# On the relative efficiency of cash transfers and subsidies.

I. INTRODUCTION

Economic theory has long argued that to raise the welfare of an individual at the lowest possible cost, cash grants are more efficient than subsidies to the consumption of specific commodities. Indeed, a demonstration of this result now constitutes part of many undergraduate price theory courses and most undergraduate courses in public finance. Put simply, cash grants do not create the deadweight losses in consumption that would result from the distortion of relative prices. (1)

As familiar as this demonstration may be, subsidies to selected commodities, notably housing, education, food and health care, represent a substantial fraction of government aid to the disadvantaged citizens of many countries. (2) It is natural for us to ask why this is so. As Gary Becker [1983; 1985] has argued, it is in no government's interest to transfer resources in a Pareto-inefficient manner. If a switch to cash grants could make both recipients and donors better off by reducing the deadweight loss, why would any government not make such a change?

To the several extant theories this paper adds another: subsidies may be more effective than cash grants at raising the welfare of all members of a family when parents put insufficient (in society's view) weight on their children's welfare in household decision making. When a government directs resources to a deserving family in which all consumption decisions are made by the head of the household, an agency problem arises. The head, as the government's agent, may not allocate resources in the way the government would wish. By constraining the transfer using subsidies or in-kind gifts, the government may be able to influence the intrafamily allocations in a desired way.

II. SUBSIDIES VS. CASH GRANTS

There can be little doubt that a large fraction of aid to low-income households in the United States comes in the form of subsidies. (3) Table I presents some statistics on the amount spent by the Federal government on the largest programs.

The reader might glean at least two striking facts from this table. The first is the magnitude of the redistribution represented here. When one adds to this list the transfers by other levels of government (for example, for education), it becomes clear that redistribution is a major function of American governments. (4) Second, a significant share of this redistribution is done using subsidies (including in-kind transfers).

The public finance literature offers a number of explanations for the popularity of subsidies. (5) Taxpayers, or their elected representatives, might simply be paternalistic; by this we mean that they evaluate the recipient's welfare using a utility function other than the recipient's. (6) Though undeniably interested in the welfare of those less fortunate, these taxpayers might not be willing to provide cash if some fraction will be applied toward the purchase of alcohol, tobacco or other luxury or disreputable commodities. It may also be that there exist certain goods that society feels should be distributed more equally than free market allocations would provide. Tobin [1970] has referred to this as "specific egalitarianism" (7)

As intuitively appealing as these explanations might seem, it is not clear why people should feel this way. (8) And, absent a theory explaining why taxpayers care about some commodities and not others, it is a difficult theory to test. (9)

It may also be that the consumption of certain commodities provides external benefits to others that have nothing to do with altruism. Subsidies to housing might eliminate the eyesore of slum dwellings. Food stamps, by improving nutrition, might reduce the demand of the poor for socially provided medical care. (10) They might also raise the incomes of some food producers (i.e. farmers), reducing their need for other forms of assistance. And the arguments regarding the positive externalities that may flow from education are very familiar.

Some recent research has focused on the inefficiencies possible when cash programs are made available to different types of recipients. Blackorby and Donaldson [1988] model the adverse selection problem that can result when it is difficult for a government to determine who is truly deserving. (11) In-kind subsidies may serve to help discourage illegitimate claims while still providing assistance to those who need it. Fallis [1986] has argued that the relative efficiency of cash grants is less clear when lump sum transfers are ruled out and recipients represent heterogeneous households facing different market wage rates and making labor supply decisions. Making the size of the cash grant depend upon the income of the recipient distorts the labor-leisure choice, creating its own deadweight loss. (12)

Let me propose another efficiency-based explanation for the choice of subsidies over cash grants, one that addresses the principal-agent problem that arises when consumption decisions for an entire household are made by the head of that household. Though taxpayers may be quite happy to let a single recipient spend cash as he sees fit, they may not like the way parents share resources with their children. Given the difficulty of giving cash directly to children, subsidies can help by distorting household consumption decisions in a way that helps the children. (13) As evidence, refer back to Table I and keep in mind that large sums of money are spent by state and local governments to subsidize primary, secondary and post-secondary education. Clearly, children are major beneficiaries of these transfers that come through subsidies.

This is paternalism of a different sort. Here, society imposes its own view of a proper household welfare function, though it accepts individuals' preferences. This theory has advantages over the more familiar version since it is more specific and offers more testable implications. Under the general theory of paternalism, any sort of subsidy program could be theoretically justified by claiming that "social preferences" were just of the form as to make that program optimal. In the theory offered here, there is less freedom to infer unlikely social preferences. Society's household welfare function respects the true individual utility functions of the household's members, as the head of the household does, but it allows for greater weight to be put on the welfare of those not making the consumption decisions. It can then be shown that subsidies of goods with specific properties may be more efficient at raising household welfare than cash grants.

III. THE MODEL

To keep things simple, consider a two-person household with one parent (the head) and one child. The parent is assumed to make all the household consumption decisions for both agents. Let x = ([x.sub.1],[x.sub.2], ..., [x.sub.n]) represent the consumption vector over n goods for the parent and z = ([z.sub.1],[z.sub.2], ... [z.sub.n]) that, over the same goods, for the child.

The parent will not necessarily be totally selfish. He will be assumed to maximize a weighted sum of his own utility, U(x), and the child's, V(z). I call this weighted sum W(x,z):

W(x,z) = U(x) + [gamma] V(z).

Society's welfare function for this household takes the same form, but may involve a different value for the relative weight [gamma]. Without any loss of generality, I assume that society chooses to weight the utilities equally, so the social welfare function for this household can be represented in the Benthamite fashion as

[W.sup.*](x,z) = U(x) + V(z).

Clearly much is arbitrary about this specification of household and social preferences. Nevertheless, it will serve as a useful and simple approach, allowing me to focus on the role of subsidies when society places a greater weight on the welfare of children than do the heads of households. (14) In fact, since I will need fairly explicit solutions to solve for household demand functions, further restrictions on the forms of U(x) and V(z) will be required to derive specific results.

The parent's problem in the absence of government intervention is to maximize W(x,z) subject to the household budget constraint Y = p(x+z), where Y is the household income and p the vector of commodity prices. Given a cash grant of G and a vector of commodity subsidies of s, this budget constraint becomes

Y + G = (p - s)(x + z).

The government's objective is to select G and s to maximize [W.sup.*] subject to two sets of constraints: (i) the first-order conditions from the parent's optimization problem, and (ii) a government budget constraint:

B = G + s(x + z).

The purpose of what follows in the next section is not to solve this problem fully, but, through the analysis of some interesting special cases, to demonstrate conditions under which the solution will involve the use of subsidies, possibly in conjunction with grants. Notice that in the absence of the child, or if [gamma] = 1, the familiar logic would instruct us to use only cash grants. Here we ask whether, in the presence of the principal-agent problem observed when [gamma] < 1, subsidies can work to redirect consumption decisions in the child's favor without causing an even greater loss to the parent.

Obviously, it is important to this analysis that society place a higher value on the welfare of children than parents do. Why this might be so is a question that deserves a better answer than is given here. It may simply represent a form of paternalism as described above. It could also be that society enjoys certain external benefits from raising disadvantaged children's welfare; for example, lower juvenile crime rates, or the development of better citizens as the children become adults. Finally, it may not be that society wants to give special assistance to children as much as it wants to keep aid programs from being too attractive to recipients who could take actions to avoid the need to draw on these programs. Presumably, there is no such moral hazard problem with children. (15)

IV. ANALYZING THE SPECIAL CASES

This section demonstrates that there are circumstances under which subsidies are more efficient on the margin than cash grants at raising [W.sup.*]. The examples also serve to illustrate some of the limitations of subsidies. (16)

CASE 1: Parent and Child have different tastes.

One must always be a little careful in defining a utility function for children. In V(z), do we have the utility function that rationalizes the choices that the child would make, given the freedom to make all her own selections? Or do we mean that V(z) rationalizes the choices she would have made for herself looking back as an adult; choices that do reflect her particular likes and dislikes, but that also recognize the long-term benefits of the consumption of certain goods, like education, and the costs of others (e.g. candy)? Here I interpret V(z) this second way, and in this first special case consider the benefits of subsidies when U(x) and V(z) differ, or more precisely, when marginal rates of substitution differ between parent and child.

Consider the effects of a first dollar of subsidy funds for the purchase of good j versus the effects of a first dollar of cash grant. From the parent's first-order condition we know that

(1) [U.sub.i](x) = [gamma][V.sub.i](z)= [lambda][p.sub.i]

where subscripts denote partial derivatives and [lambda] represents the multiplier on the household budget constraint. From society's viewpoint, the value of a dollar cash grant to this household will be:

d[W.sup.*]/dG = [summation][U.sub.i](partial derivative][x.sub.i]/ [partial derivative]G) + [summation][v.sub.i]/[partial derivative][z.sub.i] [partial derivative]G)

which, using (1), we can rewrite as

d[W.sup.*]dG = [lambda][[summation][p.sub.i]([partial derivative][x.sub.i]/ [partial derivative]G] + [summation]([p.sub.i]/[gamma]) ([partial derivative] [z.sub.i]/[partial derivative]G).

(Throughout the paper, the reader should understand that all summations are from i=1 to i=n.) The first dollar devoted to the subsidization of good j will bring a social benefit of, using (1) as before,

(2) (d[W.sup.*]/d[s.sub.j])(d[s.sub.j]/dS) = [lambda][[summation][p.sub.i] ([partial derivative][x.sub.i]/[partial derivative][s.sub.j]) + [summation]([p.sub.i]/[gamma]([partial derivative][z.sub.i]/ [partial derivative][s.sub.j])/[x.sub.j]+[z.sub.j]

where S is the total expenditure on the subsidy program (i.e., S = [s.sub.j]([x.sub.j]+[z.sub.j])). Substituting the Slutsky equation counterparts for the slopes of the demand curves in (2), and recognizing that (17)

(3) [summation][p.sub.i][([partial derivative][x.sub.i]/[partial derivative] [s.sub.j]).sup.c] = - [summation] [p.sub.i][([partial derivative][z.sub.i]/ [partial derivative][s.sub.j]).sup.c]

where the superscripted c signifies that the slope is that of a compensated demand curve, we find

(4) d[W.sup.*]/dS = [[lambda]/([x.sub.j] + [z.sub.j])] {[(1-[gamma])/[gamma]] x [summation][p.sub.i][([partial derivative][z.sub.i]/[partial derivative] [s.sub.j]).sup.c]} + d[W.sup.*/dG.

Clearly, if [gamma] = 1 the first dollar spent will be equally productive under the two programs as there is no principal-agent problem. If [gamma] < 1, then the difference depends upon the sign of [summation][p.sub.i][([partial derivative][z.sub.i]/ [partial derivative][s.sub.j].sup.c]. This sum represents the change in total expenditures devoted to the child caused by a compensated increase in the subsidy to good j. It can be positive or negative, as demonstrated by the following simple example.

Let there be two goods, 1 and 2, and assume individual utility functions that take the (Cobb-Douglas) form:

U(x) = [x.sup.a.sub.1][x.sup.b.sub.2] V(z) = [z.sup.c.sub.1][z.sup.d.sub.2].

Consider the effects of a grant of G and a per-unit subsidy toward the consumption of good 2 of [s.sub.2]. We normalize the price of good 1 to be unity. This makes the household budget constraint

(5) Y + G = ([x.sub.1] + [z.sub.1]) + ([p.sub.2] - [s.sub.2]([x.sub.2] + [z.sub.2]).

Straightforward calculations reveal that the parent maximizing W = U(x) + [gamma]V(z) subject to (5) will make the following selections:

[x.sub.1] = a(Y + G)/[a + b +[gamma](c + d)] [x.sub.2] = b(Y + G)/([p.sub.2] - [s.sub.2])[a + b + [gamma](c + d)] [z.sub.1] = [gamma]c(Y + G)/[a + b + [gamma](c + d)] [z.sub.2] = [gamma]d(Y + G)/([p.sub.2] - [s.sub.2])[a + b + [gamma](c + d)].

The effects of increases in G and [s.sub.2] on consumption are easily calculated as derivatives of these expressions.

Consider first the effect on social welfare, as measured by [W.sup.*], of an additional dollar of cash grant, in the absence of any subsidies. This will be given by

d[W.sup.*]/dG = [U.sub.1] * d[x.sub.1]/dG + [U.sub.2] * d[x.sub.2]/dG + [V.sub.1] * d[z.sub.1]/dG + [V.sub.2] * d[z.sub.2]/dG

which, in this case, yields,

(6) d[W.sup.*]/dG = (a + b + c + d)/(Y + G).

The introduction of a per-unit subsidy will, with these preferences, affect welfare only through the consumption of the second good:

d[W.sup.*]/d[s.sub.2] = [U.sub.2] * d[x.sub.2]/d[s.sub.2] + [V.sub.2] * d[z.sub.2]/d[s.sub.2].

Increasing this subsidy by a dollar per unit will raise the total cost of the subsidy program (S) by

dS/d[s.sub.2] = [s.sub.2](d[x.sub.2] /d[s.sub.2] + d[z.sub.2] /d[s.sub.2]) + ([x.sub.2] + [z.sub.2]) = ([x.sub.2]+[z.sub.2]) * [[p.sub.2]/([p.sub.2] - [s.sub.2])].

The welfare effect of an additional dollar spent on the subsidy program will be

(7) d[W.sup.*]/dS = (d[W.sup.*]/d[s.sub.2])(d[s.sub.2]/ dS) = [(b + d)/(b + [gamma]d)] {[a + b + [gamma](c + d)]/Y} [([p.sub.2] - [s.sub.2])/[p.sub.2]].

The question then becomes, under what conditions can the quantity in (7) exceed the quantity in (6)? The first thing to notice is that, if [gamma] = 1, the familiar result obtains; cash grants must be superior. Notice also that, other things equal, the relative efficiency of subsidies falls as the rate of the subsidy grows. This is due to the rising deadweight loss suffered as a result of the growing distortion in relative prices.

If we set the initial subsidy level at [s.sub.2] = 0, and assume that [gamma] < 1 however, we find that the quantity in (7) will be greater than that in (6) if

(b + d) / (b + [gamma]d) * [a + b + [gamma](c + d)] >(a + b + c + d),

which reduces to

d/c>b/a.

Thus, if the child and parent have different marginal rates of substitution (at identical quantities), it will be socially optimal to subsidize the good that the child values relatively more highly (on the margin). It will only be efficient to subsidize up to a certain level, however; as observed above, the rising deadweight loss will make cash grants more beneficial at some point.

To some extent, this result is possible because Cobb-Douglas preferences give us ordinary demand curves with cross-price elasticities all equal to zero. In general, though we expect that subsidizing the purchase of good 1 will encourage the parent to buy more of that good for his child, he might partially frustrate the program by buying her less of the other good (if the child has positive consumption of the other good). When the cross-elasticities are zero, as they are in the examples studied here, the child keeps more of the

benefit. Thus, we would expect that the higher the cross-elasticities (as long as they are positive) the less likely it is that the child will benefit significantly, and the less likely it is that subsidies will dominate cash grants. (18)

An obvious example of a good that produces more benefits for children than for their parents is education at the elementary and secondary level. Thus, we find a reason to subsidize this education rather than simply provide poorer families with the unrestricted funds sufficient to allow them to purchase it. (19) Any commodity consumed exclusively or largely by children would make an obvious candidate for such a subsidy. (20)

It should be clear that an alternative to subsidizing the consumption of certain goods that benefit children is to provide a general cash transfer, but to tax the consumption of goods that do not benefit children. Indeed, the optimal policy might include some combination of all three instruments: some cash grants, subsidies for particular commodities like education, and taxes for others, like liquor and tobacco.

CASE 2: Household public goods.

Again, suppose there are two goods. The first good we will call food, and it is purely private in the sense that units consumed by any individual are not available for any other individual. The second good, housing (denoted H), is a pure public good within the household. That is, units of housing provide benefits to all household members simultaneously. It might seem intuitive to many readers that subsidies to household public goods might help mitigate the principal-agent problem here, as the parent's ability to appropriate the benefits of the transfer scheme are somewhat restricted. In fact, under some conditions, this is indeed true and subsidies to household public goods can be efficient, even if the parent's and child's preferences are identical. These conditions may be less general than one imagines, however.

Before any transfers, the parent's problem is to maximize

W([x.sub.1],[z.sub.1],H) = U([x.sub.1],H) + [gamma]V([z.sub.1],H)

subject to the household budget constraint. The first-order conditions reveal that

(8) [U.sub.1] = [gamma][V.sub.1] = [lambda][p.sub.1]

and that

[U.sub.H] + [gamma][V.sub.H] = [lambda]h,

where h is the (before subsidy) price of a unit of housing and [lambda] is, again, the multiplier on the budget constraint.

The social value of a dollar cash grant will then be

d[W.sup.*]/dG = [U.sub.1([partial derivative][x.sub.1]/[partial derivative]G) + [V.sub.1]([partial derivative][z.sub.1]/[partial derivative]G) +([U.sub.H] + [V.sub.H])([partial derivative]H/[partial derivative]G).

Substituting from (8), this can be written

(9) d[W.sup.*]/dG = [lambda][[p.sub.1]([partial derivative][x.sub.1]/ [partial derivative]G) + ([P.sub.l]/[gamma])([partial derivative][z.sub.1]/ [partial derivative]G) + h([partial derivative]H/[partial derivative]G)] + (1 - [gamma])[V.sub.H]([partial derivative]H/[partial derivative]G).

Letting the subsidy on housing be represented by s, we can derive the effects on [W.sup.*] of a dollar expenditure through the subsidy program. Substituting from the first-order conditions and the Slutsky equations, and rearranging terms we find:

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A comparison of (9) and (10) applying (3) reveals that

d[W.sup.*]/dS > d[W.sup.*]/dG iff

(11) [lambda][p.sub.1][(1 - [gamma])/[gamma][[([partial derivative][z.sub.1]/ [partial derivative]s).sup.c] + (1-[gamma])[V.sub.H][([partial derivative]H/ [partial derivative]s).sup.c] >0.

If [gamma] = 1 both of these terms are zero and the first dollar spent is equally effective under the two programs. Also note that the second term in (11) is positive, meaning that a sufficient condition for (11) to hold, when [gamma] < 1, is that [([partial derivative][z.sub.1],/ [partial derivative]s) .sup.c] [greater than or equal to] 0. That is, if a compensated increase in the subsidy leads to an increase in the amount of the private good allocated to the child, the first dollar of the subsidy is more effective than the first dollar of a cash grant.

It might seem that this condition is unlikely to be met and, indeed, in the Cobb-Douglas case analyzed above, it is easily seen not to hold. Given the familiar result that with these preferences the ordinary demand for each good is independent of the prices of other goods, the substitution effect must be negative to offset the positive income effect. In fact, with only two goods and identical preferences, we would always expect the two goods to be substitutes under compensation. The interested reader can easily verify that (11) will not hold with these preferences.

There are preferences under which (11) will hold, however. Consider the following example in which identical preferences are assumed to avoid complicating these results with those of the previous case. Suppose the parent maximizes a function of the following form:

W = [(a[x.sub.1].sup.1/2] [(bH).sup.1/2] + [gamma][[(a[z.sub.1]).sup.1/2 + [(bH).sup.1/2]].

This yields demand curves of the form (setting [p.sub.1] = 1 for convenience)

(12) [x.sub.1] = Y/Q [z.sub.1] = [[gamma].sup.2]Y/Q H = (b/a)[[(1 + [gamma])/h].sup.2](Y/Q)

where Q [equivalent to] (1 + [[gamma].sup.2]) + (b/a)(1 + [gamma])2/h.

Straightforward calculations reveal that the social value of a single dollar of cash grant will be

d[W.sup.*]/dG = [(1/2Y)[(a[x.sub.1]).sup.1/2] + [(a[z.sub.l).sup.1/2] + 2[(bH).sup.1/2].

The value of the first dollar of subsidy to housing is

d[W.sup.*]/dS = - (1/2Y)[[(a[x.sub.1]).sup.1/2] + [(a[z.sub.1).sup.1/2] + 2[(bH).sup.1/2]] + 2[(bH).sup.1/2]/(hH).

Thus, d[W.sup.*]/dS) > d[W.sup.*]/dG if

(13) 2[(bH).sup.1/2]/hH >[[(a[x.sub.1]).sup.1/2] + [(a[z.sub.l]).sup.1/2] + 2[(bH).sup.1/2]/Y.

Inequality (13) has a simple interpretation. The left-hand side is the average social welfare generated per dollar spent on housing. The right-hand side is the overall average social welfare generated per dollar. Substituting from (12) and rearranging terms we find that inequality (13) will hold whenever ([gamma]-1).sup.2] > 0, or in other terms, whenever [gamma] [not equal to] 1, even if it is greater. Thus, it is possible for subsidies to household public goods to dominate cash grants on efficiency grounds, at least up to a point. Recall that the example employed identical preferences for parent and child. To the extent that the child's welfare function places a relatively higher value on the public good, the dominance of subsidies will be enhanced.

If we imagine the parent's optimization problem to be different from the maximization of IV, as I have it here, other possible benefits of subsidies to household public goods suggest themselves. A parent with little regard for the welfare of his child (i.e., [gamma] [approximately equal to] 0) will provide her with very little food, perhaps just enough to sustain her (and avoid trouble with the authorities). A subsidy to a household public good like housing, however, will generate spillover benefits to the child as the parent is induced to purchase more housing services. Notice, however, that to the extent that parents can recapture some of this spillover by reducing the child's consumption of other goods, this approach will be frustrated.

Although housing is a good example, and a commodity often subsidized, the list of potential household public goods is probably not a long one. (21) Somewhat related, however, will be goods and services like health care, which can produce important, within-household externalities.

CASE 3: Parent claims all marginal resources for himself

Finally, for the last case consider the following example of parent and child preferences over two purely private goods:

Parent: U(x) = aln[x.sub.1] + b[x.sub.2] Child: V(z) = aln[z.sub.1] + b[z.sub.2]

Notice that their preferences are identical and that they exhibit constant marginal utility of good 2. The parent maximizing W will in this case set [z.sub.2] = 0 if [gamma] < 1. Assuming enough resources that [x.sub.2] is not also equal to zero, the other quantities chosen will be (setting [P.sub.2] = 1 for convenience):

[x.sub.1] = a/(b[p.sub.1]) [z.sub.1] = [gamma]a/(b[p.sub.1]) [x.sub.2] = y - (1+[gamma])a/b.

With these preferences, and assuming that [gamma] < 1, any additional dollar given in the form of a cash grant will be spent exclusively on [x.sub.2]. This creates a social welfare gain of

d[W.sup.*]/dG = b.

If good 1 was subsidized instead, [x.sub.2] would not be affected, but the other quantities would change according to (assuming the initial subsidy rate was equal to zero):

d[x.sub.1]/d[s.sub.1] = [x.sub.1]/[p.sub.1] d[z.sub.1]/d[s.sub.1] = [z.sub.1]/[P.sub.1]

The social welfare effect of the first dollar spent on this program will then be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, social welfare rises more when a subsidy is used in this case and [gamma] < 1. Whereas a cash grant would be entirely devoted to the purchase of good 2 for the parent, a subsidy for the consumption of good 1, together with the unitary elasticities of demand for the good, guarantee that all the additional funds are devoted to the purchase of good I for both individuals.

If it is felt that larger cash grants will be devoted to the purchase of goods for the parent only, perhaps luxury items (e.g. entertainment, alcohol and tobacco), it may be optimal to subsidize the purchase of non-luxury items such as food and shelter.

IV. SUMMARY AND CONCLUSIONS

When parents care less than donors about the welfare of their children, a principal-agent problem arises for donors wishing to raise the household's welfare. Though donors may be nonpaternalistic in the sense that they are quite prepared to accept each family member's own utility function as the true measure of his or her actual utility, they may not be pleased with the way the head of the household allocates resources among family members. In such a case I have demonstrated that there exist conditions under which in-kind subsidies will, to a point, dominate cash transfers as a means to raise household welfare. Simply put, the subsidies can create a distortion that counteracts the distortion created by the agency relationship; they encourage the purchase of goods with a higher social value (because they are important to the child) than private value.

This is by no means the last word on these questions. The extent to which subsidies can be more efficient than cash grants obviously depends critically on the nature of the household choice mechanism, about which we have said relatively little. (22) Parents may not maximize a weighted sum of household members' individual utilities, for example; rather, they may maximize their own utility subject to certain constraints on the utility levels or consumption levels of the other members. This change in the parent's problem could have implications for the choice between subsidies and cash grants; indeed, it might mean that neither is at all effective at raising the welfare of children. (23)

Further research is clearly needed to refine the theory behind the examples studied here, to reveal the precise conditions under which subsidies can dominate cash grants in the presence of this type of agency problem. Of particular interest would be the study of the role of heterogeneity of the recipient population in the design of the optimal program. (24)

Finally, the existence of this agency relationship will have implications for other government policies, and some of these would be worth exploring. As an example, Robinson [1988] has investigated the design of optimal taxation schemes in an economy with families. He derives extensions to the familiar Ramsey rules for optimal commodity taxation for a world in which planners care about household utility that is different from the utility of the household head.

The lesson in this line of research is that recognizing the importance of transactions costs not only raises our estimates of the costs of government intervention, it can also have important implications for the design of that intervention.

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--. "The Interaction Between Public and Private Spending When Government is Responsive to the Preferences of Citizens." Mimeo, Washington University, St. Louis, September 1989.

Robinson, D. "Optimal Commodity Taxation With Families." Discussion Paper No. 02-88, Department of Economics, Laurentian University, June 1988.

Rosen, H. Public Finance. Homewood, Illinois: Richard D. Irwin, 1985.

Ross, S. "The Economic Theory of Agency: The Principal's Problem." American Economic Review, May 1973, 134-39.

Ross, T. "On the Relative Efficiency of Cash Transfers and Subsidies." Working Paper No. E-88-20, Domestic Studies Program, Hoover Institution, Stanford University, 1988.

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Stigler, G. "The Theory of Economic Regulation." Bell Journal of Economics, Spring 1971, 3-21.

Thurow, L. "Cash Versus In-Kind Transfers." American Economic Review, May 1974, 190-95.

Tobin, J. "On Limiting the Domain of Inequality." Journal of Law and Economics, October 1970, 263-77.

United States Bureau of the Census. Statistical Abstract of the United States: 1984, 104th ed. Washington, D.C.: U.S. Government Printing Office, 1983.

--. Statistical Abstract of the United States: 1988, 108th ed. Washington, D.C.: U.S. Government Printing Office, 1987.

(1.) Early proofs are found in Joseph [1939] and Hicks [1939]. See also Friedman [1962, 56-67].

(2.) On the growth of importance of this kind of aid in the United States, see Browning [1975a].

(3.) The editor has correctly pointed out that what I refer to here generally as subsidies might be more properly termed "in-kind subsidies." I assume that subsidies that specify the payment of a certain amount of money to compensate the buyer for the purchase of a specific amount of a commodity are too difficult to monitor, and are therefore not used.

(4.) In 1982, state governments spent $52.8 billion and local governments $49.8 billion on public primary and secondary education. See U.S. Bureau of the Census [1987, 119]. Out of total expenditures in 1982 of about $740 billion, almost $305 billion (41 percent) involved transfer payments to domestic recipients. These transfers represent about 10 percent of Gross National Product in 1982. U.S. Bureau of the Census [1983, 317 and 448].

(5.) Rosen [1985, 80-95], contains a nice review of the traditional explanations.

(6.) Garfinkel [1973] presents a reasonably general treatment in which taxpayers have preferences over the form of the redistribution. Pollak [1988] has recently extended paternalism arguments to explain tied transfers from parents to children.

(7.) Tobin [1970] suggested that the following may be candidate commodities for specific egalitarianism: civil rights, food, shelter, education and medical care. He suggests that in some cases society would like the distribution to be completely equal (e.g., voting rights), while in others the concern is only that everyone have a certain minimum quantity (e.g., food, shelter and education).

(8.) For a discussion of this point, see Thurow [1974].

(9.) Browning [1975b] describes the difficulties inherent in using in-kind transfers to affect Pareto optimality in the presence of consumption externalities. In Browning [1981], he argues that actual in-kind transfers do not achieve a Pareto optimum; rather they cater more to the preferences of donors than recipients.

(10.) For a recent related discussion of the interaction between altruism and moral hazard, see Bruce and Waldman's [1988] analysis of tied transfers in the presence of the Samaritan's dilemma.

(11.) See also Nichols and Zeckhauser [1982]. The idea goes back at least to Stigler [1971] who argued in the context of regulation that subsidies to favored industries will be preferred to cash grants if entry is easy.

(12.) Still another explanation for subsidies comes from the recent work of Dougan and Snyder [1990]. They model a collective choice, pie-splitting game and show that price-distorting policies (such as subsidies) can command majority support when cash transfers cannot. Such policies can create the asymmetries across agents that are essential to the formation of stable voting groups.

(13.) Further evidence that politicians worry about such agency problems comes from the "Family Allowance" (the Canadian term) programs in Canada and the United Kingdom. Both programs provide cash grants to parents based upon the number of children in the family. In both countries the checks are made out to the mother, concern having been expressed about the willingness of the father to allocate the funds properly.

(14.) The reader may recognize that this is not really a traditional principal-agent model. On the standard approach see, for example, Ross [1973] and Shavel [1979]. Specifically, there is no uncertainty in this model. This means that if the government wanted, it could dictate to the parent the desired quantities, and perfectly observe and punish disobedience. I assume that this approach is either undesirable or infeasible. Governmental uncertainty about the exact tastes of parents and children could be added to the model.

(15.) Another way to view this problem is to interpret W(x,z) as the parent's own utility function, which then exhibits some degree of altruism. Society's welfare function would then be [W.sup.*] = W + V. Then, society places a greater weight on childrens' welfare because it values the direct benefit (V) as well as the indirect benefit (through W). I am grateful to David Robinson for suggesting this interpretation.

(16.) Many of the derivations in this section are straightforward but tedious, and so some steps have been omitted from the presentation. The interested reader is directed to Ross [1988] for more detail.

(17.) Some elaboration of this point might be helpful. Each of these compensated demand derivatives is actually the sum of two derivatives, one with respect to [p.sub.j] charged for [x.sub.j] and one with respect to [p.sub.j] charged for [z.sub.j]. For example:

[([partial derivative][x.sub.i]/[partial derivative][s.sub.j]).sup.c] = -[[([partial derivative][x.sub.i]/[partial derivative][p.sup.x.sub.j]).sup.c] + [([partial derivative][x.sub.i]/[partial derivative][p.sup.z.sub.j].sup.c].

Combining this with the well-known property, derivable from the Slutsky equations,

[summation] [p.sub.i][[([partial derivative][x.sub.i]/[partial derivative] [p.sup.x.sub.j].sup.c])c + [([partial derivative][x.sub.i]/ [partial derivative][p.sup.z.sub.j]).sup.c]]

we can then derive the equation from the text.

(18.) Note 23 below contains an example in which the substitution is complete, so the child is no better off with the subsidies. In this case, cash grants will again be more efficient at raising household welfare.

(19.) In many political jurisdictions, the education of children is made compulsory as well. To the extent that governments can determine and compel the optimal levels of consumption of selected goods they avoid the agency problem modelled here. It is not difficult to believe, however, that if primary and secondary education were not so heavily subsidized, securing compliance with compulsory education laws would be more problematic.

(20.) In the province of Ontario, for example, children's clothing is exempt from provincial sales tax.

(21.) Home heating and cooling subsidies are another example. I am grateful to Frances Ruane for this suggestion.

(22.) Notice that this question is related to, but distinct from, the question of how parents allocate resources between children. On this question see, for example, Becker and Tomes [1976], and Behrman, Pollak and Taubman [1982]. Here our concern is with the allocation between parent and child.

(23.) If the parent maximized his own utility subject to a constraint that the child's utility not fall below a certain level, neither subsidies nor cash grants can raise the welfare of the child. Even subsidies to household public goods will be frustrated as the parent will only reduce expenditures for other goods given to the child. In this case, cash grants are the most efficient way to raise the utility of the parent.

(24.) Introducing heterogeneity will admit adverse selection problems, as in Blackorby and Donaldson [1988]. Heterogeneity of donors as well as recipients has proven to be an important element of Roberts" [1987; 1989] work on public versus private provision of public goods.

THOMAS W. ROSS, Department of Economics, Carleton University. The first draft of this paper was written while the author was a National Fellow at the Hoover Institution. The author gratefully acknowledges a number of invaluable discussions with Russell Cooper, the useful suggestions of Keith Acheson, Thomas Borcherding, Jonathan Kesselman, John McManus, Robert Porter, David Robinson, Frances Ruane and an anonymous referee, and the research assistance of Lynn Ross. He is also pleased to acknowledge the financial assistance provided by the Carleton Industrial Organization Research Unit and the Hoover Institution.

Economic theory has long argued that to raise the welfare of an individual at the lowest possible cost, cash grants are more efficient than subsidies to the consumption of specific commodities. Indeed, a demonstration of this result now constitutes part of many undergraduate price theory courses and most undergraduate courses in public finance. Put simply, cash grants do not create the deadweight losses in consumption that would result from the distortion of relative prices. (1)

As familiar as this demonstration may be, subsidies to selected commodities, notably housing, education, food and health care, represent a substantial fraction of government aid to the disadvantaged citizens of many countries. (2) It is natural for us to ask why this is so. As Gary Becker [1983; 1985] has argued, it is in no government's interest to transfer resources in a Pareto-inefficient manner. If a switch to cash grants could make both recipients and donors better off by reducing the deadweight loss, why would any government not make such a change?

To the several extant theories this paper adds another: subsidies may be more effective than cash grants at raising the welfare of all members of a family when parents put insufficient (in society's view) weight on their children's welfare in household decision making. When a government directs resources to a deserving family in which all consumption decisions are made by the head of the household, an agency problem arises. The head, as the government's agent, may not allocate resources in the way the government would wish. By constraining the transfer using subsidies or in-kind gifts, the government may be able to influence the intrafamily allocations in a desired way.

II. SUBSIDIES VS. CASH GRANTS

There can be little doubt that a large fraction of aid to low-income households in the United States comes in the form of subsidies. (3) Table I presents some statistics on the amount spent by the Federal government on the largest programs.

TABLE I Some Federal Programs for Low-Income Individuals, 1982 ($ billions) Cash Transfers: Aid to Families with Dependent Children $8.00 Supplemental Security Income 7.7 In-Kind and Subsidies: Medicaid 17.4 Medicare 44.8 * Food Stamps 11.0 Child Nutrition 4.0 Housing Assistance 7.9 Employment and Training 4.5 Social Services 6.7 Compensatory Education 4.0 Legal Aid 0.3 * Low-Income Energy Assistance 1.7 * Figure is for 1981 Source: Rosen [1985, 88].

The reader might glean at least two striking facts from this table. The first is the magnitude of the redistribution represented here. When one adds to this list the transfers by other levels of government (for example, for education), it becomes clear that redistribution is a major function of American governments. (4) Second, a significant share of this redistribution is done using subsidies (including in-kind transfers).

The public finance literature offers a number of explanations for the popularity of subsidies. (5) Taxpayers, or their elected representatives, might simply be paternalistic; by this we mean that they evaluate the recipient's welfare using a utility function other than the recipient's. (6) Though undeniably interested in the welfare of those less fortunate, these taxpayers might not be willing to provide cash if some fraction will be applied toward the purchase of alcohol, tobacco or other luxury or disreputable commodities. It may also be that there exist certain goods that society feels should be distributed more equally than free market allocations would provide. Tobin [1970] has referred to this as "specific egalitarianism" (7)

As intuitively appealing as these explanations might seem, it is not clear why people should feel this way. (8) And, absent a theory explaining why taxpayers care about some commodities and not others, it is a difficult theory to test. (9)

It may also be that the consumption of certain commodities provides external benefits to others that have nothing to do with altruism. Subsidies to housing might eliminate the eyesore of slum dwellings. Food stamps, by improving nutrition, might reduce the demand of the poor for socially provided medical care. (10) They might also raise the incomes of some food producers (i.e. farmers), reducing their need for other forms of assistance. And the arguments regarding the positive externalities that may flow from education are very familiar.

Some recent research has focused on the inefficiencies possible when cash programs are made available to different types of recipients. Blackorby and Donaldson [1988] model the adverse selection problem that can result when it is difficult for a government to determine who is truly deserving. (11) In-kind subsidies may serve to help discourage illegitimate claims while still providing assistance to those who need it. Fallis [1986] has argued that the relative efficiency of cash grants is less clear when lump sum transfers are ruled out and recipients represent heterogeneous households facing different market wage rates and making labor supply decisions. Making the size of the cash grant depend upon the income of the recipient distorts the labor-leisure choice, creating its own deadweight loss. (12)

Let me propose another efficiency-based explanation for the choice of subsidies over cash grants, one that addresses the principal-agent problem that arises when consumption decisions for an entire household are made by the head of that household. Though taxpayers may be quite happy to let a single recipient spend cash as he sees fit, they may not like the way parents share resources with their children. Given the difficulty of giving cash directly to children, subsidies can help by distorting household consumption decisions in a way that helps the children. (13) As evidence, refer back to Table I and keep in mind that large sums of money are spent by state and local governments to subsidize primary, secondary and post-secondary education. Clearly, children are major beneficiaries of these transfers that come through subsidies.

This is paternalism of a different sort. Here, society imposes its own view of a proper household welfare function, though it accepts individuals' preferences. This theory has advantages over the more familiar version since it is more specific and offers more testable implications. Under the general theory of paternalism, any sort of subsidy program could be theoretically justified by claiming that "social preferences" were just of the form as to make that program optimal. In the theory offered here, there is less freedom to infer unlikely social preferences. Society's household welfare function respects the true individual utility functions of the household's members, as the head of the household does, but it allows for greater weight to be put on the welfare of those not making the consumption decisions. It can then be shown that subsidies of goods with specific properties may be more efficient at raising household welfare than cash grants.

III. THE MODEL

To keep things simple, consider a two-person household with one parent (the head) and one child. The parent is assumed to make all the household consumption decisions for both agents. Let x = ([x.sub.1],[x.sub.2], ..., [x.sub.n]) represent the consumption vector over n goods for the parent and z = ([z.sub.1],[z.sub.2], ... [z.sub.n]) that, over the same goods, for the child.

The parent will not necessarily be totally selfish. He will be assumed to maximize a weighted sum of his own utility, U(x), and the child's, V(z). I call this weighted sum W(x,z):

W(x,z) = U(x) + [gamma] V(z).

Society's welfare function for this household takes the same form, but may involve a different value for the relative weight [gamma]. Without any loss of generality, I assume that society chooses to weight the utilities equally, so the social welfare function for this household can be represented in the Benthamite fashion as

[W.sup.*](x,z) = U(x) + V(z).

Clearly much is arbitrary about this specification of household and social preferences. Nevertheless, it will serve as a useful and simple approach, allowing me to focus on the role of subsidies when society places a greater weight on the welfare of children than do the heads of households. (14) In fact, since I will need fairly explicit solutions to solve for household demand functions, further restrictions on the forms of U(x) and V(z) will be required to derive specific results.

The parent's problem in the absence of government intervention is to maximize W(x,z) subject to the household budget constraint Y = p(x+z), where Y is the household income and p the vector of commodity prices. Given a cash grant of G and a vector of commodity subsidies of s, this budget constraint becomes

Y + G = (p - s)(x + z).

The government's objective is to select G and s to maximize [W.sup.*] subject to two sets of constraints: (i) the first-order conditions from the parent's optimization problem, and (ii) a government budget constraint:

B = G + s(x + z).

The purpose of what follows in the next section is not to solve this problem fully, but, through the analysis of some interesting special cases, to demonstrate conditions under which the solution will involve the use of subsidies, possibly in conjunction with grants. Notice that in the absence of the child, or if [gamma] = 1, the familiar logic would instruct us to use only cash grants. Here we ask whether, in the presence of the principal-agent problem observed when [gamma] < 1, subsidies can work to redirect consumption decisions in the child's favor without causing an even greater loss to the parent.

Obviously, it is important to this analysis that society place a higher value on the welfare of children than parents do. Why this might be so is a question that deserves a better answer than is given here. It may simply represent a form of paternalism as described above. It could also be that society enjoys certain external benefits from raising disadvantaged children's welfare; for example, lower juvenile crime rates, or the development of better citizens as the children become adults. Finally, it may not be that society wants to give special assistance to children as much as it wants to keep aid programs from being too attractive to recipients who could take actions to avoid the need to draw on these programs. Presumably, there is no such moral hazard problem with children. (15)

IV. ANALYZING THE SPECIAL CASES

This section demonstrates that there are circumstances under which subsidies are more efficient on the margin than cash grants at raising [W.sup.*]. The examples also serve to illustrate some of the limitations of subsidies. (16)

CASE 1: Parent and Child have different tastes.

One must always be a little careful in defining a utility function for children. In V(z), do we have the utility function that rationalizes the choices that the child would make, given the freedom to make all her own selections? Or do we mean that V(z) rationalizes the choices she would have made for herself looking back as an adult; choices that do reflect her particular likes and dislikes, but that also recognize the long-term benefits of the consumption of certain goods, like education, and the costs of others (e.g. candy)? Here I interpret V(z) this second way, and in this first special case consider the benefits of subsidies when U(x) and V(z) differ, or more precisely, when marginal rates of substitution differ between parent and child.

Consider the effects of a first dollar of subsidy funds for the purchase of good j versus the effects of a first dollar of cash grant. From the parent's first-order condition we know that

(1) [U.sub.i](x) = [gamma][V.sub.i](z)= [lambda][p.sub.i]

where subscripts denote partial derivatives and [lambda] represents the multiplier on the household budget constraint. From society's viewpoint, the value of a dollar cash grant to this household will be:

d[W.sup.*]/dG = [summation][U.sub.i](partial derivative][x.sub.i]/ [partial derivative]G) + [summation][v.sub.i]/[partial derivative][z.sub.i] [partial derivative]G)

which, using (1), we can rewrite as

d[W.sup.*]dG = [lambda][[summation][p.sub.i]([partial derivative][x.sub.i]/ [partial derivative]G] + [summation]([p.sub.i]/[gamma]) ([partial derivative] [z.sub.i]/[partial derivative]G).

(Throughout the paper, the reader should understand that all summations are from i=1 to i=n.) The first dollar devoted to the subsidization of good j will bring a social benefit of, using (1) as before,

(2) (d[W.sup.*]/d[s.sub.j])(d[s.sub.j]/dS) = [lambda][[summation][p.sub.i] ([partial derivative][x.sub.i]/[partial derivative][s.sub.j]) + [summation]([p.sub.i]/[gamma]([partial derivative][z.sub.i]/ [partial derivative][s.sub.j])/[x.sub.j]+[z.sub.j]

where S is the total expenditure on the subsidy program (i.e., S = [s.sub.j]([x.sub.j]+[z.sub.j])). Substituting the Slutsky equation counterparts for the slopes of the demand curves in (2), and recognizing that (17)

(3) [summation][p.sub.i][([partial derivative][x.sub.i]/[partial derivative] [s.sub.j]).sup.c] = - [summation] [p.sub.i][([partial derivative][z.sub.i]/ [partial derivative][s.sub.j]).sup.c]

where the superscripted c signifies that the slope is that of a compensated demand curve, we find

(4) d[W.sup.*]/dS = [[lambda]/([x.sub.j] + [z.sub.j])] {[(1-[gamma])/[gamma]] x [summation][p.sub.i][([partial derivative][z.sub.i]/[partial derivative] [s.sub.j]).sup.c]} + d[W.sup.*/dG.

Clearly, if [gamma] = 1 the first dollar spent will be equally productive under the two programs as there is no principal-agent problem. If [gamma] < 1, then the difference depends upon the sign of [summation][p.sub.i][([partial derivative][z.sub.i]/ [partial derivative][s.sub.j].sup.c]. This sum represents the change in total expenditures devoted to the child caused by a compensated increase in the subsidy to good j. It can be positive or negative, as demonstrated by the following simple example.

Let there be two goods, 1 and 2, and assume individual utility functions that take the (Cobb-Douglas) form:

U(x) = [x.sup.a.sub.1][x.sup.b.sub.2] V(z) = [z.sup.c.sub.1][z.sup.d.sub.2].

Consider the effects of a grant of G and a per-unit subsidy toward the consumption of good 2 of [s.sub.2]. We normalize the price of good 1 to be unity. This makes the household budget constraint

(5) Y + G = ([x.sub.1] + [z.sub.1]) + ([p.sub.2] - [s.sub.2]([x.sub.2] + [z.sub.2]).

Straightforward calculations reveal that the parent maximizing W = U(x) + [gamma]V(z) subject to (5) will make the following selections:

[x.sub.1] = a(Y + G)/[a + b +[gamma](c + d)] [x.sub.2] = b(Y + G)/([p.sub.2] - [s.sub.2])[a + b + [gamma](c + d)] [z.sub.1] = [gamma]c(Y + G)/[a + b + [gamma](c + d)] [z.sub.2] = [gamma]d(Y + G)/([p.sub.2] - [s.sub.2])[a + b + [gamma](c + d)].

The effects of increases in G and [s.sub.2] on consumption are easily calculated as derivatives of these expressions.

Consider first the effect on social welfare, as measured by [W.sup.*], of an additional dollar of cash grant, in the absence of any subsidies. This will be given by

d[W.sup.*]/dG = [U.sub.1] * d[x.sub.1]/dG + [U.sub.2] * d[x.sub.2]/dG + [V.sub.1] * d[z.sub.1]/dG + [V.sub.2] * d[z.sub.2]/dG

which, in this case, yields,

(6) d[W.sup.*]/dG = (a + b + c + d)/(Y + G).

The introduction of a per-unit subsidy will, with these preferences, affect welfare only through the consumption of the second good:

d[W.sup.*]/d[s.sub.2] = [U.sub.2] * d[x.sub.2]/d[s.sub.2] + [V.sub.2] * d[z.sub.2]/d[s.sub.2].

Increasing this subsidy by a dollar per unit will raise the total cost of the subsidy program (S) by

dS/d[s.sub.2] = [s.sub.2](d[x.sub.2] /d[s.sub.2] + d[z.sub.2] /d[s.sub.2]) + ([x.sub.2] + [z.sub.2]) = ([x.sub.2]+[z.sub.2]) * [[p.sub.2]/([p.sub.2] - [s.sub.2])].

The welfare effect of an additional dollar spent on the subsidy program will be

(7) d[W.sup.*]/dS = (d[W.sup.*]/d[s.sub.2])(d[s.sub.2]/ dS) = [(b + d)/(b + [gamma]d)] {[a + b + [gamma](c + d)]/Y} [([p.sub.2] - [s.sub.2])/[p.sub.2]].

The question then becomes, under what conditions can the quantity in (7) exceed the quantity in (6)? The first thing to notice is that, if [gamma] = 1, the familiar result obtains; cash grants must be superior. Notice also that, other things equal, the relative efficiency of subsidies falls as the rate of the subsidy grows. This is due to the rising deadweight loss suffered as a result of the growing distortion in relative prices.

If we set the initial subsidy level at [s.sub.2] = 0, and assume that [gamma] < 1 however, we find that the quantity in (7) will be greater than that in (6) if

(b + d) / (b + [gamma]d) * [a + b + [gamma](c + d)] >(a + b + c + d),

which reduces to

d/c>b/a.

Thus, if the child and parent have different marginal rates of substitution (at identical quantities), it will be socially optimal to subsidize the good that the child values relatively more highly (on the margin). It will only be efficient to subsidize up to a certain level, however; as observed above, the rising deadweight loss will make cash grants more beneficial at some point.

To some extent, this result is possible because Cobb-Douglas preferences give us ordinary demand curves with cross-price elasticities all equal to zero. In general, though we expect that subsidizing the purchase of good 1 will encourage the parent to buy more of that good for his child, he might partially frustrate the program by buying her less of the other good (if the child has positive consumption of the other good). When the cross-elasticities are zero, as they are in the examples studied here, the child keeps more of the

benefit. Thus, we would expect that the higher the cross-elasticities (as long as they are positive) the less likely it is that the child will benefit significantly, and the less likely it is that subsidies will dominate cash grants. (18)

An obvious example of a good that produces more benefits for children than for their parents is education at the elementary and secondary level. Thus, we find a reason to subsidize this education rather than simply provide poorer families with the unrestricted funds sufficient to allow them to purchase it. (19) Any commodity consumed exclusively or largely by children would make an obvious candidate for such a subsidy. (20)

It should be clear that an alternative to subsidizing the consumption of certain goods that benefit children is to provide a general cash transfer, but to tax the consumption of goods that do not benefit children. Indeed, the optimal policy might include some combination of all three instruments: some cash grants, subsidies for particular commodities like education, and taxes for others, like liquor and tobacco.

CASE 2: Household public goods.

Again, suppose there are two goods. The first good we will call food, and it is purely private in the sense that units consumed by any individual are not available for any other individual. The second good, housing (denoted H), is a pure public good within the household. That is, units of housing provide benefits to all household members simultaneously. It might seem intuitive to many readers that subsidies to household public goods might help mitigate the principal-agent problem here, as the parent's ability to appropriate the benefits of the transfer scheme are somewhat restricted. In fact, under some conditions, this is indeed true and subsidies to household public goods can be efficient, even if the parent's and child's preferences are identical. These conditions may be less general than one imagines, however.

Before any transfers, the parent's problem is to maximize

W([x.sub.1],[z.sub.1],H) = U([x.sub.1],H) + [gamma]V([z.sub.1],H)

subject to the household budget constraint. The first-order conditions reveal that

(8) [U.sub.1] = [gamma][V.sub.1] = [lambda][p.sub.1]

and that

[U.sub.H] + [gamma][V.sub.H] = [lambda]h,

where h is the (before subsidy) price of a unit of housing and [lambda] is, again, the multiplier on the budget constraint.

The social value of a dollar cash grant will then be

d[W.sup.*]/dG = [U.sub.1([partial derivative][x.sub.1]/[partial derivative]G) + [V.sub.1]([partial derivative][z.sub.1]/[partial derivative]G) +([U.sub.H] + [V.sub.H])([partial derivative]H/[partial derivative]G).

Substituting from (8), this can be written

(9) d[W.sup.*]/dG = [lambda][[p.sub.1]([partial derivative][x.sub.1]/ [partial derivative]G) + ([P.sub.l]/[gamma])([partial derivative][z.sub.1]/ [partial derivative]G) + h([partial derivative]H/[partial derivative]G)] + (1 - [gamma])[V.sub.H]([partial derivative]H/[partial derivative]G).

Letting the subsidy on housing be represented by s, we can derive the effects on [W.sup.*] of a dollar expenditure through the subsidy program. Substituting from the first-order conditions and the Slutsky equations, and rearranging terms we find:

(10) [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]

A comparison of (9) and (10) applying (3) reveals that

d[W.sup.*]/dS > d[W.sup.*]/dG iff

(11) [lambda][p.sub.1][(1 - [gamma])/[gamma][[([partial derivative][z.sub.1]/ [partial derivative]s).sup.c] + (1-[gamma])[V.sub.H][([partial derivative]H/ [partial derivative]s).sup.c] >0.

If [gamma] = 1 both of these terms are zero and the first dollar spent is equally effective under the two programs. Also note that the second term in (11) is positive, meaning that a sufficient condition for (11) to hold, when [gamma] < 1, is that [([partial derivative][z.sub.1],/ [partial derivative]s) .sup.c] [greater than or equal to] 0. That is, if a compensated increase in the subsidy leads to an increase in the amount of the private good allocated to the child, the first dollar of the subsidy is more effective than the first dollar of a cash grant.

It might seem that this condition is unlikely to be met and, indeed, in the Cobb-Douglas case analyzed above, it is easily seen not to hold. Given the familiar result that with these preferences the ordinary demand for each good is independent of the prices of other goods, the substitution effect must be negative to offset the positive income effect. In fact, with only two goods and identical preferences, we would always expect the two goods to be substitutes under compensation. The interested reader can easily verify that (11) will not hold with these preferences.

There are preferences under which (11) will hold, however. Consider the following example in which identical preferences are assumed to avoid complicating these results with those of the previous case. Suppose the parent maximizes a function of the following form:

W = [(a[x.sub.1].sup.1/2] [(bH).sup.1/2] + [gamma][[(a[z.sub.1]).sup.1/2 + [(bH).sup.1/2]].

This yields demand curves of the form (setting [p.sub.1] = 1 for convenience)

(12) [x.sub.1] = Y/Q [z.sub.1] = [[gamma].sup.2]Y/Q H = (b/a)[[(1 + [gamma])/h].sup.2](Y/Q)

where Q [equivalent to] (1 + [[gamma].sup.2]) + (b/a)(1 + [gamma])2/h.

Straightforward calculations reveal that the social value of a single dollar of cash grant will be

d[W.sup.*]/dG = [(1/2Y)[(a[x.sub.1]).sup.1/2] + [(a[z.sub.l).sup.1/2] + 2[(bH).sup.1/2].

The value of the first dollar of subsidy to housing is

d[W.sup.*]/dS = - (1/2Y)[[(a[x.sub.1]).sup.1/2] + [(a[z.sub.1).sup.1/2] + 2[(bH).sup.1/2]] + 2[(bH).sup.1/2]/(hH).

Thus, d[W.sup.*]/dS) > d[W.sup.*]/dG if

(13) 2[(bH).sup.1/2]/hH >[[(a[x.sub.1]).sup.1/2] + [(a[z.sub.l]).sup.1/2] + 2[(bH).sup.1/2]/Y.

Inequality (13) has a simple interpretation. The left-hand side is the average social welfare generated per dollar spent on housing. The right-hand side is the overall average social welfare generated per dollar. Substituting from (12) and rearranging terms we find that inequality (13) will hold whenever ([gamma]-1).sup.2] > 0, or in other terms, whenever [gamma] [not equal to] 1, even if it is greater. Thus, it is possible for subsidies to household public goods to dominate cash grants on efficiency grounds, at least up to a point. Recall that the example employed identical preferences for parent and child. To the extent that the child's welfare function places a relatively higher value on the public good, the dominance of subsidies will be enhanced.

If we imagine the parent's optimization problem to be different from the maximization of IV, as I have it here, other possible benefits of subsidies to household public goods suggest themselves. A parent with little regard for the welfare of his child (i.e., [gamma] [approximately equal to] 0) will provide her with very little food, perhaps just enough to sustain her (and avoid trouble with the authorities). A subsidy to a household public good like housing, however, will generate spillover benefits to the child as the parent is induced to purchase more housing services. Notice, however, that to the extent that parents can recapture some of this spillover by reducing the child's consumption of other goods, this approach will be frustrated.

Although housing is a good example, and a commodity often subsidized, the list of potential household public goods is probably not a long one. (21) Somewhat related, however, will be goods and services like health care, which can produce important, within-household externalities.

CASE 3: Parent claims all marginal resources for himself

Finally, for the last case consider the following example of parent and child preferences over two purely private goods:

Parent: U(x) = aln[x.sub.1] + b[x.sub.2] Child: V(z) = aln[z.sub.1] + b[z.sub.2]

Notice that their preferences are identical and that they exhibit constant marginal utility of good 2. The parent maximizing W will in this case set [z.sub.2] = 0 if [gamma] < 1. Assuming enough resources that [x.sub.2] is not also equal to zero, the other quantities chosen will be (setting [P.sub.2] = 1 for convenience):

[x.sub.1] = a/(b[p.sub.1]) [z.sub.1] = [gamma]a/(b[p.sub.1]) [x.sub.2] = y - (1+[gamma])a/b.

With these preferences, and assuming that [gamma] < 1, any additional dollar given in the form of a cash grant will be spent exclusively on [x.sub.2]. This creates a social welfare gain of

d[W.sup.*]/dG = b.

If good 1 was subsidized instead, [x.sub.2] would not be affected, but the other quantities would change according to (assuming the initial subsidy rate was equal to zero):

d[x.sub.1]/d[s.sub.1] = [x.sub.1]/[p.sub.1] d[z.sub.1]/d[s.sub.1] = [z.sub.1]/[P.sub.1]

The social welfare effect of the first dollar spent on this program will then be

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII].

Thus, social welfare rises more when a subsidy is used in this case and [gamma] < 1. Whereas a cash grant would be entirely devoted to the purchase of good 2 for the parent, a subsidy for the consumption of good 1, together with the unitary elasticities of demand for the good, guarantee that all the additional funds are devoted to the purchase of good I for both individuals.

If it is felt that larger cash grants will be devoted to the purchase of goods for the parent only, perhaps luxury items (e.g. entertainment, alcohol and tobacco), it may be optimal to subsidize the purchase of non-luxury items such as food and shelter.

IV. SUMMARY AND CONCLUSIONS

When parents care less than donors about the welfare of their children, a principal-agent problem arises for donors wishing to raise the household's welfare. Though donors may be nonpaternalistic in the sense that they are quite prepared to accept each family member's own utility function as the true measure of his or her actual utility, they may not be pleased with the way the head of the household allocates resources among family members. In such a case I have demonstrated that there exist conditions under which in-kind subsidies will, to a point, dominate cash transfers as a means to raise household welfare. Simply put, the subsidies can create a distortion that counteracts the distortion created by the agency relationship; they encourage the purchase of goods with a higher social value (because they are important to the child) than private value.

This is by no means the last word on these questions. The extent to which subsidies can be more efficient than cash grants obviously depends critically on the nature of the household choice mechanism, about which we have said relatively little. (22) Parents may not maximize a weighted sum of household members' individual utilities, for example; rather, they may maximize their own utility subject to certain constraints on the utility levels or consumption levels of the other members. This change in the parent's problem could have implications for the choice between subsidies and cash grants; indeed, it might mean that neither is at all effective at raising the welfare of children. (23)

Further research is clearly needed to refine the theory behind the examples studied here, to reveal the precise conditions under which subsidies can dominate cash grants in the presence of this type of agency problem. Of particular interest would be the study of the role of heterogeneity of the recipient population in the design of the optimal program. (24)

Finally, the existence of this agency relationship will have implications for other government policies, and some of these would be worth exploring. As an example, Robinson [1988] has investigated the design of optimal taxation schemes in an economy with families. He derives extensions to the familiar Ramsey rules for optimal commodity taxation for a world in which planners care about household utility that is different from the utility of the household head.

The lesson in this line of research is that recognizing the importance of transactions costs not only raises our estimates of the costs of government intervention, it can also have important implications for the design of that intervention.

REFERENCES

Becker, G. "A Theory of Competition Among Pressure Groups for Political Influence." Quarterly Journal of Economics, August 1983, 371400.

--"Public Policies, Pressure Groups, and Dead Weight Costs." Journal of Public Economics 28, 1985, 32947.

Becker, G. and N. Tomes. "Child Endowments and the Quantity and Quality of Children." Journal of Political Economy, August 1976, S143-62.

Behrman, J., R. Pollak and P. Taubman. "Parental Preferences and the Provision for Progeny." Journal of Political Economy, February 1982, 52-73.

Blackorby, C. and D. Donaldson. "Cash Versus Kind, Self-Selection, and Efficient Transfers*" American Economic Review, September 1988, 691-700. Browning, E. Redistribution and the Welfare System. Washington D.C.: American Enterprise Institute, 1975a.

--. "The Externality Argument for In-Kind Transfers: Some Critical Remarks." Kyklos 28(3), 1975b, 52644.

--."A Theory of Paternalistic In-Kind Transfers." Economic Inquiry, October 1981, 579-97. Bruce, N. and M. Waldman. "Transfers in Kind: Why They Can Be Efficient and Non-Paternalistic", Working Paper No. 532, Department of Economics, University of California, Los Angeles, November 1988.

Dougan, William and James Snyder. "Interest-Group Politics Under Majority Rule." Mimeo, Clemson University, February 1990.

Fallis, G. "Optimal Transfer Payments: In Cash or In Kind Revisited." Working Paper No. 86-01, Department of Economics, York University, January 1986.

Friedman, M. Price Theory: A Provisional Text, Revised ed. Chicago: Aldine, 1962.

Garfinkel, I., "Is In-Kind Redistribution Efficient?" Quarterly Journal of Economics, May 1973, 320-30.

Hicks, J. R. Value and Capital. Oxford: Clarendon Press, 1939.

Joseph, M. "The Excess Burden of Indirect Taxation." Review of Economic Studies, June 1939, 226-31.

Nichols, A. and R. Zeckhauser. "Targeting Transfers Through Restrictions on Recipients." American Economic Review, May 1982, 372-77.

Pollak, R. "Tied Transfers and Paternalistic Preferences." American Economic Review, May 1988, 24044.

Roberts, R. "Financing Public Goods." Journal of Political Economy, April 1987, 420-37.

--. "The Interaction Between Public and Private Spending When Government is Responsive to the Preferences of Citizens." Mimeo, Washington University, St. Louis, September 1989.

Robinson, D. "Optimal Commodity Taxation With Families." Discussion Paper No. 02-88, Department of Economics, Laurentian University, June 1988.

Rosen, H. Public Finance. Homewood, Illinois: Richard D. Irwin, 1985.

Ross, S. "The Economic Theory of Agency: The Principal's Problem." American Economic Review, May 1973, 134-39.

Ross, T. "On the Relative Efficiency of Cash Transfers and Subsidies." Working Paper No. E-88-20, Domestic Studies Program, Hoover Institution, Stanford University, 1988.

Shavel, S. "Risk Sharing and Incentives in the Principal and Agent Relationship." Bell Journal of Economics, Spring 1979, 55-73.

Stigler, G. "The Theory of Economic Regulation." Bell Journal of Economics, Spring 1971, 3-21.

Thurow, L. "Cash Versus In-Kind Transfers." American Economic Review, May 1974, 190-95.

Tobin, J. "On Limiting the Domain of Inequality." Journal of Law and Economics, October 1970, 263-77.

United States Bureau of the Census. Statistical Abstract of the United States: 1984, 104th ed. Washington, D.C.: U.S. Government Printing Office, 1983.

--. Statistical Abstract of the United States: 1988, 108th ed. Washington, D.C.: U.S. Government Printing Office, 1987.

(1.) Early proofs are found in Joseph [1939] and Hicks [1939]. See also Friedman [1962, 56-67].

(2.) On the growth of importance of this kind of aid in the United States, see Browning [1975a].

(3.) The editor has correctly pointed out that what I refer to here generally as subsidies might be more properly termed "in-kind subsidies." I assume that subsidies that specify the payment of a certain amount of money to compensate the buyer for the purchase of a specific amount of a commodity are too difficult to monitor, and are therefore not used.

(4.) In 1982, state governments spent $52.8 billion and local governments $49.8 billion on public primary and secondary education. See U.S. Bureau of the Census [1987, 119]. Out of total expenditures in 1982 of about $740 billion, almost $305 billion (41 percent) involved transfer payments to domestic recipients. These transfers represent about 10 percent of Gross National Product in 1982. U.S. Bureau of the Census [1983, 317 and 448].

(5.) Rosen [1985, 80-95], contains a nice review of the traditional explanations.

(6.) Garfinkel [1973] presents a reasonably general treatment in which taxpayers have preferences over the form of the redistribution. Pollak [1988] has recently extended paternalism arguments to explain tied transfers from parents to children.

(7.) Tobin [1970] suggested that the following may be candidate commodities for specific egalitarianism: civil rights, food, shelter, education and medical care. He suggests that in some cases society would like the distribution to be completely equal (e.g., voting rights), while in others the concern is only that everyone have a certain minimum quantity (e.g., food, shelter and education).

(8.) For a discussion of this point, see Thurow [1974].

(9.) Browning [1975b] describes the difficulties inherent in using in-kind transfers to affect Pareto optimality in the presence of consumption externalities. In Browning [1981], he argues that actual in-kind transfers do not achieve a Pareto optimum; rather they cater more to the preferences of donors than recipients.

(10.) For a recent related discussion of the interaction between altruism and moral hazard, see Bruce and Waldman's [1988] analysis of tied transfers in the presence of the Samaritan's dilemma.

(11.) See also Nichols and Zeckhauser [1982]. The idea goes back at least to Stigler [1971] who argued in the context of regulation that subsidies to favored industries will be preferred to cash grants if entry is easy.

(12.) Still another explanation for subsidies comes from the recent work of Dougan and Snyder [1990]. They model a collective choice, pie-splitting game and show that price-distorting policies (such as subsidies) can command majority support when cash transfers cannot. Such policies can create the asymmetries across agents that are essential to the formation of stable voting groups.

(13.) Further evidence that politicians worry about such agency problems comes from the "Family Allowance" (the Canadian term) programs in Canada and the United Kingdom. Both programs provide cash grants to parents based upon the number of children in the family. In both countries the checks are made out to the mother, concern having been expressed about the willingness of the father to allocate the funds properly.

(14.) The reader may recognize that this is not really a traditional principal-agent model. On the standard approach see, for example, Ross [1973] and Shavel [1979]. Specifically, there is no uncertainty in this model. This means that if the government wanted, it could dictate to the parent the desired quantities, and perfectly observe and punish disobedience. I assume that this approach is either undesirable or infeasible. Governmental uncertainty about the exact tastes of parents and children could be added to the model.

(15.) Another way to view this problem is to interpret W(x,z) as the parent's own utility function, which then exhibits some degree of altruism. Society's welfare function would then be [W.sup.*] = W + V. Then, society places a greater weight on childrens' welfare because it values the direct benefit (V) as well as the indirect benefit (through W). I am grateful to David Robinson for suggesting this interpretation.

(16.) Many of the derivations in this section are straightforward but tedious, and so some steps have been omitted from the presentation. The interested reader is directed to Ross [1988] for more detail.

(17.) Some elaboration of this point might be helpful. Each of these compensated demand derivatives is actually the sum of two derivatives, one with respect to [p.sub.j] charged for [x.sub.j] and one with respect to [p.sub.j] charged for [z.sub.j]. For example:

[([partial derivative][x.sub.i]/[partial derivative][s.sub.j]).sup.c] = -[[([partial derivative][x.sub.i]/[partial derivative][p.sup.x.sub.j]).sup.c] + [([partial derivative][x.sub.i]/[partial derivative][p.sup.z.sub.j].sup.c].

Combining this with the well-known property, derivable from the Slutsky equations,

[summation] [p.sub.i][[([partial derivative][x.sub.i]/[partial derivative] [p.sup.x.sub.j].sup.c])c + [([partial derivative][x.sub.i]/ [partial derivative][p.sup.z.sub.j]).sup.c]]

we can then derive the equation from the text.

(18.) Note 23 below contains an example in which the substitution is complete, so the child is no better off with the subsidies. In this case, cash grants will again be more efficient at raising household welfare.

(19.) In many political jurisdictions, the education of children is made compulsory as well. To the extent that governments can determine and compel the optimal levels of consumption of selected goods they avoid the agency problem modelled here. It is not difficult to believe, however, that if primary and secondary education were not so heavily subsidized, securing compliance with compulsory education laws would be more problematic.

(20.) In the province of Ontario, for example, children's clothing is exempt from provincial sales tax.

(21.) Home heating and cooling subsidies are another example. I am grateful to Frances Ruane for this suggestion.

(22.) Notice that this question is related to, but distinct from, the question of how parents allocate resources between children. On this question see, for example, Becker and Tomes [1976], and Behrman, Pollak and Taubman [1982]. Here our concern is with the allocation between parent and child.

(23.) If the parent maximized his own utility subject to a constraint that the child's utility not fall below a certain level, neither subsidies nor cash grants can raise the welfare of the child. Even subsidies to household public goods will be frustrated as the parent will only reduce expenditures for other goods given to the child. In this case, cash grants are the most efficient way to raise the utility of the parent.

(24.) Introducing heterogeneity will admit adverse selection problems, as in Blackorby and Donaldson [1988]. Heterogeneity of donors as well as recipients has proven to be an important element of Roberts" [1987; 1989] work on public versus private provision of public goods.

THOMAS W. ROSS, Department of Economics, Carleton University. The first draft of this paper was written while the author was a National Fellow at the Hoover Institution. The author gratefully acknowledges a number of invaluable discussions with Russell Cooper, the useful suggestions of Keith Acheson, Thomas Borcherding, Jonathan Kesselman, John McManus, Robert Porter, David Robinson, Frances Ruane and an anonymous referee, and the research assistance of Lynn Ross. He is also pleased to acknowledge the financial assistance provided by the Carleton Industrial Organization Research Unit and the Hoover Institution.

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Author: | Ross, Thomas W. |
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Publication: | Economic Inquiry |

Date: | Jul 1, 1991 |

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