# Who should be on workfare? The use of work requirements as part of an optimal tax mix.

1. IntroductionThe aim of the optimal income taxation literature is to make precise the limits to redistribution implied by behavioural responses to progressive taxation. In its modern form, as expressed by Mirrlees (1971), the optimal tax problem is essentially a problem of hidden information. The taxation authority wishes to tax individuals on the basis of ability, but is not privy to this information. The best it can do is tax labour income, a quantity that depends on both ability and effort. Mirrlees (1971) and Guesnerie (1981) have shown that the planner can do no better than to invite individuals to report their ability, and to design a mechanism such that it is in each person's interest to do so.

While the standard optimal income taxation paradigm has provided many insights into the incentive effects of tax policies, it may be criticised for not taking into account the complete arsenal of policy instruments that governments may (and actually do) use for redistributive purposes.(1) One such instrument, of current theoretical and policy interest, is workfare. The incentive effects of workfare programmes have been studied in detail by Besley and Coate (1992, 1995). They consider a world in which required work is of no productive value. Among their conclusions is that workfare may play a part in optimal income maintenance schemes. Nevertheless, work requirements are not desirable if the planner wishes to design a least-cost utility maintenance programme.

That work requirements are optimal only in non-welfarist settings appears to be somewhat paradoxical. In both the nonlinear income tax model and the income maintenance model, the labour supply of low-ability individuals is distorted downward. However, Guesnerie and Roberts (1984) have shown that it is almost always optimal to impose a small, personalised increase in a good whose allocation is subject to a downward distortion. Of course, this apparent paradox is easily resolved by recognising that market labour supply and unproductive required work are different goods. In the absence of information asymmetries, the former has marginal social value equal to the wage rate, whilst the latter has a zero shadow price. It seems reasonable, therefore, to interpret the utility maintenance results of Besley and Coate (1995) as stating that the marginal value of workfare as a screening device is insufficient to overcome the initial discrepancy between the value of market work and the value of required work.

This study has two aims: to identify how productive required work must be in order for it to become part of an optimal policy mix; and to identify who should be required to take part in workfare programmes. I employ the standard nonlinear taxation model of Guesnerie and Seade (1982), with the addition of a third good, required work.(2) Required work produces some output, but its marginal product may differ from that of market work. The taxation authority is assumed to maximise some Paretian, inequality-averse social welfare function, subject to self-selection constraints and a materials balance constraint.

After spelling out the general features of optimal nonlinear taxation with work requirements, I present a special case of the model, the two-agent economy. I show that it is optimal to implement a workfare scheme for low-ability workers when required work is sufficiently productive. When low-ability workers supply a positive amount of labour, required work need not be as productive as the market work. However, when low-productivity workers are out of the labour force, required work must produce enough output to compensate these workers for their foregone leisure. This value is determined by preferences and not by productivity. Given that much of the policy debate focuses on workfare for those out of work, this finding is potentially of much importance.

The remainder of the paper is organized as follows. The next section describes the model, and gives some general properties of its solution. An analysis of the two-agent version of the model is presented in Section 3. The fourth and final section contains some comments on the relationship of this study to the policy debate on workfare. Proofs are gathered in an Appendix.

2. The model

The most general form of economy considered in this paper consists of H individuals, who may be partitioned into n types according to their productivity. The number of individuals of type i is denoted by [[Pi].sub.i]. The productivity of an individual of type i is given by [w.sub.i]. Types are ordered such that [w.sub.1] [less than] [w.sub.2] [less than] ... [less than] [w.sub.n]. The production sector is assumed to exhibit constant returns to scale, and the labour market is perfectly competitive. Under these assumptions, the before-tax labour income of an individual i is given by

[y.sub.i]: = [w.sub.i][l.sub.i] (1)

where [l.sub.i] is the supply of market labour by person i.

In addition to market work, the planner may require an amount of workfare, [r.sub.i], from each individual of type i. Although the total amount of after-programme consumption enjoyed by an individual may depend on participation in the workfare scheme, required work is not paid a wage at the margin. One unit of required work is assumed to produce [Gamma] units of output, regardless of the individual providing the work.(3) Each person has preferences over a consumption good, x, and time spent working, t, represented by

u(x, t): = x - h(t) (2)

The function h([center dot]) is assumed to be increasing, strictly convex and twice continuously differentiable. This form of utility function is exactly the one adopted by Besley and Coate (1995), and is maintained here to allow for comparisons of the present study with theirs. Moreover, it is widely known that quasi-linear utility functions add much to the tractability of screening problems, especially when there are more than two types of individuals. For some of the results below, the following regularity condition is assumed.

Assumption 1 h[double prime](t)/h[prime](t) is non-increasing in t.

Assumption 1 restricts the curvature of h([center dot]), limiting it to grow no faster than an exponential function. It is satisfied by the often-used quadratic function.

For any individual of type i, the planner can observe [y.sub.i], but not [w.sub.i]. The planner sets a tax function and a work requirement, which jointly determine [x.sub.i] and [r.sub.i], conditional on [y.sub.i]. The actions of the planner are subject to three classes of constraints: an economy-wide materials balance constraint, non-negativity constraints, and self-selection constraints. The materials balance constraint takes the form

[summation of] [[Pi].sub.i][x.sub.i] where i = 1 to n [less than or equal to] [summation of] [[Pi].sub.i] ([y.sub.i] + [Gamma][r.sub.i]) where i = 1 to n

Notice the influence of work requirements on the constraint (F). One can recover the standard feasibility constraint of nonlinear taxation models by putting [Gamma] equal to zero. The non-negativity constraints are denoted, with obvious conventions, (N([x.sub.i])), (N([y.sub.i])) and (N([r.sub.i])). The collection of all non-negativity constraints is called (N).

Self-selection constraints are formulated in terms of the variables that the planner can observe: consumption, before-tax income, and work requirements.

Using (1) and (2), the utility that an individual of type i receives from the bundle ([x.sub.i], [y.sub.i], [r.sub.i]) is

[u.sub.i]: = v([x.sub.i], [y.sub.i], [r.sub.i] [w.sub.i]): = [x.sub.i] - h ([y.sub.i]/[w.sub.i] + [r.sub.i]) (3)

Hence, the requirement that an individual of type i weakly prefers ([x.sub.i], [y.sub.i], [r.sub.i]) to the bundle ([x.sub.j] [y.sub.j], [r.sub.j]) can be written

[x.sub.i] - h ([y.sub.i]/[w.sub.i] + [r.sub.i]) [greater than or equal to] [x.sub.j] - h ([y.sub.j]/[w.sub.i] + [r.sub.j]) (SS(i, j))

Of course, (SS(i, j)) is required to hold for all pairs of types (i, j) with i [not equal to] j. The collection of all such (SS(i,j)) is denoted (SS).

The important features of the function v([center dot]) and its constituent parts are collected in the following Lemma:

Lemma 1 (i) For any (y, r), [w.sub.j] [greater than or equal to] [w.sub.i] if any only if h[prime] (y/[w.sub.i] + r) [greater than or equal to] h[prime] (y/[w.sub.j] + r). (ii) Suppose Assumption 1 holds and let i, j, and k be types [w.sub.i] [less than] [w.sub.j] [less than] [w.sub.k], and [Mathematical Expression Omitted], [Mathematical Expression Omitted] be distinct bundles such that

[Mathematical Expression Omitted] (4)

at least one inequality in (4) being strict. Then [Mathematical Expression Omitted] implies [Mathematical Expression Omitted] for all l [greater than] k and [Mathematical Expression Omitted] implies [Mathematical Expression Omitted] for all g [less than] i.

Statement (i) of the Lemma states that, holding r fixed, preferences over (x, y) pairs satisfy the standard single-crossing property. It also implies that, holding y constant, preferences over (x, r) pairs satisfy single-crossing, with crossing from the same direction. Statement (ii) says that when Assumption 1 is satisfied preferences satisfy the double-crossing property of Matthews and Moore (1987). That is, the graphs of the function [Mathematical Expression Omitted] and [Mathematical Expression Omitted] cross no more than twice in w-space.

The taxation authority is assumed to maximise some social welfare function, W([u.sub.1], ..., [u.sub.H]), which is assumed to be increasing in all its arguments, continuously differentiable and strictly S-concave.(4) The vector ([u.sub.1], ..., [u.sub.H]) contains an element for each individual. For types with [[Pi].sub.i] [greater than] 1, it contains multiple identical entires. Thus, it is possible to formally state the planner's decision problem as

[Mathematical Expression Omitted] (P)

where [Mathematical Expression Omitted]. A solution to the problem (P) is denoted by ([x.sup.*], [y.sup.*], [r.sup.*]).

The general features of a solution to the problem (P) are collected in the following proposition:

Proposition 1 Let ([x.sup.*], [y.sup.*], [r.sup.*]) solve (P). Then:

(i) i [greater than] j implies [Mathematical Expression Omitted];

(ii) the constraint (F) holds with equality;

(iii) for all i = 1, 2, ..., n - 1, SS(i + 1, i) holds with equality, if Assumption 1 holds;

(iv) under Assumption 1, if SS(j, i) holds with equality for a pair (i, j) with i [greater than] j, then there exists a solution to (P) in which all individuals of types j, j + 1, ..., i receive the same allocation.

Clause (i) of Proposition 1 simply states that individuals of higher productivity obtain at least as much utility as individuals of a lower productivity at the solution to (P). This is a requirement of self-selection, for a highly-productive worker can earn the same amount of before-tax labour income as a lowly-productive agent whilst using less labour time. Statement (ii) is standard.

In the terminology of Matthews and Moore (1987), statement (iii) says that all downward adjacent self-selection constraints bind at a solution to (P). Given the ordering of utilities implied by clause (i), any redistribution of consumption from an individual of type i + 1 to an individual of type i results in a transfer of utility from a relatively better-off person to a relatively worse-off person. S-concavity of W([center dot]) implies that all such redistributions that cause no change in the materials balance constraint are socially desirable. The double-crossing property of preferences implies that when SS(i + 1, i) is slack a large class of such redistributions between agents of type i + 1 to agents of type i may be carried out without violating (SS).

Clause (iv) of Proposition 1 states that if an upward self-selection constraint is binding at the optimum, then, without loss of generality, several types of agents are bunched at the optimum. This is a slight weakening of a result obtained for an optimal monopoly pricing problem by Matthews and Moore (1987). In their model, the monopolist can do strictly better by bunching agents when an upward self-selection constraint binds. Such a result could be available in the present model with a more structured social welfare function. In view of clause (iii), the presence of an upward binding constraint creates what Brito et al. (1990) call a self-selection cycle. They show that when there is a self-selection cycle in a Pareto-efficient tax structure, there exists an alternative optimal tax structure in which some of the agents are bunched. The strengthening of the Brito et al. result contained in Proposition 1 results from the special structure of self-selection cycles in this model.

Proposition 1 allows us to conclude that the solution to the present problem is relatively similar to the solution of the optimal nonlinear income tax problem when the standard single-crossing and redistributive assumptions are met. All adjacent downward self-selection constraints bind at the optimum, and all implicit marginal tax rates are non-negative.(5) The only complication is that, when there are more than two types of agents, the optimal taxation scheme may not fully separate types. Indeed, Weymark (1986) considers a class of tax problems in which any pattern of bunching, save those that have bunching at the top of the ability distribution, is consistent with the standard assumptions. In his model, bunching results when the solution of the optimal tax problem with only downward adjacent self-selection constraints fails to meet certain monotonicity conditions.

It may be conjectured that the introduction of workfare may be of value in separating out types of agents for whom the tax schedule is too blunt an instrument. Proposition 1 says that the introduction of workfare is not sufficient, by itself, to rule out bunching outcomes. Yet it does, as the next proposition shows, have implications for the ordering of the amount of required work performed at the optimum.

Proposition 2 Let ([x.sup.*], [y.sup.*], [r.sup.*]) be a solution to (P) and let j [greater than] i. Then [Mathematical Expression Omitted] implies [Mathematical Expression Omitted]. In particular, [Mathematical Expression Omitted] implies [Mathematical Expression Omitted].

The first statement of Proposition 2 asserts than when an individual of a lower type performs more required work than a higher-type individual, the former receives less market income. Read contrapositively, it implies the second statement: when agents of different type earn the same amount in the market, the more productive of the two should perform at least as much required work. This result is quite striking, given that all agents are equally productive at required work.

It is interesting to interpret Proposition 2 in light of the work by Guesnerie and Roberts (1984). When persons of different type earn the same before-tax income, those of highest type have the lowest marginal rate of substitution between before-tax income and after-tax income. That is, highest-ability agents face the highest implicit marginal tax rate. Hence, these individuals are subject to the largest downward distortions of effort. Higher-ability agents are, therefore, better targets for quantity instruments.

3. The two-class model

The analysis of the preceding section does not, by itself, provide any insight into the minimum productivity of required work necessary to make its introduction welfare-improving. In order to address this issue, I consider the special case of n = 2. Many of the important features of general optimal nonlinear taxes may be ascertained from a study of the simpler, two-class model. This observation holds true in the current context as well, with one important caveat. As Stiglitz (1982) has shown, the solution of the two-class optimal nonlinear tax problem features separation of the two types of individuals. When n = 2, the solution to the problem (P) is also devoid of bunching. Moreover, with or without Assumption 1, the planner's problem reduces to

[Mathematical Expression Omitted] (PT)

I assume that the non-negativity constraints on consumption are non-binding, so that [[Theta].sub.1] = [[Theta].sub.2] = 0. The case of a binding constraint on consumption has been analysed in detail by Besley and Coate (1992). All other non-negativity constraints are of interest here, especially those that pertain to the bundle designed for individuals of type 1. Indeed, the question of whether workfare is desirable comes down to whether the non-negativity constraint on [r.sub.1] is binding at the solution to (PT).(6) The following proposition gives a lower bound on the productivity of welfare-improving required work:

Proposition 3 Suppose that [Mathematical Expression Omitted]. Then there exists a [Mathematical Expression Omitted] such that [Mathematical Expression Omitted] if and only if [Mathematical Expression Omitted].

Proposition 3 contains two important statements. First, work requirements may be part of the optimal policy mix even when required work is less productive than market work. To gain some insight into why this is the case, consider the following subset of the first-order conditions for a solution to (PT)

[[Pi].sub.1] [W.sub.1] - [[Pi].sub.1][Lambda] - [Mu] = 0 (5)

-[[Pi].sub.1][W.sub.1]h[prime] ([y.sub.1]/[w.sub.1] + [r.sub.1]) 1/[w.sub.1] + [[Pi].sub.1][Lambda] + [Mu]h[prime] ([y.sub.1]/[w.sub.2] + [r.sub.1]) 1/[w.sub.2] + [[Eta].sub.1] = 0 (6)

-[[Pi].sub.1][W.sub.1]h[prime] ([y.sub.1]/[w.sub.1] + [r.sub.1]) + [[Pi].sub.1][Gamma][Lambda] + [Mu]h[prime] ([y.sub.1]/[w.sub.2] + [r.sub.1]) + [[Kappa].sub.1] = 0 (7)

(6) and (7) imply

[[Pi].sub.1][Gamma][Lambda] + [Mu]h[prime] ([y.sub.1]/[w.sub.2] + [r.sub.1]) + [[Kappa].sub.1] = [w.sub.1][[Pi].sub.1][Lambda] + [Mu]h[prime] ([y.sub.1]/[w.sub.2] + [r.sub.1]) [w.sub.1]/[w.sub.2] + [w.sub.1][[Eta].sub.1] (8)

Solving (8) for [[Kappa].sub.1], yields

[[Kappa].sub.1] = [Pi][Lambda] ([w.sub.1] - [Gamma]) + [Mu] [([w.sub.1] - [w.sub.2])/[w.sub.2] h[prime] ([y.sub.1]/[w.sub.2] + [r.sub.1])] + [w.sub.1] [[Eta].sub.1] (9)

When [Mathematical Expression Omitted], so that

[[Kappa].sub.1] [less than or equal to] 0 if and only if

[Pi][Lambda] ([w.sub.1] - [Gamma]) + [Mu] [([w.sub.1] - [w.sub.2])/[w.sub.2] h[prime] ([y.sub.1]/[w.sub.2] + [r.sub.1])] (10)

When [Lambda] [greater than] 0, (10) is equivalent to

[Gamma] [greater than or equal to] [w.sub.1] + [Mu]/[[Pi].sub.1][Lambda] [([w.sub.1] - [w.sub.2]/[w.sub.2]) h[prime] ([y.sub.1]/[w.sub.2] + [r.sub.1])] (11)

Evaluating (11) at [r.sub.1] = 0 gives

[Gamma] [greater than or equal to] [w.sub.1] + [Mu]/[[Pi].sub.1] [Lambda] [([w.sub.1] - [w.sub.2])/[w.sub.2] h[prime] ([y.sub.1]/[w.sub.2])] (12)

A negative value for the Kuhn-Tucker multiplier [[Kappa].sub.1] at [r.sub.1] = 0 may be interpreted as indicating that the introduction of a small increase in [r.sub.1] from zero is welfare-improving. Thus, condition (12) describes the circumstances in which workfare is part of an optimal policy package. To help interpret (12), it is useful to perform the following thought experiment. Suppose that the planner offers individuals of type 1 a small increase in required work in such a way that their market work falls by an amount that leaves them indifferent to the original bundle with no required work. The amount of the compensating reduction in before-tax income is [w.sub.1]. The left-hand side of (12) is the direct gain in output associated with an increase in required work, whereas the first term on the right-hand side of (12) is the loss of market-sector output. The second term on the right-hand side of (12) reflects the effect of the policy change on SS(2, 1). This term is negative, indicating that the policy change slackens the self-selection constraint. It is easy to see why this is the case. Agents of type 2 need a reduction in y of amount [w.sub.2] to compensate them for a one-unit increase in required work. Hence, the introduction of workfare, combined with appropriate changes in the tax schedule, makes the bundle offered to type-1 agents less attractive to type-2 individuals. Through this channel, workfare has value as a screening device.

It is interesting to compare the foregoing thought experiment with the policy perturbations analysed by Besley and Coate (1995). They argue that a one-unit reduction in workfare for agents of type 1, combined with a compensating decrease in consumption, slackens SS(2, 1). Analogous to this result is that a small increase in required work for type-1 agents, combined with a compensating increase in consumption, serves to tighten SS(2, 1). This is the source of the second part of Proposition 3: workfare is not optimal when required work is completely unproductive. Using (5) and (7), one can show that [Mathematical Expression Omitted] exactly when

[Gamma] [greater than] h[prime] ([y.sub.1]/[w.sub.1]) + [Mu]/[[Pi].sub.1][Lambda] [h[prime] ([y.sub.1]/[w.sub.1]) - h[prime] ([y.sub.1]/[w.sub.2])] (13)

Relation (13) states that workfare forms part of the optimal programme when required work is productive enough to provide an amount of the consumption good sufficient to compensate programme participants for their time plus provide resources to make up for the tightening of the self-selection constraint.

When [y.sub.1] = 0, it is impossible to use reductions in market time to compensate programme participants for their foregone leisure. It may come as no surprise, therefore, that the conditions under which positive work requirements are optimal are more restrictive when low-ability workers supply no market labour. This issue is addressed in the following proposition.

Proposition 4 Suppose that [Mathematical Expression Omitted]. Then [Mathematical Expression Omitted] only if [Gamma] [greater than] h[prime](0).

Proposition 4 results from the fact that any increases in required work for lowability individuals must either cause a reduction in their utility level, or be compensated by an increase in consumption. In the latter case, the compensation must be at least as large as the marginal rate of substitution between leisure and consumption. At an interior solution, the presence of distortionary taxation results in a marginal rate of substitution less than the wage rate. This may or may not be the case at corner solutions, depending on the marginal disutility of work. There are no gains to be had from slackening SS(2, 1) because the potential mimickers and agents of type 1 have the same utility without workfare, namely x - h(0). Thus, any compensated increase in required work also tightens SS(2, 1).

Given the focus on workfare as a policy for those out of work, Proposition 4 is particularly striking. In the standard nonlinear tax model, those individuals who supply zero market labour are exactly those for whom before-tax income is most expensive in terms of foregone leisure. Requiring additional work from these individuals comes at a high cost, either in terms of their utility or in terms of compensation to these individuals. These costs can only be recovered if the work done as part of the scheme is sufficiently productive. Interestingly, the critical level of productivity is determined entirely by preferences, and not by the ability of the workers. This is simply because the shadow value of voluntarily unemployed labour is equal to the marginal rate of substitution between leisure and consumption. It is possible that this value is relatively low for those individuals who face high implicit marginal tax rates at zero labour supply. This matter is, in the end, an empirical issue.

While the exact statement of Proposition 4 is due to the special assumptions of quasi-linear preferences and only two worker types, its basic message is not. Suppose that there are many types of workers, some of whom are bunched at zero hours of work in the absence of a workfare scheme. Self-selection requires that all such workers have the same level of consumption. If one maintains the assumption that individuals have the same preferences over consumption and leisure (not income), all out of work agents require the same amount of compensation for a small increase in required work, whatever the form of the utility function. Workfare might be used to distinguish between types of unemployed workers who differ in preferences or in their productivity in required work, but not those who differ in terms of market ability alone.

4. Conclusion

This paper has uncovered a set of conditions in which required work of less value than market work forms part of an optimal redistributive taxation scheme. However, the results hold in full generality only when the targeted are in the labour force. This point is of policy relevance, given a general presumption that workfare is a policy for the out of work. In a welfarist setting, it is necessary to compensate with consumption the previously unemployed for workfare, for they cannot be compensated by reductions in market time. The amount of compensation, quite intuitively, depends upon the shadow value of leisure. Workfare imposes large costs on those individuals who have a high value of non-market time, be that time used for domestic production or pure leisure. Such individuals are not the best targets for workfare. This accords with the simulation findings of Fortin et al. (1993). They report that workfare programmes need to be targeted to specific socio-economic groups, with less stringent work requirements for groups with high marginal valuation of non-market time, in order to be more effective than Negative Income Tax (NIT) schemes.

Nevertheless, those who face high marginal tax rates may benefit from participation in a well-designed workfare program. That is, using workfare as an alternative to a benefits clawback may prove beneficial. However, there is nothing in this analysis to suggest that workfare is always the best alternative to targeted schemes with high effective marginal tax rates. At its best, workfare can be a way to give those members of the poor who do not find work too onerous an opportunity to supplement their income.

Acknowledgements

This paper was inspired by discussions at the microeconomics lunch group at the university of British Columbia. I wish to thank Charles Blackorby, Saqib Jafarey, Mick Keen, Heather Waples, and two anonymous referees for insightful comments on an earlier version of this paper. Participants in the workshop on economic theory at the University of Essex and at a seminar at the University of Birmingham also provided helpful comments. Errors, omissions and views are entirely my own.

Appendix

1. Proof of Lemma 1

(i) Follows directly from covexity of h([center dot]).

(ii) Select ([w.sub.i], [w.sub.j], [w.sub.k]), ([Mathematical Expression Omitted]) and ([Mathematical Expression Omitted]) as described in the statement of the Lemma. Define a function [Theta](w) by

[Mathematical Expression Omitted] (14)

The function [Theta]([center dot]) is twice continuously differentiable and possesses a strict local minimum, [Mathematical Expression Omitted], on ([w.sub.i], [w.sub.k]). Therefore

[Mathematical Expression Omitted] (15)

Suppose, now, that [Mathematical Expression Omitted] and [Mathematical Expression Omitted] for some l [greater than] k. Then, [Theta]([center dot]) possesses a strict local maximum, [Mathematical Expression Omitted] on ([w.sub.j], [w.sub.l]). Thus

[Mathematical Expression Omitted] (16)

[Mathematical Expression Omitted] is distinct from [Mathematical Expression Omitted], but either of the two may be the larger. It follows from (15) and (16) that

[Mathematical Expression Omitted] (17)

Now define a function [Lambda](w) by

[Mathematical Expression Omitted] (18)

By (17), and the mean value theorem, there exists a [w.sup.*] between [Mathematical Expression Omitted] and [Mathematical Expression Omitted] such that [Lambda][prime]([w.sup.*]) = 0. It follows that

[Mathematical Expression Omitted] (19)

Now suppose that [Mathematical Expression Omitted]. (15) implies [Mathematical Expression Omitted]. Then, by (19),

[Mathematical Expression Omitted] (20)

Assumption 1 implies

[Mathematical Expression Omitted] (21)

But (15) and (16) and [Mathematical Expression Omitted] imply

[Mathematical Expression Omitted] (22)

But (21) and (22) imply [Mathematical Expression Omitted] and [Mathematical Expression Omitted], contrary to the fact that [w.sup.*] is between [Mathematical Expression Omitted] and [Mathematical Expression Omitted].

A similar argument may be used to rule out the possibility of [Mathematical Expression Omitted]. The only remaining possibility is [Mathematical Expression Omitted]. Then (15) implies [Mathematical Expression Omitted]. Then [Theta]([center dot]) is independent of w and cannot possess a strict local minimum at [Mathematical Expression Omitted]. This final contradiction establishes the first part of (ii). The second has analogous proof.

2. Proof of Proposition 1

(i) Take i [greater than] j. By SS(i, j), [Mathematical Expression Omitted]. But [w.sub.i] [greater than] [w.sub.j] implies

[Mathematical Expression Omitted]

(ii) Consider the first order necessary conditions for a solution to (PF). Summing the conditions associated with [x.sub.i], i = 1, ..., n yields

[Mathematical Expression Omitted] (23)

But (23) implies

[Lambda] = [summation of] ([[Pi].sub.i] [W.sub.i]) where i = 1 to n/H [greater than] 0 (24)

Thus, the materials balance constraint binds.

(iii)(7) Consider the problem (P[prime]), obtained from the problem (P) by deleting SS(i, j) for all pairs (i, j) with i [less than] j. The problem has only downward self-selection constraints remaining. I will show that, for each k:

(a) any solution to (P[prime]) has SS(i + 1, i) holding with equality for all i = 2, ..., k;

(b) such a solution satisfies SS(j, i) for all pairs (i, j) with i [less than] j [less than] k.

It then follows that (P[prime]) and (P) have the same solution, and statement (iii) of the Proposition follows.

Claim 1 For any pair (i, j) with i [greater than] j, there exists an [[Epsilon].sup.*] [greater than or equal to] 0 such that reducing [x.sub.i] by [Epsilon] [greater than or equal to] [[Epsilon].sup.*] units and increasing [x.sub.j] by ([[Pi].sub.i] / [[Pi].sub.j]) [Epsilon] increases social welfare.

By Statement (i), [u.sub.i] [greater than] [u.sub.j]. Hence, for sufficiently small [Epsilon] (that is, small enough so that no re-ranking of individuals occurs), the post-transfer distribution of utility Lorenz dominates the pre-transfer distribution of utility. Because W([center dot]) is strictly S-concave, Theorem 1 of Dasgupta et al. (1973) implies an increase in social welfare.

Claim 2 Statements 1 and 2 hold for k = 2.

Statement 2 is trivial. For Statement 1, suppose that SS(2, 1) does not bind at a solution to (P[prime]). By Claim 1, there is a sufficiently small transfer of x from individual 2 to individual 1 that is both production-feasible and welfare-improving. Such a transfer can be found that does not lead to a violation of SS(2, 1). Moreover, (P[prime]) has no upward self-selection constraints, so a transfer to individual 1 never violates its self-selection requirements. Hence, the supposition is incorrect, establishing the claim.

Claim 3 If Statements 1 and 2 hold for some k [less than] n, then Statements 1 and 2 hold for k + 1.

To establish Statement 1, suppose, to the contrary, that there is some i [less than or equal to] k such that [Mathematical Expression Omitted]. There exist j [less than] i such that SS(i + 1, j) binds, for otherwise a small transfer can be constructed from individuals of type i + 1 to those of type i without violating any self-selection constraints. By Claim i, it would be desirable to do so. Thus, [Mathematical Expression Omitted]. Employ the inductive hypothesis to conclude that [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. Now apply Lemma 1 (ii) to ([w.sub.j], [w.sub.i], [w.sub.i+1]) with [Mathematical Expression Omitted] and [Mathematical Expression Omitted] to conclude that [Mathematical Expression Omitted] for all l [greater than] i + 1. But each such l-type agent weakly prefers its own bundle to the one designed for agents of type j. Thus, [Mathematical Expression Omitted] for all l [greater than] i. But then a small redistribution from agents of type i + 1 to those of type i does not violate any self-selection constraints. Apply Claim 1 to conclude that this situation is not optimal. This contradication establishes that SS(i + 1, i) binds at the solution to (P[prime]), and statement 1.

For Statement 2, the following claim is useful.

Claim 3a If SS(i, j) binds at a solution to (P[prime]) then [Mathematical Expression Omitted].

Otherwise, there exists a pair (i, j) for which [Mathematical Expression Omitted] and [Mathematical Expression Omitted]. Replace [Mathematical Expression Omitted] by [Mathematical Expression Omitted]. Individuals of type i are indifferent to the change, no self-selection constraints are violated (because no one strictly envies the bundle offered to individuals of type j), and the materials balance constraint is slackened. Hence, no such pair can exist. The claim follows.

Now, suppose that there exists a g [less than] k + 1 such that SS(g, k + 1) is violated at a solution to (P[prime]). Let j be the largest such g. By Claim 3a and the first part of this claim,

[Mathematical Expression Omitted] (25)

Thus, we may replace [Mathematical Expression Omitted] by [Mathematical Expression Omitted] while maintaining production feasibility. Moreover, individuals of type j are made better off by the change. By the choice of j, no downward incentive constraints are affected by the change. Hence, the candidate solution can be improved upon. The claim follows.

Statement (iii) of the Proposition now follows by induction from Claims 2 and 3.

(iv) Suppose that there exist a pair (i, j) with i [greater than] j and SS(j, i) binding. Employ Claim 3a, along with statement (iii) of the Proposition to conclude that

[Mathematical Expression Omitted] (26)

Now replace [Mathematical Expression Omitted] by [Mathematical Expression Omitted]. By (26), this can be done without violating the materials balance constraint. An argument similar to that used to establish Claim 3a establishes that this change does not violate any self-selection conditions and leaves agents of type i equally well off. Thus, this new allocation must also be a solution to (P). If, in the original candidate solution, SS(j + 1, i) is slack then SS(j + 1, j) is slack in this new solution, a contradiction. Thus SS(j + 1, i) binds as well.

Repeat the procedure of the preceding paragraph for all types k = j + 1, ..., i, concluding with agents of types j,j + 1, ..., i receiving the bundle initially designed for agents of type i.

3. Proof of Proposition 2

Suppose that [r.sub.i] [greater than] [r.sub.j] for some i [less than] j. By SS

[x.sub.i] - h([y.sub.i/[w.sub.i] + [r.sub.i]) [greater than or equal to] [x.sub.j] - j([y.sub.i/[w.sub.i] + [r.sub.j]) (27)

and

[x.sub.j] - h([y.sub.i/[w.sub.j] + [r.sub.j]) [greater than or equal to] [x.sub.i] - h([y.sub.i]/[w.sub.j] + [r.sub.i]) (28)

Adding (27) and (28) yields

h([y.sub.i]/[w.sub.j] + [r.sub.i]) - h([y.sub.i]/[w.sub.i] + [r.sub.i]) [greater than or equal to] h([y.sub.j]/[w.sub.j] + [r.sub.j]) - h([y.sub.j]/[w.sub.i] + [r.sub.j]) (29)

By the first fundamental theorem of calculus, (29) implies

[Mathematical Expression Omitted] (30)

Convexity of h([center dot]) and [r.sub.i] [greater than] [r.sub.j] imply [y.sub.i] [less than] [y.sub.j] for otherwise the integrand on the right-hand side of (30) would be everywhere smaller than the integrand on the left-hand side, yielding a contradiction. The second statement follows directly from the first.

4. Proof of Proposition 3

Consider the problem of choosing (x, y) to maximise W([u.sub.1], [u.sub.2]), subject to the constraints of the problem (PT), except N([r.sub.1]) and N([r.sub.2]). Call this problem [Mathematical Expression Omitted]. Let V([r.sub.1], [r.sub.2]) be the value function for this problem.(8) To establish the Proposition, it suffices to analyse the conditions under which ([Delta]V/[Delta][r.sub.1])(0,[r.sub.2]) [greater than] 0.

The Lagrangean function for [Mathematical Expression Omitted] is

[Mathematical Expression Omitted] (31)

By the envelope theorem

[Mathematical Expression Omitted] (32)

The right-hand side of (32) is

-[[Pi].sub.1] [W.sub.1]h[prime]([y.sub.1]/[w.sub.1]) + [[Pi].sub.1] [Gamma][Delta] + [Mu]h[prime]([y.sub.1]/[w.sub.2]) (33)

The first order necessary condition for [Mathematical Expression Omitted] associated with [y.sub.1] implies

-[[Pi].sub.1] [W.sub.1]h[prime]([y.sub.i1]/[w.sub.1]) = -[[Pi].sub.1][w.sub.1] [Lambda] - [Mu]h[prime]([y.sub.1]/[w.sub.2]) [w.sub.1]/[w.sub.2] - [w.sub.1][[Eta].sub.1] (34)

Substituting (34) into (33) yields

[[Pi].sub.1] [Lambda]([Gamma] - [w.sub.1]) + [Mu]([w.sub.2] - [w.sub.1]/[w.sub.2]) h[prime] ([y.sub.1]/[w.sub.2]) - [w.sub.1][[Eta].sub.1] (35)

When [y.sub.1] [greater than] [[Eta].sub.1] = 0 so that simplifying the expression (35) implies

[Mathematical Expression Omitted] (36)

Because [w.sub.1] [less than] [w.sub.2], the second term on the right-hand side of (36) is negative, so that the entire right-hand side of (36) is less than [w.sub.1].

On the other hand, the first order condition associated with [w.sub.1] implies

[[Pi].sub.1][W.sub.1] = [[Pi].sub.1][Lambda] + [Mu] (37)

Substituting (37) into (33) yields

[Mathematical Expression Omitted] (38)

Thus

[Mathematical Expression Omitted] (39)

Notice that both terms on the right-hand side of (39) are positive. The first, because h([center dot]) is increasing; the second, by Lemma l(i).

5.1 Proof of Proposition 4

Evaluating (39) at [y.sub.1] = 0 yields

[Mathematical Expression Omitted] (40)

The result follows.

1 Of course, there are many papers that step outside the benchmark paradigm. Examples include Boadway and Marchand (1995), Boadway and Keen (1993) and Brito et al. (1990).

2 The addition of the third good renders the formal model similar to that of Matthews and Moore (1987). Indeed, the optimal policy mix possesses many of the characteristics of their optimal monopoly provision of warranties and quality.

3 Allowing individuals to differ in productivity of required work would result in a screening model with two-dimensional uncertainty, of the type analysed by Rochet (1995). While there are potentially important lessons to be drawn from this extension to the analysis (including a study of the incentives of workers taking part in a workfare scheme) they are excluded.

4 See Dasgupta et al. (1973) for a formal definition of S-concavity and a description of the salient properties of S-concave social welfare functions.

5 If individuals of type i [less than] n faces a negative marginal tax rate, reducing their hours worked and consumption in such a way to leave them indifferent saves resources and slackens SS(i + 1, i); hence, such a situation cannot be optimal. See Guesnerie and Seade (1982) for a full treatment of this issue.

6 Because agents of type 2 face no distortions at a solution to (PT), [Mathematical Expression Omitted] if and only if [Gamma] [greater than] [w.sub.2]. In this case, required work is more productive than the market work of either individual, rendering the problem uninteresting.

7 This proof is inspired by the proof of Theorem 2 in Matthews and Moore (1987).

8 Dependence of V([center dot]) upon other parameters of the problem is suppressed from the notation.

References

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Author: | Brett, Craig |
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Publication: | Oxford Economic Papers |

Date: | Oct 1, 1998 |

Words: | 6944 |

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