convex functon (Equaton (1)) because t s progressvely more dffcult to ncrease the effort, whch s lmted by capacty. Based on our sequence of events, the nonproft problem s solved frst followed by that of the funder by backward nducton. The nonproft maxmzes hs utlty based on the gven allocaton: { } max u = max A e2 e 0 e y where y = 2 e A Substtutng the producton functon, we obtan u = A e 2/2 + 2 e A. The effort, e, that maxmzes the nonproft s utlty can easly be found as e = A 1/3. Through substtuton of e, reformulatons of the nonproft s utlty and output functons as well as the funder s utlty functon are obtaned: u = A A 2/3 (nonproft utlty) (4) y = 2 A 2/3 (nonproft output) (5) U f = N E 2c A 2/3 A + B =1 (funder utlty) (6) In ths paper, we do not work wth complex contracts n the tradtonal sense of prncpal agent models. The dea here s to understand the stuaton and develop contracts that can be applcable gven the current operatonal polces, practces, and trends of the nonproft sector. Ths prncpal agent model s dfferent from others n that t ncorporates both shared and conflctng portons of the respectve objectves of the prncpal and the agents as well as unobservable characterstcs. The utlty and objectve functons here n the nonproft settng dffer from those more famlar functons of the for-proft world. Our nonproft objectve functon ncorporates several perspectves: allocaton from a manageral fnancal sustanablty perspectve, effort from an employee perspectve, and producton from the clent and funder perspectve. Ths heterogenety of nterests and objectves, partly shared wth the prncpal or funder, s ndeed a new avenue of exploraton for both nonproft research and suppler manufacturer type game theoretc models. However, t s these dfferences that lead to unque results. In the model analyss and remander of the paper, we frst assume an unconstraned budget n 4 6. That s, we assume N =1 A < B so that we omt the budget terms n the above formulatons. In 7.1, we explore and extend our results to the case of budget constrant,.e., N =1 A = B. In the followng secton, we begn nvestgatng the funder nonproft relatonshp n order to answer our orgnally posed research questons. We start by descrbng the basc fundng stuaton wth the report-based contract as well as establsh a benchmark, frst-best contract for comparson purposes. These wll motvate us to study audt contracts n Report-Based Contract The funder would lke to make the most of her money and, therefore, would lke to allocate funds based on the effcency types of nonprofts,.e., A. Gven Equatons (4) and (6) and usng the revelaton prncple (Laffont and Martmort 2002) as s common n such models, the funder s problem can be formulated as follows: max U f = max A A subject to { N =1 } E 2c A 2/3 A (7) u (IR) (8) u u ˆ ˆ 0 1 (IC) (9) A (nonnegatvty) (10) where s nonproft s true effcency type, ˆ s nonproft s announcement of effcency type, and u ˆ s nonproft s utlty when hs true effcency type s but hs announced effcency type s ˆ ; u ˆ = A ˆ + 3/2 A ˆ 2/3. The ndvdual ratonalty (IR) constrants ensure nonnegatve utlty for each nonproft wthn the contract. The ncentve compatblty (IC) constrants constran each nonproft to truthfully report ther effcency type by use of a utlty-based ncentve. Consequently, the funder s problem s expressed as an adverse selecton problem where the funder cannot observe the amount of effort, e, the nonproft wll exert nor the nonproft s effcency type,. Note that n ths formulaton the funder allocates funds n response to the nonprofts reported effcences. Thus, the resultng report-based contract descrbes the stuaton where the nonprofts effcency types are not verfed: Theorem 1. The report-based contract takes the form A R = A R = 64/27 3 E 2/3 3, where = c / and = 1 N. Ths soluton ndcates an ntutve relaton: In a scenaro where the true effcency type s unknown, the funder would want to allocate more funds to the nonproft that s more lkely to be a hgher effcency type. The result also ndcates that the funder s unable to dfferentate among the nonprofts effcency types as she bases her allocaton solely on her expectaton

7 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS 477 of each nonproft s type, the only relable nformaton she possesses. The nonprofts do not have an ncentve to truthfully report ther type, and, therefore, ther reports are unrelable. The report-based contract, whch s based merely on reportng wthout verfcaton, consequently offers the funder no observablty of effcency types and no operatonal transparency wthn ths contractng context. Theorem 1 s ntutve n yet another way: Nonproft wll receve more fundng as the funder values hs output more (.e., as c ncreases), but wll receve less fundng as the outsde opportunty becomes more lucratve (.e., as ncreases). In other words, when the outsde opportunty s more valuable compared to nonproft output, the funder wll choose to allocate less funds to the nonproft n order to gan from the outsde opportunty. The report-based contract succeeds n modelng the basc fundng stuaton where there s no relablty n effcency reports and no transparency n nonproft operatons. Bradley et al. (2003) fnd nonprofts msreportng ther effcences n stuatons where potental funders are nvestgatng ths measure to make decsons about gvng and fund allocaton. Msreports of effcency may be due to manageral motvaton, executve compensaton ncentves, use of effcency n fundng decsons, charty ratngs, meda attenton, or reputatonal pressures (Frumkn and Keatng 2003, Trussel 2003, Jones and Roberts 2006, Krshnan et al. 2006, Keatng et al. 2008). A myrad of other lterature related to unrelable IRS data and expense shftng also supports our theoretcal result of a lack of observablty and relablty of effcency types n the current fundng stuaton. The IRS Form 990 s the chef source of the fnancal data from whch the effcency measure s derved. However, the 990 s typcally not verfed. In fact, Schwnn and Wllams (2001) assert that the IRS only examned 1.3% of 990s n Furthermore, the IRS Form 990 allows nonprofts to record certan admnstratve and fundrasng costs as offsets to revenue rather than as expenses. As a result, nonprofts may portray ther operatons as more effcent n the Form 990 (Frumkn and Keatng 2003). Jones and Roberts (2006, p. 161) report that chartes do ndeed employ expense shftng to offset 16% to 28% of potental changes n program rato [effcency]. Herzlnger (1996) clams that ths nonexstence of accountablty perpetuates neffectve and neffcent organzatons as well as other problems. Froelch and Knoepfle (1996, p. 49) summarze that overall, the nonproft stuaton [s] characterzed by extensve reportng, but very weak montorng. Ths underlnes the lack of relablty and, consequently, the need to make the relatonshp between the funder and nonprofts more relable through stronger montorng. More than a decade later, the stuaton has not changed consderably. Even today, by and large, funders have nfluence, but not control [over nonprofts] (Meredth 2009). Therefore, wth our next set of contracts detaled n 5, we wll consder montorng by funders as a way to ncrease accountablty n the nonproft sector. Before movng nto the next secton, we analyze a specal case of the report-based contract, whch s a benchmark model where effcency types,, are observable. Although ths perfect observaton of effcency types s unrealstc, t s for ths reason the funder s frst-best soluton as t elmnates the neffectveness due to the nformaton asymmetry and allows for complete transparency. Corollary 1. The frst-best contract takes the form where = c /. A FB = (11) We note that the funder allocates to all nonprofts wth a nonzero effcency ( > 0). Each nonproft s allocaton s based on hs effcency type; f the nonproft s of a very low effcency type, hs fundng s neglgble. Ths stuaton of allocatng money to every nonproft may be unrealstc as allocatng a few dollars would certanly not be worth the admnstratve costs. However, the funder may prescreen nonproft applcatons, assgnng c = 0 to nonprofts wth too low of an effcency report. Also the funder may want to have a mnmum fundng level where she only funds nonprofts whose allocatons surpass ths level, thus excludng some nonprofts from consderaton. 5. Audt-Based Contracts Whereas the frst-best contract supposes the funder can observe all nonproft effcency types (complete transparency), the report-based contract results n the funder observng none of the nonproft effcency types (no transparency). Now we turn our attenton to nvestgatng contracts that wll enable the funder to poston herself anywhere between and ncludng both the frst-best and report-based contracts n terms of both observablty of nonproft effcency types as well as contract performance. In ths settng, the funder allocates her funds to the nonprofts based on ther announced effcency type. However, we know from the analyss of the report-based contract n 4 that the funder wll not be able to dfferentate;.e., a nonproft s announcement of effcency type s not relable. In ths audtng scenaro, the funder can verfy the nonproft s effcency type by audtng after allocatng funds and then mpose a penalty upon nonprofts who msreport. Note that dealng wth ex post audtng s dfferent from dealng wth ex ante observable effcency types because

8 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector 478 Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS the funder must provde an ncentve (here a penalty) to the nonprofts to observe ther effcency wthn the audtng framework. The funder s objectve does not change from that formulated n Equaton (7) n 4. In partcular, an audtng cost s not ncluded n the funder s objectve functon because there are several stuatons where ths would not be approprate. For example, nonprofts may choose to self-audt usng a credble thrd-party audtor. Also, funders may not have specfc audtng costs or may have audtng costs and budgets separate from the contracts themselves. However, n 7.3 we extend ths analyss to nclude audtng costs. The constrant set also contnues to nclude the same IR and nonnegatvty constrants found n Equatons (8) and (10) of our orgnal model. However, the IC constrant n Equaton (9) s updated for the audt contracts. It now ncorporates an mposed penalty, P 0, for msreports, and s as follows for all and for all ˆ, 0 1 : u u ˆ P A ˆ (audt IC). (12) The left-hand sde of the audt IC constrant s stll the nonproft utlty when reportng truthfully. However, the rght-hand sde s the nonproft utlty from msreportng mnus a penalty mposed on the nonproft s allocaton. We saw n 4 that wthout ths penalty modfcaton, the nonproft has no ncentve to report truthfully. Consequently, the goal of ths penalzaton for msreportng s to create an ncentve for the nonproft to report truthfully. The funder has two levers she can use to guarantee nonproft ncentve compatblty: she can adjust allocaton, A, or penalty, P. Accordngly, she wll offer contracts ncludng both allocaton and penalty, whch take the form P A. The best stuaton for the funder s to attan frst-best allocatons wth ncentve compatblty for all effcency types, whch can be acheved by the optmal penalty detaled n Equaton (13). The correspondng optmal contract, whch utlzes ths optmal penalty and frst-best allocatons to acheve frst-best performance and observablty, s also outlned n Theorem 2. Theorem 2. The optmal audt-based contract takes the form P, where AFB P = (13) enforces ncentve compatblty for all possble effcency types of nonproft usng frst-best allocatons, A FB, gven by Equaton (11). The optmal audt-based contract enforces ncentve compatblty by settng a hgh penalty (P > ) for all types and decreasng the rght-hand sde of Equaton (12). As the penalty P s greater than, each nonproft wll need to repay more than ther valuaton of what was orgnally granted to them n the event that they msreport. Ths arses because the nonproft not only values the allocaton but also the output he would get wth that allocaton; hence, he should be penalzed accordngly. However, ths penalty can also be thought of as, at least n part, a loss of future funds due to a loss of reputaton. The optmal penalty ncreases n. As the nonproft values allocaton more, the funder penalzes more severely to enforce ncentve compatblty. The optmal penalty ntutvely decreases n = c /. As the funder values nonproft output more, she penalzes less severely agan reflectng her ncreased regard for even the less-effcent nonproft output. We summarze these observatons n Corollary 2. Corollary 2. The followng are true for the ncentve compatblty enforcng penalty, P : () P > ; () P s ncreasng n ; and () P s decreasng n = c /. Although the optmal contract s penalty, P, enforces ncentve compatblty for all types and acheves benchmark performance, t may have undesrable, f not unrealstc, characterstcs. Under ths contract, the funder s requred to audt all types of nonprofts, whch may create mplementaton problems. Furthermore, the funder mposes a penalty greater than and thus requres the nonprofts to repay more than the value granted to them. Instead, the funder may want to specfy a less severe penalty, P P, to mplement and as a result may choose not to audt low types. Factors such as funder preference, polces, or audtng resources may dctate ths penalty choce. Another factor mght be audtng costs, whch are further explored n 7.3. Utlzng a suboptmal penalty, P P, requres the funder to exercse her other lever of allocaton to enforce ncentve compatblty. Because the lower effcency types have the greatest ncentve to msreport, the funder must now compensate these types to prevent such msreports and guarantee ncentve compatblty. The audt-based contract wth a specfed penalty P allows a more lberal choce of penalty, 0 < P P, and uses the lever of allocaton to enforce ncentve compatblty. We defne the effcency type P as a cutoff type for a gven penalty P for whch the ncentve compatblty constrant n Equaton (12) holds wth equalty when mplementng frst-best allocatons;.e., for P the ncentve compatblty constrant wll hold and for < P t wll not. The followng theorem shows that a cutoff effcency type, P, always exsts and outlnes the audtbased contract.

9 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS 479 Theorem 3. The audt-based contract wth a specfed penalty P takes the form P A FB P A A A P for < P = P A FB for P where gven a penalty 0 < P <, the cutoff effcency type of nonproft, P, s P 1 = 9 6 1/3 5/3 + 8 P /3 9 (14) wth = c /, = 9 + 8, = 4 P P , and AFB gven n Equaton (11). For P P, P does exst, but n a dfferent closed-form expresson than that of Equaton (14). Usng her lever of allocaton, the funder gves effcency types below the cutoff P a constant allocaton to nduce ncentve compatblty whereas hgh types receve frst-best allocaton. Because of ths constant allocaton, the funder loses some utlty and observablty n over-allocatng funds to nonprofts wth low effcency types. The extent of ths loss s dctated by P and ultmately the funder s choce of penalty P. The funder can recover utlty by explotng the lever of her penalty choce to poston herself. Specfcally, as she ncreases her penalty P (lowers P ), the funder ncreases the range of types she can observe as well as her utlty. However, when the penalty s poorly chosen and nsuffcently hgh, ths contract can result n negatve utlty for the funder, whch would ndcate that she would not use t under the partcular condtons. Also, credblty ssues may arse when the funder chooses a penalty greater than one as ths requres the nonproft to repay more than the actual amount of ther orgnal allocaton. Nonetheless, the audt-based contract wth a specfed penalty may stll be a preferred opton for funders as t does not over-penalze any nonprofts yet stll rewards the hgher effcency nonprofts wth frst-best allocatons. Ths preference and ts assocated condtons are further explored n 6. Followng from Theorem 3, we can make two observatons on the behavor of the cutoff type: Corollary 3. The cutoff type, P, s () decreasng n the penalty, P ; and () decreasng n = c /. The frst pont s ntutve: As the funder mposes a more harsh penalty, ths larger penalty enforces ncentve compatblty for more nonproft effcency types. Thus, fewer types need to be cut off. Because = c / measures the value the funder places on nonproft s output compared to her outsde opportunty, as the funder values nonproft s output more, she cuts off fewer nonproft effcency types reflectng her ncreased regard for even the less-effcent nonproft output. In summary, the optmal audt-based contract employs a strct but optmal penalty, n fact penalzng all types to repay amounts beyond ther orgnal grant, whch enables frst-best allocatons for all types. The audt-based contract wth a specfed penalty, on the other hand, allows the funder to specfy a more lenent penalty. However, the funder pays for her lenency wth a lmted loss of observablty and performance. The lenent penalty causes only effcency types above a cutoff, P, to be ncentve compatble. Thus, types below P gan rent as they are allocated ncreasngly neffcent allocatons the further they are below P. The choce of penalty does allow the funder to at least poston herself n terms of observablty and utlty wth a hgher penalty movng her closer to the frst-best stuaton by decreasng the cutoff P. 6. Effect of Contract Type on Performance Usng our theoretcal results wth numercal examples as llustraton, we nvestgate the effect of each contract on the funder, the nonprofts, and the nonproft sector overall. All results are n expectaton, and thus, the results labeled as nonproft are ndeed the results for the populaton of nonprofts. We focus on the audt-based contract wth a specfed penalty (Theorem 3), because the optmal audt-based contract always acheves frst-best. We use bold captal letters for vectors;.e., P = P 1 P N and P = 1 P 1 N P N. We use superscrpts FB, R, and A to denote the frst-best contract, the report-based contract, and the audt-based contract, respectvely Funder Expected Utltes Examnng the funder s expected utlty from Equaton (7) reveals the followng conclusons: Proposton 1. Regardng contract effects on the funder s expected utlty, U f : () U FB f max U R f U A f ; () there exsts a threshold such that U A f U R f for P 0 and U R f U A f for P 1 ; and () U A f s decreasng n the cutoff type, P, and ncreasng n the penalty, P. These conclusons are llustrated n Fgure 1, whch shows the percentage of expected frst-best (benchmark) utlty captured by each contract and s plotted over c /, whch descrbes how the funder values the nonproft output over the outsde opportunty. In ths fgure, nonprofts are a symmetrc populaton wth U 0 1, = 1, c = c, and P = P for all. Note

10 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector 480 Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS Fgure Frst-best captured (%) Percentage of Expected Frst-Best Captured (a) Funder Frst-best P = 1 P = 0.75 Report-based P = 0.5 thus expected utlty under the audt-based contract are ncreasng n the penalty, P. Fnally, we observe that for reasonably large values of c /, the optmal penalty (Equaton (13) n Theorem 2) s almost equal to. Though not llustrated here, Corollary 2() analytcally shows that the optmal penalty s ncreasng n ;.e., as the nonprofts value ther allocaton more, the funder must penalze more to mantan ncentve compatblty Nonproft Populaton Expected Utltes Lkewse, examnng the nonproft populaton s expected utlty derved from Equaton (4), Frst-best captured (%) c/ 50 (b) Nonprofts P = 0.5 P = 0.75 P = 1 Frst-best Report-based c/ Note. The audt-based contract s ndcated by dashed lnes and the sold lnes show the report-based contract and frst-best. that the results reman unchanged for heterogeneous nonprofts and varyng type spaces. In Fgure 1(a) two regons can be dentfed for the funder. Although the frst-best contract s always superor to the report-based contract and audt-based contract (Proposton 1()), the funder s preference between the report-based contract and the audtbased contract depends on the value of c / and the choce of penalty (Proposton 1()). When the penalty s too low (P = 0 5 n Fgure 1), the funder s overly lenent on low effcency types and she suffers from her lenency; ndependent of the value of c /, the report-based contract performs better than the audtbased contract. However, for hgher values of penalty, as long as c / s not too low, the funder prefers the audt-based contract to the report-based contract. For low values of c /, the funder s beneft from fundng nonprofts s low compared to the return on the outsde opton; therefore, even small neffcences n fundng (.e., lenency toward the low effcency types) make the funder relatvely worse off. We also note that, consstent wth Proposton 1(), the funder s percentage of expected frst-best captured and u N = N [ E A + 3 A 2 2/3] =1 reveals the followng conclusons: (nonproft populaton utlty) Proposton 2. Regardng contract effects on the nonproft populaton s expected utlty, u N : () u A N ufb N ur N ; and () u A N s ncreasng n the cutoff type, P, and decreasng n the penalty, P. The audt-based contract gves a larger, constant allocaton to the low effcency types (.e., types below the cutoff P ) whle also gvng frst-best allocatons to hgh effcency types (.e., types above the cutoff P ) as shown by Theorem 3. That s, compared to the frst-best, the nonprofts are awarded hgher allocatons by the audt-based contract, whch s why the percentage of frst-best captured utltes greater than 100% can be observed wth the audtbased contract n Fgure 1(b). Consstent wth Proposton 2(), a very ntutve nonproft preference for the audt-based contract s therefore clear. However, ths preference s decreasng as the penalty ncreases (Proposton 2()); that s, as the cutoff P decreases and more types are forced to report truthfully and are awarded frst-best allocaton nstead of the larger, constant allocaton. An mportant concluson drawn here s that hgh effcency type nonprofts clearly prefer the audtbased contract over the report-based contract because they receve effcent allocatons. In fact, these hgher effcency type nonprofts may want to work wth funders on cultural changes that wll ncrease audtng and thus ncrease utlty for themselves. Ths can be seen at work n the nonproft sector push to ncrease voluntary self-audtng and legslatve efforts. Consderng the government as a promnent funder, legslaton amng to elevate the standards of accountablty and audtng are certanly evdence of funders seekng to create audt-based contract stuatons. An example of such legslaton s the Calforna Nonproft

11 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS 481 Fgure 2 65 Expected sector effcency (%) Expected Sector Effcency: Nonprofts Are a Symmetrc Populaton wth U 0 1, = 1, c = c, and P = P for All Audt-based Frst-best Report-based audt-based contract. In partcular, hgh effcency type nonprofts are clearly better off by the audtbased contract because they receve effcent allocatons. They may want to work wth funders on nonproft sector cultural change to ncrease audtng, thus ncreasng ther own utlty. Through expected sector effcency analyss, t was observed that the audt-based contract also has sgnfcant potental to mprove the effcency of the nonproft sector overall. Because funders, nonprofts, and the sector may favor audtng under approprate condtons, audtng and effcency ncreases can and may become an ndustry effort * (P) Integrty Act of 2004, whch requres larger nonprofts to conduct objectve, regular audts (Calforna Government Code 2007) Sector Expected Effcency Turnng attenton to the nonproft sector as a whole, sector expected effcency s defned as = E A / A. Ths s the expected percentage of allocatons gong toward the producton of output for the nonproft sector as a whole, whch s consstent wth the defnton of effcency dscussed n 1. Proposton 3. Regardng sector expected effcency,, assumng a common dstrbuton over nonproft types,, and 0 P P, the followng holds: () FB A R ; () as P 0, A FB and as P 1, A R ; and () A s decreasng n the cutoff type, P, and ncreasng n the penalty, P. Fgure 2 plots the expected sector effcency aganst the cutoff P. The fgure llustrates that through the choce of penalty (0 P P ) the funder can poston the effcency of the entre sector as stated n Proposton 3(). In fact, the sector can be postoned anywhere between and ncludng both the frst-best and report-based contract scenaros as made explct by the equalty possbltes n Proposton 3(). Thus, the audt-based contract wth penaltes up to the optmal penalty, P, has sgnfcant potental to mprove the effcency of the sector overall. In ths fgure, nonprofts are a symmetrc populaton wth U 0 1, = 1, c = c, and P = P for all. Note that the results reman unchanged for heterogeneous nonprofts and varyng type spaces. In revew, the funder prefers the audt-based contract under reasonable parameters and well-chosen penaltes and can, n fact, acheve frst-best performance. The nonproft populaton strctly prefers the 7. Extensons In ths secton, we study three extensons of our orgnal model. We frst extend the results of ths paper to the constraned budget settng n 7.1. We then analyze uncertanty n measurng output n 7.2. Fnally, n 7.3, we study a settng wth audt costs The Budget Constraned Case When funders face lmted budgets, strategc allocaton s even more crtcal. As stated by Brest and Harvey (2008, p. x), whether you are gvng away $100,000 or $1 bllon a year, your funds are not unlmted, and a good strategy can multply ther mpact many tmes over. In ths secton, we extend our model to ncorporate a budget constrant for the funder. Now the funder must allocate her resources to maxmze her utlty wthn the contractual framework consderng both the nonprofts effcency announcements and the constrant on her budget. The funder s objectve functon from Equaton (7) thus becomes { N } maxu f = max E 2c A 2/3 A + B A A =1 (15) wth the constrants detaled n Equatons (8) (10) of our orgnal model plus the addtonal budget constrant of N B A (budget) (16) =1 In the rest of ths secton, we wll extend the results of ths paper for the settng wth a budget constrant. Note that for ths extenson we defne a constraned budget as N =1 A = B; otherwse the system effectvely operates wth an unlmted budget. We use superscrpts FBC, RC, and AC to denote the frst-best contract, the report-based contract, and the audt-based contract, respectvely, under a constraned budget.

12 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector 482 Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS Report-Based Contract. The problem formulaton of the report-based contract follows the formulaton of the problem outlned above and prevously n 4 wth the funder s objectve functon found n Equaton (15) and constrants detaled n Equatons (8) (10) and (16). Proposton 4. The report-based contract under a constraned budget takes the form A RC = A RC = B/ RC c 3E 2/3 3, where RC = N 3. =1 c3 E 2/3 Ths soluton results n the same conclusons drawn from Theorem 1 where the funder s unable to dfferentate among the nonprofts effcency types and bases her allocaton solely on her expectaton of each nonproft s type. Thus, the report-based contract stll offers the funder no observablty of effcency types and no operatonal transparency wthn ths contractng context. For ths budget-constraned case, we agan analyze the frst-best, benchmark model where effcency types,, are observable. Ths analyss results n Corollary 4: Corollary 4. The benchmark frst-best contract under a constraned budget takes the form A FBC = B FBC c3 2 (17) where FBC = N =1 c3 2 and = 1 N. We note that the funder stll allocates to all nonprofts wth a nonzero effcency ( > 0) based on ther effcency types. Notce also that under a constraned budget, the allocaton s dependent upon the effcency types of all nonprofts. Therefore, each nonproft ntroduces an externalty to the others by hs effcency type. For nstance, n the case of two nonprofts sharng the funder s tght budget, the frst-best allocaton to one nonproft decreases as the other nonproft s type ncreases Audt-Based Contracts. The audtng scenaro remans consstent wth 5 wth the updated funder objectve functon (Equaton (15)) and constrants (Equatons (8), (10), (12), and (16)). In explorng audt contracts under a budget constrant, the dscusson runs parallel to 5. As such, an optmal audt-based contract analogous to Theorem 2 s as follows: Proposton 5. Under a constraned budget, the optmal audt-based contract stll takes the form P C A FBC where there exsts a P C that enforces ncentve compatblty for all possble effcency types of nonproft under a constraned budget usng frst-best allocatons, A FBC, gven by Equaton (17). For P C, the followng are true: () P C > ; () P C s ncreasng n ; () as the budget, B, decreases, the penalty, P C, ncreases; and (v) as the number of nonprofts, N, goes to nfnty, so does the constraned penalty, P C. The optmal contract dsplayed n Theorem 2 tself does not change apart from ts components as detaled n Corollary 4 and Proposton 5. Thus, the nterpretaton and conclusons n 5 stll contnue to hold. Just as n Corollary 2(), Proposton 5() ndcates that all nonprofts must be over-penalzed n order to enforce ncentve compatblty for all types, mplyng that each nonproft wll need to repay more than ther valuaton of what was orgnally granted to them n the event that they msreport. Furthermore, ths penalty should ncrease n (Proposton 5()). The penalty ntutvely decreases n B (Proposton 5()). When the funder s budget s tght, she needs to manage her money more effectvely, whch dctates a hgher penalty. Such a scenaro can occur, for example, when the number of nonprofts s large (Proposton 5(v)). Thus, budget certanly has a profound effect on fundng scenaros and decsons. The sheer number of nonprofts applyng for grants, let alone monetary awards, can constran a funder s budget. Analogous to Theorem 3, Proposton 6 below gves the cutoff effcency type, C P, for the budgetconstraned case. Because of the complexty of the expressons, we lmt our analyss to penaltes less than : Proposton 6. Under a constraned budget, the audtbased contract wth a specfed penalty P stll takes the form { P A A A P = A FBC P for < P P A FBC for P where gven a penalty 0 < P 1 N, the cutoff effcency type, C P, s as follows: { C Bc 3 P = 1 N 2 + 3B2/3 c /3 where = = Bc3 P c 3 + Y N c 3 max P 2 =1 Y = c 3 j max j j P j 2 j + 3c2 B2/3 2/3 2 c 3 + Y 2/3 and The audt-based contracts dsplayed n Theorems 2 and 3 can be extended to the budgetconstraned case by replacng ther components wth }

13 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS 483 A FBC max C P from Equaton (17) and Propostons 5 and 6, respectvely, where max s taken component-wse. Smlar to ts counterpart n the unconstraned case, the cutoff type n Proposton 6 s stll decreasng n penalty, P. As wth both the frst-best and optmal audt-based contract allocatons, the other nonprofts types ntroduce an externalty to nonproft brought to bear n C P Effect of Contract Types on Performance. Analogous to 6, we use numercal examples to llustrate the effect of each contract on the funder, the nonprofts, and the nonproft sector overall. All results are n expectaton, and thus, the results labeled as nonproft are agan the results for the populaton of nonprofts. We focus on the audt-based contract, because the optmal audt contract always acheves frst-best. For the numercal examples, we assume that nonprofts are a symmetrc populaton wth U 0 1, = 1, c = 25, and P = P for all ; = 1. Note that the results reman unchanged for heterogeneous nonprofts and varyng type spaces. Fgure 3 ncludes a range of budget values. Consequently, as the budget ncreases, the stuaton moves from the constraned budget case to an effectvely Fgure Percentage of Expected Frst-Best Captured Under Constraned Budget (a) Funder Frst-best P = 1 unconstraned budget case. Therefore, for hgh budget values, Fgure 3 s comparable to Fgure 1 n 6. Fgure 3 shows that all our observatons for the funder and the nonprofts from 6 contnue to hold when the budget s constraned. We further observe that a tght budget mproves the performance of both the report-based and the audt-based contracts. Even the frst-best does not have much room to dstrbute a tght budget effectvely; hence, all contracts perform smlarly. As the budget ncreases, we observe that the performances of the contracts start to dverge. In partcular, the performance of the report-based contract sgnfcantly deterorates. Hence, the value of the audt-based contract ncreases as the budget ncreases. All n all, ncorporatng a budget constrant nto the basc model s relatvely straghtforward, though more computatonally demandng. Though we fnd that the audt-based contract s most benefcal n the unconstraned budget cases, the ntutons ganed n ths analyss can help even smaller funders to best understand ther contractual optons Uncertanty n Producton The assumpton that nonproft producton, that s, nonproft output or outcome, can be measured unambguously may be unrealstc n some stuatons, especally gven the dscusson n 2 and 3 concernng the challenges of outcome measurement and evaluaton n the nonproft sector. Ths assumpton s now relaxed through the followng updated producton functon: Frst-best captured (%) Frst-best captured (%) P = 0.75 Report-based P = Thousands Budget (b) Nonprofts P = 0.5 P = 0.75 P = 1 Frst-best Report-based Budget Thousands Note. The audt-based contract s ndcated by dashed lnes and the sold lnes show the report-based contract and frst-best. y = 2 e A (uncertan nonproft output producton) (18) where s a random varable wth support 0 and mean > 0. Ths random varable can be nterpreted as an unmeasurable porton of the producton, more specfcally random varaton n the producton process, random varaton or error n the accuracy of output measurement, or a combnaton of these. Under ths new model, the updated model equatons are as follows: u = A 1 2 A 2/3 1/3 4 (nonproft utlty) y = 2 A 2/3 1/3 (nonproft output) N U f = E 2c A 2/3 1/3 A (funder utlty) =1 where each nonproft has maxmzed hs expected utlty. The followng propostons result from analyss smlar to that whch led to the theorems and corollares of 4 and 5: Proposton 7. Under uncertan producton where the uncertanty s defned as n Equaton (18), the

14 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector 484 Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS report-based contract takes the form à R = à R = 3 E 2/3 3, where = c / and = 1 N It s straghtforward to show that when producton s uncertan for N nonprofts each wth an observable effcency type, the frst-best contract takes the form à FB = 4AFB for nonproft where the uncertanty n producton s agan defned by Equaton (18) and = 1 N. Thus, n key components of the report-based contract from 4, the uncertanty of producton s nconsequental when ths uncertanty s multplcatve. The same holds for the audt-based contract of 5. Specfcally, the optmal penalty, P, and cutoff type, P for a gven P, are unchanged under a multplcatve uncertanty n producton; only the allocatons change accordng to the frst-best allocatons detaled above. Ergo, these components follow from prevous analyss as detaled n Proposton 8. Proposton 8. Under uncertan producton where the uncertanty s defned as n Equaton (18), for nonproft the mnmum penalty that enforces ncentve compatblty for all possble effcency types, P, and the cutoff effcency type, P, for a gven penalty 0 < P P are unchanged from ther defntons n Theorems 2 and 3. Under ths multplcatve form of uncertanty, several observatons can be made. Frst, because of the updated forms of allocaton, we can note that the expected producton uncertanty affects the allocaton exponentally. Specfcally, the percentage of the certan allocatons captured s 4 for the frst-best allocaton and for the report-based allocaton when and are ndependent. Ths also holds true for frstbest and report-based producton/output, respectvely. Because uncertan producton does not affect the penaltes nor cutoffs of the audt contracts, these are affected smlarly. Thus, sector effcency results n Proposton 3 of 6.3 should hold unchanged. When and are dependent, there may be slght changes n the percentages of the certan allocatons captured, but these general nsghts wll reman the same. In summary, one of the man challenges for allocatng grants s the dffculty of measurng effcences. Ths s a struggle faced by not only funders but also nonproft organzatons. In ths secton, we extend the basc model from 3 to ncorporate a multplcatve uncertanty n producton (such as a random varaton n the producton process, random varaton or error n the accuracy of output measurement, or a combnaton of these). We show that ths extenson does not sgnfcantly change our results. That s, the conclusons drawn from the analyss of the basc model and ts resultng contracts are robust to uncertanty n producton under the specfc assumptons detaled n ths secton, namely assumptons about the functonal form of the output functon and randomness Audtng Costs As brefly mentoned n 3, an audtng cost s not ncluded n the funder s objectve functon formulaton because there are several stuatons where ths would not be approprate. For example, nonprofts may choose to self-audt usng a credble thrd-party audtor. Also, funders may not have specfc audtng costs or may have audtng costs and budgets separate from the contracts themselves. In ths extenson secton, however, we explore the mpact of ncludng the cost of audtng n the funder s objectve functon. It can frst be noted that such an ncluson does not affect the report-based or frst-best contracts from 4 because these assume no audtng. For the audtbased contracts from 5, the funder s utlty functon can be updated to nclude audtng costs as follows: U f = = N E c y A I A =1 N E 2c A 2/3 A I A (19) =1 In ths formulaton, the cost of audtng an ndvdual nonproft s. Because the funder may not choose or need to audt all nonprofts, I A s the ndcator functon of audtng nonproft. Whereas ths formulaton does not affect the form of the frst-best allocatons, t does affect the funder s choces of whch nonprofts to fund and whch nonprofts to audt. Note that the optmal audt-based contract detaled n Theorem 2 s no longer optmal when there s an audtng cost, as dscussed later n ths secton. However, f the funder s commtted to audtng all nonprofts that she funds, a smlar contract can be characterzed as follows: Proposton 9. Wth audtng costs > 0, the optmal audt-based contract acheves full transparency { P AFB for P A = P 0 27 otherwse where P = + 9/ 8, = 32 2 /c 3, = c /, and A FB s gven n Equaton (11). Thus, the funder wll only offer the optmal contract to certan effcency types, namely hgh effcency types that satsfy, so that types below ths receve no allocaton. However, the hgher penalty, P > P, ensures that all types wll stll reman ncentve compatble, even those wth zero allocaton. As such, ths contract enables complete transparency, but potentally at a hgh cost, because all nonprofts are audted. Alternatvely, the funder may choose not to audt some of the nonprofts that she funds and use a

15 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS 485 contract correspondng to the audt-based contract detaled n Theorem 3 where only the hgh effcency type nonprofts must be audted. Consequently, ths revsed audt-based contract wth funder specfed penalty s as follows: Proposton 10. Wth audtng costs of > 0, there exsts a such that the audt-based contract wth a funder specfed penalty, 0 < P P, takes the form { P A Ā A P = A FB for < max P P A FB otherwse where P f P < and = P otherwse, P s defned by Theorem 3, by Proposton 9, and A FB by Equaton (11). It stll remans that the funder s dedcated to fundng low effcency type nonprofts n ths contract. They do not present any addtonal costs of audtng, but nstead only the cost of ther neffcency manfest as reduced output. The man dfferences between the settngs wth and wthout audt costs are due to the constant payment and the cutoff type. If the audt cost s low, then P would most lkely be hgher than, and the two settngs are equvalent. However, f the audt cost s hgh,.e., P <, then the funder needs to ncrease the cutoff type to and consequently the constant allocaton to guarantee ncentve compatblty for moderate effcency types,.e., the types she prefers to award frst-best allocaton but cannot because of the audt cost. Once agan, f the audt cost s hgh, there may be unfavorable stuatons where the funder gves a hgh constant allocaton under ths contract, whch would decrease contract performance and transparency. In the presence of audtng costs, dentfyng the optmal penalty s not as straghtforward as n the case of 5 where there are no such costs. However, one can use Propostons 9 and 10 to search over possble penalty values for the value that yelds the hghest funder utlty. For example, when = 1, c = 2, = 1, and = 1, the funder would prefer a penalty of Notce that ths penalty s strctly less than one, ndcatng that grantng allocatons to all types and only audtng some (Proposton 10) s better than grantng allocatons to some types and audtng all (Proposton 9). Many dfferent reasons may motvate a funder to select specfc penalty values as dscussed n 5, ncludng such consderaton of audtng costs. As evdent n Propostons 9 and 10, for both the optmal and audt contracts, the funder s decson of whether to fund (offer the contract to) each nonproft hnges on the cost of audtng,, relatve to the value presented to the funder by the nonproft s output, c. The more hghly valued the nonproft output, the more lkely the nonproft s to be funded. As expected, the performance of the audt-based contract deterorates as the audt cost ncreases relatve to the value that the funder can generate. When the audt cost s too hgh, smply relyng on reports of nonproft effcency,.e., the report-based contract, may be a better approach for the funder. 8. Dscusson and Concluson To conclude, our analyss has revealed that reportbased fundng methods do not facltate effcent allocaton of funds as they do not result n operatonal transparency n nonprofts nor resolve the asymmetrc nformaton ssue. Audt-based contracts wth sensbly chosen parameters, on the other hand, can acheve both performance and transparency comparable to the frst-best. Furthermore, both the nonproft populaton and the sector overall beneft from audtng. Therefore, t may not be surprsng to see audtng on the rse wthn the sector, especally n the wake of the Sarbanes Oxley Act. The Calforna Nonproft Integrty Act of 2004 requres ndependent audts for large nonprofts (Calforna Government Code 2007). The Panel on the Nonproft Sector (2007) recommends nonprofts to have ndependent audts. Furthermore, organzatons lke Independent Sector and Board- Source are recommendng nonprofts to self-audt, self-regulate, and take proactve actons to sustan trust and confdence (BoardSource and Independent Sector 2006). Thus, ths research offers new understandng of how funders can use audtng nformaton n ther contractual relatonshp wth nonprofts, that s, how funders can structure ther grants and/or contracts effcently and effectvely around the nformaton provded by audts, ncludng thrd-party or ndependent audts. Furthermore, our research sheds lght on the mplcatons of audtng and ts effectve use for nonprofts, funders, and the sector overall. Indeed, our conclusons regardng both funder and nonproft preference for audtng not only uncover funders potental poston as change agents n the nonproft sector, but also ndcate a potental for collaboraton to change nonproft culture from merely heavy reportng to effcent montorng. Such a culture change wll ncrease the use of audtng, transparency, and, thus, effcency for the sector overall. Carman s (2009, p. 387) recent emprcal fndngs support what we fnd theoretcally, suggestng that funders mght [ ] be able to change the way they make fundng decsons [ ] by explctly usng evaluaton and performance nformaton to make fundng decsons. We do not make any restrctons regardng the funders or nonprofts that we target n ths work; however, our models and results would be most applcable and benefcal to large fundng organzatons. Such funders have both the power and means

16 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector 486 Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS to ntate change both n the nonprofts they fund and the sector overall. Furthermore, as our results show, these types of funders have the most to gan by movng from ther current fundng methods to emergng strateges, such as audtng. Where funders engage wth a potentally growng and changng pool of nonprofts, requrng them to constantly learn more about the organzatons they fund, our model should prove most effectve. Regardng the nonprofts funded, our model does not dscrmnate; t s sutable for both nascent and more establshed nonprofts. Our paper s thus complementary to the current research n operatons management on the subjects of emergency relef and humantaran logstcs (Thomas and Kopczak 2005, Tomasn and Van Wassenhove 2009), where recent work focuses on effcency and performance measurement (Beamon and Balck 2008, Van der Laan et al. 2009). For fundng decsons n ths area, audtng would be an addtonal step of montorng wth compellng benefts. Our analyss has several lmtatons. Frst, t s lmted to theoretcal and numercal analyss. The nature of contractng makes t dffcult to obtan data, especally on nonproft true effcences n the mdst of ther reports. Ths underscores the ncentves necessary n the contracts to obtan truthful data. As such, emprcally establshng the mpact of these contracts would be very dffcult. However, data could stll be useful n provdng nsghts and enhancng numercal analyss. Also, we do make certan theoretcal assumptons regardng producton and utlty functons. These assumptons, detaled n 3, are some frst attempts to model fundng dynamcs n the nonproft sector. Whereas prelmnary analyss shows that the hgher-level qualtatve nsghts and conclusons, partcularly the answers to the ntal three research questons, wll stay ntact even under less restrctve assumptons, the functonal form assumptons made here allow closed-form, clear results, comparsons, and conclusons. For nstance, under the assumptons of ths paper, optmal contract forms can be explctly stated, whch allows for nsghts nto optmal penaltes and contract comparsons. Such straghtforward clarty would be compromsed under less restrctve assumptons. Admttedly, though, some results are less robust and follow from specfc assumptons made. For example, n 7.2, the multplcatve nature of the uncertanty allows for seamless extenson of the contracts to a settng wth uncertanty. One should expect to have dfferent observatons under dfferent forms of uncertanty, and such an extenson of our results would be valuable. Lastly, although we used the most common defnton and measure of effcency n the nonproft sector, ths s a recognzed lmtaton. A dfferent measure of operatonal effcency, such as outcome per unt of funds, could be nsghtful. Although such outcome measures mght be more dffcult to quantfy n some contexts, when properly mplemented, they are ndeed true operatonal measures. Audtng by no means s lmted to the effcency defnton used n ths paper as other operatonal effcences can also be montored through audtng. There are several potental extensons for exploraton. One such extenson s the use of long-term or multyear contracts. Although not commonplace n nonproft fundng, the use of a long-term contract may enable the funder to more precsely develop belefs about the nonprofts effcency types as well as ncentvze the nonprofts through future allocatons. Issues such as commtment, renegotaton, and breach of contract wll need to be consdered when analyzng these contracts. The current economc downturn provdes new opportuntes to evaluate neffcency n the nonproft sector. However, proper consderaton should be gven to the addtonal stresses these contracts may mpose on already fnancally burdened nonprofts. In ths research, we characterze effcency as a type as opposed to a decson varable for the nonproft because, although nonprofts can ncrease ther effcences, these changes happen only n the long term. Thus, another potental extenson s to consder the effcency as a decson varable. Multyear contracts can be desgned to encourage such longterm mprovements on effcency. Yet a thrd potental extenson s the use of sgnalng. Perhaps the nonproft proposal can be a sgnal of the nonproft s type, enablng the funder to have more precse belefs about the effcency type of the nonproft. A sgnalng game s a bt of a dverson from the lne of contractual analyss studed here, but t may prove productve. Fnally, the evaluaton of reputaton mpacts and effcency announcements wth a dynamc model, along wth long-term contracts, would be frutful n sheddng lght on sector dynamcs. The metaphor of a nonproft as a seller of servces to a funder (Brest and Harvey 2008) captures much about ther relatonshp and lends naturally to vewng grant agreements as contracts: Funders have goals and contract wth nonprofts to perform the actvtes necessary to, n part, acheve these goals. As such, our nonproft problem and a more tradtonal operatons management problem are not so dstant. Corporatons manufacture products and servces, and nonprofts produce outputs and outcomes; operatons management has a crucal role n both settngs. There also exst dfferences between nonproft organzatons and for-proft corporatons. Hghlghted n ths research s that unlke a for-proft corporaton that s presumed to dstll the objectves of ther shareholders, managers, employees, and clents nto a sngle measure of shareholder value, a nonproft

17 Prvett and Erhun: Effcent Fundng: Audtng n the Nonproft Sector Manufacturng & Servce Operatons Management 13(4), pp , 2011 INFORMS 487 organzaton has numerous and dverse stakeholders ncludng funders, staff, volunteers, benefcares, and the publc. Therefore, a proper apprecaton for the dffculty that nonprofts face n provdng socally mportant servces combned wth sound understandng of operatons management technques provde excellent opportuntes for cross-sector learnngs. Possbltes abound for research n areas nvolvng and beneftng nonproft organzatons. Ths study of effcent fundng s just one of these many prospects. Electronc Companon An electronc companon to ths paper s avalable on the Manufacturng & Servce Operatons Management webste ( Acknowledgments Ths research was partally supported by Natonal Scence Foundaton Grant NSF/CAREER and the Natonal Scence Foundaton Graduate Research Fellowshp Program. 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