Which Firms are Left in the Periphery? - Spatial Sorting of Heterogeneous Firms with Scale Economies in Transportation

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1 Whch Frms are Left n the Perphery? - Spatal Sortng of Heterogeneous Frms wth Scale Economes n Transportaton Rkard Forsld Toshhro Okubo February 2013 Abstract Ths paper ntroduces scale economes or densty economes n transportaton n a trade and geography model wth heterogeneous frms. Ths relatvely small change to the standard model produces a new pattern of spatal sortng among frms. Contrary to the exstng lterature, our model produces the result that frms of ntermedate productvty relocate to the large core regon, whereas hgh and low productvty frms reman n the perphery. Trade lberalsaton leads to a gradual relocaton to the core, wth the most productve frms remanng n the perphery. JEL Classfcaton: F12, F15 Keywords: heterogeneous frms, transportaton costs, scale economes Fnancal support from the The Swedsh Research Councl and The Handelsbanken Research Foundaton s gratefully acknowledged by Forsld. Stockholm nversty and CEPR; emal: rf@ne.su.se. Kobe nversty; emal: okubo@reb.kobe-u.ac.jp. 1

2 1 Introducton Ths paper ntroduces scale economes or densty economes n transportaton n a trade and geography model wth heterogeneous frms. It s shown how densty economes n transportaton dramatcally change the locaton pattern of frms. Baldwn and Okubo (2006) show how the most productve frms tend to agglomerate n the large central regon, when heterogeneous frms are added to a trade and geography framework. 1 Thus, ther model produces a spatal sortng pattern wth the most productve frms n the centre and the least productve frms n the perphery. The sortng pattern s amplfed by trade lberalsaton. Lower trade costs nduce more frms to relocate to the central regon, and t s always the most productve frm n the perphery that has the strongest ncentve to relocate towards the centre. The result by Baldwn and Okubo (2006) underscores the concerns that small regons or countres could end up as losers when economc ntegraton leads to an agglomeraton of frms to large central regons. Not only does the perphery lose ndustry n general, t s the most productve frms that leave. Our paper shows how scale economes n transportaton can overturn the result by Baldwn and Okubo (2006). The generc case n our model s that frms of ntermedate productvty have the strongest ncentves to agglomerate to the larger regon, and t s even the case that the least productve frms relocate to the core for strong enough scale economes n transportaton. Thus, our model produces the opposte spatal sortng pattern as compared to Baldwn and Okubo. The reason for our result s that transportaton costs become relatvely less mportant for the large and most productve frms n our model, snce they obtan low freght rates because of ther large shpments. Trade lberalsaton accentuates the spatal sortng pattern, and the last frm to leave the perphery s the most productve one. Our results mply that dfferent degrees of scale economes among ndustral sectors lead to dfferent spatal sortng patterns among the sectors. Ths s a qualfcaton for the well-known result from the trade and geography lterature that dfferent sectoral levels of trade costs wll lead to dfferent degrees of agglomeraton among sectors. The trade and economc geography lterature (see e.g. Fujta et al. 1999, Baldwn et al. 2003, Fujta and Thsse, 2002) treats trade costs as a constant parameter, whch may be proportonal to geographcal dstance or other. However, as ponted out by McCann (2005), ths s at odds wth wde-rangng emprcal work on transport rate structures n regonal economcs whch almost unversally has found economes of dstance and scale n transportaton. There are a few trade and geography papers that model transport costs n an explct manner. Behrens et al. (2006) ntroduce densty economes n transportaton n a trade and geography model. They show how ths may gve rse to multple equlbra and a catastrophc agglomeraton of ndustry. Behrens and Pcard (2011) study the locaton choce of frms when 1 Smlar results are found by Okubo et al. (2010) usng the lnear model of Ottavano et al. (2002) and by Sato et al. (2010) and Forsld and Okubo (2012) usng a three-country model. Also the vertcal lnkage model produces smlar results (Okubo 2009). 2

3 transportaton costs are endogenous. They show how the back-haul problem ncreases the freght rates on exports from the central country, and that ths dampens the tendences for agglomeraton. However, none of these papers analyse spatal sortng of frms, as they work wth models wth homogenous frms. 1.1 Scale economes n transportaton We wll here relate frms varyng producton volumes to dfferences n transportaton costs. It s qute natural that freght rates per unt declne wth the sze of a shpment due to scale economes n transportaton. Scale economes n transportaton are well establshed n regonal economcs (see e.g. McCann 2005) and e.g. Hummels and Skba (2004) fnd that a 10 percent ncrease n product weght/value leads to a 4-6 percent ncrease n shppng costs usng.s. data. 2 However, almost all types of transportaton such as trucks, arfreght and shps have freght rates that are declnng n the volume transported. It s also the case that the transportaton costs reman substantal n spte of the globalzaton process durng the last decades. In fact, the tramp freght rates have roughly remaned constant snce the 1950 s, whle the arfreght rates have fallen (Hummels 2007), and Anderson and van Wncoop (2004) fnd that the total trade costs n hgh ncome countres amount to an ad valorem tax equvalent to 170%. 2 Model Here, we ntroduce scale economes n transportaton n the Baldwn and Okubo (2006) model. Ths model s based on the Meltz (2003) heterogeneous frms model combned wth the footloose captal new economc geography model by Martn and Rogers (1995). Our model nvolves frm heterogenety n transport costs as well as n labour productvty. In essence, productve frms export more and ther trade costs per unt are therefore lower than those of less productve frms due to scale economes n transportaton. 2.1 Bascs There are two regons wth an asymmetrc populaton (market sze). Regon j s the larger core regon and Regon k s a smaller regon. There are two types of factors of producton, captal and labour, and each country has the same proporton of captal to labour. That s, regons are dentcal except for sze and regon j has a share s j of both factors of producton. Captal, whch s sector specfc, can move between regons but captal owners do not. Labour can move freely between sectors but are mmoble between regons. A homogeneous good s produced wth a constant-returns technology only usng labour. Dfferentated manufactures are produced wth ncreasng-returns technologes usng both captal and labour. 2 Another example s Mor and Nshkm (2002) who fnd that the cost of transportaton by shp declnes by 0.31% when the shppng volume ncreases by 1%. 3

4 All ndvduals have the utlty functon = C µ M C1 µ A, where C M = l Ψ c (σ 1)/σ l dl σ/(σ 1) (1) where µ (0, 1), σ > 1 are constants, and Ψ s the set of consumed varety. C M s a consumpton ndex of manufacturng goods and C A s consumpton of the homogeneous good. c l s the amount consumed of varety l. Regon subscrpts are suppressed when possble for ease of notaton. Each consumer spends a share µ of hs ncome on manufactures. The total demand for a domestcally produced varety s x = l Ψ p σ p 1 σ l dl where p l s the prce of varety l and Y s ncome n the regon. µy, (2) Ownershp of captal s assumed to be fully regonally dversfed; that s, f one regon owns X percent of the world captal stock, t wll own X percent of the captal n each regon. Therefore, the ncome of each regon s constant and ndependent of the locaton of captal. Total expendture equals total factor ncome. Frms fxed factor of producton s captal and the varable factor s labour. The return to captal therefore equals frms operatng proft n equlbrum. Thus, the total equlbrum expendtures can be wrtten E = wl + rk = wl + µe/σ. Wthout loss of generalty, we choose unts so that L 1, whch gves E = 1 1 µ/σ. The ncome of regon j s equal to ts share of total expendtures gven by σ Y j = s j E = s j σ µ. (3) Y j s thus constant rrespectve of the locaton of captal;.e. also out of long-run equlbrum. Turnng to the supply sde, the homogeneous good sector s a constant returns and perfect competton sector. The unt factor requrement of the homogeneous good s one unt of labour. The good s freely traded and snce t s also chosen as the numerare, we have p A = w = 1, (4) where w s the wage of workers n all regons. In the producton of dfferentated goods, frms have a frm-specfc unt labour nput coeffcent (a) and uses one unt of captal, as n the standard footloose captal model. There s a fxed world captal endowment, mplyng that the world mass of frms, N W, s constant, whlst nternatonal captal moblty allows frms to move between regons. We normalse the world mass of dfferentated frms to one, N W 1. The total costs for frm are specfed as T C = π + a x, (5) 4

5 where π s return to captal. The varable cost conssts of labour. Importantly, frms are heterogeneous and ther frm-specfc margnal producton costs a are dstrbuted accordng to the cumulatve dstrbuton functon G(a). Geographcal dstance s represented by trade costs. Shppng the manufactured good nvolves a frctonal trade cost of the ceberg form: for one unt of good from regon j to arrve n regon k, τ jk > 1 unts must be shpped. The trade costs are symmetrc between regons τ jk = τ j, k. Proft maxmsaton by manufacturng frms leads to a constant mark-up over margnal cost and the export prce s p τ. p = σ σ 1 a. (6) We assume that there are scale economes n transportaton, mplyng that the freght rate falls wth the quantty shpped. Frms are heterogeneous n labour productvty, a, and productve frms produce more and export more. Productve frms therefore have lower freght rates. To preserve the analytcal tractablty of the model, we assume the followng smple relatonshp between transport costs and a: τ = ta γ, (7) where t 1 and γ 0 are parameters. t represents the fundamental level of the ceberg trade cost, whereas γ s the degree of scale economes of transportaton. A larger γ mples larger scale economes. We assume that γ > 0, mplyng that dτ da > 0. Note that our specfcaton generates the standard case of a constant trade cost (τ = t) when γ = 0. Frm heterogenety n labour requrements, a, s probablstcally allocated among frms. To analytcally solve the model, we follow Forsld and Okubo (2012) and assume the followng cumulatve densty functon of a: where ρ s a shape parameter and a 0 s a scalng factor. G(a) = aρ a ρ a ρ 0 aρ (8) We assume the dstrbuton to be truncated at a, whch s exogenously gven, meanng that a a a 0. Ths mples that the productvty of frms s bounded. Wthout loss of generalty, we can normalse a 0 1. For convenence, the lower bound for the margnal cost s specfed as a t 1 γ, whch mples that τ = 1 when a = a Short-run equlbrum In the short-run equlbrum, the allocaton of captal n each regon s taken to be fxed. s j denotes the share of captal and the number (mass) of frms n regon j snce one unt of captal corresponds to one frm. 3 The assumpton that the most productve frm has zero trade costs s convenent when solvng the model, and t rules out negatve trade costs. 5

6 The return to captal of a frm n regon j s the frm s operatng proft, ( ) π j (a ) = a1 σ (σ µ) µ sj 1 s j + φ, (9) j k where φ τ 1 σ, the rght-hand sde follows from the demand functons n (2), and j s j 1 a 1 a 1 σ dg(a) + (1 s j ) a φ a 1 σ dg(a). (10) The object φ, rangng between 0 and 1, stands for frm "free-ness" of trade between regon j and k (0 s autarky and 1 s zero trade costs). It s assumed that the labour stock s suff cently large so that the homogenous sector, whch pns down the wage, s actve n all regons. Furthermore, usng (7) and defnng ϕ t 1 σ [0, 1] gves, φ = τ 1 σ = ϕa γ(1 σ), where s a frm-specfc component and ϕ s a measure of the overall degree of trade freeness. a γ(1 σ) Consder now what would happen f frms were allowed to move between regons. Frms choose locaton n ther search of the hghest captal returns. sng (9), a frm wll move from Regon j to k when π j (a ) π k (a ) χ = ( a1 σ (σ µ) (1 φ sj )µ 1 s ) j χ > 0, (11) j k where χ s a per-unt of captal fxed relocaton cost and trade costs are assumed to be symmetrc, τ jk = τ kj. 2.3 Relocaton tendences Before movng to the full long-run equlbrum of the model, we consder the relocaton ncentves faced by frms startng out from the ntal equlbrum. sng (7), the proft dfferental s gven by π j (a ) π k (a ) χ = Dfferentatng the proft gap functon n terms of a gves: d(π(a ) π(a )) da ( µ (1 ϕaγ(1 σ) )a 1 σ sj 1 s ) j χ. (12) (σ µ) j k = (1 ϕ(1 + γ)a γ(1 σ) )a σ µ(σ 1) (σ µ) ( sj 1 s ) j. (13) j k From ths expresson, t appears that the ncentves of frms depend on the level of scale economes n transportaton, γ. Wth no scale economes, γ = 0, we have returned to the Baldwn and Okubo (2006) case as depcted n Fgure 1. Only the most productve frms (the low a frms) wll have ncentves to relocate n ths case. The most productve frms are better equpped to handle the strong competton n the large market, and only these frms wll expand the sales and ncrease the profts suff cently to overcome the relocaton cost. γ = 0 therefore tends to lead to spatal sortng wth the most productve frms sortng to the larger market. The relocaton tendences are very dfferent when there are scale economes n transportaton, γ > 0. Hgh scale economes and not too low trade costs (γ > 1 ϕ 1) wll lead to the opposte case 6

7 Proft gap (Regon j Regon k) Relocaton cost 0 1 Devaton tendency (ntal eq) Fgure 1: No scale economes n transportaton. of a postvely slopng proft dfferental curve as seen from (13). Fgure 2 llustrates ths case. Here, t s the least productve frms that have the strongest ncentves to relocate to the larger regon. Hgh-productvty frms enjoy low transportaton costs, and locaton therefore becomes relatvely unmportant for them. Low-productvty frms, n contrast, face hgh transportaton costs, and therefore they have large gans from movng. Thus, ths case tends to lead to spatal sortng wth hgh-productvty frms n the perphery. Fnally, the generc case, when scale economes n transportaton are not too large, s a hump-shaped proft dfferental curve as shown n Fgure 3. The exact condtons for ths case are stated n the followng proposton: P 1 The frms of ntermedate productvty have the strongest ncentves to start relocatng to the large regon f the condtons γ < 1 ϕ 1, and γ > 0 hold. Proof: Frst, when a = a t 1 γ, d(π(a ) π(a )) da > 0 always holds from (13). On the other hand, from (13), when a = 1, γ > 0, and γ < 1 ϕ 1, d(π(a ) π(a )) da < 0. Ths mples that the proft gap curve, π(a ) π(a ), must be hump-shaped. We wll assume that the suff cent condtons n Proposton 1 hold n the followng. Intutvely, the least productve frms have dff cultes survvng n the large regon, whle the most productve frms have such low transportaton costs that they do not fnd t proftable to pay the relocaton cost n order to move to the core. Therefore, t s the frms wth ntermedate productvty that have the strongest ncentves to move from the perphery to the core. Ths 7

8 Proft gap (Regon j Regon k) Relocaton cost 0 1 Devaton tendency (ntal eq) Fgure 2: Hgh scale economes n transportaton relocaton tendency stands n sharp contrast to Baldwn and Okubo (2006) and other studes n ths ven, where the most productve frms always have the largest ncentve for relocaton and are the frst movers to the large market. Havng analysed the relocaton tendences startng from a short-run equlbrum where captal s fxed, we now turn to the long-run equlbrum after frms and captal have relocated. 3 The Long-run Equlbrum Captal s fully moble between regons n the long run. To smplfy the analyss, we assume that there s a frm (captal) relocaton cost à la Baldwn and Okubo (2006) that s related to the mgraton pressure. More specfcally, we assume the cost of movng to be gven by: ( df (a ) χ = κ df (a ) L), (14) dt dt where κ s a constant, and a and a L are the two cut-off productvtes for the hump-shaped proft dfferental curve (see Fgure 3). The relocaton cost s hgh when many frms move at the same tme, but gradually declnes to zero as the mgraton pressure falls as we approach the equlbrum. The frm wth the most to gan from relocatng has the hghest wllngness to pay the cost for relocaton. As a consequence, frms relocate n order of productvty. The relocaton process s llustrated by Fgure 4. 8

9 Proft gap (Regon j Regon k) Relocaton cost 0 1 Devaton tendency (ntal eq) Fgure 3: The generc case of ntermedate scale economes n transportaton (γ < 1 ϕ 1). Frms move as long as the proft gap s larger than the relocaton cost, and the long-run equlbrum s therefore defned by: π j (a ) π k (a ) = ( µ (1 ϕaγ(1 σ) )a 1 σ sj 1 s ) j = 0 (15) (σ µ) j k The frst thng to note, from (15), s that the proft dfferental s zero for a = a t 1 γ. Ths mples that a L = a t 1 γ n a long-run equlbrum. a must also be solved. The long-run equlbrum value of a s defned by: ( µ π j (a ) π k (a ) = (1 ϕaγ(1 σ) )a 1 σ sj 1 s ) j = 0 (16) (σ µ) j k where j (a R ) s j 1 a 1 k (a R ) (1 s j ) 1 a 1 σ df (a) + φ(1 s j ) a 1 σ df (a) + φs j a 1 a 1 σ df (a) + (1 s j )(1 φ) a a L a 1 σ df (a) + (1 s j )(φ 1) a a 1 σ df (a),(17) a 1 σ df (a).(18) a R a a L Solvng (17) and (18) gves 9

10 Frm s sg(a) Regon j 1 s 0 a (1 s)g(a) Regon k 0 1 Fgure 4: The relocaton pattern wth two cut-offs. j = λ{s(1 a β ) + (1 s)(a β aβ L )} + θϕ{(1 s)(aα L a α ) + (1 s)(1 a α )}, (19) k = θϕ{s(1 a α ) + (1 s)(a α a α L)} + λ{(1 s)(a β L aβ ) + (1 s)(1 a β )}, (20) where α (γ + 1)(1 σ) + ρ, β 1 σ + ρ, λ ρ β a ρ, and θ ρ 0 aρ α (snce σ > 1) and thus λ < θ. Moreover, we note that βλ = αθ. The relevant soluton for a s determned by s j 1 s k condton that determnes the upper cut-off, a : 1 1 a ρ. 0 aρ Note that α < β = 0. Manpulaton gves the followng α β aβ ϕaα = α β β t γ ϕ s α 1 s t γ ϕ(1 s 1 s ) = 0 (21) Ths condton cannot be solved analytcally n general but we can use t to derve comparatve statstcs on a, as seen below. We next turn to the effect of trade lberalsaton. 3.1 Trade lberalsaton Although we cannot derve a closed form soluton for the cut-off, we can nvestgate the mpact of trade lberalsaton,.e. the mpact of an ncrease n ϕ on the cut-off, by mplct dfferentaton. sng that ϕ = t 1 σ n (21) gves where k 1 γ(σ 1). F α β aβ ϕaα = α β ϕkβ ϕ 1 kα s 1 s ϕ(1 s 1 s ) = 0 (22) 10

11 Proft gap φ 0 1 Fgure 5: The effect of trade lberalsaton Frst snce β > α and ϕ < 1. Next df = α da β βaβ 1 ϕαa α 1 = αa β 1 ϕαa α 1 > 0 (23) df = aα αkϕ kβ 1 + (1 kα)ϕ kα s 1 s + 1 2s 1 s All terms on the rght-hand sde of ths expresson are negatve assumng that kα > 1. Ths condton may be rewrtten as γ < σ 1 ρ σ+1, where we have used that α (1 + γ)(1 σ) + ρ. We assume ths to hold, whch mples that df Fnally, usng the mplct functon theorem (24) < 0. (25) df da = > 0. (26) df da Thus, we have shown that, as the trade costs fall, more frms move from Regon k to Regon j leavng a smaller and smaller set of very hgh productvty frms n the perphery: P 2 Trade lberalsaton always promotes frm relocaton from small to large regons. The most productve frms are the last movers. The effect of trade lberalsaton s llustrated by Fgure 5. 11

12 3.2 Welfare mpact of Trade lberalsaton sng the Cobb-Douglas utlty functon, socal welfare n our model n Regon j and Regon k s gven by j = shown n Appendx 5.1 s µ/(1 σ) j and k = 1 s µ/(1 σ) k. Our central nterest s j and k. Frst, as d j = d j da da + d j da L da L + d j > 0. (27) Thus, frm relocaton mproves socal welfare n Regon j. Then, the same nvestgaton s conducted for Regon k. The welfare effects are ambguous n ths case (see Appendx 5.1). d k = d k da da + d k da L da L + d k 0 The ndrect effect s always negatve,.e. d k da da + d k da L da L < 0, whereas the drect effect s always postve d k > 0. Ths ndcates that f trade lberalsaton does not relocate frms to any large extent,.e. because of large relocaton costs, welfare s lkely to rse. For low relocaton costs, the opposte case s lkely to occur. P 3 Trade lberalsaton always mproves welfare n the large regon, whle the welfare effect s ambguous for the small regon. 4 Concluson Ths paper ntroduces scale economes n transportaton n a trade and geography model wth heterogeneous frms. Ths relatvely small change to the standard set-up has large effects on the locaton pattern of frms. The semnal model by Baldwn and Okubo (2006), as well as other models n ths ven, predcts that the most productve frms wll sort themselves to the core regon. Our paper overturns ths result. Frms of ntermedate productvty typcally sort themselves to the core, whereas hgh and low productvty frms reman n the perphery. The result of Baldwn and Okubo (2006) only apples when the scale economes n transportaton are very low. We show how trade lberalsaton accentuates the sortng, gradually leavng only the most productve frms n the perphery. We also show that, whereas the central regon gans from trade lberalsaton, the perphery may lose. An mplcaton of our results s that the spatal sortng of frms n a sector wll depend on the degree of scale economes n transportaton n that sector. Ths has several polcy mplcatons. Suppose e.g. that the government prompts an ndustry to change from road transport to trans for envronmental reasons. Accordng to our model, ths wll affect the spatal sortng of the ndustry, f road transport and trans have dfferent degrees of scale economes. If scale economes n transportaton are more pronounced when t comes to ral transports, the large and most productve frms have less of an ncentve to move to the core after swtchng to ral 12

13 transport. Ths mples a larger number of large frms n the perphery and thus, a geographcally more dspersed producton. However, ths tends to lead to more transportaton, whch may be detrmental to the envronment. Thus, our paper ponts out a new general equlbrum effect that may be relevant for several polces drected towards the transportaton sector. The trade and geography lterature wth homogenous frms shows how the agglomeraton pattern of a sector depends on the level of transport costs. The ntroducton of heterogeneous frms mples that t s possble to analyse whch frms agglomerate; that s, t s possble to analyse the spatal sortng of frms. Our paper adds to ths lterature by showng that scale economes of transportaton can have an mportant effect on the spatal sortng of a sector. 13

14 5 Appendx 5.1 Appendx. Welfare effects of trade lberalsaton Frst we show that From (17) (18) d j = d j da da + d j da L da L + d j > 0. (28) d j = (1 s)λβa β 1 da d j = (1 s)λβa β 1 L da L d j ϕ(1 s)αθa α 1 > 0, (29) + ϕ(1 s)αθa α 1 L < 0, (30) = θ{(1 s)(a α L a α ) + (1 s)(1 a α )} > 0, (31) snce β > α, βλ = αθ and ϕ < 1. From (26) and snce a L = t 1 σ, n the long-run equlbrum we have da da L > 0, (32) < 0. (33) Together these dervatves mply that d j > 0. Next, we llustrate that From (17) (18) d k = d k da da + d k da L da L + d k 0. (34) d k = ϕ(1 s)αθa α 1 da d k = ϕ(1 s)αθa α 1 L da L d k (1 s)λβa β 1 < 0, + (1 s)λβa β 1 L > 0, (35) = θ{s(1 a α ) + (1 s)(a α a α L)} > 0, (36) snce β > α, βλ = αθ and ϕ < 1 and from (32) and (33), we have da these dervatves mply that the ndrect effect s always negatve,.e. whereas the drect effect s postve, d k > 0. > 0 and da L d k da da < 0. Together, + d k da L da L < 0, 14

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