KNOWLEDGE MISALLOCATION AND GROWTH

Transcription

1 Macroeconomc Dynamcs, 214, Page 1 of 25. Prnted n the Unted States of Amerca. do:1.117/s x KNOWLEDGE MISALLOCATION AND GROWTH XAVIER RAURICH, FERNANDO SÁNCHEZ-LOSADA, AND MONTSERRAT VILALTA-BUFÍ Unverstat de Barcelona and Centre de Recerca en Economa del Benestar (CREB) We develop a growth model where knowledge s emboded n ndvduals and dffused across sectors through labor moblty. The exstence of labor moblty costs constrans moblty and thus generates labor msallocaton. Dfferent levels of labor msallocaton mply dfferent levels of explotaton of avalable knowledge and therefore dfferent total factor productvty across countres. We derve a postve relatonshp between growth and labor moblty, whch s consstent wth the emprcal evdence, by assumng aggregate constant returns to captal. We also analyze the short- and long-run effects of labor moblty costs n the case of decreasng returns to captal. It turns out that changes n moblty costs have larger economc effects when dfferent types of workers have small rather than large complementartes. Fnally, we show that dfferent labor ncome taxes or labor market tghtness mply dfferent rates of labor moblty and therefore can explan dfferences n Gross Domestc Product across countres. Keywords: Economc Growth, Labor Moblty, Knowledge Msallocaton 1. INTRODUCTION The emprcal growth lterature has establshed that cross-country dfferences n ncome per capta cannot be accounted for only by dfferences n the accumulaton of producton factors [see McGrattan and Schmtz (1999) and Parente and Prescott (24)]. Several authors argue that ncome per capta dspartes are manly explaned by dfferences n total factor productvty (TFP), whch are often consdered to be the result of dfferences n the msallocaton of nputs [see Klenow and Rodrguez-Clare (1997); Prescott (1998); and Hall and Jones (1999)]. Hseh and Klenow (29) corroborate the mportance of nput msallocaton n explanng TFP. They fnd that reducng labor and captal msallocaton n Chna and Inda to the U.S. level would ncrease ther TFP by 3% to 5% n Chna and by 4% to 6% n Inda. 1 The lterature has manly centered on the effects of the jont We acknowledge the comments of Petros Mlons and two anonymous referees. Fnancal support from the Spansh Mnstry of Educaton through Grant ECO and from the Government of Catalona through the Barcelona GSE Research Network and Grant 29SGR151 s gratefully acknowledged. Address correspondence to: Xaver Raurch, Unverstat de Barcelona, Departament de Teora Econòmca, Facultat d Economa Empresa, Avnguda Dagonal 696, 834 Barcelona, Span; e-mal: xaver.raurch@ub.edu. c 214 Cambrdge Unversty Press /14 1

2 2 XAVIER RAURICH ET AL. msallocaton of nputs on ndustry dynamcs [see Lagos (26); Restucca and Rogerson (28); Hseh and Klenow (29, 21); Poschke (29); Barseghyan and DCeco (211); or Moscoso Boedo and Mukoyama (212), among many others). The exstence of frm entry costs, for nstance, lowers aggregate TFP by keepng low-productvty frms n the market. Instead, Banerjee and Moll (21) and Fernald and Neman (211) focus on the effects of the msallocaton of captal. In ths paper, we contrbute to the lterature on nput msallocaton by developng a growth model where knowledge s emboded n ndvduals and dffused across sectors through labor moblty. 2 The exstence of labor moblty costs constrans moblty and, thus, generates labor msallocaton. Thus, dfferent levels of labor msallocaton mply dfferent levels of explotaton of avalable knowledge and, therefore, dfferent TFP across countres. In the labor economcs lterature, worker reallocaton occurs manly for two reasons. On one hand, frctons n the labor market create msmatches. Ths gves ncentves to workers to change jobs n order to mprove ther matches and, consequently, ther wages. Therefore, labor moblty s assocated wth hgher productvty n these models [see Jovanovc and Mofft (199)]. On the other hand, shfts n sectoral labor demand may force ndvduals to change sectors, whch usually nvolves a loss of sector-specfc human captal, and therefore, a potental productvty loss [see Moscarn (21)]. Shfts n labor demand mght be due to preference shocks, busness cycle shocks, or technology shocks [see Rogerson (1987) and Dxt and Rob (1994)]. In contrast, n our model, worker reallocaton s explaned by the complementarty between dfferent types of workers and the dynamcs of sector-specfc knowledge, whch s based on a contnuous process of sector-specfc learnng. These two elements generate worker reallocaton even n the absence of frctons and shfts n the sectoral labor demand. We assume that workers are heterogeneous and not perfect substtutes. 3 Worker heterogenety s a consequence of sector-specfc learnng-by-dong. As mentoned prevously, knowledge s dffused across sectors through labor moblty, whch s constraned by labor moblty costs. We assume that these costs arse only when frms hre workers from a dfferent sector. 4 These costs may represent tranng costs of hrng workers from other sectors, productvty loss when knowledge s sector-specfc, hgher hrng or search costs of hrng n a dfferent sector, reallocaton expenses, or tme-consumng labor moblty. More generally, they could embody nsttutonal factors such as labor market regulatons and the exstence of unons. Obvously, these moblty costs drectly affect frms hrng decsons. Thus, dfferences n labor moblty costs mply dfferent levels of explotaton of avalable knowledge and, therefore, dfferent productvty growth across countres. The purpose of ths paper s to study the effects of moblty costs on Gross Domestc Product (GDP). Table 1 reports a measure of labor moblty, GDP per capta level, and GDP per capta growth for nne 5 European countres and the Unted States n the perod We can dstngush between three groups of countres accordng to ther labor moblty level. At one end, Portugal, Span, Belgum, France, and Italy have a very mmoble labor force, wth less than one out of ten employees

3 KNOWLEDGE MISALLOCATION AND GROWTH 3 TABLE 1. Labor moblty, GDP per capta n 1994, and GDP growth between 1994 and 28 Country Labor moblty GDP per capta 1994 GDP per capta growth (%) Italy France Belgum Span Portugal Germany Netherlands US Denmark UK Average Sources: Data on labor moblty from Jolvet et al. (26) [(% of job-to-job moblty n a 3 year spell/ % of nonmoble workers n a 3-year spell) 1] and on GDP per capta and GDP growth from Penn World Table 7.1 [Heston et al. (212)]. changng jobs n a three-year spell. At the other end, the Unted Kngdom and Denmark are hghly moble, wth at least one out of fve employees changng jobs n a three-year spell. In between, there are the Unted States, the Netherlands, and Germany. The correlaton between labor moblty and GDP growth s.52 and the coeffcent of labor moblty on GDP growth controllng for ntal GDP level s postve and sgnfcant (adjusted R 2 s close to 15%). We buld on the semnal paper of Arrow (1962), where learnng-by-dong s the man determnant of frms productvty growth. Arrow assumes that knowledge learned n a frm gets freely and nstantly spread to the whole economy. However, the assumpton of nstant dffuson and utlzaton of new knowledge across all sectors seems unrealstc. We assume nstead that dffuson of knowledge across sectors s costly and requres channels of transmsson. In partcular, we assume that frms learn by hrng workers from other sectors (learnng-by-hrng). Moreover, we assume that productvty ncreases wth the exchange of knowledge by ntroducng complementartes among dfferent types of workers. 7 Hence, hrng decsons (and, thus, worker moblty) affect knowledge explotaton and the TFP. As n Arrow (1962), the learnng n one sector s a functon of the nvestment made last perod n that sector. As a result, the productvty (or emboded knowledge) of a worker hred from a partcular sector s drectly related to the accumulated stock of physcal captal n that sector. Hence, we dstngush among types of workers by the sectors n whch they have been employed. Moreover, the possblty of learnng-by-hrng allows for dfferent degrees of substtuton among the dfferent worker types, whch we nclude by defnng the labor nput as a constant-elastcty-of-substtuton (CES) aggregator of the dfferent workers. We consder two scenaros. In the frst scenaro, we assume aggregate constant returns to captal so that endogenous growth arses, as n Arrow (1962). The equlbrum does not exhbt transton, and consumpton, captal, and GDP grow

4 4 XAVIER RAURICH ET AL. at the same constant rate, whereas labor moblty s constant. We show that hgher moblty costs mply both lower labor moblty and lower GDP growth, whch explans the emprcal evdence reported n Table 1. We llustrate these results numercally, fndng that dfferences n labor moblty costs can explan up to 53% of dfferences n growth rates. Therefore, our model suggests that cross-country dfferences n GDP growth rates can be explaned partally by labor msallocaton due to dfferences n labor moblty costs. Because each sector s dentfed wth a type of knowledge, we also fnd that the number of sectors affects growth. We dstngush two dfferent effects. On one hand, an ncrease n the number of sectors drectly rases TFP. On the other hand, t reduces the number of workers of each type and, therefore, reduces TFP. The trade-off between these two effects depends on the ease wth whch one worker can substtute for another and, therefore, s drven by the elastcty of substtuton between them. When workers are hghly complementary, the negatve effect domnates and an ncrease n the number of sectors reduces the GDP growth rate. The opposte happens when workers are easly substtutable. 8 In the second scenaro, we assume aggregate decreasng returns to captal, mplyng exogenous growth and nonconstant labor moblty durng the transton. In ths scenaro, not only the standard decreasng returns to captal but also learnngby-hrng drves the transton. The equlbrum s unque and converges to a steady state where both labor moblty and GDP reman constant. We show that a permanent reducton n labor moblty costs causes a permanent ncrease n labor moblty, TFP, and ncome per capta. The effects are much larger when workers are easly substtutable. In ths case, frms more frequently replace workers from ther own sector wth workers from another sector, so that a reducton n the labor moblty cost causes a greater ncrease n the long-run level of GDP. We conclude that labor moblty costs, by causng labor msallocaton, explan dfferences n ncome per capta across countres. Thus, a crucal ssue s to explan the sources of dfferences n labor moblty costs across countres. 9 To ths end, we further the understandng of moblty costs through the analyss of two wellknown economc ssues: labor ncome taxes and labor search costs. We show that dfferences n ether taxes or search costs can explan dfferences n labor moblty across countres and, therefore, n ther ncome per capta. In partcular, labor ncome taxes amplfy the dstorton generated by labor moblty costs and, thus, ndrectly affect TFP through labor moblty. As a result, labor ncome taxes not only reduce employment, as shown n Prescott (24), but also reduce TFP. 1 When consderng labor search costs, we show that vacancy costs amplfy labor moblty costs and, thus, ndrectly affect TFP. But, as they are a type of moblty cost, they drectly affect TFP even n the absence of other moblty costs. 11 We fnd that countres wth hgh labor market tghtness suffer a larger GDP reducton because of an ncrease n moblty costs than countres wth low market tghtness. The paper s organzed as follows. In Sectons 2 and 3 we descrbe the model and derve the equlbrum, respectvely. In Secton 4 we analyze the scenaro wth constant aggregate returns to captal, whch s drectly comparable to Arrow (1962). In Secton 5 we study the effects of labor moblty on the transtonal

5 KNOWLEDGE MISALLOCATION AND GROWTH 5 dynamcs of an otherwse standard Ramsey Cass Koopmans model. We expand the dscusson on moblty costs n Secton 6, where the role of labor ncome taxes and search costs s analyzed. Fnally, we conclude n Secton 7. All the proofs are n the Appendces. 2. THE MODEL Consder a contnuous-tme economy wth a perfectly compettve fnal goods sector and a contnuum of ntermedate sectors wth a constant measure S. The technology n the fnal goods sector s defned by the followng CES producton functon: 12 S ) Y = (S 1 μ 1 Y μ μ d, (1) where Y denotes fnal goods producton, Y s the amount of ntermedate goods of sector used n the producton of the fnal good, and the elastcty of substtuton between two ntermedate goods s measured by 1/ (1 μ), wth μ 1. Note that ths producton functon does not exhbt scale effects. Any fnal goods frm solves the followng proft maxmzaton problem: max {Y } S Y p Y, subject to (1), where p s the prce of the ntermedate good of sector n unts of the fnal good. From the frst-order condtons, we obtan the nverse demand for any ntermedate good as ( ) Y 1 μ p = S μ 1. (2) Y Intermedate goods sectors use labor and physcal captal as nputs. Workers are nfntely lved and n each perod learn the knowledge of the sector n whch they are employed wthout any cost (learnng-by-dong). Frms are nterested n the knowledge a worker has learned n the last perod. Therefore, we dstngush among types of worker by the sectors n whch they have been employed last perod. Ths assumpton mples that workers have the same sector-specfc knowledge after workng one perod n the same sector. Followng Arrow (1962), knowledge of sector s related to the stock of captal n that sector. In partcular, we assume that the knowledge of sector concdes wth the average per-worker stock of physcal captal n that sector, k. 13 Because learnng s emboded n workers, t s not nternalzed by frms when they make ther own nvestment decsons. 14 However, the amount of knowledge accumulated by each type of worker s a determnant of the hrng decsons of the frm. In fact, by hrng workers from other sectors, frms learn from the nvestment decsons made n other sectors. Denote by λ j the number of workers from sector j that are hred n sector. As stated earler, they have emboded knowledge of sector j, k j. We call them poached workers.

6 6 XAVIER RAURICH ET AL. Smlarly, let η be the number of workers from sector hred by the same sector, who have knowledge of sector, k. We call them retaned workers. There s a contnuum of frms of measure 1 n each sector. We assume that the producton functon of a frm of sector s Y = [ ( ) σ S ( η k ξ + q λ j ) ] α σ σ ξφ k j dj K 1 α, (3) where K s the stock of physcal captal, ξ [, 1] measures the return from learnng, 1/ (1 σ ) s the elastcty of substtuton among dfferent types of worker, wth σ 1 and σ, and α (, 1) measures the labor ncome share. The parameter q measures the ablty of learnng-by-hrng from other sectors and φ (, 1/ξ] measures the dfferences n the return from learnng between poachng and retanng. We also assume that φ 1/ξ to have nonncreasng returns to captal at the aggregate level. The producton functon (3) can be rewrtten as Y = ψ N αk1 α, where N s the number of workers and the varable ψ = [ ( ) σ S η k ξ + q N ( ) j σ α σ λ k ξφ j dj] N s the standard measure of TFP. When q =, external knowledge s not productve, so that there s no labor moblty (η = N and λ j = ) and ψ = k αξ. In ths case, ψ ncreases wth the captal stock as n the semnal papers of the endogenous growth lterature [see Arrow (1962); Romer (1986); Barro (199); and Rebelo (1991), among many others]. In contrast, when q>, the TFP depends on the hrng decsons and, specfcally, on the ablty both to retan workers and to hre workers from other sectors. Ths ntroduces a relevant dfference: frms change ther TFP when they choose between retanng and poachng. Frms maxmze profts n a perfectly compettve market;.e., max }p Y (r + δ) K { η,λ j,k S w j λj dj w η, subject to (3) and (4), where r s the rental cost of captal, δ (, 1) s the deprecaton rate, w j s the salary pad n sector to workers poached from sector j, and w s the salary pad n sector to the retaned workers.15 The frst-order condtons wth respect to η,λ j, and K are αp ψ α σ α αp ψ α σ α N α σ N α σ q ( λ j η σ 1 k ξσ (4) K 1 α w, (5) ) σ 1 k ξφσ j K 1 α w j, (6) (1 α)p ψ N α K α = r + δ, (7) where (5) and (6) hold wth equalty when η > and λ j >, respectvely.

7 KNOWLEDGE MISALLOCATION AND GROWTH 7 We assume that swtchng among sectors s costly and that these moblty costs are proportonal to the wage. 16 Because any poached worker has to pay these moblty costs, n order to hre an external worker, the frm has to pay her at least the same wage as n the worker s ntal frm plus moblty costs,.e., w j mw j j, (8) where m 1 > measures moblty costs as a percentage of the wage. These labor moblty costs ntroduce an neffcency nto the labor market: the margnal productvty of a worker does not equalze across sectors, thereby gvng rse to labor msallocaton. Perfect competton mples that (8) holds wth equalty,.e., w j = mw j j for all j. In fact, the labor ncome net of moblty costs obtaned by a poached worker s w j j. Ths mples that the net labor ncome wj j does not depend on the sector n whch a worker s employed, but on the sector n whch she was employed durng the last perod, as ths determnes the worker s specfc knowledge. The economy s populated by a large famly of N members. Populaton sze s constant and each member supples nelastcally one unt of labor. Thus, the famly labor ncome net of moblty cost s W = S w η d+ S S wj j λj ddj. Ths labor ncome can ether be consumed or be nvested. Then, the budget constrant of the famly s C + Ȧ = ra + W, (9) where C s aggregate consumpton and A are fnancal assets. Each member s utlty functon s u(c) = c1 θ 1 1 θ, (1) where c = C/N s ndvdual consumpton and 1/θ > measures the ntertemporal elastcty of substtuton. The famly maxmzes Ne ρt u(c)dt subject to (9) and (1), where ρ s the subjectve dscount factor. From ths maxmzaton problem, we obtan that the consumpton growth rate satsfes ċ c = r ρ. (11) θ The captal market clearng condton s A = K, where K = S K d s the aggregate captal stock. The fnal goods market clearng condton mples that Y = C + K + δk + M, (12) where Y s GDP and M measures the aggregate moblty costs n unts of fnal goods. In other words, total producton s devoted to consumpton, nvestment,

8 8 XAVIER RAURICH ET AL. and moblty costs. 17 These moblty costs are defned as M = S S (m 1)w j j λj ddj. (13) Recall that N denotes total employment n sector. Thus, the labor demand n ths sector s S N = η + λ j dj, (14) and the labor market clearng condton mples that S Fnally, from (14) and (15), we obtan S η d + N d = N. (15) S S λ j ddj = N. (16) 3. EQUILIBRIUM In ths secton we characterze the symmetrc equlbrum, whch s defned as an equlbrum path where all ntermedate good sectors produce usng the same amount of nputs: K = K/S, η = η, and λ j = λ j for all. Condton (14) mples that n a symmetrc equlbrum λ j = λ for all j and (15) mples that N = N/S workers are employed n each sector. 18 The per capta stock of captal satsfes k = K/N = SK /SN = k = k for all and, hence, ψ = ψ = [( Sη k ) σ ( ξ N + qs Sλ k ξφ) ] σ α/σ N. From the producton functon, we obtan that Y = Y j for all and j. Then, usng equaton (1), we obtan that GDP s Y = SY and, usng equaton (2), the prce level of the ntermedate good s p = 1 for all. Fnally, equaton (5) mples that w = wj j = w. We measure labor moblty usng the rato between poached and retaned workers, x = Sλ/η. Then, n a symmetrc equlbrum, the frst-order condtons (5), (6), and (7) become αψ α σ α S σ 1 N 1 σ η σ 1 k 1 α+ξσ w, (17) αψ α σ α S σ 1 N 1 σ η σ 1 k 1 α+ξσφ qs 1 σ x σ 1 mw, (18) (1 α)ψk α = r + δ, (19) where (17) and (18) hold wth equalty when η>and λ>, respectvely. Moreover, the labor market clearng condton (16) mples that ( )( ) N 1 η =, (2) S 1 + x

9 KNOWLEDGE MISALLOCATION AND GROWTH 9 and, from (1), (3), (2), and the defnton of ψ, GDP per capta, y = Y/N, s equal to ( 1 + qs y = 1 σ x σ k ) ξ(φ 1)σ 1 α σ k 1 α(1 ξ). (21) 1 + x Fnally, combnng (12) and (13), and notng that k = K/N, we obtan ( ) S k = y c δk (m 1)wxη. (22) N DEFINITION 1. A dynamc equlbrum s a path {x,η,c,k,w,r,y} that, gven the ntal stock of captal k, solves the system of dfferental equatons (11) and (22), and satsfes (17) (21) and the transversalty condton lm t e ρt c θ k =. We clam that the equlbrum path s nteror when the numbers of retaned and poached workers are both postve. In ths case, (17) and (18) hold wth equalty and we obtan the followng nontrval expresson for labor moblty: [( q x = S k m) ξσ(φ 1)] 1 1 σ. (23) The followng proposton characterzes the condtons that make the path of the dynamc equlbrum nteror. PROPOSITION 1. When σ<1, the path of the dynamc equlbrum s nteror and labor moblty s measured by (23), and when σ = 1, the path of the dynamc equlbrum s nonnteror and there exsts a value of the per capta stock of captal, k = (m/q) 1/ξ(φ 1), such that (a) If k> k then η =, λ= N/S 2, and x. (b) If k< k then η = N/S, λ =, and x =. (c) If k = k then η>, λ>, and x s ndetermnate. Ths proposton shows that the equlbrum s nteror when dfferent types of worker are not perfect substtutes. Table 1 reveals that only a postve fracton of workers are moble, mplyng that the relevant case s that of an nteror equlbrum (σ <1). Therefore, hereafter, we allow any degree of substtuton between types of worker except for the emprcally nonrelevant case of perfect substtuton. When φ 1, there s a dfferental return from learnng between poachng and retanng. As follows from (23), ths dfferental return mples that captal accumulaton causes a substtuton effect between the two types of worker and, thus, labor moblty changes as captal accumulates. If φ<1, as captal ncreases, the productvty of poached workers decreases wth respect to the productvty of retaned workers. Ths drves two dfferent effects: a substtuton effect and a

10 1 XAVIER RAURICH ET AL. producton effect. The frst effect mples that less productve workers should be replaced by more productve workers, whereas the second effect mples that a larger number of less productve workers should be hred to mantan the level of producton. The frst effect domnates when workers are hghly substtutve (σ >) and the second effect domnates when workers are hghly complementary (σ <). As a result, captal accumulaton ncreases (decreases) labor moblty when the elastcty of substtuton s lower (hgher) than one. Obvously, the same ratonale explans that f φ>1 then, as captal accumulates, labor moblty ncreases when σ> and decreases when σ<. Equaton (23) also shows that labor moblty s proportonal to the number of sectors and decreases wth the labor moblty cost. Usng (19), (2), (23), and the defnton of ψ, the nterest rate can be wrtten as r + δ = (1 α)h (x) k α(ξ 1), (24) where [ ] (1 + mx) 1 α σ h (x) =. (25) 1 + x Combnng (2), (21), and (23), we obtan y = h (x) k 1 α(1 ξ), (26) and from (17), (21), (23), and the defnton of ψ, wehave ( ) 1 + x w = α y. (27) 1 + mx Usng (11) and (24), we obtan ċ c = (1 α)h (x) kα(ξ 1) δ ρ, (28) θ and substtutng (26) and (27) nto (22) gves [ ( )] k (m 1)x k = 1 α h (x) k α(ξ 1) c δ. (29) 1 + xm k DEFINITION 2. Assume that σ < 1. Gven k, a dynamc nteror equlbrum s a path {c, k, x, y} that solves the system of dfferental equatons (28) and (29), and satsfes equatons (23) and (26) and the transversalty condton lm t e ρt c θ k =. As follows from (24), on ths nteror equlbrum path, dynamcs s drven by two dfferent forces: the decreasng returns to captal and the learnng-byhrng. The frst force arses when ξ<1and the second when labor moblty s not constant, whch requres q>and φ 1. The propertes of the dynamc equlbrum wll depend heavly on whch of these two sources of transton s

11 KNOWLEDGE MISALLOCATION AND GROWTH 11 actve. In ths paper, we wll consder two very dfferent scenaros. In Secton 4, we assume that ξ = φ = 1, whch mples that nether of the two forces drvng the transton s actve. In ths case, the equlbrum wll exhbt sustaned endogenous growth. We use ths case to make a drect comparson wth the classcal endogenous growth models [see Arrow (1962); Romer (1986); Rebelo (1991)]. In Secton 5, we assume that ξ < 1, whch mples that the equlbrum exhbts a transton drven at least by the dmnshng returns to captal, and growth s exogenous. We use ths case to study the effects of labor moblty along the transton ENDOGENOUS GROWTH In ths secton we characterze the dynamc equlbrum when ξ = 1 and φ = 1. As follows from Proposton 1 and equaton (23), when φ = 1, labor moblty s constant and equal to ( q 1 x 1 σ = S. (3) m) In ths case, both the nterest rate and the consumpton growth rate are also constant. Moreover, from (26), we obtan that the producton functon smplfes to y = h ( x ) k, (31) whch s an AK functon. TFP s measured by h (x ) k α and t s endogenous, as t depends on labor moblty x. Fnally, as s standard n AK models, the fnal goods market clearng condton and the transversalty condton mply that the equlbrum does not exhbt transton and that captal, consumpton, and GDP grow at the same constant rate. PROPOSITION 2. Assume that ξ = 1, φ= 1 and σ < 1. The equlbrum does not exhbt transton, and GDP, captal, and consumpton grow at the same rate, g = (1 α)h (x ) δ ρ. θ It s worth hghlghtng that f q = then there s no labor moblty, h () = 1, and g = [(1 α) δ ρ] /θ. Thus, the long-run growth rate concdes wth the rate obtaned n Arrow s economy. In contrast, f q>, then labor moblty affects the growth rate through the TFP, as stated n the followng proposton. PROPOSITION 3. Assume that ξ = 1, φ= 1, σ<1, and q>. Then (a) An ncrease n the moblty cost m reduces the GDP growth rate. (b) An ncrease n the number of sectors S ncreases the GDP growth rate f the elastcty of substtuton among worker types s hgher than one (σ >) and decreases the GDP growth rate f ths elastcty s lower than one (σ <). An ncrease n the moblty cost m reduces labor moblty and, thus, both the TFP and the GDP growth rates decrease. Note that when moblty costs are

12 12 XAVIER RAURICH ET AL. zero (m = 1), frms are not constraned when hrng and, thus, GDP growth s maxmum. 2 In ths model, an ncrease n the measure of sectors S also affects the GDP growth rate. Ths effect s well known n the growth lterature [see the growth model based on the varety of products n Romer (199), for example]. However, n these models ths effect s postve because t s caused by a scale effect. In contrast, we have assumed that there are no scale effects and, therefore, an ncrease n S affects growth only because t ncreases labor moblty. We can dentfy two dfferent forces. On the one hand, an ncrease n S rases TFP by ncreasng the number of types of worker. On the other hand, because populaton sze s constant, t reduces the number of workers of each type and, therefore, reduces TFP. The result of these two forces depends on the ease of worker substtuton and, therefore, s drven by the elastcty of substtuton. When ths elastcty of substtuton s hgher than one, the postve effect domnates and a larger S ncreases the GDP growth rate. The opposte happens when the elastcty of substtuton s lower than one. In ths case, workers are hghly complementary and hgher growth s acheved f there are fewer types of worker and a larger number of workers of each type. In what follows we llustrate these results numercally. To ths end, we fx the value of the parameters as follows: θ = 2 and α =.65, mplyng that the ntertemporal elastcty of substtuton (IES) s.5 and the labor ncome share equals 65%, respectvely. δ = 6.47% and ρ =.176, mplyng that the annual nterest rate equals 5.2% and the growth rate s 1.72%, as ndcated n Table 1. Fnally, we set m = , σ =.1659, and Sq 1/(1 σ) =.2165 to jontly determne the followng targets: rato of captal to GDP equals 3, rato of consumpton to GDP equals.7, and the level of labor moblty equals 18%, whch s the value of labor moblty n the Unted States. 21 Note that the value of σ s negatve, mplyng a value of the elastcty of substtuton lower than one. Ths mples that an ncrease n the number of sectors reduces the growth rate of the economy, as t reduces the number of workers n each sector. Obvously, ths negatve result s a consequence of the assumed technology wthout scale effects. Note also that the calbrated measure of moblty costs (6% of annual wage) s consstent wth the results of Lee and Wolpn (26), who estmate drect moblty costs for the Unted States to range between 5% and 75% of average annual earnngs. Fgure 1 llustrates the effects of changng moblty costs on the GDP growth rate accordng to our model (sold lne), together wth the actual data and the lnear regresson (dashed lne). An ncrease n moblty costs causes a reducton n both labor moblty and GDP growth. Ths postve relatonshp between growth and labor moblty s consstent wth the emprcal evdence. Therefore, our model suggests that cross-country dfferences n GDP growth rates can be partally explaned by dfferences n labor moblty costs. In our numercal exercse, m takes values between 1 (no moblty costs) and 5.7 (moblty costs are 4.7 tmes the annual wage), mplyng n the latter case a moblty rate of 6%, close to the moblty rate n Italy, whch s the country wth the lowest moblty rate n our

13 KNOWLEDGE MISALLOCATION AND GROWTH 13 GDP per capta growth Span Netherlands Belgum Unted States Portugal France Germany Italy Denmark Unted Kngdom Labor moblty Data Calbrated GDP growth Lnear regresson FIGURE 1. Labor moblty and GDP per capta growth. Data on labor moblty from Jolvet et al. (26) [(% of job-to-job moblty n a 3 year spell/% of nonmoble workers n a 3-year spell) 1] and on GDP per capta growth from Penn World Table 7.1. sample. Ths range explans up to 53% of the dfferences n GDP growth rates. 22 Therefore, we conclude that labor moblty costs are a potentally relevant source of cross-country dfferences n growth rates. 5. EXOGENOUS GROWTH In ths secton we analyze the equlbrum when ξ<1. In ths case, the producton functon exhbts decreasng returns to captal and the economy converges to a steady state where captal and consumpton reman constant. The unqueness of the steady state s guaranteed only when φ Thus, n what follows we assume that ths condton s satsfed. PROPOSITION 4. Assume that ξ < 1, φ 1, and σ < 1. Then there s a unque and saddlepath-stable steady state. We must dstngush between the followng two cases. When φ = 1, captal accumulaton does not change the dfferental productvty between poached and retaned workers and, as a result, labor moblty s constant. In ths case, transtonal dynamcs s drven only by the decreasng returns to captal and, hence, t s dentcal to the transtonal dynamcs n the Ramsey Cass Koopmans model. In contrast, when φ<1, two forces drve the transton: decreasng returns to captal and nonconstant labor moblty. In ths secton, we frst numercally compare

14 14 XAVIER RAURICH ET AL moblty rate GDP growth rate K/Y nterest rate K GDP per capta = =.5 =1 FIGURE 2. Transtonal dynamcs when the elastcty of substtuton s.1 and the ntal stock of captal s 5% of ts long-run value. these two dfferent cases to show the effects of labor moblty on the dynamc equlbrum. And, second, we consder the effects of a reducton n labor moblty costs. We fx the value of the parameters as follows: θ = 2 and α =.65, mplyng an IES of.5 and a labor ncome share of 65%. The annual nterest rate equals 5.2%, mplyng that ρ =.52. And δ = 6.47%, mplyng that the rato of captal to GDP equals 3. We set the value m = , whch mples that the rato of consumpton to GDP equals 76%. 24 The value of Sq 1/(1 σ) s set so that n the long run x = 18%, whch s the level of labor moblty n the Unted States. Fnally, ξ =.6, mplyng a speed of convergence equal to 2% when φ = 1. Note that, n contrast to the prevous secton, the elastcty of substtuton among the dfferent types of worker s now a free parameter. In the followng we compare the case of workers beng complements (σ = 9 and the elastcty of substtuton equals.1) wth the case of workers beng substtutes (σ =.9 and the elastcty of substtuton equals 1). Ths comparson allows us to ascertan the mportance of ths elastcty of substtuton. Fgure 2 shows the transtonal dynamcs when the ntal stock of captal s 5% of ts long-run value and the elastcty of substtuton among the dfferent types of worker s.1 (σ = 9). The contnuous, dotted, and dashed lnes dsplay the economy when φ = 1, φ =.5, and φ =, respectvely. As mentoned earler, when φ = 1, the labor moblty rate s constant and, hence, the transtonal dynamcs s dentcal to that obtaned n the neoclasscal growth model wthout labor moblty, where convergence depends exclusvely on the decreasng returns to captal. In the ntal perods, the nterest rate s hgh because captal s below ts

15 KNOWLEDGE MISALLOCATION AND GROWTH x 1-3 moblty rate 1.5 GDP growth rate K/Y nterest rate K GDP per capta = =.5 =1 FIGURE 3. Transtonal dynamcs when the elastcty of substtuton s 1 and the ntal stock of captal s 5% of ts long-run value. long-run value. Hgh nterest rates cause a rapd accumulaton of captal, whch explans the ntally hgh GDP growth rates. As captal accumulates, the nterest rate decreases, whch reduces the growth of both captal and GDP. In contrast, when φ<1, labor moblty s not constant and affects the transtonal dynamcs of the man economc varables, ncludng the GDP growth rate. As captal ncreases, the relatve productvty of poached workers decreases. But, because poached and retaned workers are hghly complementary (σ <), frms have to ncrease the number of poached workers to compensate for ther lower relatve productvty and thus labor moblty ncreases durng the transton. These changes n labor moblty mply a hgher nterest rate n the ntal perods when φ<1. Obvously, ths mples a faster captal accumulaton. Indeed, the asymptotc speed of convergence s 2.7% when φ = and 2.4% when φ =.5. These speeds of convergence are clearly hgher than the speed of convergence when φ = 1. Ths faster convergence explans why n the ntal perods the GDP growth rate s hgher when φ<1 and eventually becomes lower. Fgure 3 shows the transtonal dynamcs when the ntal stock of captal s 5% of ts long-run value and the elastcty of substtuton among the dfferent types of worker s 1 (σ =.9), mplyng that workers are hghly substtutable. The transtonal dynamcs s very smlar to that obtaned n Fgure 2, and the conclusons from the comparson between the economy wth a constant moblty rate and the economy wth a nonconstant moblty rate are the same. However, there are two man dfferences wth respect to the transtonal dynamcs obtaned when workers are complementary. Frst, when poached and retaned workers are hghly substtutable, frms decrease the number of poached workers durng the transton.

16 16 XAVIER RAURICH ET AL. FIGURE 4. Transtonal dynamcs due to a permanent 2% reducton n the labor moblty cost when the elastcty of substtuton s.1. Note that ths mples that the tme path of labor moblty s the opposte of the one obtaned n Fgure 2. Interestngly, these dfferences n the transtonal dynamcs of labor moblty do not cause qualtatve dfferences n the transtonal dynamcs of the other economc varables, as follows from the comparson between Fgures 2 and 3. Ths happens because labor moblty affects the other economc varables only by changng the TFP and the effects of labor moblty on the TFP depend on the elastcty of substtuton: TFP ncreases wth labor moblty f and only f σ>, as follows from (25). In Fgure 2, workers are complements (σ <) and labor moblty s lower when φ<1. As σ<, the lower labor moblty causes a larger TFP and a larger GDP. In contrast, n Fgure 3, workers are substtutes (σ >) and labor moblty s larger when φ<1. However, as σ>, ths hgher labor moblty causes a larger TFP and a larger GDP. Ths explans why the tme paths of captal, nterest rate, and GDP are qualtatvely the same n Fgures 2 and 3. Second, the speed of convergence s hgher when workers are hghly substtutable. 25 The speed of convergence s 4.4% when φ = and 3.1% when φ =.5. Ths hgher speed of convergence when the elastcty of substtuton s large s a consequence of a faster change n labor moblty as captal accumulates. Obvously, the faster change n labor moblty causes larger dfferences n GDP growth between the economy wth constant labor moblty (φ = 1) and that wth nonconstant labor moblty (φ <1). In Fgures 4 and 5, we study the effects of a permanent shock n labor moblty costs, assumng that the economy s ntally n a steady state. We analyze the shock of a 2% reducton n moblty costs, whch mples a drop of moblty

17 KNOWLEDGE MISALLOCATION AND GROWTH 17 FIGURE 5. Transtonal dynamcs due to a permanent 2% reducton n the labor moblty cost when the elastcty of substtuton s 1. costs from 5% to 4% of annual average ncome. A reducton n these costs ntally ncreases labor moblty and TFP. As a result, the nterest rate ntally grows, mplyng an ncrease n the stock of captal durng the transton. Because captal ncreases and TFP has jumped, the GDP growth rate ntally ncreases. Decreasng returns to captal mply that both the nterest rate and the GDP growth rate decrease as captal ncreases durng the transton. In the long run, the rato of captal to GDP, the nterest rate, and the GDP growth rate converge toward the same steady state values, whereas the levels of labor moblty, captal stock, and GDP converge to hgher values. Note that the long-run ncrease n GDP s a consequence of the rse n both the stock of captal and the TFP level. There are two dfferences between Fgures 4 and 5. Frst, n Fgure 4 the elastcty of substtuton s small. After the ntal jump n labor moblty, captal ncreases and the relatve productvty of poached workers decreases. Because workers are hghly complementary, frms ncrease the number of poached workers to compensate for ther lower productvty and, thus, labor moblty ncreases durng the transton, although quanttatvely by an nsgnfcant amount. Instead, n Fgure 5 the elastcty of substtuton s large. Then, as the dfferent types of worker are hghly substtutable, frms decrease the number of poached workers and thus labor moblty decreases durng the transton. Therefore, the same shock causes opposte transtonal dynamcs dependng on the value of the elastcty of substtuton. Second, from the comparson between these two fgures, we conclude that the effects of a reducton n labor moblty costs are much greater when workers are hghly substtutable. In ths case, frms are more lkely to replace

18 18 XAVIER RAURICH ET AL. retaned workers wth poached workers, thereby causng a greater ncrease n the long-run level of GDP. In our numercal example, the ncrease n GDP due to a 2% decrease n the moblty costs s by up to 12% when σ =.9 and only.9% when σ = 9. Therefore, we conclude that labor moblty costs, by causng labor msallocaton, are a more relevant source of cross-country per capta GDP dfferences when the dfferent types of worker are hghly substtutable. 6. FURTHER INSIGHTS ON MOBILITY COSTS In ths secton we examne labor moblty costs n greater depth by analyzng labor ncome taxes and labor search costs. We show that dfferences n ether taxes or search costs can boost or even be the cause of dfferent labor mobltes across countres. Thus, we show that even n the presence of small dfferences n moblty costs, dfferent labor ncome taxes or labor search costs can explan large cross-country dfferences n labor msallocaton and growth Labor Income Taxes The nonarbtrage condton n the labor market mples that poached workers must be compensated for both moblty costs and the net wage they would earn f they had not moved. Therefore, (1 τ) w j = (1 τ) w j j + n, where τ (, 1) s the labor ncome tax rate and n accounts for the labor moblty cost. Ths equaton can be rewrtten as w j = w j n j + ˆm, where m = s the 1 τ compensaton of the labor moblty cost pad by the frm. Therefore, m measures the dstorton ntroduced by labor moblty costs. Note that ths dstorton s ncreasng n the labor ncome tax rate. Therefore, taxes ndrectly affect TFP through labor moblty. 26 Assume also that government revenues are returned to consumers as a lumpsum transfer, so that there s no wealth effect. Thus, the only dstorton due to taxes s va labor moblty costs. Ths concluson and the analyss n ths paper mply that those countres wth a lower tax rate are also those countres wth hgher labor moblty and hgher growth rates. Ths theoretcal concluson s supported by emprcal evdence. The correlaton between total tax wedge and labor moblty n the perod s clearly negatve and sgnfcant, wth a value of.68. Fgure 6 shows that countres wth low labor taxes such as the Unted States or the Unted Kngdom have hgh labor moblty, whereas countres wth hgh labor taxes such as Italy, Belgum, and France have low moblty Labor Search Costs We ntroduce labor search nto our economy. Contrary to the typcal search models, we have nether unemployment nor separaton rates (frms that close down).

19 KNOWLEDGE MISALLOCATION AND GROWTH 19 Labor moblty Unted Kngdom Unted States Portugal Denmark Netherlands Span France Germany Belgum Italy Labor Taxes FIGURE 6. Labor taxes and labor moblty. Data on labor moblty from Jolvet et al. (26) and total tax wedge from OECD year 1997 (average rate n % based on two-earner marred couple wth two chldren, one earner at 1% of average earnngs and the other at 33%). Instead, we have on-the-job-search: ndvduals who work n a frm and who are actvely searchng for another job. As usual, we assume that ths search s costless for workers. In lne wth the prevous sectons, we assume that there are nether moblty costs nor search costs for hrng workers from the same sector. In ths envronment, a frm n sector opens a vacancy V j when t wants to poach a worker from sector j. Ths vacancy has an assocated cost, κ j, and a probablty of beng flled, π j. Ths means that the number of poached workers wll be a fracton π j of the posted vacances, λ j = π j V j, and that the cost of a poached worker s κ j /π j + w j. Assumng that the vacancy costs are proportonal to the hrng wage, κ j = s j wj, where sj s a constant, and that moblty costs other than vacancy costs are a proporton d of the wage, and gven that perfect competton n the labor market mples that w j = (1 + d) w j j, we obtan that the total cost of hrng a poached worker s mw j j, where ( ) m = (1 + d) 1 + sj. π j As usual n the labor search lterature, the probablty of fllng a vacancy π j decreases wth labor market tghtness. 27 Ths mples that a hgher labor market

20 2 XAVIER RAURICH ET AL. tghtness results n a hgher effectve vacancy cost and, as follows from the expresson of m, ths causes an ncrease n the total cost of hrng a worker. On one hand, vacancy costs ncrease labor moblty costs d and, thus, ndrectly affect TFP growth. On the other hand, because vacancy costs are a type of moblty cost, even n the absence of these, d =, vacancy costs ncrease the hrng costs, causng msallocaton of labor and, thus, drectly affectng TFP growth. We conclude from ths result that countres wth hgh labor market tghtness wll be more strongly affected by moblty costs than countres wth low market tghtness and, smlarly, that moblty costs wll matter more n perods of hgh labor market tghtness (economc booms) and less n recessons. 7. CONCLUDING REMARKS We have consdered labor moblty as one of the factors that account for dfferences n TFP across countres. In so dong, we have ntroduced learnng-by-hrng nto a model of learnng-by-dong, so that although frms n a sector may have free and nstant access to the knowledge developed wthn ther own sector, they can only learn from other sectors by hrng external workers (learnng-by-hrng). Hence, hrng decsons and, thus, worker allocaton affect knowledge explotaton and, therefore, TFP. We obtan a postve relatonshp between growth and labor moblty, whch s consstent wth emprcal evdence. Moreover, we fnd that the number of worker types affects growth, and that these effects depend on the elastcty of substtuton among the types of workers. Our results reveal that, apart from the tradtonal decreasng returns to captal, labor moblty mght act as a force drvng the transton. We show that changes n moblty costs matter the most when the elastcty of substtuton among the types of worker s hgh. Therefore, the mpact of dfferent moblty costs s especally sgnfcant when dfferent types of workers have small complementartes. As a consequence, polces that reduce moblty costs are more effectve when workers are hghly substtutable. Fnally, we show that dfferences n ether labor ncome taxes or labor search costs can explan dfferences n labor moblty among countres and, therefore, n ther growth rates. Labor market regulatons and the publc system of unemployment benefts affect labor moblty and, thus, ndrectly affect GDP. The am of future research s to study f the dfferent labor market regulatons account for part of the dfferences n both labor moblty and GDP levels. NOTES 1. See Banerjee and Moll (21) and Restucca and Rogerson (213) for a comprehensve revew of other recent papers that show that the msallocaton of resources mght explan large cross-country TFP dfferences. 2. The noton that moblty of ndvduals matters for dffuson of knowledge has been used n models wth nternatonal dffuson of knowledge through cross-border flows of people [see Lucas

21 KNOWLEDGE MISALLOCATION AND GROWTH 21 (1993) and Andersen and Dalgaard (211)]. The assumpton s n lne wth the fndng n Machn et al. (212) that labor moblty s postvely related to ndvduals level of educaton. 3. In the labor lterature, Lee and Wolpn (26), when studyng the growth of the servce sector, also assume that workers are heterogeneous and not perfect substtutes. Our paper dffers from thers n two man aspects: frst, ther source of labor heterogenety comes from educaton and work experence accumulated n each sector-occupaton, whereas our labor heterogenety comes from recent sector-specfc learnng-by-dong, so that n the steady state years of experence do not ncrease labor productvty n our setup. Second, we hghlght the mportance of the elastcty of substtuton among the dfferent types of worker. 4. Lee and Wolpn (26) show that moblty costs are substantal when moblty s across sectors n the Unted States (between 5% and 75% of average annual earnngs), whereas they are small when moblty s across frms of the same sector. An analyss of the mportance of moblty costs across sectors can be found n Steger (27). 5. Data on labor moblty also nclude nformaton on Ireland. We exclude t from the analyss, however, because ts hgh growth rate durng the studed perod s explaned by exceptonal crcumstances such as a large attracton of foregn drect nvestment. 6. We measure labor moblty by computng the odds of an ndvdual changng jobs. These are computed as the percentage of ndvduals who changed job n a three-year spell dvded by the percentage of ndvduals who dd not change job durng ths perod. We draw ths measure multpled by 1. We use the same measure for labor moblty n both the theoretcal model and ts calbraton. 7. Several papers report the mportance of the exchange of knowledge as an engne of nnovaton and growth [see Audretsch and Feldman (1996) and Almeda and Kogut (1999)]. Many of them emphasze the mportance of ntersectoral rather than ntrasectoral knowledge spllovers [see Glaeser et al. (1992) and Hanson (21)]. 8. A smlar ntuton explans the results n Hseh and Klenow (29), though they consder the elastcty of substtuton between sector outputs. Smlar results are also obtaned by Jones (211), who argues that problems along a producton chan can decrease output under complementarty. But, whereas he consders complementarty n ntermedate nputs (elastcty of substtuton smaller than 1), we consder all possble degrees of substtuton/complementarty n the labor nput. 9. Blanchflower and Oswald (213) fnd that a hgh homeownershp rate leads to future rses n unemployment. They dentfy three mechansms through whch ths relatonshp mght operate: lower levels of labor moblty, greater commutng tmes, and fewer new busnesses n the regon wth hgher homeownershp. Our paper provdes an explanaton for the mechansm relatng labor moblty, homeownershp (whch ncreases moblty costs), and economc performance. 1. Smlarly, Restucca and Rogerson (28) fnd that settng taxes or subsdes to establshmentlevel labor affects TFP through a reallocaton of resources among heterogeneous frms. 11. Lagos (26) shows n a search model that the level of TFP depends on all the characterstcs of the labor market summarzed by the job-creaton and job-destructon decsons. In partcular, he centers on how labor market polces can affect the aggregate TFP by affectng the pool of actve frms. In our paper, n contrast, search affects TFP by reallocatng labor across sectors. 12. Tme subscrpts are suppressed to enhance readblty. 13. Arrow (1962) assumes that learnng n sector s a functon of the nvestment made n the last perod n that sector. As workers do not move across sectors, ths assumpton mples that the accumulated learnng n a gven sector concdes wth the stock of captal. To make comparsons wth the orgnal paper by Arrow, we must assume that () frms are nterested n the knowledge a worker has learned n the last perod and () learnng s accumulated from the stock of captal. These two assumptons mply that, even wth moblty across sectors, the knowledge of workers concdes wth the average stock of captal n the last sector n whch they have been employed. Thus, these two assumptons allow drect comparson wth Arrow s orgnal paper. 14. Thoeng and Verder (21) also consder knowledge as emboded n workers. However, they study knowledge management as a strategc choce of the frm.