Managing growth-overfishing is more important
than managing recruitment-overfishing
work in progress, January 26, 2011
Florian Diekert and Tristan Rouyer
University of Oslo
Centre for Ecological and Evolutionary Synthesis (CEES), Dept. of Biology
P.O. Box 1066 Blindern, 0316 Oslo, Norway. Tel/Fax: +47 2285 8479/4001.
E-mail: [email protected].
Abstract
Avoiding overfishing and ensuring the sustainability of the resource base is of
prime importance for fishery management. Current policies predominately rely on
aggregate estimates of stock biomass to set total harvest quotas. Since many commercially exploited fish stocks consist of several age-classes, and since many fish
grow considerably with age, there might be added-value in selectively harvesting
the most valuable age-classes. Moreover, recruitment is often highly uncertain, so
that the value of basing management decisions on the size of the spawning stock is
not clear. We calibrate a detailed model on the North-East Arctic cod fishery, investigating whether a manager would act differently when the size of today’s spawning
stock determines tomorrow’s recruitment or when there is no link between today’s
stock and tomorrow’s recruitment. We find that the manager would choose very
similar policies; namely to only target those fish that have reached their full growth
potential. This suggests that, in many cases, we can avoid recruitment-overfishing
by adequately managing growth-overfishing.
1
Introduction
The feasibility of sustainable development will not only depend on how humanity
deals with its finite resources, but also on how it manages those resources which are
– at least in principle – renewable. For example, the food supply of a growing world
population could be significantly supported by seafood production, if world fisheries
were effectively governed and adequately managed (Smith et al., 2010). According
to current estimates (FAO, 2009), more than half a billion people rely – directly or
indirectly – on fish as a source of income. However, few fisheries realize more than a
fraction of their social, economic, and biological potential. Many fisheries are plagued
by overfishing, leaving some of them severely depleted and at the brink of collapse
(Worm et al., 2006).
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GRO-draft-v02.tex
January 26, 2011
Conceptually, there are three processes linking a fish stock at a given time t to the
fish stock at a future date: recruitment, growth, and mortality. Recruitment concerns
fish which are not yet around at time t. It is a process linking the reproductive part
of the stock at time t (often summarized by the spawning stock biomass, or SSB for
short) to the number of newborn fish at a future date. Growth and mortality are processes that concern fish that already exist at time t. Most fish exhibit indeterminate
growth, so that an individual continues to gain weight with age, thereby contributing
more to the biomass of the stock at a future date – provided, of course, the individual
has survived up to that time.
However, these processes cannot be strictly isolated. For example, the contribution to recruitment may depend on growth. Not only is the onset of maturation
generally conditional on having reached a certain weight, also the fecundity and
quality of offspring is believed to increase more than proportionally with weight in
many species. Similarly, recruitment and mortality are often linked, be it because
natural mortality is higher after spawning (as in the case of salmon, which generally
dies after reproduction), be it because fishing mortality is higher during spawning (as
in the case of Atlantic cod where a significant part of the fishing effort is concentrated
on the spawning grounds).
Recruitment overfishing can then be defined as depleting the reproductive part of
the stock by so much that recruitment is impaired. Growth overfishing can be defined
as depleting the young part of the stock before it has reached its full biological and
economic potential. Both forms of overfishing constitute a negative inter-temporal
externality. They are lumped together in the classical bio-economic models explaining the “tragedy of the commons” in fisheries. Indeed, as recruitment, growth, and
mortality are confounded processes, so are growth- and recruitment-overfishing. It
is nevertheless necessary to try to distinguish these aspects in order to inform adequate management responses. Identifying and emphasizing those problems that
have the most adverse consequences and can also be solved most effectively, is a
crucial but difficult step. Therefore, management could benefit from assessing the
contribution to the overall performance of avoiding growth-overfishing on the one
hand and avoiding recruitment-overfishing on the other hand.
The common perception is that growth-overfishing is the more wide-spread form
of overfishing, whereas recruitment-overfishing has more disastrous consequences
since it directly impedes the future viability of fish stocks (Gulland, 1983). While this
is true, the economic and biological damage from harvesting fish before they have
grown to a decent size could still be very high. This is particularly the case for species
which grow significantly in weight and value with age. North East Arctic cod for example are currently targeted at an age where the fish would still grow with 20% in
value per year. Over and above, growth-overfishing is increasingly seen as a serious
biological problem (Hsieh et al., 2006; Beamish et al., 2006; Ottersen, 2008). Even –
and perhaps especially – in those fisheries where the overall biomass is reasonably
well managed. Due to the selective property of fishing gears very few fish survive to
grow old and large, implying a pronounced shift of the age-composition of harvested
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January 26, 2011
stocks. This effect is commonly referred to as “age truncation”. Since old fish are better able to buffer adverse environmental fluctuations (Ottersen et al., 2006), growthoverfishing can lead to magnified fluctuations of abundance and decreased biological
stability (Anderson et al., 2008). If harvesting has evolutionary consequences (Guttormsen et al., 2008; Jørgensen et al., 2009; Eikeset et al., 2010a), these changes
may be irreversible (Stenseth and Rouyer, 2008).
Even though most management schemes do in fact include some sort of gear
regulation or minimum age/size limits, these are often set ad hoc and far from being
optimal (Froese et al., 2008). The preferred tool of fisheries management in most
developed fisheries is the setting of total allowable catch (TAC) quotas (Froese et al.,
2010). Based upon an estimate of the aggregated stock biomass, stock managers
answer the question:“how much should be harvested?” Acknowledging the fact that
fish stocks are not a uniform mass but consist of individual fish leads to a second
question: “which fish should be harvested?”
This paper has two aims. First, we seek to illustrate the value of answering this
additional question. Second, we seek to answer whether it could be that, by avoiding
growth-overfishing, we also avoid recruitment-overfishing. We address these questions in the context of a multi-cohort species which grows significantly over age. To
this end, we calibrate a detailed age-structured1 bio-economic model on the NorthEast Arctic (NEA) cod fishery. We then compute
(a) the maximum Net-Present-Value which is obtainable when only the harvested
amount is controlled. In our model, this is achieved by setting the appropriate
amount of effort.
(b) the maximum Net-Present-Value which is obtainable when both the amount and
the composition of the harvest is controlled. In our model, this is achieved by
determining which age-classes should be spared from being harvested (i.e. by
choosing the selectivity) and by setting the appropriate amount of effort.
In order to elucidate the effect that considerations of recruitment-overfishing have
on the character of the optimal policies, we contrast two scenarios:
(1) In the first scenario, (a) and (b) are calculated for endogenous (deterministic) recruitment. Here, the manager has complete control over next year’s recruitment
by keeping the spawning stock biomass at a given level.
(2) In the second scenario, recruitment is completely exogenous (random). There is
no link between the size of the current spawning stock and future recruitment in
this scenario. Consequently, the optimal policies cannot be influenced by considerations of recruitment-overfishing.
If now the optimal policies between the first and the second scenario are similar, but
the performance between (a) controlling only the total harvested quantity and (b)
1 Of course, fishing is in effect a size-selective process (fish whose girth is smaller then the diameter of
the mesh may escape through netting, while larger fish may not). However, fish size is closely related to
age. The latter is more convenient to use as it moves at the same speed as time (one year later, a given
fish will be one year older). Moreover, ICES data for fish stocks is routinely reported in age, not in length.
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GRO-draft-v02.tex
January 26, 2011
controlling both quantity and quality of the harvest are large, then this indicates that
it is more important to avoid targeting the wrong fish (manage growth-overfishing)
rather than controlling the overall size of the spawning stock (manage recruitmentoverfishing).
There is a large literature on the optimal control of renewable resources.2 Strictly
speaking, this is not what we do. Rather, we specify a set of rules for either (a) controlling harvesting effort or (b) controlling both effort and selectivity. Each rule spans
a grid of policies and we repeatedly simulate the development of our model fishery.
Finally, we pick the rule and policy which, on average, yields the highest Net-PresentValue. We apply this procedure both for (1) the scenario is a deterministic function
of the current stock and (2) for the scenario where recruitment is independent of the
current stock. This allows us to investigate the effect that the possibility to control
recruitment has on the design of optimal policies. For a conceptual sketch of our
method see Figure 1. The simulation procedure and the rules are specified in more
detail in section 3.
Bio-economic model
calibrated on NEA cod
Scenario (1): Deterministic Recruitment
Control (a): Effort
 For management rule I,II,...:
- For policy 1,2,..:
- simulate model:
- recruitment, mortality,
growth, and harvest
- repeat for 100 time steps
- repeat simulations
- record performance of policy
Scenario (2): Random Recruitment
Control (b): Effort and Selectivity
Control (a): Effort
 For management rule I,II,...:
- For policy 1,2,..:
- simulate model:
- recruitment, mortality,
growth, and harvest
- repeat for 100 time steps
- repeat simulations
- record performance of policy
 For management rule I,II,...:
- For policy 1,2,..:
- simulate model:
- recruitment, mortality,
growth, and harvest
- repeat for 100 time steps
- repeat simulations
- record performance of policy
Control (b): Effort and Selectivity
 For management rule I,II,...:
- For policy 1,2,..:
- simulate model:
- recruitment, mortality,
growth, and harvest
- repeat for 100 time steps
- repeat simulations
- record performance of policy
Compare best policy under best management rule
Difference in outcome between (a) and (b)?
Difference in outcome between (1) and (2)?
Figure 1: Sketch of method
Growth overfishing was clearly on the agenda during the rise of modern fishery
science after the second World War (Allen, 1953; Beverton and Holt, 1957; Turvey,
1964), but the attention to it has subsequently dwindled. Even though age- and
2 The standard textbook reference is Clark (1990). Nøstbakken and Conrad (2007) provide an overview
of accounting for uncertainty in bio-economic models.
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GRO-draft-v02.tex
January 26, 2011
stage-differentiated modeling has recently gained traction in the resource economics
literature (Smith et al., 2008; Tahvonen, 2009; Quaas et al., 2010), current state-ofthe-art management strategy evaluations (Dankel et al., 2008; Kjærsgaard and Frost,
2008; Eikeset et al., 2010b,c; Brunel et al., 2010; Froese et al., 2010) do not explicitly
consider changes in selectivity. Those yield-per-recruit analyses that do (Ulltang,
1987; Kvamme and Frøysa, 2004; ?) neither include economic information nor seek
to find the optimal harvesting pattern.
In addition to the policy relevance of our work, we contribute to the literature by
providing an “empirical estimate on efficiency gains from optimal harvesting compared to currently applied biological reference points.” as requested in Tahvonen
(2009, p.297). Moreover, we introduce an estimated age-specific harvest function,
which to our knowledge has not been done before.
The article proceeds as follows: The next section describes our model of the NorthEast Arctic cod fishery. Section 3 discusses the numerical implementation of the
simulations. The main results are presented in section 4.Section 5 concludes.
2
Model and calibration
The North-East Arctic (NEA) cod stock is the largest cod stock in the world, supporting
one of the most valuable fisheries (FKD, 2011). Due to its importance, the fish stock
and its fishery is thoroughly researched.3 It is jointly managed by Russia and Norway
and fished by both a conventional coastal fleet and an ocean-going trawler fleet. The
total annual harvest is currently around 600 000 tons. Since on average more than
70% of this harvest is taken by trawlers (ICES, 2010), we do not specifically model
the annual migration of the spawning stock from the feeding grounds in the Barents
Sea to the spawning grounds along the Norwegian coast. Moreover, we concentrate
on the Norwegian trawler fleet for the calibration of the economic part of our model
because of data availability. We calibrate the model on the period 1990-2005.
2.1
The biological part of the model
We model the sequence of events within one year as recruitment, natural mortality,
growth, and harvest.4 To introduce some notation, let n,t be the number of fish of
age-class  at time t. As NEA cod are recruited to the fishery when they are three
years old, the age-classes run from  = 3 to  = A where A is the oldest age-classes
collecting all individuals of age 13 and above. Furthermore, denote the probability
of surviving a given age-class by ϕ , the weight of a given age-class by  , and the
proportion of a given age-class which have matured by mt .
3 A search of {cod AND ’North East Arctic’ OR ’Barents Sea’} returned over 7500 hits on google scholar
and over 5500 hits on ISI web of knowledge. For a general overview of the fishery see Nakken (1998).
Recent bio-economic analyses include Diekert et al. (2010b,a); Eikeset et al. (2010b,c) and Richter et al.
(2011).
4 The order of events is of little consequence (Deacon, 1989). Instantaneous harvesting is introduced
mainly for convenience, which is common in economic but also in some ecological models (for a discussion
see Tahvonen (2009, p.284)). Also ICES (2010) set mortality prior to spawning to zero.
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GRO-draft-v02.tex
January 26, 2011
The observed relationship between the number of recruits (in millions) and spawning stock biomass three years before (SSB, in thousand tonnes) is shown in Figure 2
0
500
1000
Recruits in millions
1500
2000
(The data is from 1946 to 2009, see table 3.25 in ICES, 2010, p.209).
0
200
400
600
800
1000
1200
SSB in thousand tonnes
Figure 2: Scatterplot of Recruitment vs SSB and fitted linear relationship
There is a large variability in recruitment and no stock-recruitment relationship
is directly discernible. We deliberately overstate the case of deterministic recruitment (in which the manager can control future abundance by controlling the current
spawning stock) by assuming that recruitment is proportional to SSB over the domain of observed values and constant at its highest level thereafter. Hence, we fit
the data to a linear regression forced to pass (0,0) (Table 1). For the case of random
recruitment, the number of fish that enter the fishery is a random number R. It is an
iid. draw from all observed recruitment values between 1946 and 2009.
Table 1: Regression results for deterministic recruitment function
Estimate
Std. Error
t value
1.2182
0.1213
10.04
SSB
Pr(>|t|)
1.83e-14
In summary, recruitment – according to the respective scenario – is described by
equation (1).

n3,t =

 R


1.22· SSBt−3
random recruitment
(1)
deterministic recruitment
The development of the subsequent age-classes from one year to the next is given
by equation (2) and (3). The fish in age-class  + 1 at time t + 1 are those from the
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January 26, 2011
previous age-class that survive year t (first term on the right-hand-side of equation
2), minus those that have been harvested (the second term on the right-hand-side
of equation 2; since harvest h will later be specified in terms of biomass, it has to
be divided by the age-specific weight  to be given in terms of numbers). Equation
(3) describes the cohort dynamics of the oldest age-class A. It collects all fish that
newly enter this age-class from age-class A − 1, as well as those fish that are already
present in this age-class and survive the current year.
n+1,t+1 = ϕ n,t −
1

h,t
nA,t+1 = ϕA−1 nA−1,t −
(2)
1
A−1
hA−1,t
+
ϕA nA,t −
1
A
hA,t
(3)
We assume that all age-classes but the oldest face the same annual risk of dying
from natural causes, so that the survival probability is ϕ = 0.8 for  = 3, ..., A−1.5 We
set the survival probability of the oldest age-class to ϕA = 0.5 to account for senescence and to prevent it from accumulating unreasonable amounts of biomass. The
parameters for weight-at-age ( ) and the proportion of mature fish at a given age
(mt ) are taken to be the average values from 1990-2005,6 so that the biological
and economic model are calibrated on the same time period (see Table 2).
Table 2: Biological Parameters
Age 
 (in kg)
mt
2.2
3
4
5
6
7
8
9
10
11
12
13+
0.27
0.6
1.29
2.15
3.29
4.76
6.71
9.25
10.85
12.73
14.31
0.001
0.008
0.07
0.31
0.64
0.85
0.96
0.99
1
1
1
The economic part of the model
The basis of the economic data, which is provided by the Norwegian Directorate of
Fisheries, is the annual survey of the Norwegian trawler fleet catching cod North of
62-degree latitude (Fiskeridirektoratet, several years ). One part of the data has prior
been analyzed by Sandberg (2006) and Richter et al. (2011). It spans the years from
1990 to 2000. The other part of the data is newly acquired and basically an extension
of the Sandberg data to the years 2001-2005.7 To construct a cost variable, the 14
5 ICES uses a natural mortality of 0.2 in its stock assessment and Jørgensen and Fiksen (2006) use a
value of 0.25 in their detailed model of cod life-history. Recent state-space modeling (Aanes et al., 2007)
show that there is large uncertainty around the point estimate, but that natural mortality fluctuates more
through time than over age. Using advanced statistical methods, Brinch et al. (2011) confirm that a guess
of 0.2 is indeed not far off the mark.
6 The data is from table 3.11, pp.179, and 3.12, pp.182, in (ICES, 2010) respectively.
7 In contrast to the Sandberg data, the newly acquired data had no information on the days that a boat
spent fishing for a particular species. Also in the 1990-2000 data set, the variable “days spent fishing cod”
(or coddays from now on) is not an original part of the surveys, but a construction using monthly catch
statistics (see Sandberg, 2006). We do not have monthly catch statistics for the period 2001-2005. This
is a problem, since the days spent fishing for cod is an important part of the variable “effort”. Instead
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GRO-draft-v02.tex
January 26, 2011
different entries of cost-components in the data (fuel, insurance, maintenance, etc.),
given in Norwegian Kroner, were summed and corrected for inflation.8 A summary of
the main variables in the full dataset is given below (Table 3).
Table 3: Summary statistics
Variable
Length (in meter)
Gross Tonnage (TE)
Boat age
Days of operation
Mean
Std. Dev.
Min.
Max.
N
47.1
11.5
17.6
75.5
864
971.2
693.8
175
3186
817
16.8
9.2
0
51
864
284.2
59.6
68
365
864
18 762.9
12 142.0
427.5
71 086.9
864
Total harvest (in tonnes round weight)
2 910.0
1 555.3
50.0
8 304.2
864
Cod harvest (in tonnes round weight)
885.8
636.8
2.5
4 495.1
861
Cost (in thousand NOK (year 2000))
The cod fishery of the Barents Sea is in fact a multispecies fishery. After cod,
saithe and haddock are the most important species in terms of harvested volume.
The boats can therefore be characterized as joint-input, multi-output firms, where
a common mix of input is used to produce several products (Squires, 1987; Jensen,
2007). We assume that the production of cod harvest is independent and can be separately regulated without affecting other production processes (output-separability).
We then define “effort” as the set of inputs directed towards catching cod. By inspecting the distribution of the percentage that cod contributes to total harvest (Figure 3),
it becomes clear that, for the main part of the boats in the sample, cod makes up less
than half the harvest. We drop all observations where cod makes up less than 2% of
0
50
Frequency
100
150
the harvest (30 observations out of 864).
0.0
0.2
0.4
0.6
0.8
1.0
Figure 3: Histogram: share of cod in total harvest
Several candidates exits when selecting the best proxy for “effort”. As it measures of the intensity of fishing (Gulland, 1983), it should include an element of the
we create coddays as the operation days of a boat weighted by the annual average share of cod in total
harvest. In fact, the annual averages were the best predictor of the monthly averages.
8 The Commodity price index for the industrial sectors from the Norwegian bureau of statistics (SSB) were
used. http://statbank.ssb.no/statistikkbanken/Default_FR.asp?PXSid=0&nvl=true&PLanguage=
1&tilside=selecttable/hovedtabellHjem.asp&KortnavnWeb=vppi
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GRO-draft-v02.tex
January 26, 2011
time that is spent harvesting. Moreover, if size is a significant determinant of the
harvesting process, also measure of size should be included in the effort proxy. Boat
size could be captured by either length or tonnage. We choose the latter as it correlates closer with harvest and costs (see Table 4 and 5 below). One unit of effort
is therefore one unit of boat tonnage effectively catching cod for one day. As there
were 48 observations without information on tonnage, we are left with a panel of 786
observations from 141 different boats over a time period of 16 years (on average 5.6
observations per boat, some boats were observed in all years).
Table 4: Correlation between harvest and effort candidates
codharv
codharv
1
coddays
length-coddays
coddays
0.7375
1
length-coddays
0.8360
0.9480
1
tonnage-coddays
0.7805
0.5636
0.7620
tonnage-coddays
1
Table 5: Correlation between cost and effort candidates
cost
cost
coddays
length-coddays
tonnage-coddays
1
coddays
0.6279
1
length-coddays
0.7564
0.9480
1
tonnage-coddays
0.8526
0.5636
0.7620
1
Harvest function
Harvest h is a function of effort e and the existing stock, which we denote by the
variable . A commonly used harvest function in aggregate biomass models is the
Cobb-Douglas function h = qeα β , where q is the “catchability coefficient”, α is the
effort-output-elasticity and β is the stock-output elasticity. The value of α tells by
how much harvest increases when effort increases by one unit. The value of β tells
how much harvest increases when the stock increases by one unit. If the fish stock
follows an ideal free distribution (β = 1) always occupying a given area, the density
of fish declines at the same rate as the stock gets depleted. Often it is argued that
this is an adequate description for a demersal species such as cod (Clark, 1990).
However, empirical studies have shown that this does not hold true for the NEA cod.
Richter et al. (2011) for example provide a thorough study of this fishery and find
that β ranges between 0.22 for longliners and 0.58 for trawlers (see also Hannesson
(1983); Eide et al. (2003)).
At this point, it is important to note that we are not interested in the elasticity of
the aggregate stock biomass β, but in the age-class specific parameters β . There
is no way of inferring the age-class specific parameter values from aggregate data
9
GRO-draft-v02.tex
since
P

β
January 26, 2011
β
β
β
6= 33 + 44 + ... + AA , unless β = β = 1 for all . Moreover, catch-
ability becomes confounded with selectivity in the age-structured case (Millar and
Fryer, 1999): A fish may have not been caught because the fisherman did not find
it, because the fisherman found it but the fish avoided the gear, or because the fish
had contact with the gear but was not retained (e.g. because it was so small that it
could slip through the net).
Since we want to explicitly model the choice which age-classes are harvested, we
interpret effort as producing an amount of water which has been screened for fish,
irrespective of age (Gulland, 1983). Further, we introduce a selectivity parameter
s ∈ [3, A], so that all age-classes at least as old as s are targeted. The status quo is
that all age-classes which recruit to the fishery are targeted. Changing the selectivity
pattern (choosing s > 3) then means that all age-classes younger than s are spared
from being harvested. This can be thought of as a technical modification of the gear
so that a fish, even if it were to have contact with the gear, would not be retained.
Hence, we implicitly assume a knife-edge selectivity pattern. Although this is clearly
not very realistic, it is also not very wrong as a general description of trawl retention
pattern, and it greatly enhances the transparency of our work.
In summary, the total harvest is the sum of all selected age-classes and the contribution of a given age-class  ≥ s is then captured by the parameter β . To arrive at
a general description of the harvest function (4), we construct a vector of indicator
variables  which take the value of zero for  < s and the value of one for  ≥ s.

h=
A
X
=s
h =
3
...
A



qeα 


β3
..
.
βA







(4)
The crux is that age-specific data is not available at the boat level, which means
that it is not possible to estimate equation (4) directly.9 We take the following approach to overcome this: Let J be the total number of boats in the fishery. The harvest
P
from a given boat j is aggregated over all age-classes: hj = s hs,j . The ICES data is
P
aggregated over all boats, but available in age-specific format hs = j hs,j as well as
P P
in total h = s j hs,j (see Table 3.9 and 3.10 in ICES, 2010, pp.175). By assuming
that all boats have the same selectivity pattern and using the share of each boats
harvest in total harvest, we are then able to calculate the individual age-specific
harvest as hs,j =
hs
h .10
h j
Since we have a panel of boats, we can account for individual heterogeneity.11
9 This is probably the reason why – to the best of our knowledge – an age-specific harvest function has
not been empirically estimated yet. However, it should be noted that Sumaila (1997) employs a similar
idea but he assumes a fixed selectivity pattern, linearity in effort and uses data from only one year.
10 In fact, thanks to Bjarte Bogstad and Kjell Nedreaas (both at IMR) we were able to obtain age-specific
harvest data from only the Norwegian trawler fleet (pers.comm.). However, due to the small sample size
(∼ 13 boats) the estimates from this dataset were significantly less precise and the model could only
explain a fraction of the variance explained by the regressions on ICES data.
11 Most importantly, the (unobserved) ability of the fishermen is omitted from the definition of effort
(Squires and Kirkley, 1999). A “random-effects” model would then assume that these individual factors
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January 26, 2011
We therefore estimate equation (4) by a random-effects model (the routine lme in in
the statistical program R). By using a dummy variable in front of each age-specific
biomass, we can estimate the age-class specific parameters β , but the effort elasticity α and the boat-specific intercept q remain the same for all age-classes.
ln h,j,t = q∗
+ α ln ej,t + Dβ3 ln 3,t + Dβ4 ln 4,t + ... + ϵ,j,t
j
(5)
qj ∼ IID(0, σq2 ),
ϵs,j,t ∼ IID(0, σ 2 ),
ϵs,j,t ⊥ qj ⊥ ej,t ⊥ s,t
Table 6: Regression result for harvest function
Estimate
Std. Error
t value
q
Pr(>|t|)
-16.701
0.090
-184.9760
0
log e
0.902
0.007
122.8770
0
log x3
0.733
0.006
127.4131
0
log x4
0.875
0.006
158.8186
0
log x5
0.932
0.005
173.3542
0
log x6
0.949
0.005
175.4130
0
log x7
0.956
0.006
172.7147
0
log x8
0.957
0.006
165.4993
0
log x9
0.948
0.006
154.6800
0
log x10
0.937
0.007
142.9521
0
log x11
0.922
0.007
130.6130
0
log x12
0.916
0.008
120.8221
0
log x13+
0.928
0.008
116.9850
0
The regression results are given in Table 6. As harvest is not found to be linear
in effort, we cannot aggregate from the boat level to the fleet level. Therefore, we
calibrate the model with the estimates for the average boat in sample and scale the
effort in the simulation so that it replicates the size of the actual harvest (that is, we
assume that there are 200 boats in the fishery).
Profits, costs, and prices
Fishing effort causes costs according to some function c(e), so that the fishery profits
in a given year can be written as equation (6), where p is the vector of age-class
specific prices.
are normally distributed around a common mean, whereas a “fixed-effects” model dispenses of this assumption by including a dummy variable for each individual. The first approach is consistent only if the
assumption of normality is indeed met, while latter approach is consistent regardless. However, since
the panel at hand is broad (141 individuals), but short (16 years), a large part of the information is lost by
looking only at the variation within an individual. Hence a fixed-effects model would over-emphasize largesample consistency for estimation efficiency, not the least because we are not interested in the harvest
function for the boats in this specific sample, but in the harvest function of the whole population.
11
GRO-draft-v02.tex
January 26, 2011
πt = pht − c(et )
(6)
Similar to effort above, we are only interested in the share of costs which is caused
by catching cod. Therefore, the total cost are weighted by the boat-specific share of
cod in the total harvest. Note that with this definition, costs are linear by construction.12 Yet by inspecting the scatter plot of total effort (tonnage times operation days)
against the total cost (Figure 4) we see that the underlying relationship shows no immediate signs of non-linearity. The regression results for the cost function (where the
intercept was suppressed since it would have the unwanted effect of fixed- or set-up
cost in the model simulations) are given in Table 7.
Table 7: Regression result for cost function
Estimate
Std. Error
t value
63.4611
0.8368
75.84
7e+07
tonnage-days
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4e+07
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3e+07
2e+07
1e+07
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ton_operdays
Figure 4: Scatter plot of tonnage-operation days against cost
In the most recent study of the NEA cod fishery, Richter et al. (2011) elaborately
estimate how prices depend on the quantity landed on an aggregate level. We are
interested in the age-specific prices, not the least because larger fish get a higher
price per kg. Prices at age (or – more precisely – prices at weight) are in principle
publicly obtainable from the Norwegian fishermen’s sales organization.13 However,
the issue is plagued with problems of identification. Moreover, 90% of the cod products are exported to the larger world market for whitefish, and the first hand sales are
12 In spite of a constant marginal relationship between costs and effort, the marginal relationship between
costs and harvest is increasing: For β > 0, the stock dependency makes it excessively costly to harvest
the last fish in the ocean.
13 Råfiskelag, for the database see http://www.rafisklaget.no/portal/pls/portal/PORTAL.RPT_
VAREPRIS_SLUTTSEDDEL.show_parms.
12
GRO-draft-v02.tex
January 26, 2011
furthermore regulated by minimum prices (Asche et al., 2001). We therefore take the
average values from 1997-2004 as a ballpark estimate of the vector of dock prices
(see Table 8).
Table 8: Price at age
Age
p in NOK
3
4
5
6
7
8
9
10
11
12
gp
11
11
14.5
14.5
16
16
18.5
18.5
18.5
18.5
18.5
Equation (7) finally describes the Net-Present-Value of the fishery for a given sequence of effort and stock values. The net-present-value is the sum over all annual
profits, discounted by a factor δ. We take that maximizing (7) subject to the biological model (equation (1)-(3) and parameters in Table 2) and the economic model
(equation (4)-inst-profits, and the parameters in Table 7 & 8) is the objective of the
manager.
NPV(et , t , t ) =
T
X
δt πt
(7)
t=0
2.3
Validation
In order to validate our bio-economic model, we compare the actual development
in the historic fishery with the model predictions. Figure 5a shows the development
of NEA cod biomass from 1990-2005, while the blue line shows the predictions from
the biological part of our model. These predictions were obtained by simulating the
stock, starting from the initial biomass-at-age in 1990, and reading in the values for
recruitment and harvest-at-age from the data. Clearly, there is some error in taking
weight-at-age to be at its average values and using the natural mortality pattern of
0.2 for all age classes except the oldest, but the overall trend could be captured.
Similarly, figure 5b shows the development of the harvest from 1990-2005 (black
line), compared with the predictions when reading in the recruitment values and
taking the effort in a given year to be the average from the sample in that respective
year times 200 (the assumed number of boats in the fishery). The error in the first half
of the validation period could stem from the inaccuracy in the biological part of the
model or from the fact that the sample size of the economic panel was significantly
smaller in 1990-1995 than in 2000-2005. Still, also the complete bio-economic model
succeeded in roughly replicating the overall development.
Nevertheless, it should be emphasized that we do not aim at calibrating a model
of the NEA cod to fit past observations as accurately as possible, but rather that we
aim to calibrate a generic model on the NEA cod to make inferences about the future implications of different policy options. To provide a feeling for the magnitude of
changes that could be implicated, we contrast the development of the biomass of a
cohort over age when experiences no fishing mortality (open circles) to the development of a cohort’s biomass that experiences the current fishing mortality (black dia-
13
January 26, 2011
8e+05
6e+05
harvest.sample
4e+05
2e+05
1500000
0
0e+00
500000
stock.sample
2500000
1e+06
GRO-draft-v02.tex
5
10
15
5
10
Index
15
Index
(a) Stock, observed vs. predicted
(b) Harvest, observed vs. predicted
Figure 5: Model validation
monds).14 Whereas under no fishing mortality, the age of maximum biomass would
be 10, the current selectivity pattern implies that a cohort has reached its peak at an
5
4
3
2
0
1
Biomass relative to age 3
6
7
age of 5.
3
4
5
6
7
8
9
10
11
12
13
Age
Figure 6: Cohort biomass development under no fishing and current fishing pressure
3
Simulations
In order to answer our research question, we calculate the Net-Present-Value of the
model fishery from simulating it over a period of hundred years, when either (a) only
effort is controlled, or (b) both effort and selectivity are controlled. We do this for two
scenarios: (1) when recruitment is deterministic and (2) when recruitment is random.
14 For comparison, the biomass was scaled relative to the biomass of a cohort at the age of recruitment,
the data for the current cohort development is the average stock biomass from 1990-2005.
14
GRO-draft-v02.tex
January 26, 2011
The control variables are chosen according to a set of pre-defined rules. By “rule”
we mean a general way of determining the exploitation pattern, which subsumes a
number of different policies. A “policy” is then the specific embodiment of a rule.
For example, the rule could be to harvest a share of the existing biomass, or to set a
fixed selectivity pattern. A corresponding policy could then be to harvest 20% of the
existing biomass, or to select all fish that are older than 5 years. For each policy, the
simulations of the model fishery are replicated 200 times so that the average NPV
will not be influenced by the random events (initial values and also recruitment in
scenario 2). The policies will be chosen from a grid of 20 possible values and we then
examine whether the policies yielding the three best Net-Present-Values are in the
vicinity of each other (in other words, we check whether the surface of the objective
is acceptably smooth). If this is the case, we “zoom in” and chose again a grid of 20
possible control values. We continue this procedure until the difference between the
best three NPVs is less than 5%.
As mentioned in the introduction, this is strictly speaking not the maximization of
an objective function over a general domain. The dimensionality of the state-space
(11 state variables) prohibits the use of conventional optimal control methods (e.g.
dynamic programming). Nevertheless, we explore the performance of a large set of
rules and policies from which we pick that combination of control policies which, on
average, yields the highest NPV. In the end, we are not so interested in finding the
solution to one specific optimal control problem, but we are interested in comparing
the performance of policies in two different optimal control problems. I.e. rather than
asking how should we optimally steer a specific system, we ask how does steering
system 1 according to a specific policy compare to steering system 2 according to
the same policy.
In the case where selectivity is a control variable, the rules for choosing it are:
A.) To set a fixed selectivity pattern, st = s ∈ [3, 4, ..., A]
B.) To select that age-class which is at distance θ from the age-class of maximum
biomass.
Rule A.) is a rigid exploration of the effect of selecting a given age-class. Rule
B.) is an essay to install a feedback between state and control and requires some
elaboration: Since the number of fish is continually declining, but the weight per
individual is increasing, there will be one age-class that collects the largest biomass,
denoted by m (recall Figure 1). Moreover, under the random recruitment scenario,
this age-class of maximum biomass will change over time. The selection pattern
under rule B.) are then specified in relation to this age-class of maximum biomass.
For example, when θ = 1 we have that the age class which is one year older than
m is the first age-class to enter the harvest. Formally, this rule can be expressed
as st = m,t + θ, where m,t and consequently st may very with time, but θ is
constant. It can take positive or negative values as long as s ∈ [3, A].
The control variable effort is chosen according to one of these five rules:
15
GRO-draft-v02.tex
January 26, 2011
1. To have a given value of fishing mortality at spawning stock levels above 460
thousand tonnes (this is called the Bp reference point) and linearly decreasing
to zero at a SSB of 0.
2. To employ a fixed level of effort.
3. To harvest a fixed amount.
4. To set effort proportional to total stock biomass.
5. To set harvest proportional to total stock biomass.
Rule 1 is close to the current harvest control rule (HCR) agreed upon by the Joint
Russian-Norwegian commission (ICES, 2010) when the fishing mortality is set at F.
Additionally, the current HCR specifies that F should not drop beneath a level of
0.3 at SSB levels above Bp and the calculated total quota should vary by no more
than 10% from year to year. We dispense of these additional qualifiers, which are
mainly politically motivated. The fishing mortality rate F does not feature directly in
our model. Rather we calculate the TAC according to a procedure described in the
appendix. Each simulated policy will be one fishing mortality value between 0.05
and 1. The effect of choosing different HCR for the NEA cod fishery for the current
selectivity pattern has been extensively studied by Eikeset et al. (2010b).
Rule 2 and 3 are then rigid explorations of the constant policies. Effort can take
values between between the minimum effort observed in the sample (emn = 1500)
and 2-times the maximum effort in the sample (em = 500000). Since harvest is not
a direct control variable in our model, we use the root-finding procedure optimize to
find the value of effort that gives the prescribed total harvest at a given biomass and
selectivity. The implications of setting constant harvest quotas vs. setting constant
effort quotas has been studied by Hannesson and Steinshamn (1991).
Rule 3 and 4 yield feedback policies, where the chosen control depends directly
on the state of the fish stock. The factor of proportionality between effort and stock
in rule 3 could take 20 values between 0.005% and 50%. The harvest share ranged
between 5% and 95%.
4
Results
An overview over the resulting optimal policies under no change in selectivity and
under the rule where selectivity was chosen, but constant through time are given in
Table 9 and 10. We do not report the results under the variable selectivity rule as
they were generally below the obtainable NPV from constant selectivity pattern. For
completeness, we note that the maximum obtainable NPV was 2.1 billion NOK under
random recruitment. It was the result of setting θ = 0 (so that in fact the age-class of
maximum biomass was targeted), which yielded an average selectivity of 6.9.
The best outcome for the status quo selectivity pattern was the result of setting
effort proportional to biomass (rule 4). This yielded a NPV of 1.84 billion NOK under
16
GRO-draft-v02.tex
January 26, 2011
random recruitment and 3.98 billion NOK under deterministic recruitment. The effort
level was set at 2.9% and 1.5% of the total biomass under random and deterministic
recruitment respectively. The best outcome for an optimally chosen selectivity was
again under rule 4. The optimal policy was to target all fish of age 9 and above and to
set the effort-level at 5.2% of the total stock biomass under random recruitment and
3.8% in the deterministic case. This yielded, on average, a NPV of 2.59 billion NOK
under random recruitment and 6.13 billion NOK under deterministic recruitment.
Table 9: Overview of results when (a) selectivity was not a control variable
Rule
scenario
policy
effort
select.
NPV in billion NOK
1.) HCR
deter.
F = 0.21
120 000
-/-
3.81
random
F = 0.21
114 000
-/-
1.79
2.) fixed effort
deter.
-/-
133 000
-/-
3.81
-/-
106 000
-/-
1.83
3.) fixed harvest
deter.
h = 621000 tons
225 000
-/-
0.91
random
h = 621000 tons
90 000
-/-
1.32
random
4.) effort prop. to biomass
5.) harv. prop. to biomass
deter.
e = 1.5%
124 000
-/-
3.98
random
e = 2.9%
110 000
-/-
1.84
deter.
h = 19%
132 000
-/-
3.86
random
h = 19%
127 000
-/-
1.85
Table 10: Overview of results when (b) selectivity was a control variable
Rule
scenario
1.) HCR
2.) fixed effort
policy
effort
select.
NPV in billion NOK
deter.
F = 0.21
367 000
9
5.37
random
F = 0.24
236 000
8
2.29
deter.
-/-
421 000
9
5.89
random
-/-
413 000
9
2.57
652 000
10
5.26
3.) fixed harvest
deter.
h = 2736000 tons
random
h = 1026000 tons
418 000
9
1.93
4.) effort prop. to biomass
deter.
e = 3.8%
451 000
9
6.13
random
e = 5.2%
300 000
9
2.59
5.) harv. prop. to biomass
deter.
h = 18%
367 000
9
5.47
random
h = 17%
220 000
8
2.29
4.1
Is there a difference between (a) controlling only effort
and (b) controlling effort and selectivity?
We set out to to illustrate the value of answering the question “which fish should
be harvested” in addition to answering the question “how many fish should be harvested”. The results show clearly that the efficiency loss from not controlling selectivity in an adequate fashion is large. Throughout, the NPV was almost twice as high
when selectivity was controlled compared to when it was not.
17
GRO-draft-v02.tex
January 26, 2011
In general, the obtainable NPV was very similar for the different rules. The comparably bad performance of constant harvest quotas is in line with Hannesson and
Steinshamn (1991), and also the comparably good performance of rule 1, which is
fashioned after the current HCR is in line with the findings from Eikeset et al. (2010b).
However, the target fishing mortality of around 0.2 (which is regardless of the scenario) is lower than today’s target value of 0.4. Nevertheless, the simulations with
a fishing mortality of 0.4 yielded an average NPV of 1.61 billion in the random case,
which is not so different.
Except in three cases, all rules yielded the same optimal selectivity parameter: no
fish younger than 9 years should be harvested. This is a dramatic change from the
current selectivity pattern.
4.2
Is there a difference between (1) deterministic and (2)
random recruitment?
The second aim of our study was to investigate whether it could be that, by avoiding
growth-overfishing, we also avoid recruitment-overfishing. Whether or not the manager could control recruitment made no difference for the choice of optimal selectivity pattern and also the effort levels are qualitatively similar under the two scenarios.
Under random recruitment, the fishery is effectively subsidized, no matter how small
the spawning stock, the expected number of recruits would be the same. Still this did
not lead to a less cautious harvesting strategy. Targeting only fish that have reached
their growth potential was of overriding importance.
There is, of course, a large difference in the obtainable NPV, which is much higher
under deterministic recruitment. The reason for this is the linear form of the recruitment function which we have imposed. It leads to an ever increasing feedback between spawning stock and recruitment until it hits a the plateau at the extreme point
of the domain (the largest observed spawning stock of roughly 1.2 million tonnes).
By inspection Figure 7, which contrasts the best obtainable NPV for a given selectivity pattern under random and deterministic recruitment, one can see that they follow
the same general pattern and that the difference between them becomes larger, the
later the first-age-at-capture.
4.3
Sensitivity analysis
[... to be written ...]
4.4
Discussion
Often the contrary argument to protect the old fish is brought forward, mainly out of a
concern for big, old, fecund females that contribute over-proportionally to reproduction.Although sparing the young fish indeed implies the entire harvest is concentrated
on relatively few old age-classes, it is important to realize that changing the selectivity pattern also implies that there are many more old fish around. Every age-class
18
January 26, 2011
4e+11
2e+11
0e+00
Obtainable NPV
6e+11
GRO-draft-v02.tex
3
4
5
6
7
8
9
10 11 12 13
Age
Figure 7: Comparison of obtainable NPV at age
is more abundant under the optimally changed selectivity pattern than under the
current non-selective harvesting pressure (see Figure 8, where the relative biomass
development of a cohort under the optimally changed selectivity pattern is marked
with blue squares). Nevertheless, one interesting aspect is that such a drastic change
of exploitation could lead to an evolutionary response where cod grows slower and
matures earlier, such that it can complete its entire life-cycle without ever entering the zone where it would be subject to fishing pressure (Jørgensen et al., 2009).
However, our model neither includes the necessary detail nor covers the necessary
5
4
3
2
0
1
Biomass relative to age 3
6
7
time-horizon to fruitfully discuss fisheries-induced-evolution.
3
4
5
6
7
8
9
10
11
12
13
Age
Figure 8: Cohort biomass development under no, current, & optimal fishing pressure
There are several limitations to our study that call for future work. First of all, we
19
GRO-draft-v02.tex
January 26, 2011
consider the fishery of a species which is characterized by several age-classes, and
whose individual fish grow significantly in weight and value with age. Although this
description is arguably valid for many commercially harvested species, there are also
many species for whose life-histories our model is not applicable. Pacific salmon, to
take a stark example, are mainly caught just before they enter the rivers in which
they spawn. For these fisheries, the best age-at-capture is not a choice, whereas
allowing a sufficient number of fish to pass upstream so that they can reproduce is
a question of vital importance. More generally, a pattern of slow growth and late
maturity decreases the likelihood that recruitment-overfishing could be avoided by
growth-overfishing. Furthermore, our model is probably not adequate for species that
exhibit schooling behavior as this can have a significant impact on the harvesting
technology.
Furthermore, our study is not so much a statement about the optimal management of the North-East Arctic cod fishery in particular, but rather a more general
thought-experiment on the problem of distinguishing growth- and recruitment overfishing. Especially, our study did not account for two of the most imminent biological
facts: First, there is no such thing as a single-species fishery. Although the Barents
Sea food-web consist of rather few trophic levels, the intra-species interactions have
an important effect on the stock dynamics (Hjermann et al., 2007). Additionally, the
fishery itself is a multi-species fishery as mentioned in section ??. For the coastal
fishery at least, it appears that the fishermen would not target cod the way they
do were it not also for other species (Jensen, 2007). Moreover, as cod, saithe, and
haddock are to some extent substitutable products, intra-species interactions may
also exist in the marketplace. Second, there is no such thing as a constant environment. It is indeed very likely that the fish react to a changing harvesting pattern,
either through adapting their behavior (Jørgensen and Fiksen, 2006) or through an
evolutionary response (Jørgensen et al., 2009; Eikeset et al., 2010a).
Last but not least, we have not explicitly studied the ultimate reason why fishery managers are concerned with recruitment: stock collapse. The linear stockrecruitment relationship did not feature critical depensation or the like, nor have we
assumed or estimated a hazard rate for crossing a potentially disastrous threshold.
However, by assuming that recruitment is a random draw from past observations, we
have effectively subsidized the fishery since no matter how low the stock would have
been, positive recruitment would still be realized. The fact that the optimal policies
between the random and the deterministic scenario are so similar, can therefore be
interpreted that the concern about stock collapse reinforces our argument. In general, crossing a threshold below which recruitment is significantly or even irreversibly
impaired is believed to happen at low stock sizes. Yet one of the main implications of
avoiding growth overfishing is that overall abundance increases significantly. Still, it
has to be highlighted that we do not make a statement that recruitment as a process
is not important. To the contrary, it is the basis for the continued existence of our
fish stocks. Understanding the factors and remains a key scientific challenge (Olsen
et al., 2011). However, we argue that its practical importance for fisheries manage-
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January 26, 2011
ment has been over-emphasized. It is more important to manage growth-overfishing.
5
Conclusion
Fisheries, as most renewable resources, involve both an important human and an
important natural dimension. Their management should strive to take both dimensions adequately into account, highlighting the need for interdisciplinary studies. By
combining biology with economics, we are able to point to the overriding importance
of managing growth overfishing. Our findings suggest, that at least for species like
the North-East-Arctic cod, too much emphasis is placed on avoiding recruitmentoverfishing. Targeting the right age-class could not only lead to a considerable economic gain, but it could also contain the problem of recruitment-overfishing, so to
say, en passant.
The remaining question then is, if the potential gains are as large as our study
suggests, why hasn’t selectivity been optimally controlled for the longest time? This
is, in the end, a descriptive question, but there is good reason to believe that adequate age-specific harvesting does not emerge spontaneously: The “race to fish”
extends to the dimension of age (Diekert, 2010). Moreover, even a sharing of the
stock between two sovereign nations (such as the NEA cod is shared between Russia
and Norway) is often sufficient to dissipate a large part of the rent (Diekert et al.,
2010b). The crucial role of the institutional setting for fisheries management has
recently obtained high-profile attention (Costello et al., 2008; Heal and Schlenker,
2008; Worm et al., 2009; Gutierrez et al., 2011). Solving collective action problems
still remains the biggest challenge for sustainable development.
Acknowledgements
We would like to thank Anne Maria Eikeset (CEES, University of Oslo) and Per Sandberg (Fiskeridirektoratet, Bergen) for help in constructing and obtaining the panel
on the Norwegian trawler fleet. The paper has benefited greatly from the discussions with Anne Maria Eikeset, Dag Hjermann, Andries Richter, Christian Jørgensen,
Mikko Heino, Stein Ivar Steinshamn, Fabian Zimmermann, Kjell Arne Brekke, and Nils
Chr. Stenseth. Of course, all remaining errors are our own. FKD gratefully acknowledges advice and support from Kjell Arne Brekke and Nils Chr. Stenseth and funding
from the Norwegian Research Council, through the project ”Socio-economic effects
of harvest-induced evolution”.
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January 26, 2011
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