*
：
The Alonso-Muth spatial income gradie nt model explains housing location in both the
United States where the richer households live farther away from CBD, and the European and the third world cities where the poor usually inhabit in the peripheral areas while the rich and middle class live centrally. Alonso-Muth model ignores social constraints and the consequent welfare changes to the households. This paper introduces land consumption maximum lot size restraint into the model to investigate the spatial income gradient under the restraint. We f ind that the maximum lot size restraint steepens the slope of the bid -rent function of the rich and the poor might be bid out of the central city area. The housing location, thus, is contrary to the
American cities. On the other hand, we find that the poor are better off if the maximum lot size is more strict, while the poor will be worse off if the minimum lot size zoning is also more strict.
* Associate Research Fellow, ISSP, Academia Sinica, Taiwan. I am very much indebted to professor Bruce
Hamilton of the Johns Hopkins University for his comments. Thanks are also appropriate to Miss
Tzu-heng Huang for typing. i

1. Introduction
It has been shown that once the utility function and the budget constraint equality are specified, the bid-rent criterion to examine the spatial distribution of households according to income can be applied to determine the spatial ordering of households according to income can be applied to determine the spatial ordering of households according to income.
Applying von Th
 nen’s (1826) agricultural location theory and Isard’s (1956) urban land use theory, Alonso (1964) shows that if the slope of the bid -rent function of household j is steeper than that of household i, then in market equilibrium household j will be located closer to the center. More specifically, if one finds that the slope of the bid -rent function is always negatively affected by income, then richer households will live far ther away from the center; if the slope of the bid-rent function is always positively affected by income then the richer households will live closer to the city center. But if the direction of this effect is not consistent, the locational pattern of households according to income is ambiguous [Pines, 1975]. There are many studies on the income effect on the flattening of the slope of the bid -rent function of the richer households. It is capable of explaining the flight to the suburban of the richer househol ds in the United States (Muth 1969; Mills 1972; Mills and Hamilton 1989). However, negative income effect on the slope of the bid -rent function is closely related to the assumption that households consume land freely or under minimum lot size zoning regula tion. Therefore, an argument with binding land consumption maximum restraint on households is interesting to urban economists. Section 2 of these notes sets forth a simple model to show that the maximum lot size land use restraint for housing steepens the bid-rent function of the richer households.
Section 3 compares the welfare change of the different income groups. We find that the release of land consumption constraint will make the rich better off and the poor worse off in both the maximum lot size constraint and the minimum lot size zoning case. In the last section we summarize the findings.
2. The Model
Considering one urban area includes central business district (CBD), residential area, and agricultural area. For the protection of agricultural lan d for agro-production the land in the agricultural area can only build farmhouse for farmers. In other words, residents can not build
1

housing in the rural area unless he changes his job status as a farmer, or the land in the rural area is transformed into a residential area by law due to the expansion of the city size. For the shortage of the urban land the land tax law encourages, with a low tax rate, households to consume land less than or equal to the maximum lot sizes; and discourages, through high tax rate, land consumption beyond this maximum lot size 1 .
Thus, in the absence of migration among jurisdictions the total demand for housing land under land consumption restraint is far less than the total demand without restraint. The bid price of land in the residential area will decrease. To the poor, whose land consumption is small, and the land consumption restraint is not binding, will be better off to consume more land farther away from the CBD. To the rich, whose land consumption is large beyond restrai nt, will be worse off either subject to binding of land consumption restraint or paying higher taxes.
Since the higher income households cannot flight freely to the rural area to consume more land they tend to locate closer to the CBD to live in the high q uality house. So that, under the land consumption restraint the rich might have a steeper slope of bid -rent function and the poor have a flatter one, and the spatial distribution of households according to income is quite different to what the United States has.
Suppose the utility level of a household depends on the quantity of land (which monontonically transforms into the housing space) H, and the quantity of composite goods Z, i.e.
(1) U = U (H, Z)
The household is assumed to maximize its utility le vel subject to its budget constraints and land consumption restraints ：
(2) y = R(x)H+PZ+ t(x, y)
(3) H ≦ H where
R(x)
P
= land rent as a function of distance, x, from the CBD
= given price of the composite goods t(x, y) = transportation costs of the household as a function of
distance from CBD and its income as Muth (1969)
assumed, where


t
x

0 and


t
y

0 for wage income,


t
y

0 for non-wage income. y = income of the household
Besides, we define the bid-rent function of the household in any given location x
as B
2

which solves the maximization problem below ：
(4)
H, Z
subject to
U － U(H, Z) ≦ 0
BH+PZ+ t( x , y) － y ≦ 0
H, Z ≧ 0
H
≦ H where
U
and x
are given level of utility and distance respect ively. Solving this problem for every x
with constant
U
we obtain a bid-rent function B (x, U) of the household.
Following Casetti (1971) the market equilibrium implies ：
( 5 ) R ( x ) ≧ B (x, U

) , f o r 0 ≦ x ≦ r
[ R ( x ) － B (x, U

) ]
‧
H ( x ) = 0
R ( x ) ≧ R
A
; R ( r ) = R
A where
U
*
= utility level of household in equilibrium
R
A = rent of agricultural land r = distance of the city boundar y from CBD
Since H(x) ≠ 0, equation (2) becomes
( 2 ) ' y =
B (x, U

)
H + P Z + t ( x , y )
Applying Kuhn-Tucker Theorem to equation (1), (2)' and (3) we obtain the necessary conditions for such an optimum 2 . Since y － B (x, U

) H － PZ － t(x, y)=0 totally different this necessary condition we get ：
( 6 )
 B

x
 
H
1  t

x

0 where
 t

x
is the marginal cost of commuting for an extra unit of distance. This equation delineates the negatively sloped bid -rent function. Further differentiating this equation with respect to income yields
3

( 7 )
 2
 
B
x y
 
H
1
(
 2
  t
x y



t

x y
0 where
 2
  t x y
is positive as shown by Wheaton (1977) and

= income elasticity of demand for housing land. The sign of the right hand side of equation ( 7) is indeterminate. Now suppose the land consumption constraint is binding for the higher income households, namely,
H i

H then
 i will be zero and we obtain
( 8 )
 2
 
B
x y i
 
H
1
 2
  t i

0
Thus, the slope of the bid -rent function of the rich households is positively affected by income, and the rich households will live closer to the city center 3 .
Suppose work hours are institutionally fixed to households. The leisure, a normal good in the utility function, de pends on the time spent in commuting and thus is a function of the distance of housing location from CBD. In other words, the distance x is one of the arguments in the utility function since the shorter the distance from CBD the more the leisure and the hi gher the level of satisfaction.
So that the equation (1) can be rewritten as
( 9 ) U = V ( H , Z , L ( x ) ) = U ( H , Z , x ) where L(x) = total hours minus constant work hours and commuting hours which is the function of distance. Equation (9), (2)', and (3) thus cons titute a new model which is the same as what
Alonso (1964) and Pines (1975) have done. Forming Lagrangean and using Kuhn -Tucker
Theorem we obtain the first order conditions and get where
( 1 0 )


B
x

1
H
(
R(x)U
U
H
 x




t
x
)

0

is the Lagrangean multiplier and is interpreted as the marginal cost of land consumption constraint. It is zero if the constraint is not binding, while it is negative if it is binding. Thus, the larger the H the smaller the absolute value of

. U x is the marginal disutility to have an additional mile from CBD. It is negative since x is the bad ‘good’ in the utility function. Equation (10) also displays the negative slope characteristic of the bid -rent function. Differentiating equation (10) with respect to household income we obtain
4

(11)
 2
 
B
x y

1
H
[ (U
H

 

1
) R(x)
 2
  t
x y

( dU x dy
R(x)U
U
H
 x


R(x)Ux(U
H



t

)
x y
]
 
)

2 dU
H dy
It is reasonable to assume that dU x dy
and dU
H dy
are zero while the system is in equilibrium, since x and H are determined and the marginal disutili ty of distance and marginal utility of housing land consumption are unambiguous negative and positive respectively. The equation (11) can be rewritten as
( 1 2 )
 2
 
B
x y
 
H
1
[
 2
  t
x y

(
R(x)U
U
H
 x




t
)

x y
]


Again the sign of the right hand side of equation (12) is indeterminate. But with binding land consumption constraint to the higher income household i, the income elasticity of demand for housing land is zero and we get the same result as equation (8). Thus, imposing land consumption restraint on Alonso -Pines model we can clearly get a more definite effect of income on the housing location. And, we find that the effect of land consumption constraint is to induce the higher income household to live closer to the CBD in the case of constant labor supply.
Suppose labor supply is variable. The work hours are not institutionally fixed for households. Leisure as a choice variable does not depend on distance only. It also depends on the wage rate which determines work hours denoted by k, and on commuting time which is the function of distance and denoted by c(x). In this case, we might formulate a utility function for the households as
( 1 3 ) U = U
（
H , Z , K
） where K = k + c(x) is the hours spent on work and commuting. In equilibrium, we can see that the marginal rate of substitution between distance x and the numeraire good Z is eequal to －
Wc'(x) since
( 1 4 )
U x
U
Z



U /


K
U / Z
 
Wc x
5

where W is the wage rate. － Wc'(x) is the marginal cost (value) of living an extra unit of distance away from CBD. Suppose we measure bid -rent function in terms of unit of time we can define the shadow price of time as
( 1 5 )
 

 x


U /

L

U /

Z

The last equality hold only if it is in equilibrium. Then equation (15) is the locational equilibrium condition in terms of benefits and costs of time. Thus we can try to introduce shadow value of commuting time into the model. Equations (13), (2)',and (3) together with system (4) we obtain
( 1 6 )


B
x

1
H
[P
U
K
U
Z c' (x)



t
x
]
 
H
1
(PWc' (x)



t
x
)

0 which again yields the negatively sloped characteristic of the bid -rent function. Differentiating equation (16) with respect to income we get
( 1 7 )
 2
 
B
x y
 
H
1
[Pc' (x)(1

W
 y
)

 2
  t
x y



t

x y
0
The sign of the right hand side of equation (17) is indeter minate. If (i)

=1, as Muth
(1969) has assumed, and W ＜ y or (ii) wage worker is unemployed, namely W=0, the right hand side of this equation is negative. Otherwise the sign is ambiguous. Again, with land consumption constraint binding to the higher income household,
 i
=0 and the sign of the right hand side is definitely negative since equation (17) becomes
( 1 8 )
 2
 
B
x y i
 
H
1
[Pc' (x)

 2
  t i
]

0 and the rich households will have housing location close to central city. The lower inc ome households inhabit the peripheral areas, as in the European and Latin American cities [Alonso
1964].
3 . W e l f a r e C h a n g e o f t h e L o t S i z e C o n s t r a i n t
The above analysis bases on the maximum lot size land consumption constraint which is quite different to the minimum lot size zoning in the suburban areas in the United States. Both maximum and minimum lot size will lower the bid -rent function in the area which is subject to the constraint. In the maximum lot size case, the total demand for land consumption is less than that of no constraint. The land price, thus, is lower all over the area. In the minimum lot size
6

case, the area subject to constraint will have lower capital -to-land ratio and population density.
Only the higher income households can afford th e costs of housing in those zoning areas even if the unit price of land is low but the quantity is large. Therefore, the total demand for land in zoning suburban areas are also restrained. People who might have been able to live in these areas are displaced due to the large-lot zoning regulation. They are forced to move to the central city (in our close city model) where bid -rent function becomes higher due to higher capital-to-land ratio than before [Fischel 1985, p. 260 -261].
Thus, minimum lot size zoning restrains the lower income household from living in the zoning areas and maximum lot size land use control, on the contrary, limits the higher income households not to consume too much land. Obviously, in the former case the poor are worse off because they have to crowd in the central city and consume higher unit costs of land; and the rich are better off since they consume more land with lower unit costs. In the latter case, the poor whose land consumption is smaller than maximum lot size are better off b ecause the over all land price in the city is lowered; and the rich whose land consumption is larger than maximum lot size are worse off because they might have to cut down their demand and decrease their welfare level.
We first analyze in a diagram the we lfare change of both the maximum and minimum lot size constraints and provide a mathematical proof thereafter.
Figure (1) shows that households make choice between land H and composite good Z under maximum lot size constraint. A B j and
A B i are the budget line of lower and higher income households respectively. Without maximum land consumption restraint H j and
H i are the quantity of land consumption of the poor and the rich resp ectively. Since the price of land decreases after imposing the maximum lot size restraint the budget lines shift to
A B
' j j and
A B
' i
. The land consumption of the low income households becomes larger. It is
H
' j . The welfare level of the poor thus shift upward. Subject to the constraint the high income households consume less land than before. It is H . The welfare level of the rich shift downward. The welfare loss of the higher income households is amount to i
in terms of composite goods
Z. Thus, the poor are better off and the rich are worse off.
Insert Figure 1 here
Figure (2) reflects the welfare change under minimum lot size zoning constraint.
Before imposing large-lot zoning constraint, the lower and higher income households consume
7

as much as H j and
H i of land respectively. Let
H represent minimum lot size constraint imposed in the suburban areas. The higher income households are better off since the land price in suburban areas goes down due to the large -lot zoning. They consume more land H
' i
. The lower income households are worse off since the land price in the central city goes up due to the increase in demand for land and higher capital -to-land ratio than before. They consume less land
H
' j
.
Insert Figure 2 here
The higher income households with maximum lot size constraint and the lower income households with minimum lot size constraint are worse off because of rationing quantity of land to the former and rationing distance of land to the latter.
Following equation (9), (2), and (3) we can postulate that housing location depends on transportation costs t, price of composite good P, maximum lot size H , and given level of utility U , i.e.
( 1 9 ) x = x ( t , P , H , U ) and the restrained cost function is
( 20 ) C
*
(P, H , R( x ), t, U )

C
*
(P, H , R( x (t, P, H , U )), t, U )

R(x) H
 min (PZ; U( H , Z, x)

U )
 y where x is a given distance. We then obtain
(21)


C
*
H

R(x)

R
*
(x)
>
= 0
< not binding binding
H
to
=
the
H poor
to the rich i j where
 
 
which is derived from the minimization of restrained cost function subject to U(H, Z, x)

U .


U
C
is the marginal utility of total spending. Suppose
>R(x) the shadow price of rationing larger than the market land price and the consumption of land is binding to the rich. Then the increase of
H
will decrease costs and increase the welfare level of the higher income households. The poor households consuming less
8

land than the maximum lot size will be worse off since the increase of H will increase their costs and they will suffer welfare loss.
For the general equilibrium analysis, we totally differentiate equation (20) with respect to H and get
(22) ( dC
 dH
) i , j

(

C


H
) i , j


C


R(x)
(

R(x)

H
) i , j



C

R(x)

R(x)

x


x
( )
H i, j
The first term of equation (22) is the direct effect of H on C

. The second term is the indirect effect with given distance, thus, it shows the shift of the unit cost of land due to the change of H . The third term is also the indirect effect with endogenous distance. We have
R(x)

P(U
H
 
) U
Z
from the first-order condition of the exogenous labor supply model.
Assuming Z as the numeraire we obtain
(23)

R(x)

H

1
U
Z
(

U
H

H
)

(U
H
 
)
1
U
Z

U

H
Z for j
0 for i
Since to the higher income household i, when
H
is increased they will consume more H i and less Z (assume y constant) so that (

U
H

H ) i is negative and
(

U
Z

H ) i is positive. Conversely, to the lower income household j, raising H will decrease its H j and increase Z as the substitute thus (

U
H

H ) j is positive and
(

U
Z

H ) j is negative.
Then we have (

R(x)

H
) i

0 which implies
(

R(x)

H
) i

0
, namely, the land prices are everywhere cheaper to the higher income households because sh adow price of rationing is lowered and


x
( )
H i



x
H i

H

H i 
0 then we have
(24) ( dC dH

) i

(

C


H
) i


C


R(x )
(

R(x

H
)
) i



C

R(x)

R(x)

x


x
( )
H i
< 0
On the other hand, (

R(x)

H
) j

0 implies (

R(x)

H
) j

0 , namely, the land prices are everywhere more expensive to the poor households bec ause of the increase of total demand for land induced by the release of rationing. And


x
( )
H j



x
H j

H

H j 
0 , i.e., the poor will
9

bid in the central city to live in the squatter settlement if the H is increased. Then we get
(25) ( dC
 dH
) j

(

C


H
) j


C


R(x )
(

R(x )

H
) j



C

R(x)

R(x)

x 

x
( )
H j
> 0
These results coincide with that of the above partial equilibrium analysis in equation
(21). Thus, the increase of maximum lot size constraint will better off the rich and be worse off to the poor. Conversely, the decrease of maximum lot si ze will push the rich to live closer to the central city and worse off the poor live farther away from the central city and better off.
In the minimum lot size restraint case the minimum lot size H (s) (where s ≧ x is the distance from suburban area to CBD) set a restraint to the lower income households and it is not binding to the higher income households. The equation (19) becomes
(26) s
 s (t, P, H , U ) when the H is small enough the lower income households c an also afford to live in the suburban areas. The restrained cost function is
(27) C

( P , H , R( s ), t, U )

C

( P , H , R( s ( t, P, H , U )), t, U )

R(s) H
 min [ PZ; U( H , Z, x)

U ]
 y
Where s is a given distance. We also obtain
(28)


C

H

R(s)


)



0 binding to the poor j
H = H
not binding to the rich i
Since minimum lot size is analogous to the minimum order in buy ing the commodity, to those who (household j ) cannot afford the minimum order the

R (s )

 
 
is negative because the more the H , the less the H j and
U H, Z, x )
but to those who
(household i ) can afford it

) is positive. Again, in the general equilibrium analysis
(

R(s)

H
) i

0 and (


s
H
) i

0 because (

R(s)

H
) i

0 implied by equation (23). In the meantime, (

R(s)

H
) j

0 implies
(

R(s)

H
) j

0
and (


s
H
) j

0 . Then both equation (24) and equation (25) hold for higher income and lower income households respectively, i.e.,
10

(29) (


C

H
) i

0 and
(30) (


C

H
) j

0
Thus in the minimum lot size case the richer households are better off while the poor households are worse off if we raise the lot size restraint H .
Therefore, we can summarize that the increase of the lot -size constraint makes the rich better off and the poor worse off in both cases. For the case of the maximum lot size land consumption restraint in the urban area the more strict the restraint ( i.e. the less the H ) the more welfare improvement to the lower income households, while in the case of minimum lot size zoning in the suburban area the more strict the restraint (i.e. the more the H ) the less welfare improvement to the lower income households.
4. Concluding Remarks
The Alonso-Muth spatial income gradie nt model explains housing location in both the
United States where the richer households live farther away from CBD, and the European and the third world cities where the poor usually inhabit in the peripheral areas while the rich and middle class live centrally. Alonso-Muth model ignores social constraints and the consequent welfare changes to the households. This paper introduces land consumption maximum lot size restraint into the model to investigate the spatial income gradient under the restraint. We f ind that the maximum lot size restraint steepens the slope of the bid -rent function of the rich and the poor might be bid out of the central city area. The housing location, thus, is contrary to the
American cities. On the other hand, we find that the poor are better off if the maximum lot size is more strict, while the poor will be worse off if the minimum lot size zoning is also more strict.
For future research we might have to consider tax scheme and public services under the open city model.
11

Z
A i welfare
loss
W E i
0
A j
E i
E j
H j
H j

E j

H
B j
U j

U j
B j

H i
U i
U i

U i

B i
B i

1

Z
A i
A j
0
E j

H j

H j
E j
E i

H
U j
U j

B j
E i
H i
H i

B i
U i

U i
H
1

1.
According to land tax law in Taiwan, the urbanites are encouraged to consume land not more than one twelfth acre in the urban area. Under this land consumption restraint the land value tax rate is 0.2%, and tax rate will be at least 1% to the land household consumed more than one twelfth acre. How the different tax burden affect residential choice amon g urban and rural areas has been inquired in Tsai (1982).
2 .
We can formulate Lagrangean function from (1), ( 2 ) ' and (3) as :
L

( , )
 
[y


B(x, U )H

PZ
 t(x, y)]
 
(H

H)
and have first-order conditions as below :


L
H

U
H
 


L
Z

U
Z
 
0
  
0

L
  y B(x, U )H

PZ

( , )

0

L


H

H

0
λ is the marginal utility of income y and μ the marginal disutility of land consumption restraint
H
.
μ =0 if land consumption is not binding, μ <0 if it is binding. In this analysis we postulate that μ <0 for the higher income households a nd μ =0 for the lower income households.
3.
From ( 2 ) ' we can get


B
y

1

H


t
y

0
if we reasonably assume 1



t
y
. The bid-rent function of the rich household will move upward due to income increases. So that there is a trend for the ric h to squeeze the poor to live further away from CBD.
15

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