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URBAN LAND VALUE FUNCTIONS AND THE
PRICE ELASTICITY OF DEMAND FOR HOUSING
Jataes B.
Kau and
C.
F.
Simians
#371
College of
Commerce and
Business Administration
University of Illinois at Urbana -Champaign
FACULTY WORKING PAPERS
College of Commerce and Business Administration
University of Illinois at Urbana-Charapaign
January 25, 197 7
URBAN LAND VALUE FUNCTIONS AND THE
PRICE ELASTICITY OF DEMAND FOR HOUSING
James
B.
Kau and
C.
F.
Sirmans
#371
URBAN LAND VALUE FUNCTIONS AND THE
PRICE ELASTICITY OF DEMAND FOR HOUSING
(LAND VALUE FUNCTIONS)
James B. Kau
Visiting Associate Professor
University of Illinois
and
Associate Professor
University of Georgia
C. F. Sirmans*
Assistant Professor
Department of Finance
University of Illinois
320 David Kinley Hall
Urbana, Illinois 61801
*To whom correspondence should be addressed
ABSTRACT
The structural characteristics of an urban area
are determined to a great extent by the spatial pattern of
land values.
It is thus important to develop urban land
value functions which accurately describe the spatial
structure of cities.
The purpose of this paper is to
provide some empirical evidence on the functional form
of the relationship between land values and distance from
the center of the city.
Using historical data for Chicago,
the Box and Cox transformation technique is employed to
examine the land value function.
The functional form
parameter can be used to determine the price elasticity of
demand for housing.
URBAN LAND VALUE FUNCTIONS AND THE
PRICE ELASTICITY OF DEMAND FOR HOUSING
1
.
I n troduction
The spatial pattern of land values determines to a
great extent the structural characteristics of an urban area.
Changes in exogenous factors such as income, transportation
cost and migration affect land values which in turn alters
production factor ratios and the corresponding urban
Hence, it is important to develop urban
structural patterns.
land value functions which accurately describe changes in
spatial structure.
Muth [10] and Mills
[8]
using Cobb-
,
Douglas demand and supply functions for housing, derived the
conditions necessary for the existence of an exponential
function between land values and distance.
One of the
conditions necessary for an exponential function is the
existence of a price elasticity of demand for housing equal
to minus one
The purpose of this paper is to provide some empirical
evidence on the functional form of the relationship between
land values and distance from the center of the city and to
re-evaluate the results obtained by Mills
urban land values.
[5,
6]
in his study of
Statistical transformation procedures
developed by Box and Cox
and Lee
[7]
[1]
and techniques developed by Kau
for determining the price elasticity of demand
for housing services are used.
technique introduces
a
The Box and Cox transformation
functional form parameter to generalise
the land value function.
This same parameter can be used to
-
test whether the negative exponential function provides an
accurate description of land value patterns.
The functional
form parameter can also be used to determine price elasticity
of demand for housing.
This study is divided into five section.
In the
second section, Mills' derivation of the negative exponential
The third section provides a
function will be reviewed.
discussion of the general relationship between urban land
In the fourth section, data similar to
values and distance.
that used by Mills
[7]
for the Chicago area over the time
period 1830 to 1930 is used to test both the negative
exponential and generalized functional relationships
discussed in the third section.
Finally the fifth section
will summarize the results.
2.
The Exponential Land Value Function
Following Kills
distance u, Xs
(u)
X (u)
where K{u) and L
[8]
,
the output of housing services at
is defined as
= AL(u)
(u)
a
K(u)
1_a
(1)
represent inputs of capital and land in
the production of housing services u miles from the center;
and A and a are scale and distribution parameters, respectively,
for a Cobb-Douglas production function.
In deriving a negative exponential function to describe
the relationship between the land rent R(u) and the distance u,
Mills has also defined
at u, Xd
(u)
,
as
a
demand function for housing services
X
d
= Bw Gl p(u) 02
(u)
(2)
where
B = a scale parameter and depends upon the
units which housing services are measured;
w = income for workers;
= price of housing services at distance u;
p (u)
= income elasticity; and
Gi
2=
price elasticity.
Using the first order conditions of equation
aggregate demand derived from equation
equilibrium conditions, Mills
,
the
and other
83-84]
pp.
[8,
(2)
(1)
has derived the
following relationship between land rents R(u) and distance.
i
R(u)
-
R(u)
- 5e
[R
6
+
i?tE(u-u)]
3
if
M
(3)
and
tE(ri - u)
^
- °
(4)
where
R(u) = land rents at distance u;
= a (1+9?.)
6
B- 1 -
aV
1
tA«
(5)
a
a-*)
1 "B
r = rental rate for
t -
r :(1+ei) ' (1
" 0)(1+9l)
housing capital;
the commuting cost per mile;
u = the distance from the center to the edge
of the urban area;
R - rent on non-urban uses of land;
R(u)
Equation
and
= R.
(4)
demonstrates that rents decline exponentially
with distance if and only if
p
is equal to zero.
This result
also implies that the population density declines exponentially
with distance if and only if the price elasticity,
2
,
is
minus one
The Generalized Rent Function
3.
Following the procedure developed by Kau and Lee
and after some rearrangements, equation
R
3
——
when
3
,,
.
shown that equation
be.
approaches zero.
6]
can be rewritten as
._,(u)-1 = R B -1
+ tE(u-u)
a
It can
(4)
(3)
[5,
v
(6)
(6)
will become equation
Since there exists only two
observable variables, R(u) and
u in
equation (6), it can be
written as
X
R (u)-l =
R -YU
(7)
A
where
Y = 6,
=
R
o
Equation
(7)
B
P
'
-i
Q
p
+ yu and y = tE
belongs to one of the cases that Box and Cox
have derived for determining the true functional form.
X
approaches zero, then equation
(7)
[1]
If
will become
log R(u) = Ro-YU
(8)
This implies that the negative exponential function is only
a special case cf equation
(7)
;
therefore, equation
(7)
can be
regarded as a generalized functional form to be used to
determine the true relationship between rent and distance in
an urban area.
into equation
An additive stochastic term can be introduced
(7)
^M' 1
and the relationship defined as,
= R -yu +
e,
(9)
where
a
2
with zero mean and variance
is normally distributed
e.
and R c and y are regression parameters with
X
a
functional
form parameter.
An alternative functional form, with less theoretical
foundation than equation
(7)
can be used to incorporate
,
the double-log function use by Mills
[7]
.
The generalized
functional form is
(U
Rfu^'-l
RAH2
i
\
where y c
,
Yi
'
'
= v - Yl
—
i
-i
X, "
1)
+ e!
X
'
are regression parameters,
parameter and the disturbance term
with zero mean and variance g 2
For equation
on distance; if
X
(9),
if
(10)
x
•
a
functional form
is normally distributed
e!
respectively.
,
= 1,
X
X
linear rent is regressed
approaches zero the dependent variable is
Similarly, equation
the natural logarithm of rent.
reduce to the linear form when
double logarithmic form when
will
is equal to one and to a
X'
X'
(10)
approaches zero.
Under the assumptions of normality, the probability
rent function for
f(e.)
=
e.
in equation
(2TTa
2
)"
1/2
is
(9)
EXP(- l/2
(e
2
i
/a 2 ))
(11)
If the e.s are identically and independently distributed,
the log likelihood function for (11) is written as
M
M
log R(u)~
Log L = -| log 2ro 2 + (X~1)I
2 2
2|
i=l
i=l
5-^lzA
_ Yo
+ YlU
(12)
The logarithmic likelihood is maximized with respect
to
c
2
,
Ys a nd Yi
given
»
The maximum likelihood estimates
X.
2
if a 2 for the given X, o
is then the estimated variance
(X)
of the disturbance:- of regressing
Replacing
2
a
by
for equation
o
(X),
(',)
MAX.
the maximum log likelihood [LMAX(X)]
= "§ log 27raU)
(1,
p.
W"
" L MAX (A)
Equations (13) and (14) with
<
+
j
X
1/2 * 2
X'
( '
= X ' 92
05)
by a. similar procedure.
Equation
estimates of the optimum point estimate,
X
(5)
,
U4)
(14)
is used
and
X
X'
are
Note that
provides an
From the relationship between
defined in equation
{13)
can also be derived for
significantly different from zero and/or one.
£.
(u)
•*
t
is obtained from
to tsst whether the functional form parameters,
estimate of
log R.
(X-l)Z
indicate that an approximate
216)
95 percent confidence region for
(10)
§
£
-
£
Box and Cox
equation
.
.
is
(8)
L MAV
2
-1)/X on u
(R
8
and G 2
the assumption of unitary price
elasticity can also be tested statistically.
4.
Empir ic al Re s ults
Initially three regressions were estimated for each
year; ea^h representing the specific functional form^ of
linear, exponential and double logged with land values as the
dependent variable and distance as the independent variable.
The data for this analysis are land values
2
for the
Chicago metropolitan area for the time periods 1836, 1857, 1873,
1892, 1910 and 1928 from a study by Homer Hoyt
This
[4].
sample was selected so that comparison could be made with the
and because there is no other comparable
Mills study
historical series that allows comparing land value patterns
Land values are dollars per acre
over long periods of time.
and distance is airline miles from the intersection of State
and Madison Streets.
This initial set of regressions is
.
provided to exemplify the flexibility of the functional form
The coefficients are presented in Table
approach.
most cases are similar with Mills' results.
and in
1
4
The functional form parameter is determined by
transforming R{u) and
(10)
in accordance with equation
u.
and
(7)
using A's between -0.50 and 1.50 at intervals of 0.1.
Twenty-one different regressions are run for each time period.
The L,„.,(A)
MAA
equation
!
s
(13)
of the six time periods
are calculated by
J
5
The maximum likelihood estimates of
A
for the six
time periods for Chicago are listed in Table 2.
In general,
is not symmetrically distributed with respect to its
L.,.,,(A)
maximum value; therefore, the lower bound and upper bound are
calculated respectively.
Using equation (14), the point
estimate and the 95 percent confidence region for
calculated and listed in Table
A
are
2.
The results of the functional form analysis as presented
in Table
estimated
indicate that for
2
A
4
out of the
6
time periods the
is significantly different from zero at the
percent level.
Since
A
=
3
and because
B
=
ad
+ 02)
,
5
it
can be concluded that the price elasticity of demand for
10
TABLE
CHI CA GO L AN
-.103
(27.168)
.781
208
32)
7.224
(61.900)
-2.472
(42.305)
.896
208
5674.752
(7.931)
-576.87
(6.919)
.182
211
-.513
(35.724)
.858
211
(7 0.886)
10.533
(66.811)
-3.023
(38.625)
.876
211
-2361.37
(6.721)
.177
206
(8. 090)
Exponential
9.980
(71.655)
-.34U
(21.011)
.682
206
Log-Log
11.263
(67.632)
-2.075
(24.897)
.751
206
31569.6
(9.800)
-2865.35
(7.701)
.253
173
10.043
2.558)
.418
173
(.11.169)
11.519
(46.011)
.540
173
(1 4.254)
.301
12
-.319
(12.673)
.566
123
11.879
(44.120)
-1.368
(14.265)
.624
123
94411.90
(9.393)
-7055.42
(6.061)
.206
139
Exponential
11.736
(72.390)
-.2203
(11.735)
.497
139
Log-Linear
12.455
(49.40 7)
-1.267
(10.161)
.425
139
361.591
(8.120)
5.632
Exponential
Exponential
Log-Log
Linear
Linear
Exponential
Log-Log
Linear'
Exponential
Mi.
8
8.748
24173.00
32316.7
(9.497)
10.S84
(52.0'
Log-Log
Linear
1928
a
a
208
Linear
1910_
1928
.187
Log-Log
1892
-
-36.565
(6.982)
Distance
Constant
(
1873
1836
N
Linear
1857
VALUE GRADIENTS:
-2
R
Regression
1836
j_
1
c-values are in parentheses
I
-.246
-1.758
-3100.61
3
(7.332)
11
TABLE
2
THE PRICE ELASTICITY OF DEMAND FOR HOUSING
CHICAGO, 18 36-19
70
Price Elasticity
Year
18 36
3- Vali
-.20)
Elastic
-2.25
(-1.67 ~ -3.00)
-.0 7
(-.08 ~ -.05)
Elastic
-1.35
(-1.17 ~ -1.5 3)
Elastic
-1.50
(-1.17 ~ -2.07)
Elastic
-1.40
(-1.07 ~ -2.00)
-.25
(-.30
1857
1873
~
-.10
(-.16
1892
of Demand
-.05)
~
-.09
(-.15
-.02)
..
19.10
1928
1970°
Unitary
.00
(-.15
-
+.05)
Unitary
.
(-.12 - +.05)
0.00
(-1.22 ~ +.24)
95% confidence interna]
-1.00
~ -2.00)
-1.00
(-.83 ~ -1.80)
-1.00
(-.83
Unitary
in parentheses
The point estimate oi the elasticity of demand for housing is
earliest estimate of a from Hoyt [4]
based on a = .20.
indicated a = .27.
The eDasti< ities in parentheses are based
on estimates of a of
30 and .15, using the largest and smallest
estimates of 3, respectively.
density gradients by
This result was raker, from
Kau and Lee [51.
.
-
.
•
-
12
housing services is significantly different from minus one
when the
3
is
downward sloping demand curves for housing, then
3
>
C.
Thus when
inelastic.
Given
significantly different from zero.
When
f:
>
3
when
the price elasticity of demand is
>
3
a
and since a
<
>
the demand curve for
0,
housing services is elastic.
In all significant cases for equation
(7),
'3
is
less
than zero implying an elastic demand for housing services
For 1910 and 1928,
in the earlier years.
3
was not
significantly different from zero thus implying unitary
elasticity.
Kau and Lee's study
functional forms indicates
a
housing in 1970 for Chicago.
of density gradient
[5]
minus
one.
price elasticity for
Their study also indicated
that for the total 50 cities analyzed no cases resulted
in a significant elastic demand for housing and approximately
50% of the cities had inelastic demands.
Thus over time,
there appears to be a shift away from elastic to unitary
and inelastic demands for housing services.
demonstrated in Figure
I
where the L„ AX
(A)
This is
for the three
time periods 1836, 1892 and 1928 are plotted and reveal a
movement towards
a
X
Note that the cases
value of zero.
provided are significantly different from each other.
The results for equation (10) indicate that the
X
values are significantly different from zero in three of the
cases.
Thus the double-log form is inappropriate.
the previous result with equation
but confirms the necessity to use
(7)
a
,
Given
this is not unexpected
more generalized
functional form when analyzing urban spatial structure.
-400
13
-500
-600
-700
-800
g
d
e
•H
X
1836
v
cn
Ot
CO
.H
-900
tf
<I<
*
\o
O"
o
3
m
00
£
r-H
1928
The generalized rent gradient is derived by rewriting
equation
(7)
as
Riu)
=
[XRo-1-Xyu]
1/X
(15)
and taking the derivative of equation (15) with respect to u,
and rearranging terms, the generalized rent gradient is,
R\
,9R
(16)
The elasticity of rent with respect to distance is
3R
8u
If
X
u
(17)
-yu
R
= 0, then ny - -yu which is the elasticity for the
exponential rent gradient.
The generalized rent gradient
and its elasticity are a function not only of distance
but also of R(u)
(u)
14
The effect of this additional term, R(u)
is to reduce
exponential
or increase the elasticity relative to the
gradient as
Thus
the.
X
is greater or less than zero, respectively.
conclusion for Chicago is that during the 1330 to
gradient has
1930 time period the elasticity of the rent
If it is accepted that past
been decreasing over tine.
historical rental patterns affect current rents, then
equation
(17)
allows measurement of the impact of past
development on current rental graaients.
5.
Su mmary and Conclusions
empirical
The purpose of this paper was to provide some
evidence on the functional form of the relationship between
Using
land values and distance from the center of the city.
historical data for Chicago, the Box and Cox transformation
technique was used to test alternative forms of the land
value gradient.
These estimates help to clarify the
previous work by Mills
[1]
.
for
The results indicated that the price elasticity
housing tended to be shifting from elastic to unitary
elasticity of demand.
The negative exponential
in the theoretical model by Muth
[10]
and Mills
form,
[8]
derived
proved
periods
to be the correct form in only two of the six time
tested.
The generalized rent gradient derived in this
paper indicates that the elasticity of rent with respect to
distance is a function not only of distance but also of rent.
This result has important implications for analyzing the
importance of the impact of historical rental patterns on
current rates.
15
FOOTNOTE'S
1
This technique has been used to test the functional
form of the population density gradient (Kau and Lee [5, 6]),
the money demand function (Zarembka
[11]), and the
earning- schooling relationship (Heckman and Polachek
2
[3]).
Following Mills [7], the data are on urban land
It is assumed that the capitalization rate, at any
values.
given time, differs little from one place to another in an
urban area.
Hence values and rents will be a constant-
ratio
3
It should be noted that Mills
[7]
did not include
the land value data for 1892 in his study.
4
These results are available from the authors.
The slight variations in the estimated parameters
as compared to Mills
[7]
is
probably
due to inexact
methods of measuring distance to the CBD.
6
The estimates indicated that the double-log form
was inappropriate for the three earlier time periods.
7
See Harrison and Kain
[2]
for a discussion of the
cumulative impact of time on the spatial structure of an
urban area.
16
BIBLIOGRAPHY
1.
6.E.P. Box and D.R. Cox. An Analysis of Transformations,
J. Royal Stat. Societ y, 26,211-243 (Series B, 1964).
2.
Cumulative Urban Growth
Harrison, D. and J.F. Kain.
1, 61-98
and Urban Density Function, J Urban Econ.
.
(1974)
3.
,
.
Empirical Evidence on the
Hechman, J. and S. Polachek.
Functional Form of the Earnings-Schooling Relationship,
J A S
69, 350-354 (1974).
.
.
.
.
,
"One Hundred Years of Land Values in Chicago,"
University of Chicago Press, Chicago (1933).
4.
Hoyt, H.
5.
Functional Form, Density
Kau, J.B. and C.F. Lee.
Gradient and Price Elasticity of Demand for Housing,
Urban Studies 13, 193-200 (1976)
,
6.
The Functional Form in
Kau, J.B. and C.F. Lee.
Estimating the Density Gradient: An Empirical InJ. A.S.A. , 71, 326-327 (1976)
vestigation.
7.
The Value of Urban Land in "The Quality
Mills, E.S.
of the Urban Environment" (Perloff and Wingo, Eds.),
•John Hopkins, Baltimore (1971).
8.
"Urban Economics," Scott, Foresrnan and
Mills, E.S.
(1972).
Glenview
Company,
9.
"Studies in the Structure of the Urban
Mills, E.S.
Hopkins Press, Baltimore (1972).
John
Economy,"
10.
"Cities and Housing," University of
Muxh, R.F.
Chicago Press, Chicago (1969).
11.
Functional Form in the Demand for Money,
Zarembka, P.
J.A.S.A. 63, 502-511 (1968).
WW
L.,