Urban Structures with Forward and Backward Linkages

Transcription

1 Urban Structurs with Forward and ackward Linkags Pascal Mossay, y Pirr M. Picard, z and Takatoshi Tabuchi x Dcmbr 12, 2017 Abstract W study urban structurs drivn by dmand and vrtical linkags in th prsnc of incrasing rturns to scal. ndividuals consum local urban varitis and rms us ths varitis to produc a national good. W prov th xistnc of a spatial quilibrium and obtain an invarianc rsult according to which mor intns dmand or vrtical linkags hav th sam ct on th urban structur as lowr commuting costs. Various urban con gurations can mrg xhibiting a monocntric, an intgratd, a duocntric, or a partially intgratd city structur. W discuss th rol of commuting and transport costs, intnsitis of dmand and vrtical linkags, and urbanization in a cting ths pattrns. W show that multipl quilibria may aris in quilibrium involving th monocntric city and up to a coupl of duocntric and partially intgratd structurs. Kywords: Urban spatial structur, dmand and vrtical linkags, monopolistic comptition, land us J.E.L.: R12, R14, R31 1 ntroduction n this papr w study spatial urban structurs drivn by dmand and vrtical linkags in th prsnc of incrasing rturns to scal. ndividuals consum local urban varitis producd by rms undr W wish to thank Marcus rliant, Yashar louri, Masahisa Fujita, Koichi Fukumura, Jacqus Thiss, Ping Wang and participants of th Osaka Confrnc and Vancouvr NARSC for valuabl commnts and suggstions. y Nwcastl Univrsity and CORE. pascal.mossay@ncl.ac.uk z CREA, Univrsity of Luxmbourg and CORE, Univrsité catholiqu d Louvain. pirr.picard@uni.lu x Faculty of Economics, Univrsity of Tokyo, Hongo 7-3-1, unkyo-ku, Tokyo , Japan. ttabuchi@.utokyo.ac.jp. 1

2 monopolistic comptition. Firms also us ths varitis as intrmdiat componnts to produc a homognous national good undr prfct comptition, which is sold within and outsid th city. Our aim is to study how commuting and transport costs, intnsitis of dmand and vrtical linkags, and urbanization a ct th urban structur. For this purpos, w stablish quilibrium conditions in th product and land markts. Our rsults ar th following. First w prov th xistnc of a spatial quilibrium and obtain an invarianc rsult according to which mor intns dmand or vrtical linkags hav th sam ct on th urban structur as lowr commuting costs. W thn study svral urban structurs and discuss thir condition of xistnc and proprtis. For th monocntric and intgratd citis, w provid xplicit conditions that can b rprsntd in trms of th paramtrs of intrst. As for duocntric and partially intgratd citis, w driv implicit conditions that dtrmin th bordrs of businss/rsidntial districts. Ths rstrictions on district bordrs mak part of th quilibrium conditions dtrmining th mrgnc of duocntric and partially intgratd pattrns. Our analysis con rms that a monocntric city mrgs for vry low commuting costs whil spatial intgration of businss and rsidntial districts occurs whn goods ar shippd at low transport costs within th city. As mor intns dmand or vrtical linkags hav th sam ct as lowr commuting costs, thy mak th monocntric structur mor likly. Furthrmor, for intrmdiat commuting/transport costs and modrat intnsitis of dmand/vrtical linkags, many urban structurs may mrg and coxist in quilibrium. W show th xistnc of multipl quilibria involving th monocntric city and up to a coupl of duocntric and partially intgratd urban structurs. Urbanization via an incras in city siz inducs transitions from th monocntric con guration to full intgration possibly passing by duocntric or partially intgratd structurs dpnding on th magnitud of th transport cost. Whn transport costs ar high, th conomy undrgos discontinuous transitions r cting largr bn ts gaind by rms whn production locations sparat into svral businss districts. Th prsnt papr xtnds th xisting litratur on th ndognous formation of urban aras by uncovring th rol of backward and forward linkags on th formation of citis. Whras Fujita and Ogawa (1982) and Lucas and Rossi-Hansbrg (2002) discuss th implications of agglomration bn ts arising from fac-to-fac communication and xognous tchnological spillovrs rspctivly, our papr studis agglomration forcs that ar microfoundd in th xistnc of imprfct comptition and rms incrasing rturns to scal. Accordingly, rms oprating in th city hav markt powr and arn a 2

3 positiv pro t bfor procding to land rnt and labor wag paymnts. Th prsnc of ths intrnal incrasing rturns maks our analysis mor appaling in an urban contxt. As such, our work lls a gap in th litratur. Not that th rducd form of our modl ncompasss th modl structur proposd by Fujita and Ogawa (1982). Namly, in th absnc of dmand linkags and pro t in th production of urban varitis, th rducd form of our modl is rminiscnt to thir sminal work. This suggsts that vrtical linkags hav similar agglomration cts on th urban structur as thos gnratd by fac-to-fac communication and tchnological spillovrs. W thrfor con rm th gnrality of th lattr approachs. As in Lucas and Rossi-Hansbrg (2002), our papr provids an xistnc proof of spatial quilibrium. Howvr, ours is basd on Carlir and Ekland (2007) and applis to Fujita and Ogawa (1982). Rgarding th analysis and computation of multipl quilibria, our approach contrasts with mthods rlying on xd point algorithms. W propos an innovativ way of dtrmining urban structurs, which consists in dtrmining th bordrs of potntial businss/rsidntial districts rst, and thn in chcking whthr ach structur candidat actually constituts a spatial quilibrium or not. Th papr is structurd as follows. Sction 2 prsnts th modl and discusss th product and land markts quilibrium. Sction 3 stablishs th xistnc of a spatial quilibrium and th invarianc rsult btwn dmand/vrtical linkags and commuting costs. Sction 4 studis various spatial quilibria xhibiting a monocntric, an intgratd, a duocntric, or a partially intgratd city structur. Sction 5 prsnts numrical computations and rsults rgarding th structur of ths urban pattrns. Sction 6 concluds. 2 Th modl W considr a linar city hosting a continuum of N individuals and N rms along th intrval [ b; b] R with unit width. Th location supports of individuals and rms ar dnotd rspctivly by X and Y, with X [ Y =. ndividuals occupy a unit of rsidntial land, commut to thir workplac, and consum di rntiatd varitis supplid in th local urban markts as wll as an homognous good supplid in th national (i.. intrcity) markt. n contrast to th national good, trad of local varitis incurs icbrg transport costs. Ths varitis ar also usd as componnts (inputs) to produc th national good. For ths two production activitis, ach rm hirs a workr 3

4 and uss s units of land in th production procss. Using th functional form proposd by P ügr (2004), an individual rsiding at x 2 X and working at z 2 Y is ndowd with th utility function U(x; z) = ln Y c(z; y) 1 1 (y)dy + c0 (x) whr c(z; y) is hr consumption at workplac location z of a varity producd by a rm locatd at y 2 Y, c 0 (x) hr consumption of th national good at rsidnc location x, (y) th dnsity of rms at y 2 Y, > 1 th lasticity of substitution btwn local varitis, and > 0 th xpnditur intnsity on local varitis. Hr budgt constraint writs as p(y)c(z; y) jz yj (y)dy + p 0 c 0 (x) + t jx zj + R(x) w(z) + c 0 Y whr p(y) is th mill pric of a local varity producd at y, th icbrg cost, c(z; y) jz yj th quantity of a varity to b purchasd at th mill so that consumption is c(z; y), p 0 th pric of th national good, t th commuting cost pr unit of distanc btwn rsidnc and work placs, R(x) th land rnt at th workr s rsidnc, w(z) hr wag at th workplac, and nally c 0 som initial ndowmnt of th national good. On th production sid, ach rm hirs a unit mass of workrs, ach of thm supplying a unit of labor. Each rm simultanously producs th national good and a singl componnt varity. To highlight th ct of vrtical linkags, w assum that th national good is mad of th componnt varitis availabl in th urban markts. For instanc, think of a city xporting wood and mtal furnitur: woodcraft makrs could b xporting wood furnitur with stl ornamnts mad by local stlmakrs whil stlmakrs would b xporting stl furnitur with wood ornamnts mad by local woodcraft subcontractors. Similarly, th city of London hosts law rms that us othr law rms and banks xprtis to provid law srvics in th national markt whil banking rms us othr banking and law rms comptncis to provid national banking solutions. Thrfor, th productions of th national good and local componnts ar intgratd within th sam plant or o c. This is bcaus rm s xprtis or suprvision is too costly to manag ovr di rnt plants. Also additional discussion on outsourcing would bring th modl out of focus. Finally, th national good is assumd to b homognous across rms (and citis) and rms ar pric takrs in this markt. 1 n contrast, ach rm 1 Product di rntiation and imprfct comptition in th national markt would not add much to our urban structur analysis. 4

5 is a pric sttr in th markt of its own urban varity. Mor spci cally, production taks plac as follows. On th on hand, th national good is mad of local inputs with th output of a rm locatd at y 2 Y givn by th production function q(y) = ln Y i(y; z) 1 1 (z)dz whr i(y; z) is rm y s local componnt us of a varity producd by a rm locatd at z 2 Y and > 0 th intnsity of vrtical linkags. caus of icbrg transport costs, rm y purchass i(y; z) jx (1) yj units of a varity producd in z and sold at mill pric p(z). On th othr hand, th rm producs its spci c componnt using th national good as input. Evry componnt rquirs units of th national good. Thrfor, rm y s pro t is givn by (y) = p 0 q(y) p(z)i(y; z) jy zj (z)dz Y {z } national product markt +[p(y) ] [c(z; y) + i(z; y)] (z)dz Y {z } urban product markt w(y) sr(y) (2) n this pro t xprssion, th rst two trms rprsnt th oprating pro t in th national markt (i.. rvnu minus th cost of inputs), th third trm th oprating pro t in th local componnt markt (i.. th componnt markup tims th dmand from consumrs and rms), and th two last trms th labor and land costs. Th national good is chosn as th numrair so that its pric can b normalizd to p 0 = 1. W now turn to th product markt quilibrium whr product markts clar. For this purpos, w considr th locations of rms and individuals as xd. Assuming c 0 (x) > 0, th dmand for local varitis by an individual rsiding at y and working at z is givn by c(z; y) = p(y) jz yj P (z) 1 (3) whr th local pric indx is givn by P (z) R Y p(y)1 (1 )jz yj (y)dy 1 1. Hr spnding on local varitis is qual to p 0 and hr dmand for th national good is givn by c 0 (x) = [w(z) t jx zj R(x) + (c 0 )]. On th othr hand, a rm locatd at y simultanous chooss its input mix i(y; ) and its own componnt pric p(y) that maximiz its pro t subjct to rlation (1). Sinc th rm has a zro mass, it is too small to in unc th othr rms local componnt prics and consumrs dmand. caus th pro t in th national markt has th sam functional form as th individual utility, th optimal input 5

6 mix is qual to i(y; z) = (=) c(y; z) whr c(y; z) is givn by xprssion (3). Th rm s spnding on local varitis is qual to. Th oprating pro t on th national markt can thrfor b writtn as [ln (=P (y)) 1] so that a rm supplis th national markt as long as > P (y). Also, bcaus th total dmand for a local varity producd in y, R [c(y; z) + i(y; z)] dz, has a constant pric lasticity Y, th rm sts its componnt pric to p(y) = =( 1) >, which lads to a positiv oprating pro t in th local componnt markt that turns out to b th sam across rms. Th abov dmand functions and pro t-maximizing prics allow us to driv th utility of housholds and th pro t of rms in th product markt. First, th pric indx bcoms whr T (z) P (z) = 1 T (z) Y 1 (1 )jz yj 1 (y)dy is an accss masur of location z corrsponding to th aggrgat dlivry cost from all locations on Y to location z, which incrass with th transport cost and th disprsion of support Y. Consumrs and rms dmands for local varitis ar givn rspctivly by c(z; y) = i(z; y) = 1 jz T 1 As th dmand for th national good is c 0 (x) = w(z) t jx zj R(x) + (c 0 ), th indirct utility writs as whr 1 U(x; z) = ln yj (z) ln T (z) + [w(z) + c 0 t jx zj R(x)] (6) 1= and is th Eulr numbr. n this xprssion, th trm in th rst squard brackt rprsnts th nt consumr surplus from local urban varitis whil th trm in th scond squard brackt r cts th nt consumr surplus from th national good. Th lattr is indd qual to th incom lft aftr paying for local urban varitis, commuting cost and land rnt. Th rm s output in th national markt is givn by q (y) = ln (=) Using th pro t-maximizing pric p(y) = =( ln T (y). local varity to consumrs and rms can b writtn as [p(y) ] [c(z; y) + i(z; y)] (z)dz = ( + ) G(y) whr pro t G is d nd by Y G(y) 1 Y (4) (5) 1), th gross oprating pro t from th sals of a (z) T 1 (z) jz yj dz (7) 6

7 Th xprssion (2) of th rm s pro t at location y can thn b writtn as (y) = ln ln T (y) + ( + ) G(y) w(y) sr(y) (8) Th trm in th squard brackt is th pro t drivd in th national markt and th scond trm that drivd in th markt of th local varity. Th rm has an incntiv to produc th national good if th formr pro t is positiv, i.. ln ln T (y). This holds if is su cintly small, which w assum from now on. 2 Th third trm ( + ) G(y) aggrgating th oprating pro ts from th sals of its varity to consumrs and rms, incras with th xpnditurs intnsitis on local varitis of consumrs () and rms (). Not that this oprating pro t clarly dpnds on th rm s location y. W now turn to th land markt quilibrium still assuming th distributions of consumrs and rms as givn. W assum that land is usd ithr as consumption or rms input so that th opportunity cost of land is nil. n comptitiv land markts, landownrs allocat thir plots of land to th highst biddrs (i.. rsidnt-workrs or rms). y using (6), th bit rnt of an individual rsiding at location x and working at z = z(x) is givn by r(x) = ln ln T (z) + [w(z) t jx zj + c 0 ] U (9) whr U dnots th utility sh can obtain outsid th city. Du to fr ntry and rlation (8), as rm at location y can bid for land up to a lvl which maks its pro t (y) qual to zro, its bid rnt is givn by b(y) = 1 s ln ln T (y) + ( + ) G(y) w(y) (10) As total land consumption (1 + s) N is qual to th total availabl land 2b, th city dg is givn by b = (1 + s) N=2. 3 Spatial quilibrium n this sction, w d n and study th spatial quilibrium of our conomy whr both consumrs and rms xploit spatial arbitrag opportunitis. For this purpos, w rwrit th workr s and th rm s bid rnts as follows. Th workr s bid rnt in location x corrsponds to th maximum bid a workr 2 That is, if (1 1=) =( max y2y T (y)). Givn that max y2y T (y) > 2b, a su cint condition is < max (1 1=) 2b 1. 7

8 can mak on a parcl of land in x whil rciving a wag from a rm locatd in y 2 Y willing to hir hr undr fr ntry. Using (9) and (8), w gt r(x) = max ln ln T (y) + w(y) t jx yj + c 0 U y2y;w(y)2r + s.t. (y) = ln ln T (y) + ( + ) G(y) w(y) sr(y) 0 Similarly, th rm s bid rnt corrsponds to th maximum bid that it can mak on a parcl of land in y whil th wag it pays to a workr locatd in x 2 X inducs hr to supply hr labor to th rm givn hr outsid utility U. Using (10) and (6) 1 b(y) = max ln ln T (y) + ( + ) G(y) x2x ;w(y)2r + s s.t. U(x; y) = ln w(y) ln T (x) + w(y) t jx yj R(x) + c 0 U W now mak th connction with Fujita and Ogawa (1982). n thir sminal papr, urban structurs ndognously mrg from agglomration cts gnratd by fac-to-fac communication btwn rms managrs and/or mploys. (11) (12) This fac-to-fac communication gnrats a production surplus that dpnds ngativly on th logarithm of th avrag distanc btwn rms. On could thn intrprt th paramtr xprssion( 1) in ln T (y) as Fujita and Ogawa s paramtr for facto-fac communication. This mans that from a rducd form point of viw, vrtical linkags hav th sam rol as fac-to-fac communication in Fujita and Ogawa. Howvr, our vrtical linkags also stm from th production of urban varitis undr imprfct comptition, which gnrats a pro t G(y) that is absnt in th fac-to-fac communication modl. To sum up, th rducd form of our modl ncompasss th structur of Fujita and Ogawa in th absnc of dmand linkags and pro t from production of urban varitis (that is, = 0 and G(y) = 0). Th abov problms can b solvd for th wag so that r(x) = max ln y2y b(y) = 1 s max x2x + ln + ( + ) S(y) t jx yj + c 0 U sr(y) ln + ln + ( + ) S(y) t jx yj + c 0 U R(x) whr S(y) G(y) ln T (y) is th conomic surplus in location y. W can now d n a comptitiv spatial quilibrium. D nition 1 A comptitiv spatial quilibrium is d nd by th rsidnts and rms dnsity functions :! [0; 1] and :! [0; 1=s], a land rnt R :! R +, rsidnts and rms bid rnt functions 8 (13) (14)

9 r, and b :! R + and a utility lvl U 2 R that satisfy (i) mass consrvation R = N, (ii) land markt claring (z) = (1 (z))=s for all z 2, (iii) land allocation to highst biddr for all z 2 8 f1g if r > b >< 2 [0; 1] if r = b >: f0g if r < b and R = maxf r; b; 0g, and (iv) zro land rnt R at city bordrs z = b. 3.1 Economic gains and invarianc Th urban conomic activity gnrats gains for workrs and landlords, rms making zro pro t du to fr ntry. t is instructiv to xamin thos gains and disntangl th agglomration forc stmming from dmand and vrtical linkags from th disprsion forc du to transportation, and in particular, commuting costs. Considr a workr rsiding at x and its rm producing at y at th comptitiv land markt quilibrium. Land rnt should corrspond to thir rspctiv bid rnts: r(x) = R(x) and r(y) = R(y). Thrfor, from (13) or (14), th total gains from thir urban conomic intraction ar U + R(x) + sr(y) = ln + ln + ( + ) S(y) t jx yj + c 0 (15) Whil th lft hand sid xprsss workrs utility and landlords rnts, th right hand sid r cts th consumr surplus from local varitis and th producr surplus from both consumrs and rms and from th national markt (through trms in and ). Th surpluss ar xplicitly diminishd by commuting costs. Obsrv that th spatial structur (; ) has an impact only on th trms ( + ) S(y) t jx yj. This mans that it is possibl to maintain th sam quilibrium spatial structur ( ; ) by multiplying both paramtrs ( + ) and t by th sam factor k > 0. To maintain th abov quality for all x and y, quilibrium rnts nd to b multiplid by this factor k and th quilibrium utility U incrasd to th lvl k ln k + k ln k. This huristic argumnt is con rmd blow in a formal way. Suppos that a spatial quilibrium xists and dnot th mass of commutrs moving to th right and crossing location x by n(x) = R [(z) (z)] dx. n Appndix A, w show that th spatial xb 9

10 quilibrium (; ) satis s th idntity 8 >< (z) (z) 2 >: f1g if b(z) r(z) < 0 [ 1 s ; 1] if b(z) r(z) = 0 f 1 s g if b(z) r(z) > 0 (16) whr b(z) r(z) = and bz 2 is a solution to (bz) z bz 1 ds ( + ) s dx 1 + s t sign (n(x)) dx; (17) s (bz) = 0 and dsignats a location whr commuting stops or changs dirction. Thr always xists is at last on such location. id rnts quat in thos locations bz sinc workrs and rms sttl ithr on th sam land plot or on two in nitly clos rsidntial and industrial land plot. Th rst trm in th intgral yilds th conomic surplus S (x) (up to a constant) gnratd at location x. Th scond trm r cts th travl cost of commutrs crossing location x. Of cours, w can us th rlationship = 1 s to liminat a dnsity pro l ithr or. Th abov idntity tlls that workrs and rms sttl in th locations whr thy rspctivly plac th highst bid rnts. Whn thy o r th sam bid, any shar of rms and rsidnts is possibl. n this comptitiv allocation, th spatial quilibrium dpnds on th sign of th bid rnt di rntials but not on thir amplituds. Hnc multiplying both th paramtrs ( + ) and t by any scalar k > 0 has no ct on th sign of th bid di rntials and thrfor is compatibl with th spatial quilibrium structur. This yilds th following main rsult: Proposition 1 Citis with a sam paramtr ratio ( + )=t hav an idntical spatial structur. Th ratio shows that strongr dmand or vrtical linkags and highr commuting costs hav opposit cts on th urban structur. Th conomic surplus gnratd by any pair of workr and a rm is balancd with th commuting cost btwn th workr s rsidnc and hr rm. A proportional incras in th intnsity of dmand or vrtical linkags and in commuting cost dos not chang this spatial balanc though ach ct bcoms mor intns. As a rsult, strongr linkags hav th sam impact as lowr commuting costs. 3.2 Existnc Rlationships (16) and (17) d n a xd point problm for th rms spatial distribution. ndd, th rms dnsity dtrmins th valus of th surplus S which in turn dtrmins. n Appndix 10

11 , w show th xistnc of a xd point using Shaudr s thorm. Proposition 2 A spatial quilibrium xists. Th proof of spatial quilibrium xistnc rlis on th dtrmination of th stock of commutrs owing ovr ach urban location, n(x). This is rfrrd as th stock of unhousd workrs by Lucas and Rossi-Hansbrg (2002). As shown in (16), th main di culty in stablishing th xistnc of spatial quilibrium lis in th discontinuity of th land allocation procss across rms and workrs. As in Carlir and Ekland (2007), w rst show th xistnc of a xd point for smoothr land allocation procsss by rplacing (16) by an quation whr th land allocation dcision across rms and workrs is a continuous function of thir bid rnt di rntial b r. n conomic trms, this may b intrprtd by th prsnc of xognous uncrtainty in landlords choics. W thn prov th convrgnc of thos xd points to that d nd by (16) and (17) for lss and lss smooth allocation procsss. Using th sam intrprtation, th xd point taks plac at th limit of zro uncrtainty in th land allocation procss. t must b notd that th abov xistnc rsult applis to Fujita and Ogawa s (1982) modl which lacks a gnral xistnc proof and displays paramtr valus for which no quilibrium could b constructd. As mntiond arlir, thir quilibrium conditions ar structurally th sam as ours if on sts and G to zro and intrprts ( 1) as a paramtr for fac-to-fac communication. 4 Urban structurs Our xistnc rsult in Sction 3 says nothing about uniqunss of quilibrium. n this sction, w prsnt and discuss a varity of urban structurs and dtrmin thir condition of xistnc. 4.1 Monocntric city W considr th businss district Y = [ b 1 ; b 1 ] surroundd by th rsidntial districts X = [ b; b 1 ) [ (b 1 ; b], s Figur 1 (a). Th dnsitis of rms and housholds ar (y) = 1=s and 1 rspctivly. Th businss district dg is givn by b 1 = sn=2. n th monocntric city, workrs commut from th rsidntial district to th businss district. 11

12 Th monocntric quilibrium conditions ar as follows: (i) rms outbid housholds in th businss district ( b(y) r(y) for all y 2 Y), (ii) housholds outbid rms in th rsidntial district ( r(x) b(x) for all x 2 X ); (iii) th rsidntial and th businss bid rnts qualiz at th businss district bordr ( b(b 1 ) = r(b 1 )); (iv) th zro opportunity cost of land at th city bordr b rquiring r(b) = 0. As of xprssions (4) and (7), w d n th surplus at location z as S M (z) G M (z) ln T M (z), 3 whr th accss masur T M and th pro t G M ar givn by T 1 M (z) = 1 s G M (z) = 1 s b1 b 1 (1 )jy zj dy (18) b1 jy b 1 T 1 M zj (y) dy (19) Proposition 3 Th monocntric con guration is a spatial quilibrium if and only if t t M = min(bt M ; t M ) whr t M = + S M (0) 1 + s S M (sn=2) sn=2 Whn s! 0, t M = ( + ) N < bt M. and bt M + S M (sn=2) 1 + s S M ((1 + s) N=2) N=2 Proof. Givn th symmtry of th city, our analysis focuss on th right sid of th city [0; b]. Undr linar commuting costs, workrs can arbitrag btwn th di rnt workplacs. As a rsult, a rm locatd at y must o r individuals a wag w(y) that is as attractiv as that availabl in othr work locations. Dnoting U 0 by [ln (=) arbitrag condition for any 0 y < x: ln ln T M (0)] +w(0) + c 0, w can writ th following wag ln T M (y) + w(y) + t y + c 0 = U 0 so that th quilibrium wag w(y) satis s w(y) = U 0 ln ln T M (y) t y c 0 y using this wag xprssion, th houshold s bid rnt (9) can thn b writtn as r(x) = (U 0 U ) t x (20) 3 Th surplus S M (z) corrsponds to th pro t in th markt of a local varity accounting for th cost of accssing all componnts in ordr to produc th national good. 12

13 Similarly, th rm s bid rnt (10) writs as b(y) = 1 ln + ln s U 0 + c 0 + ( + )S M (y) + t y At th land markt quilibrium, landlords rap th surplus drivd by rms and housholds. As th district bordr is givn by b 1 = sn=2, w nd to impos conditions (i), (ii), (iii), and (iv) on th bid rnt functions (20) and (21) of rsidnts and rms. t is shown in Appndix C that conditions (i) and (ii) rduc to (i ) b(0) r(0) and (ii ) r(b) b(b) rspctivly. Condition (i ) b(0) r(0) can b rwrittn as 1 ln + ln s + c 0 U s ( + )S M(0) (U 0 U ) (22) Th quilibrium conditions (iii) and (iv) ( b(b 1 ) = r(b 1 ) and r(b) = 0) dtrmin U 0 and U. n particular, w hav U 0 = ln + ln + c 0 + ( + )S M (b 1 ) + (1 + s) t b 1 s t b y plugging th abov xprssion of U 0 into (22) and using b 1 = sn=2,w gt Whn s! 0, w hav t t M (21) 2( + ) (1 + s) sn [S M(0) S M (b 1 )] (23) lim t S M (sn=2) S M (0) M = ( + ) lim s!0 s!0 sn=2 Furthrmor, w also hav b(b) = 1 ( + )S M (b) + t b + s = 1 s ln + ln ( + ) [S M (b) S M (b 1 )] + (1 + s) N t 2 so that th condition (ii ), b(b) r(b) = 0, can now b rwrittn as t bt M Gathring xprssions (23) and (24) lads to t min(bt M ; t M ). Whn s! 0, w gt = ( + ) N U 0 + c 0 2( + ) (1 + s) N [S M(b 1 ) S M (b)] (24) lim [ln T M(sN=2) ln T M ((1 + s) N=2)] = N=2 s!0 lim [G M(sN=2) G M ((1 + s) N=2)] = 1 N=2 s!0 13

14 Thrfor lim bt M 2( + ) lim [S M (sn=2) S M ((1 + s) N=2)] s!0 s!0 = 2( + ) lim [G M (sn=2) G M ((1 + s) N=2) ln T M (s=2) + ln T M ((1 + s) N=2)] s!0 = ( + ) 2 1 N=2 + N > ( + )N = lim t M s!0 so that lim s!0 bt M < lim s!0 t M. Proposition 3 says that lowr commuting costs favor th monocntric con guration as thy allow rsidnts to commut longr distancs to th businss district. Largr nal dmand or vrtical linkags also mak th monocntric con guration mor likly as thy incras th oprating pro t in th markt of local varitis du to incrasing rturns to scal, thus allowing rms in th businss district to pay highr wags and making popl mor willing to commut. Figur 2 illustrats th shap of bid rnts in a monocntric city. Th rms bid rnt curv, rprsntd in rd, is abov (rsp. blow) th rsidnts bid rnt curv (in blu) around th city cntr (rsp. th city dg). id rnts ar qual to th zro opportunity cost of land at th city bordr and intrsct at th location y = 0:5, which sparats businss and rsidntial districts. Workrs bid rnts ar linar in th distanc to th cntr as longr commuts ar compnsatd by lowr rnts. As rms bn t from agglomration conomis, thir bid rnt is ndognous and dpnds on th location of othr rms. Typically it is non-linar in th distanc to th city cntr. A priori, it is not obvious to guarant a singl intrsction btwn bid rnts. Nvrthlss, according to th abov Proposition, it turns out that only th bid rnt proprtis at th city cntr and bordr mattr. Whil th condition t bt M guarants that rms do not ovrbid workrs at th city bordr ( b(b) r(b)), th condition t t M guarants that thy ar not ovrbid by rsidnts at th city cntr ( b(0) r(0)). Th last statmnt in Proposition 3 implis that, vn if th manufacturing sctor maks no us of land (s! 0), th monocntric city structur may not b a spatial quilibrium. This quali s th standard discussion basd on th monocntric city modl whr rms ar assumd to b locatd in a city businss cntr using no spac. This papr highlights th rol of th trad-o btwn workrs commuting cost and goods transport costs. Finally, not that, whn transport costs vanish (! 0), th conomic surplus S M (z) bcoms location indpndnt and th abov thrsholds bcom t M = bt M = 0. Thn, th monocntric city is nvr a spatial quilibrium for any commuting cost t > 0. n this cas, th dmand for nal goods 14

15 and intrmdiat goods dmand is unrlatd to spac. Thr is no agglomration forc and spatial concntration brings no bn t. 4.2 ntgratd city W considr rms and housholds locating in X = Y = [ b; b], s Figur 1 (b). Givn th unit dmand for land of housholds and th s unit dmand of rms, th dnsity of rms is (y) = 1=(1 + s) and th city dg is still at b = (1 + s)n=2. Th intgratd con guration involvs no commuting. Th intgratd quilibrium conditions ar as follows: (i) th businss and rsidntial bid rnts qualiz ( r(z) = b(z), 8z 2 X = Y); (ii) th no-arbitrag rlocation implying j 0 r (z)j t; (iii) th zro opportunity cost of land at th city bordr b rquiring (b) = 0. As of xprssions (4) and (7), w d n th surplus at location z as S (z) G (z) th accss masur T and th pro t G givn by T 1 (z) = 1 G (z) = 1 + s b 1 (1 + s) ln T (z) with (1 )jz yj dy (25) b b jy Proposition 4 Th intgratd city is a spatial quilibrium if b T 1 zj dy (26) (y) t t = s S0 (b) > 0 Whn s! 0, t = ( + ) S 0 (b). Proof. As bfor, w focus on [0; b]. W nd to impos th abov conditions (i)-(iii) on th bid rnt functions. Th quilibrium condition (i), r = b, lads to w(y) = s (s ) ln T (y) s G (y) + C (27) whr C s=(1 + s) fu c 0 ln[=()] + =s ln [=()]g. From quilibrium condition (iii), (b) = 0, w hav ln ln T (b) + w(b) + c 0 U = 0 (28) From rlations (27) and (28), th constant C can b dtrmind C = s 1 + s ( + ) S (b) + ln 15

16 y plugging th wag xprssion (27) into (10) (or (9)), w gt th quilibrium land rnt R(z) = s [S (z) S (b)] According to th no-arbitrag condition (ii), an urban con guration is an quilibrium if workrs hav no incntiv to commut to any othr location, i.., if jr 0 (z)j = ( + )=(1 + s)s 0 (z) t, for 8z 2 [0; b]. n Appndix D, it is shown that S 0 (z) S 0 (b), 8z 2 [0; b], so that th no-arbitrag condition rducs to jr 0 (b)j = ( + )=(1 + s)s 0 (b) t. Proposition 4 stats that highr commuting costs fostr th intgratd con guration. ntuitivly, vry larg commuting costs ntic workrs to tak rsidnc as clos as possibl to thir workplac. Also, wakr nal dmand or vrtical linkags mak th intgratd con guration mor likly. Thy indd dcras th oprating pro ts from th production of local varitis, which rducs workrs wags, and in turn th incntivs to commut. Finally, it turns out that t is an incrasing function of (s Appndix D). So, lowr trad costs nlarg th domain of conomic paramtrs for which rms intgrat with rsidnts. For lowr trad costs, input costs dpnd lss on distanc btwn rms and vrtical linkags lad to a wakr agglomration forc for th rms. Thrfor, rms do not bn t much from clustring with othr rms and mak land rnt bids that can b matchd by workrs. Figur 3 illustrats th shap of bid rnts in an intgratd city. Th rms and th rsidnts bid rnt curvs coincid at any location in th city. id rnts vanish at th city bordrs du to th zro opportunity cost of land. As workrs do not commut, rnts do not r ct linar commuting costs but rathr non-linar agglomration conomis coming from dmand and input-output linkags. n th abov Proposition, th quilibrium condition t t guarants that ach workr has no incntiv to accpt a job away from hr rsidntial location. This proprty obtains if bid rnts from ar not too stp. t is nsurd as long as dmand and input-output linkags ar wak nough (low + ). Whn th transport cost bcoms ngligibl (! 0), th conomic surplus S (z) is location indpndnt so that S 0 (z) = 0 and t = 0. Th intgratd city is thn a spatial quilibrium for any commuting cost t > 0. n this cas, th dmands for nal goods and intrmdiat goods ar indpndnt of spac and th agglomration forc vanish. Finally, w prov th following corollary: Corollary 1 For s! 0, th monocntric and th intgratd urban con gurations nvr coxist. That is, t M < t. 16

17 Proof. S Appndix E. Whn th manufacturing sctor maks no us of land, a high commuting cost is compatibl only with th intgratd structur. This rsults from th tnsion btwn backward and forward linkags and commuting costs. Whn backward and forward linkags ar too wak, rms ar unabl to post land bids that ar high nough to forclos th workrs to rsid clos to thm. Th abov rsult forbids th possibility of multipl quilibria, whr th monocntric and intgratd structurs coxist for th sam st of paramtr valus. To th bst of our knowldg, this proprty is not wll mphasizd in th litratur (.g. Fujita and Ogawa 1982, and followrs). 4.3 Partially intgratd city Nxt, w considr a city whr workrs and rms locat with ach othr in th cntral district ( b 1 ; b 1 ), which is surroundd by businss districts ( b 2 ; b 1 ) [ (b 1 ; b 2 ), thmslvs bordrd by rsidntial districts ( b; b 2 ) [ (b 2 ; b), s Figur 1 (c). Th dnsity of rms is (y) = s=(1 + s) in th intrval ( b 1 ; b 1 ) and (y) = 1=s in th intrvals ( b 2 ; b 1 ) [ (b 1 ; b 2 ), whil that of workrs is (x) = 1=(1 + s) in th intrval ( b 1 ; b 1 ) and (x) = 1 in th intrvals ( b; b 2 ) [ (b 2 ; b), with 0 < b 1 < b 2 < b. caus th total mass of rms is N, w gt that is, 2 (b 2 b 1 ) 1 s + 2b s = N b 2 = b s + sn 2 Th partially intgratd quilibrium conditions ar as follows: (i) th bid rnts of housholds and rms qualiz in th intgratd district ( r(z) = b(z), 8z 2 ( in th intgratd district implying j 0 (z)j t, 8z 2 ( (29) b 1 ; b 1 )); (ii) no-arbitrag rlocation b 1 ; b 1 ); (iii) rms outbid housholds in th businss districts ( b(y) r(y) 8y 2 [ b 2 ; b 1 ] [ [b 1 ; b 2 ]); (iv) housholds outbid rms in rsidntial districts ( r(x) b(x), 8x 2 [ b; b 2 ] [ [b 2 ; b]); and (v) th zro opportunity cost of land at th city bordr b rquiring r(b) = 0. As of xprssions (4) and (7), w d n S P (z) G P (z) ln T P (z) whr th accss masur T P 17

18 and th pro t G P ar givn by T 1 P (z) = 1 s G P (z) = 1 s b1 b2 + b 2 b 1 b1 b2 + b 2 b 1 (1 )jz yj dy + 1 jz 1 + s yj T 1 P (y) dy + 1 b1 (1 + s) b 1 (1 )jz yj dy (30) b1 jz b 1 T 1 Proposition 5 Th partially intgratd city is a spatial quilibrium if and only if yj dy (31) P (y) bt P t t P whr th thrsholds bt P and t P ar givn by with t P s Th district bordrs (b 1 ; b 2 ) satisfy bt P s max jsp 0 (z)j ; t P min t P ; t P z2[0;b 1 ) S P (b 2 ) whr b 2 is givn by (29) and 0 < b 1 < b 2 < N. S P (b) ; t P min b b 2 y2[b 1 ;b 2 ] s S P (y) S P (b 2 ) b 2 y t = + S P (b 1 ) S P (b 2 ) (32) 1 + s b 2 b 1 Proof. As bfor, w focus on [0; b]. n th intgratd district of th city, condition (i) writs R(z) = r(z) = b(z). From xprssions (9) and (10), w gt R(z) = s S P (z) + 1 ln + ln 1 + s + c 0 w(z) = s 1 + s ln T P (z) s G P (z) s 1 + s ln ln s + c 0 U ; z 2 [0; b 1 ) (33) U ; z 2 [0; b 1 ) (34) n th rsidntial district, w hav R(x) = r(x); x 2 [b 2 ; b]. Howvr, sinc th rsidntial bid rnt r(x) givn by (9) is indpndnt of th workplac y in businss district [b 1 ; b 2 ], it can b writtn as R(x) = r(x) = A tx; x 2 [b 2 ; b] whr A = ln ln T P (y) + w(y) + ty + c 0 U ; y 2 [b 1 ; b 2 ] 18

19 is constant with rspct to y. Condition (v) says that th land rnt is nil at b, from which w gt A = tb. So, from th last two xprssions, w can writ th quilibrium land rnt in th rsidntial district and th wag in th businss district as R(x) = t(b x), x 2 (b 2 ; b] (35) w(y) = t (b y) ln ln T P (y) c 0 + U ; y 2 [b 1 ; b 2 ] (36) nsrting th wag (36) into R(y) = b(y) as givn by rlation (10) yilds th land rnt in th businss district R(y) = 1 s ( + ) S P (y) t (b y) + ln + ln + c 0 As land rnts ar continuous, R(b 1 0) = R(b 1 + 0) and R(b 2 0) = R(b 2 + 0), w gt U = ( + ) S P (b 2 ) t (1 + s) (b b 2 ) + ln U ; y 2 [b 1 ; b 2 ] (37) + ln + c 0 and ( + ) [S P (b 1 ) S P (b 2 )] = t (1 + s) (b 2 b 1 ) which is quation (32) and dtrmins b 1 as b 2 is givn by rlation (29). W still hav to impos conditions (ii), (iii), and (iv). First, condition (ii) nsuring no commuting within any two locations in th intgratd district writs as ( + ) jsp 0 (z)j t (1 + s), z 2 [0; b 1 ) that is, t > bt P s max z2[0;b 1 ) js0 P (z)j (38) Scond, condition (iii) says that rms outbid workrs in th businss district: R(y) r(y) 8y 2 [b 1 ; b 2 ]. Subtracting (35) from (37), w gt ( + ) [S P (y) S P (b 2 )] t (1 + s) (b 2 y), y 2 [b 1 ; b 2 ] which givs t t P + min y2[b 1 ;b 2 ] 1 + s S P (y) S P (b 2 ) b 2 y Finally, condition (iv) nsuring that workrs outbid rms in th rsidntial district (R(x) b(x), 19 (39)

20 8x 2 [b 2 ; b]) givs ( + ) [S P (b 2 ) S P (x)] t (1 + s) (x b 2 ) 0, x 2 [b 2 ; b] (40) which is similar to rlation (39). Only signs and supports di r. Not that in th intrval (b 2 ; b), S P (x) is convx bcaus G 00 P (x) = 2 2 G P > 0 and (ln T 1 P (x)) 00 = 0: Sinc th LHS of inquality (40) is zro at x = b 2, this inquality is satis d for all x 2 [b 2 ; b] if and only if ( + ) [S P (b 2 ) S P (b)] t (1 + s) (b b 2 ) 0 Thus, inquality (40) is quivalnt to t t P s S P (b 2 ) S P (b) (41) b b 2 To sum up, th spatial quilibrium is givn by (b 1 ; b 2 ) that solv quations (29) and (32) and satisfy (38), (39), and (41). Proposition (5) shows th partially intgratd con guration can only xist for intrmdiat valus of th commuting cost. t also maks xplicit th condition that district bordrs nd to satisfy. y inspction of quation (32), th commuting cost from b 2 to b is to b proportional to th surplus di rnc btwn locations b 1 and b 2. Figur 4 illustrats th shap of bid rnts in a partially intgratd city. Panl a) prsnts th cas of balancd us of spac, whr rms and workrs us th sam amount of land (s = 1). Around th city cntr, rms and rsidnts populat an intgratd district whr th rms and th rsidnts bid rnts coincid. This intgratd ara is surroundd by two businss districts whr rms outbid rsidnts. Rsidntial districts mrg at city dgs whr rsidnts outbid rms. Th condition t bt P nsurs su cintly high commuting cost so that workrs hav no incntiv to commut within th intgratd district. Th conomic intuition is similar to that prvailing in th intgratd city. Th condition t t P guarants that rsidnts outbid rms in priphral districts. n particular it xprsss th su cint condition that rms producing at y = b 2 hav no incntivs to rlocat production at th city dg y = b. Firms indd balanc th cts of such a rlocation on thir rvnus and wag bills. On th on hand, thy los rvnus at th city dg bcaus of th lowr accss to thir customrs. On th othr hand, thy pay lowr wags at th city dg bcaus workrs nd not b compnsatd for long commuting. Th lattr ct gts smallr and is dominatd by th formr whn commuting costs 20

21 bcom lowr. To sum up, partially intgratd citis thrfor xist only for intrmdiat commuting costs and, by Proposition 1, only for intrmdiat valus of dmand and input-output linkags. W provid two Corollaris xhibiting additional proprtis of partially intgratd quilibria. Th rst proprty rlats to th contiguity and intrsction of th sts of paramtrs supporting th intgratd and th partially intgratd citis. n particular, w show that th partially intgratd city is a spatial quilibrium for t = t and > 0 small nough. Corollary 2 Th paramtr sts that support th intgratd and th partially intgratd citis do not intrsct and ar contiguous. Proof. S Appndix F. Th scond proprty is about th xistnc of partial intgratd citis vn whn rms do not us spac. Th litratur gnrally xtnds Alonso s (1964) modl to multipl businss cntrs locatd on points (.g. Fujita, Krugman and Mori 1999). Th following corollary suggsts that, although insightful, this stratgy dos not covr th xhaustiv st of urban con gurations. ndd, considr in nitsimally small businss districts (s! 0). y Corollary 1, th commuting cost intrval (t M ; t ) is not mpty. For such commuting costs, monocntric and intgratd city structurs ar not spatial quilibria. n th following corollary, w show that spatial quilibria includ partially intgratd citis whn businsss us no land. Corollary 3 Suppos rms us an in nitly small amount of land (s! 0). Thn, thr xists a partially intgratd quilibrium with an intgratd district surroundd by two in nitly small businss districts and rsidntial districts in th priphry whn commuting costs satisfy maxft M ; bt 0 P n o g t min t ; t 0 P ; t 0 P. Proof. Towards this aim, w rwrit quation (32) as t = f s (b 1 ) + S P (b 1 ) S P (b 2 ) 1 + s b 2 b 1 whr b 2 is givn by (29). As s! 0, on can vrify lim f s(0) = t M and lim f s (b) = t s!0 s!0 caus t > t M > 0 and f s (b 1 ) is continuous in [0; b=2], w can apply th intrmdiat valu thorm. Thr xists at last a valu of b 0 1 that solvs t = f s (b 1 ) for all t 2 (t M ; t ). On thn computs th 21

22 thrsholds Thn, if bt 0 P t min nt 0 P ; t 0 P businss districts at y = b 0 1: t 0 P = ( + ) S P (b 0 1) S P (b) b b 0 1 t 0 P = ( + ) S 0 P (b 0 1) bt 0 P = ( + ) max z2[0;b 0 1 ) js 0 P (z)j o, thr xists a partially intgratd quilibrium with two in nitly small Panl b) in Figur 4 prsnts a scond xampl of bid rnts in a partially intgratd city whr th us of spac is unbalancd. n such a cas, rms do not us much spac unlik rsidnts and businss districts hav a vry small spatial xtnt (s = 0:01). Additional numrical xrciss show that th city con guration rmains idntical as s! 0: n that cas, th two businss districts bcom so small that th cntral intgratd district bcoms contiguous to th priphral rsidntial districts. Th paramtrs ful ll th conditions of th abov corollary, which thrfor dtrmin non-mpty sts of paramtrs. To sum up, th partially intgratd city xists vn in th absnc of spac rquirmnt by rms. So, although vry insightful, prvious discussions on city subcntrs concntratd on spatial points do not ncompass th full st of urban con gurations. 4.4 Duocntric city Hr, th city consists of two businss districts Y = [ b 3 ; b 1 ) [ (b 1 ; b 3 ] and thr rsidntial districts X = [ b; b 3 )[[ b 1 ; b 1 ][(b 3 ; b], s Figur 1 (d). Housholds living in [ b; b 3 ) commut to [ b 3 ; b 2 ), thos living in [ b 1 ; 0) to [ b 2 ; b 1 ), thos living in [0; b 1 ] to [b 1 ; b 2 ), and thos living in (b 3 ; b] to [b 2 ; b 3 ). Givn th total masss of rms and consumrs, w hav b 1 = 1 s (b 2 b 1 ) ; b b 3 = 1 s (b 3 b 2 ) (42) Th dnsity of rms is (y) = 1=s on Y and district dgs satisfy b 2 = (1 + s) b 1 and b 3 = b 1 + sn=2. As bfor, w focus on [0; b]. Th duocntric quilibrium conditions ar as follows: (i) rms outbid housholds in th businss districts ( b(y) r(y) for all y 2 Y); (ii) housholds outbid rms in th rsidntial districts ( r(x) b(x) for all x 2 X ); (iii) th businss and th rsidntial bid rnts qualiz at district bordrs 22

23 ( r(b 1 ) = b(b 1 ) and r(b 3 ) = b(b 3 )); and (iv) th zro opportunity cost of land at th city bordr rquiring r(b) = 0. As of xprssions (4) and (7), w d n S D (z) G D (z) and th pro t G D ar givn by T 1 D (z) = 1 s G D (y) = 1 s b1 (1 )jz yj dy + b 3 b1 T 1 dz + (z) b 3 D jz yj b3 ln T D (z), whr th accss masur T D b 1 (1 )jz yj dy T 1 dz (z) b3 b 1 D jz yj Proposition 6 Th duocntric city is a spatial quilibrium if and only if t max(bt D ; t D ) b(y) r(y) for all y 2 Y whr bt D + S D (b 1 + sn=2) S D (0) 1 + s (1 + 2s) b 1 sn=2 with b 1 2 (0; N=4] satisfying and t D s S D (b 1 + sn=2) N=2 b 1 S D (b) Proof. t (N 4b 1 ) (1 + s) s=2 = ( + ) [S D (b 1 ) S D (b 1 + sn=2)] (43) As in arlir sctions, workrs arbitrag btwn th di rnt workplacs up to thir commuting cost and land rnt paymnts. As a rsult, a rm locatd at z must o r individuals a wag w(z) that is as attractiv as in othr workplacs. Sinc workrs commut from th lft and right hand sids of th businss district locatd on (b 1 ; b 3 ), it is convnint to dnot th businss cntr of th businss district by b 2 2 (b 1 ; b 3 ). This is th uniqu location whr a rm attracts workrs from ach sid of its location. Sinc th labor markt divids about y = b 2, th lft hand sid s labor dmand and supply balanc so that b 2 b 1 = sb 1 whil its right hand sid clars so that b 3 b 2 = s (b b 3 ). Dnoting th utility of a workr working at b 2 by U 2 [ln (=) writ th following wag arbitrag condition for any y 0: ln ln T D (y) + w(y) + t jy b 2 j + c 0 = U 2 ln T D (b 2 )] +w(b 2 ) + c 0 ; w can so that th quilibrium wag w(y) satis s w(y) = U 2 ln ln T D (y) 23 t jy b 2 j c 0 (44)

24 This is also th wag that a rm should pay to attract a workr whn it rlocats in a rsidntial district. y plugging th wag xprssion into (9) and stting y = b 2, th houshold s bid rnt can b rwrittn as r(x) = U 2 U t jx b 2 j Similarly, th rm s bid rnt (10) writs as b(y) = 1 ln + ln s U 2 + c 0 + ( + ) S D (y) + t jy b 2 j Landlords rap th surplus of th production and consumption of ach pair of individuals and rms up to th lvl of th individual s utility. As bfor, w focus on [0; b] whn imposing th quilibrium conditions (i)-(iv) on th bid rnts from housholds and rms. Using th wag (44) from th wag arbitrag condition and applying it for x > b 3 > y > b 2, on can xprss th rlations r(b) = 0 and r(b 3 ) = b(b 3 ) as functions of U 2, U, b 2 and b 3. Using th sam wag (44) and applying it for x < b 1 < y < b 2, on can also xprss th rlation r(b 1 ) = b(b 1 ) as function of th sam variabls. Thrfor, conditions (iii) and (iv) togthr with th idntitis b 2 = (1 + s) b 1 and b 3 = b 1 + sn=2 provid v quations for th v variabls U 2, U, b 1, b 2 and b 3. Th solution for U 2 and U is givn by U 2 = ln + ln + c 0 + ( + ) S D (b 1 ) + t (b 2 b 1 ) st (b 2b 2 + b 1 ) U = ln + ln + c 0 + ( + ) S D (b 1 ) (1 + s) t (b 2b 2 + b 1 ) whil b 1 solvs (43). Th quilibrium inquality condition (ii) r(x) b(x) for all x 2 X, writs as r(x) b(x) = U 2 U 1 ln 1 + s + ln U 2 + c 0 + ( + ) S D (x) t jx b 2 j 0 s s whr S D (x) can b shown to b convx on th intrvals (0; b 1 ) and (b 3 ; b): SD 00 = G00 1 D ln T D h i whr G 00 D = 2 2 G D > 0 and (ln T 1 (T 1 D )00 T 1 D (T 1 2(1 ) D )02 =TD ; th last xprssion is D )00 = qual to 0 on (b 3 ; b) and to 2( 1)=(sT 1 D ) 2 R b1 b 3 (1 )(y z) dz R b 1 b 3 (1 )(z y) dz > 0 on (0; b 1 ). Thn, r(x) b(x) is concav on thos two intrvals. Sinc r(b 1 ) b(b 1 ) = 0, w hav r(x) b(x) 0 for all x 2 (0; b 1 ) if and only if r(0) b(0) 0, which can b rwrittn as t bt D ( + )[S D (b 3 ) S D (0)]=[(2b 2 b 3 ) (1 + s)]. Similarly, sinc r(b 3 ) b(b 3 ) = 0, w hav 24

25 r(x) b(x) 0 for all x 2 (b 3 ; b) if and only if r(b) b(b) 0, which can b rwrittn as t t D ( + ) [S D (b 3 ) S D (b)]= [(b b 3 )(1 + s)]. This mans that condition (ii) rducs to t max(bt D ; t D ). Th quilibrium inquality condition (i) rmains as it is: b(y) r(y) for all y 2 Y. Finally, it can asily b shown that b 1 should not xcd N=4. Othrwis, th rsidntial bid rnt would b ngativ at x = 0. To undrstand, th quilibrium condition in Proposition (43),w rwrit it as t (b 3 b 2 ) t (b 2 b 1 ) = + (1 + s) s [S D(b 1 ) S D (b 3 )] Th LHS corrsponds to th di rnc in th commuting cost to workplac b 2 from district bordrs b 3 and b 1, whil th RHS is proportional to th surplus di rnc btwn locations b 3 and b 1. This mans that ths commuting costs and surplus di rncs should balanc. Figur 5 illustrats th shap of bid rnts in a duocntric city. Rsidntial districts ar locatd at th city cntr and priphris whr th rsidnts bid rnt (in blu) is abov th rms bid rnt curv (in rd). n btwn ths rsidntial districts, two businss districts gathr rms outbidding rsidnts. Panl a) and b) of Figur 5 show two di rnt duocntric city structurs for th sam st of conomic paramtrs, for which quation (43) accpts two solutions. t can b chckd on Figur 5 show that b(y) r(y) for all y in th businss districts. So, this mans that multipl duocntric quilibria can aris. n Proposition 6, th ndognous bordr b 1 is th solution to (43), which accpts no closd-form solution. Proposition 6 also imposs th bid rnt condition b(y) r(y), for any location y in th businss districts Y. As th purpos will b to dtrmin th duocntric con gurations by us of numrical computations, w shall rplac th condition b(y) r(y), for all y 2 Y, by th ncssary condition b(b 2 ) r(b 2 ), which is quivalnt to t t D whr t D + S D (b 2 ) S D (b 1 ) 1 + s b 2 b 1 Not that w did not nd any xampl whr this condition was not also su cint. So, th thrshold valu of th commuting cost is givn by t t D = max(bt D ; t D ; t D ) 25

26 5 Numrical analysis Th thortical analysis of urban structurs in th prvious sction allows us to dtrmin and study th natur of city structurs in trms of conomic paramtrs. Givn th complx natur of duocntric and partially intgratd con gurations, w dtrmin thm by numrical computations. This is don in two stps. First, givn som xd paramtr valus, district bordrs ar dtrmind by solving (43) and (42) for th duocntric con guration (rsp. (32) for th partially intgratd con guration). Not that up to two roots ar found in ithr cas. Scond, additional rstrictions r cting rnt conditions at district bordrs ar imposd. Th rol of ths rstrictions is to rtain only th candidats that actually lad to an quilibrium con guration. Equilibrium city structurs Figur 6 gathrs th cntral rsults of our numrical analysis. t prsnts th quilibrium structurs drivd in Propositions 3, 4, 5, and 6 in th plan of transport and commuting costs (; t) for xd valus of othr paramtrs, i.. = = = s = 1 and = 3. Th gur displays th monocntric "m", th intgratd "i", th partially intgratd "p", and th duocntric "d" con gurations. Th concatnation of svral lttrs indicats th prsnc of multipl quilibria with di rnt urban structurs for th sam st of paramtrs. For instanc, "mdd" indicats th co-xistnc of a monocntric quilibrium and two di rnt duocntric quilibria. For som paramtr con gurations, up to four quilibria can co-xist (s "dmpp" at th cntr of Figur 6). n Sction 2, xistnc of a spatial quilibrium was provd. ndd, numrical computations always dlivr at last an quilibrium for any point in (; t) plan. This contrasts with Fujita and Ogawa (1982) whr som paramtr con gurations wr shown to support no quilibrium. Figur 6 displays th curv corrsponding th thrshold t M blow which paramtrs support th monocntric city. Th monocntric city ariss only if commuting costs ar rlativly low. y d nition of t M, rsidnts and businsss mak th sam land bid at th city cntr for any valu of t lying on th curv t M. For slightly highr commuting costs -and as long as transport costs ar low nough-, rsidnts ar willing to match th rms bid rnt at th city cntr. Som rsidnts and rms can thn gathr at th city cntr, whil many othr workrs ar still willing to commut. Hnc th curv t M rfrs to a transition btwn th monocntric and th partially intgratd con gurations. Figur 6 also displays th curv corrsponding to th thrshold t abov which paramtrs support th intgratd city. n th intgratd city, rsidnts work and rsid in th sam location. y d nition 26

27 of thrshold t, whn commuting costs quat t, som individuals ar indi rnt btwn working at thir rsidnc location and commuting to a rm locatd furthr away. For commuting costs slightly blow t, rsidnts locatd at th city bordr hav an incntiv to commut and choos a workplac btwn th city cntr and th city bordr, so that a partially intgratd city structur mrgs. Hnc th curv t rfrs to a transition btwn th th partially intgratd and th intgratd con gurations. Th partially intgratd con guration xists only for intrmdiat commuting costs. Considr such a partially intgratd structur. Whn th commuting cost ris bcoms too high, th businss cntrs and th rsidntial priphris shrink, and ultimatly vanish whn th commuting cost rachs th thrshold t. Hnc th partially intgratd con guration convrgs smoothly to th intgratd city. For low transport costs (i.., whn only on partially intgratd quilibrium xists), th two businss cntrs gt closr to ach othr to nally join ach othr as th commuting cost approachs t M from abov: th partial intgratd structur thn convrgs smoothly to th monocntric con guration. For highr transport costs (i.., whn th partially intgratd and th monocntric citis co-xist), on partially intgratd con guration convrgs continuously to th monocntric structur as th commuting cost approachs t M from blow. Duocntric citis ar quilibria only for high transport costs and intrmdiat commuting costs (s th right-hand sid of Figur 6). ndd, whn shipping goods ar costly, rms bn t from clustring thir production activitis in two sparat businss cntrs so as to dcras th avrag distanc to supplirs and consumrs. Considr commuting costs rising from zro (i.., a rising path in th right-hand sid of Figur 6). Whn commuting costs ar vry low, workrs accpt long journys to thir workplac and bn t from a concntratd businss district, i.. th monocntric pattrn "m". Howvr, whn commuting costs rach intrmdiat valus, an urban structur with two activity cntrs bcoms bn cial to thm bcaus long commuts can b avoidd. Whn this happns, two duocntric con gurations mrg (s transition from "m" to "mdd" in Figur 6). As commuting costs kp on incrasing, on duocntric con guration prsists whil th othr disappars (transition from "dmd" to "dm"). Finally, for vn highr commuting costs, th rmaining duocntric city pattrn cass to xist as housholds want to locat closr to thir workplac, which givs ris to a transition to a partially intgratd con guration. 27

28 Multipl quilibria Figur 6 also illustrats th multiplicity of quilibria. n many rgions of th paramtr spac, a uniqu quilibrium mrgs (s "m", "i", "p" and "d"). Thr ar also larg sts of paramtrs for which svral structurs among th monocntric, th duocntric, or th partially intgratd con gurations co-xist. Not that all structurs can co-xist for a narrow st of paramtr valus. Th fully intgratd structur cannot co-xist with any othr con guration. As mntiond in th abov paragraph, multipl quilibria can involv di rnt urban structurs of th sam typ: ithr two partially intgratd structurs lik in "mpp" and "dmpp" or two duocntric structurs lik in "dmd" and "mdd". This can only happn for intrmdiat commuting costs. For highr commuting costs, on of th two quilibria always disappars. t is intrsting to dscrib how two quilibria of th sam typ mrg. Whn commuting costs incras, it can b shown that th transition from "m" to "mpp" (rsp. from "m" to "mdd") implis that th monocntric city kps th sam structur whil a partially intgratd city (rsp. a duocntric city) initially mrgs and immdiatly sparats into two distinct partially intgratd structurs (rsp. a duocntric) with businss cntrs volving at di rnt locations. Wlfar ranking mportantly, th ordring of lttrs in Figur 6 indicats th ranking of quilibria in trms of aggrgat wlfar (i.., th sum of rsidnts utility plus aggrgat land rnts). For instanc, th ordring "mdd" indicats that th monocntric con guration has th highst wlfar whil th two duocntric con gurations hav lowr and distinct wlfar valus. Th wlfar ranking can b obtaind from Figur 7, which plots th aggrgat wlfar in trms of th transport cost for nin valus of commuting cost. Th top lft panl shows that for t = 0:2, th aggrgat wlfar falls with highr transport costs as thy indd rduc consumption and thrfor pro ts, wags and utility. Th panl shows continuous wlfar transitions btwn th intgratd, th partially intgratd and th monocntric citis as transport costs ris. For high transport costs, two duocntric con gurations mrg but yild lowr wlfar lvls. Th nxt two top panls show that for highr commuting costs t = 0:32 and 0:56 rspctivly, duocntric structurs can bring highr wlfar. Th cntral panl illustrats quilibria and th wlfar ranking for t = 0:64: two partially intgratd quilibria rank blow th monocntric city quilibrium. n th nxt panls, commuting cost ar so high that th monocntric con guration is not part of quilibria. t can b obsrvd that th duocntric city abruptly mrgs as commuting costs incras and yilds a highr aggrgat wlfar. 28

29 Population growth Finally, w look at th impact of urbanization on spatial structurs by considring gradual incrass in th city siz N for givn commuting and transport costs. According to our numrical analysis, thr cass aris dpnding on transport costs. (i) For low transport costs, transitions ar smooth. Th con guration is initially monocntric for small N (i.. t < t M ), thn it bcoms partially intgratd for intrmdiat N (i.. t M < t < t ). (ii) For intrmdiat transport costs, th con guration is initially monocntric. Thn it bifurcats into ithr a partially intgratd or duocntric city. Ths pattrns ar quit di rnt from th monocntric city so that th transition is discontinuous, th rsulting path bing undtrmind. n th cas of a duocntric city, it will partially intgrat at a latr stag, and will nally approach full intgration. (iii) For larg transport costs, th con guration is initially monocntric. Whn it bifurcats, it bcoms a duocntric city. Thn, it bcoms partially intgratd, and nally fully intgratd. Figur 8 plots this cas with = 3 and t = 0:7 for an incrasing population N. t shows that th con guration is initially monocntric (black). Whn it braks, it bcoms duocntric (rd), and thn partially intgratd (blu). Ths rsults show that transitions ar continuous (rsp. discontinuous) for small (rsp. larg) transport costs. This r cts th fact that whn transport costs ar larg, rms can gain mor whn rms locat in svral businss districts. 6 Conclusion W hav studid urban structurs drivn by dmand and vrtical linkags in th prsnc of incrasing rturns to scal, which in contrast to th xisting litratur on th topic, ar intrnal to th rm. Existnc of quilibrium has bn provd and an invarianc rsult has bn obtaind rgarding th quivalnt rol of mor intns dmand and vrtical linkags and lowr commuting costs. Various urban con gurations can mrg in quilibrium xhibiting a monocntric, an intgratd, a duocntric, or a partially intgratd city structur. W hav discussd th rol of commuting and transport costs, intnsitis of dmand/vrtical linkags, and urbanization in a cting ths pattrns. W hav nally prsntd a numrical approach to discuss th mrgnc of multipl quilibria. 29

30 Appndix A: Equilibrium conditions Lt = [ b; b], b 2 R + 0, b th support of th city. Th dnsitis of rsidnts and rms ar givn by :![0; 1] and :![0; 1=s]. Th land markt claring imposs that + s = 1 whil labor markt claring imposs that th mass of workrs is qual to mass of rms R = R = N. With th city bordr b N(1 + s)=2, vry workr and vry rm nd a job match. So, w just nd to impos R = N. Whil workr s bid rnt is th maximum bid a workr maks on a parcl of land as of (13), th rm s bid rnt is th maximum bid that it can mak on land as of (14). Th ncssary and su cint conditions for spatial quilibrium ar givn by idntitis (16) and (17). Rmind that S(y) = G(y) ln T (y), whr T (y) = R (1 )jz yj (z)dz 1 1 R and G(y) = 1 jz yj (z)dz. Not that G, T and thrfor S : T 1 (z)!r + ar intgrals and thrfor continuous functions of y. Functions G and T ar also strictly positiv bcaus thir intgrands ar positiv as has positiv masur. No cross commut At vry location, th ow of commutrs is unidirctional. Th proof can b found in Ogawa and Fujita (1980, Proprty 1). Th authors show that workrs commuting in opposit dirctions hav no incntivs to cross ach othr in quilibrium. City dgs Spatial quilibrium involvs workrs rsiding at city dgs. Suppos this is not th cas: (b) = 0 and (b) = 1=s: Thn, thr would b workrs at x < b facing a positiv land rnt and commuting to a rm locatd at b. Ths workrs would bn t from rlocating to b +, with > 0 in nitly small, bcaus of a zro land rnt and an in nitly small commuting cost t, which violats quilibrium. Th sam argumnt applis at x = b. Mor gnrally, spatial quilibrium involvs mor rsidnts than rms at city dgs. Othrwis, (b) < (b), and th abov argumnt still applis as thr would b workrs at x < b facing a positiv land rnt and commuting to a rm in b, who would bn t from rlocating to b+, with > 0 in nitly small. Lmma 1 City dgs host mor rsidnts than rms: > 0; for z = b. Land rnt continuity Th bid rnts r and b ar th maximum valus of programs (13) and (14), which involv not only th continuous functions S and jj, but also th land rnt function R which 30

31 is not a priori continuous. Thrfor, th Maximum Thorm cannot b applid in ordr to show th continuity of bid rnts r and b. nstad, th continuity of bid rnts is shown by th fact that thy ar linar functions of th land rnt R. Lmma 2 Th bid rnts r and b in (13) and (14) ar continuous functions! R. So r b and R = maxf r; b; 0g ar also continuous functions! R. Proof. ndd, ach bid rnt can b writtn as th function (x) = max z f 1 (x) + f 2 (z) + f 3 (x z), whr f 1 :! R and f 3 : [ 2b; 2b]! R ar continuous functions and f 2 :! R is not ncssarily a continuous function. For = r, tak f 1 = 0, f 2 = ln + ln + ( + ) S + c 0 U sr and f 3 (x z) = t jx zj. For = b, tak f 1 = ( ln + ln + ( + ) S + c 0 U )=s, f 2 = R=s and f 3 (x z) = t jx zj =s. Rmmbr that S is a continuous function. n contrast, th land rnt R is not a priori a continuous function. Lt y and y 0 b th maximizrs of rspctivly at x and x 0. y th d nition of a maximum, w hav (x) = f 1 (x) + f 2 (y) + f 3 (x y) f 1 (x) + f 2 (y 0 ) + f 3 (x y 0 ) and (x 0 ) = f 1 (x 0 ) + f 2 (y 0 ) + f 3 (x 0 y 0 ) f 1 (x 0 ) + f 2 (y) + f 3 (x 0 y). Using th conditions, w hav f 3 (x y 0 ) f 3 (x y) f 2 (y) f 2 (y 0 ) f 3 (x 0 y 0 ) f 3 (x 0 y) (45) Thn, w comput th di rnc (x) (x 0 ) = f 1 (x) f 1 (x 0 ) + f 2 (y) f 2 (y 0 ) + f 3 (x y) f 3 (x 0 y 0 ) and thrfor f 2 (y) f 2 (y 0 ) = (x) (x 0 ) (f 1 (x) f 1 (x 0 )) (f 3 (x y) f 3 (x 0 y 0 )) Plugging this in (45), w gt f 1 (x) f 1 (x 0 ) + f 3 (x y 0 ) f 3 (x 0 y 0 ) (x) (x 0 ) = f 1 (x) f 1 (x 0 ) f 3 (x 0 y) + f 3 (x y) Thn, bcaus f 1 and f 3 ar continuous functions, lim x 0!x jf 1 (x) f 1 (x 0 )j = 0 and lim y 0!y jf 3 (x y 0 ) f 3 (x 0 y 0 ) lim x 0!x jf 3 (x 0 y) f 3 (x y)j = 0, w hav that lim x 0!x j (x) (x 0 )j = 0. Commuting ows Lt n :!R b th numbr of commutrs owing in th right hand dirction (i.. toward positiv x). Sinc thr is no rsidnt and no rm at th lft of x = b and at th right of 31

32 x = b, w hav n(b) = 0. This ow incrass with th mass of rsidnts who start thir commuting trip in th right dirction and dcrass with th mass of rms that hir commutrs locatd on thir lft: n =. Givn th d nition of spatial quilibrium, w hav 8 f1g if r > b >< n 2 1 [ ; 1] if s r = b >: f 1g if s r < b Givn th absnc of cross commuting, a location is a sourc of commutrs if n > 0 and a sink of commutrs if n < 0. Th ow of commutrs points to th right dirction (x < y) whn n > 0 and to th lft dirction (x > y) if n < 0. y th rst fundamntal thorm of calculus, th ow of commutrs n(z) = R z b ndz is a continuous function. At city dgs, bcaus at z = b, w hav n(b) 0. Also, whn n(b) > 0, th function n is qual to 0 at last twic in ordr to satisfy n(b) = R b b (46) ndz = 0. t is usful to introduc som trminology about th sts of sourcs and sinks of commutrs moving to right or lft dirction. Lt th st of sourcs S = fz 2 : n > 0 and n > 0g and th st of sinks (dstinations) D = fz 2 : n < 0 and n > 0g whr = 1 if commutrs ow to th right dirction and = 1 othrwis. For xampl, S 1 includs sourc locations of commutrs owing to th right dirction and S 1 thos owing to th lft dirction. Obviously, commutrs dparting from S and arriving in D ow to th right dirction (rsp. th lft dirction) if = 1 (rsp. ar nithr origin nor dstination of commutrs d n th st P = fz 2 : 1). Locations that n = 0g. Locations that involv a chang in commuting dirction ar dnotd by th st Q = fz 2 : n = 0 and n 6= 0g. caus n is a continuous function and ithr n > 0 or n < 0 for z 2 Q, any location z 2 Q has zro masur, i.. it is an isolatd point. Finally, not that thos d nitions dtrmin six disjoint sts: D +1, D 1, S 1, S 1, P and Q. W now study th land rnt proprtis on ths sts. Wag arbitrag in S [ D. Considr two disjoint, opn and convx intrvals, S 0 S and D 0 D, such that (i) workrs rsiding in S 0 commut to rms locatd in D 0 and labor ows in th sam dirction within ach intrval (S 0 \ D 0 =?). Sinc n (x) and n (y) hav th sam sign for x 2 S 0 and y 2 D 0, on radily chcks that jx yj = (y x) sign(n (x)) = ysign(n (y)) xsign(n (x)) whr sign(n) = 1; 0; 1 dpnding on whthr n is positiv, zro, or ngativ. Sinc n has th sam sign in ach intrval, ach function sign(n) is a constant. Th rst ordr conditions for th maximum 32

33 problms (13) and (14) ar givn by ( + ) S t sign(n) s R = 0, y 2 D 0 (47) t sign(n) R = 0, x 2 S 0 (48) Whil th rst idntity includs only rsidnts irrspctiv of thir workplacs, th scond on involvs only rms irrspctiv of workrs rsidncs. Ths idntitis apply for any in nitsimal intrval whr workrs start to commut in th sam dirction and rms hir from th sam sid of th city. Sinc ths conditions hold for all S 0 S and D 0 D, thy hold a.. in S and D, whr a.. (almost vrywhr) mans for all points xcpt on a zro-masur st. Lmma 3 Th land bid rnt di rntial is givn by b r = 1 s ( + ) S 1 + s tsign(n); s a.. z 2 S [ D Proof. Considr again th two disjoint, opn and convx intrvals S 0 S and D 0 D. Thn, conditions (47) and (48) imply that th quilibrium land rnt R is qual to workrs and rms bid rnts. Applying th nvlop thorm to (13) and (14) yilds r = t sign (n) ; x 2 S 0 b = 1 ( + ) S t sign (n), y 2 D 0 s W construct th highst losing bids r on D 0 and b on S 0. W rst look at th st S 0. caus workrs dpart from thir rsidnc x 2 S 0 to work, thy must rsid on S 0 and win th land markt comptition. caus workrs rsid in S 0, hr bid r must b qual to th quilibrium land pric R. Considr thn a rm that would produc at x 2 S 0 and hir a rsidnt at x 0 2 S 0. caus of (48), any rsidnt in S 0 has th sam incntiv to work in th rm. So, w can say that th rm hirs a workr rsiding at x 0 = x. Th rm s highst (losing) bid b is givn by (14) valuatd at y = x. Using this, th bid rnt di rntial writs as b r = 1 ln + ln s + ( + ) S + c 0 U Di rntiating this xprssion and using (48), w hav th bid rnt di rntial 1 + s R (49) s b r = 1 s ( + ) S s t sign (n), z 2 S 0 (50) s

34 which holds for any S 0 in S, and thus for any masurabl st in S. W thn look at th st D 0. caus D 0 hosts rms, it must b that b = R. Considr a rsidnt who would choos to rsid at y 2 D 0 and b hird by a rm in D 0. caus of (47), th workr is indi rnt to any rm producing at y 0 2 D 0, so that w can focus on th cas whr th hiring rm producs at y 0 = y. Th workr s highst (losing) bid is givn by (13) valuatd at x = y. Th bid rnt di rnc writs as b r = ln + ln Di rntiating this xprssion and using (48), w gt th bid rnt di rntial ( + ) S c 0 + U + (1 + s) R (51) b r = 1 s ( + ) S 1 + s t signn; y 2 D 0 s which holds for any D 0 in D, and thus for any masurabl st in D. Wag arbitrag in P. Considr a location z 2 P, which is nithr a sourc nor a sink of commutrs. Sinc thr is no mpty hintrland, this location must host both workrs and rms, so that land markt claring imposs R = b = r. So, hav and R = s ln + ln + ( + ) S + c 0 U ; z 2 P Furthrmor, lt P b th intrior st of P so that z 2 P is not an isolatd point in P. Thn, w R = r = b = s ( + ) S, z 2 P b r = 0, z 2 P (52) id rnt di rntial at city bordrs. Th zro opportunity cost of land imposs R = 0 at th city bordr z = b. y Lmma 1, thr must b workrs with rsidncs at z = b so that th highst bid is that of workrs: r = R = 0 at z = b. Th dg location z = b also blongs to th st S 1 sinc ( b) > 0 and workrs thr cannot commut to th lft dirction. W can thrfor apply (49) and gt th land rnt di rntial b r = b = 1 ln + ln s + ( + ) S + c 0 34 U ; z = b (53)

35 Equilibrium condition W summariz th abov rsults as follows. Lt us d n :! R, (z) = 1 s ( + ) S (z) 1 + s t sign(n (z)) (54) s Not that n and may not b continuous in z. Howvr is boundd bcaus S is boundd. A spatial quilibrium is d nd by th rnt di rntial function b r and th commuting function n such that b r is continuous vrywhr on, b r = b and n 0 = n at z 2 f b; bg, and z 2 S [ D ) b z 2 P ) b r = (z) r = 0 a.. a.. (55) Rmind that Q has zro masur and thrfor is not a ctd by th abov implications. Lmma 4 A spatial quilibrium is such that z 2 S [ D () (z) 6= 0 a.. z 2 P () (z) = 0 a.. Not that any quilibrium commuting function n has two possibl pattrns: it is ithr such that n = n = = 0, 8z 2 (fully intgratd city) or such that th city spac is partitiond with n > 0 for som sourc locations z and n < 0 for som sink locations z. n th lattr cas, w hav > 0 for som z and < 0 for som othr z. This mans that always accpts a zro. Lt b th st of zros or changs in th sign of, = fz 2 : lim "!0 (z + ") (z ") 0g. Thn, th spatial quilibrium is givn by th function n :! R, which is th solution to 8 f1g if (z) < 0 >< n(z) 2 1 [ ; 1] if (z) = 0 s >: f 1 g if (z) > 0 s with n 0 = n at z = b and (z) R z (x) dx: Using n(x) = R x ( ) dz yilds condition (16) bz2 b in th txt. nvarianc To chck invarianc, on can rplac t and (+) by t = mt and (+ ) = m(+); m 2 R 0. This yilds (n) = m (n), which givs th sam st of roots b = fz 2 : (n) = 0g = fz 2 : (n) = 0g. As a rsult, th sign of (n) is th sam as that of (n), and thus, th quilibrium must b th sam. W thrfor hav invarianc with rspct to changs in paramtrs as long as ( + )=t rmains constant. 35

36 Appndix : Existnc of a spatial quilibrium Formulation of a xd point W dnot th st of continuous functions on =[ b; b] by C 0 () and th st of functions with continuous rst drivativ by C 1 (). To formulat th xd point problm, w work with th commuting ow function n which is continuous: that is, n 2 N whr N fn 2 C 0 () : n( b) = n(b) = 0, n 2 [ 1=s; 1] a..g whr a.. mans almost vrywhr, that is vrywhr xcpt on a zro masur st. W also us a concis notation for oprators whr M n xprsss th application of oprator M : N! C 0 () on a function n 2 N and Mn (z) xprsss th valu rturnd by Mn at z 2. W rd n th conomic surplus as th oprator S : N! C 1 () and its slop S : N! C 0 () so that Sn (z) = Sn ( b) + R Sn(x) 1 fz>xg dx whr 1 fag is th indicator function that rturns 1 if A is tru and 0 othrwis. Thn, w dnot th rnt di rntial by : N! C 0 (), ( ) (z) = R b r 1 fz>xg dx. Using th valu for b r in (54) and (55), intgrating from z = b, and using rnt di rntial (53) at this location, w gt n (z) = 1 s ln + ln + c 0 U W rwrit th bid rnt di rntial in a mor compact form as + 1 s ( + ) Sn (z) 1 + s t s z b sign(n (x))dx = 1 s ( + 0 U) whr th oprator : N! C 0 () is d nd by = ( + ) S (1 + s) tl, th oprator L : N! C 0 () by Ln(z) = R sign(n (x)) 1 fz>xgdx, and th scalar 0 by s ln + ln + c 0. A spatial quilibrium is thn d nd by th function n : [ th solutions to with n 0 = n at z = b. n 2 8 >< >: f1g if n + 0 U < 0 [ 1 s ; 1] if n + 0 U = 0 f 1 s g if n + 0 U > 0 b; b]! R and th scalar U which ar W now formulat th xd point problm in trms of th continuous function n rathr than th discontinuous function n. Towards this aim, w intgrat th prvious condition (56). To tak car of th thr subconditions, w d n th corrspondnc F : R! [ 1=s; 1] such that F (z) = 1=s if (56) 36

37 z < 0, F (z) = [ 1=s; 1] if z = 0, and F (z) = 1 if z > 0. W d n th functional U : N! R that for any function n, givs th scalar U that solvs 0 = F [U 0 n (z)]dz and th oprator O : N! N 0, (On) (z) = F [U n 0 n (x)] 1 fz>xg dx (57) Th d nition of U guarants that n(b) = 0 whil n( b) = 0 is trivially satis d. y construction, th oprator O rturns a function n such that n 2 [ 1=s; 1]. So, O maps N into itslf. Lmma 5 Th xistnc of a spatial quilibrium is quivalnt to th xistnc of a xd point n 2 On. Squncs of xd points Th Schaudr thorm applis on a st of continuous mappings of continuous functions. So, w us continuous approximations of F and sign functions and thn prov convrgnc. W d n th continuous, di rntiabl and strictly incrasing functions F k : R! [ 1=s; 1], k 2 N, such that F k convrgs to F as k! 1 and 0 < Fk 0 k, and th continuous, di rntiabl and strictly incrasing function H l : R! [ 1; 1], l 2 N, such that H l convrgs to sign as l! 1 and 0 < Hl 0 l. n particular, w considr logistic functions with imags on [ 1=s; 1] and [ 1; 1]: F k (z) = 1 + (1 + s) = [1 + xp( 4kz= (s + 1)] and H l (z) = 1 + 2= [1 + xp( 2lz)]. Th typ of convrgnc will b d nd formally blow. W us th subst of continuous functions N and th sup norm knk 1 = sup z2 jn(z)j. W d n th oprators l and L l : N! C 0 () by l n (z) = ( + ) Sn (z) (1 + s) tl l n(z) L l n(z) = H l (n (x)) 1 fz>xg dx Furthrmor w d n th functional Ukl : N! R as bing th solution U to R F k[u 0 l n (x)]dx = 0 and th oprator O kl : N! N by O kl n(z) = F k [Ukln 0 l n (x)] 1 fz>xg dx 37

38 Not that Ukl is a uniqu solution bcaus F k is strictly incrasing. Our stratgy is to prov th xistnc of xd points n kl to n kl = O kln kl 8k; l 2 N and prov that th limit n = lim k!1 lim l!1 n kl satis s n = On. W rst prov that th st N is a closd convx st in th anach spac C 0 () and th mapping O kl is wll d nd and continuous. Lmma 6 Th st N is a closd convx st in th anach spac C 0 (). Proof. Th spac C 0 () is a vctor spac of continuous scalar functions on th ral compact intrval =[ b; b] for th opration of addition and scalar multiplication. t is quippd with th sup norm knk = sup z2 jnj. Evry Cauchy squnc fn k g 1 k=1 with n k 2 N convrgs to som function n 2 N : t is thus a anach spac. Th st of continuous functions N = fn 2 C 0 () : n( b) = n(b) = 0; n 2 [ 1=s; 1]g is closd and convx. ndd, any convx combination n + (1 )m, n; m 2 N, blongs to N. t is also closd bcaus vry squnc fn k g 1 k=1 with n k 2 N convrgs to som n 2 N. W now chck that O kl is a wll-d nd continuous mapping. W rmind that an oprator is wll d nd if it actually maps its domain into th d nd st of imags. An oprator M : X! Y, with X; Y 2 C 0 (), is a continuous mapping if for all > 0, thr xists k > 0 such that for any n; m 2 X; kn mk 1 < 1=k ) kmn Mmk 1 <. t must b notd that th composition of continuous mappings is also a continuous mapping. Also, th convolution M : X! Y, Mn(z) = R f(z is a continuous mapping for any boundd valud function f : R! R, jfj < 1. x)n(x)dx Lmma 7 S is a wll d nd continuous mapping. R Proof. W rst prov th following usful statmnt: An oprator M : N! C 0 (R), Mn(z) = f(x; n)g(x z) n(x)dx is a continuous mapping whn w considr f : N!R as a continuous function in m with boundd imag (i.. jfj < f max ) and g : R!R + with boundd imag (i.. jgj < 38

39 g max ). W indd succssivly gt kmn Mmk 1 = sup f(x; n)g(x z) n(x)dx f(x; m)g(x z) m(x)dx z2 R f(x; n)g(x = sup z) R n(x)dx f(x; m)g(x z) m(x)dx R z2 f(x; n)g(x z) m(x)dx + R f(x; n)g(x z) m(x)dx sup n(x) f(x; n)g(x z) m(x) dx + [f(x; m) f(x; n)] g(x z) m(x)dx z2 n(x) f max g max m(x) dx + gmax [f(x; m) f(x; n)] m(x)dx Not that R n(x) m(x) dx = R bb dn(x) R b dm(x) = n(b) n( b) [m(b) m( b)] = 0 sinc b n(b) = m(b) = 0 for any n; m 2 N. Also, for any m 2 N, on has that m 2 [ 1=s; 1] and thrfor m maxf1; 1=sg. Using this, [f(x; m) f(x; n)] m(x)dx maxf1; 1 s g [f(x; m) f(x; n)] dx sup z2 maxf1; 1 s g 2b sup jf(z; m) z2 Finally, by continuity of f, for any > 0; thr xists > 0 such that km f(z; n)j nk < implis jf(z; m) f(z; n)j < for all z, or quivalntly, sup z2 jf(z; m) f(z; n)j <. Thus, for any 0 w can us th sam such that km nk < ) kmn Mmk 1 < 0 whr w st = 0 = [2bg max maxf1; 1=sg]. This provs th continuity of M. Scond, th oprator T 1 : N! C 0 (R) with T 1 n(z) = R (1 )jz xj (x)dx is a wll-d nd continuous oprator. ndd, using = 1 n =(1 + s), w hav T 1 n(z) = R (1 )jz xj dx=(1 + R s) (1 )jz xj ndx=(1 + s). Th rst trm is indpndnt of n. Rgarding th scond trm, just st f(x; n) = 1 and g (z x) = (1 )jz xj, which hav boundd imags for x; z 2. Th oprator is wll d nd as it rturns continuous functions having imags in a strictly positiv rang: T 1 n () (2b (1 )2b = (1 + s) ; 2b=s). For th highst bound, w simply plug n = 1=s in th rst lin of th abov xprssion. For th lowst bound, w comput T 1 n(y) = R (1 )jz yj 1 n(z) dz 1+s R (1 )2b 1 n(z) 2b(1 )2b dz = > 0 whr w us jz yj < 2b in th inquality and R ndz = 0 in 1+s 1+s th last quality. Thrfor, 1= [T 1 n(x)] is also a continuous function having imags in a strictly positiv rang whil ln [T 1 n(x)] is a continuous functions having ral boundd imags. R Third, th oprator G : N! C 0 (R) with Gn (z) = 1 jz xj (x)dx is a wll-d nd contin- (T 1 n)(x) uous oprator. ndd, on just has to st f(x; n) = 1= [T 1 n(x)] and g (z x) = jz xj in th 39

40 abov statmnt. t is clar that th oprator rturns a function with boundd positiv imags. Finally, sinc S = G ln T = G ln T 1, it is a linar combination of wll d nd continuous oprators. Thus, w can conclud that S : N! C 0 () is a wll-d nd continuous mapping. Lmma 8 L l is a wll-d nd continuous mapping. Proof. For any > 0; w must nd such that km nk 1 < ) kl l n L l mk 1 <. W succssivly gt kl l n L l mk = sup z2 = sup z2 = sup z2 H l (n (x)) 1 (z > x) dx H l (m (x)) 1 fz>xg dx [H l (n (x)) H l (m (x))] 1 fz>xg dx [n (x) m (x)] Hl 0 (p(x)) 1 fz>xg dx whr p(x) 2 [min(n (x) ; m (x)); max(n (x) ; m (x))] by th man valu thorm and th continuity and th di rntiability of H l. Using th fact that Hl 0 l and that n( b) = m( b) = 0, w hav z kl l n L l mk 1 l sup [n (x) m (x)] dx z2 l sup z2 b z b jn (x) = 2bl sup jn (z) z2 = 2bl kn mk m (x)j dx m (z)j So, for any, choos < =l. Thrfor, L l : N! C 0 () is a continuous mapping. Sinc H l has imag on [ 1; 1], th imag of L l lis in th intrval [ 2b; 2b]. So, it is wll d nd. Sinc l : N! C 0 () is composition of wll-d nd continuous mappings with boundd imags, it is also a wll d nd continuous mapping. W nally nd to show th corrsponding rsult for O kl which dpnds on U kl. Lmma 9 U kl is a wll-d nd continuous functional. Proof. For radability w hr tmporarily writ U kl as U. Thn, U : N! R is th uniqu solution to R F k[u 0 l n (x)]dx = 0 bcaus F k is strictly incrasing. Also, it is clar that th 40

41 solution U must li btwn min x2 [ 0 + l n (x)] and max x2 [ 0 + l n (x)]. Sinc l n has boundd imag, Un has also boundd imags and is wll d nd. Lt us considr two commuting functions n and m : N! R. Lt also p min and p max : N N! R b th minimum and th maximum functions of Un l n and Um l m. y th man valu thorm and th di rntiability of F k, w hav F k [Un 0 l n (x)] F k [Um 0 l m (x)] = Fk(p(x)) 0 [Un l n (x) Um + l m (x)] whr p :! R such that p 2 [p min, p max ]. This yilds F 0 k(p(x)) (Un U kl m) = F k [Un 0 l n (x)] F k [Um 0 l m (x)] + F 0 k(p(x)) [ l n (x) l m (x)] gt ntgrating, applying R F k[u 0 l n (x)]dx = 0 to n and m, and subtracting xprssions, w (Un Sinc Fk 0 > 0, w hav Um) Fk(p(x))dx 0 = F 0 k(p(x)) [ l n (x) l m (x)]dx R jun Umj = F k 0(p(x)) [ ln (x) l m (x)]dx R F 0 k R (p(x))dx F k 0(p(x)) j ln (x) l m (x)j dx R F 0 k R (p(x))dx F k 0(p(x)) sup z j l n (z) l m (z)j dx R F 0 k (p(x))dx = sup j l n (z) z = k l n l mk 1 l m (z)j n othr words Un is a continuous mapping of l n. Sinc l n is a continuous mapping of n, Un is also a continuous mapping of n. Lmma 10 For any k; l 2 N, O kl : N! N is a continuous mapping. Proof. Th oprator O kl is d nd by O kl n(z) = F k [Ukln l n (x)] 1 fz>xg dx (58) 41

42 whr U kl n was discussd in th prvious lmma. y construction, th imag functions m 2 O kln ar continuous and satisfy m( b) = m(b) = 0 and m 2 [ 1=s; 1]. Th oprator O kl is wll d nd as it is a mapping N! N. For radability w tmporarily dispns U kl and O kl with th subscripts kl and suprscript. Using th man valu thorm and th di rntiability of F k and p d nd in th prvious lmma, w gt kon Omk 1 = sup ff k [Un l n (x)] F k [Un l n (x)]g 1 fz>xg dx z2 = sup Fk(p(x)) 0 [Un l n (x) U(m) + l m (x)] 1 fz>xg dx z2 sup Fk(p(x)) 0 [ l n (x) l m (x)] 1 fz>xg dx z2 + jun Umj sup Fk(p(x)) 0 1 fz>xg dx z2 sup z2 W succssivly hav jfk(p(x))[ 0 l n (x) l m (x)]j 1 (x < z) dx sup jfk(p(x))j 0 j l n (x) l m (x)j 1 fz>xg dx z2 jfk(p(x))j 0 j l n (x) l m (x)j dx k j l n (x) l m (x)j dx 2bk sup j l n (z) z2 = 2bk k l n l mk 1 l m (z)j and jun Umj sup z2 Fk(p(x)) 0 1 (x < z) dx k ln k l n 2bk k l n l mk 1 sup jfk(p(x))j 0 1 (x < z) dx z2 l mk 1 jfk(p(x))j 0 dx l mk Finally, kon Omk 1 4bk k l n l mk 1 For any, w can thn choos = =(4bk) so that k l n l mk 1 < implis kon Omk 1 <. Hnc, putting back th subscripts kl, O kl is a continuous mapping of l. Sinc l is a continuous mapping of n, w hav that O kl is also a continuous mapping of n. 42

43 Lmma 11 Th imag O kl (N ) is a rlativly compact subst of N. Proof. y Arzla-Ascoli thorm, this is tru if n 2 O kl (N ) ar uniformly boundd and quicontinuous on. To say that n 2 O kl (N ) ar uniformly boundd mans that thr xists M > 0 such that knk 1 = sup z2 jn(z)j < M for all n 2 O kl (N ). This is tru sinc F k has imag on [ 1=s; 1]. Thrfor, O kl gnrats functions n :! R such that n( b) = n(b) = 0 and n 2 [ 1=s; 1]. Ths functions hav imag in [ 2n; 2b]. To say that n is quicontinuous on mans that for vry > 0 thr xists an such that for vry x; y 2 and vry n 2 N, w hav jx yj < ) jn(x) n(y)j <. This follows from th continuity of n 2 C 0 (). To sum up, th st N is a closd convx st in th anach spac C 0 (). Th oprator O kl : N! N is a continuous mapping such that O kl (N ) is a rlativly compact subst of N. W can thn apply Schaudr xd point thorm. Lmma 12 For ach k; l 2 N, thr xists a xd point n kl 2 N such that n kl = O kln kl. Convrgnc W now turn to th qustion about whthr th squnc of xd points fn kl g k;l2n convrgs to a function that is also a xd point of (57). caus N is rlativly compact and any n kl blongs to N, th squnc fn kl g k;l2n convrgs uniformly on to a function n 1 2 N. Not that F is a corrspondnc bcaus F (0) = [ 1=s; 1]. n turn, th oprator O : N! N is a corrspondnc. W thrfor want to chck whthr n 1 2 On 1. Formally, w ask whthr for any > 0, thr xists a function n 2 On 1 and two positiv intgrs k and l such that w hav ko kl n 1 n k 1 < for k > k and l > l: W hr assum that th squnc fh l g l2n convrgs to th sign function undr th norm kk 1. Formally, for any > 0, thr xists l 2 N such that R jh l () sign()j d <. Similarly, w assum that ff k g k2n convrgs to F undr th norm kk 1. That is, for any > 0, thr xists k 2 N such that R jfk () F ()j d <. Th last intgral rturns a scalar bcaus th corrspondnc F rturns a st only on th zro masur st at = 0. Lmma 13 l convrgs to uniformly undr th norm kk 1. 43

44 Proof. Th mapping l uniformly convrgs to th mapping if for any > 0, w can nd l such that k l k 1 for all l > l. Not that k l k 1 = k( + ) S (1 + s) tl l ( + ) S + (1 + s) tlk 1 = (1 + s) t kl l Lk 1. So, w just nd to chck th uniform convrgnc of th mapping L l : N! R, L l n(z) = R H l(n (x)) 1 fz>xg dx to th mapping Ln(z) = R sign(n (x)) 1 fz>xgdx. y th d nition of th convrgnc for H l, for any > 0, w can nd l such that R jh l () sign()j d for all l > l. Furthrmor w hav kl l Lk 1 = sup [H l (n (x)) sign(n (x))] 1 fz>xg dx z2 jh l (n (x)) sign(n (x))j dx = jh l () sign()j M(x : n (x) < + d)d 2b jh l () sign()j d 2b whr M(E) dnots th masur of th st E so that M() = 2b. So, for any 0, w can us th abov intgr l for = 0 =2b and gt kl l Lk 1 < 0 for all l > l. Thrfor, L l convrgs uniformly to L. Th sam conclusion applis to l and. That is, th squnc f l ng l2n convrgs to n. Lt th function F b : R! [ 1=s; 1] b such that F b () = 1 if > 0; F b () = 0 if = 0, and bf () = 1=s if < 0. t is clar that F b 2 F and that ff k g k2n convrgs to F b undr th norm kk 1. For any > 0, thr indd xists k 2 N such that R 1 1 F k () F b () d <. This is tru bcaus ff k g k2n convrgs to F and R 1 F k () F b R 1 () d = jf 1 k() F ()j d sinc F and F b di r only on a zro masur st. 1 Lmma 14 For any oprator : N! C 0 () and function n 2 N, R F k [n(x)] 1 fz>xg dx convrgs uniformly to R b F [n(x)] 1 fz>xg dx undr th norm kk 1 and R b F [n(x)] 1 fz>xg dx 2 R F [n(x)] 1 fz>xg dx. Proof. W rst prov that for any > 0 thr xists a positiv intgr k 2 N such that k = bf (n(x)) 1 fz>xg dx F k [n(x)] 1 fz>xg dx < 1 44

45 for k > k. W indd succssivly hav n k = bf (n(x)) Fk [n(x)]o 1 fz>xg dx 1 n = sup bf (n(x)) Fk [n(x)]o 1 fz>xg dx z2 b F (n(x)) F k [n(x)] dx = F b () Fk () M fx:n(x)<+dg d whr M E dnots th masur of a st E so that M = 2b. W can us th fact that th masur is always smallr than 2b and gt k 2b F b () Fk () d Finally, using th d nition of convrgnc of F k to F b, w can nd a positiv intgr k such that R F b () Fk () d < =2b, so that k for any k > k. W nally prov that R F b [n(x)] 1 fz>xg dx 2 R F [n(x)] 1 fz>xgdx. For any z 2, w hav bf [n(x)] 1 fz>xg dx = bf () M fx:x<z;n(x)<+d;n(x)6=0g d + 0 M fx:x<z;n(x)=0g F [n(x)] 1 fz>xg dx = F () M fx:x<z;n(x)<+d;n(x)6=0g d + F (0) M fx:x<z;:n(x)=0g d whr M E dnots th masur of a st E : For any z 2, F [n(x)] 1 fz>xg dx = bf [n(x)] 1 fz>xg dx + F (0) M fx:x<z;:n(x)=0g d Th last trm in th RHS is qual to ithr f0g if M fx:x<z;:n(x)=0g = 0 or th intrval [ m=s; m] whr m = M fx:x<z;:n(x)=0g if M fx:x<z;:n(x)=0g > 0. t is thn clar that R F [n(x)] 1 fz>xgdx 3 R b F [n(x)] 1 fz>xg dx. Givn ths lmmas, for any n 2 N and U 2 R, w gt lim lim O kln (z) = lim lim F k [U 0 l n (x)] 1 fz>xg dx k!1 l!1 k!1 l!1 U 0 lim h = lim F k k!1 l!1 ln = lim F k [U 0 n (x)] 1 fz>xg dx k!1 = bf [U 0 n (x)] 1 fz>xg dx 45 i (x) 1 fz>xg dx

46 whr th scond quality stms from th continuity of F k, th third on from Lmma 13, and th last on from Lmma 14. Also, from Lmma 14, w hav R b F [U 0 n (x)] 1 fz>xg dx 2 R F [U 0 n (x)] 1 fz>xg dx. So, lim k!1 lim l!1 O kl n (z) 2 On (z). Applying this to th xd point solution n 1 = lim k!1 lim l!1 n kl and U 1 = lim k!1 lim l!1 U kl n kl, w gt n 1 = lim k!1 lim l!1 O kl n 1 2 On 1. Hnc, n 1 is a xd point of O, which provs Proposition 2. Appndix C: Monocntric city Th proof maks uss of th following lmma. Lmma 15 (i) S M is convx in [0; b) thn concav in [b; b 1 ) for som b 2 [0; b 1 ) with SM 0 (0) = 0. (ii) S Y is convx in (b 1 ; b]. (iii) S M 2 C 1 in th intrval of [0; b] with S 0 M (b 1) < 0 and S 00 M (b 1 0) < 0 < S 00 M (b 1 + 0). [(T 1 Proof. (i) Th accss masur T M givn by (18) can b writtn as T 1 M (z) = 2 ( 1) s f1 (1 )b 1 cosh[( 1)z]g (59) caus cosh (z) is strictly concav and dcrasing for z 0, so is T 1 1 M. As (ln TM )00 = M )00 T 1 M (T 1 M )02 2(1 ) ]=T M < 0, ln T 1 M is strictly concav and dcrasing for z 0, and so is ln T M (z). Di rntiating G M as givn by (19), w succssivly gt G 0 M(z) = 1 z (z y) b1 (y z) s b 1 T 1 dy + M (y) z T 1 dy M (y) G 00 M(z) = z (z y) b1 (y z) s b 1 T 1 dy + M (y) z T 1 M (y) dy 2 T 1 M (z) = s 2 2 G M (z) s (z) Givn th abov xprssion of G 00 M T 1 M and th fact 2=T 1 M (60) is strictly incrasing for z 0, th curv s 2 G M is convx (rsp. concav) whn abov (rsp. blow) th curv 2=T 1 M. Two possibilitis actually aris: s 2 G M is ithr concav and blow th curv 2=T 1 M or thr xists som b 2 (0; b 1) such that s 2 G M is convx on [0; b] and concav on [b; b 1 ] with b ncssarily corrsponding to th singl intrsction point of s 2 G M with th curv 2=T 1 also b shown that S 0 M (0) = G0 M (0) = ( ln T M) 0 (0) = 0. M. S th illustration in Figur A1. t can 46

47 (ii) W now prov that S M is convx in (b 1 ; b]. For x 2 (b 1 ; b], w hav G M (x) = 1 s G 0 M(x) = s G 00 M(x) = 2 s b1 (x y) b 1 T 1 M b1 (x y) b 1 T 1 M b1 (x y) b 1 T 1 M (y) dy (y) dy < 0 and ln T M (x) = x b1 1 ln (1 s b 1 which is linar in x. Hnc, S M is convx in (b 1 ; b]. (y) dy > 0 (61) )y dy (iii) W showd that ln T M (z) is concav, dcrasing and C 1 on [0; b]. For z! b 1 0, w hav b1 G 0 M(b 1 0) = s G 00 M(b 1 0) = s (b1 y) b 1 T 1 M b1 (b1 y) b 1 T 1 as G M is concav in z = b 1 by part (i). For z! b 1 + 0, w hav G 0 M(b 1 + 0) = s G 00 M(b 1 + 0) = 2 s dy < 0 (y) M (y) dy 2 T 1 M (b < 0 1) b1 (b1 y) b 1 T 1 M b1 (b1 y) b 1 T 1 M (y) dy < 0 (y) dy > 0 by using xprssion (61). Thrfor, S 0 M (b 1 0) = S 0 M (b 1 + 0) but S 00 M (b 1 0) < 0 < S 00 M (b 1 + 0). So, S M is C 1 on [0; b]. t also appars that S 0 M (b 1) < 0. NSERT FGURE A1 HERE Figur A1: Concavity and convxity of G M : Not: Th gur rprsnts s 2 G M and 2=T M with plain and dashd curvs rspctivly. Paramtrs: lft panl: = 3, = 1, s = 1; right panl = 6, = 2; s = 1. W now show that th quilibrium conditions b(y) r(y) for all y 2 [0; b 1 ] and r(x) b(x) for all x 2 [b 1 ; b] can b rducd to b(0) r(0) and b(b) r(b) = 0 rspctivly. Proof. n th intrval of [0; b], th bid rnts of rsidnts and rms ar givn by r(y) = r(0) ty b(y) = 1 ( + )S M (y) + ty + ln + ln s U 0 + c 0 47

48 0 Givn Lmma 15(i), b is convx in [0; b) thn concav in [b; b 1 ) for som b 2 [0; b 1 ) with b (0) = t. Thrfor b(y) r(y) is quasiconcav on [0; b 1 ]. As b(b 1 ) = r(b 1 ), th quilibrium condition b(y) r(y) for all y 2 [0; b 1 ] rducs to b(0) r(0). According to Lmma 15(ii), b(z) is convx for z 2 [b 1 ; b]. As r(z) is linar and b(b 1 ) = r(b 1 ), th quilibrium condition b(x) r(x) for all x 2 [b 1 ; b] rducs to b(b) r(b) = 0. Appndix D: ntgratd city W show that js 0 (b)j js0 (z)j, 8z 2 [0; b]. Proof. Givn th similarity of xprssion of T (25) with that corrsponding to th monocntric cas (18), th argumnt usd in Appndix A showing that ln T 1 M is strictly concav and dcrasing in z in Appndix A, can b applid hr. Thus, ln(t ) is strictly concav and dcrasing for y 0. y di rntiating G (z) as givn by xprssion (26), w gt G 0 1 z (z y) b (z) = (1 + s) b T 1 dy + (y) z T 1 = z (z y) b (y z) 1 + s b T 1 (y) dy + z T 1 (y) dy 2 = z (z y) (1 + s) G (z) s b T 1 (y) dy G 00 (z) = (1 + s) 2 2 G (z) 1 + s T 1 (z) t su cs to show that G 0 (b) + G0 (z) 0 in ordr to hav (y) dy (y z) z b T 1 (z y) (y) dy jg 0 (b)j jg 0 (z)j, 8z 2 [0; b] (62) 48

49 as G 0 (b) < 0. To show that G0 (b) + G0 (z) 0, w dvlop G0 (b) + G0 (z) into b (b y) z (z y) (1 + s) [G (b) + G (z)] s b T 1 (y) dy 2 b T 1 (y) dy = b (b y) z (z y) 1 + s b T 1 (y) dy + b (y z) b T 1 (y) dy + b (b y) z T 1 (y) dy 2 b T 1 (y) dy 2 = b (y z) b (b y) 1 + s T 1 (y) dy z (z y) T 1 (y) dy T 1 (y) dy = 1 + s = 1 + s As T 1 z (y z) b z T 1 (y) dy b (y z) T 1 (y) z b z b T 1 (b y) (b y) dy (y) dy z b T 1 b b z T 1 (b y) (b y) (y) dy (y) dy z b T 1 z (z y) (y) dy (y) dy b T 1 (z y) z b T 1 (z y) (y) dy is dcrasing for z 0, w hav R b ( (y z) (b y) )=T 1 z (y)dy 0, 8z 0, bcaus th incrasing function (y z) is th r ctd imag of (b y) ovr th lin y = (z + b)=2. This provs G 0 (b) + G0 (z) 0. Finally, using th fact that ln (62) lads to Also, not that S 0 (b) < 0 as both G0 (b) and (ln dcrasing in th nighborhood of z = b: T is strictly concav and dcrasing and rlation js 0 (b)j js 0 (z)j, 8z 2 [0; b] W nxt prov that t is an incrasing function of. T ) 0 (b) ar ngativ. Thrfor, S (z) is concav and Proof. W nd to show that S 0 (b) is a dcrasing function of. First, it can b shown that d (ln T ) 0 (b) =d = 1. Scond, by using xprssion (26) w hav G 0 (b) = =(1+s) R b and (b z) =T 1 b (z)dz d d G0 (b) = 1 b 1 + s b T 1 (b z) (z) b dz s b (b z) (z) T 1 (b z) dz b (b 1 + s b z) d d T 1 (z)dz 49

50 As d[t 1 (z)]=d < 0, it su cs to show that th sum of th rst two trms is ngativ. W hav s = s = = b b T 1 b b 1 (1 + s) 1 (1 + s) (b z) (z) dz s b b (b z) T 1 (z) [(b z) 1] (b z) T 1 (z)dz 2b T 1 [b 1=()] (1 + s) (b z) dz (r 1) r T 1 [b r=()]dr 0 1 2b + (r 1) r T 1 [b r=()]dr b + (r 1) r dr < 0 as T 1 [b r=()] a dcrasing function of r and R R 0 (r 1) r dr < Appndix E: Cut-o s t M and t W prov that lim s!0 t M < lim s!0 t. As s! 0, t M = t M = ( + ). W nd to chck that (ln T ) 0 (b) G 0 (b) > whn s! 0. Givn th xprssions (25), (26) of T 1 and G 0, w hav (ln T ) 0 (b) = 1 d 1 dz ln 1 (1 )b cosh[( 1)z] (1 )b sinh[( 1)z] = 1 (1 )b cosh[( 1)z] = and G 0 2 ( 1) b (b y) (b) = 2 b 1 (1 )b cosh[( 1)y] dy whr G 0 (b) is ngativ bcaus th dnominator of this intgrand is positiv as 1 (1 )b cosh[( 1)z] 1 (1 )b ( 1)b ( 1)b + 2 2( 1)b 1 + = 1 2 2( 1)b 1 = 2 Hnc, w gt (ln T ) 0 (b) G 0 (b) >. > 0 50 z=b z=b

51 Appndix F: Proof of Corollary 2 W show that th partially intgratd city is a spatial quilibrium for t = t and small nough > 0. y continuity, S P (z) tnds to S (z) for any z whn b 1 tnds to b. This is also tru for S 0 P (z) and S 00 P (z). t is shown in Appndix D that S (z) is dcrasing and concav in th nighborhood of z = b and js 0 (b)j > js0 (z)j, z 2 [0; b]. Hnc, w hav S0 P (z) < 0 and S00 P (z) < 0 for z clos nough to b whil js 0 P (b)j > js0 P (z)j, 8z 2 [0; b]. y continuity, js0 P (b 1)j > js 0 P (z)j, 8z 2 [0; b 1] whn b 1 tnds to b. n this cas, S P (z) is dcrasing and concav in z within th intrvals [b 1 ; b] for small > 0. So, by concavity of S P (z), w comput whil + t P min y2[b 1 ;b 2 ] 1 + s b 2 t P + S P (b 2 ) S P (b) 1 + s b b 2 S P (y) S P (b 2 ) y bt P s max jsp 0 (z)j = + z2[0;b 1 ) 1 + s js0 P (b 1 )j t = + S P (b 1 ) S P (b 2 ) 1 + s b 2 b 1 = + S P (b 1 ) S P (b 2 ) 1 + s b 2 b 1 W now chck that bt P t min(t P ; t P ). ndd, t = t P and bt P t t P by concavity of S P (z). Rfrncs [1] Alonso, W. (1964) Location and Land Us, Harvard Univrsity Prss, Cambridg, MA. [2] rliant, M., S.-K. Png, and P. Wang (2002) Production xtrnalitis and urban con guration, Journal of Economic Thory 104, [3] Carlir, G. and. Ekland (2007) Equilibrium structur of a bidimnsional asymmtric city, Nonlinar Analysis: Ral World Applications 8, [4] Fujita, M. and H. Ogawa (1982) Multipl quilibria and structural transition of non-monocntric urban con gurations, Rgional Scinc and Urban Economics 12, [5] Lucas, R. and E. Rossi-Hansbrg (2002) On th intrnal structur of citis, Economtrica 70,

52 [6] Fujita, M., P. Krugman, and T. Mori (1999). On th volution of hirarchical urban systms, Europan Economic Rviw, 43, [7] Ogawa, H. and M. Fujita (1980) Equilibrium land us pattrns in a nonmonocntric city, Journal of Rgional Scinc 20, [8] P ügr, M.P. (2004) A simpl, analytically solvabl Chambrlinian agglomration modl, Rgional Scinc and Urban Economics 34,

53 -b -b 1 b 1 b -b b (a) Monocntric city (b) ntgratd city -b -b 2 -b 1 b 1 b 2 b -b -b 3 -b 2 -b 1 b 1 b 2 b 3 (c) Partially intgratd city (d) Duocntric city b Figur 1: Spatial urban structurs ( : firms and : housholds)

54 Figur 2: Monocntric city: rsidntial (blu) and businss (rd) bid rnts.

55 Figur 3: ntgratd city: rsidntial (blu) and businss (rd) bid rnts.

56 Panl a: balancd us of spac, s=1 Panl b: unbalancd us of spac, s=0.01 Figur 4: Partially intgratd city: rsidntial (blu) and businss (rd) bid rnts.

57 Panl a): Duocntric quilibrium 1 Panl b): Duocntric quilibrium 2 Figur 5: Duocntric city: rsidntial (blu) and businss (rd) bid rnts.

58 t t t M τ Figur 6: Monocntric (m), partially intgratd (p), intgratd (i), and duocntric configurations in (τ,t) plan (α=β=γ=s=1 and σ=3)

59 Figur 7: Aggrgat wlfar of urban configurations in trms of transport cost τ for various valus of commuting cost t

60 Figur 8: City structur as population siz incrass Th horizontal axis is population and th vrtical axis location. Diagonal straight lins display city bordrs. Rsidntial districts ar rprsntd in whit, monocntric businss districts in gry, duocntric businss districts in rd, and th businss and th intgratd districts of th partially intgratd city in plain and dash blu rspctivly. Ovrlaps indicat multipl quilibria.