Welfare Policy: Cash Versus Kind, Self-Selection and Notches.

• Author: Singh, Nirvikar

Nirvikar Singh [*]

Ravi Thomas [+]

This paper investigates second-best allocations where the government lacks full information about consumer types, and how such allocations may be implemented through notch schemes. Thus, we provide another rationale for notches in addition to that investigated by Blinder and Rosen (1985). We use a model of Blackorby and Donaldson (1988), extending their results to more general preferences and to more general tax-subsidy instruments (piecewise linear, rather than linear). We argue that observed policies are sometimes of this nature: In-kind subsidies that are available only if consumption equals or exceeds a particular amount have been used in practice, in housing, and medical care.

1. Introduction

   Observed welfare policies are often perceived as complex and unduly restrictive of individual choices. Public assistance in the United States, for example, is provided for by several programs, administered separately by various state and federal agencies [1]. Being poor is neither sufficient (recipients must pass a status test) nor necessary to receive public assistance.2 In 1990 over 37% of the households with income below the poverty level received neither cash nor in-kind aid while 52% of households with income above the poverty level received cash and at least one in-kind benefit (see Table 1). Choice is restricted by the fact that three quarters of the transfers are in-kind--in the form of medical care, food, housing, energy, and child care (see Table 2). The inefficiency induced by restricting individual choices and the costs of administering the various programs under the current system is obvious, but is the current system more inefficient than any of the alternatives?

   The economic theorist's standard answer is based on the second theorem of welfare economics, which shows that in convex economies, transfers of purchasing power alone are sufficient to achieve a given Pareto optimal allocation. This translates into a policy prescription for cash rather than in-kind transfers in discussions of welfare policy. [3] Public policy prescriptions such as rationing, subsidized/free provision, and subsidies in excess of a certain level of consumption contradict the intuition we develop from first-best analysis. These exceptions to the rule have relied on factors such as interdependent preferences for theoretical justification [see Friedman (1984), chapters 3 and 4, for this argument and a basic survey of the area], [4] and the argument that providing the cash equivalent of health care or food stamps would destroy the justification for the policies in the first place, the right to life, and a minimum standard of living (Kelman 1986).

   Blackorby and Donaldson (1988), henceforth RD. have shown that an economic argument for in-kind transfers can be developed in a world without interdependent preferences, but where the government has incomplete information about individual preferences. This issue of information has been raised as an important one in practice. For example, it has been suggested that welfare policy in the United States may in fact be driven by such a need. That is, the need to identify and help the poor who are the "proper objects of pity" and to deny help to those who are not (Burtless 1990). To a certain extent, this is reflected in policymakers' indecision between the use of universal and targeted benefits and when targeted benefits are chosen--a predisposition to the use of in-kind transfers. Whether intended or not, the extent to which the major welfare programs are successful in identifying the deserving poor may be due to the structure of the current system, a reliance on targeted in-kind transfers. Thus the type of reas oning pursued by BD can have important practical implications.

   In BD's model, in-kind transfers enable screening of individuals with relevant differences in preferences when who has what preferences is not known to the government, but the proportion of each type is known. The government can tailor offerings so that only those with the right preferences will select the intended support package. Thus, screening is implemented through "self-election." [5] In particular, a quantity rationing policy achieves a second-best outcome. When direct limits on quantities are not possible, then a transfer-subsidy scheme must be considered. This leads to an inferior, third-best outcome. The reason is that cash transfers, with prices set by the government, require that the income equivalent of the consumption bundles of one type not be preferred by the other type of individual, and in-kind transfers only require that individuals not prefer the other type's bundle.

   In this paper, we extend and reinforce BD's arguments in two significant ways. In section 2, we show that more efficient (partial) decentralization is possible than BD's third-best solution, where in-kind transfers are ruled out, but governments may set prices. [6] In fact, the second best can be achieved through limited lump-sum or per-unit subsidies for particular goods. In section 3, we consider the case in which both types of individuals demand both goods, generalizing BD's preference structure, and derive the possible second-best allocations for this more general case. There is further novelty in the types of policy that implement such allocations: Now we have lump-sum, in-kind subsidies that are available only if consumption equals or exceeds a particular amount. The consumption may be in the form of leisure, [7] medical care, [8] or housing. [9]

   These may also be reinterpreted as notch schemes (Blinder and Rosen 1985). Hence, we show that welfare policies used in practice may have some justification in a world of asymmetric information.

   Finally, in section 4 we conclude by summarizing our results and discussing two practical problems of implementation arising with the policies we describe: the need for some degree of observation of levels of consumption, and the possibility of illegal resale or trade of subsidized goods. We also briefly touch on a third issue, the consequences of there being more than two types of individuals. This changes the details of analysis, but not the general implications for policy.

2. Simple Nonlinear Pricing in the Blackorby-Donaldson Model

   Transfer programs are generally designed for persons with some disability or extraordinary need that is caused by illness, age, or the need to care for children. In the BD model, they will be referred to as Infirm. The model may be briefly summarized as follows. There are two types of individuals, Able (A) and Infirm (I), and two goods, yams (y, the numeraire) and medical care (z).

   Able cares only about yams and has utility

   [u.sub.A] = [y.sub.A]. (1)

   Infirm's utility is given by

   [u.sub.I] = U([y.sub.I], z), (2)

   where U is increasing, twice continuously differentiable and strongly quasi-concave. Also, y and z are normal goods for Infirm. There is no labor-leisure choice in this model. This would add another dimension to the problem that we do not tackle.

   The society is endowed with k units of the input x, and the technology y = x and z = x. The production possibility frontier is thus

   [y.sub.A] + [y.sub.I] + z = k. (3)

   Maximizing [u.sub.I] with respect to [y.sub.A], [y.sub.I] and z, subject to Equations 1 and 3 and appropriate inequality constraints for feasibility gives the first-best frontier (FBF),

   [[u.sup.f].sub.I] = F([u.sub.A]), (4)

   where F is decreasing (see Figure 1).

   With full information about preferences, the government may decentralize any point on the FBF as a Walrasian equilibrium with the price of medical care set equal to one, and with lump-sum income transfers where initial income endowments sum to k. We might also think of the endowments of the individuals as being in yams, and then Infirm trades yams for medical care in the decentralized solution. Taxation is purely redistributive in this case.

   If the government cannot identify who is Able and who is Infirm, a first-best allocation [([y.sub.A], 0), ([y.sub.I], z)] may not be achievable if it either provides an incentive for Able to pretend to be Infirm, or vice versa. For example, the Supplemental Security Income program provides cash transfers to the aged, the blind, and the disabled. However, the program has become increasingly dominated by the disability aspects, a characteristic more difficult to identify than age or blindness. [10] To avoid this, the government must incorporate the following self-selection constraints in its optimization:

   [y.sub.A] [geq] [y.sub.I] (5)

   U([y.sub.I], Z) [geq] U([y.sub.A], 0). (6)

   Satisfaction of Equation 5 ensures Able will not want to lie to obtain Infirm's bundle. Satisfaction of Equation 6 ensures Infirm will not gain from lying. BD proceed to derive the second-best frontier (SBF in Figure 1), on which the self-selection constraints are satisfied, and to characterize it (their Theorem 1). This may be paraphrased and illustrated below.

   First, if [u.sub.A] = [y.sub.A] = k (Able receives the entire endowment), the inequality in Equation 6 is reversed strictly; hence, there is no second best for [[bar{u}].sub.A] [less than] [u.sub.A] [leq] k, where [[bar{u}].sub.A] is defined by

   max[[u.sub.A]: F([u.sub.A]) = U([u.sub.A], 0)]. (7)

   In words, [[bar{u}].sub.A] is the highest utility level for Able that is consistent with Infirm's self-selection constraint and being on the first-best frontier.

   Second, if [[underline{u}].sub.A], is the smallest [u.sub.A] = [y.sub.A], satisfying Equation 5, then [[underline{u}].sub.A]. [less than] [[bar{u}].sub.A]. Hence, for [[underline{u}].sub.A], [leq] [u.sub.A] [leq] [[bar{u}].sub.A], the self-selection constraints are automatically satisfied, and SBF and FBF coincide. Third, for min([u.sub.A]) [leq] [u.sub.A] [leq] [[underline{u}].sub.A] maximizing Equation 2 subject to Equations 1, 3, and 5 gives Equation 8.

   [[u.sup.s].sub.I] = S([u.sub.A]). (8)

   If [[u.sup.m].sub.A] is...