Office of Tax Analysis

U.S. Treasury Department

Washington, D.C. 20220 

Issued: January, 1977 

Applications of Optimal Tax Theory

To Problems in Taxing Families and Individuals 

Harvey S. Rosen 

Princeton University 

OTA Paper 21 November, 1976 
TABLE OF CONTENTS 

Page 

I. Introduction.................................. 1 

I1. The Optimal Tax Literature.................... 3 

A. Optimal Commodity Taxation................ 3 

B . Optimal Income Taxation................... 5 

C. Two Problems with the Optimal Tax 

Literature................................ 6 

111. Working Wives ................................. 9 

A . Is There Discrimination Against Working

Wives?.................................... 9 

B . Efficient Taxation of the Family.......... 13 

IV. Market Versus Home Production................. 15 

A . Effects on Labor Force Participation...... 15 

B . 'Neutrality' Between Home and Market 

Production................................ 17 

C. Imputing the Value of Time Spent in 

Household Production...................... 18 

V . Singles Versus Marrieds ....................... 21 

A . Marriage Neutrality ....................... 22 

B . Equity Aspects of the Marriage Tax........ 23 

VI . Treatment of Children......................... 27 

A . Children and Ability to Pay ............... 27 

B . The Optimal Child Exemption............... 30 

VI1 . Mandatory Individual Filing................... 32 

A . Labor Supply and Efficiency ............... 32 

B . Distribution of Welfare................... 34 

VI11 . Concluding Comments ........................... 36 
-1-

I. INTRODUCTION 

The appropriate tax treatment of families has been a 

-1/

matter of debate for decades. For an economist interested 

in rationalizing the tax treatment of the family, it is 

natural to consult the recent literature on optimal taxation 

-2/

(OT). With a few exceptions, however, writers concerned 

with OT have focused upon the individual rather than the 

family, The purpose of this essay is to explore the extent 

to which the insights and methodology of the OT literature 

can be applied to the problems of family taxation. 

Section I1 reviews the basic framework of OT theory. 

Attention is focused not only upon what is contained in 

the OT literature, but also upon the questions with which 

it fails to deal adequately. This framework is then used 

to discuss the following issues in Sections I11 through VII: 

(a) the tax treatment of working wives; 

(b) the effect of the tax system on incentives to 

engage in home versus market production, and 

problems in the valuation of home production; 

(c) the treatment of single people versus married 

couples; 

(d) the optimal exemption for dependents; 

(e) the implications of mandatory individual filing. 
-2-

Of course, OT theory cannot provide insights into all aspects 

of each of these problems. As necessary, other theoretical and 

empirical literatures will be relied upon, 
-3-

31 

11. THE OPTIMAL TAX LITERATUREIn general, two different problems have been studied 

in the optimal tax literature. The first is to find a set 

of commodity taxes that is optimal given certain efficiency 

and (sometimes) equity considerations. In a second strain 

of the literature, it is assumed that the revenue system 

is based upon income rather than commodity taxation, and 

the problem is to determine the optimal degree of 

progressivity (or regressivity). This section will discuss 

each of these in turn, and proceed to outline some 

limitations of the approach. 

A. Optimal Commodity Taxation. Assume that R dollars 

of revenue must be raised by the government via a set of 

commodity taxes. (Lump sum taxes are excluded from 

consideration.) If Xi is the quantity of the ith good 

purchased by the household sector and Ti is the per unit 

tax, then the revenue constraint is 

CTiXi = R (1) 

On the assumption that producer prices are fixed, setting 

the Ti is equivalent to setting consumer prices. The basic 

problem of optimal commodity taxation is to choose a set of 

consumer prices which maximize consumer utility, given the 

revenue constraint R. 
-- 

-4-

Equivalently, tax rates are to be chosen to minimize 

4/ 

excess burden- the loss in utility in excess of that 

which would have been incurred had the tax been collected 

-5/ 

as a lump sum. Under certain conditions, minimization 

of excess burden occurs in accordance with the "inverse 

elasticity rule", 

6 

tp = Err ; r = 1, ...,m (2) 

where tr is the tax rate expressed as a percentage of 

producer price, 6 is a constant, and Err is the elasticity 

of the ordinary demand function for the rth good. In 

words, (2) says that the more elastic the demand for a 

good, the smaller should be the -ad valorem tax levied

upon it. This is because a high tax placed on a relatively 

inelastic good will lead to relatively small distortions, 

other things being the same. 

This formula has been a major factor in the formation 

of economists' intuitions on tax and price regulatory 

questions. It underlies the notion of charging according 

to 'what the traffic can bear' in transportation, for example, 

and is the basis for the acceptance on efficiency grounds 

of such taxes as those on tobacco and alcohol, the demand 

for which is presumed price inelastic. 
-5-

It is important to note the conflict between equity 

and efficiency which can arise in this context. To the 

extent that price inelastic goods have low income 

elasticities, taxing them heavily may be regressive. 

This conflict is a familiar one; the all-time champion 

for efficiency purposes, the lump sum tax, is usually an 

unacceptable alternative due to equity considerations. 

B. Optimal Income Taxation. In optimal income taxation 

studies, the tax base is assumed to be income, and the 

problem is to find a tax schedule which maximizes some 

social welfare function. Most of the recent work on this 

problem stems from a 1971 paper by James Mirrlees [20], so 

his model will be used as a focus for our discussion. 

Mirrlees assumes that society is composed of 

individuals who have identical atemporal utility functions 

in after tax income and leisure, and who differ only in 

their earnings abilities. The government must collect an 

exogenously determined amount of tax revenue. The problem 

is to find an income tax schedule (tax function) which 

maximizes the sum of individuals' utilities subject to 

-6 /

this revenue constraint. The tradeoff between efficiency 

and equity is central to this problem. To the extent that 

higher tax rates distribute income to those with low utility, 
-- 

-6-

social welfare increases, but at the same time higher tax 

rates distort the work-leisure choice implicit in the 

individual utility functions, thus tending to lower social 

welfare. 

The basic thrust of the literature should now be 

clear. An exogenously determined amount of tax revenue 

must be raised by income taxes on individuals whose economic 

choices are distorted by the presence of those taxes. 

Given technological and behavioral assumptions, the optimal 

tax schedule is that which leaves social welfare at a 

maximum after the tax is collected. The literature indicates 

that the optimal tax rates are quite sensitive to the 

underlying assumptions. 

C. Two Problems with the Optimal Tax Literature. As 

OT theory is applied to the problems of the family, various 

limitations of the approach will become apparent. Therefore, 

it seems useful to mention two major difficulties at 

the outset: 1) Administrative costs are not usually included 

-71 

in the optimal tax calculus. Thus, a judgment as to 

whether or not a particular scheme is feasible for policy 

purposes must be made on a more or less ad hoc basis. For 

example, the determination of optimal commodity tax rates 

in models where individual tastes differ may require 

information on individuals' utility function parameters. 
-7-

It is hard to imagine implementing such rates. 2) The 

notion of horizontal equity is absent from the optimal tax 

literature. In none of the studies previously discussed 

does the injunction to treat people in 'equal positions! 

equally appear as a constraint or an argument in the 

objective function. Therefore, the schemes will in -8/

general fail to provide horizontal equity. 

Although the optimal taxation literature ignores 

horizontal equity, it has sparked new interest in the 

topic, and modified the vocabulary of the discussion. No 

consensus has emerged as to the proper criterion for 

measuring horizontal equity. Traditionally, individuals 

are viewed as equal if they have the same income (or in 

the case of an expenditure tax, the same level of 

consumption expenditures). This formulation is inadequate 

because individuals with identical opportunity sets but 

different tastes will have different incomes. 

Alternatively, then, equal position could be defined 

in terms of equal opportunity sets. The difficulty here 

is that characterization of opportunity sets generally 

requires a vector rather than a single number. Consider, 

for example, the problem of describing a given family's 

opportunities, At least three numbers are needed: 
-8-

husband's wage, wife's wage, and unearned income. In 

order to compare families, some aggregation of these numbers 

is necessary. A natural way to do this is via the indirect 

utility function. 

This leads to a final criterion for horizontal equity:, 

"If two individuals would be equally well off (have the 

same utility level) in the absence of taxation, they should 

be equally well off if there is a tax." [lo] The 

difficulties involved in implementing such a utility 

definition of horizontal equity are considerable. Not only 

do the shapes of indifference curves have to be estimated, 

but some cardinalization of the utility functions is 

necessary because interpersonal utility comparisons are 

needed. 

Perhaps this is not as bad as might appear at first. 

It is hard to imagine that any of the definitions of 

horizontal equity has ethical content unless it is assumed 

that individuals' welfares are at least roughly comparable. 

The utility formulation simply makes the comparisons 

explicit. The disagreement over criteria for horizontal 

equity makes it quite difficult to assess the fairness of 

any given tax statute. In the following discussions, 

several criteria are considered. 
-- 

-9-

111. WORKING WIVES 

In this section two questions are raised: a) Does the 

present tax system discriminate against working wives?; and 

b) Is there a more efficient way to tax wives' earnings? 

For the first question, optimal tax theory per -se provides

no specific answers, although the econometric literature 

on labor supply yields some interesting insights. With 

respect to the second question, application of optimal 

tax theory suggests that the present system is probably 

inefficient. 

A. Is There Discrimination Against Working Wives? 

Under current law, joint filers pool their incomes, and 

both spouses face the same marginal tax rate on their last 

dollar of earnings. It can be argued, then, that the law 

does not discriminate against working wives for the simple 

reason that it does not distinguish between the spouses. 

This reasoning ignores the institutional reality that in 

most households, it is assumed that the husband will work 

full time and the wife makes her labor force decision 

conditional on the husband's income. In effect, then, the 

first dollar earned by the wife is taxed at the same 

marginal rate as the last dollar earned by the husband 

she bears a higher tax burden on her earnings. 
-- 

- 10 -

Is it useful to describe this situation as "discrimination against working wives?" Such a characterization may 

be misleading for two reasons. First of all, it is their 

status as secondary wage earners, not as women, that is 

relevant here. If discrimination exists, it is against 

secondary workers, not wives per se. Secondly, it seems 

inappropriate to label as discrimination an effect which 

arises because the tax law treats the family as a unit, 

instead of treating spouses asymmetrically. Although it 

is important to recognize that the current system tends to 

diminish the net returns to working for married women, 

it probably only confuses public discussion to call this 

discrimination against working wives. 

Does the higher tax burden discourage wives from 

working? Bittker argues that this disincentive and others 

that put a wedge between the gross and net wage are 'I... 

obviously outweighed by economic pressure or personal 

inclination for the millions of wives who are employed 

outside the home." [3, p. 14361. This is really beside 

the point. The issue is not whether millions of wives 

are working, but rather, the change in labor force 

participation rates and hours of work which would occur 

under alternative tax regimes. 
- 11 -

Delineating the issue in this way raises two related 

questions. Do the working wives correctly perceive the 

wedge which the income and payroll taxes impose between 

the gross and net wage, and if so, is there a substantial 

labor supply response? I have argued elsewhere [23] that 

the answer to both these questions is yes. Married women 

perceive their taxes in the 'rational' manner suggested 

by economic theory, and most investigators have found 

that wives' average labor supply elasticity is substantial, 

although there is disagreement on the exact magnitude. 

Kosters [19] places the elasticity of the labor force 

participation rate for all married women with respect to 

the wage between .4 and 1.5. Hall [14] finds that when 

the net wage changes from the $1.50 - $1.75 range to the 

$2.50 - $3.00 range for white married women aged 20 to 39, 

hours of work show an increase between 520 and 1034 hours, 

depending on the number of children. Bloch's 141 hours 

and participation rate equations taken together imply an 

uncompensated supply elasticity of about 3.0. 

Thus, the tax system acts to diminish married women's 

-91 

yearly hours of work. But this is not the end of the 

story, for other dimensions of labor supply beside hours 

of work may be affected by tax laws, These include human 
- 12 -

capital investment, occupational choice and intensity of 

effort. Unfortunately, little is currently known about 

how the tax system affects these decisions; further research 

is clearly desirable. 

A second possible source of discrimination against 

women working has been suggested. This is the fact that 

when a wife chooses to enter the market place, various 

non-deductible work related expenses are incurred. 

Moreover, it may become necessary to hire someone to do the 

housework and take care of the children, expenses which 

for the most part are also non-deductible. 

As Bittker points out, the idea that only two-earner 

married couples deserve a special consideration in this 

context 'I... is curiously narrow. Everyone who works away 

from home -- not just the working wife -- must get to the 

job site, dress as the job requires, and pay for lunch 

if it is inconvenient to bring it in a brown bag. 

Similarly, everyone who works has less time and energy 

to keep house, prepare meals, and look for bargains." [3, 

p. 14351. It is thus not clear that the absence of such 

deductions can be said to 'discriminate' against working 

wives, although it no doubt affects their labor force 

behavior. 
- 13 -

B. Efficient Taxation of the Family. Contemporary 

economic theory recognizes that membership in a household 

influences economic behavior. The view is that family 

members act jointly to maximize some household utility 

function. Very little has been said about the origins of 

this utility function. Once more the familiar problems of 

preference aggregation from the literature on social 

- welfare functions appear.10/ And just as in the social 

welfare function context, these difficulties are usually 

acknowledged and then ignored. 

Assume, then, that family utility (U) is a function 

of three arguments, "husband's yearly hours of non-market 

activity?' (L1), "wife's yearly hours of non-market activity" 

-11/

(Lz), and "family consumption" (Y) : 

u = f(L1, L2,Y) . (3) 

Formally, then, L1 and L2 are just two commodities in the 

family utility function, and therefore the insights of the 

optimal commodity tax literature should be applicable. 

-

12/

Consider,then, applying the inverse elasticity rule. 

The econometric result was previously mentioned that wives' 

elasticity of supply of labor (or alternatively, demand 

for leisure) tends to be high. On the other hand, studies 

of the labor force behavior of married men have indicated 
- 14 -

that their labor supply elasticity is quite low. (See [4] 

or [14].) This suggests that in order to tax the famiJy 

unit efficiently, the wife's earned income should be taxed 

at a lower rate than the husband's. Yet we have shown that 

in effect, the tax laws are doing just the opposite! 

As emphasized in section 11, in its current state of 

development, the OT approach is not applicable to matters 

of political and administrative feasibility. Although 

it is hard to imagine Congress implementing separate 

schedules for primary and secondary earners, there are at 

least two feasible proposals which would move in the 

direction of reducing tax induced inefficiencies. One 

The is an earned income allowance for secondary workers. 

other is mandatory individual filing, which will be 

discussed in section VII. 
- 15 -

IV. MARKET VERSUS HOME PRODUCTION 

Although the model in which family utility depends only 

upon non-market activity and income yielded some interesting 

insights, it is now useful to expand it to allow explicitly 

for household production. As Gronau 113, p. 6341 suggests, 

family members can be viewed as allocating time according to 

their comparative advantages in the production of home and 

market goods, these being determined by their relative wage 

rates and their efficiency in producing household goods. 

Currently, time spent in household production is untaxed 

while remuneration from market activity is taxed. How does 

this asymmetry in the treatment of home versus market work 

affect labor force participation rates? Is it efficient? 

These questions will first be explored, and then some 

difficulties in the valuation of time spent in home production 

will be discussed. 

A. Effects on Labor Force Participation. Consider the 

problem of allocating time between market work and non-market 

B activity (See Figure 1). 

Income Assume that prior to 

imposition of the tax, the 

C individual's budget constraint 

is AB, so the wage is the 

1 slope of that line. ON is 

0 \T 

Non-market activity 

FIGURE 1 
- 16 -

the total time endowment and MN hours per year are spent on 

market activity. Now consider a tax which changes the 

budget constraint to AC. As drawn in Figure 1, the individual 

stays out of the market, devoting his or her time exclusively 

to housework and leisure. Of course, it cannot be determined 

-a priori that imposition of the tax will lead to a diminution 

of hours of market activity because of the familiar conflict 

between the income and substitution effects. However, with 

respect to labor force participation rates, a stronger 

statement can be made. A tax induced decrease in the net 

wage will never induce an individual into the market. The 

result is unambiguous because at zero hours of labor market 

work, there is no income effect to work against the 

substitution effect. 

Given all this, one can say that the tax system biases 

secondary earners against work in the market. It should be 

noted, however, that this is not exactly the same thing 

as saying the individual is biased toward housework --

non-market activity increases, but it has two components, 

leisure and household production. To determine the impact 

of taxes on hours spent in household production, we require 

data on hours of work, hours of leisure, and hours of 

housework along with appropriate information on the 

individual's market characteristics. Bloch [4] found that 
- 17 -

when a woman's market wage increases by $1, her yearly hours 

of regular housework fall by about 640. This result is 

consistent with the preceding theory. Of course, it must be 

viewed with some caution in light of likely reporting errors 

in hours of housework. In addition, there are difficult 

conceptual problems in defining hours of housework. Is 

playing with an infant 'leisure' or 'home production'? 

B. 'Neutrality' Between Home and Market Production. 

I have suggested that failure to tax the imputed wage in 

home production results in fewer labor resources being 

devoted to the market sector than would otherwise be the 

case. Some students of the tax system, noting this 

phenomenon, have argued that ideallz, home production and 

market work should be taxed at the same rate. The feeling 

is that such a system would be 'neutral' between household 

and market activity, and therefore more efficient. However, 

a straightforward application of optimal tax theory 

indicates that this assertion is wrong. Since there will 

in general be different supply elasticities of time spent 

in the market and time spent in household production, 

differential taxation of time spent in the home and in the 

market place is likely to be necessary to secure efficient 

tax treatment. Even if it were administratively feasible 

to tax time spent in home production at the same rate as 
- 18 -

that in market production, it would probably be sub­

-131 

optimal . (This result is derived more rigorously in 

Appendix B.) 

C. Imputing the Value of Time Spent in Household 

Production. It has been argued in this paper that: 

a) although failure to tax imputed income from household 

production may be unfair, it is no more discriminatory 

-14/ 

against working wives than anyone else; and b) efficient 

treatment of the family does not necessarily require 

taxing time spent in the home and time spent in the market 

at the same rate. So far, the question of how spouses' 

time spent in the home should be valued has not been 

dealt with adequately. This question is of some interest 

even if one has no intention of actually taxing time spent 

in home production. For example, one might want to include 

imputed income from housework in the tax base in order to 

see how it affects progressivity calculations. 

For family members who participate in the labor force, 

the value of time spent in home production is the wage rate. 

For such individuals, the value of time spent in housework 

could be calculated simply by taking the number of hours of 

housework and multiplying by the wage, providing that the 

number of hours of housework could be ascertained. 
- 19 -

A major difficulty occurs when a spouse does not 

participate in the labor force. The fact that a spouse 

opts out of market work indicates that the value of time 

in the home exceeds the potential wage rate. This has 

two implications: a) the housespouse's value of time 

cannot be imputed from any observed market value; and 

b) the value of the housespouse's time 'I... is not exogenously 

given, but rather determined by the maximization process. 

A change in the parameters changes both the optimum solution 

and the price placed by the family on the wife's time ... 11 

[13, P. 6403. 

Since the value of time spent in the household for 

such spouses cannot be observed, various imputation procedures 

have been developed (see [13], C141, and L-161). The issue 

to be focused upon is the reliability of such estimates 

for the policy maker who is interested in using them for 

purposes of income imputation. Even if the underlying 

assumptions of the various models are believed, the outcomes are not very stable in a statistical sense. For 

example, the estimated standard deviation of Heckman's 

[16] equation for the logarithm of the askir,g wage is 

.532. Considering that the log of a wage like $1.75 is 

only .559, this is a large standard error. I conclude that 

given the current state of the art, any use of imputed wages 
- 20 -

to augment the tax base must be done with great caution. 

Several different values for housespouses' time should be 

used in any given problem in order to assess the sensitivity 

-l5/ of the substantive results to differences, 
- 21 -

V. SINGLES VERSUS MARRIEDS 

Consider the following three propositions: 

a) The income tax should be progressive. 

b) Married couples with equal incomes should, other 

things being equal, pay equal taxes. 

c) An individual's tax burden should not change 

when he/she marries. 

It is easy to show that a tax system which satisfies any 

two of these propositions will not in general satisfy the 

third. (See, for example, [3].) Given what seems to be 

a firm societal commitment to progressivity, then, the 

real choice is between b) and e). Under current law, b) 

is the prevailing principle, and much controversy has arisen 

over the consequent violation of e) -- many individuals' 

-16/ 

tax burdens increase when they marry. 

As it stands, objections to proposition c) can be 

raised on the grounds that due to scale economies in 

housekeeping, two married individuals with a given income 

have more 'ability to pay' than if they were single. 

Therefore, their tax liability should be greater. It 

will be discussed in section B whether or not such 

economies of scale should be considered by tax policy 

makers. In the meantime, we merely observe that if one 

adheres to this view, e) should be modified to read 
- 22 -

"The only change in an individual's tax burden when he/she 

marries should be that which corrects for economies of scale 

Even with this version of e), in general there will be a 

conflict with b). 

The first part of this section deals briefly with 

the concept of 'marriage neutrality', and whether or not 

it is a desirable feature of a tax system. In the second 

part, the choice of an appropriate unit of taxation is 

discussed. 

A. Marriage Neutrality. A tax system will be defined 

as 'marriage neutral' if, other things being equal, an 

individual's tax burden is the same whether or not he/she 

I17/

is married. Alternatively, a tax system is marriage 

neutral if it leaves unchanged the price of 'marriage 

services'. It could easily be argued that marriage 

neutrality is inefficient -- the demand for marriage - services is probably quite inelastic, 

18/ 

and therefore, 

relatively little excess burden would be generated by 

taxing it. However, I think that proponents of marriage 

neutrality view it as an equity rather than an efficiency 

criterion -- it is unfair to tax those who have a 'taste' 

for marriage more than those who don't share this taste, -191 

or have not been able to indulge it. 
- 23 -

B. Equity Aspects of the Marriage Tax. The natural 

way for a public finance economist to think about the 

ethical implications of a given statute is in terms of 

vertical and horizontal equity. The straightforward use 

of this traditional framework is impossible in this Context, 

however, because it presupposes a unit of' observation. 

It begs the fundamental question: should one look at equity 

among individuals or among families? 

One of the most cogent arguments for the family as 

the appropriate unit of taxation is found In the wellknown report of the Carter Commission [25]: "We believe 

firmly that the family is today, as it has been for many 

centuries, the basic economic unit in society" (p. 123). 

'!Taxation of the individual in ... disregard of his 

inevitably close financial and economic ties with the 

other members of the basic social unit of which he is 

ordinarily a member, the family, is in our view [a] striking 

instance of [a] lack of a comprehensive and rational pattern 

in [a] tax system" (p. 122). 

The case for the family unit is not as compelling as 

the Carter Commission report suggests. Traditional family 

arrangements are not as omnipresent today as they once were. 

Marriage is certainly the predominant institution, but 

individuals can choose other ones. As Blttker argues, 
- 24 -

"If married couples are taxed on their consolidated income, 

for example, should the same principle extend to a child 

who supports an aged parent, two sisters who share an 

apartment, or a divorced parent who lives with an adolescent 

child? Should a relationship established by blood or 

marriage be demanded, to the exclusion, for example, of 

unmarried persons who live together, homosexual companions 

and communes?" [3, p. 13981. 

It might be useful to examine these two different 

views in terms of the analytical structure used in 

section 111. The Carter Commission implicitly assumes, as 

has been assumed in this paper, that there exists a family 

utility function, U = U(Ll,L2, Y) where the variables 

are as previously (see equation (3)) defined. It is 

important to note that with this formulation, it is a matter 

of indifference how Y is distributed within the family. 

Property rights of the spouses are viewed as absolutely 

irrelevant. 

On the other hand, if property rights within the 

family are important and spouses are more like individuals 

who care about one another than a single unit, then two 

utility functions are involved: 

u1 = U1(LpY,U2) (4) 

and 

(5) 
- 25 -

where a is the first spouse's share of family income. 

Even this story is too simple, however, because many of the 

goods consumed by the household have a public goods aspect 

(for example, housing). Therefore, Y should be broken 

up into three components: "his1', "hers", and "theirs". 

The problem of efficient taxation is solvable either with 

formulation (3) or formulation (4)-(5). But OT theory 

says nothing about which formulation is to be preferred. 

Another issue which arises with respect to the treatment of singles versus marrieds is economies of scale. 

Two may not be able to live as cheaply as one, but they 

may be able to live as cheaply as (say) one point five. 

Thus, the argument goes, a married couple has more 'abilityto-pay' than two singles with the same total income, and 

should therefore pay more taxes. Some have gone so far as 

to suggest that the whole matter of horizontal and 

vertical equity should be examined in terms of per adult 

equivalents in order to take such economies explicitly 

into account. (See, for example [18]). 

I find this line of reasoning unconvincing. Nonmarrieds generally have the opportunity to join together 

with others in order to enjoy the economies of scale of 

living together. If they do not, it simply shows that 

the value of living apart more than compensates for such 
- 26 -

economies. Neither is it a universal rule that marriage 

results in economies of scale; in some two earner families, 

it is necessary to maintain separate residences. 

This section can only be concluded with the hope that 

our analysis has helped clarify the nature of the issues 

surrounding marriage neutrality, although it has provided 

no definite answers. Solutions here are even more dependent 

on one's views of how society should operate than is 

usually the case. 

I 
- 27 -

VI, TREATMENT OF CHILDREN 

Currently, the family is allowed a tax exemption of $750 

per child. This can be rationalized in several ways: 1) It 

adjusts for the effect of the number of children on ability to pay: 

2) It provides more progressivity at the bottom end of the income 

scale; and 3) It may be an instrument of optimal population policy. 

In this section, focus is centered only on 1) and 2). Optimal 

population policy is an exceedingly difficult problem which is just 

20/

beginning to receive attention. (See, for example, [21].) -

It will be argued that the number of children is an 

inappropriate way to adjust ability to pay. However, in view of 

the widespread social acceptance of the exemption, we will then 

discuss how the optimal child exemption might be determined given 

that it is to be used. 

A. Children and Ability to Pay. The basic argument for 

the exemption as an adjustment in ability to pay can be condensed 

-21/ 

as follows: Taxable income should include only that income 

in excess of subsistence. The addition of a child increases the 

level of income required for family members to survive, and 

therefore an exemption equal to this change is needed. 

There are some who agree with the basic idea that the number 

of children affects ability to pay, but object to correcting for 

it with an exemption. The dollar value of an exemption, they 

note, is an increasing function of the family's marginal tax rate. 

Therefore, rich families gain more from it than the poor. The 

proposed solution is either a vanishing exemption (an exemption 

which diminishes as income increases), or a tax credit for each 
- 28 -

child (a dollar amount which does not vary with income). 

It has been argued for many years that all such approaches 

are misdirected (see, for example, [27]). If a husband and wife 

choose to devote family resources to the rearing of children, this 

is their business, but there is no particular reason to award a 

tax reduction for such behavior. Indeed, it can be argued that 

such a policy is an outright violation of horizontal equity. Sets 

of spouses who have identical opportunities, but different tastes 

regarding family size, are treated unequally under such a regime. 

The extent to which non-economists ridicule the view taken 

in the last paragraph is striking. The typical bemused reaction 

is that of Bittker[3]: "..,the decision to have children is 

irreversible: they cannot be abandoned, like a hobby that has become 

burdensome or boring. Moreover, society feels an obligation to 

support the children--but not the hobby--if the taxpayer is unable 

to do so. No one but a tax theorist, it might be asserted, could 

fail to see the difference (p. 1448) ...I doubt that even the 

most computerized econometrician weighs the value of a dependency 

exemption before procreation" (p. 1449). 

Part of this reaction is probably due to a failure to view 

decision-making in a life-cycle context--the dependents simply 

exist and have to be taken care of. Little thought is given to 

the fact that children are components of some long-term consumption 

plan. But part of the reason for the rejection of the idea of 

'children as consumption', I think, is the way it is 'packaged' 

by economists. The argument is often framed in terms like 'children 

are just another good, no different from apples or yachts'. This 
is misleading--although children's services appear as arguments in 

the utility function like apples or yachts, they are regarded 

differently by parents. .Indeed, unlike apples or yachts, it is 

possible that children can change the form of the utility function 

by influencing the household decision making process Perhaps the 

idea might be more palatable if it were emphasized that children 

are a choice, rather than that children are a good. 

In any case, a fundamental equity problem arises if the 

'children as a choice' approach is taken. Although this approach 

would help secure horizontal equity between sets of spouses, it 

would not necessarily secure equality in opportunity between children. 

Similarly, a policy designed to subsidize child rearing might 

foster equality of opportunity for children, but at the expense of 

creating inequities between the parents. This appears to be an 

insoluble dilemma. The provision of certain child-care services 

by the public sector might be a partial solution, but raising the 

funds to run them would, of course, induce economic distortions. 

The proper mix of subsidies and direct expenditures needed to 

arrive at the socially determined amount of child care can be 

analyzed as an application of optimal tax theory, but is beyond 

the scope of this paper. E/ 

Finally, if the argument were accepted that children are 

just another commodity, how might children's services be taxed most 

efficiently? Currently there is not enough consensus as to the 

appropriate elasticities to venture a guess.z/ It is not even clear 

how to define the price of children's services. At this time, then, 

there is a lack of sufficient empirical information to implement 
----- 

- 30 -

efficient taxation of children's services. For reasons similar to 

those discussed with respect to marriage services, most people 

would view such a tax as inequitable and unacceptable anyway. 

B. The Optimal Child Exemption. There are some who view the 

child exemption less as a subsidy than as an instrument for gaining 

progressivity at the bottom end of the income scale. Suppose the 

number of children per family is given exogenously, and it has been 

decided that some exemption per child is to be given. What should 

be the size of the exemption? 

This is similar in structure to the basic problem of the 

optimal income tax literature. Assume as before that each family 

has a utility function which depends on family income and time 

spent in non-market activity. Assume further that each family's 

tax bill depends upon its income, but that taxes are reduced a given 

amount for each child. The optimal tax problem is to choose the 

income tax rate and child care exemption so as to maximize social 

welfare subject to the constraint 'that a given amount of tax revenue 

be collected. 

Given appropriate assumptions on specific functional forms 

and parameter values, this problem presumably could be solved 

either analytically or numerically.-24/ For the purposes of this 

paper, however, it is sufficient to call attention to some general 

characteristics of the solution. 

a) In general, the optimal child exemption must be 

determined jointly with the optimal tax rate. 
b) The exemption depends upon the relationship between 

number of children and earnings abilities of parents. 

c) Value judgments on the proper degree of egalitarianism 

influence the optimal exemption. If, for example, the social 

welfare function weighs increments to the incomes of the poor more 

than to those of the rich, and poor families tend to have more 

children than the rich, then this will tend to increase the exemption. 

d) The notion that children increase the 'subsistence 

needs' of a family is absent from this model. 'Poverty lines' need 

not be calculated in order to generate a positive exemption. 

If the tax function allowed for increasing marginal tax 

rates, the question of exemptions versus credits could be explored. 

I have suggested elsewhere [24] that such changes are unlikely to 

have much influence one way or another as far as their overall 

impact on a utilitarian social welfare function. The differences 

involved are not substantial enough tosignificantly matter fron 

either an efficiency or equ-ity point of view. 
------ 

- 32 -

VII, MANDATORY INDIVIDUAL FILING 

Some criticisms of joint filing as it is done under current 

law have previously been asserted: it imposes a relatively large 

burden on secondary workers, fails the test of marriage neutrality, 

assumes that property rights within the family are irrelevant, 

and ignores social changes which are lessening the primacy of the 

family as the basic social unit. In light of these considerations, 

it has been suggested that joint filing be eliminated, and that all 

individuals be required to file separately. In this section, a 

few implications such a move might have for labor supply, economic 

efficiency, and the distribution of income will be discussed. 

A. Labor Su-lyand - -------I---­ Efficiency. When speculating about 

mandatory individual filing, a primary problem involves the 

treatment of unearned income. Would high earnings individuals 

transfer capital to their mates in order to lower family tax bills 

(“bedchamber transfers of property”)? It is difficult to predict. 

My guess is that property rights within families do_ matter in many 

cases. After all, significant amounts of estate tax could be 

avoided if more inter vivos gifts were made, but they are not-­

individuals appear to be willing to pay high prices to retain 

control of their property.- 25’ To the extent inter-family property 

transfers to avoid taxes did take place, the result would be a lower 

rate of taxation on capital income. It is unclear whether or not 

this would be desirable from a social point of view. It should be 

noted, however, that the overall effect of mandatory filing cannot 
- 33 -

be determined a priori. It depends upon how the rate structure 

would be changed to adjust for revenue changes. 

In a recent attempt to ascertain the effects of mandatory 

individual filing,-I26/ assumed that half of non-earned income is 

imputed to each spouse for purposes of computing taxable income, 

and the spouse with the higher adjusted gross income takes all the 

deductions and exemptions to which the family is entitled. Note 

that if each spouse faces the same rate schedule as the family faced 

prior to the introduction of individual filing, total tax revenues 

would change. Therefore, in this exercise, the marginal tax rates 

for each bracket were adjusted proportionately to keep revenue 

collections constant. In this way, the degree of progressivity of 

the rate schedule was maintained. 

When this change in the tax law was simulated using a sample 

of middle-aged white families, the following results emerged: 

a) Hours per year of market work increase for wives in all 

income brackets, and the change in hours of work is greatest for 

women for whom other family income is the highest. This result 

is not hard to understand. Since the elasticity of married women's 

labor supply is positive, the change in the number of hours of work 

is an increasing function of the increase in the net wage. Because 

women with high earnings husbands faced the highest tax rates 

under joint filing, the move to individual filing gives them the 

largest inducement to work more. The change in tax regime is assumed 

to have no effect on husbands' yearly hours of work. This assumption 
- 34 -

simplifies the analysis greatly, and probably is not a bad 

approximation to reality. For example, in Hall's [I41 careful study 

of the labor supply behavior of married males, no significant wage 

effect is isolated. 271 -

b) The excess burden of the tax system decreases by about 

thirty percent. This is a consequence of the fact that the tax 

burden has been moved from a set of workers with high elasticities 

of supply (wives) to those with low elasticities (husbands). These 

results are consistent with the earlier discussion of efficient 

taxation of secondary earners. 

Since these calculations are based upon a sample which is 

not representative of the entire population, the figure of thirty 

percent must be viewed as only a rough approximation of the efficiency 

gain in moving to mandatory single filing. A further complication 

arises if the wage varies with the number of hours of work, that 

is, if there are economies of scale of work in the market place. 

In this case, there would be efficiency losses due to the 

discouragement of the division of labor. Not much is known about 

whether or not this is an important phenomenon empirically. (See 

[23] for some preliminary results.) There can be little doubt, 

however, that such a move would lower the excess burden of the 

tax system. 

B* --------,----------. Distribution of Welfare As just mentioned, a consequence 

of adopting mandatory individual filing is that wives from wealthier 

families tend to work more hours. Therefore, the incomes of these 

wealthier families would tend to go up by greater dollar amounts 

than those of the poar, and the distribution of family income 
- 35 -

(according to some measures) would become more unequal. 

However, the change is less inegalitarian than might first 

appear. Households that gain the most in terms of income are also 

losing the most in terms of either leisure or household production-­

the distribution of family utility changes less than the distribution 

of income. Nevertheless, some inequality is introduced by the 

move, and the question is whether it outweighs the benefits of 

increased efficiency. My calculations indicate that for a wide 

variety of egalitarian preferences, changing to mandatory individual 

filing leads to a small positive change in social welfare. If 

these results are at least qualitatively correct, they suggest that 

such a move would be beneficial. Note, however, that this is 

conditional on a view that does not attach normative significance 

to the fact that families with the same total incomes can pay 

different amounts of taxes. 
VIII. CONCLUDING COMMENTS 

Many problems associated with the taxation of the family have 

not been discussed. For example, no mention has been made of the 

interesting problems which arise with respect to the treatment of 

gifts and bequests. If the bequest motive induces some amount of' 

saving, then optimal taxation of bequests would have to balance off 

equity (redistribution of wealth) and efficiency using tools 

similar to those of optimal income tax theory. Presumably, the 

elasticity of bequests with respect to the 'price of bequests' 

would be a key parameter in the problem. 

A different set of problems arises when considering giftgiving within the family unit. Whether gifts should be taxed at 

all depends on whether individuals or families are viewed as the 

taxable unit. These problems were analyzed in thediscussion of the 

marriage tax. 

Neither has any attention been devoted to the question Of 

alternative bases for family taxation. It has been suggested, for 

example, that a consumption based persona1tEi.X might be superior 

to the present one. (See, for example, [lo].) The analyses of 

this paper transfer easily from an income to a consumption tax, 

although some details would change.-28' Whatever the tax base, 

however, policy makers will still have to worry about such problems 

as the unit of taxation, marriage penalties, and child-care 

deductions. 
In conclusion, it should be emphasized again that OT theory 

cannot provide an escape from making certain basic value 

judgments. However, on a variety of issues which involve efficiency 

(for example, tax treatment of secondary earners) or tradeoffs 

between equity and efficiency (for example, determination of the 

optimal child care exemption), a very theoretical literature has 

yielded some extremely useful insights. 
- 38 -

FOOTNOTES 

* The author would like to thank D. Bradford, M. Feldstein, 

K. Small, and members of the Office of Tax Analysis Seminar for 

useful comments. 

1/ For a history of the family's tax treatment in America, 

see Bittker [31. 

21 See, for example, the paper by Boskin [51. -

For a more extensive discussion of the OT literature, 

see D,

3/ 

Bradford and H. Rosen, "The Optimal Taxation of Commodities and Income," American Economic Review, May 1976. 

4/ For 'small' taxes, it can be shown (see Harberger [I51

that Tf taxes on m goods are being considered, then the excess 

burden is 

where Sij is the Slutsky coefficient (the compensated change 

in consumption of good i with respect to a change in the price 

of good j) and dpj is the tax induced change in the price of 

the jth good.

There are a number of ways to solve this problem (see 

Sandmo [26]. The basic result which emerges is that if tax 

rates are excess burden minimizing, 

k=l, . . . ,m, 

where B is independent of k, and other variables are as defined 

above. The left hand side gives an estimate of the change in 

demand for the kth good which would occur if taxes were removed. 

Thus, the equation suggests that for maximum efficiency, the 

proportional change in demand should be the same for all 

commodities. 

5/ A loss in utility in excess of the tax receipts arises 

because the taxes lead to a mimation of resources. 
- 39 -

6/ Assuming that utility functions are Cobb-Douglas and 

the earnings ability distribution is lognormal or pareto, the 

following results emerge: a) The optimal tax function is 

approximately linear with a negative intercept; and bj it is 

characterized by 'low' marginal tax rates which fall. somewhat 

with income. (Atkinson's [2] interpolations of Mirrleesr 

results indicate rates in the neighborhood of 20 percent for 

a zero-revenue purely redistributive tax,) 

-7/ There has been some work on tax evasion, See [l], 

8/ It can be shown that if all individuals have identical 

tastes and there is only one type of ability, then horizontal 

equity will be satisfied by virtually any broad-based tax. 

(See Feldstein [lo].) Such assumptions, as we have seen, are 

built into a number of the optimal tax studies. (For an 

exception, see Diamond and MirrleesC81.) 

9/ Alternatively, we can think of the tax system as 

increasing the yearly hours of non-market activity. (This 

aspect of the problem will be considered later.) 

-10/ For a discussion, see Ishikawa [17]. 

-11/ In this atemporal model there is no saving,

12/ The inverse elasticity rule applies strictly only when 

the Esbandrs and wirers labor supplies are independent of each 

other. The more general case with interdependent labor supplies

is discussed in Appendix A. 

13/ Although such a program would not be optimal, it might 

be superior to the present system in which household production

is entirely untaxed. It would be necessary to compute the excess 

burdens associated with each of these suboptima in order to 

ascertain which was superior, 

-14/ Although this provision is not particularly unfair to

wives as a group, it might be thought of as discriminating against 

households in which the wife has a relatively strong taste for 

engaging in market activity. 

15/ As an alternative to calculating the value of time spent

inhome production, one might consider calculating the value of 

market substitutes for housewives' time. This approach also 

involves serious estimation problems. 
- 40 -

16/ For a discussion of how the size of the 'marriage tax' 

varierwith the spouses1 incomes and number of dependents, see [22]. 

l7/ Bittker argues that neutrality cannot be defined without 

prioragreement on what constitutes taxpaying capacity: "Neutrality...is a destination, not a guidepost" [3, p. 14491. It is 

hard to see the basis for this argument, 

18/ Note, however, that changes in the price of marriage

mightlead to divorce, separation or postponement of the date of 

marriage. 

19/ In addition, there are some who believe marriage should 

not bediscouraged because of moral principles, 

20/ To the extent that unplanned children are a concern of 

tax policy makers, a fourth rationalization is possible. The 

exemption can be viewed as a form of insurance against unplanned

children. 

-

21/ For a more detailed defense, see [7].

22/ For a general theory of the optimal mix of "tax 

expenditures" and direct subsidies to achieve set goals, see 

Feldstein [ll]. 

23/ For an interesting study of the determinants of the 

demandfor children, see [12), 

24/ The mathematical structure of the problem is as follows. 

Let the ith family's utility function be Ui = U(Lli, L2i, q), 

where the variables are as previously defined. Assume a linear 

tax function Ti = a + bYi -<Ci, where a is the (positive or 

negative) transfer payment per family, b is the marginal tax rate, 

5 is the exemption per child, and Ci is the number of children in 

the ith family. The family maximizes utility subject to the budget 

constraint Yi = (l-b)Wli (T-Lli) + (1-b)W2i(T-L2i) - a + 5Ci* 

Given a revenue constraint, R, the optimal exemption is part of 

the solution to the problem max W = W(U,, . . . ,U,), subject to 

a,b&

C Ti = R, where k7 is the social welfare function. 
- 41 -

25/ This ignores the existence of 10-year grantor trusts,

in which an individual can transfer a trustts income to his/her 

spouse for 10 years, at the end of which time the principle 

reverts back to the individual, It is not known how widespread the use of such trusts is. 

-26/ The study uses data from the year 1967. See [24].

27/ This should not be interpreted as an assertion that 

taxeshave no impact on work effort. Other dimensions of labor 

supply such as human capital investment, occupational choice 

and time of retirement may very well be affected by the tax 

system. 

28/ Interestingly, part of the case for the consumption 

tax relies on the application of optimal tax theory, When certain 

separability conditions on the intertemporal utility function are 

obtained, it can be shown that a consumption tax has a smaller 

excess burden than an equal yield income tax. See [lo]. 
--- 

- 42 -

REFERENCES 

[l] Allingham, M.G. and A. Sandmo, "Income Tax Evasion: A 

Theoretical Analysis," Journal of Public Economics, 1972. 

[2] Atkinson, A.B., "HOW Progressive Should the Income Tax Be," 

in Longmans, Essays on Modern Economics, London, 1973. 

[3] Bittker, B.I., "Federal Income Taxation and the Family," 

Stanford Law Review, July 1975. p] Bloch, F., "The Allocation of Time to Market and Non-Market 

Work Within a Family Unit, Stanford University, mimeo, 1973. 

[5] Boskin, M.J., "Optimal Tax Treatment of the Family," Stanford 

University, mimeo, 1975. 

[6] Bowen, W.G. and T.A. Finegan, The Economics of Labor Force 

Participation, Princeton, New Jersey: Princeton University 

Press, 1967. 

[7] Brannon, G.M. and E.R. Morss, "The Tax Allowance for 

Dependents: Deductions Versus Credits," National ----,-,,,,,~,Tax Journal

December 1973. 181 Diamond, P. and J. Mirrlees, "Optimal Taxation and Public 

Production: 11,'' American ---,----,---,---? Economic Review June 1971. 

[g] Feldstein, M.S., "On the Optimal Progressivity of the Income 

Tax," Journal of Public Economics, 1973. 

[101 ,,-,-------- , "On The Theory of Tax Reform," Journal of 

,-,-Public Economics July, 1976. 

m1 ---I-------- I "The Theory of Tax Expenditures, Harvard 

University, mimeo, 1975. 

[12] Gardner, B., "Economics of the Size of North Carolina Rural 

Families," Journal of Political Economy, March-April 1973. 

[13] Gronau, R., "The Intrafamily Allocation of Time: The Value 

of the Housewives Time, It American Economic Review, 1973. [14] Hall, R., "Wages, Income and Hours of Work in the U.S. Labor 

Force," in Cain, E. and H. Watts (eds.), Income Maintenance 

and Labor Supply, Rand McNally, 1973. 

[l5] Harberger, A.C., "Taxation, Resource Allocation and Welfare," 

in J. Due (ea.), The Role of Direct and Indirect Taxes in the --I_------- Federal Revenue System, Brookings, 1964. 
--------- 

-------- 

- 43 -

[16] Heckman, J., "Shadow Prices, Market Wages, and Labor Supply,"

Econometrica, July 1974. [l7] Ishikawa, T., "Family Structures and Family Values in the 

Theory of Income Distribution, " Journal of Political Economy,

October 1975. 

[IS] Kondor, Y., "Optimal Deviations from Horizontal Equity: The 

Case of Family Size," Public Finance, 1975. 

[l9] Kosters, M.H., "Effects of an Income Tax on Labor Supply,"

in Harberger, A.C. and M.J. Bailey (eds.1, The Taxation of 

Income from Capital, Washington, DC: The Brookings Institution, 

1969. 

[20] Mirrlees, J., "An Exploration in the Theory of Optimum Income 

Taxation," Review of Economic Studies, 1971. 

mi ___-______ , "Population Policy and Taxation of Family Size," 

Journa 1 of Public Economics, 1972. 

[22] Nussbaum, J.M., "The Tax Structure and Discrimination Against

Working Wives," National Tax Journal, June 1972. 

[23] Rosen, H.S., "Taxes in a Labor Supply Model with Joint Wage-Hours

Determinations," Econometrica, May 1976. 

[24 1 t "A Methodology for Evaluating Tax Reform Proposals,"

Journal of Public Economics, July, 1976. 

[25] Royal Commission on Taxation, Report of the Royal Commission 

,,-..,-,- on TaxationI volume 3, Ottawa, 1966. 

[261 Sandmo, A., "A Note on the Structure of Optimal Taxation," 

American ,-,--,---J Economic Review September 1974.

f271 Simons, H., Personal Income Taxation, University of Chicago

Press, 1938. 

[28] Stern, "On the Specification of Models of Optimum Income 

Taxatiof;:' Journal of Public Economics, July, 1976. 
- 44 -

APPENDIX A 

The purpose of this appendix is to generalize the analysis of 

Section I11 to the case where the labor supply behaviors of the 

husband and wife are interdependent. The problem of efficiency is 

to choose a tax rate for the wife (t2)and for the husband (tl) to 

minimize excess burden: 

Isubject to a revenue constraint 

where: w Ig and w 

253 

are the gross wages of the husband and wife, 

respectively; S is the compensated derivative of the wife’s 

L2’w2 

hours of work with respect to her net wage, and the other elements 

- of the Slutsky matrix are defined analogously; and T is the time 

endowment. 

In an interesting paper, Michael Boskin [5] has solved this 

problem using his estimates of the Slutsky matrix from econometric 

analysis of cross section data. According to his calculations, the 

leisure of husbands and wives should not be taxed at the same rate. 

Even given cross substitution effects, efficiency would be gained 

if wives were taxed less heavily than their spouses, the same 

general conclusion as was found in Section 111. 
- 45 -

APPENDIX B 

In this Appendix it is shown that in general, efficiency will 

not require equal rates of tax on household and market activity. 

To do this, one version of Gronau's [ls] model of family economic 

activity is used. Let M = market goods, H = home goods, 

L1 = husband's leisure and L2 = wife's leisure. Then family 

utility U is 

u = u(M,H,L~,L~). (1) 

Market goods are produced bycombining market inputs into the home 

production function process, X , and the time of the spouses. 

Assume that the husband devotes no time to household work, both 

spouses participate in the labor force, and home production is 

governed by a Leontief technology. These assumptions make the 

discussion easier, and for our purposes, result in no important 

loss of generality. 

Under these assumptions, 

T 

H = min( x_2 ,,-a) 

where l/Y is the marginal product of market inputs in home 

production and 1/6 is the marginal product of wife's time in 

home production ( T2). Inputs thus vary proportionately with amounts 

of home goods produced. 

X = 'YH 

T2 = 6H , 

so that the price of home goods, II , is given by 

II = Yp, + 6w2 y 
- 46 

where PX is the price of market inputs into home production, and 

w 2 is the wife's price of time (equal to her market wage since she 

participates in the market]. 

Finally, note that the family budget constraint can be written 

pm M + pXX = W 1(T-L,) + W2(T-T2 -L 2 ) + V , (6) 

where pm is the price of market goods, T is the time endowment, 

w1 is the husband's price of time, and V is non-labor income. 

Suppose the husband's earned income is taxed at rate tl ' 

wife's earned market income at t2, and the value of wife's time 

spent in the home at tH. This changes the prices of the items 

in the utility function in the following way: the price of 

husband's leisure goes from w1 to wl(l-tl) : the price of wife's 

leisure goes from w 2 to w2(1-t2); and the price of home goods 

goes from Ypx + 6w2 to Ypx + 6w2(l+tH). Therefore, the excess 

burden of the tax system can be written as 

t2W2 S 

H, fl 

sL 

1, fl 

S 

L2' 

where the 'ij 

are the components of the appropriate Slutsky matrix. 

If ( ) is minimized subject to the revenue constraint 

t 111 (T-L )W t + (T-L~-T~)w + T ~ = Rw , ~ ~ ~ 22 

we find 
--- 

- 47 -

- S ww s

L2,W2 W 2 

A( T-L2 -T2 )W2 

L2Jw1 1 2 2 

-S AT2w2 

n 

t*2 - 2 

1 

J(T-L1)W1 -SL1,W1 w6w 1 2 2

- 2 62 

A( T-L~-T~)w~ -s L2,nW2 2 (9) 

W2 62 T2W2 ’H,H 2 2 

n 

where A = I sL1>wT1 

2 

SL2, w lw lW2 - L1w T1 2w2 

and is the Lagrangian multiplier associated with the revenue 

constraint. It is clear from a glance at the expressions for t*g 

and t*H that only in very special cases will they be equal-­

‘neutrality’ is in general inefficient. 
- 48 -

In the absence of reliable estimates of elements of the Slutsky 

matrix and the parameters of the household production function, is 

there anything else to be said about the neutrality issue? If it 

is assumed that household demand for home goods is relatively 

inelastic and cross effects are all 'small,' then the inverse 

elasticity rule suggests that tH should be higher than t2 at 

the optimum. Suppose, however, that for reasons of administrative 

feasibility t2 must be zero. Can anything be done to reach the 

optimum? In this model, since market inputs are perfect complements 

to wife's time, they can be taxed instead. More specifically, 

suppose that the value of tH which minimizes ( 7) is 

tg , that is, IT*= Ypx + 6w2( l+t&).Then a tax on market inputs, 

tg, can be found such that (l+t;)ypX + 6w2 = Ypx + b2(l+tk) . 

In a model with a more general specification of the household 

production process, we might still want to tax household inputs to 

gain efficiency, but without perfect complementarity, the 

relationship between t& and t; would be more complex.