- Email: [email protected]
Econometric estimation for a model with dynamic housing tenure choice
~?
ATHEMATICS AND COMPUTERS N SIMULATION
ELSEVIER
Mathematics and Computers in Simulation 39 (1995) 359-365
Econometric estimation for a model with dynamic housing tenure choice Mitsuo Takase Faculty of Economics Fukuoka University, Nanakuma Jonan-ku, Fukuoka 814-01, Japan
1. Introduction
The objective of this paper is to show a method for estimating the parameters of a dynamic utility maximization model which includes discrete choices and to take housing d e m a n d as an example. The utility maximization problem introduces here attempts to determine how households decide when to purchase housing and how they decide to allocate their resources between housing and non-housing consumption. The discrete choice here is concerned with whether they own a house or rent it. [1-3] show the estimation methods with the discrete choice of rented and owner-occupied housing in a 'static' framework. Cross-sectional data are employed in their studies. This study is an attempt to develop an estimation method of a dynamic model emphasizing the d e m a n d for rental and owner-occupied housing and the timing of owner-occupied housing purchase. The likelihood function in this estimation is expressed in multi nominal probit formation. U n d e r a given income and wealth distribution, this model reveals the aggregate d e m a n d for rental and owner-occupied housing. This estimation method makes it possible to exploit the longitudinal character of panel data.
2. Utility maximization model
The model solves a dynamic utility maximization problem which includes the discrete choice of rental and owner-occupied housing. However, there is no general method to solve a constrained maximization problem where one of the variables in the problem is not continuous. Therefore, this paper takes the following steps: In the first step the utility function is maximized subject to the budget constraint for each possible housing tenure pattern. Then the indirect utility function for each tenure pattern is derived. In the second step the optimal tenure pattern is chosen by comparing all the values of the indirect utility functions derived in the first step. 0378-4754/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 7 5 4 ( 9 4 ) 0 0 0 8 4 - W
360
M. Takase / Mathematics and Computers in Simulation 39 (1995) 359-365
W e assume that a household lives for n years and purchase a house when the household is z years old. A household born in the j-th year maximizes its utility subject to its budget constraint and this is expressed as the problem (1) in the following: &
Maximize qt
C i j t -.1-j~t-Iijta - TeijQijt
/=12"a
subject to Kit +
(1 "1-~)t-2+j
qz+jHijz
(1 + r ) t - z + j
= L Pt+jCijt "1- {pt+t(1
-
Q.ijt) -~- RUCzjtQijt)I~ijt
(1 + r) ,-2+t
,=1 where
Wqo
Kij
(1
(1)
~-,
q-r) j - I
'
l i t , - Tit `
q- t=l /_a (1
+r
Cit , = the non-housing consumption of the i-th household which is t years old and was born in the j-th year, Hiz = the housing services consumed by the household, eit = the taste p a r a m e t e r relating to the owner-occupied housing of the household, a = the discount rate of the future utility, Wq0 = the initial e n d o w m e n t of the household, Iitt = the labor income of the household, Tq, = the income tax of the household, P,+t-1 = the price of non-housing consumption in the (t + j - 1)-th year, Pt+t-1 = the unit price of rental housing in the (t + j - 1)-th year, RUCzt , = the unit cost of owner-occupied housing of t year-old household which was born in the j-th year and bought the house at the time when the household was z years old, qz+t-1 = the price of owner-occupied housing in (z + j - 1)-th year,
Qitt
= 1 if a dwelling is owned, 0 if a dwelling is rented =
and r = the interest rate. The solutions for the problem (1) are as follows: For z = 1, we have
Hiit = Hijz = Cit t = GttJit
Jij
for 1 ~< t ~< n, for 1 ~< t < n,
where
4t
( + =Kit/
qi
1 r=lL (1
+
1
r) "-2+fPr+j-IGO + r=lL (1-{-r) r - 2 + j R U c l j r
)
M. Takase/ Mathematicsand Computersin Simulation39 (1995)359-365
361
and
1 "t- 6 ) t-
flPt+j-1/
(1 +
r) ,-2+jRUClj¢
r) j-1 "
(1 +
For z >t 2, w e have t
Hijt = H AkjEij
for I < t < z,
k=2
for z ~< t ~< n,
Hij t = Hii z Cij t
=
BtjEij
for 1 ~< t < z ,
-1/l+a
Pz+j-1 ) Pz-2+j Cijt =
fi
k=z+l
z-1
for l = z,
H AkjEij k=2
O k j l l ~ - r r )[3
Pz-2+j
for z < t <<.n,
HZkjEi)
k=2
where
Pk+j_l ]l-1/l+a
l+t~
t 1-I A~,j = 1 k=2
Hiiz =
[(
for t = 1, _
1
L
"F=Z
qz +j- 1
(1 + r) ¢-2+iRUCzi¢ - (1 + r) z-2+j ljl+
Btj= /3
/
Pk+j-1
i-iAkj
Pk+j-1
k=2
for t = l ,
Pi
Pk-2+j)
for l ~ < t < z ,
(1 + 3)(1
+
r) z-3+j
Pz-2+j
1/l+a
z-1 H AkjEij,
k=2
362
M. Takase/ Mathematics and Computersin Simulation39 (1995)359-365
and
z-1
-1/l+a
1
(1
+
z-1
r) r-2+jPr+t-lBrt + n
+
Pz-2+j
1
X k=2Hakj+ r=z+lE (l + r ) r-e+tpr+t-1 )<
fi
1+6 ~
Dkj
k=z+l
Pz+j-1 Pz-2+j
~
l--I Akj
k=2
z-1 1 r + ,=,Y~ (1 + r) "-2+tpr+j-' kI~I=2AkJ= 1
1
+ r=z ~-" (l+r) r-2+tRuczjr
_
qz+j-1
(l+~)(l+r)Z-3+J}} -'/'+a Pz-a+j
(1 + r) z-a+j
{{
X 1-I Akj k=2
~
+
1
r=z (1 + r) r-2+tRUCzjr
(1 + 6)(1
qz +j - 1
(1
= (l+r) r-2+tRUCzjr
r) z-2+j
+
r) z-3+j
Pz-2+j
If the household never owns a house, this utility maximization problem will be expressed as the problem (2) in the following: Maximize U/j
Ci77 + 13Hits"
=
~'~ t=l (1 + •)t-2+j
-
subject to Kij =
~
Pt+jCijt + pt+jI-Iijt
t=l
(1 + r) '-2+t
The solutions for the problem (2) are as follows: for t= l,
Hit ' = Fit ,
t
Hit t = Fit kiwi__ 2A kj
for 2~t-%
Cij t = FijBtj
for l ~ t ~ < n ,
(2)
M. Takase/ Mathematicsand Computersin Simulation39 (1995)359-365
363
where
Fij = Kij /
~ Pr+j_lBrj+Prl-Irk=2Akj } r=2 (1 -F r) r-2+j + PjBlj + pj "
3. Indirect utility functions The indirect utility function of the maximization problem (1) is obtained as a function of z and eii as follows: For z = 1 we have
1
-,~
Vij(Z' Eij) --- -- t=l (l + a)t-2+j (GtjJij)
n
1
- .
- /3 Et=I (1--F ¢~)t-2+j (Jij)
n
-+-'y Et=I (1+6) t-2+i
For z > 2 we have
Vo(z, eij)= -
BtjEii)-"+/3
/(1 + •)t-l+j
AkjEij
t=l
-
A~jEij
+/3
×
.--~z (1 + r) *-i+fRUCzj¢ (1 + r) ~-'+j
(1 + 6)(1 + r) z-2+j - l / l + a z _ l
]-a
H AkjEij k=2
Pz - 1+j
_yeii]/( 1 + ¢~)z-l+j t=z+l
k=z+l
-1/l+az_l ×
e
+13
)<
• ~= z
Pz-,+,
(1 + r)- ~-'
(1 + 6)(1
+
r) z-2+j
Pz 1+j -
~-a
klTI=2A Eij kj
+iRUCzj~ -1/l+a
Zl
qz +j (1 +
HAkjEij k=2
r) z-l+]
]
-~/sij
1/(1
ei j
"l- ~) t-l+j
M. Takase/ Mathematics and Computers in Simulation 39 (1995) 359-365
364
The indirect utility function of the maximization problem (2) is obtained as
(Fij) -a
(FijBtj) -a Vii=-
r),-2+j
t=l (1 +
J~
"
(1 + r ) ' -
n 1
1
~=2
Y'~ (1 + r ) ,-2+i t=2
=
(FijAki)-""
Therefore, the optimal timing of purchasing a house for the i-th household born in the j-th year, zij, can be generally written as
zij = cb(eij ) where zii = 1, 2 ..... n, since
Vii(z*, ei~)= MaxVi~(z , Eij ). z
This implies that the timing of purchasing a house depends on the level of taste parameter, e~y. Hence, we can obtain the threshold level of e~y for respective timing of housing purchase. For instance, we call eli ~ as the threshold level of 6ij which concerns w h e t h e r the household born in the j-th year purchase a house in the (k + j - 1)-th year or in the (k + j ) - t h year. elm is the threshold level of eiy which concerns w h e t h e r the household purchase a house in the (n + j - 1)-th year or never own a house. These threshold points are obtained from the indirect utility functions derived above.
4. The likelihood function
The probability of a household purchasing a house at the time when the household was (k + 1) years old is written as
c1)ijk
:
foofe,, C~(Eij' Kij) dEij dKij
for k = 1, 2 . . . . . ( n -
1),
"0 "ely,k+1 where b(eij) is the joint probability density function of the taste parameter, eij and the present value of life-time income and assets. The probability of a household purchasing a house at its first period is written as
flP(sij' Kij) deij dKij '
q ) i J ° - - fo f~ ijl
A n d the probability of a household never owing a house is written as oo
~ijn
*ijn= fo fo (~(Eij, Kij) dEij dKij. Therefore, the likelihood function for the households which were born m o r e than n years ago is obtained as y--n+l
L.=
vj f i
I-I
1-I
j=l
i=1
q~iil,"
k=O
where y = n u m b e r of years of observation, and vj = n u m b e r of households born in the j-th year
M. Takase / Mathematics and Computers in Simulation 39 (1995) 359-365
365
The likelihood function for the households which are r years old at present is obtained as
L,=I-I
f o r , - - O, l , 2 . . . . .
i=1 k=O
where j = y - ~- + 1, ~
1/t'+l(Kij)
~ij,r+l
= fo fo
(a(eiJ'Kij) deij dKij"
~+~(Ki~)
represents the probability of the r year-old household which does not own a house yet at present. Therefore, the likelihood function of our model is obtained as n
L=
FILm
.
m=0
By maximizing this likelihood function with respect to the parameters of the model, we can obtain the maximum likelihood estimators for our model.
5. Concluding remarks This paper shows an estimation m e t h o d for a dynamic maximization model which includes a discrete choice concerning the housing tenure. Empirical analysis has not been done in this study. Mainly because no suitable data for this study are available at present. The estimation result of our model will directly enable us to predict effects of exogenous changes on the timing of housing purchase and the housing demand, and indirectly enable us to analyze effects of exogenous changes on savings rate of the economy as a whole and savings rate of each generation. Here, exogenous changes include price inflation of commodities and housing, income and wealth changes, and so forth. This, however, are left for further research.
References [1] L.F. Lee and R.P. Trost, Estimation of some limited dependent variable models with applications to housing demand, J. Econom. 8 (1978) 357-382. [2] H.S. Rosen, Housing decisions and the U S income tax: an econometric analysis, J. Public Econom. 11 (1979) 1-23. [3] M. King, An econometric model of tenure choice and demand for housing as a joint decision, J. Public Econom. 14 (1980) 137-159.
ATHEMATICS AND COMPUTERS N SIMULATION
ELSEVIER
Mathematics and Computers in Simulation 39 (1995) 359-365
Econometric estimation for a model with dynamic housing tenure choice Mitsuo Takase Faculty of Economics Fukuoka University, Nanakuma Jonan-ku, Fukuoka 814-01, Japan
1. Introduction
The objective of this paper is to show a method for estimating the parameters of a dynamic utility maximization model which includes discrete choices and to take housing d e m a n d as an example. The utility maximization problem introduces here attempts to determine how households decide when to purchase housing and how they decide to allocate their resources between housing and non-housing consumption. The discrete choice here is concerned with whether they own a house or rent it. [1-3] show the estimation methods with the discrete choice of rented and owner-occupied housing in a 'static' framework. Cross-sectional data are employed in their studies. This study is an attempt to develop an estimation method of a dynamic model emphasizing the d e m a n d for rental and owner-occupied housing and the timing of owner-occupied housing purchase. The likelihood function in this estimation is expressed in multi nominal probit formation. U n d e r a given income and wealth distribution, this model reveals the aggregate d e m a n d for rental and owner-occupied housing. This estimation method makes it possible to exploit the longitudinal character of panel data.
2. Utility maximization model
The model solves a dynamic utility maximization problem which includes the discrete choice of rental and owner-occupied housing. However, there is no general method to solve a constrained maximization problem where one of the variables in the problem is not continuous. Therefore, this paper takes the following steps: In the first step the utility function is maximized subject to the budget constraint for each possible housing tenure pattern. Then the indirect utility function for each tenure pattern is derived. In the second step the optimal tenure pattern is chosen by comparing all the values of the indirect utility functions derived in the first step. 0378-4754/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 8 - 4 7 5 4 ( 9 4 ) 0 0 0 8 4 - W
360
M. Takase / Mathematics and Computers in Simulation 39 (1995) 359-365
W e assume that a household lives for n years and purchase a house when the household is z years old. A household born in the j-th year maximizes its utility subject to its budget constraint and this is expressed as the problem (1) in the following: &
Maximize qt
C i j t -.1-j~t-Iijta - TeijQijt
/=12"a
subject to Kit +
(1 "1-~)t-2+j
qz+jHijz
(1 + r ) t - z + j
= L Pt+jCijt "1- {pt+t(1
-
Q.ijt) -~- RUCzjtQijt)I~ijt
(1 + r) ,-2+t
,=1 where
Wqo
Kij
(1
(1)
~-,
q-r) j - I
'
l i t , - Tit `
q- t=l /_a (1
+r
Cit , = the non-housing consumption of the i-th household which is t years old and was born in the j-th year, Hiz = the housing services consumed by the household, eit = the taste p a r a m e t e r relating to the owner-occupied housing of the household, a = the discount rate of the future utility, Wq0 = the initial e n d o w m e n t of the household, Iitt = the labor income of the household, Tq, = the income tax of the household, P,+t-1 = the price of non-housing consumption in the (t + j - 1)-th year, Pt+t-1 = the unit price of rental housing in the (t + j - 1)-th year, RUCzt , = the unit cost of owner-occupied housing of t year-old household which was born in the j-th year and bought the house at the time when the household was z years old, qz+t-1 = the price of owner-occupied housing in (z + j - 1)-th year,
Qitt
= 1 if a dwelling is owned, 0 if a dwelling is rented =
and r = the interest rate. The solutions for the problem (1) are as follows: For z = 1, we have
Hiit = Hijz = Cit t = GttJit
Jij
for 1 ~< t ~< n, for 1 ~< t < n,
where
4t
( + =Kit/
qi
1 r=lL (1
+
1
r) "-2+fPr+j-IGO + r=lL (1-{-r) r - 2 + j R U c l j r
)
M. Takase/ Mathematicsand Computersin Simulation39 (1995)359-365
361
and
1 "t- 6 ) t-
flPt+j-1/
(1 +
r) ,-2+jRUClj¢
r) j-1 "
(1 +
For z >t 2, w e have t
Hijt = H AkjEij
for I < t < z,
k=2
for z ~< t ~< n,
Hij t = Hii z Cij t
=
BtjEij
for 1 ~< t < z ,
-1/l+a
Pz+j-1 ) Pz-2+j Cijt =
fi
k=z+l
z-1
for l = z,
H AkjEij k=2
O k j l l ~ - r r )[3
Pz-2+j
for z < t <<.n,
HZkjEi)
k=2
where
Pk+j_l ]l-1/l+a
l+t~
t 1-I A~,j = 1 k=2
Hiiz =
[(
for t = 1, _
1
L
"F=Z
qz +j- 1
(1 + r) ¢-2+iRUCzi¢ - (1 + r) z-2+j ljl+
Btj= /3
/
Pk+j-1
i-iAkj
Pk+j-1
k=2
for t = l ,
Pi
Pk-2+j)
for l ~ < t < z ,
(1 + 3)(1
+
r) z-3+j
Pz-2+j
1/l+a
z-1 H AkjEij,
k=2
362
M. Takase/ Mathematics and Computersin Simulation39 (1995)359-365
and
z-1
-1/l+a
1
(1
+
z-1
r) r-2+jPr+t-lBrt + n
+
Pz-2+j
1
X k=2Hakj+ r=z+lE (l + r ) r-e+tpr+t-1 )<
fi
1+6 ~
Dkj
k=z+l
Pz+j-1 Pz-2+j
~
l--I Akj
k=2
z-1 1 r + ,=,Y~ (1 + r) "-2+tpr+j-' kI~I=2AkJ= 1
1
+ r=z ~-" (l+r) r-2+tRuczjr
_
qz+j-1
(l+~)(l+r)Z-3+J}} -'/'+a Pz-a+j
(1 + r) z-a+j
{{
X 1-I Akj k=2
~
+
1
r=z (1 + r) r-2+tRUCzjr
(1 + 6)(1
qz +j - 1
(1
= (l+r) r-2+tRUCzjr
r) z-2+j
+
r) z-3+j
Pz-2+j
If the household never owns a house, this utility maximization problem will be expressed as the problem (2) in the following: Maximize U/j
Ci77 + 13Hits"
=
~'~ t=l (1 + •)t-2+j
-
subject to Kij =
~
Pt+jCijt + pt+jI-Iijt
t=l
(1 + r) '-2+t
The solutions for the problem (2) are as follows: for t= l,
Hit ' = Fit ,
t
Hit t = Fit kiwi__ 2A kj
for 2~t-%
Cij t = FijBtj
for l ~ t ~ < n ,
(2)
M. Takase/ Mathematicsand Computersin Simulation39 (1995)359-365
363
where
Fij = Kij /
~ Pr+j_lBrj+Prl-Irk=2Akj } r=2 (1 -F r) r-2+j + PjBlj + pj "
3. Indirect utility functions The indirect utility function of the maximization problem (1) is obtained as a function of z and eii as follows: For z = 1 we have
1
-,~
Vij(Z' Eij) --- -- t=l (l + a)t-2+j (GtjJij)
n
1
- .
- /3 Et=I (1--F ¢~)t-2+j (Jij)
n
-+-'y Et=I (1+6) t-2+i
For z > 2 we have
Vo(z, eij)= -
BtjEii)-"+/3
/(1 + •)t-l+j
AkjEij
t=l
-
A~jEij
+/3
×
.--~z (1 + r) *-i+fRUCzj¢ (1 + r) ~-'+j
(1 + 6)(1 + r) z-2+j - l / l + a z _ l
]-a
H AkjEij k=2
Pz - 1+j
_yeii]/( 1 + ¢~)z-l+j t=z+l
k=z+l
-1/l+az_l ×
e
+13
)<
• ~= z
Pz-,+,
(1 + r)- ~-'
(1 + 6)(1
+
r) z-2+j
Pz 1+j -
~-a
klTI=2A Eij kj
+iRUCzj~ -1/l+a
Zl
qz +j (1 +
HAkjEij k=2
r) z-l+]
]
-~/sij
1/(1
ei j
"l- ~) t-l+j
M. Takase/ Mathematics and Computers in Simulation 39 (1995) 359-365
364
The indirect utility function of the maximization problem (2) is obtained as
(Fij) -a
(FijBtj) -a Vii=-
r),-2+j
t=l (1 +
J~
"
(1 + r ) ' -
n 1
1
~=2
Y'~ (1 + r ) ,-2+i t=2
=
(FijAki)-""
Therefore, the optimal timing of purchasing a house for the i-th household born in the j-th year, zij, can be generally written as
zij = cb(eij ) where zii = 1, 2 ..... n, since
Vii(z*, ei~)= MaxVi~(z , Eij ). z
This implies that the timing of purchasing a house depends on the level of taste parameter, e~y. Hence, we can obtain the threshold level of e~y for respective timing of housing purchase. For instance, we call eli ~ as the threshold level of 6ij which concerns w h e t h e r the household born in the j-th year purchase a house in the (k + j - 1)-th year or in the (k + j ) - t h year. elm is the threshold level of eiy which concerns w h e t h e r the household purchase a house in the (n + j - 1)-th year or never own a house. These threshold points are obtained from the indirect utility functions derived above.
4. The likelihood function
The probability of a household purchasing a house at the time when the household was (k + 1) years old is written as
c1)ijk
:
foofe,, C~(Eij' Kij) dEij dKij
for k = 1, 2 . . . . . ( n -
1),
"0 "ely,k+1 where b(eij) is the joint probability density function of the taste parameter, eij and the present value of life-time income and assets. The probability of a household purchasing a house at its first period is written as
flP(sij' Kij) deij dKij '
q ) i J ° - - fo f~ ijl
A n d the probability of a household never owing a house is written as oo
~ijn
*ijn= fo fo (~(Eij, Kij) dEij dKij. Therefore, the likelihood function for the households which were born m o r e than n years ago is obtained as y--n+l
L.=
vj f i
I-I
1-I
j=l
i=1
q~iil,"
k=O
where y = n u m b e r of years of observation, and vj = n u m b e r of households born in the j-th year
M. Takase / Mathematics and Computers in Simulation 39 (1995) 359-365
365
The likelihood function for the households which are r years old at present is obtained as
L,=I-I
f o r , - - O, l , 2 . . . . .
i=1 k=O
where j = y - ~- + 1, ~
1/t'+l(Kij)
~ij,r+l
= fo fo
(a(eiJ'Kij) deij dKij"
~+~(Ki~)
represents the probability of the r year-old household which does not own a house yet at present. Therefore, the likelihood function of our model is obtained as n
L=
FILm
.
m=0
By maximizing this likelihood function with respect to the parameters of the model, we can obtain the maximum likelihood estimators for our model.
5. Concluding remarks This paper shows an estimation m e t h o d for a dynamic maximization model which includes a discrete choice concerning the housing tenure. Empirical analysis has not been done in this study. Mainly because no suitable data for this study are available at present. The estimation result of our model will directly enable us to predict effects of exogenous changes on the timing of housing purchase and the housing demand, and indirectly enable us to analyze effects of exogenous changes on savings rate of the economy as a whole and savings rate of each generation. Here, exogenous changes include price inflation of commodities and housing, income and wealth changes, and so forth. This, however, are left for further research.
References [1] L.F. Lee and R.P. Trost, Estimation of some limited dependent variable models with applications to housing demand, J. Econom. 8 (1978) 357-382. [2] H.S. Rosen, Housing decisions and the U S income tax: an econometric analysis, J. Public Econom. 11 (1979) 1-23. [3] M. King, An econometric model of tenure choice and demand for housing as a joint decision, J. Public Econom. 14 (1980) 137-159.