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Housing prices and tenure choice with asymmetric taxes and progressivity
Journal
of Public
Economics
19 (1982) 271-290.
North-Holland
Publishing
Company
HOUSING PRICES AND TENURE CHOICE WITH ASYMMETRIC TAXES AND PROGRESSIVITY Peter ENGLUND
and Mats PERSSON*
Stockholm School of Economics, S-113 83 Stockholm, Sweden Received January
1981, revised version
received
February
1982
This paper studies the effects of tax schedule changes on prices and tenure choice in the housing market. It is shown that, given the present asymmetric treatment of owner-occupants vs. renters, an increase in the degree of progressivity is likely to lead to an increase in the prices of both owner-occupied and rental housing. A numerical example indicates that the effects may be quite large. Equilibrium prices are calculated based on the actual Swedish income tax schedules for 1971 and 1979. According to these simulations the tax changes that took place between these years caused the price of owner-occupied houses to increase by around 30 percent, and the rent level to increase by 2 or 3 percent.
1. Introduction During the 1970s the market for owner-occupied houses boomed in many Western countries. In Sweden the real price of owner-occupied houses increased by 32 percent from 1971 to 1979, and this development was accompanied by a significant increase in the rate of construction. At the same time, the market for rental dwellings was depressed. Rates of construction went down and rents followed consumer prices fairly closely. This general development is due to a combination of factors, the relative strengths of which vary from country to country. The two most commonly stated factors are both related to the increased rates of inflation. One is that this increase has been accompanied by a decrease in the real rate of interest and hence by an increase in property values. The other factor is related to the tax system. In most countries the imputed rent from owner-occupied housing is untaxed, or taxed at a much lower rate than capital income from most other sources. In particular the interest rate used for imputation tends to be fixed, perhaps around what is regarded as a ‘real’ interest rate of a couple of percentage points, whereas the interest paid on loans is deductible by its nominal *This research was supported by research grant 7908763 from the Swedish Council for Building Research, and by the Bank of Sweden Tercentenary Foundation. We wish to thank Richard C. Arnott, Karl G. Jungenfelt, Karl-Giiran Maler, Eytan Sheshinski, and Yoram Weiss for valuable comments on earlier drafts.
0047~2727/82/000Cr0000/$02.75
0
1982 North-Holland
272
P. Englund and M. Person,
Housing prices and tenure choice
amount and interest income on bank accounts is taxed nominally. Hence, when inflation rates and nominal interest rates climb, there is an increasing asymmetry in the tax system and a house will become a more attractive item in the household’s portfolio. The two factors mentioned above may be sufficient to explain even quite significant increases in the owner-occupiers’ demand for housing. [See Hendershott and Hu (1979) for calculations on U.S. data.] They also explain why some households have moved from rental dwellings to owner-occupied houses. Hendershott and Shilling (1980) estimate that the decline, over the period 1955-79, in the cost of owner-occupied housing relative to rental housing has caused the U.S. homeownership rate to be 65 percent rather than 60 percent. It is quite evident that changes in inflation will have the effects stated above. It is less frequently discussed that changes in the income tax schedule may also have major effects if the tax system is asymmetric in the sense described above and if marginal tax rates increase with income. The main purpose of this paper is to study these effects both analytically and by simulating equilibrium prices for the actual Swedish income tax schedules of 1971 and 1979. The analysis is done within a short-run partial equilibrium framework where the supply of housing is fixed. The effects on prices of changes in the tax schedules are then interesting for two reasons. First, an increase in the price of owner-occupied houses will obviously affect resource allocation. Resources will be diverted to construction of new houses. Second, our results suggest that changes in the income tax schedules that are intended to be egalitarian may have quite opposite effects on the income distribution. An increase in the degree of progressivity of the tax schedules might very well cause an increase in the market price of owner-occupied houses. Since high-income earners tend to own their homes, while lowincome earners tend to rent theirs, such a price increase will materialize in the form of capital gains for the high-income earners, thereby neutralizing the ‘egalitarian’ change in the income tax schedules.
2. Consumers’ choice The consumer faces a two-stage decision problem. First, he chooses the mode of tenure. Then, he chooses the quantity of housing. The choice of mode of tenure can be explained by various factors. Owner-occupied houses and rental dwellings may be seen as different goods and differences in tastes may hence be regarded as an explanatory factor. This view typically lies behind econometric studies, as for example the ones by Li (1977) and Rosen (1979). Another view is that what enters people’s utility functions is housing services. This good can either be purchased directly in the market for rental dwellings, or it can be produced using as inputs the house owned, labour and
P. Englund and M. Person,
Housing prices and tenure choice
213
fuel, etc. Differences in productivity in this household production process may then explain why some choose to rent and some to own their home. This is the approach taken by Weiss (1978). One may also focus on differences in risk attitudes. Prospective capital gains are an important part of the economic calculations of a homeowner, and since these are uncertain people will value them differently. In this paper we will abstract from all these factors, i.e. we will assume that there is a homogeneous good named ‘housing services’ that can either be rented or produced by owner-occupants, in which case all households are equally productive. Instead, we will focus on the fact that people typically pay different after-tax prices for owner-occupied housing. We assume that the imputed rent from an owner-occupied house is not taxed at all, whereas interest payments are deductible from taxable income by their full amount. This implies that the price of owner-occupied housing is inversely related to the individual’s marginal tax rate. Hence, high-income earners with high marginal tax rates will tend to own their homes, whereas low-income earners will be renters.’ Consumers maximize a quasi-concave utility function U(h,z), where h is the amount of housing stock (= housing services) consumed, and z is the amount consumed of other goods. For renters the budget constraint is straightforwardly written as Rh+z=y(x), where R is the per unit rent, and where the price of other goods is normalized to equal unity. y(x) is the disposable income out of pre-tax income x, which for simplicity is regarded as exogenous. It is assumed that y’(x) >O and that y”(x)
In (1) it is assumed that all housing costs2 are deductible from taxable income and can be expressed as a constant r times the value of the house poh. The parameter Y will here be interpreted as an interest rate, and other housing costs including expected capital gains will be disregarded.3 The ‘This conforms with empirical studies by Li (1977) and Rosen (1979) which show that income significantly affects the choice of mode of tenure, and in particular King (1980). where taxinduced price differences is the only factor which explains why some rent and some and some own. ‘par is the pre-tax rental cost of owner-occupied housing. We will often talk about this as the price of owner-occupied housing without pointing out at every instance that it is a pre-tax price and that housing costs vary across individuals according to their marginal tax rates. .‘Scc the discuGon at the end of section 3 below.
274
P. Englund
and M. Persson, Housing
prices and tenure choice
equation then says that the expenditure on other goods equals the disposable income left for consumption other than housing.4 The renter’s maximization problem will give rise to a Marshallian demand function for rental housing, h,. The owner-occupant’s solution of the optimization problem will yield a demand function for owner-occupied housing, h,, which is a function of income, price and the parameters of the tax system (which we express by a vector 0)’ The choice whether to rent or to own is made by comparing the maximum utility obtainable from renting with the maximum utility obtainable from owner-occupancy. The renter’s indirect indirect utility function is V(y(x), R), whereas the owner-occupant’s utility can be written as P(x, 8, pot+), Hence, individual demand is given by the system h, = Q(x),
R),
ho = t;(x, 6 nor),
Y(x) 5 0, Y(x) 2 0,
(2)
Y(x) = B(x, 6, par) - V(y(x), R).
Under certain conditions Y(x*) = 0 represents a unique switching-point between owning and renting;’ everybody with an income above x* will own and everybody else will rent. We will assume that these conditions are fulfilled, as they can be shown to be, for example, for a logarithmic utility function combined with any concave disposable income function, y(x). Problems of multiple switch-points may arise if there are intervals over which the income elasticity of housing is high and the marginal tax rate is falling sharply with taxable income, and hence with housing consumption. It would seem that homotheticity of preferences alone is a sufficient condition. It appears, however, that it has to be combined with some regularity condition on y(x). One such condition is that it is exponential like the one we use in subsection 4.2 and the numerical simulations below. The situation at x* is depicted in fig. 1. Whereas the owner’s budget constraint is strictly convex and the renter’s is linear, the joint budget ‘I:q. (1) presumes that there is no taxation of imputed rent. Although this approximation, it is not entirely true for some countries, e.g. Sweden. Assuming instead a fraction 6 of housing costs is deductible, the equivalent budget constraint is
is a fair that only
In the following, we will assume that 5 is sufficiently close to unity to be disregarded. ‘With the assumptions made concerning the tax function y(,), the owner-occupant’s budget set will be strictly convex. The demand h, will therefore be unique for each vector (x, 0, par). ‘.x* is the unique switch-point income if &y’[x*-rp,h,(x*)]>&y’(x*), where I?o is the Lagrange multiplier associated with the owner’s maximization problem and 1, is the corresponding multiplier for the renter. For a derivation of this condition and further discussion of the uniqueness issue, see Englund and Persson (1981).
P. Englund and M. Person,
275
Housing prices and tenure choice
Fig.
1
hh
>X
Fig. 2
constraint (as depicted by the heavy curve) is non-convex. This implies that the demand for housing is a discontinuous function of pre-tax income, as illustrated by fig. 2. It also implies that a lump-sum transfer of post-tax income da given at the switch-point will lead the household to prefer renting;’ av/ac( > aV/acr for X=X *. The explanation for this is simply that the ‘Note that this is opposite to the effect that is the main message transfer of pre-tax income will lead the household to prefer owning.
of the model,
namely
that a
marginal utility of money is higher for the renter, since he faces a lower marginal price of housing than the owner (and the same marginal price of the numeraire). This is directly seen by comparing the slopes of the budget lines at ho and h, in fig. 1.
3. Market equilibrium Individual demand is represented by the system (2). Having assumed a unique switch-point between owning and renting, and assuming a continuum of households, the aggregate demand for owner-occupied housing, Ho, and by integrating over the individual demand rental housing, H,, is obtained functions:
Ho = j-ho 4(x) dx X*
(3)
and
H, =
70 h, 4(x) dx,
(4)
where 4(x) is the density function of the income distribution. The switch-point between owning and renting, x*, is given by Y(x*) = 0.
(5)
The supply of housing can in the short run be regarded as exogenously given. There are normally considerable costs, legal and others, associated with transforming rental housing into owner-occupied housing or vice versa. Therefore it is reasonable to treat not only the total housing stock, but also its components, i.e. rental and owner-occupied housing, as exogenously given.8 Denote these supplies by Hs and Hi, respectively. We then have the equilibrium conditions
H,=Hs,
(6)
H,=H;,
(7)
and
where aggregate demands H, and H, are given by (3H5). Thus, par, R, and of the utility .x* are determined as functions of H& Hi, the parameters “On the other hand, we assume the stocks of II, and H, to be perfectly malleable. This of course is a stringent assumption. Ideally, one would let short-run supply consist of a number of indivisible dwellings.
P. Englund and M. Person,
Housing prices and tenure choice
,
211
function, the tax function, and the income distribution. This can be referred to as short-run equilibrium since the stocks of Ht and HE are treated as exogenous, and it will be the main topic of this paper. In the longer run, the mode of tenure will be flexible. If total supply is still . * regarded as fixed, the equilibrium solution is given by H,+H,=HS,
(8)
together with the switch-point equation (5). In this equilibrium the price of owneroccupied housing must be equal to the price of rental housing, i.e. po = pR = p. Since landlords regard the holding of rental housing as an ordinary investment, a perfect capital market requires pr=R.
(9)
Eqs. (8) and (9) thus determine the values of pr and R which, by the demand functions (3) and (4), give us the distribution of housing stock between owned and rented dwellings. It is easy to see that in such a case, which may be termed medium-run equilibrium, all housing will be transformed into owneroccupancy. By (9) we have R -> y’(W
1,
(10)
for all y’()< 1. Thus, renting is less favourable than owning, at the margin, which means that in equilibrium all dwellings will be transformed into owner-occupancy. This property is a direct consequence of the asymmetry of the tax system and the assumption that the mode of tenure does not enter the utility function. The short-run prices will also affect the rate of housing investment until a long-run equilibrium is reached, where prices equal production costs. It is important to notice that the rates of supply adjustment both transformation of tenure and new construction - will have an impact on the size of the short-run price changes. This is clear if proper account is taken of expected capital gains (losses). In such a case the budget constraint (1) should be written as
z = Y(X-w-h + Iio), with expectations assumed to be formed with perfect foresight - instead of, as in our case, being assumed static with Ijo = 0. Also, the landlords’ portfolio choice should depend not only on rent incomes, but also on expected capital gains, rjR. Assuming that the rate of construction is a function of the
P. En&md
278
and M. Persson.
Ifousin~
priers
and tenure choice
difference between the market price (p. and pR, respectively) and the construction cost, we would have a dynamic equilibrium model which will have essentially the same qualitative properties as our model. For numerical calculations of the difference between the two assumptions about expectations formation, see Poterba (1980). As seen above, the present asymmetry in the tax system implies a corner solution: in the medium and long run only owner-occupied dwellings will exist. In reality, such an extreme solution will of course not obtain. People differ in tastes and abilities, and different modes of tenure do not provide exactly the same type of services. Nevertheless, the corner solution points at an important tendency towards owner-occupancy that is created by the tax system. And since such a tendency will be more pronounced, the higher is the tax rate, this provides a plausible explanation of an observed feature in many countries today: parallel with an increase in tax rates, we observe how rental dwellings are transformed into owner-occupied ones (condominiums) at an increasing pace. However, in what follows we will confine ourselves to analyzing the short-run equilibrium.
4. Comparative
static analysis of tax schedule changes
We are concerned with the effects of changes in the income tax schedule, given the asymmetric treatment of housing costs. A change in the income tax schedule affects housing demand in two ways. First, it affects disposable income. Secondly, a tax change normally is not a simple shift of the tax function, but is also a change in the marginal tax rates. Thus, it affects the marginal cost of owner-occupied housing, y/(x-rp,h,). A tax increase will therefore lower the individual demand for rental housing, if housing is a normal good, whereas the individual demand for owner-occupied housing may be affected in either direction. Analyzing the short-run market equilibrium (3)-(5) shows that very little can be said about the effects of changes in the tax schedule. One reason for this ambiguity is the non-convexity of the budget constraint. As pointed out above, this implies that, at the switch-point, the marginal utility of income is higher for the renter than for the owner-occupant. This property contributes to make even an increase in lump-sum taxation ambiguous. For example, one would guess that the introduction of a lump-sum tax (which does not affect the slope of the budget constraint, but only reduces disposable income for everybody) would depress demand so that equilibrium prices would fall for both types of housing. This is however not necessarily true; some renters, facing a lower disposable income, might move into owning.’ Thus, aggregate “This follows section 2 above.
from
the differences
in the marginal
utility
of money
discussed
at the end of
P. Englund und M. Person,
Housing prices and tenure choice
219
demand for owner-occupied housing may actually increase, although all previous owners certainly reduce their demands, and the result will be an increase in the price level. To be able to say something more definite about comparative statics, we have to put some restrictions on our model. We have done this in two ways. First, we study the model with an unspecified utility function and a very general tax function, but linearize the budget constraint in a suitable way. Second, we accept the non-convex budget set, but assume that the utility function is logarithmic in h and z, and that disposable income is an isoelastic function of pre-tax income. 4.1. Linearization
of the budget
constraint
By treating the budget constraint of the owner-occupant as though it were linear we can focus on the impact of the utility function on the various effects of changes in the tax schedules. The correct procedure would be to linearize around equilibrium housing consumption [see, for example, Hall (1973)]. If we did this we would still have to cope with the problem that the marginal utility of money at the switch-point x* would differ between an owner and a renter. To avoid this we linearize instead around h = 0, which implies that at the switch-point the budget constraints for owner and renter coincide. Clearly, this may be misleading for a commodity like housing which occupies a large share of the household’s budget. We write disposable income as
Y(X)= @l+/k(x),
(11)
where g(x) is any non-negative, increasing and concave function. This fairly general formulation allows us to distinguish between changes in the level of taxation and changes in the degree of progressivity. A change in the level of taxation can have either a lump-sum character, da, or imply an equiproportionate change in marginal tax rates, dfi. An increase in progressivity can here be modelled by increasing a and making an offsetting reduction in p so as to leave total tax revenue unaffected. With the disposable income function (1 l), we can write the market equilibrium equations (3)(5) in this simple form:
(12) (13) bg’(x*)p,r
= R.
(14)
280
The comparative
P. En&ud
and M. Persson, Housing
prices und trmrre
statics of this model are summarized
diwl W dx*/
+ &
choir
below:”
C-1 + ‘,
The effects of a lump-sum income increase on both prices are unambiguous. An increase in p will lead to an increase in R but, in general, to a decrease in per. Sufficient conditions for the latter effect (dpor/da < 0) is that housing and other goods are gross substitutes and that not even a household with zero pre-tax income will have a negative disposable income (i.e. ~20). Let us next regard the effects of an increase in progressivity, where a reduction in /I is compensated by an increase in CI so as to leave total tax revenue unaffected. Under the above assumption this will lead to an increase in the price of owner-occupied housing, par. The rent level R will also increase unless the marginal outlay on housing is much larger for highincome than for low-income groups; a sufficient condition is that the Engel curve is linear, i.e. that Zh/c7y is constant irrespective of income. Finally, we have the effects on the switch-point x*. An increase in cx will in general, e.g. for a logarithmic utility function, lead to an increase in x*, i.e. people will move over to renting from owning. This is so because the equilibrium price of housing at x* can be shown to increase more for the owner than for the renter. The effect of a change in j3 is ambiguous. The crucial factor is the relative size of the price and income elasticity for owners as opposed to renters. If owners have a high income elasticity and a low price elasticity relative to renters, dx*jdp is positive, i.e. an increase in marginal tax rates (dp
budget set
The results in the previous section were obtained at the cost of making a linear approximation of the budget constraint. To take proper account of the “‘I-or the formal derivation and Person (I 98 I ).
of this and the following
comparative-statics
results, see Englund
P. Englund and M. Person,
Housiq
prices and tenure choice
non-linearity of the budget set, we will now parameterize the model suitable way. We thus assume that the utility function is given by U(h,z)=alogh+blogz, while disposable Y(X) =
income
w,
281
in a
(15)
is given by j>O,
O
(16)
Here the parameter y can be directly interpreted as a measure of progressivity. Given any pre-tax income distribution and two tax systems of Lorenz curve associated with y2 the form (16), with yi >yZ, the post-tax strictly dominates the Lorenz curve associated with y1 [see Jakobsson (1976)]. The utility function (15) and the disposable income function (16) give rise to the individual demand functions
/,,=a?
a+by
par
(17)
and h,=p
a
/?x’
a+b’R’
As expected, h, has unitary elasticities with respect to disposable income and rent, while h, exhibits unitary elasticities with respect to pre-tax income and pre-tax price. In fact ho is independent of the level of taxation, /I; the price and income effects exactly cancel. As for an increase in progressivity’ ’ (dy < 0), we see that it makes h, decrease. This is hardly surprising, since it makes everybody’s disposable income fall. The effect on ho is more interesting. For the high-income earners (i.e. the home-owners) we get the slightly paradoxical result that their demand for housing is actually stimulated by an increase in progressivity. Although their disposable incomes fall, the price effect outweighs the income effect so that a decrease in y makes ho increase. The market equilibrium conditions are:
H;=
(19)
“Note that this is an uncompensated change, i.e. an increase in taxation via the elasticity parameter. For compensated changes, we will in general get the same effect as with the tax schedule (11); both demands will increase.
282
P. Englund
and M. Person,
ffousing
prices and tenure choiw
and
(20) \vith the switch-point
x* determined
by
.,,(~~)+b log(&Bx’-) (21) lhc
effects on the market
equilibrium
are summarized
below:
There is a striking contrast between the effects in the two markets. An increase in taxation not affecting progressivity (d/I
to the Swedish tax system
In the preceding section affect the prices of housing
we have seen how changes in the tax schedules and the choice of tenure. While the direction of
P. Englund and M. Persson, Housing prices and tenure choice
283
these effects can be analytically determined, at least for some reasonable specifications of the utility and the tax functions, their magnitudes can be evaluated only by numerical methods. In this section we will use the model to simulate equilibrium housing prices for the Swedish tax system in 1971 and in 1979 to evaluate the impacts of the tax changes that took place in the 1970s. Fig. 3 shows the disposable income function y(x) for 1971 and 1979, and fig. 4 displays the marginal tax rates 1 -y’(x) for the two years.” The tax
60 000 50 000 40008 30 000 20 000
I
10000 i
IX
50 000
100 000
150 000
Fig. 3. Disposable income (including housing allowances) as a function of pre-tax income according to the 1971 tax system (solid line) and the 1979 tax system (broken line). 1971 SW. crowns.
1- Y’(X)
0.80 0.700.60 -
50 000 Fig. 4. Marginal
tax rates
100000
150000
for Sweden 1971 (solid line) and 1979 (broken intervals of 5,000 SW. crowns. 1971 prices.
line), computed
at
“The figures for 1979 are deflated to the 1971 price level by the consumer price index. Data are taken from a study of the Swedish income tax system by Blanck and Matthiessen, recently updated by Herthelius and Lind (1980).
284
P. Englund
and M. Persson. Housing
prices and tenure choice
schedules include the effect of housing allowances. These are related to housing consumption, income and family size. However, for most households the level of housing consumption is so high that housing allowances are not affected at the margin. Fixing family size, it can hence for these households - be assumed that housing allowances only depend on income. To simplify we assume that this is valid for all. The tax schedule used applies to a family with one income-earner and two children. Fig. 4 shows that the marginal tax rate has a tendency to increase with income. It is not a monotone increasing function, however, as is presumed in our previous analysis. This is due to the housing allowances which are dependent on income and therefore give very high marginal rates in some brackets a bit above the average. Applied to the model this may give several switch-points between owning and renting. The question arises of whether one should use some simple approximation to the tax function, or the actual schedules depicted in figs. 3 and 4. We have chosen to approximate the actual schedule by the constant-elastic function (16) for two reasons. First, it provides some simplifications that cannot be obtained if the actual tax schedules are used. In combination with a logarithmic utility function it implies that the ratio of the prices for owner-occupied houses between two tax systems is homogeneous of degree zero in the supplies Hi and Hi. Thus, we can reduce the number of parameter values that we must fix for our simulations; with the actual tax system we must know both Hg and H& while with the parametric form we need only their ratio Hz/Hi. Second, it is not really certain that the households regard the actual tax schedules as permanent, and thereby entirely relevant to their housing decisions. Middleincome earners, facing marginal tax rates of 80 and 90 percent over narrow brackets, will hardly expect these rates to prevail over their entire planning horizon. And while the housing allowances obviously play some role in the families’ decisions, it is difficult to say exactly which role they play. In the choice between excluding them completely and including them but fitting a monotone, iso-elastic function to the system, we have chosen the latter alternative. Fitting the function (16) to the actual tax schedules by least squares regression’ 3 we obtain the following values for J and y: 1971: y= 1o9.7.xo.53,
R2 = 0.979,
1979: y = 547.3 xo,37,
R= = 0.990.
The correlation coefficient is of course a questionable There is nothing stochastic about the tax schedules. “The
regressions
are made for the case of an interval
measure in this case. Nevertheless, it gives
of 1.000 SW. crowns.
P. Englund and M. Persson, flousin~ prices and tenure choice
285
some indication about the goodness of tit of the curves. Also, the observations that are included in the regressions are at our discretion.14 Comparing these estimates it seems clear that the tax system of 1979 is more progressive than that of 1971. At the same time fig. 3 shows that the level of taxation was higher in 1979 than it was in 1971.” We will now calculate the ceteris paribus effect of this tax schedule change for the isoelastic formulation of the y(.) function and a logarithmic utility function, i.e. the case analyzed in subsection 4.2 above. During the 1970s at least five things have happened in the Swedish housing market. (i) The supplies, Hi and Hz, have increased with Hg distribution $(x) has increasing much faster than Hz. (ii) The income changed. (iii) The real rate of interest, r, has fallen. (iv) The tax function, i.e. the parameters p and y, have changed. (v) The prices p. and R have changed. Now, our aim is to study the relation between points (iv) and (v). In order we regard supplies and income to concentrate on this main point, distribution as unchanged. The role played by a decrease in I is immediately seen from the model, which only determines the product rp,; a decrease in r is in this model fully capitalized in house values due to the assumption of static expectations. To fix figures we approximate the actual income distribution by a log-normal distribution, where the logarithms of gross income are normally distributed with mean 9.84 and variance 0.66.1h The solution of the system (19x21) gives us R, rp, and x* as functions of the stocks of housing Hi and HE, of the ratio of the parameters of the utility function a/b, and of the parameters of the tax system /I and y. By (21) we can express the switch-point as a function of the relative pre-tax prices x*(p,r/R). Furthermore, it can be shown that the ratio (p,r/R) is homogeneous of degree zero in H; and HE. If we are only interested in this relative price, we only need information about the relative housing stocks. Likewise, the ratios of the same price for different tax systems (e.g. the ratio between R for the 1979 system and R for the 1971 system) are homogeneous of degree zero in Hz and Hi. Choosing the value of a/b in the utility function we can hence compute the cctcris paribus effect on par and R that arises in our model as a result of the “We have excluded the very lowest incomes (below 10,000 SW. crowns). We have also excluded the very highest incomes (above 160,000 SW. crowns in 1971 prices) on the ground that these people are so few that they should properly have a very low weight in the regressions. 15This is so in particular if account is taken of the increase in real pre-tax income from 1971 to 1979, which means that all people have moved into higher tax brackets. t6This is the distribution of reported, taxable income over individuals in 1975, a year chosen since it is right between our years of study, 1971 and 1979. The figures are deflated by the consumer price index to the 1971 level. Source: Statistisk .&&ok (1977, p. 415). Since the figures only show reported income, which contains only a minor fraction of actual capital gains, the figures probably underestimate the income in the higher brackets. We have therefore made alternative assumptions about the mean and the variance of the distribution, but the results turn out to be quite robust with regard to the income distribution.
actual tax schedule change between 1971 and 1979. We have taken a/h=0.3. This implies that for renters housing occupies 23 percent of the budget, a figure that is roughly in accord with the true data. rp,(79) and R(79) denote the solution values of R and rp, with the tax parameters of 1979. The ratios in table 1 thus give us information about the changes in prices of housing that have occurred in the 1970s that may be attributed to tax changes. It should be noted that an assumption of static expectations probably leads us to overestimate the price change. Simulations by Poterba (1980) indicate that the short-run price change assuming perfect foresight is about half the size of those resulting from static expectations. Obviously the magnitude of this difference depends crucially on the rate of supply adjustment. The last two variables, D(79) and 0(71), are the proportions of the households that are owner-occupants for the two sets of tax parameters, defined by ~(79) E 100’
‘j
&c) dx.
x*(79)
We thus ‘see from table 1 that the changes in income taxation which took place between 1971 and 1979 caused, ceteris paribus, the prices of owneroccupied houses to increase by around 30 percent in real terms, and that this figure does not change very much with the assumed value of Hi/H& This increase is a consequence of the fact that interest deductions are more profitable the higher is the marginal tax rate, so that the increase in taxation that took place during the decade actually stimulated demand for owneroccupied housing. At the same time, due to the rise in progressivity, disposable income increased for the low-income earners, thereby stimulating demand for rental housing too. The effect on the rent level is however much smaller; between 1 and 4 percent. Finally, the last two columns show the proportion of the population living in owner-occupied houses. Although the absolute values of D(79) and D(71) of course reflect the assumed value of Hi/H& their d@erence is very robust.
Table Solutions
of the
model
flS,IH:,
Vo(79VVo(7
0.4 0.6 0.8 1.0 I.2
1.27 1.29 1.30 1.31 1.32
1
with a/h=O.3 H;/H;. 1)
for
varying
values
of
W79YW7 1)
D(79)
D(71)
1.04 1.03 1.02 I .02 1.01
55.4 46.9 40.6 35.8 32.0
52.7 43.8 37.5 32.7 28.9
P. En,qlund and M. Persson, llousing
prices and tenure c,hoiw
287
Thus, the tax changes occurring between 1971 and 1979 caused, according to the model simulations, the proportion of owner-occupants to increase by around 3 percentage points. This means that the average housing consumption increased for the renter and decreased for the owner-occupant. Incidentally, the results of our simulations turn out to conform pretty well to observed reality; between 1971 and 1979 prices of owner-occupied houses increased by 32 percent in real terms. Rents increased by 1 percent on the average, between December 1971 and December 1978.” However, one should keep in mind that ours is a pure ceteris paribus experiment only intended to illustrate the tendencies in prices and housing patterns to change in response to changes in income taxation, keepin!: other,f&ors constunt. In order to check the sensitivity of the model, we have also calculated the solutions for different values of a/b, ranging from 0.1 to 0.5, i.e. implying budget shares for renters from 9 to 33 percent. The effects on rp, are reported in table 2. The effect on other variables does not vary much. R goes up between 0.9 percent (a/b=OS; Hi/H;= 1.2) and 5.4 percent (a/b=O.l; Hi/H; =0.4), and the share of owners increases by between 2.2 (a/b = 0.1; Hi/H:, =0.4) and 3.3 (n/h = 0.5; Hi/Hi = 1.2) percentage points. Table 2 Values of rpo(79)/rpo(71)
for different
values of a/h and of
HS,/HS,.
alh
H”,/Hs,
0.1
0.2
0.3
0.4
0.5
0.4 0.6 0.8 1.0 1.2
1.37 1.38 1.39 1.40 1.41
1.31 1.33 1.34 1.35 1.36
1.27 1.29 1.30 1.31 1.32
I .24 1.26 1.27 1.28 1.29
1.22 1.23 1.24 1.25 1.26
These numerical examples take into account that the budget constraint for In the analytical treatment of the an owner-occupier is non-linear. comparative statics in subsection 4.1 we were helped by making the simplifying assumption that the marginal tax rate is independent of the size of the interest deductions. To check the accuracy of such an approximation, we have computed numerical solutions of the model (12H14), still with a and a constant-elastic disposable income logarithmic utility function function. It is easy to demonstrate that for the ‘approximate’ model, the price ratios are independent of a/b, which we see from table 2 are of significance in “Sources: Meddelanden:
Sandelin Bo.
(1977)
and
The
Swedish
Central
Bureau
of
Statistics,
Statistiska
288
Table 3 Solutions
of the linearized
H;IH;
vo(79)
0.4 0.6 0.8 1.0 1.2
1.46 1.47 I .48 I .50 I .50
model for different
l’Po(71 1
values of Hi/H&
R(79)/R(71)
D(79)
w7u
1.07 1.05 1.04 1.03 I .02
4x 39 33 30 26
46 37 31 27 23
the ‘true’ model. In table 3 we show the solutions of the linearized model: the results show that the approximation overestimates the effects of the tax changes, particularly on rpo. The implication for resource allocation of our results is clear. Resources will be diverted away from investments in industry and other sectors of the economy towards housing, in particular owner-occupied housing, and there will be an increase in the rate at which rental dwellings are changed into owner-occupancy. As the model is stated - with all consumers regarding h, and h, as perfect substitutes ~~ there will be no renters left in the long run. However, all the main conclusions will hold if there are differences across households in the taste for owning vs. renting. Then some households may remain as renters in the long run and there will be a welfare cost of the tax system since consumers face different prices for two goods with equal costs of production. Finally, a few words about income distribution. We have seen that the 1979 tax schedule is considerably more progressive than the 1971 schedule. The distributional effects of this can be seen by calcultating the ‘equivalent gains’ of the tax schedule change for various income groups.18 Denote the 1971 price system by P,, and the 1971 disposable income of a particular consumer by JJ~,(x), with a similar notation for 1979. Then the equivalent gain G of the consumer is the lump-sum transfer that makes his 1971 indirect utility equal to his 1979 utility:‘”
where disposable income y E /Ix? + G. The numerical solutions for G, for different income groups, are reported in table 4. We see that for a given pre-tax income distribution low-income
P. Englund and M. Persson, Housing prices and tenure choice
289
Table 4 Equivalent gains for various groups. 1971 SW. crowns.
x 10,ooo 15,000 20,ooo _____________ 25,000 30,ooo 40,Wo 50,ooo 75,ooo 100,000
Renter, G 1,991 1,188 381
income
Owner, G
~---~ 174 - 777 - 1,948 - 3,075 - 5,726 -8,184
groups (which coincide with renters) have gained and others have lost. At the extreme ends of the income distribution these gains and losses are quite considerable. However, the point is that the pre-tax income distribution is not unchanged. Home-owners have made a once-and-for-all capital gain, and they are earning imputed rent on a larger equity in the houses they own. The average value of an owner-occupied house traded in 1970 was 105,000 SW. crowns.2o A 30 percent price increase thus means a capital gain equivalent to one year’s income for an average home-owner (pre-tax income ~~30,000). To this should be added the imputed rent, which as a percentage of pre-tax income equals the interest rate used for imputation. When we start to think of these effects it is no longer clear which groups have gained and which have lost from the increase in the progressivity of the tax schedules that took place in Sweden during the 1970s.
6. Concluding comments The model analyzed in this paper is based on the same observation as the tax clientele hypothesis in finance: with varying marginal tax rates, different households will face different price systems. Just as this hypothesis seems to go some way towards explaining differences in portfolio composition across income groups [see Feldstein (1976)], it has been shown empirically to have some relevance for tenure choice decisions; low-income groups tend to rent while high-income households tend to own their homes. Whereas this impact may be modest when marginal tax rates vary between, say 25 and 50 percent, it is likely to be more dramatic when, as in the Swedish case, they vary between 50 and 80 percent. “See
Sandelin
(1977, p. 100).
290
P. En,qlund cd
M. Prrsson,
Housing priers and tenure choice
While our approach provides an explanation of why some people own and others rent, our main concern has been to study how the income tax schedule affects prices in the housing market and to see how housing price changes may counteract attempts to redistribute income via changes in the income tax progressivity. We tind our results pretty striking on this point. Not only can we show that an increase in progressivity in general tends to lead to an increase in the values of owner-occupied houses, thereby distributing wealth back to the rich in the form of capital gains, but we also see that these capital gains can be quite large in the Swedish case. Indeed, our calculated equivalent losses for high-income earners are mostly significantly below the increase in imputed rent from an average-sized owneroccupied house. However, the magnitude of this effect is critically dependent on our assumption of short-run equilibrium with static expectations. With perfect foresight and a rapid supply adjustment, the price effects would be much smaller.
References t.nglund, P. and M. Persson, 1981, Housing prices and tenure choice with asymmetric taxes and progressivity, Stockholm School of Economics, Research Paper no. 6212. Feldstein, M., 1976, Personal taxation and portfolio composition: An econometric analysis, Econometrica 44, 63 l-650. Hall, R.E., 1973, Wages, income and hours of work in the U.S. labor force, in: Cain and Watts, eds., Income maintenance and labor supply (Chicago). Hendershott, P.H. and J.H. Hu, 1979, Inflation and the benefits from owner-occupied housing, National Bureau of Economic Research, Working Paper no. 383. Hendershott, P.H. and J.D. Shilling, 1980, The economics of tenure choice, 1955-79, National Bureau of Economic Research, Working Paper no. 543. Herthelius, C. and J. Lind, 1980, Ett progressivt skatte- och bidragssystem i en inflationsekonomi (Progressive income taxes and allowances in an inflationary economy). mimeo. Stockholm School of Economics. Jakobsson, U., 1976, On the measurement of the degree of progression, Journal of Public Economics 5, 161-168. King, M.A., 1980, An econometric model of tenure choice and the demand for housing as a joint decision, Journal of Public Economics 14, 137-159. King, M.A., 1981, Welfare analysis of tax reforms using household data, University of Birmingham, mimeo. Li, M., 1977, A logit model of home ownership, Econometrica 45, 1081-1098. Poterba, J.M., 1980. Inflation, income taxes and owner-occupied housing, National Bureau of Economic Research, Working Paper no. 553. Rosen, H., 1979, Housing decisions and the U.S. income tax: An econometric analysis, Journal of Public Economics 11, l-24. Sandelin, B., 1977, Prisutveckling och kapitalvinster pi bostadsfastigheter (Price developments and capital gains in the housing market), Department of Economics, University of Gothenburg.
of Public
Economics
19 (1982) 271-290.
North-Holland
Publishing
Company
HOUSING PRICES AND TENURE CHOICE WITH ASYMMETRIC TAXES AND PROGRESSIVITY Peter ENGLUND
and Mats PERSSON*
Stockholm School of Economics, S-113 83 Stockholm, Sweden Received January
1981, revised version
received
February
1982
This paper studies the effects of tax schedule changes on prices and tenure choice in the housing market. It is shown that, given the present asymmetric treatment of owner-occupants vs. renters, an increase in the degree of progressivity is likely to lead to an increase in the prices of both owner-occupied and rental housing. A numerical example indicates that the effects may be quite large. Equilibrium prices are calculated based on the actual Swedish income tax schedules for 1971 and 1979. According to these simulations the tax changes that took place between these years caused the price of owner-occupied houses to increase by around 30 percent, and the rent level to increase by 2 or 3 percent.
1. Introduction During the 1970s the market for owner-occupied houses boomed in many Western countries. In Sweden the real price of owner-occupied houses increased by 32 percent from 1971 to 1979, and this development was accompanied by a significant increase in the rate of construction. At the same time, the market for rental dwellings was depressed. Rates of construction went down and rents followed consumer prices fairly closely. This general development is due to a combination of factors, the relative strengths of which vary from country to country. The two most commonly stated factors are both related to the increased rates of inflation. One is that this increase has been accompanied by a decrease in the real rate of interest and hence by an increase in property values. The other factor is related to the tax system. In most countries the imputed rent from owner-occupied housing is untaxed, or taxed at a much lower rate than capital income from most other sources. In particular the interest rate used for imputation tends to be fixed, perhaps around what is regarded as a ‘real’ interest rate of a couple of percentage points, whereas the interest paid on loans is deductible by its nominal *This research was supported by research grant 7908763 from the Swedish Council for Building Research, and by the Bank of Sweden Tercentenary Foundation. We wish to thank Richard C. Arnott, Karl G. Jungenfelt, Karl-Giiran Maler, Eytan Sheshinski, and Yoram Weiss for valuable comments on earlier drafts.
0047~2727/82/000Cr0000/$02.75
0
1982 North-Holland
272
P. Englund and M. Person,
Housing prices and tenure choice
amount and interest income on bank accounts is taxed nominally. Hence, when inflation rates and nominal interest rates climb, there is an increasing asymmetry in the tax system and a house will become a more attractive item in the household’s portfolio. The two factors mentioned above may be sufficient to explain even quite significant increases in the owner-occupiers’ demand for housing. [See Hendershott and Hu (1979) for calculations on U.S. data.] They also explain why some households have moved from rental dwellings to owner-occupied houses. Hendershott and Shilling (1980) estimate that the decline, over the period 1955-79, in the cost of owner-occupied housing relative to rental housing has caused the U.S. homeownership rate to be 65 percent rather than 60 percent. It is quite evident that changes in inflation will have the effects stated above. It is less frequently discussed that changes in the income tax schedule may also have major effects if the tax system is asymmetric in the sense described above and if marginal tax rates increase with income. The main purpose of this paper is to study these effects both analytically and by simulating equilibrium prices for the actual Swedish income tax schedules of 1971 and 1979. The analysis is done within a short-run partial equilibrium framework where the supply of housing is fixed. The effects on prices of changes in the tax schedules are then interesting for two reasons. First, an increase in the price of owner-occupied houses will obviously affect resource allocation. Resources will be diverted to construction of new houses. Second, our results suggest that changes in the income tax schedules that are intended to be egalitarian may have quite opposite effects on the income distribution. An increase in the degree of progressivity of the tax schedules might very well cause an increase in the market price of owner-occupied houses. Since high-income earners tend to own their homes, while lowincome earners tend to rent theirs, such a price increase will materialize in the form of capital gains for the high-income earners, thereby neutralizing the ‘egalitarian’ change in the income tax schedules.
2. Consumers’ choice The consumer faces a two-stage decision problem. First, he chooses the mode of tenure. Then, he chooses the quantity of housing. The choice of mode of tenure can be explained by various factors. Owner-occupied houses and rental dwellings may be seen as different goods and differences in tastes may hence be regarded as an explanatory factor. This view typically lies behind econometric studies, as for example the ones by Li (1977) and Rosen (1979). Another view is that what enters people’s utility functions is housing services. This good can either be purchased directly in the market for rental dwellings, or it can be produced using as inputs the house owned, labour and
P. Englund and M. Person,
Housing prices and tenure choice
213
fuel, etc. Differences in productivity in this household production process may then explain why some choose to rent and some to own their home. This is the approach taken by Weiss (1978). One may also focus on differences in risk attitudes. Prospective capital gains are an important part of the economic calculations of a homeowner, and since these are uncertain people will value them differently. In this paper we will abstract from all these factors, i.e. we will assume that there is a homogeneous good named ‘housing services’ that can either be rented or produced by owner-occupants, in which case all households are equally productive. Instead, we will focus on the fact that people typically pay different after-tax prices for owner-occupied housing. We assume that the imputed rent from an owner-occupied house is not taxed at all, whereas interest payments are deductible from taxable income by their full amount. This implies that the price of owner-occupied housing is inversely related to the individual’s marginal tax rate. Hence, high-income earners with high marginal tax rates will tend to own their homes, whereas low-income earners will be renters.’ Consumers maximize a quasi-concave utility function U(h,z), where h is the amount of housing stock (= housing services) consumed, and z is the amount consumed of other goods. For renters the budget constraint is straightforwardly written as Rh+z=y(x), where R is the per unit rent, and where the price of other goods is normalized to equal unity. y(x) is the disposable income out of pre-tax income x, which for simplicity is regarded as exogenous. It is assumed that y’(x) >O and that y”(x)
In (1) it is assumed that all housing costs2 are deductible from taxable income and can be expressed as a constant r times the value of the house poh. The parameter Y will here be interpreted as an interest rate, and other housing costs including expected capital gains will be disregarded.3 The ‘This conforms with empirical studies by Li (1977) and Rosen (1979) which show that income significantly affects the choice of mode of tenure, and in particular King (1980). where taxinduced price differences is the only factor which explains why some rent and some and some own. ‘par is the pre-tax rental cost of owner-occupied housing. We will often talk about this as the price of owner-occupied housing without pointing out at every instance that it is a pre-tax price and that housing costs vary across individuals according to their marginal tax rates. .‘Scc the discuGon at the end of section 3 below.
274
P. Englund
and M. Persson, Housing
prices and tenure choice
equation then says that the expenditure on other goods equals the disposable income left for consumption other than housing.4 The renter’s maximization problem will give rise to a Marshallian demand function for rental housing, h,. The owner-occupant’s solution of the optimization problem will yield a demand function for owner-occupied housing, h,, which is a function of income, price and the parameters of the tax system (which we express by a vector 0)’ The choice whether to rent or to own is made by comparing the maximum utility obtainable from renting with the maximum utility obtainable from owner-occupancy. The renter’s indirect indirect utility function is V(y(x), R), whereas the owner-occupant’s utility can be written as P(x, 8, pot+), Hence, individual demand is given by the system h, = Q(x),
R),
ho = t;(x, 6 nor),
Y(x) 5 0, Y(x) 2 0,
(2)
Y(x) = B(x, 6, par) - V(y(x), R).
Under certain conditions Y(x*) = 0 represents a unique switching-point between owning and renting;’ everybody with an income above x* will own and everybody else will rent. We will assume that these conditions are fulfilled, as they can be shown to be, for example, for a logarithmic utility function combined with any concave disposable income function, y(x). Problems of multiple switch-points may arise if there are intervals over which the income elasticity of housing is high and the marginal tax rate is falling sharply with taxable income, and hence with housing consumption. It would seem that homotheticity of preferences alone is a sufficient condition. It appears, however, that it has to be combined with some regularity condition on y(x). One such condition is that it is exponential like the one we use in subsection 4.2 and the numerical simulations below. The situation at x* is depicted in fig. 1. Whereas the owner’s budget constraint is strictly convex and the renter’s is linear, the joint budget ‘I:q. (1) presumes that there is no taxation of imputed rent. Although this approximation, it is not entirely true for some countries, e.g. Sweden. Assuming instead a fraction 6 of housing costs is deductible, the equivalent budget constraint is
is a fair that only
In the following, we will assume that 5 is sufficiently close to unity to be disregarded. ‘With the assumptions made concerning the tax function y(,), the owner-occupant’s budget set will be strictly convex. The demand h, will therefore be unique for each vector (x, 0, par). ‘.x* is the unique switch-point income if &y’[x*-rp,h,(x*)]>&y’(x*), where I?o is the Lagrange multiplier associated with the owner’s maximization problem and 1, is the corresponding multiplier for the renter. For a derivation of this condition and further discussion of the uniqueness issue, see Englund and Persson (1981).
P. Englund and M. Person,
275
Housing prices and tenure choice
Fig.
1
hh
>X
Fig. 2
constraint (as depicted by the heavy curve) is non-convex. This implies that the demand for housing is a discontinuous function of pre-tax income, as illustrated by fig. 2. It also implies that a lump-sum transfer of post-tax income da given at the switch-point will lead the household to prefer renting;’ av/ac( > aV/acr for X=X *. The explanation for this is simply that the ‘Note that this is opposite to the effect that is the main message transfer of pre-tax income will lead the household to prefer owning.
of the model,
namely
that a
marginal utility of money is higher for the renter, since he faces a lower marginal price of housing than the owner (and the same marginal price of the numeraire). This is directly seen by comparing the slopes of the budget lines at ho and h, in fig. 1.
3. Market equilibrium Individual demand is represented by the system (2). Having assumed a unique switch-point between owning and renting, and assuming a continuum of households, the aggregate demand for owner-occupied housing, Ho, and by integrating over the individual demand rental housing, H,, is obtained functions:
Ho = j-ho 4(x) dx X*
(3)
and
H, =
70 h, 4(x) dx,
(4)
where 4(x) is the density function of the income distribution. The switch-point between owning and renting, x*, is given by Y(x*) = 0.
(5)
The supply of housing can in the short run be regarded as exogenously given. There are normally considerable costs, legal and others, associated with transforming rental housing into owner-occupied housing or vice versa. Therefore it is reasonable to treat not only the total housing stock, but also its components, i.e. rental and owner-occupied housing, as exogenously given.8 Denote these supplies by Hs and Hi, respectively. We then have the equilibrium conditions
H,=Hs,
(6)
H,=H;,
(7)
and
where aggregate demands H, and H, are given by (3H5). Thus, par, R, and of the utility .x* are determined as functions of H& Hi, the parameters “On the other hand, we assume the stocks of II, and H, to be perfectly malleable. This of course is a stringent assumption. Ideally, one would let short-run supply consist of a number of indivisible dwellings.
P. Englund and M. Person,
Housing prices and tenure choice
,
211
function, the tax function, and the income distribution. This can be referred to as short-run equilibrium since the stocks of Ht and HE are treated as exogenous, and it will be the main topic of this paper. In the longer run, the mode of tenure will be flexible. If total supply is still . * regarded as fixed, the equilibrium solution is given by H,+H,=HS,
(8)
together with the switch-point equation (5). In this equilibrium the price of owneroccupied housing must be equal to the price of rental housing, i.e. po = pR = p. Since landlords regard the holding of rental housing as an ordinary investment, a perfect capital market requires pr=R.
(9)
Eqs. (8) and (9) thus determine the values of pr and R which, by the demand functions (3) and (4), give us the distribution of housing stock between owned and rented dwellings. It is easy to see that in such a case, which may be termed medium-run equilibrium, all housing will be transformed into owneroccupancy. By (9) we have R -> y’(W
1,
(10)
for all y’()< 1. Thus, renting is less favourable than owning, at the margin, which means that in equilibrium all dwellings will be transformed into owner-occupancy. This property is a direct consequence of the asymmetry of the tax system and the assumption that the mode of tenure does not enter the utility function. The short-run prices will also affect the rate of housing investment until a long-run equilibrium is reached, where prices equal production costs. It is important to notice that the rates of supply adjustment both transformation of tenure and new construction - will have an impact on the size of the short-run price changes. This is clear if proper account is taken of expected capital gains (losses). In such a case the budget constraint (1) should be written as
z = Y(X-w-h + Iio), with expectations assumed to be formed with perfect foresight - instead of, as in our case, being assumed static with Ijo = 0. Also, the landlords’ portfolio choice should depend not only on rent incomes, but also on expected capital gains, rjR. Assuming that the rate of construction is a function of the
P. En&md
278
and M. Persson.
Ifousin~
priers
and tenure choice
difference between the market price (p. and pR, respectively) and the construction cost, we would have a dynamic equilibrium model which will have essentially the same qualitative properties as our model. For numerical calculations of the difference between the two assumptions about expectations formation, see Poterba (1980). As seen above, the present asymmetry in the tax system implies a corner solution: in the medium and long run only owner-occupied dwellings will exist. In reality, such an extreme solution will of course not obtain. People differ in tastes and abilities, and different modes of tenure do not provide exactly the same type of services. Nevertheless, the corner solution points at an important tendency towards owner-occupancy that is created by the tax system. And since such a tendency will be more pronounced, the higher is the tax rate, this provides a plausible explanation of an observed feature in many countries today: parallel with an increase in tax rates, we observe how rental dwellings are transformed into owner-occupied ones (condominiums) at an increasing pace. However, in what follows we will confine ourselves to analyzing the short-run equilibrium.
4. Comparative
static analysis of tax schedule changes
We are concerned with the effects of changes in the income tax schedule, given the asymmetric treatment of housing costs. A change in the income tax schedule affects housing demand in two ways. First, it affects disposable income. Secondly, a tax change normally is not a simple shift of the tax function, but is also a change in the marginal tax rates. Thus, it affects the marginal cost of owner-occupied housing, y/(x-rp,h,). A tax increase will therefore lower the individual demand for rental housing, if housing is a normal good, whereas the individual demand for owner-occupied housing may be affected in either direction. Analyzing the short-run market equilibrium (3)-(5) shows that very little can be said about the effects of changes in the tax schedule. One reason for this ambiguity is the non-convexity of the budget constraint. As pointed out above, this implies that, at the switch-point, the marginal utility of income is higher for the renter than for the owner-occupant. This property contributes to make even an increase in lump-sum taxation ambiguous. For example, one would guess that the introduction of a lump-sum tax (which does not affect the slope of the budget constraint, but only reduces disposable income for everybody) would depress demand so that equilibrium prices would fall for both types of housing. This is however not necessarily true; some renters, facing a lower disposable income, might move into owning.’ Thus, aggregate “This follows section 2 above.
from
the differences
in the marginal
utility
of money
discussed
at the end of
P. Englund und M. Person,
Housing prices and tenure choice
219
demand for owner-occupied housing may actually increase, although all previous owners certainly reduce their demands, and the result will be an increase in the price level. To be able to say something more definite about comparative statics, we have to put some restrictions on our model. We have done this in two ways. First, we study the model with an unspecified utility function and a very general tax function, but linearize the budget constraint in a suitable way. Second, we accept the non-convex budget set, but assume that the utility function is logarithmic in h and z, and that disposable income is an isoelastic function of pre-tax income. 4.1. Linearization
of the budget
constraint
By treating the budget constraint of the owner-occupant as though it were linear we can focus on the impact of the utility function on the various effects of changes in the tax schedules. The correct procedure would be to linearize around equilibrium housing consumption [see, for example, Hall (1973)]. If we did this we would still have to cope with the problem that the marginal utility of money at the switch-point x* would differ between an owner and a renter. To avoid this we linearize instead around h = 0, which implies that at the switch-point the budget constraints for owner and renter coincide. Clearly, this may be misleading for a commodity like housing which occupies a large share of the household’s budget. We write disposable income as
Y(X)= @l+/k(x),
(11)
where g(x) is any non-negative, increasing and concave function. This fairly general formulation allows us to distinguish between changes in the level of taxation and changes in the degree of progressivity. A change in the level of taxation can have either a lump-sum character, da, or imply an equiproportionate change in marginal tax rates, dfi. An increase in progressivity can here be modelled by increasing a and making an offsetting reduction in p so as to leave total tax revenue unaffected. With the disposable income function (1 l), we can write the market equilibrium equations (3)(5) in this simple form:
(12) (13) bg’(x*)p,r
= R.
(14)
280
The comparative
P. En&ud
and M. Persson, Housing
prices und trmrre
statics of this model are summarized
diwl W dx*/
+ &
choir
below:”
C-1 + ‘,
The effects of a lump-sum income increase on both prices are unambiguous. An increase in p will lead to an increase in R but, in general, to a decrease in per. Sufficient conditions for the latter effect (dpor/da < 0) is that housing and other goods are gross substitutes and that not even a household with zero pre-tax income will have a negative disposable income (i.e. ~20). Let us next regard the effects of an increase in progressivity, where a reduction in /I is compensated by an increase in CI so as to leave total tax revenue unaffected. Under the above assumption this will lead to an increase in the price of owner-occupied housing, par. The rent level R will also increase unless the marginal outlay on housing is much larger for highincome than for low-income groups; a sufficient condition is that the Engel curve is linear, i.e. that Zh/c7y is constant irrespective of income. Finally, we have the effects on the switch-point x*. An increase in cx will in general, e.g. for a logarithmic utility function, lead to an increase in x*, i.e. people will move over to renting from owning. This is so because the equilibrium price of housing at x* can be shown to increase more for the owner than for the renter. The effect of a change in j3 is ambiguous. The crucial factor is the relative size of the price and income elasticity for owners as opposed to renters. If owners have a high income elasticity and a low price elasticity relative to renters, dx*jdp is positive, i.e. an increase in marginal tax rates (dp
budget set
The results in the previous section were obtained at the cost of making a linear approximation of the budget constraint. To take proper account of the “‘I-or the formal derivation and Person (I 98 I ).
of this and the following
comparative-statics
results, see Englund
P. Englund and M. Person,
Housiq
prices and tenure choice
non-linearity of the budget set, we will now parameterize the model suitable way. We thus assume that the utility function is given by U(h,z)=alogh+blogz, while disposable Y(X) =
income
w,
281
in a
(15)
is given by j>O,
O
(16)
Here the parameter y can be directly interpreted as a measure of progressivity. Given any pre-tax income distribution and two tax systems of Lorenz curve associated with y2 the form (16), with yi >yZ, the post-tax strictly dominates the Lorenz curve associated with y1 [see Jakobsson (1976)]. The utility function (15) and the disposable income function (16) give rise to the individual demand functions
/,,=a?
a+by
par
(17)
and h,=p
a
/?x’
a+b’R’
As expected, h, has unitary elasticities with respect to disposable income and rent, while h, exhibits unitary elasticities with respect to pre-tax income and pre-tax price. In fact ho is independent of the level of taxation, /I; the price and income effects exactly cancel. As for an increase in progressivity’ ’ (dy < 0), we see that it makes h, decrease. This is hardly surprising, since it makes everybody’s disposable income fall. The effect on ho is more interesting. For the high-income earners (i.e. the home-owners) we get the slightly paradoxical result that their demand for housing is actually stimulated by an increase in progressivity. Although their disposable incomes fall, the price effect outweighs the income effect so that a decrease in y makes ho increase. The market equilibrium conditions are:
H;=
(19)
“Note that this is an uncompensated change, i.e. an increase in taxation via the elasticity parameter. For compensated changes, we will in general get the same effect as with the tax schedule (11); both demands will increase.
282
P. Englund
and M. Person,
ffousing
prices and tenure choiw
and
(20) \vith the switch-point
x* determined
by
.,,(~~)+b log(&Bx’-) (21) lhc
effects on the market
equilibrium
are summarized
below:
There is a striking contrast between the effects in the two markets. An increase in taxation not affecting progressivity (d/I
to the Swedish tax system
In the preceding section affect the prices of housing
we have seen how changes in the tax schedules and the choice of tenure. While the direction of
P. Englund and M. Persson, Housing prices and tenure choice
283
these effects can be analytically determined, at least for some reasonable specifications of the utility and the tax functions, their magnitudes can be evaluated only by numerical methods. In this section we will use the model to simulate equilibrium housing prices for the Swedish tax system in 1971 and in 1979 to evaluate the impacts of the tax changes that took place in the 1970s. Fig. 3 shows the disposable income function y(x) for 1971 and 1979, and fig. 4 displays the marginal tax rates 1 -y’(x) for the two years.” The tax
60 000 50 000 40008 30 000 20 000
I
10000 i
IX
50 000
100 000
150 000
Fig. 3. Disposable income (including housing allowances) as a function of pre-tax income according to the 1971 tax system (solid line) and the 1979 tax system (broken line). 1971 SW. crowns.
1- Y’(X)
0.80 0.700.60 -
50 000 Fig. 4. Marginal
tax rates
100000
150000
for Sweden 1971 (solid line) and 1979 (broken intervals of 5,000 SW. crowns. 1971 prices.
line), computed
at
“The figures for 1979 are deflated to the 1971 price level by the consumer price index. Data are taken from a study of the Swedish income tax system by Blanck and Matthiessen, recently updated by Herthelius and Lind (1980).
284
P. Englund
and M. Persson. Housing
prices and tenure choice
schedules include the effect of housing allowances. These are related to housing consumption, income and family size. However, for most households the level of housing consumption is so high that housing allowances are not affected at the margin. Fixing family size, it can hence for these households - be assumed that housing allowances only depend on income. To simplify we assume that this is valid for all. The tax schedule used applies to a family with one income-earner and two children. Fig. 4 shows that the marginal tax rate has a tendency to increase with income. It is not a monotone increasing function, however, as is presumed in our previous analysis. This is due to the housing allowances which are dependent on income and therefore give very high marginal rates in some brackets a bit above the average. Applied to the model this may give several switch-points between owning and renting. The question arises of whether one should use some simple approximation to the tax function, or the actual schedules depicted in figs. 3 and 4. We have chosen to approximate the actual schedule by the constant-elastic function (16) for two reasons. First, it provides some simplifications that cannot be obtained if the actual tax schedules are used. In combination with a logarithmic utility function it implies that the ratio of the prices for owner-occupied houses between two tax systems is homogeneous of degree zero in the supplies Hi and Hi. Thus, we can reduce the number of parameter values that we must fix for our simulations; with the actual tax system we must know both Hg and H& while with the parametric form we need only their ratio Hz/Hi. Second, it is not really certain that the households regard the actual tax schedules as permanent, and thereby entirely relevant to their housing decisions. Middleincome earners, facing marginal tax rates of 80 and 90 percent over narrow brackets, will hardly expect these rates to prevail over their entire planning horizon. And while the housing allowances obviously play some role in the families’ decisions, it is difficult to say exactly which role they play. In the choice between excluding them completely and including them but fitting a monotone, iso-elastic function to the system, we have chosen the latter alternative. Fitting the function (16) to the actual tax schedules by least squares regression’ 3 we obtain the following values for J and y: 1971: y= 1o9.7.xo.53,
R2 = 0.979,
1979: y = 547.3 xo,37,
R= = 0.990.
The correlation coefficient is of course a questionable There is nothing stochastic about the tax schedules. “The
regressions
are made for the case of an interval
measure in this case. Nevertheless, it gives
of 1.000 SW. crowns.
P. Englund and M. Persson, flousin~ prices and tenure choice
285
some indication about the goodness of tit of the curves. Also, the observations that are included in the regressions are at our discretion.14 Comparing these estimates it seems clear that the tax system of 1979 is more progressive than that of 1971. At the same time fig. 3 shows that the level of taxation was higher in 1979 than it was in 1971.” We will now calculate the ceteris paribus effect of this tax schedule change for the isoelastic formulation of the y(.) function and a logarithmic utility function, i.e. the case analyzed in subsection 4.2 above. During the 1970s at least five things have happened in the Swedish housing market. (i) The supplies, Hi and Hz, have increased with Hg distribution $(x) has increasing much faster than Hz. (ii) The income changed. (iii) The real rate of interest, r, has fallen. (iv) The tax function, i.e. the parameters p and y, have changed. (v) The prices p. and R have changed. Now, our aim is to study the relation between points (iv) and (v). In order we regard supplies and income to concentrate on this main point, distribution as unchanged. The role played by a decrease in I is immediately seen from the model, which only determines the product rp,; a decrease in r is in this model fully capitalized in house values due to the assumption of static expectations. To fix figures we approximate the actual income distribution by a log-normal distribution, where the logarithms of gross income are normally distributed with mean 9.84 and variance 0.66.1h The solution of the system (19x21) gives us R, rp, and x* as functions of the stocks of housing Hi and HE, of the ratio of the parameters of the utility function a/b, and of the parameters of the tax system /I and y. By (21) we can express the switch-point as a function of the relative pre-tax prices x*(p,r/R). Furthermore, it can be shown that the ratio (p,r/R) is homogeneous of degree zero in H; and HE. If we are only interested in this relative price, we only need information about the relative housing stocks. Likewise, the ratios of the same price for different tax systems (e.g. the ratio between R for the 1979 system and R for the 1971 system) are homogeneous of degree zero in Hz and Hi. Choosing the value of a/b in the utility function we can hence compute the cctcris paribus effect on par and R that arises in our model as a result of the “We have excluded the very lowest incomes (below 10,000 SW. crowns). We have also excluded the very highest incomes (above 160,000 SW. crowns in 1971 prices) on the ground that these people are so few that they should properly have a very low weight in the regressions. 15This is so in particular if account is taken of the increase in real pre-tax income from 1971 to 1979, which means that all people have moved into higher tax brackets. t6This is the distribution of reported, taxable income over individuals in 1975, a year chosen since it is right between our years of study, 1971 and 1979. The figures are deflated by the consumer price index to the 1971 level. Source: Statistisk .&&ok (1977, p. 415). Since the figures only show reported income, which contains only a minor fraction of actual capital gains, the figures probably underestimate the income in the higher brackets. We have therefore made alternative assumptions about the mean and the variance of the distribution, but the results turn out to be quite robust with regard to the income distribution.
actual tax schedule change between 1971 and 1979. We have taken a/h=0.3. This implies that for renters housing occupies 23 percent of the budget, a figure that is roughly in accord with the true data. rp,(79) and R(79) denote the solution values of R and rp, with the tax parameters of 1979. The ratios in table 1 thus give us information about the changes in prices of housing that have occurred in the 1970s that may be attributed to tax changes. It should be noted that an assumption of static expectations probably leads us to overestimate the price change. Simulations by Poterba (1980) indicate that the short-run price change assuming perfect foresight is about half the size of those resulting from static expectations. Obviously the magnitude of this difference depends crucially on the rate of supply adjustment. The last two variables, D(79) and 0(71), are the proportions of the households that are owner-occupants for the two sets of tax parameters, defined by ~(79) E 100’
‘j
&c) dx.
x*(79)
We thus ‘see from table 1 that the changes in income taxation which took place between 1971 and 1979 caused, ceteris paribus, the prices of owneroccupied houses to increase by around 30 percent in real terms, and that this figure does not change very much with the assumed value of Hi/H& This increase is a consequence of the fact that interest deductions are more profitable the higher is the marginal tax rate, so that the increase in taxation that took place during the decade actually stimulated demand for owneroccupied housing. At the same time, due to the rise in progressivity, disposable income increased for the low-income earners, thereby stimulating demand for rental housing too. The effect on the rent level is however much smaller; between 1 and 4 percent. Finally, the last two columns show the proportion of the population living in owner-occupied houses. Although the absolute values of D(79) and D(71) of course reflect the assumed value of Hi/H& their d@erence is very robust.
Table Solutions
of the
model
flS,IH:,
Vo(79VVo(7
0.4 0.6 0.8 1.0 I.2
1.27 1.29 1.30 1.31 1.32
1
with a/h=O.3 H;/H;. 1)
for
varying
values
of
W79YW7 1)
D(79)
D(71)
1.04 1.03 1.02 I .02 1.01
55.4 46.9 40.6 35.8 32.0
52.7 43.8 37.5 32.7 28.9
P. En,qlund and M. Persson, llousing
prices and tenure c,hoiw
287
Thus, the tax changes occurring between 1971 and 1979 caused, according to the model simulations, the proportion of owner-occupants to increase by around 3 percentage points. This means that the average housing consumption increased for the renter and decreased for the owner-occupant. Incidentally, the results of our simulations turn out to conform pretty well to observed reality; between 1971 and 1979 prices of owner-occupied houses increased by 32 percent in real terms. Rents increased by 1 percent on the average, between December 1971 and December 1978.” However, one should keep in mind that ours is a pure ceteris paribus experiment only intended to illustrate the tendencies in prices and housing patterns to change in response to changes in income taxation, keepin!: other,f&ors constunt. In order to check the sensitivity of the model, we have also calculated the solutions for different values of a/b, ranging from 0.1 to 0.5, i.e. implying budget shares for renters from 9 to 33 percent. The effects on rp, are reported in table 2. The effect on other variables does not vary much. R goes up between 0.9 percent (a/b=OS; Hi/H;= 1.2) and 5.4 percent (a/b=O.l; Hi/H; =0.4), and the share of owners increases by between 2.2 (a/b = 0.1; Hi/H:, =0.4) and 3.3 (n/h = 0.5; Hi/Hi = 1.2) percentage points. Table 2 Values of rpo(79)/rpo(71)
for different
values of a/h and of
HS,/HS,.
alh
H”,/Hs,
0.1
0.2
0.3
0.4
0.5
0.4 0.6 0.8 1.0 1.2
1.37 1.38 1.39 1.40 1.41
1.31 1.33 1.34 1.35 1.36
1.27 1.29 1.30 1.31 1.32
I .24 1.26 1.27 1.28 1.29
1.22 1.23 1.24 1.25 1.26
These numerical examples take into account that the budget constraint for In the analytical treatment of the an owner-occupier is non-linear. comparative statics in subsection 4.1 we were helped by making the simplifying assumption that the marginal tax rate is independent of the size of the interest deductions. To check the accuracy of such an approximation, we have computed numerical solutions of the model (12H14), still with a and a constant-elastic disposable income logarithmic utility function function. It is easy to demonstrate that for the ‘approximate’ model, the price ratios are independent of a/b, which we see from table 2 are of significance in “Sources: Meddelanden:
Sandelin Bo.
(1977)
and
The
Swedish
Central
Bureau
of
Statistics,
Statistiska
288
Table 3 Solutions
of the linearized
H;IH;
vo(79)
0.4 0.6 0.8 1.0 1.2
1.46 1.47 I .48 I .50 I .50
model for different
l’Po(71 1
values of Hi/H&
R(79)/R(71)
D(79)
w7u
1.07 1.05 1.04 1.03 I .02
4x 39 33 30 26
46 37 31 27 23
the ‘true’ model. In table 3 we show the solutions of the linearized model: the results show that the approximation overestimates the effects of the tax changes, particularly on rpo. The implication for resource allocation of our results is clear. Resources will be diverted away from investments in industry and other sectors of the economy towards housing, in particular owner-occupied housing, and there will be an increase in the rate at which rental dwellings are changed into owner-occupancy. As the model is stated - with all consumers regarding h, and h, as perfect substitutes ~~ there will be no renters left in the long run. However, all the main conclusions will hold if there are differences across households in the taste for owning vs. renting. Then some households may remain as renters in the long run and there will be a welfare cost of the tax system since consumers face different prices for two goods with equal costs of production. Finally, a few words about income distribution. We have seen that the 1979 tax schedule is considerably more progressive than the 1971 schedule. The distributional effects of this can be seen by calcultating the ‘equivalent gains’ of the tax schedule change for various income groups.18 Denote the 1971 price system by P,, and the 1971 disposable income of a particular consumer by JJ~,(x), with a similar notation for 1979. Then the equivalent gain G of the consumer is the lump-sum transfer that makes his 1971 indirect utility equal to his 1979 utility:‘”
where disposable income y E /Ix? + G. The numerical solutions for G, for different income groups, are reported in table 4. We see that for a given pre-tax income distribution low-income
P. Englund and M. Persson, Housing prices and tenure choice
289
Table 4 Equivalent gains for various groups. 1971 SW. crowns.
x 10,ooo 15,000 20,ooo _____________ 25,000 30,ooo 40,Wo 50,ooo 75,ooo 100,000
Renter, G 1,991 1,188 381
income
Owner, G
~---~ 174 - 777 - 1,948 - 3,075 - 5,726 -8,184
groups (which coincide with renters) have gained and others have lost. At the extreme ends of the income distribution these gains and losses are quite considerable. However, the point is that the pre-tax income distribution is not unchanged. Home-owners have made a once-and-for-all capital gain, and they are earning imputed rent on a larger equity in the houses they own. The average value of an owner-occupied house traded in 1970 was 105,000 SW. crowns.2o A 30 percent price increase thus means a capital gain equivalent to one year’s income for an average home-owner (pre-tax income ~~30,000). To this should be added the imputed rent, which as a percentage of pre-tax income equals the interest rate used for imputation. When we start to think of these effects it is no longer clear which groups have gained and which have lost from the increase in the progressivity of the tax schedules that took place in Sweden during the 1970s.
6. Concluding comments The model analyzed in this paper is based on the same observation as the tax clientele hypothesis in finance: with varying marginal tax rates, different households will face different price systems. Just as this hypothesis seems to go some way towards explaining differences in portfolio composition across income groups [see Feldstein (1976)], it has been shown empirically to have some relevance for tenure choice decisions; low-income groups tend to rent while high-income households tend to own their homes. Whereas this impact may be modest when marginal tax rates vary between, say 25 and 50 percent, it is likely to be more dramatic when, as in the Swedish case, they vary between 50 and 80 percent. “See
Sandelin
(1977, p. 100).
290
P. En,qlund cd
M. Prrsson,
Housing priers and tenure choice
While our approach provides an explanation of why some people own and others rent, our main concern has been to study how the income tax schedule affects prices in the housing market and to see how housing price changes may counteract attempts to redistribute income via changes in the income tax progressivity. We tind our results pretty striking on this point. Not only can we show that an increase in progressivity in general tends to lead to an increase in the values of owner-occupied houses, thereby distributing wealth back to the rich in the form of capital gains, but we also see that these capital gains can be quite large in the Swedish case. Indeed, our calculated equivalent losses for high-income earners are mostly significantly below the increase in imputed rent from an average-sized owneroccupied house. However, the magnitude of this effect is critically dependent on our assumption of short-run equilibrium with static expectations. With perfect foresight and a rapid supply adjustment, the price effects would be much smaller.
References t.nglund, P. and M. Persson, 1981, Housing prices and tenure choice with asymmetric taxes and progressivity, Stockholm School of Economics, Research Paper no. 6212. Feldstein, M., 1976, Personal taxation and portfolio composition: An econometric analysis, Econometrica 44, 63 l-650. Hall, R.E., 1973, Wages, income and hours of work in the U.S. labor force, in: Cain and Watts, eds., Income maintenance and labor supply (Chicago). Hendershott, P.H. and J.H. Hu, 1979, Inflation and the benefits from owner-occupied housing, National Bureau of Economic Research, Working Paper no. 383. Hendershott, P.H. and J.D. Shilling, 1980, The economics of tenure choice, 1955-79, National Bureau of Economic Research, Working Paper no. 543. Herthelius, C. and J. Lind, 1980, Ett progressivt skatte- och bidragssystem i en inflationsekonomi (Progressive income taxes and allowances in an inflationary economy). mimeo. Stockholm School of Economics. Jakobsson, U., 1976, On the measurement of the degree of progression, Journal of Public Economics 5, 161-168. King, M.A., 1980, An econometric model of tenure choice and the demand for housing as a joint decision, Journal of Public Economics 14, 137-159. King, M.A., 1981, Welfare analysis of tax reforms using household data, University of Birmingham, mimeo. Li, M., 1977, A logit model of home ownership, Econometrica 45, 1081-1098. Poterba, J.M., 1980. Inflation, income taxes and owner-occupied housing, National Bureau of Economic Research, Working Paper no. 553. Rosen, H., 1979, Housing decisions and the U.S. income tax: An econometric analysis, Journal of Public Economics 11, l-24. Sandelin, B., 1977, Prisutveckling och kapitalvinster pi bostadsfastigheter (Price developments and capital gains in the housing market), Department of Economics, University of Gothenburg.