Dynamic analysis of transport - location interrelationships : theory and models

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C.S. Bertuglia, G. Leonardi, R. Tadei

To cite this version:

C.S. Bertuglia, G. Leonardi, R. Tadei. Dynamic analysis of transport - location interrelationships :

theory and models. [Research Report] Institut des mathématiques économiques ( IME). 1985, 54 p.,

bibliographie. �hal-01543634�

EQUIPE DE RECHERCHE ASSOCIEE AU C.N.R.S.

DOCUMENT DE TRAVAIL

INSTITUT DE MATHEMATIQUES ECONOMIQUES

UNIVERSITE DE DIJON

FACULTE DE SCIENCE ECON OMIQUE ET DE GESTION 4, BOULEVARD GABRIEL - 21000 DIJON

C.S. Bertuglia, G. Leonardi, R. Tadei.

1. Introduction

This paper describes a dynamic model for an urban system.

The subject might not seem very up-to-date, nor appealing to £ conomists.

Indeed the overall structure of our model parallels quite clo­ sely the traditional one proposed by Lowry (1964) in his celebrated, but non-economic, A Model of Metropolis. In his pioniering work Lo wry captured what we still consider as the main interactions deter­ mining the urban structure, among housing, labour and services (or, more generally, non-basic activities). However, his model was la­ cking several features we would consider as standards to be met in modern urban economics, in particular:

a) the unfolding over time of the urban pattern change is obsent,

b) no economic underpinnings are provided for the behaviour of each actor involved in the urban scene (households, workers, firms,

landlords),

c) no economic variables (e.g.: prices and wages) appear explicitly and are generated endogenously.

Dynamic analysis of transport - location interrelationships: theory and models

It is a claim of the authors that the model proposed in this paper gives a consistent answer to points a) , b) and c) . Namely, an attempt has been made to combine modern theory on dynamic systems (Weidlich and Haag, 1983), modern theory on discrete-chpoce models (Domencich and McFadden, 1975, Anas, 1982, Leonardi, 1983 and 1984), and modern theory on bid-price markets (Ellickson, 1981, Lerman and Kern (1983), Leonardi, 1985). In addition, classic urban economic theory (Alonso, 1964, Beckmann, 1973) has been kept as a benchmark for consistency.

If dynamics is dropped, our modelling style has an antecedent in Anas (1984a). The Anas model is based on a Lowry-type framework and uses discrete choice models extensively. However, it is a gene­ ral equilibrium model and it does not use bid-price theory to gene­ rate market signals. The way we treat dynamics and price-formation are therefore the two main original features in our contribution.

2. Outline of the accounting framework

The detailed equations for the time-accounting of the model will be given in a later section. Here we will limit ourselves to give the general principles which guided the model-building.

Two alternatives have been considered: discrete time and cont^L nuous time. In the discrete-time version, the time horizon is sub­ divided into a countable sequence of periods of equal length (typi­ cally one year).

3

the system (population, jobs, housing stock) are updated according to a transition scheme of a Markov type. Transition probabilities a re non-homogeneous, and depend both on state variables and economic signals (prices, wages).

Economic signals are computed according to the assumption that they clear the market within each period. This is a partial equili­ brium assumption, implying market signal dynamics is much faster than quantity adjustment. The overall result is that prices and wa­ ges clear the market at every point in time, but quantities do not. This assumption, although admittedly simplistic, is widely used in modelling such phenomena in the urban economic literature. Referen­ ce can be made to Anas (1984a,b) for a recent account on the power of the method.

In the continuous-time version, differential equations repla­ ce the difference equations typical of discrete time. The state va­ riables counting quantities are computed according to a non-homoge­ neous Markov process or, according to the terminology introduced by Weidlich and Haag (1983), to a master equation. The economic signa­ ls cannot be assumed to be in local equilibrium in this case, and Walras-type equations are derived for them. It should be underlined they are derived, and not assumed: a micro-economic theory is deve­ loped, consistent with the behavioural assumptions about demand and

supply, which leads to them as an aggregate result. The theoretical approach we use is quite new in the economic literature, and it is based on the theory of stochastic extremal prqcesses developed by Leonardi in Bertuglia, Leonardi and Wilson (eds.) (1986, Chapter 5).

ded for the discrete-time model while the continuous-time one will only be outlined at the end of the paper.

3. Demand and supply behaviour in the discrete-time model

As we said, three interacting markets are considered in our mo del, namely housing, labour and services. For the sake of explaining the method, reference will be made here to the housing market. It has to be understood that the same approach carries over to the la­ bour and service market, "mutatis mutandis".

Let us define:

i, j, k subscript labelling zones in the urban area under conside­ ration, i, j, k = l,...,n.

and assume the following state variables are known at the be­ ginning of each time period:

p 1 population (households) having a job in zone i and a dwel-j

ling in zone j, i, j = l,...,n,

0

P unemployed population (households) having a dwelling in zo j

5

w, number of vacant dwellings in zone j, j = l,...,n.

In addition, we assume the following quantities are given exo­ genously or determined by other subsystems:

v. t average utility of a dwelling in zone j for a household ha^ ving a job in zone i (or no job, if i = 0) , net of housing costs, i = 0, l,...,n, j = l,...,n.

Specifications of v^j will be given in later sections. Typical^ ly this term would include the fixed expenditures a household incu­ rs (transport costs, expenditures in goods and services).

Concerning demand behaviour, we assume households maximize uti­ lity in their choices. Moreover, we assume demand is heterogeneous in preferences, and heterogeneity can be described in probabilistic terms. More precisely, let:

f 1 be the set of households having a job in zone i (or no jcb, j

if i = 0) and a dwelling in zone j, i = 0,...,n, j := 1 f • • • / n

T = U r 1 be the total set of households . . j

i,]J

Ü be the set of vacant dwellings in zone j, j = l,...,n. j

The cardinality of the sets T"*" and Q t are of course:

i l l x

|r I = P , i = 0 , . . . , n

j = 1 ,. . ., n

The utility a household h associates to a dwelling z is assumed of the form:

u = v + e - r , h E I*1 , z E ft , (1)

hz ij hz z j j

i = 0,...,n j = 1,...,n,

where i ^ z } is a sequence of independent identically distributed (i.i.d.) random variables with common distribution function:

P {e < x} = F ( x ), h , z, r hz —

and r is the price of dwelling z. In other words, equation (1) de- z

fines a random utility model (Domencich and McFadden, 1975),and the random utility term reflects the heterogeneity of preferences ci mong demand units. Due to demand heterogeneity, also prices r will

z

be heterogeneous within each zone (being determined by the demand- supply bargaining process), i.e. they can be treated as random va­ riables. Let us define:

R (x) = P {r < x| z € ft. } the distribution of housing prices in

j r z — 1 J

7

In each period, demand is supposed not to have perfect informa tion of the choice set. Each demand unit h does not know the whole set £2 , for all j . He rather knows a sample drawn out of it random

j “

ly. However, each demand unit h E r is equally exposed to the info£ mation about vacant dwellings. Let:

n

hj be the sample of vacant dwellings in zone j known to a hou sehold h, h G T , j = l,...,n,

and assume the average sample size is:

\q I = X w , V" h e r, j = l,...,n,

hj j

0

< A < 1.

Since demand maximizes utility, within each period each house­ hold h will scan the list of available alternatives, i.e. the sam­ ple of vacant dwellings he knows, Q , j = l,...,n, and his

cur-hj

rent dwelling. If any vacant dwelling has a higher utility than the current one, a move takes place. Otherwise, the household stays in the current dwelling.

_ i

For each h £ T define: k

u the utility associated with the best alternative in zone j hj

j = 1, . .. , n

hk hk h h k

Then the following equations hold:

u =

hj

v + max (e - r ) i j hz z

z G fl,

.

v + max (e - r ) ik hz z _

*

z g

n

hk j = k. (2)

If one defines the distributions:

h1 . (x) = p {u . ^ x| h e r 1} , i = 0,...,n

kj r h j k

kf j — 1, . ..,n,

then standard probability calculations yield (for the detailed der_i vation see Leonardi, 1985):

i r - -i Aw • H (x) = fib. (x - v ..)J 3 ,

k]

'-j

i j

(3) H* (x) = [D (x - v )]XWk + 1 kk u k ik (4) where D (x) = P {e - r < x l h G T , z E Q • } j r hz z — J oo = J * F i x + y) d R_.(y), j = 1,— ,n.

9

Once distributions (3) and (4) are known, the probability of moving

can be computed. Define:

i

q the probability that a household having a job in zone i and *j

living in zone k moves to a new dwelling in zone j, i = 0,...,n, k, j = l,...,n.

This probability, according to the utility maximizing assumption,is equal to the probability that the best dwelling available in zone j is better than all the other alternatives, including the currently occupied dwelling in zone k. This probability is given by:

00

q,1 . = I n (x) d H 1 . (x) i = 0,...,n (5)

*3

J

kl

kn

-°° k, j = 1,. . . ,n.

Note that, for the case k =j, equation (5) includes both moving to

a new dwelling in the same zone k and staying in the current dwel­ ling.

Equations (3), (4) and (5) completely describe demand behaviour,

once the price distributions are known.

Let us now turn to the analysis of the behaviour of supply,i.e. the landlords. Define the bid-price as the highest price a house­ hold is willing to pay for a dwelling, without decreasing the utili^ ty level offered by all the alternatives in the market. The maximum utility level for a household h 6 T1 is:

u = m a x u tr\

h hj (6)

1< j < n

and this is a random variable with distribution:

l

H 1 (x) = P {u < x lh

e r } =

k r h ~ k

= TT H . (x) , i = 0,...,n (7)

. . I kj

Consider a vacant dwelling z E ft^ and assume a price rhz is bid for it by a household h E . The total utility associated with z

JC

is, according tC equation (1):

u = v + e - r (8)

hz ij hz hz

The highest price h is willing to pay for z is the one which equa

tes (6) to (8), that is the one wihich yields for z a utility at

least as high as the maximum utility offered in the market. Clearly

any higher value for the price would yield a utility lower then the

one offered by the other alternatives, and the h ousehold would have

no reason to consider the dwelling z as an alternative. This leads

to the equation:

v + e - r = u ij hz hz h

Equation (9) defines the bid-price of a household h for a dwelling

z, and is completely consistent with the bid-price (or bid-rent)con

cept as introduced by Alonso (1964),

We assume that landlords maximize profits, which in this case

is equivalent to maximizing revenues, i.e. to sell (or rent) their

dwelling to the household offering the highest bid. Of course the

transaction occurs only if the highest bid is non-negative; otherwi

se, the dwelling will remain vacant at the end of the period.

Let:

T be the sample of potential buyers in T1 known to the ow

zk k —

ner of the dwelling z, Vz,

and assume the average sample size is:

j r | = X P. , V z, i = 0,. .., n

ZK K

k = l , . . . , n *

i i

r the highest bid-price from households h C r , i=0,..,,n,

kz k

k ~ !,•••/n,

and introduce the distributions;

R

1

. (x) = P {r

1

< x |Z £

8

.} / i ? 0..,, r 'n kj r kz — 3 i f . k,j * 1 » r r • * G 1 (x) = P {e - u | Vz, h € r_{1 }} k r hz h k w

• /

F (x + y) dHk (y), (*0)

The following equations, analogous to equations (?) to (4), hold:

kz v +

_{h e rJ}

max ,i (e hz v z C il zk (ll) R . (x)

k]

G, (x - v. .) k i j xp. (1 2)

The distribution for the highest bid^price in zone j is thus given by:

13

n n 1 , % r. (x) = n n R . . (x),

]

i « k-1

k;1

(13)

We recall, however that this is not the distribution of the actual market price. The actual price is equal to the bid-price when this is non-negative. When it is negative, the actual price is set equal to zero. The market price distribution is thus:

R .(x)= 3 R. (x) 3 x > 0 x < 0 (14)

Note also that R.(0) is the probability that a dwelling in j remains vacant.

4. Asymptotic approximations

Let us now introduce some additional assumptions which will greatly simplify the equations given in section 3. A general result will be required. Its proof is found in Leonardi (1985),but it will repeated here for clarity.

Theorem 1. Let F(x), R(x) and D(x) be distributions satisfying the properties:

1

- F(x + y) -Sx

l x m --- ---- = e / g > o

y-*» ! _ f (y)

(15)

oo

D U , -

J

F(x + y) dR(y) ₍₁₆₎

and define the sequence a as root of the equation:

N 1 - F (a ) = 1/N , N » 1, 2(... r (17) Then, for w > 0;

lim

N-x®

D(x + aN ) UN

_exp

+ r) ₍₁₈₎

where r is defined by the equation

(19)

Proof - First notice çhat, from equation (17)*

lim a = ». N

N-*°°

Substitution from equation (17) and some rearrangements yield:

-ex

e lim F (x + a ) = lim (1 - ---) . N N “*» N~*°° N

Substitution of this result in equation (16) yields, on account of definition (19): lim D(x + a ) = lim N N-x» . N-*»

1

--3 (x + r) N

From this result and the well-known limit

a N -a lim (1 - — ) = e

N

N-x»

statement (18) follows. Q.E.D.

Theorem 1. provides a typical resijlt from the asymptotic theo­ ry of extremes (Galambos, 1978). It can be used to simplify the e- quations in section 3. Let N be the total housing stock (or any o t her measure of the total size of the system) , and assume N becomes large in such a way that the ratios:

u). Xw_,/N t j — l/«.*, n (2 0)

e

,

1

= x

p x/n ,

i = o,..., n

k

k

k r= 1 , . . . ,n (2 1 ) remain constant.

We also assume that the distribution Fix) for the random utiljL ty terms satisfies property (15). Since a change in the origin of the utility scale does not affect demand behaviour, we can arbitra­ rily subtract the constant a„, define«? by (17)# from the terms v. ..

N

x

j

using theorem l., and assuming N is large enoughf one can work out the following approximations:

D (x - v + a ) j ij N

Awj

D . (x t v . .+ a )

3

iD

N

WjN

exp - w e -

3

(x - Vj ■*3 + r.)

3

L 3 (2 2) G 1 (x _ k - v ij + a ) N _ X p k » exp i -6 (x - 0 e k - v ij ' G* (x - v . . + a ' ^

8

5tN

H

0.3

-i „ i

v T 1"

(23)

17

where r. and u are defined by the equations:

j * i -$r . P -Bx e D = / e dR, (x) (24) -&u e k e PX dH^(x) k (25)

An approximation of the distribution R.(x) can be computed sub

stituting result (23) in equations (12) and (13), yielding:

R (x) 'v j exp - e ip . (u) (26) where n n ^ . (u) = E £ 01 e 3 i=0 k=l 1 15(Vij - Uk ) (27) j = 1, — , n.

uses result (22) and the fact that for N large enough property (15) allows to writes -

8

(x - vik + rk ) D (x - v + a ) ^ 1 k ik N N exp

.

i

6(11

- vi k + <*>

N (28)

Therefore, substituting (22) and (23) in (3), (4) an<l (7) one has:

H (x) ^ exp k -

6

x,i ,

-e

<

)>

k (r)

(29) where: , i . . _

6

(vii ir x+)

1

6

(Vjv -<p (r) = Z w e J + — e liC * (30) k . - j N 3 =

1

1

—

0

,«• •, n ^ k —

1

^ f n f

The above approximations applied to equation (5) easily yield for the choice probabilities:

19 B(vii - r.) x to-; a -> J q. . ^ — 3--- :--- i = 0,., . ,n (31)

k j

* V >

k k, j s

It is remarkable that equation (31) defines a Logit model, al­

though no specific assumption on the form of F(x) has been made, ex­

cept for property (15). Note that for k = j equation (31) gives the

probability of moving within the same zone. The probability of sta­

ying in the same dwelling can be obtained as a complement, and is g i ­

ven by:

n

1

“ £ qLX . .

In this sense, the probabilities (31) are slightly differently defi­

ned from those of equation (5), where the case k = j included bpth

moving and staging. Definition (31) is more convenient in many ways

and will be adopted from now on, with no loss in generality.

Ir remains to compute the arrays r = {r } and u = ( u f K To do

j k

this, we replace the approximations (26) and (29) in equations (21),

(24) and (25) and compute the integrals. Some calculations yield:

e"3uk = 1/<j)1(r)

(32)

r

- Br,

e j 1 - e

-iM u)

j

/'I'. (u)

3 (33)

1

(34)

B(vij - r-j) + e 3(vik - rk )

5. Market clearing conditions

Equations (24) and (25) define the arrays r and u as average pri. ces and utilities, although they are exponential, rather than arith­ metic averages, in this section we will shqw that the array r sol­ ving equations (32) and (33) is a vector of majrket^clearing prices, and prove its uniqueness.

Substituting equation (32) in definition (27) we obtain

'i'.(r) = I

D

_i

₌₀

_k

n

j =

1

(35)

a function of r only.

Substituting definitions (21) and (30) in equations (35) and rearrari ging:

21

n n j = Z Z P, k i

=0

k=l X e6v«

»

1

w e6<Vij " rj'+ e8(Vlk

3-1 3

r k )

(36)

Comparison of (36) with (34) shows that:

i M r ) = 3 n n i i"

E

z pv ^ ■

k ki i

=0

k=l J - $r • /(w. e P 3)

D

(37)

Substituting (37) in (33) and rearranging we obtain:

n n r 1 1 Z Z P q = w . 1 -k -kn i ! i

=0

k=l J J L

-iM r)

e 3 (38)

We recall now that the probability that a dwelling in zone j remains vacant is R (0) or, from (26):

j

5 (

0

) - . ' ♦ i 1* ’ .

D

Hence

— Jp • (jr)

1

- e J is the probability that a vacant dwelling in j is sold (or rented) within the period, j =

1

,... ,n

and the term on the right-hand side of (38) is the expected number of dwellings sold in j, i.e. it is the expected supply. The term on the left-hand side of (38) is the expected number of moves from all zones to vacant dwellings in zone j, i.e. it is the expected demand. Equations (38) are therefore market-clearing conditions,equating de­ mand and supply in each zone. They are completely equivalent to e- quations (32) and (

33

), and can be solved for the market-clearing pri_ ces r, provided a solution exists and is unique.

For the proof, consider the functions:

f . (r) 3

1

1

-e TJ

in

---(r) (39) j =

1

where ^ (r) is defined by (36), and the mapping: j

(40)

where

f(r) = if (r)r • • • tf (r)} .

n

The vector equation:

23

is clearly equivalent to the market-clearing conditions (38).The fol lowing result shows that (40) has a unique fixed point.

Theorem 2. The mapping f defined by (40) is a contraction.

Proof. Let us first observe that the inequality:

-'l'j(r)

1

- e J

0

< --- <

1

» j = l,...,n (r)

3

holds for any value of r, since (r) > 0 by construction. Hence,frcm (39), the inequality:

f (r) >

0

, j =

1

____,n

holds for any value of r. This implies that equation (41) has a non­ negative solution (if any).

From the mean-value theorem we can write:

n

f (x) = f (y) + I f_.^ (£) (x^ - y^), j =

1

,__ ,n (42)

for some £ on the line-segment joining x and y, x, y,C G R 1, where

f. (£) = --3.--.5j— , j, =

1

, — ,n.

31

« 1

max z , z G R

i

1

< i < n

Some lengthy calculations yield for f

1

-(

5

) 3 l-e

3

n n

.3

-1 i=0 k = 1 i=0 j=l (43) where :

1

1

pk 9k j

? S » V i - O k - l k kj i =

0

,... ,n k , ^ —* l,»*«,n«

It is clear from the definition that:

n n

I

l y

-

1

.

i

=0

k=l and

0 < y. < i.

— ik — Therefore, since

< i

and

25 n 1 - E q 1 < 1 3=1

13

" it follows: n n 0 < E E y

3

q

,1

+ E y

3

(1- E q

1

) - *k Tc ll Hlj i

=0

k=l i

=0

j=l J n n n n = E E y

3

q

3

+ E y

3

(1 - E q

1

) < . / i k kl . il , lj -i

=0

k=l i

=0

j=l k/ j / n n < E E y

3

= 1, — ik i

=0

k=l

Substitution of this result in equation (43) yields the inequality:

( ? )

0 < f . (?) £ 1 --- . (44)

3 1 , (?)

1

-e ■>

Easy calculations also show that:

n ip , (?)

E f, (?) = 1 ---

3

--- . (45)

1=1

3

■,

-4>.

(?)

l-e j

0

< l----L,--- < l. (46)

The strict inequality > 0 holds since only non-negative £ are considered, and the minimum value for (£) is:

ip (

0

) >

0

j

from equation (36).

Combining (44), (45) and (46) one concludes it exists a constant a,

0

< a <

1

, such that

0 < f (?) < a (47)

~ j l

Z f (?) < a (48)

1 = 1 jl

On account of (42), (47), and (48) one can thus write:

| f (x) - f (y)

1

£ a |x - y|

0

< a <

1

27

Due to theorem 2, resort can be made to the Banach (1922)fixed- point theorem for equation (41), or equivalently equation (38). This ensures a unique fixed point exists, which can be computed with the recursion:

for r given.

^ • The discrete-time model

6.1 Accounting framework

As we said befpre, the accounting framework for population mobi lity is made consistent with the master equations approach (Haag in Bertuglia, Leonardi and Wilson, eds., 1986, Chapters 4., 8. and 14.). This will be more clear in the section on the continuous time ver­

sion, where the master equations for the means appear in their stan­ dard form. Here we shall develop a discrete-time version of the equa_ tions for the means, that is difference equations that later on will be shown to tend to the standard differential equations in the limit. The difference equations proposed here have the same structure of tho

se of a discrete-time Markov chain, a model widely used in multista­ te demography (Rogers, 1975). However, unlike in conventional multi­ state population models, we assume the process is non-homogeneous,

i.e. the transition probabilities are not constant over time (indeed they are endogenously determined as a function of the state vari«a bles).

A distinctive feature of the framework is the main accounting _u nit for the quantities, which is the residence-place of work pair. This means the model describes the trajectory of the whole origin-de^

stination matrix for journeys-to-work, rather than the residences and jobs counts separatly. More precisely, let:

i,j,k be subscripts (or superscripts) labelling zones in which the urban area under study is conveniently subdivided, i,j,k =

Then the main state variable in our accounting equations is:

P (t) the population having place of work in zone i and residence ij

in zone j at time t, i,j=l,...,n, t = 0, A, 2 A, . . .

Here A is a sufficiently small time interval (e.g.: a year), used as a unit for discrete time.Although no generality would be lost by set. ting conventionally A = 1, we prefer to keep the more general nota — tion, in order to make the passage to the limit easier for the conti nuous-time version.

The variable P (t) conveys all the information needed both for ij

the population counts and for the stocks of economic activities. It can be interpreted as in the definition given above, but also as the number of workplaces in zone i occupied by workers living in zone j.

29

Both interpretations will be used, according to whether emphasys is placed on population choices (like changing residence) or on choices of firms (like changing residence) or on choices of firms (like fi­ ring or hiring workers) .

The matrix {P (t) } is augmented by an additional row accoun- ij

ting for unemployment:

P (t) unemployed with residence in zone j at time t, j=l,...,n. Oj

The dynamics of the process are specified by assuming the exi­ stence of suitable transition probabilities.

The additional assumption is made that the time interval A is small enough to neglect multiple events, i.e. in an interval(t, t+A) an individual unit can undergo only one change (if any), but not mo­ re. For istance, a resident might change residence or be fired(if he is employed), but not both. This assumption is of course only appro ximately true in discrete time, but it will become exactly true in continuous time, as A shrinks to 0.

We define therefore:

q

1

(t,A) the probability that people working in zone i and living in zone k at time t move to a dwelling in zone j in (t, t + A ) , i,j,k —

1

,...,n;

^.(t,A) the probability that an unemployed living in zone k at time t moves to a dwelling in zone j in (t, t + A ) , k,j=l,...,n;

p

. (t,A) the probability that a workplace in zone i occupied by a worker living in k at time t is occupied by a worker living in j at time t + A; i.e. the worker living in k is fired and an unemployed living in j is hired in (t, t +A), i, j, k = =

1

,...,n;

the probability that a workplace in zone i occupied by a worker living in zone k at time t is closed down in(t,t+A), l , k —

1

, .. ., n ;

p (t,A) the probability that a new workplace is open in zone i and ij

occupied by a worker living in j in (t, t + A),i,j = l,... ,n; this probability is applied to a stock of potential workpla^ ces which can be open in zone i, which is assumed to be exo genously given from land-use constraints.

In the version discussed here, the model uses some simplifying assumptions, which can be easily relaxed in future versions:

a) the total population is assumed constant, i.e. no births, deaths and migrations to and from the rest of the world occur in the a- rea;

b) the total housing stock is assumed constant for each zone in the area, i.e. no new constructions and demolitions take place;

c) all the economic activities are assumed to be endogenous, i.e. non- basic;

i

31

d) a land-use constraint is exogenously given for the development of economic activities in each zone, expressed in terms of maximum number of workplaces which can be open.

According to the above assumptions, the following exogenous va­ riables are defined:

P total population in the area;

Q^ total housing stock (both vacant and occupied dwellings) in zo ne j, j = 1, ...,n;

S. maximum number of workplaces in zone i, i = l,...,n.

We need further to define:

n

T (t) = S. - E P (t) number of new workplaces which can be open 1 3-1 lj

in zone i at time t, i = l,...,n.

Combining all the state variables, transition probabilities and exogenous quantities defined above, and considering all the changes of state which can occur in the interval (t, t + A ) , we obtain the following difference equations for the employed and unemployed popu­ lation , respectively:

n

i i

P (t+A)

+ P. .(t)

ID

n n i - £ q 1, (t,A) - t p

1

(t,A) k-1 * k=0 * + T.(t) p. .(t,A),

i

13

n P (t+A) = E Prtl (t) a . (t,A) + P . (t) Oj Ok Ten Od Jv — X n 1 - E q., (t,A) k=l J n E T. (t) p . . (t,A) , i

- 1

1

13

(49) n n n + E P (t) E p

1

(t,A) - E P., (t) p .(t,A) -i-1 13 k=0 jk i-1 lk ^ (50)

It is easy to check that equations (49) and (50) imply the conserva­ tion of the total population, by summing them over all zones, i.e. if one defines:

n n

P(t) = I Z P (t) the total population in the whole area at ti^ i

=0

j=l ±j

me t,

then summation of equations (49) and (50) yields with some rearrange ments:

33

6.2 Behavioural models and transition probabilities

It is a distinctive feature of our model that the transition pro babilities have a structure derived from micro-economic justifica­ tions. More precisely, they are assumed to be given by Logit models. Such models are by now well established and accepted in urban econo­ mic applications, due to their combination of theoretical rigor and ease of computation. The derivation of Logit models from classic ran, dom utility theory can be found in Domencich and McFadden (1975).The ir extensive use in urban economics has been thoroughly explored in Anas (1982). Their derivation in a dynamic setting, using the asymp­ totic properties of stochastic extremal processes, is developed by Leonardi in Bertuglia, Leonardi, Wilson (eds.) (1986, Chapter 5.). The models for the transition probabilities given in this Section a- re based on the latter references and the theory presented in Se<c tions 3., 4. and 5., although of course they bear similarities with the approach of Anas.

Let us define the following variables:

c cost of a return trip between zones i and j, i,j=l,...,n; ij

c (t) average per-capita expenditure in services for people living j

in zone j at time t, j = l , . . . , n ;

r (t) average dwelling price in zone j at time t, j=l,— ,n? j

y (t) average disposable income for people living in zone j at ti- j

tt^ (t) average revenue for a unit of economic activity (i.e. a work place) in zone i at time t, • i = l,...,n;

W j t ) = Q_. - Z t (t) vacant dwellings in zone j at time t,

j = 1,... ,n.

With the exception of transport costs c , all the variables defined ij

above are time-varying, and indeed endogenously determined according to laws to be specified in the next section.

In order to derive the transition probabilities for people in state (i, j) we will proceed by steps, first considering housing and labour mobility separately, and them combining them. For the housing choice, we assume demand tends to minimize total cost, that is the sum of transport, service and housing expenditures. The utility fun ction of a dwelling in zone j for people working in zone i is there­ fore:

u . .(t) =

13

L l 3 c + c (t) + r (t)j j

and a standard Logit specification for the choice probability,without considering labour changes, would b e :

XA Z W (t) e 6ui j (t) , eeU i k (t)

3-1 3

35

number of vacant dwellings in zone j known to each demand unit in a time interval (t, t + A) . Equation (51) therefore implies the assum­ ption that moves can be made only to vacant dwellings; moreover, it

implies that not all the choice set is known to demand units, but on ly a fraction XA of it, and this fraction is smaller the smaller the interval length A. The parameter X is therefore a rate of diffusion of information about vacant dwellings available.

Similarly, for the choice of a worker, we assume demand (which in this case is a firm, not a household) tends to maximize total pro fit, which neglecting other costs is given by the difference between revenue and wage paid to the worker. We assume each worker is subje­ ct to a budget constraint, so that he would not be willing to accept a wage which does not cover his expenditures. The part of the wage which is left after expenditures are deduced is by definition the di sposable income, therefore the wage a workplace in zone i pays for a worker living in zone j turns out to be:

c + c . (t) + r . (t) + y . (t)

I D D D D

and the utility function for a firm hiring a worker living in zone j for a workplace in zone i (that is the profit) is:

v . .(t) =

7

r .(t) c . , + c t (t) + r (t) + y (t) .

I D i L I D D j D

Again a standard Logit specification for the choice probability of the firm, without considering housing changes, would be:

P*.(t,A) = $Vij(t) yA jjAPqj (t) e I p (t) eBvii(t)+ r 3-1 03 (52) + e$vik(t)

The parameter y is a rate measuring the intensity of firing-hiring cte cisions. The assumption implicit in equation (52) is that only unem­ ployed can be hired, and for an already occupied workplace the p r e — vious worker remains unemployed. Firing can also occur without any turnover, and with no profit for the firm, when the workplace is clo^ sed down. This is reflected in the additional unit term appearing in the denominator of (52). More specifically, the probability that a worker in i with residence in k is fired because the workplace is clo sed down would be:

¿k 0 ( t ' A) = yA yA Z P (t) e Bvi^ (t) + 1 3-1 ° j (53) + e

6

v ik ( t >

We shall also need the probabilities of making no change, which for the residential and labour mobility are given respectively by the fol^ lowing equations, easily derived from equations (51) and (52):

n q*(t,A) =

1

- Z q

1

.(t) = k j=l

k3

frUifc(t )

(54) XA I W (t) eeUi3 (t) + e3Uik(t) 3 = 1 j

37 n p

1

(t,A) =

1

k

eBvikit) (55) +

1

+ eBvik(t)

We can now combine the two processes. Due to the assumption that each individual can undergo at most one change in (t, t + A),the si­ multaneous occurrence of housing and job changes is excluded. There­ fore the probability of moving given by (51) can only by applied to those who kept their workplace in (t, t + A). Similarly the probabi­ lity of being fired (or of hiring a new worker, from the firm's point of view) given by (52) can only be applied to those who kept their residence in (t, t + A). Under the above assumptions, the actual tran sition probabilities to be used in the difference equations (

49

) and

(50) are given by:

Substituting from equations (51) - (55) and neglecting the terms of

2

order A equations (56) and (57) finally yield:

q^_.(t,A) = q^ (t , A) p^(t,A) ₍₅₆₎

p^.itjA) = p 1 , (t,A) q

1

(t,A).

kj kj k (57)

XA Wj(t) ee[ u±j(t) + vik (t)]

(58)

©t (t,A) k

„4 P..(t> eBCvi3 (t) * i Ui p (t,A) = --- (

59

)

3

i

D

1

(t , A)

k guik(t)

1

, .. pA e pko (t' } --- (60)

D.1 (t , A)

k

where the denominator in given by:

dV . A ) > X4 £ W (t) eeuij(tl +

J-l 3

+ yA eBui k (t) n

I

3 =

1

Pn .ct) e 6Vli (t)

O

d

+

1

+ eS Eu ik(t) + vik(t)] i, k =

1

,...,n. (6 1 )

The transition probabilities for the unemployed and for the new jobs are somewhat simpler then the ones above, since they refer to a sin- gle-actor, rather than a two-actors, decision. Namely, the unemployed have no firm (no job) associated with them, and they can only make re

39

sidential changes. Similarly, the new jobs have no worker associated with them,and they can only make labour changes (i.e. be open and h^L re somebody) . The unemployed have no commuting cost, therefore the utility function for their residential choice is:

u (t)= - c .(t) + r (t)

OD

L ]

D

_

and the probability that an unemployed moves from zone k to zone j is given by the following Logit model:

k, j ~ 1,...,n.

The firm opening a new workplace has the same expected profit as that associated to already existing jobs, therefore the probability that a potential workplace in zone i is open and occupied by a worker li­ ving in zone j is given by the following Logit model:

0

qk j (t'A) j (6 2) Pij(t, A) (63) if j — 1,..., n.

E q u a t i o n s (58) to (63) s p e c i f y all the t r a n s i t i o n p r o b a b i l i t i e s r e ­ q u i r e d in t h e d i f f e r e n c e e q u a t i o n s (49) a n d (50). It is e a s i l y c h e ­ c k e d f r o m t h e i r d e f i n i t i o n s t h a t t he y ar e c o n s i s t e n t p r o b a b i l i t i e s , i.e. : q . (t,A) < 1

k]

P, . (t, A) < 1 a 1 . (t,A) + p,1 . (t,A) < 1 K] kj p k o (t'A) < 1 q^..(t,A) < l p . . (t,A)< 1

ID

a n d m o r e o v e r : n Z

D =

1

a . (t,A) + p . (t,A) _

TtD

k j

(64)

n Z j = l q ^ , (t,A) < 1 (65)

41

n

I

p . . ( t , A) < 1

(66)

j=l 13

i , k 1, . .. , n •

E q u a t i o n s (64) to (66) state the c o n s e r v a t i o n p r o p e r t i e s of the sy­ stem. T h e t o t a l p r o b a b i l i t y of a c h a n g e of sta te in (t , t + A), for a n y i n i t i a l state, is less than 1. Its c o m p l e m e n t to u n i t y is of c o u £ se t h e p r o b a b i l i t y of m a k i n g no change, t h a t is of r e m a i n i n g in the i n i t i a l state. 6 . 3 T h e f o r m a t i o n of p r ices, i n c o m e s a n d r e v e n u e s In s e c t i o n 6.2. se ve r a l e n d o g e n o u s t i m e - v a r y i n g s i g n a l s h a v e be en i n t r o d u c e d , w h i c h e n t e r in th e d e f i n i t i o n of the u t i l i t y functions u s e d in t h e t r a n s i t i o n p r o b a b i l i t i e s . T h e y a r e the d w e l l i n g prices, the d i s p o s a b l e incomes, the u n i t r e v e n u e s for f irms a n d the a v e r a g e p e r - c a p i t a e x p e n d i t u r e s in services. In t h i s s e c t i o n m o d e l s ar e p r o ­ v i d e d to c o m p u t e t h e s e vari abl e s. In the d i s c r e t e - t i m e v e r s i o n of the m o d e l , it is a s s u m e d t h a t a l l t he m a r k e t s u n d e r c o n s i d e r a t i o n (hou­ sing, la b o u r , services) ar e c l e a r e d w i t h i n o n e p e r i o d . T h a t is, the p r i c e s a t th e e n d of ea c h p e r i o d (say a t t i m e t + A) a re c o m p u t e d in s u c h a w a y a s to b a l a n c e d e m a n d a n d s u p p l y a s d e t e r m i n e d b y the c o n ­ d i t i o n s a t t h e b e g i n n i n g o f the p e r i o d (say at ti m e t) . D u r i n g the t i m e i n t e r v a l (t, t + A) d e m a n d - s u p p l y i n t e r a c t i o n s t a k e p l ace ; h o ­ w e v e r A is a s s u m e d la rge e n o u g h f o r the i n t e r a c t i o n s to set t l e down

a t p a r t i a l e q u i l i b r i u m at the end.

T he u n d e r l y i n g a s s u m p t i o n b e h i n d t hi s a p p r o a c h is t h a t the pri_ ce a d j u s t m e n t p r o c e s s is f a s t e n o u g h c o m p a r e d to c h a n g e s in q u a n t i — t ies ( p o p ul a tio n a n d stocks) , so th a t its d y n a m i c s is v e r y short-term. T h i s a s s u m p t i o n m i g h t n o t b e f u l l y j u s t i f i e d in som e case s, a n d i n d e e d it w i l l b e abandoned in t he continuous-time version, w h i c h d o e s n o t r e ­ q u i r e it (on t he c o n t r a r y , it w o u l d n ot w o r k w i t h it). H o w e v e r , in di_ s c r e t e t i m e t his is the m o s t p r a c t i c a l w a y to k e e p c o n s i s t e n c y in the a c c o u n t i n g f r a m e w o r k a n d e n s u r e fea s i b l e t r a j e c t o r i e s for t h e q u a n t i ties. T he a p p r o a c h of p e r i o d m a r k e t c l e a r i n g is e x t e n s i v e l y u s e d b y A n a s (1982, 1984b), u n d e r t h e a s s u m p t i o n tha t b o t h d e m a n d a n d s u p p l y b e h a v e a c c o r d i n g to a L o g i t model. H e r e a s l i g h t l y d i f f e r e n t a p p r o ­ a c h w i l l b e used, b a s e d o n the t h e o ry of s e c t i o n s 3.,4. a n d 5.. W h i l e the d e m a n d b e h a v i o u r is s t i l l d e s c r i b e d by L o g i t m o d e l s (the t r a n s i ­ t i o n p r o b a b i l i t i e s in s e c t i o n 6.2.), the s u p p l y b e h a v i o u r is a s s u ­ m e d to b e b i d - p r i c e m a x i m i z i n g , as d e s c r i b e d in t h e g e n e r a l t h e o r y of s t o c h a s t i c e x t r e m a l p r o c e s s e s (Leonardi in B e r t u g l i a , L e o n a r d i a n d W i l s o n , eds., 1986, C h a p t e r 5.). F o r t h e h o u s i n g mar ke t , let:

H .(t,A) b e the t o t a l d e m a n d f or h o u s i n g in zone j in (t, t + A ) , t h a t

D

is t he p e o p l e l o o k i n g for a d w e l l i n g in z o n e j in (t, t +

A),

U s i n g e q u a t i o n s (58) a n d (62), t h is is g i v e n by:

H. (t,A) = I

43

A c c o r d i n g to t h e b i d - p r i c e m a x i m i z i n g a s s u m p t i o n , a l a n d l o r d o w n i n g a v a c a n t d w e l l i n g is w i l l i n g to sell it if h e h a s r e c e i v e d at least o n e n o n - n e g a t i v e bid. F r o m the t h e o r y in s e c t i o n 4. the p r o b a b i l i t y t h a t t h i s e v e n t o c c u r s in (t, t + A) is: w h e r e $.(t,A) is d e f i n e d as: fir ( t ) . (t,A) = e D H . ( t , A ) / A W i ( t ) , j = l,...,n. H e n c e th e t o t a l s u ppl y of h o u s i n g in zone j in (t,t + A) is g i v e n by: W. (t) 3 . <!>+ (t,A) "I, j — lf*»*,n, 1 - e J I (68)

a n d t h e m a r k e t - c l e a r i n g e q u a t i o n s for the h o u s i n g p r i c e s r^(t) are:

H (t,A) = W .(t) 3 3 - < M t , A ) 1 - e J j—I , • • • ,n. (69) S i m i l a r a r g u m e n t s a p p l i e d to the l a b o u r m a r k e t l e a d t o the f o l ­ l o w i n g m a r k e t - c l e a r i n g e q u a t i o n s for t h e d i s p o s a b l e in c o m e s y ( t ) : L. (t,A) = P n J t )

3

Oj

-ijj. (t,A) 1-e J (70) w h e r e :

L_.(t,A) is the to t a l d e m a n d f o r w o r k e r s l i v i n g in z o n e j i n(t , t + A ) , j = 1 , . . . ,n, a n d u s i n g e q u a t i o n s (59) a n d (63) this is g i v e n by: n n . n L . (t ,A ) = E E P (t) P ;\( t,A) + E T.(t) p . . (t, A) , (71)

3

H

W

lk

i-1 1

15

j — 1, • • . , n ; w h e r e :

V

* ’

- i M t , A ) 1 - e J is th e total s u p p l y of l a b o u r f r o m (72) the u n e m p l o y e d l i v i n g in z o n e j, j ^ 1 , . . . ,n,

a n d the q u a n t i t y ip (t,A) is d e f i n e d as: j <P . (t, A) = e 8 y 3 (t) L . (t, A) /A P„.(t), j = l, ... ,n. J J O j It s h o u l d b e n o t e d t h a t in ou r labour m a r k e t m o d e l t he a v e r a g e d i s p o s ab l e i n c o m e p l a y s the r o l e of the e q u i l i b r a t i n g p r i c e . T h i s is b e ­ c a u s e a p p l i c a n t s to o f f e r e d jobs ar e a s s u m e d to b e d i s p o s a b l e - i n c o m e m a x i m i z e r s : t h e y e v a l u a t e a l t e r n a t i v e jobs a c c o r d i n g to t h e d i s p o s a ­ ble in c o m e t h e y p r o v i d e , o n c e f i x e d c o n s u m p t i o n s a r e d e d u c e d , a n d p i ­ c k u p the o n e p r o v i d i n g the h i g h e s t d i s p o s a b l e inc ome, if it is n o n ­ n e g a t i v e (the n o n - n e g a t i v i t y c o n d i t i o n b e i n g i m p l i e d b y t h e a s s u m p ­ ti o n of a b u d g e t c o n s t r a i n t for the hou s e h o l d s ) . If not, t h e y r a t h e r r e m a i n u n e m p l o y e d a n d d o n o t c o n t r i b u t e to the l a b o u r supply.

45

Some additional assumptions need to be introduced in order to de

rive equations for the revenues of the firms. It has been said a l ­

ready that only endogenous (i.e. market oriented, or non-basic)econo mic acti v i t i e s are considered in this version of the model. This a-

mounts to focus our attention on the service sector, and indeed to

a s sume that our urban area has only service activities. This is by far the most interesting sector from the point of view of the urban)* structure and dynamics, and in many instances it is also the pr e v a i ­ ling one. However it should not be forgotten that a basic industrial sector, whose production is not oriented towards the local consump­ tion, is still important in most cities. We leave this as a subject for future generalizations, which can be easily introduced without a l t ering the structure of the model.

A s a second simplification, we do not disaggregate services by

sectors. Such a disaggregation could also be easily introduced wi­

thout disr u p t i n g the structure of the model, bu t it would require a c u mbersome notation, which would obscure the basic structure of the m o d e l .

As a third assumption, demand for services is considered in e- q u i l i b r i u m at every instant in time. In other words, demand adjusts to changes in service location in a time span negligible compared to other changes in the process (like housing and labour m o b i l i t y ) .

A s f o r the d e m a n d for s e r v i c e s a m o d i f i e d H a r r i s a n d W i l s o n (1978) m o d e l h a s b e e n used, of t h e form:

A° (t) e-8 [<=ij * *j< ‘ >]

F. .(t) = G. (t) --- --- , i,j = l , . . . , n (73)

I

a V ) e-6[ciJ + x i (t)]

w h e r e :

F. (t) is the flow of customers from zone i to services in zone i

ID

at time t, i, j = 1 , . . . ,n,

G^(t) is the total flow of customers from zone i, i=l,,..,n; it is assumed to be of the form

n

G . (t) = 'P I P . (t) + * I P., (t)

l 1 ki 2 lk

k=0 k=l

where and ^ are the frequencies of trips gene r a t i n g from residences and workplaces, respectively,

A .(t) = I P (t) is the total size of service acti v i t i e s in zone j, 3

k -i

measured in terms of employment, j = n,

(*)

x_.(t) is the average price for an average unit of service red in zone j, j = l,...,n,

is a non-negative parameter.

F r o m equation (73) the revenues n t (t ) can be computed: £ F • • (t) x-j(t)

/ ,

i=l

J

V fc) = ---n --- ' 3 = 1 ... n (74)

3 A-;(t)

(*) Having just one sector, a unit of service here can only be in­ terpreted as an average bundle of goods and services consumed per unit time.

47

as well as the average per-capita expenditure in service consumption (including transport costs) for people living in zone i, c (t):

i n _a ■ e [cCii + (t)] £ Aa (t) j = l 3 (75)

Eq ua tion (75) is not an arithmetic mean, but an inclusive v a l u e , in the sense d e fined by Domencich and McFadden (1975) and Williams(1977).

a

Let us interpret the term A, (t) as a production fun tion of services in zone j , i.e.;

a A is the total production of service units in zone j at time j

t, a > 0, j = l,...,n.

Then arguments similar to those used for the housing and labour mar­ k ets lead to the following market-clearing equations for the service p rices x (t): j F . (t) = a A. (t) 3 3 -0,(t) 1 - e 3 j=l,...,n (76) where:

F (t) = F (t) is the total demand for services in j at time t,

j

j i ‘ s

0 (t) = e gXj(t) F (t)/a a“(t ) , j = l ,...,n.

j 3 3

7. Outline of a continuous-time model

In order to build a continuous-time version of the difference £ quations (49) and (50) we simply take their limit as A-K). Ass u m e the following limiting transition rates exist:

49

Then equ a t i o n s (49) and (50) tend to the following differential equa tions: P. (t) = E P . (t) 13 k-i lk q 1 (t) + p 1 . (t)

L kj

k3 .

- p

ID

. .(t) _{- k -1 kD}E q .(t) + n + E p (t)

k=o * ■

+ T. (t) p . . (t),

i

ID

(77)

n

n

o P (t) = E P (t) o . (t) - P (t) E q (t) + 0

,

k„ 0k

*1

k-1 3k

n n n n + E P (t) E p 1 (t) - E E P., (t)

p,X .(t)

I D D k . , . l k k D

i—1

J

k=0

i=l k=l

(78) n E T . (t) p ..(t) i=l 1 1D if j = 1 f m m • f n.

The limiting transition rates do actually exist, and from equations (58) to (63) they are given by:

q 1 (t) = X W (t) e 6 » i 3 < t > -

kj

j

0

q (t) = A W (t) e kj j g[ u 0 j(t) - u^t t)] (80) p ^ . (t) = yP . (t) e

k }

0

j

B[vij(t) - v i k (t)] (81) - 6 v i k (t) (82) (83) x ^ ^ , ]c ~ 1 , . • • , n • E q u a t i o n s (79) to (83) d e s e r v e a remark. A l t h o u g h t h e f u n c t i o n a l form of the t r a n s i t i o n p r o b a b i l i t i e s in the d i s c r e t e - t i m e m o d e l l o o k s qui_ te d i f f e r e n t , th e c o r r e s p o n d i n g t r a n s i t i o n r a t e s in t he c o n t i n u o u s ­ t i m e m o d e l h a v e a f o r m w h i c h is i d e n t i cal to the o n e p r o p o s e d in the m a s t e r e q u a t i o n s a p p r o a c h b y W e i d l i c h a n d H a a g (1983) a n d H a a g in Ber tuglia, L e o n a r d i a n d W i l s o n (eds.) (1986, C h a p t e r s 4., 5., 11. a n d 14.), a l t h o u g h t h e y h a v e b e e n d e r i v e d f r o m d i f f e r e n t a s s u m p t i o n s .

In o r d e r to f i n d the d y n a m i c e q u a t i o n s for p r i c e s a n d incomes, d e f i n e t h e f o l l o w i n g limits:

$. (t) = l i m (f> . (t, A) j A-H) 3

ip (t) = l i m ip . (t,A) , j _A-+0 J

51

T h e n f r o m the t h e o r y d e v e l o p e d b y L e o n a r d i in B e r t u g l i a , L e o n a r d i and W i l s o n (eds.) (1986, C h a p t e r s 5., 9. a n d 15.) one h a s t he f o l l o w i n g d i f f e r e n t i a l e q u a t i o n s for the h o u s i n g p r i c e s a n d d i s p o s a b l e incomes:

v * » ■

t

i

-*.< t) e-

6

r j ( t >

:

(84)

■ i

1 - e j i t ) -I

1

--By

. (t) i^.(t) e J 3 (85)

a s i m i l a r e q u a t i o n h o l d s for the s e r v i c e prices:

V l) ' ê

1 - e J - g X - j ( t ) 0 . (t) e 3 (86) j — l,...,n, where y is a non-negative parameter.

Leonardi shows in Bertuglia, Leonardi and Wilson (eds.) (1986, C h a p ­