F E M M Faculty of Economics and Management Magdeburg

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1 OTTO-VON-GUERCKE-UNVERSTY MAGEBURG FACULTY OF ECONOMCS AN MANAGEMENT Target-Group ad Quality ecisios of equity-averse Etrepreeurs Steffe Burchhardt Christoph Starke FEMM Workig Paper No. 11, April 010 F E M M Faculty of Ecoomics ad Maagemet Magdeburg Workig Paper Series Otto-vo-Guericke-Uiversity Magdeburg Faculty of Ecoomics ad Maagemet P.O. Box Magdeburg, Germay

2 Target-Group ad Quality ecisios of equity-averse Etrepreeurs Steffe Burchhardt *, Christoph Starke 1 epartmet of Ecoomics ad Maagemet Otto-vo-Guericke Uiversity Magdeburg, Germay Abstract Limited doatios force oprofit etrepreeurs to ratio eedy idividuals by decidig o who is served at what quality level. We propose a positive model of this allocatio for applicats with differig icomes uder the assumptio of perfect user-fee discrimiatio. By followig recet experimetal ecoomic research o social prefereces, we assume that etrepreeurs behave iequity averse, i.e. they care about the relative cosumptio possibilities of others. We fid that less iequity-averse etrepreeurs prefer to serve wealthier idividuals at high referece quality. cotrast, more iequity-averse etrepreeurs care for the poorest idividuals but offer miimum quality. Furthermore, as iput costs icrease, etrepreeurs with low iequity aversio chage the target group, while etrepreeurs with high aversio do ot. Keywords: iequity aversio, oprofit, quality, ratioig, social etrepreeur, user fees JEL Classificatio: L31, H41, 45 * epartmet of Ecoomics ad Maagemet, Otto-vo-Guericke Uiversity, P.O. Box 410, Magdeburg, Germay. Steffe.Burchhardt@ovgu.de 1 This paper beefited greatly from extesive discussios with Matthias G. Raith. Commets by uiversity colleagues were also extremely helpful.

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4 1. troductio The quality with which oprofit orgaizatios provide eedy idividuals with goods ad services is subject to large variatios, eve withi the same braches of the same regio: The provisio of shelter rages from a low-quality emergecy stay to a log-term accommodatio at market stadard; food is supplied o a oprofit basis by soup kitches as well as higher quality uiversity cafeterias. The choice of quality level follows a specific patter related to the icome of the target group. cases where oly the ability to pay defies the eediess of idividuals, the good or service provided to the poorest is frequetly of sigificatly lower quality tha comparable market offers. Accordig to the World Bak (003, i low- ad middle-icome coutries services for poor people are ofte of low quality characterized by iadequately skilled workers, lackig resources, facilities i diepair etc. More specifically, for micro-isurace schemes addressig the poor i developig coutries a survey by McCord (001 shows that these isuraces coverage of health risks is very limited. Similar fidigs are reported for food assistace programs, which ofte supply low-quality food. 3 From these observatios oe may questio why oprofits do ot alteratively use their icome from doatios to lift the service quality to market level at the cost of a lower quatity of recipiets. Our iterest i this paper is to provide a theoretical foudatio to explai how oprofits geerally choose the quality/quatity mix of social goods ad services. Existig explaatios for the low quality of services to the very poor are limited to the role of govermetal provisio. For example, Glazer ad Niskae (1997 highlight the importace of a poor majority i a public choice settig while Besley ad Coate (1991 study govermetal measures for redistributig icome from the rich to the poor. However, due to the iability of raisig taxes, these approaches caot be adapted to private oprofit orgaizatios. A survey of the correspodig literature reveals three differet patters to implemet service quality ad quatity ito the objective fuctio of private oprofit decisio makers. Newhouse (1970 ad Rose-Ackerma (1987 follow the established covetio that idifferece curves betwee service quality ad quatity have the usual covex shape. Alog a secod lie, or ad Farley (1996 as well as Frieser ad Rosema (004 argue i favor of The study pipoits major exclusios ad limitatios i the coverage of micro-isurace schemes. Moreover, most of the schemes operate with reimbursemet limitatios. 3 Food for Survival (000 studied 971 New York soup kitches ad food patries ad foud that the majority of offered food cosists of cheap o-perishable goods (rice, pasta, beas, powdered milk, caed foods etc. while the supply of fresh food is relatively rare. 1

5 service itesity-adjusted output, where quality (characterized by service itesity ad quatity are multiplicably depedet withi the oprofit s utility fuctio. A third specificatio is give by Blau ad Moca (00, who apply a Cobb-ouglas objective fuctio i a childcare settig. However, all approaches lack a profoud motivatio for the specific iteractio of quality ad quatity withi the decisio maker s utility fuctio. Specifically, the ituitio of the assumed depedecy betwee the margial utility of service quality ad the absolute level of provided quatity remais uclear. The preset paper fills this gap by assumig that oprofits are iequity averse i makig their decisios. We thereby implemet oe of the major isights of recet experimetal ecoomic research o social prefereces obtaied from distributio games. 4 Accordigly, i our theoretical model we assume that the decisio maker cares about the relative payoff of others ad experieces a disutility if the cosumptio possibilities of a idividual deviate egatively from a social referece level. We show that this characterizatio provides a clear uderstadig of how oprofits beefit from service quality, quatity, ad the compositio of recipiets with regard to their iitial cosumptio edowmet. Moreover, we show withi our theoretical framework that allocatios which correspod to the empirical observatios metioed above ca be explaied. We fid the followig patters: Weakly iequity-averse etrepreeurs choose to serve the least eedy idividuals at (imum social referece quality. cotrast, highly iequity-averse etrepreeurs provide the poorest idividuals at miimum quality. Allocatios betwee both extremes occur oly for etrepreeurs with moderate aversio. The orgaizatio ad mai results of the paper are give as follows. Sectio itroduces a model of the etrepreeur s allocatio problem accoutig for applicats with differig icomes ad exogeously give doatios. Additioally, we allow the etrepreeur to charge perfectly discrimiated user fees, which goes i lie with commo oprofit practices. 5 Sectio 3 aalyzes how a variatio i doatios ad iput costs impacts the ratioig behavior of oprofits. We show first that a icrease i doatios leads to a extesio of the target group for all etrepreeurs ad additioally to a improvemet of service quality for highly iequity-averse etrepreeurs. Secod, a icrease i iput costs icites decisio makers with less iequity aversio to serve eve wealthier idividuals at costat (social referece quali- 4 Semial work i this field has bee doe by Fehr ad Schmidt (1999 ad Bolto ad Ockefels ( Theoretical aspects of price discrimiatio by oprofits are studied by Le Grad (1975 ad Steiberg ad Weisbrod (005. A discussio of istrumets eablig user-fee discrimiatio is give by Steiberg ad Weisbrod (1998.

6 ty. More averse etrepreeurs leave the target group uchaged but decrease service quality. We coclude i sectio 4 with a discussio of these results.. The Model We cosider a cotiuum of idividuals = [, ] R N seekig to satisfy a basic huma eed. Examples of such eeds are food, shelter, clothig, health etc. Each idividual N is willig to sped a budget b ( o purchasig oe uit of a eed-specific good. We assume that idividuals are ordered accordig to their willigess to pay, such that db ( d < 0 ad b ( = 0. ifferet product qualities of the good are available o perfectly competitive markets where firms face zero profits, ad the price of the good icreases with its quality level. We distiguish idividuals oly by their budget ad, therefore, assume that cosumers prefereces are idetical. Moreover, their margial utility of quality is strictly positive. The latter assumptios reflect the basic-huma-eed character of the good. tuitively, for this type of goods cosumer prefereces are similar ad relatively itesive util a miimum quality level is reached. For example, the miimum level for food might be give by a balaced periodical utritio. Together, our specificatios of cosumer prefereces allow us to treat the terms willigess to pay ad paymet ability equally ad, thus, to differetiate idividuals by their icome, i.e. poverty level. Accordigly, the idividual is the poorest whereas mi represets the wealthiest idividual. Suppose a social etrepreeur is able to perfectly observe idividual budgets. This assumptio is supported by oprofit practices, implyig that it is quite commo to differetiate the fiacial situatio of eedy people either through icome verificatio sheets or through appropriate idicators. 6 Moreover, Steiberg ad Weisbrod (005 argue that idividuals may be willig to reveal their paymet willigess to oprofit but ot to for-profit orgaizatios. The social etrepreeur compares the idividual budgets with a subjective social referece level b, which might be equal to her ow cosumptio budget or might be deduced from mi + 6 Steiberg ad Weisbrod (1998 provide a geeral discussio of these idicators. More specifically, FAO (001 surveys ad discusses the applicatio of idicators of several utritio programs i developig coutries (e.g. socio-ecoomic status, educatio level, age, household size, umber of childre etc.. Although such practices are supposed to cause so-called targetig costs, we simplify by igorig them for the followig reaso: These costs maily arise due to the idetificatio of suitable icome idicators ad the screeig of idividuals. However, sice the social etrepreeur must scree all applicats to detect the targeted idividuals, targetig costs are idepedet of the quatity ad compositio of recipiets. Hece, they are fixed costs that simply reduce the amout of doatios. A variatio i doatios is aalyzed i sectio 3. 3

7 scietific or regulatory guidelies. 7 This referece level determies the idividuals the etrepreeur cosiders eedy. For reasos of simplicity, we assume that all idividuals ow a mi budget edowmet equal or below this level, i.e. b = b(. Cosequetly, the social etrepreeur observes a budgetary iequity of qea ( : = b b( 0 for the th idividual, which will be referred to i the followig as ex-ate iequity. order to mitigate the ex-ate iequity the oprofit etrepreeur offers oe uit of a eed-specific social good to ay preselected idividual. This selectio is based o two related decisios: Which product quality should be offered ad which eedy subgroup should be targeted? We make three assumptios about the quality of the social good. First, the good is provided to all recipiets at uiform quality, i.e. we do ot cosider quality discrimiatio. Secod, the margial costs of producig a additioal uit of the social good c R + are idepedet of the supplied quatity but positively correlated to the product s quality level. 8 the followig, we do ot distiguish betwee quality ad margial productio costs ad deote quality equivaletly by c. Third, for reasos of simplicity, it is assumed that the quality of the social good is produced with the same techology as the market good. order to illustrate the settig we have i mid, cosider the followig applicatio to food-cosumptio. Here, the good is viewed as a budle of staple foods of specific quatity ad quality. Ay chage i the compositio of the budle that icreases eed satisfactio is modeled as a icrease i the good s quality. Hece, a icrease i the umber ad scope of meals through additioal food as well as a icrease i the quality of a sigle item ehaces the overall quality. The secod decisio of the social etrepreeur cocers the compositio ad size of the target group. As will be argued by the followig assumptios, this decisio solely requires the choice of the margially poorest recipiet N. First, we defie [, as the margially wealthiest recipiet ad we assume that the group of served idividuals lies i the closed iterval [,], with the quatity of recipiets give by. Furthermore, we allow the etrepreeur to perfectly discrimiate prices. The differetiatio of user fees accordig to paymet ability, which is ofte observed i practice, is a basic assumptio i models of oprofit mi 7 Exemplarily, the UK govermet (School Food Trust 007 defied a miimum quality for school food by pipoitig items that have to be offered withi a specific period. 8 A differet approach is take by Rose-Ackerma (1987, who argues that the margial costs of quality for the provisio of social goods are zero. Although sharig the opiio that there exist some factors improvig quality without additioal costs, e.g. chagig school teachig from frotal to iteractive mode, we accout for the majority of dimesios where improvemets i quality are costly. 4

8 firms. 9 this regard, Hasma (1980 as well as Steiberg ad Weisbrod (1998 provide umerous examples of oprofit idustries frequetly chargig slidig-scale fees for differet users. our model, the social etrepreeur charges the th idividual a user fee that exactly correspods to the budget edowmet b (. The idividual purchases the social good, if its quality c does ot fall short of the user-fee level, i.e. c b(, or, i other words, if its quality is at least as high as the affordable quality of the market good. 10 Cosequetly, the etrepreeur s total user-fee reveues F are give by (1 F = b( d. additio to these reveues, the etrepreeur receives a exogeously give level of doatios ( 0,, with = [( b ] b( d mi mi as the imum level at which all idividuals are served at social referece quality. lie with the orgaizatio s oprofit status user-fee reveues ad doatios have to be spet completely o fiacig the allocatio of the social good to eedy idividuals, i.e. ( F + = c (. The oprofit-coditio ( shows that for give levels of doatios ad idividual budgets b (, the etrepreeur s choice of the good s quality c ad the margially poorest recipiet determies the margially wealthiest recipiet = ( c, ad, likewise, the size of the target group ( c,. These depedecies are depicted i figure 1. Give the idividual edowmets b (, the social etrepreeur is cofroted with the status-quo budgetary iequity q ( = b b(. With doatios at had, she decides o the ea quality level c of the social good ad determies the specific target group by choice of the poorest recipiet. ue to the oprofit-coditio, she completely speds doatios to cover 9 Theoretical aspects of price discrimiatio by oprofits are studied i Le Grad (1975 ad Steiberg ad Weisbrod ( Recall that firms face zero profits i perfectly competitive markets ad use the idetical productio techology as oprofit orgaizatios. Cosequetly, the quality a idividual purchases from the market equals the budget which is spet. 5

9 b,c b c ( b ( F c, mi ( c, Figure 1: Allocatio effects of the choice of quality ad target group. the differece betwee margial costs c ad idividual cotributios. Startig with the poorest recipiet the fuds suffice to subsidize ( c, idividuals. Sice recipiets have to pay a user fee equal to their paymet abilities, total user-fee reveues amout to F ( c,. Subsequet to the allocatio of the social good, there remais a iequity with served idividuals amoutig to q ( c : = b c, which will be referred to as ex-post iequity i the followig. ep With the choice of her allocatio the etrepreeur simultaeously shows two types of ratioig. First, by choosig the target group she completely ratios all idividuals [ ( c,,]. Secod, her determiatio of a quality level partially ratios all recipiets sice they do ot receive the social referece level. As idicated i the itroductio, we characterize the social etrepreeur as a iequityaverse decisio maker. Specifically, she draws a egative utility from a deviatio of a idividual s cosumptio possibilities b ( from the social referece level. By providig eedy idividuals with the social good she reduces the iequity ad, hece, her ow disutility. We thereby build o recet experimetal ecoomic research which ivestigates geeral social prefereces by meas of simple distributio games, e.g. dictator ad ultimatum games, where oe idividual decides o the distributio of a exogeously give amout of moey betwee herself ad other players. their semial work Fehr ad Schmidt (1999 as well as Bolto ad Ockefels (000 aalyze the results of several experimets ad coclude that the iequity-aversio motive is able to explai the observed behavior. Exemplarily, Fehr ad Schmidt (1999 thereby use the followig defiitio: equity aversio meas that people resist 6

10 iequitable outcomes; i.e., they are willig to give up some material payoff to move i the directio of more equitable outcomes. We apply this motive to our model for two reasos. First, the aalyzed distributio games are closely related to the decisio cotext of the social etrepreeur i that a exogeously give amout of third-party fuds has to be allocated betwee differet idividuals. 11 Secod, give that the priciple of iequity aversio costitutes a buildig block i uderstadig the geeral fairess prefereces of idividuals, we ca expect it to characterize the motivatio of social etrepreeurs, i particular these, whose raiso d être lies i the mitigatio of existig iequitable allocatios. However, we use a broader defiitio of iequity aversio tha Fehr ad Schmidt (1999, who model the prefereces of the distributor as self-cetered iequity aversio, meaig that she cares about her ow payoff relative to the payoff of others. cotrast, we do ot restrict the referece outcome (i our paper: the social referece level b to be the etrepreeurs ow budget edowmet but, as previously argued, also allow for alterative referece levels, e.g. societal stadards. The iequity-aversio motive is itroduced ito our model through the parameter α R+. t determies the social etrepreeur s disutility from iequity by expoetially weightig q ea ( ad ( c, respectively. The fuctioal form of her disutility ca be writte as q ep { } α (3 v ( q = q, with q q (,q ( c. ea ep The parameter α thereby determies the level of the costat elasticity of margial disutility ε = α 1 ad is likewise a measure for the curvature of value fuctio (3. 1 Additioally, as with the class of Cobb-ouglas utility fuctios, α characterizes the etrepreeur s itesity of disutility. Margial disutility is decreasig with α ( 0,1, costat with α = 1, ad icreasig with α ( 1,. 13 More specifically, a etrepreeur with α = 0 does ot care about differeces i budgetary iequity betwee idividuals ad values q ea ( ad q ep ( c idetically. cotrast, for ay positive α the etrepreeur draws a icreased disutility from idivid- 11 Although we do ot accout for efficiecy cocers i our model, the distributio game closest to our model specificatio is aalyzed as treatmet R i Egelma ad Strobel (004. Here, the decisio maker is the wealthiest idividual ad is likewise ot able to extract ay rets for herself. 1 The elasticity of margial disutility is defied as ε = [ d v ( q dq] [ q v ( q ]. 13 With these specificatios of margial disutility we broade the scope of Fehr ad Schmidt (1999, who itegrate α multiplicatively ito the utility fuctio ad, hece, restrict their aalysis to liear iequity aversio, i.e. costat margial disutility. However, they also observe a oegligible fractio of people who exhibit oliear iequality aversio i dictator experimets (p

11 uals beig subject to higher iequity. This icrease i disutility is the larger the higher the value of α is, ad it becomes ifiite with α. 14 As will be show later, etrepreeurs with extreme iequity aversio care oly for the poorest target group idividuals. Based o the itroduced disutility cocept, we ow characterize the social etrepreeur s utility from allocatig oe uit of the social good to a target group idividual by the followig fuctioal form: ( ( (4 u( c, = v( q ( v q ( c = [ b b ] α ( b c α. ea ep Her utility equals the differece betwee the weighted ex-ate ad ex-post iequity, i.e. the reductio of disutility through provisio of the social good. As ituitive result, a o-iequityaverse etrepreeur ( α = 0 receives o utility from allocatig the good idepedet of the type of recipiet. Hece, she does ot egage i the social-good provisio. As previously argued, by simultaeously choosig the quality level c of the social good ad the poorest recipiet, the etrepreeur, due to oprofit coditio (, idirectly determies the wealthiest recipiet ( c, ad, hece, also the quatity of served idividuals, ( c,. Aggregatig the utility values of equatio (4 for each recipiet the yields the followig total utility level: α [ ] α (5 U ( c,,( c, = [ b b( ] ( b c ( c, For reasos of tractability, the otatio of utility fuctio (5 icludes the etrepreeur s decisio variables c ad as well as their ifluece o the value of the wealthiest recipiet ( c,. We thereby allow for a precise characterizatio of the etrepreeur s scope of alteratives: Uder cosideratio of oprofit-coditio (, the etrepreeur ca (directly or idirectly vary two of the variables with the third kept costat. The imizatio problem of the etrepreeur is give by d. (6 c, U ( c,,( c, s.t. [ c b( ] d = Note that the case α = correspods to imi-prefereces. 8

12 the followig, we prove the existece of corer ad iterior solutios to imizatio problem (6. 16 Propositio 1: Weakly iequity-averse etrepreeurs ( α (0,1 choose the imum quali- ty ( c = b ad provide oly the wealthiest idividuals ( c, = (. O the other had, highly iequity-averse etrepreeurs ( α ( 1, serve oly the poorest applicats ( = ( c b at the lowest feasible quality ( c = b ( c, ad Proof: See Appedix. oly exist if α = 1. ( mi. Fially, iterior optima f doatios are isufficiet to serve all eedy idividuals, the social etrepreeur chooses the mix of quality ad recipiets that imizes her utility from reduced iequity uder the fulfillmet of oprofit-coditio (. As propositio 1 shows, a first imum is give for weakly iequity-averse etrepreeurs ( α (0,1. Their margial utility of servig the ext poorer recipiet is always lower tha both their margial utility of a improvemet i quality (give a costat wealthiest recipiet ad their margial utility of servig the ext wealthier recipiet (give a costat quality. Cosequetly, the etrepreeur imizes the social- good quality ( c = b ad serves oly the wealthiest recipiets ( c, = (. tuitively, weakly iequity-averse etrepreeurs show the highest margial disutility of iequity for margial deviatios of idividual budgets from the social referece level. As immediate cosequece, the first uit of doatios (i form of the social good is used to completely elimiate the iequity of the wealthiest eedy idividual ( mi which requires the etrepreeur to choose the imum quality for the good. Util the etire doatios are spet, idividuals are successively supplied accordig to the ext higher iequity. The characterized corer solutio is depicted i figure, pael (a. mi 15 Employig equatio (1 ito oprofit-coditio ( ad rearragig it with respect to yields [ c b( ] = d. 16 Utility fuctio (5 is similar to the ormative poverty measure put forward by Foster et al. (1984. Applyig this measure Bourguigo ad Fields (1990 aalyze optimal govermetal subsidies to idividuals. Their fidigs resemble the results of propositio 1. 9

13 b,c c = b b,c b b,c b c ( b ( b (, F c ( c, mi = ( b (, F c ( c, c F ( c, ( c, = (a (b (c Figure : Corer allocatios ad a arbitrary iterior solutio. Secod, iterior optima ( c b ad exist for moderately iequity-averse etrepreeurs ( α = 1. Their margial utility of a chage i each of the three variables is equally large, which allows for ay values that satisfy oprofit-coditio (. Etrepreeurs i this category show a costat margial disutility of iequity ad, thus, do ot care for which applicats ad to what level iequity is reduced. A arbitrary iterior solutio is characterized i figure, pael (b. Throughout the rest of the paper the case of α = 1 will o loger be aalyzed. depedet of the subsequetly cosidered parameter variatios it ca be show that the margial utilities of quality, the wealthiest ad the poorest recipiet remai equally large. Cosequetly, ay allocatio satisfyig oprofit-coditio ( is optimal ad, therefore, α =1 has o further explaatory value. Third, the margial utility of highly iequity-averse etrepreeurs ( α ( 1, is lower for a improvemet i quality tha for a provisio of both the ext poorer ad the ext wealthier recipiet. The resultig allocatio is depicted i figure, pael (c. Here, oly the poor- est recipiets ( = are served at the miimum quality ( c = b( ( c,. 17 The ituitio rus cotrary to that of pael (a. Sice the margial disutility from iequity is largest for the highest iequity level, utility is imized, if doatios are trasferred to the poorest i- dividuals ( ( c,, such that the ex-post iequity is equal across recipiets but highest across all eedy idividuals. This procedure determies the low quality level of the social good. 17 terestigly, this is also the optimal allocatio uder imi-prefereces. 10

14 additio to these fidigs, figure 1 (pael (c idicates that highly iequity-averse e- trepreeurs choose to serve the largest quatity of eedy idividuals ( ( c,. However, this result oly holds if the fuctio of budget edowmets b ( is covex. More specifically, differeces i the chose target-group quatity deped o both the social etrepreeurs iequity aversio ad the curvature of the b( -fuctio, as we show formally with the followig propositio. Propositio : Highly iequity-averse etrepreeurs ( α ( 1, serve the imum quatity of idividuals ( c,, if the b( -curve is covex. cotrast, if b ( is cocave, the the quatity of recipiets is largest for weakly iequity-averse etrepreeurs ( α (01,. However, both types of etrepreeurs choose the same ad likewise imum quatity of recipiets if b ( is a liear fuctio. Proof: See Appedix. tuitively, the imum quatity of idividuals is served if the required average subsidy margi, i.e. the average differece betwee costat margial productio costs c ad the perfectly discrimiated user fee b (, is lowest. There are two requiremets to a miimal average subsidy. First, sice margial productio costs are assumed to be equal across idividuals, ad b ( is a decreasig fuctio i, ay target group is served with the lowest possible amout of doatios, if the wealthiest recipiet receives o subsidy. Otherwise, ay positive subsidy to this idividual would have to be likewise grated to each other recipiet, implyig icreased spedig of doatios. Secod, a miimum average subsidy margi arises amog those recipiets whose budgets are most uiformly distributed. For those idividuals the gap betwee costs ad user fee c b( is smallest o average. Followig propositio 1, the first requiremet is met for all etrepreeurs with α ( 0, \{ 1 }. However, the fulfillmet of the secod requiremet depeds o the curvature of the fuctio of budget edowmets b (. Give that b ( is covex, idividual budgets vary least amog the poorest idividuals, such that highly iequity-averse etrepreeurs ( α ( 1, serve the imum quatity of recipiets. cotrast, give a cocave b( - fuctio, budgets are most uiformly distributed amog the wealthiest idividuals which are supplied by weakly iequity-averse etrepreeurs ( α ( 01,. Cosequetly, they serve the imum quatity of recipiets. Fially, there exist o such differeces i the distributio of 11

15 idividual budgets, if the b( -curve is liear, which implies a equal ad imum targetgroup quatity for all etrepreeurs with α ( 0, \{ 1 }. 3. Variatios i oatios ad put Costs As argued i sectio, the determiats of the social etrepreeur s allocatio decisio iclude available third-party fuds ad productio costs. These fiacial coditios are likely to chage durig the lifetime of a social busiess. A door might withdraw or exted aouced fuds or might simply termiate a log-term relatioship. put costs might vary due to periodic shortages or shocks o resource markets. this sectio, we aalyze the impact of those variatios o the etrepreeur s choice of target group ad social-good quality. priciple, the social etrepreeur ca alteratively use additioal doatios to serve more or differet idividuals, or to improve the quality of the social good. The ext propositio shows that, o the oe had, etrepreeurs react differetly o variatios i doatios but, o the other had, the classificatio of corer ad iterior solutios by level of iequity aversio remais uaffected. 18 Propositio 3: Give a icrease i doatios, etrepreeurs with α ( 0, \{ 1 } elarge the quatity of served idividuals ( ( c, > ( c,. particular, weakly iequity-averse etrepreeurs ( α ( 0, 1 keep servig the wealthiest idividuals ( ( c, = ( c, = at the social referece level ( c = c = b mi ad expad their target group toward the ext poorer idividuals ( >. cotrast, highly iequityaverse etrepreeurs ( α ( 1, still focus o the most eedy idividuals ( = =, improve the social-good quality ( c > c ad serve the ext wealthier applicats ( ( c, < ( c,. Proof: See Appedix. tuitively, a icrease i doatios does ot affect the etrepreeur s margial disutility of ex-ate iequity as obtaied from equatio (3. Hece, there is o effect o her decisio o how to reduce this iequity optimally, i.e. the order of her margial utilities of quality c, 18 the followig the etrepreeur s decisio variables are superscripted by to accout for the state of icreased doatios. 1

16 margially poorest recipiet, ad margially wealthiest beeficiary ( c, remais uchaged. Cosequetly, etrepreeurs with α ( 0, 1 still have the highest margial disutility for the lowest levels of iequity which icites them to serve the wealthiest idividuals ( ( c, = ( c, = at social referece quality ( c = c = b mi. These recipiets ow comprise the ex-ate target group ad, additioally, the ext poorer applicats ( >. Etrepreeurs with α ( 1,, o the other had, elimiate the imum disutili- ty of iequity, if they keep o servig the poorest idividuals ( = = at miimum quality. Additioal doatios are spet o servig the ext wealthier applicats. However, these idividuals are oly willig to purchase the social good, if its quality is at least equal to their budget edowmet. Hece, the etrepreeur, likewise, improves quality uless the weal- thiest recipiet is idifferet betwee the market ad the social good ( = b( ( c, c. Cosequetly, the model predicts a icrease i both the quatity of recipiets ad the socialgood quality as reactio to a icrease i third-party fuds. As a secod variatio, cosider a geeral icrease i iput costs (i the followig idexed by superscript. Note that i sectio we assumed perfectly competitive for-profit markets ad idetical quality-productio techologies of for- ad oprofit firms. These assumptios imply that, for a costat quality, the icrease i iput costs equally icreases the price of the market good. Additioally, it still holds that ay idividual owig a budget equal or below the quality level c applies for the social good ad idividuals with b ( > c demad the market good. The icrease i iput costs is reflected by a chage of two parameters. First, the social referece budget icreases ( b b > because higher expeditures are required to purchase the correspodig cosumptio quality. Secod, we assume that the total quatity of eedy idividuals elarges by those people who are o loger able to afford the social referece cosumptio. As a result, the set of eedy idividuals is ow characterized by [, ] N = mi R with mi mi + < ad b( mi b =. Give that the social etrepreeur does ot chage margial productio costs ( c = c, she is restricted to use qualitatively lower or less iputs per uit of the social good, which deteriorates its quality. Alteratively, she could icrease c to keep the quality costat, but this, accordig to oprofit-coditio (, would imply a decrease i the quatity of served idividuals. As propositio 4 shows, a icrease i iput costs leads to cotrary reactios of social etrepreeurs depedig o their level of iequity aversio. 13

17 Propositio 4: For weakly iequity-averse etrepreeurs ( α ( 0, 1 a icrease i iput costs leads to a provisio of wealthier idividuals ( ( c, = at (uchaged social refer- ece quality ( c = b mi. cotrast, highly iequity-averse etrepreeurs ( α ( 1, keep servig the status-quo target group ( = = ( c, = ( c, margial costs ( c c = b( ( c, Proof: See Appedix. =, i.e. lower quality. ad at costat Weakly iequity-averse etrepreeurs ( α ( 0, 1 show the highest margial disutility of ex-ate iequity for margial deviatios of idividual budgets b ( from the social referece level. A icreased budget b required to cosume the social-referece quality ad a simul- taeously elarged quatity of eedy idividuals ( > iitial choices of margial costs ( c = b etrepreeur reacts by icreasig margial costs to the ew wealthiest applicats ( ( b, = mi mi ad target group ( ( b, = mi mi, thus, reders the suboptimal. The b ad shiftig the target group toward. This way, she elimiates the fractio of iequity with the highest disutility. As figure 3 idicates, a complete shift i the target group occurs, if b is such that doatios are isufficiet to allocate the good to more tha the ew applicats at social referece quality, i.e. mi [ b b( ] d. mi No iitially served idividual is further cosidered by the etrepreeur. cotrast, the margial disutility of highly iequity-averse etrepreeurs ( α ( 1, icreases with the iequity level. As show i sectio, they choose to serve the poorest idi- viduals ( = at miimum quality ( c = b( ( c,. Sice a icrease i iput costs exerts o effect o the relative poverty of idividuals, i.e. the idividuals withi the set [ ( c,, ] ( = = ( c, = ( c, are still poorest, the etrepreeur either chages the target group ( c c = b( ( c, ad or the margial productio costs =. However, quality ecessarily drops due to icreased iput costs. 14

18 b,c c = b c = b ( b mi mi Figure 3: A complete shift of the target group as a weakly iequity-averse reactio o a icrease i iput costs. Additioally, figure 3 idicates that a weakly iequity-averse social etrepreeur ot oly chages the compositio of recipiets but also their quatity. The ext propositio shows that this chage uambiguously depeds o the curvature of the budget fuctio b (. Propositio 5: Give a cocave (covex fuctio of budget edowmets b (, weakly iequity-averse etrepreeurs ( α ( 01, icrease (decrease the quatity of served idividuals, i.e. ( c, > ( c, ( ( c, ( < c,, as a reactio to a icrease i iput costs. Give a liear budget fuctio, they do ot chage the quatity of recipiets. Proof: See Appedix. From propositio, we kow that the quatity of recipiets is egatively correlated with the average subsidy margi required to serve the targeted idividuals. Sice the wealthiest recipiet receives o subsidy idepedet of the iput costs, this margi is oly coditioal o the distributio of idividual budget edowmets, i.e. the curvature of the b( -fuctio. The average subsidy is thereby the smaller the more uiformly budgets are distributed. Give that b ( is cocave, the dispersio is lowest amog the highest budgets. Cosequetly, the target group is larger after iput costs icreased, because recipiets are wealthier o average. However, the ex-post quatity is smaller if b ( is covex, which is exemplarily depicted i figure 3. Here, idividual budgets are least uiformly distributed amog the wealthiest applicats. Fially, due to the same reasoig, o differeces occur if b ( is liear. 15

19 4. Coclusio Our objective i this paper was to develop a positive model of a oprofit etrepreeur s allocatio decisio, which icludes the selectio of the target group ad the quality of the social good, i the light of limited third-party fuds. By assumig that a social etrepreeur`s decisio is characterized by iequity aversio, we follow recet results of experimetal ecoomic research o social prefereces. We demostrate how this preferece assumptio coveys a better uderstadig of how the good s quality, the quatity of recipiets as well as their icome distributio iteract withi the objective fuctio of private oprofit decisio makers. Specifically, a improvemet of service quality icreases the cosumptio level of beeficiaries ad, hece, reduces iequity. cotrast, a elargemet of the target group reduces the iequity for additioal recipiets. both cases the etrepreeur beefits through a reductio of her disutility from iequity. Fially, the compositio of recipiets eters the decisio calculus through the margial disutility of iequity. With icreasig (decreasig margial disutility the etrepreeur prefers to reduce a give amout of iequity of a poorer (wealthier idividual. We fid that weakly iequity-averse etrepreeurs choose to provide wealthier idividuals at high social referece quality. cotrast, highly iequity-averse etrepreeurs care for the poorest idividuals but offer miimum quality. These results allow for two explaatios of the low quality of services to the very poor. First, the goods or services cosidered i these studies were provided by highly iequity-averse etrepreeurs ad/or, secod, they were supplied by weakly iequity-averse etrepreeurs applyig a low subjective referece quality. Whether social etrepreeurs apply subjective referece levels or rather a societally stadardized orm remais a empirical questio. As a further result, we show that the quatity of supplied idividuals depeds o the curvature of the budget fuctio. Give covexity (cocavity, highly (weakly iequity-averse etrepreeurs serve the imum umber of eedy people. Moreover, we fid that etrepreeurs react differetly with regard to variatios i doatios ad iput costs. rrespective of the cosidered variatio, etrepreeurs with low aversio ever chage the quality of the social good. cotrast, etrepreeurs with high aversio improve quality if additioal fuds are available, ad they lower quality whe iputs used for productio become more expesive. Commo to both types of decisio makers is the provisio of more idividuals if doatios icrease. However, give a sufficietly high icrease i iput costs, highly iequity-averse 16

20 etrepreeurs do ot chage the target group while weakly iequity-averse etrepreeurs serve a completely differet (viz. wealthier group. Our results yield implicatios for stakeholders of oprofit orgaizatios whose objectives are related to quality, quatity ad the compositio of recipiets. More specifically, doors or govermets aimig at imizig the umber of served idividuals with give fuds should fud etrepreeurs who focus o the poorest people, if the majority of eedy idividuals is relatively poor (suggestig a covex budget-fuctio i the model. cotrast, stakeholders geerally iterested i miimizig the umber of eedy idividuals, through a provisio of imum service quality, should support etrepreeurs servig less poor idividuals. Those stakeholders do ot eve eed to chage their cotributio if iput costs icrease. Fially, the framework developed i this paper costitutes a basis for aalyzig additioal issues of social etrepreeurial behavior. Specifically, it merits further ivestigatio of how the differet allocatio patters chage if stakeholders exert a ifluece o the social etrepreeur s decisio. Especially, so-called lead doors, typically gratig a sigificat ad ofte the largest part of the iitial fiacial eed of oprofit orgaizatios, might wish to regulate if the etrepreeurial behaviour iadequately reflects their ow objectives. Appedix: Proof of Propositios 1-5 Proof of Propositio 1: For otatioal clarity, we temporarily expad the term U ( c,,( c, to U ( c,,( c, ;α to emphasize the ifluece of the etrepreeur s iequity aversio. However, we simplify the explicit otatio by use of U. By isertig user-fee reveues (1 ito oprofit-coditio ( ad applyig the implicit fuctio theorem, oe obtais the partial depedecies d dc = [ ( c, ] [ c b( ] < 0, d ( c, dc = [ ( c, ] [ c b( ( c, ] > 0 ad d ( c, d = [ c b( ] [ c b( ( c, ] > 1. Give that ( c, is costat, the social etrepreeur icreases c at the cost of, or vice versa, if her total utility level is icreased. She leaves both decisio variables uchaged if the utility imum is reached. Equivalet cosideratios apply for the pairwise variatios of c ad ( c,, while keepig costat, as well as ad ( c,, with c costat. Cosider the variatio of c ad for a costat ( c,. The correspodig coditio for margial utilities ca be writte as 17

21 (A1 c < => d dc. Specifically, the etrepreeur icreases (decreases c ad likewise decreases (icreases if (A1 holds with > (<. Both variables are left uchaged if (A1 holds with equality. sertig the partial derivatives ito coditio (A1 ad rearragig it yields (A α 1 [ b b( ] [ ( b b( ( b c ] ( b c α < =>. [ b b( ] ( b c As a first result, coditio (A holds with equality for α = 0 ad α = 1. Sice ay etrepreeur with α = 0 draws o utility from ad, hece, does ot egage i the allocatio of the social good, a iterior utility imum is solely give for α = 1. Furthermore, the right term of coditio (A is covexly icreasig i α. Combiig the two results gives U c > ( ( d dc, if α ( 0, 1, U c = ( ( d dc, if α = 1, ad U c < [ ] ( d dc, if α ( 1,. The same reasoig applies to the pairwise variatio of c ad ( c, for a costat. Formulatig the coditio o margial utilities yields (A3 d( c, c < => ( c, dc. ts rearragemet gives a similar expressio as show i coditio (A: (A4 α 1 [ b b( ( c, ] [ ( b b( ( c, ( b c ] ( b c α < =>. [ b b( ( c, ] ( b c Agai, coditio (A4 holds with equality for α = 0 ad α = 1 ad its right term is covexly icreasig i α. Hece, U c > [ ( c, ] [ d ( c, dc] if α ( 0, 1, U c = [ ( c, ] [ d ( c, dc] if α = 1, ad U c < [ ( c, ] [ d( c, dc] if α ( 1,. Fially, cosider the pairwise variatio of ad ( c, for a costat c. Here, the coditio o margial utilities is writte as (A5 d( c, < =>, ( c, d 18

22 or, equivaletly, [ ] α α α α [ b b( ( c, + ϕ] ( b c ( b b( ( c, ( b c (A6 x( ϕ, α : = c b( ( c, + < =>, ϕ c b( ( c, with ϕ : = b( ( c, b( > 0 ad x( ϕ, α [ b b( ( c, + ϕ] = ϕ ( c b( ( c, + ϕ α 1 xˆ ( α 0 < =>, α 1 b c. b( ( c, + ϕ with xˆ ( α : = α [ c b( ( c, + ϕ] [ b b( ( c, + ϕ] + ( b c b For α = 0 ad α = 1, coditio (A6 holds with equality ad xˆ ( α = 0 ad, hece, x ( ϕ,α ϕ = 0. For α { 0, 1}, x ( ϕ, α ϕ ad d xˆ ( α dα are idetermiate. However, sice d xˆ ( α dα > 0, it follows that x ( ϕ,α ϕ < 0 ad, hece, U < [ ( c, ] ( d( c, d if α ( 0, 1. U = [ ( c, ] ( d( c, d if α =1. Fially, x ( ϕ,α ϕ > 0 ad U > [ ( c, ] ( d( c, d if α ( 1,. The results of the pairwise comparisos show that, for ay give α, the orderig of margial utilities is idepedet of the levels of c,, ad ( c,. Hece, with exceptio of the special case α = 1, the social etrepreeur directly or idirectly chooses the imum levels of those two variables that show the highest margial utility. Thus, combiig the previous results, oe obtais c > d dc ad < d ( c, ( c, d if α ( 0, 1, c = d dc = d ( c, ( c, dc if α = 1, ad > d( c, ( c, d ad c < d( c, ( c, dc if α ( 1,. Cosequetly, c = b satisfy oprofit-coditio ( if = 1 Q.e.d. if α ( 0, 1, c ad ad ( c, = mi α, ad = b( ( c, c ad = ca adopt ay values that if α ( 1,. 19

23 Proof of Propositio : Let l idex the optimal choices for α ( 0, 1 ad h for α ( 1,. The imum quatity of recipiets is give if the average subsidy margi to served idividuals, c b( d [ ( c, ], is miimal. Sice db ( d < 0 ad c is costat for all ( c, [ ( c,, ] wealthiest recipiet, i.e. (A7 b( ( c, = 0 c,, a miimum average margi implies o-subsidizatio of the margially which is, followig the proof of propositio 1, fulfilled for α 1. Furthermore, for ay two pairs c i, it holds that i ad c j, j fulfillig (A7 ad with (A8 b( ( c, + μ c b( ( c, + μ c c ad for all μ 0, ( c,, i > c i i i < => j j j if d b( d 0 < =>. j ( ] Cosequetly, if d b( d > 0, the the average idividual subsidy margi is miimal for the choices c h ad h ( = which implies the imum quatity of served idividuals ( c,. cotrast, if d b( d < 0 the the choices c ( = ad h imum quatity of recipiets ( b, l ( c, = ( b, h l l l l b j l imply the. Fially, if d b( d = 0, the we have. Q.e.d. j Proof of Propositio 3: From the proof of propositio 1, the order of the margial utilities of c,, ad ( c,, as give i (A, (A4, ad (A6, is uiquely determied by α, ad cose- quetly idepedet of. Thus, for α ( 0, 1 a icrease i leads to c = c = b ad ( c, = ( c, mi =. Give these values, oprofit-coditio ( is fulfilled if > which implies ( c, > ( c,. cotrast, for α ( 1, the etrepreeur chooses = ad = b ( c, = ( c which implies ( c, < ( c, ad, hece, ( c, > ( c,. Q.e.d. c > c ad Proof of Propositio 4: the proof of propositio 1 we showed that the order of the margial utilities of c,, ad ( c,, as give i equatios (A, (A4, ad (A6, is uiquely deter- 0

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