MARKET EQUILIBRIUM UNDER THE CIRCUMSTANCES OF SELECTABLE ECONOMIC CONDITIONS. Osamu Keida

Transcription

1 MARKET EQUILIBRIUM UNDER THE CIRCUMSTANCES OF SELECTABLE ECONOMIC CONDITIONS Osamu Keida WP-AD Correspondence: Kumamoto Gakuen University Editor: Instituto Vaenciano de Investigaciones Económicas, S.A. Primera Edición Febrero 2006 Depósito Lega: V IVIE working papers offer in advance te resuts of economic researc under way in order to encourage a discussion process before sending tem to scientific journas for teir fina pubication.

2 MARKET EQUILIBRIUM UNDER THE CIRCUMSTANCES OF SELECTABLE ECONOMIC CONDITIONS Osamu Keida ABSTRACT Tis paper presents an anaysis of market equiibrium under te circumstances wit severa discrete economic conditions by using pure excange economy mode. First, as preiminary anaysis, it wi sow te tempora market equiibrium under a given distribution of popuation over te different circumstances in section 2. Next, in section 3 our study wi prove te existence of market equiibrium in te case tat economic agents can coose teir economic conditions freey for teir utiity maximization. Finay our researc tries to approximate our mode to te residentia ocation mode troug te specified assumptions on initia endowments and agent s preference, and it derives some properties of equiibrium consumptions and prices. Keywords: equiibrium, oca goods, excess utiity, JEL cassification: D51, R13, R20 1 Introduction We sa try to extend te ordina market equiibrium anaysis to a more genera case: te market consisted of seectabe economic circumstances by using pure excange economy mode, and ten to approximate tis mode to te urban economic mode by making assumptions specific. In te ordina market mode an economic agent determines te optima beavior to te prices of goods, and te difference between te market demand and te market suppy of eac good affects its price and oter prices, wic infuences eac agent s beavior of demand or suppy again. An equiibrium soution is obtained by tis repetition if it exists. By contrast, in urban economic modes and regiona economic modes te prices of some goods ike and rent differ depending on te ocations of economic activities, and ten te consumption quantities of suc goods are different at distinct ocations. Te suppies of suc goods aso differ at every ocations. Eac agent of te economy seects a ocationa point in order to determine is optima economic beavior for maxima utiity or profit. Tus, te eminent feature of urban economic modes and regiona economic modes is te coice of a ocationa point wic means te coice of prices at wic an agent faces on is ocation. Te agent performs utiity maximization or profit Te autor wises to tank te anonymous referee of IVIE for epfu comments, and te autor aso wises to tank Professor Otani, Y. of Kyusu Sangyo University for usefu comments on West Japan Economic Conference(2003 and 2005). Kumamoto Gakuen University (E-mai:keida@kumagaku.ac.jp) and University of Aicante (visiting scoar from September 2004 to February 2005) 1

3 maximization on is ocationa point. If te utiity or profit is smaer tan te one derived in oter point, ten te agent wi cange is ocation for a better eve of utiity or profit. In te end an equiibrium is attained so tat te eve of eac agent s utiity or profit is indifferent in te same group type even if is ocation is different from oter agents. We sa extend an anaysis of te typica market mode to sow te existence of market equiibrium of an extended mode in wic te coice of ocation means te coice of te prices of goods dependent on ocations. Tere as been made a ot of works on te genera equiibrium mode so far. Among tem we find some patform researces wit a rigid and strict framework: Arrow and Han [2], Debreu [3] and Hidenbrand [3]. On te oter and we encounter some papers of axiomatic approac in urban economics as in Turnbu[8] and [9] wic spur us to make an anaysis in tis direction. Tus we try to appy te resuts of te genera equiibrium teory to te case of seectabe economic circumstances by reying on a newer researc of Viar [10] wo gives us a cear-cut procedure of proof. To tis end we sa introduce a new key concept for equiibrium: excess utiity, wic is a distinctive feature of tis paper. We can review a ot of spendid works suc as Aonso[1], Mis[6] and Weaton[11] in te area of urban economics. Te structure of a city: ousing demand, rent, popuation and city size were we anayzed wit comparative anaysis of equiibrium. However, since te interests of tese researces oriented toward te structure of urban area, te equiibrium mecanism toward equiibrium in an urban area was not specified ceary ike te genera equiibrium teory. Here we try to reinforce equiibrium mecanism in te urban economic modes to empoy a new concept for equiibrium: excess utiity wic corresponds to excess demand in te genera equiibrium teory and as an important roe in equiibrium anaysis. Te basics of our mode is as foows. An economy consists of different economic circumstances tat eac agent can coose freey. Te economy as two kinds of goods; one type of good of wic price canges wit te circumstances and te oter type of good of wic price is constant over te circumstances. Tese goods wic are given to a agents in certain amounts at an initia state are traded among te agents as an pure excange economy. Te market cearing condition of a goods does not necessariy yied a permanent but a temporary equiibrium of te economy. Tis is because te agent distribution over te circumstances is given wen te market condition becomes ceared. If te utiities of agents are different in te circumstances and some agents obtain smaer eves of utiity, ten tose agents wi cange teir ocation in seeking a better utiity eve, wic breaks te temporary equiibrium. However suc repetition seeking for a new temporary equiibrium attains at a permanent equiibrium in te end. Te permanent equiibrium is te situation in wic agents wi not cange teir ocation at temporary equiibrium. A new concept of agent s excess utiity away from average utiity pays a significant roe for te permanent equiibrium in Section 3 corresponding to te market cearing condition of goods. Once te economy satisfies te new condition tat agent s excess utiity becomes zero, te economy attains an equiibrium of agent distribution wic resuts in a permanent equiibrium of te economy. Anoter purpose of tis paper is to sow te property of equiibrium in te economy of seectabe economic circumstances. Te property is obtained in te mode specified wit additiona assumptions on te agent s initia odings of goods and preference for goods, wic approximate our mode to te residentia ocation mode. Te main feature of 2

4 te residentia ocation mode is tat consumption of residentia service or and increases wit te distance from city center, te consumption of composite goods decreases wit te distance, and te price of residentia service decines wit te distance. Tis paper derives te same kind of properties of goods and price ike te urban economic mode troug te specification of te mode. Te paper as four sections as foows. In Section 2, a basic mode setting is made foowing Viar[10] and Debreu [3]. After mode buiding, we sa make a preiminary anaysis of a temporary market equiibrium wen te ocation of eac agent is given. Section 3 treats te main task of tis paper to prove te market equiibrium wen eac agent coose its ocation freey. Here is given a new condition of excess utiity over average utiity. Section 4 gives furter anaysis of equiibrium wic provides some properties of equiibrium in a specified economy wit additiona assumptions on initia endowments and preference. Te resut of tis section coincides wit te resuts of te residentia ocation mode. Finay te concusion foows in Section 5. 2 Mode setting and temporary market equiibrium First, we wi make a mode ere to perform a preiminary anaysis of temporary equiibrium in seectabe economic circumstances. Economic circumstance and goods An economic circumstance is defined as te pace of economic activity tat brings te prices of goods and te income of an agent to cange. In tis paper it is assumed tat economic circumstances are discrete, and tat te tota number of te circumstances is j, indexed by j = 1, 2,, j. Eac economic agent is assumed to coose ony one from te economic circumstances. Tus te eac circumstance is mutuay excusive as a candidate for agent s ocation. Two types of goods: Loca goods dependent on economic circumstances: te eac market of oca goods is formed in eac economic circumstance one by one, and ten different markets ave different prices depending on te circumstances. For exampe different vaues of rent are bided on different points. Te number of oca goods is assumed to be 1, and Good 1 is te oca goods. Genera goods independent of economic circumstances: ony one market is constituted wit a te circumstances for eac gnenera good. Ten a common price eve of a genera good is formed over te economy. Here, te number of te genera goods is assumed to be 1, and te goods are indexed by = 2,,. Te prices of goods are denoted by a vector p. Since te oca goods form different markets based on te economic circumstances, different prices are formed in tose circumstances. Ten, te price of a oca good at economic circumstance j is denoted by p 1j, and te price of genera good 2 to, by p. p = (p 11, p 12,, p 1j,, p 1 j, p 2, p 3,, p ) P R j+ 1 + }{{}}{{} prices of oca goods prices of genera goods 3

5 Economic agent (consumer) and consumption pan It is assumed tat tere exist ĩ types of agents, and tat I is te set of agents. Agent type i is te continuous entity wic is a point in te interva I i [a i, b i ], wic is divided into te intervas I ij tat type i consumers ocate in j at te initia state. Denote te number of agents of I i, n(i i ) as te size of te interva n i = b i a i > 0, and te number of agents of I ij as n(i ij ) n ij. j n ij = n i. Let us denote furter te size of te type i agents wo cange is ocationa point from j in te initia state to in te termina state as n ij. ñ ij (n ij1, n ij j ) is te distribution of agent i at a initia state, and ñ ij is defined on te simpex N ij R j ñij + n ij = n ij. =1 Te distribution of a agent types is n (ñ 11, ñ 12,, ñĩ j ). Te space N is defined as N ĩ j i=1 j=1 N ij. Consumption of goods and its utiity An economic agent seects a consumption pan wic maximizes utiity under is budget constraint. Tis consumption pan ere incudes te seection of one economic circumstance. However, Section 2 supposes tat eac agent makes consumption decision on an economic circumstance given to im. Te agents of te same type are assumed to be identica in terms of taste in consumption of goods, and ten tose agents ave a identica utiity function. Te utiity function u ij of te type i agent of te initia circumstance j is defined on a consumption set of R j+ 1. Suppose tat type i agent of te initia economic circumstance j as moved to te circumstance. Te consumption vector of te agent at te circumstance is x ij = (0,, 0, x ij, 0,, 0, xij 2,, xij ) R j+ 1, (1) were x ij is te consumption of goods 1 of te agent type i of te initia circumstance j at te economic circumstance, and x ij ( = 2,, ) is te consumption of te genera goods of te agent. Te parts of oca goods and genera goods are aso expressed as x ij (0,, 0, x ij, 0,, 0), xij (x ij 2,, xij ). Te consumptions of oca goods at te circumstances oter tan are aways zero for te agent wo ocates at te circumstance. Terefore, te ony (x ij, xij ) determine te utiity of te agent at circumstance. Te utiity function u ij (x ij ) can be written as u ij (x ij, xij ), 4

6 were te consumption eement is (x ij, xij ) X ij R. Consumption pan and anaysis of market equiibrium We wi make an anaysis of market equiibrium in two steps in tis paper. Consumption panning in two steps: Step1: Optima consumption and market equiibrium under te economic circumstance given to eac agent: an optima consumption is made wen eac agent s ocation is given at some economic circumstance. (anaysis in Section 2) Step2: Optima consumption by seection of economic circumstance: an optima consumption is made under seection of economic circumstance. (anaysis in Section 3) In Step1 we consider an optima consumption pan wen an economic circumstance is given to eac agent. Since an agent consumes te oca good ony at is own circumstance, te consumption of te good is zero at oter circumstances. For exampe, wen a consumer is te type i agent wo moves from te initia economic circumstance j to te circumstance, is consumption takes pace at. Hence is consumption wi be x ij 0, xij 0( = 2,, ) and x ij 1m = 0(m = 1, 2,, j, m ). In Step2 eac agent wi attain is optima consumption by seection of te most preferabe economic circumstance. A circumstances are te candidates for is ocation, and ten eac agent tries to maximize utiity by reseecting te most preferabe circumstance. If te type i agent of initia circumstance j canges is ocation from to, ten is consumption vector becomes (0,, 0, x ij, 0,, 0, x ij 2,, x ij ) from (0,, 0, x ij, 0,, 0, xij 2,, xij ). Initia endowment of goods and budget set Te agents of a same type are assumed to ave an identica initia endowments of goods, if teir initia circumstances are te same ocation. Atoug te agents beong to te same type, if teir initia circumstances are different ten teir initia endowments migt be different. As for te type i agent of initia circumstance j, is initia oding of good 1 is denoted as ω ij 1j, and te initia odings of oter goods, as ωij. Hence te initia odings of goods of te agent of type i at initia circumstance j is ω ij = (0,, 0, ω ij 1j, 0,, 0, ωij 2,, ωij ), wic is decomposed wit oca goods ω ij (0,, 0, ω ij 1j, 0,, 0) and genera goods ω ij (ω ij 2,, ωij ). A consumer is supposed ere to decide is optima consumption wit te initia odings of goods wic is sod to make is incime of budget for consumption pan. Consumption budget and demand correspondence Te type i agent of te initia economic circumstance j makes purcase of is necessary goods by using te initia odings of goods ω ij. Let us denote te budget set of goods wic te consumer of tis type can buy wit te initia odings at te circumstance as β ij (p 1j, p, p, ω ij 1j, ωij ) {( x ij, xij) X ij p x ij + p xij p 1j ω ij 1j + p ωij}. 5

7 Te optima consumption is te consumption tat attains a maxima utiity under te budget set. Since te budget set of an agent depends on te prices of goods and initia odings, te optima consumption, i.e., te consumer s demand of goods is determined by tose prices and te initia odings. Looking at te type i agent of te initia circumstance j wo makes consumption pan at te circumstance, te agent s demand x ij is determined by te demand correspondence. Tat is, x ij ξ ij ( p 1j, p, p, ω ij 1j, ωij), were te oca goods and te genera goods are ( x ij ξ ij p 1j, p, p, ω ij 1j, ωij), and ( ( ( x ij ξ ij 2 p 1j, p, p, ω ij 1j, ωij),, ξ ij p 1j, p, p, ω ij 1j, ωij)). By denoting te genera goods as ξ ( ( ( ( ij p 1j, p, p, ω ij 1j, ωij) = ξ ij 2 p 1j, p, p, ω ij 1j, ωij),, ξ ij p 1j, p, p, ω ij 1j, ωij)), te demand correspondence is ξ ij ( p j, p, p, ω ij 1j, ωij) ( ξ ij (p j, p, p, ω ij 1j, ωij ), ξ ) ij (p j, p, p, ω ij 1j, ωij ). Te correspondences of a agents are combined into a vector. ( ( ) ξ p, ω ij 1j, ωij) ξ 111 (p j, p 1, p, ω ij 1j, ωij ),, ξĩ j j (p j, p j, p, ωij 1j, ωij ) Axioms on consumption It is assumed tat te consumption set, te utiity function, and te initia endowment of goods satisfy te foowing conditions. Axiom 1 As for te type i agent of a initia economic circumstance j wo makes a consumption pan at a circumstance (i) X ij (ii) u ij : X ij R is a non-empty cosed and convex set bounded from beow. R is a continuous, quasi-concave and non-satiabe utiity function. (iii) Te eements of initia endowments are ω ij X ij, and tere exists xij0 X ij suc tat x ij0 < ω ij. If we take a arge enoug positive scaar α > 0, ten te consumption x ij can be smaer tan te vector αe, were e = (1,, 1). Let us denote te set of consumption vectors tat satisfy tis reation as X ij (α). X ij X ij (α) = {xij X ij xij αe} (α) is a compact set, and te image wic Xij (α) projects xij to R is aso compact. A budget set β ij (p 1j, p, p, ω ij 1j, ωij ) is non-empty from Axiom (iii). Ten te budget set is expressed as a non-empty and continuous correspondence form te price set P j 6

8 to te goods set Xij (α).1,were P j is te set of prices of oca goods at j and and prices of genera goods. Since tis is a restricted correspondence to goods set X ij (α), te budget set is aso restricted one. Let us denote tis restricted budget set as ˆβ ij (p 1j, p, p, ω ij 1j, ωij ). Te utiity function as a maximum on te budget set from Axiom (ii). Terefore te foowing teorem is derived from Berge s maximum teorem in Appendix. Tis is te direct appication of te typica genera equiibrium teory to tis mode. R +2 + Teorem 1 Suppose tat (i), (ii) and (iii) of consumption axioms od. Let α be a arge enoug positive constant suc tat X ij (α) Xij is non-empty. Ten, for te type i agent of te initia economic circumstance j wo makes a consumption pan at circumstance, its restricted demand correspondence ˆξ ij is a non-empty, compact and upper-emicontinunous correspondence. Excange economy Let us denote te distribution of te type i agents over on te economic circumstances at an initia state as I ij (i = 1, 2,, ĩ, j = 1, 2,, j). Since eac economic agent is caracterized by te initia endowments of goods, consumption goods set and utiity function, we can define an excange economy as foows. E = {(((n(i ij )(X ij, uij, ω ij ))ĩi=1) j j=1 ) j =1 } were n(i ij ) is te number of te eements of te set I ij, tat is, n(i ij ) = n ij. Aocation and feasibiity Te aocation of goods in an excange economy E is expressed by an eement of product of consumption sets of a economic agents, i.e., a point (((x ij ) j ĩ j j i=1 j=1 =1 Xij. We say tat a point (((xij ) j te point satisfies te next two conditions. (a) (((x ij ) j ) j =1 j=1 )ĩi=1 is an aocation, (b) ĩ j i=1 j=1 n ijx ij ĩ j i=1 j=1 n ijω ij ) j =1 j=1 ( = 1, 2,, j), ĩ j j i=1 j=1 =1 n ijx ij ĩ j j i=1 j=1 =1 n ijω ij ) j =1 j=1 )ĩi=1 in )ĩi=1 is a feasibe aocation if ( = 2, 3,, ). Waras equiibrium in an excange economy Waras equiibrium in an excange economy is te sate of a feasibe aocation wit its supporting price system wic is (((x ij ) j ) j =1 j=1 )ĩi=1, p ). Tis equiibrium is obtained wen te agent distribution n = ((ñ ij )ĩi=1 ) j j=1 is given. If te agent distribution canges, ten te aocation wit its supporting price system is not in equiibrium any onger. Hence te equiibrium (((x ij ) j ) j =1 j=1 )ĩi=1, p ) is temporary because te agent distribution is given for tis equiibrium in te economy.we ca tis Waras equiibrium as temporary market equiibrium. 1 See page 106 to 109 of G, Debreu An Axiomatic Anaysis of Economic Equiibrium(1959). 7

9 Excess demand correspondence An agent decides is optima consumption or demand based on te initia odings of goods, and for tis purpose te agent makes an excange of is goods in te markets. Eac economic circumstance forms te demand of Good 1 differenty. At te circumstance te demand of Good 1 of te type i agent is ĩ Since te tota endowments of Good 1 at is ĩ Goods 1 at te economic circumstance is ζ (p) i=1 j=1 n ij ξ ij (p j, p, p, ω ij 1j, ωij ) As for te genera goods te excess demand is ζ (p) j=1 i=1 =1 n ij ξ ij (p j, p, p, ω ij i=1 1j, ωij ) j i=1 j=1 n ijξ ij (p j, p, p, ω ij 1j, ωij ). j j=1 n ijω ij, te excess demand of i=1 j=1 n ij ω ij ( = 1, 2,, j). (2) j=1 i=1 n ij ω ij. Combine (2) and (3) in order to express as te excess demand vector ( = 2,, ). (3) ζ(p) (ζ 11 (p),, ζ 1 j (p), ζ 2(p),, ζ (p)) (4) Te excess demand ζ(p) is a mapping ζ : P Z of R j+ 1 on R j+ 1. A temporary equiibrium is obtained under some agent distribution n being given. Wat we need for te proof of te temporary equiibrium is te reation between P and Z in te mapping. Te next teorem is derived from te above preparations for te temporary equiibrium. Teorem 2 :Existence of temporary market equiibrium under te economic circumstances being given to a agents Let Z R j+ 1 be a compact set, and ζ : P Z be an upper emi-continuous correspondence wit non-empty, compact and convex vaue. Suppose furtermore tat for a z ζ(p), and for a ˆp ˆP, ˆpẑ = 0 under n being given. Ten, tere exist ˆp ˆP and ẑ ξ(p ) suc tat ẑ i 0, wit ˆp i = 0 if ẑ i < 0. (Proof) 1. Proof of Feasibiity (b): ĩ j j=1 j =1 n ijx ij j i=1 j=1 n ijx ij j j=1 ĩ j i=1 j =1 n ijω i j=1 n ijω i ( = 1, 2,, j) and ĩ i=1 ĩ i=1 ( = 2, 3,, ) to be satisfied. If eac agent performs is consumption witin te budget, ten te foowing inequaity ods. As for te type i agent of te initia circumstance j wo make consumption pan at circumstance, is budget constraint is tat is, px ij pω ij, p x ij + p xij p 1j ω ij 1j + p ωij. 8

10 By summing up te inequaities for a te agents of type i of te initia circumstance j and te present circumstance we get n ij p x ij + n ij p x ij n ij p 1j ω ij 1j + n ij p ω ij. Furter summing te above inequaities for a agents in te woe economy gives n ij p x ij + n ij p x ij n ij p 1j ω ij 1j + n ij p ω ij. Te eft-and side is te product of te price vector (p 11,, p, p 1 j 2,, p ) and te vector of tota sum of demands ( j ĩ j=1 i=1 n ij1x ij 11,, j ĩ j=1 i=1 n ij j xij, j j 1 j =1 j=1 n ij x ij 2,, j j ĩ n ijx ij vector (p 11,, p, p 1 j 2,, p ) and vector of tota sum of initia endowments ( j n i ω ij 11,, j ĩ j=1 i=1 n i j ωi1 1 j, j =1 Terefore, te feasibiity (b) is satisfied. ĩ i=1 ). Te rigt-and side is aso te product of price j ĩ j=1 i=1 n ijω ij 2,, j j ĩ =1 j=1 j=1 ĩ i=1 i=1 n ijω ij 2. Proof of tat te mapping of price adjustment is non-empty,compact and convex; Define a mapping µ : Z P as foows: µ(z) = {p P p z p z, p P } Since P is compact, te continuous function p z as a maximum on P wen z is given. µ(z) is bounded because P is compact. µ(z) is aso cosed from te continuous function p z. If p and p µ(z) ten λp + (1 λ)p µ(z), for a λ [0, 1] Tus te µ(z) is convex. 3. Proof of upper emi-continuousness of µ(z). p z is continuous on P Z. Take a constant correspondence ν(z) = P for a z Z. Ten te Maximum teorem concudes tat µ(z) is upper emi-continuous. 4.Proof of existence of fixed points under te restricted demand correspondence. Since te restricted goods set X ij (α) is a compact set, te eements of tis set form a compact set. It foows from te teorem 1 tat te demand correspondence of eac agent, ˆξ ij is an upper emicontinunous correspondence wit convex vaues. Ten, te market demands summed up troug a economic agents ave an upper emicontinunous correspondence wit convex vaues, and te excess demand ζ (p) and ζ (p) defined by (2) and (3) aso as te same property. Make a new mapping φ : Z P P Z suc tat for a (z, p) Z P, φ(z, p) = ζ(p) µ(z). Z P is a non-empty,compact convex set, and φ(z, p) is an upper emi-continuous correspondence wit non-empty, compact and convex vaues. Tus Kakutani s fixed point teorem concudes te existence of a fixed point under te restricted demand correspondence. ). 9

11 5. Proof tat x ij maximize utiity of te agent under X ij. Let x ij maximize utiity under X ij β ij (p 1j, p, p, ω ij 1j, ωij ) (α), tat is, xij maximize te utiity under {( x ij, xij) X ij p x ij + p xij p 1j ω ij 1j + p ωij}. Now suppose tat a different point (x ij, xij ) in X ij coud attain bigger utiity tan x ij, tat is, u ij (x ij, xij ) > u ij (x ij, xij ). Since (x ij, xij ) is an inner point of [ αe, αe], tere exist some λ : 0 < λ < 1 suc tat Since X ij (x ij, xij ) = λ(x ij, xij ) + (1 λ)(x ij, xij ) [ αe, αe] is convex, (xij, xij ) X ij (α). Tus, it foows from te convexity of preference tat u ij (x ij, xij ) > u ij (x ij, xij ). Tis contradicts tat (x ij, xij ) maximizes te utiity on te X ij (α). Terefore, xij maximizes te utiity on te X ij. Teorem 2 ensures tat a temporary market equiibrium exists wen te agent distribution is given. 3 Market equiibrium under seectabe economic circumstances Wen economic circumstances are given for a economic agents, teir maxima utiity eves may become different at eac circumstance because of te different consumptions of goods at equiibrium. So ong as an economic circumstance is given for an agent, tis situation wi continue. But, if eac agent can coose te best economic circumstance for is utiity maximization, wat wi appen? Probaby eac agent tries to coose suc an economic circumstance so as to acieve a igest utiity eve. For tis reason te number of agents in eac circumstance wi cange and ten affect te consumptions of agents, wic creates an anoter caenge for equiibrium. Terefore we sa anayze te market equiibrium in te case tat eac agent can coose an economic circumstance, tat is, in te case of te seectabe economic circumstances. According to Teorem 2, wen an aocation of economic agents in te economic circumstances, i.e., n (ñ 11, ñ 12,..., ñĩ j ) is given, tere exist some prices of market equiibriums. If te distribution of agents canges, ten some different tempora equiibrium prices wi be determined correspondingy. Tus te tempora equiibrium prices correspond to te distribution of te numbers of agents respectivey. Let us denote te tempora equiibrium prices as ˆp (n), and ere we ave te next emma. Lemma 1 ˆp (n) is upper emi-continuous wit respect to n. (Proof) Suppose tat ˆp 0 (n 0 ) is a tempora equiibrium price vector at n 0, and tat ˆp q ˆp (n q ) is a tempora equiibrium price vector corresponding to n q wic converges to te n 0. Furter suppose tat im n q n 0 ˆp q = ˆp / ˆp 0 (n 0 ). Ten since ˆp 0 (n 0 ) is a tempora equiibrium price, te excess demand is ( p 0 ˆp 0 (n 0 )) : ζ(p 0 ) 0. We get te next reation. ζ(ˆp ) > 0 ζ(p 0 ) (5) 10

12 On te oter and wen n q n 0, te excess demand of oca goods and genera goods are im ζ (ˆp q ) = im n q n 0 = n q n 0{ i=1 j=1 i=1 j=1 n ij ξ ij (p q ) n ij ξ ij (ˆp ) i=1 j=1 i=1 j=1 n ij ω i } n ij ω i ( = 1, 2,, j) (6) im ζ (ˆp q ) = im n q n 0 = n q n 0{ n ij ξ ij n ij ξ ij (ˆp ) (p q ) n ij ω i } n ij ω i ( = 2,, ) (7) Te rigt-and side of (6) and (7) is positive form (5). Terefore, for n coser to n 0 te excess demands of oca goods and genera goods become j=1 i=1 n ij ξ ij (p (n)) n ij ξ ij j=1 i=1 (ˆp (n)) n ij ω i ( = 1, 2,, j) and (8) n ij ω i ( = 2,, ) (9) wic are bot positive. But since ˆp (n) is a tempora equiibrium price, te above reations are ζ(ˆp (n)) wic must be non-positive in equiibrium. Tat (8) and (9) are positive contradicts ζ(ˆp (n)) 0. Terefore we get te reation im ˆp q ˆp 0 (n 0 ). n q n 0 Wen te type i agent of te initia economic circumstance j makes consumption pan at is budget set β ij (p) is a non-empty and continuous correspondence. Te utiity function u ij (x ij ) is aso a continuous function of x ij from Axiom 1-(ii). Te demand correspondence ξ ij (p) is upper emi-continuous on P. Terefore te indirect utiity function u ij (p) = u ij (ξ ij (p)) is continuous on P form Berge s maximum teorem in te appendix. For saving symbos, et us use te same symbo to denote te indirect utiity function of type i agent as u ij. Wen te type i agents of te initia circumstance j ocate over a te circumstances to make teir consumption, te utiities of te type i agents are u ij (p) (u ij1 (p), u ij2 (p),, u ij j (p)). From Lemma 1, te tempora equiibrium price vector is upper emi-continuous wit respect to te distribution of agents n, and te indirect utiity function u ij (p) is continuous wit respect to te price p. Ten we obtain te next emma. Lemma 2 Wen a price vector p is a tempora equiibrium price, te utiity u ij (p(n)) is upper emi-continuous wit respect to n. 11

13 Te utiities of a agent types are expressed as u(p) (u 11 (p), u 12 (p),, u ij (p),, uĩ j (p)), wic are upper emi-continuous wit respect to n. Excess utiity and permanent equiibrium states of economy Wen eac agent is freey abe to coose is ocation for is consumption of goods, e wi try to coose a better circumstance for bigger utiity. Wat wi be te coice criterion of economic circumstance by an economic agent? Tere migt be conceived severa ways for it. Here we sa try to construct a mecanism based on difference from average utiity of agents. If te utiity of an agent is bigger tan te average utiity of te same group of agents, e wi tink imsef in a better state, and if is utiity is smaer tan te average utiity, e wi tink tat e is in a inferior state and e soud take some action for better utiity. Te average utiity can be a standard eve wen eac agent is to take is action for better utiity. Hence we first define te average utiity of te same type agents as average utiity, and next we define te difference of eac agent s utiity from te average utiity as excess utiity. As ong as tere exists excess utiity, eac agent wi cange is ocation to get a better circumstance for im, and ten te distribution of one type group wi continue to fuctuate accordingy. Once te distribution of popuation ceases to cange and remains constant in te end, an agent wi no onger ave an incentive to cange is ocation in te economy, tat is, te agent wi attain te same utiity eve wit te agents in te same group of te economy. Tis can be viewed as an equiibrium state of agent distribution. In wat foows we wi formuate tis mecanism. Te average utiity ū ij is a weigted average of utiities of te type i agents of te initia circumstance j. Tat is, te average utiity ū ij is te weigted average utiity wit weigts n ij ( = 1,, j) wen te number of agents wo move to ( = 1,, j) is n ij. ū ij (n) 1 n ij =1 n ij u ij (p(n); ω ij 1j, ωij ) (10) Define te difference of eac agent s utiity from te average utiity of te agent type as excess utiity. Te excess utiity v ij of te type i agent of te initia circumstance j at te circumstance is v ij (n) u ij (p(n); ω ij 1j, ωij ) ū ij (n). (11) Te excess utiities of te type i at a circumstances are expressed by a vector, v ij (n) (v ij1 (n),, v ij j (n)). Te excess utiities of a agent types are v(n) (v 11 (n),, vĩ j (n)). Te set V ij of te excess utiities of te type i agents of te initia economic circumstance j can be defined so as to be compact and convex. V ij R j 12

14 We can find te compactness and convexity of te set V ij as foows. First, take any two v ij and v ij of V ij and make a convex combination λv ij +(1 λ)v ij, (λ 0). Te excess utiities v ij and v ij of V ij correspond to te consumptions ((x ij 11, xij1 ),, (x ij, xij j )) 1 j and ((x ij 11, xij1 ),, (x ij, xij j )) respectivey. Since te consumption set X ij 1 j ( = 1, 2,, j) wit (x ij, xij ) is convex, j =1 Xij is aso convex. Tere exits an price p to support a convex combination λ((x ij 11, xij1 ),, (x ij, xij j )) +(1 λ)((x ij 1 j 11, xij1 ),, (x ij, xij j )), 1 j (λ 0) wic beongs to j =1 Xij. Terefore, λvij +(1 λ)v ij, (λ 0) can be acieved under te price p to be in te V ij, tat is V ij is convex. Next, te consumption set X ij (α) is compact, and te utiity function uij is continuous wit respect to consumption. Ten te image of consumption set X ij (α) by utiity function u ij is aso compact, wic in turn makes te excess utiity set V ij compact. Terefore, te excess utiity set V ij is compact and convex. Define te set of excess utiity sets V ij as V ĩ j i=1 j=1 V ij Rĩ j j. Te mapping of excess utiity is v : N V. Since te average utiity u ij (n) is upper emi-continuous wit respect to n, te excess utiity v ij (n) is aso upper emi-continuous wit respect to n from (10) and (11). Terefore te excess utiity v is upper emicontinuous wit respect to te agent distribution n. Lemma 3 A mapping v(n) is upper emi-continunous on te popuation distribution set N. Equiibrium under te economy wit seectabe economic circumstances In Section 2, te tempora economic equiibrium is defined under te economy wit a given distribution of agents. Here we define te equiibrium in te case tat eac agent can coose is ocation for consumption freey. Define te equiibrium of excange economy as te feasibe aocation wit a supporting price vector and a feasibe distribution of agents (((x ij ) j ) j =1 j=1 )ĩi=1, p, ((ñ ) j j=1 )ĩi=1 ). It is te tempora equiibrium in wic eac agent wi not cange is economic circumstance for better utiity. Once te equiibrium (((x ij ) j ) j =1 j=1 )ĩi=1, p, ((ñ ) j j=1 )ĩi=1 ) is obtained, te distribution of agents ((ñ ) j j=1 )ĩi=1 ) wi not cange any more. Te corresponding tempora equiibrium (((x kij ) ) j k A ij j=1 )ĩi=1, p ) wi not cange eiter. Terefore, tis equiibrium is a permanent equiibrium in te meaning tat te equiibrium wi not cange so ong as te distribution of agents remaining in te same state. Here we ca tis permanent equiibrium as market equiibrium. Define a mapping g ij (v ij ) : V ij N ij of te type i agents of te initia circumstance as g ij (v ij ) {ñ ij N ij ñ ij v ij ñ v ij, ñ N ij } Lemma 4 ñ ij v ij = ñ ij v ij (n)= 0 for n = (ñ 11,, ñ ij,, ñĩ j )Cvij = v ij (n). 13

15 (Proof) ñ ij v ij = (n ij1, n ij2,, n ij j )(vij1 (n), v ij2 (n),, v ij j (n)) = (n ij1, n ij2,, n ij j )(uij1 ( ) ū ij, u ij2 ( ) ū ij,, u ij j ( ) ū ij ) = = = = 0 =1 =1 =1 n ij u ij ( ) ū ij n ij =1 n ij u ij ( ) ū ij n ij n ij u ij ( ) 1 n ij =1 n ij u ij ( ) n ij Lemma 5 g ij (v ij ) is non-empty, compact and convex. (Proof) Since N is compact, and ñ ij v ij is continuous on N ij N, ñ ij v ij necessariy as its finite vaue. Te g ij (v ij ) is bounded because of compactness of N and cosed from te definition of g ij ( ). Ten g ij (v ij )is compact. Te convexity of g ij (v ij ) can be sown as foows. Make a convex combination λñ ij + (1 λ)ñ ij of ñ ij, ñ ij g ij (v ij ) for 0 < λ < 1. We conduct te next cacuation to te convex combination. (λñ ij + (1 λ)ñ ij)v ij = λñ ij v ij + (1 λ)ñ ij v ij Tus λñ ij + (1 λ)ñ ij g ij (v ij ). λñ ij v ij + (1 λ)ñ ij v ij, ñ ij N ij = ñ ij vij. Lemma 6 Te correspondence g ij (v ij ) is upper emi-continuous wit respect to v ij. (Proof) Te emma is proved by using Berge s Maximum teorem. n ij v ij is continuous on N ij V ij. Next, we make te correspondence δ ij : V ij N ij so tat δ ij (v ij ) = N ij ods for eac v ij. Tis constant correspondence δ i is continuous. Terefore δ ij (v ij ) = N ij is upper emi-continuous form Maximum teorem. Let us define te correspondence g(v) by use of g ij (v ij ). g(v) (g 11 (v 11 ), g 12 (v 12 ),, gĩ j (vĩ j )) tat is, g : V N. 14

16 Lemma 7 Te correspondence g(v) is upper emi-continuous wit respect to v. (Proof) Take any point v 0 of V. Next If we take v so tat its eement v ijq v ij0 (i = 1,, ĩ, j = 1,, j), ten v q v 0. Now tat ñ q ij gij (v ijq ) from te upper emi-continuousness of g ij (v ij ) and ñ q ij ñ0 ij, we get ñ 0 ij g ij (v ij0 ) for eac ñ 0 ij of n 0 = (ñ 0 11, ñ 0 12,, ñ 0 ĩ j ). Tus we obtain n0 g(v 0 ) wen v q v 0. Teorem 3 : Existence of market equiibrium Suppose tat V R is compact and convex set, v : N V is upper emi-continuous correspondence wit non-empty compact and convex vaue, and furter tat vn = 0 for a v v(n) and a n N. Ten, tere exist n N,v v(n ) suc tat v 0, wit n = 0 if v < 0. (Proof) Define te mapping µ : V N N V as foows. µ(v, n) = g(v) v(n) for (v, n) V N Ten V N is a non-empty compact set because V and N are bot non-empty compact sets. Since ξ is upper emi-continuous correspondence wit non-empty compact convex vaue from Teorem 1, v as aso te same property. g is aso upper emicontinuous correspondence wit non-empty compact convex vaue from Lemma 5 and 7. Terefore µ as te same property. Ten tere exists (v, n ) µ(v, n ) from Kakutani s Fixed Point Teorem. Tat is, n g(v ), v ζ(n ), 0 = n v nv n N Since n 0 for a n N and 0 / V, we get v 0. If v k < 0 ten, n k = 0. According to Teorem 3 tere exist some utiity eve tat agent distribution wi not cange and remain constant, wen eac agent is abe to coose an economic circumstance for is utiity maximization. Suc vaues are te utiity and te agent distribution of permanent equiibrium. Te corresponding equiibrium vaues of consumptions and prices wic are determined by Teorem 2 are te permanent equiibrium of prices and consumptions. Hence Teorem 3 wit Teorem 2 guarantees permanent market equiibrium wen eac agent is abe to coose an economic circumstance. In oter words, tese teorems ave extended te existence of market equiibrium to te case of te seectabe economic circumstances. Te economic modes in wic te price of some goods canges in different ocations suc as residentia ocation and pubic service consumption ocation can be unified and studied by our mode, wic migt be contribution to economics. 15

17 4 Properties of te equiibrium under seectabe economic circumstances Wat is te properties of equiibrium soutions? Our interests in te properties awakes te investigation in te study of equiibrium anaysis. For tis purpose te mode needs to be specified in more detai, because it is too genera to investigate suc properties. Since te mode of tis paper copes wit seectabe economic circumstances, te specification of te mode is made in te direction of te urban residentia mode tat consumers face to different prices of and rent in a city. Ten we wi anayze te properties of te equiibrium soutions wit an approac to urban residentia mode. Urban residentia ocation mode First we review te urban economic mode wit a monocentric city 2. Te monocentric urban city mode as a circuar city tat as a centra business district(cbd) in te city center. Te residents commute to te city center to work and to earn same income. Tey make consumption of goods and service wit te disposabe incomes after commuting cost. Since te consumers are assumed to be identica in taste, tey attain te same eve of utiity in equiibrium. Te given rent at city boundary and te exogenous urban popuation enabes us to obtain te utiity eve of consumers in te city. It is cear from te anaysis of te residentia ocation mode tat te quantity of ousing service (and) is increasing wit te distance from te city center, and tat te price of ousing service become decreasing wit te distance. Generay te interest of te anaysis reates to te effects of te cange of parameters on economic variabes. Te parameters are te exogenous popuation size, and rent at city boundary, consumer s income and commuting cost parameter tat infuence te quantity and price of ousing service, te city size and consumer s utiity eve. Te assumptions of urban residentia ocation mode Te setting of a basic residentia ocation mode are: U1. Eac ocation point is distinguised according to te distance from te city center. U2. A consumers reside in a city and commute to te CBD to work. U3. A consumers earn te same nomina income. U4. Te income after te commuting cost, wic is spent on te consumption of te goods and service, is different depending on te distance to te city center from te residence. U5. A consumer as te same preference on goods and service and te utiity function is identica. U6. Te consumer s taste to goods and service is unreated to te residentia ocation. U7. Te goods and service for consumption are ousing service (and) and composite goods. U8. Te urban popuation is given in a fixed size for a cosed city mode. 2 For exampe, see Brueckner, Te Structure of Urban Equiibria: A Unified Treatment of te Mut- Mis Mode (1987). 16

18 U9. Te and rent (agricutura and rent) at a city boundary is given exogenousy. Consistency wit te assumptions of te urban residentia mode To reside in a ocationa point in te city described in U1 and 2 corresponds to ocate a different economic circumstance of tis mode. However, our mode is fundamentay different from te urban residentia ocation mode on te assumption of income. Wie te initia endowments of goods is te source of income in our mode, te money income is given in te urban residentia mode. From U4 te diposabe income after commuting cost wic is spent on goods and service becomes smaer wit te distance from te city center. Te probem is ow to correspond te initia endowments to te disposabe income. Te utiity function of U5 is basicay consistent wit our mode. Te U6 is te typica setting in te urban residentia mode, and te taste of goods can be aso modeed to be indifferent from economic circumstances in a basic mode. Te ousing service or te and rent of wic price differs in te distance from te city center in U7, corresponds to te oca goods of te mode of tis paper, and te composite goods correspond to te genera goods. Te fixed popuation size in U8 matces te given size of agents in te economy of our mode. Wie te city size is determined by U9, te size of economic environments is given in our mode. Tus te essentia points to approximate our mode to te urban residentia ocation are : 1. Te money income is a common vaue in te entire city. 2. te disposabe income deducted after commutation expense decreases successivey wit te distance form te city center. Here we empoy te foowing assumptions on te initia endowments of goods. Assumption 1 Eac agent of te same type as te same quantities of oca goods and genera goods as initia odings, wic are irrespective wit economic circumstances, tat is, (i) Loca goods : ω i1 11 = ωi2 12 = = ωij 1j = = ωi j 1 j (i = 1, 2,, ĩ) = ω i2 (i = 1, 2,, ĩ, = 2, 3,, ) (ii) Genera goods : ω i1 = = ω ij = = ω i j Tis assumption of initia endowments matces wit te constant money income of te residentia ocation mode. Te income to be spent on goods soud be different according to te economic circumstances. It is pausibe tat te income for consumption canges depending on economic circumstances because of some kind of transportation ike te urban economic mode. We assume ere tat wie eac agent can od te initia odings of oca good for consumption or income witout oss at initia state, e can ony use smaer quantities of genera goods because of some kind of oss ike transportation. Ten we set as foows. Assumption 2 Loca goods and genera goods for consumption; (i) Loca goods can be used in te initia endowments witout oss of quantities. 17

19 (ii) Te quantities of initia endowments of genera goods can be used ess easiy as te number of an economic circumstance is arger.te degree of ease for consumption appears as reduction of te amount of te initia possession of genera goods, and te degree is expressed by te residua coefficient of te quantities of initia endowments. Te residua coefficient in eac economic circumstance a j is 3 1 = a 1 > a 2 > > a j > 0. An agent s preference for goods wit respect to economic circumstance is assumed in te foowings. Assumption 3 An agent s preference doesn t cange by te difference of te economic circumstance. Budget constraint and equiibrium soution We sa sow tat te mode under Assumption 1, 2 and 3 as market equiibrium. Te difference from Section 2 is te budget tat is specified in more detai in tis section. Wen a type i agent of te initia circumstance j makes consumption pan, is budget for consumption of goods canges depending on is economic circumstance. His disposabe income wi be p 1j ω ij 1j + a p ω ij if is circumstance is, and it wi be p 1j ω ij 1j + a p ωij if is circumstance is. Wen a type i agent of te initia circumstance j makes a consumption pan at, is budget set becomes β ij (p 1j, p, p, ω ij 1j, ωij ) {( x ij, xij) X ij p x ij + p xij p 1j ω ij 1j + a p ω ij}. Wie te feasibe condition of market equiibrium is te same as Section 2 as for oca goods, te conditions for genera goods cange into i=1 j=1 =1 n ij x ij i=1 j=1 =1 n ij a ω i ( = 2, 3,, ). In addition te excess demand correspondence of oca goods is te same as Section 2, but te excess demand correspondence of genera goods become ζ (p) j=1 i=1 =1 n ij ξ ij (p j, p, p, ω ij 1j, ωij ) j=1 i=1 =1 ( = 1, 2,, ). n ij a ω i. We can appy te anaysis of Section 2 to te oca goods, and it is ony necessary for us to sow tat te feasibe condition of genera goods is satisfied. If te conditions are fufied, ten te argument on te proof of te teorem 2 in Section 2 wi appied to tis section as it is, and market equiibrium wi exist. We sa exibit tat te feasibe conditions of te genera goods are fufied. Wen a type i agent of te initia circumstance j makes a consumption pan at, te budget constraint is p x ij + p xij p 1j ω ij 1j + a p ω ij. 3 Here we assume te iceberg type transportation cost. 18

20 Te number of te type i agents of te initia circumstance j wo cange is circumstance form j to is n ij. By summing up te budget constraints of tose agents, n ij p x ij + n ij p x ij n ij p 1j ω ij 1j + n ija p ω ij. Furter summing up te LHS and RHS of te above inequaities in te woe economy eads to n ij p x ij + n ij p x ij Te LHS and te RHS are arranged as foows. LHS = (p 11,, p 1 j, p 2,, p ) ( j=1 i=1 n ij1 x ij 11,, RHS = (p 11,, p 1 j, p 2,, p ) ( =1 i=1 n i ω i1 11,, j=1 i=1 =1 i=1 n ij p 1j ω ij 1j + n ij j xij 1 j, n i j ω1 j 1 j, n ij a p ω ij. n ij x ij 2,, n ij a ω ij 2,, n ij x ij ) Comparison of te corresponding eac eements of quantity vectors in LHS and RHS brings to sow tat tose are te feasibe conditions (b). Terefore te economy of tis section as market equiibrium. Teorem 4 Te mode of Section 2 as market equiibrium under Assumption 1, 2 and 3. Properties of market equiibrium We sa anayse te properties of market equiibrium. In te foowings, we attac asterisk to denote te quantities in equiibrium. For exampe, te equiibrium consumptions are x i 1j and x ij (, j = 1,, j, = 2,, ). First, we find tat te consumptions cannot be te same as te initia odings, tat is, te equiibrium consumptions at j j cannot be ω ij 1j = xij 1j = xij 1j = ω ij 1j,ω ij = x ij j = x ij j = ω ij. Tis is because te initia odings for te consumption of genera goods wi decrease depending on te increase in number of circumstances from Assumption 2, and ten te same type agent wi consume different genera goods, wic ead to different to utiity eves. Terefore te equiibrium consumption must be different from te initia odings. We wi carify te properties of market equiibrium troug te anaysis of equiibrium quantities of te type i agent of te initia circumstance j. Lemma 8 Under Assumption 1,2 and 3 te consumptions of te oca goods at economic circumstances and ( < ) by type i agent of te initia circumstance j are x i 1 x i n ij a ω ij )

21 te consumptions of te oca goods at j and j (j > j ) is x 1j 1 x1j 1. (Proof) Tis is proved by contradiction. Suppose tat te consumptions of oca goods at bot circumstances are equa, tat is, x ij = xij. From te quasi-concavity of preference we obtain p x ij + k p x ij + k, and we aso get from te quasi-concavity of preference p x ij + k p x ij + From tese two inequaities we obtain p k (x ij k x ij k ) 0 k. p k (x ij k x ij k ). It is needed for tis inequaity tat x ij k = xij k, (k = 2,, ). Tis means tat te consumptions of oca goods and genera goods at and are equa respectivey. But, as we aready ave argued, te consumptions at different economic circumstances cannot be equa as for eac goods. Terefore te consumptions of oca goods at equiibrium is x ij xij. From Lemma 8 te consumptions of te oca goods at any and are imited to te cases of x ij > xij or x ij < xij. Te possibe combination of genera goods and te oca goods in te equiibrium woud be k < k > k in correspondence wit x ij > xij, (12) k in correspondence wit x ij < xij. (13) Te reason is as foows. If x ij > xij and p kx ij k > p kx ij k ten te agent at can consume more tan te agent at, wic makes te utiity of agent at bigger tan tat of te agent at. Tis case cannot be in equiibrium. Te case x ij < xij and p kx ij k < p kx ij k cannot be in equiibrium eiter. Terefore possibe combination of consumptions are (12) and (13). Furter anaysis sows tat te ony atter case, (13) ods in an equiibrium. Property 1 Under Assumption 1, 2 and 3 te equiibrium consumptions of te oca good and te genera goods are x ij < xij and k > k. 20

22 (Proof) We sa sow tat te case : x ij 1j contradicts te reation of disposabe incomes p 1j ω ij 1j + a p ω ij > p 1j ω ij 1j + a p ωij. > x ij and p kx ij k < p kx ij k Te prices corresponding to x ij and xij k (k = 2, 3,, ) are p and p k. Te next reation foows from te quasi-concavity of preference. p 1j ω ij 1j + a p ω ij = p x ij + k=1 k < p x ij + k=1 k (14) On te oter and, wen te consumptions at te circumstance meet wit te prices p and p k respectivey, we aso obtain te next reation from te quasi-concavity of preference. p x ij + k=1 Tis is transformed into p (x ij xij ) > and (14) is canged into k < p x ij + 0 < p (x ij xij ) < k=1 k p k (x ij k x ij k ), (15) It foows from te reations (15) and (16) tat p (x ij xij ) < p (x ij xij ). p k (x ij k x ij k ). (16) Tus p < p. Finay te price reation p < p eads us to te properties of tis Property 1. We obtain p x ij + k=1 k < p x ij + (14) and (17) give us te reation k=1 p 1j ω ij 1j + a p ω ij < p 1j ω ij 1j + a p ωij. k = p 1j ω ij 1j + a p ωij (17) Tis contradicts te premise of budgets at economic circumstance and p 1j ω ij 1j + a p ω ij > p 1j ω ij 1j + a p ωij. Tis means tat te case: x ij > xij and p kx ij k < p kx ij k cannot be in an equiibrium. Tus te consumptions of te oca and te genera goods are x ij and p kx ij k > p kx ij k in an equiibrium. 21 < xij

23 Property 2 Under Assumption 1, 2 and 3 te equiibrium prices of te oca goods at < are p > p. (Proof) Te property is proved from te quasi-concavity of preference as in te proof of Property 1. p x ij + p x ij + k < p x ij + k < p x ij + Transformation brings te next reations p (x ij x ij ) > 0 < p (x ij x ij ) < k k p k (x ij k xij k ), (18) p k (x ij k xij k ). (19) Tus from (18) and (19) we get p (x ij x ij ) < p (x ij x ij ), wic means p < p. Wat we ave sown is in property 1 and 2; If an agent s initia odings of goods and te initia genera goods are identica but decreases ike iceberg wen e canges is ocation, ten 1: among te agents wo are te same type and te same initia economic circumstance, te agent s consumption of oca goods increases wit te number of circumstance and te expenditure on te circumstance genera goods decreases wit te number of circumstance, and 2: te price of te oca goods decreases wit te number of circumstance. It migt be viewed tat tis resuts correspond wit te residentia ocation mode in urban economics and te economic mode of tis section approximates it we. However tere sti exists difference between te two modes. Property 1 of tis paper insists tat te properties of goods consumption in equiibrium ods ony among te same type agents wo are in te same initia economic circumstance. By contrast, in te residentia ocation mode te consumption of residentia service increases and te consumption of composite goods decreases wit distance in an equiibrium. Tat is, te consumption of tose goods at a ocation coser to te city center is arger tan tat at a ocation farter from te city center. Te difference comes from weter a mode assumes te initia ocation of agents or not. Te residentia mode does not assume te initia ocation of agents and it eads te genera property of consumption of goods. But te mode of tis paper assumes te initia ocation of agents, and ten te property of consumption of goods od ony te same agents wo are in te same initia circumstance. So if we restrict te mode of tis paper to aow ony one type of agents wo ocate in 22