Estimation of a composite fish stock index using data envelopment analysis

Fisheries Research 69 (2004) 91–105

Estimation of a composite fish stock index usingdata envelopment analysis

Sean Pascoea,∗, Ines Herrerob,1a Centre for the Economics and Management of Aquatic Resources (CEMARE), University of Portsmouth,

Boathouse No. 6, College Road, HM Naval Base, Portsmouth PO1 3LJ, UKb Department of Economy and Business, University Pablo de Olavide, Seville, Spain

Received 26 March 2003; received in revised form 13 February 2004; accepted 23 March 2004

Abstract

The level of fisheries production is generally assumed to be a function of the level and quality of inputs employed (effort) andthe level of the stock. Estimating this relationship requires measures of both effort and stock, but measures of stock abundanceare often unavailable, requiring some proxy measure to be employed. Further, the estimation of production functions inmulti-species fisheries requires a composite stock index of all species caught. This index must reflect the relative impact thatchanges in the abundance of each species has on the overall composite measure of output. In this study we propose a new methodto calculate a stock index based on the changes in the DEA efficiency scores over time. The DEA analysis is configured suchthat within-period variations in efficiency are independent of the underlying stock, and between-period differences in efficiencyare thereby assumed to be directly proportional to changes in stock abundance. The method is validated though applicationto an artificial data set, and is also applied to two Spanish fisheries—a single-species fishery and a multi-species fishery.© 2004 Elsevier B.V. All rights reserved.

Keywords:Stock index; Data envelopment analysis; Stochastic production frontier; Methodology

1. Introduction

The examination of technical efficiency has beenundertaken in a wide range of industries, although rel-atively few attempts to measure technical efficiency infisheries have been undertaken (Kirkley et al., 1995,1998; Campbell and Hand, 1998; Sharma and Leung,1999; Grafton et al., 2000; Pascoe et al., 2001; Pas-coe and Coglan, 2002; Fousekis and Klonaris, 2003).These have largely focused on the use of stochastic

∗ Corresponding author. Tel.:+44-23-9284-8484;fax: +44-23-9284-4037.E-mail addresses:[email protected] (S. Pascoe),[email protected] (I. Herrero).

1 Fax: +34-954-349-339.

production frontiers to estimate efficiency, althoughsome attempts have been made to estimate efficiencyin fisheries using data envelopment analysis (DEA)(Felthoven, 2002). The econometric approach hasgenerally been favoured due to the (believed) highlevel of stochasticity in the fishing industry (Kirkleyet al., 1998).

The stochastic production frontier approach re-quires the initial estimation of a frontier productionfunction, where the level of output is expressed as afunction of a number of inputs. As these are gener-ally derived using panel data, account must be takenof changes in outputs due to changes in the underly-ing stock conditions between time periods. Failure totake into account the effects of changes in stock sizeon catch will lead to these effects being captured in

0165-7836/$ – see front matter © 2004 Elsevier B.V. All rights reserved.doi:10.1016/j.fishres.2004.03.003

92 S. Pascoe, I. Herrero / Fisheries Research 69 (2004) 91–105

the inefficiency component of the model. Hence, theresultant estimates of efficiency will be biased.

In some cases, information on stock abundance isavailable through regular stock surveys undertaken bybiologists. From this, a stock index can be derived thatcan be used in the production function (Pascoe et al.,2001, 2003). However, detailed stock assessments arenot undertaken for all species. As a result, some alter-native approach is required to estimate the stock mea-sure.

One approach that has been used is to assume thatchanges in the average level of catch per unit of effort(CPUE) over time is indicative of changes in stocksize, and hence may be used as an stock index. Forexample,Kirkley et al. (1995, 1998)used the CPUEof a reference fleet as an indicator of stock abundanceand incorporated this index directly into the translogproduction frontier as an explanatory variable. This isalso a common assumption underlying most biologicalestimates of stock abundance.

There are a number of potential problems with thisapproach. Foremost of these are the implicit assump-tion of constant returns to both effort and stock, andthe absence of any inefficiency. In any given period(and ignoring random fluctuations for the purposesof simplicity), the level of output from a given boatcan be estimated asCt = qFβ1

t Sβ1t uvt whereC is the

catch,q a proportionality constant (the ‘catchability’coefficient),F the level of fishing effort (which maybe comprised of several variables in practice),S thestock size,u the level of efficiency of the boat (as-sumed constant over time for simplicity),vt randomerror andβ1, β2 the production elasticities associatedwith the effort and stock inputs. CPUE is only propor-tional to stock size if the catch-effort relationship isof the formCt = qFtSt . Under these circumstances,Ct/Ft = qSt and CPUE (i.e.Ct /Ft) will therefore beproportional to stock size. However, from previouslycited studies of production functions and frontiers infisheries, constant returns to either effort or stock (i.e.β1 = β2 = 1), and the absence of inefficiency, aregenerally unrealistic assumptions.

This paper presents a method for deriving a stockindex from the available data that does not involve anyrestrictive assumptions regarding the relationship be-tween catch, effort and stock size. The method is basedon that used for estimating the technical change com-ponent of the Malmquist index, derived using DEA

(Caves et al., 1982; Färe et al., 1994). The methodwas applied to an artificial data set for the purposesof validation, and two Spanish fisheries based in theSouth Atlantic region. The index was used to estimatethe frontier production function over the same periodfor the fishery, and the results compared with otherindicators of stock abundance (i.e. CPUE).

2. Methodology

Changes in the level of output for a given level ofdiscretionary inputs can arise in fisheries through ei-ther changes in stock abundance, changes in the levelof technology employed or changes in efficiency (e.g.skill of skipper and crew). Empirical estimates of tech-nical change in fisheries suggest that the introductionof new technologies has had a relatively small im-pact on fishery production relative to changes in stock.Fitzpatrick (1996)calculated a 270% increase in theaverage fishing technology coefficient of ‘world’ fish-eries between 1965 and 1995, representing, on aver-age, a 3% cumulative annual growth rate. Much of thisincreases in technological change occurred in devel-oping countries through increased use of mechanisedfishing techniques. In European fisheries,Banks et al.(2001) and Kirkley et al. (2001)found that techno-logical changes increased productivity by only around1% a year on average over the period 1985–1999.

In contrast, stocks can change by substantiallygreater proportions both across the year (i.e. changesin seasonal abundance due to migratory or schoolingbehaviour) and between years (e.g. as a result of en-vironmental fluctuations or a consequence of fishingpressure itself). Further, stock levels can both increaseand decrease over time, whereas pure technologicalchange would be expected to increase productivityonly. For example, cod stocks in the North Sea expe-rienced inter-annual variations ranging from+24%to −27% over the period 1985–2002, with a 70% re-duction over the period as a whole (ICES, 2002). Bycomparison, pure technological change is relativelyinsignificant.

Given the above, apparent changes in productivitybetween periods are most likely to reflect changes instock abundance and efficiency, and this informationcan be used to derive a stock index to estimate produc-tion functions and frontiers. While the measure will

S. Pascoe, I. Herrero / Fisheries Research 69 (2004) 91–105 93

also include the effects of technological change, thisis likely to be small. In any case, capturing the effectsof any technological change that may exist from thedata will result in more robust estimates of efficiencyand input elasticities.

2.1. Measuring productivity change

As noted above, the change in stocks from oneperiod to the next can be effectively measured asproductivity change.Caves et al. (1982)defined theMalmquist index of productivity change between agiven base time period (e.g. time 1) and time periodtas

M10 = D1

0(xt, yt)

D10(x1, y1)

(1)

whereDt0(xt, yt) = min{δ : (xt, yt/δ) ∈ St}, xt the setof inputs used in periodt, yt the set of outputs pro-duced in periodt, and St the production possibilityset (i.e. technologically feasible and technically effi-cient output set for a given input set) in time periodt.The subscript “0” indicates that the measure refers toa particular DMU under consideration. Alternatively,the index can be determined with reference to the cur-rent period, such that

Mt0 = Dt0(xt, yt)

Dt0(x1, y1)(2)

Färe et al. (1994)developed a version of the Malmquistindex as the geometric mean of both measures (i.e.Eqs. (1) and (2)), and demonstrated how by using bothindexes, changes in productivity between time periodscan be further disaggregated into efficiency change(both pure technical efficiency and scale efficiency)and pure technical change assuming constant returnsto scale. The measures used in this analysis, however,are based on the original specification of productivitychange with reference to a base time period, i.e. as rep-resented byEq. (1), following the approach ofCaveset al. (1982)as we are concerned only with techni-cal change. The technical change component using theCaves et al. (1982)approach is identical to that de-rived using theFäre et al. (1994)approach. However,as noted byFäre et al. (1994), the ratio of the dis-tance functions inEq. (1)will be affected by changesin efficiency between the periods as well as techni-cal change. If the decision making unit (DMU, which

in the case of fisheries is the fisher) under consider-ation is efficient (i.e.Dt0(xt, yt) = 1), then the indexreduces to

M1∗0 = D1

0(xt, y∗t )

D10(x1, y

∗1)

= D10(xt, y

∗t ) (3)

wherey∗t is the efficient level of output given the level

of inputs and the technology employed. While in prac-tice not all DMUs will be efficient, an equivalent setof efficient DMUs can be derived from the observeddata byy∗

t = yt/Dt0(xt, yt). The efficiency level of the

DMU may differ from period to period (for example,a poor skipper could be replaced by a better skipper),but by making each DMU ‘efficient’ in each time pe-riod through the appropriate radial expansion of theiroutput levels, all ‘efficient’ DMUs will sit on the fron-tier relating to that time period. With the removal ofthe efficiency component from both time periods, theapproach is essentially equivalent to that proposed byFäre et al. (1994)for separating the technical changefrom the efficiency change in their approach. The re-sultant index represents technical change, which inthe case of fisheries is assumed to primarily reflectchanges in stock abundance.

2.2. The stock effect in fisheries production functions

Given that the main shift in production from oneperiod to the next in fisheries is most likely to be dueto changes in stock abundance rather than changes intechnology per se, the measure of technical changeprovides a proxy measure of stock change that can beused to estimate stochastic fisheries production func-tions or frontiers.

If the effects of the stock on production were neutral(i.e. have the same impact on all vessels irrespectiveof their size), then the underlying production functionwould effectively have a Cobb–Douglas functionalform (i.e. Ct = qFβ1

t Sβ2t u), and the values ofM1∗

0would be almost equal for all vessels. In such a casederived estimate ofM1∗

0 would representSβ2, as theany increasing or diminishing returns to stock sizeeffects would be implicit in the estimated value basedon observed production levels, and the productionfunction for a given vessel, would therefore be givenby C0,t = qFβ1

0,tSβ2t = qFβ1

0,tM1∗0 , whereM1∗

0 has a

94 S. Pascoe, I. Herrero / Fisheries Research 69 (2004) 91–105

value of 1 in the first period and varying values forsubsequent periods.

Most of the previously cited studies of productionfunctions and efficiency in fisheries have adopteda translog production function, the general form ofwhich is given by

ln Yit = β0 +∑i

βi lnXit

+∑i

∑j

βi,j lnXit lnXjt − uit + vit (4)

where Yit is the level of output,Xit the inputs,uitthe level of inefficiency andvit the random errorcomponent. The translog production function is gen-erally preferred to the Cobb–Douglas productionfunction as it allows greater flexibility in terms ofelasticities—both production and substitution. TheCobb–Douglas production function assumes that pro-duction elasticities are constant (and therefore cannotbe influenced by, say, the size of the boat), and alsothat the elasticity of substitution between inputs isconstant with a value of 1. This precludes the possibil-ity that some inputs may be used in fixed proportions,while others may be readily substituted. Both theseassumptions are relaxed in the translog specification,which allows for variability in both production andsubstitution elasticities between producers.

In the case of fisheries production, the output iscatch and the inputs are the levels of effort andstock. An implicit assumption of this functionalform is that the effects of changes in stock canvary with the level of effort employed, such thatthe effect of stock on the lnCt can be given byβS ln St + βS,S ln2 St + βS,F ln St lnFt . As a result,the impact of the stock change on output will bynon-neutral (i.e. will vary by boat, depending on itslevel of effort). In such a case, variations in the indi-vidualM1∗

0 scores may represent both random fluctu-ations in catch (which presumably could be separatedout using a stochastic production frontier) as well asthe effects of changes in stocks between time periodsspecific to the activity of the boats.

In either case, the measure of stock change alreadyincorporates the possibility of diminishing (or increas-ing) returns to stock. Incorporation of this measure intoa production function as an explanatory variable cre-ates difficulties, as the coefficient associated with the

variable will need to be restricted to 1 (i.e.(Sβ2t )

β2∗ =Sβ2β2∗t → β2∗ = 1). This is even more problematic in

the translog, as the measure also includes the effectsof interactions with the other variables. To overcomethese problems, the index is not used as an explanatoryvariable per se, but is used to adjust the output mea-sure in order to allow for the effects of stock changeon output to be incorporated into the analysis. That is,the model is specified, in the case of a Cobb–Douglas,asC0,t/S

β2t = C0,t/M

1∗0 = qFβ1

0,t . Similarly, in thecase of the translog, the model is specified as

lnC0,t − (βS ln St + βS,S ln2 St + βS,F ln St lnF0,t)

= ln

(C0,t

M1∗0

)= ln β0 + βF lnF0,t + βF,F ln2F0,t

(5)

Given this, the index is a measure of the combinedeffect of changes in the set of stocks persecuted byfishers on the level of their production. Hence, it is de-scribed as a stock ‘effect’, and is an appropriate singlecomposite stock index for use in estimating fisheriesproduction functions and frontiers that also use a sin-gle composite output index. Although the index maybe based on a set of stocks, it is not possible to dis-entangle the effects of any one stock on production(unless it was estimated for a single-species fishery).However, a major difficulty in estimating productionfrontiers is the definition of an appropriate compos-ite stock measure, even if stock information is avail-able. The DEA approach overcomes this difficulty bydirectly estimating the effects of changes in a set ofstocks on production.

2.3. Estimating the index using DEA

An output-oriented DEA model is used to estimatethe efficient level of output and the set of frontiers.The output-oriented model is more consistent with theproduction processes in the fishery (i.e. inputs are ap-plied to produce a set of outputs). As with any DEAmodel, a range of assumptions can be made regardingreturns to scale. If the technical change was consid-ered neutral (i.e. has the same impact on all firms irre-spective of their size) then constant returns to scale areappropriate. In contrast, if technical change is consid-ered non-neutral (i.e. the effects differ by boat size),

S. Pascoe, I. Herrero / Fisheries Research 69 (2004) 91–105 95

then some form of variable returns to scale is moreappropriate.

As mentioned previously, most fisheries productionfunctions and frontiers have been estimated usinga translog production function, which includes theassumption of interactions between stock and vesselinputs. Tests against a standard Cobb–Douglas pro-duction function (which implicitly implies neutraltechnical change with respect to the stock variable)have generally rejected this functional form (seeKirkley et al., 1995, 1998; Campbell and Hand, 1998;Sharma and Leung, 1999; Pascoe et al., 2001; Pascoeand Coglan, 2002). Given this, a variable returns toscale assumption is more appropriate.

In this study, a non-increasing returns to scale(NIRS) assumption was adopted, although other scaleassumptions could be imposed. This has the featureof constant returns to scale for vessels below the‘optimal’ scale, and decreasing returns for vesselsabove this level. This assumption has been adopted toavoid infeasibilities, as will be explained further fol-lowing the presentation of the models. However, theresults from most of the previously cited empiricalstudies undertaken suggest that diminishing returnsto input use exists in most cases, and hence an NIRSis an appropriate assumption. An implicit assumptionof this approach is that vessels below the optimalscale, if not on the frontier, are technically inefficient(rather than just scale inefficient). It also imposes theimplicit assumption that the effect of stock changeon production is neutral for all vessels below the‘optimal’ scale, but is non-neutral above this level.These assumptions are not necessarily ideal, but arenecessary to avoid infeasibilities.

A DEA model incorporating NIRS is first used toestimate the technical efficiency of each DMU, givenby

Maximize θ1,

Subject to xi,j0,t ≥ ∑ntj=1λjxi,j,t, i = 1, . . . ,M,∑nt

j=1λjyr,j,t ≥ θ1yr,j0,t , r=1, . . . , S,∑ntj=1λj ≤ 1,

λj ≥ 0, j = 1, . . . , nt(6)

wherexi,j,t corresponds to the level of inputi used byDMU (decision making unit, such as a vessel)j in time

t, andxi,j0,t to the inputi of the DMU under evaluationin time t. Analogously,yr,j,t represents outputr ofthe DMU j in time t, and yr,j0,t represents outputrof the DMU under evaluation in timet. The nt unitsin the reference set are those operating in the sametime periodt as vessel under evaluation. The value ofθ1(θ1 ≥ 1) corresponds to the proportional increasein the outputs that would make DMUj0 efficient. TheTE score associated with DMUj0 is 1/θ1 which variesbetween 0 and 1.

In the second step of the analysis, the units are trans-formed onto their projected (radially efficient) unitson the frontier. The efficient level of outputy of theDMU j0 in time t is given byy∗

r,j0,t= θ1yr,j0,t . This

estimate of the efficient level of output is then usedin a second DEA model to estimate the shift in thefrontier. Each DMU is compared with a common ref-erence set operating in a base time period (time 1).The third step of the analysis is thereby given as

Maximize θ2,

Subject to xi,j0,t �=1 ≥ ∑n1j=1λjxi,j,t=1, i = 1, . . . ,M,∑n1

j=1λjy∗r,j,t=1 ≥ θ2y

∗r,j0,t �=1, r = 1, . . . , S,∑n1

j=1λj ≤ 1,

λj ≤ 0, j = 1, . . . , n1,

(7)

where TEi,t,1 = 1/θ2 represents change in (efficient)output if DMU i between timet = 1 (the base timeperiod) and timet due to the technological change (i.e.M1∗

0 ). The value of TEi,t,1 andθ2 will be 1 in the baseperiod (timet = 1), as all DMUs are ‘efficient’ in thesecond stage, and when compared with themselves allDMUs will have a value ofθ2 = 1. By allowing forsuper-efficiency, feasibility is assured asθ2 may begreater or less than 1 in time periodst > 1 and theλmay take very small values. Values ofθ2 greater than1 suggest that the stock has declined compared to thebase period, while values less than 1 indicate that thestock has increased.

This is illustrated inFig. 1 for a single input and(composite) output production process. In the firststage, the efficiency estimates derived from DEA areused to determine the efficient output for each vessel.For example, inFig. 1(a), E is inefficient. The levelof output thatE could produce if operating efficientlyis E∗, given by a radial expansion to the frontier (de-fined by vessels A, B, C and D). The expansion factor,θ1, is equivalent to E∗/E, while the level of technical

96 S. Pascoe, I. Herrero / Fisheries Research 69 (2004) 91–105

A

B

C D

E

E*

Inputs

Out

puts

Inputs

Out

puts

Frontier

Z

X

Frontier t=1

Fig. 1. Single input and output production process.

efficiency of vessel E is given by TE1 = 1/θ1. An ef-ficient vessel has a value of the expansion factor ofθ1 = 1, and inefficient vessels have valuesθ1 > 1.

In the second stage, the efficient levels of output ofvessels in timet �= 1 are compared with the efficientfrontier in some base period (time 1). As the vesselsare operating under different stock conditions in timet, it is possible for the efficient catch to be either aboveor below the frontier level in time 1. For example, theefficient output of vessel Z in timet is greater than theefficient frontier in time 1. In this case,θ2 < 1, andTE2 > 1. In contrast, the efficient output of vessel X(assumed to come from a different time period thanvessel Z) is below the frontier, such thatθ2 < 1, andTE2 < 1.

The input constraint in Model 7 (i.e.xi,j0,t �=1 ≥∑λjxi,j,t=1∀i) would be infeasible if the levels of an

input xi of all DMUs in time t = 1 were greater thanthat of DMU0 in time t > 1 had a variable returnsto scale (VRS) model (i.e.

∑λj = 1) been imposed.

However, by allowing for NIRS (i.e.∑λj ≤ 1), the

restriction on the inputs will always be feasible. Forexample, if the inputxi of DMU0 was a fractionz(<1) of the equivalent input of the next smallest obser-vation DMUj which formed its peer, thenxi,j0,t �=1 =zxi,j,t=1, and

∑λj = z ≤ 1. This is an important con-

sideration especially when using unbalanced paneldata, as information on some small firms may not beavailable in the reference period. Imposition of VRSwould require limiting the set of observations to onlythose DMUs larger or equal in size to those of the ref-erence period. In the case of fisheries analysis whereinformation is often scarce, this restriction wouldgreatly restrict the (already limited) data that could beused.

For the smaller vessels, the assumption of NIRSimposes a common stock effect. While some informa-tion on the interactions between boat size and stockabundance is lost for these vessels, the assumptionof VRS may result in other distortions and informa-tion loss that are potentially greater. Previous studieshave generally concluded that smaller boats are gen-erally less efficient than their larger counterparts (e.g.Sharma and Leung, 1999; Pascoe and Coglan, 2002).With a VRS model, the smallest boats would be iden-tified as efficient. Hence, shifts in the frontier at thebottom end of the scale may reflect changes in effi-ciency rather than the stock effect.

3. Analyses

3.1. Method validation

The method was initially tested using an artificialdata set with a known stock index. The artificial dataset represents a single-species fishery, with a fleet com-posed of 24 vessels operating over 24 time periods.Three effort-related inputs were used in the produc-tion frontiers: engine power (in kW), boat volume (inGRT) and days fished, with production elasticities of0.1, 0.1 and 0.8, respectively. These were generatedfor each vessel using a random number generator as-suming a uniform distribution, with days fished alsovaried across each of 24 time periods. The stock sizewas also generated in each time period assuming anormal distribution. Stock size was assumed to haveunitary output elasticity. The boats were also assumedto vary in their level of technical efficiency, assumedto be min(1, N[0.8,0.4]).

S. Pascoe, I. Herrero / Fisheries Research 69 (2004) 91–105 97

The catches of each observation were generatedusing a Cobb–Douglas standard production function,given by

Cj,t = qGRTβ1j,tHPβ2

j,tTRIPSβ3j,tSTOCKtTEj (8)

A number of separate analyses were undertaken totest the methodology. The analysis was run with bothbalanced and unbalanced panel data. In the latteranalyses, observations were excluded using a randomnumber generator. Two levels were examined—onewith 5/6th of the observations present on averagein each time period and the other with 50% of theobservations present on average in each time period.The sensitivity of the method to random variation wasalso examined by introducing a stochastic componentto the catch. Two levels of random variation wereconsidered. The catch was multiplied by a randomnumber with a distributionN[1,0.1] in the first setof simulations, andN[1,0.2] in the second set. Theanalyses was run 50 times with each of the randomshock levels, generating 1150 stock estimates (i.e.50×23 time periods, excluding the first period whichhad a constant stock index of 1).

3.2. Case studies and data

The method was applied to two Spanish fisheriesoperating in the South Atlantic. The two fisheries havedifferent characteristics, the first consisting of an arti-sanal fleet targeting essentially a single species (withrelatively small quantities of bycatch), while the sec-ond is a multi-species fishery exploited by relativelylarger vessels.

The first fishery, which targets octopus, is highlyseasonal, running from November to May. During theremainder of the year, the vessels participate in alter-native fisheries characterised by lower priced species.The vessels are relatively small, and operate on a dailybasis (i.e. trip length is 1 day). Monthly data on thefishery were available for the period 1992–1997. Overthis period, catches peaked in 1994 (733,113 tonnes),after which they decreased to negligible levels in1997 (8256 tonnes). This reduction in catch is be-lieved to be due to low stock levels (possibly due to arange of reasons such as overexploitation, migration,weather or other biological factors) rather than dueto demand, as the price paid for octopus increased

substantially between the two periods. Fishing effortwas diverted to the alternative fisheries as the avail-ability of the octopus decreased, and catches in thealternative fisheries correspondingly increased.

The second case study is based on a sample of 59vessels from the Spanish Andalusian fleet that oper-ated during 1985. While these data are relatively old,they are suitable for the purposes of demonstratingthe methodology. These are deep-water trawl vesselsthat fished in Moroccan Waters North of the 28◦44′parallel. The main target species in the fishery arecrustaceans (deep water rose shrimp, scarlet shrimp,and scampi) and European hake, with some otherspecies caught as incidental bycatch. The species areoften caught together (as they are all bottom dwellingspecies), but the proportion of each in the catch variesby area fished. Hence the fishers are able to altertheir catch composition to some degree through fish-ing in different areas at different times, although thepotential for this is limited.

The inputs used in the analyses were a physicalmeasure of boat capacity (measured as gross regis-tered tonnage, or GRT), engine power (measured inhorsepower, HP) and number of trips per month. Infor-mation on other inputs, such as labour, was not avail-able. However, labour is generally highly correlatedwith boat size in trawl fisheries.2 A summary of thefleet characteristics is presented inTable 1. While thenumber of boats participating in the octopus fisherydeclined over the period examined as the stock sizedeclined, the average characteristics of the participantswere fairly constant over time, varying on average only3–4% over the period of the data (Fig. 2). In contrast,the average size and engine power declined consider-ably over the period of the data in the multi-speciesshrimp fishery.

Only one output—the amount of octopus catch—wasconsidered for each observation in the first case study.Hence, the derived index is effectively an indicatorof changes in the octopus stock. For the second case

2 Pascoe et al. (2003)questioned the use of labour as an input intrawl production functions. The main role of labour in large trawlvessels is often to sort the catch, with only some minimum levelof labour required to shoot the nets and operate the vessel. Mostother activities are mechanized. As a consequence, the level oflabour employed is a function of the expected catch, which itselfis a function of net size, boat speed (both a function of enginepower) and capacity utilisation.

98 S. Pascoe, I. Herrero / Fisheries Research 69 (2004) 91–105

Table 1Technical characteristics of the case study fleets

Octopus fishery Multi-species shrimp fishery

Mean Standard deviation Mean Standard deviation

Gross registered tonnage (GRT) 2.95 0.5 85.4 32.7Horsepower (HP) 31.4 7.5 322.1 94.7Trips per month 8.3 4.8 1.6 0.6Number of vessels 70 59

study, six different outputs were considered—thefive most important species (representing 80% oftotal landings by value) and a sixth category in-cluding the rest of the species (around 20% of to-tal). This study was also conducted on a monthlybasis.

The stock indicators derived from the DEA anal-ysis were used in a stochastic frontier productionfunction and the elasticity associated with each input

3.10

3.15

3.20

3.25

3.30

1991 1992 1993 1994 1995 1996 1997

Year(a)

(b)

aver

age

Gro

ss T

on

nag

e

34.0

34.5

35.0

35.5

36.0

aver

age

eng

ine

po

wer

(H

P)

engine power

gross tonnage

75

80

85

90

95

0 2 4 6 8 10 12

Month

Ave

rag

e g

ross

to

nn

age

300

310

320

330

340

350

Ave

rag

e en

gin

e p

ow

er(H

P)

engine power

gross tonnage

Fig. 2. Changes in average technical characteristics of the fleets over time: (a) octopus fleet; (b) multi-species shrimp fishery.

was estimated. A production frontier was used ratherthan a production function, as the former accountsfor un-measurable differences in quality of inputs aswell as other boat-specific characteristics not incorpo-rated into the function in the inefficiency estimated.With a production function, exclusion of these fac-tors may result in mis-specification bias. The analysiswas also undertaken using CPUE as a measure of thestock index. The objective of this was to compare the

S. Pascoe, I. Herrero / Fisheries Research 69 (2004) 91–105 99

effects of using different stock indexes on the effi-ciency scores and input elasticity estimates, and testthe hypotheses that the elasticity of the effort measurewas equal to 1—essential if CPUE is to be a valid stockindex.

4. Results

4.1. Validation

The derived stock index estimates using the methodwere highly correlated with the ‘actual’ stock mea-sures even with unbalanced panel data, although thecorrelation declined as the number of observationsin each time period declined (Table 2). The errorassociated with the estimates also increased as thenumber of observations in each time period declined.With all boats operating in each year, the coefficientof variation of the error (i.e. the standard deviationof the error divided by the mean stock level) wasaround 3%. This increased to around 5% when onlyhalf the boats were present, on average, in each timeperiod.

When random variation is added to the data, themethod also appears to perform well. Although thereis an increase in the error, as would be expected, theerror is still proportionally less than the amount of ran-dom variation imposed in the system. The coefficientof variation of the error was found to be 6 and 11%, re-spectively when random components with coefficientsof variation of 10 and 20% are imposed on the catch

0%

5%

10%

15%

20%

25%

30%

35%

<-30

-29 to -25

-24 to -20

-19 to -15

-14 to -10

-9 to -5

-4 to 0

1 to 5

6 to 10

11 to 1

5

16 to 2

0

21 to 2

5

>25

Error (%)

Prop

ortio

n of

obs

erva

tions

(%)

10%

20%

Fig. 3. Distribution of error from stochastic analyses (years 2 to 24).

Table 2Comparison of estimated stock to ‘actual’ stock

Average proportion of boatsin each time period

100% 84% 50%

Correlation with‘actual’ stock

0.996 0.993 0.980

Coefficient of variationof error (%)

2.9 3.2 5.0

data. Around one-third of the estimated stock valueswere within 5% of the ‘true’ value for the analyseswith the higher amount of variation, and almost 60%within 5% of the ‘true’ value for the analyses with thelower amount of variation (Fig. 3).

4.2. Estimated stock effect indexes for case studyfisheries

In the analysis of the octopus fishery, the resultsroughly conform to the a priori expectations in termsof stock changes. As noted above, catches peaked in1994. This corresponds to an estimated higher thanaverage stock level (Fig. 4a). Following 1994, catchesdeclined, the assumption being that the stock had beenoverexploited. This is also apparent in the DEA results,as the stock effect decreased over time, until it reacheda very low level in 1997.

The distribution of the stock effect across the seasonis also in line with a priori expectations. The octopusis effectively an annual species, and the available stockis effectively mined over the season. As a result, it

100 S. Pascoe, I. Herrero / Fisheries Research 69 (2004) 91–105

00.20.40.60.8

11.21.41.61.8

2

11992

71992

11993

71993

11994

71994

11995

71995

11996

71996

11997

71997

DE

A a

nd C

PUE

ind

ex DEA index

CPUE index

00.20.40.60.8

11.21.41.61.8

22.22.42.6

1 2 3 4 5 6 87 9 10 11month

DE

A a

nd C

PUE

inde

x

DEA index

CPUE index

(a)

(b)

Fig. 4. Estimates of the DEA and CPUE indexes: (a) octopus fishery; (b) multi-species shrimp fishery.

would be expected that the stock abundance woulddecrease over the season, and this is also apparent inthe DEA estimates of the stock effect (Fig. 4a). Thestock effect generally peaks in November each season,and decreases over the remainder of the season. TheCPUE index (based on average catch per trip) alsofollows a similar trend over the period of the availabledata (Fig. 3a). However, while the two indexes followa similar pattern, the CPUE index is generally greaterthan that estimated using DEA.

For the multi-species fishery, the estimates of theDEA and CPUE indexes also diverged (Fig. 4b), withthe CPUE index providing a lower estimate of thestock abundance. This may largely be an artefact ofthe change in fleet composition over the year. As catchis generally assumed to be a function of boat charac-teristics as well as time fished, then a change in boatcharacteristics will change the average relationship be-tween catch and nominal effort (i.e. expressed solelyin terms of time and not adjusted for changes in boat

characteristics). For example, an increase in the av-erage size of the vessels would be expected to resultin an increase in CPUE,ceteris paribus. The averagephysical capacity for some of the months was up to12% lower than for month 1, and the average enginepower was up to 11% lower. As a result, part of thedecline in CPUE is due to a change in the fleet charac-teristics. This was not a problem for the octopus fish-ery, as the fleet was very homogeneous throughout theperiod of the data.

4.3. Stochastic production frontiers

The input elasticities and efficiency distribution ofthe fleet were estimated using a stochastic productionfrontier, with the dependent variable (catch) adjustedusing the derived stock effect measure. In the first casestudy, the output used was the standardised catch ofoctopus (i.e. the catch divided by the stock effect mea-sure as given inEq. (5)). For the second case study

S. Pascoe, I. Herrero / Fisheries Research 69 (2004) 91–105 101

Table 3Estimated stochastic production frontier models

Octopus fishery Multi-species shrimp fishery

DEA stock measure CPUE stock measure DEA stock measure CPUE stock measure

Coefficient t-Statistic Coefficient t-Statistic Coefficient t-Ratio Coefficient t-Ratio

Constant 4.346 4.588∗∗∗ 5.1872 6.0242∗∗∗ −0.2289 −0.2377 14.3985 14.9400∗∗∗ln grt −2.601 −2.666∗∗∗ −3.5257 −3.7508∗∗∗ 0.1584 0.1952 −2.9506 −3.5415∗∗∗ln hp 1.325 2.033∗∗ 0.8243 1.4322 2.3776 3.7025∗∗∗ −2.8295 −4.3565∗∗∗ln trip 0.994 8.984∗∗∗ 1.2088 13.4651∗∗∗ 1.4915 1.5063 −5.0349 −4.6426∗∗∗ln 2grt −0.242 −0.467 −0.2842 −0.3909 0.0884 0.1699 0.3629 1.0286ln 2hp −0.337 −2.400∗∗ −0.2932 −2.2034∗∗ −0.1619 −0.5290 0.2425 1.0959ln 2trip −0.113 −10.646∗∗∗ −0.0265 −2.9194∗∗∗ −0.4030 −0.7491 0.0603 0.2376ln grthp 0.920 2.230∗∗ 1.2534 2.6420∗∗∗ −0.0834 −0.1079 0.0534 0.0998ln grttrip 0.044 0.949 −0.0149 −0.3763 0.2488 1.7001∗ 0.9230 5.4154∗∗∗ln hptrip 0.063 1.917∗ −0.0239 −0.9122 −0.2531 −1.2273 0.4450 2.1674∗∗∗Sigma-squared 0.130 12.344∗∗∗ 0.1561 9.6913∗∗∗ 0.1665 5.1605∗∗∗ 0.3084 8.5156∗∗∗Gamma (γ) 0.278 5.678∗∗∗ 0.5460 15.0037∗∗∗ 0.4198 4.1307∗∗∗ 0.2939 4.5477∗∗∗Mu (µ) 0.381 8.118∗∗∗ 0.5839 11.0630∗∗∗ 0.5288 2.9436∗∗∗ 0.6021 7.0032∗∗∗Eta (η) 0.006 4.783∗∗∗ – – −0.1862 −10.9856∗∗∗

Log likelihood −591.559 −329.989 −180.6766 −397.3557LR test of frontier 595.788 851.110 136.7273 76.6249

∗ Significant at 10% level.∗∗ Significant at 5% level.∗∗∗ Significant at 1% level.

fishery, the six outputs considered in the DEA modelhad to be aggregated before being standardised.3 Forcomparison, the model was also estimated using anindex of stock abundance based solely on CPUE. Forconsistency, the dependent variable was again modi-fied to adjust for different stock conditions (i.e.C′ =C/cpue, where cpue is an index with value of 1 in thebase period).

The models were estimated using FRONTIER(Coelli, 1996) assuming random, rather than fixed, ef-

3 There are numerous approaches to aggregating multiple out-puts, including the use of prices to derive revenues, or revenueshares to derive a composite measure expressed in terms of weight(i.e. kg) but weighted to reflect the importance of each species inthe catch. In this study, the catch weight only was added. This hasimplications for the implicit assumptions underlying the producerbehaviour (i.e. they attempt to maximise catch weight rather thancatch volume). However, for the purposes of this example, this isnot considered an issue.Herrero and Pascoe (2003)applied themethod to a more recent data set for the same fishery using botha quantity and value based aggregation of outputs (and a corre-sponding quantity and value based stock effect index). Readersinterested in the effects of the output aggregation technique on thederived elasticities and efficiency distributions are referred to thispaper.

fects. In order to separate out the inefficiency elementfrom the stochastic component of the error term, a dis-tributional assumption was necessary. For the initialanalyses, it was assumed that the inefficiency wouldfollow a truncated normal distribution with meanµ(i.e. uj ≈ |N(µ, σu)|), and that efficiency may betime variant, such thatuj,t = uje

−η(t−T), whereη isthe average rate of change in inefficiency over time.These distributional assumptions were tested againstalternative distributions (i.e.η = 0 andµ = 0) andthe most appropriate model selected. In both fisheries,the most appropriate functional form when usingDEA as a stock index was the time variant efficiencymodel with a truncated normal distribution (i.e.η �= 0andµ �= 0). For the octopus fishery, the value ofηwas greater than zero, suggesting that efficiency in-creased over time, while for the multi-species fishery,average efficiency appeared to decrease over the year(possibly related to the change in fleet composition).A value ofη greater than zero suggests that efficiencyincreased over time. In contrast, when CPUE wasused as a stock index, the most appropriate functionalform was the time invariant model with a truncatednormal distribution (i.e.η = 0 andµ �= 0) (Table 3).

102 S. Pascoe, I. Herrero / Fisheries Research 69 (2004) 91–105

Table 4Estimated input elasticities at the mean input value

Mean input level DEA stock measure (time variant model) CPUE stock measure (time invariant model)

Elasticity t-Statistic Elasticity t-Statistic

Octopus fisheryln(grt) 1.063 0.104 0.766 0.118 0.694ln(hp) 3.411 0.123 0.090 0.112 0.072ln(trip) 1.864 0.836 65.599 1.013 93.405

Multi-species fisheryln(grt) 4.3683 0.5515 6.9533 0.8917 7.3390ln(hp) 5.7232 0.0591 0.0206 0.3560 0.1790ln(trip) 0.3964 0.8100 18.4719 1.5918 27.1872

When the DEA estimates of the stock index wereused, all estimated inputs were inelastic (i.e.<1), asexpected (Table 4). For the octopus fishery, the lackof significance of the elasticities for the physical in-puts (GRT and horsepower) is more likely a result

0%

5%

10%

15%

20%

25%

<0.4 0.40-0.45

0.45-0.50

0.50-0.55

0.55-0.60

0.60-0.65

0.65-0.70

0.70-0.75

0.75-0.80

0-80-0.85

0.85-0.90

0.90-0.95

0.95-1.00

Average efficiencyscore

prop

ortio

n of

the

flee

t

DEA index

CPUE index

0%

5%

10%

15%

20%

25%

<0.40 0.40-0.45

0.45-0.50

0.50-0.55

0.55-0.60

0.60-0.65

0.65-0.70

0.70-0.75

0.75-0.80

0.80-0.85

0.85-0.90

0.90-0.95

0.95-1.00

Average efficiencyscore

Prop

ortio

n of

the

flee

t

DEA index

CPUE index

(a)

(b)

Fig. 5. Distribution of average TE of the fleet from the stochastic production frontier: (a) octopus fishery; (b) multi-species shrimp fishery.

of the high level of homogeneity of the fleet. In bothfisheries, the elasticity for trips was greater than 1when CPUE was used as the stock index, suggesting(unrealistically) increasing returns to days fished. Theassumption of unitary elasticity for trips was tested

S. Pascoe, I. Herrero / Fisheries Research 69 (2004) 91–105 103

and rejected, suggesting that the CPUE measure isbiased as a unit elasticity for trips was one of theCPUE model assumptions.

The distribution of the TE score from the two mod-els were also different (Fig. 5). In both fisheries ex-amined, a higher proportion of inefficient vessels werefound when CPUE was used as the proxy for stock.This is consistent with the mean value of the ineffi-ciency term (µ), which was significantly greater in theCPUE based models than the models using the DEAestimate of the stock index (Table 3).

5. Discussion and conclusions

The estimation of efficiency in fisheries is becom-ing increasingly important as the distribution of effi-ciency has a considerable impact on the effectivenessof effort controls. The ‘failure’ of many effort reduc-tion programmes to achieve their desired impact onthe level of output in the fishery is most likely relatedto changes in average efficiency of the remaining fleet(seePascoe and Coglan (2000)for an example). Thestochastic nature of the fishing industry has resultedin a tendency for researchers to estimate efficiency us-ing the stochastic production frontier approach. Rel-atively few examples of the use of DEA in fisheriesexist for the purpose of estimating efficiency, althoughthe technique is becoming increasingly popular forthe measurement of capacity and capacity utilisation(e.g.Vestergaard et al., 2003; Tingley et al., 2003). Inthis paper, DEA has been used to estimate changes instock over time by estimating the shift in productionfrontiers between different periods.

The use of DEA to derive an index to account fordifferences in stock conditions between periods is the-oretically more robust than the use of simple measuressuch as CPUE. The DEA approach is not constrainedby the assumptions of constant returns to stock andeffort. In contrast, the use of CPUE as a stock indexassumes that stock changes are reflected in changesin the catch per nominal unit of effort regardless ofthe characteristics of the vessels operating in the timeperiod considered. Changes in the average technicalcharacteristics of the fleet over time can be account forin the DEA measure (as all inputs used are includedin the analysis), although not in the CPUE measure(based on catch and one element of effort: days fished).

While these problems can be overcome to some ex-tent by using a constant reference sub-fleet, the choiceof the sub-fleet may affect the resultant CPUE index.In contrast, DEA uses information on all of the boatsthat participated in the fishery.

Biologists often attempt to overcome the problemof change in fleet characteristics in their estimateof CPUE by deriving a measure of effort that in-cludes both time and some characteristics of the fleet(e.g. days times engine power, such that CPUE=catch/(kW × trips)). However, this imposes evengreater restrictive assumptions on the production pro-cess, as it assumes constant returns to both nominaleffort (e.g. trips) and the characteristics of the fleet,such thatC = q(TP)S, whereT is the measure ofnominal effort (i.e. time fished, such as days, trips orhours) andP is the physical characteristic used in the‘standardisation’ process (e.g. engine power or grosstonnage). In such a model, the level of effort is as-sumed to be represented byTP. From the results above(and also the results of other studies), constant returnsto these inputs is generally not a valid assumption.

As mentioned previously, the measures are not‘pure’ measures of stock changes as they also incor-porate other factors that affect changes in productivityover time. Improvements in technology will resultin higher catch rates, which could be interpreted asan increase in stock abundance in both the DEA andCPUE measures. It is theoretically possible to sepa-rate the effects of technological change from the stockeffect if a set of reference boats can be identifiedthat have not undertaken any technological changebetween periods. The difference between the “stock”effects for these vessels and those where technologi-cal change has taken place provides a measure of theimpact of the technological change.

For the purposes of this analysis, it was assumedthat technological change was zero. In the case studyfishery, no significant technological developments oc-curred that would have distorted the stock estimates.In theory, crowding pressures (i.e. increased com-petition as a result of increased fleet size) may alsoinfluence catch rates, again affecting both measures.The extent to which crowding affects the output of in-dividual vessels has not been adequately estimated forany fishery, so it is not possible at this stage to accountfor the effects of changes in fleet size the analysis.The main effect of crowding, however, is likely to be

104 S. Pascoe, I. Herrero / Fisheries Research 69 (2004) 91–105

a reduction in individual fishing effort (i.e. the level ofinputs employed by an individual) rather than a reduc-tion in the level of output for a given level of inputs.

A common problem with fisheries data is unreli-able catch information, as incentives often exist tounder-report the quantity landed. Using the DEA ap-proach, shortfalls in catch due to under-reporting willbe captured in the inefficiency measure in the firststage of the analysis. As a result, the frontier will beless effected by inaccurate data, assuming those re-porting the greatest levels of output are reporting ac-curately, and the resulting stock index estimate alsoless distorted. In contrast, the CPUE measure will bedirectly biased by mis-reporting as no adjustments tothe outputs are made.

The method presented in this paper allows morerobust estimates of production functions, frontiers andefficiency when information on stock abundance is notknown. As illustrated in the case study to which themethod was applied, use of simple proxy measuressuch as CPUE can result in a distortion in the measuresof technical efficiency and production elasticities. Theuse of the DEA index is free of distributional andproduction related assumptions in its derivation, andis therefore not subject to the same potential for bias.

The DEA measure, however, does not take into ac-count random variations in output. The effects of thison the stock estimate, however, may not be great, asdifferences in efficiency and ‘luck’ are removed ineach period by using the fully efficient level of catch.Assuming that ‘luck’ is relatively constant over theyear, then differences between periods should also beminimally affected by differences in random events.From the stochastic analyses using the artificial dataset, the method was found to be relatively robust evenin the light of considerable random variation. In anycase, differences in random events between time pe-riods would also be reflected in the use of CPUE in-dexes, so the method is no worse in this regard thanthe other potential measures of stock changes.

Acknowledgements

Earlier versions of this paper were presented at theAnnual Conference of the European Association ofFisheries Economists, Salerno, Italy, 2001, and the An-nual conference of the Operational Research Society,

Bath, UK, 2001. The authors would like to thank theparticipants at these meetings as well as the two anony-mous reviewers for their useful comments. The studywas carried out with the financial support of the Com-mission of the European Communities Fifth Frame-work programme, QLK5-CT1999-01295, “Technicalefficiency in EU fisheries: implications for monitoringand management through effort controls”.

References

Banks, R., Cunningham, S., Davidse, W.P., Lindebo, E., Reed,A., Sourisseau, E., de Wilde, J.W., 2001. The impact oftechnological progress on fishing effort. LEI Report PR.02.01.Agricultural Economics Research Institute (LEI), The Hague.

Campbell, H.F., Hand, A.J., 1998. Joint ventures and technologytransfer: the Solomon Islands pole-and-line fishery. J. Dev.Econ. 57, 421–442.

Caves, D.W., Christensen, L.R., Diewert, E., 1982. The economictheory of index numbers and the measurement of input, outputand productivity. Econometrica 50, 1393–1414.

Coelli, T.J., 1996. A guide to FRONTIER4.1: A computer programfor stochastic frontier production functions and cost functionestimation. CEPA Working Paper 96/07. University of NewEngland, Armidale, NSW.

Färe, R., Grosskopf, S., Lovell, C.A.K., 1994. Production Frontiers.Cambridge University Press, Cambridge.

Felthoven, R.G., 2002. Effects of the American Fisheries Act oncapacity, utilisation and technical efficiency. Mar. Res. Econ.17, 181–206.

Fitzpatrick, J., 1996. Technology and fisheries legislation, In: FAO(Ed), Precautionary Approach to Fisheries. Part 2. ScientificPapers. FAO Fisheries Technical Paper 350/2. FAO, Rome.http://www.fao.org/docrep/003/w1238e/w1238e00.htm.

Fousekis, P., Klonaris, S., 2003. Technical efficiency determinantsfor fisheries: a study of trammel netters in Greece. Fish. Res.63, 85–95.

Grafton, Q., Squires, D., Fox, K.J., 2000. Private property andeconomic efficiency: a study of a common pool resource. J.Law. Econ. 43, 671–714.

Herrero, I., Pascoe, S., 2003. Value versus volume in the catch ofthe Spanish south-Atlantic trawl fishery. J. Agric. Econ. 54 (2),325–341.

International Council for the Exploitation of the Sea (ICES),2002. Report of the ICES Advisory Committee on FisheryManagement. ICES Cooperative Research Report No. 255.ICES, Copenhagen.

Kirkley, J.E., Squires, D., Strand, I.E., 1995. Assessing technicalefficiency in commercial fisheries: the mid-Atlantic Sea scallopfishery. Am. J. Agric. Econ. 77 (3), 686–697.

Kirkley, J.E., Squires, D., Strand, I.E., 1998. Characterizingmanagerial skill and technical efficiency in a fishery. J. Prod.Anal. 9, 145–160.

S. Pascoe, I. Herrero / Fisheries Research 69 (2004) 91–105 105

Kirkley, J., Morrison Paul, C.J., Cunningham, S., Catanzano, J.,2001. Technical progress in the sete trawl fishery, 1985–1999.Agricultural and Resource Economics Department (ARE)Working Paper 01-001. University of California, Davis, USA.

Pascoe, S., Coglan, L., 2000. Implications of differences intechnical efficiency of fishing boats for capacity measures andreduction. Mar. Pol. 24 (4), 301–307.

Pascoe, S., Coglan, L., 2002. Contribution of unmeasurable factorsto the efficiency of fishing vessels: an analysis of technicalefficiency of fishing vessels in the English Channel. Am. J.Agric. Econ. 84 (3), 45–57.

Pascoe, S., Andersen, J.L., de Wilde, J.W., 2001. The impact ofmanagement regulation on the technical efficiency of vesselsin the Dutch beam trawl fishery. Eur. Rev. Agric. Econ. 28 (2),187–206.

Pascoe, S., Hassaszahed, P., Andersen, J., Korsbrekke, K., 2003.Economic versus physical input measures in the analysis oftechnical efficiency in fisheries. Appl. Econ. 35 (15), 1699–1710.

Sharma, K.R., Leung, P., 1999. Technical efficiency of the longlinefishery in Hawaii: an application of a stochastic productionfrontier. Mar. Res. Econ. 13, 259–274.

Tingley, D., Pascoe, S., Mardle, S., 2003. Estimating capacityutilisation in multi-purpose multi-métier fisheries. Fish. Res.63 (1), 121–134.

Vestergaard, N., Squires, D., Kirkley, J., 2003. Measuringcapacity and capacity utilisation in fisheries: the caseof the Danish Gill-net fleet. Fish. Res. 60 (2-3), 357–368.