1
6 Supporting material: It’s the economy, stupid! Projecting the fate of fish populations
2
using ecological-economic modeling
3
In the first part of this section we describe the general set-up of the bio-economic model. We
4
put special emphasize on the specification of (i) the demand functions, as these determine the
5
(economic) species interaction, as well as (ii) management effectiveness. The second part
6
deals with the parameterization of the model. It describes biological and economic data used,
7
and statistical methods to derive parameter values and the projected trends for the economic
8
driving factors (demand, costs, and aquaculture production). The final part describes our
9
numerical simulation strategy, and contains the programming code.
10
6.1 Bioeconomic model
11
The objective function builds on Quaas and Requate (2013), assuming that the different
12
species of fish are considered to be imperfect substitutes in consumption. Substitution means
13
that if the price for one species goes up, the demand for this species shifts to other species.
14
The extent of this substitution effect is captured by the elasticity of substitution between these
15
species. We assume the following inverse demand functions
salmon pS 
Y
1
S H S 
v
  1
1
16
I
 1  1 

Y
1 
tuna species i pTi  T Ti H Ti  Ti H Ti 
v
 i 1

1
Y

seabass pB   B H B 
v
cod stock j pCj 
Y
C Cj H Cj
v
1
(1)
  1
1
 1 
 1 
 J

 Cj H Cj 
 j 1

17
with scaling parameters with k  0 , S  T  B  C  1 , iTi  1 and  jCj  1 . Here,
18
HS is used to denote the consumed quantity of farmed salmon, analogously HB is the
1
19
consumed quantity of farmed sea bass; HTi is consumption of tuna species i, and HCj
20
consumption of cod from stock j. Total expenditures for fish are given by Y, which are
21
assumed to be independent of fish supply, and
  1
  1
 1

 1 
 1  1 
 1 
 J
 1
 I




 T Ti HTi 
  B H B  C Cj H Cj 
 i 1

 j 1

22
v  S H S
23
is a measure of utility derived from aggregate fish consumption. Parameters of particular
24
interest are the (constant) elasticities of substitution: σ for the substitution between different
25
types of fish (salmon, tuna, sea bass, cod), ψ for different species of tuna, and ξ for cod from
26
different stocks.
27
Fishing effort targeted at tuna species i is ETi, effort targeted at cod stock j is ECj, with
28
constant marginal costs cTi and cCj, respectively, as typically assumed in the literature. We
29
assume that the harvesting functions are given by the Baranov (1913) catch equations
ST
H Ti (t )  (1  exp( ETi (t )) qTi ( s) wTi ( s) xTi ( s, t )  (1  exp(  ETi (t )) BTi (t )
s 1
30
(2)
SC
H Cj (t )  (1  exp( ECj (t )) qCj ( s) wCj ( s) xCj ( s, t )  (1  exp(  ECj (t )) BCj (t )
s 1
31
Here, xTi(s) denote stock numbers of tuna stock i, and xCj(s) stock numbers of cod stock j, of
32
age s; qTi(s) and qCj(s) denote the catchability coefficients, and wTi(s) and wCj(s) denote the
33
weights of an individual fish of age s of tuna stock i and cod stock j, respectively. The
34
variable BTi   qTi ( s ) wTi ( s ) xTi ( s, t ) measures the “efficient” biomass of tuna stock i (an
STi
s 1
35
analogous definition applies to cod stock j), i.e. the biomass that is susceptible to fishing
36
given the age-specific catchability coefficients.
37
The age-structured fish population dynamics model is an extension of the single-species
2
38
model developed by Tahvonen (2009). For tuna (analogously for cod)
ST
ssbTi (t )   Ti ( s) wTi ( s) xTi ( s, t )
s 1
xTi (1, t  1)  Ti (ssbTi (t ))
39
xTi ( s, t  1)  Ti ( s  1) 1  qTi ( s  1) (1  exp( ETi (t )))  xTi ( s  1, t )
for s  2,
(3)
, ST  1
xTi ( ST , t  1)  Ti ( ST  1) 1  qTi ( ST  1) (1  exp( ETi (t )))  xTi ( ST  1, t )
 Ti ( ST  1) 1  qTi ( ST ) (1  exp( ETi (t )))  xTi ( ST , t )
40
Here,  Ti ( s) are age-specific maturities. Recruitment is described by the stock-recruitment
41
function Ti (ssbTi (t )) . For all stocks we use Ricker (1954) stock-recruitment functions, i.e.
42
we specify
43
 (ssb)  1 ssb exp(2 ssb)
(4)
44
The parameters Ti ( s ) are age-specific natural survival rates. They are calculated from
45
age-specific natural mortality rates M Ti ( s) as Ti ( s)  exp(  M Ti ( s)) .
46
An important part of our analysis relies on a parameter that quantifies the effectiveness of
47
fisheries management. For this sake, we conceptualize management effectiveness as the
48
fraction of the external costs of fishing that are internalized in the fishermen’s decisions on
49
fishing effort. These external costs of fishing can be determined by the shadow price, or
50
co-state variable, k , of stock k, which is derived from the dynamic optimization problem to
51
maximize the present value of the sum of consumer surplus and fishing profits subject to the
52
population dynamics. Once the shadow price of the stock is known, optimal harvesting effort
53
is determined by the condition that the price of fish from stock k is equal to the marginal costs
54
of fishing, which are composed of the direct (private) harvesting costs and the shadow price
55
of the stock, which captures the external costs of fishing:
3
56
pk 
ck
 k
exp( Ekå ) Bk
(5)
57
To obtain a dimensionless measure of the external costs of harvesting, we consider the
58
value-added shadow price
59
 k  k / pk
(6)
60
Using the harvesting function (2) we can express optimal supply of fish as a function of its
61
market price and the value-added external costs as follows
62
H kå  Bk 
ck
pk (1   k )
(7)
63
We conceptualize the degree of management effectiveness as the fraction  k of value-added
64
external costs that is taken into account when setting the total allowable catch (TAC) for stock
65
k. Thus, the TAC can be written as
66
H kTAC  Bk 
ck
pk (1  k  k )
(8)
67
It is important to note that in equation (8) the value-added external costs of harvesting are
68
determined from the dynamic optimization problem, but the price pk is formed on the fish
69
market for given total allowable catches. This means that pk, as given by the inverse market
70
demand (1), depends on the TACs for all stocks, and equation (8) only implicitly defines the
71
TAC for stock k. Obviously, the TAC depends on the degree of management effectiveness µk.
72
Perfect management effectiveness k  1 corresponds to the optimal TAC, as given by
73
equation (8). If no external costs of fishing are taken into account, i.e. if k  0 , the
74
management would not restrict harvesting at all and thus catches would be equal to open
75
access catch quantities, generating zero profits.
4
76
For our simulations we use values in between these extremes. Specifically, we use estimates
77
of management effectiveness in the different fishing areas from Mora et al. (2009). In order to
78
not underestimate the stock sizes in 2048, we take a slightly more optimistic view and assume
79
that management effectiveness is 40% for tuna and 60% for the cod stocks.
80
6.2 Parameterization
81
Data from stock assessment
82
Data on age-specific maturity and natural mortality rates, as well as stock numbers for 2008
83
are taken directly from the stock assessments (Tab. 1, 2). For weight-at-age (in stock) we use
84
mean values for the period 2005-2008.
85
Age-specific catchabilities are obtained by dividing harvest-at-age by the stock biomass in the
86
different age groups, which yields the annual fishing mortalities of the different age groups.
87
We take the mean of these values for the period 1993-2008 and divide the resulting average
88
annual fishing mortalities by the maximum of these numbers. Thus, by normalization, the
89
maximum of the catchabilities is equal to one.
90
Northeast Arctic cod fishery
age class
weight
s
wneac(s)
1
0
2
0
3
0.259
4
0.66
5
1.2746
6
2.1578
7
3.276
8
9
4.590
6.287
6
6
8.606
0.962
0.996
0.675
10
11
12
10.696
12.73
4
1
maturity
γneac(s)
0
0
0
0.0026
0.0676
0.3526
8
0.877
6
4
1
1
catchability
qneac(s)
0
0
0.026
0.196
0.496
0.725
0.884
0.982
0.993
0.993
0.948
1
0
0
0.32
0.23
0.21
0.2
0.2
0.2
0.2
0.2
0.2
0.2
357.90
583.9
851.32
650.70
325.25
162.28
60.61
18.39
4
5
8
9
6
3
4
7
6.994
0.827
0.293
Mneac(s
mortality rate )
stock
numbers
5
64.92
91
92
North Sea cod fishery
age class
s
1
2
3
4
5
6
7
weight
wnsc(s)
0.3122
0.8924
2.082
3.835
5.597
7.5272
10
maturity
γnsc(s)
0.01
0.05
0.23
0.62
0.86
1
1
catchability
qnsc(s)
0.42
0.957
0.954
1
0.956
0.929
0.981
mortality rate
Mnsc(s)
0.468
0.264
0.386
0.26
0.26
0.332
0.2
87.728
33.323
31.948
3.88
2.017
0.499
0.331
stock numbers
93
94
Eastern Baltic cod fishery
age class
s
1
2
3
4
5
6
7
8
Weight
webc(s)
0
0.173
0.493
0.818
1.183
1.742
2.613
4.306
Maturity
γebc (s)
0
0.13
0.36
0.83
0.94
0.96
0.96
0.98
Catchability
qebc (s)
0
0.166
0.548
0.842
1
0.904
0.891
0.907
mortality rate
Mebc (s)
0
0.2
0.2
0.2
0.2
0.2
0.2
0.2
198.143
204.938
124.999
58.493
23.986
8.579
2.325
1.018
stock numbers
95
96
Table 1: Parameter values for cod fisheries from ICES stock assessment reports.
6
97
Bigeye tuna
age group
S
1
2
3
4
5
6
7
8
weight
wbet(s)
0
4
13
26
41
58
76
101
maturity
γbet (s)
0
0
0
0
1
1
1
1
catchability
qbet (s)
0
1
mortality rate Mbet(s)
0.8
0.8
0.4
0.4
59.834
9.053
4.877
3.19
stock numbers
0.507 0.675
0.846 0.863 0.755 0.358
0.4
0.4
0.4
0.4
2.256 1.809 0.459 0.759
98
99
Yellowfin tuna
age group
s
1
2
3
4
5
6
weight
wyft (s)
1
2
11
35
62
87
maturity
γyft (s)
0
0
0
1
1
1
catchability
qyft (s)
0.202
0.46
0.296
0.48
0.895
1
Myft(s)
0.8
0.8
0.6
0.6
0.6
0.6
48.054
12.478
1
2
3
4
5
6
mortality
rate
stock numbers
4.827 2.696
0.96 0.191
100
101
Eastern Bluefin tuna
age group
s
weight
wbft (s)
5.6
11
20.2
33.6
52
73.6
maturity
γbft (s)
0
0
0
0
1
1
catchability qbft (s)
0.422
1
mortality rate Mbft(s)
0.14
0.14
0.142
0.242
stock numbers
0.78 0.567
0.14
0.14
0.273 0.267
7
9
10
95.4 119.8 146.6
215.4
1
1
1
0.401 0.395 0.359 0.403
0.65
0.711
0.14
0.14
0.336 0.281 0.176 0.172 0.139
0.317
0.14
0.14
1
8
0.14
0.14
102
103
Table 2: Parameter values for tuna fisheries from ICCAT stock assessment reports.
104
7
105
Stock-recruitment functions
106
Following Cook et al. (1997), we assume log-normal auto-correlated errors to estimate the
107
parameters of the stock-recruitment function. We use ICES and ICCAT stock assessment
108
estimates for the number of recruits and spawning stock biomasses to estimate the parameters
109
of the stock-recruitment functions. Using sk to denote the age of recruitment for stock k, we
110
thus estimate the following AR(1) model by means of OLS
ln( xk ( sk , t ) / ssbk (t  sk ))  ln(1 )  2 ssbk (t  sk )   t with  t    t 1   t 1,
111
112
where  t  IIDN(0,  2 ) is an independently and identically normally distributed series of
113
random errors. We first estimate the correlation coefficient v̂ by means of nonlinear least
114
squares and then estimate
115
116
 xk  sk , t 
xk  sk , t  1 
ln 
 vˆ
  (1  vˆ) 1  2 ln ssbk  t  sk   vˆ ssbk  t  1  sk    t
 ssbk  t  sk 

ssb
t

1

s


k
k



117
118

by means of OLS with Newey-West (1987) robust estimation of the covariance matrix to
account for heteroscedasticity and autocorrelation. Results are given in Table 3.
ln(1 )
2

 2
Northeast Arctic cod
0.683
(0.217)
0.0011
(0.00029)
0.370
0.191
North Sea cod
1.699
(0.083)
0.00391
(0.00110)
0.189
0.322
Eastern Baltic cod
0.530
(0.325)
0.00182
(0.00091)
0.759
0.108
Bigeye tuna
-1.735
(0.231)
0.0011
(0.00024)
0.672
0.029
Yellowfin tuna
-0.467
(0.154)
0.0039
(0.00050)
0.629
0.026
1
For the density dependence of the North Sea cod recruitment, current stock assessment data does not allow a
reliable estimate, presumably because the stock has been at very low levels for the last two decades. To
overcome this difficulty, we use the estimate for φ2 from Cook et al. (1997) and estimate ln(φ1), and ν contingent
on this value of φ2.
8
Bluefin tuna
-4.494
(1.180)
0.0041
(0.00293)
0.931
0.147
119
120
Table 3: Estimates and Newey-West (1987) standard errors for Ricker stock-recruitment
121
functions
122
9
123
Demand functions
124
For the elasticities of substitution we assume
125
  1.7
  2.5
  4.3
(9)
126
The parameter value   1.7 is taken from Asche et al. (1996) and Quaas and Requate
127
(2013), and reflects the elasticity of substitution between types of fish as different as salmon
128
and crustaceans. Also the assumed values for the substitution elasticities for different cod
129
stocks and different tuna species are based on empirical evidence. The demand elasticity for
130
Baltic cod has been estimated to be 1/   0.23 (Nielsen 2006). For our calculations, we thus
131
use   4.3 . An estimate for the demand elasticity for different types of tuna – i.e. fresh and
132
frozen tuna supplied by long-line fleets – is   2.53 , as reported in Bertignac et al.
133
(2000:163) and Miyake et al. (2010:107). The values for the elasticities of substitution satisfy
134
the following properties of the demand functions: substitution among fish species is more
135
elastic than substitution of fish by other commodities (σ > 1); tuna species among themselves
136
are better substitutes than different types of fish species (ψ > σ), and finally cod from
137
different stocks is better substitutable than different tuna species (ξ > ψ).
138
To calibrate the demand parameters ηk, we use data on (export) prices and data on supply of
139
farmed salmon and sea bass from FAO fishstat (http://www.fao.org/fishery/statistics/en
140
accessed April 1, 2014), and data from the ICES and ICCAT stock assessments for the
141
harvest of cod and tuna for the period 1993-2011.
142
Equations how to compute the values for ηk from the price and harvest quantity data are
143
derived from the inverse demand functions (1), as follows.
10
S 
 v

T Y
 1 
 1

pTi H Ti 

i 1

I
1
v
pS H S
Y
I
 1
 Ti H Ti
i 1
1 
v
v
Ti 
pTi H Ti 
T Y
T Y
 1 
1
 1

p
H

Ti
Ti 
i 1

I
  1
 1 
1 
 I

v I
T    pTi H Ti 
pTi H Ti 


Y  i 1

 i 1

1
v
 B  pB H B
Y
 v

C Y
 1 
 1

pCj H Cj 

j 1

J
J
 1
 Cj H Cj
j 1
1 
v
v
Cj 
pCj H Cj 
C Y
C Y
145
 1 
1 
 J

v J

C    pCj H Cj 
  pCj H Cj 
Y  j 1

 j 1

1 1 1
(10)
Using ηS + ηT + ηB + ηC = 1, we have
  1
146
 1 
1
 1

pCj H Cj 

j 1

J
  1
144
11 1
11 1
 1 
1 
1
 I
 I

v
 pS H S   pTi H Ti 
pTi H Ti 


Y
 i 1

 i 1



1
 pB H B   pCj H Cj 
 j 1

J
1
  1
 1 
1 1 1


  pCj H Cj 
 j 1

J
(11)
147
Thus, for each year’s observation of prices and quantities, we obtain values for ηk. To
148
calibrate the demand functions we use the mean values (standard errors in brackets)
149
S
T
B
C
 0.2525(0.0237)
 0.3756 (0.0471)
(12)
 0.0751(0.0141)
 0.2969 (0.0257)
150
11
bet  0.2991(0.1048)
 yft  0.2724 (0.0830)
151
(13)
bft  0.4285(0.0834)
152
neac  0.4519(0.0222)
nsc  0.2692(0.0274)
ebc  0.2789(0.0141)
153
(14)
154
155
For total expenditures, we use the value of 2008, Y = 10.15 billion USD. Estimating the
156
equation
157
expenditures on fish exports (cod, the three tuna species, salmon and sea bass) in the FAO
158
data from 1993-2011, we obtain ln(1+gY) = 0.0569 (0.000052),  2  0.0295 ; using aggregate
159
expenditures on the wild fish only (cod and tuna), we obtain ln(1+gY) = 0.0256 (0.000050),
160
 2  0.0285 . The reason is that the supply of farmed fish substantially increased, while the
161
increase in expenditures for wild fish was purely driven by increasing prices.
162
Cost functions
163
For the cod stocks, cost data is available from the literature. For Northeast Arctic cod we use
164
the estimate cneac = 1.564 billion USD (Arnason et al. 2004);2 for North Sea cod we use cnsc =
165
0.155 billion USD from Froese and Quaas (2012); and for Eastern Baltic cod we use cebc =
166
0.135 billion USD from Froese and Quaas (2011).3
ln(Yt )  ln(Y0 )  ln(1  gY )t   t
with  t  IIDN(0,  2 )
2
,
using
aggregate
This figure is obtained by converting the estimate of 8.824 million NOK into US dollars.
These figures are obtained by converting the estimates of 106 million euros (for North Sea cod) and 92 million
euros into US dollars, based on the 2008 conversion rate.
3
12
167
For the tuna stocks, no similar data was available. We rather adopt an indirect approach to
168
estimate the cost parameters, using the method described in Quaas et al. (2012). We assume
169
that the stocks have been fished under a regime of de-facto open access in the past, such that
170
τk = 0 in Error! Reference source not found.. We thus can use observations on the
171
harvestable biomass and fish prices to estimate the cost parameter. Using data from FAO
172
fishstat for prices and stock assessment data for harvestable biomasses, we obtain the
173
following mean values (over the period 1993-2008) for the three stocks: cbet = 3.024, cyft =
174
0.775, and cbft = 1.669.
175
Aquaculture production
176
For the first optimization we keep fixed the supply of farmed fish at 900,000 tons for salmon
177
and 50,000 tons for sea bass, which are roughly the quantities in 2008 according to FAO data.
178
Estimating the equation ln( H jt )  ln( H j 0 )  ln(1  g Fj )t   t with  t  IIDN(0,  2 ) , with data
179
on export quantities Hjt for the period 1993-2011 we obtain ln(1+gFS) = 0.0795 (0.000019),
180
 2  0.0125 for salmon and ln(1+gFB) = 0.0933 (0.000081),  2  0.0540 for sea bass.
181
The discount rate is set to 5% per year in all simulations
182
For the numerical calculation we employ the interior-point algorithm of the Knitro (version
183
9.1) optimization software (Byrd et al. 1999; 2006). All programming codes were
184
implemented in AMPL, and are available as supporting material.
185
Minimum acceptable biomass levels are defined on basis of the historic stock trends. We use
186
the threshold of 80% of the mean of the last 10 years of available data (see Table), i.e.
187
1999-2008, to set SSB limits for each species.
Bluefin tuna
171.006
13
Bigeye tuna
384.043
Yellowfin tuna
159.734
Northeast Arctic cod 523.434
North Sea cod
40.929
Baltic cod
89.993
188
189
6.3 Numerical simulation approach
190
The numerical challenge for the simulation is the computation of total allowable catches
191
under imperfect management effectiveness. Computing the shadow price requires to solve the
192
dynamic optimization problem for the age-structured population dynamics which are coupled
193
by demand-side interactions. We solve the dynamic optimization problem for the expected
194
development of recruitment, expenditures and farmed fish supply, as it is numerically
195
infeasible to solve a stochastic dynamic optimization problem of this size, and as the error of
196
using the shadow price derived from the deterministic optimization problem is negligible
197
(Kapaun and Quaas 2013). We employ the interior-point algorithm of the Knitro (version 9.1)
198
optimization software (Byrd et al. 1999; 2006) to solve this large-scale optimization problem.
199
All programming codes were implemented in AMPL, the programming codes are provided
200
below. Since management is imperfect, shadow prices have to be computed newly every time
201
step. As population dynamics, expenditure growth, and farmed fish supply are stochastic, we
202
use a Monte Carlo simulation with 1000 samples of the stochastic dynamics to obtain reliable
203
estimates for the mean future development as well as for the standard deviation of the time
204
development of spawning stock biomasses, assuming that they are log-normally distributed.
205
In each run, and for each time step, once the shadow prices are determined for given initial
206
stock numbers of all six fisheries, we determine the total allowable catches by numerically
14
207
solving equation (8) for all stocks. The next year’s initial stock numbers are derived from the
208
age-cohort model (3) with stochastic stock-recruitment functions; aggregate expenditures,
209
fishing costs and supply of the two farmed fish species are updated according to the stochastic
210
processes governing the dynamics of the respective variable. We newly compute the shadow
211
prices of the different stocks and repeat the simulation for each year between 2008 and 2048.
212
Figures 1 (for the likely growth rate of aggregate fish demand) of in the main text, S1 for
213
constant economic parameters at 2008 levels, and S2 for a low growth rate of aggregate fish
214
demand, show the results.
215
Figure S1 shows that current management effectiveness would perform reasonably well if
216
economic drivers would stay constant at present-day levels: All stocks, perhaps except
217
Bluefin tuna, would tend to increase in the next decades compared to 2008 levels.
218
While the model has not been calibrated to fit past development of spawning stock biomasses,
219
it is useful to compare the model projection starting at some point in the past with the
220
development of stock estimates obtained from stock assessments. Figure S3 shows the stock
221
sizes from assessments and the model output starting at 1988 stock sizes for the 20-year
222
period from 1988 to 2008; using the baseline scenario (likely demand growth) and assumed
223
management effectiveness. The figure shows a sample of stochastic development (empty
224
circles), and most likely development for the baseline scenario (solid line). Red shaded areas
225
show the most likely development +/- one standard deviation; green shaded areas the 95%
226
confidence interval. It is evident that our model overestimates the past sock sizes. This shows
227
that we use an optimistic scenario with respect to management effectiveness. For most of the
228
stocks a simulation with zero management effectiveness would fit the stock assessment data
229
much better (results not shown). Yet, the trends of stock size development are reproduced
230
well for the majority of stocks, in particular for North Sea cod, Baltic cod, Bigeye tuna and
231
Yellowfin tuna. The model performs somewhat worse for Northeast Arctic cod, in particular
15
232
for the last years in the observations, and Bluefin tuna. For Bluefin tuna this is due to the bad
233
quality of the estimated stock-recruitment function, which results in a high variability and
234
uncertainty of model output (cf. Figure 2). For Northeast Arctic cod, the actual recruitment
235
has recently been much higher than predicted by the Ricker stock-recruitment function
236
assumed here. This is due to strongly improved environmental conditions, which have
237
reduced the density-dependence of recruitment (Kjesbu et al. 2014) and led to a strong
238
increase in spawning stock biomass, which significantly exceeds the stock sizes projected by
239
our model. Such a strong increase of stock sizes of Northeast Arctic cod has been simulated
240
before in single-species bio-economic models using a Beverton-Holt type of stock recruitment
241
function and a constant economy (Diekert et al. 2010), which leads to the conjecture that the
242
smaller stock sizes from our model are due to the assumed stock-recruitment function. Even
243
with our more conservative assumption of a Ricker stock-recruitment function, however, the
244
Northeast Arctic cod stock is projected to stay at relatively healthy sizes in the next decades.
16
245
246
Figure S1. Past development (according to ICES/ICCAT stock assessments) in filled circles, sample of
247
stochastic future development (empty circles), and most likely future development for a scenario with economic
248
parameters set constant at 2008 levels, under present day management effectiveness. Shaded areas show the most
249
likely development +/- one standard deviation. Stock dynamics are interlinked by market interactions.
17
250
251
Figure S2. Past development (according to ICES/ICCAT stock assessments) in filled circles, sample of
252
stochastic future development (empty circles), and most likely future development for the baseline scenario of
253
future spawning stock sizes, as in Figure 1 in the main text, except for a low growth rate of aggregate fish
254
demand (2.56 %/y), under present day management effectiveness. Shaded areas show the most likely
255
development +/- one standard deviation. Stock dynamics are interlinked by market interactions.
256
18
257
258
259
260
261
Figure S3. Past development (according to ICES/ICCAT stock assessments) in filled circles, and model runs
starting in 1988 with sample of stochastic development (empty circles), and most likely development for the
baseline scenario. Red shaded areas show the most likely development +/- one standard deviation; green shaded
areas the 95% confidence interval.
262
19
263
264
Programming code: AMPL run file
265
266
267
268
269
270
271
272
273
274
275
276
277
278
279
280
281
282
283
284
285
286
287
288
289
290
291
292
293
294
295
296
297
298
299
300
301
302
303
304
305
306
307
308
309
310
311
312
313
314
315
316
317
318
319
reset;
# load model description
model 2048.mod;
# load parameter values
data 2048.dat;
option solver "/home/martin/Ziena/knitro-9.1.0-z/knitroampl/knitroampl";
option knitro_options "maxit=3000 opttol=1.0e-7";
#option knitro_options "maxit=3000 opttol=1.0e-7 multistart=1
ms_maxsolves=5";
# base scenario
let gY:=0.0576; # growth rate of expenditures for fish
let varlnY:=0.0295; # variance of log expenditures
# let gY:=0.0259; # growth rate of expenditures for fish
# let varlnY:=0.0285; # variance of log expenditures
let
let
let
let
gF[1]:=0.0795; # growth rate of farmed salmon supply
varlnF[1]:=0.0125; # variance in log supply farmed salmon
gF[2]:=0.0933; # growth rate of farmed sea bass supply
varlnF[2]:=0.0540; # variance in log supply farmed sea bass
let gC:=0.02; # rate of technical progress in fishing
# write parameter values to file Out.csv
printf "\n\n">Out.csv;
printf "#management effectiveness Northeast Arctic cod=\t%f
\n",feefactor[1]>Out.csv;
printf "#management effectiveness North Sea cod=\t%f \n",feefactor[2]>Out.csv;
printf "#management effectiveness Baltic cod=\t%f \n",feefactor[3]>Out.csv;
printf "#management effectiveness Bigeye tuna=\t%f \n",feefactor[4]>Out.csv;
printf "#management effectiveness Yellowfin tuna=\t%f
\n",feefactor[5]>Out.csv;
printf "#management effectiveness Bluefin tuna=\t%f
\n\n",feefactor[6]>Out.csv;
printf "#growth rate expenditures:gY=\t%f \n", gY>Out.csv;
printf "#growth rate salmon supply: gF(salmon)=\t%f \n", gF[1]>Out.csv;
printf "#growth rate sea bass supply: gF(sea bass)=\t%f \n\n", gF[2]>Out.csv;
printf "#growth rate technical progress: gC=\t%f \n", gC>Out.csv;
printf
"#year\tssb_neac\tssb_nscod\tssb_baltic_cod\tssb_bet\tssb_yft\tssb_bft\n">O
ut.csv;
for {run in 1..1000} {
printf "# run\t%f\n",run>Out.csv;
# assign values to parameters that exogenously change over time
let Y[0] := 10.15;
let c0[0]:=1;
let farmsupply[1,0]:= 900;
let farmsupply[2,0]:=50;
20
320
321
322
323
324
325
326
327
328
329
330
331
332
333
334
335
336
337
338
339
340
341
342
343
344
345
346
347
348
349
350
351
352
353
354
355
356
357
358
359
360
361
362
363
364
365
366
367
368
369
370
371
372
373
374
375
376
377
378
379
380
381
let {i in 1..6} epsilon[i]:=0;
for {t in 1..T-1} {
let Y[t] :=(1+gY)*Y[t-1]; # growing expenditures for fish
let c0[t]:=c0[t-1]/(1+gC); # decreasing harvesting cost
let farmsupply[1,t]:=(1+gF[1])*farmsupply[1,t-1]; # growing salmon farming
let farmsupply[2,t]:=(1+gF[2])*farmsupply[2,t-1]; # growing sea bass
farming
}
# initial stock numbers
let {i in 1..6, s in 1..n[i]} x0[i,s]:=x2008[i,s];
# determine optimal fishing first
# hence keep fixed all variables relevant only in market equilibrium
fix H0TAC;
drop zeroprofit_1; drop zeroprofit_2; drop zeroprofit_3; drop zeroprofit_4;
drop zeroprofit_5; drop zeroprofit_6;
# hence allow to vary - and optimize over - variables relevant for optimization
unfix E; unfix x;
restore population_dynamics_1; restore population_dynamics_2; restore
population_dynamics_3; restore initial_condition;
# optimize
objective pvprofit;
solve;
# now determine development of fishery under imperfect management.
# this requires to determine shadow prices of the fish stocks in each time step,
which requires to solve the optimization problem
# then compute the market equilibrium in that time step given that not the full
shadow price is taken into account
# loop over time
for {years in 0..41} {
# store initial starting values for stock numbers
let {i in 1..6, s in 1..n[i]} xstart[i,s] := x0[i,s];
for {t in 1..T-1} {
let Y[t] :=(1+gY)*Y[t-1]; # growing expenditures for fish
let c0[t]:=c0[t-1]/(1+gC); # decreasing harvesting cost
let farmsupply[1,t]:=(1+gF[1])*farmsupply[1,t-1]; # growing salmon farming
let farmsupply[2,t]:=(1+gF[2])*farmsupply[2,t-1]; # growing sea bass
farming
}
# determine optimal fishing first
# hence keep fixed all variables relevant only in market equilibrium
fix H0TAC;
drop zeroprofit_1; drop zeroprofit_2; drop zeroprofit_3; drop zeroprofit_4;
drop zeroprofit_5; drop zeroprofit_6;
# hence allow to vary - and optimize over - variables relevant for optimization
unfix E; unfix x;
restore population_dynamics_1; restore population_dynamics_2; restore
population_dynamics_3; restore initial_condition;
# optimize
objective pvprofit;
solve;
# now determine market equilibrium for limited management effectiveness
# hence allow to vary all variables relevant for market equilibrium
21
382
383
384
385
386
387
388
389
390
391
392
393
394
395
396
397
398
399
400
401
402
403
404
405
406
407
408
409
410
411
412
413
414
415
416
417
418
419
420
421
422
423
424
425
426
427
428
429
unfix H0TAC;
restore zeroprofit_1; restore zeroprofit_2; restore zeroprofit_3; restore
zeroprofit_4; restore zeroprofit_5; restore zeroprofit_6;
# hence keep fixed all variables relevant for optimization problem
fix E; fix x;
drop population_dynamics_1; drop population_dynamics_2; drop
population_dynamics_3; drop initial_condition;
# determine equilibrium solution satisfying the market clearing / zero profit
conditions
objective no_objective;
solve;
430
Programming code: AMPL mod file
431
432
433
434
435
436
437
438
439
440
param T;
#time horizon (years)
param r;
#annual interest rate
param n {i in 1..6};
#number of age classes
param w {i in 1..6, s in 1..n[i]};
#weight; unit kg per individual in
age class
param g {i in 1..6, s in 1..n[i]};
#maturity
param q {i in 1..6, s in 1..n[i]};
#catchability coefficents
param a {i in 1..6, s in 1..n[i]};
#survivability
param x0 {i in 1..6, s in 1..n[i]}; # initial state, number of individuals;
unit 10^6
# compute resulting stock dynamics from current to next time step
# to obtain the next period's starting values
let {i in 1..6}
epsilon[i]:=nu[i]*epsilon[i]+sqrt(varrecruitment[i])*Normal(0,1);
let {i in 1..6}
x0[i,1]:=(phi[i,1]*SSB[i,0]*exp(-phi[i,2]*SSB[i,0]))*exp(epsilon[i]-varrecr
uitment[i]/2); # stochastic growth of fish stock i;
let {i in 1..6, s in 1..n[i]-2}
x0[i,s+1]:=a[i,s]*(1-q[i,s]*H0TAC[i]/B[i,0])*xstart[i,s];
let {i in 1..6}
x0[i,n[i]]:=a[i,n[i]-1]*(1-q[i,n[i]-1]*H0TAC[i]/B[i,0])*xstart[i,n[i]-1]+a[
i,n[i]]*(1-q[i,n[i]]*H0TAC[i]/B[i,0])*xstart[i,n[i]];
let Y[0] :=Y[1]*exp(sqrt(varlnY)*Normal(0,1)-varlnY/2); # stochastic
expenditure growth
let c0[0]:=c0[1];
let
farmsupply[1,0]:=farmsupply[1,1]*exp(sqrt(varlnF[1])*Normal(0,1)-varlnF[1]/
2); # stochastic growth of famed salmon supply;
let
farmsupply[2,0]:=farmsupply[2,1]*exp(sqrt(varlnF[2])*Normal(0,1)-varlnF[2]/
2); # stochastic growth of famed sea bass supply;
# write output to file Out.csv
printf "%f\t", years > Out.csv;
for {i in 1..6}{
printf "%f\t", SSB[i,0] > Out.csv;
}
printf "\n">Out.csv;
} # end loop over time
} # end loop over scenarios
22
441
442
443
444
445
446
447
448
449
450
451
452
453
454
455
456
457
458
459
460
461
462
463
464
465
466
467
468
469
470
471
472
473
474
475
476
477
478
479
480
481
482
483
484
485
486
487
488
489
490
491
492
493
494
495
496
497
498
499
500
501
502
param xstart {i in 1..6,s in 1..n[i]};
param x2008 {i in 1..6,s in 1..n[i]};
param phi {i in 1..6,j in 1..2};
#parameters of Ricker stock-recruitment
function
param epsilon {i in 1..6};
# autocorrelation recruitment
param nu {i in 1..6};
# autocorrelation recruitment
param varrecruitment {i in 1..6};
# variance in log recruitment
# elasticities of substitution
param sigma;
param psi;
param xi;
# parameters for demand function
param etastop {i in 1..4};
param etastuna {i in 1..3};
param etascod {i in 1..3};
param Y {t in 0..T-1};
param gY;
param varlnY;
param c {i in 1..6};
param c0 {t in 0..T-1};
param gC;
fishing
# expenditures for fish
# growth rate of expenditures
# variance in log expenditures
# parameter of fishing cost function
# parameter of fishing cost function
# rate of technical progress in
param farmsupply {i in 1..2, t in 0..T-1}; # supply of 2 farmed fish species
param gF {i in 1..2};
# growth rate of farm supply
param varlnF {i in 1..2};
# variance in log farm supply
param feefactor {i in 1..6};
# management effectiveness
# variables for dynamic optimization problem
var E {i in 1..6, t in 0..T-1} >=0;
#fishing effort
var x {i in 1..6, s in 1..n[i], t in 0..T} >= 0;#number of individuals [millions]
var B {i in 1..6, t in 0..T-1}=sum{s in 1..n[i]} q[i,s]*w[i,s]*x[i,s,t];
#biomass [1000 tons]
var SSB{i in 1..6, t in 0..T-1}=sum{s in 1..n[i]} w[i,s]*g[i,s]*x[i,s,t];
#spawning stock [1000 tons]
var H {i in 1..6, t in 0..T-1} = (1-exp(-E[i,t]))*B[i,t]; #total harvest [1000
tons]
#
# equations for computing optimal fishing and shadow prices of fish stocks
#
# utility from fishing
var v {t in 0..T-1} =
etastop[1]*farmsupply[1,t]^((sigma-1)/sigma)+etastop[2]*(etastuna[1]*H[4,t]
^((psi-1)/psi)+etastuna[2]*H[5,t]^((psi-1)/psi)+etastuna[3]*H[6,t]^((psi-1)
/psi))^(psi*(sigma-1)/((psi-1)*sigma))+etastop[3]*farmsupply[2,t]^((sigma-1
)/sigma)+etastop[4]*(etascod[1]*H[1,t]^((xi-1)/xi)+etascod[2]*H[2,t]^((xi-1
)/xi)+etascod[3]*H[3,t]^((xi-1)/xi))^(xi*(sigma-1)/((xi-1)*sigma));
# objective to maximize present value of net revenues / profit
maximize pvprofit: sum{t in 0..T-1}
(1/(1+r))^t*(-c0[t]*(c[1]*E[1,t]+c[2]*E[2,t]+c[3]*E[3,t]+c[4]*E[4,t]+c[5]*E
[5,t]+c[6]*E[6,t])+(Y[t]*sigma/(sigma-1))*log(v[t]));
# age-structured population dynamics for 6 stocks of wild fish
subject to population_dynamics_1 {i in 1..6, t in 0..T-1}:
x[i,1,t+1]=(phi[i,1]*SSB[i,t]*exp(-phi[i,2]*SSB[i,t])); # Ricker
stock-recruitment function
23
503
504
505
506
507
508
509
510
511
512
513
514
515
516
517
518
519
520
521
522
523
524
525
526
527
528
529
530
531
532
533
534
535
536
537
538
539
540
541
542
543
544
545
546
547
548
549
550
551
552
553
554
555
556
557
558
559
560
561
562
563
564
subject to population_dynamics_2 {i in 1..6, s in 1..n[i]-2, t in 0..T-1}:
x[i,s+1,t+1]=a[i,s]*(1-q[i,s]*(1-exp(-E[i,t])))*x[i,s,t];
subject to population_dynamics_3 {i in 1..6, t in 0..T-1}:
x[i,n[i],t+1]=a[i,n[i]-1]*(1-q[i,n[i]-1]*(1-exp(-E[i,t])))*x[i,n[i]-1,t]+a[
i,n[i]]*(1-q[i,n[i]]*(1-exp(-E[i,t])))*x[i,n[i],t];
subject to initial_condition {i in 1..6, s in 1..n[i]}: x[i,s,0] = x0[i,s];
#
# market equilibrium for limited management effectiveness
#
# variables for management in current year
var H0TAC {i in 1..6}; # harvest
minimize no_objective: 0;
# market equilibrium / zero profit conditions
subject to zeroprofit_1:
B[1,0]-H0TAC[1]-c0[0]*c[1]/(etastop[4]*etascod[1]*(Y[0]/v[0])*H0TAC[1]^(-1/
xi)*((etascod[1]*H0TAC[1]^((xi-1)/xi)+etascod[2]*H0TAC[2]^((xi-1)/xi)+etasc
od[3]*H0TAC[3]^((xi-1)/xi)))^(xi*(sigma-1)/((xi-1)*sigma)-1)*(1-feefactor[1
]*(1-c0[0]*c[1]/(etastop[4]*etascod[1]*(Y[0]/v[0])*H[1,0]^(-1/xi)*((etascod
[1]*H[1,0]^((xi-1)/xi)+etascod[2]*H[2,0]^((xi-1)/xi)+etascod[3]*H[3,0]^((xi
-1)/xi)))^(xi*(sigma-1)/((xi-1)*sigma)-1)*(B[1,0]-H[1,0])))))=0;
subject to zeroprofit_2:
B[2,0]-H0TAC[2]-c0[0]*c[2]/(etastop[4]*etascod[2]*(Y[0]/v[0])*H0TAC[2]^(-1/
xi)*((etascod[1]*H0TAC[1]^((xi-1)/xi)+etascod[2]*H0TAC[2]^((xi-1)/xi)+etasc
od[3]*H0TAC[3]^((xi-1)/xi)))^(xi*(sigma-1)/((xi-1)*sigma)-1)*(1-feefactor[2
]*(1-c0[0]*c[2]/(etastop[4]*etascod[2]*(Y[0]/v[0])*H[2,0]^(-1/xi)*((etascod
[1]*H[1,0]^((xi-1)/xi)+etascod[2]*H[2,0]^((xi-1)/xi)+etascod[3]*H[3,0]^((xi
-1)/xi)))^(xi*(sigma-1)/((xi-1)*sigma)-1)*(B[2,0]-H[2,0])))))=0;
subject to zeroprofit_3:
B[3,0]-H0TAC[3]-c0[0]*c[3]/(etastop[4]*etascod[3]*(Y[0]/v[0])*H0TAC[3]^(-1/
xi)*((etascod[1]*H0TAC[1]^((xi-1)/xi)+etascod[2]*H0TAC[2]^((xi-1)/xi)+etasc
od[3]*H0TAC[3]^((xi-1)/xi)))^(xi*(sigma-1)/((xi-1)*sigma)-1)*(1-feefactor[3
]*(1-c0[0]*c[3]/(etastop[4]*etascod[3]*(Y[0]/v[0])*H[3,0]^(-1/xi)*((etascod
[1]*H[1,0]^((xi-1)/xi)+etascod[2]*H[2,0]^((xi-1)/xi)+etascod[3]*H[3,0]^((xi
-1)/xi)))^(xi*(sigma-1)/((xi-1)*sigma)-1)*(B[3,0]-H[3,0])))))=0;
subject to zeroprofit_4:
B[4,0]-H0TAC[4]-c0[0]*c[4]/(etastop[2]*etastuna[1]*(Y[0]/v[0])*H0TAC[4]^(-1
/psi)*(etastuna[1]*H0TAC[4]^((psi-1)/psi)+etastuna[2]*H0TAC[5]^((psi-1)/psi
)+etastuna[3]*H0TAC[6]^((psi-1)/psi))^(psi*(sigma-1)/((psi-1)*sigma)-1)*(1feefactor[4]*(1-c0[0]*c[4]/(etastop[2]*etastuna[1]*(Y[0]/v[0])*H[4,0]^(-1/p
si)*(etastuna[1]*H[4,0]^((psi-1)/psi)+etastuna[2]*H[5,0]^((psi-1)/psi)+etas
tuna[3]*H[6,0]^((psi-1)/psi))^(psi*(sigma-1)/((psi-1)*sigma)-1)*(B[4,0]-H[4
,0])))))=0;
subject to zeroprofit_5:
B[5,0]-H0TAC[5]-c0[0]*c[5]/(etastop[2]*etastuna[2]*(Y[0]/v[0])*H0TAC[5]^(-1
/psi)*(etastuna[1]*H0TAC[4]^((psi-1)/psi)+etastuna[2]*H0TAC[5]^((psi-1)/psi
)+etastuna[3]*H0TAC[6]^((psi-1)/psi))^(psi*(sigma-1)/((psi-1)*sigma)-1)*(1feefactor[5]*(1-c0[0]*c[5]/(etastop[2]*etastuna[2]*(Y[0]/v[0])*H[5,0]^(-1/p
si)*(etastuna[1]*H[4,0]^((psi-1)/psi)+etastuna[2]*H[5,0]^((psi-1)/psi)+etas
tuna[3]*H[6,0]^((psi-1)/psi))^(psi*(sigma-1)/((psi-1)*sigma)-1)*(B[5,0]-H[5
,0])))))=0;
subject to zeroprofit_6:
B[6,0]-H0TAC[6]-c0[0]*c[6]/(etastop[2]*etastuna[3]*(Y[0]/v[0])*H0TAC[6]^(-1
/psi)*(etastuna[1]*H0TAC[4]^((psi-1)/psi)+etastuna[2]*H0TAC[5]^((psi-1)/psi
)+etastuna[3]*H0TAC[6]^((psi-1)/psi))^(psi*(sigma-1)/((psi-1)*sigma)-1)*(1feefactor[6]*(1-c0[0]*c[6]/(etastop[2]*etastuna[3]*(Y[0]/v[0])*H[6,0]^(-1/p
si)*(etastuna[1]*H[4,0]^((psi-1)/psi)+etastuna[2]*H[5,0]^((psi-1)/psi)+etas
tuna[3]*H[6,0]^((psi-1)/psi))^(psi*(sigma-1)/((psi-1)*sigma)-1)*(B[6,0]-H[6
,0])))))=0;
24
565
566
567
568
569
570
571
572
573
574
575
576
577
578
579
580
581
582
583
584
585
586
587
588
589
590
591
592
593
594
595
596
597
598
599
600
601
602
603
604
605
606
607
608
609
610
611
612
613
614
615
616
617
618
619
620
621
622
623
624
Programming code: AMPL dat file
param T := 59;
param n :=
1 12
2 7
3 8
4 8
5 6
6 10;
param gY := 0.056; # growth of expenditures for wild fish
param gC := 0.02;
param r := 0.05;
param feefactor :=
1 0.7
2 0.7
3 0.7
4 0.4
5 0.4
6 0.4;
param sigma := 1.7;
param psi := 2.5;
param xi := 4.3;
param etastop :=
1 0.2525 # salmon
2 0.3756 # tuna
3 0.0751 # sea bass
4 0.2969; # cod
param etastuna :=
1 0.2991 # bet
2 0.2724 # yft
3 0.4285; # bft
param etascod :=
1 0.4519 # neac
2 0.2692 # nsc
3 0.2789; # ebc
param phi
1 1
1 2
2 1
2 2
3 1
3 2
4 1
4 2
5 1
5 2
6 1
6 2
:=
1.98
0.0011
5.468
0.0039
1.6989
0.00182
0.1764
0.0011
0.627
0.0039
0.0112
0.0041;
# data from 1975-2012
param nu :=
25
625
626
627
628
629
630
631
632
633
634
635
636
637
638
639
640
641
642
643
644
645
646
647
648
649
650
651
652
653
654
655
656
657
658
659
660
661
662
663
664
665
666
667
668
669
670
671
672
673
674
675
676
677
678
679
680
681
682
683
684
685
686
1
2
3
4
5
6
0.370
0.759
0.189
0.672
0.629
0.931;
param varrecruitment :=
1 0.191
2 0.108
3 0.322
4 0.029
5 0.026
6 0.147;
param c
1
2
3
4
5
6
:=
1.564 # neac: Arnason et al. 2004
0.155 # nsc: Froese/Quaas 2012
0.135 # ebc: Froese/Quaas 2011
3.024
0.775
1.669;
param w:=
1 1 0 # Northeast Arctic cod
1 2 0
1 3 0.2590
1 4 0.6600
1 5 1.2746
1 6 2.1578
1 7 3.2760
1 8 4.5906
1 9 6.2876
1 10 8.6060
1 11 10.6964
1 12 12.7310
2 1 0.3122 # North Sea cod
2 2 0.8924
2 3 2.0820
2 4 3.8350
2 5 5.5970
2 6 7.5272
2 7 10.0000
3 1 0 # Eastern Baltic cod
3 2 0.173
3 3 0.493
3 4 0.818
3 5 1.183
3 6 1.742
3 7 2.613
3 8 4.306
4 1 0 # Bigeye tuna
4 2 4
4 3 13
4 4 26
4 5 41
4 6 58
4 7 76
4 8 101
5 1 1 # Yellowfin tuna
5 2 2
5 3 11
26
687
688
689
690
691
692
693
694
695
696
697
698
699
700
701
702
703
704
705
706
707
708
709
710
711
712
713
714
715
716
717
718
719
720
721
722
723
724
725
726
727
728
729
730
731
732
733
734
735
736
737
738
739
740
741
742
743
744
745
746
747
748
5
5
5
6
6
6
6
6
6
6
6
6
6
4 35
5 62
6 87
1 5.6000 # Bluefin tuna
2 11.0000
3 20.2000
4 33.6000
5 52.0000
6 73.6000
7 95.4000
8 119.8000
9 146.6000
10 215.4000;
param q:=
1 1 0
1 2 0
1 3 0.0260
1 4 0.1960
1 5 0.4960
1 6 0.7250
1 7 0.8840
1 8 0.9820
1 9 0.9930
1 10 0.9930
1 11 0.9480
1 12 1.0000
2 1 0.4200
2 2 0.9570
2 3 0.9540
2 4 1.0000
2 5 0.9560
2 6 0.9290
2 7 0.9810
3 1 0.000
3 2 0.166
3 3 0.548
3 4 0.842
3 5 1.000
3 6 0.904
3 7 0.891
3 8 0.907
4 1 0
4 2 1.0000
4 3 0.5070
4 4 0.6750
4 5 0.8460
4 6 0.8630
4 7 0.7550
4 8 0.3580
5 1 0.2020
5 2 0.4600
5 3 0.2960
5 4 0.4800
5 5 0.8950
5 6 1.0000
6 1 0.4220
6 2 1.0000
6 3 0.7800
6 4 0.5670
6 5 0.4010
6 6 0.3950
27
749
750
751
752
753
754
755
756
757
758
759
760
761
762
763
764
765
766
767
768
769
770
771
772
773
774
775
776
777
778
779
780
781
782
783
784
785
786
787
788
789
790
791
792
793
794
795
796
797
798
799
800
801
802
803
804
805
806
807
808
809
810
6
6
6
6
7 0.3590
8 0.4030
9 0.6500
10 0.7110;
param g:=
1 1 0
1 2 0
1 3 0
1 4 0.0026
1 5 0.0676
1 6 0.3526
1 7 0.6758
1 8 0.8770
1 9 0.9626
1 10 0.9964
1 11 1.0000
1 12 1.0000
2 1 0.0100
2 2 0.0500
2 3 0.2300
2 4 0.6200
2 5 0.8600
2 6 1.0000
2 7 1.0000
3 1 0.000
3 2 0.130
3 3 0.360
3 4 0.830
3 5 0.940
3 6 0.960
3 7 0.960
3 8 0.980
4 1 0
4 2 0
4 3 0
4 4 0
4 5 1
4 6 1
4 7 1
4 8 1
5 1 0
5 2 0
5 3 0
5 4 1
5 5 1
5 6 1
6 1 0
6 2 0
6 3 0
6 4 0
6 5 1
6 6 1
6 7 1
6 8 1
6 9 1
6 10 1;
param a:=
1 1 1.0000 # Northeast Arctic cod
1 2 1.0000
1 3 0.7261
28
811
812
813
814
815
816
817
818
819
820
821
822
823
824
825
826
827
828
829
830
831
832
833
834
835
836
837
838
839
840
841
842
843
844
845
846
847
848
849
850
851
852
853
854
855
856
857
858
859
860
861
862
863
864
865
866
867
868
869
870
871
872
1
1
1
1
1
1
1
1
1
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
4 0.7945
5 0.8106
6 0.8187
7 0.8187
8 0.8187
9 0.8187
10 0.8187
11 0.8187
12 0.8187
1 0.6263 # North Sea cod
2 0.7680
3 0.6798
4 0.7711
5 0.7711
6 0.7175
7 0.82
1 1 # Eastern Baltic cod
2 0.82
3 0.82
4 0.82
5 0.82
6 0.82
7 0.82
8 0.82
1 0.4493 # Bigeye tuna
2 0.4493
3 0.6703
4 0.6703
5 0.6703
6 0.6703
7 0.6703
8 0.6703
1 0.4493 # Yellowfin tuna
2 0.4493
3 0.5488
4 0.5488
5 0.5488
6 0.5488
1 0.8694 # Bluefin tuna
2 0.8694
3 0.8694
4 0.8694
5 0.8694
6 0.8694
7 0.8694
8 0.8694
9 0.8694
10 0.8694;
param x2008:=
1 1 357.9040
1 2 583.9500
1 3 851.3280
1 4 650.7090
1 5 325.2560
1 6 162.2830
1 7 64.9200
1 8 60.6140
1 9 18.3970
1 10 6.9940
1 11 0.8270
1 12 0.2930
29
873
874
875
876
877
878
879
880
881
882
883
884
885
886
887
888
889
890
891
892
893
894
895
896
897
898
899
900
901
902
903
904
905
906
907
908
909
910
911
2
2
2
2
2
2
2
3
3
3
3
3
3
3
3
4
4
4
4
4
4
4
4
5
5
5
5
5
5
6
6
6
6
6
6
6
6
6
6
1 87.7280
2 33.3230
3 31.9480
4 3.8800
5 2.0170
6 0.4990
7 0.3310
1 198.143
2 204.938
3 124.999
4 58.493
5 23.986
6 8.579
7 2.325
8 1.018
1 59.8342
2 9.0535
3 4.8768
4 3.1898
5 2.2556
6 1.8093
7 0.4586
8 0.7592
1 48.0540
2 12.4777
3 4.8269
4 2.6958
5 0.9605
6 0.1914
1 0.1421
2 0.2420
3 0.2735
4 0.2672
5 0.3362
6 0.2808
7 0.1756
8 0.1721
9 0.1387
10 0.3174;
30
912
References
913
Arnason, R., Sandal, L., Steinshamn, S., Vestergaard, N., 2004. Optimal feedback controls:
914
Comparative evaluation of the cod fisheries in denmark, iceland, and norway. American
915
Journal of Agricultural Economics 86 (2), 531-542.
916
Baranov, T.I. (1918). On the question of the biological basis of fisheries, Nauch. issledov.
917
iktiol. Inst. Izv., I, (1), 81-128. Moscow. Rep. Div. Fish Management and Scientific Study of
918
the Fishing Industry, I(1).
919
Baumgärtner, S., Quaas, M.F. 2009. Ecological-economic viability as a criterion of strong
920
sustainability under uncertainty. Ecological Economics 68, 2008–2020.
921
Bertignac, M., Campbell, H. F., Hampton, J., Hand, A. J., 2000. Maximizing resource rent
922
from the Western and Central Pacific tuna fisheries. Marine Resource Economics 15 (3),
923
151-177.
924
Beverton, R. J. H., Holt, S. J., 1957. On the Dynamics of Exploited Fish Populations.
925
Blackburn Press, Caldwell, New Jersey.
926
Branch, T. A. 2008. Not all fisheries will be collapsed in 2048. Marine Policy 32:38–39.
927
Byrd, R., Hribar, M., Nocedal, J., 1999. An interior point method for large scale nonlinear
928
programming. SIAM Journal of Optimization 9 (4), 877-900.
929
Byrd, R., Nocedal, J., Waltz, R., 2006. Knitro: An integrated package for nonlinear
930
optimization. In: di Pillo, G., Roma, M. (Eds.), Large-Scale Nonlinear Optimization.
931
Springer, p. 35
932
Cheung, W.W.L., Lam, V.W.Y., Sarmiento, J.L., Kearney, K., Watson, R., Zeller, D., Pauly,
933
D.. 2010. Large-scale redistribution of maximum fisheries catch potential in the global ocean
31
934
under climate change. Global Change Biology 16, 1: 24–35.
935
Cook, R., Sinclair, A., Stefánsson, G., 1997. Potential collapse of North Sea cod stocks.
936
Nature 385 (6616), 521-522.
937
Costello, C., Gaines, S. D., and Lynham, J. 2008. Can Catch Shares Prevent Fisheries
938
Collapse? Science 321, 1678–1681.
939
DeLara, M., Doyen, L. 2008. Sustainable Management of Natural Resources. Springer.
940
Diekert, F., Hjermann, D., Nævdal, E., Stenseth, N., 2010. Non-cooperative exploitation of
941
multi-cohort fisheries-the role of gear selectivity in the North-East Arctic cod fishery.
942
Resource and Energy Economics 32 (1), 78-92.
943
Drinkwater, K.F., 2005. The response of Atlantic cod (Gadus morhua) to future climate
944
change. ICES Journal of Marine Science 62(7), 1327-1337.
945
Eide, A., Skjold, F., Olsen, F., and Flaaten, O. (2003). Harvest functions: the Norwegian
946
bottom trawl cod fisheries. Marine Resource Economics, 18(1), 81–94.
947
Essington, T.E., Moriarty, P.E., Froehlich, H.E., Hodgson, E.E., Koehn, L.E., Oken, K.L.,
948
Siple, M.C., Stawitz, C.C., 2015. Fishing amplifies forage fish population collapses. PNAS
949
published ahead of print April 6, 2015, doi:10.1073/pnas.1422020112
950
FAO (2014). The State of World Fisheries and Aquaculture: Opportunities and challenges.
951
Food and Agriculture Organization of the United Nations, Rome.
952
Froese, R., Quaas, M., 2012. Mismanagement of the north sea cod by the european council.
953
Ocean & Coastal Management 70, 54-58.
954
Froese, R., Quaas, M. F., 2011. Three options for rebuilding the cod stock in the eastern
32
955
Baltic Sea. Marine Ecology Progress Series 434, 197-200.
956
Greenberg, P., 2010. Four Fish: The Future of the Last Wild Food. Penguin, London.
957
Hilborn, R., 2007. Reinterpreting the state of fisheries and their management. Ecosystems 10,
958
1362–1369.
959
Kapaun, U., Quaas, M.F., 2013. Does the optimal size of a fish stock increase with
960
environmental uncertainties? Environmental and Resource Economics 54(2): 293-310.
961
Kjesbu O. S., Bogstad, B, Devine, J. A., Gjøsæter, H., Howell, D., Ingvaldsen, R. B., Nash, R.
962
D. M., Skjæraasen J. E., 2014. Synergies between climate and management for Atlantic cod
963
fisheries at high latitudes. PNAS 111 (9), 3478-3483.
964
Merino, G., Barange, M., Blanchard, J. L., Harle, J., Holmes, R., Allen, I., Allison, E. H., et
965
al., 2012. Can marine fisheries and aquaculture meet fish demand from a growing human
966
population in a changing climate? Global Environmental Change, 22, 795–806.
967
Metian, M., Pouil, S., Boustany, A., Troell, M. 2014. Farming of Bluefin Tuna –
968
Reconsidering Global Estimates and Sustainability Concerns. Reviews in Fisheries Science &
969
Aquaculture 22(3): 184-192.
970
Miyake, M. P., Guillotreau, P., Sun, C.-H., Ishimura, G., 2010. Recent developments in the
971
tuna industry: Stocks, fisheries, management, processing, trade and markets. FAO fisheries
972
and aquaculture technical paper 543, FAO, Rome.
973
Mora, C., Myers, R. A., Coll, M., Libralato, S., Pitcher, T. J., Sumaila, R. U., Zeller, D.,
974
Watson, R., Gaston, K. J., Worm, B., 2009. Management effectiveness of the world's marine
975
fisheries. PLoS Biology 7 (6), e1000131.
976
Newey W. K. and West K.D., 1987. A simple, positive semi-definite, heteroskedasticity and
33
977
autocorrelation consistent covariance matrix. Econometrica 55(3):703-708.
978
Nielsen, M., 2006. Trade liberalisation, resource sustainability and welfare: The case of east
979
Baltic cod. Ecological Economics 58 (3), 650-664.
980
OECD, 2011. Rates of conversion. In: OECD Factbook 2011-2012: Economic,
981
Environmental and Social Statistics. OECD Publishing.
982
Quaas, M. F., Froese, R., Herwartz, H., Requate, T., Schmidt, J. O., Voss, R., 2012. Fishing
983
industry borrows from natural capital at high shadow interest rates. Ecological Economics 82,
984
45-52.
985
Quaas, M. F., Requate, T., 2013. Sushi or fish fingers? Seafood diversity, collapsing fish
986
stocks and multi-species fishery management. Scandinavian Journal of Economics 115 (2),
987
381-422.
988
Ricker, W. E., 1954. Stock and recruitment. Journal of the Fisheries Research Board of
989
Canada 11, 559-623.
990
Smith, M. D., Roheim, C. A., Crowder, L. B., Halpern, B. S., Turnipseed, M., Anderson, J.
991
L., et al. (2010). Sustainability and Global Seafood. Science, 327(5967), 784–786.
992
Squires, D., and Vestergaard, N. (2013). Technical Change and The Commons. Review of
993
Economics and Statistics, 95(5), 1769–1787.
994
Tahvonen, O., 2009. Economics of harvesting age-structured fish populations. Journal of
995
Environmental Economics and Management 58 (3), 281-299.
996
Voss, R., Quaas, M.F., Schmidt, J.O., Kapaun, U. 2015. Ocean acidification may aggravate
997
social-ecological trade-offs in coastal fisheries. PLoS ONE 10(3): e0120376.
34
998
Worm, B., Barbier, E., Beaumont, N., Duffy, J., Folke, C., Halpern, B., Jackson, J., Lotze, H.,
999
Micheli, F., Ralumbi, S., Sala, E., Selkoe, K., Stachowicz, J., Watson, R., 2006. Impacts of
1000
biodiversity loss on ocean ecosystem services. Science 314, 787-790.
35