Boadway e Marchand 1995

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The Use of Public Expenditures for Redistributive PurposesAuthor(s): Robin Boadway and Maurice MarchandReviewed work(s):Source: Oxford Economic Papers, New Series, Vol. 47, No. 1 (Jan., 1995), pp. 45-59Published by: Oxford University PressStable URL: http://www.jstor.org/stable/2663663 .

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46 PUBLIC EXPENDITURES FOR REDISTRIBUTIVE PURPOSES

being provided could vary in quality. The public sector provided a uniform

quality of the good to all, but lower than the quality that the higher income

persons would have chosen. There would be some income level beyond which

persons would choose to substitute a higher quality for that provided by the

government and forgo the freely-available public provision entirely. Besley and

Coate showed that uniform provision by the public sector financed by

proportional taxation could improve social welfare. In both Usher and

Besley-Coate, the use of the tax system for redistribution is assumed away.

Given that individual incomes are exogenous in their models and the redistribu-

tion via expenditures generates some inefficiency, redistributive taxation would

clearly dominate expenditures as devices for redistribution.

The second strand of the literature was initiated by Arrow (1971) who

investigated optimal public expenditure policy when expenditures could be

targeted to particular individuals.Using his notation, households were assumedto be distributed by a characteristic x and to obtain utility U(x, y) where y is

government expenditure. (Utility of income was abstracted from entirely.) The

government could observe x and provide a different amount of expenditure y

to each household. Arrow studied whether optimal expenditure policy under a

utilitarian social welfare function should be progressive or not, and he applied

his analysis to the cases of education and health. However, given that household

characteristics are observable, tax policy would dominate expenditure policy

as a redistributive device. That is, if the government could use the information

at its disposal to make income transfers among households, it should do soand allow households to purchase whatever quantities of y they desire. Thus,

the use of expenditures for redistribution purposes is unnecessary.

Our purpose is to investigate whether the use of expenditure policies can be

justified on redistributive grounds. We do so in the context of a model in which

public expenditures have similar properties to those in Arrow, but in the spirit

of the optimal income tax literature, the government does not have full

information about individual characteristics. We allow the government to

implement the optimal income tax in the sense of Mirrlees (1971), and ask

whether social welfare could be improved by instituting universal provision ofsome service by the government. Given our assumption of non-observability

of individual characteristics, government expenditures must be provided in the

same amount for all. However, households are allowed to supplement public

expenditures with their own private provision. We first provide a general

analysis of the problem, and then apply it to two prototypical types of

government expenditures both of which are quasi-private goods. The first,

which corresponds with education, allows the good to interact with ability and

affect the wage rate received by the household. The second is simply the public

provision of a quasi-private good which enters the utility function directly and

cannot be resold, namely public pensions (future consumption).

In the optimal income tax problem underlying our analysis, we consider two

household types, following Stiglitz (1982) and Stern (1982). The restriction to

two ability types allows the principles to be analysed most clearly. In these



R. BOADWAY AND M. MARCHAND 47

models, the self-selection constraint restricts the amount of redistribution that

can be achieved. Government expenditures will be social-welfare-improving to

the extent that they serve to relax the self-selection constraint. Our analysis will

be aimed at determining when that is possible.'

The interaction of optimal income taxation and education decisions has been

analysed in models similar to ours by Hare and Ulph (1979) and Tuomala

(1990). As Arrow did, Hare and Ulph assume that ability to benefit from

education can be observed by the government for the purposes of allocating

education expenditures among households. However, this same information

cannot be used for determining taxes. Their purpose is to examine whether

introducing optimal income taxation in Arrow's model makes public provision

of education less regressive. Tuomala focusses on the other side of the coin,

that is, how education choices affect the progressivity of the optimal income

tax. As in our analysis, one of his models considers the optimal choice of auniform provision of public education, but he does not look for circumstances

which make public provision desirable in the first place.

2. The self-selection model of public expenditures and optimal income taxation

The analysis will be conducted using the standard optimal income tax

assumption of Mirrlees (1971), and it will concentrate on the case of a two-class

economy as in Stiglitz (1982). In this standard model, there are two types of

individuals (i = 1, 2), who differ only in their labour productivitywi.

This is

the number of efficiency units of labour they supply per unit of working time.

By convention, w2 > wl. Both types of individuals share the same utility

function U(ci, 1i)where ci is the consumption of a private good and it is labour

supply. The technology is linear and transforms one efficiency unit of labour

into one unit of the private good. The private good is used as the numeraire.

The government cannot observe either the wage rate of a household (wi) or his

labour supply (li). It can only observe gross income which we denote by

yi = wili. From the latter, it can infer after-tax income, denoted xi, from the tax

schedule. In the standard model, that income is only used for consumption ci(ci = xi). As we shall see soon, this will not be true in our model where in

addition to ci, other private (or quasi-private) expendituresare accounted for.

Following Stiglitz (1982), it is useful to rewrite the household's utility function

in terms of the variables that the government can actually observe or infer. This

is done by substituting for 1ifrom yj = wili. Thus, utility is given by

V~i M U Cio i!

' The spirit of the analysis is similar to that of Blackorby and Donaldson (1988) who investigatedpublic provision in a slightly different context. They considered the case for in-kind provision ofpublic services in an example of a two-person economy in which the persons had differing tastes(needs) for the good provided. Public provision screened those who needed the good most andthereby provided a form of redistribution which could not be achived by cash transfers. Moregenerally, Guesnerie and Roberts (1984) have argued the case for quantitative restrictions(rationing) as policy devices in a second-best world of imperfect information with optimal taxes.




Vi yields the indirect utility function

v'(xi, yi, g) _ (Xi-Zi(Xii, yi, g), yi, Zi(Xi,yi, g) + 9) (3)

Note that the government can observe or decide upon all variables in (3). Some

properties of (3) will be useful in what follows. By the envelope theorem, we have

VI= VI vI = VI and v'= Vy

Also, it straightforward to show that as g increases, the demand for private

purchases zi declines if ci is a normal good. Eventually private purchases are

completely crowded out. Denote by gi the level of g at which private purchase

zi is just crowded out.

The above problem applies for both types of households. We will presume

that the government wishes to transfer income from the high-ability to the

low-ability persons. As is well-known (see Stiglitz, 1982), such redistribution is

limited by a self-selection constraint which precludes the high-ability persons

from mimicking the low-ability persons, and such a constraint will typically be

binding. We need to characterize the behaviour of the mimicking high-ability

persons. We denote the mimicking person by a bar so -2 = Yi and -2 =

The choice of private purchasesZ2by the mimickeris determined by the solution

to the following problem

max V2(x- Z2, Y1,Z2 + 9) (4)Z2

The Kuhn-Tucker conditions are

-V_2 + VF2 SO and Z2(-V ?V42 -2 0 (5)

The solution to this problem gives rise to the indirect utility function for the

mimicker v2(x1, y1, g). It has analogous properties to the above indirect utility

functions for person i (i = 1, 2). Again, we define 92 as the level of public

provision at which the mimicker's Z2would just become crowded out.

The government is assumed to use an optimal non-linear income tax for

redistributive purposes as well as to finance whatever public provision of g is

chosen. For purposes of analysing government behaviour, we proceed in twosteps. In the first step, the government chooses an income tax schedule

optimally, given the level of government expenditure per household 9. The

second step involves evaluating changes in g with taxes always set optimally.We assume for concreteness that the government uses a utilitariansocial welfare

function, though any quasi-concave social welfare function would do as well.

Indeed, the problem could be set up as a Pareto-optimizing problem as in

Stiglitz (1982), thus emphasizing that what is at stake is the design of an efficient

systemforredistribution.The issue is whether the use of governmentexpenditurescan improve the efficiency of the redistributive mechanisms even when fullynon-linear income taxes are in place.

The determination of the optimal non-linear income tax system for a givenlevel of g follows directly from the analysis of Stiglitz (1982). The governmentis modelled as if it could directly control the level of gross and net income (yi




and xi), subject to a budget constraint and the self-selection constraint,

v'(X2, Y2, 9) ) P2(x1,Yi, g).2 Assuming that the only revenue requirement is to

finance g and denoting by ni the number of persons of type i (i = 1, 2), the

government's problem is

max n1vl(xl, Yl,g) + n2v2(x2,Y2, 9)Xi, Yi

subject to

v2(x2, Y2,g) V2(x1,y1, g)

il(yl- x1) + n2(y2 -x2)- (n1 + n2)g = 0

The Lagrangian expression of this problem may be written

Q(xi, yi,,M t; g) = n1vl(xl, y1, g) + n2v2(x2,Y2, 9)

+ ,u[v2(x2, Y2, 9)-

v2(x, Yi g)]

+ y[n1(yl - x1) + n2(y2 -x2)- (n1 + n2)9] (6)

The first-order conditions are

1 -2nlvx Lvx ynl = 0

n1vY - 1iVY+ yn1 = 0

(n2 + t)V2 -yn2 = 0

(n2+

,i)V2+ yn2 = 0

These conditions and their interpretation are identical to those of Stiglitz (1982)

and we need spend no time discussing them here. Note only that they imply

that the marginal tax rate on the high-ability person is zero, while that on the

low-ability person is positive.3

Our real interest is in the effect of increasing g given that taxes are always

adjusted to be at their optimum. By the envelope theorem, the effect of

increasing g on social welfare can be obtained by differentiating the Lagrangian

expression (6) partially with respect to g to give

dQ = n1v + (n2 + _v - y(nl + n2)dg

Using the first-order conditions above on x1 and x2, this may be rewitten

d = nl(vl - v1) + (n2 + _)(v - v)-,u(P -v P) (7)dg

2

2 Stiglitz analyses the efficiency of optimal income taxes and allows the self-selection constraint

to apply to either individual depending on the circumstances. Since we are considering the case inwhich the planner wants to redistribute from the high to the low income person, applying theself-selection constraint to the high income person is appropriate. The low income person wouldnever choose to mimic the high. This is what Stiglitz refers to as the 'normal' case.

3 These first-order conditions are only necessary conditions. We also require that the single-crossing property holds. We return to a consideration of that property as well as the form of themonotonicity property in looking at the specific applications below.




From (7) we can readily inferthe circumstances in which increases in g will be

welfare-improving. Note that from the first-orderconditions (2) and (5) for the

households that if z1, z2, Z2 > 0, dQ/dg = 0. In this case, the change in g is

inframarginal to everyone and is equivalent to an equal lump-sum transfer.

Thus, the effect of g can be replicated by the tax system so g is redundant.

However, as g increases, private provision Zifalls, and eventually it will fall to

zero. Private purchases of different persons will become crowded out at

different levels of g and, depending on the order of crowding out, the welfare

consequences will differ.

To see this, note that persons 1 and 2 are crowded out when the non-

negativity constraint on zi binds; the first two terms in (7) are then negative by

(2). When person 2 gets crowded out while mimicking person 1 (z2 = 0), the

last term in (7) becomes positive rather than zero by (5). Recall that g1, g2,

and -2 are the levels of g at which the private purchases of persons 1, 2, andthe mimickerjust become crowded out by public provision. Then the following

result is apparent:

Proposition 1 If and only if Y-2< min(gl,

g2), public spending beyond Y2will

be welfare-improving up to some amount strictly above g = min (1, 92).

The intuition behind this result is as follows. Increasing g beyond the point at

which person 1 or person 2 gets crowded out simply makes those persons worse

off by constraining their choice, and it thus reduces social welfare. However,

pushing g beyond the point at which Z2 is crowded out makes the mimicking

person worse off and therefore relaxes the self-selection constraint. If the latter

occurs before persons 1 and 2 are crowded out, a gain in social welfare can be

achieved.

To determine when uniform public provision g would be welfare-improving,

we need to investigate conditions under which -2 < min (g1, 92). This depends

upon the nature of zi and the way it enters into the utilityfunction.As mentioned

earlier, in what follows we consider two cases. In the first, what we refer to aseducation, zi, affects the wage rate faced by person i. In the second, called the

pension case, zi simply enters the utility function as an ordinary consumption

good.

3. The case of education

Here, we make the wage rate for a person of given ability endogenous and

dependent on education expenditures, with the high-ability person getting a

higher wage from given education expenditures than a low-ability person. So

7i stands here for education expenditures made privately by household i, and

g for education expenditures provided uniformly by the public sector, where

g >?0 and zi > 0. Thus we allow households to supplement public with privateprovision. Denote total education by ei=zi + g. For example, public provision

could be associated with mandatory schooling and private provision with




further education. Since the government is unable to observe ability, public

expenditures cannot be conditional on ability. Also, we assume that they cannot

be conditional on incomes either. This is because the planner has to commit

to such expenditures before knowing the incomes that will be produced. If the

government could observe abilities, it would be possible to reach any point onthe utility frontier through appropriate lump-sum transfers. Private education

expenditures would be first best, and there would be no reason for the

government to provide public education. In the real world there might be

imperfect indicators of ability such as test scores that we do not take into

account. We assume there is a wage function for each person given by wi(ei)

where w' > 0. A different wage function applies for persons of different ability

such that w2(e) > w1(e).Other relationships between the wage functions of high

and low ability persons are critical to the case for using public expenditures

for redistributive purposes, so we postpone consideration of them untildiscussing our results.

Households obtain utility from the private good ci and disutility from labour

li. In this case, the utility function of the household Vi( ) is defined from its

primitive U(ci, 1i)as follows

V(xi- zi, yi, zi + g)UXi - zi, W(z g) (8)

The first-order conditions to the household's problem (1) may be written

- U - Ul I < ? zi-W- Yi ()2) = 0 (9)

This yields the demand for private education zi(xi, yi, g) and the indirect utility

function vt(xi, yi, g). The envelope theorem gives the derivatives of this indirect

utility function to be v' = V'= U' and vt = Vy= U'/wi. As well, van- Uyiwi/(wi)2. Analogous expressions hold for the mimicker.

The conventional optimal income tax results obtained from problem (6)

apply here as well. However, we need a monotonicity property to ensure that

the single-crossing propertyholds in (y, x) space. Denote by Si(y, x) =-(vl/v')the slope of person i's indifferencecurve for g < min (g1, g2). As in the standard

model, we assume that S1(y, x) > S2(y, x) at any point (y, x).4 From the

envelope properties given above and first-order conditions (9), we infer

Si(y, X) = _ _= wi(ei)

wiU' w'(ei)

'In this model with education as a choice variable, the montonicity condition could applyeverywhere in the other direction, so S1(y, x) < S2(y, x) for all (y, x), and the standard optimalincome tax results would go through. However, this would imply that at the optimal income tax

equilibrium,Y2 < Y, and x2 < x1. Since this seems unlikely, we restrictour analysis to the standardform of the monotonicity assumption, S1(y, x) > S2(y, x).




where the wage function is evaluated at ei = zi + g, with zi being individual i's

optimal choice at (y, x). Therefore,the standard monotonicity property requires

wj(e>) W2(e2)

w' (e) W (e)

where e1 and e2 reflect the choice of z1 and Z2at the common point (y, x). Define

the elasticity of the wage function as si(ei) = w'(ej)ei/wi(ei). Then the standard

monotonicity property requires

c2(e2)> s1(e) (10)

e2 el

where e1 and e2 depend upon (y, x). In what follows, we assume that thisproperty is satisfied at all points in (y, x) space.

Given the standardmonotonicity property,the optimal income tax equilibrium

(given the value of g < min (g1, g2)) is depicted in Fig. 1 by the solid lines. The

indifference curves reached at this equilibrium by the two individuals are there

drawn. This diagram is similar to that in Stiglitz (1982) except that the vertical

axis measures xi + g (= ci + ei) which is not identical to consumption. At all

points in the diagram individuals are choosing zi optimally. A useful propertyin what follows is that if households are unconstrained in their choice of

zi,

xi ? g2

+ g)~~~~~~~~~y, 2+g/ ~ I

/~ /

///

//

)450/ . ~~~~~~~~~~~~~

Yi

FIG. 1.




then zi will increase as one moves up an indifference curve if consumption c

and leisure are normal goods.5

Proposition 1 above applies; that is, public provision of education will be

welfare-improving if and only if the mimicker gets crowded out before either

household I or household 2. We can readily find sufficient conditions for thisto be the case. First, if e1(el) > E2(e2) at point (x1, Yl) in Fig. 1, then e1 > e2

in orderto satisfy the standard monotonicity condition (10). From this it follows

that -2 <ai. Furthermore, since Z2 is increasing as one moves up household

2's indifference curve in (x, y) space when c and leisure are normal, z2 > Z2

when both are unconstrained. This implies that g-2<g2. Therefore,we can state

the following result:6

Proposition 2 Public spending on education up to some amount strictly above

g = min(g1, g2) will be welfare-improving if:(i) the standard monotonicity property holds,

(ii) c1(e1)> c,(g2) at (yi, x1) when g =92,

(iii) private good c and leisure are normal goods.

This result can be given a graphical interpretation using Fig. 1. As before,

(yl, x1 + g) and (Y2, x2 + g) represent the optimal allocations when g is notbinding for any household (including the mimicker). Suppose now we set g

such that g-2<g < min (g1, g2). This implies that the mimicker will require

more net income x2 to compensate for being forced to consume too much

education. Recall that if consumption and leisure are normal goods, zi falls as

one moves down an indifferencecurve. This implies that the shape of household

2's indifference curve for the given value of g will be as shown by the broken

line. There will be a point a between (yl, x1 + g) and (Y2' x2 + g) at which the

' The proof of this is as follows. Without loss of generality, suppose g = 0. Also, delete subscripti to simplify notation. From the solution to first-order conditions (9) we obtain z(.x,y). Totallydifferentiating(9) yields

dz yw' dx I

dy W dy W IV3)

If we restrict x and y to be along the same indifferencecurve, differentiating (8) yields

dx it

Substituting this in the expression for dZ/dv and using (9) yields

dz / U it, U I

-=(u11-U + ( Cu

+-U)?y + U yw' U, )

From the standard static analysis of the consumer, the first term is positive if c is a normal good,and the second term is positive if leisure is normal.

6 While we have assumed the standard monotonicity property, it is clearly not required.We haveshown in a backgrounddiscussion paper(Boadway and Marchand, 1990)that a sufficient conditionfor public spending to be welfare-improving is "?1(g2)> E2(92) with c and leisure being normal,whether or not the monotonicity property holds. The use of the standard monotonicity propertyis reasonable and makes the demonstration of sufficiency much easier.




household would be just crowded out. As one moves southwest along 2's

indifference curve from point a, the household becomes more and more

constrained so the new indifference curve deviates more and more from the

unconstrained one. The incentive constraint no longer binds, and therefore

social welfare can be improved; indeed, a Pareto improving allocation can be

achieved.

Proposition 2 states that, under the stated hypotheses, social welfare can be

improved by providing public education until at least one person (either 1 or

2) gets crowded out. Unfortunately, it is not possible to say which person will

be crowded out first since, given these sufficiency conditions, it cannot be

assessed whether z1 ] Z2 when households are unconstrained.7 Once either 1

or 2 gets crowded out, further increases in g will have conflicting effects.

Increases in g will continue to weaken the self-selection constraint thereby

improving welfare (the third term in(7)), but will worsen welfare

as 1 or 2 is

forced to consume more education than is desired (the first two terms of (7)).

The optimal level of public education is that which equates these two effects at

the margin.8

4. The case of pensions

In the case of education, public spending entered indirectly into the utility

function through the wage function. In this section we consider the case of the

public spending being on a private good which enters directly into the utility

7To see this, suppose for some value of g, household 1 is choosing z1 optimally. Then evaluatethe change in household 2's utility at z1 from a change in z2. Using (9), we obtain

dU2 if -Ui2 2'(e1)

dz2 vW2(el)

Since (9) applies with equality for household 1, this requires:

-U Y2 t'(el) -Ul Yi wv(el)

U,,V2W22(el)

Uliv, wl(el

At the optimal tax equilibrium

Y2> Y1 and - =1>U2 W2 U W

Therefore,if

v'v(el) w2(e1)

ivl(el) "72 (e1)

(which is consistent with the sufficiency condition, but not identical) we cannot be sure whetherthis inequality applies.

8 In Boadway and Marchand (1990), we also analyzed the case of an ad valorem subsidy oneducation as an alternative policy instrument. The same conditions which are sufficient for g to be

welfare-improving are also sufficient for a subsidy on zi to be welfare-improving. Furthermore,public provision and the subsidy are substitute instruments in the sense that if either one were setoptimally, the optimal value of the other would be zero. Different local optima are obtained withthe two instruments, Whether the local optimum with a positive subsidy (and g = 0) is superiorto that with g > 0 (and no subsidy) depends upon a global comparison between the two allocations.




function. Given that the public sector will be providing what is essentially a

private good to the households, it is critical that the good in question cannot

be resold. Otherwise, it would be equivalent to providing a lump-sum transfer

to all households.9 This case, which is similar to that analysed by Usher (1977)

and Besley and Coate (1991), is somewhat simpler and also more familiar since

it is related to the problem of optimal taxation in a multi-commodity world.

Persons differonly by an ability parameter normalized to equal the wage rate.

We use the example of public pensions for this case, where the good provided

can be thought of as future consumption. The pension is assumed to be fully

funded so as to concentrate on intra-generational redistribution and thereby

avoid the dynamic complications that arise from intergenerationalredistribution.

Other interpretations of the publicly-provided good are possible however. For

example, the case of health care could be analysed as a private consumption

good. In this case we might want to modify the analysis to allow persons tovary not only by an ability parameterbut also by a second characteristic,health

status. While ability is not observable,health status is at least to some extent;so

differing amounts of health care can be provided to persons.of different health

status, independently of ability (wage rate). The analysis for this case is a

straightforward extension of the pension case, except that a different amount

of public spending is provided to each health class. Similar conditions for public

provision to each class to be welfare-improving will apply in this case. Also,the optimal income tax rates will vary by health status.'

The household is thus assumed to consume two goods, c and z, and to supplylabour. Good c can be thought of as present consumption and z as future

consumption. The government may also supply an amount g of future

consumption uniformly to all persons and finance it by an optimal income tax.

The utility function of household i is U(ci, lj, zi + g). Therefore, the derived

utility function Vi( ) is defined as follows

V(xi- Zi, yi, Zi+ g) U -z,~ , Zi+ g) (11)

Wi

As in the standard case a la Mirrlees, wi is exogenous and w2 > wl. The

first-order conditions to the household's problem (1) may be written

-U' + U' <0, zj(-U,+ U') = 0 (12)

This yields the demand for private pensions zi(xi, yi, q) and the indirect utility

function v'(xi, yi, g). The envelope theorem gives the derivatives of this indirect

utility function to be vi = VI = U, vy= V, = U'/wi, and van V' = U1.

Analogous expressions hold for the mimicker.

9 Blackorby and Donaldson (1988) have avoided this problem by assuming that the good is onlydemanded by one of two types of persons in the economy.

10The case for an optimal linear income tax whose parameters vary with health status was madeby Blomqvist and Horn (1984).




As for the education case, we assume that the standard monotonity condition

applies so that S1(yx) > S2(y,x) for any point in (y, x) space where the

non-negativity conditions on z does not bind. Using the envelope theorem, this

requires

U1 U12

U 'w1 U 2

Since w2 > w1, a sufficientcondition for this is that - -U 1 > U /U2 . This

condition will be satisfied if total consumption (c + z) is non-inferior (i.e. if

c + z does not fall as exogenous income rises).

Once again, Proposition 1 applies. Public pensions will be welfare-improving

if the mimicker gets crowded out before either household 1 or household 2

(2 < min (g1, 2)).The question then becomes whether this will occur. Consider

q1 relative to 92 first. At g = g2, z2 = 0 and, from (12), U2/U' = 1. Then, g1will be greater than -2 if and only if at g = -2, increasing z1 from z1 = 0is welfare improving for person 1. This will be so if v'/az1z -U 1 + U 1 > 0

at z1 = 0 and g = g2. Thus, we require

U (C , 11,Y2)> 1 U c,1 2

Ucl(Cl,11,92) U2c, 122)

Since 11> 12, this will be satisfied if and only if U 1/U 1 is increasing in labour,

thatis,

ifand only

if zis less complementary with leisure than c is."Next consider g2 relative to g2. The effect on household 2's utility from

increasing z2 will be 8v2/8z2 =- 2 + U2. This will be positive at z2 = 0 and

g = Y2 if

U(C2, 12, Y2)> 1 (c 1,

U (C2, 12, 92) U(1 2 2

Since 12 > 12 and c2 > c1 for self-selection to operate under the standard

monotonocity condition, this would also be satisfied if z is relatively less

complementary with leisure than c is. It is because then both the higher valuesof 12 and c2 (given g) will tend to increase the value of the left-hand side.

These results are summarized in the following proposition, which assumes the

standard monotonicity property:

Proposition 3 A necessary and sufficient condition for public provision up to

at least g = min (1, 92) to be welfare-improvingis that z be less complementary

with leasure than c is.

This result may be compared with the finding of Atkinson and Stiglitz (1976)that the absence of separability between goods and leisure is enough to justify

differential commodity taxation alongside an optimal income tax. In our

" Of course, it should not be so much less complementary so as to the violate the standard

monotonicity property.




context, absence of separability is also required in order to justify public

provision. In optimal tax theory, one typically finds the higher tax being

imposed on the commodity which is more complementary with leisure (see

Corlett and Hague 1953; and Harberger 1964). That is, the tax system favours

the other commodity. This is consistent with our finding that public provisionof the commodity which is less complementary with leisure can be welfare-

improving.1

5. Conclusion

Governments apparently accomplish a good deal of theirredistributiveobjectives

through the expenditure side of the budget using such instruments as public

education, public health provision, and public pensions. We have investigated

whether a theoretical case can be made for using expenditures for redistributivepurposes. This has been done in a model in which the government is able to

pursue redistribution fully through an optimal non-linear income tax, and in

which the only role for expendituresis redistributive.That is, public expenditures

are on items which are otherwise purely private and could be allocated by

markets. Doing so abstracts from the fact that there may be significant

externalities associated with their use in practice which would justify public

provision on market failure grounds. By adopting these assumptions, we are

forced to make the strongest case for using expenditures for redistributive

purposes. We have considered two types of expenditures one which affects the

wage rate and we have identified with education, and one which is like an

ordinary private good and which we have called future consumption (public

pensions). The latter case could represent health care with some minor

amendments. In each case we assumed that the public provision was of a good

(or service) which could not be re-traded among households.

For both these instances of potential public provision, a similar set of results

is obtained. When an optimal income tax is in place, uniform public provision

which crowds out the private provision of at least one household is welfare-improving if the following condition holds: the level of public provision which

crowds out the private provision of the high income person when mimicking

the income of the low income person is less than the level of public provision

which crowds out either person individually. In the case of education, this would

be the case if the elasticity of the wage function with respect to education

expenditures is higher for the low wage person. For public pensions and health,

leisure and the good in question must be substitutes.

Our analysis has been restricted to a two-person economy and to analysing

12 We have also shown in our background discussion paper, Boadway and Marchand (1990), thata policy of subsidizing commodity z would be welfare-improving under the conditions ofProposition 3. As with the education case, the subsidy and public provision are alternativeinstruments, each giving a separate local social optimum. In the optimum with the subsidy, public

provision is zero; in that with public provision, the optimal value of the subsidy is zero. The

comparison between the two optima involves a global comparison.




individual types of expenditure. It would be useful to extend the analysis to a

more complicated setting. For example, the methodology of Guesnerie and

Seade (1982) could be used to consider a multi-person economy. A priori, it

seems apparent that similar arguments could be extended to this case. If the

single-crossing property holds throughout, Proposition 1 would apply to least

able individuals and also locally to mimicking most able ones.

ACKNOWLEDGEMENTS

The authors would like to thank Claude d'Aspremont, Victoria Barham, Timothy Besley, Nicolas

Marceau, Pierre Pestieau, and two anonymous refereesfor helpful comments on an earlier version.

Financial support from SSHRCC, CIM, and CORE is gratefully acknowledged.

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Boadway e Marchand 1995

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