- Email: [email protected]
An open-city model with nonmalleable housing
JOURNAL
OF URBAN
7,248-277
ECQNOMICS
An Open-City
Model
(1980)
with Nonmalleable
Housing’
NEIL VOUSDEN Department of Economics, Fad@ of Economics, Australian National Uniw?rsi@, P. 0. Box 4, Canberra. A. C. T. 2600 Australia Received May 24, 1977; revised July 24, 1978 Previous city and housing models are extended to allow for the nonmalleability of housing and two classes of residents. The model, which is framed in the context of a monocentric circular city, assumes an individual housing unit to be defined in terms of attributes (quality and residential density). The quality of a given housing unit can be varied without adjustment costs, but changes in residential density on a particular site require prior demolition of the existing structure on the site. Producers of housing and consumers are assumed to be myopic. By assuming that the city is in short-run equilibrium at each point of time, it is shown that the pattern of land use observed at any stage depends on the past history of the city and the current rates of population growth. The possibilities for filtering of houses from one income group to another are also discussed briefly.
INTRODUCTION Until recently it was convenient to model urban land use in a context in which the housing consumed by an individual was a perfectly divisible flow of services which could be adjusted instantaneously and infinitesimally.2 The city represented in such models was a long-run equilibrium city (or equivalently, a city in which the entire housing stock was perfectly malleable). Recently there have been several attempts to go beyond this simple framework and allow for the durability and nonmalleability of housing in a spatial model (see Anas [ 11, Fujita [3, 41). Once it is acknowledged that significant adjustment costs are incurred in reshaping the existing housing stock long-run static analysis is no longer appropriate. The city at any point of time must necessarily be regarded as the outcome of a sequence of “short-run” equilibria. Furthermore, the time-honored ‘This research was completed while the author was visiting the Department of Economics, University of British Columbia. The author wishes to thank Yoshitsugu Ranemoto, David Riefer, Ngo Van Long, Frank Milne, Ted Sieper, Peter Swan Les Witchard, and an anonymous referee for helpful comments and discussion. ‘See Beekmann [2], Muth [6], Oron et al. [8], and Solow [lo]. 248 0094-l 190/80/020248-30$02.00/0 Copyright0 19&l by Academic Rms, All Ii&J
of reprocldon
in any form
Inc. r.zsmvcd.
NONMALLEABLE
CITY
249
equality between land rent and the marginal product of land in housing will no longer necessarily hold, because, as Anas points out, “at any one time only a small part of the urban area operates as the arena for competitive land bidding. The rest of the urban land is occupied by durable structures built in the past under different economic conditions.“3 It becomes clear that the pattern of land use predicted by short-run (or nonmalleable) models will no longer be a simple function of current parameter values (as it is in static long-run models) but will also depend on the previous history of the city.4 The main aim of this paper is to consider the implication for urban land use of nonmalleable housing in a framework which allows for (i) two income groups and (ii) demolition of preexisting structures as the city grows. This expansion of the opportunity set makes the problem considerably more complex, because, at any point of time, the owner of a building is faced with the alternatives of (a) retaining the building and the tenants, (b) retaining the building but letting it “filter” to the other income group, (c) demolishing the building and replacing it with a better one, (d) abandoning the building. From the point of view of the city as whole, the pattern of land use observed will depend on which of the above altematives is chosen at each location at each point of time. Obviously both the spatial distribution of the two income groups and the rate at which the city boundary expands will be sensitive to decisions of the above kind. This, in turn, will have implications for the form of the city’s residential density profile. To facilitate analysis and exposition we shall employ a hedonic formulation of the problem, assuming that both suppliers and consumers of housing perceive housing “quality” and residential density as separate attributes. This formulation will provide us with a convenient means of analyzing filtering of preexisting structures from one income group to another. We shall also assume that landlord/developers and households are myopic (in the sense that they have static price expectations)5 and that households are perfectly mobile. The city we are analyzing will be an open circular city (open in the sense of Wheaton [ 131). The outline of the paper is as follows. In Section 1 we derive the conditions satisfied by each income group’s rent offer function. Section 2 is devoted to the solution of the landlord-developer’s optimization problem. ‘Reference [ 1, pp. 66-671. ‘The idea that observed density profiles may be better explained as the outcome of a historical growth process is at the heart of a seminal paper by Harrison and Kain [5]. %hile it is clearly unrealistic to exclude foresight from the model, the assumption of myopia is probably no less appropriate than the alternative convenient assumption of perfect foresight. We shall be content here with identifying those aspects of the urban structure which are the outcome of myopic behavior.
250
NEIL VOUSDEN
In Section 3 our notion of short-run equilibrium is defined and the main equilibrium conditions are given. Section 4 analyzes the properties of the equilibrium. Section 5 is concerned with characterizing one possible growth pattern for the city and deriving conditions for the continuance of this pattern. Section 6 is devoted to a verbal discussion of some aspects of the model and Section 7 summarizes the main conclusions and suggests possible extensions. 1. THE CONSUMER-RESIDENTS
PROBLEM
First we assume that there are two distinct income groups in the city (rich and poor) with income levels Y’ and Y2, respectively (Y’ > Y’). Throughout we shall let a superscript denote the relevant income group. As foreshadowed in the Introduction we also assume that afl residents (actual or potential) of the city possess the same utility function, u = u( C, H, N),
where C = a nonhousing consumption good (assumed to be numeraire); N = residential density; u E C”) is assumed to be a strictly concave function of C, H, and N with uc > 0, u, > 0, u, < 0, H E housing quality; uHC
=
uCH
>
0,
u,,
=
u,,
<
0,
UCN
=
UNC
<
0.
Each resident is assumed to rent a single residential unit (which for group i will have attributes (H’, N’), i = 1, 2). In addition to selecting the amount of C he consumes and the attributes (H, N) of his housing unit the consumer selects his residential location, defined by the radial distance, X, from the center of the city. We assume that x does not enter into the utility function, but that the individual makes one return trip to the CBD daily and the transport cost per 2 units of distance is I (assumed constant and the same for both income group8). A consumer resident with income Y’ can be thought of as selecting C’, Hi, N’, and x so as to Max u(C’, Hi, N’) s.t. Y’ = C’ + R’(H’, N’, x) + tx,
(1)
where R ‘(Hi, N i, X) is the group i rental rate on a unit with attributes (Hi, N’) at distance x from the CBD. %hile this is obviously unrealistic, it is a natural outcome of ouy assumption that x does not enter the utility function. If all’that separates different forms of transport is money cost then everyone will opt for the cheapest one and so everyone will have the same unit transport cost. To put is slightly differently, the only reasonable way to have different transport costp for different income groups is to include leisure in the utility function as a normal good, and that goes beyond our simple purpose here.
NONMALLEABLE
251
CITY
Equivalently, the above optimization problem can be thought of as defining an equilibrium hedonic offer function for group i, R’(H’, N’, x).~ The necessary conditions satisfied by a solution to (1) are (i = 1, 2) R;; = u;;/u&
(2)
R,& = u;/u&
(3)
R; = -t,
(4)
Y’ = C’ + R’(H’, Equation
N’, x) + tx.
(5)
(4) implies R’ can be written in the form R’(H’,
N’, x) =f’(H’,
N’) + A’ - tx,
where A’ is a constant which (without arbitrary) may be expressed in the form R’(H’,
N’, x) = P’(H’,
loss of generality
because A is
N’) + t(x, - x),*
(6b)
where x0 is the radius of the city. We shall use form (6b) for R’ in what follows. Then we can rewrite the necessary conditions as Pjf = ur;/u;,
(7)
p; = u;/ u;,
(8)
Y’ - txxo= C’ + P’(H’,
N’).
(9
Note that expressing R’(H’, N’, x) in form (6b) is valid for given x,,, but that the solution for P’(H’, N’) will in general depend on the value of x0 and would therefore normally be expected to change as the city grows. We shall return to this question in Section 4. To complete the demand side of the model we need one further equation to reflect the fact that we are considering a city which is open (in the sense defined by Wheaton [13]; i.e., there is free mobility in and out of the city determined by whether the welfare level in the city is greater or less than the exogenously given level outside the city). Accordingly we add the condition u(C’, Hi, N’) = U’, (10) ‘See. Rosen [9]. sit is not gcncrally valid to express a hedonic price function in this separable form; the separability here is a consequence of the fact that x is not an argument in the utility function (or equivalently, that commuting is costly in money but not in time). I am grateful to Lea Witchard for suggesting the usefulness of this separability.
252
NEIL VOUSDEN
where 17’ is the level of utility (exogenously given).
obtainable
outside the city by group i
2. LANDLORD-DEVELOPERS PROBLEM In this section we shall specify the supply side of the city’s housing market. The city is assumed to consist of a number of sites of equal area. Residential density is defined in terms of the number (N) of residential units on a given site. The representative landlord-developer can choose both the number of residential units (N) in a structure on a given site and the quality (H) of each unit (H is the same for all units in a given structure on the site). Casual observation would suggest that within the framework of a given structure it is more feasible to change H than N. Existing units can be upgraded, etc., but adding additional units to a completed apartment building (for example) is not feasible beyond certain narrow limits.’ With this in mind, and in an attempt to capture the essential nonmalleability of housing without at the same time sacrificing the more malleable aspects, we shall adopt the simplifying assumption that for a given unit, H can be varied in either direction instantaneously” but that N can only be varied by demolishing the structure and rebuilding to the desired density. It is also assumed that the process of demolition-reconstruction can be effected instantaneously and without cost. The cost function associated with a residential unit with attributes (H, N) is of the form
+(H) +
u(N)“>
where 9’ > 0,
@” > 0,
Y’ > 0,
y” > 0.
We assume that + and y are sufficiently convex to ensure the existence of a unique solution in all the relevant maximizing problems. 90bviously there is the very real possibility of crowding more tenants into existing units, a phenomenon which is frequently observed in low-income housing areas. For the sake of analytical simplicity we shall abstract from this possibility here. “‘The author has worked through the case where adjustment costs prevent instantaneous adjustment of H to a desired level. The details of this more general case and the conditions under which it reduces to the simple static case studied here are contained in an earlier draft of this paper, which may be obtained from the author by request. ‘%mdially y(N) is some basic fixed cost associated with the construction of N units to some minimal level (H = 0). All productive effort beyond that point is treated as upgrading quality from zero to H (costing H(H)). Note that the cost of a structure is independent of the number of structures built. To this extent we are assuming constant returns to scale.
NONMALLEAI?LE
253
CITY
We shall also assume that landlord-developers are myopic and wish to maximize current profits. Then for a preexisting structure occupied by group i, and containing N units, the relevant optimization problem is Max Ni[ P’(H’, j, Hi
N’) +
x0 - x) - q$H’)
t(
- y(N’)].
(11)
The value of Hj which solves (11) for given j must satisfy
Ph(H’, N’) = +‘(H’).
(12)
In addition, in the choice of j, the landlord has the option of deciding which income group will occupy the structure. In the event that the structure passes from group i to group j (j # i), then what is commonly known as filtering has occurred. We formalize this in the following: DEFINITION 1. If 3 Hj (j # i): Ni[ P’(H’,
N’)
+ t(x,
> Ni[ P’(H’,
- x) - +(H’) N’)
+ t(x,
- y(N’)]
- x) - +(H’)
- y(N’)]
V Hi,
then a preexisting structure containing N’ units previously occupied by group i will be said to filter’= to group j. In view of our definition only preexisting units can filter. The problem to be solved by a landlord-developer deciding the best new structure to be placed on a vacant site at distance x from the CBD will involve the choice of quality and density as well as the choice of income group to occupy the structure. Specifically the problem is to Max N’[ P’(H’, i, H’,
N’) + t( x,, - x) - c$(H’) - y(N’)].
N’
(13)
The solution(s) to (13) must satisfy
Pf;(H’, N’) = #(Hi), Pi + t(x, Ni[ P’(H’,
- x) - &Hi) N’)
2 Nj[ P’(Hj,
(14) - y(N’)
= N’[y’(N’)
+ t( x,, - x) - +(H’) N’)
+ t(x,
- P;],
(15)
- y(N’)]
- x) - +(H’)
- y(Nj)]
V (H’, N’),j
= I, 2.
(16)
Note that (15) implies a positive value for N’ if and only if the proposed “For a rigorous treatment of filtering in a nonspatial setting see Sweeney [I 1, 121.
254
NEIL VOUSDEN
new structure earns positive profits. We shall assume that the appropriate second-order conditions for a maximum are satisfied for problem (13) (which amounts to assuming + and y are sufficiently convex and makes no assumption about the form of P) and for given P’, i, x,,, x, we denote the resulting unique solutions to (14) (15) as (fi’, fi’). Because even a preexisting structure can always be instantaneously adjusted to the values (@, fii) if housing is perfectly malleable, we introduce the following terminology to facilitate comparison of our model with the malleable case. DEFINITION2. (i) The unique value (fi’, $) satisfying (14) and (15) for given i, Pi, x,,, x is termed the malleable optimal structure for group i at x associated with given (Pi, x0); (ii) The unique value of (ii, Z?) satisfying (14), (15), and (16) for given pj 0’ = 1, 2), x0, x is termed the malleable optimal structure at x associated with given (P’, x0) (j = 1, 2). It remains for us to specify the conditions under which it will be optimal to demolish a preexisting structure with attributes (iY’, N’) (where Hj satisfies (12)) and replace it with the malleable optimal structure (Ai, Ai). Such demolition-reconstruction will occur if and only if Ay Pi@,
Ai) + t(x, - x) - +(z?)
- v(Ai)]
2 Nj[ P’(H’,
IV’) + t(x, - x) - +(H’)]. (17)
3. EQUILIBRIUM While we can define what we mean by short-run equilibrium it is not possible to immediately present all of the equilibrium conditions. In particular the condition that all of the city’s population (both income groups) be housed can only be formulated after we know where each income group is located. But, as we have suggested, this, in itself, is the outcome of the city’s past history and thus the demand = supply condition can only be given explicit form after we have either discovered or assumed something about the city’s previous pattern of growth. For the time being we content ourselves with the following definition: DEFINITION 3. A short-run equilibrium for the two-class, nonmalleable open city consists of a consumption-housing profile for grq i (C’(x), H’(x), N(x)) (i = 1,2), a hedonic price function Pi (i = 1, 2), and a value for x,, (the city boundary), such that: (i) all individual consumers and producers are in equilibrium; (ii) site rents are zero at x,, (competitive land and housing markets); (iii) supply of residential units for group i in the city = demand for residential units in the city by group i.
NONMALLEABLE
255
CITY
Requirement (i) of the above definition is seen to be satisfied if the following conditions are fulfilled (see Sections 1 and 2) at x: f(C’, Hi, N) z $(Ci,
Hi, N) = Pk(H’,
N) = +‘(H’),
g(C’, Hi, N) E p,
Hi, N) = Ph(H’,
N) $ y’(N)
- P’(H’,
i = 1,2,
(18)
N) + t(x, - x) - c#J(H’) - y(N) N
I
>ifNfi’
) (19) I (20)
u(C’, Hi, N) = ii’, Y’ - tx, = C’ + P’(H’,
(21)
N).
In the above equations_ y is historically given unless the structure is new, in which case: (a) N = N’, (b) N is endogenous, (c) there is equality on the RHS of (19). In addition, from (11): A preexisting structure (Hi, N) satisfying the above conditions will be occupied by a resident of group i if and only if N[ P’(H’,
N) - +(H’)]
2 N[ P’(H’,
N) - cp(H’)]
(22)
V Hj, j = 1, 2.
Also, from (16): A new structure at x will be occupied by group i if and only if fii[ P(A’, 2 iq
9)
+ t( x0 - x) - +(zF) - y(e)]
P’(z?,
fij) + t( x0 - x) - +(Z?) - y(tij)],
j = 1,2, (23)
where (Z?, @ represents the malleable ture for group j at x (j = 1, 2).
optimal
struc-
Finally all new structures replacing preexisting structures must satisfy a demolition condition of form (17). Conditions (17)-(23) complete the specification of equilibrium for individual suppliers and consumers. It is worth noting that the possibility of strict inequality on the RHS of (19)
256
NEIL VOUSDEN
reflects the discrepancy between land rent and the marginal product of land in housing which we would expect to find when the land is being used either more or less intensively than is “optimal” (“optimal” being used in the sense of what would be optimal if it were possible to put a new structure on the site without incurring any adjustment costs). In a growing city we would generally expect to find preexisting structures using land less intensively than is optimal, in which case we would observe the first inequality in (19) (i.e., the marginal product of land in housing < profits/unit land). We also note at this stage that if housing were perfectly malleable then the malleable optimal structure will always be found at each x and there will be equality on the RI-IS of (19). Then our model reduces to the static long-run model I la Muth and Solow. Requirement (ii) of our definition of short-run equilibrium is expressed as Mfx+(x,))[
P’(+(xJ,
ii(
- @(x0))
- @(x0))]
= RA (24)
(where ($(x,,), Ai( satisfies (18)-(20) and (19) with equality at x0>. Thus, if there is housing for group i at the city boundary (xg), then it will have attributes (Hi,, ZV,$ satisfying AC& Hi,, N;) = $(HA), g( C;, Hi,, A$) = y’(N;)
(25) - RA/ (N;)‘,
u( C;, H;, IV;) = ii’.
(26) (27)
Our assumptions on the utility and cost functions ensure that (c,$ HA,,N,$) = (ei(xO), A’(x,J, #(x0)) will be unique for given U’. Finally, as we have already noted, requirement (iii) of the definition cannot be formulated explicitly until we know the city’s past pattern of growth. Nevertheless we can say a great deal about the properties of a city in short-run equilibrium without writing down an explicit equation for (iii). The next section is devoted to deriving some of these results. 4. EQUILIBRIUM In what follows we shall let
GROWTH
PATTERNS
r’(N, x) E P’(H’, N) + t(x, - x) - @(Hi) - y(N)
for given x0, Pi, where (Hi, N) satisfies (18)-(20), and let P(x) E P’(A’,
Ai) + t(xo - x) - c#@) - y(e)
NONMALLEABLE
257
CITY
for given x,,, where (Z?‘, I?‘) satisfies (lS)-(20) can now prove:
with equality
in (19). We
LEMMA 1. For a city in short-run equilibrium: (a) -f--[j+(x)-
fi(x)]
(b) -$ [ AI(
i = 1, 2,
< 0,
9(x*) - 3*(x*). i*(x*)] > 0,
where x* is a value of x for which i+(x) Proof.
(a) By definition
- i’(x)
= C(x).
of (I?‘(x),
F’(x).
Z?(x)):
2 [i+(x) - ii(x)] = -k’(x)
< 0.
Now, at a point x* where A’(x*) the following conditions
* 9(x*)
= z?(x*)
* i’(x*)
(28)
must apply for each income group:
f(C, ii, A) = qqfi), q y’fi)
= Y - c - tx* - c#@) - y(A).
- g(& ii, fi)]
Also, from (27) A(Y-Ltx*-C+(B)-y(fQ)=k, where k is independent w.r.t. Y, we obtain
of Y. Differentiating
fc 1 dN -=dY
l-I&
f, - +"
0
f f-N&
1
A
1
the above three conditions
evaluated at x*,
(29)
258
NEIL VOUSDEN
where
fc AZ
l-&c 1
f, - v f-i&, f
fN 2y’ + icy” - g - I?gN < 0.
g
The RHS of (29) is negative. Hence Z?‘(x*) > fi’(x*) g
[if ‘(x*) * 9(x*)
(30)
- $2(x*)
It is now easy to prove the following malleable housing.
* ;‘(x*)]
and we obtain > 0.
Q.E.D.
result for the case of a city with
PROPOSITION1. If housing is perfectly malleable then when the ci@ is in short-run equilibrium, 3 x* E (0, x,-J such that for x < x* all housing is consumedby the poor and for x > x* all housing is consumed@ the rich. Proof. If housing is malleable, then in equilibrium there must exist an E*. such that j?‘(x*~i’(x*) = s2(x*)F2(x*); otherwise, by the continuity of N’(x)?(x) either N’(x)?‘(x) > A2(x)i2(x) V,, or vice versa, and one of the two groups would be unsupplied. Proposition 1 is then a trivial consequence of Lemma l(b). Q.E.D.
Proposition 1 is a generalization of a result obtained by Muth [6] and Beckmann [2], who found that in long-run static equilibrium, given the constant unit transport cost assumed here, the rich would choose to live further from the city center in lower-density housing and the poor would live in the inner areas in higher-density housing. In our model there is the additional possibility of using higher quality to compensate for the higher density of inner locations and it is interesting to note that the traditional results are robust to this generalization. Proposition 1 will serve as a basis for comparing our results for the nonmalleable city with results of traditional models. Now let us note that the following is a trivial consequence of (25)-(27): LEMMA 2. Zf the city is in short-run equilibrium, then Ci E ei(xO), Hh E Z?(x,), Ni s i?(x,,) are independentof x,, for given ii’.
Obviously if (CA., Hi, Ni) is x,-invariant, then so is P’(H& Ni), for (provided there is housing for group i at x,,) it is given by the zero profits condition Pi(H;,
N;) = R,/N;
+ @(Hi) + y(N;).
NONMALLEABLE
259
CITY
In order to make P’(H& NA) x,-invariant we shall make the following assumption. Assumption A. U’ remains constant as the city grows. This assumption is made for two reasons. First, it is a reasonable approximation to reality in certain specific cases (such as cities in less developed countries where the city is surrounded by a rural hinterland at a subsistence level of ii’). Second, and more to the point, we are concerned with isolating those features of urban equilibrium which are independent of changes in parameters outside the city. It should then be an easier task to determine the role of changes in these parameters in the growth process. In particular (as we shall see) constant U’ leads to one pattern of growth which proceeds entirely through demolition-reconstruction without filtering of residential units from one group to the other. For such a growth pattern the phenomenon of filtering is entirely due to the price effects (changes in P’(H& A$,)) caused by changes in ii’. We shall denote the x,-invariant value of P’(H& Ni) as P; z P’(H&
Equilibrium
P’(H’,
(where a’(H’, P’(H&
N;).
N) will satisfy the partial differential P;; = f(“i(H’,
N), Hi, N),
Pj, = g(a’(H’,
N), Hi, N),
N) is the inversion
i=
equation
1,2
of (20) for Ci) with initial
(31)
condition
N;) = P;.
To simplify matters we make the following additional assumptions. B. The partial differential equation (31) has a unique solution.*3 Assumption C. The city is in short-run equilibrium at each point in time. Assumption
Assumptions
(A)-(C)
are assumed in all of what follows. We have:
LEMMA 3. Equilibrium P’( H i, N) occupies some housing at x0.
is x,-invariant
as long as group i
The following results are also useful. “To see that this assumption is not vacuous we need only check the following u’ - A’(CaHb/NC), A’, (I, b, c constants. It is easily seen that for this utility has the unique solution pi(~i,
N)
= [ pi + (pi)‘/“.
(~6)~“1”.
(N;)‘/“]
- ($)I/“.
(Hi)-b/a.
simple function
NC/O.
case: (31)
260
NEIL VOUSDEN
PROPOSITION 2. Residential units constructed for group i with attributes (Hi*, N’*) will retain those attributes until demolished as long as they are occcupied by group i. Proof. First, let us observe that all preexisting units which were built with attributes (Hi*, N’*) must have density N’* until they are de molished. C’ and Hi at all locations x where N = N’* must satisfy f(C’, Hi, N’*)
u(C’, Hi, N’*)
= #(Hi), = ii,
which are x- and x,-invariant. Given our convexity assumptions on ui, +, and y, Hi* solution of the above equations for Hi.
is the unique Q.E.D.
LEMMA 4. The malleable oftim$ structure replacing an (H j, N) structure will have unique attributes (Hj, NJ) which are invariant w.r.t. x0 provided both groups are occupying some units at xW ProojIfi, Lemma 3 guarantees the invariance Now, (HJ, NJ) must satisfy
of Pj (j = 1,2) w.r.t xe.
where x* is the value of x at which replacement
is taking place. Also,
I?[ Pj(z?,
I?) + t( x0 - x*) - (p(rij) = Ni[ P’(H’,
Differentiating
- y(fij)] N’) + t(x, - x*) - +(H’)].
both sides of this equation gives t(fij
- N’)(dx’
- dx,) = 0.
It is routine that I?j > N’; thus kc, = uk, and (x0 - zt) is, independent of the value of %T ‘DJUS the system of equations for (HJ, NJ) is independent of x0 and (NJ, NJ) is similarly independent. Q.E.D. PROPOSITION 3. At all stages of the city’s growth there are on& three possible growth patterns for new housing: I. All inner redevelopment is occupied by the poor; new peripheral housing is occupied by the poor and the rich.
NONMALLEABLE
261
CITY
II. Inner redevelopment is occupied by the poor and the rich; new peripheral housing is occupied by the rich. III. All inner redevelopment is occupied by the poor; new peripheral housing is occupied by the rich. ProojI
If the poor occupy new peripheral
?‘(xo)A2(xJ
housing (i.e., at x0):
2 i’(x,)z?(x,).
Lemma l(b) can then be invoked to show that all inner redevelopment housing will be allocated to the poor. It is then easy to see that I-III above exhaust all the possible cases. Q.E.D. Note that a simple consequence of Proposition 3 is that group 1 must always occupy some housing at x0. This combined with the assumption of competition implies
N@“(Kj,
Nd) - +(f&j) -
u(Nd)]
= RA.
(32)
In what follows we shall have reason to be particularly interested in the first of the above patterns. There are two reasons for this. First, our assumptions that t is the same for both groups and that U’ is constant as the city grows imply that the pattern of land use resulting from uninterrupted pattern I growth is relatively simple and regular. Second, and more important, is the fact that pattern I growth implies a land-use pattern markedly different from that predicted by existing models and we are therefore interested in obtaining conditions under which it would be the prevalent growth pattern. Accordingly we derive the properties of a city which grows according to pattern I. 5. PATTERN
I GROWTH
PROPOSITION4. Zf pattern I has prevailed at all times since the city’s inception up to and including time r, then no filtering can occur at r.
Proof.
See the Appendix.
The proof of Proposition 4 does not depend on the x,-invariance of Pi. It does, however, depend on (a) the invariance of (Hi, N,$ (i = 1, 2) as the city expands (which is in turn a consequence of constant i?), (b) the fact that t is the same for both groups. The intuition of the proof of Proposition 4 is illustrated as follows: If one income group were to outbid the other group for an inner (Hi, N$ structure then because of (a) locational equilibrium, (b) the identical slopes of the rent profiles (- t), and (c) the invariant nature of new dwellings at
262
NEIL
VOUSDEN
the boundary, such a possibility implies that the same group could also outbid at x0 and everywhere else for that matter. The assumed prevalence of pattern I would then be contradicted. The intuition behind the rest of the proof is similar. The following result is then a simple consequence of Propositions 2 and 4. PROPOSITION 2a. Zf pattern I prevails at all times -T E [0, r*] for some r*, then residential units constructed with attributes (Hi*, N’*) at some time 7 E [0, r*] will retain those attributes V r E [0, r*] until they are demolished.
We now establish structure.
the “timing”
of the demolition
of a preexisting
PROPOSITION 5. Assume the city grows according to pattern I at all times after its inception. Then for a structure with attributes (Hi, N’) constructed when x,, = xz, 3 a unique X0 > x$ + e s.t. when x,, > &, it will be optimal for landlords to demolish th_e (ZZ!(, N’) strycture and replace it with the malleable optimal structure (HJ, NJ), where NJ > N’. Such a demolition will not be optimal for x,, < &.
Proof.
See the Appendix.
Proposition 5 tells us simply that once a building has been constructed on a site, a more than infinitesimal (and generally quite large) increase in x0 will be needed to induce demolition and replacement by the malleable optimal structure on the same site. This is because the commitment implicit in the construction cost of the preexisting structure is a bygone (here included as a sunk flow cost, NY(N)) and x0 must rise sufficiently to push up rents to the point where it is worthwhile to incur another such commitment. We also prove: LEMMA 5. As x,, increases while the city is growing according to pattern I, land rent from an (Hi, N,$ structure at x ( < x0) is greater than and increases more than that from an (Hd, NJ) structure at x. In addition, if an (Hd, N,‘) structure satisfies the demolition condition (17) with equality at x, then it is not optimal for the landlord to demolish the (Hi, Ni) structure at x at that time. Proof
See the Appendix.
The following proposition is an easy consequence of the above results and describes the form of a city which has grown without interruption according to pattern I. PROPOSITION 6. Assume that the equilibrium nonmalleable open city with two income groups grows V r 2 0 according to pattern I. Then it will consist
NONMALLEtAEJLE
263
CITY
of a series of concentric rings with radii 0 = <
XZ(n+l)
5 X1(n+l)
. * *
<
X2n
<
Xln
<
X2(“--1)
<
Xl(n-1)
such that (a) within the ring (x~(~+~), xzi] housing has attributes (Hi, N$ (i = 19 * * * 9 n), where (Hi, Ni) is the malleable optimal structure for group 2 at x2i; (b) within the ring (xlCi+,), xii] housing has attributes (H& Nh), where (HE, NE) is the malleable optimal structure for group 2 at xii; (c) within the ring (x2t, x0] housing has attributes (Hi, N$, and within the ring (x,,, x0] housing has attributes (Hd, N,$. >N;,-,) >N:,,-,, >-->N; >N; (d) N&+1) >N,z, >N:, > -.- >N;>N;. Thus, the “pattern I city” consists of a sequence of successive replacements of (Hd, Nd) structures overlapping with a corresponding sequence arising from (Hi, Nt) structures-in any interval (xlCi+ij, x2], for example, there will be two types of housing (each occupied by group 2) with attributes (H& Nk) and (H& N$ respectively. Having obtained a complete picture of the structure of a city which has been growing according to pattern I, we are almost in a position to establish conditions under which such a growth pattern will be observed. First we introduce the following notation, consistent with that used in the statement of Proposition 6. This notation, in addition to being essential in the proof of Proposition 7, will be useful in formulating requirement (iii) of our equilibrium definition in a dynamic context.
W:‘,, x2i(r)
xii(r)
B&(7)
Niol
Attributes of structure occupied by group j, which is a replacement for a structure with attributes (H& N$ (i = 0, . . . , (n - 1);j = 1, 2) W,zs N,z) Attributes of structure occupied by group j, which is a replacement for a structure with attributes (HE, N$ (i = 0,. . . , n,j = 1,2) (Nd, Hi> Point of demolition of (H& ,), N&- ,> structure at time 7 (i = 1, . . . , n) Point of demolition of (H&-,), N&-,,) structure at time 7 (i = 1, . . . , (n + 1)) Number of new structures with attributes (H&+,), N&+ ,$ built at time 7 (i = 0, . . . , (n - 1); j = 1, 2)
264
NEIL
B:‘iW B&d 4it7)
Dli(?)
F2i(7)
Fli(r)
Wj( WI
7)
VOUSDEN
Number of new structures with attributes (H{(i+i), A$+,,) built at time r (i = 1, . . . , n;j = 1, 2) Number of new structures with attributes (Hi, Ni) built atx,attimer(j= 1,2) Number of structures with attributes (Hi, Nii) demolished at time 7 (i = 0, . . . , (n - 1)) Number of structures with attributes (Hi, NFi) demolished at time r (i = 0, . . . , n) Number of (Hi, Ni) structures filtering up to group 1 at time 7 (i = 0, . . . , n) Number of (H& Nt) structures filtering up to group 1 at time 7 (i = 0, . . . , (n + 1)) Population of group j at time r (assumed exogenous)14 Proportion of sites at x originally occupied by (Hd, ZVd) structures
Now, noting that the malleable siie;rent functiorr’?(x) . i’(x) will depend on x0 and hence 7, and writing N’(x)?‘(x) E (N’ * i’)(x, T) to reflect this we prove the following result, which assures us that in a city which is growing continuously a la pattern I there will be no jumps in the malleable site-rent function. LEE 6. If pattern I has prevailed at all times r E [0, r*] for some P, then (N’ - i’)(x, 7) is continuous at r*, if W-~(T) is a continuous monotonically increasingfunction of 7 (i = 1, 2;j = 1, 2). Proof: See the Appendix. While the proof of Lemma 6 is somewhat long, the idea is simple enough. The zero profit condition means essentially that the malleable site-rent profile (($ * F’)(x)) is the lowest such profile consistent with housing all of the city’s inhabitants. The proof essentially shows that a downward shift of the (A’. i2) profile in the time interval [T*, r* + T)] would imply that a lower such equilibrium profile would be feasible at r*, in contradiction to the definition of the profile at r*. On the other hand, the (&’ +F2) profile is bounded above by (8’ - i’) in the interval [T*, T* + s] in a way which limits any upward shift of the (fi2. $) schedule and ensures continuity at 7*. Lemma 6 can be used to obtain the following result: “Assuming an exogenous population for the city would seem to run counter to Wheaton’s definition of an “open” city. However, in both cases “open” means that movement into and out of the city is free so that the utility level in the city will be equated to that enjoyed elsewhere. It is immaterial whether we assume that the level of income in the city rises, thus attracting immigrants, or whether we set it up as a problem in which the population growth is taken as given and the level of income (Y’) required to induce that growth is endogenous.
NONMALLEABLE
CITY
265
LEMMA 7. If pattern I has prevailed at all times r E [0, r*] for some r* then pattern I cannot switch into pattern II at r*.
ProofI
See the Appendix.
So now we know that the only switch possible from pattern I is to pattern III. We are now in a position to derive conditions for the continuance of pattern I. To make the subsequent analysis more tractable we shall confine our attention to the case where P,‘(T), x0(7), xii(r), xzi(7) are differentiable functions of 7 at r*, which is not unreasonable since we have at least shown them to be continuous. PROPOSITION 7. Assuming that Wj(r), P,‘(r), x,,(r), xii(r), xzi(r) (i,j = 1, 2) are differentiable functions of r at r*, and that the city has grown according topattern I V r E [0, r*] the city will continue to grow according to pattern I at r* if and onb if
dW’
A
dW*
-*------xdr xolv;
where
+
ProojI
iso
CNGi+l)
-
NZfd)x2(i+1)(1
-
6(x*(i+1))
Seethe Appendix.
COROLLARY. Assuming that w(r), P&r), x0(r), xii(r), xzi(r) (i,j = 1, 2) are differentiable functions of r the city wilI grow according to pattern I VrE[O,r*]ifandonbif Vr E[O, r*].
Proof. Noting that the condition is trivially satisfied in the initial stage of the city’s growth before any demolition takes place (since then A = 0), we also note that pattern I is in any case the only possible pattern in the predemolition state. The result then is a trivial consequence of Proposition 7. Q.E.D. Proposition 7 and its corollary give us a simple condition showing how the growth pattern of the city is dependent on the relative population
266
NEIL VOUSDEN
growth of the two groups (dW’/dr and &V2/dr) and the historical land-use pattern in the city (as reflected in the values of A(r) and 8(x,,(r)). In the next section we shall make a few more detailed observations on various aspects of the growth process implicit in Proposition 7. 6. DISCUSSION
OF THE
MODEL
(i) Population Growth
The manner in which the outcome depends upon relative population growth is simple enough. The larger the growth rate of the poor population relative to the rich population, the greater the likelihood of pattern I. This is because a rapid growth of the population of the poor will require some of the new residents in that group to locate at the urban periphery in order to induce demolition-reconstruction at inner locations for the purposes of housing the rest of their number. In the polar case where the population of the rich stays constant and the population of the poor grows, new construction of houses for the poor on the periphery is both necessary and sufficient for induced replacement housing for the poor on inner sites. (ii) Previous Histoly of the City
The manner in which the pattern at r depends on the city’s history up to r is somewhat complex, but the mechanism of the dependence may be better understood if we look at the case where (Hd, Nd) structures are being demolished, but it is not yet economic to demolish any (H,f, N,f) structures. Then the condition for pattern I reduces to
-. In such a case the lower the value of @(xi,), the greater the likelihood of pattern I. This is explained as follows: the lower the historically given proportion of sites at x,, originally occupied by the rich, the fewer the rich who will be unhoused by the replacements at x,,, so that more poor must be housed at the edge of the city because (a) there are fewer rich to augment housing at the periphery and thereby induce demolition-reconstruction on inner sites by pushing up t(x, - x); (b) there is less demolition of housing induced by a given increase in x0 because there were fewer (H,‘, N,‘) structures at x,, to start with. (iii) Population Density
It is certainly clear that in the event that pattern I predominates, it is possible to have a density profile which differs from the standard downward-sloping one. The possibility of an upward-sloping profile is particularly strong in the city’s predemolition phase; for example, if W’ and W2
NONMALLEABLE
267
CITY
are each growing at a constant rate then the density profile will be upward sloping if the growth rate of IV2 is higher than that of W r. In general, when pattern I is followed, the fact that group 2 is housed at the periphery will tend to mean at least a flatter, and possibly an upward-sloping, density profile. (iv) Spatial Distribution
of Income
Groups
We have shown that there are perfectly reasonable conditions under which pattern I will be the predominant growth pattern for a city. If those conditions are satisfied then we would expect to find a spatial distribution of different income groups which is very different from that predicted by long-run models (see Beckmann [2] and Proposition 1 in this paper). Rather than having the poor entirely accommodated in crowded housing near the center and the rich in low-density housing further out we find that the poor may be distributed fairly evenly throughout the city with a possible concentration at the periphery if there is a sudden large rise in their population. This is the sort of pattern we observe in many cities in less developed countries, where a steady large influx from a subsistence rural sector is accommodated in shantytowns at the edge of the city.” It is worth noting, however, that one element in the spirit of preexisting results carries over here. Despite the possibility that some of the lowerincome group may occupy peripheral land, there is no pattern of equilibrium growth in which this income group does not occupy the inner land. (v) Filtering We have shown (Proposition 4) that pattern I proceeds without filtering (that is, provided it proceeds without being interrupted by pattern II and/or pattern III), and we have observed that this uncomplicated growth is the result of assuming constant Pi and identical t for both groups. It is possible, however, to envisage a city in which U’ is steadily rising and iz is constant or falling. I6 Then one would expect that: (a) pattern I would be less likely, with the rich opting for more spacious accommodation at the periphery (and thereby inducing more inner redevelopment for the poor) and the poor accepting higher-density inner housing; (b) rising U’ would imply that rich people occupying inner housing would vacate it in favor of more spacious peripheral housing and the vacated inner housing would filter down to the poor. ‘%uth [7j has commented on this possibility model. 16Anas [l] considers the effect of variations and a single income group.
in the context
of a nonspatial
in ii for the case of infinitely
vintage
housing
durable
housing
268
NEIL VOUSDEN
Despite the obvious technical difficulties involved, it would indeed be interesting to analyze this case. It is misleading, however, to suppose that our own assumptions have completely ruled out the possibility of filtering. Indeed, it is a very real possibility for either of patterns II and III. Since Pi will be falling for some interval of time after a switch-out of pattern I, we have the possibility that for some ~a, 3 H’:
Y(&?)] = N,z[P2(H;,N,z,70)+ f(+,(d - x) - cp(H,z) - Y(%)]
N,2[P’(H’, Wf) + f(&,)
- x) - W’)
-
vx
with the (Hi, iVt) structure filtering up to group 1. This would occur because the site-rent function for group 1 is being pushed up sufficiently in excess of that for group 2 to induce upward filtering. It would be primarily the result of relatively rapid growth in the group 1 population and is a phenomenon frequently observed in modem cities where there is a rapid growth in the size of the middle/upper class. When pattern III is operative there is the additional possibility that redevelopment structures occupied by the rich (with attributes (H,& NjJ (k = 1, 2) will subsequently filter down to the poor. 7. CONCLUSION The purpose of this paper has been to generalize existing circular-city land-use theories to allow for the nonmalleability of the housing stock. A model of an open city with two income groups was developed in which the housing stock was assumed to be durable, but housing units could be demolished and replaced by new structures which utilize inner land more efficiently. Such replacement involves substantial adjustment costs (the sacrifice of the cost sunk in the existing building) so that for a time land will be used inefficiently. Land will only be used efficiently (with land rent equal to the marginal product of land) for newly constructed buildings. Not surprisingly it was found that, if certain not unreasonable conditions were satisfied, the pattern of land use (in particular the density profile and the spatial distribution of income groups) observed in a nonmalleable city would differ markedly from those predicted by long-run static models. This paper has proceeded under the assumption that the utility level of each income group is exogenously fixed. While this assumption has great analytical value in enabling us to isolate that part of the growth process which is independent of exogenous factors, it is equally important to explore the effects of changes in U’, especially given the probable implications of such changes for filtering. We have also assumed that landlords are myopic, and obviously extensions of this model should pay some attention to the role of foresight in the
NONMALLEABLE
269
CITY
growth process. Finally it is important that the model be placed in a normative context so that policy questions arising from nonmalleability in cities may be properly defined and discussed. APPENDIX Proof of Proposition
4
Since all redevelopment housing must be occupied by the poor under pattern I there is no possibility of downward filtering of inner redevelopment units (unless of course it occurs after some upward filtering). We therefore have to rule out upward filtering of these units: To this end suppose that a redevelopment structure built at x for group 2 with attributes (H*, N*) (as a replacement for a structure with attributes (H*, N*)) is occupied by group 1 at some later time at which its attributes change to (H’, N*). Lemma 4 tells us that all replacements for structures with the same attributes will themselves have the same attributes. Then the replacement for an (H*, N*) structure built at time 7 (at, say, xi) will also have attributes (Hz, N*). Now, we must have N’[
P’(H’,
N2) + t(x, > N’[
- x> - $4~‘) P*(H*,
- Y(N2)]
A+) + t( x0 - x) - W2)
- VW*)].
This implies N’[
P’(N’,
N2) + t( x0 - x1) - cpW) > N”[ P2(H2,
Iv)
-
Y(N2)]
+ t( x0 - $1 - W2)
- YW3]
and group 1 could outbid group 2 at xi. This contradicts the assumption that pattern I is ruling at time 7. Thus upward filtering of redevelopment structures is ruled out. The only other possibility of filtering arises in the case of the (Hi, ZVA) structures at x < x0. Let us therefore suppose that a structure built at x for group i with attributes (H& NA) is occupied by groupj (j # i) at some later time at which its attributes change to (Hj, Ni). For this to happen, clearly we must have
> N;[ P’(H;,
iv;) - &Hi)
- y(&)]
> RA
(i = 1, 2)
so that group j could outbid group 1 at x0, by opting for units with attributes (H’, Nh). But then either (a) u(Cj, H’, Nh) < ii-j or (b) the
270
NEIL VOUSDEN
uniqueness of (Hi, N&) is contradicted
(since Nd # N,$ in general).
Proof of Proposition 5
For given x, define V’(xo> as y’(x,)
s I+[ pq9,
f+) + t(x, - x) - +(I?9 - Y(+)]
- Ni[ P’(H’, N’) + t(x, - x) - +(H’)].
Now, Lemma 3 together with (18) and (19) implies dl/‘/dxo Noting
G t(tij
- N’).
that V x0 C$ + P’( H& N$) = cj + P’(lj’,
we can differentiate
fij),
system (18)-(20) and use Lemma 2 to obtain dl? 1-s dxo
t(fh - f&f
- ry(Gj)) A
3
where A is as defined in (30). Thus we can write vi(xo)
= p(xo) - N’tx, + <(Hi, N’),
where /3(x,) s r;i[ ~j(lj’,
+)
+ t(x, - x) - +(I@) - y(W)]
and [(Hi, N’) E Ni[ P(H’, N’) - tx - $@‘)I.
From the above j3’ > 0, p” > 0. Also, (a) V’(xo*) = - Niy(Ni) (b) dYi/dxo = 0
>o (c) d2Vi/dx;
< 0;
when x0 = xg when x0 > x0*;
= p”(xJ
> 0.
Q.E.D.
NONMALLEABLE
271
CITY
It then follows that 3 g)
> x0* +
z s.t.
d%o) dxo
> o
,
xpx.
so X0 is unique for giyen xg (or for given (H’, N’), which amounts to the same thing). Clearly NJ > N’. Q.E.D. Proof of Lemma 5 { N,z[ P2(H;,
N;)
+ t(x,
- { N;[ P’(H,‘,
N;)
- x) - +@,z) + t(x,
- Y(%)]
- xl - +(ffd) = t(x, - x)(N,z
(since Ni > N,-j). The derivative Also, [N,2[ P2(H,z,
N;)
+ t(x,
- h}
- Y(%)] - N;)
- RA} > 0
for x < x0
of this expression w.r.t. x,, is also > 0.
- x) - $@;)I
-Nd[
P’(Hd,
N;)
+ t(x,- x) - +W)l I = 0, - x)(N,z- Nd)+ No%(%)-
N;Y(N;) > o,
so that if the demolition condition (17) for the (Hi, Nd) structures is satisfied with equality then it is not satisfied for the (Hi, Ni) structures. Q.E.D. Proof of Lemma 6
We begin by proving that x0 is a continuous function of T (it is trivial that it is monotonically increasing). We know that”
“This
inequality
simply
says that the number
edge of the city ~lKkbh)>2 population
(I( d(7,)
-
W’(Q)l).
- (~&JYll)
of new residential units for group i at the 1s . no greater than the increase in the city%
272
NEIL VOUSDEN
Now continuity
of W’ * V c > 0, 3 6:
Now (Z?’ . i’)(x, 7) is easily seen to be continuous in T since it is continuous in x0 (because of the differentiability of the functions and the invariance of P’). (Z?’ * i’)(x, 7) can be obtained from the system P’(ri2,
A’)
+ e* = p*(L?,2, N,z) + c;
(from W)),
f(t?,
ii*, 3’) = P;(I?*,
I?‘) = r#i(ri’),
g(P,
22, 9)
fi’)
= P;(I?*, - P*(ii*,
= y’(Af2)
fi”)
+ f(X&)
- x)g@)
- y(i?)
fi’ up,
ii*, iv)
7
= ii*,
where (C,“, Hi, A$) is the solution to (25)-(27). Thus (I?*, i*) will be continuous in r if it can be shown that P:(T) E P*(H,f, Ni) is continuous. We let P;(T) E P’(H& Ni), i = 1, 2, and define a’(q, x) 3 N(g P@
+ 77)+ t(x,(7*
+ 77)- x) - 4@;)
- Y(Nli)], n 2 0.
We will show that (Y* is continuous at n = 0. First we show that a*(~, x0(7*)) 2 0. Suppose otherwise for 11 > 0. Then, if P*(H*, N*, r) denotes the equilibrium for group 2 at time 7 we have that: (i) Since P*(H*, A’*, T* + 7) satisfies equilibrium (7* + 7) we can construct P**(H*,
N*, T*) = P*(N*, N*,
7*
hedonic price function
+ 7) + t(x&*
conditions (18)-(20) at + T/) -
x0(7*))
(33)
and it is easily shown that this price function will satisfy conditions (18)-(20) at 7*. (ii) The population of group 2 at (7* + 9) is entirely housed at locations x0(7* + q < x0(7*) since a*(~, x0(7*)) < 0 * a*@-/, x) < 0 v x E [x0(7*),
41.
NONMALLEABLE
273
CITY
Therefore the price function P2* we have constructed (Eq. (33)) will house the entire population of group 2 at time r* within the boundary x0(7*) (since this population is smaller than that at (r* + 77)). (iii) C,Z + P2(H,2, N,Z, 7* + 17) + t(xO(,* + 77) - x0(7*)) + txxo(7*) < C,z + P’(H,z, N;, T*) + tx,, = Y2(7*) (since we are assuming a’(r), x0(7*)) < 0 for n > 0, whereas (~~(0, x0(7*)) = 0). Thus price function P2* given by (33) is an equilibrium price function for r = T* which requires a lower level of Y2 for its realization (or equivalently will yield a higher ii2 for the same Y2). It does, however, imply site rents at x0(7*) at time T* equal to (~~(77,x0(7*)), which we have assumed negative. This contradicts the prevalence of pattern I at r*; :. a2(q, x0(7*)) 2 0, Now we know from Proposition
-q 2 0.
3 that
a2hxl@ + 1)))5 ayq,x()(7*+ 17)) and this, together with the above result, implies
0 5 a2h x,(7*)) 5 a’(99 x&*)) and this, together with the continuity of (Y’ in n, is easily seen to imply that (~~(7, x0(7*)) is continuous at q = 0. It is a trivial consequence of this that P,‘(T) is continuous at 7*. Q.E.D. Proof of Lemma 7 When pattern I is iuling at 7*:
$K fi2 - i’)(x, T*) - (I? *9)(x, 7*>] > 0, since i’(x) SE
> Z?‘(x).” min [x,1(7*),
X0(7*)1
Now, define
(
$2. i’)(x, ii
T*) - (A’ * i’)(x,
T*)]
[
‘*This may be shown by differentiating Eqs. (18)-(20) 6’ + ii(sii, I’?‘) = C,j + Pi& A$) (from (21)), Eqs. P’(Zf& N&).
((19) with (25)-(27),
equality) and the
1
I
w.r.t. IT, using invariance of
NEIL VOUSDEN
274 Then, at r* we have
(A2 * ?‘)(x, 7*) - (A’ - i’)(x, T*) 2 s(x&*) - x) > 0, v x E (x,,(~*>, x0(7*) (A’ - i2)(xo(T*)7*) - (A’ *i’) * (x0(7*), T*) = 0. (34) Suppose now that a switch to pattern II takes place at P, so that at time 7* + q (rl > O),
(fi2 * P)(x,,(r* + q), 7* + Tj) - (A’ - i’)(x,,(7* + q), r* + 1) I 0. (35) Now,fromLemma6weknowthatV~>0,36:q<6* l{(i’*
i2)xll(7*
+ 1)), 7* + Tj) - (I?‘* i’)(X,,(T’
- {(z?‘. i’)(x,,(7*
+ q), 7* + 17))
+ q), +T*) - (if’ - i’)(x&*
+ q), 7*>}1 < Q.
If e = s(x~(T*) - x1,(7* + 7))) then we have a contradiction (35).
to (34) and Q.E.D.
Proof of Proposition 7 1. Necessity. Assume that pattern I holds. Then there is no filtering, and dW’ x=
dW2 -= dr
N’B’
-
00
N’D
0
10
19
7
N,2B,z + 5 B;. N;(,+,)
+ “$’ B;N&+
,)
i-0
i-0 n-l
-
2 DziNi i-0
(iii)
- $ DliN;, i=l
‘%is and the equation for dW2/h are merely of our definition of short-run equilibrium.
a dynamic
representation
of requirement
NONMALLEABLE
275
CITY
Lemma 3 guarantees that Pi is invariant w.r.t r so we can differentiate demolition conditions at xii to show that dx,i -=-
dr dxzi -=dr
Then we can manipulate
dx,
i = 1, . . . ) (Tr + 1).
dr ’ dx, dr ’
i=l
,...,a
the above equations to obtain
&,%I)
B,2
the
xdh)
X0
---.
X0
dw’ h
A x&;
For pattern I to hold we must have Bi > 0. The coefficient of Bi in the above equation is easily seen to be positive. The proof of necessity follows. 2. Sufficiency. Suppose pattern I does not continue at r*. By Lemma 7, pattern III must hold. If pattern III holds then there may be upward filtering but no downward filtering is possible since: (a) there are no inner redeveloped units to filter down since pattern I has obtained V r E [0, r*]; (b) if any (Hei, Nci) units were to filter down then it would be optimal for all such units to filter down and group 1 would be unhoused (see proof of Proposition 4). Pattern III will then satisfy the equations dW’ dr
-=
n+l
N;B,j - N;D,,
- 2 DliNL i=l
+ 2
F,iN,Zi + i
i=l
i-0
- ni’ D2iNi. i-0
- “i’F,iNi i-l
B; = D2i = 2rX2(i+ l)e(X2(i+ 1))7d+(i+ I) 7
F2iN2i,
- 2 F2iNiy i-0
i = 0, . . . , (n - 1).
276
NEIL VOUSDEN
Now, the demolition conditions at xii and xzi can be differentiated to give
= N,2i. W(H,Zi,arN& 7)
t(N;(,+,) - Nh) Jg! - 3p) (
ap2(Hi(i+1),
N&i+l)~
7)
(i = 0, . . . ) (n - l)),
a7
-%(i+l)
Now, a given (HL, Ni) (k = 1, 2) must satisfy P2(H& N;) + c; = P2(H,z, N;) + c,z,
u( C;, HA, N;)
k = 1,2,
= u( C,2, H,z, N;) = U2.
(37)
And, when (Hz, N,$ is given, (37) =+ C$ given; when (Hi, (37) =$ Ct given. Therefore, from (36):
aP’(Hk,
N& r) = W’(H,f, a7
a7
aP2(H& N$, r) _ @i(r) 37 dr
dPi(r) Ni, r) =dr
’
(36)
Ni) is given,
i = 1, . . . ) (n + l),
i= 0 ,...,a
’
Thus, we can manipulate the equations for pattern III to obtain
n+l
+
x i-
F,,NFi + 5 F 21 .N2.2,)( 1 + 1
&i).
(38)
i-0
Now, a switch from pattern I to pattern III requires that site rents for an (Hi, N,$ structure at the city boundary not become positive, and this in
NONMALLEABLE
CITY
277
turn requires that: dP,z/dr
I 0.
Since the last of the three terms on the RI-IS of (38) is nonnegative we have
This completes the proof of sufficiency.
Q.E.D.
REFERENCES 1. A. Anas, Dynamics of urban residential growth, J. Urban Econ., 5.66-87 (1978). 2. M. J. Beckmann, On the distribution of urban rent and residential density, J. Econ. Theoty, 1, 60-67 (1969). 3. M. Fujita, Spatial patterns of optimal growth, J. Urban Econ., 3, 193-208 (1976). 4. M. Fujita, Toward a dynamic theory of land-use, Pqv. Reg. Sci. Assoc., 37, 133-165 (1976). 5. D. Harrison, Jr., and J. F. Rain, Cumulative urban growth and urban density functions, J. U&an Econ., 1, 68-98 (1974). 6. R. F. Muth, “Cities and Housing,” Univ. of Chicago Press, Chicago (1969). 7. R. F. Muth, A vintage model of the housing stock, Pq. Reg. Sci. Assac., 30, 141-156 (1973). 8. Y. Oron, D. Pines, and E. She&in&i, Optimum vs. equilibrium land use pattern and congestion toll, Bell J. Econ. Manag. Sci., 4, 619-636 (1973). 9. S. Rosen, Hedonic prices and implicit markets: Product differentiation in pure competition, J. Pal. Econ., 82, 34-55 (1974). 10. R. Solow, Congestion cost and the use of land for streets, Bell J. Econ. Manag. Sci., 4, 602-618 (1973). 11. J. L. Sweeney, Quality, commodity hierarchies and housing markets, E&wmetrica, 42, 147- 167 (1974). 12. J. L. Sweeney, A commodity hierarchy model of the rental housing market, J. Urban Econ., 1, 288-323 (1974). 13. W. Wheaton, A comparative static analysis of urban spatial structure, J. Econ. Theory, 9, 223-237 (1974).
OF URBAN
7,248-277
ECQNOMICS
An Open-City
Model
(1980)
with Nonmalleable
Housing’
NEIL VOUSDEN Department of Economics, Fad@ of Economics, Australian National Uniw?rsi@, P. 0. Box 4, Canberra. A. C. T. 2600 Australia Received May 24, 1977; revised July 24, 1978 Previous city and housing models are extended to allow for the nonmalleability of housing and two classes of residents. The model, which is framed in the context of a monocentric circular city, assumes an individual housing unit to be defined in terms of attributes (quality and residential density). The quality of a given housing unit can be varied without adjustment costs, but changes in residential density on a particular site require prior demolition of the existing structure on the site. Producers of housing and consumers are assumed to be myopic. By assuming that the city is in short-run equilibrium at each point of time, it is shown that the pattern of land use observed at any stage depends on the past history of the city and the current rates of population growth. The possibilities for filtering of houses from one income group to another are also discussed briefly.
INTRODUCTION Until recently it was convenient to model urban land use in a context in which the housing consumed by an individual was a perfectly divisible flow of services which could be adjusted instantaneously and infinitesimally.2 The city represented in such models was a long-run equilibrium city (or equivalently, a city in which the entire housing stock was perfectly malleable). Recently there have been several attempts to go beyond this simple framework and allow for the durability and nonmalleability of housing in a spatial model (see Anas [ 11, Fujita [3, 41). Once it is acknowledged that significant adjustment costs are incurred in reshaping the existing housing stock long-run static analysis is no longer appropriate. The city at any point of time must necessarily be regarded as the outcome of a sequence of “short-run” equilibria. Furthermore, the time-honored ‘This research was completed while the author was visiting the Department of Economics, University of British Columbia. The author wishes to thank Yoshitsugu Ranemoto, David Riefer, Ngo Van Long, Frank Milne, Ted Sieper, Peter Swan Les Witchard, and an anonymous referee for helpful comments and discussion. ‘See Beekmann [2], Muth [6], Oron et al. [8], and Solow [lo]. 248 0094-l 190/80/020248-30$02.00/0 Copyright0 19&l by Academic Rms, All Ii&J
of reprocldon
in any form
Inc. r.zsmvcd.
NONMALLEABLE
CITY
249
equality between land rent and the marginal product of land in housing will no longer necessarily hold, because, as Anas points out, “at any one time only a small part of the urban area operates as the arena for competitive land bidding. The rest of the urban land is occupied by durable structures built in the past under different economic conditions.“3 It becomes clear that the pattern of land use predicted by short-run (or nonmalleable) models will no longer be a simple function of current parameter values (as it is in static long-run models) but will also depend on the previous history of the city.4 The main aim of this paper is to consider the implication for urban land use of nonmalleable housing in a framework which allows for (i) two income groups and (ii) demolition of preexisting structures as the city grows. This expansion of the opportunity set makes the problem considerably more complex, because, at any point of time, the owner of a building is faced with the alternatives of (a) retaining the building and the tenants, (b) retaining the building but letting it “filter” to the other income group, (c) demolishing the building and replacing it with a better one, (d) abandoning the building. From the point of view of the city as whole, the pattern of land use observed will depend on which of the above altematives is chosen at each location at each point of time. Obviously both the spatial distribution of the two income groups and the rate at which the city boundary expands will be sensitive to decisions of the above kind. This, in turn, will have implications for the form of the city’s residential density profile. To facilitate analysis and exposition we shall employ a hedonic formulation of the problem, assuming that both suppliers and consumers of housing perceive housing “quality” and residential density as separate attributes. This formulation will provide us with a convenient means of analyzing filtering of preexisting structures from one income group to another. We shall also assume that landlord/developers and households are myopic (in the sense that they have static price expectations)5 and that households are perfectly mobile. The city we are analyzing will be an open circular city (open in the sense of Wheaton [ 131). The outline of the paper is as follows. In Section 1 we derive the conditions satisfied by each income group’s rent offer function. Section 2 is devoted to the solution of the landlord-developer’s optimization problem. ‘Reference [ 1, pp. 66-671. ‘The idea that observed density profiles may be better explained as the outcome of a historical growth process is at the heart of a seminal paper by Harrison and Kain [5]. %hile it is clearly unrealistic to exclude foresight from the model, the assumption of myopia is probably no less appropriate than the alternative convenient assumption of perfect foresight. We shall be content here with identifying those aspects of the urban structure which are the outcome of myopic behavior.
250
NEIL VOUSDEN
In Section 3 our notion of short-run equilibrium is defined and the main equilibrium conditions are given. Section 4 analyzes the properties of the equilibrium. Section 5 is concerned with characterizing one possible growth pattern for the city and deriving conditions for the continuance of this pattern. Section 6 is devoted to a verbal discussion of some aspects of the model and Section 7 summarizes the main conclusions and suggests possible extensions. 1. THE CONSUMER-RESIDENTS
PROBLEM
First we assume that there are two distinct income groups in the city (rich and poor) with income levels Y’ and Y2, respectively (Y’ > Y’). Throughout we shall let a superscript denote the relevant income group. As foreshadowed in the Introduction we also assume that afl residents (actual or potential) of the city possess the same utility function, u = u( C, H, N),
where C = a nonhousing consumption good (assumed to be numeraire); N = residential density; u E C”) is assumed to be a strictly concave function of C, H, and N with uc > 0, u, > 0, u, < 0, H E housing quality; uHC
=
uCH
>
0,
u,,
=
u,,
<
0,
UCN
=
UNC
<
0.
Each resident is assumed to rent a single residential unit (which for group i will have attributes (H’, N’), i = 1, 2). In addition to selecting the amount of C he consumes and the attributes (H, N) of his housing unit the consumer selects his residential location, defined by the radial distance, X, from the center of the city. We assume that x does not enter into the utility function, but that the individual makes one return trip to the CBD daily and the transport cost per 2 units of distance is I (assumed constant and the same for both income group8). A consumer resident with income Y’ can be thought of as selecting C’, Hi, N’, and x so as to Max u(C’, Hi, N’) s.t. Y’ = C’ + R’(H’, N’, x) + tx,
(1)
where R ‘(Hi, N i, X) is the group i rental rate on a unit with attributes (Hi, N’) at distance x from the CBD. %hile this is obviously unrealistic, it is a natural outcome of ouy assumption that x does not enter the utility function. If all’that separates different forms of transport is money cost then everyone will opt for the cheapest one and so everyone will have the same unit transport cost. To put is slightly differently, the only reasonable way to have different transport costp for different income groups is to include leisure in the utility function as a normal good, and that goes beyond our simple purpose here.
NONMALLEABLE
251
CITY
Equivalently, the above optimization problem can be thought of as defining an equilibrium hedonic offer function for group i, R’(H’, N’, x).~ The necessary conditions satisfied by a solution to (1) are (i = 1, 2) R;; = u;;/u&
(2)
R,& = u;/u&
(3)
R; = -t,
(4)
Y’ = C’ + R’(H’, Equation
N’, x) + tx.
(5)
(4) implies R’ can be written in the form R’(H’,
N’, x) =f’(H’,
N’) + A’ - tx,
where A’ is a constant which (without arbitrary) may be expressed in the form R’(H’,
N’, x) = P’(H’,
loss of generality
because A is
N’) + t(x, - x),*
(6b)
where x0 is the radius of the city. We shall use form (6b) for R’ in what follows. Then we can rewrite the necessary conditions as Pjf = ur;/u;,
(7)
p; = u;/ u;,
(8)
Y’ - txxo= C’ + P’(H’,
N’).
(9
Note that expressing R’(H’, N’, x) in form (6b) is valid for given x,,, but that the solution for P’(H’, N’) will in general depend on the value of x0 and would therefore normally be expected to change as the city grows. We shall return to this question in Section 4. To complete the demand side of the model we need one further equation to reflect the fact that we are considering a city which is open (in the sense defined by Wheaton [13]; i.e., there is free mobility in and out of the city determined by whether the welfare level in the city is greater or less than the exogenously given level outside the city). Accordingly we add the condition u(C’, Hi, N’) = U’, (10) ‘See. Rosen [9]. sit is not gcncrally valid to express a hedonic price function in this separable form; the separability here is a consequence of the fact that x is not an argument in the utility function (or equivalently, that commuting is costly in money but not in time). I am grateful to Lea Witchard for suggesting the usefulness of this separability.
252
NEIL VOUSDEN
where 17’ is the level of utility (exogenously given).
obtainable
outside the city by group i
2. LANDLORD-DEVELOPERS PROBLEM In this section we shall specify the supply side of the city’s housing market. The city is assumed to consist of a number of sites of equal area. Residential density is defined in terms of the number (N) of residential units on a given site. The representative landlord-developer can choose both the number of residential units (N) in a structure on a given site and the quality (H) of each unit (H is the same for all units in a given structure on the site). Casual observation would suggest that within the framework of a given structure it is more feasible to change H than N. Existing units can be upgraded, etc., but adding additional units to a completed apartment building (for example) is not feasible beyond certain narrow limits.’ With this in mind, and in an attempt to capture the essential nonmalleability of housing without at the same time sacrificing the more malleable aspects, we shall adopt the simplifying assumption that for a given unit, H can be varied in either direction instantaneously” but that N can only be varied by demolishing the structure and rebuilding to the desired density. It is also assumed that the process of demolition-reconstruction can be effected instantaneously and without cost. The cost function associated with a residential unit with attributes (H, N) is of the form
+(H) +
u(N)“>
where 9’ > 0,
@” > 0,
Y’ > 0,
y” > 0.
We assume that + and y are sufficiently convex to ensure the existence of a unique solution in all the relevant maximizing problems. 90bviously there is the very real possibility of crowding more tenants into existing units, a phenomenon which is frequently observed in low-income housing areas. For the sake of analytical simplicity we shall abstract from this possibility here. “‘The author has worked through the case where adjustment costs prevent instantaneous adjustment of H to a desired level. The details of this more general case and the conditions under which it reduces to the simple static case studied here are contained in an earlier draft of this paper, which may be obtained from the author by request. ‘%mdially y(N) is some basic fixed cost associated with the construction of N units to some minimal level (H = 0). All productive effort beyond that point is treated as upgrading quality from zero to H (costing H(H)). Note that the cost of a structure is independent of the number of structures built. To this extent we are assuming constant returns to scale.
NONMALLEAI?LE
253
CITY
We shall also assume that landlord-developers are myopic and wish to maximize current profits. Then for a preexisting structure occupied by group i, and containing N units, the relevant optimization problem is Max Ni[ P’(H’, j, Hi
N’) +
x0 - x) - q$H’)
t(
- y(N’)].
(11)
The value of Hj which solves (11) for given j must satisfy
Ph(H’, N’) = +‘(H’).
(12)
In addition, in the choice of j, the landlord has the option of deciding which income group will occupy the structure. In the event that the structure passes from group i to group j (j # i), then what is commonly known as filtering has occurred. We formalize this in the following: DEFINITION 1. If 3 Hj (j # i): Ni[ P’(H’,
N’)
+ t(x,
> Ni[ P’(H’,
- x) - +(H’) N’)
+ t(x,
- y(N’)]
- x) - +(H’)
- y(N’)]
V Hi,
then a preexisting structure containing N’ units previously occupied by group i will be said to filter’= to group j. In view of our definition only preexisting units can filter. The problem to be solved by a landlord-developer deciding the best new structure to be placed on a vacant site at distance x from the CBD will involve the choice of quality and density as well as the choice of income group to occupy the structure. Specifically the problem is to Max N’[ P’(H’, i, H’,
N’) + t( x,, - x) - c$(H’) - y(N’)].
N’
(13)
The solution(s) to (13) must satisfy
Pf;(H’, N’) = #(Hi), Pi + t(x, Ni[ P’(H’,
- x) - &Hi) N’)
2 Nj[ P’(Hj,
(14) - y(N’)
= N’[y’(N’)
+ t( x,, - x) - +(H’) N’)
+ t(x,
- P;],
(15)
- y(N’)]
- x) - +(H’)
- y(Nj)]
V (H’, N’),j
= I, 2.
(16)
Note that (15) implies a positive value for N’ if and only if the proposed “For a rigorous treatment of filtering in a nonspatial setting see Sweeney [I 1, 121.
254
NEIL VOUSDEN
new structure earns positive profits. We shall assume that the appropriate second-order conditions for a maximum are satisfied for problem (13) (which amounts to assuming + and y are sufficiently convex and makes no assumption about the form of P) and for given P’, i, x,,, x, we denote the resulting unique solutions to (14) (15) as (fi’, fi’). Because even a preexisting structure can always be instantaneously adjusted to the values (@, fii) if housing is perfectly malleable, we introduce the following terminology to facilitate comparison of our model with the malleable case. DEFINITION2. (i) The unique value (fi’, $) satisfying (14) and (15) for given i, Pi, x,,, x is termed the malleable optimal structure for group i at x associated with given (Pi, x0); (ii) The unique value of (ii, Z?) satisfying (14), (15), and (16) for given pj 0’ = 1, 2), x0, x is termed the malleable optimal structure at x associated with given (P’, x0) (j = 1, 2). It remains for us to specify the conditions under which it will be optimal to demolish a preexisting structure with attributes (iY’, N’) (where Hj satisfies (12)) and replace it with the malleable optimal structure (Ai, Ai). Such demolition-reconstruction will occur if and only if Ay Pi@,
Ai) + t(x, - x) - +(z?)
- v(Ai)]
2 Nj[ P’(H’,
IV’) + t(x, - x) - +(H’)]. (17)
3. EQUILIBRIUM While we can define what we mean by short-run equilibrium it is not possible to immediately present all of the equilibrium conditions. In particular the condition that all of the city’s population (both income groups) be housed can only be formulated after we know where each income group is located. But, as we have suggested, this, in itself, is the outcome of the city’s past history and thus the demand = supply condition can only be given explicit form after we have either discovered or assumed something about the city’s previous pattern of growth. For the time being we content ourselves with the following definition: DEFINITION 3. A short-run equilibrium for the two-class, nonmalleable open city consists of a consumption-housing profile for grq i (C’(x), H’(x), N(x)) (i = 1,2), a hedonic price function Pi (i = 1, 2), and a value for x,, (the city boundary), such that: (i) all individual consumers and producers are in equilibrium; (ii) site rents are zero at x,, (competitive land and housing markets); (iii) supply of residential units for group i in the city = demand for residential units in the city by group i.
NONMALLEABLE
255
CITY
Requirement (i) of the above definition is seen to be satisfied if the following conditions are fulfilled (see Sections 1 and 2) at x: f(C’, Hi, N) z $(Ci,
Hi, N) = Pk(H’,
N) = +‘(H’),
g(C’, Hi, N) E p,
Hi, N) = Ph(H’,
N) $ y’(N)
- P’(H’,
i = 1,2,
(18)
N) + t(x, - x) - c#J(H’) - y(N) N
I
>ifN
) (19) I (20)
u(C’, Hi, N) = ii’, Y’ - tx, = C’ + P’(H’,
(21)
N).
In the above equations_ y is historically given unless the structure is new, in which case: (a) N = N’, (b) N is endogenous, (c) there is equality on the RHS of (19). In addition, from (11): A preexisting structure (Hi, N) satisfying the above conditions will be occupied by a resident of group i if and only if N[ P’(H’,
N) - +(H’)]
2 N[ P’(H’,
N) - cp(H’)]
(22)
V Hj, j = 1, 2.
Also, from (16): A new structure at x will be occupied by group i if and only if fii[ P(A’, 2 iq
9)
+ t( x0 - x) - +(zF) - y(e)]
P’(z?,
fij) + t( x0 - x) - +(Z?) - y(tij)],
j = 1,2, (23)
where (Z?, @ represents the malleable ture for group j at x (j = 1, 2).
optimal
struc-
Finally all new structures replacing preexisting structures must satisfy a demolition condition of form (17). Conditions (17)-(23) complete the specification of equilibrium for individual suppliers and consumers. It is worth noting that the possibility of strict inequality on the RHS of (19)
256
NEIL VOUSDEN
reflects the discrepancy between land rent and the marginal product of land in housing which we would expect to find when the land is being used either more or less intensively than is “optimal” (“optimal” being used in the sense of what would be optimal if it were possible to put a new structure on the site without incurring any adjustment costs). In a growing city we would generally expect to find preexisting structures using land less intensively than is optimal, in which case we would observe the first inequality in (19) (i.e., the marginal product of land in housing < profits/unit land). We also note at this stage that if housing were perfectly malleable then the malleable optimal structure will always be found at each x and there will be equality on the RI-IS of (19). Then our model reduces to the static long-run model I la Muth and Solow. Requirement (ii) of our definition of short-run equilibrium is expressed as Mfx+(x,))[
P’(+(xJ,
ii(
- @(x0))
- @(x0))]
= RA (24)
(where ($(x,,), Ai( satisfies (18)-(20) and (19) with equality at x0>. Thus, if there is housing for group i at the city boundary (xg), then it will have attributes (Hi,, ZV,$ satisfying AC& Hi,, N;) = $(HA), g( C;, Hi,, A$) = y’(N;)
(25) - RA/ (N;)‘,
u( C;, H;, IV;) = ii’.
(26) (27)
Our assumptions on the utility and cost functions ensure that (c,$ HA,,N,$) = (ei(xO), A’(x,J, #(x0)) will be unique for given U’. Finally, as we have already noted, requirement (iii) of the definition cannot be formulated explicitly until we know the city’s past pattern of growth. Nevertheless we can say a great deal about the properties of a city in short-run equilibrium without writing down an explicit equation for (iii). The next section is devoted to deriving some of these results. 4. EQUILIBRIUM In what follows we shall let
GROWTH
PATTERNS
r’(N, x) E P’(H’, N) + t(x, - x) - @(Hi) - y(N)
for given x0, Pi, where (Hi, N) satisfies (18)-(20), and let P(x) E P’(A’,
Ai) + t(xo - x) - c#@) - y(e)
NONMALLEABLE
257
CITY
for given x,,, where (Z?‘, I?‘) satisfies (lS)-(20) can now prove:
with equality
in (19). We
LEMMA 1. For a city in short-run equilibrium: (a) -f--[j+(x)-
fi(x)]
(b) -$ [ AI(
i = 1, 2,
< 0,
9(x*) - 3*(x*). i*(x*)] > 0,
where x* is a value of x for which i+(x) Proof.
(a) By definition
- i’(x)
= C(x).
of (I?‘(x),
F’(x).
Z?(x)):
2 [i+(x) - ii(x)] = -k’(x)
< 0.
Now, at a point x* where A’(x*) the following conditions
* 9(x*)
= z?(x*)
* i’(x*)
(28)
must apply for each income group:
f(C, ii, A) = qqfi), q y’fi)
= Y - c - tx* - c#@) - y(A).
- g(& ii, fi)]
Also, from (27) A(Y-Ltx*-C+(B)-y(fQ)=k, where k is independent w.r.t. Y, we obtain
of Y. Differentiating
fc 1 dN -=dY
l-I&
f, - +"
0
f f-N&
1
A
1
the above three conditions
evaluated at x*,
(29)
258
NEIL VOUSDEN
where
fc AZ
l-&c 1
f, - v f-i&, f
fN 2y’ + icy” - g - I?gN < 0.
g
The RHS of (29) is negative. Hence Z?‘(x*) > fi’(x*) g
[if ‘(x*) * 9(x*)
(30)
- $2(x*)
It is now easy to prove the following malleable housing.
* ;‘(x*)]
and we obtain > 0.
Q.E.D.
result for the case of a city with
PROPOSITION1. If housing is perfectly malleable then when the ci@ is in short-run equilibrium, 3 x* E (0, x,-J such that for x < x* all housing is consumedby the poor and for x > x* all housing is consumed@ the rich. Proof. If housing is malleable, then in equilibrium there must exist an E*. such that j?‘(x*~i’(x*) = s2(x*)F2(x*); otherwise, by the continuity of N’(x)?(x) either N’(x)?‘(x) > A2(x)i2(x) V,, or vice versa, and one of the two groups would be unsupplied. Proposition 1 is then a trivial consequence of Lemma l(b). Q.E.D.
Proposition 1 is a generalization of a result obtained by Muth [6] and Beckmann [2], who found that in long-run static equilibrium, given the constant unit transport cost assumed here, the rich would choose to live further from the city center in lower-density housing and the poor would live in the inner areas in higher-density housing. In our model there is the additional possibility of using higher quality to compensate for the higher density of inner locations and it is interesting to note that the traditional results are robust to this generalization. Proposition 1 will serve as a basis for comparing our results for the nonmalleable city with results of traditional models. Now let us note that the following is a trivial consequence of (25)-(27): LEMMA 2. Zf the city is in short-run equilibrium, then Ci E ei(xO), Hh E Z?(x,), Ni s i?(x,,) are independentof x,, for given ii’.
Obviously if (CA., Hi, Ni) is x,-invariant, then so is P’(H& Ni), for (provided there is housing for group i at x,,) it is given by the zero profits condition Pi(H;,
N;) = R,/N;
+ @(Hi) + y(N;).
NONMALLEABLE
259
CITY
In order to make P’(H& NA) x,-invariant we shall make the following assumption. Assumption A. U’ remains constant as the city grows. This assumption is made for two reasons. First, it is a reasonable approximation to reality in certain specific cases (such as cities in less developed countries where the city is surrounded by a rural hinterland at a subsistence level of ii’). Second, and more to the point, we are concerned with isolating those features of urban equilibrium which are independent of changes in parameters outside the city. It should then be an easier task to determine the role of changes in these parameters in the growth process. In particular (as we shall see) constant U’ leads to one pattern of growth which proceeds entirely through demolition-reconstruction without filtering of residential units from one group to the other. For such a growth pattern the phenomenon of filtering is entirely due to the price effects (changes in P’(H& A$,)) caused by changes in ii’. We shall denote the x,-invariant value of P’(H& Ni) as P; z P’(H&
Equilibrium
P’(H’,
(where a’(H’, P’(H&
N;).
N) will satisfy the partial differential P;; = f(“i(H’,
N), Hi, N),
Pj, = g(a’(H’,
N), Hi, N),
N) is the inversion
i=
equation
1,2
of (20) for Ci) with initial
(31)
condition
N;) = P;.
To simplify matters we make the following additional assumptions. B. The partial differential equation (31) has a unique solution.*3 Assumption C. The city is in short-run equilibrium at each point in time. Assumption
Assumptions
(A)-(C)
are assumed in all of what follows. We have:
LEMMA 3. Equilibrium P’( H i, N) occupies some housing at x0.
is x,-invariant
as long as group i
The following results are also useful. “To see that this assumption is not vacuous we need only check the following u’ - A’(CaHb/NC), A’, (I, b, c constants. It is easily seen that for this utility has the unique solution pi(~i,
N)
= [ pi + (pi)‘/“.
(~6)~“1”.
(N;)‘/“]
- ($)I/“.
(Hi)-b/a.
simple function
NC/O.
case: (31)
260
NEIL VOUSDEN
PROPOSITION 2. Residential units constructed for group i with attributes (Hi*, N’*) will retain those attributes until demolished as long as they are occcupied by group i. Proof. First, let us observe that all preexisting units which were built with attributes (Hi*, N’*) must have density N’* until they are de molished. C’ and Hi at all locations x where N = N’* must satisfy f(C’, Hi, N’*)
u(C’, Hi, N’*)
= #(Hi), = ii,
which are x- and x,-invariant. Given our convexity assumptions on ui, +, and y, Hi* solution of the above equations for Hi.
is the unique Q.E.D.
LEMMA 4. The malleable oftim$ structure replacing an (H j, N) structure will have unique attributes (Hj, NJ) which are invariant w.r.t. x0 provided both groups are occupying some units at xW ProojIfi, Lemma 3 guarantees the invariance Now, (HJ, NJ) must satisfy
of Pj (j = 1,2) w.r.t xe.
where x* is the value of x at which replacement
is taking place. Also,
I?[ Pj(z?,
I?) + t( x0 - x*) - (p(rij) = Ni[ P’(H’,
Differentiating
- y(fij)] N’) + t(x, - x*) - +(H’)].
both sides of this equation gives t(fij
- N’)(dx’
- dx,) = 0.
It is routine that I?j > N’; thus kc, = uk, and (x0 - zt) is, independent of the value of %T ‘DJUS the system of equations for (HJ, NJ) is independent of x0 and (NJ, NJ) is similarly independent. Q.E.D. PROPOSITION 3. At all stages of the city’s growth there are on& three possible growth patterns for new housing: I. All inner redevelopment is occupied by the poor; new peripheral housing is occupied by the poor and the rich.
NONMALLEABLE
261
CITY
II. Inner redevelopment is occupied by the poor and the rich; new peripheral housing is occupied by the rich. III. All inner redevelopment is occupied by the poor; new peripheral housing is occupied by the rich. ProojI
If the poor occupy new peripheral
?‘(xo)A2(xJ
housing (i.e., at x0):
2 i’(x,)z?(x,).
Lemma l(b) can then be invoked to show that all inner redevelopment housing will be allocated to the poor. It is then easy to see that I-III above exhaust all the possible cases. Q.E.D. Note that a simple consequence of Proposition 3 is that group 1 must always occupy some housing at x0. This combined with the assumption of competition implies
N@“(Kj,
Nd) - +(f&j) -
u(Nd)]
= RA.
(32)
In what follows we shall have reason to be particularly interested in the first of the above patterns. There are two reasons for this. First, our assumptions that t is the same for both groups and that U’ is constant as the city grows imply that the pattern of land use resulting from uninterrupted pattern I growth is relatively simple and regular. Second, and more important, is the fact that pattern I growth implies a land-use pattern markedly different from that predicted by existing models and we are therefore interested in obtaining conditions under which it would be the prevalent growth pattern. Accordingly we derive the properties of a city which grows according to pattern I. 5. PATTERN
I GROWTH
PROPOSITION4. Zf pattern I has prevailed at all times since the city’s inception up to and including time r, then no filtering can occur at r.
Proof.
See the Appendix.
The proof of Proposition 4 does not depend on the x,-invariance of Pi. It does, however, depend on (a) the invariance of (Hi, N,$ (i = 1, 2) as the city expands (which is in turn a consequence of constant i?), (b) the fact that t is the same for both groups. The intuition of the proof of Proposition 4 is illustrated as follows: If one income group were to outbid the other group for an inner (Hi, N$ structure then because of (a) locational equilibrium, (b) the identical slopes of the rent profiles (- t), and (c) the invariant nature of new dwellings at
262
NEIL
VOUSDEN
the boundary, such a possibility implies that the same group could also outbid at x0 and everywhere else for that matter. The assumed prevalence of pattern I would then be contradicted. The intuition behind the rest of the proof is similar. The following result is then a simple consequence of Propositions 2 and 4. PROPOSITION 2a. Zf pattern I prevails at all times -T E [0, r*] for some r*, then residential units constructed with attributes (Hi*, N’*) at some time 7 E [0, r*] will retain those attributes V r E [0, r*] until they are demolished.
We now establish structure.
the “timing”
of the demolition
of a preexisting
PROPOSITION 5. Assume the city grows according to pattern I at all times after its inception. Then for a structure with attributes (Hi, N’) constructed when x,, = xz, 3 a unique X0 > x$ + e s.t. when x,, > &, it will be optimal for landlords to demolish th_e (ZZ!(, N’) strycture and replace it with the malleable optimal structure (HJ, NJ), where NJ > N’. Such a demolition will not be optimal for x,, < &.
Proof.
See the Appendix.
Proposition 5 tells us simply that once a building has been constructed on a site, a more than infinitesimal (and generally quite large) increase in x0 will be needed to induce demolition and replacement by the malleable optimal structure on the same site. This is because the commitment implicit in the construction cost of the preexisting structure is a bygone (here included as a sunk flow cost, NY(N)) and x0 must rise sufficiently to push up rents to the point where it is worthwhile to incur another such commitment. We also prove: LEMMA 5. As x,, increases while the city is growing according to pattern I, land rent from an (Hi, N,$ structure at x ( < x0) is greater than and increases more than that from an (Hd, NJ) structure at x. In addition, if an (Hd, N,‘) structure satisfies the demolition condition (17) with equality at x, then it is not optimal for the landlord to demolish the (Hi, Ni) structure at x at that time. Proof
See the Appendix.
The following proposition is an easy consequence of the above results and describes the form of a city which has grown without interruption according to pattern I. PROPOSITION 6. Assume that the equilibrium nonmalleable open city with two income groups grows V r 2 0 according to pattern I. Then it will consist
NONMALLEtAEJLE
263
CITY
of a series of concentric rings with radii 0 = <
XZ(n+l)
5 X1(n+l)
. * *
<
X2n
<
Xln
<
X2(“--1)
<
Xl(n-1)
such that (a) within the ring (x~(~+~), xzi] housing has attributes (Hi, N$ (i = 19 * * * 9 n), where (Hi, Ni) is the malleable optimal structure for group 2 at x2i; (b) within the ring (xlCi+,), xii] housing has attributes (H& Nh), where (HE, NE) is the malleable optimal structure for group 2 at xii; (c) within the ring (x2t, x0] housing has attributes (Hi, N$, and within the ring (x,,, x0] housing has attributes (Hd, N,$. >N;,-,) >N:,,-,, >-->N; >N; (d) N&+1) >N,z, >N:, > -.- >N;>N;. Thus, the “pattern I city” consists of a sequence of successive replacements of (Hd, Nd) structures overlapping with a corresponding sequence arising from (Hi, Nt) structures-in any interval (xlCi+ij, x2], for example, there will be two types of housing (each occupied by group 2) with attributes (H& Nk) and (H& N$ respectively. Having obtained a complete picture of the structure of a city which has been growing according to pattern I, we are almost in a position to establish conditions under which such a growth pattern will be observed. First we introduce the following notation, consistent with that used in the statement of Proposition 6. This notation, in addition to being essential in the proof of Proposition 7, will be useful in formulating requirement (iii) of our equilibrium definition in a dynamic context.
W:‘,, x2i(r)
xii(r)
B&(7)
Niol
Attributes of structure occupied by group j, which is a replacement for a structure with attributes (H& N$ (i = 0, . . . , (n - 1);j = 1, 2) W,zs N,z) Attributes of structure occupied by group j, which is a replacement for a structure with attributes (HE, N$ (i = 0,. . . , n,j = 1,2) (Nd, Hi> Point of demolition of (H& ,), N&- ,> structure at time 7 (i = 1, . . . , n) Point of demolition of (H&-,), N&-,,) structure at time 7 (i = 1, . . . , (n + 1)) Number of new structures with attributes (H&+,), N&+ ,$ built at time 7 (i = 0, . . . , (n - 1); j = 1, 2)
264
NEIL
B:‘iW B&d 4it7)
Dli(?)
F2i(7)
Fli(r)
Wj( WI
7)
VOUSDEN
Number of new structures with attributes (H{(i+i), A$+,,) built at time r (i = 1, . . . , n;j = 1, 2) Number of new structures with attributes (Hi, Ni) built atx,attimer(j= 1,2) Number of structures with attributes (Hi, Nii) demolished at time 7 (i = 0, . . . , (n - 1)) Number of structures with attributes (Hi, NFi) demolished at time r (i = 0, . . . , n) Number of (Hi, Ni) structures filtering up to group 1 at time 7 (i = 0, . . . , n) Number of (H& Nt) structures filtering up to group 1 at time 7 (i = 0, . . . , (n + 1)) Population of group j at time r (assumed exogenous)14 Proportion of sites at x originally occupied by (Hd, ZVd) structures
Now, noting that the malleable siie;rent functiorr’?(x) . i’(x) will depend on x0 and hence 7, and writing N’(x)?‘(x) E (N’ * i’)(x, T) to reflect this we prove the following result, which assures us that in a city which is growing continuously a la pattern I there will be no jumps in the malleable site-rent function. LEE 6. If pattern I has prevailed at all times r E [0, r*] for some P, then (N’ - i’)(x, 7) is continuous at r*, if W-~(T) is a continuous monotonically increasingfunction of 7 (i = 1, 2;j = 1, 2). Proof: See the Appendix. While the proof of Lemma 6 is somewhat long, the idea is simple enough. The zero profit condition means essentially that the malleable site-rent profile (($ * F’)(x)) is the lowest such profile consistent with housing all of the city’s inhabitants. The proof essentially shows that a downward shift of the (A’. i2) profile in the time interval [T*, r* + T)] would imply that a lower such equilibrium profile would be feasible at r*, in contradiction to the definition of the profile at r*. On the other hand, the (&’ +F2) profile is bounded above by (8’ - i’) in the interval [T*, T* + s] in a way which limits any upward shift of the (fi2. $) schedule and ensures continuity at 7*. Lemma 6 can be used to obtain the following result: “Assuming an exogenous population for the city would seem to run counter to Wheaton’s definition of an “open” city. However, in both cases “open” means that movement into and out of the city is free so that the utility level in the city will be equated to that enjoyed elsewhere. It is immaterial whether we assume that the level of income in the city rises, thus attracting immigrants, or whether we set it up as a problem in which the population growth is taken as given and the level of income (Y’) required to induce that growth is endogenous.
NONMALLEABLE
CITY
265
LEMMA 7. If pattern I has prevailed at all times r E [0, r*] for some r* then pattern I cannot switch into pattern II at r*.
ProofI
See the Appendix.
So now we know that the only switch possible from pattern I is to pattern III. We are now in a position to derive conditions for the continuance of pattern I. To make the subsequent analysis more tractable we shall confine our attention to the case where P,‘(T), x0(7), xii(r), xzi(7) are differentiable functions of 7 at r*, which is not unreasonable since we have at least shown them to be continuous. PROPOSITION 7. Assuming that Wj(r), P,‘(r), x,,(r), xii(r), xzi(r) (i,j = 1, 2) are differentiable functions of r at r*, and that the city has grown according topattern I V r E [0, r*] the city will continue to grow according to pattern I at r* if and onb if
dW’
A
dW*
-*------xdr xolv;
where
+
ProojI
iso
CNGi+l)
-
NZfd)x2(i+1)(1
-
6(x*(i+1))
Seethe Appendix.
COROLLARY. Assuming that w(r), P&r), x0(r), xii(r), xzi(r) (i,j = 1, 2) are differentiable functions of r the city wilI grow according to pattern I VrE[O,r*]ifandonbif Vr E[O, r*].
Proof. Noting that the condition is trivially satisfied in the initial stage of the city’s growth before any demolition takes place (since then A = 0), we also note that pattern I is in any case the only possible pattern in the predemolition state. The result then is a trivial consequence of Proposition 7. Q.E.D. Proposition 7 and its corollary give us a simple condition showing how the growth pattern of the city is dependent on the relative population
266
NEIL VOUSDEN
growth of the two groups (dW’/dr and &V2/dr) and the historical land-use pattern in the city (as reflected in the values of A(r) and 8(x,,(r)). In the next section we shall make a few more detailed observations on various aspects of the growth process implicit in Proposition 7. 6. DISCUSSION
OF THE
MODEL
(i) Population Growth
The manner in which the outcome depends upon relative population growth is simple enough. The larger the growth rate of the poor population relative to the rich population, the greater the likelihood of pattern I. This is because a rapid growth of the population of the poor will require some of the new residents in that group to locate at the urban periphery in order to induce demolition-reconstruction at inner locations for the purposes of housing the rest of their number. In the polar case where the population of the rich stays constant and the population of the poor grows, new construction of houses for the poor on the periphery is both necessary and sufficient for induced replacement housing for the poor on inner sites. (ii) Previous Histoly of the City
The manner in which the pattern at r depends on the city’s history up to r is somewhat complex, but the mechanism of the dependence may be better understood if we look at the case where (Hd, Nd) structures are being demolished, but it is not yet economic to demolish any (H,f, N,f) structures. Then the condition for pattern I reduces to
-. In such a case the lower the value of @(xi,), the greater the likelihood of pattern I. This is explained as follows: the lower the historically given proportion of sites at x,, originally occupied by the rich, the fewer the rich who will be unhoused by the replacements at x,,, so that more poor must be housed at the edge of the city because (a) there are fewer rich to augment housing at the periphery and thereby induce demolition-reconstruction on inner sites by pushing up t(x, - x); (b) there is less demolition of housing induced by a given increase in x0 because there were fewer (H,‘, N,‘) structures at x,, to start with. (iii) Population Density
It is certainly clear that in the event that pattern I predominates, it is possible to have a density profile which differs from the standard downward-sloping one. The possibility of an upward-sloping profile is particularly strong in the city’s predemolition phase; for example, if W’ and W2
NONMALLEABLE
267
CITY
are each growing at a constant rate then the density profile will be upward sloping if the growth rate of IV2 is higher than that of W r. In general, when pattern I is followed, the fact that group 2 is housed at the periphery will tend to mean at least a flatter, and possibly an upward-sloping, density profile. (iv) Spatial Distribution
of Income
Groups
We have shown that there are perfectly reasonable conditions under which pattern I will be the predominant growth pattern for a city. If those conditions are satisfied then we would expect to find a spatial distribution of different income groups which is very different from that predicted by long-run models (see Beckmann [2] and Proposition 1 in this paper). Rather than having the poor entirely accommodated in crowded housing near the center and the rich in low-density housing further out we find that the poor may be distributed fairly evenly throughout the city with a possible concentration at the periphery if there is a sudden large rise in their population. This is the sort of pattern we observe in many cities in less developed countries, where a steady large influx from a subsistence rural sector is accommodated in shantytowns at the edge of the city.” It is worth noting, however, that one element in the spirit of preexisting results carries over here. Despite the possibility that some of the lowerincome group may occupy peripheral land, there is no pattern of equilibrium growth in which this income group does not occupy the inner land. (v) Filtering We have shown (Proposition 4) that pattern I proceeds without filtering (that is, provided it proceeds without being interrupted by pattern II and/or pattern III), and we have observed that this uncomplicated growth is the result of assuming constant Pi and identical t for both groups. It is possible, however, to envisage a city in which U’ is steadily rising and iz is constant or falling. I6 Then one would expect that: (a) pattern I would be less likely, with the rich opting for more spacious accommodation at the periphery (and thereby inducing more inner redevelopment for the poor) and the poor accepting higher-density inner housing; (b) rising U’ would imply that rich people occupying inner housing would vacate it in favor of more spacious peripheral housing and the vacated inner housing would filter down to the poor. ‘%uth [7j has commented on this possibility model. 16Anas [l] considers the effect of variations and a single income group.
in the context
of a nonspatial
in ii for the case of infinitely
vintage
housing
durable
housing
268
NEIL VOUSDEN
Despite the obvious technical difficulties involved, it would indeed be interesting to analyze this case. It is misleading, however, to suppose that our own assumptions have completely ruled out the possibility of filtering. Indeed, it is a very real possibility for either of patterns II and III. Since Pi will be falling for some interval of time after a switch-out of pattern I, we have the possibility that for some ~a, 3 H’:
Y(&?)] = N,z[P2(H;,N,z,70)+ f(+,(d - x) - cp(H,z) - Y(%)]
N,2[P’(H’, Wf) + f(&,)
- x) - W’)
-
vx
with the (Hi, iVt) structure filtering up to group 1. This would occur because the site-rent function for group 1 is being pushed up sufficiently in excess of that for group 2 to induce upward filtering. It would be primarily the result of relatively rapid growth in the group 1 population and is a phenomenon frequently observed in modem cities where there is a rapid growth in the size of the middle/upper class. When pattern III is operative there is the additional possibility that redevelopment structures occupied by the rich (with attributes (H,& NjJ (k = 1, 2) will subsequently filter down to the poor. 7. CONCLUSION The purpose of this paper has been to generalize existing circular-city land-use theories to allow for the nonmalleability of the housing stock. A model of an open city with two income groups was developed in which the housing stock was assumed to be durable, but housing units could be demolished and replaced by new structures which utilize inner land more efficiently. Such replacement involves substantial adjustment costs (the sacrifice of the cost sunk in the existing building) so that for a time land will be used inefficiently. Land will only be used efficiently (with land rent equal to the marginal product of land) for newly constructed buildings. Not surprisingly it was found that, if certain not unreasonable conditions were satisfied, the pattern of land use (in particular the density profile and the spatial distribution of income groups) observed in a nonmalleable city would differ markedly from those predicted by long-run static models. This paper has proceeded under the assumption that the utility level of each income group is exogenously fixed. While this assumption has great analytical value in enabling us to isolate that part of the growth process which is independent of exogenous factors, it is equally important to explore the effects of changes in U’, especially given the probable implications of such changes for filtering. We have also assumed that landlords are myopic, and obviously extensions of this model should pay some attention to the role of foresight in the
NONMALLEABLE
269
CITY
growth process. Finally it is important that the model be placed in a normative context so that policy questions arising from nonmalleability in cities may be properly defined and discussed. APPENDIX Proof of Proposition
4
Since all redevelopment housing must be occupied by the poor under pattern I there is no possibility of downward filtering of inner redevelopment units (unless of course it occurs after some upward filtering). We therefore have to rule out upward filtering of these units: To this end suppose that a redevelopment structure built at x for group 2 with attributes (H*, N*) (as a replacement for a structure with attributes (H*, N*)) is occupied by group 1 at some later time at which its attributes change to (H’, N*). Lemma 4 tells us that all replacements for structures with the same attributes will themselves have the same attributes. Then the replacement for an (H*, N*) structure built at time 7 (at, say, xi) will also have attributes (Hz, N*). Now, we must have N’[
P’(H’,
N2) + t(x, > N’[
- x> - $4~‘) P*(H*,
- Y(N2)]
A+) + t( x0 - x) - W2)
- VW*)].
This implies N’[
P’(N’,
N2) + t( x0 - x1) - cpW) > N”[ P2(H2,
Iv)
-
Y(N2)]
+ t( x0 - $1 - W2)
- YW3]
and group 1 could outbid group 2 at xi. This contradicts the assumption that pattern I is ruling at time 7. Thus upward filtering of redevelopment structures is ruled out. The only other possibility of filtering arises in the case of the (Hi, ZVA) structures at x < x0. Let us therefore suppose that a structure built at x for group i with attributes (H& NA) is occupied by groupj (j # i) at some later time at which its attributes change to (Hj, Ni). For this to happen, clearly we must have
> N;[ P’(H;,
iv;) - &Hi)
- y(&)]
> RA
(i = 1, 2)
so that group j could outbid group 1 at x0, by opting for units with attributes (H’, Nh). But then either (a) u(Cj, H’, Nh) < ii-j or (b) the
270
NEIL VOUSDEN
uniqueness of (Hi, N&) is contradicted
(since Nd # N,$ in general).
Proof of Proposition 5
For given x, define V’(xo> as y’(x,)
s I+[ pq9,
f+) + t(x, - x) - +(I?9 - Y(+)]
- Ni[ P’(H’, N’) + t(x, - x) - +(H’)].
Now, Lemma 3 together with (18) and (19) implies dl/‘/dxo Noting
G t(tij
- N’).
that V x0 C$ + P’( H& N$) = cj + P’(lj’,
we can differentiate
fij),
system (18)-(20) and use Lemma 2 to obtain dl? 1-s dxo
t(fh - f&f
- ry(Gj)) A
3
where A is as defined in (30). Thus we can write vi(xo)
= p(xo) - N’tx, + <(Hi, N’),
where /3(x,) s r;i[ ~j(lj’,
+)
+ t(x, - x) - +(I@) - y(W)]
and [(Hi, N’) E Ni[ P(H’, N’) - tx - $@‘)I.
From the above j3’ > 0, p” > 0. Also, (a) V’(xo*) = - Niy(Ni) (b) dYi/dxo = 0
>o (c) d2Vi/dx;
< 0;
when x0 = xg when x0 > x0*;
= p”(xJ
> 0.
Q.E.D.
NONMALLEABLE
271
CITY
It then follows that 3 g)
> x0* +
z s.t.
d%o) dxo
> o
,
xpx.
so X0 is unique for giyen xg (or for given (H’, N’), which amounts to the same thing). Clearly NJ > N’. Q.E.D. Proof of Lemma 5 { N,z[ P2(H;,
N;)
+ t(x,
- { N;[ P’(H,‘,
N;)
- x) - +@,z) + t(x,
- Y(%)]
- xl - +(ffd) = t(x, - x)(N,z
(since Ni > N,-j). The derivative Also, [N,2[ P2(H,z,
N;)
+ t(x,
- h}
- Y(%)] - N;)
- RA} > 0
for x < x0
of this expression w.r.t. x,, is also > 0.
- x) - $@;)I
-Nd[
P’(Hd,
N;)
+ t(x,- x) - +W)l I = 0, - x)(N,z- Nd)+ No%(%)-
N;Y(N;) > o,
so that if the demolition condition (17) for the (Hi, Nd) structures is satisfied with equality then it is not satisfied for the (Hi, Ni) structures. Q.E.D. Proof of Lemma 6
We begin by proving that x0 is a continuous function of T (it is trivial that it is monotonically increasing). We know that”
“This
inequality
simply
says that the number
edge of the city ~lKkbh)>2 population
(I( d(7,)
-
W’(Q)l).
- (~&JYll)
of new residential units for group i at the 1s . no greater than the increase in the city%
272
NEIL VOUSDEN
Now continuity
of W’ * V c > 0, 3 6:
Now (Z?’ . i’)(x, 7) is easily seen to be continuous in T since it is continuous in x0 (because of the differentiability of the functions and the invariance of P’). (Z?’ * i’)(x, 7) can be obtained from the system P’(ri2,
A’)
+ e* = p*(L?,2, N,z) + c;
(from W)),
f(t?,
ii*, 3’) = P;(I?*,
I?‘) = r#i(ri’),
g(P,
22, 9)
fi’)
= P;(I?*, - P*(ii*,
= y’(Af2)
fi”)
+ f(X&)
- x)g@)
- y(i?)
fi’ up,
ii*, iv)
7
= ii*,
where (C,“, Hi, A$) is the solution to (25)-(27). Thus (I?*, i*) will be continuous in r if it can be shown that P:(T) E P*(H,f, Ni) is continuous. We let P;(T) E P’(H& Ni), i = 1, 2, and define a’(q, x) 3 N(g P@
+ 77)+ t(x,(7*
+ 77)- x) - 4@;)
- Y(Nli)], n 2 0.
We will show that (Y* is continuous at n = 0. First we show that a*(~, x0(7*)) 2 0. Suppose otherwise for 11 > 0. Then, if P*(H*, N*, r) denotes the equilibrium for group 2 at time 7 we have that: (i) Since P*(H*, A’*, T* + 7) satisfies equilibrium (7* + 7) we can construct P**(H*,
N*, T*) = P*(N*, N*,
7*
hedonic price function
+ 7) + t(x&*
conditions (18)-(20) at + T/) -
x0(7*))
(33)
and it is easily shown that this price function will satisfy conditions (18)-(20) at 7*. (ii) The population of group 2 at (7* + 9) is entirely housed at locations x0(7* + q < x0(7*) since a*(~, x0(7*)) < 0 * a*@-/, x) < 0 v x E [x0(7*),
41.
NONMALLEABLE
273
CITY
Therefore the price function P2* we have constructed (Eq. (33)) will house the entire population of group 2 at time r* within the boundary x0(7*) (since this population is smaller than that at (r* + 77)). (iii) C,Z + P2(H,2, N,Z, 7* + 17) + t(xO(,* + 77) - x0(7*)) + txxo(7*) < C,z + P’(H,z, N;, T*) + tx,, = Y2(7*) (since we are assuming a’(r), x0(7*)) < 0 for n > 0, whereas (~~(0, x0(7*)) = 0). Thus price function P2* given by (33) is an equilibrium price function for r = T* which requires a lower level of Y2 for its realization (or equivalently will yield a higher ii2 for the same Y2). It does, however, imply site rents at x0(7*) at time T* equal to (~~(77,x0(7*)), which we have assumed negative. This contradicts the prevalence of pattern I at r*; :. a2(q, x0(7*)) 2 0, Now we know from Proposition
-q 2 0.
3 that
a2hxl@ + 1)))5 ayq,x()(7*+ 17)) and this, together with the above result, implies
0 5 a2h x,(7*)) 5 a’(99 x&*)) and this, together with the continuity of (Y’ in n, is easily seen to imply that (~~(7, x0(7*)) is continuous at q = 0. It is a trivial consequence of this that P,‘(T) is continuous at 7*. Q.E.D. Proof of Lemma 7 When pattern I is iuling at 7*:
$K fi2 - i’)(x, T*) - (I? *9)(x, 7*>] > 0, since i’(x) SE
> Z?‘(x).” min [x,1(7*),
X0(7*)1
Now, define
(
$2. i’)(x, ii
T*) - (A’ * i’)(x,
T*)]
[
‘*This may be shown by differentiating Eqs. (18)-(20) 6’ + ii(sii, I’?‘) = C,j + Pi& A$) (from (21)), Eqs. P’(Zf& N&).
((19) with (25)-(27),
equality) and the
1
I
w.r.t. IT, using invariance of
NEIL VOUSDEN
274 Then, at r* we have
(A2 * ?‘)(x, 7*) - (A’ - i’)(x, T*) 2 s(x&*) - x) > 0, v x E (x,,(~*>, x0(7*) (A’ - i2)(xo(T*)7*) - (A’ *i’) * (x0(7*), T*) = 0. (34) Suppose now that a switch to pattern II takes place at P, so that at time 7* + q (rl > O),
(fi2 * P)(x,,(r* + q), 7* + Tj) - (A’ - i’)(x,,(7* + q), r* + 1) I 0. (35) Now,fromLemma6weknowthatV~>0,36:q<6* l{(i’*
i2)xll(7*
+ 1)), 7* + Tj) - (I?‘* i’)(X,,(T’
- {(z?‘. i’)(x,,(7*
+ q), 7* + 17))
+ q), +T*) - (if’ - i’)(x&*
+ q), 7*>}1 < Q.
If e = s(x~(T*) - x1,(7* + 7))) then we have a contradiction (35).
to (34) and Q.E.D.
Proof of Proposition 7 1. Necessity. Assume that pattern I holds. Then there is no filtering, and dW’ x=
dW2 -= dr
N’B’
-
00
N’D
0
10
19
7
N,2B,z + 5 B;. N;(,+,)
+ “$’ B;N&+
,)
i-0
i-0 n-l
-
2 DziNi i-0
(iii)
- $ DliN;, i=l
‘%is and the equation for dW2/h are merely of our definition of short-run equilibrium.
a dynamic
representation
of requirement
NONMALLEABLE
275
CITY
Lemma 3 guarantees that Pi is invariant w.r.t r so we can differentiate demolition conditions at xii to show that dx,i -=-
dr dxzi -=dr
Then we can manipulate
dx,
i = 1, . . . ) (Tr + 1).
dr ’ dx, dr ’
i=l
,...,a
the above equations to obtain
&,%I)
B,2
the
xdh)
X0
---.
X0
dw’ h
A x&;
For pattern I to hold we must have Bi > 0. The coefficient of Bi in the above equation is easily seen to be positive. The proof of necessity follows. 2. Sufficiency. Suppose pattern I does not continue at r*. By Lemma 7, pattern III must hold. If pattern III holds then there may be upward filtering but no downward filtering is possible since: (a) there are no inner redeveloped units to filter down since pattern I has obtained V r E [0, r*]; (b) if any (Hei, Nci) units were to filter down then it would be optimal for all such units to filter down and group 1 would be unhoused (see proof of Proposition 4). Pattern III will then satisfy the equations dW’ dr
-=
n+l
N;B,j - N;D,,
- 2 DliNL i=l
+ 2
F,iN,Zi + i
i=l
i-0
- ni’ D2iNi. i-0
- “i’F,iNi i-l
B; = D2i = 2rX2(i+ l)e(X2(i+ 1))7d+(i+ I) 7
F2iN2i,
- 2 F2iNiy i-0
i = 0, . . . , (n - 1).
276
NEIL VOUSDEN
Now, the demolition conditions at xii and xzi can be differentiated to give
= N,2i. W(H,Zi,arN& 7)
t(N;(,+,) - Nh) Jg! - 3p) (
ap2(Hi(i+1),
N&i+l)~
7)
(i = 0, . . . ) (n - l)),
a7
-%(i+l)
Now, a given (HL, Ni) (k = 1, 2) must satisfy P2(H& N;) + c; = P2(H,z, N;) + c,z,
u( C;, HA, N;)
k = 1,2,
= u( C,2, H,z, N;) = U2.
(37)
And, when (Hz, N,$ is given, (37) =+ C$ given; when (Hi, (37) =$ Ct given. Therefore, from (36):
aP’(Hk,
N& r) = W’(H,f, a7
a7
aP2(H& N$, r) _ @i(r) 37 dr
dPi(r) Ni, r) =dr
’
(36)
Ni) is given,
i = 1, . . . ) (n + l),
i= 0 ,...,a
’
Thus, we can manipulate the equations for pattern III to obtain
n+l
+
x i-
F,,NFi + 5 F 21 .N2.2,)( 1 + 1
&i).
(38)
i-0
Now, a switch from pattern I to pattern III requires that site rents for an (Hi, N,$ structure at the city boundary not become positive, and this in
NONMALLEABLE
CITY
277
turn requires that: dP,z/dr
I 0.
Since the last of the three terms on the RI-IS of (38) is nonnegative we have
This completes the proof of sufficiency.
Q.E.D.
REFERENCES 1. A. Anas, Dynamics of urban residential growth, J. Urban Econ., 5.66-87 (1978). 2. M. J. Beckmann, On the distribution of urban rent and residential density, J. Econ. Theoty, 1, 60-67 (1969). 3. M. Fujita, Spatial patterns of optimal growth, J. Urban Econ., 3, 193-208 (1976). 4. M. Fujita, Toward a dynamic theory of land-use, Pqv. Reg. Sci. Assoc., 37, 133-165 (1976). 5. D. Harrison, Jr., and J. F. Rain, Cumulative urban growth and urban density functions, J. U&an Econ., 1, 68-98 (1974). 6. R. F. Muth, “Cities and Housing,” Univ. of Chicago Press, Chicago (1969). 7. R. F. Muth, A vintage model of the housing stock, Pq. Reg. Sci. Assac., 30, 141-156 (1973). 8. Y. Oron, D. Pines, and E. She&in&i, Optimum vs. equilibrium land use pattern and congestion toll, Bell J. Econ. Manag. Sci., 4, 619-636 (1973). 9. S. Rosen, Hedonic prices and implicit markets: Product differentiation in pure competition, J. Pal. Econ., 82, 34-55 (1974). 10. R. Solow, Congestion cost and the use of land for streets, Bell J. Econ. Manag. Sci., 4, 602-618 (1973). 11. J. L. Sweeney, Quality, commodity hierarchies and housing markets, E&wmetrica, 42, 147- 167 (1974). 12. J. L. Sweeney, A commodity hierarchy model of the rental housing market, J. Urban Econ., 1, 288-323 (1974). 13. W. Wheaton, A comparative static analysis of urban spatial structure, J. Econ. Theory, 9, 223-237 (1974).