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Comparative statics of a residential economy with several classes
JOURNAL
OF ECONOMIC
Comparative
THEORY
13, 396-413 (1976)
Statics of a Residential Several Classes*+
Economy
with
JOHN HARTWICK Department of Economics, Queen’s University, Ontario, Canada URS SCHWEIZER Department of Economics, M.I.T., Cambridge, Massachusetts 02139 AND
PRAWN VARAIYA Decision and Control Sciences Group; Electronic Systems Laboratory, Department of Electrical Engineering and Computer Science, M.I. T., Cambri&e, Massachusetts 02139; Electronics Research Laboratory, University of California, Berkeley, California 94720 Received June 18, 197.5; revised May 3, 1976
1. INTRODUCTION
AND SUMMARY
The geography and economic structure of the city model under consideration are orthodox. The city possessescircular symmetry and at its center is the CBD (centra! business district) of radius p. People live at distances x 3 p and they commute to work in the CBD. Commuting cost is solely a money cost and depends only upon the distance of the residences from the CBD. Individuals have fixed money wages and they use it to occupy land (housing) and to consume other commodities. These latter are available at fixed prices which are constant throughout the city. The land market is competitive and so land rents, in equilibrium, coincide with the maximum of individual bid rents and an exogenously specified agricultural rent rA . Land is owned by absentee landlords. * A version of this paper was presented at a conference on Mathematical Models of Land Use at McMaster University, April 1975. + Research partially supported by National Science Foundation Grants GK-41647 and ENG 74-01551-AOl, by the Swiss National Science Foundation, and by the Canada Council. We are grateful for helpful comments from members of the Urban Workshop, Department of Economics, M.I.T., Cambridge, Massachusetts 02139.
396 Copyright All rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
STATICS
OF
A
RESIDENTIAL
397
ECONOMY
There are n classes of people, class i is characterized by the number of people in the class Ni and, for each individual in the class, his wage wi and utility function Ui . Vi is assumed to be smooth, strictly quasiconcave and such that housing is a normal good. Furthermore, it is assumed that preferences and incomes between classes are related in such a way that an individual in class i will occupy more land than one in class i + 1. In particular, this holds if all individuals have the same utility function and wi > wi+r . Under these assumptions, in equilibrium, people in different classes reside in characteristic concentric rings around the CBD with individuals in class i -- 1 living closer to the CBD than those in class i. Suppose class i occupies the ring Ji = [x~+~, xi] where x1 > ... > x,+~ = p, the CBD radius. Let u1 ,..., u, denote the equilibrium utility levels. The dependence of these equilibrium values on the exogenous parameters can be expressed in a functional form, Xi
=
x,(N~
)...)
N, ) WI ,..., WIT)
Zli = tri(Nl )...) Nn ) ~1 )...) ~1,)
i = I,..., tt, i = l,..., n.
Our first set of results shows that the signs of the various partial derivatives are unambiguous. The specific statements are these. axJaN, > 0 if,j < i,
axJaN, < 0 if j > i, and
au,/ahr, < 0 all j, (i)
i.e., if the ith class increases in size, then the outer classes are pushed away from the CBD whereas the inner ones are squeezed towards it, and everyone’s real income is reduced. This is not at all surprising. axjjawi
> 0 all,j,
and
au,/&,
> 0 ifj 3 i,
auj/anli < 0 ifj < i, (ii)
i.e., if the ith class’s income rises, then all classes are more suburbanized, and the outer classes suffer a reduction in their real income whereas the inner classes enjoy an increase. The total asymmetry of (i) and (ii) is of a striking simplicity. Many people may find (ii) counter-intuitive since it would appear at first sight that, since increases in Ni or Wi both create increased demand pressures in the land market, therefore they would have the same impact on the real income of the other classes. These qualitative results, while telling us that increases in income or size of an outer class have opposing impacts on an inner class, do not provide any clue as to the relative magnitudes of these two effects. Our second set of results is the outcome of some numerical exercises designed to reveal what magnitudes of these effects might plausibly be expected. Choosing n = 2, and the same Cobb-Douglas utility function for both classes,
398
HARTWICK,
SCHWEIZER AND VARAIYA
we show that the negative effect on the poor due to an increase in the size of the richer class is far greater than the positive effect due to an increase in the latter’s income. When this paper was first written, the only work related to ours was the comparative analysis carried out by Wheaton [2] for the case of a single class, n = 1. Since then two papers have appeared. Miyao [4] proves the simple special case of(i) and (ii), ax,/aN, > 0, axJaw, > 0 forj 1 l,..., n, and Wheaton [5] proves the result (ii) above for the case 12 = 2 when the utility function is the same for all. We are indebted to the referee for pointing out these references and for several suggestions.
2. THE MODEL. AND ITS EQUILIBRIUM
Aside from the notation introduced already we need the following. The money cost of commuting from x is t(x) and is such that ASSUMPTION
1. t, = dtjdx > 0.
Rent of land is denoted by r, and the amount of land available for housing at x is exogenously given to be 6(x)x (for a circular city we have 0(x)x = 27rx.) The amount of land occupied by an individual is denoted h and the bundle of other goods he consumes is the (column) vector c. The exogenously fixed prices of these other goods is the (row) vector p. The expenditure function for an individual of class i can then be defined in the usual way as P(r, U) = minfpc
+ rh 1uI(c, h) = u>
and this is related to his compensateddemandfunctions Ci, Hi by Ei(r, u) = pCi(r, u) + rHi(r, u);
U”(G(r, u), HYr, u)) = 24. (1)
Ei has the following well-known properties: E,j = aEil& > 0,
E: = aEil&- = H* > 0, (2)
Ei, = a2Ei/ar2 = HPi x 0.
Let ri(u, I) be the function which gives the maximum rent he can offer and still achieve a utility level u when his disposable income (net of commuting costs) is I, ri(u, I) = Max(h-l(Z
- pc) / Ui(c, h) = u>.
399
STATICS OF A RESIDENTIAL ECONOMY
From this definition it follows that Ei(r$4, I), u) = I.
(3)
Define the demand for land when ri(u, I) is the rent by hi(u, I) = Hi(ri(u, I), u).
(4)
We assumethat housing is a normal good, and further, if individuals from classi - 1 and i face the same rent then the former demands more housing, i.e., 2.
ASSUMPTION
H,: = awlaM > 0, (a) (b) if for any r, xE”(r, uk) = wk - f(x), Hi-l(r, ui-J > Hi(r, q).
(5) k = i - 1, i, then
Note that (5) together with (2) yields
= ~,i~ = a2Eija,a, > 0.
E:,
(6)
Next, from (2) and (3), r; = (Et)-’
> 0,
rui = -Eui(E,i)-’
< 0,
(7)
hui = Hvirui + Hui > 0.
63)
while, from (2), (4), (5), and (7), ht = H,ir,i < 0,
Finally, since the wage of an individual of classi is wi and since t(x) is his commuting cost, his bid-rent function is ri(u, wi - t(x)), considered as a function of w and x. From Assumption 1 and (7), rXi = W/ax = -fzrIi
-c 0.
(9)
The next result, which is used repeatedly, shows that the difference in two bid-rent functions varies strictly monotonically in x. LEMMA
1.
Suppose7 = @(us, wi - t(Z)) = rj(uj , wj - t(Z)) and i
Then Y,j(Uj
Proof. K?(b),
, Wj
-
t(X))
<
rzi(ui
, Wf
-
t(X))
By (1) Ek(Y, u,) = ullc- t(x),
<
0
k = i,j
Hi( F, ui) > H’(F, u,).
for
x < X.
(10)
and since i < j, by (11)
400
HARTWICK,
SCHWEIZER AND VARAIYA
From (3), Ei(ri(zri , w, - t(x)), ui) = wi - t(x), and if we differentiate this with respect to x and then use (2), we obtain Hi(ri(ui , wi - t(x)), ui) r,i(ui , wi - t(x)) = -t,
.
(W
H’(rj(Uj
.
(13)
Similarly , Wj
-
t(X)),
Uj)
r,‘(Uj
, Wj
-
t(X))
=
-t,
From these relations and (11) we conclude that (10) holds for x = X. Next, as long as both x < E and ri(ui , wi - t(x)) < r’(Uj , Wj - t(x)), we can show using (2) and Assumption 2 that Hi(ri(ui , Wi - t(X)), Ui) > Hi(rj(uj , Wj - t(X)), Ui) > fP(rj(uj , wj - t(x)), uj), and so again using (12), (13) we can conclude that (10) holds. Using this fact and knowing that (10) holds at least for x = X the assertion follows quite readily. As a corollary to the lemma we have COROLLARY
1.
ri(ui , Wj ~
Under the assumptionsof Lemma 2, t(X))
5
rj(Uj
, W -
according as x 5 .U. (14)
t(X))
Next we define the envelope of the individual bid-rent functions, Rh, >...>21,, WI - t(x),..., w, - t(x)> = Max{P(u, , w1 - t(x)) ,..., P(u, , w, - t(x)), rAj. Recall that 0(x)x dx is the amount of land available for residencesin the ring [x, x + Ax]. We can now define the notion of an equilibrium. DEFINITION. An equilibrium consists of a set of utility levels zil ,..., ~1, and a set of residence rings J1 ,..., J, contained in [p, CO]such that
x E Ji if and only if ri(ui , wi - t(x)) = R(u, ,..., u, , w1 - t(x) ,..., IV, - t(x)),
i = I,..., n,
R(u, ,..., u, , w1 - t(x) ,..., M’, - t(x)) is continuous in x, e(x) x Ni = j,, hi(Ui ) l,zi ~ t(x))
dxl
i = I,..., n.
(15) (16) (17)
Equation (15) ensures that land is occupied by the highest bidders. Equation (16) saysthat the equilibrium rent must be continuous acrossthe
401
STATICS OF A RESIDENTIAL ECONOMY
boundary between land occupied by adjacent classes, that is, no individual pays more than he has to. Equation (17) is simply the land market clearing condition. We derive some properties of an equilibrium. First of all from (14) and (15) we see that individuals in class i + 1 live closer to the CBD than those in class i. Hence the rings must have the form x1 > x2 > ..’ > x,+1 = p, Ji = [Xi+12xi1 where (18) so that (16) can be rewritten more directly as rl(ul
, iv1 -
t(xl))
ri(ui , wi - t(xi))
= rA , = ri-l(ui-l
, wpl
for
~ r(xi))
i = l,...,n.
(19)
From (19) we can solve for u1 as a function of x1 and w1 . Then, by induction, we can solve for ui as a function of xi ,..., x1 and wi ,..., w1 . The signs of the partial derivatives of these functions can be determined as follows. LEMMA
2.
au,jax, < 0 Proof.
and
Fu,/2wj
> 0 for j < i.
We proceed by induction.
Differentiating
rul(aul/awl)
r,l@~,/ax,)
(20)
the first equation in
(19) gives + rll = 0,
- tzrI1 = 0,
and so, together with (7) and (9) this yields iiu,/&v, > 0, &,/ax, < 0. Hence (20) is true for i = 1. Now suppose (20) holds for i - 1. Just as before we still have, after differentiating the second equation in (19) with respect to wi , that rui(ui , wi -
tcxi))(au,/aw,) +
rf(ui
, wi - t(x$))
so that again aui/&vi > 0; whereas differentiation us rTti(ui , wi = rkl(uiel
t(xJ)(&,/~.ul) , wpl
+ rzi(uj -
= 0
(21)
with respect to xi gives , wi -
t(xi))
t(xJ).
(22)
From (7) (IO), (22), we conclude that aui/axi < 0. Let us also note from (7) and (21) that 8% awj -
-
rIi(ui , wi - t(x$)) rui(ui , wi - t(.u,))
=
Eui(ri(ui
1 , wi - t(xi)),
uf) > O*
(23)
402
HARTWICK,
SCHWEIZER
AND
VARAIYA
To evaluate the derivative for j < i, we again differentiate the second equation in (19),
(using (23)) = r-f &-l [ l -
E;-l(Yi-l(ui--l E~-yr~-yUi-l
) N’i-1 - W), ) u’i-l - r(x,-J),
4-l) ut-J
I
(using (21), (23)). The expression in brackets is negative because of (6), and so using (7) we get 8ui/&vi_, > 0. Now, from (19) again, r,yau,/au,-,)
= I;‘--’
so that au,/au,, > 0. Hence, using the induction hypothesis, we get
Hence (20) holds for i, and the lemma is proved. From (17) and the remark preceding Lemma 2 we note that Ni is (mathematically) determined by xi+1 ,..., x1 and wi ,..., w1 . The derivatives in the next lemma are with respect to this functional dependence. 3.
LEMMA
aN. -& > 0, 3 Proof.
< 0 ’
From (17) and (18) we obtain
aNi - _ a.xi+,
alv, axi,,
j < i;
@i+d hyz.$
) wi
Xi+1 -
t(xi+1))
<
O3
-f$$ < 0,
j < i.
403
STATICS OF A RESIDENTIAL ECONOMY
xi atdi alvi w x hui(ui , wi - t(x)) axi dx - axi = - J’Ti+l [hi& ) w, - t(X))y > 0 w4 xi + hi(ui , wj - t(xi)) ’
by (8) and (20). Furthermore, xi aNf
bXj
-_
B(x)
I
again using (17) and (20), we get forj
< i,
aui
x
3ci+lW(ui , ~'i - W)Y
hUi(uj , uvi - t(x)) axj dx > 0.
Tt only remains to prove the last assertion, which forj < i follows in the same way as above from (17) and (20). Finally, for j = i, we get
“Ni al$,i = -
xi O(x)x au, scri+l[Wui >wi - W)l” IhUi aM=t t
1
hIi dx.
(24)
Now, from (2), (4) and (8) we have h,i 2
+ ,I; = CA&4 &yryu, ‘1
7 M’i - W)? 4 ) M’[ - t(xJ), Ui)
+ I?;,(... x . ..) , _ E,i(... x . ..) [
l&y... x . ..) E,i(... xi . ..) I ’
which is positive since each of the terms on the right is. Hence, from (24), aNi/awi > 0 also. Lemmas 2 and 3 are crucial for the remainder of the analysis. Both follow from Assumption 2. However, if one is willing to assume (5), (lo), and (18) instead of Assumption 2 then Lemmas 2 and 3 continue to hold. This is Miyao’s approach. However, if Assumption 2 does not hold, then the only way to check whether (18) holds is to explicitly compute the equilibrium. In the case of identical utility functions our analysis shows that Assumption 2(a), together with w1 > w2 ... > W, implies Assumption 2(b), an important special case which seems to have been overlooked by Miyao. 3. COMPARATIVE
STATICS
If we express the functional dependence of Ni on xi+1 ,..., x1 and M’i ,...) w1 in differential form we get i-cl
dNi = c (aN&xj)
j=l
642/13/3-s
dxi + i
j=l
(aNJaw,) dwj
i = l,..., II,
404
HARTWICK,
SCHWEIZER AND VARAIYA
or in matrix notation dN = A dx + a dx,+l - B dw.
(25)
By Lemma 3 the vector a and the 12x n matrices A, B have the sign pattern shown below
a=
B==
6’6) Here dx = (dx, ,..., dx,)‘, dw = (dw, ,..., dw,)‘. Now to obtain the results announced in the introduction we will first obtain the sign pattern of A-‘, as well as some information about the relative magnitudes of some of its coefficients. With this knowledge and the results obtained so far we will be able to reach the desired conclusion. A portion of the sign pattern of A-l can already be determined from (26). LEMMA 4. If A has the pattern of (26), then det A > 0 and A-l has the pattern shown below
A-1 =
(27)
where ? means the sign is unknown. Proof. See Lancaster [3, p. 1191. As a corollary we can seethe effect of a change in the size of the innermost class, N, . COROLLARY
2. 2xj/aNya > 0, all j;
aujjaiv, -=c0, afl j.
ProoJ From (25) and (27) we can see that the vector dx = A-l(dNn)e, > 0 if dN% > 0, where e, = (O,..., 0, l)‘, thus giving the first result. The second assertion follows from the first one and Lemma 2.
STATICS
OF
A RESIDENTIAL
ECONOMY
405
To determine the effects of a change in Ni, i < n, it is necessary to proceed somewhat circuitously. Instead of considering an exogenously fixed agricultural rent rA as in (19), we consider as exogenously given the outer radius of the city, x1 . This will cause some simple changes in the various functional dependencies. Specifically, u1 is no longer determined by x1 and w1 via (19). As a consequence ui becomes a function of Xi ,..., x2 and u1 via (19). (In the following argument w1 ... W, are fixed throughout.) Hence Ni is a function of xi+1 ,..., x, and zll, so that
After dividing both sides by (alv,/ax,) we can express this in matrix form,
(which
The sign pattern of A” can be shown, Lemma 3, to be of the form
in a way similar to the proof
+.:.
0 .
.
. .. . .
-1
. .
is positive by Lemma 3)
of
0
.I : .
.
.
.o . -
.
+
and using an argument similar to that of Lancaster
[3], we get (29)
Next, we observe that if we fix xz and consider only classes 2,..., n we are in the same situation as the one considered above (where we fixed x1 and considered classes I,..., n). Thus (29) stated for this situation is of the form
[ 21 = [;] dx,= [I;] dx,,
406
HARTWICK,
SCHWEIZER
AND
VARAIYA
where the last equality follows from (29). Proceeding inductively we obtain
1 +La=r1 12
; +
dx,
and
[ :I]
= [i]
dx,,
(30)
an intuitive result, since it states that if the city boundary is shifted outwards then all classesget more suburbanized and everyone’s welfare improves. We can now prove result (i) stated in the introduction. THEOREM
1. Under Assumptions I and 2,
i3x,/lJN, > 0 j < i; Proof.
axj/aNi < 0 ,j > i;
and
au,/aNf < 0 all j.
Suppose dN, = 0 j # i, dwj = 0 all j. Then (25) reads
By Lemma 4 and then by Lemma 2 we get
The behavior of the remaining classes,i + I,..., n, is the same as if they lived in a city of radius xi+r which changed by dxi+, . Hence from (30) and (31) (32)
The result follows from (31), (32). We now evaluate the effects of income changes. Once again this is easy to seewhen only the innermost class’sincome, w, , changes. From Lemmas 2-4, (25) (26), it is easy to show that (331
STATICS OF A RESIDENTIAL ECONOMY
Let us show that au,/&,
407
> 0, i.e., du, = [+] dw, .
From (17) we get
since dN,, = dx,+l = 0. Since the second term on the right is positive by (33), this relation can hold only if there exists x E (x,+r , x,), where hun(~u,/aw,)
+ h," > 0,
which implies (34) becauseof (8). Thus the casewhere only w, changesis completely settled. For later purposes, however, we need to determine how the rent at the CBD changeswith w, , i.e., we need the sign of &,+,/awn where Y~+~= P(u, , w, - t(x,+,)). We will show that
arn+llaM'n< 0, provided that the following assumption holds. ASSUMPTION
3.
[~(x)xl-l(d/dx)(~(x)x) > [t,ix)l-‘(d/dx)(t,(x))
all x.
We will discussthis condition 1ater.l In addition to Y,+~, also define r, = rA ,
ri+l = ri(ui , wi - t(~~+~))
i=l
,..., n.
(35)
These rents are interrelated by the formulas
The formula is obtained by differentiating the identity Ei(ri(u, , wi - t(x)), ui) = wi - t(x), to obtain l?ir,Ti = -t,
,
1 Note that if 6(x) = 257, Assumption 3 reads 1 > (xjr(x))(d/dx) t,(x), which is the condition imposed in [2, p. 2321, whereas if t, is constant then Assumption 3 requires only that 0(x)x is increasing in x.
408
HARTWICK,
SCHWEIZER AND VARAIYA
so that
which, upon integration by parts, yields (36).2 Now differentiate (36)
Now in this relation (d/dx)[@)x/t,(x)] (a/aw,) ri(ui , wi - t(x)) = rui(lk,/aw,) ~(xi+J -xi+l ari, 1 <; t.r(.G+J
4x2)
vyi
f,( Xj)
aw,
Starting with the fact that (&,/awn) (38) to conclude that ar,/hv,
> 0 by Assumption 3 and > 0 for i < n by (33) so that %ri
,
i < 11.
zw,
(38)
= 0 we can proceed recursively using
< O,..., &,/%w,
< 0.
An argument similar to the one used in Lemma 3 shows that
E,,“(...
.Y . . .)
h”(rr, , II’, -
1 ----. zpqr.Y ...)
iill, &I',
1
In the last expression the term in square brackets is increasing in x by virtue of (6) so that if (ajaw,) P(u, , w, - t(x,,,)) = (hnt.l/h,,) 3 0, then we must have (a/&,) rn(u, , w, - t(x)) > 0 for x > x,+~ which contradicts (37). Thus we have proved arn+Jawn < 0,
(39)
as desired. Equation (39) implies that the rent at the CBD decreasesas the income of the innermost class increases. While it is obvious that the density at the CBD must decreasewith increasesin w, . since people in class n will demand more land, it does not follow that the reduction in density will be so large that rents will decreasealso. To reach this conclusion Assumption 3 is crucial. 2 This formula is interesting in its own right since it directly relates the population to the rent function and transportation cost.
STATICS
OF
A RESIDENTIAL
409
ECONOMY
We will make critical use of (39) to study the effects of a change in wi , i < n. But first let us note that if dN = 0, dw = 0 but d~,+~ # 0 in (25) then we have A dx = -a dx,+l so that from (26) and (27) we get 1
+
72
+
El=r1 [:I [I !
and, using Lemma 2,
(404
-
1
=
n
dxn+l
; -
d-Y,+1 .
(4Ob)
Equation (40b) is obvious; it asserts that everyone is worse off (due to increased transport cost) if the CBD size grows; (40a) is perhaps less obvious; it asserts that all classes are more suburbanized (even without any increase in income) if the CBD grows. Now consider a change in wi , for some i < n. Setting dx,+l = 0, dN = 0 and dwj = 0 j # i, in (25), we see using (26) that
,
(41)
where exactly the first i - 1 components are zero. From (41) and (27), and then from Lemma 2, we can deduce that 1
+ ;
dwj ,
and
[dTyj
ryd& 1 = [ + 1
= [I]
dwi
z
Thus classes living further than i get more suburbanized and suffer a loss in real income when the income of class i increases. As regards the classes i + I,..., n, their allocation is the same as if they lived in a city of radius xi+1 and the other classes did not exist at all. Since class i + 1 lives farthest amongst these classes we can use (30) to conclude that [;r]
= [;]
dxicl,
[;;:I
so that we must determine the sign of dx,+l .
= [i]
dxx,+l,
(42)
410
HARTWICK,
SCHWEIZER
AND
VARAIYA
But before we do this let us note that classes I,..., i receive the same allocation as if they lived alone in a city whose CBD radius is xi+1 . In this sense ui is determined by xi,1 and wi , and we may express this as ui = mi+1 7 Wi) and since class i is the innermost among these classes we can use (40) and (34) respectively to conclude that
aFlaw, > 0.
c3F/8xi+, < 0,
(43)
To determine the sign of d~,+~ we differentiate the equilibrium (19) ri+yui+l , wi+l - ~(Xi,l)> = e4 7 wi - ~(Xi,l)),
condition
to obtain (recall that below ui+i is a function only of xi+*) i+l dui+l r, r dx<+l + rZ1 dx,+l 2+1
rUi [A dx,+l + g ax,+1
dw,) + rli dwi + rei dx,+* ,
.
(44)
which can be rearranged as
Now rt+’ < 0 by (7), dui+l/dxi+l < 0 by (42); (rz’ - rzi) < 0 by Lemma 1; -rUi(8F/8xi+& < 0 by (7) and (43). Thus the coefficient multiplying dx,+l is negative. On the other hand, by (39), we know that
hl __ = awi [
(45)
Thus we have shown that dx,+l = [+] dwi .
(46)
Only the sign of dui remains. From (44) we note that
au. dwli = r,,i rui -2 [E dwi + 5 ItI &+l] awi =r 2’” *
dx,+l + (rc’
- rxi) bl
dxi+l
-
?+l
=r
i+l u
dui+l ~~7
= [-]dwi
7+1
+
(rf+l
rTi)
dx,+l
- rIi dw,
411
STATICS OF A RESIDENTIAL ECONOMY
whence, since rUi < 0 by (7), we get dui = [+] dwi .
(47)
THEOREM 2. Under Assumptions1-3, axJaw, > 0 allj; and au,/aw, > 0 if j 2 i, &.+(awi < 0 zfj < i. For i = n Assumption 3 is not needed.
Proof. The first assertion follows from (41), (42), and (46), whereas (42), (46), and (47) imply that &,/8wi > 0 if j > i. The remaining assertion has already been proved. The critical step in proving Theorem 2 was the establishment of (46), or (45), which assert that an increase in wi causes not only a suburbanization of the ith class but a reduction in the rent faced by the (i + 1)st class. To reach this conclusion Assumption 3 is imposed. Indeed if Assumption 3 is violated, which can happen if at some distance x transportation cost increases very rapidly or land available for housing increases very slowly, then it is always possible to choose a set of parameters N1 ,..., N, and w1 ,..., w, at which the signs in (45) and (46) are reversed so that Theorem 2 no longer holds. In the course of deriving Theorem 1 and Theorem 2 we have had to determine how the equilibrium rent function shifts due to changes in the Ni and wi . Since this comparative statics result has some independent interest we state it separately here. Let r(x) = r(x, NI ,..., N, , w1,..., w,) denote the equilibrium rent function. THEOREM
3. Under Assumptions 1 and 2,
(at-/W&x) > 0
G+~ < x < x1,
all i.
Under Assumptions 1-3,
(ar/aw,)(x) > 0,
xi
and (ar/aw,)(x) -c 0,
x,+~ < x <
4. A NUMERICAL
Xi+1
,
all i.
EXAMPLE
We consider here a city model with two population classes. The parameter values chosen are adapted from [l]. There is only one other good besides land. Every individual has the same utility function, U(c, h) = c”~‘vzo~25.
412
HARTWICK, SCHWEIZER AND VARAIYA TABLE I Income w1 = $12,000.00 79,000
97,000
116,000
149,000
171,000
207,000
246,000 3.25
---
2.46
2.57
2.68
2.85
2.95
3.10
1.60
1.58
1.55
1.53
1.50
1.48
1.45
58.1
57.6
57.1
56.2
55.7
54.9
54.1
153.2
151.2
149.2
146.2
144.4
141.7
139.1
29,400
31,300
33,300
36,600
38,800
42,300
46,000
54,700
56,700
58,700
62,700
64,700
68,700
72,700
0.00
45
89
174
214
293
368
0.00
39
77
152
189
260
329
0.00
152
302
547
693
930
1161
0.00
135
265
470
587
770
942
TABLE II Population N1 = 200,000 97,000
65,000
25,000
11,500
6000
8.95
7.21
4.43
3.02
2.30
1.6
1.58
1.53
1.48
1.45
58.1
57.6
56.2
54.9
54.1
1308
853
320
136
66
29,400
31,300
36,600
42,300
46,000
54,700
56,700
62,700
68,700
72,700
0.00
45
174
293
368
0.00
39
152
260
329
The population of the poor class is fixed at Nz = 135,000 and their annual income is fixed at ~2~ = $5000. The unit of distance (corresponding roughly to 2 miles) is such that the radius of the CBD is x, = 1. The price of the other good is $1. Transportation cost per annum per individual is linear in distance and is given by l(x) = $1200 x. Agricultural rent per annum per unit area is $20,000. A constant proportion of the area of each circular ring is available for housing and it is given by e(x) = 0.47r rad. Table I shows how various equilibrium values change as the population of the rich class, N1 , grows from 79,000 to 246,000. The same notation as before is maintained with the exception of the row labels C,(X) and C,(X). C,(x) is the compensation in dollars per annum necessary for one rich
STATICS OF A RESIDENTIAL ECONOMY
413
individual at x to achieve the utility level u1 = 153.2; whereas C,(x) is the corresponding compensation for a poor individual at x to maintain the utility land u2 = 58.1. Thus the C,(x) is a simple measureof welfare loss. A comparison of these two tables reveals that the loss incurred by a poor individual due to a threefold increase in the population of the rich from 79,000 to 246,000 is the same as a sixteen-fold decrease in their income from $97,000 to $6,000. REFERENCES
I. R. M. 2. 3. 4. 5.
SOLOW, Congestion cost and the use of land for streets, Bell J. Econ. Management (1973), 602-618. W. C. WHEATON, A comparative static analysis of urban spatial structure, J. Econ. Theory 4 (1974), 223-237. K. LANCASTER, The scope of qualitative economics, Rev. Econ. Studies (1962), 99-123. T. MIYAO, Dynamicsand comparative statics in the theory of residential location, J. Econ. Theory 11 (1975), 133-146. W. C. WHEATON, On the optimal distribution of income among cities, J. Urban Econ. 3 (1976), 31-44.
OF ECONOMIC
Comparative
THEORY
13, 396-413 (1976)
Statics of a Residential Several Classes*+
Economy
with
JOHN HARTWICK Department of Economics, Queen’s University, Ontario, Canada URS SCHWEIZER Department of Economics, M.I.T., Cambridge, Massachusetts 02139 AND
PRAWN VARAIYA Decision and Control Sciences Group; Electronic Systems Laboratory, Department of Electrical Engineering and Computer Science, M.I. T., Cambri&e, Massachusetts 02139; Electronics Research Laboratory, University of California, Berkeley, California 94720 Received June 18, 197.5; revised May 3, 1976
1. INTRODUCTION
AND SUMMARY
The geography and economic structure of the city model under consideration are orthodox. The city possessescircular symmetry and at its center is the CBD (centra! business district) of radius p. People live at distances x 3 p and they commute to work in the CBD. Commuting cost is solely a money cost and depends only upon the distance of the residences from the CBD. Individuals have fixed money wages and they use it to occupy land (housing) and to consume other commodities. These latter are available at fixed prices which are constant throughout the city. The land market is competitive and so land rents, in equilibrium, coincide with the maximum of individual bid rents and an exogenously specified agricultural rent rA . Land is owned by absentee landlords. * A version of this paper was presented at a conference on Mathematical Models of Land Use at McMaster University, April 1975. + Research partially supported by National Science Foundation Grants GK-41647 and ENG 74-01551-AOl, by the Swiss National Science Foundation, and by the Canada Council. We are grateful for helpful comments from members of the Urban Workshop, Department of Economics, M.I.T., Cambridge, Massachusetts 02139.
396 Copyright All rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
STATICS
OF
A
RESIDENTIAL
397
ECONOMY
There are n classes of people, class i is characterized by the number of people in the class Ni and, for each individual in the class, his wage wi and utility function Ui . Vi is assumed to be smooth, strictly quasiconcave and such that housing is a normal good. Furthermore, it is assumed that preferences and incomes between classes are related in such a way that an individual in class i will occupy more land than one in class i + 1. In particular, this holds if all individuals have the same utility function and wi > wi+r . Under these assumptions, in equilibrium, people in different classes reside in characteristic concentric rings around the CBD with individuals in class i -- 1 living closer to the CBD than those in class i. Suppose class i occupies the ring Ji = [x~+~, xi] where x1 > ... > x,+~ = p, the CBD radius. Let u1 ,..., u, denote the equilibrium utility levels. The dependence of these equilibrium values on the exogenous parameters can be expressed in a functional form, Xi
=
x,(N~
)...)
N, ) WI ,..., WIT)
Zli = tri(Nl )...) Nn ) ~1 )...) ~1,)
i = I,..., tt, i = l,..., n.
Our first set of results shows that the signs of the various partial derivatives are unambiguous. The specific statements are these. axJaN, > 0 if,j < i,
axJaN, < 0 if j > i, and
au,/ahr, < 0 all j, (i)
i.e., if the ith class increases in size, then the outer classes are pushed away from the CBD whereas the inner ones are squeezed towards it, and everyone’s real income is reduced. This is not at all surprising. axjjawi
> 0 all,j,
and
au,/&,
> 0 ifj 3 i,
auj/anli < 0 ifj < i, (ii)
i.e., if the ith class’s income rises, then all classes are more suburbanized, and the outer classes suffer a reduction in their real income whereas the inner classes enjoy an increase. The total asymmetry of (i) and (ii) is of a striking simplicity. Many people may find (ii) counter-intuitive since it would appear at first sight that, since increases in Ni or Wi both create increased demand pressures in the land market, therefore they would have the same impact on the real income of the other classes. These qualitative results, while telling us that increases in income or size of an outer class have opposing impacts on an inner class, do not provide any clue as to the relative magnitudes of these two effects. Our second set of results is the outcome of some numerical exercises designed to reveal what magnitudes of these effects might plausibly be expected. Choosing n = 2, and the same Cobb-Douglas utility function for both classes,
398
HARTWICK,
SCHWEIZER AND VARAIYA
we show that the negative effect on the poor due to an increase in the size of the richer class is far greater than the positive effect due to an increase in the latter’s income. When this paper was first written, the only work related to ours was the comparative analysis carried out by Wheaton [2] for the case of a single class, n = 1. Since then two papers have appeared. Miyao [4] proves the simple special case of(i) and (ii), ax,/aN, > 0, axJaw, > 0 forj 1 l,..., n, and Wheaton [5] proves the result (ii) above for the case 12 = 2 when the utility function is the same for all. We are indebted to the referee for pointing out these references and for several suggestions.
2. THE MODEL. AND ITS EQUILIBRIUM
Aside from the notation introduced already we need the following. The money cost of commuting from x is t(x) and is such that ASSUMPTION
1. t, = dtjdx > 0.
Rent of land is denoted by r, and the amount of land available for housing at x is exogenously given to be 6(x)x (for a circular city we have 0(x)x = 27rx.) The amount of land occupied by an individual is denoted h and the bundle of other goods he consumes is the (column) vector c. The exogenously fixed prices of these other goods is the (row) vector p. The expenditure function for an individual of class i can then be defined in the usual way as P(r, U) = minfpc
+ rh 1uI(c, h) = u>
and this is related to his compensateddemandfunctions Ci, Hi by Ei(r, u) = pCi(r, u) + rHi(r, u);
U”(G(r, u), HYr, u)) = 24. (1)
Ei has the following well-known properties: E,j = aEil& > 0,
E: = aEil&- = H* > 0, (2)
Ei, = a2Ei/ar2 = HPi x 0.
Let ri(u, I) be the function which gives the maximum rent he can offer and still achieve a utility level u when his disposable income (net of commuting costs) is I, ri(u, I) = Max(h-l(Z
- pc) / Ui(c, h) = u>.
399
STATICS OF A RESIDENTIAL ECONOMY
From this definition it follows that Ei(r$4, I), u) = I.
(3)
Define the demand for land when ri(u, I) is the rent by hi(u, I) = Hi(ri(u, I), u).
(4)
We assumethat housing is a normal good, and further, if individuals from classi - 1 and i face the same rent then the former demands more housing, i.e., 2.
ASSUMPTION
H,: = awlaM > 0, (a) (b) if for any r, xE”(r, uk) = wk - f(x), Hi-l(r, ui-J > Hi(r, q).
(5) k = i - 1, i, then
Note that (5) together with (2) yields
= ~,i~ = a2Eija,a, > 0.
E:,
(6)
Next, from (2) and (3), r; = (Et)-’
> 0,
rui = -Eui(E,i)-’
< 0,
(7)
hui = Hvirui + Hui > 0.
63)
while, from (2), (4), (5), and (7), ht = H,ir,i < 0,
Finally, since the wage of an individual of classi is wi and since t(x) is his commuting cost, his bid-rent function is ri(u, wi - t(x)), considered as a function of w and x. From Assumption 1 and (7), rXi = W/ax = -fzrIi
-c 0.
(9)
The next result, which is used repeatedly, shows that the difference in two bid-rent functions varies strictly monotonically in x. LEMMA
1.
Suppose7 = @(us, wi - t(Z)) = rj(uj , wj - t(Z)) and i
Then Y,j(Uj
Proof. K?(b),
, Wj
-
t(X))
<
rzi(ui
, Wf
-
t(X))
By (1) Ek(Y, u,) = ullc- t(x),
<
0
k = i,j
Hi( F, ui) > H’(F, u,).
for
x < X.
(10)
and since i < j, by (11)
400
HARTWICK,
SCHWEIZER AND VARAIYA
From (3), Ei(ri(zri , w, - t(x)), ui) = wi - t(x), and if we differentiate this with respect to x and then use (2), we obtain Hi(ri(ui , wi - t(x)), ui) r,i(ui , wi - t(x)) = -t,
.
(W
H’(rj(Uj
.
(13)
Similarly , Wj
-
t(X)),
Uj)
r,‘(Uj
, Wj
-
t(X))
=
-t,
From these relations and (11) we conclude that (10) holds for x = X. Next, as long as both x < E and ri(ui , wi - t(x)) < r’(Uj , Wj - t(x)), we can show using (2) and Assumption 2 that Hi(ri(ui , Wi - t(X)), Ui) > Hi(rj(uj , Wj - t(X)), Ui) > fP(rj(uj , wj - t(x)), uj), and so again using (12), (13) we can conclude that (10) holds. Using this fact and knowing that (10) holds at least for x = X the assertion follows quite readily. As a corollary to the lemma we have COROLLARY
1.
ri(ui , Wj ~
Under the assumptionsof Lemma 2, t(X))
5
rj(Uj
, W -
according as x 5 .U. (14)
t(X))
Next we define the envelope of the individual bid-rent functions, Rh, >...>21,, WI - t(x),..., w, - t(x)> = Max{P(u, , w1 - t(x)) ,..., P(u, , w, - t(x)), rAj. Recall that 0(x)x dx is the amount of land available for residencesin the ring [x, x + Ax]. We can now define the notion of an equilibrium. DEFINITION. An equilibrium consists of a set of utility levels zil ,..., ~1, and a set of residence rings J1 ,..., J, contained in [p, CO]such that
x E Ji if and only if ri(ui , wi - t(x)) = R(u, ,..., u, , w1 - t(x) ,..., IV, - t(x)),
i = I,..., n,
R(u, ,..., u, , w1 - t(x) ,..., M’, - t(x)) is continuous in x, e(x) x Ni = j,, hi(Ui ) l,zi ~ t(x))
dxl
i = I,..., n.
(15) (16) (17)
Equation (15) ensures that land is occupied by the highest bidders. Equation (16) saysthat the equilibrium rent must be continuous acrossthe
401
STATICS OF A RESIDENTIAL ECONOMY
boundary between land occupied by adjacent classes, that is, no individual pays more than he has to. Equation (17) is simply the land market clearing condition. We derive some properties of an equilibrium. First of all from (14) and (15) we see that individuals in class i + 1 live closer to the CBD than those in class i. Hence the rings must have the form x1 > x2 > ..’ > x,+1 = p, Ji = [Xi+12xi1 where (18) so that (16) can be rewritten more directly as rl(ul
, iv1 -
t(xl))
ri(ui , wi - t(xi))
= rA , = ri-l(ui-l
, wpl
for
~ r(xi))
i = l,...,n.
(19)
From (19) we can solve for u1 as a function of x1 and w1 . Then, by induction, we can solve for ui as a function of xi ,..., x1 and wi ,..., w1 . The signs of the partial derivatives of these functions can be determined as follows. LEMMA
2.
au,jax, < 0 Proof.
and
Fu,/2wj
> 0 for j < i.
We proceed by induction.
Differentiating
rul(aul/awl)
r,l@~,/ax,)
(20)
the first equation in
(19) gives + rll = 0,
- tzrI1 = 0,
and so, together with (7) and (9) this yields iiu,/&v, > 0, &,/ax, < 0. Hence (20) is true for i = 1. Now suppose (20) holds for i - 1. Just as before we still have, after differentiating the second equation in (19) with respect to wi , that rui(ui , wi -
tcxi))(au,/aw,) +
rf(ui
, wi - t(x$))
so that again aui/&vi > 0; whereas differentiation us rTti(ui , wi = rkl(uiel
t(xJ)(&,/~.ul) , wpl
+ rzi(uj -
= 0
(21)
with respect to xi gives , wi -
t(xi))
t(xJ).
(22)
From (7) (IO), (22), we conclude that aui/axi < 0. Let us also note from (7) and (21) that 8% awj -
-
rIi(ui , wi - t(x$)) rui(ui , wi - t(.u,))
=
Eui(ri(ui
1 , wi - t(xi)),
uf) > O*
(23)
402
HARTWICK,
SCHWEIZER
AND
VARAIYA
To evaluate the derivative for j < i, we again differentiate the second equation in (19),
(using (23)) = r-f &-l [ l -
E;-l(Yi-l(ui--l E~-yr~-yUi-l
) N’i-1 - W), ) u’i-l - r(x,-J),
4-l) ut-J
I
(using (21), (23)). The expression in brackets is negative because of (6), and so using (7) we get 8ui/&vi_, > 0. Now, from (19) again, r,yau,/au,-,)
= I;‘--’
so that au,/au,, > 0. Hence, using the induction hypothesis, we get
Hence (20) holds for i, and the lemma is proved. From (17) and the remark preceding Lemma 2 we note that Ni is (mathematically) determined by xi+1 ,..., x1 and wi ,..., w1 . The derivatives in the next lemma are with respect to this functional dependence. 3.
LEMMA
aN. -& > 0, 3 Proof.
< 0 ’
From (17) and (18) we obtain
aNi - _ a.xi+,
alv, axi,,
j < i;
@i+d hyz.$
) wi
Xi+1 -
t(xi+1))
<
O3
-f$$ < 0,
j < i.
403
STATICS OF A RESIDENTIAL ECONOMY
xi atdi alvi w x hui(ui , wi - t(x)) axi dx - axi = - J’Ti+l [hi& ) w, - t(X))y > 0 w4 xi + hi(ui , wj - t(xi)) ’
by (8) and (20). Furthermore, xi aNf
bXj
-_
B(x)
I
again using (17) and (20), we get forj
< i,
aui
x
3ci+lW(ui , ~'i - W)Y
hUi(uj , uvi - t(x)) axj dx > 0.
Tt only remains to prove the last assertion, which forj < i follows in the same way as above from (17) and (20). Finally, for j = i, we get
“Ni al$,i = -
xi O(x)x au, scri+l[Wui >wi - W)l” IhUi aM=t t
1
hIi dx.
(24)
Now, from (2), (4) and (8) we have h,i 2
+ ,I; = CA&4 &yryu, ‘1
7 M’i - W)? 4 ) M’[ - t(xJ), Ui)
+ I?;,(... x . ..) , _ E,i(... x . ..) [
l&y... x . ..) E,i(... xi . ..) I ’
which is positive since each of the terms on the right is. Hence, from (24), aNi/awi > 0 also. Lemmas 2 and 3 are crucial for the remainder of the analysis. Both follow from Assumption 2. However, if one is willing to assume (5), (lo), and (18) instead of Assumption 2 then Lemmas 2 and 3 continue to hold. This is Miyao’s approach. However, if Assumption 2 does not hold, then the only way to check whether (18) holds is to explicitly compute the equilibrium. In the case of identical utility functions our analysis shows that Assumption 2(a), together with w1 > w2 ... > W, implies Assumption 2(b), an important special case which seems to have been overlooked by Miyao. 3. COMPARATIVE
STATICS
If we express the functional dependence of Ni on xi+1 ,..., x1 and M’i ,...) w1 in differential form we get i-cl
dNi = c (aN&xj)
j=l
642/13/3-s
dxi + i
j=l
(aNJaw,) dwj
i = l,..., II,
404
HARTWICK,
SCHWEIZER AND VARAIYA
or in matrix notation dN = A dx + a dx,+l - B dw.
(25)
By Lemma 3 the vector a and the 12x n matrices A, B have the sign pattern shown below
a=
B==
6’6) Here dx = (dx, ,..., dx,)‘, dw = (dw, ,..., dw,)‘. Now to obtain the results announced in the introduction we will first obtain the sign pattern of A-‘, as well as some information about the relative magnitudes of some of its coefficients. With this knowledge and the results obtained so far we will be able to reach the desired conclusion. A portion of the sign pattern of A-l can already be determined from (26). LEMMA 4. If A has the pattern of (26), then det A > 0 and A-l has the pattern shown below
A-1 =
(27)
where ? means the sign is unknown. Proof. See Lancaster [3, p. 1191. As a corollary we can seethe effect of a change in the size of the innermost class, N, . COROLLARY
2. 2xj/aNya > 0, all j;
aujjaiv, -=c0, afl j.
ProoJ From (25) and (27) we can see that the vector dx = A-l(dNn)e, > 0 if dN% > 0, where e, = (O,..., 0, l)‘, thus giving the first result. The second assertion follows from the first one and Lemma 2.
STATICS
OF
A RESIDENTIAL
ECONOMY
405
To determine the effects of a change in Ni, i < n, it is necessary to proceed somewhat circuitously. Instead of considering an exogenously fixed agricultural rent rA as in (19), we consider as exogenously given the outer radius of the city, x1 . This will cause some simple changes in the various functional dependencies. Specifically, u1 is no longer determined by x1 and w1 via (19). As a consequence ui becomes a function of Xi ,..., x2 and u1 via (19). (In the following argument w1 ... W, are fixed throughout.) Hence Ni is a function of xi+1 ,..., x, and zll, so that
After dividing both sides by (alv,/ax,) we can express this in matrix form,
(which
The sign pattern of A” can be shown, Lemma 3, to be of the form
in a way similar to the proof
+.:.
0 .
.
. .. . .
-1
. .
is positive by Lemma 3)
of
0
.I : .
.
.
.o . -
.
+
and using an argument similar to that of Lancaster
[3], we get (29)
Next, we observe that if we fix xz and consider only classes 2,..., n we are in the same situation as the one considered above (where we fixed x1 and considered classes I,..., n). Thus (29) stated for this situation is of the form
[ 21 = [;] dx,= [I;] dx,,
406
HARTWICK,
SCHWEIZER
AND
VARAIYA
where the last equality follows from (29). Proceeding inductively we obtain
1 +La=r1 12
; +
dx,
and
[ :I]
= [i]
dx,,
(30)
an intuitive result, since it states that if the city boundary is shifted outwards then all classesget more suburbanized and everyone’s welfare improves. We can now prove result (i) stated in the introduction. THEOREM
1. Under Assumptions I and 2,
i3x,/lJN, > 0 j < i; Proof.
axj/aNi < 0 ,j > i;
and
au,/aNf < 0 all j.
Suppose dN, = 0 j # i, dwj = 0 all j. Then (25) reads
By Lemma 4 and then by Lemma 2 we get
The behavior of the remaining classes,i + I,..., n, is the same as if they lived in a city of radius xi+r which changed by dxi+, . Hence from (30) and (31) (32)
The result follows from (31), (32). We now evaluate the effects of income changes. Once again this is easy to seewhen only the innermost class’sincome, w, , changes. From Lemmas 2-4, (25) (26), it is easy to show that (331
STATICS OF A RESIDENTIAL ECONOMY
Let us show that au,/&,
407
> 0, i.e., du, = [+] dw, .
From (17) we get
since dN,, = dx,+l = 0. Since the second term on the right is positive by (33), this relation can hold only if there exists x E (x,+r , x,), where hun(~u,/aw,)
+ h," > 0,
which implies (34) becauseof (8). Thus the casewhere only w, changesis completely settled. For later purposes, however, we need to determine how the rent at the CBD changeswith w, , i.e., we need the sign of &,+,/awn where Y~+~= P(u, , w, - t(x,+,)). We will show that
arn+llaM'n< 0, provided that the following assumption holds. ASSUMPTION
3.
[~(x)xl-l(d/dx)(~(x)x) > [t,ix)l-‘(d/dx)(t,(x))
all x.
We will discussthis condition 1ater.l In addition to Y,+~, also define r, = rA ,
ri+l = ri(ui , wi - t(~~+~))
i=l
,..., n.
(35)
These rents are interrelated by the formulas
The formula is obtained by differentiating the identity Ei(ri(u, , wi - t(x)), ui) = wi - t(x), to obtain l?ir,Ti = -t,
,
1 Note that if 6(x) = 257, Assumption 3 reads 1 > (xjr(x))(d/dx) t,(x), which is the condition imposed in [2, p. 2321, whereas if t, is constant then Assumption 3 requires only that 0(x)x is increasing in x.
408
HARTWICK,
SCHWEIZER AND VARAIYA
so that
which, upon integration by parts, yields (36).2 Now differentiate (36)
Now in this relation (d/dx)[@)x/t,(x)] (a/aw,) ri(ui , wi - t(x)) = rui(lk,/aw,) ~(xi+J -xi+l ari, 1 <; t.r(.G+J
4x2)
vyi
f,( Xj)
aw,
Starting with the fact that (&,/awn) (38) to conclude that ar,/hv,
> 0 by Assumption 3 and > 0 for i < n by (33) so that %ri
,
i < 11.
zw,
(38)
= 0 we can proceed recursively using
< O,..., &,/%w,
< 0.
An argument similar to the one used in Lemma 3 shows that
E,,“(...
.Y . . .)
h”(rr, , II’, -
1 ----. zpqr.Y ...)
iill, &I',
1
In the last expression the term in square brackets is increasing in x by virtue of (6) so that if (ajaw,) P(u, , w, - t(x,,,)) = (hnt.l/h,,) 3 0, then we must have (a/&,) rn(u, , w, - t(x)) > 0 for x > x,+~ which contradicts (37). Thus we have proved arn+Jawn < 0,
(39)
as desired. Equation (39) implies that the rent at the CBD decreasesas the income of the innermost class increases. While it is obvious that the density at the CBD must decreasewith increasesin w, . since people in class n will demand more land, it does not follow that the reduction in density will be so large that rents will decreasealso. To reach this conclusion Assumption 3 is crucial. 2 This formula is interesting in its own right since it directly relates the population to the rent function and transportation cost.
STATICS
OF
A RESIDENTIAL
409
ECONOMY
We will make critical use of (39) to study the effects of a change in wi , i < n. But first let us note that if dN = 0, dw = 0 but d~,+~ # 0 in (25) then we have A dx = -a dx,+l so that from (26) and (27) we get 1
+
72
+
El=r1 [:I [I !
and, using Lemma 2,
(404
-
1
=
n
dxn+l
; -
d-Y,+1 .
(4Ob)
Equation (40b) is obvious; it asserts that everyone is worse off (due to increased transport cost) if the CBD size grows; (40a) is perhaps less obvious; it asserts that all classes are more suburbanized (even without any increase in income) if the CBD grows. Now consider a change in wi , for some i < n. Setting dx,+l = 0, dN = 0 and dwj = 0 j # i, in (25), we see using (26) that
,
(41)
where exactly the first i - 1 components are zero. From (41) and (27), and then from Lemma 2, we can deduce that 1
+ ;
dwj ,
and
[dTyj
ryd& 1 = [ + 1
= [I]
dwi
z
Thus classes living further than i get more suburbanized and suffer a loss in real income when the income of class i increases. As regards the classes i + I,..., n, their allocation is the same as if they lived in a city of radius xi+1 and the other classes did not exist at all. Since class i + 1 lives farthest amongst these classes we can use (30) to conclude that [;r]
= [;]
dxicl,
[;;:I
so that we must determine the sign of dx,+l .
= [i]
dxx,+l,
(42)
410
HARTWICK,
SCHWEIZER
AND
VARAIYA
But before we do this let us note that classes I,..., i receive the same allocation as if they lived alone in a city whose CBD radius is xi+1 . In this sense ui is determined by xi,1 and wi , and we may express this as ui = mi+1 7 Wi) and since class i is the innermost among these classes we can use (40) and (34) respectively to conclude that
aFlaw, > 0.
c3F/8xi+, < 0,
(43)
To determine the sign of d~,+~ we differentiate the equilibrium (19) ri+yui+l , wi+l - ~(Xi,l)> = e4 7 wi - ~(Xi,l)),
condition
to obtain (recall that below ui+i is a function only of xi+*) i+l dui+l r, r dx<+l + rZ1 dx,+l 2+1
rUi [A dx,+l + g ax,+1
dw,) + rli dwi + rei dx,+* ,
.
(44)
which can be rearranged as
Now rt+’ < 0 by (7), dui+l/dxi+l < 0 by (42); (rz’ - rzi) < 0 by Lemma 1; -rUi(8F/8xi+& < 0 by (7) and (43). Thus the coefficient multiplying dx,+l is negative. On the other hand, by (39), we know that
hl __ = awi [
(45)
Thus we have shown that dx,+l = [+] dwi .
(46)
Only the sign of dui remains. From (44) we note that
au. dwli = r,,i rui -2 [E dwi + 5 ItI &+l] awi =r 2’” *
dx,+l + (rc’
- rxi) bl
dxi+l
-
?+l
=r
i+l u
dui+l ~~7
= [-]dwi
7+1
+
(rf+l
rTi)
dx,+l
- rIi dw,
411
STATICS OF A RESIDENTIAL ECONOMY
whence, since rUi < 0 by (7), we get dui = [+] dwi .
(47)
THEOREM 2. Under Assumptions1-3, axJaw, > 0 allj; and au,/aw, > 0 if j 2 i, &.+(awi < 0 zfj < i. For i = n Assumption 3 is not needed.
Proof. The first assertion follows from (41), (42), and (46), whereas (42), (46), and (47) imply that &,/8wi > 0 if j > i. The remaining assertion has already been proved. The critical step in proving Theorem 2 was the establishment of (46), or (45), which assert that an increase in wi causes not only a suburbanization of the ith class but a reduction in the rent faced by the (i + 1)st class. To reach this conclusion Assumption 3 is imposed. Indeed if Assumption 3 is violated, which can happen if at some distance x transportation cost increases very rapidly or land available for housing increases very slowly, then it is always possible to choose a set of parameters N1 ,..., N, and w1 ,..., w, at which the signs in (45) and (46) are reversed so that Theorem 2 no longer holds. In the course of deriving Theorem 1 and Theorem 2 we have had to determine how the equilibrium rent function shifts due to changes in the Ni and wi . Since this comparative statics result has some independent interest we state it separately here. Let r(x) = r(x, NI ,..., N, , w1,..., w,) denote the equilibrium rent function. THEOREM
3. Under Assumptions 1 and 2,
(at-/W&x) > 0
G+~ < x < x1,
all i.
Under Assumptions 1-3,
(ar/aw,)(x) > 0,
xi
and (ar/aw,)(x) -c 0,
x,+~ < x <
4. A NUMERICAL
Xi+1
,
all i.
EXAMPLE
We consider here a city model with two population classes. The parameter values chosen are adapted from [l]. There is only one other good besides land. Every individual has the same utility function, U(c, h) = c”~‘vzo~25.
412
HARTWICK, SCHWEIZER AND VARAIYA TABLE I Income w1 = $12,000.00 79,000
97,000
116,000
149,000
171,000
207,000
246,000 3.25
---
2.46
2.57
2.68
2.85
2.95
3.10
1.60
1.58
1.55
1.53
1.50
1.48
1.45
58.1
57.6
57.1
56.2
55.7
54.9
54.1
153.2
151.2
149.2
146.2
144.4
141.7
139.1
29,400
31,300
33,300
36,600
38,800
42,300
46,000
54,700
56,700
58,700
62,700
64,700
68,700
72,700
0.00
45
89
174
214
293
368
0.00
39
77
152
189
260
329
0.00
152
302
547
693
930
1161
0.00
135
265
470
587
770
942
TABLE II Population N1 = 200,000 97,000
65,000
25,000
11,500
6000
8.95
7.21
4.43
3.02
2.30
1.6
1.58
1.53
1.48
1.45
58.1
57.6
56.2
54.9
54.1
1308
853
320
136
66
29,400
31,300
36,600
42,300
46,000
54,700
56,700
62,700
68,700
72,700
0.00
45
174
293
368
0.00
39
152
260
329
The population of the poor class is fixed at Nz = 135,000 and their annual income is fixed at ~2~ = $5000. The unit of distance (corresponding roughly to 2 miles) is such that the radius of the CBD is x, = 1. The price of the other good is $1. Transportation cost per annum per individual is linear in distance and is given by l(x) = $1200 x. Agricultural rent per annum per unit area is $20,000. A constant proportion of the area of each circular ring is available for housing and it is given by e(x) = 0.47r rad. Table I shows how various equilibrium values change as the population of the rich class, N1 , grows from 79,000 to 246,000. The same notation as before is maintained with the exception of the row labels C,(X) and C,(X). C,(x) is the compensation in dollars per annum necessary for one rich
STATICS OF A RESIDENTIAL ECONOMY
413
individual at x to achieve the utility level u1 = 153.2; whereas C,(x) is the corresponding compensation for a poor individual at x to maintain the utility land u2 = 58.1. Thus the C,(x) is a simple measureof welfare loss. A comparison of these two tables reveals that the loss incurred by a poor individual due to a threefold increase in the population of the rich from 79,000 to 246,000 is the same as a sixteen-fold decrease in their income from $97,000 to $6,000. REFERENCES
I. R. M. 2. 3. 4. 5.
SOLOW, Congestion cost and the use of land for streets, Bell J. Econ. Management (1973), 602-618. W. C. WHEATON, A comparative static analysis of urban spatial structure, J. Econ. Theory 4 (1974), 223-237. K. LANCASTER, The scope of qualitative economics, Rev. Econ. Studies (1962), 99-123. T. MIYAO, Dynamicsand comparative statics in the theory of residential location, J. Econ. Theory 11 (1975), 133-146. W. C. WHEATON, On the optimal distribution of income among cities, J. Urban Econ. 3 (1976), 31-44.