A toolbox for calculating and decomposing Total Factor Productivity indices

ContentslistsavailableatScienceDirect

Computers and Operations Research

journalhomepage:www.elsevier.com/locate/cor

A toolbox for calculating and decomposing Total Factor Productivity indices

Bert M. Balk

^a

, Javier Barbero

^b,c,∗

, José L. Zofío

^d,e

a Rotterdam School of Management, Erasmus University, Netherlands

b European Commission, Joint Research Centre (JRC), Spain

c Oviedo Efficiency Group, University of Oviedo, Spain

d Universidad Autónoma de Madrid, Spain

e Erasmus Research Institute of Management, Erasmus University Netherlands

a rt i c l e i n f o

Article history:

Received 7 February 2019 Revised 16 September 2019 Accepted 9 November 2019 Available online 27 November 2019 Keywords:

Total factor productivity Malmquist

Moorsteen–Bjurek Fisher

Törnqvist

Data envelopment analysis

a b s t r a c t

TotalFactorProductivityToolboxisanewsetoffunctionstocalculatethemainTotalFactorProductivity (TFP)indicesandtheirdecompositions,basedonShephard’sdistancefunctions,andusingDataEnvelop- mentAnalysis(DEA)programmingtechniques. Thepackageincludes codeforthe standardMalmquist, Moorsteen–Bjurek,price-weightedand share-weightedTFPindices, allowingforthe choiceoforienta- tion(inputoroutput),referenceperiod(base,comparison,geometricmean),returnstoscale(variableor constant),andspecificdecompositions(aggregate,oridentifyingscaleeffects,aswellasinputandout- putmixeffects).ClassicdefinitionsofTFPcorrespondingtotheLaspeyres,Paasche,Fisher,orTörnqvist formulascanalsobecalculatedasparticularcases.Thispaperdescribesthemethodologyandimplemen- tationoftheproductivityfunctionsinMATLAB.Wecomparetheresultscorrespondingtothedifferent definitionsbystudyingproductivitytrendsintheUSagricultureattheindividualstatelevel.

© 2019TheAuthor(s).PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBYlicense.(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Total factor productivity (TFP) change is an important con- cept in economics because it measures the ability of firms, in- dustries, and national economies to increase the aggregate vol- ume of outputs they yield, relative to the aggregate volume of inputsthey use.TFP constitutesa standard instrumentfor moni- toringandbenchmarkingobservations,andrepresentsthecorner- stoneinmultilateralstudiesoftechnologicalandeconomicperfor- mance,OECD(2001).Definedastheratioofanoutputquantityin- dextoan inputquantity index,TFP changecanbe calculatedand decomposedinvariousways.

TFP measurement has grown in importance over the past decades due to the increasing availability of data to study the productive performance of units,regardless of their market, gov- ernmental, ornot-for-profitorientation. However, there aremany waystomeasureTFPdependingonwhetherquantitiesandprices are available.TFP measurementrequires theaggregationofquan- tities throughsuitable functions. Ifprices are available, itis pos-

∗ Corresponding author.

E-mail address: [email protected] (J. Barbero).

URL: http://www.javierbarbero.net (J. Barbero)

sible to aggregate output and input quantities by index formu- las of Laspeyres, Paasche, Fisher, or Törnqvist. If only quanti- tiesare available, the concept of distance function asintroduced by Shephard emerges as the building block in the definition of MalmquistandMoorsteen–Bjurekindices.Aggregationthen relies on optimization—mathematicalprogramming—techniques such as thenon-parametricDataEnvelopmentAnalysis(DEA,Cooperetal., 2007).

There are various ways to decompose TFP change to identify thecomponents. Thisapplies both tothe classical definitionsus- ing prices as aggregators, and those relying only on quantities through distance functions. These components are (technical)ef- ficiencychange,technological change,scaleeffect,andchanges in the mix of inputs and outputs. The various definitions of these terms that have been proposed over the years gave rise to a healthydebatebetweenauthors,including(Balk,2001;2003;Färe etal.,2008;1994;Lovell,2003;RayandDesli,1997;Zofío,2007), and, more recently, (O’Donnell, 2018). Balk and Zofío (2018) ex- amine how to identify those terms that allow a meaningful in- terpretation and decomposition of TFP in a general framework.

Althoughthe presentpaperis self-contained,the toolbox follows theseauthors,wheremoredetailedtheoreticalandempiricalcon- siderationscanbefound.Thetoolboximplementsseveralfunctions https://doi.org/10.1016/j.cor.2019.104853

0305-0548/© 2019 The Author(s). Published by Elsevier Ltd. This is an open access article under the CC BY license. ( http://creativecommons.org/licenses/by/4.0/ )

to calculate the main quantity-only and price-based TFP indices proposed in the literature, along withtheir associated decompo- sitions. The toolbox allows for a choice of orientation (input or output),referenceperiod(base,comparison,geometricmean),re- turnsto scale (variable or constant),andspecific decompositions (aggregateor identifying scale effects, as well as input and out- putmix effects).Thecodecalculates Shephard’sinput andoutput distancefunctionsapproximatingaproductiontechnologyempiri- callythroughDEA.ThetoolboxalsocalculatesclassicTFPmeasures thatdonotrelyondistancefunctionsfortheirdefinition,including Laspeyres,Paasche,Fisher,andTörnqvist.

Quantity-only TFP indices based on distance functions, corre- spondingmainlytotheMalmquistindex,andcoupledoccasionally withtheMoorsteen–Bjurekindex, canbe found instandard soft- ware packages like Stata (StataCorp, 2015); through user-written commands as Lee et al. (2011); in LIMDEP (Econometric Soft- ware (2014); in dedicated non-commercial software accompany- ing academic handbooks such as in Cooper et al. (2007) and Bogetoft and Otto (2011) (implemented in R); in commer- cial software, including trial versions (O’Donnell, 2011 and Emrouznejad and Thanassoulis, 2011); in free-ware programs (Coelli,1998); andevenin tutorials forspreadsheets(Zhu, 2014).

FortheclassicTFPindicesusingpricesasaggregators,Shehataand Mickaiel(2015)developedaStatamoduletocalculatevalueindex numbers,while Coelli (1999) offered a stand-alone program that calculates Fisher and Törnqvist indices, including transitive ver- sionsusing the EKSmethod.O’Donnell (2011)expanded thislast optiontoLowe-typeindicesunderacommerciallicense.Recently, Dakpoet al.(2018) havecoded an R-basedproductivitypackage, based on O’Donnell (2011, 2012, 2018). However, while it covers manyofthedefinitionsconsideredinthistoolbox,thefactorscor- respondingto technological change,aswell asthe scale andthe input-andoutput-mix effects varyas resultofdifferencesinthe underlyingtheoretical models. As remarked by O’Donnell (2012), thereisapotentiallyinfinitenumberofexhaustivedecompositions ofaTFP index. Forexample,one familyofthese decompositions, identifiedbyDakpoetal.(2018)andO’Donnell(2011,2012,2018), definestechnological changeglobally asthe changein maximum productivitybetweentwotimeperiods,whileinthistoolboxalo- cal measure of technological change is considered. Consequently, therestoffactorsmustaccommodatethenumericaldifferencebe- tweenthe two. However,all these decompositionscan be rightly interpreted, while complementing each other. Hence, researchers can rely on one or another depending on their preferred defini- tions, and compare them using all the available packages. Addi- tionally,asanoveltyofthistoolbox,weofferthepossibilityofde- composingshare-weightedgeometricindices,whichisunavailable inpreviouspackages.Finally,acomprehensivereviewoftheavail- ablegeneralpurposeanddedicatedsoftwareoptionsforefficiency andproductivityanalysiscanbefoundinDaraioetal.(2019).

AlthoughthesepackagesimplementthemainTFPindices,there is a lack of a full set of functions for MATLAB, and none of them includes a complete decomposition of productivity change according to its multiple components. Thus, besides implement- ing the basic TFP definitions based on quantities and prices in the MATLAB environment, our toolbox calculates a large array ofindexnumberscapturing (technical)efficiencychange,techno- logicalchange,aswell asscaleeffects, andinput andoutputmix effects.Quantities-only Malmquist andMoorsteen–Bjurek indices, as well as price-based Fisher and Törnqvist indices, are decom- posedintomutuallyexclusivefactorswithmeaningfulinterpreta- tions interms of economic indexnumber theory.The Total Fac- tor Productivity Toolbox introduces a set of functions, covering a wide rangeof TFP indices, andreportsnumerical results using a common example to ease comparability and to illustrate their use.Thetoolboxisavailableasfreesoftware,undertheGNUGen-

eralPublic License version 3,andcan be downloaded fromhttp:

//www.tfptoolbox.com,withallthe supplementarymaterial (data, examples, andsource code) to replicate all the results presented inthispaper.The toolboxiscompatiblewiththe2017b(8.5)ver- sionofMATLAB,(TheMathWorks,Inc.,2017).TheU.S.agricultural dataemployedinthisarticle,aswellthecodeusedtocalculatethe variousproductivityindicesareavailableassupplementaryon-line material.Thetoolbox isalsohostedonan opensourcerepository onGitHub.¹

Thispaperisorganizedasfollows.Section2presentsthedata structurescharacterizingtheproductionpossibilitysets,thestruc- tureofthefunctions,results,andtheU.S.agriculturaldatathatis usedasreal-casestudyandtoillustratethetoolbox.Section3cov- ers the Malmquist productivity indices, relying on radial output or input distance functions. We show how these indices can be decomposedintofactors withmeaningfulinterpretations, suchas technicalefficiencychange,technologicalchange,scale effect,and inputandoutputmixeffect.Wealsorelateandinterpretthesefac- torsintermsoftheoutputcodethatisobtainedwhenrunningthe specificfunctions.Malmquistproductivityindicestakeintoconsid- eration onlyone ofthetwo sidesoftheproductionprocess, out- put or input. In Section 4 we consider the class of Moorsteen–

Bjurekindices,definedasratioofanoutput quantityindexto an input quantityindex, whichinturncan be expressedintermsof output andinput distancefunctions, respectively.We presentthe decompositionoftheseindicesusingvariousalternatives toiden- tifythe above effects.If input and output prices are available, it is possible to calculate and decompose classical indices such as Fisher andTörnqvist. The price-weighted productivityindices are consideredinSection5.Ratherthanmultiplyinginputandoutput quantitiesbytheirprices,onecanaggregateindividualquantityra- tios by ageometric mean,weighing withcost orrevenue shares.

Section 6 deals with the class of share-weighted indices, whose bestknownrepresentativeistheTörnqvistproductivityindex.Ad- vancedoptions,includingdisplayingandexportingresultsaredis- cussedinSection2.2.Section8concludes.

2. Datastructures,outputtablesandempiricalanalysis 2.1. Datastructures

TotalFactor Productivity changemeasures the temporal varia- tion in the productive performance of a DMU (decision making unit such as a plant, firm, industry, or economy) over time. In time period t=0,1,...,T each DMU transforms a vector of in- puts x^kt∈R^N₊₊ into a vector of outputs y^kt ∈R^M₊₊ (^k=1,...,K)^. Piecewiselinearapproximationsoftheperiod-specifictechnologies can be obtained through Data Envelopment Analysis techniques.

Specifically, the constant returnsto scale (CRS)production possi- bilitysetintroducedbyCharnesetal.(1978)(CCR),correspondsto S^t=



x^t,y^t



|

^xtX^t

λ

, y^tY^t

λ

,

λ

0



, where X^t=(^xkt)∈R^N×K andY^t=(^ykt)∈R^M×Karematrices,and

λ

=(

λ

¹,...,

λ

^K)^isasemi- positive vector. Alternatively, the variable returns to scale (VRS) productionpossibilitysetintroducedbyBankeretal.(1984)(BCC), corresponds to S^t=



x^t,y^t



|

^xtX^t

λ

,y^tY^t

λ

,e

λ

=1,

λ

0



, where e is a row vector withall elements equal to 1. The only difference withtheCCRmodel istheadjunction ofthecondition

K

k=1

λ

k=1. If input and output prices are available, w^kt ∈R^N₊₊

andp^kt ∈R^M₊₊,we havethe followingpanel data structure: (w^kt, x^kt,p^kt,y^kt)(^k=1,...,K; t=0,1,...,T)^{.Dataaremanagedasreg-} ularMATLABvectorsandmatrices,constitutingtheinputsofthe estimationfunctionsthataredescribedinwhatfollows.

1 The address of the repository is https://github.com/joselzofio/TFPMATLAB .

All estimation functions return a structure

tfpout

^{that con-}

tainsfieldswiththeestimationresultsaswellastheinput ofthe estimation function.Fieldscan be accesseddirectlyusing thedot notation,andthewholestructurecanbeusedasaninputtoother functionsthatprintorexportresults(e.g.,

tfpdisp

).Someofthe fieldsofthe

tfpout

structurearethefollowing:²

•

X

and

Y

:Containtheinputandoutputquantities,respectively.

•

W

^and

P

^{:Containtheinputandoutputprices,}respectively.

•

n

:Numberofobservations.

•

m

and

s

:Numberofinputandoutputvariables,respectively.

•

orient

^,

period

^,

decomp

^{: Stringscontainingthe}orientation oftheTFPmodel,referenceperiod,anddecompositiondepend- ingonscaleandmixeffectsassumptions.

•

tfp.M

,

tfp.MB

,

tfp.PROD

, and

tfp.GPROD

: Computed Malmquist, Moorsteen–Bjurek, price-weighted, and share- weightedproductivityindices.

•

tfp.EC

, and

tfp.TC

: Computed technical efficiency change andtechnologicalchangefactors.

•

tfp.SEC

,

tfp.OME

,

tfp.IME

and

tfp.RTS

:Computedscale effect,output mixeffect,input mixeffect,andreturnstoscale factors.

•

names

:NamesoftheDMUs.

2.2. Displayingandexportingresults

2.2.1. Customdisplay

Throughoutthetext,weillustratehowtodisplayoutputtables bycallingthe

tfpdisp(out, dispstr)

functionaftercomput- ingacertaindecomposition.Inthetableheaderappropriateinfor- mationconcerningtheestimatedmodelisdisplayedonthescreen.

This setting can be changed to display bespoke information by specifyingin the

tfpdisp

function thestring

dispstr

(display string)asasecondparameter.

For example, the default

dispstr

^{after using the}

deatfpgprod

function with the geometric mean option is

names/tfp.GProd/tfp.EC/tfp.TC/tfp.IME/tfp.SEC

. The fields displayed in the output table must be separated by a

/

andinclude the names corresponding to the field names of the

tfpout

structure.The availablefieldsare thosepresentedin Table1.

2.2.2. Exportingresults

Results can easily be exported to various file formats for posterior analysis and sharing. First, the

tfpout

structure should be converted to a MATLAB

table

data type by us- ing the

tfp2table(out, dispstr)

^{function withthe desired}

dispstr

. If the

dispstr

parameter is omitted, the default is used.

The table can then be exported by using the function

writetable

^.3

2.3. AgriculturalproductivityintheUS

To illustrate the toolbox we compare the various productiv- ity trends in US agriculture obtained with the existing TFP def-

2 For a full list see the help of the function typing help tfpout in MATLAB .

3 See the official documentation for this function at http://www.mathworks.com/

help/matlab/ref/writetable.html .

Table 1

Fields of the tfpout structure available for the dispstr string.

Common fields

Field Data

names DMU names

EC Technical efficiency change

TC Technological change

Productivity indices

tfp.M Malmquist productivity index (after deatfpm ) tfp.MB Moorsteen–Bjurek productivity index (after deatfpmb ) tfp.Prod Price-weighted productivity index (after deatfprod ) tfp.GProd Share-weighted productivity index (after deatfpgprod )

Output orientation tfp.SEC Scale effect (geometric mean) tfp.OME Output mix effect (geometric mean) tfp.SEC_100 SEC o^t(x ¹ , x ⁰ , y ⁰)

tfp.OME_110 OME ^t ( x ¹ , y ¹ , y ⁰ ) tfp.SEC_101 SEC _o^t(x ¹ , x ⁰ , y ¹) tfp.OME_010 OME ^t ( x ⁰ , y ¹ , y ⁰ )

Input orientation tfp.SEC Scale effect (geometric mean) tfp.IME Input mix effect (geometric mean) tfp.IME_100 IME ^t ( x ¹ , x ⁰ , y ⁰ )

tfp.SEC_110 SEC _i^t(x ¹ , y ¹ , y ⁰) tfp.IME_101 IME ^t ( x ¹ , y ⁰ , y ¹ ) tfp.SEC_010 SEC _i^t(x ⁰ , x ¹ , y ⁰)

initions. The data has been collected and tabulated by the Eco- nomic ResearchSection(ERS) ofthe UnitedStatesDepartmentof Agriculture (USDA) in an effort to study long term productivity trends.Itcorresponds toa subset ofthestate-leveltablesinclud- ing price indicesand implicitquantities of farm outputs and in- puts. It is readily available in MATLAB format as supplemental on-linematerial to this article.We calculateproductivity growth between1960and2004,correspondingtothefirstandlastavail- ableyearsinthedataset.Thefulldata,correspondingtoTable#23, canbe downloaded fromhttps://www.ers.usda.gov/data-products/

agricultural-productivity-in-the-us/. Our dataset consists of three outputs(livestock,crops,andother farmrelatedoutput),andfour inputs(capital, land, labor, andintermediate inputs). More infor- mation on the data,including descriptive statistics anda discus- sionoftheagriculturalproductivitygrowthintheUScanbefound intheabovelink.Duetoitscomprehensivenessandreliability,this datasethasbeenusedovertheyearsbymanyauthors,whosere- sultsarecomplementedwiththoseobtainedbysolvingthemodels includedinthistoolbox;see,tocitebutafew,Färe etal.(2008), Balletal.(2001), andZofíoandLovell (2001).Once downloaded, dataoninputandoutputquantitiesandpricescanbebroughtinto theworkspaceby runningthefollowingcode includedintheex- amplefile:

3. Malmquistproductivityindex

Whenonlyquantitiesareavailable themostpopularmeasure- ment instrument is the Malmquist productivityindex (MPI). The

Fig. 1. Visualization of the structures for model parameters and results.

MPI requires calculation of the output- or input-orientated dis- tanceofobservation(x^kt,y^kt)intwoconsecutiveperiods(say,base periodt=0andcomparisonperiodt=1)tothefrontierofacer- tainbenchmarktechnology exhibitingCRS.The impositionofCRS is necessary for the distance function to satisfy the homogene- ityconditions that ensure that theMPI hasthe required propor- tionalityproperties,seeBalkandZofío(2018).Thus ingeneralthe benchmarktechnologywilldifferfromanyactualtechnology.

Itisusualtotaketheconetechnologyofacertainperiodt,Sˇ^t, asbenchmark.ItsoutputdistancefunctionisdefinedbyDˇ^t_o(^x,y)≡ inf

{ δ | δ

>0,(^x,y/

δ

)∈Sˇ^t

}

^. Operationally, within the DEA frame- work,thiscan be calculated bysolving the program Dˇ^t_o(^x,y)⁻¹= max_φˇ

,λ

{ φ |

^{ˇ x}X^t

λ

,y

φ

ˇY^t

λ

,

λ

0

}

^{. Then} (^x,y/Dˇ^t_o(^x,y)) ^{is the} pointon the frontierof the periodt cone technologythat is ob- tainedbyholdingtheinputquantityvectorxconstantwhileradi- allyexpandingtheoutputquantityvectory.

The input distance function is defined as Dˇ^t_i(^x,y)≡ sup

{ δ | δ

>0,(^x/

δ

,y)∈Sˇ^t

}

, and can be calculated by solving the pro- gram Dˇ^t_i(^x,y)⁻¹=min_θˇ,λ

{ θ |

^{ˇ x}

θ

^ˇX^t

λ

, yY^t

λ

,

λ

0

}

^{. Then} (^x/Dˇ^t_i(^x,y),y) ^{is the point on the frontier of the period t cone} technologythat isobtainedby holdingtheoutputquantityvector y constant whileradially contracting theinput quantity vector x. NoticethatDˇ^t_i(^x,y)=1/Dˇ^t_o(^x,y)^.

The counterpartsoftheaboveoutput andinput distancefunc- tions,definedontheactualtechnologyS^twhichingeneralexhibits VRS,denoted byD^t_o(^x,y)^andDt_i(^x,y),arecomputedin thesame way,withe

λ

=1asadditionalconstraint.

3.1.Theoutput-orientatedMPI

The output-orientated MPI, conditional on the period t cone technology,foracertainDMU,isdefinedby⁴

Mˇ^t_o

(

^x1,y¹,x⁰,y⁰

)

≡Dˇ^t_o

(

^x1,y¹

)

Dˇ^t_o

(

^x0,y⁰

)

^{. (1)}

4 Here and in the sequel the superscript k , designating a specific DMU, is deleted to simplify presentation.

Selecting the base period cone technology then leads to Mˇ⁰_o(^x1,y¹,x⁰,y⁰),andselectingthecomparisonperiodconetech- nologyleadstoMˇ¹_o(^x1,y¹,x⁰,y⁰)^{.TheTFPtoolboxcalculatesboth,} aswellastheirgeometricmean.Letusstartwiththefirstoption.

3.1.1. Thebase-period-output-orientatedMPI

FollowingBalkandZofío(2018),who providemeaningfulthe- oreticalinterpretations forthedifferentfactors,the firstextended decomposition of the base-period-output-orientated MPI(termed

‘PathA’)is

Mˇ_o⁰

(

^x1,y¹,x⁰,y⁰

)

=ECo

(

^x1,y¹,x⁰,y⁰

)

× TCo^1,0

(

^x1,y¹

)

× SECo⁰

(

^x1,x⁰,y⁰

)

× OME⁰

(

^x1,y¹,y⁰

)

. (2) Inthisexpression thereare fourmutuallyindependentfactors, withthefollowinginterpretation:

• Efficiency change: ECo(^x1,y¹,x⁰,y⁰)=D¹_o(^x1,y¹)/D⁰o(^x0,y⁰), representingthechangeinthetechnicalefficiencyoftheDMU, alsoknownasthecatch-upeffect.

• Technological change: TC_o^1,0(^x1,y¹)=D⁰_o(^x1,y¹)/D¹o(^x1,y¹), capturingthe change in the actual technological frontier, also knownasthefrontier-shifteffect.

• Radial scale and input mix effect: SEC_o⁰(^x1,x⁰,y⁰)= [Dˇ⁰_o(^x1,y⁰)/D⁰o(^x1,y⁰)^]× [D⁰_o(^x0,y⁰)/Dˇ⁰_o(^x0,y⁰)^], corresponding toscaleefficiencyimprovements associatedtoradial increases intheinputquantities,andtheadditionaleffectfromchanges intheinputquantitymix.⁵

• Outputmix effect:OME⁰(^x1,y¹,y⁰)=[Dˇ⁰_o(^x1,y¹)/D⁰o(^x1,y¹)^]× [D⁰_o(^x1,y⁰)/Dˇ⁰_o(^x1,y⁰)^], showing the counterpart effect associ- atedtochangesintheoutputquantitymix.

Factors with values greater than 1 contribute to productivity growth (e.g., through technical efficiency gains or technological progress),whilefactorswhosevaluesaresmallerthan1aredetri- mental.

5Balk and Zofío (2018) show that SEC ⁰o(x ¹ , x ⁰ , y ⁰) can be decomposed into a ra- dial scale effect and an input quantity mix effect. The implementation in a DEA framework is however too complicated to be practically useful.

Analternative decompositionofthebase-period-output-orientatedMPIreversesthe orderinwhichchangesintheinput andoutput spacetakeplaceinthelasttwofactorsinexpression(2).Thisyields

Mˇ⁰_o

(

^x1,y¹,x⁰,y⁰

)

= ECo

(

^x1,y¹,x⁰,y⁰

)

× TC_o^1,0

(

^x1,y¹

)

× SEC⁰o

(

^x1,x⁰,y¹

)

× OME⁰

(

^x0,y¹,y⁰

)

, (3) which is termed‘Path B’. The differencesbetweenthis decompositionand that inexpression (2) are subtlebutnoteworthy. The parts capturingefficiencychangeandtechnologicalchangeareidentical.Thefactorcapturingtheradialscaleeffectandtheinputmixeffectis conditionalony⁰inexpression(2),butony¹inexpression(3).Thereversehappenswiththeoutputmixeffect;thiseffectisconditional onx¹ inexpression(2),butonx⁰inexpression(3).

Consequently,therearetwo,equallymeaningful,decompositionsoftheMalmquistproductivityindexMˇ⁰_o(^x1,y¹,x⁰,y⁰)^.Ifthereisno preferenceforoneofthemthenthegeometricmeanofexpressions(2)and(3)canbetaken,reading

Mˇ⁰_o

(

^x1,y¹,x⁰,y⁰

)

= ECo

(

^x1,y¹,x⁰,y⁰

)

× TC_o^1,0

(

^x1,y¹

)

× [SEC_o⁰

(

^x1,x⁰,y⁰

)

× SECo⁰

(

^x1,x⁰,y¹

)

^]1/2

× [OME⁰

(

^x0,y1,y0

)

^{× OME0}

(

^x1,y1,y0

)

^{]1/2. (4)}

TocomputetheMPIinMATLABtheusercallsthe

deatfpm(X, Y, ...)

functionwithi)the

orient

parametersettotheoutput orientation

oo

;2)the

period

parametersetto

base

;andiii)thedecompositionparameter

decomp

setto

complete

.Withtheoptional parameter

names

wecanspecifyacellstringwiththenamesoftheDMUs,whichinthiscasearethenamesoftheU.S.states.⁶

Themodelparametersandalltheancillaryinformationcanbefoundintheeditorunder

tfpm_oo_base_complete

^{.Theresultsare}

displayedinthe‘command window’,andthestructurewithall thevariables andresultscan be foundintheworkspace byclicking on

tfpm_oo_base_complete.tfp

.Fig.1showsthestructuresforthemodelparametersandresultsintheMATLABeditor.

TheoutputofthefunctionsuccessivelyshowstheMPI(

M

^{),thetechnicalefficiencychangefactor(}

EC

^),thetechnologicalchangefactor (

TC

), thegeometricmeanofthescaleeffectfactors(

SEC

),thegeometric meanoftheoutput mixeffectfactors(

OME

),andtheseparate valuesofthetwolastfactorscorrespondingto‘PathA’and‘PathB’above.Thesevaluesareidentifiedbythetimesuperscriptsoftheinput andoutputargumentsineachfactor;forexample,

SEC_100

correspondstoSEC_o⁰(^x1,x⁰,y⁰)ⁱⁿexpressions(2)and(4)above.Thefunction alsoreturns themaindescriptive statisticsofthe variousfactors.Finally,a relevantclarificationconcerns thenature ofreturnstoscale.

The Malmquistindexis definedwithrespecttoa conetechnology, independentlyofwhetherthe actual technologyischaracterized by globalCRSornot.ThusthedefaultdecompositionisbasedonVRStechnologies,whichisidentifiedbythecorrespondingreturns-to-scale header below;i.e., “

Returns to scale: vrs (Variable)

”. Thecaseofatechnologycharacterized by globalCRSis consideredin thefollowingsubsection.

We exemplify individual results by discussing the productivity trends of the first andlast three states. Moststates increased their agriculturalproductivity,Illinois(IL)leadingwitha592.00%increase,correspondingtothepercentagechangeofthemaximumMPIvalue.

ExceptionsexhibitingproductivitydeclineareOklahoma(OK),unreported,andWestVirgina(WV),witha18.23%reduction(theminimum value oftheMPI).The maincomponentofproductivitygrowth,on average215.99%,istechnological progress,whichgenerallyoutpaces efficiency gainsby far(on averagethey contribute 199.44% and11.01%, respectively). Changesin thescale ofproduction orthe output

6 If the optional parameter names is omitted, the DMUs will be numbered from 1 to n .

mixdonotplay asignificant role,implyingthat therehavenobeenrelevantchangesintheproductionscaleandtheinput andoutput structureofthestatesovertheforty-fouryearperiod1960–2004.Similarcommentscanbemadeforeachindividualstate.⁷

3.1.2. Relateddecompositionsofthebase-period-output-orientatedMPI

Theabovefour-factordecompositionscanberelatedtosimplerproposalsintheliterature.First,theygeneralizetheearlierproposalby RayandDesli(1997).Bymergingthescaleeffectandtheoutputmixeffectinexpressions(2)or(3),bothdecompositionsreduce tothe samethree-factordecomposition:

Mˇ⁰_o

(

^x1,y¹,x⁰,y⁰

)

= ECo

(

^x1,y¹,x⁰,y⁰

)

× TC_o^1,0

(

^x1,y¹

)

× RTS⁰

(

^x1,y¹,x⁰,y⁰

)

, (5)

whereRTS⁰(^x1,y¹,x⁰,y⁰)≡ [Dˇ⁰_o(^x1,y¹)/D⁰o(^x1,y¹)^]/[Dˇ⁰_o(^x0,y⁰)/D⁰o(^x0,y⁰)^{].Lovell(2003)suggeststhatthisfactormeasuresthe contribu-} tionof returns-to-scaleto productivitychange. Following thisinterpretation, ifit isgreater than 1, increasing returns toscale result in productivitygrowth,whileifit issmaller than1,decreasing returnsto scale aredetrimental. Ifequalto one,constant returns toscale prevailsattheobservedinput-outputvalues.

Toobtainthissimplerdecomposition,the syntaxisthesameasintheextendedone butwiththeparameter

decomp

setto

rts

.In theoutputtablethecolumn

RTS

containsthecontributionofreturns-to-scaletoproductivitychange.

Noticethatthecontributionofreturns-to-scaletoproductivitygrowth,consistentwithsmallvaluesofthescaleandoutputmixeffects, isquitelimited.InthecaseofWestVirginia(WV)theexistenceofdecreasingreturnstoscale,alongwiththesubsidenceoftheproduction frontier,associatedtovaluesoftechnologicalchangesmallerthanone,resultsinthepreviouslyreportedproductivitydeclinetothetune of−18.23%.

Second, the technicalefficiency change andtechnological change terms commonto expressions (2),(3), and(4) correspond to the base-period-output-orientatedindexproposedbyCavesetal.(1982)(CCD),

M⁰_o

(

^x1,y¹,x⁰,y⁰

)

≡ ECo

(

^x1,y¹,x⁰,y⁰

)

× TC¹_o^,0

(

^x1,y¹

)

=D⁰_o

(

^x1,y¹

)

D⁰_o

(

^x0,y⁰

)

^{. (6)}

Substitutingexpression(6)inanyoftheabovedecompositionsyieldsalternativeexpressions,

Mˇ⁰_o

(

^x1,y1,x0,y0

)

^{= M0}o

(

^x1,y1,x0,y0

)

^{× SEC}o⁰

(

^x1,x0,y0

)

^{× OME0}

(

^x1,y1,y0

)

⁽⁷⁾ Mˇ⁰_o

(

^x1,y¹,x⁰,y⁰

)

= M⁰_o

(

^x1,y¹,x⁰,y⁰

)

× SECo⁰

(

^x1,x⁰,y¹

)

× OME⁰

(

^x0,y¹,y⁰

)

⁽⁸⁾

Mˇ⁰_o

(

^x1,y¹,x⁰,y⁰

)

=M⁰_o

(

^x1,y¹,x⁰,y⁰

)

× [SEC_o⁰

(

^x1,x⁰,y⁰

)

× SECo⁰

(

^x1,x⁰,y¹

)

^]1/2× [OME⁰

(

^x0,y¹,y⁰

)

× OME⁰

(

^x1,y¹,y⁰

)

^]1/2. (9) Toobtainthisdecompositiontheparameter

decomp

mustbesetto

ccd

.

7 Descriptive statistics, productivity distributions, and associated graphs, can be easily calculated and plotted in MATLAB .

In the extant literature on productivity measurement, the CCD index frequently figures under the name ‘the (output orientated) Malmquist productivity index’. However, the CCDindex, expression(6), cannot be regarded asa productivity index. First, itcannot be writtenasameasure ofaggregateoutput growthdivided bya measureofaggregateinput growth,the correspondingaggregatorsbeing non-negativenon-decreasinglinearlyhomogenousscalarfunctions.O’Donnell(2012)usestheterm“multiplicativelycomplete” toreferto TFPindexesconstructedinthisway.Moreimportantly,theCCDindexdoesnotsatisfytheproportionalityproperty,asinitiallyshownby Grifell-Tatjé andLovell(1995),unlesstheactualbaseperiodtechnologyexhibitsCRS.Inthespecificcaseofa technologysatisfyingCRS, alltheSECandOMEtermsinexpressions(7),(8),and(9)becomeidenticalto1,andtheMPIMˇ⁰_o(^x1,y¹,x⁰,y⁰)^{reducestotheCCDindex} M⁰_o(^x1,y¹,x⁰,y⁰)^{.TheMPIisthendecomposedasinitiallyproposedbyFäreetal.(1994),8}

Mˇ⁰_o

(

^x1,y¹,x⁰,y⁰

)

= ECˇo

(

^x1,y¹,x⁰,y⁰

)

× ˇTC¹_o^,0

(

^x1,y¹

)

. (10)

Thetwo-factordecompositionoftheMPIforaCRStechnologycanbecalculatedbysettingtheparameter

decomp

to

crs

,

8 The toolbox offers the possibility of testing the hypothesis that a certain technology is characterized by VRS or CRS. The function implements the algorithms developed by Simar and Wilson (2002) , following Bogetoft and Otto (2011) , to determine whether the VRS and CRS distance function values are significantly different or not. The test can be performed by calling the deatestrtsm(X, Y, ...) function, with the orient parameter set to the same orientation as the MPI.

Notice that, as the MPI does not depend on an actual technology, its outcomes are identical to those in the previous tables. The decompositions inEC andTC factors, however,are different. The differencesare not large, asfor mostof the DMUsthe RTS factor in expression(5)residesintheneighbourhoodof1.WeagainseetheprominentroleplayedbytechnologicalchangeinmostoftheAmerican states,exceptOklahoma(OK),notreportedinthetableabove,andWestVirginia(WV).

3.1.3. Thecomparison-period-output-orientatedMPI

Asanticipated,wecanalsoselectthecomparisonperiodtechnology,Sˇ¹,asbenchmark.Thecomparison-period-output-orientatedMPI isthengivenbyMˇ_o¹(^x1,y¹,x⁰,y⁰)^.Thecompletefour-factordecompositionanalogoustoexpressions(2)and(3)are,respectively,

Mˇ¹_o

(

^x1,y¹,x⁰,y⁰

)

= ECo

(

^x1,y¹,x⁰,y⁰

)

× TC¹o^,0

(

^x0,y⁰

)

× SECo¹

(

^x1,x⁰,y⁰

)

× OME¹

(

^x1,y¹,y⁰

)

, (11) and

Mˇ¹_o

(

^x1,y¹,x⁰,y⁰

)

= ECo

(

^x1,y¹,x⁰,y⁰

)

× TC¹o^,0

(

^x0,y⁰

)

× SECo¹

(

^x1,x⁰,y¹

)

× OME¹

(

^x0,y¹,y⁰

)

, (12) whichare identified by ‘Path C’ and‘Path D’ inBalk andZofío(2018).Again, ifthere isno preferencefor eitherof thetwo then it is advisedtotakethegeometricmeanofthesetwodecompositions,

Mˇ¹_o

(

^x1,y¹,x⁰,y⁰

)

= ECo

(

^x1,y¹,x⁰,y⁰

)

× TC_o^1,0

(

^x0,y⁰

)

× [SEC¹_o

(

^x1,x⁰,y⁰

)

× SECo¹

(

^x1,x⁰,y¹

)

^]1/2

×[OME¹

(

^x0,y1,y0

)

^{× OME1}

(

^x1,y1,y0

)

^{]1/2. (13)}

Thecomparison-period-output-orientatedCCDindexisdefinedby M¹_o

(

^x1,y¹,x⁰,y⁰

)

≡ ECo

(

^x1,y¹,x⁰,y⁰

)

× TC¹_o^,0

(

^x0,y⁰

)

=D¹_o

(

^x1,y¹

)

D¹_o

(

^x0,y⁰

)

^{. (14)}

The complete decomposition of the comparison-period-output-orientated MPI is calculated by calling the

deatfpm(X, Y, ...)

function,replacing

base

by

comparison

inthe

period

parameter,andleavingtheotherparametersunchanged:

Theoutputhasthesamestructureasthebaseperiodanalogueandisthereforenotreportedhere.Also,obtainingtherelateddecom- positionsisdonebysettingtheparameter

decomp

^toeither

rts

^,

ccd

^,or

crs

^.

3.1.4. Thegeometric-mean-output-orientatedMPI

In generaltheoutput-orientated MPIswithbaseandcomparisonperiodbenchmarkswilldeliverdifferentresults.Acompromise be- tweenthetwoMPIsistheirgeometricmean,

Mˇo

(

^x1,y¹,x⁰,y⁰

)

≡ [Mˇ⁰_o

(

^x1,y¹,x⁰,y⁰

)

× ˇM¹_o

(

^x1,y¹,x⁰,y⁰

)

^]1/2=



_ˇ

D⁰_o

(

^x1,y¹

)

Dˇ⁰_o

(

^x0,y0

)

Dˇ¹_o

(

^x1,y¹

)

Dˇ¹_o

(

^x0,y0

)



1/2

. (15)

Thisindexcanbedecomposedintothefollowingfactors,

Mˇo

(

^x1,y¹,x⁰,y⁰

)

=ECo

(

^x1,y¹,x⁰,y⁰

)

× [TC_o^1,0

(

^x0,y⁰

)

^TCo^1,0

(

^x1,y¹

)

^]1/2

× [SEC_o⁰

(

^x1,x⁰,y⁰

)

^SECo⁰

(

^x1,x⁰,y¹

)

^SECo¹

(

^x1,x⁰,y⁰

)

^SECo¹

(

^x1,x⁰,y¹

)

^]1/4

× [OME⁰

(

^x0,y¹,y⁰

)

^OME0

(

^x1,y¹,y⁰

)

^OME1

(

^x0,y¹,y⁰

)

^OME1

(

^x1,y¹,y⁰

)

^]1/4. (16) Thisdecompositioniscomputedbysettingthe

period

^parameterto

geomean

^inthe

deatfpm(X, Y, ...)

^function.

Thecorrespondingindividualtermsassociatedto‘PathA’,‘PathB’,‘PathC’,or‘PathD’canberecoveredbyrunningtheprevious

base

or

comparison

options.It isthenpossibletoconfirmthatthetwo decompositionsareindeeddifferentandtheir geometricmeanisa usefulcompromise.Also,therelateddecompositionsareobtainedbysettingtheparameter

decomp

^to

rts

^,

ccd

^,or

crs

^.

TheimportanceofthebenchmarkperiodisclearinthecaseofUSagriculture.Thegeometricmeanofthebaseandcomparisonperiod benchmarksdelivers a54.94% increasein productivityoverthe44 yearperiod.Thisissubstantially lower thanthe previouslyreported resultfromthebaseperiodbenchmark,215.99%, suggestingthatthecomparisonperiodbenchmarkdeliversaratherlimitedproductivity change.Sincethe

EC

factoriscommontobothdecompositions(11.01%),thereductioninproductivitygrowthcorrespondstolower

TC

to thetuneof33.52%.Finally,regardlessofthechoiceofthebenchmarktechnology,inputandoutputscaleandmixeffectsbarelycontribute toproductivitygrowth,theirmagnitudesbeingclosetoone.

3.2. Theinput-orientatedMPI

RecallthataconetechnologyexhibitsCRS,thusDˇ^t_o(^xt,y^t)=1/Dˇ^t_i(^xt,y^t)^.Thustheoutput-orientatedMPIdefinedbyexpression(1)can alsobewrittenas

Mˇ^t_o

(

^x1,y¹,x⁰,y⁰

)

=Dˇ^t_i

(

^x0,y⁰

)

Dˇ^t_i

(

^x1,y¹

)

^{≡ ˇM}

t

i

(

^x1,y¹,x⁰,y⁰

)

, (17)

thatis,asinput-orientatedMPI,conditionalontheperiodtconetechnology,Sˇ^t.

The optionsfor decomposinganinput-orientated MPIrun parallel to thosealreadypresented fortheoutput-orientated counterpart, andtherefore itis sufficient todescribe the changes that needto be introduced inthe MATLABfunctionsto obtain thedifferent de- compositions.Forformaldefinitions anda graphicalrepresentationoftheinputdecompositions seeBalkandZofío(2018,Section 4).In general,allthatisrequiredistosubstitute

io

for

oo

inthe

orient

parameterintheMPIfunction.

Thebase-period-input-orientatedMPI,definedasMˇ⁰_i(^x1,y¹,x⁰,y⁰),iscalculatedanddecomposedbyrunningthefollowingcode:

Followingproceduresidenticaltothosefortheoutput-orientatedMPItherelateddecompositionscanbe obtained.Mergingtheinput mix andscale effectsinto thereturns-to-scale factoryields the three-factordecompositioncounterpart to expression(5). Toobtainthis simplerdecompositiontheparameter

decomp

mustbechangedto

rts

.