On Optimal Taxation of Housing

JOURNAL OF URBAN ECONOMICS ARTICLE NO.

43, 315]335 Ž1998.

UE972044

On Optimal Taxation of Housing Helmuth Cremer IDEI and GREMAQ, Uni¨ ersity of Toulouse, Toulouse, France

and Firouz Gahvari Department of Economics, Uni¨ ersity of Illinois at Urbana-Champaign, Champaign, Illinois 61820 Received April 16, 1996; revised, February 12, 1997 This paper studies the question of optimal taxation of housing, when the set of tax instruments at the government’s disposal is not artificially restricted. There are two groups of persons, who differ in earning abilities and in tastes, and two types of housing goods Žhigh- and low-quality.. The paper characterizes the Pareto-efficient allocations that are attainable through the tax policy. It demonstrates that optimality calls for differential tax treatment of housing and that the required tax rates are nonlinear. It derives conditions under which consumption of housing by the poor must be subsidized. It also notes the circumstances under which taxation, rather than subsidization, is the required policy. Q 1998 Academic Press

1. INTRODUCTION There is a longstanding debate among academics and policy makers alike regarding government intervention in housing markets. The recent literature on this issue has seriously questioned the efficacy of the two most important federal policies toward housing. ŽSee w20x for a survey.. The first consists of certain provisions of the U.S. federal income tax codes which result in preferential tax treatment of owner-occupied housing.1 The 1 There are also favorable tax provisions for rental housing; albeit at a much lower scale. The often-advanced argument in favor of this policy is based on a purported externality in the housing market. ŽOne person’s expenditure on housing, increases property values in the neighborhood.. However, while certain types of housing expenditures may contain some measure of externality, the bulk of the expenditures do not. Another argument is that the favorable federal tax treatment is an offset to the local property taxes. The problem here is that property taxes are generally viewed as benefit taxes whose proceeds are used by the locality to provide local public goods.

315 0094-1190r98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

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second concerns provision of housing to the ‘‘poor’’ at below market rents.2 Starting with the pioneering study of Laidler w14x, a number of researchers have calculated the size of the efficiency loss associated with the preferential tax treatment of housing.3 Given the enormous size of residential capital in the United States Žaccounting for nearly half of all private fixed capital., the magnitudes of the estimated welfare costs tend to be quite significant. At the same time, other studies have questioned the policy of public provision of housing to the poor. The criticism of this policy has centered around the presumed inefficiency of in-kind in comparison to cash transfers. 4 From the perspective of optimal tax theory, the twin questions of differential tax treatment of housing in general, and its provision to the poor in particular, can be studied as a single question within a unified framework. Will a government that chooses its tax instruments optimally, tax housing differently from other goods? And if yes, will the tax rate be related to income? It is the aim of the current paper to find answers to these questions. The question we are asking is thus essentially that of the usefulness of non-linear commodity taxes. In this context, one is immediately reminded of Atkinson and Stiglitz’s w2x celebrated result which denies a role for commodity taxes Žlinear as well as nonlinear.. The result has been applied by Atkinson w1x to the very question of housing allowances. However, Atkinson and Stiglitz’s result is crucially dependent on the assumption that consumers have identical tastes. Once this assumption is relaxed, one can no longer rely on the result. 5 The paper proves that, when consumers differ in their tastes, differential tax treatment of housing and its subsidization to the poor may indeed become desirable. The distinctive feature of our study is that we do not restrict the set of available tax instruments at the outset. On the contrary, we choose them optimally on the basis of public availability of information in the economy. The underlying informational structure is the one most commonly used in 2 The rationale for this policy is of course income redistribution; see w1x. The most notable such program in the United States has been the Section 8 Program; see w18x for an empirical investigation. 3 See, among others, w19x, w11x, w12x, w10x, w4x, w17x, and w22x. The loss stems from not taxing the implicit income from home-ownership, while taxing other investment incomes. With the exception of w10x, these studies are invariably based on the assumption that the returns to housing assets must be taxed at the same rate as the returns to non-housing assets. On this particular question, see w13x, w8x, w9x, and w24x. 4 In this context, one must make a distinction between outright provision by the government and price subsidies. See w5x. 5 To be sure, Atkinson emphasizes the fact that the foregoing result holds when ‘‘the only intrinsic differences between individuals are those related to earnings’’ w1, p. 13x.

OPTIMAL TAXATION OF HOUSING

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the optimal taxation literature. An individual’s type Žearning ability and taste. and labor supply are not publicly observable. Hence differential lump-sum taxes are not available. On the other hand, individual incomes are observable so that non-linear income taxes are feasible. Regarding commodity taxes, we assume that the tax administration has information on anonymous transactions but Žwith one exception. not on the identity of the consumers. That is, the administration observes the total sales of a commodity but not who buys how much. This is the standard assumption in the literature, so much so that it has been used as part of the very definition of indirect taxation; see w3, p. 427x. This assumption precludes imposition of non-linear commodity taxes. If, for instance, the tax rate is linked to the quantity purchased, the buyer can avoid higher taxes by splitting the transactions. As a rule, only linear commodity taxation is feasible. The exception we allow is in housing consumption. In this case, we recognize that governments typically have information on personal consumption levels. This is particularly true for owner-occupiers. That individuals purchase housing only very infrequently over their lifetime makes it much easier for the government to obtain information on their personal purchases. Indeed, in the current institutional structure, to assess property taxes, the tax administration does estimate the market value of all owneroccupied houses. This allows the government to levy non-linear taxes on housing.6 We posit a two-group model in which preferences depend on two housing goods Žhigh- and low-quality., n other consumption goods, and labor supply. Consumers exhibit different tastes for the two types of housing goods. In particular, we will assume that tastes for high-quality housing and wages Žor, more loosely, tastes for high-quality housing and incomes. go hand-in-hand. We characterize Pareto-efficient allocations that are constrained, in addition to resource balance, by the standard self-selection constraints as well as the linearity of commodity taxes Žon non-housing goods.. To do this, we derive an optimal ‘‘revelation mechanism.’’ For our purpose, a mechanism consists of a set of type-specific housing consumption levels, before-tax incomes, aggregate expenditures on non-housing goods, and a vector of commodity tax rates Žsame for everyone. associated with non-housing goods. This procedure determines the commodity tax rates right from the outset. A complete solution to the 6 Our results below, concerning the usefulness of non-linear taxesr subsidies on housing, will go through even if we assume that non-linear taxation of other goods are also feasible. We have opted for linearity only because it is a more accurate reflection of reality. See Section 4 for detailed discussion of this point.

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optimal tax problem per-se then requires only the design of an implementing transfer function Žconditioned on both incomes and personal consumption levels of housing.. We prove that differential tax treatment of housing is a useful instrument for achieving a Žconstrained. Pareto-efficient allocation, and demonstrate that if, in comparison to the rich, the poor have a higher marginal willingness to pay for low-quality housing, their consumption must be subsidized. On the other hand, their consumption should be taxed if their marginal willingness to pay Žfor low-quality housing. is lower. The intuition is the same in both cases. When the tax system entails redistribution from high- to low-wage persons, the extent of redistribution is limited by the possibility of high-wage mimicking low-wage individuals. The suggested tax differentiation weakens this otherwise binding self-selection constraint and allows further redistribution. 2. THE MODEL Consider an economy consisting of a large group of individuals who differ in earning ability: Individuals indexed by l are less skilled, earning a lower wage than those indexed by h. There are N l persons in the first group and N h in the second. Each person, regardless of the group he belongs to, is endowed with one unit of time. There are two classes of Žproduced. goods: consumption and housing. There are n consumption and two housing goods. The latter are classified as ‘‘high’’ and ‘‘low’’ quality. All goods are produced by a linear technology with constant producer prices.7 Denote labor supply by L, low-quality housing by y,

7 The constant-producer-price is the standard assumption in the optimal tax literature. It is no more restrictive here than in other optimal tax models, as long as one treats housing as structures only. This is in fact what most models of optimal taxation of housing do; see, e.g., w1x and w10x. Introduction of land poses two added problems. First, the optimal tax problem must take explicit account of land markets. Second, land makes the linearity assumption more problematic. Given a fixed supply of land, one may expect to see decreasing returns in production of housing. If land is homogeneous Ži.e., if it is not differentiated according to size and location., one can think of housing as consisting of structures plus a uniform plot of land. If all individuals are endowed with one plot of land, or if land is owned by absentee owners, there will be no need to change the formal structure of our model. The price of land will simply be a constant in the consumers’ choice problem. In this case, all our results will go through without any modifications. With heterogeneous land, the problem becomes more complicated. The simplest way to generate results similar to those of this paper, is to assume that the government can tax away all land rents and pure profits in housing production. This of course requires certain modifications in our model. In particular, we have to ensure that the resource constraints will encompass the land markets. The analysis of such a model will take us beyond the scope of the current paper.

OPTIMAL TAXATION OF HOUSING

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high-quality housing by Y, and the vector of consumption goods by x. Normalize the producer prices of Y and x at 1, and denote the Žgross of tax. wage by w, and the producer price of y by p. Consumers exhibit different tastes for high- and low-quality housing. However, taste variations are correlated with ability types; within each ability group tastes are similar. In particular, we assume that tastes for high-quality housing and wages Žor, more loosely, tastes for high-quality housing and incomes. are positively correlated. That is, the relative valuation of high- to low-quality housing is higher for the high-wage persons Žin comparison to low-wage persons.. Moreover, it must generally be the case that e¨ ery person would consume one type of housing only. To ensure this, we assume that high- and low-quality housing are ‘‘perfect substitutes’’ for all consumers Žthough not one for one..8 Consequently, every individual will have a constant marginal rate of substitution between high- and low-quality housing. This rate is the same for all individuals within an ability group but different across groups. We further assume that preferences are separable in consumption, housing, and labor supply. This is analytically convenient. It will also provide a basis for understanding our results. Atkinson and Stiglitz w2x have shown that, in the absence of taste variations, if preferences are separable between labor supply and other goods, Pareto-efficient tax structures require uniform taxation of all produced goods. In our context, this means that there should be no tax or subsidy on housing Žas well as on consumption goods.. The separability assumption allows one to isolate the significance of variations in tastes for this result. Preferences are represented by U s u Ž x. q w Ž Y q u j y . y f Ž L . ,

j s l, h,

Ž 1.

where U is twice continuously differentiable and strictly increasing in x, Ž Y q u j y ., and L; u is strictly quasi-concave, and w 0 - 0, f 0 ) 0. The parameter u shows the intensity of preferences for low- versus high-quality housing, with u l ) u h.9 8

A more general specification would be to allow preferences between the two types of housing to be non-convex. As long as the individuals continue to be at one corner, this added generality will not change the results of the paper. 9 A simpler way to introduce taste differentiation is to postulate that relati¨ e to other goods, housing Žas one composite good. is valued less Žor more. by high-wage persons Žin comparison to low-wage persons.. While nonlinear housing taxes can also be shown to be useful here, the assumption is more restrictive than need be. Our assumption requires only that the relative valuation of high- to low-quality housing be higher for the high-wage persons Žin comparison to low-wage persons.. It allows for high-wage persons to have a marginal rate of substitution for low-quality housing Žin terms of non-housing goods. which is greater than, equal to, or smaller than that of the low-wage persons.

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3. FULL INFORMATION It is instructive to start by characterizing the full information optimum. Assume individuals’ types as well as x j, Y j, y j, and L j Ž j s l, h. are publicly observable. Pareto-efficient allocations can then be described by the solution to the following problem. Maximize u Ž x h . q w Ž Y h q u h y h . y f Ž Lh . q m u Ž x l . q w Ž Y l q u l y l . -f Ž Ll . ,

Ž 2. with respect to x h , Y h , y h , Lh , x l , Y l , y l , and Ll ; subject to the resource constraint N h w h Lh y

ž

Ý x ih y Y h y py h i

/ q N ž w L y Ý x y Y y py / G R l

l

l

l i

l

l

i

Ž 3. and the non-negativity constraints 10 Y G 0,

y G 0,

Ž 4.

where R is the government’s external revenue requirement and m is a positive number. Assume that the problem has an interior solution for x j, Ž Y j q u j y j ., and L j. However, it is clear from our setup that consumers must consume one type of housing only. That is, for an individual of type j, either Y j ) 0 or y j ) 0 Žbut not both.. The following lemma, which is proved in the Appendix, helps sort out the possible cases. LEMMA 1. Indi¨ iduals of type j should exclusi¨ ely consume low-quality housing if u j ) p, and high-quality housing if u j - p. Lemma 1 indicates that there are three possible cases of interest.11 First, u ) u h ) p. In this case, everyone should consume only low-quality housing. Second, u l ) p ) u h ; high-ability persons must consume highquality, and low-ability persons low-quality housing. Third, p ) u l ) u h ; everyone must consume only high-quality housing. For later references and comparisons, from Lemma 1 and the first-order conditions of problem Ž2. ] Ž4., we can derive the following expressions for l

10

We have singled out the two housing goods because, given our setup, the non-negativity constraints on other goods may safely be assumed to be automatically satisfied. 11 In what follows, and throughout the paper, we will ignore the intermediate cases of j u s p. This is for conciseness only. Citing these possibilities will add nothing to our discussion. Readers can easily see for themselves the implications of these cases.

OPTIMAL TAXATION OF HOUSING

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a j-type person’s marginal rates of substitution between y j and x ij, Y j and x ij, x kj and x ij, and L j and x ij, for j s l, h, and i, k s 1, 2, . . . , n,

u jw X Ž u j y j . ­ u Ž x j . r­ x ij wX Ž Y j . ­ u Ž x j . r­ x ij ­ u Ž x j . r­ x kj ­ u Ž x j . r­ x ij fX Ž L j . ­ u Ž x j . r­ x ij

s p,

if u j ) p,

Ž 5a .

s 1,

if u j - p,

Ž 5b .

s 1,

Ž 5c .

s w j.

Ž 5d.

That the above marginal rates of substitution must all be set equal to their respective Žproducer. price ratios is obvious. The second theorem of welfare economics tells us that to achieve pareto-efficiency, market prices should not be distorted and redistribution must be done via differential lump sum taxes only. Given public observability of types as well as x j, Y j, y j, and L j, these tax instruments are feasible here. 4. IMPERFECT INFORMATION We now turn to the imperfect information case. As in the literature on optimum income taxation, we shall assume that an individual’s type Žearning ability and taste. and labor supply are not publicly observable. This makes differential lump-sum taxes unavailable; the government must use distortionary taxes in order to effect the desired redistribution. On the other hand, individual incomes, I s wL, are assumed observable. Hence nonlinear income taxes are feasible. Turning to commodity taxes, it is plain that information on personal consumption levels is not typically available to the tax administration, at least not for all commodities. What may reasonably be assumed available is information on anonymous transactions. That is, the government has information on total sales of a commodity but not on who purchases how much. This is the standard assumption in the literature, so much so that it has been used as part of the very definition of indirect taxation. In discussing direct versus indirect taxation, Atkinson and Stiglitz w3, p. 427x write ‘‘the essential aspect of the distinction wisx the fact that direct taxes may be adjusted to the individual characteristics of the taxpayer, whereas indirect taxes are levied on transactions irrespective of the circumstances of buyer or seller.’’

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The assumption that the government observes only anonymous transactions implies that it lacks the required information to levy nonlinear commodity taxes. As a rule, only linear commodity taxes are feasible.12 Note that the informational requirement for non-linear commodity taxation is much more stringent than for nonlinear income taxation. The latter type of taxes require information only on each person’s aggregate expenditure Žor equivalently income.. Non-linear commodity taxes, on the other hand, require information on each person’s expenditure on every single good. The linearity of commodity taxes is thus a direct implication of the informational structure typically available in the economy. In the case of housing, we recognize that governments typically have information on personal consumption levels. This is particularly true for owner-occupiers who constitute about 65% of households in the United States. There are many reasons for this. Individuals purchase housing only very infrequently over their lifetime. This makes it much easier for the government to obtain information on their personal purchases. Moreover, most people finance their housing purchases through borrowing. The fact that mortgages must be officially registered also contributes to the public observability of housing.13 In any event, in the current institutional structure, to assess property taxes, the tax administration does estimate the market value of all owner-occupied houses. Consequently, with personal consumption levels of housing being publicly observable, it is feasible for the government to levy non-linear taxes on it.14 Given this informational structure, one may proceed to characterize Pareto-efficient allocations that are constrained, in addition to resource balance, not only by the standard self-selection constraints but also by the linearity of commodity taxes Žon privately produced goods.. To do this, we derive an optimal revelation mechanism. For our purpose, a mechanism consists of a set of type-specific housing consumption levels Ž Y j s and y j s., before-tax incomes Ž I j s., aggregate expenditures on non-housing goods Ž c j s., and a vector of tax rates Žsame for everyone. associated with 12 For example, any attempt to tie-in commodity tax rates to the quantity purchased can easily be foiled by multiple or grouped purchases. See w5x. 13 The foregoing arguments also apply to automobiles and other consumer durables, albeit not to the same degree. At the same time, the usual arguments pertaining to implicit housing subsidies also equally apply to other consumer durables. 14 Our asymmetric treatment of housing and other goods, with respect to public availability of information, is only due to our belief that it is the more accurate reflection of reality. The following points must be emphasized. First, whether personal purchases of other goods are publicly observable or not have no bearings on our results. Second, the public availability of housing is important only in that it allows housing to be taxed in a non-linear fashion. Without it, given taste variations, one can still prove that differential tax treatment of housing is desirable. In this case, optimal housing taxes will be linear and the extent of redistribution is further restricted.

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OPTIMAL TAXATION OF HOUSING

non-housing goods. This procedure determines the commodity tax rates t s Ž t 1 , t 2 , . . . , t n . right from the outset. A complete solution to the optimal tax problem per-se then requires only the design of an implementing transfer function Žconditioned on I, Y, and y .. Note that instead of commodity taxes, the mechanism may equivalently specify the consumer prices q s Ž q1 , q2 , . . . , qn ., where qi s 1 q t i , Ž i s 1, 2, . . . , n.. To proceed further, it is necessary to consider the optimization problem of an individual for a gi¨ en mechanism Žq, Y, y, c, I .. This is necessitated by the fact that the mechanism determines personal consumption levels Žon non-housing goods. only indirectly, namely, through prices. The mechanism assigns Žq, Y j, y j, c j, I j . to an individual who reports type j. The consumer then allocates c j between the non-housing commodities. Formally, given any vector Žq, Y, y, c, I ., an individual of type j solves max u Ž x . ,

Ž 6a .

x

n

subject to

qi x i s c.

Ý

Ž 6b .

is1

The resulting demand functions are denoted by x i Žq, c . and the indirect utility function by ¨ Ž q, c . ' u Ž x Ž q, c . . .

Note that the functional forms of x i and ¨ are identical for both types of individuals. This is due to the separability of preferences. For ease of notation, we will also define x ij ' x i Žq, c j . and ¨ j ' ¨ Žq, c j . for j s l, h. 4.1. Pareto-Efficient Ž Constrained. Allocations Constrained Pareto-efficient allocations can be described as follows. Maximize ¨ Ž q, c

h

. q w ŽY

qu y

h

q m ¨ Ž q, c

h

l

h

. yf

. q w ŽY

l

Ih

ž / wh

qu y l

l

. yf

Il

ž / wl

,

Ž 7.

with respect to q, c h , Y h , y h , I h , c l , Y l , y l , and I l ; subject to the resource constraint Nh Ih y

ž

Ý x ih y Y h y py h i

/ q N ž I y Ý x y Y y py / G R , l

l

l i

i

l

l

Ž 8.

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CREMER AND GAHVARI

the self-selection constraints ¨ Ž q, c h . q w Ž Y h q u h y h . y f

Ih

ž / wh

G ¨ Ž q, c l . q w Ž Y l q u h y l . y f ¨ Ž q, c l . q w Ž Y l q u l y l . y f

Il

ž / wh

,

Ž 9a .

Il

ž / wl

G ¨ Ž q, c h . q w Ž Y h q u l y h . y f

Ih

ž / wl

,

Ž 9b .

and the non-negativity constraints Y G 0,

y G 0.

The appearance of self-selection constraints Ž9a. ] Ž9b. in this problem, in comparison with problem Ž2. ] Ž4., is due to the public non-observability of individuals’ types. The maximization with respect to q is necessitated because of the non-observability of personal consumption levels x j; they are determined only indirectly. In characterizing the ‘‘solution’’ to the government’s optimization problem, we shall concentrate on one particular configuration of binding constraints. This is the regime where only the ‘‘downward’’ self-selection constraint associated with a high-ability type mimicking a low-ability type binds. That is, to achieve self-selection, the government must make sure only that a high-wage person does not want to mimic the behavior of a low-wage person. This case is ‘‘more interesting’’ in that it intuitively means public policy entails redistribution from high- to low-wage persons.15 Denote the Lagrangean expression by L , the Lagrangean multipliers associated with the self-selection constraint Ž9a. by l, and with the resource constraint by g . Further denote the marginal utility of income of a 15

One must of course also ensure that the ‘‘upward’’ incentive constraint is not binding when the downward constraint binds. When individuals differ in more than one characteristic and with more than two goods, as in our model, one cannot derive simple conditions to ensure this. However, it is intuitively clear that if wage rates are sufficiently different and if the social welfare function does not attach too much weight to the rich, the considered regime will be the only outcome. Indeed, when the social welfare function is such that this regime is the only possible outcome under identical tastes, this regime will continue to be the only possible one when u l ) u h. In this case, taste differentials reinforce the need for redistribution from high- to low-wage persons. Low-ability persons should receive positive transfers for two reasons. First, they have a lower income level. Second, in comparison to high-ability persons, and in terms of non-housing goods, they value low-quality housing more Žand high-quality housing the same.. See w7x.

OPTIMAL TAXATION OF HOUSING

325

j-type individual by a j ' ­ ¨ jr­ c j. Assume that the problem has an interior solution for x j, Ž Y j q u j y j ., and L j. However, it is clear from our setup that consumers will generally consume one type of housing only. Consequently, the Kuhn]Tucker conditions for this problem are ­L s y Ž 1 q l . a h x ih y Ž m y l . a l x il ­ qi y g NhÝ

ž

­L ­ ch ­L ­ cl ­L ­Ih ­L ­Il ­L ­Yh ­L ­ yh ­L ­Yl ­L ­ yl Yh y

­L

­Yh ­L h ­ yh

Yl y

­L ­Yl ­L l ­ yl

­ x kh

k

­ qi

s Ž 1 q l. a y g N h

­ x kl

q NlÝ

­ qi

k h

/

s0,

­ x kh

Ý ­ c h s0,

Ž 10a . Ž 10b .

k

s Ž m y l. a l y g N l Ý

s0,

Ž 10c .

q g N hs0,

Ž 10d.

k

sy sy

1ql wh

m wl

fX

f9

Ih

­ x kl ­ cl

ž / ž / ž / wh

Il

wl

q

l

Il

w

wh

fX h

q g N ls0,

Ž 10e .

s Ž 1 q l . w 9 Ž Y h q u h y h . y g N hF0,

Ž 10f .

s u h Ž 1 q l . w X Ž Y h q u h y h . y pg N hF0,

Ž 10g.

s mw X Ž Y l q u l y l . y lw X Ž Y l q u h y l . yg N l F 0,

Ž 10h .

X l s u mw Ž Y l q u l y l . y u hlw X Ž Y l q u h y l . y pg N lF0,

Ž 10i .

s 0,

Y h G 0,

Ž 10j.

s 0,

y h G 0,

Ž 10k.

s 0,

Y l G 0,

Ž 10l .

s 0,

y l G 0,

Ž 10m.

where we have made use of Roy’s identity in these derivations.

326

CREMER AND GAHVARI

Equations Ž10a. ] Ž10m., along with the equality versions of Ž8. and Ž9a., characterize the Pareto-efficient allocations Žassuming second-order conditions are satisfied.. They will be implemented through a general tax or transfer function. It is important to point out what this may entail. Recall that the allocations are constrained only by the informational structure we have assumed. The only restriction this imposes on the availability of tax instruments is the linearity of taxes on non-housing goods. Consequently, it may be the case that the implementation would require the imposition of a single tax or transfer function on individuals, conditioned jointly on I, Y, and y. This implies that the marginal tax a consumer faces on his consumption of housing would depend on his income as well as on his consumption of housing. Similarly, his marginal income tax rate would depend not only on income but also on his housing consumption. In practice, and in the context of housing, a unified tax function should not be deemed ‘‘undesirable.’’ Some housing subsidy schemes are in fact implemented by reducing the recipients’ taxable income. However, one may also wish to have an implementing function with the property that the housing tax or subsidy would depend only on Y and y, and the income tax on I. If this is the case, one will have to impose these further restrictions on the government’s optimization problem right at the outset. This may restrict the set of constrained Pareto-efficient allocations further. However, it is easy to show that in our two-group model, given the separability of preferences assumption, this will impose no further restrictions. The optimal mechanism we have derived can indeed be implemented by a transfer function which is separable in housing consumption and income. 5. HOUSING AND TAXATION In this section, we first ascertain which consumers consume what type of housing and then study the nature of the taxes that will have to be levied on their consumption. We will see below that, with asymmetric information, the question of who consumes what is determined not exclusively by how the consumer’s relative marginal valuation of low- to high-quality housing, u j Ž j s l, h., compares to the relative marginal costs of these goods, p. The other parameters of the model will also play a part in determining the answer to this question. The following lemma, the counterpart of Lemma 1 under full information, helps sort out the possible cases. The proof of the lemma is given in the Appendix. LEMMA 2. High-ability persons will exclusi¨ ely consume low-quality housing if u h ) p, and high-quality housing if u h - p. Low-ability persons also will exclusi¨ ely consume low-quality housing if u l ) p Ž gi¨ en u l ) u h .. But, if u l - p, they may consume either type of housing.

OPTIMAL TAXATION OF HOUSING

327

Lemma 2 tells us that the type of housing Žlow- or high-quality. that high-ability persons should consume must not be distorted. On the other hand, there are circumstances under which low-ability persons’ choice of housing types will have to be altered.16 Note that even if an individual’s choice of housing types is not distorted, his consumption level might be. Propositions 1]3 below shed light on this issue. There, we shall also develop an intuition for why introducing such ‘‘distortions’’ may be helpful. We next give another lemma which will facilitate our later discussion of the required tax rates on housing goods. This lemma is also proved in the Appendix. LEMMA 3. The optimal commodity tax rates on non-housing goods are uniform. The intuition behind Lemma 3 comes from Atkinson and Stiglitz’s w2x result; see also w23x. Commodity taxation may be used as a basis of separation, in an attempt to bring about more redistribution, if, for the same bundle of goods, marginal rates of substitution between non-housing goods are different for high- and low-ability persons. In our setting, this will be the case if the relevant marginal rates of substitution depend on individuals’ labor supply Žwhich vary with ability. andror on their tastes for housing. The separability assumption ensures that neither of these characteristics will affect one’s marginal rates of substitution between non-housing goods. Note also that because there is one extra degree of freedom in setting the tax rates, one may, without any loss of generality, normalize and set the uniform tax rate on non-housing goods at zero. We shall use this normalization. This means that qk s 1 for all k s 1,2, . . . , n. We are now in a position to consider all possible cases that may arise. Given our stipulation of u l ) u h , there are three possible cases.17 5.1. Case Ž a.: u l ) u h ) p The following proposition gives our first main result. PROPOSITION 1. Assume u l ) u h ) p. Then, Ži. all consumer types will consume only low-quality housing; Žii. the high-wage person’s marginal consumption of housing should neither be taxed nor subsidized; and Žiii. the low-wage person’s marginal consumption of housing should be subsidized Ž his l XŽ l l. marginal willingness to pay be set at less than p . if u hw X Ž u h y l .ru w u y - 1, h XŽ h l. l XŽ l l. and taxed if u w u y ru w u y ) 1. 16

See the proof of Lemma 2 in the Appendix for sufficient conditions under which this type of distortion should or should not occur. 17 Again, we ignore the intermediate cases of u j s p. See footnote 11.

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Proof. The proof of Ži. follows immediately from Lemma 2. To prove Žii. and Žiii., note that Y j s 0 and write the j-type individual’s marginal rate of substitution between y and x i Žhis marginal willingness to pay for y j in terms of the non-housing goods. as

u jw X Ž u j y j . aj

,

Ž 11 .

where in this formula we have made use of qi s 1 and of 18

­ uŽ x j .

s a j qi .

­ x ij

Ž 12 .

Next substitute qk s 1 in Ž6b. and partially differentiate it with respect to c j. It yields

Ý k

­ x kj ­cj

s 1,

j s l, h.

Ž 13 .

In turn, substituting Ž13. into Ž10b. ] Ž10c. and simplifying results in

Ž 1 q l . a h y g N h s 0,

Ž 14 .

Ž m y l . a l y g N l s 0.

Ž 15 .

It now immediately follows from the equality version of Ž10g. and eq. Ž14. that

u hw X Ž u h y h . ah

s p.

This proves that the marginal price of y to the high-ability person Žrelative to x i s. must be set at p. We have thus proved Žii.. Turning to Žiii., manipulating the equality version of Ž10i. and eq. Ž15., yields l uw 9Ž u l y l .

a 18

l

sp

myl h X

l X m y lŽ u w Ž u h y l . . r Ž u w Žu l yl ..

,

Ž 16 .

Equation Ž12. is derived from first-order conditions of the individual’s problem Ž6a. ] Ž6b..

OPTIMAL TAXATION OF HOUSING

329

where, from Ž15., m y l ) 0. To complete the proof, it is then sufficient to show that

u hw X Ž u h y l . l X uw Žu l yl .

can take values both greater and smaller than 1. This is equivalent to having the function uw 9Ž u y . be increasing as well as decreasing in u . This in turn is possible due to concavity of w Ž?..19 Proposition 1 is interesting in two ways. First, it tells us that, contrary to Atkinson and Stiglitz’s result w2x and their subsequent argument Žsee w3x., there is a case for differential tax treatment of housing. Second, it tells us that if, in comparison to the rich, the poor have a higher marginal willingness to pay for low-quality housing, their consumption must be subsidized. On the other hand, a lower marginal willingness to pay calls for a tax rather than a subsidy. The result pertaining to high-ability persons is another facet of the famous ‘‘no-distortion at the top’’ result of the optimal income tax literature Žsee, e.g., w15x, w21x, and w23x.. The intuition is the same. Distorting one’s choices may be beneficial only if it relaxes a self-selection constraint. Now because in equilibrium nobody wants to mimic the high-ability person, there will be no point in taxing his marginal income or consumption. This is also why we found in Lemma 2 that one does not want to distort the high-ability persons’ choice of housing types. Turning to the low-ability individuals, note that one can rewrite the condition in Proposition 1 as u hw 9Ž u h y l .ra l being smaller or greater than l XŽ l l. uw u y ra l. In other words, the question hinges on whether the mimicker’s marginal willingness to pay for y l is smaller or greater than that of the low-ability person’s. If it is smaller, the proposition tells us, the low-ability person must face a subsidy on his marginal consumption of housing; and if it is greater, a tax. The first possibility corresponds to the high-ability person’s indifference curves between y and x i Ž i s 1, 2, . . . , n., being flatter than the low-ability person’s; the second, to their being steeper. This makes quite a bit of sense. When the tax system entails redistribution from high- to low-wage persons, the extent of redistribution is limited by the possibility of high-wage mimicking low-wage individuals. The suggested tax differentiation would make the low-ability person’s package less appealing to the mimicker. This weakens the otherwise binding self-selection constraint thus allowing further redistribution. X X Y The derivative of uw Ž u y . with respect to u is equal to w q u yw . This is positive X Žnegative. if and only if Žyu yw 0rw . is smaller Žgreater. than 1. Thus the condition may be X expressed as having the absolute value of the elasticity of w be smaller or greater than 1. 19

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5.2. Case Ž b .: u l ) p ) u h Lemma 2 implies that, in this case, high-ability persons will consume only high-quality housing, and low-ability persons only low-quality housing. Regarding the question of a housing tax or subsidy, consider high-ability persons first. With y h s 0, it immediately follows from the equality version of Ž10f. and eq. Ž14. that

wX Ž Y h . ah

s 1.

This implies that, as far as high-ability persons are concerned, one should not distort their marginal price of Y Žrelative to x i s.. Turning to the low-ability persons, given that they consume only lowquality housing, it is easy to check that their situation is identical to the one described in case Ža.. Consequently, their tax treatment is governed by Eq. Ž16. and the result in Proposition 1 continues to apply to them. The results in this case are summarized as: PROPOSITION 2. Assume u l ) p ) u h. Then: Ži. high-wage persons will consume only high-quality housing, and low-wage persons only low-quality housing; Žii. the high-wage person’s marginal consumption of housing should neither be taxed nor subsidized; and Žiii. the low-wage person’s marginal l XŽ l l. consumption of housing should be subsidized if u hw X Ž u h y l .ru w u y - 1, h XŽ h l. l XŽ l l. and taxed if u w u y ru w u y ) 1. 5.3. Case Ž c .: p ) u l ) u h In this case, Lemma 2 implies that high-ability persons will consume only high-quality housing, but low-ability persons may consume either lowor high-quality housing. It is again easy to check that the tax treatment of high-ability persons remain the same as in case Žb.. Turning to the low-ability persons, it matters if they consume low- or high-quality housing. Under the former, their tax treatment will be as described in cases Ža. and Žb. above. The second possibility will have them consume only high-quality housing. In this case, from the equality version of Ž10h. and eq. Ž15., one has

wX Ž Y l . al

s 1.

Hence no taxation of Y l is required. The results in this case are summarized as: PROPOSITION 3. Assume p ) u l ) u h. Then: Ži. High-wage persons will consume only high-quality housing, and their marginal consumption of hous-

OPTIMAL TAXATION OF HOUSING

331

ing should neither be taxed nor subsidized. Žii. Low-wage persons may consume either low- or high-quality housing. If they consume low-quality housing, their marginal consumption of housing should be subsidized if l XŽ l l. l XŽ l l. u hw X Ž u h y l .ru w u y - 1, and taxed if u hw X Ž u h y l .ru w u y ) 1. If they consume high-quality housing, their marginal consumption of housing should neither be taxed nor subsidized. As a final observation, we point out the link between our results and the lessons from optimal tax theory. Stiglitz in his discussion of the usefulness of commodity taxes points out ‘‘whether commodity j should be taxed or subsidized relative to k depends on whether the more able individual’s marginal rate of substitution of j for k exceeds that of the low-ability person, or conversely’’ w23, p. 1026x. Stiglitz w23x and Atkinson and Stiglitz w2x have examined economies where individuals differ only in earning abilities. Consequently, whenever preferences are separable in labor and other goods, the two types will have identical marginal rates of substitution Žfor the same consumption bundles.. This in turn implies that no goods, including housing, should be taxed or subsidized. In our model, on the other hand, individuals differ in both earning abilities and tastes. One can then have differing marginal rates of substitution even with separability. It is this property which drives our results on the desirability of differential tax treatment of Žlow-quality. housing. The reason that non-housing goods Žand high-quality housing. go untaxed is due to the particular way we have introduced variation in tastes. Even with our formulation, the high- and the low-ability persons continue to possess identical marginal rates of substitution between these goods Žfor the same consumption bundles..20 6. CONCLUDING COMMENTS This paper has studied the twin questions of differential tax treatment of housing and its subsidization to the poor, within a unified model of optimal income and commodity taxes. Its distinctive feature has been not to restrict the set of tax instruments available to the government arbitrarily. They are determined optimally on the basis of public availability of information in the economy. The main lesson that has emerged is that the differential tax treatment of housing, which is often criticized, may in fact be justified on grounds of optimal tax policy. It can be rationalized under circumstances more general than hitherto assumed. In particular, it is not the case that differential treatment ‘‘is not desirable where the weak separability condition wbetween labor and other goodsx applies’’ w3, p. 439x. 20 With the exception of w16x, the optimal tax literature has generally ignored the implications of taste variations. For another application, see w6x.

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The specific results of the paper depend on the specification of preferences and particularly the separability of preferences between labor, housing, and non-housing goods. Given the orientation of the paper, this was the natural formulation to adopt. Were it not for the separability Žof labor and other goods., the question of ineffectiveness of differential tax treatment would not even arise. Moreover, the aim of the paper has been to concentrate on the tax treatment of housing versus non-housing goods. This focus is sharpened by the ability to treat the non-housing goods as one category while abstracting away from differential taxation of goods within it. Our specification of preferences has allowed us to do this by requiring that the optimal tax rates on non-housing goods be uniform. An alternative way to ensure a single tax rate per non-housing category would be to restrict the tax instruments Žrather than the preferences .. That is, one could simply assume that the tax rate on non-housing goods must be uniform. All non-housing goods can then be aggregated into a single composite commodity. It is easy to see that all our results will continue to hold under this alternative formulation. No modifications are necessary. Finally, for precise policy prescriptions, it will be desirable to undertake two further studies. First, one may consider a more general specification of preferences allowing for differential tax treatment of non-housing goods. This would make the characterization of optimal tax rates more complicated. But one should be able to derive conditions for subsidizing housing, along the lines of this paper. Second, one must empirically investigate if such conditions are in fact satisfied. These issues are left for future research. APPENDIX Proof of Lemma 1. Denote the Lagrangean expression associated with problem Ž2. ] Ž4. by V and the Lagrangean multiplier associated with the resource constraint Ž3. by g . We have

­V ­Yh ­V ­ yh ­V ­Yl ­V ­y

l

s wX Ž Y h q u h y h . y g N h ,

Ž A1.

s u hw X Ž Y h q u h y h . y pg N h ,

Ž A2.

s mw X Ž Y l q u l y l . y g N l ,

Ž A3.

X l s u mw Ž Y l q u l y l . y pg N l .

Ž A4.

OPTIMAL TAXATION OF HOUSING

333

We know from Kuhn]Tucker conditions for problem Ž2. ] Ž4. that ­ Vr­ y j F 0, and that Y j ) 0 « ­ Vr­ Y j s 0. Now assume u j ) p. It follows from equations ŽA1. ] ŽA4. that

­V ­Y

s0«

j

­V ­yj

) 0,

j s l, h.

This rules Y j ) 0 out. Next assume u j - p. In this case, eq. ŽA1. ] ŽA4. imply

­V ­y

j

s 0«

­V ­Yj

) 0.

A similar argument will then rule y j ) 0 out. Proof of Lemma 2. Consider high-ability persons first. From Kuhn]Tucker condition Ž10j., Y h ) 0 « ­ Lr­ Y h s 0. While from Ž10f. ] Ž10g., one can write

­L ­y

h

yuh

­L ­Yh

s Ž u h y p.g N h .

Ž A5.

If u h ) p, from ŽA5., it follows that

­L ­Y

h

s0

«

­L ­ yh

) 0,

contradicting Kuhn]Tucker condition Ž10g.. This rules Y h ) 0 out. Next, from Kuhn]Tucker condition Ž10k., y h ) 0 « ­ Lr­ y h s 0. While, from ŽA5., if u h - p,

­L ­y

h

s0

«

­L ­Yh

) 0.

This contradicts Kuhn]Tucker condition Ž10f.. Consequently, in this case, y h ) 0 is ruled out. Now consider low-ability persons. From Ž10h. ] Ž10i., one can write

­L ­y

l

yul

­L ­Y

l

s Ž u l y u h . lw X Ž Y l q u h y l . q Ž u l y p . g N l . Ž A6.

If u l ) p, the right-hand side of ŽA6. will be positive. Following the exact steps as for u h ) p, one can immediately rule Y l ) 0 out. On the other

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hand, if u l - p, the left-hand side of ŽA6. can take both positive as well as negative values. A positive expression rules Y l ) 0 out, while a negative expression rules y l ) 0 out.21 Proof of Lemma 3. Substitute from eq. Ž10b. for Ž1 q l. a h and from Ž10c. for Ž m y l. a l into Ž10a. and simplify. This yields

ž

N h Ý x ih k

­ x kh ­ ch

q

­ x kh ­ qi

/

ž

q N l Ý x il k

­ x kl ­ cl

q

­ x kl ­ qi

/

s 0.

Ž A7.

Using the Slutsky equation, one can further simplify equation ŽA7. to NhÝ k

­˜ x kh ­ qi

q NlÝ k

­˜ x kl ­ qi

s 0,

Ž A8.

where ˜ x kj denotes the j-type person’s compensated demand for good k. Next rewrite Eq. ŽA8. by substituting qkrŽ1 q t k . for 1. This results in NhÝ k

qk

­˜ x kh

1 q t k ­ qi

q NlÝ k

qk

­˜ x kl

1 q t k ­ qi

s 0,

i s 1, 2, . . . , n. Ž A9.

From the property Ý k qk Ž ­ ˜ x kj r­ qi . s 0, it immediately follows that t k s t Ž k s 1, 2, . . . , n. is a solution to ŽA9.. ACKNOWLEDGMENTS We thank Jan Brueckner and two anonymous referees for helpful comments.

21 Technically speaking, it is also possible that the right-hand side of ŽA6. will be zero, so that both ­ Lr­ Y l and ­ Lr­ y l are zero. Of course, this can happen only for a particular set of parameter values. Under this particular circumstance, one or both variables Y l and y l may take positive values. Consumers, however, do not ‘‘mix’’ their consumption of housing. Consequently, we will interpret this to mean that, while one cannot say if Y l ) 0 or y l ) 0, only one of the two will hold.

REFERENCES 1. A. B. Atkinson, Housing allowances, income maintenance and income taxation, in ‘‘The Economics of Public Services’’ ŽM. S. Feldstein and R. P. Inman, Eds.., Macmillan, London Ž1977.. 2. A. B. Atkinson and J. E. Stiglitz, The design of tax structure: Direct versus indirect taxation, Journal of Public Economics, 6, 55]75 Ž1976.. 3. A. B. Atkinson and J. E. Stiglitz, ‘‘Lectures on Public Economics,’’ McGraw-Hill, New York Ž1980..

OPTIMAL TAXATION OF HOUSING

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4. J. Berkovec and D. Fullerton, A general equilibrium model of housing, taxes, and portfolio choice, Journal of Political Economy, 100, 390]429 Ž1992.. 5. H. Cremer and F. Gahvari, In-kind transfers, self-selection and optimal tax policy, European Economic Re¨ iew, 41, 97]114 Ž1997.. 6. H. Cremer and F. Gahvari, Nonlinear pricing, redistribution and optimal tax policy, mimeo Ž1996.. 7. H. Cremer and P. Pestieau, Redistributive taxation and social insurance, International Tax and Public Finance, 3, 281]295 Ž1996.. 8. F. Gahvari, Incidence and efficiency aspects of differential taxation of residential and industrial capital in a growing economy, Journal of Public Economics, 25, 211]233 Ž1984.. 9. F. Gahvari, The optimal taxation of housing, Public FinancerFinances Publiques, 39, 213]225 Ž1984.. 10. F. Gahvari, Taxation of housing, capital accumulation and welfare; a study in dynamic tax reform, Public Finance Quarterly, 13, 132]160 Ž1985.. 11. P. H. Hendershott and S. C. Hu, Government-induced biases in the allocation of the stock of fixed capital in the United States, in ‘‘Capital Efficiency and Growth’’ ŽG. M. Von Furstenberg, Ed.., Ballinger, Cambridge, MA Ž1980.. ¨ 12. M. A. King, The distribution of gains and losses from changes in the tax treatment of housing, in ‘‘Social Science Research Council Programme, No. 20, London’’ Ž1981.. 13. L. J. Kotlikoff, The optimal imputation of rent, mimeo Ž1978.. 14. D. Laidler, Income tax incentives for owner-occupied housing, in ‘‘The Taxation of Income from Capital’’ ŽA. D. Harberger and M. J. Baily, Eds.., Brookings Institution, Washington, DC Ž1969.. 15. J. A. Mirrlees, An exploration in the theory of optimal income taxation, Re¨ iew of Economic Studies, 28, 175]208 Ž1971.. 16. J. A. Mirrlees, Optimal tax theory: A synthesis, Journal of Public Economics, 6, 327]358 Ž1976.. 17. Y. Nakagami and A. M. Pereira, Budgetary and efficiency effects of housing taxation in the United States, Journal of Urban Economics, 38, 68]86 Ž1996.. 18. W. J. Reeder, The benefits and costs of the section 8 existing housing program, Journal of Public Economics, 26, 349]377 Ž1985.. 19. H. S. Rosen, Housing decisions and the U.S. income tax: An econometric analysis, Journal of Public Economics, 11, 1]23 Ž1979.. 20. H. S. Rosen, Housing subsidies: Effects on housing decisions, efficiency, and equity, in ‘‘Handbook of Public Economics’’ ŽA. Auerbach and M. Feldstein, Eds.., Vol. 1, North-Holland, Amsterdam, Ž1985.. 21. J. Seade, On the shape of optimal tax schedules, Journal of Public Ecnomics, 7, 203]236 Ž1977.. 22. J. Skinner, The dynamic efficiency cost of not taxing housing, Journal of Public Economics, 59, 397]417 Ž1996.. 23. J. E. Stiglitz, Pareto efficient and optimal taxation and the new new welfare economics, in ‘‘Handbook of Public Economics,’’ ŽA. Auerbach and M. Feldstein, Eds.., Vol. 2, North-Holland, Amsterdam Ž1987.. 24. S. J. Turnovsky and Okuyama, Taxes, housing, and capital accumulation in a two-sector growing economy, Journal of Public Economics, 53, 245]267 Ž1994..