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The optimal taxation of families Brett, Craig 2005

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The Optimal Taxation of Families by Craig Brett B.A., Mount Allison University, 1991 M.A., The University of British Columbia, 1992 A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF , DOCTOR OF PHILOSOPHY in THE FACULTY OF GRADUATE STUDIES (Department of Economics) We accept this thesis as conforming t^the recpihred standard  THE UNIVERSITY OF BRITISH COLUMBIA May 1996 © Craig Brett, 1996  In presenting  this  degree at the  thesis  in  University of  freely available for reference copying  of  department  this or  publication of  partial fulfilment  of  British Columbia,  I agree  and study.  his  or  her  Department The University of British Columbia Vancouver, Canada  DE-6 (2/88)  that the  representatives.  may be It  this thesis for financial gain shall not  permission.  requirements  I further agree  thesis for scholarly purposes by  the  that  advanced  Library shall make it  by the  understood be  an  permission for extensive  granted  is  for  that  allowed without  head  of  my  copying  or  my written  ABSTRACT  T h i s thesis presents an analysis of two classical problems in the theory of optimal taxation: commodity tax reform and nonlinear income taxation. Economic behavior is modeled as arising out of a family decision making process rather than owing to individual utility maximization. T h e taxation authority is assumed to have no direct control over intra-family allocations of ^resources. In this way, family interactions change the nature of the second-best constraints the planner faces. T h e analysis focuses on the impact of these constraints on optimal policy choices.  Attention is focused on families with two  members, whom the planner can (in most situations that are modeled) tell apart.  In the chapters dealing with commodity tax reform, behaviour is modeled as the Pareto-efficient outcome of a family decision process.  Conditions for the existence of  a feasible, Pareto-improving tax change are presented and contrasted with those that obtain in the individualistic case. T h e consequences of treating households as a single individual are also discussed. It is shown that treating families as if they were individuals can lead to misleading conclusions.  A n example is presented to demonstrate that the  traditional analysis may go wrong even when families behave as if they are individuals. Moreover, it is shown that household budget data alone is insufficient to address this issue. T h e model is then put to use to address question of temporary inefficiencies in tax reform. I present how the circumstances under which temporary inefficiencies can arise vary with the structure of poll taxes.  ii  T h e problem faced by a planner choosing an income tax schedule for families is modeled as a multi-dimensional screening problem. Families are described by a twodimensional vector of characteristics, interpreted as the labour productivities of their members. T h e planner cannot observe these characteristics directly. Furthermore, families are free to redistribute the after-tax incomes of their members. T h e planner must take this behaviour into account when choosing the tax schedule. A description of the possible Pareto-efficient mechanisms is given. T h e implications of a standard redistributive assumption on the sign of marginal tax rates are explored.  In contrast to unidimen-  sional taxation models, the redistributive assumption does not imply that marginal tax rates are everywhere non-negative.  For much of the analysis, the usual assumption of  quasi-linear preferences is jettisoned, allowing an exploration of the implications of this additional structure. T h e qualitative features of optimal tax- schedules are discussed. It is concluded that neither individual-based taxation nor taxation based solely on total family income is optimal.  iii  CONTENTS  Abstract  ii  List of Figures  vi  Acknowledgement  vii  Chapter 1: Family Economics and Family Taxation  ,  1  1.1: Introduction  1  1.2: Models of Family Decision-Making  3  1.3: Issues of Ethics and Information  8  1.4: A n Overview of T h i s Study  11  Chapter 2: T a x Reform and Collective Family Decision-Making  17  2.1: Introduction  17  2.2: Collective Family Decision-Making  20  2.3: General Equilibria  23  2.4: Optimal Policy Changes  26  2.5: Implementation  37  2.5.1: Consequences of Ignoring Family Interactions Chapter 3: Temporary Inefficiencies and Demogrants  38 42  3.1: Introduction  42  3.2: Unrestricted Poll Taxes  43  3.3: Restricted Poll Taxes  47  iv  Chapter 4: Optimal Non-linear Taxes for Families  59  4.1: Introduction  59  4.2: T h e Model  64  4.3: Self-Selection  71  4.4: Properties of Optimal Tax Schedules  86  Chapter 5: Optimal Non-linear Taxes: Some Special Cases  96  5.1: Introduction  96  5.2: Redistributive Taxation  97  5.2.1: T h e Usual Cases  99  5.2.2: T h e Unusual Cases  101  5.2.3: A Summary  103  5.3: T h e Consequences of Asymmetric Family Decisions  105  Conclusion  110  References  113  Appendix A  118  Appendix B  144  v  List of Figures  Figure 1. T h e Space of Family Types  144  Figure 2. Monotonicity Properties Implied by Self-Selection  145  Figure 3. Monotonicity Properties Implied by Self-Selection  146  Figure 4. Partial Monotonicity for Families H H and L L  147  Figure 5. T h e Lack of Attribute Ordering  148  Figure 6. Allocations for Families not Ordered by >j?  149  Figure 7. A n Implication of a Binding Self-Selection Constraint  150  Figure 8. A n Example of W h e n A Zero Marginal Tax Rate is O p t i m a l  151  vi  ACKNOWLEDGEMENT  I would like to thank my supervisory committee, Charles Blackorby, John Weymark and  D a v i d Donaldson, for all the help they have provided to me in writing this thesis.  Charles Blackory kept this thesis on track. I have benefited greatly from his comments, suggestions and encouragement. John Weymark has provided detailed comments on this thesis, adding much to the breadth of the analysis and to the clarity of exposition. D a v i d Donaldson has been a seemingly limitless source of ideas and encouragement.  Some of the work on this thesis was done while I visited D E L T A , Paris, during the Fall of 1994. I am indebted to its director, Roger Guesnerie, for the hospitality he has shown to me. While at D E L T A , I received helpful comments and encouragement from Pierre-Andre Chiappori.  M a r c Duhamel, Guofu T a n and Michael Smart have all read and provided useful comments on earlier drafts of parts of this study. M y views on the importance of family interactions for tax policy have been shaped by numerous discussions with Chris Worswick.  I have also had stimulating conversations about the economics of families with  Siwan Anderson, Denise Doiron, Jiyoung O h and Terry Wales.  I would also like to thank my parents, Arthur and E t h e l Brett, for the support they have given me throughout my time at the University of British Columbia.  Financial  assistance from the Social Sciences and Humanities Research Council of Canada, in the form of a doctoral fellowship, is gratefully acknowledged.  vii  C H A P T E R 1: Family Economics and Family Taxation  1.1. Introduction  A n understanding how individuals interact with each other for the provision of their wants and needs is among the primary goals of economic analysis.  M a n y of these  interactions take the form of market transactions. Indeed, the detailed study of individual decisions on the basis of market signals (prices) and of the concomitant equilibria has become the basis upon which both modern welfare economics and the theory of optimal taxation rests. Individual interactions are not limited to exchanges in anonymous markets. Often, agents agree to bring (at least some of) their resources into group relationships and to engage in activities governed by forces other than the market.  One need look no further than the family for an example of this type of  arrangement. T h i s analysis represents an attempt to bring family decision-making into models of optimal taxation.  It has two complementary aims: uncovering the  aspects of individual-based models of taxation that are robust to introducing family interactions, and uncovering some shortcomings of these theories in the family context.  Families engage in many activities: child-rearing, the care of seniors, home production, and consumption. In this study, I make no attempt to capture all behaviours or to discern the effects of policy on family formation and composition. T h e objective of this study is more modest. In line with the literature on optimal taxation spawned  1  by Ramsey (1927), I consider the problem of a taxation authority wishing to design a tax system for a set of agents who act as both consumers and suppliers of labour. Unlike much of the literature, I take the family as the unit of decision making, not individuals.  Even in the restricted setting proposed in this analysis, the problem of designing optimal policies for families raises many questions that cannot arise when all decisionmaking agents are individuals, along with some issues that have solutions that are not easily transferred to the family setting.  One would expect families to behave  differently than individuals. T h a t is, individual-based consumer theory may be an inappropriate way to view how consumption decisions are made within a family. Moreover, the process of deriving statements about individual well-being from family behaviour is rather involved. Indeed, there may even be a notion of family well-being that is an important consideration in the problem of taxing families. These three sets of issues - behavioural, informational and normative - are central in this study.  T h e remainder of this introduction is devoted to expounding the special features of models of family behaviour and family taxation. T h e next section surveys models of family behaviour and their links to the analysis of tax policy. Section 3 provides a discussion of normative and informational issues. T h e remainder of this thesis is sketched in Section 4.  2  1.2. Models of Family Decision-Making  While substantial agreement exists among economists on how to model the behaviour of individual consumers (at least in the absence of uncertainty), numerous ways to think about families have been suggested. One approach is to simply treat families as individuals, positing that total family consumption is the consumption of some aggregate agent. Hoddinott and H a d d a d (1993) have termed this approach the "unitary" model of family behaviour. Unlike the treatment of consumption goods, the labour supplies of individuals within families are often dealt with as separate goods in this framework. Killingsworth (1983) provides a survey of attempts to use this model in the analysis of family labour supply behaviour, showing how its empirical implications are rejected by most studies. These findings are hardly surprising, given the strong assumptions on individual preferences required for a collective to behave as if it were an individual (cf. G o r m a n (1953), Deaton and Muellbauer (1980)).  A natural way to abandon the unitary model is to endow each family member with their own preferences over consumption and leisure. However, once this is done, it is necessary to make statements about how these possibly differing objectives are reconciled in family decisions. A wide range of possibilities now emerges.  One class of models uses non-cooperative game theory to describe interactions among family members.  Leuthold (1968) introduced a simple Cournot-type model  of labour supply decisions for families with two members. E a c h member is assumed  3  to maximise his or her own utility subject to the family budget constraint, taking the choices of the other as given. T h e resulting decisions give rise to a pair of labour supply functions that depend on prices (including wages) and the actions of the other. T h e family is at an equilibrium if these labour supplies are compatible; that is, if both members are actually maximising their preferences simultaneously, given the actions of the other. Ashworth and U l p h (1981) have tested the unitary model against the Leuthold model. Their data support a rejection of the unitary model. Woolley (1992) extended this model to study the provision of a household public good and the impact of income taxation on these decisions. She reports the possibility of a tradeoff between reducing intra-family inequality and negative effects on the provision of the household public good.  W h e n family members engage in a non-cooperative game, there is no guarantee that the equilibrium outcome is Pareto-efficient for the family. T h a t is, there may a rearrangement of family resources that makes both members better off. T h i s feature of non-cooperative models has been criticised (cf. Kooreman and K a p t e y n (1990)). Given the repeated nature of family interactions and the degree of communication that it possible among partners, it seems reasonable to expect efficient outcomes to emerge. O f course, efficient outcomes may arise from an ostensibly non-cooperative decision process. E v e n in a one-shot game, Becker (1974) shows how the actions of  4  a benevolent patriarch may induce self-interested family members to choose actions that result in efficient outcomes.  1  M a n y studies treat within-family efficiency as a maintained hypothesis. A m o n g this class of models are those that make explicit use of (cooperative) bargaining theory. T h e early work in applying bargaining theory to analyse family decisions was carried out by Manser and Brown (1980) and M c E l r o y and Horney (1981). In both of these works, labour-consumption outcomes depend on the feasible set of allocations for the family and a pair of threat points. Threat points are mieant to capture the utilities of the family members in the absence of a cooperative agreement.  There  remains some debate over the appropriate specification of disagreement outcomes. Should they correspond to outside options (divorce) or to some sub-optimal (noncooperative) outcome within the family?  2  A s M c E l r o y (1990) has pointed out, if  outside options influence threat points, there is scope for variables that reflect the state of the marriage market or divorce settlements to influence the intra-family distribution of resources.  T h e bargaining approach has gained acceptance, in part, because it allows for family behaviour to violate the income-pooling hypothesis, which states that only total family (exogenous) income matters in family decisions.  T h a t is, for a fixed  total family income, behaviour is invariant to redistribution of that total among This is the essence of Becker's famous "rotten kid" theorem. For a detailed discussion of the importance and limitations of this theorem see Bergstrom (1989). See Lundberg and Pollak (1993) for a discussion of how the sort of actions to which family members can credibly commit affects the way threat, points ought to be interpreted. 1  2  5  family members. T o see that income pooling need not hold for a bargaining solution, consider an increase in the exogenous income of one family member in a two-member household. T h i s expands the consumption possibilities of both members. It may also improve the threat position of the person with the increased income. changes in behaviour reflect both of these effects.  In general,  Suppose, instead, that the same  increase in exogenous income had fallen to the other family member. T h e effects on the family budget would be the same, while threat-point effects would be different. Notice that in the unitary model changes in exogenous incomes influence the budget constraint alone, so that income pooling is satisfied. Income pooling is, in principle, a testable hypothesis. A m o n g the empirical studies that have rejected it are those by Thomas (1990) and Phipps and Burton (1993). Indeed, this series of evidence is one of the major motivations for abandoning the unitary model of family behaviour.  T h e scope of theories of efficient family decision-making reaches beyond cooperative bargaining models.  Indeed, it has already been mentioned that  non-cooperative procedures may lead to efficient outcomes.  ostensibly  Forms of cooperative  decision-making other than the bargaining models usually employed are also conceivable. It is interesting to ask, then, if there are features common to all decisionmaking procedures that generate efficient allocations within families.  T h i s is the  research agenda of the "collective" school of modeling the family. ^ Lundberg and Pollak (1993) argue that this need not be the case. It depends on the form of the income gain and how the marriage market responds to such changes.  6  A n early contribution to the collective paradigm is the work of Samuelson (1956). He posits the existence of a fixed social welfare function, defined over the utilities of family members.  Abstracting from the details of budgeting decisions, he assumes  that there is a "family consensus" that gives rise to the family objective.  Families  are then assumed to choose consumption bundles as if to maximise this social welfare function, subject to the family budget constraint. He shows how such a family can be viewed as engaging in a two-stage budgeting procedure. In the first stage, the family budget is allocated among the family members. In the second stage, each individual maximises his or her own utility, subject to the constraint that he or she can spend no more than the "allowance" granted him or her in the first stage.  T h e division  of resources depends on prices and total family income, so that Samuelson's model generates behaviour consistent with the income pooling hypothesis.^  More recent work in the collective approach has jettisoned the notion of an agreed-upon family social welfare function. has been adopted.  Instead, the "sharing-rule" approach  In his analysis of household labour supply decisions, Chiappori  (1988) was the first to make explicit use of household sharing rules. He shows that when family members have preferences over bundles of own-consumption, all efficient decision-making procedures can be modeled as two-stage budgeting procedures. U n like the situation that obtains in Samuelson's model, the income sharing rule may ^ It is more usual to group Samuelson's contribution with the unitary model, because there is a well-defined family objective, and income pooling holds. However, it has two important features of the collective model: family members have their own preferences; and there is a notion of incomesharing in the model.  7  depend on income sources and the state of the marriage market. T h i s interpretation also holds for families in which individuals show altruism in the form of caring about the utility of other family members, not about their consumption per se.  Besides the generality of the model, the collective approach has a number of empirical advantages. First, it leads to testable restrictions on household labour supply and consumption behaviour. (Chiappori (1988), Browning and Chiappori (1994)) Moreover, the tests that have been carried out to date show that there is insufficient evidence to reject these restrictions, at least not those restrictions placed on consumption (Bourguignon et al.  (1992), Browning et al.  (1994), Browning and Chiappori  (1994)). Second, a good deal of information about the sharing rule can be recovered from family budget data alone. In particular, Chiappori (1988, 1992) has shown that the derivatives of the sharing rule with respect to prices and individual incomes are identifiable. From the viewpoint of tax theorists, this means that information is available on how changes in consumption and income taxes affect sharing with the family. T h i s represents a potentially powerful tool to be in used in analysing the effects of taxes on families and on their constituent members.  1.3. Issues of Ethics and Information  T h e link between applied welfare economics, including the theory of optimal taxation, is both well-known and often-exploited. For an individual, well-being is usually identified with preferences.  Once this notion of well-being is accepted, net market  8  transactions contain quite a bit of welfare information. Roy's Identity, a standard result in consumer theory, states that the effects on welfare of changes in consumer prices are negatively proportional to an individual's net demand vector. T h e consequences of this result for the theory of taxation are profound. It says that a taxation authority need observe no more than net market transactions to decide if a change in commodity taxes leads to a local welfare improvement for an individual.  It is  not surprising, then, that much of the modern theory of tax reform rests on Roy's Identity.  5  Unfortunately for the applied welfare economist, market transactions are not always recorded at the individual level. Instead, most data sources contain, at best, records of family transactions.  T h e direct link between observed transactions and  individual welfare is broken. One might ask: Is there any welfare information in net family transactions? Because this is one of the central questions of Chapter 2 of this thesis, I provide only preliminary remarks on it here.  If the unitary model of family decisions holds, Roy's Theorem can be used along with market data to identify changes in consumer prices that increase the optimised value of the family criterion function.  T h e normative significance of such changes  is unclear. If there is an ethical notion of making families better off, then one can substitute the word "family" for the word "individual" and carry out the standard Guesnerie (1977) is the classic reference. This theory is expounded in greater detail in Guesnerie (1995). There is another branch of the tax reform literature, based on compensated demand functions and Shephard's Lemma, owing much to Hatta (1977) and Diewert (1978). The two approaches yield equivalent results. 5  9  procedure.  Such a substitution is in violation of the individualistic principles usually  held by economists. Multi-person families can be viewed as mini-societies.  As such,  they have no more claim to being units of ethical account than countries do.  A more generous interpretation is available within the Samuelson framework. It can be argued that a family social welfare function summarises a set of agreedupon ethics that govern the intra-family allocation of goods.  A taxation authority  interested in increasing the value of household welfare may be said to be respecting the ethics of families. T w o caveats to this interpretation are worth keeping in mind. First, respecting the ethics of the family may involve sacrificing the welfare of one family member for the benefit of another. Second, it is often difficult to distinguish between agreed-upon family ethics and patently unjust relationships within families.  7  Family decisions result in informational problems more profound than just those faced by researchers or policy analysts wishing to identify and uncover relevant data. W h e n families possess more tax-relevant information than the planner does, it is often necessary that the planner take account of this asymmetry. T h i s is especially true when the tax system features non-linearities, or different rates for different individuals. A common example of a non-linear tax structure is income taxation. Mirrlees ^ The somewhat ambiguous term "households" is often used by economists to indicate decisionmaking units, be they individuals or families. I would argue that it is important to account for which usage of the term "households" is more appropriate i n specific settings. This caveat was recognised by M i l l (1859, p. 238), who presents an argument against accepting the notion family ethics based on the "almost despotic power of husbands over wives." 7  10  (1971) was the first to recognise the importance of asymmetric information in designing non-linear income tax schedules.  He envisions a world in which, if it could,  the taxation authority would like to tax on the basis of innate ability. Individuals alone know their ability. T h e best the planner can do is to tax earned income.  If  the planner chooses to ignore the fact that individuals have an informational advantage, workers may choose to "hide" their ability by working less than they otherwise might. T h i s approach to modeling the effects of income taxation on work effort has been used often, and has generated a form of conventional wisdom on the qualitative features of optimal income tax schedules.  8  M u c h of this work has relied heavily on  the assumption that there is a single tax-relevant characteristic the planner does not know. E v e n if one accepts the notion that ability is the only such individual trait, it is difficult to imagine that there is some notion of "family ability" that can take the place of individual ability in these models. It seems more natural to allow family members to differ in ability, resulting in a model of decision-making units that differ along more than one dimension.  A priori,  it is not clear how much of the conventional  wisdom applies to this more complex setting.  1.4- An Overview of This Study  I have already stated that the focus of this thesis is to study the optimal taxation of families as units of consumption and labour supply. T h i s is a natural point from which 8  Chapter 4 of this thesis gives an account of this literature.  11  to begin adding models of family decisions to the traditional analysis of commodity tax reform and optimal non-linear taxation. T h e remainder of this thesis is devoted to exactly this task. T h e assumption that all families consist of two members, each of whom may participate in the decision-making process, is maintained throughout. It is often maintained that these two individuals can be distinguished on the basis of some demographic characteristic, like gender.  T h e analysis of commodity tax reform begins with Chapter 2. In it, I assume that each member of the family has preferences over own-consumption of a set of private goods.  In line with the collective approach, I assume that family decisions  are Pareto-efficient for the family. T h e taxation authority is assumed to have at its disposal a full set of linear commodity taxes, including a tax on leisure time, and poll taxes. It cannot, however, effect l u m p - s u m redistributions across families, nor impose the division of resources within families. A simple aggregate production sector is posited, with all pure profit taxed away. Under these assumptions, I characterise the directions of policy reform that are both feasible and Pareto-improving at the individual level. T h i s characterisation is compared to the results that would obtain if family budget data were treated as if the family were actually an individual. It is shown that the necessary and sufficient conditions for such pseudo-Pareto-improving changes are necessary for an actual Pareto-improving direction to exist, but not sufficient. Thus, changes in policy that appear to be Pareto-improving when family interactions are ignored may fail to be actual Pareto-improvements.  12  It is shown  by means of any example that this may be the case even when family consumption behaviour is observationally equivalent to that of an individual. Moreover, I show that family budget data alone is insufficient to identify actual Pareto-improving directions of policy reform.  Chapter 3 continues the analysis of tax reform, showing how the characterisation of Pareto-improving directions of reform is influenced by the types of demogrants available to the planner. Three demogrant structures are compared: poll taxes varying by the demographic characteristic, identical poll taxes for each individual, and redistribution of a fixed total family demogrant between the two family members. T h e chapter also contains a discussion of the related issue of temporary inefficiencies in tax reform procedures. In the context of an individual-based model of tax reform, Guesnerie (1977) noticed that under some circumstances, all feasible Pareto-improving directions of policy reform may require the economy be moved inside its production frontier. I present conditions under which this somewhat anomalous occurrence cannot arise. These conditions are restrictions on aggregate demand behaviour and the derivatives of the family sharing rule. Thus, they can be checked using family budget data.  T h e focus shifts to the question of optimal non-linear income taxes for families in Chapter 4. In line with the work of Mirrlees (1971), I allow the individuals within families to differ in labour productivity. I focus attention on the case of only two productivity types.  T h e taxation authority can observe before-tax labour income,  13  but cannot separate the contributions of hours worked and skill to the total. L a b o u r consumption decisions are made by families, assumed to maximise a weighted sum of their members' utilities. T h e weights are assumed to be independent of incomes. In this way, the family decision process can be described by a single parameter, the relative weight of person 2.  T h i s parameter is known to the planner.  Because  families are assumed to maximise a fixed social welfare function, an income pooling result holds. T h e allocation of consumption within the family depends on total family after-tax income alone.  W i t h this characterisation of family behaviour, it is possible to view the taxation problem as a particular mechanism design problem. T h e planner offers bundles of goods composed of the before-tax incomes of the two family members and total family after-tax income. T h e planner is assumed to maximise some social objective function subject to an economy-wide materials balance constraint and a set of self-selection constraints. T h e self-selection constraints are formulated in such a way that families have no incentive to mis-report the types of their members.  Because each family  has two members (possibly of different individual productivity), a two-dimensional screening problem arises.  In order to highlight the role of the added dimension to  the analysis, I assume that the social objective is defined over the family criterion functions, not over individual utilities. T h a t is, despite the caveats mentioned in the previous section, I allow the planner to respect the ethics of the family.  14  Were the  planner to care about individual utility, some qualitative features of optimal n o n linear income taxes in the individual setting would not hold merely because of a desire to "correct" for family decisions.  9  I wish to explore the robustness of the  individual-based results in a more comparable situation - where the planner respects the criterion functions of the decision-making agents (here, families).  A variety of qualitative features of all Pareto-efficient tax structures are outlined in the chapter. It is shown that self-selection requires that, when two individuals in distinct families have equally productive partners, the individual with higher productivity must receive at least as much before-tax income.  T h i s is, however, the only  strong monotonicity result owing to self-selection alone. In particular, it cannot be shown that self-selection implies that both members of a family of highest type be allocated more before-tax income than the corresponding members of other families. Further results are available once conditions of optimality are added to the analysis. It is shown that, at an optimum, the materials balance constraint must always bind, that no two families receive the same allocation, and that there exists a family that faces no marginal distortions.  Chapter 5 contains a discussion of two issues related to the problem of non-linear income taxation:  redistributive taxation, and the'influence asymmetries in family  decision-making. It is common in non-linear income taxation to assume that, in the Seade (1980) shows how optimal tax schedules with non-standard features can arise when the planner has non-welfarist objectives. 9  15  absence of self-selection constraints, the planner would like to redistribute consumption from the more able to the less able. Assumptions of this form are often sufficient to guarantee that marginal tax rates must lie between zero and one (Guesnerie and Seade (1982), Roell (1985)). I formulate the analogue to this criterion in the family context, stating that whenever two families can be unambiguously ordered on the basis of ability, the planner would like to redistribute after-tax income from the more able to the less able. It is shown that this assumption is not sufficient to guarantee non-negative marginal tax rates. Negative marginal tax rates are shown to arise when there is a tension between the redistributive assumption and the ability of the planner to distinguish between the (different) families with one member of each productivity type.  It is also shown that when families face linear tradeoffs between the labour  supplies of their members, knowledge of the parameter of the family decision-making process can be used to infer which families have least incentive to imitate others. T h i s highlights both the importance of the special assumptions on family behaviour maintained throughout the analysis and the usefulness of information about family decisions for setting tax policy.  16  C H A P T E R 2 : Tax Reform and Collective Family D e c i s i o n - M a k i n g  2.1.  Introduction  Welfarist evaluations of tax policy are concerned with the effects of these policies on the well-being of the individuals that comprise a society. T h e standard literatures on both optimal taxation in private good economies (Atkinson (1977), Stiglitz (1987)) and tax reform (Guesnerie (1977), Diewert (1978), Weymark (1979)) identify wellbeing with preferences as revealed by market behaviour. There is an important, a n d rarely addressed, issue to be resolved when implementing second-best taxation models. Market data often presents itself at the family level, whereas welfarist evaluations are carried out at the individual level. Standard results in normative taxation theory can be reinterpreted to suit market data only if family preferences are well defined and family well-being is an appropriate ethical concept.  In order to bring the analysis back to the level of the individual we must be able to assess the well-being of each household member separately. If we wish to maintain the family as the basic unit of consumption decisions then we must make some statements about how the possibly conflicting interests of family members are reconciled within the household. I draw on the literature spawned by Becker (1974) on efficient household decision-making processes to make these statements. I find that household budget data are insufficient to calculate Pareto-improving directions of tax change, even under circumstances in which these data can be used to identify changes  17  in the intra-family allocation of resources. Moreover, I show that the necessary and sufficient conditions for a Pareto-improving change found by acting as if families were a single person fail to be sufficient in the family setting, although they remain necessary.  Indeed, as long as there is at least one good for which  individual net  demands are not observable, a rote application of traditional tax reform formulae may lead to erroneous policy prescriptions.  T h e approach of this analysis is very much in the spirit of second-best taxation problems. T h e planner cannot impose the division of resources within the household. It can control the family only indirectly by effecting changes in the environment it faces.  Control over the environment is described by the set of available policy  instruments:  a full set of linear commodity taxes, including a tax on leisure time,  and poll taxes. L u m p - s u m redistribution between and within families is not feasible. Policy is thus constrained by both the material balance constraints for the economy and the behavioural responses of individuals.  Unlike the economies described in  traditional second-best problems, these behavioural responses occur on two fronts: the allocation of resources within families and the interaction of families in competitive markets. I also choose to consider the problem of tax reform rather than that of tax design. F r o m an initial, possibly sub-optimal tax policy I seek small changes in the rates that are both feasible and Pareto improving at the  individual level. If no such  directions exist then this initial tax structure is a local second-best optimum.  18  I focus attention on an economy made up of two-person families. It is assumed that each member of the household has preferences over own-consumption of a set of private goods, and that household decisions are Pareto-efficient for the family with respect to these preferences. In order to derive a characterisation of Pareto improving policy changes, I assume the social planner can observe the intra-family allocation of all commodities. T h e planner can also see how this allocation varies with changes in the economic environment. T h i s is more information than is included in standard budget surveys, but this assumption is made so that I can focus on the normative question of how one ought to use information on intra-family allocation apart from the question of how one might obtain this information.  I show that some of this  additional information is actually needed to calculate Pareto-improving directions of tax change, demonstrating the deficiency of household budget data alone.  It should be pointed out that the families considered in this study conform to the Chiappori (1988,1992) collective model of decision-making.  Moreover, there is  no household production. T h e work of A p p s and Rees (1996) and Chiappori (1994) has underlined the difficulty in identifying the parameters of the household decision process when household production is incorporated into the analysis. T h e addition of household production to the current model can only limit the information available to the planner, strengthening the conclusions of this analysis.  T h i s chapter is organised as follows.  Section 2 presents the model of family  decision-making. Section 3 provides a description of the production and government  19  sectors of the economy and the basic structure of general equilibria. Section 4 provides a characterisation of directions of tax reform that are both feasible and Paretoimproving. T h i s furnishes a fortiori a characterisation of second-best optima. T h e consequences of ignoring family interactions are considered in Section 5.  2.2. Collective Family  Decision-Making  I begin by setting out the notation. dexed by h — 1, . . . , H.  There are H households in the economy, in-  Each family consists of two members, indexed by i = 1, 2.  Individuals have preferences over their own consumption, x , lh  goods.  of vectors of n private  Consumer prices are given by the vector q. I assume that the preferences of  person i in household h can be represented by a continuous, increasing and quasiconcave function U  th  :  —* M, a l H , h. I allow family members to have an initial  endowment of goods. Let these endowments of goods be denoted by u  lh  €  The  total resources available to any family are its total endowment plus any transfers from the planner. There is no household production process.  Because the planner must  take the endowments as parametric, they play a limited role in computing optimal directions of tax change. I assume that each family allocates its resources so that the final allocation is Pareto efficient for the family. It is helpful to think of the family members as the agents in a two-person exchange economy with the final allocation lying somewhere on the contract curve.  20  Let us now consider a typical family in more detail. For notational convenience I omit the superscript h for the remainder of this section. income of person  i.  Let m  1  denote the total  T h e assumption of Pareto efficiency implies that the family  allocates resources as if to solve  q(x + x ) < m + m . 1  2  1  2  Note that the utility level u is not exogenous, but may change with incomes and prices. Later on, I allow the incomes of the individuals to be under the control of the planner via tax instruments. Problem (P) models the decision process of a family conditional on its tax environment. I follow the lead of Chiappori (1988, 1992) and give a sharing rule interpretation of this decision process. For this I need the following result, due to Chiappori (1988, p. 68).  Lemma  2.1.  Let f (q, m , m ) , s ( m , m ) solve (P). Then there exists a function 1  1  2  2  1  2  ip(q, m , m ) such that 1  2  i.) f ( q r , m , m ) solves: 1  1  2  max U (x ) s.t. qx < <p(q, m , m ) . l  (PI)  1  1  1  2  X ii.) x (q,m , 2  (P2)  l  m ) solves: 2  m a x f 7 ( x ) s.t. qx 2  2  2  < m  1  X  21  + m  2  — <p(q, m , 1  m ). 2  x(q, m , m ) := x^(q, m ,m ) +  Total demand of the family is denoted  1  1  2  2  x (q, m , m ) . T h e function <> / may be interpreted as a household sharing rule, indi2  1  2  cating the value of household resources spent on the consumption of goods by person 1. W i t h this interpretation in mind, denote by  p , p? the effective incomes of family 1  members 1 and 2, respectively. T h a t is,  p (q,m ,m ) 1  l  := y?(<?, m , m );  2  1  2  (2.1)  p (q, m , m ) := m + m — <£>(<?, m , m ). 2  1  2  l  2  1  2  L e m m a 2.1 is the analogue in the family context to the second theorem of welfare economics. Under this interpretation (p = m  1  —t  1  , where t  1  is the l u m p - s u m transfer  of income from person 1 to person 2 required to decentralise the Pareto efficient allocation (x , x ) 1  at prices q. Notice that the sharing rule depends on the incomes  2  of the individuals rather than on total family income.  T h a t is, I am not imposing  the income pooling assumption. I am allowing individual incomes to affect household decisions in ways other than their effects on total income.  T h i s may be due to  bargaining effects, as in M c E l r o y and Horney (1981), or compensating transfers, as in Chiappori (1992). I remain agnostic on this account.  The levels of well-being attained by the family members are given by the value functions for the programs (PI) and (P2). Specifically, let V  be the indirect utility  1  function dual to U ,i l  = 1, 2. T h e n  u = L7 (£ ( , 1  1  1  g  1 m  V (q .fj}{q,m ,m )) 1  ,m )) = 2  1  2  i  (2.2)  u = U {x {q,m ,m )) 2  2  2  l  2  = V (q, p (q, m , m )). 2  22  2  1  2  Expression (2.2) displays the role of price and income changes on the well-being of the individuals. Consumer price changes play two roles. T h e usual price effects are present, and changes in consumer prices serve to change the effective incomes of the family members through sharing-rule effects. Income changes affect well-being in a complicated way. A one—unit increase in own-income does not necessarily increase indirect utility by the marginal utility of income V . %  i  in  It is true that a one-unit change  effective income has this impact on utility. However, changes in money income need  not be translated one-for-one into changes in effective income. It is also important to notice that an increase in the income of another family member can have a positive or negative effect on an individual's welfare.  2.3.  General Equilibria  I now describe the environment the families face.  There is a production sector  characterised by an aggregate technology set Y. T h e aggregate firm faces a vector of producer prices p, and acts so as to maximise profit at these prices. T h e assumption of an aggregate firm can be made without loss of generality under competitive conditions (Bliss (1975, p. 68)). I assume that the solution to the profit maximisation problem defines a net supply function  y = rj(p), where n :  are interpreted as input demands.  { 0}  (Full rank).  lR . Negative supplies n  I assume that n is differentiable. In addition, I  make the following assumption:  Assumption F  —>  V n(p) is of rank n — 1. p  23  Homogeneity of the supply function implies that is a maximal rank assumption.  V n(p).  is singular, so that this  p  Assumption F rules out the possibility of kinks or  ridges in the aggregate production frontier. A t any kink or ridge in the production frontier the set of supporting prices is non-unique: Thus, sufficiently small changes in producer prices may fail to have any effect on production. W h e n Assumption F is satisfied the planner has full local control over the production sector via changes in producer prices.  A m o n g the formal consequences of Assumption F is that the  V w(p) : JR+\ { 0 }  mapping  p  —>  M  n  p  is invertible in the subspace orthogonal to  (Guesnerie (1995)).  T h e government sector is characterised by a set of available tax instruments. For simplicity, there is no public sector production. T h e planner has at its disposal a full set of per-unit commodity taxes t\, . . . ,t . n  Consumer prices are thus the sum  of producer prices and taxes. T h e planner can also use a poll tax or subsidy. I allow this demogrant to vary by the index i, but not by household.  T h i s requires that  this index corresponds to some easily observable characteristic.  For a 'traditional'  family, gender would be such a characteristic.  Denote these transfers R  1  and it! . 2  T h i s formulation of the demogrant structure admits two important restrictions on the powers of the planner as special cases. W h e n the planner cannot make changes in the demogrant conditional on the index i, I can write dR  1  dR} + dR  2  = dR . 2  T h e restriction  = 0 corresponds to the redistribution of a fixed total demogrant between  persons of different indicies.  I also assume that the planner taxes away all pure  24  profit, so that the profits of the production sector do not affect the consumption sector through distributed profit. W i t h this policy structure, the income of a typical individual is given by m  i h  = R* + J .  (2.3)  h  q  A n examination of (2.3) reveals that the family decision process described by the programme (P) is not the most general process that generates Pareto-efficient outcomes. I could have permitted the utility level u to depend on endowments and l u m p - s u m incomes separately, instead of on total incomes. There are two justifications for this simplification. First, it is quite reasonable to expect whatever influence the ownership of endowments has on family decision-making to depend on their value. Second, the possibility arises of non-homogeneity of net demands when endowments exert effects independent of their values.  I am now in a position to describe the equilibria in this economy. First, aggregate demand for goods is given by  H  x{q,R\R ) 2  H : = ^ V f o f l ^ ) = ^x {q,qu h=l h=l 1  h  2  lh  + R ,qu l  2h  + R ). 2  (2.4)  T h e dependence of aggregate demand on the endowment is suppressed from the notation. T h e aggregate endowment is denoted by w, and is given by u :=  zZlih ^00  A n equilibrium for this economy exists when aggregate demand is satisfied by the  25  combination of production and aggregate endowment.  Thus, an equilibrium is said  to exist at (q, R , R ,p) if 1  2  x(q,R},R )  <rj(p)+u-  2  (-) 2  A n equilibrium is said to be tight if (2.5) holds with equality.  5  If families exhaust  their budgets on consumption, Walras' Law guarantees that the government budget is in balance at any tight equilibrium.  It should be pointed out that, contrary to  the situation that obtains in a general equilibrium setting with no distortions, n o n tight equilibria are consistent with all goods having positive prices. T h e existence of a government sector implies that the value of private excess demand (measured at either consumer or producer prices) need not be zero. Indeed, the government never runs a deficit at any equilibrium (Guesnerie (1995)). Thus, the value of private excess demand can be negative, so that non-tight equilibrium are possible.  2.4-  Optimal Policy Changes  Assume that the economy is initially in a tight equilibrium at to avoid boundary problems, I assume (q, R , R ,p) 1  2  > >  are defined to be marginal changes [dq , dR , dR , dp ] T  1  are both feasible and Pareto i m p r o v i n g . interpretations.  10  2  T  T  0.  (q, R , R ,p). 1  2  In order  O p t i m a l policy changes  in the tax instruments that  Feasibility can be given two equivalent  T h e first is common i n public economics:  the government budget  position is not worsened by the change. T h e second is that the change is equilibrium 1 0  As a notational convention, all vectors (both row and column) are enclosed in square brackets. 26  preserving. T h e equivalence follows from the fact that moves from a tight equilibrium to any other equilibrium cause the government budget to change from balance to a non-deficit.  D e f i n i t i o n : A direction of policy change [dq~^, dR , dR , dp ] 1  2  T  is said to be equilib-  T  rium preserving if for initial policy (q, R , R , p) it satisfies 1  V x{q, R , R )dq + V ix(q, R , R )dR 1  2  1  q  2  + V 2x(q, R , R )dR  l  1  R  where V x(q, R , R ) 1  2  q  2  2  < V r,(p)dp,  2  R  p  (2.6)  is the Jacobian of aggregate demand with respect to consumer  prices.  T h e assumptions made on the aggregate technology provide an alternate representation of feasible directions of policy change. This is given in the following result, due to Guesnerie (1977, p. 187).  L e m m a 2.2.  Suppose that, the aggregate-technology satisfies the fidl rank assumption.  Let (q, R , R ,p) 1  2  be an initial tight equilibrium. Then for any [dq ,dR ,dR } T  1  that sat-  2  isfies  p (V x(q, R ,R )dq T  l  2  q  + V ix{q, R , R )dR 1  2  R  x  + V 2x{q, R , R )dR ) 1  2  < 0,  2  R  (2.7)  there exists a direction of producer price change dp such that [dq , dR , dR , dp~^] is T  equilibrium preserving.  27  1  2  L e m m a 2.2 states simply that a policy reform is feasible if and only if it induces a marginal change in demand that remains inside the production set. the characterisation of equilibrium preserving directions of change.  I use (2.7) as  It is helpful to  rewrite (2.7) as  dq dR dR 1  (2.8)  >0,  2  where  $ := -p V x(q,  R , R ) and $  T  q  1  q  := -p V x(q,  2  R , R ), i = 1,2.  T  Ri  1  Ri  2  Consider welfare-improving directions of policy change.  I search for directions  that are strictly Pareto-improving; that is, changes that make everyone better off. In what follows let f  denote the partial derivative of the function / with respect to the  z  argument z. Differentiating the top line of (2.2), and using (2.1) and (2.3), I obtain .lft __ V 7 T T / 1 A J „ 1 T / l f t V 7 T . _ f t , J „ l T / l f t , - f t , .lft,,,, du" = V V" dq + Vft V~Z<p dq + Vfi^ico dq + 1  n  h  q  in  „  ,,,  ,, „,, Vfiip^^dq  h  (2.9)  + V\ *p M h  h  1  h  Now let a  lh  := V ^, A*  + V^  dR .  1  2  2  the marginal utility of effective income for person lh.  1  B y Roy's  Identity,  -a x lh  lhT  =  V^V . lh  (2.10)  Substituting (2.10) into (2.9) and rewriting in matrix notation yields  A, 1ft  au  ,1ft  ^ 1ft =a  Because a  lh  -X  1  ,  -rjT +V f qt  ft , ft ,1ft , ft ,2ft +¥ lUJ m  ft  +(f 2U  dq dR dR  ft  1  ,<P l,<P 2  m  m  m  (2.11)  2  is positive, the directions of change [dq , dR , T  1  dR ] 2  T  that make person  lh better off are exactly those changes for which  1ft  1  , vvT ft , ft , ,1ft , ,„ft , ,2ft  ft  ,„ft  dq dR dR 1  2  28  > 0.  (2.12)  T h i s is the characterisation that I need in what follows.  It is possible to rearrange (2.12) to reinterpret it in terms of the change in the budget of person lh caused by the policy change. Notice that  du  lh  > 0 «-  x dq lhT  < V^ dq h  +  (^iUJ  l h  + <p^ w )dq. + ^ 2h  d  R  2  l  +^  d  R  -  2  (- ) 2  13  T h e left-hand side of the second inequality of (2.13) is the change in the cost of the initial consumption bundle.  T h e right-hand side is the change in effective income  brought about by two sources.  T h e first three terms capture the effect of relative  price changes on intra-family allocation. T h e fourth and fifth terms give the portion of the changes in the demogrants that are added to (or subtracted from) effective income. Hence, the consumer is made better off if the net increase in the cost of the initial consumption bundle is less than the net increase in effective income brought about in the family decision-making process.  Next, consider the effects of a policy change on person 2h. I follow the same procedure and notational conventions as in the analysis of person lh to conclude that the condition for [dq , dR , dR , dp ] T  1  2  T  T  to bring about a welfare improvement  for person 2h is  du  2h  >0  „2h  T  V7T h t  ft  , ,1ft  ft  , ,2ft , , ,1ft , , ,2ft  ft  -i  ft  dq dR dR 1  > 0.  2  (2.14)  29  A n equivalent condition is  x dq 2hT  <  -Vjip dq  + {-^  h  - V *u )dq + (co + u )dq + (1 - ip^dR  l h  h  2h  lh  2h  1  m  + (1 -  ^ .)dR . 2  m  (2.15) T h e interpretation of (2.15) is exactly the same as that of (2.13), keeping in mind that >p gives the effective income of person lh. Person 2h spends the remainder of h  the family budget. I am now in a position to state a simple, but important, proposition.  Proposition 2 . 1 . A necessary condition for the welfare of both members of any family h to be improved by a policy change is that the cost of total initial family consumption increase by less than household income.  Proof: Suppose that [dq^, dR}\ dR ] 2  T  improves the welfare of both members of house-  hold h. T h e n both (2.13) and (2.15) are satisfied for this household. A d d i n g these inequalities yields  (x  lhT  +x )dq 2hT  < (u + lh  cu )dq + dR + dR . 2h  1  2  (2.16)  B u t the left-hand side of (2.16) is just the change in the cost of the initial aggregate consumption bundle of the family, while the right-hand side is the.net change in full family income.  •  Proposition 2.1 has an important (and immediate) corollary.  30  C o r o l l a r y 2.1.1  There are no directions of policy change [dq , dR , dR ] satisfying  dq = 0 and dR + dR = 0 which make both members of any family better off. 1  2  Corollary 2.1.1 speaks to the debate on the identity of the recipient of income transfers from the government to households. It says: when household decisions are made efficiently, a marginal redistribution of a fixed l u m p - s u m between family members cannot make both members better off. Note that this conclusion holds even in the absence of income-pooling as a behavioural hypothesis. T h u s any arguments in favour of marginal redistributions between family members must presuppose inefficiencies in the household, or be based on distributional judgments.  Note that the condition expressed in the Proposition 2.1 is not sufficient for the improvement of the welfare of are  both family members. Because intra-family allocations  efficient, ameliorating the family budget position implies that at least one family  member is made better off. However, this improvement may come at the expense of the other member of the family. T h e following example makes this point clear.  E x a m p l e 2.1  Consider a household whose members have utility functions  C/^.y ) = 1  lnx - H n y ; l  1  U {x , y ) = \nx + 2 1 n y . 2  31  2  2  2  2  (2.17)  Suppose that the family acts so as to maximise the sum of the utilities of its members. Assume also that the family has no endowment, so that all income is l u m p - s u m . It is easy to deduce that the demand functions of the family are: x\q ,  q, R \ R)  = (R  + R )/5q ;  y\q , q, R, R)  = (R  x {q , q , R \ R )  = (R  + R )l$q ;  y (q ,  = 2{R  2  x  1  y  2  1  x  x  2  +  l  y  R )/5q 2  y  (2.18) 2  2  x  y  l  2  q, R, R)  2  x  1  x  2  l  y  +  R )/5q . 2  y  (2.18) entails that the indirect utility functions of the individuals can be written as V\q ,  q, R \ R)  = \n((R} + R )/5q ))  2  x  + l n ^  2  y  x  +  1  R )/5q )); 2  y  (2.19) V (q ,q ,R ,R )=ln((R 2  1  x  2  -\- R )/5q ))  1  +2ln(2(R  2  y  +  1  x  Now consider a direction of policy reform (dq , dq , dR ,  2  y  dR ).  1  x  R )/5q )). It follows from (2.19)  2  y  that l  du  d  _  dq  dqy  ~~  q  q  2_  x  x  2(dR  + R , 3(dR -\-dR ) 2  1  y  T J T  ~~Tx  2  1  2dq  x  dR )  R  v  dq  +  1  R +R 1  +  (2-20)  2  2  If the initial state of the economy is characterised by R  + R  1  = 5, q  2  x  = q  y  = 1, then  the initial consumption vector is  (x\y ,x ,y ) 1  2  = (1,1,1,2).  2  (2.21)  Furthermore, (2.20) becomes du  1  = -dq  - dq  x  + 2(dR  l  y  +  dR )/5 2  (2.22) du  2  = -dq  - 2dq + 3(d,R +  dR )/5  1  x  2  y  Now consider the direction of policy reform 7 := (dq , dq , dR , 1  x  (-1.9, 1 , 0 , 0 ) .  11  y  T h e direction 7 clearly satisfies (2.16).  direction yields du  1  = 0.9 and du  2  dR ) 2  =  Evaluating (2.22) for this  = —0.1.  The length of the vector 7 is of no consequence in this example. Any vector pointing in the same direction is feasible and induces changes in the utilities of the individuals of the same signs. 1 1  32  Example 2.1 brings out an important point.  E v e n if aggregate household be-  haviour is identical to that of a single person, using that behaviour for normative analysis may be inappropriate. T h e family depicted in the example behaves like a single consumer with Cobb-Douglas preferences. However, directions of policy change that improve the welfare of this 'constructed' consumer may reduce the welfare of one of the actual consumers in the family. Alternatively, we may view the family as having Utilitarian ethics, but the planner does not necessarily respect these ethics when making decisions concerning policy change.  T h i s conclusion is similar in spirit to a finding of A p p s and Rees (1988).  In  their model, the planner may wish to use a linear income tax to redistribute income within a family if the ethics of the planner and of the family differ. In E x a m p l e 2.1 it is changes in relative consumer prices that are acting as redistributive tools rather than changes in demogrants.  T h i s demonstrates the importance of considering the  intra-family effects of changes in all possible tax instruments.  I am now in a position to characterise the feasible strictly Pareto-improving directions of change. First some notation is introduced that allows one to consider (2.8),(2.12) and (2.14) jointly.  r  r  .la.  ,2ft  lft T  —x  2ft T  i V7T „ft i  „h  ,lft  T  ift T  ,  ft  ,2ft  T  2ft T  ft  +W  lft T  ft  2ft T  (2.23)  33  T h e n a direction 7 : = [dq , dR , T  1  dR }'  is both feasible and Pareto-improving if and  2  only if T  i  /  l  T 7  > 0, h = 1, ...,H,  i =  1,2;  (2.24)  $ 7 > 0, where 3> := [<& , $ i, g  R  Qfp)^.  I make some use of the mathematics of cones in the  sequel. Hence, I require the following definition.  Definition:  Let  (x ) be a collection of vectors in lR . T h e n the cone generated by l  (x ), denoted K((x )), 1  1  k  is defined by:  K({x ')) := { z G R \z j  k  =  ^ > 0} .  (2.25)  Before stating a central proposition, I introduce two assumptions.  Assumption A: There exists a 7 such that T  lh  7 > 0, h = 1,. . ., H, i = 1, 2.  T h a t is, there exists a strictly Pareto-improving direction of policy change, ignoring feasibility constraints. T h i s assumption is generally maintained in the literature (Diewert et al. (1989), Guesnerie (1977)), and is a minimal condition for making the problem interesting.  Assumption B: $ 7^ 0.  34  A typical component of the vector $ is the marginal cost, measured at initial producer prices, of meeting the changes in demand induced by a change in the corresponding policy instrument. Assumption B simply states that not all of these marginal costs are zero. T h i s assumption rules out the possibility that all directions of policy reform are tight-equilibrium preserving. If this assumption were violated, the requirement of Assumption A renders the problem uninteresting.  T h e planner  could implement the change 7 mentioned therein (or any other for that matter) while maintaining tight equilibrium.  T h e next proposition gives a characterisation of the local second-best optima in this environment.  Moreover, read contrapositively, it characterises the feasible  strictly Pareto-improving directions of policy reform.  Proposition 2 . 2 . Let Assumptions A and B hold. Then $ G K((—T ))  if and only  ih  if there exist no feasible strictly Pareto-improving directions of policy change.  Proof: 7 is feasible and Pareto-improving if and only if it satisfies (2.24). B u t (2.24) is satisfied if and only if  jBp  ih  > 0, not all/3  ih  = 0 and A > 0 such that ^ j 0 " T  i f c  + A$ = 0 „  + 2  ,  (2.26)  i,h by Motzkin's transposition theorem.  Equivalently, there are no feasible strictly  Pareto-improving directions exactly when there are /3  lh  > 0, V i , V7i, (not all equal  Motzkin's Theorem states: for given matricies A, B and C, with A nonvacuous, exactly one of the systems of relations a) and b) below has a solution: a){j4x >> 0, Bx > 0,Cx = 0}; b) {A + B y + C y = 0,yi>0 (but ^ 0), y > 0} (Mangasarian (1969, pp. 28-29)). T  r  yi  T  2  3  V  l  2  35  zero) and A > 0 such that  ^ P  i  r  h  h  + A$ = Q  n + 2  .  (2.27)  i,h  Suppose that A = 0.  Then £  )0 'T i  i j f t  = 0.  i h  T h e n , by Motzkin's Theorem, the  top half of (2.24) has no solution. T h i s contradicts Assumption A . Thus, in view of Assumption B , A > 0. Hence, I may rearrange (2.27) to give that $ G is clear that <P £  K((—T )) lh  K({-r )).  implies the existence of a solution to (2.27).  K({—F )) lh  Although a somewhat technical condition, $ €  ih  It  •  states that the vector  $ can be written as a negative linear combination of the vectors  (T ). lh  A revealed preference version of the argument may shed some light on the condition (2.26). Consider the indirect utility functions of the agents defined in terms of policy variables  V (q, R.\ R ) := V (q, rf(q, qu ih  2  ih  lh  + R\ qoj + R )). 2h  2  (2.28)  Employing the notation of (2.23), write  r  ~rih  Ah  \rih  £ :=l/a . ih  ih  Suppose there are no feasible Pareto-improving directions. (2.26) is satisfied. T h e n for any 7 = [dq , dR , T  7 (j2P  ihFih  T  1  + )  (2.29)  T h e n the equation in  dR } 2  7  A$  i,h  36  = °-+2-  ( - °) 2  3  Expanding (2.30) using (2.29) yields  (2.31)  i,h Now, assume 7 is equilibrium preserving so that it satisfies (2.7). T h e n A > 0 implies  (2.32)  i,h Because there is least one positive /3 , the term in braces in (2.32) must be non-positive for at least one individual i, h. However, these terms correspond to the changes in utilities brought about by the policy change 7. W h e n this quantity is non-positive a consumer cannot be made better off. Hence, there must be at least one individual who is not made better off by the change.  2.5. Implementation  One of the most attractive features of the standard literature on optimal tax changes is that the information requirements of implementing the procedure are not prohibitive.  Knowledge of net market transactions and aggregate demand and supply  elasticities suffice. "* In the family context, once one has the information needed to 1  calculate the vectors V  lh  and $ the problem of computing optimal policy changes  reduces to finding a solution to (2.24). T h i s can be done by standard linear programming techniques. T h e vector $ can be constructed with knowledge of producer prices ^ Guesnerie (1977) and Wibaut (1987) have excellent discussions of information requirements i n some specific settings.  37  and aggregate demand elasticities with respect to consumer prices, male income and female i n c o m e . differ.  14  It is important to note that the two sets of income elasticities may  T h e calculation of the vectors T  lh  is more difficult.  One needs to know the  derivatives of each family sharing rule, and the initial transactions of each Bourguignon  individual.  et.al. (1992) demonstrate how the derivatives of the sharing rule can be  calculated from demand systems estimated on family-level data. Family budget data is not sufficient, however, to tell one how much of each commodity each member of the household consumes.  2.5.1. Consequences of Ignoring Family Interactions  In light of the difficulty in obtaining sufficient data to compute optimal policy changes in the family setting, it is worth asking what penalty is paid when family interactions are ignored in applied work. One source of error is inaccurate calculation of aggregate demand elasticities, because ignoring family interaction amounts to imposing income pooling in the aggregate. There is, however; a more serious problem. Suppose that, in line with the standard literature, one takes the amelioration of the family budget position as a necessary and each family member.  sufficient condition for improvement in the welfare of  Rearrange (2.16) for each household and form the system of  inequalities  *  f t T  7 > 0, h = 1,...,H, (2.33)  $ 7 T  1 4  > 0,  I interpret the index i as gender to facilitate discussion. 38  where T  ) + (u +LU ), lh  2h  1, 1  (2.34)  Now, the condition for 7 to be an 'optimal' direction is that it satisfy (2.33).  In view of Example 2.1, satisfying (2.33) is not sufficient for a direction to be Pareto-improving.  Thus, policy based on such a recommendation may send the  economy in the wrong direction.  If the analysis finds no solutions to (2.33), one  reports that the economy is at a local second-best optimum. T h i s conclusion is not in error, because, under the regularity conditions assumed throughout this analysis, satisfying (2.33) is a necessary condition for a direction to be truly optimal.  It is  also interesting to note that (2.33) is exactly the system that characterises feasible directions of welfare improvement if the family has a well-defined value function and the planner respects family ethics. It is not surprising that family level data suffices for such a planner to make decisions.  T h e limitations of family-level data can be best exemplified by considering the search for Pareto-improving directions of consumer prices alone, ignoring feasibility constraints. Once again employing Motzkin's theorem, such a direction can be found if there is no solution to  p  ih  39  > 0 (not all zero).  (2.35)  One sufficient condition for (2.35) to have no solution is that each TQ has a strictly positive entry in (for example) the first p o s i t i o n .  15  Recalling (2.23), this is the case  when, for all / i , lft  .  ,  ft  , lh ,  ft  , ,2h ,  n  (2.36)  9 1  „2ft  P  d( _  , /1  ,„ft v ,1ft , ..ft + U ~ n  v ,2ft  n  > U.  Combining the two inequalities in (2.36) yields  1 1ft 1 2ft\ t„lh . „2ft\ ^ 2ft (w + w ) + x ) > -x x  x  t  x  dy>  ft  . /-t ft \, ,1ft 1 / 1 ,„ft \. ,2ft v . n + (1 - V i ) w i + (1 - <P 2)Ui > 0. m  m  Q7^ (2.37)  (2.37) can be compared to the Diamond-Mirrlees (1971) sufficient condition for the existence of a strictly Pareto-improving direction of price change in the i n d i v i d u a l based model.  T h e latter holds when there exists a good that is in net supply by  all individuals. Thus, increasing its consumer price leads to a budget amelioration for each individual. Notice that (2.37) is satisfied when the combined net supply of good 1 is sufficiently greater than zero for each family, so that a rote application of the Diamond-Mirrlees condition to combined family net supplies is inappropriate. Although increasing the consumer price of good 1 leads to an amelioration of the family budget position, Example 2.1 indicates that this need not bring about a Paretoimprovement.  Two words of caution are in order at this stage. First, the magnitude by which net supply must exceed zero varies with h. Furthermore, calculation of the middle term of (2.37) requires that the division of consumption and endowments of good 1 1 5  A similar argument can be made for the case of strictly negative entries. 40  within the household be known. For most goods this is clearly more information th is contained in a family budget survey.  t  41  C H A P T E R 3: Temporary Inefficiencies and Demogrants  3.1. Introduction  One of the curious features of the standard tax reform model is that under certain circumstances, feasible Pareto-improving directions must fail to be tight equilibrium preserving.  (See Guesnerie (1977).) T h a t is, the planner may have to choose direc-  tions of price change that move the economy inside the production frontier. However, the Diamond-Mirrlees (1971) result indicates that full optimality requires production efficiency.  For this reason the above phenomenon is termed temporary inefficiency.  Smith (1983) has pointed out that temporary inefficiencies disappear if a poll tax or subsidy is among the instruments available to the planner and if aggregate demand satisfies the H a t t a (1977) normality conditions.  T h e planner considered here has potentially two l u m p - s u m transfers available, so we might expect temporary inefficiencies to be ruled out on similar grounds. However, the agents in the present model interact differently than they do in the standard model. In a sense, pairs of agents are forced to behave cooperatively. It is conceivable that this may result in non-standard responses to demogrants. In this Chapter, I turn my attention to this issue.  It is also helpful at this point to recall that a change is non-tight equilibrium preserving if the bottom line of (2.24) holds with strict inequality.  42  3.2. Unrestricted Poll Taxes  Suppose that the planner can make independent changes in the poll subsidies. T h e n the following restatement of the theorem of Guesnerie (1977, Proposition 4, pp. 189190) on this matter obtains in the present context.  Proposition 3.1 Under the Assumptions A and B, the following statements hold: i)  £ K((T ))  if and only if there exist strictly Pareto-improving  lh  directions, all  of which are non-tight equilibrium-preserving. ii) $ G K((r )) lh  n K((—T' ))  c  lh  if and only if there exist strictly Pareto-improving  c  directions that are tight equilibrium-preserving.  Proof: i) $ E K({T )) lh  exactly when there exist (3  lh  > 0 satisfying  (3.1)  Assumption B ensures that at least one f5  lh  A = [r ,.. .,r ] 1 1  2H  T  and  is positive. B y Motzkin's theorem (with  B = — $ ) , (3.1) implies the following has no solution  r  h  7 > 0,Vi,V/i;  $'7 < 0.  (3.2)  In particular, there are no strictly Pareto-improving tight equilibrium-preserving directions of reform. In order for Assumption A to be satisfied, there must exist a 7 for which  r  h  7 > o,v*,v/i.  43  (3.3)  B y (3.2), 3> 7 > 0 for any such 7. T h e 'only i f part of Statement i) follows. T  Conversely, let r  i / l T 7  > O.Vi.Wi;  $  T  7 > 0  (3.4)  have a solution and let  r  f t T  7 > 0,Vi,V/i;  $ 7 = 0 T  (3.5)  have no solution. A p p l y Motzkin's Theorem to (3.5) to conclude that there exists pih •> 0 ^  n o  £ ji q a  e  U a  i  z e r o  )  a i l (  i a A G JR such that  J2l3 r ih  By A /  ih  + A$ = 0.  (3.4) and Proposition 2.2, $ g K((-T )). ih  0.  (3.6)  Hence, A < 0 . B y Assumption A ,  (The argument is identical to the one used in the proof of Proposition 2.2.)  Thus, A < 0 . Rearranging (3.6) yields $ G  K((T )). ih  ii) T h i s follows from Statement i) and Proposition 2 0  Smith (1983) rules out case (i) of Proposition 3.1 by assuming that the H a t t a (1977) conditions are satisfied; that is, an increase in a l u m p - s u m transfer leads to a positive change in the cost of (net) demand, evaluated at the original producer prices.  T h i s assumption is obviously satisfied when producer and consumer prices  coincide, provided that consumers are nonsatiated. T h e intuition behind this result is clear. Suppose that the planner changes consumer prices in such a way that everyone ^ The original Hatta conditions impose this restriction on compensated aggregate demand. In the sequel, I impose similar conditions on uncompensated aggregate demand.  44  is better off and the resulting equilibrium is non-tight. T h e n there exists an excess supply of goods.  Suppose now that the planner redistributes this surplus with a  l u m p - s u m transfer. Everyone is made still better off, and by the restriction placed on aggregate demand, some of the surplus is consumed. T h e planner could choose an increase in the poll subsidy large enough to get rid of the entire surplus.  In the present context, it is not guaranteed that an increase in a specific demogrant makes all consumers better off.  Something must be said about sharing  within families before such a conclusion can obtain.  A s s u m p t i o n C : 0 < ip^ < 1, V7i and 0 < </^ < 1, V/i. 2  T h a t is, additions to the l u m p - s u m grants are 'split' in the usual sense of the word. T h i s property need not hold in general, due to the presence of endowments. A s long as the planner may increase the demogrant afforded to each person, Assumption C entails Assumption A of the preceding section because an increase in either demogrant makes all consumers better off.  In what follows I have occasion to use the following assumptions, each of which is in the spirit of Smith's restriction.  A s s u m p t i o n N I : p V ix(q,  R , R ) > 0.  A s s u m p t i o n N 2 : p V 2x(q,  R , R ) > 0.  T  R  T  R  1  1  2  2  45  Like the H a t t a conditions, Assumptions N I and N2 are a form of normality conditions on aggregate demand. It turns out that temporary inefficiencies may be ruled out when either N I or N2 holds. T h i s is the content of the next proposition.  Proposition 3.2 Let Assumptions B and C hold. Then strictly Pareto-im,proving directions of policy change with temporary inefficiencies cannot arise if either Assumption NI or N2 holds.  Proof: Proposition 3.1 states that temporary inefficiencies occur exactly when there exist  P >0 ih  -p V ix(q, T  a  pT. ih  such that $ = £ \  h  R \ R) 2  ih  T h e last two rows of this equality are:  = 5 > % ^ +$> h  - P V r f . s f a - R\ T  R) 2  2 / l  (l - A ) , (3.7)  h  = X > % *  a  +  -  ft ft Assumption C allows me to conclude that the right-hand sides of each equation in (3.7) is nonnegative.  N I precludes the top line of (3.7) from holding. N2 precludes  the second.D  T h a t only one of N I and N2 is required to rule out temporary inefficiencies is not surprising. T h e planner needs but one instrument to redistribute the surplus to individuals.  Note the role played by Assumption C in this framework.  It ensures  that an increase in either poll subsidy is unanimously preferred by everyone, thereby allowing the surplus to be distributed in a Pareto-improving way.  46  3.3. Restricted Poll Taxes  It may be argued that the planner cannot make l u m p - s u m transfers contingent on the index i. T h i s may be because there is no easily observed characteristic to which it corresponds, or it may be deemed inappropriate to 'discriminate' on the basis of that characteristic.  One may also take the view that it is difficult to enact new policies  that aim at reducing existing differences in l u m p - s u m payments. circumstances can be viewed as imposing the restriction dR  1  Either of these  = dR? on the directions  of policy change available to the planner. It is, therefore, interesting to investigate the conditions under which temporary inefficiencies may arise in this context.  It is necessary to introduce some new notation at this point. Let 0 denote the 2N-  dimensional zero vector. Define also the following sets:  0  x G M \x 2N+2  K :=K\  = v " 0"  " 0"  1  -1  -1  _ 1 _  (-r >, i h  K := K I (T )  1  V,  (3.8)  (3.9)  " 0"  " 0" ih  v £ 1R  1  >  -1  (3.10)  _ 1 _ £ is the negative 45-degree line in the subspace of  IR  2N+2  spanned by the last two  elements of the standard basis. Given Assumption C , the sets K and K are supersets of  K({—T )) lh  and / C ( ( r ) ) , respectively. T h e first 2N components of any vector in K l / l  must be generated as a semi-positive linear combination of the first 2N components of vectors (—V ). TH  Furthermore, the projection of any vector in K onto its last two 47  components lies off the negative 45-degree line.  K({T )). lh  K bears an analogous relation to  Bearing these notations in mind, the following obtains.  Let Assumptions B and C hold. Then the following hold.  P r o p o s i t i o n 3.3  i) $ G K if and only if there are no feasible strictly Pareto-improving directions of policy reform satisfying dR = dR . ii) <P G K if and only if there exist strictly Pareto-improving  directions of policy  change satisfying dR = dR?, all of which are necessarily non-tight equilibrium1  preserving. iii) <P G K  c  nK  c  if and only if there exists tight equilibrium-preserving directions  of policy change satisfying dR = dR . 1  2  Proof: i) There are no feasible strictly Pareto-improving directions of change satisfying dR  = dR  1  2  if and only if the following has no solution  r <y  > O.Vt.fc;  ihT  $ 7>0; T  [0 , 1,-1] T  7  = 0.  (3.11)  But, by Motzkin's theorem, (3.11) has no solution exactly when  J2/3 r ih  ih  +\ $ +K  0  1  = 0; (3  ih  > 0,  some f3  ih  > 0, A > 0, K G  1R  (3.12)  -1  i,h  has a solution. Suppose, by way of contradiction, that A = 0. T h e n the last two rows of (3.12) become  h  h  (3.13)  h  h 48  Now,  by Assumption C , the top line of (3.13) implies n < 0, whereas the bottom  line of (3.13) implies K > 0. A contradiction ensues. Therefore, A > 0. Rearranging (3.12) yields that <& G K.  It is a matter of straightforward computation to show that  <P G K implies the existence of a solution to (3.12) .  ii) $ G K if and only if there exist f3  lh  > 0 (not all zero), and K G M satisfying  0 $ = ^p  i h  T  +K  i h  i,h Now,  (3.14)  1 -1  by Motzkin's Theorem, it must be the case that  T  7 > 0,Vi,/i;  ih  $'7<fJ;  [0T ,1, - 1 ]  0  (3.15)  [ 0T\ 1 , - 1 ] 7 = 0  (3.16)  =  1  7  has no solution. In particular, there is no solution to  r  ifcT  7 > 0,Vt,V/i;  $'7 = 0;  Note that Assumption C implies that there is a solution to  r  (Pick 7 =  ifc  ' 7 > 0,Vi,/i;  T  [O',l,-l]7  (3.17)  =0  [ 0 , 1 , 1 ] ; that is, increase both poll subsidies by the same amount.) T  T  Hence, there must be a solution to  T  ih  7 > 0,Vi,V/i;  J  $'7>0;  The 'only i f part of Statement ii) follows.  49  [0T , 1 , - 1 ] = 0. 1  7  (3.18)  Conversely, suppose there is a solution to (3.18), but that there is no solution to (3.16). A p p l y Motzkin's Theorem to (3.16) to conclude that there exists f3  lh  > 0  (not all zero), and K, A G JR satisfying 0  +\ $ +K  J2(3 r ih  ih  0.  1  (3.19)  -1  i,h  W h e n A = 0, the last two rows of (3.19) reduce to (3.13). T h i s violates Assumption C . A > 0 implies <P G K.  In view of Statement i), this violates condition (3.18). Thus,  A < 0. Rearranging (3.19) now yields that $ G K.  iii) T h i s statement follows directly from i) and ii).  •  It is interesting to compare Proposition 3.3 with Propositions 2.2 and 3.1 when Assumptions B and C (and, hence, A) hold. Because  K((—V )) th  is contained in  K,  statement (i) of Proposition 3.3 indicates that there are fewer values of the vector $ for which feasible strictly Pareto-improving directions of policy reform exist when poll taxes are restricted. T h i s is hardly surprising. in K,  Because i f ((P' )) is contained 1  statement (ii) of Proposition 3.3 indicates that there are more values of the  vector $ for which temporary inefficiencies arise when the planner operates within the restricted set of poll taxes.  Some insight into circumstances giving rise temporary inefficiencies is afforded by considering a necessary condition for temporary inefficiencies.  Suppose $ G K.  T h e n it must be the case that  p V ix(q R\R )^p V 2x(q,R\R ). T  2  R  t  T  2  R  50  (3.20)  In the presence of Assumption N I (or N2), it is clear that temporary inefficiencies cannot arise when (3.20) is violated. In that case, the population would act as two sub-populations, each inducing the same change in the value (measured at producer prices) of aggregate demand to changes in the demogrant. T h e intuition underlying Proposition 3.2 would apply.  Given that the restricted planner still has some effective means of using poll subsidies, one might suspect that temporary inefficiencies may be easily ruled out. T h i s can be shown by appealing to the following assumption.  Assumption N 3 : p (V ix(q, T  R  R , R ) + V 2 x ( q , R ,R )) l  2  l  R  2  > 0.  Notice that Assumption N3 implies that one of N I or N2 must hold, but not necessarily both. It is also consistent with condition (3.20) . T h e following proposition may come as no surprise.  Proposition 3.4 Let Assumptions B, C and N3 hold. Then strictly Pareto-improving directions of policy reform with temporary inefficiencies cannot arise when dR = 1  dR . 2  51  Proof: B y Proposition 3.3, temporary inefficiencies can hold only when there exist  pih > g  ^ £ ij q j no  a  e  zero),K  U a  R satisfying (3.14). A d d i n g the last two lines of  €  (3.14) yields  p (^ xx(q,R\R ) T  2  a  + V 2x(q R ,R )) l  R  1  2  =^ ^ " ( ^ i + ^ O + E ^ ^ - ^ i - ^ ) h  h (3.21)  Assumption N3 implies that the left-hand side of (3.21) is negative, whereas Assumption C implies that the right-hand side of (3.21) is nonnegative. ensues.  A contradiction  •  T h e role played by assumption N3 in Proposition 3.4 is clear. It is sufficient to ensure that any surplus generated by a price change will be 'eaten up' if each poll subsidy is increased by the same amount.  I also wish to investigate the power of a planner who can merely redistribute a fixed l u m p - s u m between the family members.  Corollary 2.1.1 indicates that the  intuitive argument for the elimination of temporary inefficiencies breaks down, since redistribution of a fixed total alone cannot achieve strict Pareto-improvements. However, a planner endowed with such a power is not identical to one who has no l u m p sum taxation power at all. In order to state the analogue to Proposition 3.3 in this context, I require some additional notation:  (3.22)  52  "o" K := K  " 0" (3.23)  -1  1 1 _  K := K I (r ) i/l  "0"  " 0"  1  -1  (3.24)  _1_ T h e line £ is the 45-degree line in the plane spanned by the last two elements of the standard basis. T h e sets K and K are analogous to K and K, respectively, and can be given similar interpretations. Let $ Q , TQ denote the vectors $, F  L H  ,  respectively,  with their last two components deleted.  P r o p o s i t i o n 3.5  Let Assumptions A, B and C hold. Then the following statements  hold. i) There are no feasible strictly Pareto-improving directions of policy reform that 0 E K((T )) satisfy dR + dR = 0 if and only if $ € K or 1  2  IH  ii) Inhere are strictly Pareto-improving directions of policy reform that satisfy dR^ + dR = 0, all of which are non-tight equilibrium preserving, if and only if <P G K 2  (£K({r )).  and  ih  iii) There exist tight equilibrium preserving strictly Pareto-improving policy reform that satisfy dR + dR 0\ 1  '  2  directions of  = 0 if and only if $ G K  DK  c  c  and  iK{{Y )). ih  iv) Moreover, if there exists h such that<p\ ^ <p\, then whenever p V i x ( q , R , R ) R  andp V 2x(q, 1  R  R  L  , R) 2  p V ix{q,R ,R ) T  1  are distinct, there exists $  | G K.  2  R  .pTV^te,/* ,*? ) 1  2  53  Q £  K((T Q)) 1  such that  Proof: i) There are no feasible strictly Pareto-improving directions of change satisfying dR  1  + dR? = 0 exactly when there is no solution to  T 'J  > 0,Vi,h;  ih  [0T' , 1,1] = 0.  $'7>0;  (3.25)  7  B y Motzkin's Theorem, (3.25) has no solution if and only if there exist f3  m  p  ih  > 0 (some  > 0), A > 0,K G R satisfying  (3.26)  = 0. i,h W h e n (3.26) holds with A > 0, $ G K.  If A = 0, the last line of (3.26) becomes  (3.27)  It follows from Assumption C that K < 0. J calculation confirms that either [ 0 , 1 , 1 ]  Hence, [0 , 1, if T  G K ({T )) lh  G K((F )).  Direct  ih  or $ G K implies the existence  of a solution to (3.26).  ii) Let $ G K and [0 , 1,1] T  T  £ K((T )).  $ G K exactly when there exist p  ih  ih  > 0  (not all zero) and K G JR such that  $ = ^2p r ih  ih  +K  (3.28)  i,h B y Motzkin's Theorem, (3.28) implies that there is no solution to  r  7 > 0, Vi, h; $ 7 < 0; [0 T' , 1,1] = 0.  ift  1  7  (3.29)  In particular, there is no solution to  r  .-I.T  lft  7  >  0,Vi,/i;  $ 7 = 0; T  54  [0T, 1 , 1 ] = 0 1  7  (3.30)  [0 ,1,1] T  T  £ K((r )) ih  implies that there does not exist  p  ih  >  0 (not all zero) and  K < 0 satisfying To 0.  (3.31)  i.h  B y Assumption A , (3.31) has no solution with semi-positive f3 and K — 0. T h e last lh  row of (3.31) is exactly (3.27), so that Assumption C rules out the possibility of a solution to (3.31) with semi-positive f3 and K > 0. Thus, by Motzkin's Theorem, lh  there exists a solution to  r  h  T 7  >  0,Vi,h;  [0 ,1,1]7 = 0.  (3.32)  T  T h e n , by (3.29), there is a solution to  r  i f c T  7 > 0,Vi.fr;  $ 7>0; T  [0 , 1,1] T  7  = 0.  (3.33)  Conversely, let (3.33) have a solution, but let there be no solution to (3.30). A p p l y Motzkin's Theorem to the first and third components of (3.33) to conclude that [0 ,1, 1 ] T  T  £  K({T )).  implies the existence of  ih  fj > lh  Because (3.30) has no solution, Motzkin's Theorem 0 (not all zero) and real numbers A and K satisfying  (3.26). In view of (the proof of) Statement i), A > 0 contradicts (3.33). Thus, (3.26) holds with A < 0. Hence, $ G K.  iii) This statement follows from i) and ii).  55  iv) Take arbitrary p V g i x ( q , R ,^ ) T  1  ^ p V 2 x ( q , R , R ).  2  T  1  T h e existence of an h as  2  R  described in the statement ensures that 1 - c A  (3.34)  1 - ^ 2  since the set of generators contains at least three noncollinear vectors.  T h e n there  > 0, at least one nonzero and K G IR such that  exist P  lh  -p V x(q, T  Rl  + E^  R\ R ) = E^V^i 2  h  -p v ix(q, T  R  Now select $ Q : = Y,i,hP  R \ R)  =  2  T  >  a  n  -  1  2/l  t  K  h  jyy^+E^ (i d  +  h  e  r  e  s  u  l  t  follows.  (3.35)  +«  •  A few words of comment on Proposition 3.5 are in order. First of all, suppose that there are no Pareto-improving directions of change in consumer prices alone, ignoring feasibility. T h e n , by Motzkin's Theorem, 0 G K((FQ)). C holds, this condition is equivalent to [ 0 , 1, 1 ] T  T  G  K((r }). lh  W h e n Assumption T h i s is a strengthening  of Corollary 2.1.1: Whenever there are no strictly Pareto-improving directions of change in consumer prices alone, adding purely redistributive transfers affords no new Pareto-improving directions of reform. Moreover, <3> G K implies $ Q G  K({—V' )). h  Hence, whenever the planner considered in Proposition 3.5 can find no feasible strictly Pareto-improving directions of policy reform, neither can a planner who does not have the power to use any demogrants. This conclusion should come as no surprise, since dR  1  + dR  2  — 0 is satisfied whenever dR  1  = dR  2  = 0. However, purely redistributive  transfers can be used to induce demand responses that make feasible some changes in consumer prices that would otherwise be infeasible. 56  Statement (iv) of Proposition 3.5 indicates that the restricted planner may face the prospect of temporary inefficiencies regardless of the demand responses to l u m p sum taxation.  Purely redistributive transfers may induce demand responses that  lead the economy toward the production frontier (indeed such a transfer can be found whenever the conditions expressed in statement iv) are satisfied), but, in view of Corollary 2.1.1, these transfers by themselves can bring about no P a r e t o improvements.  Notice that Assumption A must be included in the hypothesis of Proposition 3.5, because it is not implied by Assumption C when only purely redistributive transfers are available.  To see this, let $  E I.  T h e n [dq , dR , T  1  dR ] 2  •$  =  0 for all  dq and for all purely redistributive changes in l u m p - s u m transfers dR ,dR . l  2  There  exists a feasible Pareto-improving direction of policy change if and only if there exists a Pareto-improving change in consumer prices alone.  p V ix(q, R , R ) = —p V 2x(q,R ,R ) T  1  2  R  T  l  2  Moreover, whenever  purely redistributive changes in the de-  R  mogrants have no effect on the value of aggregate demand (measured at initial producer prices).  Hence, the feasibility condition reduces to feasibility of directions of  price changes alone.  Assumption C also fails to imply Assumption A when there is  income pooling in the presence of purely redistributive taxation. Indeed, as A p p s and Rees (1988) have shown, when households act as if they are maximising a price and income independent social welfare function, marginal purely redistributive changes  57  in demogrants have no effect at all on intra-family allocations as long as boundary constraints do not bind.  58  C H A P T E R 4 : Optimal Non-linear Taxes for Families  4-1. Introduction  Since the work of Mirrlees (1971) it has been recognised that the design of income tax policy must take into account the asymmetry of information between agents and the planner. W h e n agents have private information about their characteristics upon which the planner wishes to base a taxation scheme, they may have an incentive to mis-report these characteristics to the planner.  T h e planner must design the  taxation scheme to prevent this sort of misrepresentation.  It is customary in the  literature to assume that the hidden differences among agents can be summarised by a single p a r a m e t e r .  17  T h i s assumption has been questioned as a complete description  of individuals. It seems even more dubious when decision-making units are comprised of more than one individual.  A n example of a planner designing optimal policies for heterogeneous groups is the problem of family income taxation.  W i t h i n a family, individuals may differ  in labour productivity or they may have unequal say in family decisions.  These  differences influence labour supply behaviour, which, in turn, has consequences for the design of a tax system.  In this study, I trace the effects of family interactions  Guesnerie and Seade (1982) provide a characterisation of optimal tax schedules for a finite economy under this assumption. Weymark (1986a, 1986b, 1987) shows how the problem can be decomposed into simpler sub-problems when preferences are quasi-linear, and provides a more detailed description of optimal tax schedules. 1 7  59  on optimal tax schedules, paying particular attention to the role played by diversity within the family.  Some classic questions of public finance can be addressed by considering the non-linear tax problem for families. For instance, the analysis of this problem can contribute to the debate over whether the base for income taxation ought to be family income or individual income. Indeed, the issue of whether all members of the same family should be taxed at the same rate is one of the central questions of this study. I show that taxing distinct members of a family at the same rate is not always optimal. T h i s can also be viewed as a contribution to the debate over the desirability of a uniform "flat" tax.  I consider an economy inhabited by two-person families. Attention is restricted to the case of workers of two productivity types. T h i s results in four possible family compositions. Each family member has preferences over leisure and a consumption good. Families are assumed to act so as to maximise a weighted sum of their members' utilities.  T h e weights are assumed to be independent of incomes. In this way, the  family decision process can be described by a single parameter. T h i s parameter is known to the planner.  T h e planner designs a tax schedule for these families. Families are then free to re-allocate their after-tax incomes to maximise their objectives.  Because decisions  are made at the family level, self-selection constraints are formulated in such a way that families have no incentive to mis-report the types of their members. T h a t is, 60  families are viewed as decision-making units, differing along two dimensions, namely the productivities of their members.  Under the maintained hypothesis that there  exists a family social welfare function, there is no loss of generality in considering tax schedules that specify the total tax liability for a family as a function of the pair of before-tax incomes of its members. Hence, the model economy studied here has three goods in it. T h i s contrasts with the work of Guesnerie and Seade (1982), who consider non-linear taxation in a two-good economy.  Because there are three goods in the economy, family indifference surfaces are two-dimensional.  Thus, if the indifference surfaces of distinct families cross, their  intersection usually occurs along a curve. In particular, it is unreasonable to expect a standard single-crossing property, such as the one posited by Guesnerie and Seade (1982, Assumption B , p. 168), to hold. Nevertheless, the family objectives specified in this analysis possess some special geometric properties. W h e n two families differ along exactly one dimension, certain projections of their indifference surfaces satisfy a single-crossing property. Moreover, the indifference surfaces of any such pair such families intersect along a line parallel to one of the coordinate axes. I demonstrate that these two features of preferences have implications for the structure of optimal tax schedules.  A minimal amount of structure is placed on the objectives of the taxation authority. Only Pareto efficiency with respect to family objectives is posited.  Family  objectives are assumed to be additively separable, but not quasi-linear. E v e n with  61  this minimal amount of structure, it can be shown that individuals in the same family may face different marginal tax rates, so that using total family income as a tax base is not optimal.  T h e formal analysis is an example of mechanism design with two-dimensional uncertainty, of which the work of Rochet (1995) is the most complete to d a t e .  18  He  considers the problem faced by a monopolist wishing to maximise profits by offering a non-linear price schedule for two goods. H e derives all the possible optimal pricing mechanisms for the case of linear preferences and quadratic costs. Mechanisms with the qualitative features he describes are possible in the present context. Given the relatively unstructured environment considered here, some additional possibilities arise. Drawing attention to these possibilities demonstrates the strength of the assumption of linearity.  Like the non-linear pricing problem, the taxation problem can be described as a choice among alternatives that satisfy self-selection, criteria. There are, however, fundamental differences between the two problems. There is a natural group of consumers whom a monopolist wishes to identify and extract surplus from, those who have a greater taste for its product.  A priori,  there is no group of workers that a  planner wishes to tax more heavily than others.  It is common to assume that the  planner wishes to redistribute income from more able workers to less able workers. Among the contributions to multi-dimensional income tax problems in the continuous case are Mirrlees (1976) and Seade (1979). Wilson (1995) provides an overview of the mechanism design problem and the associated computational issues. 1 8  62  Nevertheless, it is important to recognise that such an assumption is a value judgment. Moreover, non-linear pricing problems are usually presented in a partial equilibrium context, with the amount of total surplus extracted by the monopolist constrained by the voluntary participation of consumers.  A n economy-wide materials balance  constraint limits the scope of the taxation authority.  Another important distinction between optimal taxation problems and monopoly pricing models is that the objective function of the taxation authority is usually assumed to be increasing in the welfare of each agent, whereas monopolists are modeled as being concerned about profits rather than the welfare of their customers.  19  As  Brito et al. (1990) have shown, many qualitative features of optimal non-linear tax schedules follow directly from the planner giving positive consideration to the utility of every agent.  Their analysis is not restricted to the two-good world, nor to the  case of agents who differ in only one characteristic. Indeed, they do not use any parameterisation of the differences among agents to derive their results. B y exploiting the special structure of the family objectives in the present model, I can make some statements about the tax rates faced by specific families. These kinds of statements cannot be derived in the more general framework of Brito et al. (1990).  T h e remainder of the chapter is organised as follows. Section 2 gives an outline of the economy.  T h e implications of the self-selection constraints are presented in  Another related class of problems, regulation design, features a planner with an objective function that takes both consumers' and producers' surplus into account (cf. Dana (1993)). 1 9  63  Section 3. Section 4 summarises the implications of the Pareto-efficiency assumption. Proofs for this chapter and the following one are collected in an Appendix.  4.2.  The Model  Individuals in the economy are assumed to differ according to their productivity. Specifically, there are two types of individuals, indexed by WL,WJJ with WL < WJJ. T h i s index corresponds to the efficiency units of labour per unit of labour time supplied by the individual. I assume constant returns to scale in the production sector and a perfectly competitive labour market so that the before-tax income of an individual is given by yi  :=  (4.1)  Wik,  where k is the labour supplied by person i. There is a single consumption good, c. Individuals have preferences over the consumption good and labour supply given by  Ui(ci,  k)  •=  U(ci)  -  h(k).  (4.2)  T h e function U(-) is assumed to be continuously differentiable, increasing and strictly concave.  It is also assumed that U'(c)  tends to positive infinity as c tends to zero.  T h e function h(-) is assumed to be continuously differentiable, increasing and strictly convex.  Families consist of two individuals. There are four types of families: L L , H L , H H and L H . L e t T denote the set of families.  64  Family decisions are assumed to be  consistent with the maximisation of a weighted utilitarian household social welfare function  W(u\,  112)  •= u\ +  7 > 0.  7U2,  (4.3)  Note that W(-) is not a symmetric function, so that a family of type H L is not identical to one of type L H . T h e function W(-) is symmetric only when 7 = 1. Even in this special case, it is not possible to identify families H L and L H a priori. A s long as the index i attached to an individual is observable, the planner can use this information in setting the tax schedule. One would need to show whether it is indeed optimal to ignore this information by allocating the same bundle of goods to, say, individual 1 in family H L and individual 2 in family L H .  Let x\, X2 denote the after-tax incomes of the two individuals in the family and let x :— x\ + X2- T h e n consumption decisions for a typical household arise as the solution to max  (P)  Ci,C2  U( )-h(^)  y  U(c )-h(^) 2  +7  Cl  2  U>1'J  V  L  subject to  w  2  c\ + C2 < x\ + x 2  I now turn to a description of the important features of the solutions to ( P ) .  Proposition 4.1 Let £i(x\,  X2)  and C2(x\,  ci{xi,x ) = c\(x); 2  X2)  2 0  be the solution functions for (P). Then  C2{xi, x ) = C2(x). 2  (4.4)  20 Throughout this analysis it is assumed that the non-negativity conditions (which are not stated explicitly) are satisfied.  65  Condition (4.4)  states that for a given total family after-tax income, the alloca-  tion of consumption within the family does not depend on the identity of the family member who receives the income. T h i s is know as the income pooling condition on family behaviour.  It holds when families maximise any Bergson-Samuelson social  welfare function, not just for the additive form specified here.  Its major implica-  tion for this study is the reduction of the number of planner's choice variables per family from four to three.  Because the choice variables are consumption levels, the  formulation of (P) may lead one to believe that the family is not making any labour supply decisions. T h i s suspicion is untrue. T h e problem (P) describes how after-tax income is allocated between family members. Labour supply choices are modeled as the choice from the tax menu offered by the planner. Families take the division of consumption and its effects on family welfare into account when choosing how much to work.  Using (4.4), it is possible to define the function  V(x) :=U{c (x))+-yU{c {x)), 1  (4.5)  2  the component of family welfare owing to after-tax income. W i t h this notation, I can now state some further properties of solutions to the problem (P).  Proposition 4.2 Let ci(-) and 62O) be as defined in Proposition i.) c' (x) + c' (x) = l. 1  2  ii.) ci(-) and c~2{') are increasing.  66  Then  iii.) V(-) is increasing and strictly concave.  Condition i) is a direct result of the increasingness of U(-). It states that a oneunit increase in after-tax income increases total family consumption by one unit. Statement ii) indicates that the family considers the consumption of each member to be a normal good. T h i s follows from the separability of preferences.  Increasingness  of V(-) follows from the increasingness of individual utility in consumption. While the function V(-) is the sum of concave transformations of the optimal consumption choices, the choice functions need not be concave. Indeed, condtion i) of the proposition implies that no more than one of the choice functions can be strictly concave. However, the same condition prevents both of the choice functions from being convex, and results in the strict concavity of V ( - ) .  In what follows I need to consider the value function for (P), which depends on the before-tax incomes of the family members and on joint family after-tax income. Denote the value function for a family of type i by W . l  Then  I take the consumption good as numeraire and assume that the producer price of effective labour is one. Let 7 r be the proportion of families of type i in the population. l  T h e n the materials balance constraint for the economy can be written as  (F)  E^^E^+5>u i  i  i  67  T h e lack of complete information prohibits the use of optimal l u m p - s u m taxation.  Instead, the planner must design an allocation of goods that satisfies the  self-selection conditions  (SS)  W^yiyi)  > W^x^ylyi),  Vi ^ j .  T h a t is, each family must (weakly) prefer the bundle of goods designed for it to the bundle intended for any other family. T h e relations (SS) represent the natural incentive-compatibility constraints in this environment, as decisions are made by families.  It follows from the taxation principle (cf.  Guesnerie (1981)) that the problem  of designing a tax schedule for these families is equivalent to offering a menu of alternatives satisfying (SS). Because of income pooling, there is no loss of generality in considering tax functions that specify the total tax liability of a family for a given  ordered pair of before-tax incomes. For fixed before-tax incomes it is merely a transformation of variables to consider total family after-tax income rather than total tax liability as the decision variable of the taxation authority. Such a tax schedule must be anonymous in that each family faces the same budget set.  It need not be  anonymous at the individual level, because the tax paid by one member of a family may depend on the choices of her partner.  Clearly, when families choose labour-  consumption bundles to maximise their welfare from an anonymous tax schedule the outcomes satisfy (SS).  68  Furthermore, it has been shown by Guesnerie (1981) that for any allocation that satisfies (SS) and (F) a tax schedule can be constructed that induces the families to choose that allocation. T h e tax schedule constructed by Guesnerie requires the possibility of offering an infinitely negative amount of after-tax income to a family with before-tax incomes other than those that arise from a truth-telling game. W h e n negative after-tax incomes are infeasible (as they are assumed to be here), these "punishment" allocations cannot be used.  In the current model, it is possible to  support any allocation which satisfies (SS) and (F) with a tax schedule that does not require negative consumption. T o see this, take an allocation that satisfies (SS) and (F). Let <S denote the set of family bundles in that allocation. Define the sets  £ •= {(yi,y2)\{yi,y2,x)  For any ordered pair  x (3/1,3/2)  e s}-,  (3/1,3/2), x(2/l>2/2)  is  a  :=  e S}.  {x\(yi,V2,x)  (4.7)  singleton, so with a slight mis-use of  notation, I consider x( ) t ° be a function from C into M+. -  Now, construct a tax  function T: IR.\ - » M by  ^  U  T(yun):=  J y i + J/2-xk/1.3/2),  + y2,  if (2/1,2/2) e £ ;  if  ,  ,  (48)  Given the tax function T ( - ) , the budget set faced by each family is  £ :=<SU  Given that U'(c)  {(3/1, y , 0)10,1,2/2) 2  (4-9)  tends to positive infinity as c tends to zero, family indifference  surfaces do not cross the (yi, y2)-plane. Thus, families choose only the bundles from B that are contained in S.  Moreover, because the bundles in S satisfy (SS), each  69  family chooses the bundle intended for it in the allocation (SS). Hence, the resulting choices from the tax schedule also satisfy (F).  T h e planner seeks to maximise a social objective. There are two possible sets of arguments for a welfare-based objective: individual utilities and family welfare levels. T h e former is more closely related to the standard value judgments of individualism and welfarism. However, the arguments of the planner's objective function then fail to coincide with the criterion functions of the agents (here, families) in the economy. Hence, the formal analysis of this case is closely related to the work on income taxation with non-welfarist objectives (Seade (1980), Kanbur, Keen and Tuomala (1994)). T h e latter class of planner's objectives arise inevitably from treating a family as a homogeneous unit.  Using family welfare levels as the arguments of a planner's  objective can also be interpreted as the planner respecting the ethics embodied in the family social welfare function. T h e resulting analysis is formally related to the recent work on optimal non-linear policies in environments of two-dimensional uncertainty. (Dana (1993), Rochet (1995), Armstrong (1996)).  In order to place this study in the more familiar context of multi-dimensional screening I choose to take the family as the basic unit of welfare analysis. Specifically, I analyse the solutions to the problem  (PF)  max Z(W , W , W , LL  yi,y2,x  HL  HH  W ) LH  70  subject to (SS) and (F),  where x := ( x  L L  ,x  H L  ,x  H H  ,x  L H  ),  y  {  :=  (yf ,  yf ,  L  L  yf ,  yf ),  H  H  4 =  1,2 and the  function Z(-) is assumed to be increasing, continuously differentiable and concave.  4-3.  21  Self-Selection  T h e structure of family objectives, summarised by (4.6), influences the nature of the self-selection constraints. I now turn to an elucidation of the important features of family objectives and the implications these features have for allocations that satisfy the constraints (SS).  Consider, first, the marginal rates of substitution between before-tax  incomes  and after-tax income, given by  MRSi y i  '  x x  :=  h'A) . ] . . W  t  w\V'(x )  MRS*  %  '  y2  x  =  7^(4) .  io V'(x ) l  l  (4.10) V  ;  2  It is clear from (4.10) that any differences among the marginal rates of substitution of different families arise from the structure of the function h(-). For this analysis, two properties of h(-), both of which are direct consequences of its convexity, are crucial. These properties are given in the following two lemmas.  Lemma 4.1 For ally,  ±h'(£)  >  ^h'(^).  The problem (PF) may also be interpreted as the problem faced by a planner who wishes to design a tax scheme for individuals with multiple characteristics and preferences represented by 2 1  (4.6).  71  A n important implication of L e m m a 4.1 is that holding y\, say, constant, preferences in (y2> a;)-space satisfy the single-crossing property. T h a t is, the projections of indifference surfaces onto (3/2, x)-space are flatter for families with a person of higher ability in position 2. It is natural to expect such a property to hold, as individuals of low type must give up more leisure time than their high-type counterparts to gain an equal amount of before-tax income. Hence, families with low-type individuals would require more additional consumption to compensate them for increases in before-tax income.  L e m m a 4.2 For any (y,y), y > y if and only if  (4.11)  L e m m a 4.2 states that, viewed as a function of y and w, /?.(•) has decreasingdifferences in y (cf. holds for  Topkis (1978)). It is important to note that no such property  W(-).  A standard feature of optimal income tax models is that more able individuals receive both a higher before-tax and a higher after-tax income (Weymark (1986a)). One would not expect this result to obtain in the present context given that the notion of a "more able" family is not immediate. A t least, the immediate concept of a more able family does not completely order the families. It is, however, possible to define a partial ordering on the set of families in a natural way.  72  Definition: only if w\  >p  T h e relation  > w\ and w\  component of  on  > w2  T', the  set of families, is defined by:  i >p j  if and  T h e relation >p is defined to be the asymmetric  >p.  Figure 1 provides a diagrammatic representation of the set of family types. relation > ^ orders all families but the pair {HL, LH},  The  located at cross-corners and  off the 45-degree line.  T h e self-selection constraints place some structure on the pattern of before-tax incomes, especially for those families that can be compared using the partial order >F- T h i s is the content of the next two propositions.  Proposition 4 . 3 Any allocation that satisfies (SS) also satisfies: i. ) y^  H  > y\ . R  then x HL  x  HL  iii.) y  HH  LH  > y\ . L  >x  LL  >y . HL  then x vk  LH  X  H  L  LH  then x  > y^ , then x L  (with equality if and only if y H  >y , HL  >y , L  2  e  U  73  L 2  >x  HH  then x  LH  H  and if y  = y , then L  2  L  then x  2  H  2  H  and if yf  HL  H  =  yf , L  = y J-  H  L  2  H  y ,  = y\ )• LL  (with equality if and only if y  Moreover, if y^  =  H 2  = V\ )-  RL  Moreover, if yf  and if y  LH  >x  HL  > L L (yjiffo q ality if and only if y X  >x  HH  HH  L  HL  > vk •  >y ,  (with equality if and only if y  Moreover, if y^  >x  HH  iv'•)  H  >x  HH  ii. ) y  Moreover, if y^  2  >x  LL  = y ). L  2  and if y  H 2  = y , then J jL 2  Proposition 4.3 states that given an equally productive partner, a person of high productivity will earn at least as much before-tax income as a low-productivity person.  Abstracting from differences among individuals 2 (by, say, considering an  economy populated entirely by families of types H H and L H ) produces a three-good economy with one-dimensional uncertainty. T h a t is, when focusing on families along an edge of the type space depicted in Figure 1, the planner is facing what is essentially a one-dimensional screening problem with multiple instruments.  Proposition 4.3  states that self-selection imposes a monotonicity property on the allocation of y\, the good over which agents differ in a way that is unknown to the planner. Such worlds have been studied in an environmental regulation context by van Egteren (1996). He reports that self-selection requires a monotonicity property on pollution control standards, the cost of which is the only source of private information in his model.  Proposition 4.3 has a geometric interpretation, which I now give for clause i). Denote by (y^ , H  y , H  2  x ) LH  the bundle designed for family L H . Consider the indiffer-  ence surface of family L H through this point. In order for the L H - H H self-selection constraint to be satisfied, the bundle offered to family H H must lie on or below this surface. T o satisfy the H F I - L H self-selection constraint, the bundle designed for family H H must lie on or above the indifference surface of family H H passing through the point (y\ , H  y , H  2  x ). LH  According to Proposition 4.3, all points lying between  these two surfaces have the feature that y\ > y\ . R  surfaces is the line with equation: y\ —  y\ . H  74  In fact, the intersection of these  Figures 2 and 3 provide a two-dimensional representation of Proposition 4.3. T o understand these figures, it is useful to think in terms of "pseudo-indifference curves" for the families. A pseudo-indifference curve is a level set of a partial welfare function, showing points that give equal amounts of welfare derived from two of the three goods (say, yi and a;). Given the additively separable form of family objectives, it may also be interpreted as a slice of a family indifference surface (drawn for a constant 3/2)-  First, I consider Figure 2. Let u  LH  denote the welfare level of family L H at the  allocation designed for it. T h e point D represents (y^ ,  x ).  H  u  LH  u  •= V(x ) LH  - h(^~) = u  LH  Now let  LH  +7 M — ) •  (4.12)  is the label on a pseudo-indifference curve for family L H i n (y\, a;)-space, namely  LH  the one that depicts the set of bundles that give family L H welfare level u ,  holding  LH  y$  H  at y\ . M  T h i s curve is denoted by L H in Figure 2. T h e value in parentheses  next to the label L H , y , H  2  serves as a reminder that the curve L H is the y slice H  2  the indifference surface of family L H through  (yf , j 7 11  f f 2  ,  x ). LH  of  Clearly, the curve L H  passes through the point D . T o analyse self-selection, it important to consider the welfare of family L H at the allocation designed for some  other  family (here, family H H ) .  W i t h this purpose in mind, denote the allocation of family H H by  (yf , y > 11  H  2  x ) HH  and define u  LH  := u  LH  + 7/i(^—-).  75  (4.13)  The L H - H H self-selection condition implies  u >V(x )-h(^-). LH  T h a t is, u (y\ i  L  x )  H  H  HH  (4.14)  is the label of a pseudo-indifference curve for family L H that all  pairs that satisfy the L H - H H constraint must lie below. T h i s curve is the  HH  slice of the indifference surface of family L H through Note that u  H  corresponds to the u  L  L  H  H  > u  L  than y  L  if and only if y \ H  > yf .  H  x ) LH  (y~i , H  at y =  indifference curve under the assumption that y f ^ is greater  .  the welfare level of family H H at (y[ , y H  H 2  , x ).  u H  H  H  := V(x )  H  —LH  - h(^-)  LH  denote  H  = u  H  H  + 7>i(—)•  (4.15)  is the label on the pseudo-indifference curve for family H H in (yi, x)-space  depicting the [y\ , yf  H  H  Define  LH  —LH  H  •  H 2  T h e curve labeled L H ' in Figure 2  H  Consider now the possibility of family H H mimicking family L H . Let u  u  y  2  x )  pairs that provide family H H the welfare level u  HH  H  H  , holding  at y% . This curve is denoted by H H in Figure 2. Notice that it passes through H  the point D . But the before-tax income of person 2 in family H H is y  B 2  \ not y  H 2  •  Thus, when analysing the behaviour of family H H it is more appropriate to consider the pseudo-indifference curve given by the equation &  B H .  =  j  i  H H  +  h  i V 2 ^ Y  (  4  1  6  )  The H H - L H self-selection constraint requires:  V  {  x  H H  }  _  h  r V l _ ^  _  76  7 /  ,(^2_) >  u  H  H  .  (4.17)  Conditions (4.16) and (4.17) imply —HH  u  <V(x )-h{^-).  HH  Thus, we may interpret u  as the label on the pseudo-indifference curve for family  BH  H H above which all (y  HH  x ) HH  t  pairs satisfying the H H - L H constraint must lie. F r o m  (4.15) and (4.16) it follows that u  > u  HH  assumption that y  > y .  LH  HH  we may draw the u  LH  in Figure 2 as the curve H H ' .  if and only if y  HH  is greater than y ,  HH  (4.18)  HH  LH  Under the  indifference curve  Comparing (4.13) with (4.16), we can see that the  vertical distance between the curves L H and L H ' is identical to the vertical distance between the curves H H and H H ' .  Thus, L H ' and H H ' intersect at a point like D ' ,  directly above D .  The  discussion of the preceding two paragraphs has shown that when y  HE  equal to y , HH  all combinations of yf  H  and x  HH  is  that satisfy both the L H - H H and  H H - L L self-selection constraints lie in the wedge between the curves L H ' and H H ' , assuming y  HH  > y H  2  T h a t is, this wedge is the y s\\ce HH  of the region in 3-space  corresponding to the bundles for family H H that satisfy both the H H - L H and L H - H H self-selection constraints, given the bundle of family L H . A l l points in that wedge are to the right of the point D and have at least as much y\ and more x than the point D.  Figure 3 illustrates the argument for y  HH  < y~ '. H  T h e point of intersection of  the relevant pseudo-indifference curves is directly below the point D , with the selfselection slice depicted by the shaded region. 77 '  Notice that there are points in this  region that have a lower x than the allocation D has. Hence, by itself, Proposition 4.3 does not provide enough information to order after-tax  incomes.  In order to derive the validity of Proposition 4.3 from these diagrammatic arguments, it is enough to note that for any choice of y y^^-pseudo-indifference  ,  h r 2  the intersection of the  curves for families L H and H H lies on the vertical line in  (y\, x)-space through the point D .  2 2  Proposition 4 . 4 Let (SS) hold. Then i.) y  HH  y  HH  < y^  implies y  L  >y  L  E H  > y[  L  >x ,  and y  > y^  ii. ) y  HH  < y  implies y  y  HH  > y^  and x  y  HH  >y  and y  iii. ) y  HL  < y\  LL  L  LL  iv. ) y  H 2  < yf  > y^  L  either: a) y  HL  HL  > y[  H  d  L  implies x  H H  = yf  or b) y  >x , LH  H  L H  78  Furthermore,  LL  LL  >x ,  HL  =x .  > x .  LH  HL  L  HH  and x  E  = y% implies either: a)  and x  L  >x .  > y{ . and x  Furthermore,  LL  LL  and x HL  = x .  HH  > x .  HL  and x  x  an  >y  LH  implies either: a)  L  HH  HH  HH  H  = y\  Moreover, y  L  This is also true for the slice at y = y\ , through (y~i , x ) need not be shifted. 2 2  H H  or b) y  L  implies y  1  J  > y{ LH  LH  > y{.  R H  LL  implies y  either: a) y  implies x  >x ,  H H  HH  H  HH  L  = y\  H  or b) y  LL  HH  Moreover, yf  L  and x  HH  2  y  > y% •  HH  Moreover, y  HL  LH  >x . L H  or b) yf  L  = y^  L  R L  and x  Moreover, y  — y^  H  L H  H  and x  x . HL  = y§  H  L  2  = y{  =  H L  =  implies  implies x . L H  the case for which the pseudo-indifference curves  Statements i.) and ii.) of Proposition 4.4 are consistent with a general notion of more productive individuals receiving a higher before-tax income, as at least one member of family H H must earn more before-tax income than the corresponding member of family L L . T h e only exception to this tendency occurs when families H H and L L receive exactly the same bundle.  T h e geometric intuition behind statement ii.) of Proposition 4.4 is presented in Figure 4. L e t ( y f , y , L  L  2  x )  denote the bundle designed for family L L and let the  LL  point A be the projection of this point onto the (yi, x) plane. T h e curves L L and H H are the pseudo indifference curves through the bundle designed for family L L at for families L L and H H , respectively. Suppose y  < y .  H  L 2  Following the notational  LL  2  y  conventions used in the discussion of Proposition 4.3, define the number  " "  fl  :=fi  +  / i ( ^ ) .  7  (4.19)  WL  T h e L L - H H self-selection condition implies  &™>V(x )-h(p—). Ba  (4.20)  w  L  T h a t is, the pseudo-indifference curve with the label u  is the upper boundary of  LL  the region of points in (yi, x)-space satisfying the L L - H H self-selection constraint, fixing y  at y .  HH  H  2  T h i s curve is labeled L L ' in Figure 4.  the welfare level of family H H at (y[ , L  symbol u , HH  y  L 2  , x ). LL  Now, let u  HH  denote  W i t h this reinterpretation of the  equation (4.16) describes the pseudo-indifference curve above which all  points satisfying the H H - L L constraint must lie, given 79  y  H 2  — y '• E  2  In Figure 4,  this curve is denoted H H ' . In view of the relation WL < WH, (4.16) and (4.19) imply that the vertical distance between the curves L L and L L ' is greater than the vertical distance between H H and H H ' . Thus, the intersection of L L ' and H H ' occurs to the right of the point A . Notice that the point A ' may lie below the point A , so that no conclusion may be drawn about the ordering of x  L L  and  x  H H  .  Some high-productivity individuals may earn less than corresponding lowproductivity individuals at an allocation consistent with self-selection.  T h i s possi-  bility is illustrated in Figure 5. Let point A denote the projection of the allocation of family L L onto (yi, a;)-space. Suppose that {V\ i  x  H  H H  )  y  L 2  <y • HH  Given  y  HH  =y > H  2  ah"  pairs that satisfy the L L - H H self-selection constraint lie below the curve  labeled L L ' . T h e curve H H ' delimits the lower boundary of the region in (y\, x)-space that all (y ,  x  HH  assuming  y  H 2  H H  )  pairs satisfying the H H - L L self-selection constraint must lie in,  =y • H  2  (The argument is identical to that used in the discussion of  Figure 4, save that curves shift in the opposite direction.)  As Figure 5 illustrates,  the intersection of these two curves may occur at a point to the southwest of the point A . W h e n this is the case, no conclusion can be drawn about the ordering of and  x  H H  ,  nor can yf  L  and y  Intuitively, an increase in y  HH  H 2  be ordered by self-selection  x  L L  considerations alone.  is less costly in terms of labour time for family H H  than it is for family L L viewed as a mimicker of family H H . Hence, such a change induces a tightening of the H H - L L self-selection constraint of a magnitude smaller than its slackening effect on the L L - H H self-selection constraint.  80  Statements iii.)  and iv.)  of Proposition 4.4 indicate that there are cases for  which the self-selection constraints place restrictions on the ordering of the after-tax incomes of families L H and H L . There is, however, a caveat to this apparently strong result about after-tax incomes. T h e conditions under which this ordering is available appear to be rather strong. For instance, when y  H 2  is less than y ,  it is the case  L  2  that an individual of lower ability is working less than a high-ability individual.  Figure 6 gives a graphical illustration of statement iv.). Let the point D denote the projection of family L H ' s bundle onto the (yi, a;)-plane. T h e curve L H is a slice of the indifference surface for family L H through ( y j y . H  2  JjH  , y , H  2  % ), LH  drawn for y  T h e y ^ - s l i c e of family H L ' s indifference surface through ( y f ^ , y , H  2  noted by H L . Suppose y  H 2  < y . L  2  T h e n the corresponding y  L 2  H 2  x ) LH  =  is de-  pseudo-indifference  curves are shown by the curves L H ' and H L ' in Figure 6. Like family L L , family H L has a person of lower ability in position 2, while (like family H H ) family L H has a person of higher ability in position 2. Thus, the vertical distance between the curves H L ' and H L is greater than the vertical distance between the curves L H ' and L H . Consequently, the intersection of H L ' and L H ' must occur to the northeast of the point D . Hence, family H L receives more y\ and more x than family L H .  T h e indeterminacy of the ordering of after-tax incomes at this stage of the analysis is perhaps the most important obstacle to overcome in characterising the optimal solution.  It is also one of the most important distinctions between m u l t i -  dimensional optimal tax mechanisms and their single-dimensional counterparts. T h i s  81  indeterminacy is not the direct result of the two-dimensional uncertainty faced by the planner. Rather, as Proposition 4.3 illustrates, it results from the multiplicity of instruments.  Nevertheless,  when hidden information can be summarised by a one-  dimensional characteristic, the use of multiple instruments leads to solutions that bear striking resemblance to the standard unidimensional problem with two goods as long as sufficient structure is imposed to ensure that agents of higher type receive more of all goods.  A m o n g the contributions of D a n a (1993) and Rochet (1995) has been to draw attention to self-selection constraints between agents that differ in more than one dimension. B o t h present screening models in which such constraints can bind at an optimum. Neither devotes attention to which properties of the solution are directly attributable to these binding constraints. A s a step in that direction, I now turn to the problem of describing the characteristics of allocations that satisfy (SS) with one of these constraints binding.  Proposition 4.5 Let (SS) hold, i.) If, in addition, the HH-LL V\  H  < V\  L  self-selection constraint holds with equality, then  (with equality only if the HH-LH  straints also hold with equality) andy  HL  and HL-LL  < y\  L  and LH-LL  self-selection con-  (with equality only if the  HH-HL  self-selection constraints also hold with equality).  ^ This property is known as "attribute ordering" (cf. Matthews and Moore (1987)). Besley and Coate (1995) provide a model i n which a monotonicity property different from, but in the same spirit as, attribute ordering is sufficient to render the problem "well-behaved."  82  ii.) If, in addition, the LL-HH yHH  <• yHL  self-selection constraint holds with equality, then  q lity  e  only if the LL-HL  ua  straints also hold with equality) andy  <y  H 2  and LH-HH  and HL-HH  self-selection con-  (with equality only if the  H 2  LL-LH  self-selection constraints also hold with equality).  The geometric intuition behind clause i.)  of Proposition 4.5 is illustrated in  Figure 7. Let the point A represent the projection of the allocation of family L L onto (yii x)-space. T h e curve L L is the pseudo-indifference curve, drawn at y  L 2  = y  L 2  •,  for family L L through A . W h e n the H H - L L self-selection constraint is binding, given y  H 2  = y , H  2  labeled u , HH  family H H must have an allocation along the pseudo-indifference curve where r.LL  u  :=  HH  When y  H 2  > y ,  V(x ) LL  \ - h( -l-) + 7 M y2 — ) V  = §2 )-  H  L  2  C U R V E  N  o  w  •  (- ) 4  21  this pseudo-indifference curve lies above the pseudo-indifference  L  2  curve of family H H through A (which is drawn for y  u~HH  u(V2  H - )  SU  p p o s e that y  H 2  > y . L  2  (y  H 2  > y  L 2  Let H H ' denote the  by Proposition 4.3.) T h e n ,  in order for the L H - L L self-selection constraint to be satisfied, it must be the case that the (projection of the) allocation of family L H must lie on or above a pseudoindifference curve such as the one labeled L H , assuming y2  H  =y • H  2  Notice that the  curve L H is above the curve L L , due to the fact that family L H must be compensated for the higher before-tax income of its person 2. T h e label on the curve L H is given  83  by LL  :=V(s")-/i(2M+7 W ^ fin T '  hC*-)-h02-) 11 TT ^ /)/!  L  .  (4.22)  rr '  Now consider family H H faced with the possibility of mimicking family L H . T h e H H - L H self-selection constraint is satisfied for any (y{ , x ) H  pseudo-indifference curve H H " (a y -slice).  H H " has utility label  LH  V  {  ^  H  )  _  pair on or below the  LH  LH +  7  h  1 /1 T T WH  ( ^ - \- { ^ - ) . 111 J • ' ^ 11} J I ' WH WH h  (4.23)  V  T h e vertical distance between the curves H H and L L is equal to the sum of the last term in (4.21) and the last term in (4.23). T h e final term of (4.22) is the vertical distance between L H and L L . Comparing (4.22) with (4.21) and (4.23), it can be seen that the intersection of the curves L H and H H " lies directly above the point A . Thus, all points that satisfy (SS) for y  LH  7, and, hence, y f  f f  < y{ . L  =y  LH  2 4  must lie in the shaded area of Figure  Note that this shaded region contains both points above  A and points below A . Thus, Proposition 4.5 is silent about the ordering of after-tax incomes.  Proposition 4.6 Let (SS) hold, i.) If, in addition, the LH-HL V\  L  < V\  H  self-selection constraint holds with equality then  (with equality only if the LH-HH  straints also hold with equality) and y  LL  and LL-HL  <y  and HH-HL  self-selection con-  (with equality only if the  RL  LH-LL  self-selection constraints also hold with equality).  The argument given here implicitly assumes y% > y ' • When y% < y , the curve H r l " lies below the curve HH'. It is still the case, however, that H H and L H intersect directly above the point A . 2  4  H  n  2  84  H  H  2  ii.) If, in addition, the HL-LH V\  L  < V\  L  self-selection constraint holds with equality then  (with equality only if the HL-LL  straints also hold with equality) andy  H 2  and EE-LE  The  <y  H 2  and LL-LH  self-selection con-  (with equality only if the  EL-EE  self-selection constraints also hold with equality).  geometric intuition of Proposition 4.6 is identical to the intuition under-  lying Proposition 4.5, except for some relabeling of indifference curves. Indeed, the symmetry in the statements of Propositions 4.5 and 4.6 is striking.  T h i s may be  somewhat surprising, given that the families H H and L L are ordered by the relation >p, while the families L H and H L are not. Nevertheless, there are reasons to expect some symmetry between the two propositions. Like Proposition 4.5, each statement of Proposition 4.6 uses information contained in the self-selection constraints relating three distinct families. However, while only "downward" self-selection conditions are used in deriving Proposition 4.5, Proposition 4.6 requires the use of "upward" conditions.  Another view of Figure 1 shows that the pair { H H , L L } has something  in common with the pair { L H , H L } .  One may always proceed along the edges of  the type space given in Figure 1 from the family L H to the family H L if we allow ourselves to travel "up the partial order >p."  It is the symmetric treatment of up-  ward and downward self-selection constraints that results in the similarity between Propositions 4.5 and 4.6.  A comparison of Propositions 4.5 and 4.6 reveals that the presence of certain binding constraints implies orderings on components of allocations that cannot be 85  ordered by appealing to Proposition 4.3.  Note, however, that a binding H H - L L  self-selection constraint and a binding H L - L H constraint have almost opposite implications for the ordering of y  HL  together only if y  HL  and y  LL  and y\ . L  Indeed, these two constraints can bind  are equal.  4-4- Properties of Optimal Tax Schedules  The  results of the last section followed directly from the self-selection  constraints.  In particular, none of the analysis relied on the assumption that the planner seeks to maximise a social objective function.  In this section I incorporate optimising  behaviour on the part of the taxation authority into the analysis to derive some additional properties of solutions to the problem ( P F ) . T h e first of these is standard in optimal income tax theory.  Proposition 4.7 At any solution to (PF), the constraint (F) binds with positive multiplier.  Proposition 4.7 can be interpreted as stating that any surplus output can always be distributed among the families in such a way that self-selection  constraints are  not violated.  Although simple, Proposition 4.7 is an important building block for further anlaysis of the problem. In particular, it allows us to conclude that if, starting from an  86  initial allocation, a change can be made that slackens the resource constraint, does not violate the self-selection constraints and makes no family worse off, then the initial allocation is not a solution to ( P F ) . T h i s is exactly the type of reasoning behind the next proposition.  Proposition  4.8  Suppose that at a solution to (PF), W (x ,y\,y2) l  l  =  ^'(s^j/j,^)  for a pair of families i, j. Then vi+yi-x'^yi+vi-x*.  (4.24)  Proposition 4.8 is the restatement of a result of Brito et al. (1990, Proposition 1, p. 66) in the current context.  One would expect this result to hold here, for it  states simply that a family never wishes to mimic a family that has a higher total tax bill than itself. In particular, the statement of Proposition 4.8 contains no reference to the labeling of families adopted in this analysis. It is well-known that, in the twogood model with unidimensional differences among agents, the pattern of binding self-selection constraints determines the qualitative features of optimal marginal tax rates (Roeil (1985)). Proposition 4.8 tells us that the pattern of binding self-selection constraints can also be used to make statements concerning total tax liabilities.  W h e n discussing the relationship between the self-selection constraints and tax functions, I considered a wide class of tax functions. In particular, the tax functions were allowed to be non-differentiable. A t kinks in the tax schedule, it is impossible  87  to define marginal tax rates as (partial) derivatives of the tax function. It is possible, however, to define implicit marginal tax rates at any allocation by:  t\ := 1 - MRS* , = 1  =—±—,  tl := 1 - MRS'  x  =l  .  ,  2 N  .  (4.25)  It is clear from their definitions that implicit marginal tax rates are positive when the marginal rate of substitution between labour and consumption is less than one, which is the producer wage for an efficiency unit of labour in this model. Marginal wage subsidies (negative marginal tax rates) correspond to marginal rates of substitution in excess of the producer wage.  Marginal tax rates serve as an important summary statistic of the distortions arising from the planner's lack of information. In the current context, the following proposition demonstrates that the sign of marginal tax rates depends on the pattern of binding self-selection constraints.  Proposition 4 . 9 i.) For any family i and any family member k, if w = w^, then k faces a nonl  k  negative marginal tax rate t . l  k  Furthermore, t >0if k  and only if there is a  family j with w = wj{ and the j-i self-selection constraint binds with a positive 3  k  multiplier. ii.) For any family i and any family member k, if w = WJJ, then k faces a nonk  positive marginal tax rate t . k  Furthermore, t <0if k  88  and only if there is a  family j with w = WL and the j-i self-selection constraint binds with a positive J  k  multiplier.  It follows directly from Proposition 4.9 that if, for all j , the j-i  self-selection  constraint does not bind at a solution to ( P F ) , then both members of family i face a zero marginal tax rate. This is a direct analogue of a result due to Guesnerie and Seade (1982, Proposition 2, p. 164), which states that if there is an individual with a bundle that all other individuals view as strictly inferior to their own, then that individual faces a zero marginal tax rate. However, Proposition 4.9 is more than a restatement of the result of Guesnerie and Seade (1982) in the current context.  It  says: no distortions are introduced on the labour supply behaviour of individual 1, say,  when a self-selection constraint binds between families with identical persons  1. Because two such families view trade-offs between y\ and x identically, any such trade-offs can be made without violating the self-selection constraints involving these two families. Were it not for the presence of other families, the planner could vary the allocations of y\ and x for these two families until an undistorted bundle is achieved.  Proposition 4.9 speaks to the debate over the choice of tax base. In particular, it suggests that a tax based solely on total family income will fail to be optimal in many circumstances. Notice that the implied marginal tax rates for the individuals in family H L are, in general, different. Indeed, these rates can coincide only if both are zero. T h e person of lower productivity faces a higher marginal tax rate at the optimum. A n analogous result holds for family L H . It seems reasonable that the planner might want 89  to apply different tax rates within a family, given that total production in the economy is determined by how much individuals decide to work. Like one-dimensional taxation models, the choice of marginal tax rates for higher ability individuals is dominated by efficiency considerations. T h e planner wishes to extract effort from these individuals, since they are the ones whose work effort provides the most output. B y using different marginal tax rates within the household, the planner is able to identify which of the two individuals in a "mixed" household is the one with higher productivity, enabling it to provide sufficient incentives for that individual to provide work effort.  If the  planner is forced to use a "flat" tax, then its power to identify high-productivity individuals is limited.  Proposition 4.9 has an important corollary.  Proposition 4.10 No two families of different type receive the same allocation at a solution to (PF).  Phrased in the language of screening models, there is no bunching at the optimum.  Proposition 4.10 is an analogue to the no-pooling result of Stiglitz (1982)  for economies with two types of consumers.  It shows that it is not the existence  of only two types of consumers per se that is important in deriving the no-pooling result.  A s long as there are only two families along each dimension of uncertainty  there must be separation at the optimum. It should be stressed at this point that the argument establishing Proposition 4.10 relies on the assumption that all individuals  90  receive a positive before-tax income at the solution to ( P F ) . Later on, I present a special case of the model for which some individuals receive zero before-tax income. In that environment, some families are pooled at the optimum. (See Proposition 5.5.)  Further progress in understanding the solutions to ( P F ) requires that something be said about the pattern of binding self-selection constraints at the optimum. One can expect this pattern to be influenced by the form of the social objective function Z(-).  T h e question arises: C a n some patterns may be ruled out at any solution? I  now turn to the task of answering this question.  In order to make some of the discussion easier, I adopt some terminology of Wilson (1995).  Definition: Family i attracts the family j if the j-i self-selection constraint is binding.  Proposition 4.11  At a solution to (PF), any pair of families i and j, except possibly  the pair { HH, LL }, are mutually attracted to each other only if they receive the same allocation.  T h e one-dimensional analogue to Proposition 4.11 is an immediate consequence of the usual single-crossing property, and holds for all allocations.  In the present  context indifference surfaces intersect at more than one point. Hence, it is important to emphasise that the conclusion of Proposition 4.11 holds at the optimum.  91  The  exclusion of the pair { HH, LL } at this point is due to the.fact that there is insufficient structure on the general form of ( P F ) to ensure that the solution is attribute ordered (cf.  Matthews and Moore (1987)). In particular, we cannot conclude that  both members of family H H receive more before-tax income than the corresponding members of family L L .  The intuition behind Proposition 4.11 can be illustrated by considering families H H and L H once more.  Given the specification of family objectives adopted here,  mutual attraction of these two families is equivalent to y  HH  = y\ , H  so that self-  selection requires they be placed on the same indifference curve in (y , a;)-space. 2  Figure 8 illustrates the case of different allocations for the two families, drawn at the same level of y\.  T h e curve L L is the yf  H  -slice of the indifference surfaces of  families L L and H L through the bundle designed for family H H . T h e curve H H is the analogous slice for families H H and L H . T h e points C and D correspond to the (y , X) pairs offered to families H H and L H , respectively, under the assumption that 2  y  HH  < y  H 2  • F ° this pair of allocations it turns out that neither L L nor H L can be r  attracted to L H , for otherwise the L L - H H and H L - H H constraints would be violated. To see this, notice that for any pseudo-indifference curve for the families H H and L H , points such as C must lie to the left of points such as D . If the pseudo-indifference curve for family L L defining the upper boundary of the set of points that satisfy the L L - L H constraint passes through a point such as D , then the point corresponding to C must be above that boundary. T h e n , the L L - H H self-selection constraint must be  92  violated.  Now, by Proposition 4.9,  MRSjrf*. = 1. T h e n MRS™  < 1. Consider a  move along the indifference curve from point C toward point D . T h i s has no effect on the H H - L H and L H - H H constraints.  Furthermore the move is resource saving and  can only slacken the L L - H H and H L - H H  constraints.  Proposition 4.11 is closely related to a result of Brito et al. (1990, Proposition 2, p.  67).  T h e y show that when there are cycles of self-selection constraints,  the  planner can do no worse by pooling the agents involved in the cycle. T h e additional structure of this model allows this conclusion to be strengthened for most pairs of families to say that the planner can always do better by pooling. However, in view of Proposition 4.10, the planner can do better still. T h i s statement is formalised in the next proposition.  Proposition 4.12  No pair of families is mutually attracted at a solution to (PF).  For all pairs save { H H , L L } Proposition 4.12 follows directly from Propositions 4.10 and 4.11. and  For the pair { H H , L L } the proposition follows from Proposition 4.10  the aforementioned result of Brito et al.  (1990). Indeed, Proposition 4.12 is a  special case of a more general result, itself a direct consequence of Proposition 4.10 and the work of Brito et al. (1990). In order to state this result, I need the following definition.  93  Definition: A self selection cycle is said to exist when there is a set of families  {«!,...,  in} such that family i\ attracts family i  n  and for all  k < n, family ik+i  attracts family  Proposition 4.13 There are no self-selection cycles at a solution to (PF).  Another standard feature of optimal non-linear tax schedules is the existence of an agent who faces no distortions. In one—dimensional models with single-crossing and two goods, this is often the agent of highest ability, and, for some continuous models, the agent of lowest ability as well (cf. Seade (1977)). A version of this result also holds in the present context.  Proposition 4.14 For any family i, if  y[ +yi~  x  {  > y{ + y i -  x\  for all j ^ i,  (4.26)  then family % faces no marginal distortions at a solution to (PF).  It follows from Proposition 4-14 that there is a family whose allocation is not subject to marginal distortions. Moreover, it identifies undistorted families : those that pay the highest taxes. T h i s is exactly the finding of Brito et al. (1990). O f course, a family that faces no marginal distortions is not pooled with any other family, just as was found by Guesnerie and Seade (1982).  A t this point, it is useful to define two classes of constraints.  94  Definition: T h e H H - L L and L L - H H self-selection constraints are called diagonal. The H L - L H and L H - H L self-selection constraints are called transverse.  Combining Propositions 4.5 and 4.6 shows that when diagonal and transverse self-selection constraints bind simultaneously there are allocations which are identical in some components. Given this tendency toward pooling, at least for some goods, the following proposition may come as little surprise.  Proposition 4.15 At a solution to (PF), it cannot be the case that a diagonal constraint and a transverse constraint bind simultaneously.  Proposition 4.15 greatly simplifies the search for solutions to ( P F ) . Moreover, it shows that it is not by coincidence that no solutions discovered by D a n a (1993) a n d by Rochet (1995) in their models feature a binding diagonal constraint and a binding transverse constraint.  Note that quasi-linearity (which is assumed in these earlier  studies) is not needed to generate this result.  95  C H A P T E R 5: Optimal Non-linear Taxes: Some Special Cases  5.1. Introduction  T h e requirement of Pareto-efficiency has yielded quite a bit of information about the properties of optimal tax schedules.  Further progress in characterising the solutions  to the problem ( P F ) requires that some structure be placed on the objectives of the planner. There are two ways to provide this structure. One is to posit a form for the objective function of the planner, say a weighted sum of family objectives. T h e other method involves making a redistributive assumption. In the next section, I formulate an analogue to a standard redistributive assumption. Unlike the situation that usually occurs in one-dimensional income tax problems (cf. Guesnerie and Seade (1982)), it is shown that the redistributive assumption is not strong enough to rule out negative marginal tax rates for some individuals. It does, however, have an implication for the ordering of the optimal before-tax incomes of some families.  T h e general characterisations provided in the preceding chapter do not make the role of asymmetries within the family transparent. In the final section of this chapter, I specialise the model to the case of families who trade off the labour supplies of their members in a linear fashion. In this case, differences within the family are reflected in labour-force participation decisions. A n analysis of these decisions provides a clearer picture of the role of differences between individuals within the same family.  96  5.2. Redistributive Taxation  It is usual in the literature to assume that, when incentive effects are ignored, the planner wishes to transfer consumption from the more able to the less able (Guesnerie and Seade (1982), Chambers (1989)).  In the spirit of the literature, I employ the  following redistributive assumption.  Assumption R  Suppose that i >p j for a pair of families {i,j).  Then there exists  some sufficiently small e > 0 such that, at the optimum., it is socially desirable to transfer e units of after-tax income from family i to family j if the constraints (SS) are ignored.  Assumption R is not equivalent, in general, to a desire to increase the welfare of families with less able individuals (Dixit and Seade (1979)). It is also important to note that Assumption R places no restriction on how the planner views the consumption of family H L  vis-a—vis the consumption of family L H . Despite this, Assumption  R has implications for the structure of solutions to ( P F ) . It is not sufficient, however, to ensure that all marginal tax rates are non-negative.  It is also necessary to say something about the distribution of families in order to classify more succinctly the optimal taxation mechanisms. E m p l o y i n g Assumption R is much more straightforward when all of the redistributions it declares to be desirable  97  are production-feasible.  T h i s condition is guaranteed by the following assumption  about the distribution of families.  Assumption U n  L L  = n  H L  = ir  HH  =  n  L H  .  Notice that Assumption U is stronger than the assumption of statistical independence between the two components of the space of family types. T h e analysis can also be carried out under the independence assumption, with additional care taken to assess which redistributions of after-tax income can be made without violating the materials balance constraint.  Unlike the situation that obtains in the model of D a n a (1993), Assumption U does not rule out asymmetric solutions in this model. Asymmetries may arise due to the form of the planner's objectives or due to the asymmetry of the family decision process.  T h e remainder of this section is devoted to cataloguing the qualitative features of possible optimal configurations. In order to compare the current results with others in the literature, it is necessary at times to impose additional structure on family objectives. T h e following assumption will sometimes be cited.  A s s u m p t i o n Q V(x) = x.  98  Assumption Q is consistent with the model of family decision making only if U(x)  = x?^  T h i s specification can lead to one member of the family consuming  everything. Assumption Q may be more palatable in other screening contexts. T h e main reason for its use in this study is to compare the more general results obtained without it to the optimal mechanisms that arise when it is satisfied. In this way, the full force of this common assumption can be ascertained.  One more piece of notation is required. D e f i n i t i o n : T h e set C is defined by saying that an ordered pair of indices (i, j) is an element of C if and only if the i-j  self-selection constraint binds at the solution to  (PF).  5.2.1. The usual cases  A s emphasised by D a n a (1993) and Rochet (1995), much of the analytical difficulty in multi-dimensional screening problems arises from the possibility of binding diagonal or transverse constraints. Tn this subsection, patterns of binding constraints that can occur when diagonal constraints are assumed not to bind are presented. There is a surprisingly large number of configurations that are compatible with Assumption R. T h e main qualitative implication of Assumption R is given by the next proposition. 2  ^  S t r i c t l y s p e a k i n g , A s s u m p t i o n Q is inconsistent w i t h the b o u n d a r y c o n d i t i o n s r e q u i r e d o n pref-  erences r e q u i r e d to guarantee t h a t , for a n y a l l o c a t i o n satisfying (SS), the p l a n n e r c a n c o n s t r u c t a n equivalent tax schedule.  99  Proposition 5.1 i. ) Let Assumptions R and U hold and let (HH, LL) 0 C. Then { (HH, HL), (LH, LL) } Q C or {(H H, LH), (H L, LL)} C C. ii. ) Let Assumptions R, U and Q hold and let (HH, LL) 0 C. Then { (HH, HL), (HL, LL), (LH, LL), } C C or { (IIH, LH), (LH, LL), (H L, LL)} C C. Moreover, (LL, HH) <£C.  T h e implication of Proposition 5.1 for tax rates is that either all low-productivity individuals occupying position 1 in a family or all low-productivity individuals in position 2 in a family face a strictly positive marginal tax rate.  Assumption R implies  that family H H is attracted to some other family, say family H L . T h e n individual 2 of family H L has its labour supply distorted downward. family L L must attract some other family.  Also by Assumption R,,  If family H L is attracted to family L L  (while family H L attracts family H H ) , satisfying the H H - L L self-selection constraint requires that y  HL  > y^?  7  T h e n , individual 2 of family L L must also have its labour  supply distorted downwards.  Statement ii.)  of Proposition 5.1 shows the force of  Assumption Q . W h e n Assumption Q is maintained, both members of family L L face a strictly positive marginal tax rate at the optimum. For the remainder of the discussion, it is assumed that all binding constraints bind with positive shadow values. A proof of this assertion requires a slight strengthening of Assumption R to state that the social gains from a downward transfer of after-tax income are of the first order. Guesnerie and Seade (1982, p. 174) provide the argument. 97  This assertion can be checked using a calculation similar to the one used to prove Proposition 4.5, without imposing that the C - A constraint binds.  100  Proposition 5.1 is silent about the pattern of binding constraints when family L L attracts family H H . However, Proposition 4.12 ensures that family L L cannot be attracted to family H H when the H H - L L self-selection constraint binds. statement ii.) of Proposition 5.1 permits the conclusion that the L L - H H  Then,  self-selection  constraint can never bind under Assumptions R, U and Q . Notice, however, that the combination of Assumptions R and Q is not strong enough to rule out a binding L L - H H constraint at the optimum.  Given Assumption R, any redistribution of after-tax income to family L L from any  other family is socially desirable. A binding H H - L L self-selection constraint is  an impediment to this kind of redistribution. However, transfers to family L L can only slacken the L L - H H self-selection constraint. Although it is possible for the L L H H self-selection  constraint to bind at the optimum, the planner can, for a given  pattern of other binding self-selection  constraints, carry out the same downward  redistributions regardless of whether family H H attracts family L L or not. there is no interesting analogue to Proposition 5.1 for the case of (LL, HH)  Hence, C.  5.2.2. The unusual cases  U p to this point, little has been said about families who are unordered by >p.  It  is the presence of these families that makes the current model substantively different from a one-dimensional model. T h e possibility of the L H - H L or H L - L H constraint binding implies the possibility of negative marginal tax rates for high-productivity  101  individuals in families H L and L H . T h i s is a major departure from the standard results of the literature on non-linear redistributive taxation. In this subsection, I show that Assumption R does not provide sufficient structure to rule out such phenomena.  It  does, however, place some restrictions on the pattern of marginal tax rates when a transverse constraint binds.  T h e next proposition shows that a negative marginal rate to person 1 in family HL  (resp. person 2 in family L H ) arising from a binding transverse constraint must  be accompanied by a negative marginal tax rate for an individual 1 (resp. 2) in family HH.  Proposition 5.2 Let Assumptions R and U hold. Then i. ) (LH, HL) e C implies { (HH, HL), (LH, LL), (LH, HH) } C C. ii. ) (HL, LH) e C implies { (HH, LH), (HL, LL), (HL, HH) } C C.  Proposition 5.2 implies that when one of the transverse self-selection constraints binds, the pattern of binding constraints for family H H is completely determined. For instance, when the L H - H L self-selection constraint binds, family H H is attracted to family H L and family L H is attracted to family H H . Because a transverse constraint binds, it follows from Proposition 4.15 that neither of family H H or family L L is attracted to the other. Notice, however, that the pattern remains undetermined for family L L . T h i s asymmetry is a direct result of Assumption R and the concomitant tendency for high-ability families to be tempted to under-report their productivity at  102  the optimum. T h i s indeterminacy vanishes, however, when Assumption Q is satisfied. T h i s is the content of the next proposition.  Proposition 5.3 Let Assumptions R, U and Q hold. Then i.) (LH, HL) E C implies C = { (HH, HL), (HL, LL), (LH, LL), (LH, HH) }. n.) (HL, LH) e C implies C = { (HH, LH), (HL, LL), (LH, LL), (HL, HH) }.  28  The  only possibility not considered thus far is that the H H - L L self-selection  constraint may bind. In previous work (Dana (1993), Rochet (1995)), this case has arisen only in the so-called "separable" context, in which family i attracts family j if and only if j >p i. In the present model, separability means that, at the optimum, all low-productivity individuals receive the same before-tax income. Income effects once again destroy the simplicity of the solution. However, in view of Proposition 4.5, it is clear that the "separable" case can occur only if the j-i binds for all pairs of families with j >p  self-selection constraint  i.  5.2.3. A summary  At first glance, Assumption R appears to provide only limited structure on the nature of optimal tax mechanisms. While it is true that the situation is much more no  It is interesting to compare Propositions 5.2 and 5.3 with the "singular" case of Rochet (1995, Proposition 3.4). The introduction of income effects leads to some additional indeterminacy. For example, Rochet's analogue to statement i) of Proposition 5.2 features the L H - L L self-selection constraint binding as well. This is neither ruled out nor required i n the absence of Assumption Q. Proposition 5.3 demonstrates that the assumption of quadratic cost functions employed by Rochet is not essential in deriving the singular case, but that the singular case is the only possible outcome in this class under Assumption Q. 103  complex than is usually found in the literature, some basic properties are common to all solutions.  First, as the next proposition shows, when high-ability individuals  occupy the same position in a household, the individual with a low-productivity partner works at least as hard as the person with a high-productivity partner.  Proposition  y  H L >  y  H H  5.4  Let Assumptions R and U hold. Then, at any solution to (PF),  andy% >yl . H  IH  In view of Proposition 5.4, any marginal wage subsidies offered to individuals in family H H must not be large enough to induce them to work more than h i g h productivity individuals in other types of families. Phrased in the language of screening models, the solution fails to be attribute ordered. for this violation to occur.  Moreover, it seems natural  We would expect part of the gain to having a h i g h -  productivity partner to be consumed as leisure.  Second, in view of Proposition 5.2, whenever family L L is attracted to no other family, the use of wage subsidies for high-productivity individuals is concentrated on individuals with high-productivity partners. W h e n no family attracts family L L , high-productivity individuals with low-productivity partners receive wage subsidies only if individuals in the same position in a family with high-productivity partners also receive marginal subsidies. Suppose, for the sake of concreteness, that family H L attracts family L H , so that individual 1 in family H L faces a negative marginal tax  104  rate. In the quasi-linear context it is clear that this can happen only if marginal increases in the welfare of family H L are weighted more heavily than marginal increases in the welfare of family L H at the optimum. T h i s desire to redistribute resources away from family L H conflicts with the need to prevent family L H from masquerading as family H H . T h e only way to prevent this occurring, it turns out, is to distort the labour supply of family H H upward. In this case, it is precisely the desire to make "transverse" redistributions that upsets the standard non-negative marginal tax rate result. Notice that Assumption R is silent about the desirability of such redistributions. A characterisation of the class of social evaluation functionals for which both downward and transverse redistributions are desirable at the optimum is unknown to me.  T h i r d , status in the household can exert an independent influence on optimal tax policy. T h a t is, two equally-productive individuals with equally productive partners can face different marginal tax rates on the basis of observed demographics that have a known effect on household decisions. Once again, let family H L attract family L H . It follows from Proposition 5.2 that the low-productivity individual in family H L faces a positive marginal tax rate, while, because no family is attracted to family L H , the low-productivity individual in family L H faces a zero marginal tax rate.  5.3.  The Consequences  of Asymmetric  Family  Decisions  In this section, I present a special case of the model in which the role played by the  105  relative weights each individual has in family decision making is made transparent. The  special case corresponds to the following assumption on preferences.  Assumption SQ h(l) = I.  Given weighted-utilitarian decision making in the household, Assumption S Q implies that families trade off the labour supplies of their members in a linear fashion. Thus, it is impossible to assume that all individuals will supply a strictly positive amount of labour at the optimum. Indeed, it is the case that only one person per family will work for most values of 7. T h e identity of the person who does not work in a family depends on 7. It is this extreme behavioural response to variations in relative say in family decisions that provides a stark picture of the role of the internal workings of the family.  Proposition 5 . 5 Let Assumption SQ hold, and let 7 solution to (PF), y[  H  If in addition, y  HH  1.)  y\  L  =y  HL  = 0.  = 0 , then  = 0.  ii.) Families HH and LH receive the same allocation, iii.)  (HH, LL) G C if and only if (LH, LL) G C.  106  G (WL/WJJ,!).  Then, at the  Proposition 5.5 states how families sort themselves according to participation decisions.  W h e n 7 < 1, person 1 's utility gets more weight in family decisions.  In  particular, the labour supply of person 1 is considered more distasteful to the family. For family L H , the effect of relative say in family decisions is exacerbated by the fact that person 1 is less productive, so the family has an additional reason to concentrate work effort with person 2. For family H L , productivity considerations lead the family to prefer that person 1 work.  W h e n 7 > WL/WJJ,  the extra weight given to the  leisure of person 1 is not sufficient to undo the productivity effect.  T h i s has two  important consequences for the way the planner trades off the before-tax of individuals within certain families.  incomes  First, because the objective function of the  planner is increasing in the welfare of family H L , it wishes to substitute the effort of person 1 in family H L for that of person 2. Second, a one-unit increase in combined with a one-unit decrease in y\ , H  y , H  2  reduces the welfare of family H L at the  allocation designed for family L H , slackening the H L - L H self-selection constraint. Hence, self-selection considerations and concerns for the welfare of family L H work in the same direction.  The importance of the assumption that the planner knows the value of 7 ought to be stressed at this point. Proposition 5.5 indicates exactly how the planner uses this information. For the range of 7 considered in the proposition, the planner knows that family H L is the only one that prefers for its individual 1 to supply labour. Without prior knowledge of 7, the planner would not be able to disentangle the effects  107  of "pure" efficiency considerations within the family from the consequences of family distributional ethics. In particular, it would not be able to assess how rearrangements of before-tax incomes affect the welfare of family H L viewed as a mimicker of family LH.  For families with equally productive members ( L L and H H ) , the preference for the work of person 2 is enough to make the family want to substitute the effort of person 2 for that of person 1. In the absence of binding self-selection constraints, this process can be stopped only when person 1 is supplying no labour at all. Consequently, if no family is attracted to family H H , then person 1 in family H H supplies zero labour. Thus, the assumption that  y =0 HH  can be interpreted as saying that it  is not optimal to distort the labour supply behaviour of family H H enough to induce its individual 1 to participate in the labour market.  Statement ii.) no-pooling result.  of Proposition 5.5 appears to contradict Proposition 4.10, the It should be noted, however, that the no-pooling conclusion is  derived under the assumption that all individuals have a positive before-tax income. A binding non-negativity constraint is a source of upward distortion on labour supply. For an individual of low-type, it is the only possible source of upward distortion. T h e argument for Proposition 4.10 rests on a non-negative marginal tax rate for all individuals of low-ability.  T h e presence of a binding non-negativity constraint  destroys this argument.  108  Statement iii.)  of Proposition 5.5 has implications for how the planner can  feasibly redistribute consumption to family L L . Suppose that the planner wishes to increase the consumption of family L L at the expense of families H H and L H . Given the assumptions underlying statement iii.), yf  L  = y\  H  = yf . H  Hence, the problem  reduces to the one-dimensional case with two agents of high type ( H H and L H ) and one of low type ( L L ) . If, instead, the planner would rather carry out redistributions from family H L to family L L , then it is possible that the process of redistribution could be stopped by concern over family H L mimicking family L L , say, before families H H and L H would be tempted to masquerade as family L L . In that case, neither the H H - L L nor the L H - L L constraints binds.  109  CONCLUSION  There is mounting evidence that families do not behave as if they are a single person (see, for example, Thomas (1990), Phipps and Burton (1992)), but that their decisions are compatible with household efficiency (Browning e £ . a / . ( 1 9 9 4 ) Browning and Chiappori (1994)). T h i s presents a challenge to applied welfare economists: how can we use the new insights in the economics of family behaviour to improve our normative analyses? T h i s thesis has presented a view on how we might consider tax policy in a setting that takes the interaction of family members seriously.  T h i s analysis of tax reform has also uncovered some formal similarities between the problem of tax reform in a family setting and tax reform in an individual-based model.  For example, in both cases temporary inefficiencies may be ruled out by  suitable normality conditions and sufficiently flexible powers of l u m p - s u m taxation. Despite these similarities it may be a grave error to apply the individual-based model to the family setting. Misleading policy prescriptions may follow, even if the family is behaviourally equivalent to an individual.  The preceding analysis has also put the debate on identifying household sharing rules in some perspective by showing that, although useful in descriptive analysis of household behaviour, knowledge of the derivatives of the sharing rule is not sufficient for calculating Pareto-improving directions of tax reform. Thus, the model proposed here is not implementable with family budget data alone. T h e analysis can be seen  110  as reinforcing an idea that ought to be self-evident: In order to accurately assess the welfare impact of policies on individuals it is necessary to have individual-level data.  T h i s study has also presented a model of family income taxation based on the notion of multi-dimensional screening. Viewing the problem in this way gives arguments for members of the same family to face different marginal income tax rates, casting doubt on total family income as appropriate income tax base.  T h e results  show that using the individual as the basis of income taxation is also not sufficient. B o t h the productivity of the partner and the relative position of a person in the household have some bearing on the marginal tax rate faced by an individual.  It is interesting to reflect upon the possibility of negative marginal tax rates in this model. It was noted in the discussion following Proposition 5.4 that, when Assumptions R and U hold, this important difference from standard one-dimensional tax models arises when the planner has a reason to transfer consumption among families that differ in both characteristics. In the current model, the desirability of such redistributions depends on the social welfare function.  W h e n Assumptions R  and U are violated, asymmetries in the distribution of the population or a desire to offset "undesirable" effects of family interactions may provide motivations for such redistributions. 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Rust, eds.,  Handbook of Computational Economics, Volume 1,  Elsevier  Science Publishers , Amsterdam (1995). Woolley, F . , 'Taxing the Family: L u m p - S u m Transfers and O p t i m a l Linear Income Taxes,' W.P. 92-04 (1992), Carleton University.  117  A P P E N D I X  A  In this appendix I present the proofs of the propositions and lemmas contained in Chapters 4 and 5. First, I introduce some notation that will be used throughout. Relabel the families so that the symbols A , B , C and D correspond to families L L , H L , H H and L H , respectively.  Proof of Proposition 4-1' T h e statement follows immediately from the way that x\ and X2 enter the constraint of problem (P) and only the constraint. Q  Proof of Proposition 4-2:  i) follows from the fact that the resource constraint binds at any optimum for the family, since £/(•) is increasing. Differentiating this equality constraint gives i).  T h e first-order necessary conditions for the problem (P) are: U'(ci)  -  K =  0,  7C/'(c ) - K = 0,  (A.l)  2  x — c\ — c  2  = 0,  where K is the multiplier associated with the budget constraint. Differentiating the system ( A . l ) yields  U ' \ c \ ) d c \ — dK =  <yU"{c )dc 2  dx  -  2  — dc\  dK = 0,  — dc'2 =  118  0,  0.  (A.2)  A p p l y Cramer's rule to the system (A.2) to conclude  dci(x)  .-yU"(c )  dx  *7"(ci)+7tf"(c )'  2  2  dc (x) _  Strict concavity of £/(•)  A  3  )  U"( )  2  dx  (  C1  U"(ci) +  iU"(c )' 2  implies that in each line of (A.3) both numerator and denom-  inator are negative. Statement ii.) follows.  Increasingness of V(-) is established by the calculation:  V'(x) = U'ihWftix)  > 0.  + U'(c (x))c' (x) 1  2  2  (A.4)  It follows from (A.4) that  V'\x)  = U'\h(x))[c[(x)}  + U (c (x))cUx)  2  +  ,  1  (A.5)  By  ( A . l ) , we may group the second and fourth terms of (A.5) to yield  V"(x) = U"(c (x))[5[( )f  2  1  X  +  7  t/"(5 (x))[c (x)] 2  2  2  +  U'{ci(x))  + %(x)].  (A.6)  B y statement i.), the part of the last term of (A.6) in square brackets is zero, so that  V"(x) = U"{ci{x))[^(x)]  + lU"(t (x))[c' (x)}  2  2  2  establishing the strict concavity of V(-)-  2  < 0,  (A.7)  •  Proof of Lemma 4-1 '• ™H  > wi  implies that for all y, V/WL > y/wjj. It follows from strict convexity  of h(-) that h'{^)  >  fr'(^r)-  L e m m a 4.1 follows immediately.  119  •  Proof of Lemma 4-2 B y the first fundamental theorem of calculus, (4.11) holds if and only if  (A.8)  h(-) strictly increasing implies that the integrands on each side of (A.8) are positive. L e m m a 4.1 implies that the integrand on the left-hand side of (A.8) is everywhere greater than the integrand on the right-hand side. Thus, (A.8) holds if and only if  y>y-U  Proof of Proposition 4-3: The proof of statement i) is presented. Statements ii)—iv) are proven analagously to statement i).  Let the C - D and D - C self-selection constraints be satisfied. T h e n  V(x ) C  -h(&-)-  jh(&.)  - V(x ) + hA  + ih(&-)  D  > 0  (A.9)  and  V(x ) - h(&-) - M^-) wi wu D  K  - V(x ) c  J  + / » ( ! £ ) + f c ( j £ ) > 0. WL WH 7  (A.10)  A d d i n g (A.9) and (A. 10) yields  (A.ll)  A p p y L e m m a 4.2 to conclude that y{ >  120  .  Now let y$ > y§. Because y f > y f , (A.9) implies V(.<r ) > V ( a ; ) . B u t V(-) c  is increasing, so x  c  >  x. D  Suppose, instead, that y  ~ y  2  and, hence,  V(x ), D  = V(x  V(x ) c  D  x ° >  Then  ).  D  x  x  c  D  =  • T h e n y f > y.y a n d (A.9) imply V(x ) c  2  >  O n the other hand, y f = y f , (A.9) and (A.10) imply  .  x  D  .  •  Proof of Proposition 4-4 '•  T h e proof of statements i) and iii.) are presented. Statement ii) can be proved in an anlagous manner to statement i.); statement iv.), to that of statement iii.).  Let the C - A and A - C self-selection constraints be satisfied. T h e n  V  (  X  C )  _  h  _ ft(i£) _  fV?_)  7  WE  WH  -h(^-)~  ihA  K  V  ( x  A  )  +  hA + jhA > 0, WH  (A.12)  VJH  and  {x ) A  V  WL  W  -  V(x ) c  +  h(^-) + ihO&.) > W'  J  Y  L  0.  (A.13)  Wl,  L  A d d i n g (A.12) and (A.13) yields  <yL)  h  <"L  -h(£) + hA WL  W  H  -V2 \ -u^y2\,,^y \ 1,(^2 - f t A u iH^). h(^-)+h(^)~ (^) > o. A  WE  WL  WL  WE  h  WE  J  (A.14) Now let y f < y . A  A p p l y L e m m a 4.2 to conclude that the first term in square brackets  in (A.14) is negative. Hence, the second term is positive. A p p l y L e m m a 4.2 once more to conclude y  2  > y. 2  121  Suppose y f = y .  T h e n , by (A.14) and L e m m a 4.2, y% > y .  A  (A. 12) and (A. 13) can both be satisfied only if of V(-), this implies x  = x.  A  V(x ) c  V(x ); A  >  If y% = y  A  If y  c  > y  A  2  x  c  that is, when  V (x )  >x. A  A  =  V(x ). c  A  then  Given increasingness  then (A.12) can be satisfied only if  T h e final sentence of i.)  follows directly  from (A.12).  Let the B - D and D - B self-selection constraints be satisfied. T h e n  -fcA- A - V(x ) +hA+ yh(^-)  V(x ) B  7  > 0,  D  /i  (A.15)  and  V(x ) D  hA  -  7  f r ( ^ ) -  A h(^-)  V(x ) B  +h  +1  > 0.  (A.16)  A d d i n g (A.15) and (A.16) yields  ^ i '  -  ^w//  ^w ' H  V  WJJ'  +M*h) toi  -»(£) toi  w/r  > 0. (A.17)  Now let y  < yf.  B  B y L e m m a 4.2, the first term in square brackets in (A.17) is  negative.  T h e n , the second term must be positive.  that y f >  y . Furthermore, by (A.16), V(x ) > V(x ). D  B  A p p l y L e m m a 4.2 to conclude B  Hence,  Suppose y f = y f . T h e n , by L e m m a 4.2 and (A.17), y f > y . B  (A.16) can be satisfied only if x  D  x =x. D  Let ¥ i  B  > x.  If y  B  •  := W*{x\  y\, y\) -  ^(^,y|,y^).  122  = y, B  2  x  D  > x". ~ B  If y  > y, B  2  then  then (A.15) and (A.16) imply  Proof of Proposition 4-5: Once again, the proof of statement i.)  is presented,  ii.)  can be proved in  analagous fashion.  Notice that  *CA _ yCD _^DA  =  _  fVt)  + h(^-) + (x ) - ftA - hOS-) D  y { x A ) +h  _ y^D)  V  y_?_\  +  (y£-) + (x ) - h A - h(^-y A  h{  +  h  V  wi'  WE  WH (A.18)  F r o m (A.18) it follows that  yCA  _ yCD _ ^DA  wJ/lN _ hM)  =  +h(^-) - h A .  WI'  WL'  WH  Let the C - A self-selection constraint bind. T h e n fy  (A.19)  WH  = 0, so that the left-hand side  CA  of (A.19) must be non-positive when (SS) holds. A p p l y L e m m a 4.2 to the right-hand side of (A.19) to conclude that y f < y , A  T h e argument establishing y  < y  A  2  with y f = y  only if f  is analogous.  •  A  C  = V  D  = 0.  DA  Proof of Proposition 4-6: T h e proof of i.) is presented. Statement ii.) has a similar proof  It can be shown that  _ ^DC  _  =  (yl)^  (?jL)  h  h  wr,'  K  123  +  w£  (yjL)  h  _  WH  uyL). H  W  (A  .2o)  Now let the D - B self-selection constraint bind, so $ $>  CB  D  = 0. B y (SS), ^  B  D  C  > 0 and  > 0. Thus, the left-hand side of (A.20) is non-positive. A p p l y L e m m a 4.2 to  ^  T h e argument establishing y  > y  B  Let  = ty = 0.  DC  conclude that y f > y f , with equality only if  CB  is analogous.  A  •  denote the Lagrange multiplier associated with the i-j  self-selection  constraint, and let A denote the Lagrange multiplier associated with the constraint (F). T h e first order necessary conditions for a solution to ( P F ) can be written as:  + E ^ ' ~ Mi)  w*  z  v  \ )  =  x i  u,v\ wil -V(4) E ^ 7 ' (wi4 ^ l l w\ 1  zw*+E ^  w  w  J wi  —  wi  *-r {  2  2  (A.22)  Vi;  ini  x  w  w  JW  = Xix%  £<*-V(4) = ^ -  z > +E H X h ' $ ) %  /i  ini  (A.21)  Vi;  A 7 r  v?;  (A.23)  2  W  tfi  Proof of Proposition 4-7: Consider the first order necessary conditions for a solution to ( P F ) . Suppose A = 0. T h e n , the first-order necessary condition associated with y  A  A  ZA W  + PAB + PAC + PAD - PBA —h'(—) \w  PC A +  PDA  w'  K L  becomes  JLfc'(J£_) W  =  o.  (A.24)  WH  H  L  Suppose that p c A + PDA > 0- If follows from L e m m a 4.1 that  ZA W  + PAB + PAC  + PAD -  1 PBA -  PCA -  PDA  \w  2  H  (—)  w  y L  124  , ,y \  7  —  ; L  < 0.  (A.25)  Because h(-) is increasing,  ZA W  + UAB + PAC  + PAD  - PBA  ~ PC A ~ PDA  However, the first-order condition associated with x  A  + UAB + VAC  ZA W  Hence, V'(x ) A  + PAD  - PBA  (A.26)  is  ~ PC A ~ PDA  — 0, a contradiction. Thus, PCA + PDA  < 0.  =  V'' (x ) = 0.  (A.27)  A  0.  A similar argument, using the first-order necessary condition associated with y, A  PCA  establishes that p c A + P-BA =  PDA  — 0. Because all the multipliers are non-negative,  — PBA ~ 0- T h e n (A.27) becomes  ZA W  B u t Z A + PAB W  + PAC  + PAB + PAC + PAD + PAD  V'(x ) A  = 0.  (A.28)  > 0, since Z(-) is increasing. Thus, V'(x ) A  = 0, a  contradiction, f j  Proof of Proposition 4-8:  T h e argument given here is essentially due to Brito et al. (1990), except that they analyse a "dual" problem that any solution to ( P F ) must also be a solution to. T h e proof below makes reference to ( P F ) alone.  Suppose, contrary to the statement of the proposition, that at a solution to ( P F ) there exists families i and j such that W (x , y\, y ) = W (x , y l  l  l  2  yl+yi-x^vi+yi-x*125  3  3 v  y ) and 2  (A.29)  Now construct a new allocation identical to the original solution, except replace the allocation of family i by that of family j (leaving family j with its original allocation). Clearly, vT/y = vT/J = 0 at the new allocation. Furthermore, because \fi ' > 0 for all 1  fcj  k ^ i,j at the old allocation, ty > 0 for all k ^ i, j in the new allocation. In addition, kl  ¥  k  > 0 for all k ^ i , j  at the new allocation because in the original allocation family  i was indifferent between its own allocation and that of family j and weakly preferred its own bundle to that of other families.  Thus, the new allocation satisfies all self-  selection constraints, makes no family worse off than in the original allocation and, by (A.29), slackens the feasibility constraint. Hence, the original allocation could not have been a solution to ( P F ) .  •  Proof of Proposition  T h e proof of statement i) is presented.  T h e proof of statement ii.)  is almost  identical.  Pick a family i with w\ Partition J into J := {j first-order  = w^.  £ J : w\  Let J be the set of indices not equal to i.  — WH} and J := {j  £ J :  necessary conditions for a solution to ( P F ) include  3£J  and  126  = WL}.  The  If we replace each term in the sum over J in (A.30) by P j i ( h ' / L ^ j i w  then we  may use L e m m a 4.1 to conclude  Zi + Y>v -  L  W  —h'(—) < ATT*. iw w '  tri  (A.32)  y  L  L  with (A.32) holding with equality if and only if pji — 0 for all j G J. B y Proposition 4.7, A is positive. Hence we may divide (A.32) by (A.31) to yield  h'(&) W  L  WT,V'(X'  K < 1 ,  (A-33)  R  with (A.33) holding with equality if and only if pji = 0 for all j G J.  •  Proof of Proposition 4-10:  Suppose, by way of contradiction, that there exist two families i and j of different type that receive the same allocation, (x,yi,y~2),  k  exist  MRS  y  G {1,2}  such that  (x yi y~2).  x  t  w  l k  — WJJ  and  w  3 k  =  WT,.  a  t a solution to ( P F ) . There Then  MRSy (x, ktX  yi,  2/2) >  T h a t is, person A; in family j faces a lower marginal tax rate  t  than person k in family i. B u t , by Proposition 4.9, the former faces a non-negative marginal tax rate while the latter faces a non-positive marginal tax rate. A contradiction ensues.  •  Proof of Proposition 4-11  T h e result is proven for the pair of families ( C , D ) and for the pair of families (B,D). T h e proofs for the pairs ( A , B ) , ( A , D ) and ( C , B ) are analogous to the demonstration for the pair ( C , D ) . 127  Let the C - D and D - C constraints bind simultaneously. T h e n (A.9), (A. 10), and (A.11) all hold with equality. B y L e m m a 4.2, y f = y f . Thus, (A.10) reduces to  _ h A  ( »)  7  V x  - V(x ) + ^ ( — ) = 0. C  (A.34)  7  WH  W' H  B y definition,  h(^) -  = V(x ) A  WL  7  W'  Y  V(x )+hA hA D  /i(^-) -  L  WL  WL'  WL  W  +1  WL  (A.35)  W  J  K  L  J  L  A d d i n g (A.34) and (A.35) yields  =  v  {  x  A  )  _ ,vt\  _  h  7 / l  WL  +7  .  WL  =V  +7  AC  /  WL ( ^ £ ) _ WL  l  .  ntt\  (*2_) _ WL  /  + h  WJJ l  ( y £ ) WL  +  h  WL  +  /  l  A \ WL (A.36)  WJJ'  ( y £ ) _ , ( ^ £ ) WJL WH J /  Analogous calculations can be used to show  yBD  =  ^BC  +  7  /  . Suppose that y  > y.  D  2  i  ( y £ ) _ ^ ( y £ ) WL  +  /  l  ( y £ ) _ WH  /  l  ( ^ £ ) WH  (A.37)  T h e n L e m m a 4.2, (A.36) and (A.37) imply that both the A -  D and B - D constraints are slack. A p p l y i n g Proposition 4.9 yields that M RSy  2X  and  MRS^  > 1. However, (A.34) implies a ; > x , so that MRS®  2X  a contradiction. T h e case of y  2  c  y  2  c  0  2>X  >  = 1  MRS^ , X  > y§ is ruled out by a symmetric argument. Thus,  . In order for (A.34) to hold, it must be the case that families C and D  receive the same bundle.  Suppose that the D - B and B - D constraints both bind. T h e n  ( D)  V X  _  _ WL  K  tvL) -  yh  (x ) B  V  WH  +h(£)+ 7ftA WL  128'  W  H  =  0,  (A.38)  and  V^B) _ h(?£_) _  ( l.) WL' y  l h  Wff'  K  _  v ( a ;  0)  +hA  +  WH  7  M — ) = 0. WL  (A.39)  J  Adding (A.38) and (A.39) gives  -w ' L  ^w '  "w  L  -w '  H  It follows from L e m m a 4.2 that  Assume that y f > y f . Furthermore,  y  B  ^£  H  Wjf  H  W  > y f if and only if y f > y f  T h e n y f > y f , and self-selection requires z  fw^ > yf'/WJJ.  Thus,  MRS  tion 4.9 implies that at a solution to ( P F ) ,  > MRSy  B 2X  MRS  x  2X  >  x. D  . However, Proposi-  < 1 and MRS^  B  s  >  1.  A  contradiction ensues.  The case of y f > y f can be ruled out by similar arguments, involving M RSy and  MRSy . lX  Hence, y f = y f and y f = y f .  that families B and D receive the same bundle.  Self-selection requires  x  B  — x, D  so  •  Proof of Proposition 4-12:  For all pairs of families except the pair { C , A } the proposition follows directly from Propositions 4.10 and 4.11.  Suppose that families C and A are mutually at-  tracted at a solution to ( P F ) . T h e n , by Proposition 4.8,  A y  + y  A  - x  A  = yC + y C - x  c  .  (A.41)  Now construct a new allocation by giving family A the allocation of family C , leaving the bundles received by all other families unchanged. 129  B y the argument outlined  in the proof of Proposition 4.8, this new allocation satisfies all of the self-selection constraints. (A.41) ensures that the new allocation is production-feasible. Moreover, all families are indifferent between the new allocation and the original. Because the original allocation solves ( P F ) , the new one does as well. T h a t is, there is a solution to (PF) at which families A and C receive the same allocation, contradicting Proposition 4.10.  •  Proof of Proposition 4-13:  Suppose, by way of contradiction, that there is a self-selection cycle at a solution to ( P F ) . In view of Proposition 4.8, all families in the cycle have the same total tax liability. Now select a pair of families, (i, j), from the cycle for which family j attracts family i. T h e n , we may replace the bundle of family i by the bundle of family j without making any family worse off and without violating the materials balance constraint. T h i s results in a solution to ( P F ) in which families i and j are pooled, contradicting Proposition 4.10.  square  Proof of Proposition 4-14'  Suppose that relation (4.26) of the text holds for family i.  If no families are  attracted to family i, then the result follows immediately from Proposition 4.9.  By  Proposition 4.8, any families attracted to family i must pay the same total tax bill as family i. Let family j be such a family. Replicating the argument of Proposition 4.12 allows one to conclude that replacing the allocation of family i by that of family j  130  results in another solution to ( P F ) that features pooling. T h i s contradicts Proposition 4.10.  •  Proof of Proposition 4-15:  I show that the C - A and D - B self-selection constraints cannot bind simultaneously. A l l other cases have identical proofs.  Suppose, by way of contradiction, that the C - A and D - B self-selection straints both bind at a solution to ( P F ) . B y Proposition 4.5, y f Proposition 4.6, y  A  < yf - Hence, y f = y .  < VA  con-  But, by  Now apply Propositions 4.5 and 4.6 once  A  more to conclude that the B - A and A - B self-selection constraints must both bind. T h i s contradicts Proposition 4.12.  •  Proof of Proposition 5.1:  T h e proof employs the following claim:  Claim 1: If (C, A) <£ C and if { (C, B), {B,A)}CC,  then (£>,  A) e C.  Suppose, by way of contradiction, that the D - A constraint is slack. It can be shown that  yCA _ yCB _ yBA  ,B  =  7  „.A  (^)_ (^)_ (y^) (?tJL)  h  .  h  h  WT,'  WT,  Now, by the hypothesis of the claim,  „,J3  „,A  WJJ'  WJJ  .  +h  =  =  0.  Thus, by L e m m a 4.2,  the C - A self-selection constraint can be satisfied only if y f > y . A  131  (A.42)  'i  B y Proposition  4.3,. y f > y . A  MRSy .  Moreover,  2X  MRS  < MRS  A  =  B  }X  yt  ^  an<  x  B  T h e n self-selection requires that  =  x  B  > x  A  (because  2X  yA  A  y f > y ). A  or  Hence, if  A  y  B  < MRS .  4.9 implies MRS  A  A  B  2X  2X  A  MRS  A  x B  iX  A  x, A  violating the n o -  Because the C - A constraint  — 1. B u t , by Proposition 4.9, MRS  B  %X  <  then  2>X  = x,  2X  >  B  When  = MRS ,  A  B  M RS  Hence,  > y.  MRS  then self-selection implies x  pooling condition. Consequently, is slack, Proposition  yf > y  if either  > x.  2X  <1  at a solution to ( P F ) , a contradiction. Hence, the D - A self-selection constraint must bind. T h i s establishes the claim.  Suppose that Assumptions R and U hold. Consider a pair of families (i, j) with i >F j- B y Assumption R, a sufficiently small transfer of x from family i to family j is socially desirable and does not lead to a violation of the materials balance constraint (by Assumption U ) . Moreover, such a redistribution will not lead to a violation of the constraints (SS) as long as, at the before-transfer allocation, the "donor" family is not attracted to any other family and the "recipient" family attracts no other family.  Now, let the C - A constraint be slack. Suppose that the C - D constraint is slack as well. I will show that the C - B constraint must bind. Suppose otherwise. T h e n a small redistribution of x from family C to family B would not violate the materials balance constraint (by Assumption U ) and would be socially desirable (by Assumption R ) . Because family C is not attracted to any family at the original candidate solution, the posited redistribution can be infeasible only if either the D - B constraint or the A - B constraint binds at the candidate solution.  132  Case 1: D - B binds and A - B binds.  T h e D - A self selection constraint is slack, for otherwise there would be a cycle, contradicting Proposition 4.13. B y Proposition 4.12, the B - A self-selection constraint is slack. T h e n a small redistribution from family C to family A is both feasible and desirable. Thus, the original allocation is not a solution to ( P F ) .  Case 2: D - B binds and A - B does not bind.  B y Proposition 4.12, the B - D self-selection constraint is slack. Suppose the A D constraint is slack as well. Because the C - D constraint is slack by assumption, a small redistribution of x from family C to family D is feasible and socially desirable, contradicting the optimality of the original allocation. Now suppose the A - D constraint binds. T h e n the B - A constraint must not bind, for otherwise there is a cycle  ( {(D,B), {B,A), (A, D)}). B y Proposition 4.12, the D - A constraint must be slack. Consequently, a small redistribution from family C to family A is feasible and socially desirable. Hence, the candidate solution is not optimal.  Case 3: D - B does not bind and A - B binds.  T h e B - A constraint does not bind (by Proposition 4.12). If the D - A constraint is slack, a small redistribution of x from family C to family A is both feasible and socially desirable. If, instead, the D - A constraint is binding, then Proposition 4.12 ensures that the A - D constraint is slack. Now, a binding B - D constraint produces the cycle { ( £ , D ) , (D, A), (A, B)}.  Hence, the B - D constraint is slack. Thus, a small  133  redistribution from family C to family D is feasible and socially desirable, violating the optimality of the original allocation.  Thus, it has been shown that the C - B constraint must bind whenever the C - A and C - D constraints are both slack.  It is also the case that either the D - A constraint or the B - A constraint must bind (possibly both). Otherwise a small redistribution of x from family C to family A would be socially desirable (by Assumption R ) , production-feasible (by Assumption U) and will not lead to a violation of the self-selection C - A constraint is slack).  constraints (because the  If the D - A constraint binds, we have immediately that  { (C, B), ( D , A) } C C. Now, suppose that the D - A constraint does not bind.  Then  the B - A constraint must bind. E m p l o y the claim to conclude that the D - A constraint must bind as well, a contradiction.  Now suppose that the C - D constraint is binding. Again, we must have either the D - A or the B - A constraint (or both) binding. If the B - A constraint binds, we are done. If the D - A constraint binds, an argument symmetric to the one establishing C l a i m 1 applies, and the proof of statement i) is complete.  Now suppose that Assumptions R, U and Q hold and that the C - A and C - D constraints do not bind. T h e n , by statement i) of the proposition, the C - B and D - A constraints must bind.  Suppose that the B - A constraint does not bind.  Assumption Q , we can transfer sufficiently small and  134  equal  amounts of  T h e n , by  x  from C to  D and from B to A without affecting the C - B or D - A constraints. B y Assumption U , this redistribution does not affect the production constraint, and it is socially desirable (by Assumption R ) . There are now two cases to consider.  1: B - D does not bind.  Case  B y Proposition 4.12 and the D - A constraint binding, the A - D constraint is slack. Thus, no family is attracted to family D . Because family A attracts family D alone, the posited redistribution does not violate (SS).  Case  2: B - D binds.  W h e n ty  BD  = 0, it can be shown that  CB  (A.43)  Given that the C - B constraint binds, L e m m a 4.2 implies ^ yf  > yf.  Because the C - D constraint is slack, yf  follows that  MRS  B 2X  > MRSy , 2X  C  D  > 0 if and only if  > yf • F r o m Assumption Q , it  violating Proposition 4.9. Hence, this case cannot  arise.  Now suppose the C - D constraint binds. We may replicate the argument in the last paragraph of the proof of statement i) to conclude that the B - A constraint must bind as well. If, in addition, the C - B constraint binds, it follows from C l a i m 1 that the D - A constraint is binding, establishing statement ii). If the C - B constraint does  135  not bind, then an argument symmetric to the one used when it was assumed that the C - D constraint is slack can be employed.  Now  suppose that the A - C constraint binds. T h e n , if the C - D constraint binds,  {(C, D), (D, A), (A, C)} C-B  is a self-selection cycle. If the C - D constraint is slack, the  constraint must bind.  Hence,  {(C, B), (B,A), (A,C)}  Thus, in both cases, Proposition 4.13 is violated.  is a self-selection cycle.  •  Proof of Proposition 5.2:  Only the proof of statement i) is presented. may  Proposition 4.15 implies that we  assume that the C - A and A - C self-selection constraints are slack throughout  this proof.  Assume that the D - B constraint binds. T h e n W  = *  CB  m  CB  =  v(x ) - h(£-) c  -  7  M — ) -  WH  W£  yCB  =  J  -  or  iK—) H  W  (AAA)  v(x°)+M—)+iK—y WL  WH  /  +  H  B  W  7  X  A d d and subtract h[^-)  + M — )  ft A -  -k(£)-  B  B  WH  K  +V(X )  v(* )  C  WH  to the left-hand side of (A.44) to conclude that  yCD  +  ,v£s  - (Vl-)+  h  h  WL  J  WH  (VL)  h  _ WH  (?l-\  h  (A.45)  « L  C a s e 1: y f > y f .  L e m m a 4.2 and (A.45) imply that the C - B constraint must be slack, so, by Proposition 5.1, the C - D and B - A constraints bind. It follows from Proposition 4.12  136  that the D - C and A - B constraints are slack. Thus, by Proposition 4.9, and  MRSy  x  MRSy^  =1  x  > 1. However, by Proposition 4.3, y f > y f , so that y f > y f . A p p l y  the M R S conditions to conclude that x  B  > x.  B u t , by Proposition 4.3, y f >  c  yf.  T h e n the C - B constraint must be violated, a contradiction.  C a s e 2: y f = y f .  T h e n , by L e m m a 4.2,  (A.45), and Proposition 5.1, both the C - D and C - B  MRSy  constraints bind. Thus,  = 1 and M RS  B  lX  1X  > 1. F r o m this, we may replicate  the argument of Case 1 to deduce a contradiction.  C a s e 3: y f > y f .  Then, by L e m m a 4.2 and (A.45), the C - D constraint is slack. Thus, by Proposition 5.1, both the C - B and D - A constraints bind. Now, it must be the case that  yDC  =  ^DC _ yDB  +  yCB^  HfDC  =  o r  vU  h{  WL  _  h  ,V?\ WL  +hA  Hence, the D - C constraint is satisfied only if y f Proposition 4.3, y f > y . B  Thus,  MRSy  > MRS  B  lX  lX  - h(£-). WH  WH  > yf.  Suppose y f  (A.46)  > yf.  Hence, the C - B constraint can be satisfied only if x  c  By >  x. B  > 1, where the second inequality follows from Proposition  4.9 and D - B binding. Because the A - C constraint is slack, we may apply Proposition 4.9 to conclude that the D - C constraint must bind.  So that, by (A.46), y f =  yf,  contradicting y f = y f . Thus, we may conclude that y f = y f and, in view of (A.46), that the D - C constraint binds.  137  T h e result follows from Case 3 being the only possibility that is not contradictory.  • Proof of Proposition 5.3:  In view of Proposition 4.15, this proposition follows immediately from Propositions 5.1 and 5.2.  •  Proof of Proposition 5-4 '•  I give the proof of the first inequality.  T h e second is proven in a symmetric  fashion.  MRSJ*  I first show that if  > M RSy  uX  lX  Suppose, to the contrary, that MRSy  lX  at a solution to ( P F ) , then y f >  > M RS^  and y f < y f . T h e n , x  B  lX  >  yf.  x. c  But, by Proposition 4.3, y f > y f . Thus, the C - B constraint is violated.  Now suppose 4.9,  MRS  B lX  MRS  < MRS^ .  B UX  > 1. Thus,  uX  MRSy  lX  (Otherwise, we are done.) B y Proposition  > 1. Hence, by Proposition 4.9, it must be the  case that either the A - C constraint binds or the D - C constraint binds (or both). But, by Proposition 4.5, when the A - C constraint binds, y f > y f .  It remains to consider the case of D - C binding. T w o possibilities need to be considered.  C a s e 1: C - A does not bind.  138  B y Proposition 4.12, the C - D constraint is slack. In view of Proposition 5.1, this implies that the C - B constraint binds. B y the argument used to establish (A.46), we know that  ^DC _ yDB  +  yCB  =  (VL) -h(^-).  wVfx  +h  WL  WL  WH  (A.47)  WH  Because the C - B and D - C constraints bind, (A.47) and L e m m a 4.2 imply that  ty  DB  >  0 if and only if y f > y f . C a s e 2: C - A binds.  It can be shown that  yCA  +  ^DC  =  yDA  _ ,yt\ h  +  WL  +h(^-) -  WL  K  h A .  WH  (A.48)  WH  Because both the C - A and D - C constraints are assumed to bind, the left-hand side of (A.48) is zero. A p p l y L e m m a 4.2 and (A.48) to conclude that if Vi > y f • However, by Proposition 4 . 3 , y f > y . A  fy  DA  Thus, y f > y f .  > 0 if and only •  Proof of Proposition 5.5: In addition to the notation already used, let ry^ denote the multiplier on the constraint y\ > 0, for j = A, B, C, D and % = 1, 2.  Under Assumption S Q , the first-order necessary conditions associated with the choices of before-tax incomes for family D are  (Z  wD  +  p )/w Dj  L  -  E 139  PJD/W{ -  r)i = ATT , d  (A.49)  l(Z D W  Eliminating Xn  —)  - 7 E VJD/wi - V2 = j^D  H  X  ^  D  (A.50)  '  from (A.49) and (A.50) yields  D  (  + E PDj)/w J^D  ( WO Z  + Y]  PDj)  I+ r  7  -ii  PCD  [7-1"  +  w  PAD  +  " 7  L  «L  1 " P-BD-  —  (A.51) Next, I show that the right-hand side of (A.51) is positive, so that nf and,  hence, that j / f = 0. Now, 7 > W^/WH  W  (A.52)  > 0.  W  Tj  > 0,  implies  7  1  > rjf  H  Thus, the last term of (A.51) is non-negative. Consequently, it suffices to show that the first term, which is non-negative, exceeds the second and third terms in absolute value. B u t ,  [1-7]  PCD  +  [1-7] W  PAD  <  L  since wj_, < WH- However, wi  first-order  7  W  (PCD + PAD),  (A.53)  L  < WH also implies that  1-7  The  n- i  <  1  —  7  (A.54)  necessary condition associated with the choice of after-tax income for  family D is  ZD W  + J2  3±D  VDj -  E  PJD] V'(X ) D  j^D  140  +r,S-  Xn  D  = 0,  (A.55)  where the i]  is the multiplier associated with the non-negativity constraint on  x  x. D  — 0. B y Proposition  Because U'(c) tends to positive infinity as c tends to zero, 4.7, A > 0. Hence,  z  wD  Because V'(x )  +  m - J2  > °-  *  '-  (A  56)  > 0, it may be canceled from (A.56) without changing the sense of  D  the inequality. Using HBD ^ 0» it  n  o  follows that  w  W +E  ^ ' '  >  Z D  CD  fJ AD  (A.57)  Combining (A.53), (A.54), and (A.57) yields the result that y f — 0.  Using the analogues to (A.49), (A.50), and (A.51) for family B , we can show that  ) { WB  + E  Z  [7-1]  VBj)  w  PCB  ll  +  W  H  1]  PAB  +  'j_  1_"  DB-  n  L  (A.58) But the first term on the right-hand side of (A.58) is negative, while all the others are non-positive. Hence, i]  < i] .  B  Thus, r]  B  B  > 0, and y f = 0.  F r o m the assumption y f = 0 (= y f ) , along with (A.9) and (A.10), it follows that ^DC  =  _^CD  =  V  [  X  D  )  _  y £ _  Substituting (A.59) into the definition of $  qCA  =  qDA  C A  v  ,  x  C  )  +  y£_  =  0  (A.59)  and using y f = 0, yields  +  141  {  yf__yt_.  (A.60)  Suppose that y  > 0. T h e n , by (A.60), the D - A constraint is slack. Using the  A  analogues to (A.49), (A.50), and (A.51) for family A and the fact that UDA — 0,  (1-7)  ( Z A + ^2^Aj) +  ' 7  W  1 "  —  WL  PBA +  r  7  - i i  (A.61)  PC A-  T h e first term in (A.61) is positive, while the second is non-negative. PC A = 0, we have n  > n  A  A  Hence, if  > 0. Now suppose UQA > 0. In this case, (1 — ^)/WH <  (1 — 7 ) / W L , SO that we have  [1-7 WH  1  PC A <  [1-7] w  (A.62)  PCA  L  B u t the first-order necessary condition for the allocation of after-tax income to family A implies ZA W  -I- E  PAj > PCA-  (A.63)  T h e n , the first term on the right-hand side of (A.61) exceeds the final term in absolute value, so that n  A  follows that y  A  > n  A  > 0 in this case as well. Thus, y  A  = 0, a contradiction. It  = 0. T h i s proves L ) .  Statement iii.) now follows directly from (A.60).  B y (A.59), both the C - D and D - C constraints bind. Suppose, by way of contradiction, that yf  > y§• T h e n , by the argument given in the proof of Proposition 4.11,  both the A - D and B - D constraints are slack. Also, Notice that  yf > 0,  so that  M RSy  2X  = 1.  enters the analogue to (A.50) for family C with a negative sign, just  142  like UJC does for j ^  C.  Thus, by an argument similar to the one used to estab-  lish Proposition 4.9, M RSy  MRSy  2X  < MRSy  x  > 1.  Self-selection requires x  D  > x. c  B u t then,  , a contradiction. A symmetric argument can be used to rule  out y f > y f .  Thus, y f = y f , and, because y f = y f , Proposition 4.3 implies x  c  establishes ii.).  •  143  = x. D  This  APPENDIX B Figure 1. T h e Space of Family Types  w  2  • LH  w  L  • HH  o LL  • HL  144  Figure 2. Monotonicity Properties Implied By Self-Selection  145  igure 3. Monotonicity Properties Implied By Self-Selection  146  Figure 4. Partial Monotonicity for Families H H and L L  x  147  Figure 5. T h e Lack of Attribute Ordering  2/1  148  II!)  igure 7. A n Implication of a Binding Self-Selection Constraint  x  2/1  1:50  Figure 8. A n Example of When A Zero Marginal Tax Hale is Optimal  V'l  151  

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