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1 Docuento de Trabao/Working Paper Serie Econoía Productivity and efficiency ith dicrete variable and quadratic cot function by Eduardo Martínez-Budría Juan J. Díaz-Hernández and Sergio Jara-Díaz May DT-E-- ISSN: Intituto Univeritario de Dearrollo Regional, Facultad de iencia Econóica y Epreariale, Univeridad de La Laguna, aino de la Hornera /n La Laguna, Santa ruz de Tenerife, Spain

2 PRODUTIVITY AND EFFIIENY WITH DISRETE VARIABLES AND QUADRATI OST FUNTION Eduardo Martinez-Budria a Juan Joe Diaz-Hernandez b Sergio Jara-Díaz c a,b Univeridad de La Laguna c Univeridad de hile Abtract We propoe an inde of productivity baed on a quadratic cot function and developed for dicrete data including technical and allocative inefficiency, ointly ith technical change and return to cale, a deterinant of Total Factor Productivity. Thi ne inde i applied to Spanih tevedoring indutry o a to identify the ource of change in the productivity of a ulti-productive activity, here oe copanie do not produce oe of the output and/or do not ue oe input. In thi contet, the functional quadratic for and the productivity inde propoed prove particularly ueful. Keyord: productivity; quadratic cot function; inefficiency; dicrete data JEL laification: D4 orreponding author: Eduardo Martinez-Budria. Intituto Univeritario de Dearrollo Regional and Departaento de Análii Econóico. Univeridad de La Laguna. aino de la Hornera /n. 387 La Laguna. Santa ruz de Tenerife. Spain. Eail: ebudria@ull.e; Phone: ; Fa: odiaz@ull.e; aradiaz@ing.uchile.cl

3 .- Introduction Increaed productivity i regarded a one of the ot relevant factor to eplain econoic groth. Since the 96 the aount of econoic literature on the ethod for eauring it ha uhrooed. Likeie, there ha been an interet by analyt and public official alike in eauring output, in claifying the unit analyzed a a function of the productivity achieved, and in aeing the uitability of econoic policie through their effect on productivity. Due to the ultitude of product and factor preent in every production activity, inde nuber theory ha played a pivotal role in the aggregation of product and factor, ince change in productivity are calculated a the difference beteen the rate of change in an output and input indice. Dieert (976, 978) gave a deciive ipetu to the tudy of productivity ith hi theory on eact and uperlative inde nuber, hich deontrated the eitence of a unique correpondence beteen the inde nuber utilized and the underlying technology, hence the ue of the ter eact. By accepting a pecific functional for for the technology and auing an optial behavior for the agent in a copetitive environent, productivity can be calculated ithout having to reort to etiating the paraeter that characterize aid technology. If the functional for choen doe not properly repreent the actual technology, the reulting productivity eaure ill be keed. To iniize thee error, Dieert propoed the ue of fleible functional for, reulting in the o-called eact and uperlative inde nuber. In addition to eauring change in productivity, any tudie on productivity literature have focued on the decopoition of the deterinant of it. With thi obective have been ued both nonparaetric and econoetric technique for calculating or etiating the technology to eaure the different coponent of productivity. The ue of econoetric technique With the ai of offering a practical ethod that iplifie the eaureent of productivity by anager of fir, Hannula () preent a ethod to eaure total productivity baed on partial productivity ratio rather than inde nuber.

4 raie the decopoition of productivity ithin a tochatic approach that take into account the eitence of tatitical noie in the data. Moreover, the ue of a fleible functional for to repreent the underlying technology help reduce the proble aociated ith the need to chooe a pecific functional for 3. In thi line, Denny, Fu and Waveran (98) developed an inde that allo for the effect of a technical change and of econoie of cale on productivity to be identified. Thi inde a developed for continuou variation in data, herea the inforation available on ot econoic variable only allo for calculation of it variation rate in dicrete ter. To overcoe thi proble, they have generally ued the fleible tranlog functional for that underlie the Törnqvit Inde. Thi functional for, although it i conitent ith a fleible repreentation of technology, ha the diadvantage of not being ell defined in a ultioutput fraeork here oe copanie do not produce oe output and/or ue oe input. Martinez-Budria, Jara-Díaz, and Rao-Real (3) etended the ethod of the Denny et al odel to the quadratic functional for and dicrete data. The quadratic functional for i fleible and it ha the advantage that it i ell defined for null output, hich i relevant in ulti-output contet. Thee tudie aue efficient behavior by the deciion aker, o only to ource of productivity gain, technical change and a better utilization of productive cale are conidered. If the efficiency auption i not appropriate, then decreaing inefficiency hould be conidered a a third ource of productivity gain. Beide providing a better and ore coplete vie of productivity per e, introducing inefficiency ha three benefit: i) it avoid confuing technical change ith inefficiency change, ii) production technology i correctly etiated and iii) it help avoiding erroneou entrepreneurial deciion. 3 There i a vat literature that ha copared the eaure of productive efficiency fro nonparaetric and econoetric technique. For intance, Murillo-Zaorano, L. and Vega-ervera, J. () ho epirical evidence that indicate that, in their tudy, the choice beteen both technique could be irrelevant to rank fir according to their efficiency core.

5 In order to incorporate change in efficiency a an additional ource of productivity, Bauer (99) introduced inefficiency into the odel of Denny et al (98). Thi application ha alo been ipleented uing the tranlog functional for, ithout taking into account the eitence of non-produced output or input not eployed. Thi iue i addreed in Tekoura et al (4) here a duy variable technique uggeted by Battee (997), i etended to a tranlog pecification of the input ditance function for the etiation of Malquit productivity change inde, hen data contain obervation ith zero value. In thi ork, e propoe a odel baed on the quadratic cot function that i particularly adequate in ulti-product activitie here oe copanie do not produce oe output and/or do not ue oe input. In addition, thi ne inde include technical and allocative inefficiency, and i developed for dicrete data. In the net ection, e build pecific indee for input and output for thi functional for. Hence, the evolution of productivity i decopoed a a function of technical change, return to cale, variation in technical and allocative efficiency, and the bia that te fro the anner by hich input have been aggregated. In the third ection e apply the odel developed uing data fro the Spanih tevedoring indutry to calculate the ource of productivity ith fir and tie variability. The final ection contain a ynthei of the ot relevant concluion..- The odel The rate of change in productivity i defined a TFP Q F [] here TFP i the Total Factor Productivity Inde, Q and F are aggregate indee of input and output repectively, and dot variable indicate rate of change. 3

6 4 We no obtain and decopoe F. Let u tart by epreing the actual cot in period,, a a function of price,, and quantitie of input,, ued in period and, and their increent. ) )( ( [] Taking coon factor of the lat to ter of the previou epreion, the rate of variation in the actual cot beteen to conecutive period can be epreed in ter of dicrete variation, a: t G [3] here: G, and i the hare of input in the cot. Rearranging equation [3], e obtain the folloing epreion: G [4] In the ae ay, if e take the coon factor in [] of the econd and fourth ter, e obtain: [5] Adding [4] and [5] give: ( ) G F ) ( [6]

7 here F i an iplicit and aggregate inde of input that can be interpreted in to ay. So, the firt ter of epreion [6] indicate that F i a eighted u of the quantitie of input ued in both period, herea the econd eleent highlight the fact that thi iplicit input inde repreent the change in the level of actual cot that i not eplained by variation in the price of input. We no conider the deterining factor of F in greater depth. More pecifically, our interet i focued in decopoing in epreion [6]. To thi end, actual cot,, i epreed a: [7] E here i iniu cot and E Farrell cot efficiency inde. Epreing [7] in variation rate, ( ) E ( E E) E E E EE E E E E E EE E ( ) E E EE E E E E E E [8] Finally, E E E [9] 5

8 Hence, the actual cot variation rate,, depend on the level of cot efficiency in the to conecutive period, on the rate of change of the optiu cot, inde,, and on the efficiency E. Epreion [9] ho that either an increae (reduction) in optiu cot or a fall (increae) in the level of efficiency ill lead to an increae (reduction) in actual cot. We then decopoe the rate of change of both optiu cot and the efficiency inde E, a a function of the dicrete variation rate of their repective deterining factor. To thi end, in the decopoition of ued, hich give the folloing epreion:, the reult obtained by Martinez-Budria et al (3) i ( G ) H q T T ( t ) t [] here: G, and are the optiu hare of input in the period and H q ε ε, ε q c, q c, q, q i the cot-product elaticity of output, q i it variation rate, and T t i the rate of technical change. We then decopoe don the variation rate of Farrell cot efficiency inde, E. E E ( ET EA ) E E ET EA EA ET EA ET EA ET EAET EA ET EA ET [( ET ET ) ( EA EA )] ET EA [] 6

9 7 Thi epreion ho that change in cot efficiency depend on the dicrete variation rate of the technical efficiency inde, ET, and allocative efficiency inde, EA. If e introduce the reult found in [9], [], and [] in epreion [6], e get an iplicit and aggregate inde of input, F, that contain four additive ter: ( ) ( ) ( ) G G E E EA ET ET EA E E t t T T E E q H E E F [] The firt of thee ter depend on the degree of econoie of cale, the econd i related to technical change, the third i a function of change in the indee of technical and allocative efficiency, hile the final ter i a function of the difference beteen optiu hare,, and actual hare, to capture the biae in the iplicit inde of input caued by uing oberved hare in cot a a eighting hen there i allocative inefficiency. Martinez-Budria et al (3) propoe an aggregate inde of output hich ue cot-product elaticitie for eighting the rate of change of product, that e ue belo: ε ε ε ε,q,q,q,q H q H ) q q ( )q q q ( M [3] Finally, by introducing [] and [3] in epreion [], e get:

10 8 ( ) ( ) ( ) G G E E EA ET ET EA E E t t T T E E H E E H q H F M TFP [4] Thi for of decopoing productivity can be interpreted a follo:.- The firt coponent account for the effect of the return to cale on productivity and conit of to ter: an aggregate inde of output ( M ) and the unit inu the eighted ean of the cot product elaticitie in period and. If e aue that the indee of efficiency are contant over the to conecutive period and the cot function ho contant return to cale, an equi-proportional increent in all output ill not affect productivity. On the other hand, a radial increae in the production vector ill lead to an increae (decreae) in productivity if there are increaing (decreaing) econoie of cale. Thi reult coincide ith the reult obtained by Denny et al (98) for the cae of continuou variation in variable..- The econd ter i a eighted average of the technical change that ha occurred in the to period. The hift of the cot boundary caued by technical change ho that it i poible to obtain any level of production at a leer (greater) cot in the cae of technical progre (regreion), hich, in turn, ill lead to an iproveent (decline) in productivity.

11 3.- The third addend ho to hat etent change in productivity are caued by variation in technical and/or allocative efficiency. So, if a better (ore) ue of reource and/or choice of productive proce i achieved, increae (decreae) in productivity ill be attained. 4.- The final coponent i different fro thoe entioned above and repreent the biae in the eaureent of the rate of change in productivity generated by chooing an input hare other than the optiu hare a a conequence of allocative inefficiency. It i not therefore a deterinant of the productivity but a bia in the eaureent of the ae one. 5.- When there are change in cot efficiency indee beteen to conecutive period E ( ), epreion [4] highlight the contribution of each of the four factor that E deterine productivity, hich ill be biaed if aid change are ignored. In thi ene, hether one erroneouly adit efficiency in the to period or hether one aue that inefficiency i identical ( E E ), etiation of the effect of technical change, return to cale, and bia related to input aggregation ill be ditorted. Finally, note hould be taken that in cae that the cot inefficiency indee are invariable beteen the to period, ( E E < ), and the indee of technical and allocative inefficiencie do not change ET EA, the contribution of technical change and of return to cale to the productivity inde ill be alo correctly calculated uing [4]. 3.- Application to Spanih tevedoring indutry The Spanih tevedoring indutry a refored in the late 98 ith the creation of State Stevedoring opanie (SEED, fro it initial in Spanih). Thi refor entailed a reduction in the nuber of tevedore and in deregulating the copoition of ork tea. A in the ret of the orld, the driving force behind thi legilative refor a the profound technological 9

12 change that the increaing ue of the container repreented for cargo handling operation (Talley, ). Thi reulted in ignificant invetent in odern echanical equipent capable of ore rapidly tranferring erchandie beteen hip and hore. Thu, the change introduced by the legilative refor hint at the eitence of change in productivity, hoe eaure and decopoition are the ain obective of thi application. A hon in Jara-Díaz et al (6, 8), a proper knoledge of a port production tructure require adopting a ulti-productive approach. In the pecific cae of cargo handling ervice, thi require differentiating the ervice provided in order to package the cargo, ince thi i hat deterine the type of operation required and, thu, their cot. To that end, thi tudy ditinguihe aong three different output type: containerized general cargo (G), noncontainerized general cargo (NG) and olid bulk handled ithout any pecialized facilitie (SB). The Annual Statitic of the public agency Puerto del Etado (Port of the State) have been ued to obtain the inforation on the aount of erchandie handled annually at each port included in the aple, aid output level being epreed in ton/year. Providing thee tevedoring ervice require the ue of to production factor: labor and capital (crane). Each port Annual Report, along ith a quetionnaire ent to the oner of the privately-held echanical reource, alloed u to copile the inforation on the crane operating hour/year, a ell a on the cot aociated ith thi type of achinery. The other data ource ued a a urvey ent to all of the SEED, hich are reponible for organizing the ork required to accoplih thee tak. Thi urvey provided u ith the relevant inforation on the labor factor a it pertain to labor cot and to the hour/year orked by the peronnel involved in thee port operation. The cot analyzed ere capital ependiture (crane) and the ork cot aociated ith handling the cargo traffic entioned above. The cot i epreed in illion of 998 peeta, deflated uing Spain conuer price inde (IP) a publihed by the Spanih National

13 Statitic Intitute. Indicator for the price of factor ere derived by calculating the average cot of each input. Thi a done by uing the inforation on the total cot of each factor along ith the nuber of hour/year a a eaure of the aount of input utilized. Uing thee data, e proceeded to contruct unbalanced panel data copriing 9 port in Spain port yte for the period fro 99 to 998, incluive. The port included in thi tudy ere: Algecira, Alicante, Bilbao, adiz, artagena, atellon, Gion, Huelva, La oruña, Malaga, Mallorca, Alcudia, Motril, Pontevedra, Tenerife, Santander, Seville, Valencia and Vigo. Thi data aple i ufficiently repreentative of the Spanih port yte alloing u to dra oe concluion about the evolution of the tevedoring indutry a a hole and the change brought by refor. Rodríguez-Álvarez et al (7) tudie cargo handling ervice, including infratructure ervice, in a port terinal level uing onthly data of one Spanih port. They etiate technical and allocative efficiency ith fir and tie-varying variability uing a paraetric ditance function. The econoetric odel etiated, hich ill perit u to identify the paraeter required to decopoe the TFP uing equation [4], i preented in greater detail in Díaz-Hernández et al (8a). The technical and allocative inefficiencie ere incorporated into the odel uing the paraetric approach developed by Atkinon and ornell (994a,b). We alo adapted the decopoition of both inefficiency type to the noralized quadratic function, a propoed by Kubhakar (997) uing the tranlog functional for. We no briefly decribe the ot relevant apect of the econoetric odel ued, hich i contructed baed on the noralized quadratic hado cot function (NQSF), epreed a n P n n αipi αrpk Yr αipi αrlpk YrY l i r i k k Pk r l ( P, Y, t) b b n n αirpi Yr αitpi t αrtpk Yr t αtpk t αttpk t i k r i k r [5]

14 here Y(Y,, Y n ) be a vector of n output, X(X,, X ) a vector of input and P the correponding price vector, ith P (k P,..., k P ) a the hado price vector for hich the cobination of actual input i allocatively efficient. k i i a paraeter that indicate ho the relative actual price ratio of input i to input deviate fro the relative hado price ratio and defined uing the k it ( η η t) i it tie variability odel, hich ill depend on the etiate for η i and η it. Alo, <b i the paraeter that correct input oriented technical inefficiency 4. Moreover, in (5) e have added a tie trend a a proy variable repreenting technical change that interact ith output level and the price of input. A to-tage econoetric etiation proce baed on the NQSF a ued. In the firt tage, the equation for the input deand ratio ere etiated, hich alloed u to identify the paraeter of the cot function that characterize the technology (α), a ell a the paraeter that account for the allocative inefficiency ith it correponding tie variability (η L,η LT ). A e ill conider only to input, the odel to be etiated in the firt tage i the ratio beteen capital (crane) and labor deand equation, ith the price of the forer being ued to noralize. Uing capital and labor deand obtained trough the application of the Shephard lea to the NSF, the econoetric odel to be etiated i X X a Kft a Lft P b( α α Y α η η t ( ) α Y Y α Y t α t α t ) n n n n Lft k r rft LL ( L Lt ) rl rft lft rt rft t tt r PKft r l r α f D f υ ft P b ( α α η η t α Y α t) n Lft L LL ( L Lt ) Lr rft Lt PKft r (6) 4 We have odelled technical efficiency uing a radial-approach, o thee eaure hold the relative proportion of input contant. An alternative non-radial technical efficiency eaure ha been conidered (ee Färe and Lovell, 978). In thi line, hen (3) decopoe the Malquit productivity inde uing Data Envelopent Analyi. Thi non-radial efficiency eaure i not invariant to the unit of eaureent and o a change in unit affect the efficiency core.

15 here D f i a duy variable for fir f introduced to control for unoberved port heterogeneity affecting deand ratio. Finally, υ ft i a tandard noie error ter ith zero ean. The reult of the etiate are hon in Appendi 5. In the econd tage of the etiation proce, actual cot i regreed veru the u of the aduted optial cot level ˆ ( ) and the aduted allocative inefficiency cot ( ˆ al ). The right hand ide i calculated fro the reult in the firt tep uing the paraeter reported in Appendi. Then the technical inefficiency paraeter b it are etiated fro the folloing equation: b D ˆ ˆ ξ [7] a al ft ( / ft ) f ft ft ft f here D f i a f-port duy variable, (/ b) ft β f β ft t in order to account for port and tie variability, and ξ ft i a tandard noie error ter ith zero ean. The reult of the etiate of thi econd tage are hon in Appendi. Note the accuracy of the fit and the ignificance of the etiated odel Decopoition of the TFP change Baed on the reult of the to-tage etiate, the different coponent in epreion [4] are calculated o a to eaure the contribution of the technical change, of the cale effect, and of the technical and allocative efficiencie to the change in the TFP ith port and tie variability. The aggregation biae ere negligible, and thu not included. Likeie, the product of AE and ET a negligible, a a reult of hich it a poible to etiate the 5 See Díaz-Hernández et al (8a) for a ore coprehenive dicuion on the analyi of the reult of thi etiation. Of particular note i the good fit obtained and the tatitical ignificance of the odel. Alo verified a the copliance ith the onotonicity and curvature propertie of the hado cot function. The noralized quadratic hado cot function correpond to a ell-behaved production function becaue it i onotonically increaing in hado input price and output quantitie and concave in hado input price. 3

16 contribution of the allocative inefficiency eparately fro the technical inefficiency in the third addend of [4]. The reult for thi decopoition of TFP for the different year analyzed are hon in Table. Table about here The accuulated productivity groth for the entire period a 4.8% at an average annual rate of 5.35%. The technical change account for 58.5% of thi increae, hile iproved ue of the production cale account for.% of the TFP total. Iproved efficiency repreent the reaining 3.4% increae in productivity, ith the technical efficiency providing approiately double the gain of the allocative efficiency. An analyi of the evolution in tie over the period in quetion ho a light variability, given the all ocillation of the value around the average. The coparatively carce contribution of return to cale over a period ith relatively high traffic fluctuation i due to the fact that the average port ha a degree of econoie of cale of. (Díaz-Hernández et al, 8b), eaning it i in the area of contant return. That i hy thi effect i not very pronounced ecept during large variation in traffic, hich only occurred during the receion. The iproveent in technical efficiency contributed a ignificant part, 9%, of the change in productivity. It a poible to identify to ub-period. Fro 99 to 99, there a a noticeable deterioration in the ue of production reource, evidence of the difficulty of adapting the labor force available during period of decreaing production. Starting in 993, hen traffic tarted to rebound, e ee a ignificant annual iproveent in the technical efficiency, hich proved better able to adapt to riing than to falling factor. A iilar pattern i een ith the allocative efficiency, though ith a reduced influence on TFP. 4

17 In uary, cargo handling operation in Spanih port iproved ignificantly in ter of productivity, attributable ainly to the effect of the technical change and, to a leer etent, to a drop in inefficiency. A detailed analyi of the productivity coponent for each of the port analyzed offer the poibility of gaining a greater undertanding of oe of the factor influencing productivity in cargo handling operation. Thi raie the quetion regarding the eitence of difference in productivity aong Spanih port. With thi obective in ind, the decopoition of the rate of change of productivity for each of the port analyzed i hon in Table, here e ee the relatively lo cattering in the TFP rate of change aong Spanih port for the tie period in quetion. The port that ehibited ignificant econoie of cale ho a coniderable contribution to productivity, reulting fro increaed activity during thi period. The productivity value for the port of Alcudia, Pontevedra, Mallorca and atellon, in particular, hoed coniderable iproveent due to better ue of their production cale. The port of adiz, Malaga, Motril and Santander ehibit dieconoie of cale (Díaz-Hernández et al 8b), hich eplain hy the increae in the aount of cargo handled negatively affected their productivity. Table about here Secondly, e note the difference that eit beteen the port in ter of contribution ade by iproved technical efficiency. It a the larger port analyzed in thi tudy (Algecira, Valencia and Bilbao) that eperienced a loer contribution fro advance in productivity aociated ith advance in technical efficiency. Thi ee to be becaue thee port, over the coure of the period in quetion, enoyed elevated level of technical efficiency that ere aintained over tie. We hould alo note that thee port have a greater preence in handling international traffic, and it i poible that the copetitive preure in thee arket 5

18 induce greater incentive for the efficient ue of the factor utilized in thee tak. We alo note the advance achieved by port uch a Malaga and Alicante that, depite having tarted ith lo technical efficiency indice, achieved ignificant iproveent in thi area. With regard to iproveent in allocative efficiency, of note i it inability to eplain the change oberved in productivity. Furtherore, no ignificant difference ere oberved on a port-byport bai. 4.- oncluion We have built a pecific inde of productivity for a quadratic cot function hich ha been developed for dicrete data through the introduction in the odel of the allocative and technical inefficiencie a coponent of the inde, beide the contribution of technical change and econoie of cale. Thi fleible functional for allo a correct treatent of the output and/or input ith zero value. Moreover, taking into account the inefficiency in the odel, e avoid ditortion on production and e can properly calculate the contribution to productivity groth of different ource. We alo applied thi inde to Spanih tevedoring indutry for the period fro 99 to 998, iediately folloing the legilative overhaul of the tevedoring indutry in the late 98. The average port a a 4.8% increae in productivity over the period analyzed. The firt eplanation for thi a the technological change, hich contributed 5 percentage point to thi increae. The econd ource behind the iproved productivity a better efficiency, both technical and allocative, in the ue of production factor. Thi accounted for 3.4%. In thi ene, both the drop in output factor a ell a the iproved long-ter adaptation by the copanie to the price of the factor contributed to thi increae in productivity. Finally, there a the ore efficient ue of the production cale, hich accounted for.% of the change in productivity. A port-by-port analyi ho a change tructure for productivity 6

19 iilar to that found for the average port, that i, ot of the iproveent a due to the technical change and, to a leer etent, to change in efficiency. Acknoledgeent We ih to thank the valuable coent and ueful uggetion ade by eber of the Group of Reearch in Efficiency and Productivity Analyi at the Univerity of Oviedo in Spain. 7

20 Appendi. Etiate of deand ratio equation paraeter Paraeter Etiate t ratio Port Duy Etiate t ratio α K. 4.5 DAlgecira α L.5.36 DAlicante α LL DBilbao α G.7.53 Dádiz α NG Dartagena α SB Datellón α T DGión α GG DHuelva.7 6. α NGNG DLa oruña α SBSB..9 DMálaga α GNG DMallorca.4.5 α NGSB DAlcudia.98.8 α GSB DMotril α GPL.9. DPontevedra α NGPL DTenerife α SBPL DSantander α TT DSevilla α TPL DValencia.4 5. α TG DVigo α TNG..454 α TSB.7.3 η L η Lt Source: Díaz-Hernández et al (8a) 8

21 Appendi. Etiate of technical efficiency paraeter Port β i Etiate t ratio β it Etiate t ratio Algecira Alicante Bilbao ádiz artagena atellón Gión Huelva La oruña Málaga Mallorca Alcudia Motril Pontevedra Tenerife Santander Sevilla Valencia Vigo Source: Díaz-Hernández et al (8a) 9

22 Reference Atkinon S., ornell., 994a. Paraetric etiation of technical and allocative inefficiency ith panel data. International Econoic Revie, 35, Atkinon S., ornell., 994b. Etiation of output and input technical efficiency uing a fleible functional for and panel data. International Econoic Revie, 35, Battee, G.E., 997. A note on the etiation of obb-dougla production function hen oe eplanatory variable have zero value. Journal of Agricultural Econoic, 48, 5-5. Bauer, P.W., 99. Decopoing TFP groth in the preence of cot inefficiency, non contant return to cale, and technological progre. Journal of Productivity Analyi,, hen, Y., 3. A non-radial Malquit productivity inde ith an illutrative application to hinee aor indutrie. International Journal of Production Econoic, 83 (), Denny, M., Fu,M., Waveran, L., 98. The eaureent and interpretation of total factor productivity in regulated indutrie, ith an application to anadian telecounication. In: oing, T.G., Stevenon, R.E., (Ed.). Productivity eaureent in regulated indutrie, Ne York: Acadeic Pre, Díaz-Hernández, J.J., Martínez-Budría, E., Jara-Díaz, S., 8a. Paraetric inefficiency in cargo handling in Spanih port. Journal of Productivity Analyi, 3, 3-3. Díaz-Hernández, J.J., Martínez-Budría, E., Jara-Díaz, S., 8b. The effect of ignoring inefficiency in the analyi of production: the cae of cargo handling in Spanih port. Tranportation Reearch, Part A, 4 (), 3-9. Dieert, W.E., 976. Eact and uperlative inde nuber. Journal of Econoetric, 4, Dieert, W.E., 978. Superlative inde nuber and conitency in aggregation. Econoetrica, 46 (4),

23 Färe,R., Lovell,.A.K., 978. Meauring the technical efficiency of production. Journal of Econoic Theory, 9, 5 6. Hannula, M.,. Total productivity eaureent baed on partial productivity ratio. International Journal of Production Econoic, 78 (), Jara-Díaz, S., Martínez-Budría, E., Díaz-Hernández, J.J., 6. Multiple output in port cot function. In ullinane, K., Talley W.K., (Ed.). Port Econoic. Reearch in Tranportation Econoic, (6), Elevier, Jara-Díaz, S., Tovar, B., Truillo, L., 8. On the proper odelling of ultioutput port cargo handling cot. Applied Econoic, 4 (3), Kubhakar, S.., 997. Modelling allocative inefficiency in a tranlog cot function and cot hare equation: an eact relationhip. Journal of Econoetric, 76, (-), Martínez Budría, E., Jara-Díaz, S., Rao-Real, F.J., 3. Adapting theory of productivity to the quadratic cot function. An application to the Spanih electric ector. Journal of Productivity Analyi, (), 3-9. Murillo-Zaorano, L.R., Vega-ervera, J. A.,. The ue of paraetric and nonparaetric frontier ethod to eaure the productive efficiency in the indutrial ector: A coparative tudy. International Journal of Production Econoic, 69 (3), Rodríguez-Álvarez, A., Tovar, B., Truillo, L., 7. Fir and tie varying technical and allocative efficiency: An application to port cargo handling fir. International Journal of Production Econoic 9 (-), Talley, W.K.,. Ocean container hipping: ipact of a technological iproveent. Journal of Econoic Iue, XXXIV (4),

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25 Table.- Decopoition of the TFP change Period Technical hange Scale effect Technical efficiency effect Allocative efficiency effect TFP change ,7,49 -,8, 5, ,8 -,66 -,83 -,4, ,6 -,75,36,56 6, ,53,3,,79 6, ,95,9,96,44 6, ,9,8,97,76 5, ,98,74,7,54 5, ,3,7,59,64 5,8 Mean 3,3,54,9,6 5,35 Table.- Decopoition of the rate of change of productivity by port Port Technical hange Scale effect Technical efficiency effect Allocative efficiency effect TFP change Algecira,87,85,59,9 4,6 Alicante 3,,8,47, 5,96 Bilbao,94,46,99,9 5,3 ádiz 3,6 -,59,,67 4,7 artagena 3,3,5,55,39 5,33 atellón 3,7,4,,47 5,89 Gión 3,3,8,6,5 5,78 Huelva 3,,3,8,79 5,85 oruña 3,7,47,4,9 5,78 Málaga 3,63 -,3,69,7 5, Mallorca 3,8,,,44 5,84 Alcudia 3,38,6,3, 6,5 Motril 3,6 -,4,34,83 5,9 Pontevedra,75,3,5,8 5,9 S/ Tenerife,6,5,8,3 4,99 Santander 3, -,37,9,9 4,65 Sevilla 3,9,4,87,47 5,3 Valencia 3,5,63,64,39 4,7 Vigo 3,,53,4, 5,8 Mean 3,3,54,9,6 5,35 3