Spatial barter economies under locational choice

Journal

of Mathematical

SPATIAL

Economics

BARTER

9 (1982) 259-274.

ECONOMIES Alexander

University

North-Holland

UNDER

Publishing

LOCATIONAL

Company

CHOICE

KARMANN

of Karlsruhe, D-7500 Karlsruhe 1, W Germany

Received

September

1980, accepted

March

1981

This paper presents a model of spatial barter economy with costly transportation to the CBD and a continuum of households. The notions of competitive equilibrium under spatial choice as well as under fixed assignment of households to their place of residence are introduced. Using a formal framework which allows for a simultaneous treatment of both types, equilibrium results will be derived under natural conditions for households’ characteristics and transportation technologies.

1. Introduction:

A critical

approach

to spatial economies

Traditionally, economists introduced spatial dimensions into the well known Arrow-Debreu-frame of equilibrium analysis by an a-prioriassignment of households to a set of possible locations [see, for instance, Samuelson (1953), Isard and Ostroff (1958), Lefeber (1968), Foley (1970) or Takayama and Judge (1971)]._But, the analysis of household’s locational choice behaviour itself in such a frame leads to considerable problems when following Debreu’s device (1959, p. 29) of indexing commodities by locations: multiplying the dimension of the ordinary commodity space times the number of possible locational sites and assuming convex preference preorderings of households over the extended commodity space, under competitive equilibrium, every household usually has to consume at every possible location r. Of course, this implication of a spatial overall presence of households is one of the major disadvantages when adopting the ArrowDebreu-frame directly to a spatial analysis. On the other hand, the union D(h,p) of households h’s preference optimal demand sets D(h,r,p) over all possible locations r under a price system p is in general not convex, as simple examples show (see fig. 1 and section 2.3 for the exact definitions). An economically meaningful interpretation of the convex hull of the demand set D(h,p) can easily be derived by assuming a continuum of households which can be distributed arbitrarily over space. This assumption well known in modern Urban Economics [Alonso (1964)] was introduced into equilibrium analysis first by Schweizer, Varaiya and Hartwick (1976) and taken up by Castello-Ruiz (1978). 0304-4068/82/0000-0000/$02.75

@ 1982 North-Holland

A. Kawnann,

260

Spatial barter economy

In the following, the model of a barter economy with a central marketor so-called CBD (central business district)-structure and finitely many locational sites will be presented. The approach will allow for a simultaneous treatment of the ideas of locational choice and of a-priori fixed assignment of households to locations. Besides, the models described in Schweizer, Varaiya and Hartwick (1976) and in Castello-Ruiz (1978) will be extended by introducing general convex transportation technologies and generalizing the definition of budget sets in the sense that households’ selling activities are restricted not any longer to the trade with own initial endowments [see also Kurz (1974)]. The existence of competitive equilibria [rather than compensated ones as in Schweizer, Varaiya and Hartwick (1976)] will be established. 2. Basic elements 2.1.

of the model

Transportation

Let R = {r E N: 15 r 5 t} be the set of possible household locations r. We distinguish between 1 mobile commodities and at each location d (locationally) fixed commodities (as housing etc.). Therefore, c := dt + 1 will be the natural candidate for the dimension of the commodity space after embedding fixed commodities at the t different locations into the t-fold product of R”+. Let P = (Pf,PJ

Ep

(1)

denote a generic element of the unit simplex P of normalized ptice systems in R’. Thereby, pf =(p,r(r))rER where p,.(r) E Rd, denotes the price system of fixed commodities at location r, and pm E R!+ is the price system of all mobile commodities. At each location r, there is available a transportation technology set T(r) in R3’ for mobile commodities with generic elements (x,,x,,g)

E T(r),

(2)

describing the amount g E R’ of real resources used up in transporting the commodity bundle x, E R’ from location r to the CBD and xb E R’ from the CBD to r (subscript s stands for selling, b for buying). The following assumptions will be met throughout [see Kurz (1974)]: (T.l) T(r) is a closed convex set in R?. (T.2) (Free disposal) If (xb,x,,g) E T(r) then Ojxb~~~,O~x:~x, and imply (x&x&g’) E T(r). (T.3) (Inaction) (O,O,O)E T(r). (T.4) (Active trade) There exists (xb,x,,g) E T(r) such that x, - g >>0.

gig’

A. Kamann,

Spatial

barter economy

261

Remark. We present two examples of transportation known in literature for which (T.l)-(T.4) are fulfilled:

technology

sets well

1) Linear

transportation [Castello-Ruiz (1978)]: Always a certain percentage 0 < hi(r) < 1 of commodity i will be usedup (at least) in transporting one unit of commodity i between location r and the CBD, i.e.,

(%&g) (ai denotes

E T(r)

if

Ri

2

the ith component

f&(Xb,Xs)

:=

k(r)(Xbi

of vector

+Xx,,),

a E R’).

2) Labour-intensive

transportation [Isard and Ostroff (1958)]: Transportation of any commodity i is possible only in the presence of a predetermined commodity, say 1 like labour. Let 0 < Af(r) < 1 be the portion of commodity I used up (at least) in transporting one unit of commodity i between location r and the CBD, then _

(xb,x,,g)E T(r) 3) A generalization Hartwick (1976)]:

of

(xb,xs,g) E T(r) where

2.2.

if

Lziil AKr>(xt,i + Xsi)*

gt2 g(xt,,x,)

1) and

if

as [Schweizer,

Varaiya

and

g 2 T’(x, +x,),

T’ is a non-negative

Households’

2) is given

(1 x Q-matrix

with elements

less than

1.

characteristics

There is given a partitioning of households into finitely many different household classes each of which is represented by a continuum of individual households. Let I be the finite set of all household classes, Si a non-empty measurable subset of complete preference preorderings > over Rp and of initial endownments e = ((efJrsR, e m) in K+,Ki a measurable subset of replica numbers K in fhe unit interval with positive Lebesgue-measure. Similarly as above, ef,. denotes the initial endownment of fixed commodities at location r. Every household is now defined as an element (> ,q,e,

;K) E H,

(3)

where H:=uierHi and Hi:=SiXK,. We assume that Si n S,=g and Ki A Kj=$J holds for i,jE I with i#j. For simplicity, let us assume that only trade with mobile commodities is costly, not the transfer of assets coming from sales of fixed commodities for example. Therefore, we adopt the convention that an individual household first chooses a location r and sells all his initial endowments of fixed

262

A.

Kamann,

Spatial barter economy

commodities. Then he can buy back some amounts of those fixed commodities available at his location and can trade in mobile ones according to the transportation technology T(r) available at r. Formally, let X,(r): ={(xf1,. . .,xf,*)EIZd,t:~i=O

for

i#d(r-l)+l,...,

dr},

then action space X(r) and consumption space C(r) at r are given by X(r):=Xf(r)xT(r)

and

C(r):=Xf(r)xR:.

For technical reasons we introduce rectangles K,, in p+31 defined by (n EN). We set X,(r): =X(r) n K,. Similarly, for T,(r), X,,(r), etc. K,: = {X:Xi~ n}

2.3. Households’

activities

Let h=(>,ef,em;lc) be a household and >, the restriction of > to C(r). An action x = (xf,xb,xS,g) E X(r) in r leads to a consumption bundle c(e,x) and to an excess demand (at the CBD) Z(x) where c(e,x):=

Xf

( e,+x,-xx,

and

Z(x):=

(,,_z+,).

)

Let af(e):= (2). For p E P, r E R we define h’s budget set, demand optimal consumption plans at r under p, respectively, by B&,r,p):

(4) and

= {x E X&r): &x) B 0, G(x) i p6#},

D,,(h,r,p): = {x E l&(h,r,p):c(e,x)

is >-,-optimal},

(5)

Ct,,,(h,r,p) : = Me,x> E C(r): x E Q,~(h,r,pN. Note that by the above definitions individual households are not restricted by x, s e,,, or g 5 e,,, as required in Schweizer, Varaiya and Hartwich (1976) and Castello-Ruiz (1978). We shall assume that for every household h = (> ,e,K) the following holds: (Cl) >I is convex and locally non-satiated (C.2) e is strictly positive.

on C(r) for every r E R.

Note that by completeness assumption of > we claim that consumption bundles c, E C(r) at different locations r are comparable in utility, but not, that households can calculate convex combinations of them. The following result is a direct generalization of lemma 2 in Kurz (1974). The proof is therefore omitted. Lemma 1.

The correspondence

p-+Q,(kw),

(6)

263

A. Karmann, Spatial barter economy

Fig.1. Optimal consumption bundles in the case of one fixed and one mobile commodity, three location sites I and preferences z , on C(r) being independent of location r. For simplicity, e, = O,T(r) linear.

from P to X,,(r) is upper hemicontinuous, every hgH, rER, ncN.

convex- and compact-valued

for

set

A EC,, = ~CG&LR~(Hi)isl) with properties (T.lHT.3) and (Cl)((2.2) will be called a (restricted) transaction structure [and with deterministic characteristics if for Hi = Si X Ki, in addition, holds ISi\ = 1 (i E I); in this case, the representative characteristics of Si will be denoted by si = (> i,ei)]. 3. Locational

choice in spatial economies

3.2. Spatial economies

Considering spatial economies with a continuum of households we introduce a probability space (Q.&F) and a measurable mapping a: 0+(P

x IF+) x [O, 11,

(7)

where 9 is the space of all preference preorderings over R”+. By (Y~= (> (.), er(.), e,(.)) we denote the respective projection of a into B x R!! x R:, by q the one of a into [O,l]. Let EC,,)be a (restricted) transaction structure. Adopting the definition of economies in Hildenbrand (1974) to our setting we get: Definition I. 8,“) = wM~~J;;E;~,n,) a(.) E H p-u.+ such that M.) dp
> 0 (i E I).

and

e,(.)s

is called a (restricted) spatial economy if k

w-a.e. for some

k E R\;

where Q(P) is Lesbesgue-continuous

and @)(Hi)

A. Karmann,

264

If in addition characteristics, characteristics. A feasible

Spatial barter economy

EC,,, is a (restricted) transaction structure with deterministic we call gCnj a (restricted) spatial economy with deterministic

action

of the

spatial

economy

b,,,

is now

a ,u-integrable

mapping, f = (f(r, .))reR = (xf(r, .), xb(rY .), x,(6 .), g(r, .))reR,

(8)

from 0 to fl,,,X,,,(r> such that x,(r, .)-x,(r, .) is p-integrable f(r, .)) 9 0 a.e. for every r E R. Let M(R) be set of all probability measures on R. Then, distribution is a tupel

v =

and c(e(.), a population

(Vi)ieI E nM(R).

(9)

icI

Thus, vi(r) represents Q :=cy -‘(Hi) for iEI. Definition distribution

the

fraction

of household

2. A pair (f,v) of feasible action is called market allocation of d,,,, if

class

of B,,,

i living

and

of population

C [ C I (Zof)(r,.)dp.vi(r)5ef,

iEI

where

rsRR,

Cf := J$(e(.))

in r. Let

(10)

1

dp.

Remark. (10) is equivalent to [xi,, co{Jni (Z of)@, .) dp: r E R} -efl By this formula, the convexifying effect of population distributions apparent. For every market allocation (f,v), a corresponding properly ment of household samples in R to locations can be found;

n RL # 8. becomes

chosen assignformally:

Proposition. For every market allocation (xv) there exists a corresponding locational decision function 6, i.e., a mapping from 0 to R with the properties: (a) 6 is compatiable with v in the sense that for every i E I, (i) if w E pi then.b(w) = r implies v’(r)> 0, (ii) vi(r) = pni({w E l2i: 6(O) = r}); (b) 6 is compatible with f in the sense that

A.

Karmann,

For fixed i E I define of LX&L) there exists a

Proof. Let (f,v) be a market allocation. R +(vi):= {r E R: vi(r) > O}. By Lesbesgue-continuity disjoint covering (Kiir)rcR+C,,S)with the property cYz(F)(Ki,)=

Vi(r)*f_Y~(~)(Ki)

For r E R\R+(v’) let K,, := $?J.Let respectively, from K,,H,, respectively, S’(S,K)=~(K)=~ Then,

if

i e I.

for every h = (s,K) E to R by

Hi.

Define

(11) a mapping

ICEK~,.

for 6 : = 8’0~~ we get by independency

=

265

Spatial barter economy

&;S’,

(12) of (or and CY_~,

S (ZOf)(s’(h>,h)a(EL)(dh) SxK.

= Doing

3.2.

,~I,~ (Zof)(r, .I c-b-u’(r).

this for every

i E I we get the desired

result.

Locational choice

As a first approach we would define a competitive equilibrium as a market allocation (f,u) and a price system p E P with the property that f leads for all household classes i to an optimal consumption choice at all those locations which are occupied by i, i.e., f(r, .)E D(a(.),r,p)

a.e. on Oi for r with

v’(r)>O.

This definition is unsatisfactory because it does not exclude the case that two household samples 01,w2 with identical characteristics s = (> ,e) are located at different sites rl, r2 E R [with v’(r,), vi(rz) > 0] where they achieve different utility levels, i.e., with

A.

266

Karmann,

Spatial barter economy

For a carefully chosen redefinition let us focus first on: Example

1. Let 8 be a spatial economy with deterministic characteristics (> i,ei)i,r. Let C(i,r,p) := C(h,,r,p) be the representative optimal consumption plans of all households of class i at r under p. It is quite natural to assume that these households try to locate at those location sites where they achieve the highest utility level available in R under p. Formally, for (c~),.~ with c, E E+ let

ML(N

rocz

i9(Cr)rER):={Cro:

%c~,(~,P):={r E R : Cdi,r,p)

R and there is no I with c,>~ c,.~}, c m(>

i,(C,,,(i,r,P)),,,)}.

Then P&,, is the set of optimal locations for i. In this case, an a-prioriselection of ‘reasonable’ locational distributions will be an element in the set &,,,(i,p) : = {vi E M(R

j:for

r E R with vi(r) > 0 holds r E 2&,(&p)}.

(13) The relation

&/II’from IX P to M(R)

will be called selection’ of optimal

locational distributions.

For any spatial economy &(,,, the following locational, distributions are meaningful:

two concepts

in selecting

Example

2. When dealing with an a priori fixed assignment rr of household classes I to locations R [see, for example, Isard and Ostroff (1958), Takayama and Judge (1971), Samuelson (1953)], we define

dr(i,p):={vi

EM(R):

~~(r)=e,(i)(r)},

where E, denotes the Dirac-measure Example Lemma

3. 2.

dR(i,p)

(14)

on {r}.

: = M(R).

The correpondences

JU”, AR and A:: are well-defined,

IXP+M(R),

upper hemicontinuous,

(15) compact- and convex-valued.

Any correspondence & from IX P to M(R) with these properties called selection of a-priori possible locational distributions.

will be

Prooj

The assertions for A”, _&YRare trivial. Let nE N and B, be a restricted spatial economy with characteristics (> i,ei)i,I. Let i E I. By completeness and continuity of I> i, the relation a)

(xr)rER

-+ML(

x,ERE+.

>

i,(xr),E

R)

is non-empty

and

upper

hemicontinuous

for

A. Karmann,

261

barter economy

is non-empty and upper hemicontinuous on P for 1 and the continuity of the mapping x-+c(e,x).

b) p-+c,(p) := C,,(i,r,p) every r E R by Lemma C) p+%,(i,p) := ML(> ous by a) and b).

Spatial

i,(c,(p))rER)

is non-empty

and

upper

hemicontinu-

d) p-W,(i,p) is non-empty by c) and upper hemicontinuous: .?&,(i,p)= {r E R:C,(i,r,p) n %‘Ji,p)# 8. Let (p,,JmoN in P with lim and

first, pm=p

(16) Consider a sequence in C,(i,r,p,) n W”(i,p,) which is bounded by boundedness of X,(r). Choose w.l.0.g. c =lim c, which, by upper hemicontinuity of C,,(i,r, .> and %,(i, .>, is an element in C,,(i,r,p>n%,,(i,p), thus (17) We establish e) non-empty

now the desired because

properties

of the relation

p+~%‘,(i,p):

for rOE 2&(&p) [see d)] we get e,,~A.C,(i,p).

f ) upper hemicontinuous: that limu,=v and

let (p,)

in P with lim pm = p, (v,,,) in M(R)

(18)

v, E KXi,p,). By the latter, R+(v,,,) is in %,,(i,p,) where R+(v,,,):={r: vm(r)>O}. r E R+(v) implies r E R+(u,) and thus r E a,(&&) for m sufficiently Hence, we have v E AtC,(i,p).

3.3.

Competitive

Then large.

(19)

g) closed-valued, because for (v,,,) with lim u, = v follows for m sufficiently large. h) convex-valued, because vl, v2 E .A’,& p) vz(r) = 0 and thus DIVE + (1 - cI)v2(r)= 0.

such

and

R+(v) = R+(v,,,)

r q!%?,(i,p) implies

vl(r) =

equilibrium

The examples l-3 motivate now the following notion of equilibrium which, besides, allows for the above promised simultaneous representation of different spatial concepts. Let A? be a selection of a-priori possible

A. Karmann, Spatial barter economy

268

locational

distributions:

Definition 3. (f, v, p) is called competitive equilibrium of &‘(“)relative to ~2’ if (f, v) is a market allocation of E,,, and p E P is such that for every i E I holds vi E .M(i,p),

(20) a.e. on .Ri for

f(r, .) E D,,(a(.),r,p) where

R+(v’):={r~

r E R+(u’),

R:v’(r)>O}.

Definition 4. A competitive equilibrium of & relative competitive equilibrium under locational choice.

to

.,&” is called

4. Results Let 8 be a spatial locational distributions. Theorem

1.

economy

and

~2’ be a selection

There exists a sequence

of a-priori

possible

(fn,u,,,pn)nEN of competitive equilibria

(f,, \I,,, p,) of 8, relative to A with the additional property that there is some k E R$ such that independent of n holds x,,(r, .>5 k

a.e.,

(21)

where again fnCr, .) = (xfn(r, .), x,,(r, .), x,,(r, .), g,(r, .)). Theorem 2. A?“).

There exists a competitive equilibrium (f; v, p) of 6 (relative to

Theorem 3. Let II be an a-priori fixed assignment of household classes to locations. Then, there exists a competitive equilibrium (f, v, p) of 8 relative to AR, i.e., with the property v’(r)=

1

iff

r =7r(i).

(22)

Thus, v concentrates i on r(i). Theorem 4. For any spatial economy d with deterministic characteristics there exists a competitive equilibrium (f,v,p) under locational choice. In addition: (f,v,p) is egalitarian, i.e., for all r E R with vi(r) > 0 holds ML(>

i2(C(i,r$)),s

R) -

ic(e(.)y

f (r, .))

a.e. On G.

Thus household samples of one class achieve equal utility independent where they live in R+(v’). Second, total expenditures, p,(x,(r,‘.)-x,(r,

.>)+p,g(r,

.)+p+f(r,

.),

(23) of (24)

269

Spatial barter economy

A. Kamann,

for trade in mobile and purchase in fixed commodites available constant for almost all samples in Oi and equal to the income

at r are (25)

w(i,p) : = pf.efi, derived from sales of all fixed commodities. 5. Proofs Let gcnj, A be given. For the proof of Theorem

1 define for ns:N,

f%,(i,r,p) := J Q,(. ,r,p) da (I*), H,

finbW,p) :={(~~(i,r,p).v’(r)),,,: viE.Mi,p)l, &W)(P)

(26)

:= ~((finW)(i,P))iEr),

where i2((xi(r))i,,)

Lemma

3.

=I Z(x’(r)). i,r

The relations

o,,, B,,(A)

and

a(&)

(27)

are well-defined, upper hemicontinuous, convex- and compact-valued pondences with respect to all variables. Moreover, p.A,(A)(p) Proof.

5 0

1) Consider

for every

the relation

corres-

p E P.

p +D,,(i,r,p)

which

is

(a) a correspondence by Hildenbrand (1974, DII4, theorem 2) because X,,(r) is bounded. (b) convex-valued by Lemma 1 and Hildenbrand (1974, DII4, theorem 4). (c) compact-valued by Hildenbrand (1974, DI14, proposition 7) because D,(. ,r,p> is integrably bounded and compact-valued by Lemma 1. (d) upper hemicontinuous by Hildenbrand (1974, DI14, proposition 8) because D,,(. ,r,p) is upper hemicontinuous on Hi by Lemma 1 and X,,(r) is bounded. 2) Consider (a) an upper

and

the relation hemicontinuous

p + a,(.A>(i,p) correspondence:

(p,)

in P with lim pm = p,

(x,)

in

X,(r)

with

x,~b,(A)(i,p,)

which is let (28) and

limx,=x.

Choose

(v,)

270

A. Karmann, Spatial barter economy

with

x,(r)E

w.l.0.g.

limv, v E

(29)

Rkr,P,)-h(r),

= V. By u.h.c.

of A(& .) follows

A(i,p).

We yet have to show that x(r) E %(i,r,p)v(r)

(30)

holds and, thus, x E fin(At)(i,p). Take r E R with v(r)>O; then boundedness, x,,(r)[v,(r)]-l E D,(i,r,p,) converges w.l.0.g. x(r)[v(r>lP’ which, by upper hemicontinuity [see l)], is an element o,(i,r,p). For r E R with v(r) = 0 relation (30) is easily verified. (b) compact-valued: the proof is similar to (a). (c) convex-valued: for j = 1,2 let xj(r)=ai(r)vj(r) For OSaSl

with

vj(r)EAd(i,p).

6,(r)E&(i,r,p),

by to of

(31)

define

0 = 0

VI(r) = vz(r) = 0,

if

= cYvl(r>[avl(r)

+ (1 - a)v2(r)]-’

Then ax,(r)+(l -u)xZ(r)=(P6,(r)+(l element in I?,(A)(i,p) by convexity 3) The assertions linearity of Z(.). 4) p.2?,,(./tl)(p)SO: p[Z(f(r, and the transformation DII4, theorem 5)].

for

the

relation

(32) otherwise.

-B)b,(r))(crv,(r)+(l-cc)v,(r)) of D,(i,r,p) and A&p). p +Zm(.&)(i,p)

hold

is

by 2) and

an

the

this holds by .)) -

$(4.>>1~ 0,

theorem

for correspondences

[Hildenbrand

(1974,

By Lemma 3 we can adopt the fixed point theorem of Gale-NikaidoDebreu [Hildenbrand (1974, CIIIlS)] for the correspondence Z,,(A): Lemma

4.

There exists p E P such that .&(At)(p)

c;IR? #$I

Thus, there exists a competitive equilibrium (f.,v.,p,,) We yet have to show that w.1.o.g. x,,(r, .) 5 k holds.

of 6.

relative

to A.

A. Kamann,

Let C#Jbe a mapping

from @+31

Spatial barter economy

to tii3’

271

defined

by

4(xf,xb,xS,g) : = (Xf’WS,d, with (33)

~~:=(Xsj-max(x,--k,O)j)l~jjI,

~,:=(x,j-max(x,-k,O)j)1~j~l.

A proper reduction of large, trade activities in the above sense is technologically feasible and does not affect the outcome of trade; formally: Lemma

5.

Let h = (> ,e,K) E I-l, x E X,,(r) with c(e,x) 2 0 and e, 5 k. Then

d(x) E X,(r)

with

c(e,x) = c(e,ddx)),

x 2 4(x),

k 2 f,,

(34)

Z(x) = Z(dJ(x)).

Proof If c(e,x)lO then x~~x,-~,~x,-R,zO. Always kzjC-,z 0. By free disposal

k by (T.2),

definition

of

H,

thus,

so 4(x) E X,,(r). The relation xb - ZS = x, - x, implies the last two properties. Especially for feasible actions f = (f(r, .))reR, Lemma 5 states that the mapping 4(f) with 4(f)(.)=((4 of)(r, .))reR is also feasible action with 2, (r, .) 5 k which preserves market allocations and preference optimality, i.e., if (f,~) is a market allocation, so (+(f),v), if (f,v,p) is a competitive equilibrium of 8, relative This completes

the proof

of Theorem

to 4,

so (4(f),v,p).

1.

In the following, let (f,,v,,,p,) be a sequence of equilibria as stated in Theorem 1. Note that the sequences (v,,) and (p,) are bounded. Given i E I, consider any r E R for which a subsequence (n,) in N exists with lim Vi,(r) > 0. Let p =lim pn,. Lemma 6. Proof:

L s(f”,(r, .)) cD(a(.), r, p)

ae. on a.

W.1.o.g. lim v;(r) > 0 and limp,, = p.

Ls(f,(r, .)) c X(r) a.e. on Q by definition of (f,). Ls(f,(r, .))=B(c~(.),r,p) a.e. on 0, by fn(r, .)=B(a(.),r,p,) and the closedness of the relation B. Ls(f,(r, .))~D(a(.),r,p) a.e. on Q: otherwise there is fii in Q with CL(ni)>O such that (c) does not hold on di. Choose 0: in fli with ~(.il{) = ~(0~) and f,,(r, .) E D,,(a(.),r,p,) on 0:.

272

A. Karmann, Spatial barter economy

Observe that p(fii no;)> 0. By continuity of preferences we get the desired contradiction for all w E di no;: by construction of 4, for x E Z,s(f,(r,o)) holds x $ D(a(o),r,p). Therefore there exists a sequence (n,) in N and a certain X’E B(a(W),r,p) such that x = lirnxnG, Moreover, that

x, =f,,(r,w),

x’~B,(a(w),r,p)

for n sufficiently

x&,E B,(cY(w),r,p,,,) Then

the continuity

for m,q sufficiently x, E D,(a!(w),r,p,)

Define Lemma

and

of > (o),

c(e(w),xA)z

(o),c(e(o),x). large.

Choose

(35) (x&) such

x’ = 1imxL.

and (35)-(36)

(36)

imply

(w),c(e(w),x,)

(37)

large; this contradicts for some m = n, to the relation which holds by definition of 0:.

Z;(r):-jfn(r,.)dg*vL(r) 7.

c(e(w),x’)>

The sequence

for

rER,

i~1,

HEN,

(XL(r))i,r,n is bounded in R$$-c3’.

Proof. By the conditions of market Z;,(r), Z:,(r), g:(r)) with _$,(r>S k,

we get for xi(r) = (x&(r),

allocations

le,(.)d~~~~(n:,(r)-_xb,(r))>C

gi(r)ZO, i,r

and so OS &(r)Sje,(.) Therefore we can adopt brand (1974), DII4L.3)]: Lemma 8. subsequence and

dp

and

OS Z;,(r) 5 (qt)k.

Fatou’s Lemma

in several

[Hilden-

There exist a feasible action f of the initial spatial economy and a (n,) of N with the properties that v = lim v,,, and p = lim pn, exists

1) (f(rJLR+(yl)E Ls((f,_(r, .))TER+~Y4 a.e. on % 2) 2 f(r, .) dp Sl’,” 3)

dimensions

(2 f,,(r, .) dp)

jx,(r,.)d~=limjx,,,(r,.)dp Q 4 a.

for every

f(r, .) = 0 elsewhere;

i E I, r E R+(v’);

for iE1, rER,

with v’(r)>O.

A. Kamann,

Spatial

213

barter economy

Proof. By boundedness of (v,), (p,) and choose (II,), in N such that the respective subsequences converge to some V, p, and xi(r). sequence Lemma to the Fatou’s every iEI adopt For ((f,,(r, .)4,(r))rAcN. By x,,,(r, .)S k, thus x&r, _)ui(r)S k for every q E N, this implies the existence of integrable functions (fl(r, _)rtR) on Q such that properties l)-3) hold for fl(r, .) [instead of f(r, .)]. Define for every i E I, f(r, .):=f”(r, := 0 and extend

f naturally

.)*[G(r)]-’

on Q if

u’(r)>O,

elsewhere, on R = IJ i&i.

(38) Then properties

l)-3) hold for f(r, .).

For (f,v,p) as given by Lemma 8 it is easily seen that (f,v) is a market allocation so that together with Lemma 6 (f,v,p) is a competitive equilibrium in & (relative to A”). This completes the proof of Theorem 2. For J4 : = A”, observing that i E I.

which is possible by Lemma 2, we derive Theorem if v=lim vn4 such that vLq= E+) then vi = E,(i) for

3 by every

For A := A/II’ similarily: first, there exists for every n EN a competitive equilibrium (j;I,v,,,pJ of b,, relative to A?‘,. Consider (f,v,p) as in Lemma 8 which turns out to be a competitive equilibrium of & under locational choice by: Lemma 9.

Zf (f,v,p) has properties 4-3) of Lemma 8 then

vk E AC,(i,p,)

for every n E N implies viE A’(i,p),

i E I.

Proof. Fix i E I. For rI E R+(2) we have to show rl E %(i,p). By Lemmas 6 and 8 there exists 0: in IRi with I= Il(ai) such that for every o E Q: holds (a) aI(w) = si> (b) fJr,o)

E D,,(a(o),r,p,,)

for r E R+(d)

(c) (f(r,a)),,R+(vXj 6 Ls((f,,(r,w)),,,+c,,,)

(q sufficiently

large),

c (Wkr~P)L~+cv~~,

(d) f(r,w) = 0 for r E R\R+(d). Case of r2 with vi(r) > 0: such that for all r E R+(d)

by (c), there exists a subsequence (n,,),, of (n,), holds f(r,w) = lim f,_(r,w). By (b) follows:

c(ei,fn,.(r,,o>> -ic(ei,f,,,.(rz~)) thus the relation

also holds

for every

q’ EN,

in the limit by continuity

of preferences.

(39)

274

A. Kamann,

Spatial

barter econom);

Case of r, with ui(r) = 0: then f(r;w) = 0. From both cases follows that tl E JIc‘(i,p) The additional assertions of Theorem

holds.

4 are easy to verify.

References Alonso, W., 1964, Location and land use (Harvard University Press, Cambridge, MA). Castello-Ruiz, J., 1978, Residential choice and general equilibrium theory: Existence and properties of a competitive equilibrium in a special case, Unpublished Ph.D. dissertation (Northwestern University, Evanston, IL). Debreu, G., 1959, Theory of value (Wiley, New York). Foley, D., 1970, Economic equilibrium with costly marketing, Journal of Economic Theory 2, 276-291. Hildenbrand, W., 1974, Core and equilibria of a large economy (Princeton University Press, Princeton, NJ). Isard, W. and D.J. Ostroff, 1958, Existence of competitive interregional equilibrium, Papers and Proceedings of the Regional Science Association 4, 49-76. Kurz, M., 1974, Arrow-Debreu equilibrium of an exchange economy with transaction cost, International Economic Review 15, 699-717. Lefeber, L., 1968, Allocation in space (North-Holland, Amsterdam). Samuelson, P.A., 1953, Spatial price equilibrium and linear programming, American Economic Review 42, 283-303. Schweizer, U., P. Varaiya and J. Hartwick, 1976, General equilibrium and location theory, Journal of Urban Economics 3, 285-303. Takayama, T. and G.G. Judge, 1971, Spatial and temporal price and allocation models (North-Holland, Amsterdam).