An Agency Model of Welfare and Disability Assistance

QEDQueen’s Economics Department Working Paper No. 949

AN AGENCY MODEL OF WELFARE AND DISABILITYASSISTANCE

Robin BoadwayQueen’s University

Nicolas MarceauUniversite du Quebec a Montreal

Motohiro SatoQueen’s University

Department of EconomicsQueen’s University

94 University AvenueKingston, Ontario, Canada

K7L 3N6

05-1997

AN AGENCY MODEL OF WELFARE AND

DISABILITY ASSISTANCE

byRobin Boadway� Queen�s University

Nicolas Marceau� Universit�e du Qu�ebec �a Montr�eal

Motohiro Sato� Queen�s University

May� ��

ABSTRACT

Welfare programs including disability bene�ts have been considered as an e�cient way ofdelivering transfers to the needy� This paper addresses the importance of administrativecost of welfare systems by focusing on an agency problem arising between the governmentand social workers� whose job is to use tagging to determine eligibility for welfare bene�ts�Tagging is imperfect in that it involves type I and II errors� and its accuracy dependson the e�ort of the social workers�which is private information� To induce positive e�ort�the government needs to monitor the social workers using a costly auditing procedure�Using the framework of optimal non�linear income taxation� we characterize second�bestredistribution policies when the government can operate a welfare program alongside anegative income tax� The welfare program contains a disability bene�t and a generalwelfare component� Welfare applicants that are tagged receive the disability bene�t� whilethose who are untagged are accepted for general welfare bene�ts� It is emphasized thatwhether or not general welfare recipients� who may be the able or disabled� should beallowed to work is substantially a�ected by the nature of the administration cost of welfareprograms and by indivisibilities that may exist in labor supply� Our ultimate objective is tocontribute to the debate concerning transferring income to the poor using welfare programsversus negative income tax systems� This involves trading o� the costs of administeringwelfare relative to the bene�ts of tagging�

KEY WORDS� tagging� welfare programs� disability bene�t� social worker

JEL CLASSIFICATION� H�� I�

We have received helpful comments on an earlier version from Donald Cox� Bruce Hansen�JimHines� Bentley MacLeod and Daniel Parent� Financial support from the Social Sciencesand Humanities Research Council of Canada� le Fonds pour la Formation de Chercheurset lAide a la Recherche du Qu�ebec� and the Japan Society for the Promotion of Scienceis gratefully acknowledged�

AN AGENCY MODEL OF WELFARE AND

DISABILITY ASSISTANCE

byRobin Boadway� Queen�s University

Nicolas Marceau� Universit�e du Qu�ebec �a Montr�eal

Motohiro Sato� Queen�s University

I� INTRODUCTION

Financial assistance to the needy is typically delivered in two � via a negative income

tax or via a welfare system� Many countries combine elements of the two systems as partof their overall programs of assisting the poor�� While much of the literature on optimalredistribution focuses on negative income taxation Mirrlees� �� Stiglitz� �� lessattention has been devoted to optimal welfare programs� The optimal design of negativeincome tax and welfare systems are likely to di�er because of the di�erent institutions thatdeliver them� Negative income taxes are administered by the income tax authorities whorely on self�reporting by taxpayers or transfer recipients� and on monitoring by randomaudits using criminal sanctions as a penalty� On the other hand� welfare systems areadministered by agencies employing social workers who are responsible for determiningwhich applicant are eligible for which types of bene�ts� Monitoring typically takes placeadministratively rather than through the legal system�

The purpose of this paper is to construct a simple model of the design of a welfareprogram that combines the standard optimal non�linear income tax framework with theadministrative features of welfare systems� One such feature is that the eligibility of appli�cants is typically assessed on the basis of personal characteristics rather than being solelydependent on reported income as in negative income systems� The relevant characteristicwill be taken to be disability status� but others can be imagined� such as state of health�employability� or asset wealth� The process of determining individual eligibility has beencalled �tagging by Akerlof �� a process which is imperfect� Two types of tagging errorscan occur Parsons� �� some of the needy may be regarded as not being eligible� whilesome of persons pretending to be eligible may be tagged� Despite this imperfection� the useof tagging does improve the information available to the government and thereby improvesthe ability to target transfers to those who need them most� To use the terminology ofthe literature on optimal redistribution� tagging relaxes the self�selection constraint whichrestricts the extent of redistribution to the disabled and thereby enhances the e�ciency ofredistribution�

In much of the previous work on tagging Akerlof� �� Diamond and Sheshinski�� Parsons� �� the accuracy of the tag� and therefore the probability of errors�has been taken as given� But the process of tagging is implemented by welfare agenciesemploying social workers� and we would expect that the accuracy of tagging depends on the

� Assistance to the poor may also include various forms of in�kind transfers� See� for example�the summary in van de Walle and Nead �� We concentrate on cash transfers in thispaper�

�

amount of resources� including e�ort� devoted to it� The social workers obviously play animportant role� since they are the ones who observe and apply the tag� But the motivationof social workers may be di�erent from the government� and their actions are privateinformation� This gives rise to a standard agency problem that contributes to the costof administering welfare programs� the government needs to monitor the social workersthrough costly auditing to manipulate their incentives� The costs of administrationmust beset against the bene�ts of tagging to determine whether welfare systems are cost�e�ectivemeans of delivering transfers to the needy�

This paper addresses the implications of there being agency costs of administering awelfare program� and analyzes how the structure of the welfare program is a�ected by itsbeing alongside a redistributive income tax system� The welfare program we consider isone which contains two components� a disability bene�t and a general welfare program�Tagging by social workers determines eligibility for the former� Applicants regarded asnot being eligible for the disability bene�t are allowed to participate in the general welfareprogram� Those who do not apply remain in the income tax system� The disabilitycomponent is intended to target the disabled� and the general welfare program those whoare not well�o� for other reasons� in our case because they have low earnings potential�

The scheme described here is similar to that in many countries� including Canada�the one we are most familiar with� In the Canadian system� there is a welfare systemadministered by the provinces which exists alongside the progressive income tax system ofthe federal government� Although the latter has a system of refundable credits associatedwith it� for the most part those who rely on transfers obtain them from the welfare system�Eligibility for welfare is virtually automatic for those in need� Within the welfare system�there are di�erent bene�t levels according to whether recipients are deemed to be disabledor not� Eligibility for disability is determined administratively and involves some medicalcerti�cation� In all provinces� the disabled receive more than those who are employable�For example� in �� a disabled person in Ontario received �� compared with ��for a single employable person� Welfare recipients are also allowed to earn minimal amountsof income without losing their welfare assistance� There is evidence that there are signi��cant tagging errors in provincial welfare programs� An �� audit of Qu�ebecs welfareprogram revealed that mistakes in determining need had occured in about �� percent ofthe cases audited� and that about four��fths of those errors involved excessive generosity type II errors below�� Similar evidence has been found in other countries� Duclos ��studies the take�up of Supplementary Bene�ts for the non�working poor administered bythe Department of Social Securityin the United Kingdom� Using estimates obtained froma maximum likelihhod procedure� he �nds that in �� conditonal on all units applying�� of individuals who were entitled to the bebe�t were rejected by the agency type Ierrors�� while �� of individuals not entitled were accepted type II errors�� Our modelis consistent with these characteristics of actual welfare programs�

Our approach shares an important feature with Parsons �� who analyzes theconsequences of imperfect tagging for redistributive tax�transfer policy� One of our main

�

concerns is whether or not welfare recipients should be allowed to work�� In Parsons ��the desirability of separating the households by allowing welfare recipients to work restssolely on the ability or�and taste for labor of the latter� and arises because of an assumedindivisibility of labor supply� In our model� including the costs of administering the taggingprocess by social workers allows us to identify other sources of bene�t and cost fromallowing the welfare recipients to work� These have to do with how the existence of workingwelfare recipients a�ects the information available to the government and the incentivesfacing social workers� Making the accuracy of tagging by social workers endogenous iscritical to our analysis�

The issue of whether welfare recipients should be allowed� or induced� to work isrelevant not only in welfare programs but in negative income tax systems as well� As iswell�known from the optimal income tax literature� it may well be e�cient to have lowerability persons not working� simply because their productivity is not enough to compensatefor the loss in their utility from having them work Mirrlees� �� Similar argumentsapply in welfare programs as well� But there are also other considerations� as our analysiswill make clear� Tagging is costly� and inducing welfare recipients to work may improvethe e�ciency of tagging processes used to sort di�erent categories of welfare recipients� Aswell� it may serve as a device for assisting in the monitoring of the amount of e�ort thatwelfare agencies put into improving the accuracy of tagging� On the other hand� inducinglow�ability persons to work in order to improve tagging may be excessively costly if thereare indivisibilities of labor supply� The upshot is that it may be e�cient to have welfarerecipients work even though they should not in an optimal non�linear income tax systemthat uses no tagging� and vice versa�

II� BASIC FEATURES OF THE MODEL

The model we construct is highly stylized and contains a number of simpli�cations intendedto focus the analysis on what we regard to be key elements� There are three ability�types of persons in the economy�high ability �� low ability �� and disabled �� Thedistribution of ability�types is uniform� there are M of each of the three types� Abilityis private information to each person� the government and the social agency know thedistribution� Let C be consumption and Y income� The able persons have utility functions

� In Parsons �� the only two categories of persons are the tagged and the untagged� wheretagging is an imperfect signal of disability� He refers to a dual negative income tax systemas one in which the tagged persons are allowed to work� Untagged persons always have thechoice between working or not� He shows that it may or may not be preferable to allow thetagged to work� In our model� the tagged persons do not work� The issue is whether or notthe untagged persons who enter the general welfare system should be allowed to work� Theexact relation between our model and that of Parsons will become apparent as we proceed�

� Adding more high ability�types adds nothing of substance to the problem we are addressing�They would simply become taxpayers under the non�linear progressive tax system� like high�ability persons� The properties of that system are well�known�

�

V i C�Y � i � �� which are increasing in C and decreasing in Y � with marginal rates ofsubstitution �V i

Y �ViC increasing in Y � Leisure and consumption are normal so that for any

bundle C�Y �� V �Y �V

�C � �V �

Y �V�C the single�crossing property�� We need not specify

precisely the nature of the di�erence in V i �� between ability�types� One interpretation�as in the optimal income tax literature� is that it is due solely to di�erences in wage rates�But we can also allow for other interpretations� such as di�erences in taste for leisure�The disabled persons are unable to work� Their utility function is V � C�� an increasingand strictly concave function� It is assumed that for a common level of consumption� thelevel of utility of a disabled person is no more than that of either type of able person�V � C� � V i C� �� i � �� But the marginal utility of consumption is larger for thedisabled� V �

C C� � V iC C� �� i � ��

All output is produced by labor and is used for consumption� Though labor supplyis variable� we assume that there is some discreteness involved in choosing to work� Aperson who works must earn at least a minimal amount of income y� so that Y � y� Suchdiscreteness� which may re�ect a minimum level of hours worked or�and some time costsrelating to traveling� is consistent with casual empirical evidence�� Whether or not thisconstraint is binding will turn out to have important implications for the design of thegeneral welfare program� i�e�� whether it is e�cient to induce welfare recipients to work�If Y � y is not binding� our analysis parallels the standard framework of optimal incometax models� If it is binding� our analysis captures some features of the existing taggingmodels of Akerlof �� Diamond and Sheshinski �� and Parsons �� all of whomabstract from the labor supply decision by assuming that the amount of labor suppliedwhen working is �xed�

The objective of the government is to redistribute income from higher to lower ability�types� Our purpose will be to discover the form of policies that achieve a given amountof redistribution in the most e�cient way� given the instruments and the informationavailable to the government� The government operates a non�linear negative income taxof the standard form and can supplement it with a welfare program� The purpose of thewelfare program is to take advantage of the information obtained by tagging� Whetheror not such a program is desirable depends upon the cost of administering the welfareprogram relative to the bene�ts of tagging� For the purposes of presentation� we assume awelfare program is in place and analyze the costs and bene�ts of administering it e�ciently� In the absence of a welfare program� a non�linear income tax system would be used byitself� and the properties of it are well�known�� Because the purpose of the welfare programis to make the bene�t to the disabled more generous than they would be under the non�linear negative income tax system� both the low�ability and the disabled will be included inthe welfare program� leaving the high�ability persons to be dealt with by the income tax�Each person who applies for welfare is automatically considered for disability bene�ts�

� It is obvious that nobody would accept a job involving very few hours worked� Holding a jobentails monetary costs �eating� traveling costs�� time costs �time to travel�� and utility costs�one would prefer being with his children� that all have some discreteness �Killingsworth��

�

those who choose not to apply stay in the income tax system� Eligibility for disabilitybene�ts is examined by the social worker using a tagging technology� Those who are nottagged are admitted to the general welfare program� The assumption that all applicantsare considered for the disability bene�t does not a�ect the generality of our analysis�

We follow the standard practice Stiglitz� �� of characterizing the non�linear neg�ative income tax by the consumption�income bundles o�ered by the government for eachability�type� The parameters of this system induce truthful revelation of preferences� Thisimplies both that it is administratively possible to eliminate tax evasion by some combi�nation of penalties and auditing� and that it is not e�cient to allow some tax evasion�� Inthe negative income tax system� the bundle intended for type i is denoted Ci� Yi�� Sincethe disabled persons cannot work� their bundle will be C��

In the welfare program� the disability bene�t for those who are tagged consists of thebundle g� �� General welfare recipients� those who are untagged� may or may not beallowed to work� We refer to the case where they are allowed to work as Regime W andthe one in which they cannot as Regime N � Given that disabled persons cannot work�two alternative bundles� c�� y�� and c�� must be o�ered under general welfare RegimeW � where lower�case letters are used to distinguish welfare from negative income taxation�The bundle c�� y�� is intended for the low�ability persons� In Regime N � the only bundleo�ered to the non�tagged welfare recipients is c�� It will turn out to be the case thatg � c� in our analysis� given our assumption that the government wishes to redistributefrom the better�o� to the less well�o�� To simplify matters� we assume that the bundles c�� y�� and c�� o�ered in the welfare scheme are the same as the bundles C�� Y�� and C�� in the income tax system� This ensures that� given g � c�� both the low�ability anddisabled persons apply for welfare� Thus� the bundles C�� Y�� and C�� are not reallyrelevant for our analysis� To induce each of the ability�types to choose the bundle intendedfor them in the relevant program� appropriate self�selection constraints must be satis�ed�These self�selection constraints are�

V � C�� Y�� V � c�� y�� V � c�� y�� V � c��

where the latter one only applies in Regime W �

The tagging technology takes the following form� Following Akerlof �� and Par�sons �� let �i be the probability that a person of ability�type i is tagged� We assumethat �� so high�ability persons cannot be tagged and thus do not apply for welfare�� along with equation �� and the single crossing property ensures that the high�ability persons do not apply for welfare� For the disabled and low�ability persons� �� and�� are endogenously determined and fall in the range � � �� Since ��is the probability that the disabled will be correctly� tagged� � � �� is the probability

� Stiglitz �� has argued that allowing tax evasion may be e�cient for one of two reasons� Itmay relax incentive constraints if lower ability persons are less risk�averse than higher abilitytypes that are mimicking them� which seems unlikely� Or� it may be a form of randomizationof the tax system to exploit the fact that the aggregate revenue function of the governmentmay be convex in tax rates�

�

that they will be incorrectly untagged� these are the type I errors� Similarly� �� is theprobability that a low�ability person will incorrectly receive a tag� or that a type II errorwill occur� The accuracy of tagging� that is� the magnitudes of �� and �� are increasingin the e�ort of the relevant social worker� To simplify matters� we assume that �� increases in �� reductions in type I errors� are accompanied by equal decreases in �� re�ductions in type II errors�� This allows us to write � � �� which simpli�es boththe algebra and the notation used in our analysis� Let e be the e�ort of a social worker�Then � e� relates e�ort to the probability of the disabled being tagged and the able notbeing tagged� where � e� is an increasing function� We assume that � �� that is� ifno e�ort is taken� nothing is learned from tagging� Recall that there are the same numberof disabled as low�ability persons� so �� is equivalent to tagging disabled andlow�ability persons randomly��

The government hires social workers� whose sole responsibility is to tag welfare ap�plicants� The caseload of each social worker is drawn randomly from the pool of welfareapplicants� which includes all �M disabled and low�ability persons� Assuming that thenumber of social workers is large� the probabilities of a given welfare applicant being low�ability or disabled are �� each� The social worker receives a salary from the governmentof S per client� but must pay a penalty of k if either a type I or type II error in tagging isdetected� The penalty k is independent of type of error and of the total number of errorsdetected� The ex post income received by the social worker for an applicant will then beeither S � k or S depending on whether or not a tagging error has been detected�

The detection of tagging errors is of crucial importance in our analysis� There are twoalternative ways that may be done� The �rst way is to take advantage of the fact thatoptimal redistribution schemes are like revelation mechanisms that induce persons to revealtheir ability�types� If we allow welfare applicants who do not succeed at being tagged towork� the welfare system can be designed such that non�tagged low�ability persons chooseto work and are therefore separated from the non�tagged disabled� who simply cannot work�The government knows which social worker is responsible for each welfare recipient in thewelfare program because both tagged and untagged persons remain part of the caseloadof the social worker who examined their eligibility� This is a key di�erence between theincome tax system and the welfare program� It provides information that the governmentcan use for the purpose of monitoring the social workers to detect errors in tagging� andpenalizing them for such errors� Given that� in RegimeW all type I errors can be detectedcostlessly� This is a major bene�t from allowing non�tagged low ability�types to work�

Second� the government might randomly audit those welfare recipients who do notreveal themselves� This will include all �M welfare recipients in RegimeN and all M taggedpersons in Regime W � Let q be the proportion of the relevant body of welfare recipientswho are audited� If audited� the ability�type of a welfare recipient is revealed� Of course�auditing is costly to the government as discussed later� Given the audit probability q� ifa social worker has committed an error� expected income will be given by S � qk� Giventhat auditing is costly� it is a standard result in the economics of crime that the cost of anygiven level of compliance can be reduced by increasing the penalty while reducing auditprobabilities� To limit the extent of penalty� we impose a maximum value for the penalty

�

by assuming that k � S� The argument is that making errors is not a criminal o�ense� sothe most we can expect is that the salary of the social worker is con�scated� Given thatauditing is costly� it is obvious that the constraint k � S will always be binding� Therefore�from now on we assume S � k and suppress S from our analysis��

The payo�s available to the social worker per welfare applicant in the various possibleoutcomes of Regimes W and N are shown in the appropriate columns in Figure �� This�gure is constructed for a given value of �� that is� for a given e�ort level by the socialworker� Since e�ort is re�ected in � through the tagging function � e�� we can suppresse�ort and suppose that � is the object of choice by the social worker� We assume socialworkers are risk�neutral so that expected utility is expected income less the cost disutility�of e�ort� Let � �� be the cost per applicant of achieving a given level of �� where �� and � �� The latter implies that if the social worker exertsno e�ort� so the tag yields no information� there is no cost� From Figure �� the expectedutility of the social worker in the two Regimes is�

EUW �� q� k� � �k � �� q�k

��

EUN �� q� k� � �k � � � �� q�k � � ��

where the superscripts W and N denote the Regime� In equations �� and �� for theexpected utility per applicant of the social worker� we are assuming that the social workerchooses to use the tagging technology� captured by �� to assign disability status to ap�plicants� In the following section� we need to consider the case where the social workerchooses not to use � to accept or reject applicants�

We need to be sure that the social worker will agree to accept employment� One wayto do so that would be to impose a participation constraint that requires expected utilityto be some minimal amount� say� zero� Instead� it turns out that� given the constraint onthe penalty function k � S� the expected utility of the social worker is always positive� Ofcourse� we could impose a stricter participation constraint� but that would only complicatethe analysis�

III� BEHAVIOR OF THE SOCIAL WORKER

The payo�s of the social worker EUW and EUN given by �� or �� depend upon theparameters set by the government k� q� and by the e�ort exerted by the social worker ineach Regime� In this section we consider the behavior of the social worker� The followingsection takes up government policy�

Regime W� Welfare Recipients Allowed to Work

Social workers must decide whether to exert any e�ort at all and� if so� how much to exert�Assume that social workers choose some positive level of e�ort� They will choose � to

� Other less stringent constraints on k would also be possible� For example� we could setk � S � for some arbitrary value of without changing the essence of the argument�

�

maximize expected utility given by �� The �rst�order condition is�

k � �� q�k

��

The second�order condition requires �� whcih we have already assumed� Equation �� which determines the e�ort chosen by the social worker� can also be solved for thevalue of k that corresponds with given levels of � and q when the social worker is choosingthe optimal level of e�ort� Solving �� for k and substituting into �� we obtain the levelof EUW attained by the social worker when optimal e�ort is exerted�

EUW �� q� �� q�

�� q

� � ��

Satisfaction of the second�order condition is not su�cient to ensure that the socialworker will supply positive e�ort� Instead� the social worker may exert no e�ort and acceptall applicants� Using the payo�s from Figure �� expected utility will be�

EUA q� k� ��

q

�

�k�

Intuitively� by accepting all applicants� no type I errors will be committed� A penaltywill be imposed only if the social worker commits a type II error and is audited withprobability q�� To induce the social worker to exert e�ort� an e�ort incentive constraint

must be satis�ed� EUW �� q� � EUA q� k�� or� using �� in EUA q� k��

�� q��

� � q� � �� q�

��

� � q� ��

Whether or not the e�ort incentive condition �� is binding turns out to be important inchoosing between the two Regimes and among other policy options�

De�ne by qW �� the audit probability that makes �� binding� When �� is an equality�we obtain�

qW ��

��

It is easy to verify using �� that qW �� is decreasing in �� Also� lim�� qW ��

�� Using �� can be restated as q � qW �� Thus for a given �� the government isconstrained to choosing only those values of q such that q � qW �� if it wants to operatea welfare program� If the social workers do not exert e�ort� there are no bene�ts fromtagging��

� An alternative strategy of the social worker may be to reject all applicants� If so� the

expected utility per client becomes EUW �reject� � k�� Obviously� this is less than EUA sothis alternative strategy would not be used�

�

Given this behavior of the representative social worker� we can determine administra�tive costs of the welfare program in Regime W � With M being large� administrative costscan be treated as deterministic� Administrative costs include the expected income of thesocial workers net of penalties� plus the cost of auditing a proportion q of the M taggedpersons� Let the total cost of auditing be C qM�� an increasing� convex function withC �� Given the behavior of social workers as re�ected in EUW �� q�� administrativecosts per welfare recipient in Regime W can be written�

AW �� q� � �� q��

� � q�C qM�

�M ��

where admissible combinations of � and q must satisfy q � qW �� as de�ned by �� above�

The properties of the administrative cost function AW �� q� are of some interest�Di�erentiating �� with respect to �� it is straightforward to show that�

�AW �� q�

�� q��

��

� � q� ��

Not surprisingly� inducing greater e�ort from the social worker requires an increase inadministrative costs� The derivative �� represents marginal administration cost whene�ort level is observable perfect information�� If information on e�ort is perfect� thegovernment does not have to audit the social workers and S would be chosen so thatthe individual rationality constraint is satis�ed� that is� S � � �� which is also theadministration cost in such a circumstance� When information is imperfect and thusthere is penalty imposed� higher � implies lower type I and II errors� which in turn leadsto lower �ne revenue� That is why the marginal cost of AW �� q� exceed that under fullinformation� Di�erentiating �� with respect to q and simplifying yields�

�AW �� q�

�q�

C � qM�

��

�

� � q��

This satis�es ��AW �� q��q� � �� implying that administrative costs are strictly convexin q as shown in Figure ��

Now consider the governments choice of q to minimize administrative costs AW �� q�for any given value of �� If �� is not binding� the government would select q at the bottomof the U�shape of AW �� q�� Denote this unconstrained level of q as q�W �� It satis�es�AW �� q��q � � in �� for q�W �� Since �� and C � �� it is easy to seethat q�W �� may result for � su�ciently close to �� The possibility of such a cornersolution is not essential to our analysis� however� When q�W �� does not exceed qW ��the government is forced to choose the latter since the incentive constraint �� would beviolated otherwise� Therefore� if we denote cost�minimizing level of audit for a given � incase W as qW �� we have qW �� maxf qW �� q�W ��g� Figure � summarizes the twopossibilities� In Panel A� the e�ort constraint is binding at the optimum� q�W �� qW ��and so� qW �� must be chosen� On the other hand� in Panel B� the e�ort constraint is notbinding so that the optimum q�W �� can be chosen�

�

We have seen that qW �� is decreasing in � and that lim�� qW � �� On the other

hand� q�W �� is monotonically increasing in �� These curves are depicted in Figure �� Letthe level of � that equates q�W �� and qW �� be �� q�W �� qW �� If � � !�� "�the e�ort constraint is binding and q �� must be chosen� while if � � ! �� "� the e�ortconstraint is no longer binding and the optimum q�W �� can be chosen��

This characterizes the cost�minimizing values of qW �� With some abuse of notation�denote by AW �� the minimum value function for the governments problem of choosingq to minimize administrative costs subject to the e�ort incentive constraint� That is�AW �� AW �� qW �� Note that when qW �� q�W �� the envelope theorem impliesdAW ��d� � �AW �� qW �� for � � �� inducing higher e�ort increasesadministrative costs� But if the e�ort constraint is binding so qW �� qW ��

dAW ��

d��

�AW �� qW ��

��AW �� qW ��

�q

d qW

d��

Since the last term is negative� the sign of this expression is generally ambiguous� It isreasonable to suppose that administration costs in Regime W are as depicted in Figure ��For example� if �� and �� are small enough at � close to �� this will be the case� Inthis case� initially in the left portion of the curve�� the second term on the right�hand sideof the previous expression dominates the �rst one so� administrative costs are decreasing�A point comes however where the �rst term becomes more important and so� the curveeventually becomes upward sloping� We will return to Figure � later�

Regime N� Welfare Recipients not Allowed to Work

Consider now the behavior of social workers in Regime N � Their expected utility is givenby �� For given k and q� the social workers choose � to maximize EUN �� q� k�� The�rst�order condition is simply�

qk � ��

The second�order condition is also satis�ed here with �� Solving �� for k andsubstituting into �� we obtain the expected utility of the social worker when optimale�ort is exerted�

EUN �� q� �� q�

�� q

� � ��

As in Regime W � a social worker may choose to provide no e�ort and to accept anyapplicant� reject any applicant� or accept applicants randomly� As before� EUA q� k� � �� q��k� But� as opposed to Regime W � a social worker providing no e�ort � � ��but using the tagging technology also obtains the same expected utility� EUN �� q� k� � � � q��k by �� This implies that the e�ort incentive constraint is never binding in

� It could be the case that �� If so� qW �� qW �� for all �� Though easy to analyze�this case is not particularly interesting� In what follows we assume that �� is interior�

��

Regime N � This di�erence turns out to be an important distinction between Regimes Nand W �

Administrative costs per welfare client in Regime N are again the expected incomeper client of the social worker plus the cost of auditing� where here a proportion q all �Mtagged and untagged welfare recipients must be audited� Given the optimizing behaviorof the social workers� administrative costs can be written�

AN �� q� �� q�

�� q

�C �qM�

�M� ��

Di�erentiating AN q� �� with respect to �� again we can see �AN q� �� As incase W � administration costs are rising in �� and asymmetric information on e�ort makesthe marginal administration cost exceed the marginal cost of e�ort to the social worker�

The government chooses audit e�ort to minimize AN q� �� for any given level of ��Since the e�ort incentive constraint is always satis�ed in this Regime� cost�minimizing levelof q is given as the solution to the following �rst�order condition�

C � �qM� ��

q��

Since ��AN��q� � �� the solution to �� qN �� is a global minimum� Moreover� qN ��is increasing in � with qN �� Comparing �� and �� we can see that qN �� canbe greater or less than q�W �� for any value of � � �� Figure � shows the case whereqN �� q�W �� This case applies� for instance� when the audit cost function C �� islinear�

Denote the minimum value function to the governments audit�choice problem inRegime N by AN �� AN �� qN �� We wish now to compare administrative costsof attaining a given level of e�ort in the two Regimes� The following Lemma� proven inthe Appendix� is useful for comparing AW �� and AN ��

Lemma �� For a given pair �� q�� the administrative costs are lower when non�taggedwelfare applicants can work than when they cannot� AW �� q� � AN �� q��

Recalling that � is the value of � such that the e�fort incentive constraint just becomesbinding� the following proposition is proven in the Appendix�

Proposition �� If � � �� then AW �� AN �� administrative costs are lower whennon�tagged welfare applicants can work than when they cannot�

In Figure �� administrative costs in both Regimes are represented� Note that AN �� is astrictly increasing curve and that given Proposition �� it must cross AW �� from belowat some � strictly smaller than �� Proposition � leads us to conclude that when thegovernment wants to implement a relatively higher level of �� it may �nd Regime W in

��

which the welfare recipients are allowed to work preferable to Regime N � But we cannotclaim that this would be always the case without examining the government optimizationwith respect to the parameters of the welfare program and the income tax system� Thetransition from one Regime to another involves a discrete change in resource allocation�We turn to this issue in the next section�

In the above discussion� we have already seen that lim�� qW � �� while qN ��

�� From �� and �� we have k � S � � in case W if � � �� Solving �� fork and substituting qN �� yields the penalty that is necessary to induce a given level of�� kN �� qN �� qN ��C � �MqN �� where we have used �� to obtain thelast equality� Obviously� kN �� This implies that AW �� approaches C M��M as� approaches �� while AN �� By the continuity of Ar �� r � W�N�� we canconclude that when targeted � is relatively close to �� RegimeW is unambiguously moreexpensive than Regime N to administer� which may induce the government to choose thelatter case�

Proposition �� For � su�ciently close to �� AW �� AN ��

IV� THE PROBLEM OF THE GOVERNMENT

In the previous section� we found that the relative magnitude of administration costs inRegimesW and N depend on the e�ort put into tagging by the social workers� But the twoRegimes also di�er in their optimal welfare and income tax policies� To determine whichof the two Regimes is superior� we need to optimize over these other government policyinstruments� Unfortunately� exact comparisons become di�cult since optimal allocationsin Regimes W and N di�er discretely� To obtain insight into the factors that favor oneRegime relative to the other� we approach the problem in stages� First� we determine whichwelfare system requires the least net revenue to achieve a given combination of transfersto the disabled g� c�� and accuracy of tagging �� Then� we consider the program thatmaximizes the weighted sum of utilities of each type of person� Finally� we examine anexample in which the objective function attributes weight to the disabled only� and inwhich the able individuals have quasi�linear preferences�

The Cost�Minimizing Welfare Scheme

Suppose g� c� and � are all given� with g � c�� Fixing � may be interpreted as having atarget level for the proportion of the disabled that are entitled to the disability bene�t� Italso allows our approach to be compared with those in which � is exogenously given� suchas Parsons �� The objective of �nding the least�cost way of delivering given levels ofsupport to the disabled is analogous to the criterion adopted by Besley and Coate ��

However� AW �� C�M��M since there is no point in auditing �setting q � �� to get noinformation �� Note that the e�ort constraint �� is satis�ed for any q if � � ��Hence� AW �� has a discontinuity at � � �� In fact� AW �� This discontinuity isrepresented by the empty point at C�� on the vertical axis in Figure ��

��

�� in analyzing the use of workfare to screen workers of di�erent ability types in welfareschemes��

In Regime W � total welfare program expenditures are given by�

TW � EW � �MAW ��

� Mg �M � � ��c� �M� c� � y�� M C� � Y�� MAW ��

where EW is the amount of transfers net of taxes required to �nance the given level ofbene�ts to the tagged welfare recipients� Recall� using our notation �� and �� that �� so that one half of welfare applicants are tagged� a proportion � of whichare disabled� Equation �� re�ects this� Similarly� in Regime N � total welfare programexpenditures are�

TN � EN � �MAN �� Mg �Mc� �M C� � Y�� MAN ��

Given g� c� and �� for each Regime the government will choose the values for itsremaining instruments� c�� y�� C�� Y�� to minimize TW or TN subject to the self�selectionconstraints�

V � c�� y�� V � c�� a�

V � C�� Y�� V � c�� y�� b�

Constraint ��a� is relevant only in RegimeW since it ensures that low�ability workers willprefer to work when participating in the general welfare program� Constraint ��b� mustbe taken into account in both Regimes W and N � although c�� y�� should be replaced by c�� in Regime N � The minimum income constraint also requires y� � y in Regime W � We assume that Y� � y is not binding in the optimum� implying that type � persons havesu�ciently high ability��

It is worth highlighting an informational implication of the restriction y� � y� Inabsence of the agency problem� the government may prefer low�ability persons not to workbecause their ability is too low or because the self�selection constraint for the high�abilitypersons is tightened� But when agency problems are present� the government may want toinduce the low�ability types to work to detect type I errors without auditing� even thoughit may be costly to do so� the disabled and the low ability individuals need to be separatedin the general welfare� We return to this issue below� First we consider the governmentspolicy choices for each of the Regimes in turn�

�� In their model� which does not contain social workers� the criterion is to minimize the costof delivering a given level of income to the poor� A slightly more general criterion whichassumes a continuous distribution of abilities and maximizes an index of the extent to whichhouseholds fall below some �poverty level� of consumption is in Kanbur� Keen and Tuomala��

��

The Lagrangian expression in Regime W can be written as�

LW c�� C�� y�� Y�� Wi � � TW�W� y� � y� � �W�

�V � c�� y�� V � c��

�� W�

�V � C�� Y�� V � c�� y��

�where TW is given by �� The �rst�order conditions with respect to the governmentspolicy instruments are�

c� � �M � �W� V �c � �W�

bV �c � � ��a�

y� � ��M � W� � �W� V �y � �W� bV �

y � � ��b�

C� � M � �W� V �c � � ��c�

Y� � �M � �W� V �y � � ��d�

where bV �c �bV �y represent the marginal utility of a high�ability mimicker with respect to argu�

ment c�� y�� A similar convention will be used below for low�ability types when mimickingthe disabled� From ��c� and ��d�� we immediately obtain that standard result that themarginal tax rate at the top should be zero� �V �

y �V�c � �� It is also straightforward to

show that low�ability individuals face a marginal tax rate that is between zero and ��percent if y� � y is not binding�� V �

y �V�c � �� Otherwise� the sign of the marginal

tax rate is ambiguous at y� � y� It may be negative� re�ecting the need for a wage subsidy

to induce low ability persons to choose y� � y��

Denote the minimum value function of the governments problem in Regime W byTW �� g� c�� EW �� g� c�� MAW �� Applying the envelope theorem yields��

�EW

�� c��y��c��M � � if y� � y�

�EW

�g� M �

�EW

�c�� M��W� bV �

c �M�

��

�� The marginal tax rate is not uniquely de�ned� Following Stiglitz �� an implicit marginal

tax rate can be de�ned by � � V �y �V

�c �

�� Combining ��a� and ��b� yields�

�W� V �c

��

V �y

V �c

�� W� bV �

c

��

bV �ybV �c

�� W�

The �rst term is always positive due to the single�crossing property� If �W� � � and thusy� � y is binding� the second term may dominate the �rst one� which causes the marginal

tax rate to be negative�

�� The �rst inequality obtains since y� � y implies MRS�yc � � when evaluated at �c�� An increase in y� beyond zero then allows the government to decrease its transfer to thelow�ability individuals without violating their self�selection constraint� The third inequalityfollows from the fact that if leisure is not inferior� V i

cy � �� we then have bV �c � V �

c � By ��a��

�W� bV �c � �W� V �

c � �M � which leads to �EW ��c� � M �

��

Thus� a one dollar increase in c� increases net costs by more than one dollar per generalwelfare recipient because such an increase tightens the self�selection constraints for thetwo types of individuals�� And� increasing � will reduce costs if y� � y is not binding� Butit could increase costs if y� � y� the net payment for the untagged able c� � y� might be

higher than c� paid to the untagged disabled��

The Lagrangian expression for Regime N is�

LN C�� Y�� N� � � TN � �N�

�V � C�� Y�� V � c��

�where TN is given by �� The �rst�order conditions with respect to the policy instruments C�� Y�� again yield ��c� and ��d�� implying a zero marginal tax rate at the top� Asabove� let TN �� g� c�� EN �� g� c�� MAN �� be the minimum value function of thegovernments problem� From the envelope theorem�

�EN

��

�EN

�g� M

�EN

�c�� M � �N�

bV �c �M� ��

From the expressions for LW �� and LN �� it is obvious that for a given �� theadministration costs have no e�ect on the cost�minimizing structure of the income taxschedule and the welfare program� On the other hand� whether the restriction y� � yis binding in Regime W does have an impact� First� suppose y� � y� This implies thatbunching is ine�cient and the low�ability individuals should be working in Regime W �Then� regardless of the di�erence between AW �� and AN �� separating the householdsin the general welfare program always reduces the cost of transfers net of taxes� that is�EW �� g� c�� EN �� g� c�� Next� consider the case where the optimum requires y� � y�

The comparison between EW �� and EN �� becomes ambiguous� y� � y may be excessivefor the low�ability persons and the government would have preferred to set y� below y� but

cannot� Therefore� EW �� g� c�� EN �� g� c�� could hold� separating welfare recipientsis more costly than allowing bunching at the bottom� To summarize�

Lemma �� Given �� g� and c�� if the minimum income constraint y� � y is not binding

in the optimum of Regime W � then EW �� g� c�� EN �� g� c�� Otherwise� it can be thecase that EW �� g� c�� EN �� g� c��

�� To illustrate conditions under which the minimum income requirement is binding� suppose

preferences are quasi�linear� V i�c� y� � c � hi�y�� Then� applying the envelope theorem tothe problem of Regime W � we obtain�

M�� LW

�y�� h��y�� h��y�� h��y��

where the second�order conditions are satis�ed if h��y� � h��y� and h��y� � h��y�� If�� h��y� � � � h��y� for the given �� then �LW ��y� is strictly positive for any y� � y�

which implies that in the optimum� y� � y�

�� In other words� �LW ��y�jy�y � ��

��

From Proposition �� we know that for � � �� AW �� AN �� On the other hand�if � is su�ciently low� we know from Proposition � that AN �� AW �� By makinguse of Lemma � along with these propositions� TW �� g� c�� can be evaluated relative toTN �� g� c�� The comparison depends on whether y� � y is e�ective in the optimum of

Regime W � If y� � y in the optimum� EW �� EN �� by Lemma �� For � such that

� � �� it must be that TW �� g� c�� TN �� g� c�� so it is cost�minimizing to induce thelow�ability persons to work in the general welfare system� For su�ciently small �� thecomparison becomes ambiguous� If the di�erence in the administration costs outweighsthe ine�ciency caused by bunching at the bottom� Regime N will be less costly thanRegime W �

Suppose now that y� � y� Then� from Lemma �� EW �� EN �� may be the case� This

possibility implies that even if � � � so that AW �� AN �� the relationship betweenTW �� and TN �� may be ambiguous� When � is su�ciently low� so AW �� AN ��and if EW �� g� c�� EN �� g� c�� then it is clearly the case that TW �� TN �� Thefollowing proposition summarizes this discussion�

Proposition ��

i� Suppose y� � y in the optimum of Regime W� Then� for � � �� Regime W is lesscostly than Regime N � For � small enough� it is possible that Regime N is less costlythan Regime W �

ii� When y� � y causes EW �� EN �� even for � � �� TW �� TN �� may be possible�while this is de�nitely the case for a su�ciently small ��

The Pareto�Ecient Welfare Scheme

We now turn to the more general problem of a government maximizing a welfare functionof the form�

SW � �V � g� � �� V � c�� V � g� �� V � c�� y��

�� V

� C�� Y��

where c�� y�� reduces to c�� in Regime N � This can be interpreted as the objectivefunction that would result in an allocation of resources along the economys second�bestutility possibilities frontier UPF�� To narrow the range of alternatives� assume that � �� implying that redistribution goes from the able to the disabled individuals�� This�in turn� ensures that the incentive constraints used in the previous sections and subsectionsare still the relevant ones� To begin with� we hold � �xed and characterize the optimalpolicy in each regime� given �� We subsequently return to the choice of � in each Regime�

�� This is analogous to assuming that the objective function SW corresponds with a socialwelfare function that is a function of expected utilities and is increasing and quasi�concave�An alternative� more complicated� formulation would treat social welfare as a function ofex post utilities and attach separate welfare weights to the utility of each household in eachstate�

��

In RegimeW � the government chooses fg� c�� c�� y�� C�� Y�g to maximize SW given by �� subject to

g � c� W� �

y� � y W� �

V � c�� y�� V � c�� W� �

V � C�� Y�� V � c�� y�� W� �

M Y� � C�� M� y� � c�� M � � ��c� �Mg � �MAW �� W� �

where the multiplier assigned to a constraint appears as its equation label� For a given�� the administration costs are already minimized and summarized in AW �� depicted asthe U�shaped curve in Figure �� From the �rst�order conditions to this problem shownin the Appendix�� the optimal policy in Regime W has the following properties� As inthe cost�minimization problem� the marginal tax rate is zero at the top� �V �

y �V�c � ��

If the constraint y� � y is not binding� the marginal tax rate is positive at c�� y�� If

it is binding W� � �� the marginal tax rate for low�ability individuals may be negativeso as to induce them to supply a positive amount of labor� The optimal values for thegovernments instruments are denoted gW �� cW� �� and so on� These yield the maximumvalue function SWW �� The following Lemma relating gW �� and cW� �� is proven in theAppendix�

Lemma �� In the optimum of case W � gW �� cW� ��

In Regime N � the government chooses fg� c�� C�� Y�g to maximize

�V � g� � �� V � c�� V � g� �� V � c��

�� V

� C�� Y��

subject to

g � c� N� �

V � C�� Y�� V � c�� y�� N� �

M Y� � C��Mc� �Mg � �MAN �� N� �

Denote the solutions to this problem as gN �� cN� �� etc� The maximum value functionfor this problem is SWN �� Using a proof similar to that of Lemma �� we can derive thefollowing�

Lemma � gN �� cN� �� for any value of ��

In the �nal stage of each Regime� the government optimizes SWW �� or SWN ��with respect to �� For Regime W � the envelope theorem allows us to obtain the �rst�order

��

condition for the optimal �� denoted by �W ��V �

�gW ��

�� V �

�cW� ��

��

�V �

�gW ��

�� V �

�cW� �� yW� ��

�� W� M

�cW� ��

�cW� �� yW� ��

�� W� �M

�AW ��

��

The left�hand side is the marginal bene�t from increasing the accuracy if tagging� whilethe right�hand side represents the marginal administrative cost from so doing� The �rstterm of the left�hand side is positive by Lemma �� The second is negative since �W� � �in the optimum� which implies V � cW� � yW� � � V � cW� � �� V � gW � �� The assumptionV �c c� � V �

c c� �� along with � � � ensures� however� that the sum of the �rst and secondterms is positive�� To evaluate the third of the left�hand side� note that in the generalwelfare program if yW� �� y� the transfer to the disabled individuals is larger than the

net� one to the low�ability individuals� Thus� in this case� cW� �� cW� �� yW� �� anincrease in � decreases general welfare payments� On the other hand� if yW� � y� it may

be the case that cW� �� cW� �� yW� �� to induce low�ability persons to work and earnat least y in the general welfare program� the increment of transfers they receive may haveto be greater than the amount of output of income they earn� The right�hand side showschange in AW when � increases� It is worth noting that if y� � y is not binding� the

left�hand side of �� is unambiguously positive� That implies that �AW �� at� � �W if yW� �W � � y� This is depicted in Figure �� where �W is assumed to be in the

rising portion of the AW �� curve�� If the minimum income constraint y� � y is binding�

it is conceivable that the left�hand side of �� is negative� in which case �W is on thefalling portion of the AW �� curve� The solution to �� yields a value of social welfareSWW �W ��

Analogously� the necessary condition for the optimal � in Regime N is given by��V �

�gN ��

��V �

�cN� ��

��

�V �

�gN ��

��V �

�cN� ��

�� N� �M

�AN ��

��

�� To see this� note that the �rst two terms in �� can be rewritten as��V �

�gW ��

�� V �

�cW� ��

��

�V �

�gW ��

�� V �

�cW� ��

��

Z gW ��

cW��

�V �c �c�� V

�c �c� ��

�dc

Obviously� the right�hand side is positive� and so is the left�hand side�

�� It is technically possible that the curve AW �� be declining throughout its range� In that

case� �W � � as long as y� � y is not su�ciently binding� We do not discuss that case

further� though it can also give rise to either Regime being preferred�

��

Since individuals are not separated in the general welfare program� a marginal change in� does not a�ect the payments in that program� Given that gN �� cN� �� by Lemma�� the �rst term of the left�hand side is positive and the second term is negative� The theright�hand side is clearly positive since administration costs are unambiguously increasingin � in this Regime see Figure �� Let �N be the solution obtained by solving �� Atthis optimum� social welfare is given by SWN �N ��

It is di�cult to compare SWW �W � and SWN �N � in general� But it is possibleto stipulate circumstances under which Regime W will be preferred over Regime N � andvice versa� For example� suppose that �N is greater than �� For this range of �� wehave seen that AW �� AN �� In addition� if y� � y would not be binding in Regime

W when � � �N � then e�ciency can be improved by separating the individuals in thegeneral welfare program� That is� starting in the optimum of Regime N � social welfare canbe improved upon by moving to Regime W � Therefore� SWW �N � � SWN �N �� And�since SWW �W � � SWW �N �� we can conclude that SWW �W � � SWN �N � and thatRegime W is superior to N �

On the other hand� there are two possible circumstances in which Regime N might bepreferred to Regime W � Suppose �rst that the minimum income constraint y� � y is not

binding� but that �N is low enough such that AN �N � � AW �N �� Then� it is may be thatAN �N � � AW �W �� so administrative costs are lower in Regime N than in Regime W �This case is depicted in Figure �� If the saving in administrative costs by adopting RegimeN exceeds the bene�ts from the better information �W � �N� and from separating thelow�ability persons from the disabled in the general welfare program in Regime W � theformer will be preferred� Alternatively� suppose that y� � y is binding in the optimum ofRegime W � This makes it more costly to separate low�ability types from the disabled� andmay make Regime N better than Regime W even if AN �N � � AW �W �� So it is possiblethat Regime N is superior to Regime W SWN �N � � SWW �W �� Generally� there willbe some y large enough to make Regime N superior to Regime W � To summarize�

Proposition �

i� If �N � � and yW� �N � � y� then Regime W is superior to Regime N SWW �W � �

SWN �N ��

ii� If �N is su�ciently low or if yW� �W � � y is su�ciently binding� it is possible that

Regime N is superior to Regime W SWN �N � � SWW �W ��

It is obviously di�cult to be more precise about when Regime W or Regime N wouldbe preferred� That depends upon the relevant parameters of the economy� To illustratethe possibility that either regime might be preferable we next provide a simple illustrativeexample�

An Illustrative Example

A simple example illustrates the properties of an optimal welfare program alongside anoptimal income taxation and provides a more concrete welfare comparison between the

��

two Regimes� Suppose that all able persons have quasi�linear preferences of the formV i c� y� � c�hi y� i � �� where hi y� is the disutility of supplying income and has thefollowing properties h�� y� � h�� y�� h

�� y� � h�� y� and h� �� h� �� This would be

consistent with both persons having the same convex disutility�of�labor function h L�� suchthat hi y� � h y�wi�� The utility function of the disabled is assumed to be logarithmic�V � c� � log c� Suppose also that the government cares only about the disabled� so theobjective function is the expected utility of disabled persons� � log g� �� log c�� Aftersome manipulation� we can obtain the maximized expected utilities of the disabled� given�� for each Regime by�

SWW �� log � � �� log

��

�

�� log

�V��H �� y� � �AW ��

� ��a�

SWN �� log� � �� log

��

�

�� log

�V�� AN ��

�� b�

where V�� maxfy � h� y�g and H �� y� � maxy��yf�y� � � � ��h� y�� h� y��g� The

termH �� y� can be interpreted as the net reduction in cost from separating the low�abilityfrom the disabled in the welfare system�

We can con�rm with numerical examples that either regime N or Regime W maybe preferred under reasonable circumstances� Let the cost of auditing a proportion qof R welfare recipients be given by C qR� � � qR�� For simplicity� we normalize thenumber of individuals of each type to unity i�e�� M � �� Also� suppose that to obtainaccuracy �� social workers have to exert e�ort costing � � � �� Finally� suppose thatto obtain income y� able individuals have to incur a cost hi y� � i��y�� with � � �re�ecting the fact that low�ability individuals are less e�cient at generating income thanhigh�ability individuals� Thus for the able persons� V i c� y� � c � i��y�� while for thedisabled V � c� � log c� By specifying parameter values for �� and y� we use

equations ��a� and ��b� to calculate the values of �W and �N that maximize SWW andSWN in the two Regimes� Consider the following three cases�

i� SWW � SWN

Suppose that the parameters take the following values� � � �� and � � �� Initially let the minimal income requirement y be very low say � �� so it is

not binding� Then� social welfare in the optimum of Regime W � reached at �W � �� iswell above social welfare in the optimum of Regime N � reached at �N � �� SWW �� SWN � �� Thus� Regime W is superior to Regime N for those parameters� Thevalue for � turns out to be �� implying that the e�ort incentive constraint is binding inthe optimum of Regime W �

ii� SWN � SWW � y� � y

Now suppose that the minimal income requirement y is increased to y � �� but all otherparameter values are kept the same� The minimal income requirement then becomes a

��

binding constraint for the government� In this case� social welfare in the optimum ofRegime N � which is unchanged� is above social welfare in the optimum of Regime W SWW � �� SWN � �� We also obtain �W � �� so the e�ort incentiveconstraint is binding�

iii� SWN � SWW � y� � y

As mentioned above� social welfare could be larger in Regime N even when y� � y isnot a binding constraint� If so� the superiority of Regime N over Regime W comes fromthe tightness of the e�ort incentive constraint when � is su�ciently low� Consider thefollowing parameter values� � � �� and � � �� Suppose againthat the minimal income requirement is not constraining y is chosen to be small�� Then�

SWN � �� compared with SWW � �� In the optimum of the two regimes� �N � ��while �W � �� It is interesting to note that the level of social welfare here�expected utility of the disabled� that would be obtained by dispensing with welfare andrelying solely on a non�linear negative income tax system is lower than SWN in this case� Social welfare is �� which is above SWW but below SWN �� This is so despite the factthat under the negative income tax system� there would be no bunching at the bottom�unlike with Regime N �

Given these various bene�ts and costs� whether or not Regime W is preferred toRegime N is obviously an empirical issue� Reasonable circumstances can be imaginedin which either Regime is the preferred one� Moreover� whether or not either Regime ispreferred to a simple negative income tax system that does not take advantage of tagging isalso an empirical question� It depends upon whether the costs of administering the taggingsystem� AW or AN � outweigh the advantages of tagging in terms of better targeting oftransfers to the needy�� One interesting point to note is the following� It is well�knownthat in a negative income tax scheme� it may be e�cient not to have the low�ability peopleworking�� But is may be the case that Regime W dominates either the negative incometax or RegimeN � That is� it may be optimal to induce low�ability persons to work in orderto implement tagging in the most e�cient way� And this may be so even if the constraintyW� �� y is binding� The reason is that forcing the low�ability persons to work allowsthe government to detect type I errors and therefore reduces the cost of monitoring the

� Note that if Y� � � in the optimum of a single negative income tax system �so there is bunch�ing at the bottom�� the introduction of a welfare program is unambiguously welfare enhanc�ing� Let �CI

� � CI� � Y

I� � �� and �CI

� � YI� � be the solution to the government optimization

when only the negative income tax is operated� Keep the bundles unchanged and introduce aRegime N welfare program with � � �� g � c� � CI

� �� Since �� the high�ability per�son will not apply for the disabled bene�t while the disabled and the low�ability person mayapply� Thus the initial allocation is still incentive compatible� Note that AN�� so theinitial allocation is feasible after the welfare program is introduced� For � � �� by Lemma�� we know g � c� in the optimum� which implies SWN �� SW �negative income tax��We have MaxfSWW ��W �� SWN��N �g � SWN�� and thus we can get the desired result�

�� This was apparent in Mirrlees �� original analysis�

��

social workers� By the same token� the converse is true� Regime N may be preferred evenif under the negative income tax system it would be e�cient for the low�ability personsto work� Thus� the various bene�ts and costs of employing a tagging technology allowvirtually anything to happen�

V� EXTENSIONS

Our analysis has been conducted in the simplest of settings� It is worth contemplatinghow the structure of welfare programs and the robustness of our results might be a�ectedby introducing some natural complications and extensions to the model� In this sectionwe brie�y consider the implications of the following factors� altruistic preferences for thedisabled among the social workers� workfare and allowing tagged persons to work� Thediscussion will be primarily intuitive and will rely on lessons that have been learned fromthe more formal modeling of the previous sections�

Altruistic Social Worker Preferences

It is reasonable to suppose that social workers may attach particular weight to the welfareof the disabled� this may have motivated the choice of social work as a profession in the �rstplace� For simplicity� suppose such altruism does not extend to the low�ability persons�The existence of altruism can a�ect both the tightness of the e�ort incentive constraintand the choice of e�ort by the social worker� Consider the former �rst� The social workermust decide whether or not to exert e�ort before knowing anything about an applicant�Ex ante� the probability of a given applicant being disabled is one�half� If an applicantis accepted without tagging� the expected altruistic bene�t to the social worker is simplyV � g�� This must be added to EUA de�ned earlier to determine the expected utilityof accepting all applicants and exerting no e�ort� Alternatively� suppose that the socialworker participates in the tagging technology� Again� ex ante the probability that a givenclient will be disabled is one�half� and therefore the probability that a tagged applicant isactually disabled is �� Similarly� the probability that an untagged applicant is disabledis � � �� Thus� the expected altruistic bene�t to the social worker per applicant is�V � g�� V � c�� This must be added to EUW given by �� to determinethe expected utility of engaging in the tagging technology� Since g � c�� the altruisticpart is larger when the social worker accepts all applicants than when he uses the taggingtechnology� This implies that altruism will cause the social worker to have more incentiveto accept them than when he is sel�sh� which in turn implies that the e�ort incentiveconstraint is tightened in Regime W and it can even make it e�ective in Regime N��

�� The presence of altruism makes the lowest levels of � non�implementable in both Regimes�

First consider Regime W and recall that �qW �� when the social worker is completelysel�sh� q � � equates the non�altruistic component of expected utility from tagging with thatfrom accepting all applicants� In the present context� however� the gain from the altruisticcomponent in EUA is still higher than the one in EUW for � � �� which leads to EUA �EUW �� even with q � �� The same is true for Regime N � the non�altruistic component is

��

Besides a�ecting the e�ort constraint� altruism for the disabled can also in�uence thesocial workers choice of e�ort in tagging� It is easy to see that an increase in � raisesthe value of the altruistic component by V � g� � V � c�� and this must be added to�EU j�� j � W�N�� This implies that q and k � S need not be set so high relative to theone in the basic model to implement a given �� which in turn implies lower administrationcosts in both Regimes for � such that the e�ort constraint is not e�ective� In addition�since �EU j�� j � W�N� is increasing in g and decreasing in c�� higher lower� g orlower higher� c� brings more less� e�ort by the social worker�

Overall e�ect of the altruistic motive on administration costs after the governmentsolves the cost minimization for given �� income tax and social policy as described insection III� is not straightforward in either Regime� To summarize� an increase in g and�ora decrease in c� tightens the e�ort constraint� while encouraging e�ort given that the socialworkers use tagging� Therefore� administration costs can be in�uenced by g and c� as wellas by ��

Workfare

As noted by Besley and Coate �� requiring welfare recipients to work may serve as ascreening device to separate the needy from other potential bene�ciaries� In our context�workfare has somewhat di�erent implications� Suppose that all untagged applicants arerequired to work a certain amount of hours� say y� � y� to qualify for the welfare transfer of

c��y�� The untagged disabled cannot work so cannot receive welfare� they are left to thenegative income tax system where they receive the bundle C�� When the self�selectionconstraint V � c�� y�� V � C�� is satis�ed� the untagged households are separated asin Regime W � But there is critical di�erence between Regime W and workfare� In theformer� by observing income y� � �� the government can detect type I errors by eachsocial worker since it can identify the social worker who is responsible for the error� In thepresent context� however� by observing Y� � � in the income tax system� the governmentcannot trace the source of the type I error without auditing because the tie between thesocial worker and the untagged disabled person is broken� Therefore� the informationalstructure is the same as in Regime N � and the minimized administration cost for a given� is AN �� under workfare� Except for the fact that c� is replaced by C�� we obtain thesame set of participation and self�selection constraints as in Regime W �

When the minimum income constraint y� � y is not binding for a given � so it ise�cient to separate the low�ability from the disabled� workfare is advantageous relativeto Regime N � it can achieve the bene�ts of separating while keeping the administrationcost unchanged from Regime N � In our example in the previous section� for a given �� the

the same for tagging as for accepting both for � � �� while the altruistic gain is higher forthe latter than for the former�

�� We assume the labor done by the participants in the workfare is productive� in contrast withBesley and Coate �� who assume it has no productive value�

��

maximized social welfare under workfare is given by�

SWWF ��

�� log � � � � �� log

��

�

�� log

�V��H �� y� � �AN ��

�The comparison between SWWF �� and SWN �� is solely dependent on the sign ofH �� y�� If H �� y� � � that is� the net gain from inducing the low�ability persons towork is positive�� workfare is preferable to Regime N � and vice versa� On the other hand�the relation between SWWF �� and SWW �� is determined by the size of AN �� rela�tive to AW �� It is immediately apparent that SWWF �� SWW �� if and only ifAN �� AW ��

Once � is optimized for each Regime� welfare comparisons are limited and generallyambiguous as in the basic model� Let �WF be the optimal level of � under workfare� We cansay that SWW �W � � SWWF �WF � holds if AW �W � � AN �W �� And� SWW �W � �SWWF �WF � if AW �WF � � AN �WF �� Thus� as in Besley and Coate �� workfarecan be a useful policy instrument in the right circumstances� Of course� the case forworkfare will be weakened if the required work is less unproductive than market work�And� it can be strengthened if the workfare requirement allows a smaller amount of workthat the minimum required to earn y in the private sector�

Tagged Persons Allowed to Work

We have so far considered the case where the welfare recipients can work in the generalwelfare program RegimeW � but tagged persons must accept the same bundle g� �� in thedisability program� The presence of type II error may justify allowing welfare recipients towork in the disability bene�t� Parsons �� who abstracts from administration costs andwho assumes that persons who work have no control over the amount they work and earn�shows that it may be e�cient to allow tagged persons to work if the cost of separating theable from the disabled persons is not too great� The latter depends upon the preferencesand ability of the tagged able persons� and upon the amount of work e�ort required� In ourmodel� we have the additional complications of administrative costs and variable earnings�Following the convention of Parsons� let us refer to the case where people are separated inall three programs disability� general welfare� and income tax� as �triple negative incometaxation and denote it by superscript �T �

The bene�t of o�ering two bundles g�� and g�� yT� � with yT� � y� in the disabilitybene�t and inducing the low�ability persons to choose the latter bundle is that it enablesthe government to detect type II errors� In RegimeW � the government can already identifytype I error� so now both types of errors can be detected without auditing� In this triplenegative income tax system� the expected utility of each social worker when using the tag�ging technology is given by EUT �� k�� Alternative strategies� such as acceptingall applicants� are not useful now since all errors can be detected� Therefore� the gov�ernment is not confronted with the e�ort incentive constraint� Minimized administrationcosts for a given � is AT ��

Despite the fact that both type I and type II errors can be traced automatically�administration costs AT �� di�ers from that under full information � �� This is due to

��

the constraint S � k� When the restriction of y�� yT� � y is either not binding in theoptimum or not so signi�cant� the triple negative income tax system is preferred to otherwelfare schemes� It should be emphasized that the advantage of this sort of income taxsystem is not only in improving the allocation by the separation of households� but also insaving the administration cost by improving information�

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APPENDIX

Proof of Lemma �� Combining �� and �� it is possible to obtain�

AN �� q� �AW �� q� � � � �� q��

q� �� q��

��

� � q

��

q � � q� � � q� � � � �� q�� q � �� q��

��

q � � q� � �� q� � � � �� q��

��

q � � q� �� q� � �

Therefore� AN �� q� �AW �� q� � �� QED

Proof of Proposition �� For � � �� qW �� q�W �� Using this�

AN �� AN �� qN �� AW �� qN �� AW �� q�W �� AW ��

The inequality comes from Lemma �� QED

Proof of Lemma �� The �rst�order conditions in Regime W are�

g � �V �c g� � � �� V

�

c � W� � �W� M � � A � ��

c� � � � ��V �c c�� W� � �W�

bV �c � �W� �� M � � A � ��

c� � ��V�c � �W� V �

c � �W�bV �c � �W� �M � � A � ��

y� � ��V�y � W� � �W� V �

y � �W�bV �y � �W� �M � � A � ��

C� � �V�c � �W� V �

c � �W� M � � A � ��

Y� � �V�y � �W� V �

y � �W� M � � A � ��

where V�

c � V � g� �� and bV ia is the marginal utility of a type i mimicker with respect to

argument a� From A � �� and A � ��

� � ��V �c c�� V

�c � W� � �W� bV �

c � V �c � � �W�

bV �c � �W� M � �

Subtracting A � �� from this equation gives�� V �

c c�� V�c

��V �

c g� � � � � �� V �c

�� W� � �W� bV �

c � V �c � � �W�

bV �c �

Since c� � c� because of the incentive constraint V � c�� y�� V � c�� and V �cy � ��

the right�hand side is strictly positive� V �c c�� y�� V �

c � bV �c � V �

c c�� Suppose thatg � c� in the optimum� Then the left�hand side is�

�� V �c c��

��V �

c c�� y�� V �c c��

��

�V �c c�� V

�c c��

��

where the inequality obtains because V �c c�� y�� V �

c c�� Because � � �� and V �

c c� � V �c c� �� it is clear that � � ��

�V �c c�� V

�c c��

�� This� in turn�

implies that left�hand side becomes non�positive� This� however� contradicts the fact thatthe right�hand side is strictly positive� QED

Proof of Lemma � In Regime N � the �rst�order conditions for g� C� and Y� are thesame as A � �� A � �� and A� �� The �rst�order condition with respect to c� is�

c� � � � ��V �c c�� V

�c � N� � �N�

bV �c � �N� M � �� A � ��

The proof is analogous to the one of Lemma �� QED

��

REFERENCES

Akerlof� G�A� �� The Economics of �Tagging as Applied to the Optimal Income Tax�Welfare Programs� and Manpower Training� American Economic Review ��

Besley� T� and S� Coate �� Workfare versus Welfare� Incentive Arguments for WorkRequirements in Poverty�Alleviation Programs� American Economic Review ��

Besley� T� and S� Coate �� The Design of Income Maintenance Programs� Review of

Economic Studies ��

Cremer� H� and F� Gahvari �� Tax Evasion and the Optimal General Income Tax�Journal of Public Economics � � ��

Diamond� P� and E� Sheshinski �� Economic Aspects of Optimal Disability Bene�ts�Journal of Public Economics ��

Duclos� J��Y� �� Modeling the Take�Up of State Support� Journal of Public Eco�

nomics ��

Kanbur� R�� M�J� Keen� and M� Tuomala �� Optimal Non�Linear Income Taxationfor the Alleviation of Income Poverty� European Economic Review ��

Killingsworth� M�R� �� Labor Supply Cambridge� Cambridge University Press��

Mirrlees� J�A� �� An Exploration in the Theory of Optimum Income Taxation� Reviewof Economic Studies ��

Parsons� D� O� �� Imperfect �Tagging in Social Insurance Programs� Journal ofPublic Economics ��

Stiglitz� J�E� �� Self�Selection and Pareto E�cient Taxation� Journal of Public Eco�

nomics ��

Stiglitz� J�E� �� Pareto E�cient and Optimal Taxation and the New New WelfareEconomics� in A�J� Auerbach and M�S� Feldstein eds�� Handbook of Public Economics�Vol� � Amsterdam� North�Holland��

van de Walle� D� and K� Nead �� Public Spending and the Poor� Theory and Evidence

Baltimore� Johns Hopkins University Press��

��

An Agency Model of Welfare and Disability Assistance

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