FAIR CT96-1778
The Management of High seas Fisheries
Partner: Fisheries Research Institute, University of Iceland, Reykjavik, Iceland
Estimation of Production Functions for
the Icelandic Purse Seine Fleet*
by
Sveinn Agnarsson
Ragnar Arnason
and
Gunnar Ó. Haraldsson
M-1.99
This document does not necessarily reflect the views of the
Commission of the European Communities and in no case
anticipates the Commission’s position in this domain.
*
Acknowledgements: This document has been produced as a part of the European Commission
research project: FAIR-CT-1778. Participants in this project are: The Foundation for Research in
Economics and Business Administration and Centre for Fisheries Economics, Norwegian School of
Economics, Bergen, Norway; Helsinki University of Technology, Helsinki, Finland; Fisheries Research
Institute, University of Iceland, Reykjavik, Iceland; Universidade Nova de Lisboa, Lisbon, Portugal.
The financial support of the European Commission FAIR programme is hereby gratefully
acknowledged.
1
1.
Introduction
This paper describes discusses the general production theory for purse seine fisheries
with an empirical application from the Icelandic purse seine fishery.
2.
Fisheries catch functions
In standard production theory it is assumed that a certain flow of inputs, x, is needed
to produce a flow of output, y, i.e. y = f(x). There is generally little uncertainty
surrounding the production process as the producer can easily calculate the amount of
output a certain input bundle will yield. It is also unlikely that the producer will wish
to put the inputs to non-productive use, i.e. utilise the inputs without producing some
output.
Fishing, the production of landings, is different. First, fishing really is combination of
several different production processes, i.e. searching, fishing, sailing to harbour and
landing. Second, due to the variable distance to the fishing grounds and the pervasive
uncertainty of catches, the relationship between inputs and outputs is much weaker
than in the classical case. Third, the use of certain inputs (e.g. fuel consumption) may
easily occur without any output, and the use of some other inputs will only occur
when there is output (e.g. landings).
Rather than model fishing as a function of inputs such as fuel, capital and labour, it
might therefore be more reasonable to write the production function, or harvesting
function as
(1)
y = Y(e, z)
where e is fishing effort and z is biomass. It should be noted that (1) assumes
landings of all catches. Effort is defined as
(2)
e = tf . E(xk)
where tf is fishing time (including search) and xk are features of the fishing vessels
(e.g. capital) that determine fishing power. Fishing time is further defined as
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2
(3)
tf = t0 - 2  d - 
where t0 is total operating time (days at sea), d the distance to the fishing grounds, 
the reciprocal of the vessel speed and  the time spent landing. Needless to say, (3)
refers to a single fishing trip.
The harvesting function can then be written as
(4)
y = Y(( t0 - 2  d - ).E(xk), z).
In this formulation, the only short run control is t0 while xk represents long run control
variables.
The harvesting function in (4) differs from traditional production functions in two
fundamental ways. On the one hand, it does not include standard inputs, such as fuel,
labour or material. Instead, production is viewed as a function of days at sea and
some vessel characteristics. On the other hand, total biomass is allowed to enter the
function as a separate argument.
3.
Data
The data consists of yearly observations of nine purse seiners during the period 19891994, and includes data on vessel characteristics, costs, sales and catches in tons, as
well as the stocks of capelin and herring. According to the Fisheries Association
(Fiskifélag Íslands) there were around 40 boats classified as purse seiners in those
years so our sample represents approximately 25% of the population.
Descriptive statistics of the data used are presented in Table 1. The data on technical
characteristics of individual vessels is obtained from the Fisheries Association and
includes information on vessel size in GRT, length in meters, engine size in
horsepowers (HP) and age of engine. As can be seen the purse seiners vary somewhat
in size, with the largest boats almost three time bigger than the smallest.
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3
Data on sales, catches and the use of various inputs are obtained from the National
Economic Institute. These data are, not surprisingly, highly confidential and are not
published except in aggregate form.
Table 1.
Descriptive statistics:
_________________________________________________________________
Mean
Std. dev.
Min.
Max.
Vessel characteristics:
Size (GRT)
Length (meters)
Size of engine (HP)
Age of engine (years)
Insurance value (mill. kr.)
582.0
50.1
2073.9
10.4
246.3
217.6
8.4
789.8
7.0
136.1
324.0
36.9
900.0
1.0
138.2
950.0
67.6
3182.0
27.0
638.4
Effort variable:
Days at sea
205.0
79.2
0.0
338.0
143.5
66.9
48.0
317.8
23.9
15.5
1.5
112.3
134.1
111.0
115.0
299.0
500.0
591.0
Output variables:
Value of sales (mill. kr.)
Total catch of pelagic
species (tons)
Fish stocks:
Capelin (tons)
Herring (tons)
386.7
431.2
_________________________________________________________________
4.
Estimation procedure
The availability of panel data makes it possible for us to take advantage of estimation
procedures specially developed for panels. In this study we apply the technique
suggested by Kmenta (1986), which allows for autoregressive errors and groupwise
heteroscedasticity, to estimate catch and cost functions for the purse seiners fleet.
The Kmenta method is as follows. Suppose we have N cross-sectional units (boats in
our case) observed over T time periods. The regression can be written as
(5)
y it  xit'    it
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4
where  is a vector of parameters to be estimated and  is a random error term. An
estimate of  is obtained by a generalised least squares procedure (GLS) which makes
the following assumptions on the error term.
(6)
E ( it2 )   2i
heteroscedasticity
(7)
E ( it  jt )  0 for i  j
cross section independence
(8)
 it   i  it 1  vit
first order autoregression
and E(vit) = 0, E (vit2 )   v2 , E(vitvjt) = 0 for all i  j, E(vitvjs) = 0 for t  s and E(it-1vjt)
= 0. These assumptions about the error structure imply the following: First, different
purse seiners may have different error variance, i.e. different degrees or operational
stability. Second, there is no tendency for common deviations from the efficiency
mean for the purse seiners.1 A relaxation to allow for cross-section correlation would
involve: E( it  jt )   ij , E( it jt )  ij and E( it js )  0 for t0. Third, the error terms of
each purse seiner tend to correlated over time so that an efficient vessel in one year
tends to remain so over time and vice versa.
The estimation of  proceeds with the following steps. Step 1: Estimate  in (13) by
OLS and obtain the estimated residuals eit. Step 2: Compute the autocorrelation
coefficients, i, as the sample correlation coefficient between eit and ei,t-1.2 Step 3: Use
the ̂ i ’s to transform the observations, including the first observation, apply OLS to
the transformed model and obtain the error covariance matrix, V. Step 4: Obtain the

~
GLS estimator as ˆ  X 'V 1 X
 X V~ Y  where V~
1
'
1
is the diagonal V matrix with
zero’s on the off-diagonals.
1
This hypothesis is somewhat doubtful and may need a further check.
T
2
̂ i
is defined as ̂ i 
e
it e i ,t 1
t 2
T
,
e e
it
t 2
for i = 1,…,N
T
i ,t 1
t 2
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5.
Results
A harvest function, based on the specification outlined in (4), was estimated as
(9)
yit   0   1 d it   2 hit   3 sit   4 ait   5 z tc   6 z th   it
were, dropping subscripts, y is total catch of each boat of all pelagic species, d denotes
days at sea, h is the size of the engine in HP, s the size of the boat in GRT, a the age
of the engine, zc and zh the stocks of capelin and herring respectively and the ’s are
parameters to be estimated. All variables are in natural logarithms. The parameter
estimates are presented in Table 2.
Table 2.
Parameter estimates of a catch function and estimated
elasticity. Dependent variable is catch of all pelagic
species. All variables in natural logarithms.
Standard errors in parenthesis.
_____________________________________________
Parameter estimates
Constant
Day
Engine size (HP)
Boat size (GRT)
Age of engine
Stock of capelin
Stock of herring
-2.600
(1.428)
0.421
(0.024)
-0.980
(0.165)
1.332
(0.208)
-0.129
(0.043)
0.029
(0.056)
0.712
(0.131)
*
*
*
*
*
Buse R2
0.932
__________________________________________
* denotes significance at the 1% level.
All the parameters except the constant and the zc-parameter are significant at the 1%
level and given the high Buse R2 value the function appears to have good explanatory
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power.3 However, the sign of the engine parameter is contrary to expectation, as one
would, a priori, certainly expect a ship with a larger engine to fish more than a ship
with a smaller engine. The age parameter is also negative, implying that having an
old engine has a detrimental effect on the total catch. The sum of the effort and vessel
characteristic parameters is 0.5, which could indicate the presence of economies of
scale.
The herring stock parameter is significantly positive while the parameter associated
with the capelin stock is insignificant. This could possibly be explained by the fact
that during our period of observation the boats have usually managed to fish all their
allocated quota of herring, while the capelin catch was sometimes less than the quota
allocated. Since the total quota each year is closely correlated with estimates of the
spawning stock of each species this could explain the results obtained.
ˆ 1 e~
e~´
where Y is an NTx1 vector of
ˆ 1 (Y  DY )
(Y  DY )´
ˆ  R(
ˆ  I ) R´ . R is the NTxNT matrix that
observations stacked by cross-section unit and 
T
ˆ 1W ) 1W´
ˆ 1
performs the autoregressive transformation. The matrix D is constructed as: D  W (W´
3
The Buse R2 statistic is calculated as: 1 
where W is a NTxN matrix such that column i has a vector of 1´s for cross-section i and 0´s
elsewhere.Therefore Y-DY transforms the observations to deviations from a weighted mean.
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References
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Demersal Fisheries. Ph.D. thesis. University of British Columbia.
Arnason, R. 1995. The Icelandic Fisheries – Evolution and Management of a Fishing
Industry. Fishing News Books. Oxford.
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Discussion. Marine Resource Economics. Vol. 5. pp. 99-123.
Haraldsson, G.Ol. 1997. The Icelandic Fish Processing Industry. Unpublished M.Sc.thesis. University of Iceland.
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the Theory and Practice of Econometrics. John Wiley & Sons.
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