1957
An endogenous bioeconomic optimization algorithm to evaluate
recovery plans: an application to southern hake
José-Marı́a Da Rocha, Santiago Cerviño, and Marı́a-José Gutiérrez
Da Rocha, J-M., Cerviño, S., and Gutiérrez, M-J. 2010. An endogenous bioeconomic optimization algorithm to evaluate recovery plans: an
application to southern hake. – ICES Journal of Marine Science, 67: 1957 – 1962.
Recovery plans were analysed by introducing social and economic behaviour and endogenous disinvestment decisions into bioeconomic models. Considering these endogenous constraints, a dynamic optimization problem was solved to find fishing mortality
(F ) trajectories that maximize discounted profits per vessel, subject to recovery of the stock to a spawning-stock biomass (SSB)
target in 2015. The algorithm developed was used to assess the southern hake recovery plan. Three scenarios were analysed: (1) represents the current plan with an annual 10% reduction in F; (2) represents the optimum trajectory where profits must be positive all
along and the SSB target is reached no later than 2015, and (3) represents the optimum trajectory allowing profits to be negative. The
results from (3) indicate that if economic and social restrictions are not considered a prior condition, the optimum solution implies a
fleet reduction in 2010 and 2011. Comparing (1) and (2), our results suggest that reducing F to 0.30 by 2010 achieves the recovery
target in 2012, increases the net present profits by 7.7% relative to the current plan, and is compatible with maintaining the current
fleet size.
Keywords: control in age-structured models, economic assessment, endogenous bioeconomic optimization algorithm, fishery management
optimization, southern hake recovery plan.
Received 6 November 2009; accepted 19 May 2010; advance access publication 8 August 2010.
J-M. Da Rocha: Universidad de Vigo, Facultad CC. Económicas, Campus Universitario Lagoas-Marcosende, CP 36200 Vigo, Spain. S. Cerviño:
Instituto Español de Oceanografı́a, Centro Oceanográfico de Vigo, Cabo Estai-Canido, 36200 Vigo, Spain. M-J. Gutiérrez: Universidad del Paı́s
Vasco (UPV/EHU), Avda. Lehendakari Aguirre, 83, 48015 Bilbao, Spain. Correspondence to J-M. Da Rocha: tel: +34 986 812400; fax: +34 986
812401; e-mail: [email protected].
Introduction
In the European context, the aim of a recovery plan is generally to
rebuild the spawning-stock biomass (SSB), to some safe minimum
level within a prescribed time frame (e.g. EC Reg. 2371/2002; EC,
2002). Different management trajectories might achieve this goal,
in which case stock-rebuilding algorithms could be used to select
the trajectory that minimizes losses in yield or any other economic
variable (Gröger et al., 2007).
Depending on the level of depletion, stock recovery might
involve drastic cuts in catches, a policy that implies great shortterm economic losses for the fisheries. To mitigate these losses,
recovery plans should try to avoid large reductions in fishing mortality (F ) and TACs from year to year. Consequently, they usually
include, along with biological targets, social and economic considerations. Economic constraints relate to a guaranteed instantaneous profit per vessel. Social constraints refer to maintaining
some minimum fleet size. Both constraints can be translated into
a maximum annual reduction in F that is acceptable across the
time-frame of the plan. This is the case for the southern hake recovery plan (EC Reg. 2166/2005; EC 2005), which is aimed at reaching
35 000 t of SSB by 2015, by reducing F annually. Based on social
and economic considerations, the annual reductions in F and
TAC have been constrained to be ≤10% and +15%, respectively.
When such restrictions are imposed, the range of alternative
options to reach the biological target might become reduced
# 2010
drastically, and the result could even be an empty set of trajectories
that achieve the SSB target at the proposed date. We sought to
introduce greater flexibility in designing recovery plans based on
social and economic variables. To meet this goal, we designed a
stock-rebuilding algorithm that does not involve prescribed cuts,
but takes into account the effects of adjustments in F upon instantaneous vessel profits and the long-term sustainability of the fleet.
Introducing social and economic behaviour and disinvestment
decisions is not novel in bioeconomic models (Martinet et al.,
2007; Hoff and Frost, 2008). However, behaviour was not predetermined in our model by exogenous rules based on lagged
average profits and/or other past economic decisions (as in
random-utility models; Bockstael et al., 1989; Kaoru et al., 1995;
McConnell et al., 1995; Lin et al., 1996; Hutton et al., 2004;
Haab et al., 2008). Quite the opposite, we assumed that disinvestment decisions are endogenously determined by the expected evolution of economic profits.
We built an endogenous optimization bioeconomic algorithm
(Arnason, 2000) to recover stocks, which may be implemented
using standard non-linear optimization methods. Because managers cannot force vessels to both fish and invest, the feasible
annual reduction in F is characterized by endogenous constraints
set by the state of the resource. Considering these endogenous constraints, a dynamic optimization problem has to be solved to find F
trajectories that maximize discounted profits per vessel subject to
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1958
J-M. Da Rocha et al.
recovery of the stock. Following Gröger et al. (2007), we used an
age-structured model and included a constant rate of discarding.
Moreover, the algorithm could be applied for solving deterministic, as well as stochastic decision problems.
We applied the algorithm to the southern hake recovery plan,
using the results from a Bayesian statistical catch-at-age model
(Fernández et al., 2010). This model is an extension of the
current ICES assessment model (ICES, 2009) taking discard data
into account.
b ¼ (1 + r)21), and cI is the investment cost. When the optimal
decision was to invest, the criterion that it is equal to 1 was
adopted. Note that the optimal investment rule is based on
rational expectations regarding future profits (Muth, 1961). The
fleet dynamics associated with the optimal decision rule is given by
The model
A feasible recovery plan
Stock dynamics and yield
The assumption was made that the recovery plan may not use sidepayments as subsidies to induce vessels to fish or as decommissioning grants to reduce the number of vessels. In this situation,
the only variable controlled by managers is the TAC: total effort
is given by the optimum decision that maximizes each year the
profits per vessel, and the number of vessels is given by the
optimum investment rule.
Under these conditions, a feasible recovery plan for a stock
exploited by a fleet equal to n0 is a fishing mortality trajectory,
{Ft }1
t=0 , such that:
We used a standard age-structured forward-projection model used
in virtual population analysis (Lassen and Medley, 2000). The
assumption was that the stock is split into A cohorts. The stock
a
a+1
dynamics are given by Nt+1
= e−zt Nta, where zta is the total
annual mortality rate affecting the numbers N of age group a
during year t. The total mortality rate is decomposed into
fishing mortality F and natural mortality m, which is assumed constant across ages. F is further decomposed into landings and discards. Formally,
zta = ( pa + da ) Fta + m,
a
(1 − it )
nt+1 = nt 1 −
.
T
(i) Recover SSB at date t ≤ T,
a
where p and d represent the fractions accounting for landings
and discards at age a, respectively, which could be estimated
empirically from the selectivity parameters. The size of a new
1
cohort (recruitment), Nt+1
is given by the flexible stock–
recruitment function proposed by Shepherd (1982) and the yield
is determined by Baranov’s (1918) equation.
Economic behaviour
Assume that the fleet is composed of nt homogeneous vessels.
Landings of each vessel v are assumed proportional to its fishing
effort, et. The assumption is that there is a fixed cost of operation
equal to cf and a fishing effort cost equal to w.
In a stable system of individual quota, the maximum effort level
for each vessel is assumed proportional to its individual quota.
However, a vessel cannot be forced to fish or to invest. That is, if
instantaneous profits are negative, the optimal effort of a vessel
is zero. Therefore, the 1-year profit per vessel, pt, is given by
A
a a
a=1 pr Yt
pt = max
− wet − cf , 0 ,
nt
where pra and Yta are the price, in E kg21, and the yield at age a,
respectively.
We also assumed that the lifetime of a vessel is a finite number
of years, t. Then, in each year t, a fraction nt/t of the vessel owners
decides to reinvest in a new vessel or to exit the fishery. The
optimal investment rule is given by
⎧
t
⎨ 1 if bt pt − cI . 0,
it =
t=0
⎩
0
otherwise
t
t
where t=0 b pt represents the discounted future profits, b is the
discount factor (discount is frequently introduced into fishery
economics using the discount rate, r, instead of a discount
factor; the former is usually applied in continuous time frameworks, whereas the latter is more commonly used in discrete
set ups; the inverse relationship between the two is given by
SSBt =
A
ma va Nta ≥ SSB,
(1)
a=1
where va and ma are the weight and the maturity fraction at
age a, respectively. Here, SSB is the target SSB at time T.
(ii) The associated sequence of TACs induces vessels to fish and to
invest,
A
pt =
a=1
pra Yta
nt
t
−
w Ft
− cf . 0,
q nt
bt pt+1 ≥ cI ,
(2)
(3)
i=0
where fishing effort is converted into F, assuming that catchability q is constant (i.e. Ft ¼ qntet).
Note that given that there is a fixed cost, a minimum level of F
must exist, where vessels are indifferent between choosing whether
or not to fish the TAC. This “break-even point” is endogenously
given by the state of the resource Nta . Formally, Ftmin satisfies that
A
pra Yta (Nta , Ftmin ) −
a=1
w min
F
= nt cf .
q t
Endogenous algorithm
The optimum F trajectory, {Ft }1
t=0 , can be obtained as the solution
of an optimization-constrained problem, where the present value
of total profits is maximized. That is,
Max
1
{Ft }t=0
1
bt nt pt ,
(4)
t=0
subject to the stock dynamics, the stock –recruitment, and the
restrictions imposed by the feasible recovery plan [Equations
(1) –(3)].
1959
Bioeconomic evaluation of southern hake recovery plan
To simplify the problem, one must first find the optimum F trajectory that maximizes Equation (4), assuming that profits are
positive and the number of vessels is constant. That is, problem
4 is solved without taking into account constraints 2 and 3.
The optimum F trajectories that solve problem 4 can be characterized by the solution of an infinite set of dynamic equations
(a full description is available from the authors upon request).
Formally, ∀t ¼ 1, . . . 1,
A
a=1
A−1
A−a
∂Yta a w a+j
a
pr
Nt − =
p
bj va pra+j Yt+j
q
∂Ft
a=1
j=1
∂wt+j
a+j
+
lt+j + ut+j ma+j va+j Nt+j
,
∂Ft
A
a
(5)
a
ba pra Yt+1+a
fat+1+a = lt+1
nt cf =
a=1
∂wt+j
−
ba
lt+1+a + ut+j ma+j va+j fat+1+a ,
∂Ft+j
a=1
A
ut
A
(7)
where lt and ut are the Lagrange multipliers associated with the
stock– recruitment relationship wt and the minimum SSB restriction [Equation (1)], respectively, and fat can be interpreted as the
survival function that establishes the probability of a recruit born
in the period t 2 (a 2 1) in reaching age a . 1 for a given F path
{Ft, Ft21, Ft22, . . . Ft2(a21)}. It is given by
a−1
i=1
1
1
e−zt−i (Ft−i ) Nt−(a−1)
a−1
if
if
pra va
a
pa F2008 w
a
1 − e−z2008 N2008
− F2008 .
a
q
z2008
(6)
1
ma va fat Nt−(a−1)
− SSB = 0,
A
a=1
a=1
fat =
However, because
catchability was assumed constant, we also had
a
w/q = (0.4989 × Aa=1 pra Y2008
)/F2008 ¼ E53 187.
Proper estimates of the economic and social restriction parameters require having information on fixed costs and the
number of vessels in the fleet and their lifetime, but such information was not available. However, if we assumed that the
current plan was feasible, we could indirectly estimate these economic restrictions. This assumption was not very heroic, because if
it were not satisfied, it would mean that the current plan could not
be selected as optimum under any circumstances. Under this
assumption, two conditions were satisfied: current profits have
to be non-negative, and vessels were induced to invest in accord
with the plan. Satisfying both conditions implied that the
maximum feasible fixed costs for the fleet could be obtained by
solving
a . 1,
.
a=1
Once the solution is found, it is verified whether condition 2 is satisfied for any t. If it is not satisfied, then Ft = Ftmin , and the optimal
a+1 min
trajectory is recalculated, using as initial condition Nt+1
(Ft ).
Finally, once Ft . Ftmin ∀t, it is verified that the investment constraint 3 is satisfied in the solution.
Application to the southern hake recovery plan
The aim of the southern hake recovery plan is to reach 35 000 t of
SSB by the year 2015 (ICES, 2009). To attain this goal, the plan
establishes that F must be reduced annually by at most 10%.
Data for the parametrize for the age-structured population in
2008 were taken from Fernández et al. (2010). The stock– recruitment relationship was estimated based on their time-series (1982 –
2007),
which
results
in
a
quasi-Ricker
stock–
recruitment relationship, with b ¼ 2, a ¼ 9.64, and K ¼ 31 540.
We also tested a density-independent Beverton –Holt relationship,
with b ¼ 1, but this generated very large recruitments at SSB values
higher than historical values. Given these apparent overestimates,
we decided not to present the results.
Time-series of price-at-age were obtained from daily sales at 54
Galician fish markets, from January 2007 to October 2008. To calculate the ratio w/q, we used the fact that total
fishing cost, wntet,
represented 49.89% of the value of yield, Aa=1 pra Yta , for 2008.
The investment cost, cI (t), was estimated assuming that for any
vessel lifetime, t, it cannot be greater than the discounted current
profits associated with the recovery plan, p2008+i (RP). This
assumption allowed us to estimate the maximum investment
cost as:
c f ( t) =
t
bt pt2008+i (RP),
i=0
where the discount factor b was set equal to 0.95.
Finally, optimum trajectories were calculated, updating the initial
conditions. For this, we assumed that according to the plan, the
resulting F in 2009 had to be 10% lower than it was in 2008.
Results
After calibrating the model, three scenarios were analysed
(Figure 1). Scenario 1 represents the trajectories under the
current recovery plan with an annual 10% reduction in F; scenario
2 represents the optimum trajectory that solves problem 4,
satisfying the economic constraint on profits and the biological
constraint of meeting the SSB target in 2015; and scenario 3 represents the optimum trajectory without considering the economic
constraint.
As expected, the optimum trajectory in scenario 3 required
reducing F drastically and recovering the SSB very quickly (in
2011). However, profits are negative in 2010 and 2011
(Figure 2). Therefore, this scenario did not satisfy the economic
constraints.
In scenario 2, F did not adjust as quickly as in scenario
3. Nevertheless, the optimum trajectory generated greater
reductions of F than those established in the current recovery
plan (scenario 1). The reason is that reducing F not only recovers
the stock faster, but also contributes to increasing profits. The
recovery path resulting from applying the algorithm increased
the discounted profits by 7.7%. Consequently, the time-frame
restriction was not invoked when profits were maximized, and
the stock recovers well before 2015. An adjustment of F to 0.30
by 2010 recovered the SSB in 2012, guaranteeing at all times
larger profits than those obtained in 2008.
1960
J-M. Da Rocha et al.
Figure 2. Net present profits and the optimal investment decision as
a function of vessel lifetime. Black indicates that vessels exit the
fishery and grey indicates no exit from the fishery.
Figure 1. Comparison of the trajectories for three scenarios of:
(a) fishing mortality (F); (b) SSB (horizontal line represents the target
of 35 000 t); and (c) profits. The scenarios are: (1) annual reduction
in F by 10%; (2) optimum F trajectory based on economic constraints
on profits, as well as the biological constraint of reaching the SSB
target in 2015; and (3) optimum F trajectory based only on the
biological constraint.
Discussion
Recovery plans are usually designed in two steps: fixing a recovery
target for a specific date; and finding the annual reduction of F that
allows the target to be reached among a reduced set of options
(210%, 215%, etc). Instead of fixing these percentages a priori,
an alternative is to look for the annual F reduction that drives
the fishery to the target (Horwood et al., 2006). Using optimization techniques represents an improvement, because this allows
selecting F trajectories that could vary over time, thereby
opening the door to new solutions for solving strategic management problems (Gröger et al., 2007). Our main contribution to
this area is the introduction of infinite horizons (no deadline in
the accounting process) in the objective function. This allows analysing possible advantages of advancing the deadline of a recovery
plan. When infinite horizons are considered, our algorithm takes
into account “all” future profits associated with a recovered
stock. For instance, the deadline that maximizes the net present
value of the profits when only biological restrictions are taken
into account (scenario 3) is advanced by 4 years compared with
the current plan (scenario 1).
One disadvantage of these optimization algorithms is that they
generate optimum trajectories that imply large changes in F.
Managers have always considered sudden changes in F undesirable, because they result in profit instability for the producer
sector (Butterworth et al., 2010). Our algorithm, instead of limiting maximum change in F, includes constraints that guarantee a
minimum annual profit per vessel (scenario 2) in a similar way
as Martinet et al. (2007) did, although they used a biomass
dynamic model. This approach allows us a greater flexibility to
adjust F; therefore, it provides more strategic options for achieving
recovery. In scenario 2, profits are used as a minimum boundary of
the feasible solution set. From the social point of view, this could
be interpreted as providing an entry for the producer sector to take
part in the design of the recovery plans by proposing admissible
limits.
Unlike Martinet et al. (2007), our algorithm does not allow for
reducing the number of vessels participating in the fishery.
Another shortcoming is that we have not explored the possibility
of pulse fishing (Hannesson, 1975; Tahvonen, 2009). The consideration in future research of such limitations should allow
even greater flexibility in selecting the feasible set of F values
that might recover the stock.
This analysis is not meant to be exhaustive in dealing with
uncertainty or conclusive regarding the success of the recovery
plan for southern hake. However, we have dealt with some
sources of uncertainty recognized by ICES (2009). In contrast to
1961
Bioeconomic evaluation of southern hake recovery plan
the current assessment model for the hake stock, our model considers a constant discard rate. This is considered more realistic,
because discards represent a substantial part of the total catch of
the younger age groups (Fernández et al., 2010; Jardim et al.,
2010). Consequently, reducing F faster should allow more fish to
survive to the age at maturity; therefore, it increases the probability
of reaching the recovery target faster than when discards are
ignored.
Apart from considering variability about the initial abundance
distribution, we have only presented results for an assumed Ricker
stock– recruitment relationship. A similar analysis assuming a
Beverton –Holt relationship was rejected, because the optimum
trajectories result in scenarios where SSB reaches levels well
above those observed in the historical record. The reason is apparently that the available data do not allow a reliable estimate of the
asymptote. Other potential sources of uncertainty not included
relate to changes in growth (De Pontual et al., 2006; Piñeiro
et al., 2007), natural mortality, cannibalism, weights and
maturity-at-age (Piñeiro and Saı́nza, 2003; Mehault et al., 2010),
prices per age, and costs of fishing. All these factors could affect
the optimum F. A possible way to treat this kind of uncertainty
is to integrate the algorithm into a management strategy evaluation framework (Kell et al., 2007, 2009; Butterworth et al.,
2010). In such a context, the algorithm would recalculate the
optimum trajectory every period, once new information from
the operating model becomes available.
The algorithm allows definition of a harvest control rule (HCR)
based on the net present value of the entire path of future catches.
Our intuition is that the use of this HCR based on optimization
techniques could be more robust in uncertain scenarios: first,
because it is more flexible in looking for an optimum F trajectory
that sustains the recovery; second, because the algorithm could
advance the deadline, which would generate a buffer to face
future uncertainties; third, because reference points based on bioeconomic models with infinite horizons are more precautionary
than, for instance, the F associated with maximum sustainable
yield based on a model with a finite horizon (Grafton et al., 2007).
We have used the algorithm to evaluate a recovery plan with an
explicit SSB target, but it could be used also to define reference
points obtained as stationary solutions of a bioeconomic model.
In this way, the different metrics used by biologists and economists
might be unified to compare various scenarios.
The southern hake recovery plan has not been evaluated fully,
although some analyses have demonstrated that the SSB target
might be achieved with the more optimistic scenarios
(Fernández et al., 2010; Jardim et al., 2010). Our results also indicate that the target might be achieved, but this might require a
greater reduction in F than prescribed as a threshold by the recovery plan. A greater reduction does not necessarily ruin the fishery
economically, but rather the opposite happens. In contrast to
general expectations, compared with a constant annual reduction
of 10%, a short-term reduction .10% might actually improve the
medium-term profits and at the same time increase the probability
of recovery.
Acknowledgements
We thank Niels Daan, Denis Bailly, Doug Wilson, Sarah Kraak,
and the participants of the UNCOVER symposium for their comments. Carmen Fernández provided the results from the Bayesian
model with discards. Financial aid from the Spanish Ministry of
Education and Science (ECO2009-14697-C02-01 and 02) and
the Basque Government (IT-241-07 and HM-2009-1-21) is gratefully acknowledged. This study was carried out with the financial
support from the Commission of the European Communities,
SSP-4 project “Understanding the mechanisms of Stock
Recovery”.
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