Modelling the folk theorem: A spatial Cournot model with explicit. increasing returns to scale. Sylvain Barde. Department of Economics

Transcription

1 Modelling te folk teoe: A spatial Counot odel wit explicit inceasing etuns to scale Sylvain Bade Depatent of Econoics Univesity of Kent Abstact Tis pape attepts to odel diectly te folk teoe of spatial econoics, accoding to wic inceasing etuns to scale ae essential fo undestanding te geogapical distibutions of activity. Te odel uses te siple stuctue of ost New Econoic Geogapy papes, wit two identical egions, a costlessly taded agicultual secto and a anufactuing secto subject to icebeg costs. Tis siple setting isolates IRS in anufactuing poduction function as te only potential aggloeating foce. Tis iplies tat an unstable syetic equilibiu eans IRS cause aggloeation Te cental esult is tat wile a CRS anufactuing secto will always stay at te syetic equilibiu, te pesence of IRS in anufactuing causes te syetic equilibiu to becoe unstable and aggloeation becoes te only long un equilibiu fo te syste JEL Classification: R0, R2, F2 Keywods: Aggloeation, inceasing etuns to scale, ipefect copetition. Addess fo Coespondence: Depatent of Econoics, Univesity of Kent, Cantebuy, Kent, CT2 7NP. Telepone: Eail: sb7@kent.ac.uk

2 Modelling te folk teoe: A spatial Counot odel wit explicit inceasing etuns to scale. Intoduction In a bencak eview of econoic aggloeation teoy Fujita and Tisse state tat inceasing etuns to scale ae essential fo explaining te geogapical distibutions of econoic activities (996, p342). Many ote eviews, suc as Kugan (998), Ottaviano and Puga (997), o Bakan et al (200) siilaly point out tat etuns to scale ae a key concept undepining aggloeation. Tis as becoe known in te field as te folk teoe of spatial econoics. Howeve, to ou knowledge, no existing spatial odel looks diectly at te ipact of explicit IRS in anufactuing on aggloeation. Existing New Econoic Geogapy odels using te Dixit-Stiglitz (977) appoac do integate etuns to scale, oweve, tey use a poduction stuctue wit fixed and vaiables costs to ceate te etuns to scale. Tis pape investigates te effect of explicit IRS in te anufactuing secto of a two egiontwo secto odel on te stability of te dispesed equilibiu. Most of te teoetical analyses of aggloeation, especially in NEG, use te two egion/two secto setting because it is te siplest one in wic te vaious centipetal and centifugal foces can be identified. In paticula, ost of te, suc as Puga (999) focus on deviations fo te syetic equilibiu. Te siulation appoac used in tis pape is teefoe siila to te one in Puga (999). Te ai is to link te pesence of explicit inceasing etuns to scale in anufactuing to deviations Te fist odel to use tis faewok is basic Kugan (99). Fujita et al (999) also povides a eview of NEG odels. Fo a NEG odel wit oe coplex inteactions, te eade is efeed to Puga (999). 2

3 fo te syetic equilibiu. Because by constuction no ote aggloeation foces ae included in te ode, canges in te stability of te syetic equilibiu can be taced diectly to te pesence of IRS in te anufactuing secto. In ode to deal wit te stategic copetition tat eeges fo including explicit IRS into te poduction function, a Counot copetition faewok is used. Futeoe, tee is an existing spatial Counot liteatue, suc as Andesen and Neven (99), Gupta et al (997) o Maye (2000), Cobes and Lafoucade (2005) wic povide econoic basis fo aggloeation. Aggloeation in ost of tese studies ests fi enty in te ost pofitable locations is te aggloeating foce. Tis study will teefoe ipose te estiction of a fixed nube of fis, so as to be able to asses te aggloeative effect of IRS independently of fi enty. Te eainde of te pape is oganised as follows: Section 2 pesents te odel used in te siulation. Section 3 discusses te beaviou of te odel at te syetic equilibiu and ow te pesence of IRS in anufactuing odifies te poduction costs. Section 4 ten analyses ow te stability of te syetic equilibiu is affected by te IRS in anufactuing, and section 5 concludes. 2. A 2 egion 2 secto Counot odel wit explicit IRS In ode to geneate a Counot setting in wic IRS is te only aggloeating foce, te following assuptions will be ade. Te two egions will be identical in endowents of land (K) and labou (L). Distance is noalised to one, as is te TFP fo bot sectos. Agicultue is feely taded and anufactuing is subject to icebeg tanspot cost e τ. In te fist siulation, anufactuing will be subject to CRS, in ode 3

4 to povide a bencak, but ten IRS will be intoduced. In tes of notation, supescipts a and indicate te agicultue and anufactuing sectos espectively. Poduction and fi beaviou Te poduction function used ee is siila to te one in Cobes and Lafoucade (200), but intoduces inceasing etuns to scale. In tes of notation, we assue tee ae two industial sectos ove two egions. Fo te pupose of te notation, =, 2 and = a,. As in Cobes and Lafoucade (200), a single Cobb- Douglas poduction function descibes bot agicultue and ining anufactuing. Tese two industies diffe only in tat te elasticities of output wit espect to inputs will teefoe be industy-specific. Te diffeence wit Cobes and Lafoucade (200) is te assuption tat fo any given industial secto, tee is a fixed nube of poduces. Te spatial Counot liteatue sows tat fee enty in te ost pofitable locations can causes aggloeation even in constant etuns to scale. Teefoe, in ode to isolate te aggloeative effect of IRS, we assue a fixed nube of poduces. Fo puposes of siplification, we coose a single poduce pe secto and egion. Fujita and Tisse (996) point out tat te pesence of IRS in a poduction function ceates non-convexities. Fo exaple, a popety of Cobb-Douglas functions unde CRS is tat te sae of poduce expenditue allocated to an input is equal to te elasticity of output wit espect to tat input. Howeve, wit IRS tis su of expenditues on individual inputs would be geate tan te total expenditue of te poduce, wic akes no sense econoically. We sow late in tis section tat ost of tese pobles can be satisfactoily addessed. In paticula, an assuption ade to coect tis poble is tat te poduces use one set of CRS elasticities and a etuns to scale paaete to odify te elasticities. Te CRS elasticities, wic su to one, ae 4

5 used to deteine te inputs saes fo expenditue. Te etuns to scale paaete ten intoduces IRS into te CRS poduction function. Te poduction function fo te t industy in location is a Cobb-Douglas of te following fo: Hee ( ) α ' ( ) β H ', ' ( ) ε, i i i= y = A K L x () A tecnology, K is te use of land, L is labou. Inteediate consuption is intoduced toug x i, wic is te use of te i t industy s good as an inteediate i, input, wee ε is te elasticity of te t industy s poduction wit espect to te input fo te i t industy. Tis allows fo vetical linkages witin te odel. Agicultue is assued to ave sole use of land, so α = 0, and does not use inteediate inputs, eaning i, i, ε = 0. Futeoe, as explained above, if Ψ is te industy-specific etuns to scale paaete, te following applies: H i, α + β + ε = (CRS elasticities) i= α' + β' + ε ' = α + β + ε Ψ =Ψ i= i= H H i, i, (IRS elasticities) As claified above, te set of CRS elasticities is used fo all te input deand deteinations, weeas te odified elasticities ae te ones used in evaluating te cost-educing effects of te existence of etuns to scale. subject to (): Te cost iniisation poble is staigtfowad, and involves iniising (2) H i i, i= C = ck + w L + p x (2) 5

6 Wee c is te ental pice of capital, w ae te wages and te final suation is te expenditue by te t industy on inteediate inputs fo ote industies. Te cost function obtained toug te iniisation is: C ( ) Ψ y = χ (3) Wee te χ, te input coponent of aginal cost, is: χ A H Ψ α' β' i ( Ψ ) ( c) ( w) ( p) H, ( α' ) ( β' ) ( ε ' ) i, ' i= = i, α' β' i ε' i= ε Ψ (4) Tanspot costs ae integated using exponential icebeg costs, as in Sauelson (952, 954). Te pesence of IRS, oweve, ceates a poble fo te calculation of tanspot costs. One can see fo te cost function (3) tat in CRS, wee Ψ = aveage and aginal costs ae equal to χ, and applying te ultiplicative icebeg costs is staigtfowad. Tis is not te case fo inceasing etuns to scale, wee Ψ >, as te divegence of aveage and aginal costs eans tat a coice needs to be ade as to wic cost te icebeg applies to. We assue tat te sipping costs elate to aginal costs of poduction. It akes econoic sense fo te incease in te cost of sipping an exta unit of output to elate to te value of tat aginal unit. Te total cost of poducing in and sipping to egion s is teefoe: Wee te aginal cost of poduction is: s τ ( ) y C = C + y C e (5) s, d s, s, y C χ = ( ) y Ψ Ψ Ψ (6) Wit tis assuption, one can see below tat te aginal cost of poducing an exta unit in and sipping it to s contains two coponents: te fist is te aginal cost 6

7 of poducing tat exta unit, wic is te sae egadless of te taget aket. Te second pat elates to te sipping of te exta unit to te taget aket s, wic depends only on te aginal cost of poduction. dc dy s, d C C e s, s τ ( ) = + (7) Te total cost of poducing in a given egion is given by te suation of all cost flows (5) ove te s taget egions: R R d Cs, C ys, C e τ s= s= s ( ) = + (8) Te total cost contains two sepaable coponents, te cost of poducing in a egion and te cost of sipping to ote egions. An additional assuption ade is tat te C te in te tanspot cost coponent is constant wit espect to any output flow. Tis is necessay to keep equation (8) sepaable. C s, = 0 y s, Wit te cost function specified, te next step is to deive te deand function fo a secto in a location, in ode to be able to close te odel. Final consuption and deand As is te case fo all te vetically linked odels, aggegate deand in eac location is te su of final consuption by wokes and inteediate consuptions by ote poduces. Te utility function fo wokes in egion is Cobb-Douglas, wit te final consuption of te t good in egion. H ( ) U = Q µ wit µ = (9) = Q 7

8 All te incoe flows, including etuns on capital and pofits, ae spent as final consuption. We assue tat entepeneus ave te sae utility function as wokes, so tat tei incoes can be pooled in te labou constaint, po ata of te location of anufactuing and agicultual fis. In two egions, tis gives κ = 0.5. Te budget constaint is: given by: H R H pq = wl + κ ( ck + π ) (0) = = = Given te exogenously fixed aounts of labou and land, te wage and ents ae w = 2 = C L β () c a a C α = (2) K Solving te utility axiisation poble gives te optial deand fo eac good: Q wl = + κ + π µ R H ( ck ) = = p (3) Te second souce of deand is inteediate consuption fo ote sectos. As explained above, in ode fo te odel to close, it is ipotant to note tat te i, elasticity used is te CRS vesion ε. Te inteediate deand of industy i inputs in fo industy in location is x C i, i, ε = i p Cobining te two souces of deand yields te aggegate deand fo te t industy s good in te t location. Futeoe, in ode fo te odel to close, te output tat elts away duing tanspot ust also be e-intoduced as final 8

9 consuption. Because of te elatively sall size of te tanspot cost and in ode to iniise te ipact on te odel s solution, te total tanspot cost fed back into te egions, po-ata of te sae of total output ϖ te egion poduces. Final deand is teefoe: D w L ck C ( ) R H H i, i = p + κ + π µ + ε = = i= H R R s d + ϖ y, sc ( e ) τ = = s= In ode to siplify te notation, tis is e-witten as: D Φ = (4) p Wee Φ, te egional expenditue in on secto, is te te in backets in te pevious equation. Equilibiu and Counot copetition Te equilibiu condition fo te odel is tat fo eac industy te su of all te output flows y s, fo vaious locations s and diected to is to be equal to deand in : D R = y (5) s= s, In ode to deteine te Counot solution fo all te poduces, we equie te pofit equation fo te t industy in location. It is ipotant to note tat tanspot costs ae taken into account ee as a pat of poduction costs. Fo equations (8) and (3), te total pofit fo a poduce of te t industy located in can be witten as: R R d py s, s C y, sc e τ s= s= s ( ) (6) Π = 9

10 Unde te Counot copetition faewok te following fist ode condition ust apply ove all taget egions s: s d ys, ps C e s, s, s, s τ ( ) dπ dp dc = + = 0 dy dy dy As is te case in te Counot liteatue and Cobes and Lafoucade (200), te poduce assues tat te quantities supplied by ote copetitos to te sae location eain constant. Te conjectual vaiation pat of te deivative is: dp dp dd dy s s s = s, dds dys, Tis assuption ensues tat dds = dys,, and teefoe: dd dy s s, =. Te conjectual vaiation ten siply becoes: dp dy s s, dp = dd s s As a esult, a siple vesion of te fist ode condition can be witten out: d Π dp = s s d y + s, p τ s Ce = 0 (7) dy dd s, s Using tis solution, as well as te picing elation in equilibiu (4) and te equilibiu condition (5), one can find te equilibiu solutions fo output flows and pice on eac aket, sown below, in equations (8) and (9). y s τ ( p C e ) Φ = (8) d s s s, 2 ( ps ) p s = R = C e τ R s d (9) 0

11 Additionally, fis in a given secto and location will only poduce fo a egional aket s if te pofits tey ake on tat flow ae positive. If tey ae not, te poduce dops out of tat paticula aket, and y, = 0. Te condition can easily be deived fo te pofit equation (6): s s ( ) d p C e τ +Ψ 0 (20) s Equations (3), (4), (6), (), (2), (4), (6), (8), (9) and (20) above descibe te equilibiu of te econoy. Equations (3), (4), (6), (), (2), (4), and (6) enable us to deteine te pice inputs and te poduction costs in eac location. Using te pice and output equations, (8) and (9), as well as te pofitability constaint (20) we can deteine ten te flow of goods fo one location to anote. Suing tose flows yields te equilibiu output pe location and secto. 3. Model beaviou at syetic equilibiu Te syetic equilibiu defined by te syste of equations sown in above was siulated using te paaete values sown below in Table. Table : Paaete values fo te Counot 2 2 test Vaiable Sybol Value Elasticity of agicultual output w..t land α 0.45 Elasticity of agicultual output w..t labou β a 0.55 Elasticity of anufactuing output w..t labou β 0.6 Elasticity of anufactuing output w..t inteediate input ε 0.4 Elasticity of utility w..t te agicultual good µ a 0.5 Elasticity of utility w..t te anufactuing good µ 0.5 Labou (pe egion) N 2 Land (pe egion) K In ode to povide a bette undestanding of te effect of including IRS in te odel, a bencak siulation was un, wit a CRS anufactuing secto. Tis is ten

12 copaed to siulations wit an anufactuing IRS paaete of.05 and. At tis point, it is ipotant to point out a tecnical issue elating to te integation of IRS in anufactuing. IRS ae added to te odel by inceasing te size of te poduction elasticities by a facto Ψ. Tis ceates a poble wit TFP, as canging te intensities of inputs canges te units in wic TFP is easued. Inceasing Ψ at a given level of tanspot cost equies te e-calculation of TFP in te new units. In an applied siulation, wee tee is data available on inputs and outputs, tis ecalibation of TFP fo a ige level of IRS is just a pat of te oveall calibation of te odel. Howeve, in te abstact siulations pesented tee is no coect level of outputs and inputs to efe to as a calibation point. Ex ante, all levels of output ae equally as valid as a efeence point. Futeoe, in ode to be able to asses te ipact of IRS in te odel, we ust be able to copae te diffeence between te CRS and IRS cases fo te entie tanspot cost ange, wic is not possible if TFP is ecalibated fo evey level of tanspot cost. Manufactuing TFP will teefoe be assued to be unitay, fo all levels of tanspot cost and etuns to scale. Te consequence of tis is tat te absolute levels of te CRS and IRS cuves in te following figues cannot be copaed diectly, and only te elative slopes ae eaningful. Table 2 sows, oweve, tat wen coecting fo te cange in TFP at diffeent levels of tanspot costs and etuns to scale, output inceases as expected wit te level of etuns to scale. Figue sows tat as one would expect, te syetic wage ate inceases as tanspot costs fall. Moe open akets encouage oe anufactuing tade and ige anufactuing output, esulting on inceased copetition between bot sectos fo te fixed labou supply. Futeoe, te pesence of IRS in te anufactuing secto does not eally ipact te labou aket. Te cange in levels is seen in te figue is due to te unitay TFP assuption, and te slopes of te cuves ae elatively uncanged. 2

13 Figue : Syetic Wage Rate, CRS vs IRS Figue 2: Eployent in Agicultue and Manufactuing, CRS vs IRS Te incease in wage as tanspot costs fall is explained by a cange te stuctue of eployent. Figue 2 sows tat as te sipping costs incued by anufactuing ae educed, labou is tansfeed fo agicultual eployent to anufactuing. Tis is because te expanding anufactuing secto copetes wit 3

14 agicultue and daws labou fo it. Te incease in wages seen in Figue is just te tace of tis sectoal eallocation of labou. Howeve, Figues and 2 sow tat te pesence of IRS does not cange significantly te elations between agicultue and te anufactuing secto in te labou aket. As fo te CRS case, as tanspot costs go down, anufactuing as a geate deand fo labou, and is teefoe able to squeeze oe labou out of te agicultual secto. Te igtwad sift of te cuves is due te fact tat TFP is not adjusted, but te intesectoal adjustent does not appea to appen at an inceased ate unde IRS. Tis sows tat witin tis odel IRS do not ave a ajo ipact on te way te labou aket functions. Te pictue is diffeent if ones look at output. Figues 3 and 4 sow te evolution of te output of bot sectos as a function of tanspot costs. As expected, te lowe te tanspot cost, te ige te level of anufactuing output, because of te gadual opening of lage akets fo anufactuing goods tat esult fo lowe tanspot costs. Te CRS dop in agicultual output wit tanspot costs is a consequence of te inceased copetition on te labou aket and te ise in wages visible in Figue. As one can see fo figues 3 and 4, at te syetic equilibiu, te agicultual output is geneally ige tan te anufactuing output, Te eason beind tis te assuption tat bot agicultual and anufactuing TFPs ae unitay and te fact tat te agicultual poduction as te exclusive use of land, weeas anufactuing as to copete wit final consues fo its inteediate inputs. As expected, te pesence of IRS in te anufactuing secto does not diectly affect te poduction stuctue of agicultue, wic is still assued to be CRS. Te appaent dop in output is due to te upwad sift of wages seen in Figue, but as explained above, ost of te sift in te level wages is itself due to te absence of 4

15 ecalibation of TFP. Te ipotant esult is tat te slope of te agicultual output cuve itself is uncanged. Figue 3: Agicultual Output, CRS vs IRS Figue 4: Manufactuing Output, CRS vs IRS Figue 4, sows tat IRS do ave an effect on te slopes of te cuves. Fo a given eduction in tanspot costs, te incease in anufactuing output is stonge te ige te level of etuns to scale. IRS inceases te slope of te output cuve toug 5

16 te cost eduction effect visible in equation 6, and te ige input intensities of te poduction function. As explained peviously, te appaent eduction in output is due to TFP not being adjusted. Wit an appopiate ecalibation of TFP, tis will lead to inceases in output as tanspot costs dop. Tis can be seen in Table 2 Figue 5 sows te pofits of bot sectos decline as tanspot costs dop. In agicultue, it is because wages incease as copetition fo labou wit an inceasingly lage anufactuing secto leading to te ising labou costs sown in Figues and 2. Tis is also tue fo anufactuing, oweve anufactuing is also faced wit te fact tat as tanspot costs dop, te copetiti, on between te two poduces fo bot egional akets inceases, diving down te pice of te anufactuing good. At syetic equilibiu, anufactuing pofits ae squeezed fo bot diections: te pice of its output falls, as can be seen in below Figue 7, wile te labou costs ise. Figue 5: Agicultual and Manufactuing Pofits, CRS vs IRS Figue 5 also sows tat te effect of IRS on pofits is coplex. Again allowing fo te diffeences in intecepts, one can see tat te equilibiu pats of agicultual 6

17 pofits ae not affected, as was te case fo agicultual output. Tee does oweve see to be an effect of IRS on anufactuing pofits, as te cuve sees to flatten out at low levels of tanspot cost. Te cost-educing effect of IRS on te poduction stuctue of te anufactuing secto is confied by te analysis of te aginal cost of poduction in Figue 6. Only te anufactuing aginal cost is sown ee, te agicultual aginal cost is fixed and equal to 0.5 egadless of tanspot costs and etuns to scale 2. Te fact tat te aginal cost of CRS anufactuing dops wit tanspot costs even toug labou costs ae ising. is due to te input-output stuctue of te anufactuing secto, As sown in Figue 7, te pice of te anufactuing good falls wit tanspot costs, and as one can see fo Figue, it falls at a faste ate tan te ate of incease of wages, teefoe causing te aginal cost to fall sligtly. Te size of tis effect teefoe depends on te elative saes of labou and inteediate inputs in te poduction function. Te intoduction of IRS into anufactuing, oweve, as a clea effect. In Figue 4 te slope of te output cuve becoes steepe, sowing tat as tanspot costs fall and anufactuing output inceases, te pesence of IRS educes poduction costs copaed to te CRS case. At ig levels of tanspot costs, output is low and as a esult, te aginal cost will be elatively ig. At low levels of tanspot costs output is ige, and te cost-educing effect of IRS inceases, causing eductions in aginal cost tat ae ige tan te eductions seen unde CRS. 2 Tis is linked to te fact tat te agicultual good is te nueaie, and is explained in oe detail in te appendix 7

18 Figue 6: Maginal cost of poduction in Manufactuing, CRS vs IRS Figue 7: Pice of te Manufactuing Good, CRS vs IRS Figue 7 indicates tat te IRS cost-eduction seen in aginal costs feeds toug to pice of te anufactuing good. Equation (7) sows tat unde Counot copetition te anufactuing pice in a location is just a weigted aveage of te aginal costs of te supplies to tat location. It is teefoe expected tat te eductions in aginal costs sould be eflected in te pice. 8

19 Having exained te syetic equilibiu of te syste unde of CRS and IRS, it is ipotant to addess te fact tat canging te intensities of inputs canges te units in wic TFP is easued. As peviously explained, tis is not a poble fo applied wok, wee data on existing output and inputs povides a efeence point fo calibation. Howeve, in te siulation caied out ee any level of tanspot cost can povide a efeence point. Fo a given level of tanspot costs, TFP is calibated by dividing te CRS output at tat level of tanspot costs by te Cobb-Douglas poduction function of CRS inputs. Ipotantly te values of te elasticities used in tis Cobb- Douglas ave to be in line wit te elevant aount of IRS desied 3. Te IRS odel can ten be siulated wit a value of TFP consistent wit te level of IRS cosen. Te TFP ecalibation fo selected levels of tanspot cost τ and te esulting outputs ae sown below in Table 2. Recalibated anufactuing TFP Table 2: Recalibated IRS TFPs and outputs Recalibated anufactuing output τ CRS IRS.05 IRS. τ CRS IRS.05 IRS Table 2 confis tat fo eac level of tanspot costs, ige levels of etuns to scale lead to ige anufactuing output, as one would expect fo an IRS odel. 3 In ote wods, A = y β' ( L ) ( xt ) wee A is TFP, y is output, L is labou and x is inteediate inputs. Siply inseting te elevant IRS values fo β and ε allows fo te calculation of te TFP fo tat level of IRS. ε' 9

20 4. Stability of te syetic equilibiu Because we ae attepting to ceck te aggloeative effect of explicit IRS, we assue only one fi pe egion is tis odel. We define aggloeation as a situation wee a single fi, epesenting a egion, inceases te size of its output wit espect to ote egions. Te nube of fis teefoe does not cange but te output of te fi in te aggloeated egion goes up elative to ote egions. Altoug te uppe bound fo te nube of poduces is te nube of egions, tei nube can be educed by te pofitability constaint. If a poduce finds iself in a situation wee all output flows ae unpofitable, te pofitability condition of te odel, equation (8), will foce i out of poduction. Total aggloeation is teefoe a situation in wic only one egion poduces, and it is unpofitable fo ote egions to do so. Unfotunately, poduces exit te aket discetely, and tei output dops staigt fo a positive nube to zeo. Tis ceates jups in te pice, output and pofit equations, and teefoe discontinuities in any analytical function tat would illustate aggloeation. It is teefoe difficult to povide a coplete analytical desciption of long un equilibia. Howeve, fo any equilibiu situation wee te nube of copetitos on a aket is set, eite at syetic equilibiu, o at any point between te discontinuities, it is possible to deteine analytical conditions tat deteine wete o not te equilibiu is stable, and if aggloeation is pofitable o not. Tis is not a coplete desciption of all te equilibia, but a set of pocedues tat allow fo te analysis of te local stability of a paticula equilibiu. Howeve, because we ae only sowing te conditions unde wic te syetic equilibiu is unstable, tis is enoug. 20

21 Tis appoac ests on te copaison of cange in pofits of bot poduces as a esult in a cange in output of one of te poduces, in ote wods dπ dy and dπ s dy. If tese fist ode conditions ae negative fo bot poduces, neite of te ave an incentive to incease tei output. If at least one is positive, ten it is possible fo at least one of te poduces to incease pofits by inceasing output, and te syste oves away fo te syetic equilibiu. Te easiest way of doing so analytically is to calculate te cange in pofits in a egion in esponse to an incease of te oe flow y,. In tis case, te tanspot cost te is equal to one, and te pofit on te flow is: ( ) Π = y p C Ψ (2),, Te te in backets is te pofitability of te flow, te aount of pofit ade on eac unit of output sold in te oe aket. Totally diffeentiating Π, wit espect to te oe output flow yields: ( ) ( ) dπ = dy p C Ψ + dp Ψ dc y (22),,, Te fist te sows te incease in pofits due to te exta output at ex ante pofitability. Tis is educed by te second te, wic sows te ipact of te incease in output on pofitability itself. Pedictably, te vaiations in pofitability boil down to a cobination of pice and aginal cost vaiation. Te total diffeential of te pice equation (9) is: τ e dp = dc + dc R R s Tis equation elation aleady eveals te basic ecanis beind aggloeation. Fo te pupose of siplifying te calculations and undelining te causes of aggloeation, we sall assue tat te ote poduce s output is uncanged. 2

22 Tis is not necessaily te case, and te siulated stability conditions sown below will take tis into account and look at te total effect 4. Tis assuption just claifies te analytical insigts. dp =Ω dc wit Ω= R (23) Ω, te atio between dp and dc, is basically te aginal ak-up between pices and aginal cost, and indicates te aket powe of any individual poduce. In te geneal case, te oe poduces R tee ae on a aket, te salle Ω will be. Looking at (22), if te aginal cost of te poduce in inceases, te pice in is oe aket will only incease by a faction Ω of tat, and te pofitability will fall. If on te ote and te aginal cost falls, pice will only fall by a popotion Ω of tat aount, and pofitability inceases. Hee, wit two egions, Ω is siply equal to one All tat is left to do is to sow unde wat conditions te aginal cost falls o ises. Totally diffeentiating te equation (6) yields: dc Ψ dy dχ = C + Ψ y χ (24) We can see ee tat tee is aleady a ole played by IRS in causing a fall o incease of aginal cost wit espect to output. One can see, tat unde CRS, wee Ψ =, te vaiation in dc collapses down to dχ, and te level of output plays no ole. Howeve unde IRS, inceasing output will educe te size of te aginal cost, ote tings equal. One would expect tis to be counte-acted by te total diffeential of te fixed coponent of aginal cost χ wic is a positive function of te cost of 4 Te siulated stability conditions in Figues 8 and 9 ae obtained using te iplicit function etodology developed in Bade (2006). Tese ae based on te nueical Jacobian of te syste of equations and teefoe contain all te infoation on patial deivatives. 22

23 inputs to poduction. In ode to get a bette desciption of tese potential inceases, oweve, we equie te total diffeential of dχ. Fo equation (4), one gets: dw d χ = χ β ' + ε ' w dp p (25) (24) gives: Replacing (25) in te total diffeential fo anufactuing aginal cost equation dc Ψ dy dw dp = C + β' + ε ' Ψ y w p (26) Replacing (23) in (22) and (24) to eliinate dp : ( ) ( ) dπ = p C Ψ dy + Ω Ψ y dc (27),,, dc C Ψ dy dw = + β ' C Ψ y w ε ' Ω p (28) Replacing in (28) in (27) to eliinate dc and assuing tat dy dy, = gives te stability condition fo te oe flow: Ω Ψ Ψ dy dπ = p C Ψ dy + y C +, ( ), ' β C Ψ y w ε ' Ω p dw (29) Te fist pat of equation (29), te cange in pofits due to te cange in output at ex ante pofitability is uncanged fo (22). Teefoe, te ipotant tes of equation (29) ae te tes in backets containing dy and dw, and te ultiplicative atio in font of it, wic deteine te evolution of pofitability. Witin te backets, an incease in output dy as a diect and negative effect unde IRS. Tis coesponds to te cost-educing effect of IRS on aginal costs above. One would expect dw dy to be positive due to te exta pessue on te labou aket. Fo siplicity, tis effect is not explicitly included ee, but te siulations in section 3 sow tat te sign is positive. A geat deal of te stability analysis teefoe ests on te elative stengt of tese two 23

24 effects and wete, fo a given incease in output, te IRS-induced eductions in aginal cost bougt by te exta output outweig te ige labou cost incued. Te atio in font of te te in backets is a ultiplie wic accounts fo te negative feedback effect on te pofit equation of a ise in pices. Te ultiplie exists because Ω, ε and te aginal cost to pice atio ae all individually salle tan one. Te nueato Ω Ψ is negative, given tat te ak up Ω is salle o equal to one, and tat te degee of etuns to scale Ψ will be equal to o geate tan one. Tis ultiplie stes fo te vetical linkages in te odel 5. Because of tis inteediate consuption, if fo any eason te anufactuing pice inceases, poduction costs (and aginal costs), will be pused up, tus pusing pices up even fute. Tis effect also woks in evese wit eductions in pice. Tis ultiplies any cange in pofitability due to wage inceases o IRS output effects. Figue 8: Stability of Syetic equilibiu, CRS 5 Wic is wy te ultiplie te depends on te elasticity of output w..t inteediate consuption ε and te ak-up Ω 24

25 Equation (29) confis te ipotance of IRS in poviding incentives to aggloeate. Unde constant etuns to scale, wit Ψ equal to one, tee is no diect effect of an incease in output dy on pofitability. Te only effect, toug te incease in labou costs dw dy, is negative. Unde CRS, output inceases can only lead to eductions in pofitability. Te fist alf of equation (29) does sow an incease, but it will nealy always be salle tan te dop in pofitability, because of te ultiplie effect in te pofitability te. Figue 8 above sows tat witout te cost-educing effects of IRS, deviating fo te syetic equilibiu is detiental to pofits, eaning tat te syetic equilibiu is always stable. Unde IRS, oweve, pofitability does not necessaily dop. Equation (29) sows tat te cost-educing effect of te exta output inceases can diectly itigate te incease in wages. Te nueical stability conditions visible in Figue 9, in appendix, confi te analytical indications given above. As te level of inceasing etuns to scale inceases, te oe effect, followed by te foeign effect becoe positive, indicating tat it becoes pofitable fo eite te oe o te foeign poduces, o bot, to incease tei output, tus oving away fo te syetic equilibiu. In doing so, tey will geneate a vaiety of aggloeation pattens wic depend on te level of IRS. In te case wee bot effects ae negative coesponds to a stable syetic equilibiu, uc like te CRS case in Figue 8. If, oweve te own effect is positive and te foeign effect is negative, as is te case in te fist five diagas, ten an incease in oe output inceases te incentive to poduce in te oe aket and educes te incentive to poduce in te foeign aket. Tis sould lead to total aggloeation in te oe aket, allowing fo a discontinuity wen te foeign poduce is caugt by te pofitability constaint and exits te aket. In te case wee 25

26 bot effects ae positive but te own one is lage, bot poduces ave an incentive to incease output and ove away fo equilibiu. 5. Conclusion Tis pape set to investigate te folk teoe of spatial econoics and investigate ow explicit IRS can influence poduction coices and aggloeation. Altoug te intoduction of IRS in a Counot odel ceates analytical pobles wit te saes of expenditue and allocations of poduction costs, we sow tat tese can be woked aound. Siulations and analytical investigations of te odel sow tat if te agicultue and anufactuing sectos ae bot CRS, ten te syetic equilibiu is stable fo all tanspot costs. Tis coesponds to dispesion of activity acoss egions, eaning tat aggloeation of poduction in one egion is not possible. If, oweve, IRS ae applied to te anufactuing secto ten te syetic equilibiu can becoe unstable. Witin te settings of te odel, tis eans tat it becoes pofitable fo one egion to incease its anufactuing output wit espect to te ote egion, tus inceasing its sae of total output. Te ipotance of tis esult is te explicit confiation of te folk teoe specified by Fujita and Tisse (996). Because we assue tee is a single poduce pe egion, te discontinuities intoduced by pofitability condition ake it difficult to povide an analytical desciption of te aggloeation pocess. Tis is wy we instead concentate on te depatue fo equilibiu, as unde te settings used in te odel it is enoug to pove tat te syetic equilibiu is unstable. Howeve, tis does point out te ipotance of fi enty in poviding te conditions fo a full undestanding of aggloeation. 26

27 Te odel descibed is not intended to be a substitute fo te Counot aggloeation odels entioned in te intoduction. Rate, it is intended as a copleent, wic accounts fo an exta aggloeation foce not linked to fi enty, but to te possible existence IRS in key sectos of te econoy. Fixing te nube of poduces just allows us to undeline te existence of tis aggloeation foce. A diection wotwile exaining would teefoe be to elax tis assuption of a fixed nube of poduces and cobine bot aggloeation foces in a single faewok. 27

28 Appendix Figue 9 - Stability of Syetic equilibiu, IRS 28

29 Appendix 2 Poof of te stability of te agicultual aginal cost Analytically, one can deive te stability of te agicultual aginal cost fo te pice equation (7) in section. Because te syste is at syetic equilibiu, all te vaiables ae equal ove egions. Futeoe, te agicultual good is te a a a a a a nueaie. Teefoe, p = ps = p* = and χ = χs = χ*. Replacing in te pice equation fo te agicultual secto, wit R = 2 gives te pice at equilibiu. a p s = 2 = χ a R a a siplifies down to p* = 2χ* Tis explains te value of 0.5 fo te aginal cost of te agicultual secto, but a does not fully explain wy it is constant, i.e. wy dχ* dτ = 0. To sow tis, one ust a a eplace p * and χ * in te output equation (6). Doing so gives y a * Φ = 4χ a * a * Wit an agicultual cost equation equal to: C Φ = χ y = 4 a a a * * * a * a Te agicultual cost function does not depend on te aginal cost χ *, and teefoe vaiations in agicultual costs do not depend on χ * a 29

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