Spatial patterns of urban growth: Optimum and market

JOURNAL

OF URBAN

ECONOMICS

3, 209-241 (1976)

Spatial Patterns of Urban Growth: Optimum and Market MASAHISA FUJITA’ Department of‘Regiona1 Science, University

of Pennsylvania,

Philadelphia, Pennsylvania 19104

Received December 16, 1974; revised September 2, 1975 The optimum spatial pattern of urban growth for a problem which is essentially a dynamic version of von Thtinen model of land use is first studied by using a notion of bid land price curves, which is a dynamic extension of bid rent curves by Alonso [l]. Then a competitive market of which equilibrium growth path coincides with this optimum one is obtained by examining demand side properties. It is hoped in this way to throw some light for the future development of dynamic theory of urban land use.

1. INTRODUCTION Since the mid-1960’s, increasing attention has been paid to the economic theory of urban land use by scholars in various fields. The economic theory of urban land use was started by Alonso [l] generalizing the von Thtinen model of agricultural land use to urban land use. Since then, it was further developed and refined by Muth [9], Mills [7], Solow [13], Yamada [lS], and others. Recent studies of urban land use have another source originated by Solow and Vickrey [14]. They analyze optimum urban land use making use of control theory or programming. Recent examples in this stream are Mills [S], Dixit [2], Sheshinski [12], Livesey [63, and Riley [l I]. These studies made great contributions to the subject. But when we try to apply these models to the study of real cities, it is recognized that they have one common severerestriction, namely, the static nature of theory. In rapidly growing cities, like most large Japanese cities, the urban area does not develop densely especially in part of the suburbs, and land price is not proportional to land rent as would be expected from static models. Thus the static theory is not adequate to explain the spatial structure of cities. It is also easy to recognize that the construction of a dynamic theory of urban land use is not a simple task since urban phenomena are complicated enough even for a static analysis and, moreover, it is difficult even to find 1This paper is an outgrowth of the author’s paper read at the 10th Meeting of the Japan Regional Science Association, Oita, December 1973. The author wishes to thank Professors Yasuhiko Oishi and Noboru Sakashita for their valuable comments. 209 Copyright 0 1976 by Academic Press, Inc. All rights of reproduction in any form reserved.

210

MASAHISA

FUJITA

out a reliable nonspatial theory of economic dynamics. It is hoped in this paper to throw light on this problem. We first formalize a normative model of urban spatial growth which is a dynamic extension of the von Thiinen model. Our main concerns in this paper are two: first to get an explicit solution for this problem so as to obtain a detailed economic characterization of the efficient urban spatial growth process; and second to investigate whether there exists a competitive urban spatial market whose equilibrium growth path coincides with the normative one. We first state the mathematical optimality conditions for the original problem. Then, instead of solving it in a purely mathematical way, we show these optimality conditions can be viewed as the equilibrium conditions in a competitive market through which the public authority tries to realize the optimum path by using an appropriate subsidy policy for construction of buildings. After we obtain an economic translation of the optimality conditions, the general solution of the original problem is obtained by purely economic reasoning. In this way, we get a detailed economic characterization of the optimum growth process. Finally we investigate the necessarycharacteristics of renters for a purely competitive market to attain the optimum path. Hence, we obtain also an equilibrium growth path for a specific dynamic urban land market. It is shown that, in contrast with the static case, general features of the spatial pattern of urban growth are the “sprawl-fashioned” suburbanization and a large mixture of activities in most areas. We get also a dynamic extension of “bid rent theory” which was one of the main contributions of Alonso [I]. 2. ASSUMPTIONS AND PROBLEM FORMULATION In this section we formulate a normative problem of dynamic urban land use whose solution is the central concern of this paper. Supposea city is to be developed on an isolated featurelessplain. We divide the area into n districts, and they are given indices dl < dz < . . . < dl < . . . < d,, O
I=l,2

,...,

n-l,

sn=OO,

(2.1) (2.2)

where dl is the distance of district I from the predetermined center of the city and szis the size of district I. Since n can be arbitrarily large, locations within each district are considered to be homogeneous. The city is to consist of m types of buildings each of which is identified not only by its structure but also by its lot size. For example, we may suppose that i = 1 and i = 2 represent, respectively, a particular type of businessbuilding and a particular type of commercial building and i = 3, . . .each represents a certain type of residential house.

SPATIAL

GROWTH

PATTERNS

211

The planning horizon is denoted by [0, T] where T is assumed finite for a while and will be made infinite in Section 5-3. Denote the total number of buildings of type i in location 1 at time t by ~~~(2);this number will be increasedor decreaseddepending on the construction or the destruction of that building. For simplicity, the destruction of any building is not considered in this paper.2 Thus, denoting the number of buildings of type i constructed in district I at time t by Qt), we get cl(t) = m(t),

Uit(f) 2 0.

The lot size of each type of building is denoted by ki which is a constant. Hence, the spatial restriction should be satisfied in each district as follows: C kixil(t) 6 st. Now we make the important assumption that the necessaryamount of each type of building in the city is predetermined as a function of time t, and this is denoted by &(t). As will be explained in Section 7, this assumption can be considered to come from another assumption that the demand for each type of building in the whole city is predetermined. Then, the following condition should be satisfied at each t:

Differentiating both sides with respect to t, this condition is restated as follows: C

uil(t)

=

oi(t>,

where “ . ” denotes the time derivative. Since we are interested in spatial patterns of urban growth in this paper, it is assumedhereafter that Al.

d,;(t)&O,

i=

1,2 ,...,

m,O5ttT.

(2.3)

The spatial pattern of urban growth depends on how we choose the value of uil(t) at each time under the above physical restrictions. Hence, we now turn to the evaluation of planning. The objective function employed in this paper is T

J0

ecBtC C [*iz(t)xidt) i 1

- Bi(t)uidt)]dt.

Here, &(t) is the construction cost per unit of building i at time t, and is assumed to be the same in all districts. Construction cost does not include * The spatial pattern of urban growth with renewal is mathematically quite complicated. For a partial solution for this case, see Fujita [3].

212

MASAHISA

FUJITA

land cost. It is enough, from the normative point of view, to interpret +i2(f) to represent a subjective monetary value attached by the planning authority to a unit of building service i in district I at time t. But, as will be explained in Section 7, it will be more interesting from a positive point of view to consider that \friz(t) represents a “bid rent” of renter group i, which has a one-to-one correspondence with the building type i, for this building. Finally, /3 is the rate of time discount. Summarizing, our problem is now formulated as follows. PROBLEM

l=l,2

)...,

A. Obtain a value of construction speed uiz(t) (i = 1, 2, . . ., m, n, 0 _I t 5 7’) to maximize the objective function T

I II

e-fit 5 $

-

[\Tri&)Xi&)

&(t)z&)ldt,

(2.4)

subject to the following restrictions: (a) building stock-construction relation &z(t) = &z(t),

(2.5)

m(t) 2 0;

(2.6)

(b) total demand restriction c

f&z(t)

=

B,(t);

(2.7)

I

(c) spatial restriction

c ksGz(t)5 sz;

(2.8)

Xii(O) = &l(O).

(2.9)

(d) initial condition In this problem, Gil(t), l&(t) and Di(t) are given functions of time. Further, it is assumedfor a mathematical reason that A2. Gil(t), B<(t), and ai

are continuously differentiable with respect to t.

In short, this problem says: allocate the new construction requirement for each type of building among districts at each time, subject to the spatial restriction of each district, so as to maximize the total net revenue obtainable from all buildings over the planning time. 3. OPTIMALITY CONDITIONS AND DECENTRALIZED PROCEDURE The optimality conditions for Problem A are first obtained in this section. It is then shown that these optimality conditions coincide with the equilibrium

SPATIAL

GROWTH

PATTERNS

213

conditions in a competitive market through which the planning authority tries to realize the optimum path of Problem A by a decentralized procedure. Such restatement of optimality conditions via market conditions has two advantages. First, it enables us to obtain the solution of Problem A by purely economic reasoning, without any knowledge on control theory. Second, it provides an answer to the question in Section 7 whether there exists a perfectly competitive market whose equilibrium path coincides with the optimum one of Problem A. Optimality conditions for Problem A can be summarized as follows by applying the maximum principle in optimal control theory.R Optimality Condition A

For nonnegative uil(t) (i = 1, 2, . . . , m, 1 = 1, 2, . . . , n, 0 S t 6 T) to be an optimum construction processfor Problem A, it is necessaryand sufficient that there exist xii(t) and a set of multipliers 4tl(t), pi(t), and qi(t) (i = 1, 23 . . . . m,l= 1,2, . . . . n, 0 5 t 5 T) such that: (i) building stock-construction relation (3.1)

*
(ii) variation of expected revenue dil(t)

=

(3.2)

-$tl(t);

(iii) equilibrium condition in land market p,(t) I 0 for each I,

(3.3)

tit(t) = 0 when C kixil(t)

< s[;

(3.4)

(iv) nonprofit condition for builders + q,(t) 5

&l(t) [+il(t>

b,(t)

+ kipl(t)

+ qi(t) - bt(f) -

kipl(t)]U,l(t)

for each i and I, = 0;

(3.5) (3.6)

(v) demand constraint c m(t) = h(t); 2

(3.7)

(vi) initial condition Xi@) = 5&(O);

(3.8)

3Sincecondition (c) in Problem A restricts the allowable range of state variables, an elementarymaximumprincipleis not enough for Problem A. Optimality Condition A was obtainedby applyingHestenes [S,Chap.8, Theorem2.11. This theorem is a restatement of Pontryagin ef al. [IO, Theorems 23, 241 in a more convenient form.

214

MASAHISA

FUJITA

(vii) terminal condition h(T)

= 0,

(3.9)

pz(T) = 0.

(3.10)

In the above, the following notation is used. h(t) = e-+&(t)

(3.11)

yTiz(t)= e-Wiz(t;.

(3.12)

The economic interpretation of each condition above will be explained hereafter. Though the above conditions are purely mathematical, we want to translate them into economic conditions. For this purpose, consider how to realize the optimum path of Problem A through a decentralized procedure in a competitive market, instead of by using computers. Suppose a perfectly competitive market composed of many builders for each type building, many land owners in each district, and the public authority. Also suppose that every member of the market has the same time discount rate 8, given in (2.4). Further assume that the time horizon for every member is the same and is given by Tin (2.4).4 The public authority has a duty to realize the optimum path of Problem A and thus intervenes in the market when necessary.This public authority is the only (i.e., representative) renter of all buildings in the city, and the authority announces at the outset that it will pay Gil(t) dollars of rent for a unit supply of building (service) i in district 1 at time t. The function Giz(t) is the same as in (2.4). Further supposethat each builder becomesthe owner and the supplier of that building thereafter. We denote the land price in district 1 at time t by P,(t). If there is no public intervention in this market, profit nil(t) for a unit construction of building i in district I at time t is L(t)

= &z(t) - h(t) - kPz(t),

where T

&z(t) =

st

e-~(~Wiz(r)d7,

(3.13)

the present value of the total expected revenue for this building. Therefore, the construction level uit(t) of this building is chosen by builders as follows: m(t)

= a = 0 = indefinite,

when @iz(t) when Q(t) when Cbiz(t)

> < =

l&(t) R(t) &(t)

+ + +

Wz(t), Wt(t), kPz(t).

4 For this assumption to be realistic, Tshould be infinite. But, for simplicity, it is supposed for T to be finite for a while, and later we assume T to be infinite in Section 5-3.

SPATIAL

GROWTH

PATTERNS

215

Hence, the next relation should be satisfied for each i at each t for the planning requirement (2.7) to be satisfied in this market. ail(t) I &(t) + kiPz(t) for each Z, and

(3.14) a,,(t) = &(t) + W’,(t)

for at least one I when h%(t) > 0.

Since +il(t) and &(t) are predetermined functions of t, and since P,(t) is determined in the land market, it is possible only by chance that the above relation is satisfied for every i and 1 at each time. This means the public authority should intervene in the market to satisfy the planning requirement (2.7). One way is a subsidy (and tax) policy in the construction market. Suppose the public authority subsidizes Qi(t) dollars per unit construction of building i at time t. Then, the profit &(t) is calculated as follows. %(t) = h(t)

+ QiW - h(t) - U’dt).

(3.15)

Hence, for the condition (2.7) to be satisfied, the value of subsidy Ql(t) should be chosen so as to meet the following condition. (ail(t) + Qi(t) 6 &(t) + M’,(t)

for each Z,

and

(3.16)

ail(t) + Qi(t) = &(t) + W,(t)

for at least one 1 when B>(t) > 0.

But, since the construction function assumes constant returns to scale, builders are indifferent as to how much to construct of building i in district I if the equality in (3.16) is satisfied. Thus, the public authority should also control the total number of new buildings of each type so as to satisfy the planning requirement (2.7), i.e., T ui~(t) = hi(t)

where IIil(t)uiL(l) = 0.

(3.17)

= e-Widt),

(3.18)

Let us put h(t)

pz(t) = e-W,(t) 2

(3.19)

qi(t) = e+Qi(t).

(3.20)

and Then, (3.16) together with (3.17) is equivalent to (3.5) and (3.6) together with (3.7). Thus, &l(t), pi(t), and qi(t) in Optimality Condition A represent, respectively, the discounted value of expected revenue for each building, the discounted land price in each district and the discounted value of subsidy for each construction. Hence, conditions (3.5) and (3.6) represent the nonprofit condition in the construction market in terms of discounted prices.

216

MASAHISA

FUJITA

From (3.13) and (3.18), condition (3.9) states the terminal condition in this market. Further, (3.2) is the restatement of (3.13) in terms of discounted prices. Now we turn to the land market. Since it is assumedthat no one considers the future further than T, there is no expectation of rent revenue from the land after T. Thus, P,(T) = 0. (3.21) Further, the land market should satisfy the next condition in equilibrium so as to prohibit speculative profit. l%t> 5 /m(t).

(3.22)

On the other hand, if R(t) < Wz(t) holds for vacant land in district 1, the land market is also not in equilibrium since the growth rate of the land price is below the expectation land owners have in this district. Therefore, Pi(t) should satisfy Pz(t) = pPz(t) when C kixil(t) < sL. I

(3.23)

Since we do not consider destruction of buildings, there cannot exist any land market in a district once it is fully occupied by buildings. Thus, (3.21), (3.22) and (3.23) give enough conditions for the equilibrium of land market.5 Substituting (3.19) into (3.21), (3.22), and (3.23), respectively, we get (3.10), (3.3), and (3.4). Hence, they represent equilibrium conditions in the land market. Since (3.1) and (3.8) state physical conditions, they should hold also in the market process. Summarizing the above discussion, we obtain PROPOSITION 1. Supposethe public authority announcesat the outset that it will pay qit(t) dollars of rent per unit supply of building i in district I at time t. We further assumethat the land and construction markets are perfectly competitive and that every member in these markets has the same time discount rate p, which is given in (2.4). Then, when the initial Iandprices P,(O) (I = I, 2, . ..) n) are appropriately given6 for the public authority to realize the optimum path of Problem A, it only has to choose the construction subsidy Qi(t) at each time so as to satisfy condition (3.16) and also has to control the 5Though they are enough conditions for the solution, we can consider a fictituous land market in each fully occupied district to investigate the relation between the land price and the land rent. For this point, see Section 5-2. 6The word, “appropriately,” means that the market process ends up satisfying terminal condition (3.21). Here remains a fundamental difficulty of decentralized procedure in this dynamic model.

SPATIAL

GROWTH

PATTERNS

217

total construction of each building i in order to satisfy condition (3.17). In this case, the equilibrium conditions coincide with the optimality conditions for Problem A.

Thus the economic meaning of Optimality Condition A is clear. It is now also possible to attach a market interpretation for each auxiliary variable there. We are especially interested in the variation of land prices Pi(t) [and hence,pi(t)] over space as well as over time. Incidentally, from (3.16), the rule for the choice of subsidy value Q,(f) for the public authority is simply stated as follows: Qi(t) = rnin [i&(t) + kiPl(t)

- ail(t)]

when B,(t) > 0,

and

(3.24)

Qi(t) is any value less than rnin C&(t) + kiPL(t) - a,,,(t)]

when B,,(t) = 0. 4. LAND PRICE CURVES AND BID PRICE CURVES The introduction of the notion of bid rent curves was a key for the development of urban land use theory in Alonso [I]. The purpose of this section is to extend the static notion of bid rent curves to our dynamic case and to examine the relations between land price curves and bid price curves at each time. The notion of bid price curves helps our intuitive understanding of the problem since it enables us to analyze the problem in “urban space” instead of in “commodity space” in formal economic theory. To obtain the explicit solution for Problem A, we hereafter employ the next simplifying assumption. A3.

G,,(t) = %‘io(t) - a,dl,

a, > 0.

(4.1)

That is, each rent function %jt(t) given in (2.4) decreasesat the rate ai with distance from the city center. Since the naming of buildings is arbitrary, let buildings be given indices in order of the steepnessof their rent functions per unit of lot as follows: A4.

al/k1 > at/k2 > . . ’ > a,/ki > . . . > a,, ‘k,.

(4.2)

The analysis in this section starts by defining a function ptl(t) for each building i as follows: pi&) = [h(t)

+ qi(f) - bd,r)llk;

(4.3)

From (3.15) and definitions (3.1 I), (3.18), and (3.20),pll(t) shows how much profit per unit of lot will result from construction of building i in district I at

MASAHISA

218

FUJITA

time t if the land price were zero.’ To put it another way, this shows the maximum value of the land price which builders of facility i can bid for a unit of land in district 1at time t. Thus, considering index I varies over districts, we call function pit(t) the “bid (land) price curve” of builders of building i at time t. Then, from (3.13) and (3.18), we seethat apil(t)/ad, = - (Ui/ki)CU(T) - u(t)], where

(4.4)

t u(t)

=

/

e-flrd7 = (1 - e-@)//?, 0 5

t

S T.

(4.5)

0

We now summarize useful relations between land price curves and bid price curves for later analysis. They are stated only in terms of discounted value to avoid repetition. First, the following three properties are translations of (3.3), (3.4), and (3.10), respectively. Property 1. The land price pi(t) never increasesover time. Property 2. The land price pi(t) does not decrease while there remains

vacant land in that district. Property 3. The land pricepl(t) is zero at the terminal time T, i.e.,pl(T) for every I.

= 0

From Properties 2 and 3, we seethat Property 4. If there remains vacant land in a district at the terminal time T, the land price in that district was zero from the beginning.

Further, from (3.5) and (3.6) and from definition (4.3), Property 5. The bid price curve pil(t) is never above the land price curve pi(t) in any district. When building i is being constructed in a district, the bid price curve pit(t) touches the land price curve pi(t) in that district.

On the other hand, from (4.4), Property 6. Under A3, bid price curve pit(t) is a linear function of distance from the city center with gradient - (u~/ki)[a(T) - u(t)]. Hence the steepness of pit(t) is maximum at t = 0, decreasescontinuously with time and is zero at t = T.

Denote the urban fringe at T by i, that is, i is the farthest district from the city center in which x&J > 0 [more correctly, x&J - &l(O) > 0] for at least one i. Then, the next two properties are easily seen from Properties 4, 5, and 6. 7 It should be noted that each “price” variable denoted by a small letter represents the discounted value of current price. Further, each phrase “the present value of” and “the discounted value of” is omitted whenever possible.

SPATIAL

GROWTH

PATTERNS

219

Property 7. Under A3, there remains no vacant land between the city center and the urban fringe at the final time T, that is, x; kixil(T) = sL for each I < i. Property 8. Under A3, ~~(0) > 0 for each I < i and ~~(0) = 0 for each

I _r 1. From Properties 2 and 5, the initial land price curve pi(O) is the upper envelope of a set of linear functions pii( i = 1, 2, . . , , m, t E [0, T] in which the vaIue of qi(t) is fixed to that of the optimum solution. The upper envelope of linear functions is a convex function. Hence, considering the latter half of Property 6, Property 9. Under A3, the initial land price curve ~~(0) is strictly convex* between the city center and district i.

Considering the above properties, the relation between the land price curve pi(t) and each bid price curve pzl(t) can be depicted as in Fig. 1. Hereafter, denote the district of the construction of building i at time t by &(t). That is, Zi(t) is the district in which uil(t) is positive at that time.g Then, Property 10. Under A3, the construction site of each building i moves toward the suburbs at each time. That is, Zi(t’) 2 Ei(t) when t’ > t.

FIG. 1. General relation between the land price curve pi

(4.6)

and each bid price curve p,&).

* This is literally true only when we take distance dt continuously from the city center and thus there is an infinite number of districts. When dl is discrete, the initial land price curve is as in Fig. 5. g It is not difficult to see from Properties 5 and 9 that there does not exist a finite period of time at which u;r(t) is positive more than one district for each i. For a mathematical proof of this fact, use (3.4), (3.6), and A3.

MASAHISA

220

FUJITA

Considering that uil(t’) > 0 implies vacancy in district I until time t’, it is not difficult to see the above property by using Properties 2, 5, 6, and 9. On the other hand, suppose&(t) > 0 for 0 5 t 5 T. Then, the bid price of building 1 should always touch to land price curve in some district with vacant land. Thus, since building 1 has the steepest bid price curve all the time, we obtain the next property by using Properties 2, 5, 6, and 9. Property 11(a). Under A4, the land price pi(t) in each district remains constant from the initial one until the bid price curve of building 1 touches and leaves the land price curve in that district. That is, pi(t) = pi(O)

for

I? L(t).‘O

(4.7)

Without using A4, this property can be restated as follows. Property 11(b). The land price pi(t) remains constant in each district when there remains vacant land between this district and the city center.

Define the “basic bid price curve” &(t) as l%(t) = [dJi&> - b,(t)-J/k.

(4.8)

Then, from (4.3) and (4.4), pil(t) moves up or down parallel to $;l(t), depending on the subsidy q%(t). Hence, the relation between pit(t) and qi(t) can be depicted as in Fig. 2. The role of the public authority is to control qi(t) so as to make the bid price curve just touch the land price curve in some district.

P,,(i)

Fm. 2. Relation between the bid price curve p%l(f> and the construction subsidy q<(r) lo Even when fil(r) = 0 for some time interval, we can require the bid price curve of building 1 always to touch the land price cruve in some district. Hence it can be seen that this property holds even in this case. For an example of land price change in such case, see Section 5-3.

SPATIAL

GROWTH

PAITERNS

221

Our next problem is to obtain the explicit solution of Problem A by using the above properties.” Since destruction of buildings is not considered in this problem, assumefor simplicity of figure presentations that %2(O), i== 1, 2, . . .) m, I = 1, 2, . . . , n.

(4.9)

This is equivalent to the assumption that each s1given in (2.2) is equal to the size of district I minus ~~~(0). Finally, define the present value of bid price, pil(t), as follows for later analysis: (4.10) Pd(t> = @pdt). 5. OPTIMUM SPATIAL PATTERN-THE CASE OF TWO BUILDING TYPES (m = 2) Assuming that the city is to be composed of only two types of buildings (i.e., m = 2), the optimum process of spatial growth for Problem A is first obtained in this section. Then, the time variation of the land price curve is examined in detail. Finally, a comparative analysis of solutions is given briefly. Suppose,to be specific, that building 1 represents office buildings and building 2 residential houses. Though all the mathematical results in this section are true under the assumption of b6(t) 2 0 (i = 1, 2), figures are drawn assuming bi(,) > 0 for 0 -I t -_< T (i = 1, 2).12 Further, all figures hereafter are drawn assuming that distance dl changes continuously except when discrete representation is necessary. 5.1. Optimum Growth Process

We obtain the optimum spatial pattern of Problem A, by using properties obtained in the previous section. We assumeA4 holds throughout, and thus, al/k1 > a2/kx.

(5.1)

We know from Properties 7 and 10 that construction of one of two building types should begin in the district nearest the city center at the outset. From (4.4), (5.1), and Properties 5 and 9, it should be type 1. Thus, the bid price curve pii of building 1 supports the land price curve pi in district 1 and the bid price curve pi supports the curve ~~(0) in some district far from the city center. That is, L(0) = 1, 1 5 12(O)I n. 11In this paper an emphasis is put on a graphical representation of the solution procedure. For a rigorous, purely mathematical solution procedure for Problem A, see Fujita [4]. ‘2 Other cases are briefly discussed in Section 5-3.

MASAHISA

222

FUJITA

Pe(1) PJf)

e (al/k,)

(c(T)-=(t))

ia,/k,)b(T)-Tit))

FIG. 3. Relation between the land price curve p&) and each bid price curve p,&).

The relation between the land price curve and bid price curves at an intermediate time t (0 5 t 5 T) can be drawn as in Fig. 3. From (5.1), building 1 is always nearer the city center than building 2. That is, from (2.1), Z,(t) =< L(t),

0 5 t 5 T.

(5.2)

Further, from Property 1l(a), the land price pi(t) is equal to the initial price ~~(0) for each I 2 l,(t) as depicted in this figure. From Properties 7 and 10, all the districts between L(t) and the city center are fully occupied by building 1 or by both types of buildings. Then, since there remains no land to be transacted and thus a land market can not exist in these districts, land prices in these districts are essentially indeterminate. But, from economic reasoning explained in Section 5-2, we let the land price in each of these districts be the same as the bid price of building 1 in that district, that is pi(t) = plz (t) for 1 s L(t), as depicted in Fig. 3. Finally, from Properties 3, 1l(a), and (5.2), we have at T that II(T) = L(T) =

i.

(5.3)

where the urban fringe i is known, from Property 7, as the district that satisfies i-1 i C sz < C kiDi $ C $1. (5.4) I=1

i

2-l

Therefore, the general process of the optimum urban spatial growth under (5.1) can be summarized as follows. (1). The construction of building 2, whose bid price curve is less steep than that of building 1, begins from some district &(O)[l s IZ(0) 6 n] far

SPATIAL

GROWTH PATTERNS

223

from the city center. It occupies part of this district and, while some vacant land remains there, moves to the next district Z2(0)+ 1. It similarly moves gradually toward the suburbs leaving some vacant land in each district, and finally reaches district i at T. (2). Construction of building 1 begins in the nearest district to the city center. Then, occupying the whole area in each district, it moves step by step toward district &(O) - 1. After this district, it gradually occupies the whole remaining area in each district after construction of building 2, and finally it is constructed in district i. Figure 10(a) depicts this process of urban spatial growth. The vertical axis represents the land use ratio 0 between two types of buildings in each district, and the horizontal axis shows the distance from the city center. One of the outstanding characteristics of this growth processis that the construction of facility 2 moves toward the suburbs while leaving a large area vacant in each district. Hence, this growth process of urban space has the property of urban sprawl, and results in a mixture of different buildings in these districts. Thus, the resulting pattern of urban land use is quite different from those in static models like von Thtinen and Alonso. We now turn to the determination of the location of Z2(0),the number of each building constructed in each district, and the time of each construction. For this purpose, we denote the time when the construction of building i moves from district I to district I + 1 by til* (I = 1, 2, . . . , i - 1 for i = 1, and Z = Zz(0),L(O) + 1, . . ., i - 1 for i = 2). Further, tllo is defined as the time to satisfy the next relation. klDl(tll')

I10 o s

0,

tl;

=

=

m

2 j-1

Sj,

if kIDI

(5.5)

< 6 si. j=l

Thus, tllo denotes the required time for building 1 alone to occupy the whole area between district I and the city center. The value of tl10 is determined directIy from (5.5) since Di(t) is a given function of time. Therefore, if we know the district Z2(0),each switching time til*[l

s z 5 Zz(O)- l]

is determined as follows. tll

*- -

tltO,

1 5 z 6 12(O)- 1.

(5.6)

When dr is continuous, as seen from Fig. 3, district Z2(0)is determined from the time at which the slope of the bid price pll(t) is equal to that of ~~~(0).But, when dl is discrete, this procedure becomesmore complicated as shown in Fig. 4. Substitute h for Z*(O)for a while for simplicity of notation.

224

MASAHISA

PC (0)

P,(f) Pi,

FUJITA

it)

Ph.,(O)‘P,., p,(O):

dl

Ph(‘;h.l)’

PhL$)

Ph+,(0)=Ph+,(f,;)

i\

O

If” Ih-I 1

d h-1 dh

d htl

%

d,

b2/k,)qtT)

FI;.

4. Determination of the initial district h[=&(O)]

for the construction of building 2.

Then, from Property 5 the bid price line pzL(0) touches the price curve ~~(0) in district h at t = 0, the bid price line plr(tlh-lO) touches the price curve pl(tlh-lO) in districts h - 1 and h at t = tlh-lo, the bid price line pll(tlh*) touches the price curve PE(~M*)in districts h and h + 1 at t = tlh*. From Property 11(a), &-l(O)

=

+-l(flh-lo),

$h(O)

=

ph(tlh-lO)

=

&(tlh*),

P/L+@)

= ph+l(hh*).

These relations are depicted in Fig. 4. Considering Property 9, the slope of ~~~(0)is smaller than that ofpll(ta-1”) and larger than that ofpll(tlh*). Thus, using (4.4), (al/kl)[a(T)

- u(tlh*)l < (az/kz)u(T) I (al/W-u(T)

- a(tl~l)J.

By definition, tlh* 5 tlho and thus o(I~*) % u(tlho). Hence,

u(tlh-lo)5 [1 -g$]um

< 41h0).

(5.7)

Therefore, calculating each value of tllO(l 5 I 5 i - 1) by (5.5) the district h, that is district L(O) is uniquely determined from (5.7). The determination of each switching time tij*[ls(0) 5 j 5 i - I] is simple. From Properties 5 and 11(a), the slope of the bid price line pzl(tzi*) should equal that of pll(tlj*) as shown in Fig. 5. Hence, using (4.4), [U(T) - u(tlj*)]al/kl

= [u(T) - u(t,j*)]aJkz, j = k(O), L(O) + 1, . . . , i - I.

(5.8)

SPATIAL

di

GROWTH

d,+, 8,

(@,/k,~(u~,T~--~It;,~~-~a~

225

PATTERNS

de k,:

<-t,‘!-w’+;,,,)

FIG. 5. Determination of the construction switching time rtj*.

Since the whole area between I = 1 and I = j should be fully occupied, because of Properties 7 and 10, by buildings 1 and 2 at time f13*, klDI(tlj*)

+ w&j*)

= k sz, j = L(O), h(0) + 1, . . ., 7- 1. (5.9) Z=l

FIG. 6. Determination of fir0 and t,t* (m = 2).

226

MASAHISA

FWJITA

Thus, construction switching time tij*(j = Z2(0),Z2(0)+ 1, . . ., i - 1) is obtained by solving the simultaneous equations (5.8) and (5.9) for each j. The economic meaning of condition (5.8) is not difficult to see from our experience with the static von Thtinen model. Figure 6 summarizesthe determination of Z2(0)(=h) and tij* in one diagram. The horizontal axis and the vertical axis represent the time for i = 1 and the time for i = 2 in proportion to the value of u(t). Of these coordinates, equations C kiDi

2

= 2 Sjy I = 1, 2, . . . , i

- 1

(5.10)

= [o(T) - a(t,)]az/kz

(5.11)

j=l

give a set of downward sloping curves, and [a(T) - a(tl)]al/kl

defines an upward sloping line, as depicted in this figure. The intercepts between the horizontal axis and curves (5.10) give the value sof tit’. Further, intersections between (5.11) and (5.10) determine the values of til*, and the intercept between (5.11) and the horizontal axis definesthe district h [ =Zt(O)]. In summary, the optimum construction process uil(t) for Problem A is uniquely determined as follows;

uim =

B,(t), tu* 5 t < tu*, 0, otherwise,

(5.12)

where switching time tir* (i = 1, 2, I = 1, 2, . . ., i) is uniquely obtained from (5.5), (5.6), (5.7), (5.8), and (5.9) and we put til* = T for I 2 i. The number of buildings constructed in each district I is given by D&l*)

- Di(til-I*),

i = 1, 2.

(5.13)

5-2. Land Price and Land Rent We here examine the time variation of the land price curve corresponding to the optimum expansion process obtained in the previous section. We know from Property 1l(a) that pi(t) = ~~(0) for I 4 Z,(t). But, for the part of the land price curve between I = 1 and Zl(t), it is known from Properties 3 and 5 only that it should be above the bid price curves of i = 1 and i = 2 at each time and that it finally should becomezero. Mathematically, all land price curves which satisfy these two conditions give the same solution for Problem A. Thus the part of the land price curve between Z = 1 and II(t) is indeterminate. This is true economically, as noted before, since there remains no land to be transacted in these districts and thus a land market can not exist in these districts. But, if we suppose each builder has an option of purchasing the land or renting it from the land owner, we can decide the land price of that part by means of economic logic.

SPATIAL

GROWTH

PATTERNS

227

We suppose that each building can be moved freely within and only within the district at each time. This assumption is theoretically reasonable since we do not distinguish one part of a district from the other part in Problem A. Hence, each building owner who wants continuous occupation of the lot must make a new contract with some land owner in that district at each time. Of course he can buy the lot from the land owner if he desires. Further, under this assumption, even new persons (especially, new builders) could bid for the land in the fully occupied district since all renters (or buyers) are the same from the land owner’s point of view under the assumption of mobility. In this way, we can retain the land market in each district even after it is fully occupied with buildings. It is also true from the assumption of free mobility that the unique land rent and the unique land price prevail in each district at each time. On the equilibrium path (that is, in the equilibrium land market corresponding to the optimum path), of course, the land price Pi(t) and the land rent Rl(t) in each district I (I = I, 2, . . . , n) should be such that at each time it does not matter to each lot owner whether his lot is rented or purchased. Otherwise some building owners will get a negative (or positive) profit eventually and this is inconsistent with the assumption of equilibrium path. Hence, the first relation between the land price and the land rent on the equilibrium path is 7

T Pl(O

=

e-B(r-t)R,(,)d,

=

/’ t

edt

e-flTR*(T)dT. .i

Differentiating both with respect to

(5.14)

f

t,

(5.15)

h(t) = Wz(t) - h(t). Or, from definition (3.19), this is rewritten as follows: &(t)

=

(5.16)

-e-~tRl(t).

From the assumption of free mobility within a district, = 0 when C k,xi(t) < .sf.

l&(t)

(5.17)

Hence, from (5.16), we get Pi(t) = 0 when C kixil(t)

< sz.

(5.18)

Which is consistent with (3.4). Then, as the second relation between the equilibrium land price and land rent, we see Z&(t) =

0, I h/kd[W)

I > h(t),

- 41, 1 5 L(t),

(5.19)

228

MASAHISA

FUJITA

and hence, from (5.16),

&6(t) =

I > h(t),

0, i -e-yal/kl>[4,(t)

(5.20)

- dz], I =< h(t).

The first part of (5.19) holds from (5.17) and Property 10. The second part is also not difficult to see from the fact that, from Assumption (4.1), e9lz(t)/adz

=

--al,

and the fact that, from (4.4) and Property 5, $12(t) = -e-~z(al/kl)Cd~,(t>

- 4 1.

(5.21)

Equation (5.19) states that land rent in each district I [I 5 Zl(t)] is the difference between the rent of building 1 on a unit of lot in that district and the rent of building 1 in the construction district A(t) at the time t. Hence, the district of construction of buildings with the steepestrent frunction per unit lot (and hence, the steepestbid price curve) plays an important role in the determination of the land price and the land rent over the time. Summarizing (5.20) and (5.21) we get Property 12. Pz(t) =

pz(O),

zz .4(t),

PldO,

I <

1

(5.22)

m.

FIG. 7. Variation of land price curve pi(t) over time (in cases of Type (a) and Type (b-l)).

SPATIAL

GROWTH

PATTERNS

229

Since, from Property 5, the bid price curve pmt) is the tangent of the initial land price curve p!(O) in the district II(t), the time variation of the land price curve pi(t) can be drawn as in Fig. 7. The straight part of the land price curve rotates around the district Z,(t) at an angle speedof e-~“(al/kl), and this construction district II(t) of building 1 moves towards the suburbs with time. Further, to examine the present value of land price Pi(t), we consider the case of T = cc..13Then, from (4.4) and (4.10) dPil(t)/ddl = -(Ui/ki)lP.

(5.23)

That is, under A3, the slope of the undiscounted bid price curve PLl(t) is constant over time. Therefore, summarizing (5.22) and (5.23) with (3.19) and (4.10) we obtain Property 13. The suburban part of the land price curve P,(t) [that is, I 2 Z,(t)] increases at the rate 0 at each t. Under A4, the center city part of the land price curve PL(t) [that is, 1 I I I Zl(t)] is a straight line, and it is tangent to the suburban part in the district Zl(t). With further assumption of T = CCI,it has a constant gradient (al/kl)/P throughout.

Hence, the relation betweenthe land price curve Pl(t)and the bid price curves Pll(t) (i = 1, 2) can be depicted as in Fig. 8. This figure corresponds to the

0<1’<1<1”< dP,,(t)

FIG.-8.-Variation

i ddE =

(c ia,/k,)

/p

of the land price curve &(r) and the bid price curves P,l(t) (T = =).

la As noted in footnote 3, T should be infinite for a realistic market interpretation of the problem.LSufficient conditions for the existence of the optimum path for Problem A under T = 00 are given in Section 5-3.

230

MASAHISA

FUJITA

case in which the expansion of the city stops ultimately at i. If this city grows at a finitely fast speed forever, the district i is infinitely far, and thus the ultimate land price P1(w ) does not exist. 5-3. Comparative Analysis of Growth Patterns

We here study briefly how the optimum spatial pattern of urban growth changes according to the values of parameters in Problem A. First, to examine how it will change according to the speed of growth a,(t) of each building, consider the next two examples for comparison. hi(t)

> 0 for 0 5 t 5 T, i = I, 2.

hi(t)

> 0 for 0 5 t 6 f, Bi(t) = 0 for 1 < t 5 T,

(5.24a) i = 1, 2.

(5.25b)

Let the first be called Type (a), the second Type (b). Then, Type (b) is the casewhen the growth of the city stops before the end of the planning horizon. As stated before, though all the mathematical statements in the previous section are valid for both types, verbal explanations and the figures are given implicity under the assumption of Type (a). Hence, we now examine the growth patterns of Type (b) referring to the figure. Growth patterns of Type (b) can be further subdivided into two types. One is drawn in (b-l) of Fig. 10 and we call this Type (b-l). The other is given in (b-2) of Fig. 10 and this is called Type (b-2). In Type (b-2), building 2 occupies the whole area between the districts &(O) and i. Hence, no mixture of two types of building happens in this case, and the pattern of urban space after time Zconsists of two Thtinen rings. That is, it coincides with the spatial pattern of the static (i.e., instantaneous) optimum allocation of D@) and D&). This type happens when l/T is relatively small. On the other hand, Type (b-l) is a mixture of Types (a) and (b-2), and this happens when l/T is relatively large. Whether Type (b-l) or Type (b-2) takes place is known by drawing a corresponding figure on Fig. 6 under the assumption of (b). Under this assumption, area curves defined by (5.10) are kinked horizontally at tl = f and vertially at tz = Z. Therefore, if the intercept of the Iine (5.11) is to the right of a(tl) = a(l), no mixtures of the two building types occur. That is, when 2 2 __ 47 Cl - :,::,I is greater than a(Z), Type (b-2) happens. When it is smaller than ~(0 we have Type (b-l). Hence, when 2/T is very small compared with 1, we always have Type (b-2). Figure 9 depicts the variation of the land price curve pi(t) over time in Type (b-2). In this case, the initial land price curve PI(O) is kinked in district

SPATIAL

O
GROWTH

< t:,
PATTERNS

231

(h=.e2(0))

FIG. 9. Variation of the land price curve p*(r) over time (in the case of Type (b-2)).

&(O). The slope of the left tangential line is (al/kl) [a(T) - u(i)] and that of the right tangential line is (a&+(T). Since

from the definition of Type (b-2), the left slope is greater than the right one. Therefore, after r,(t) has reached district k(O) (that is t = Z), the right-hand edge of the straight part of the land price curve stays in the district k(O) for a while, and begins to move again towards the suburbs when it becomes (al/kl)[a(T) - u(t)] = (an/kz)a(T). The variation of the land price curve PI(t) in the case of Type (b-l) is essentially the same as that of Type (a). Though it has been implicitly assumedso far that T is finite, let us examine the case of T = 00. From the analysis in previous sections, it is clear that the solution of Problem A, when it exists,14is obtained simply by substituting 00 for Tin all equations there.15In this case,the optimal growth pattern can be I4 Of course, the solution always exists when ,r is finite, since we assume 3%= m in (2.2). Even n is infinitely large, the solution exists under some reasonable assumptions. For example, the following is a set of such assumptions: fl > 0, x; k,D,( m) < 00,the distance

1-l

dl of the district i, which satisfies C SJ< C$ k,D,( p ) 5 C,=I~ sl, is finite from the

1=1 city center, and finally ~o@;~(t) --+ 0, e-%(t) + 0 as t ----fm . I6 When T = m terminal conditions (3.9) and (3.10) should be replaced, respectively, with q&r(t) + 0 as t 4 0~ and pi(t) + 0 as t -+ m.

232

MASAHISA

FUJITA

0 dL 1 (0)

(b-1)

(b-2)

FIG.

d, 2(0)

dl (t) I

dp ct) 2

di

6,(ti,62(t)>OforO<=t<=i,=Ofori
b,(t1,62(ti>OforO
l,xge

i/Tts

SWII

10.Spatialpatternsof urbangrowth(m = 2).

drawn as in Figure 10(c). Building type 2 is constructed always outside of type 1. 6. GENERALIZATION

TO m TYPES OF BUILDINGS

Results in the previous section are generalized in this section to the case of an arbitrary number of building types. Since the validity of the results in this section is easily demonstrated from the analyses in the previous section, explanations will be as simple as possible. All figures are depicted for m = 3 for simplicity. Suppose i = 1 corresponds to office buildings, i = 2 to highrise residencesand i = 3 to low-rise houses.Assumption A4 is made throughout this section. The location Zi(0) of initial construction of each building i is given by the tangential district between the initial land price curve ~~(0) and the initial bid price curvepit( Under A4, II(O) 5 k(O) 5 . . +I Z,(O)5 . . .5 Z,(O). Figure 11 depicts the relation between these curves at an intermediate time t (0 5 t s T). The part of land price curve pi(t) farther than Z,(t) coincides with that part of the initial curve ~~(0). The center city part is equal to that part of the bid price curve pll(t) of building 1. Thus, the variation of land price curve is dominantly influenced by the location of the construction of

SPATIAL

GROWTH

p,(t)

‘4

PATTERNS

= PL(0),

e>1,

233

(t)

/

(q/k,)

b(T)-&(t))

b,;k,)b(T)-o(t))

(0,/k,)

(a(T)-a(t))

FIG. 11. Relation between the land price curve pi(t) and bid price curves ~,~(r) (m = 3).

building 1. Further, becauseof A4, Z,(t) 5 12(t) s.. . s Zi(t) I..

. 5 Zm(t), 0 5 t s T.

(6.1)

Construction site Zi(t) of each building moves towards the suburbs at each time, as explained in Property 10. The general process of optimum spatial growth can be drawn as in Fig. 12. Except for districts between I = 1 and I = Z2(0),every district eventually has more than one type building. The further a district is from the city center, the larger the number of the types of buildings. Let us obtain the exact location of each initial district Z,(O)and the exact time of each switching tll*. First, 11(O)= 1.

(6.2)

Suppose next that districts Z2(0) and Z3(0) be known. Further, denote the switching time of construction of building i (i = 1, 2) from the districts Zto I+ l[Z,(O) 5 Z < Z,(O)] by til. Since 12(O)and 13(O)are unknown at present,

FIG. 12. Spatial pattern of urban growth (m = 3).

234

MASAHISA

FUJITA

we use tiz instead of til* for a while. Then, since the gradient of bid price curve pIi at t = tlz is equal to that of pzj(t) at t = t21, and since all the districts between the city center and the district Z are fully occupied by buildings 1 and 2 at time tlz, [a(T) - a(tld]al/kl ; kiDi

= [a(T) - u(tdlaa/kz, (6.3)

= 2 q.

i=l

j=l

Hence, the location Z2(0)is obtained as the nearest district to the city center among all districts for which the values of tlz and t2l calculated from (6.3) are both positive.16Since we now know Z2(0),we put *- - t1z, 1 s I < 12(O). t11 (6.4) by solving the equation klDl(tll) = C+I~ si for each I. Suppose next we know district &(O). Similarly, we show for awhile the switching time of building i (i = 1, 2, 3) from district Z to Z+ 1 [13(O)s Z < Z4(0)] by tiL instead of til*. Then, an analogous reasoning,

[u(T) -

u(tl~>-Jal/kl

= [u(T) - u(tzt)lQcz = [a(T) - &dl+s

; kiD&)

= i sj.

i=l

(6.5)

j=l

Hence, the location Z,(O) is given as the nearest district to the city center among the districts for which the values of tiz (i = 1, 2, 3) are all positive. Now that we know districts h(O) and Z3(0),by solving (6.3) we put tiz* = :tii,

i = 11,2,

12(O)

5

z <

13(O).

(6.6)

n,,(t)

0

FIG. 13. Construction profit function &l(t)

and construction site [i(r) (m = 3).

16When we do not have such districts, we must use (6.5) instead of (6.3). But, since each district is assumed to be sufficiently small, we neglect this problem.

SPATIAL

GROWTH

PATTERNS

235

With similar processes,we can obtain all the values of Ii(O) and ti2* one by one. Then, using these values, the optimum construction process is uniquely given as follows; a(t)

=

D,(t),

t&-1* 5 t < t,l*,

(6.7) otherwise. 1 0, The time variation of the land price curve pi(t) corresponding to this optimum growth process is given by (5.20). Thus, the properties of land price curve are those in Section 5-2. Finally, Fig. 13 depicts the relation between the construction profit function Bjl(t) defined by (3.15) and the actual construction site &(t) at each time. 7. EQUILIBRIUM

SPATIAL PATTERN OF URBAN GROWTHANALYSIS OF DEMAND SIDE

In the above, we obtained the optimum spatial pattern of urban growth for Problem A. Further, we obtained also land prices sustaining that optimum path. Though these prices are equilibrium prices in a competitive land market, the market as a whole is not fully competitive for two reasons. First, the construction market connected with that land market is partly controlled by the public authority. Second, the rent market of buildings is dominated by the public authority. The purpose of this section is, hence, to examine whether there exists a perfectly competitive market of which equilibrium growth path and land prices coincide with the optimum ones. This is equivalent to asking what characteristics are required on the side of building renters for the market solution to coincide with the above optimum solution. In the following, we present a set of assumptions to give an affirmative answer to this question. It is first assumed that all renters (i.e., firms and households) can be divided into m groups depending on their characteristics (for examples, activity types, life cycles and income levels) and these renter groups have one to one correspondence with building types. Thus we assumethat renter group i rents only building type i in some district. Further, it is supposedthat the total number of renters in each group i is exogenously known as a function of t, and is denoted by D;(t) (i = 1, 2, . . . , n, t E [0, ~0I). It is also assumed that each renter uses one and only one unit of corresponding building at each time. Therefore, we put the next assumption. Bl. The total demand for each type of building is exogenously given as a function Di(t) (i = 1, 2, . . . , n, t E [0, ccI). We next assumethat; B2. Each renter can move between, as well as within, districts without cost at each time, and he chooses his district with a short-run optimization rule at each time.

236

MASAHISA

FUJITA

For example, the behavior of a household in group i may be represented as follows:

subject to PZ(f).z(t) + &l(t) + Fi(dL) = YZ(t), (7.1) where Ui is his utility function, z is a vector of goods other than housing, and P, is a vector of correspondings prices, xi is the consumption of housing service i and xi(t) = 1 by assumption, Ril is the rental price of housing i in

the district 1, F;(&) is the transport cost for household i when he lives in district Z,and Yi is his income. Then, following the original definition of bid function in Alonso [I], we define a bid rent function Qil[t, vi(t)] for each group i as follows: qil[t, Ui(t)] is the maximum value of rent for building i in district 1 under which a renter in group i can derive a given level of utility (or profit, in the case of firms) U<(t) at time t. For example, in case of (7.1), bid rent function Sil[t, vi(t)] is derived as follows: *il[t,

Ui(t)]

=

m;x [ Yi(t) - P,(t).z(t)

- Fi(dl)],

subject to Ui[z(t), xi(t) = 1, dl, t] = vi(t),

1 = 1, 2, . . ., IZ. (7.2)

We put an important assumption on the bid rent curves such that B3.

a9il[t, ui(t)]/ad, = -ai < 0.

(7.3) That is, we assumeeach bid price curve is a linear function of distance with a negative slope which is a constant for given i regardless of the level of utility (or, profit) and time.” The slope of bid price curves depends only on building types (and hence, renter group types). Figure 14 depicts a set of bid rent lines for different levels of utility or profit. Each function 9il[t, vi(t)] decreasesas Ui(t) increases. Suppose Di(t) and ai (i = 1, 2, . . ., n) are equal to those in the previous sections. Construction cost per unit of building i in the market is given &(t) as in Problem A where we assumede+Bi(t)/dt < 0. Further, we take T = a. Then, it can be shown that the long-run equilibrium path in this market with perfect competition coincides with the optimum path of Problem A under the assumptions Al to A3. To see this, denote the equilibrium rent of building i in district I at time t by Ril(t). Then, from the assumption of free mobility of renters, all values I7 This is a strong assumption. But, for example, suppose that U%(z,xi = 1, dr, t) = U%(z,XI = 1) in (7.1), and that zis a scalar variable. Then, from (7.2), we haveCWit[r, us]/ adr = - aFi(dt)/adl. Hence, this assumption is satisfied when transport cost Fi(di) is a linear function of distance. Therefore, if we take housing type discretely depending on its size in the models of Muth [9] and Solow [ 133, this assumption is satisfied as long as commuting cost is a linear function of distance.

SPATIAL

GROWTH

PATTERNS

237

FIG.14.Bid rent function\krl[tr ut(t)l. of Ril(t) in those districts with xii(t) > 0 should be on a bid price curve \kil[t, Ui(t)] with the same level of Ui(t). Therefore, the present value @il(t, { Ui(~)j ,,tm) of total expected revenue for a unit of building i constructed in the district I at time t is calculated as follows:

U/,(T)]dT %z(t, {U,(T)) .=lrn) =sPe-(r-t)*‘,J7, 1

We put

+d(t, { U,(T)) 7=tm)= e-8t@L,(t,{ Ui(7)) +OC).

(7.4)

(7.5)

Then the discounted bid price pi2 for a unit land by builders of building i is calculated at each time t as follows:

Pidf, ( Ui(7-)1 7=tffi)= bdt, f U%(T)1 .=t9 - UOllk,

(7.6)

where hi(t) is defined by (3.11). We do not know, of course, the equilibrium level of U;(t) yet. But, from B3,

@,z(t,{ U,(T)),=t”>/& = --(al/k)C4T) - 401.

(7.7) That is, as long as the slope of bid rent functions for each building is constant, the slope of bid price functions for that building is independent of the equilibrium level of utility (or, profit) ( U,(T)} +grn, and is given by (7.7). Note the right-hand side of (7.7) is equal to that of (4.4) and that the optimum spatial pattern of Problem A and corresponding land prices have been obtained by using only two simple facts: the slope of each bid price curve changes over time as given by (4.4), and the total demand for each building i is given by Di(t) over time. These two facts are satisfied also in this market. Thus, summarizing the above discussion, we obtain PROPOSITION2. Assume Bl to B3. Then, when perfect competition prevails everywhere in the market, the equilibrium spatial pattern of urban growth and

238

MASAHISA

FUJITA

the corresponding land prices are the same as the optimum spatial pattern and the corresponding land price of Problem A with T = CCI .

On the other hand, we can seethat the equilibrium rent l&(t) in this market is given as follows: Mt)

= h(t)

+ &,o(t),

I 5 L(t)

where h(t) k’(t)

(7.8)

= P&(t) - l&(t), = aLdt&) - 41,

(7.9)

and &(t) is the construction district of building i at time t. Further, the equilibrium land price curve pi(t) is obtained by solving (5.20) with the terminal condition (3.10) [that is, limpi + 0 as t -+to]. As a matter of fact, when (7.8) and (7.9) hold, we first see that it does not matter to each renter of building i where he rents becauseof assumption B3. Second, when they hold, we can obtain the following result: 00 It

e-b(r-t)Ril(T)dT

= Bi(t) + kiPl(t).

In obtaining the above, we make use of relations (3.10), (5.20), (7.8), and (7.9) together with [u(m)

- d(tll*)]al/kl

= [u(m)

- a(til*)]ai/ki,

I2 L(O), i = 1,2, . . . , m,

which are obtained by resorting to similar steps to (6.3) and (6.5). Hence, under (7.8) and (7.9), each builder is also in equilibrium over time. Finally, the equilibrium of land market under (3.10) and (5.20) was already demonstrated in Section 5-2. It is interesting to note that the equilibrium rent for each building is divided into two parts at each time as shown in (7.8). R,(t)

FIG. 15. Variation of equilibrium building rent curve Ril(r) over time [B&)

= B,].

SPATIAL

GROWTH

PATTERNS

239

FIG. 16. Relation of equilibrium building rent functions (divided by lot size) at time (m = 2).

and &l’(t) correspond to the building cost and the site cost respectively. Further, comparing (5.19) and (7.9), the site cost I&l’(t) coincides with the land rent per unit building lot only when i = 1. The equilibrium utility (or, profit) level vi*(t) of each renter is obtained by solving (7.10) for any r[Z,(O) 5 I S L(t)]. &I(Z) = \kil[t, Ui*(t)],

i = 1, 2, . . ., m.

(7.10)

Hence, the time variation of equilibrium building rent curve I&(t) for a particular i can be depicted, from (7.8) to (7.10), as in Fig. 15 where Bi(t) is assumed to be constant over time for simplicity. Figure 16 depicts the relation of equilibrium rent (per unit lot) curves of two types of buildings. Unlike the static case, a part of each (per unit lot) rent curve locates under the other curve. Further, Fig. 17 (shows) the relation between the equilibrium PI Pi, \ P 0) 1 \e

FIG. 17. Relation between the equilibrium land price curve PI(~) and tte equilibrium bid price curve p&, ( Vi*(,)) r-fm) (m = 2).

240

MASAHISA

FUJITA

land price curvepr(t) and the equilibrium bid price curvePi2(t, ( U,*(r)) i,tm) at time t. It will be interesting to compare Figs. 2 and 14. In the optimum process of Problem A, the zero profit condition for builders was satisfied by the control of subsidy variable Qi(t) [and thus, qi(t)] by the public authority. In this market process, the equilibrium utility (or, profit) level of each renter is adjusted so as to satisfy this zero profit condition. Therefore, suppose that, by chance, the value of each subsidy variable qi(t) was zero throughout the time in the solution of Problem A. Then this means that the bid rent function (kil(t)) t,Om given in Problem A is in reality the equilibrium bid rent (&r(t)} l,Omin this market. 8. CONCLUDING

REMARKS

We have studied the optimum spatial pattern of urban growth for a specific problem. A competitive market was also obtained whose equilibrium growth path coincides with the optimum one. The growth process was explained for the most part by using the notion of a bid land price curve which is a dynamic extension of the bid rent curves in Alonso [I]. Though the model concerned in this paper is a dynamic version of ThtinenAlonso type models of land use, an outstanding characteristic of the spatial pattern in this model is the sprawl-fashioned suburbanization. That is, except for Type (b-2) in Section 5-3, the construction site of each type of buildings is decentralized at each time towards the suburbs, leaving a large mixture of different types of buildings in these suburban districts. Thus, in contrast with the static case,vacant land continues to exist in each suburban area for a long time, and von Thtinen rings do not exist at all. The characteristic of the land price curve is that each district has a positive land price even when there remains vacant area within it. This is also a quite different result from the static case. A mixture of activities takes place even in static cases, as in Mills [S], when direct interactions between these activities are considered. But, the mixtures of activities in our dynamic model come from an intertemporal efficiency of the allocation of immobile buildings instead of direct interactions between activities. Of course, these results are obtained from the analysis of a specific model with a set of assumptions. Hence, it will be important to examine to what extent these results depend on these assumptions. First, if we do not adopt the linearity assumption of bid rent functions which were employed in A3 and B3, analytical difficulties after Section 4 increase enormously. But, this will not greatly change the essential conclusions of the model, as we can guess from our experienceswith the static von Thiinen model. The most restrictive assumption in the model may te that the demand for each type building is given exogenously as a function of time. Instead, if we take substitutability of

SPATIAL

GROWTH

PATTERNS

241

building types for each activity into the model, the demand for each building type should be decided endogenously at each time. Even in this case, some important conclusions, such as the sprawl-fashioned suburbanization, will not lose validity. But, others, like a simple coincidence between the optimum solution and the equilibrium solution, can not be expectedto hold. Therefore, this generalized case should be studied in detail in the future. Another important restriction of our model is the absenceof urban renewal. That is, the destruction of buildings is not considered in the model. If the destruction of buildings is allowed in the model, there must be a possibility that the spatial pattern of urban growth as well as the land price curve will change from the present ones that we have indicated in some essential aspects. In that sense, the study of the spatial pattern of urban growth with renewal must also te an important subject to be explored in the future. REFERENCES 1. W. Alonso, “Location and Land Use,” Harvard Univ. Press, Cambridge (1964). 2. A. Dixit, The optimum factory town, Bell J. Econ. Management Sci., 4,637-654 (1973). 3. M. Fujita, Optimum patterns of urban spatial growth, Lz “Proceedings of the Japan Society of Regional Science,” 90-107 (1973). 4. M. Fujita, Optimum expansion process of urban space, in “Annual Report of Japan Society of Regional Science,” Vol. 4 (1975). 5. M. R. Hestenes, “Calculus of Variations and Optimal Control Theory,” Wiley, New York (1966). 6. D. A. Livesey, Optimum city size: A minimum congestion cost approach, J. Econ. Theory, 6, 144-161 (1973). 7. E. S. Mills, An aggregate model of resource allocation in a metropolitan area, Amer. Econ. Rev., 57, 197-210 (1967). 8. E. S. Mills, Efficiency of spatial competition, Papers, Regional Science Assoc., 25, 71-82 (1970). 9. R. F. Muth, “Cities and Housing,” Univ. Chicago Press, Chicago (1969). 10. L. S. Pontryagin, V. G. Boltyanskii, R. L. Gamkrelidge, and E. F. Mischenko, “The Mathematical Theory of Optimal Processes,” Interscience, New York/London (1962). 11. J. G. Riley, Optimal residential density and road transportation, J. Urban Economics, 4, 230-249 (1974). 12. E. Sheshinski, Congestion and the optimum city size, Amer. Econ. Rev., 63, 61-70 (1973). 13. R. M. Solow, Congestion, density and the use of land in transportation, Swed. J. Econ., 74, 161-173 (1972). 14. R. M. Solow and W. Vickrey, Land use in a long narrow city, J. Econ. Theory, 3,430 447 (1971). 15. H. Yamada, On the theory of residential location: accessibility, space, leisure, and environmental quality, Papers, Regional Science Assoc. 28, 125-t 35 (1973).