Dynamics and comparative statics in the theory of residential location

JOURNAL

OF ECONOMIC

Dynamics

THEORY

11,

133-146

and Comparative Residential TAKAHIRO

Department

(1975)

Statics in the Theory Location

of

MIYAO*

of Political Economy, Institute for the Quantitative Analysis of Social Economic Policy, University of Toronto, Toronto, Ontario, Canada

and

Received December 6, 1974

This paper introduces a dynamic adjustment mechanism of residential boundaries among a finite number of household classes in a circular city model. First we derive some meaningful conditions for all the residential boundaries to be locally stable. Then, assuming that the stability conditions are satisfied, we examine the comparative static properties of equilibrium boundary positions, e.g., the effects of changes in a land tax, an income subsidy, the number of households, etc.

1. INTR~OUCTI~N

Although equilibrium models of residential land use involving many household classes have been intensively studied by Alonso [I], Beckmann [2, 31, Mills 151, Montesano [7], Muth [8], Solow [I l] and others, virtually no attempt has been made to find the dynamic stability properties or comparative static properties of equilibrium positions in such multihouseholdclass modelsl. Recently, Mills [6] and Wheaton [12] have analyzed some comparative static properties of their models with only one household class. However, they paid no attention to the underlying dynamic stability nature of the equilibrium in question. The present paper deals especially with dynamics and its relation to comparative statics in a model with a finite number of household classes * This paper is largely based on my Ph.D. dissertation submitted to the Massachusetts Institute of Technology in May, 1974. I am grateful to my thesis supervisor Robert M. Solow for his valuable comments. I also wish to thank Franklin M. Fisher, Robert A. Mundell, Paul A. Samuelson, William C. Wheaton and an anonymous referee for their useful suggestions. 1 A notable exception is Muth [Sl, who has derived some comparative static results in his multihousehold model. However, his comparative static analysis is limited to the subjective equilibrium positions of households. 133 Copyright All rights

0 1975 by Academic Press, Inc. of reproduction in any form reserved.

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TAKAHIRO

MIYAO

including the case of one household class as a special case. In our model households in different classes are assumed to have different utility functions and/or different incomes from each other. First we introduce a dynamic adjustment process of residential boundaries among different classes, including the outer boundary of the city, and derive sufficient conditions for the equilibrium positions of the boundaries to be locally stable. Then we analyze the comparative static properties of the equilibrium, assuming that the stability conditions are satisfied. In this respect, the present study may be regarded as an application of Samuelson’s Correspondence Principle [lo] to residential location theory. The stability conditions state that at each boundary between two different classes every household in the outer class should consume more land than any household in the inner class, provided that land is a normal good. This implies that richer classes live farther from the city center in the special case of identical utility functions for all classes. Our comparative static analysis shows that, among other things, all the equilibrium boundary positions will move inward as the opportunity cost of land rises or as the rate of a residential land tax increases. It is also proved that in the case of identical utility functions, an increase in the income level or in the number of the poorest households will have an expansionary effect on all the boundaries of the city.

2. THE MODEL Let us consider a circular city with its CBD at the center where all employment takes place. The road system is radial and dense, and transport cost is a sole function of distance2. Each household spends its income on a consumption good, residential land and transportation. There are a finite number, say it, of household classes with different utility functions and/or different incomes, and each class consists of a given number of identical households. We adopt the following notation: uj = utility level of the jth class (j = I,..., n), Vi = indirect utility function of the jth class (j = l,..., n), wj = income of thejth class (j = l,..., n), rj = bid rent function of the jth class (j = I,..., n), hj = demand for land per household in the jth class (j = l,..., n), 2Here we are mainly concerned with out-of-pocket costs and ignoring congestion costs. The basic assumptions adopted in this paper are essentially the same as those of Solow [ll].

DYNAMICS

Ni = q= x = xj = x,, = x, = y0 = g =

AND

COMPARATIVE

STATICS

135

number of households in thejth class (j = I,..., n), unit transport cost, distance from the city center, boundary between thejth and the (j + 1)th classes (j = 2,..., n - l), 0 (or any given positive number), outer city boundary, opportunity cost of land, fraction of land devoted to residential use.

In our model the equilibrium conditions are as follows. Assuming that the price of the consumption good is exogenously given and normalized as unity, we can write the indirect utility function of a household living at distance x from the CBD as (j = l,..., n),

(1)

where the household belonging to the jth class must achieve the utility level uj which is common for all the households in the jth class; otherwise some household could improve its welfare by moving to a better place in the city. Here it is natural to assume 2av. ar.3
LJVj

and

(j = l,..., n),

z. 3 > O

(2)

which, together with (I), imply ar. 3

au.I
and

(j = l,..., n).

We also suppose that unit transport cost is an increasing function distance. q’(x) > 0.

of (4)

It is clear from the well-known properties of indirect utility functions that hj is a function of yj and wi - q(x); h,(X)

=

-

2/z

=

hj[rj(X),

U’j -

q(X)]

(j = l,..., n).

(5)

We shall assume that in equilibrium the city will be subdivided into n rings and each ring will be occupied exclusively by a single class of house-

136

TAKAHIRO

MIYAO

holds3. Then the “full accommodation condition” stating that all the households should be accommodated withing the city can be written as dx = iVj

(j = I,..., n),

(6)

where it is assumed that Nj > 0 (j = 1,..., n) and g is a known function. Finally, in equilibrium the overall rent function should be continuous at each boundary; rj(xj) = rj+l(xj)

(j = l,..., n - l),

(7)

and r&n)

= ro 9

(8)

where r. is assumed to be given exogenously. In this system, the equilibrium conditions (I), (5), (6), (7) and (8) can determine the equilibrium functional forms of rj and hj and the equilibrium values of ui and xj simultaneously (j = 1,..., n). Now, let us see how the equilibrium values of the boundaries xj are determined in particular. Since the condition (1) gives the rent functions as r?(x) =

rj[Wj

-

q(X),

Uj]

(j

=

l,..., n),

(9)

hj can also be expressed as a function of wj - q(x) and Uj from (5). The full accommodation condition (6), therefore, determines the utility levels Mj as functions of xjeI and xj , given wi and Nj ; Uj

=

Uj(Xj-1

3 Xj

; Wj

) Nj)

(j = l,..., n).

(10)

Then the rent conditions (7) and (8) may be written as

rj[wj - 4(-%),%(%-II Xj; wj , Nj)I = ri+lbj+l - dxA uj+l(xi) xj+l i Wi+i2Nj+l)l

(j = l,..., n- 11, (11)

and r,[w,

- 4hA G6+1 , x, ; w, , NJ1 = ro,

02)

which together will determine the equilibrium values of xi (j = l,..., n). Here we prove the following proposition which will be used later. 3 It remains to be investigated what assumptions on incomes and preferences will lead to such a “segregated” residential pattern. For a further discussion, see Mills [5, p. 65-771.

DYNAMICS

AND

COMPARATIVE

STATICS

137

LEMMA 1. Under the present assumptions, the function uj in (10) has the following properties:

2&

( 0,

2

> 0,

2 > 0,

+$ < 02

(13)

(j = I,..., n).

(14)

if land is a normal good everywhere, i.e., for Proof.

Xj-1 <

<

X

Xj

Define

If (14) is satisfied, we find

Here we are using (3) and ahj/arj < 0 which follows from (5) because the substitution term is always negative and (14) means that the income effect is reinforcing. Thus % axi-,

1

2m(xj-l)

= &

a& --axj

xj-1

<

0

h,(xj-J 1 %dxJ +f

hj(xj)

’ xj > 0 ’

where ah. ar.

(>2+2b) arj a+

3 uj=constant

because this is just a substitution term; and

auj =‘
$bj

< 0,

138

TAKAHIRO

3. DYNAMIC

MIYAO STABILITY

Suppose there is a small displacement of boundary positions from equilibrium and the overall rent function is no longer continuous at given boundaries, that is, the equilibrium rent conditions (11) and (12) do not necessarily hold for arbitrarily given values of xj , provided that all the other conditions, (l), (5) and (6), are met. We then introduce the following dynamic adjustment mechanism. A boundary will be adjusted gradually over time in such a way as to move inward or outward according as the rent for the inner land is lower or higher than that for the outer land at the boundary. This adjustment process implicitly assumes that due to some frictional elements it takes time for a class of households which can pay higher rents than the other class at a boundary to outbid the other class and move across the existing boundary. This is mainly because the movement of a residential boundary necessarily involves some changes of durable objects, e.g., houses, roads, public facilities, etc., in accordance with the difference in incomes or preferences between classes4. It is also natural to assume that the rate of boundary change with respect to time will diminish as the rent difference narrows at the boundary in question. Our dynamic adjustment process may then be formally expressed as 4 =h[rj{Wj - rj+l{Wj+l

- 4(Xj),

uj(xj-l

- q(Xj),uj+l(Xj

3xj ; Wj 2 Nj)l

3Xj+l; wj+l3Nj+l>>l(j = I,..., n - I), (15)

and

*vi=L&-7i{w,- 4(x,), %dLl 3xv2; w, I NJ) - rol, where the dot denotes differentiation

with respect to time, and

fj’( > > 0 andJ;(O) = 0

(j = I,..., n).

Then it is clear that the stationary equilibrium *j = 0

(16)

conditions

(j = I,..., n)

reduce to the equilibrium conditions (11) and (12) which will give the equilibrium positions of the boundaries Xj* (j = l,..., n). * Although frictional elements on the “supply” side are emphasized here, those on the “demand” side may be equally important, e.g., the existence of moving costs, uncertainties, lack of information, interdependency among households, etc. The referee’s comment was very helpful to clear up this point.

DYNAMICS

AND

COMPARATIVE

139

STATICS

For a small displacement from equilibrium, the dynamic system (15) and (16) can be approximated by the following linear differential equation system: i = FAz, (17) where F is a diagonal matrix whose jth diagonal element is fj’(O) (j = I,..., n), and z is a column vector whose jth element is xj - xj* (j = l,..., n); and aI

0

.

.

0

.

.

...

.

.

0

.

From (15) and (16) we can calculate aii = - (

arj(xj*) awj

ar,(x$*) +

ann = -

auj

%+1(x, *) -

1

h+,

a2dj ax,*

arj+&*)

-

4'(Xi

(18)

*I

a4+,

%+1

1 .

ax,*

(j = l,..., n - l),

au, ar,Gh*) 4'(Xn*) + ar,(x,*) au72 -$gT aurn n '

(j = 2,..., n), and ajj+l = -

arjdxi*) au?+l

atl,+l c

(j = I,..., n - l),

where all the variables are evaluated at the equilibrium. Now we prove the following theorem. THEOREM 1. The equilibrium boundary positions are locally stable for any positive speeds of adjustment fj'(0) > 0 (j = I,..., n), if in the equilibrium the demand for land per household in the outer class is greater than that in the inner class at each boundary between two classes, i.e.,

hdXj*, “‘j) < h+l(xj*t Wj+-l) (j = l,..., n - l),

(19)

140

TAKAHIRO

MIYAO

and land is a normal good everywhere, i.e., ahAx, wi) > o awj ’

f or

XjTel

<

Gxlxj

*

(j = l,..., n).

(20)

In order to prove the theorem, we shall use the following lemmas: LEMMA

2.

Zf (19) and (20) are satisfied, then ajj < 0 (j = l,..., n),

> 0 (j = 2,..., n) and aji+l > 0 (j = I,..., n - 1).

ajjwl

Proof of Lemma 2.

We find

I

ari(xi*> _ arj+l(%*)

ahtlj

1

aw,+l - - hj(xj*, wi) -

->o hj+l(xi*, w~+~)

from (19). Therefore, ajj < 0 (j = I,..., n) follows from (3), (4) and Lemma 1. Similarly, it follows from (3) and Lemma 1 that afiT > 0 (j = 2 ,..., n) and ajj+l > 0 (j = l,..., n - 1). Q.E.D. LEMMA 3. The linear deferential equation system (17) is stable for any choice of positive diagonal matrix F, tf A is indecomposable with negative diagonal elements and is “quasidominant diagonal”, i.e., there exist positive constants ci (j = l,..., n) such that

ci I ajj I > 1 ck I aik I

(j = I,..., n).

k#i

Proof of Lemma 3. See McKenzie [4]. We are now ready to prove Theorem I. Proof of Theorem 1. It is clear from (18) and Lemma 2 that A is indecomposable. All we need to show, therefore, is that there actually exist cj > 0 (j = I,..., n) such that

-wll

> wlZ ,

-ciaji

> ci-,aii-l

(21) + cj+Iaij+I

(j = 2,..., n - I),

(22)

and -wlzn

Let c1

(23)

> cn-lann-l ,

1 and (j = 2,..., n),

DYNAMICS

AND COMPARATIVE

141

STATICS

where it is obvious that ci > 0 (j = l,..., n) from Lemma 1. Define

and uji

E

auj axi*

.

Then, (21) follows from (RI - R,) q’(x,*)

- &UII

+ WJ,,

> (WU,,>

SJJ,, .

We also find (22), that is,

fi2 (-u~+l/ud[(Rj - &+I) d(xj*) - Sjuii + sj+~uj+~l j-1 j+1 >

fl 6 7;=2

ukk-l/"7c7c)(s~u~j-l)

+

n (k=2

u~~-l/"~~>(~si+luj+li+l)

(j = 2,..., n - I), where the inequality comes from the following equivalent expression:

(-u~~-Ju~j>[(&- &+I) q’(xj*) - Sjujf + Sj+lUj+lil > sj”ij-l + ~“~ij-llui~)(uj+lilui+l~+l~(~si+lui+lj+l) (j = 2,..., n - 1). Finally (23) is obtained from

(-- um-,lUnnWnq’(x,*) - Snunn)> Snunn--l.

Q.E.D.

As a corollary we have the following proposition. COROLLARY 1. In the case of identical utility functions for all classes, the equilibrium is locally stable for any positive speeds of adjustment, ;f higher income classes live farther from the CBD, i.e., wj

and land is a “strongly” wx,

4

awi

> 0

<

(j = l,..., n - l),

wi+1

(24)

normal good everywhere in the equilibrium, i.e., .for

x7?-,

<

x <

xj*

(j

=

I,...,

n).

(25)

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TAKAHIRO MIYAO

It should be noted that essentially the same condition as (24) can be found in the literature, e.g., Beckmann [2, 31, Mills [5], Montesano [7], Muth [8] and Solow [l 11, although their approach is somewhat different from ours. In the special case of “constant elasticity” utility functions Uj(b, , hi) = rj(bj)“$hi)‘J

(j = I,..., n),

where bi represents the amount of the consumption good, we have h,(x) = * 3

3

IL’jr-x;(-x) J

(j = I,..., n).

Then the condition (19) is equivalent to

&j-j+5 - 4(x?*)) < aj+lpylp,{Wj+1 3 3

q(xj*):

(j = I,..., n -

1).

3+1

In this special case, the condition (20) is always satisfied, as can easily be shown. It is clear that if all households receive identical incomes, the stability condition (19) reduces to

A 9 + Bj

<

A+1 %+I + Sj+l

(j = l,..., n - I).

4. COMPARATIVESTATICS

As we have seen, the rent conditions (I 1) and (12) give the equilibrium values of xj as functions of exogenous parameters, y0 , wj and Nj (j = l,..., n). Here we shall examine how the equilibrium boundary positions move with small changes in those parameters. First we take up a small change in the opportunity cost of land r,, at the urban boundary. From (11) and (12) we find Ax, = e, ,

(26)

where A is defined as in (18). Here x, is a column vector whosejth element is dxj*Jdr,, (j = l,..., n), and e, is a unit column vector with its nth element equal to unity and all other elements zero. Suppose that the stability conditions (19) and (20) are satisfied. Then we can utilize the following well-known property. LEMMA 4. If A is an indecomposable and quasidominant diagonal matrix with negative diagonal and nonnegative off-diagonal elements, then A-l exists and A-l < 0, i.e., ail the elements of A-l are strictly negative.

DYNAMICS

AND

COMPARATIVE

STATICS

143

Proof. See Quirk and Saposnik [9, p. 2121 for example. Then, under the present assumptions, it follows from (26) that X, = A-len < 0, since all the elements of A-‘, particularly its nth column, are negative from Lemma 4. Thus we have proved the following theorem. THEOREM 2. If the stability conditions (19) and (20) are met, then all the equilibrium boundary positions xj*(,j = I,..., II) will move inward as the opportunity cost of land increases at the urban boundary. i.e..

dx * AtO drn

(.j = I )..., n).

It can be shown that the effect of an increase in a proportional tax levied on residential land is just the same as that of an increase in the opportunity cost of land, provided that the initial level of the opportunity cost is positive. This is because, defining p,?= (I + 7) rj with tax rate 7, we obtain Vj[pj(x), w, - y(x)] = uj(,j = I,.... n) in place of (I), and the rent conditions p&xi) = pj+l(Xj) (,i = l,..., n - 1) and p,(x,) = (I f T) rO , where it is obvious that the properties of the function pi( ) are exactly the same as those of rj( ) in (11) and (12)5. Notice that the “effective” cost of land for residential use is now (1 + T) r0 , since a proportional tax will be levied on the land around the urban boundary if it is converted into residential land. Thus we have the following. THEOREM 3. If the stability conditions (19) and (20) are satisfied. (i) all the boundary position xj* (,j = I,..., n) will moL)e inward as the rate qf a proportional land tax increases in the case that r,, > 0, and (ii) such a tax increase does not have any effect on the equilibrium boundary positions at aN in the case that r,, = 0.

It is noted here that the case (ii) has already been studied by Alonso’j. Next, we shall see the effect of a small change in the income of the class living closest to the CBD, namely, the first class. It follows from (11) and ( 12) that Ax,,, = k, (27) where x,,. is a column vector with its .jth element dxj*/dw, and k is a column vector with its first element

5 This point 6 See Alonso 642/11/1-IO

was first suggested [I, p. 1161.

to me by Robert

M. Solow.

(,j = I,..., n)

144

TAKAHIRO

MIYAO

and all other elements zero, kj = 0 (j = 2,..., n). Using (6) we obtain

k,=

II* -g(s)x ____2h, irr,(x) (h# i i;r, &I,,

e[br$-;*)

+i??!$&s,

+

‘I* g(x)x 137, iirl(s) -----d*I] W2 2rl % ardx-,*) aw,

br,( x1 *) 814,

from (20). The latter expression is zero or negative, since -

arl(xl 7 aw,

i

%,(x1 *) > _ or, au,

’

-au,

ar,b-) /

for

0 < s < x1*,

which follows from

Thus, from (27) we have .I., = A-lk > 0, if the conditions (19) and (20) are met. Here it can easily be seen that if land is a “strongly” normal good for the first class, i.e.,

then k, < 0, and therefore x, = A-lk > 0. Hence the following. THEOREM 4. Zf the conditions (19) and (20) are satisfied, no equilibrium boundary positions will move inward with a small increase in the income of the,first class, the one living closest to the CBD, i.e., dx.* > o I,c/w,

(,j =

I,..., I?).

DYNAMICS

AND COMPARATIVE

STATICS

145

If, furthermore, land is a “strongly” normal good for the first class, i.e., if we have (28) instead of (20), then all the boundaries will move out;

““l-Tro dw,

(j

=

l)...,

n).

Finally we can easily determine the effect of an increase in the number of the households in the first class, NI . Here we obtain AxN = m, where xN is a column vector with its jth element dxj*/dN, (,j = l...., n) and n? is a column vector with its first element i?r,(s, *) iiu, ml = - ___ __ ku, alv,

)

and all other elements zero, mj = 0 (j = 2,..., n). Since ar,/iiu, < 0 and ~u,/~N, < 0 from (3) and Lemma 1, we find X’N= A-lrn > 0. Thus we have proved the following theorem. THEOREM 5. If (19) and (20) are satisfied, all the boundaries (.j = I..... n) will be pushed out as thefirst class increases in number; 9 dN,

>’

(J =

I,.... 77).

From Theorems 4 and 5, together with Corollary following. COROLLARY 2. In the case of identical all the boundaries xj* (j = l,..., n) will move income level (due to an income subsidy, for the poorest households. provided that the satisfied.

xi”

1, we can derive the

utility functions for all classes, out with a small increase in the example) or in the number of conditions (24) and (25) are

REFERENCES I. W. ALONSO, “Location and Land Use,” Harvard University Press, 1964. 2. M. J. BECKMANN, On the distribution of urban rent and residential density, J. Econ. Theory 1 (1969), 60-67. 3. M. J. BECKMANN, Equilibrium models of residential land use, Regional and Urban Economics 3 (1973), 361-368. 4. L. W. MCKENZIE, Matrices with dominant diagonals and economic theory, in “Proceedings of a Symposium on Mathematical Methods in the Social Science,” Stanford University Press, 1960, pp. 277-292. 5. E. S. MILLS, “Urban Economics,” Scott-Foresman, 1972.

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6. E. S. MILLS, “Studies in the Structure of the Urban Economy,” Johns Hopkins Press, 1972. 7. A. MONTESANO, A restatement of Beckmann’s model on the distribution of urban rent and residential density, J. Econ. Theory 4 (1972), 329-354. 8. R. F. MUTH, “Cities and Housing,” University of Chicago Press, 1969. 9. J. QUIRK AND R. SAPOSNIK, “Introduction to General Equilibrium Theory and Welfare Economics,” McGraw-Hill, 1968. 10. P. A. SAMUELSON, “The Foundations of Economic Analysis,” Harvard University Press, 1947. I I. R. M. SOLOW, On equilibrium models of urban location, in “Essays in Modern Economics” (M. Parkin, Ed.), Longman Group Ltd., 1973, pp. 2-16. 12. W. C. WHEATON, A comparative static analysis of urban spatial structure, J. Ecorz. Theory

9 (1974),

223-237.