The Effect of the Capital Gains Tax on the Intertemporal
Housing Demand
Wen-Chieh Jack Wu* AbstractSpecial provisions in the tax code of some countries enable homeowners to waive tax payments on the capital gain from selling their previous house if they buy a new house of equal or greater value within a short period time of when they move. Owing to this special tax treatment on the capital gain, house owners often face a kinked budget constraint. This paper incorporates this special capital gains tax treatment into the two-period intertemporal housing demand framework. We assume individuals may have a certain degree of myopia on the tax treatment on the beginning of the first period in which the first house decision is made. However, they will become knowledgeable on the tax treatment as well as the stock of previous house when the second house decision is made on the beginning of the second period. Optimal solutions obtained from traditional intertemporal optimization framework suggest that the capital gains tax rate generally has a negative effect on the optimal first-period housing consumption, while it generally has a positive effect on the optimal second-period housing consumption. However, the optimal second-period housing consumption should be modified when the information of tax treatment and previous house stock becomes available on the beginning of the second period. We find the previous house stock has a negative effect on the modified second period housing consumption. The influence of the capital gains tax rate on the modified second period housing consumption becomes ambiguous.
Keywords: Intertemporal housing demand, Capital gains tax, Kinked budget
constraint
JEL classification:D11,D91,H21,R0
*
The Department of Public Finance, National Chengchi University, Taipei, Taiwan. E-Mail: jackwu@nccu.edu.tw.
1. Introduction
Special provisions in the tax code of some countries enable homeowners to waive tax payments on the capital gain from selling their previous house if they buy a new house of equal or greater value within a short period time of when they move. For instance, rollover provisions in the US tax code enable homeowners to avoid paying tax on the capital gain from the sale of their home if they purchase another home of equal or greater value within two years of when they move. [Hoyt and Rosenthal (1992)] If the household buys down, they pay a capital gains tax, for which the taxable portion of the capital gain is, at most, equal to the difference between the value of the previous and current house. These provisions make previous homeowners face a kinked budget constraint. Taiwanese tax code also provides the similar treatment. If the households buy up within two years of when they move, they can deduct their paid capital gains tax completely from the income tax of that year. On the other hand, they cannot deduct any if they buy down. These provisions create a discontinuous budget constraint. Overall speaking, this type of special tax treatment on the capital gains creates a nonlinear budget constraint.
Previous housing studies, such as Rosen (1979) and King (1980), argue that capital gains tax provisions have no effect on housing demand because few homeowners actually buy down. Hoyt and Rosenthal (1992) simulates the effect of a simultaneous increase in the capital gains tax and lowering of federal marginal income tax rates on the housing demand. An increase in the capital gains tax raises the penalty for buying down, while lower marginal tax rate raises the user cost of owner-occupied housing. Their results suggest that this type of change in tax policy would enhance the importance of the capital gains kink, and as a result, a reduction in the capital gains tax rate would lower housing demand when some families currently at the kink would buy a less expensive home.
Previous studies have been interested in how the capital gains tax would affect the housing demand for the new (second) house when they move. [Hoyt and Rosenthal (1992)] However, we are also interested in how the future capital gains tax would affect the housing demand for their first (previous) house decision. The main purpose of the paper is to examine the effects of the capital gains tax on housing demands for their previous (first) house and the new (second) house before and after moving. We incorporate the special capital gains tax treatment into the static1 two-period intertemporal housing demand framework.
The length of stay plays an important role on influencing housing demand. [Haurin and Lee (1989), Henderson and Ioannides (1989), Zorn (1988), and Haurin and Chung (1998)] It is supposed to be endogenous. In other words, the time of relocating should be endogenous. In order to simplify the model, we assume the time of relocating is exogenous. As well as the length of stay, transaction cost is also a key factor influencing the housing demand. [Hendershott and Shilling (1982), Rosenthal (1988), and Haurin and Chung (1998)] Haurin and Chung (1998) consolidate the standard user cost of owning, transaction costs, and the expected length of stay into one measure. They develop a transaction-adjusted weighted average cost of owning. In this study, we simply define the standard user cost of owing as the transaction-adjusted weighted average cost of owning.
We assume individuals may have a certain degree of myopia on the capital gains tax treatment on the beginning of the first period in which the first house decision is made. However, they will become knowledgeable on the tax treatment as well as the stock of previous house when the second house decision is made on the beginning of the second period. We use the traditional two-period intertemporal optimization method to derive intertemporal housing demand functions under the consideration of myopia. The obtained optimal first period housing consumption will become given on the beginning of the second period. As well as the first-period housing stock, individuals become full knowledgeable on capital gains tax treatment at time of relocating. Given the available information, therefore, the optimal second-period housing consumption function can be modified in order to maximize the utility of the second period.
The remainder of the paper is laid out as follows: the next section presents a static two-period intertemporal housing model. Then, Section three discusses the capital gains tax treatment and the nonlinear budget constraint. The traditional intertemporal optimization incorporating with the capital gains tax is solved in the section four. Section five is the modified second-period optimization problem. The final section presents our conclusions.
(1998) have developed the intertemporal housing demand in a dynamic setting.
2. The Intertemporal Housing Demand Model
In this section, we establish a static two-period intertemporal housing demand model. There are several basic assumptions in the model:
1. The household’s decision maker’s planning horizon is divided into two periods. The first period is from current time to the time of moving or relocating. The second period is from the relocating time to the death time. Both the moving time and the death time are exogenous. The household chooses a dwelling unit H 1 on the beginning of the first period. It sells the housing and then moves to another owner-occupied dwelling unit H on the beginning of the second period and stay in 2 the unit until the death time.
2. Assume the decision maker has a perfect foresight on future variables except the capital gains tax treatment. The household’s decision maker has a certain degree of myopia on the future capital gains tax treatment when he makes the first house decision on the beginning of the first period. However, he will have full information on the tax treatment at the time of relocating in which the second house decision is made.
3. The depreciation of the housing is ignored. It only incurs the capital gains when the first house is sold.2 Individuals have no housing bequest motives.
A household’s two-period intertemporal utility function is
1
( , ) ) , ( 1 1 1 2 2 2 1 x H U x H U U (1)where Uj, j=1,2, is the utility function of period j. Hj is the quantity of housing
services, and x is the quantity of the residual composite good. t is the rate of
time preference.
The household’s intertemporal budget constraint is
] [ ) 1 ( ] )[ 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( 2 2 2 1 1 1 1 1 2 1 1 0 H R x r H R x T G r I t r I t W (2)
0
W is the initial endowment. Ij is the permanent income of the period j. Rj is
the standard per unit cost of owning3 in the period j. r is the real interest rate. t is the marginal income tax rate. The degree of myopia is assumed to be . When is equal to zero, the individual has the perfect foresight. On the other hand, the individual is completely myopic as is equal to 1. G is the housing capital gain and T is the capital gains tax payment. Owing to special tax code, we introduce the specification of the capital gains tax treatment in the next section.
3. Capital Gains Tax Treatment
Tax code and treatment
Rollover provisions in the tax code of some countries, such as US and Taiwan, enable house owners to waive tax payments on the capital gain from selling their previous house if they buy another home of equal or greater value within a short period time (ie, two years) of when they move. On the other hand, they cannot avoid paying tax on the capital gain if they buy down. In United States, if the household buys down, they pay a capital gains tax, for which the taxable portion of the capital gain is, at most, equal to the difference between the value of the previous and current home. [see Hoyt and Rosenthal (1992)] In Taiwan, they pay the entire capital gains tax if they buy down. Generally speaking, Taiwan case is relatively more extreme than the US case. Unlike Taiwanese homeowners pay the entire capital gains tax, US homeowners can pay partial capital gains tax even though they buy down.
The paper is interested in a mixed case and develops a new tax treatment on the capital gain from the sale of previous home. The tax treatment is like as follows: If the household buys down, they pay the complete capital gains tax. On the other hand, if they buy up, the increase in the housing value can be deducted from the taxable capital gains. In other words, the taxable portion of capital gains is equal to capital gains minus the difference between the values of the current and previous home.
3
It is similar to the transaction-adjusted weighted average cost of owning proposed by Haurin and Chung (1998). The transaction cost is taken into account.
Kinked budget constraint
Suppose the owner-occupier sells his house at a price PmH1, the family obtains
a capital gain4, G(Pm P0)H1, where P is the price per unit of housing stock in 0
the beginning of the first period and P is the price per unit of housing stock in the m
beginning of the second period. The household then buys a new home of value
2
H
Pm right after. If the households buy down, they have to pay the entire capital
gains tax k . G k is the fraction of taxable capital gains; is the capital gains tax rate. In the case of buying up, we can write the capital gains tax as below:
} 0 )], ( [ max{k G P H2 H1 T m (3)
This proposed tax policy on the capital gains creates two kinks in the budget
constraint, a convex kink at
m m P H P P H 0 1 2 ) 2 (
, and a concave kink at H2 H1.
The kinked intertemporal budget constraint for homeowners can be written as:
m m P H P P H H R x r H R x G r I t r I t W 1 0 2 2 2 2 1 1 1 1 1 2 1 1 0 ) 2 ( ], [ ) 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( m m m P H P P H H H R x r H R x H H P G k G r I t r I t W 1 0 2 1 2 2 2 1 1 1 1 1 2 1 2 1 1 0 ) 2 ( ], [ ) 1 ( )]} ( [ { ) 1 ( ) 1 ( ) 1 ( ) 1 ( 1 2 2 2 2 1 1 1 1 1 2 1 1 0 ], [ ) 1 ( ] [ ) 1 ( ) 1 ( ) 1 ( ) 1 ( H H H R x r H R x G k G r I t r I t W (4)
Which segment of the kinked budget constraint is the optimal solution located at? It depends upon the individual preference on weather buying up or buying down. Few homeowners actually buy down. In other words, most people usually buy up when they purchase the second house. According to the specification of tax treatment, the capital gains tax payment becomes zero if the value of new house is significantly higher than the value of the previous house. In this case, the capital gains tax has no effect on the housing demand. Therefore, we exclude two extreme cases and only focus on the case that the individual household prefers to buy up and the incremental
value of the new house can deduct partial capital gains tax. We rewrite the middle segment of the budget constraint as follows:
m m m P H P P H H H R x r H R x H H P G k G r I t r I t W 1 0 2 1 2 2 2 1 1 1 1 1 2 1 2 1 1 0 ) 2 ( ], [ ) 1 ( )]} ( [ { ) 1 ( ) 1 ( ) 1 ( ) 1 ( (5)
If the value of second house is equal to the convex kink, the capital gains tax payment is zero. On the other hand, if the value of second house is equal to the concave kink, the entire capital gains tax is paid. If the value of second house is between two kinks, only partial capital gains tax is paid.
4. Traditional Intertemporal Optimization with Various Degrees of
Myopia
In this section, we derive optimal intertemporal housing demand functions using the traditional optimization method. However, the problem of myopia on the capital gains tax treatment is taken into account. A household’s two-period intertemporal utility function is
1
( , ) ) , ( 1 1 1 2 2 2 1 x H U x H U U (6) The household’s intertemporal budget constraint with a certain degree of myopia is shown as follows: } ] ) 1 ( [ { ) 1 ( ] ) )( 1 )[( 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( 2 2 2 1 1 1 1 1 1 0 1 2 1 1 0 H P k R x r H R x H P k H P P k r I t r I t W m m m (7)By maximizing (6) subject to (7), the following conditions (8)~(10) should be satisfied: ] ) )( 1 )[( 1 ( ) 1 ( 1 ) , ( ) , ( 0 1 1 * 1 * 1 * 1 * 1 1 1 m m H x P k P P k r R H x U H x U (8) m H x P k R H x U H x U ) 1 ( 1 ) , ( ) , ( 2 * 2 * 2 * 2 * 2 2 2 (9) } ] ) 1 ( [ { ) 1 ( ] ) )( 1 )[( 1 ( ) 1 ( ) 1 ( ) 1 ( ) 1 ( * 2 2 * 2 1 * 1 1 * 1 * 1 * 1 0 1 2 1 1 0 H P k R x r H R x H P k H P P k r I t r I t W m m m (10)
In equation (8), the marginal substitution rate between the first-period residual composite consumption and the first house is equal to the inverse of real user cost of owning the first house. The real user cost of owning the first house comprises of three components: the standard user cost of owning R , the present value of 1 marginal distortion on the tax deduction, k , and the present value of marginal Pm
after-tax capital gains, (1k)(PmP0). The marginal distortion on tax deduction is the additional reduction value of tax exemption for an additional unit of the first house. The marginal after-tax capital gain is the additional after-tax capital gains for additional unit of the first house. If the individual has a perfect myopia on the tax treatment, the real user cost of owning the first house is simply equal to the
standard user cost of owning the first house. Otherwise, the real user cost of owning should be equal to the sum of the standard user cost and the present value of marginal distortion on the tax deduction minus the present value of marginal after-tax capital gains. According to equation (8) and the assumption of positive marginal utility of the first-period housing consumption, the real user cost of owning the first house should be positive. From equation (9), we find the real user cost of owning the second house is equal to the difference between the standard user cost of owning the second house R and its marginal incremental value on the tax deduction 2 k . Pm
The marginal incremental value on the tax deduction is the additional value of tax exemption for an additional unit of the second house. The real user cost of owning the second house should be positive, too.
The optimal first-period housing consumption of the individual is a function of the capital gains tax rate as well as other variables including income variables, the standard user cost , price of housing, the fraction of taxable capital gains income, the interest rate, the degree of myopia, the income tax rate and so on. The effect of the capital gains tax rate on the optimal first-period housing consumption can be shown using the following comparative static equation:
} ) 1 ( ] {[ ] )[ 1 ( ) 1 ( ] [ 2 2 2 2 2 2 1 1 2 2 2 2 1 1 2 2 2 2 2 2 1 * 1 x H H x H H x x x H H x x x x H H x U U U r U U U A B r U U U A H (11) ] ) )( 1 )[( 1 ( ) 1 ( ] [A R1 r 1 k PmP0 kPm )} ( ) ( { ] [B k PmP0 H1 kPm H2 H1 ]
gains tax payment of the tax rate. The real user cost is positive, while the marginal net capital gains tax payment of the tax rate is at least greater or equal to zero. Equation (11) suggests that the capital gains tax has no effect on the optimal
first-period housing consumption if the individual has a perfect myopia ( 1)on the capital gains tax treatment. Even though the individual has the perfect foresight, the capital gains tax still has no effect on the optimal first-period housing consumption as the marginal net capital gains tax of the tax rate is equal to zero.5 Except these extreme cases, the capital gains tax rate generally has a negative effect on the optimal first-period housing consumption. There are two reasons for this outcome. First, as the capital gains tax rate is raised, and ceteris paribus, the capital gains tax
payment is increased. The disposable lifetime incomes are decreased. The optimal first-period housing consumption is reduced owing to this income effect. Second, the present value of marginal distortion on the tax deduction is increased and the present value of marginal after-tax capital gains is decreased as the tax rate is raised. The net effect of the tax rate on the real user cost of owning the first house is positive. Therefore, the quantity of the first-period housing consumption should be decreased due to the substitution effect. However, the absolute value of the negative effect falls with the degree of myopia.
The optimal second-period housing consumption of the individual is also a function of the capital gains tax rate as well as other variables including income variables, the standard user cost per unit, price of housing, the fraction of taxable capital gains income, the interest rate, the degree of myopia, the income tax rate, and so on. The effect of the capital gains tax rate on the optimal second-period housing consumption can be shown in the following comparative static equation:
} ) 1 ( ] {[ ) 1 ( ) 1 ( 2 2 2 2 2 2 1 1 2 2 2 2 1 1 1 1 2 2 3 * 2 x H H x H H x x x H H x x H H x m U U U r U U U A U U kP r H (12)
From equation (12), we find the capital gains tax has no effect on the optimal second-period housing consumption if the individual has a perfect myopia ( 0). Otherwise, it generally has a positive effect on the quantity of the second-period housing consumption. As the tax rate is raised, the marginal incremental value on the tax deduction is increased. Therefore, real user cost is lowered. Owing to the substitution effect, the optimal second-period housing consumption should be increased.
5. Modified Second-period Optimization Problem
At the moving (relocating) time, individuals become full knowledgeable on the previous housing consumption and the tax code related to capital gains. Given this information, they seek for the optimal second-period housing consumption and the optimal second-period residual composite consumption in order to maximize the second period utility level. The modified second-period housing consumption should substitute for the optimal solution obtained from traditional intertemporal optimization problem. The objective function of the optimization problem in the second period is ) , ( 2 2 2 , 2 2 H x U U Max x H (13) The second-period budget constraint is
) ( ) )( 1 ( ) 1 ( ) 1 ]( ) 1 ( [ 2 2 2 * 1 * 1 0 2 * 1 1 * 1 1 0 H P k R x H P k H P P k I t r H R x I t W m m m (14)
Both x and H have been determined on the beginning of the first period, 1 so they are given on the beginning of the second period. The left hand side of the equation (14) represents the net disposable wealth of the second period. Its first term is the given accumulated wealth, and its third term is the given after-tax capital gains income, and the fourth term is the given distortion on tax deduction. The net disposable wealth can be spent on both second-period residual composite good consumption and second-period housing consumption. The per unit real user cost is still the standard user cost per unit R minus its marginal incremental value on the 2 tax deduction k . By maximizing (13) subject to (14), we find the modified Pm
optimal second-period housing consumption H2m and the optimal second-period residual composite good consumption x2m should satisfy the following conditions:
m m m H m m x P k R H x U H x U 2 2 2 2 2 1 ) , ( ) , ( 2 2 (15) 0 ) ( ) )( 1 ( ) 1 ( ) 1 ]( ) 1 ( [ 2 2 2 * 1 * 1 0 2 * 1 1 * 1 1 0 m m m m m H P k R x H P k H P P k I t r H R x I t W (16)
Both equation (15) and (9) are the same if the degree of myopia is zero. Otherwise, they are not the same. We find the modified second-period housing
consumption is a function of both the capital gains tax rate and the first-period
housing stock as well as other variables. Comparing with the optimal second period housing consumption function obtained in the previous section, we find there are two changes in the modified function. First, the given first period stock would influence the second period housing consumption. Second, the degree of myopia is not one of determinants of the optimal second-period housing consumption any more.
The impact of the first-period housing stock on the second-period housing consumption can be shown in the following equation:
2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 * 1 2 ) ( ] )[ ( ) 1 ]( [ ) ( ) 1 ]( [ x x m H H H x x H m x x m x H m U P k R U U U P k R r A U P k R r A U H H (17) ] ) )( 1 [( ) 1 ( ] [A R1 r 1 k Pm P0 kPm
The sign of [ A]is positive, so the impact of the first-period housing stock on the second-period housing consumption is negative. The negative impact attributes to three effects: the negative exemption effect as well as both negative wealth effect and positive gains effect. As the first-period housing stock is increased, the amount of deductive tax exemption is decreased and then the net after-tax gains income is
decreased. Therefore, the second-period housing consumption falls owing to a lower disposable income of the second-period. This is called negative exemption effect. When the first-period housing consumption is increased, and ceteris paribus, the household saving in the first period is decreased and then the disposable wealth in the second-period is decreased. Therefore, the second-period housing consumption should be decreased owing to the negative wealth effect. On the other hand, after-tax net capital gain income is increased, so the second-period housing
consumption is increased owing to the positive gains effect. Both negative wealth effect and negative exemption effect dominate the positive gains effect, so the impact of the first-period housing stock on the second-period housing consumption is
negative.
The impact of the capital gains tax rate on the second-period housing consumption is shown as follows: 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 * 2 * 2 ) ( ] )[ ( ] [ ) ( ] [ x x m H H H x x H m x x m m x H m U P k R U U U P k R B U P k R kP B U H (18) )} ( ) ( { ] [B* k Pm P0 H1* kPm H2m H1*
If the marginal net capital gains tax of the tax rate [B] is equal to zero, the tax rate has a positive effect on the second-period housing consumption simply because of the positive deduction effect. In other words, the marginal incremental value on the tax deduction is increased as the tax rate is raised. Therefore, per unit real user cost is lowered. Owing to the substitution effect, the optimal second-period housing
consumption should be increased. If the marginal net capital gains tax of the tax rate ]
[B is positive, the tax rate has an ambiguous effect on the second-period housing consumption depending upon income effect as well as the mentioned substitution effect. As the tax rate is raised, the capital gains tax payment is increased and then the disposable income of the second period is decreased. Therefore, the second period housing consumption is decreased owing to the income effect. If the income effect dominates the substitution effect, the influence of the tax rate on the second period housing consumption is negative, and vice versa.
By summarizing the analysis results in both section four and section five, we find the optimal solutions of intertemporal housing demand are H and 1 H2m. The modified second-period housing consumption H2m is a function of the optimal first-period housing consumption H . The impact of the first-period housing stock 1 on the second-period housing consumption is negative. The tax rate generally has a negative effect on the optimal first-period housing consumption H except several 1 special cases. However, the effect of the tax rate on the modified second period housing consumption is ambiguous. The effect is negative if the income effect dominates the substitution effect, and vice versa.
6. Conclusion
This paper incorporates the capital gains tax into the intertemporal housing demand framework. Rollover provisions in tax codes of some countries enable houseowners to avoid paying tax on the capital gain from selling their house if they buy a new house of equal or greater value within a short period time of when they move. Owing to this special tax treatment on the capital gain, houseowners often face a kinked budget constraint. Since most of individuals prefer to buy up for the second house, we ignore the case of buying down in the study.
Using the traditional intertemporal optimization framework, we find the optimal first-period housing consumption of the individual is a function of the capital gains tax rate as well as other variables including income variables, the standard user cost
per unit, price of housing, the fraction of taxable capital gains income, the interest rate, the degree of myopia, and so on. The capital gains tax has no effect on the optimal first-period housing consumption if the individual has a perfect myopia on the capital gains tax treatment. Even though the individual has the perfect foresight, the capital gains tax still has no effect on the optimal first-period housing consumption as the marginal net capital gains tax of the tax rate is equal to zero. Except these extreme cases, the capital gains tax rate generally has a negative effect on the optimal
first-period housing consumption. On the other hand, the capital gains tax rate generally has a positive effect on the optimal second-period housing consumption.
In reality, individuals can modify their optimal second-period decisions on the beginning of the second period. At this point of time, they become full
knowledgeable on the previous housing stock and capital gains tax code. Given these information, individuals choose both residual composite consumption and second-period housing consumption in order to maximize the second period utility subject to the second period budget constraint. We find the modified second-period housing consumption function is different from the traditional optimal second period housing consumption function. The modified second-period housing consumption is now a function of the given first-period housing stock, while the traditional optimal second period housing consumption function is not. The first-period housing stock has a negative effect on the modified second-period housing consumption. The effect of the capital gains tax rate on the modified second-period housing
consumption is ambiguous. The effect is negative if the income effect dominates the substitution effect, and vice versa.
This study suggests the optimal intertemporal housing consumption functions are mixed solutions from both traditional and modified optimization problem. The optimal first period housing consumption H is obtained from the traditional 1 intertemporal framework, while the modified second period housing consumption
m
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