In order to perform estimations and predictions of the bioeconomic impact derived from different management strategies, a dynamic modelling approach of the resource and the fishery as a whole is needed. In this Secetion we develop : (1) the static and dynamic versions of the Gordon-Schaefer (Gordon, 1953, 1954) model; (2) a distributed-delays fleet dynamics model based on Smith's (1969) model; (3) yield-mortality models; and (4) age-structured dynamic models (Seijo & Defeo, 1994a).

The logistic equation (Verhulst, 1838) describes population growth based on the following mathematical expression (Graham, 1935):

Where *r* is the intrinsic rate of population growth, *B(t)* is population biomass in time *t* and *K* is
the carrying capacity of the environment. Population behavior through time is described as a
sigmoid curve, where the unexploited biomass increases unitl a maximum lievel *B∞*,
constrained by *K* (Fig.2.1: see pella & Tomlinson, 1969; Schaefer, 1954 for details).

**Figure 2.1.** Population logistic growth model for K=3.5 million tonnes and r=0.36.

Under exploitation, Schaefer (1954) introduced the catch rate *Y(t)* as:

Y(t)=qf(t)B(t) | (2.2) |

Where *f(t)* is the fishing effort and q is the catchability coefficient, defined as the fraction of the
population fished by an effort unit (Gulland, 1983). Biomass changes through time can be
expressed as:

When the population is at equilibrium, *i.e., dB/dt*=0, and thus losses by natural and fishing
mortalities are compensated by the population increase due to individual growth and
recruitment. Equilibrium yield can be defined as:

Thus, the equilibrium biomass (*Beq*) as a function of fishing effort can be defined as:

A given amount of fishing effort will lead to a specific level of *Beq*, being both variables
inversely correlated. Equilibrium yield as a function of effort can be obtained by substituting
(2.7) in (2.2):

Equation (2.8) gives a parabola that represents the long-term production function of the fishery,
where the corresponding yield (*Y*) for a given level of fishing effort (*f*) in a population at
equilibrium is called sustainable yield. Equilibrium yield will increase with *f* up to the point of
Maximum Sustainable Yield (*MSY*), falling onwards as fishing effort increases.

The economic model developed by Gordon (1954) is based on Schaefer's model, and
introduced the concept of economic overfishing in open access fisheries. The model
establishes that the net revenues π derived from fishing are a function of total sustainable
revenues (*TSR*) and total costs (*TC*):

π=TSR-TC | (2.9) |

or, alternatively:

π=pY-cf | (2.10) |

where *p* is the (constant) price of the species and *c* the (constant) costs per unit of effort. The
latter includes fixed costs, variable costs and opportunity costs of labor and capital. Fixed costs
are independent of fishing operations (depreciation, administration and insurance costs),
whereas variable costs are incurred when fishers go fishing (fuel, bait, food and beverages,
etc.). Opportunity costs are the net benefits that could have been achieved in the next best
economic activity, *i.e.*, other regional fisheries, capital investment or alternative employment,
and thus must be integrated in cost estimations.

Substituting (2.2) in (2.10), π can be defined as a function of effort:

π=[pqB-c]f | (2.11) |

As in the biological model, Gordon (1954) assumes equilibrium to obtain the long-term
production function of the fishery. The open-access equilibrium yield occurs when *TSR* equals
*TC* and thus *π(t)* = 0, and there will be no stimulus for entry or exit to the fishery. If, additionally,
biomass is assumed a: equilibrium, the yield thus established will provide a simultaneous
equilibrium in both an economic and a biological sense, leading to *bioeconomic equilibrium
(BE)*. Biomass at bioeconomic equilibrium (*B _{BE}*) can be defined by solving equation (2.11) for

*B( t) will be always greater than 0, because fishing effort will be reduced or even ceased at TC
≥TSR*. Thus, the model predicts:(1)

*TSR* at equilibrium can be obtained by multiplying (2.4) by the unit price *p*:

The *TSR* curve as a function of effort will have the same form as the sustainable yield curve,
but in monetary terms (Fig. 2. 1c). *TC* is obtained from equation (2.2), as a function of fishing
effort:

The long-term function of *TC* is calculated by solving for *f* and multiplying by *c*:

Hence, the long-run sustainable biomass and production functions of the fishery can be built by
specifying the corresponding levels of fishing effort at Maximum Economic Yield (*f _{MEY}*),
Maximum Sustainable Yield (

**Figure 2.2.** Gordon-Schaefer static
model. Sustainable (a) biomass, (b)
yield, and (c) total sustainable
revenues (*TSR*) and costs (*TC*).

Under unrestricted access, the net benefit or economic rent of the fishery is positive when *f<*
*f _{BE}* and zero when

Both average *(AY)* and marginal *(MY)* yields correspond to the application of an additional unit
of effort, and will be sustainable per unit of time. The former continuously falls as effort
increases (Fig. 2.3a), until reaching 0 when the resource is exhausted. *MY* decreases more
sharply than *AY* and reaches 0 at *MSY*, beyond which it takes negative values, *i.e.*, the yield
obtained by an additional effort unit is progressively lower than the preceding one. This
happens because each fisher operates according to variations in his *AY*, and does not take into
account the external effects caused to other users and to the fishery as a whole.

**Figure 2.3.** Open access regime. (a) Sustainable average and marginal yields; (b) average and
marginal costs, and revenues, as a function of effort under open access conditions.

One biological goal is to set effort and yield levels at *MSY* through time. For this end, we can
take the first derivative of the yield function in (2.8):

So:

Operating at *f _{MEY}* maximizes the economic rent, because the difference between

Multiplying (2.22) by the average price of the species and dividing by *f*, the average value of
the fishing effort *(AVE)* is obtained:

The marginal value of fishing effort *(MVE)* is obtained by multiplying (2.23) by the average price
of the species (*p*):

Fishing effort at *MEY* (f_{MEY}) is obtained by equaling (2.24) to the unit cost of fishing effort (c),
and solving for f.

The bioeconomic equilibrium is reached when *AVE* equals the costs per unit of effort *AVE = c:*
Fig. 2.3b). The latter can be estimated by equaling (2.23) to costs (c) and solving for f. It will be
noted that *f _{BE}=2f_{MEY,} i.e., f_{BE}* is twice

The economic model developed by Gordon also takes into account the assumptions considered by Schaefer (1954) for the biological model:

The population is at equilibrium (see above). Thus, it behaves in a more or less regular fashion such that changes in the trajectory of catch and effort could be used to reflect assertions about the future behavior of the system (Caddy, 1996).

Under equilibrium, fishing mortality

*(F)*is proportional to effort (*f*), being the catchability coefficient (*q*) the constant of proportionality,*i.e.*:*F=qf***(2.26)**The catch per unit of effort

*(CPUE)*is a relative index of population abundance:The stock is constrained by a constant carrying capacity of the environment.

The stock will respond immediately to variations in the magnitude of effort exerted.

Fishing technology is constant.

Prices and marginal/average costs are constant and independent of the level of effort exerted.

*TC*are proportional to effort, and thus a change in the slope of the*TC*curve will determine changes in*BE*and*MEY*levels.

All processes affecting stock productivity (

*e.g.*, growth, mortality, and recruitment) are subsumed in the effective relationship between effort and catch.The catchability coefficient q is not always constant, and may differ due to e.g, different aggregation behavior of pelagic and sedentary resources. Factors related to differential gear selectivity by age/lengths are not taken into account.

*CPUE*is not always an unbiased index of abundance. This is especially relevant for sedentary resources with patchy distribution and without the capacity of redistribution in the fishing ground once fishing effort is exerted. Sequential depletion of patches also determines a patchy distribution of resource users, precluding model applicability (see Caddy, 1975, 1989a, b; Conan, 1984; Orensanz*et al.*,1991).Variations in the spatial distribution of the stock are usually ignored, as well as the biological processes that generate biomass, the intra/interspecific interactions, and stochastic fluctuations in the environment and in population abundance.

Ecological and technological interdependencies (see Chapter 3) and differential allocation of fishing effort in the short term (see Chapter 6) are not usually taken into account.

Improvement in technology and fishing power determines that q often varies through time.

It becomes difficult to distinguish whether population fluctuations are due to fishing pressure or natural processes. In some fisheries, fishing effort could be exerted at levels greater than twice the optimum (Clark, 1985).

Smith (1969) assumes that long-run fishing levels are proportional to profits:

If π(t)≥ O, vessels will enter the fishery; exit expected to occur ifπ(t)≤O. Parameter φ can be
empirically estimated according to variations in *π(t)*, turn will have a close relation with
the incurred costs for different effort levels (Seijo *et al.*, 1994b).

Variations in fishing effort might not be reflected immediatly in stock abundance and
perceived yields. For this reason, Seijo (1987) improved Smith's model by incorporating the
delay process between the moment fishers face positive or negative net revenues and the
moment which entry or exit takes place. This is expressed by a distributed–delay parameter
*DEL*) represented by an Erlang probability density function (Manetsch, 1976), which describes
the average time lag of vessel entry/exit to the fishery once the effect of changes in the net
revenues is manifested (see also Chapter 6). Hence, the long-run dynamics of vessel type m
(V_{m}(t)) can be described by a distributed delay function of order g by the following set of differential
equations:

where V_{m} is the input to the delay process (number of vessels which will allocate their fishing effort
to target species); γt_{g}(t) is the output of the delay process (number of vessels entering the fishery);
γ_{1}(t), γ_{2}(t),…, γ_{g-1}(t) are intermediate rates of the delay; *DEL _{m}* is the expected time of entry of vessels
to the fishery; and g is the order of the delay. The parameter g specifies the member of the Gamma
family of probability density functions.

Example 2.1. Dynamic bioeconomic model

Consider a pelagic fishery with parameters defined in Table 2.1.

Parameter/Variable | Value |
---|---|

Intrinsic growth rate | 0.36 |

Catchability coefficient | 0.0004 |

Carrying capacity of the system | 3500000 tonnes |

Price of the target species | 60 US$/tonne |

Unit cost of fishing effort | 30000US$/yr |

Initial population biomass | 3500000 tonnes |

Fleet dynamics parameter | 0.000005 |

Fig. 2.4 shows variations in biomass, yield, costs and revenues resulting from the application of
the dynamic and static version of the Gordon-Schaefer model, as a function of different effort
levels. *f _{BE}* is reached at 578 vessels and

**Figure 2.4.** Static (equilibrium) and
dynamic trajectories of biomass
(a), yield (b) and cost-revenues (c)
resulting from the application of
different fishing effort levels.

Fig. 2.5 shows temporal fluctuations in performance variables of the fishery. Yield and net revenues decrease at fishing effort levels higher than 630 vessels, followed by a dynamic entry/exit of vessels to the fishery, as the economic rent becomes positive or negative, respectively. Bioeconomic equilibrium (π=0) is reached at 1200 tonnes, after 50 years of fishing operations.

**Figure 2.5.** Dynamic trajectories of (a) biomass, (b) yield, (c) economic rent, and (d) fishing effort.

Yield-mortality models link two main outputs of the fishery system: yield Y (dependent variable)
and the instantaneous total mortality coefficient Z. Fitting Y against Z generates a Biological
Production curve, which includes natural deaths plus harvested yield for the population as a whole
(Figure 2.6). Y-Z models provide alternative benchmarks to *MSY*, based on the Maximum
Biological Production (*MBP*) concept (Caddy and Csirke, 1983), such as the yield at maximum
biological production (*Y*_{MBP}) and the corresponding mortality rates at which the total biological
production of the system is maximised (*Z _{BMBP}* and

Csirke & Caddy (1983) expressed the equilibrium yield equation of Graham (1935) in terms of the equilibrium value of annual mortality rate (see p. 45 and also Caddy, 1986), thus reducing equation (2.1) to a quadratic form:

*Y _{i} = aZ^{2}_{i} + bZ_{i} + c*

Where Y_{i} and Z_{i} are the yield and the mean total mortality coefficient for year i, respectively.

Under logistic assumptions, equation (2.32) gives a parabola passing through the abscissa to the
right of the origin. Using multiple regression, Where Z_{i} and Z_{i}^{2} are treated as two independent
variables, the “convex-downwards” curve that relates annual values of yield and total mortality can
be drawn. An estimate of the natural mortality coefficient M can be obtained by solving this equation
for Z = M (Y_{i} and F = 0). See Csirke & Caddy (1983) and Caddy & Defeo (1996) for the calculation
of this and the other parameters related to the Biological Production curve (Fig.2.5).

Csirke & Caddy (1983) suggested an alternative approach to fitting the logistic model, based on the abundance index:

The above equation was preferred owing to theoretical objections to the direct fitting procedure (Hoenig & Hoenig, 1986; Caddy & Defeo, 1996). This model is fitted by using different trial values of M, in which the best value selected is that which maximize a goodness of fit criterion (Caddy, 1986). Parameters of this logistic model can be obtained as in Caddy & Defeo (1996).

Caddy & Defeo (1996) extended the theory of production modelling with mortality estimates to include the exponential model of Fox (1970). Linear and non-linear approaches were used to fit this model. The exponential model for yield and mortality data can be summarized as:

Where B^{∞} and b' can be estimated by nonlinear regression techniques. As in the case of the
alternative logistic approach, the model is fitted for different trial values of *M*, selecting those that
maximize a goodness of fit criterion. The estimation procedure for the remaining parameters is fully
explained in Caddy & Defeo (1996). A linearised approach of the above equation can be easily
derived as:

using *In(Y _{i}/Z_{i}-M)* and

In order to obtain bioeconomic reference points (*RP _{s}*) for precautionary fishery management
(Caddy & Mahon, 1995), Defeo & Seijo (in press) developed an expression for the economic rent (π)
of a stock from the exponential version of the yield-mortality model in its linearised form:

Where p is the average price of the target species and c is the unitary cost of the fishing effort.

Differentiating the above equation, and expression that yields the marginal rent (πm) with changes in
*F* is obtained:

Solving for *F*, an expression that provides the fishing mortality rate at *MEY (F _{MEY}*) is estimated as:

By doing a= e^{1}/(p^{B}q)c, a special function of MathCad®^{1} called *W[a]*can be built as follows:

^{1} Mathcad 5.0 for Windows. 1994. Mathsoft, Inc.

**Example 2.2. Bioeconomic yield-mortality model**

The example to be given below (Defeo & Seijo, in press) is based on a hypothetical data set
used by Caddy (1986: p. 387), which is adapted to the methodology proposed (Table 2.2).
Bioeconomic information that would allow the calculations reported here on a real data set is
not available, so that the results (and the estimates of mortlity used) are only intended to
illustrate the bioeconomic model developed and the proposed methodology of fitting the data.
Input data chosen to run the model was p=$3000, q=0.0001, and c=$25. The *M* value used
as input for the model was found by iterating equation (2.35) and maximizing the goodness-of-fit-criteria.
The highest *R ^{2}* corresponded to

Year | Yield (tonnes) | Z(1/yr) |
---|---|---|

1 | 7.5 | 0.175 |

2 | 12.5 | 0.170 |

3 | 19.0 | 0.250 |

4 | 35.0 | 0.440 |

5 | 40.5 | 0.610 |

6 | 39.5 | 0.795 |

7 | 30.5 | 1.080 |

8 | 20.0 | 1.170 |

9 | 26.0 | 0.900 |

10 | 29.5 | 0.790 |

11 | 27.5 | 0.710 |

12 | 29.0 | 0.470 |

Fig. 2.6 shows the relationship between *Y* and *Z*, fitted by the linearised exponential model.
The three main *RPs*: *MSY*, *y _{MEY}* and

rates (*F _{MBP}* and

**Figure 2.6** Bioeconomic *Y-Z* model: yield and biological production curves fitted to hypothetical data.
The position of *MSY, Y _{MEY}, MBP* is shown. A

Simulations involving changes in the unit cost of fishing effort (c) resulted, as expected, in variations
in the bioeconomic *RPs* derived from the *Y-Z* derived from the *Y-Z* model (Table 2.3). For instance, a reduction in c of
40% (from $25 to $15 per unit of effort) determined a concomitant increase in the mean bootstrap
estimates of bioeconomic *RPs* of the order of 14% for *Y _{MEY}*, 38% for

The bioeconomic approach for fitting yield-mortality models developed by Defeo & Seijo (in
press) unambiguously showed that mean and confidence intervals of bioeconomic *RPs* tended
to fall in the lower bound of those corresponding to the biological model, clearly suggesting that
they constitute relatively cautious *RPs* for management. The *RPs* derived from the Biological
Production curve, such as the *Y _{MBP}* and the corresponding mortality rates (Caddy & Csirke,
1983), also constitute important benchmarks to be considered in future research on the subject,
especially if it is considered that

Sensitivity analysis on the model to variations in unit costs resulted in changes in the
bioeconomic *RPs*. As expected, they systematically increased with decreasing costs and
approached the “maximum sustainable”*RPs*. This could be important in many artisanal coastal
fisheries with relatively low total costs and high unit value of harvested stocks, such as
shellfisheries, where the bioeconomic equilibrium is often reached at high levels of fishing effort
(Seijo & Defeo, 1994b) and the corresponding *F _{MEY}* approaches

Parameter | c=$15 | c=$25 | ||||
---|---|---|---|---|---|---|

Mean | 2.5 Cl | 97.5 Cl | Mean | 2.5 Cl | 97.5 Cl | |

B_{∞} | 225 | 160 | 291 | 228 | 185 | 296 |

MSY | 36 | 31 | 41 | 36 | 32 | 41 |

F_{MSY} | 0.440 | 0.348 | 0.531 | 0.435 | 0.363 | 0.511 |

Z_{MSY} | 0.570 | 0.478 | 0.661 | 0.565 | 0.493 | 0.641 |

Y_{MEY} | 32 | 26 | 38 | 28 | 22 | 34 |

F_{MEY} | 0.258 | 0.234 | 0.283 | 0.187 | 0.168 | 0.203 |

Z_{MEY} | 0.388 | 0.364 | 0.413 | 0.317 | 0.298 | 0.333 |

MBP | 48 | 40 | 56 | 49 | 42 | 57 |

Y_{MBP} | 35 | 31 | 40 | 35 | 32 | 40 |

F_{MBP} | 0.375 | 0.283 | 0.466 | 0.352 | 0.281 | 0.429 |

Z_{MBP} | 0.505 | 0.413 | 0.596 | 0.482 | 0.493 | 0.559 |

f_{MSY} | 4,395 | 3,476 | 5,314 | 4,349 | 3,527 | 5,171 |

f_{MEY} | 2,584 | 2,339 | 2,828 | 1,867 | 1,690 | 2,044 |

A simple approach to formulation of risk-averse management strategies was explored by Defeo
and Seijo (in press), using decision theory (Schmid, 1989: see Chapter 7) jointly with the
bioeconomic Y-Z approach developed here. For this purpose, the concepts of Maximax.
Maximin and Minimax were recognised as powerful tools for choice under uncertainty, because
they seem to be well adapted for formulation of risk-averse management strategies and
precautionary fisheries management (FAO, 1995a, Pérez & Defeo, 1996). An alternative risk
analysis could be carried out by using the probability density functions of *Y _{MEY}* and

In our view, yield-mortality models have some advantages over the classic catch-effort models because (see also Caddy & Defeo, 1996; Caddy, 1996):

They can be considered as “output-output”, i.e., both

*Y*and*Z*constitute outputs of the biologic and economic subsystems. Thus, errors due to poor calibration of fishing effort (input variable) in standard catch-effort models, as well as unperceived effects of changes in*q*with fishing intensity and biomass, might be reduced using this approach.Values calculated from the logistic model for the total mortality rate at maximum biological production

*Z*_{MBP}tend to fall in the low percentiles of the*Z*_{MSY}cumulative distribution, thus constituting a relatively cautious reference point for management. Depending on the unit cost of effort,*Z*_{MBP}is even more cautious than*Z*_{MEY}Unless there have been very considerable annual changes in fishing effort, the successive annual points in a

*Y-Z*plot would not show the sharp jumps from left to right hand sides of the yield curve, characteristic of catch-effort production models with wide departures from equilibrium.The bioeconomic model shown here assumes “pseudo-equilibrium conditions” (sensu Caddy, 1996: p. 219). Nevertheless,

*Z*values derived from catch curves and multi-age group analysis more closely represent past and present impacts of fishing on all harvested year classes than do annual values of fishing effort, thus providing robustness with respect to departures from equilibrium.

A stochastic dynamic model following the systems science approach could be alternatively formulated to compare the performance of both dynamic and static approaches and to evaluate, under the light of model assumptions, which of them will prove most effective and useful for management advice. A multiple criterion optimization approach could also be developed for one or more sets of policy goals and management targets, in order to reflect the willingness of the decision maker to allow for tradeoffs among performance variables (Diaz de León & Seijo, 1992; Seijo et al., 1994c: see Chapter 5).

Age structured models consider factors affecting biomass through time, such as growth,
recruitment and mortality, in a population homogeneously distributed in space and time. These
models are based on the static model of Beverton & Holt (1957), and explicitly include the age
structure of the population. The Beverton & Holt model and subsequent variations, assume that
recruitment is independent of stock size and that it is not affected by variations in fishing
intensity. Moreover, it relies on the “Dynamic Pool Assumption”, which allows a unit stock to be
treated as perfectly mixed age groups with homogenous distribution and equal probability of
capture within the distribution area, before and after applying fishing effort. Growth and
mortality parameters are the same for the entire area, and constant for the entire life span of
the species (see Hancock, 1973 and Seijo *et al*., 1994b for details).

An alternative approach to static models considers variations in population structure through time, based on the dynamic accounting of inflows and outflows of individuals to each age of the population structure. The incorporation of recruits is dynamic, allowing for a seasonal analysis of recruitment and its distribution (Seijo, 1986; Seijo & Defeo, 1994b). In the dynamic model, changes in the number of individuals through time can be defined as:

Where S_{i} denotes the survival rate of organisms of age i and A_{i} correspond to the total
mortality rate (Gulland, 1983). Therefore, *S _{l-1}(t)* can be expressed as:

S_{t-1}(t)=1-[MR_{t-1}(t)+FR_{T-1}(T)] **(2.47)**

where *MR(t)* and *FR(t)* are the finite natural and fishing mortality rates, respectively, derived
from previous estimations of the corresponding instantaneous rates of natural and fishing
mortality.

Rearranging:

Thus, the number of individuals in each cohort (N_{i}) can be obtained by integrating in the interval
[t, t+DT], the number of individuals of age i-1 that survive and grow into a cohort in time t,
minus the total mortality rate (A_{i}) minus the rate at which organisms surviving cohort i (S_{i}) are
incorporated into cohort i+1 in time t (Seijo & Defeo, 1994b). Using Euler numerical integration
(Chenney & Kincaid, 1985), the dynamic population structure cna be expressed as:

B_{i}(t + DT) = N_{i}((t)+DT[S_{i-1}N_{i-1}(t)-N_{i}(t)] **2.50**

In this case, the von Bertalanffy growth equation and a length-weight relationship (W=a.l^{b}) are
used to estimate the biomass for each age class:

B_{i}(t + DT) = N_{i}((t + DT)W_{i} **(2.51)**

Fishing mortality (*F _{i}*) and yield

Total revenues *TR* are obtained by multiplying the unit price *(p _{i})* by the yield estimated for each
age:

Y_{i}(t) = q_{i} B_{i}(t) f (t) **(2.54)**

Total costs *(TC (t)* and net revenues *π(t)* are obtained as in the Gordon-Schaefer model.

**Example 2.3. Age-structured dynamic bioeconomic model**

In the following example, the dynamic behaviour of population biomass, yield, effort and
revenues is analyzed for a hypothetical trawl fishery with parameters defined in Table 2.4.
Simulations involve variations in the age at first capture *(tc)* and in the amount of fishing effort *f*.

Parameter/Variable | Value |
---|---|

Maximum observed age | 10 years |

Age at first maturity | 2 years |

Average fecundity | 5000 eggs |

Age at first capture | 2 years |

Sex proportion | 0.5 |

Natural mortality coefficient | 0.2/mo |

Curvature parameter of von Bertalanffy equation | 0.5/yr |

ι_{0} of von Bertalanffy equation | 0.0 |

Asymptotic length L∞ | 100 mm |

Asymptotic weight W∞ | 200 g |

Selectivity parameters | L_{50}=20 mm |

L_{75}=30 mm | |

Area swept per day | 0.1 Km^{2} |

Total distribution area of the stock | 10 Km^{2} |

Maximum observed recruitment | 20000000 |

Average price | 10000 US$/tonne |

Unit cost of fishing effort | 75000 US$/vessel/d |

Fleet dynamics parameter | 0.00005 |

The dynamic trajectories of fishery performance varibles under different *tc* values are
observed in Fig. 2.7. Biomass decreases to a minimum concurrently with highest yields, 20
years after the beginning of the fishery. Decrements in biomass are more noticeable with low *tc*
values, which in turn determines the lowest values of yields and economic rent. A long-term
equilibrium is reached after 45 years (Fig. 2.7a to d). Maximum fishing effort is about 200
vessels for a *tc* comprised between 2 and 3 yr at ca. 20 years and diminishes onwards as a
result of negative economic rent (Fig.2.7c). The number of vessels at *tc* values varying from 1
to 4 yr under bioeconomic equilbrium, are, respectively, 67, 90, 115 142 (Fig. 2.7d); *i.e.*, a
relatively high *tc* (e.g., 4 yr) allows the fishery to support a greater number of vessels. However,
an indiscriminate increase in *tc* (e.g., greater than 4 years) could not justify vessels operating
in the fishery. Yields and economic rent are highest with *tc* = 2 years. Certainly, the resulting
dynamic biomass for high *tc* values are higher and with low variations through time. Fishing
effort tends to increase proportionally to the rent generated by the fishery under different *tc*
scenarios (Fig. 2.7d). In the long run, under open access conditions and with *tc* = 4, the fishery
is able to support more than twice the number of vessels than with *tc* = 1 yr. The yield at
bioeconomic equilibrium increases from 158 tonnes with *tc* = 1 yr to 272 tonnes with a *tc* =4 yr
(Fig.2.7b).

**Figure 2.7.** Age-structured bioeconomic model: dynamic effect of different *tc* in (a) biomass; (b)
yield; (c) economic rent; and (d) fishing effort.

As can be observed from the above example, the dynamic age-structured model allows one to explore the impact of several sizes/ages at first capture. When the selectivity by size is variable, this important control variable becomes a management instrument that the global models such as the Gordon-Schaefer cannot handle.

Two key predictions were derived from the classical *static* bioeconomic model: 1) an open
access regime leads to stock overexploitation and dissipation of the economic rent; and 2)
*MEY* will occur at a lower exploitation rate than *MSY*. However, *this model ignores the time
dimension in the estimation of optimal yield and effort levels.* In this section we introduce the
price of time to explain the bioeconomic dynamic behaviour of a fishery. We also show that the
optimal exploitation rate of a fish stock could be greater or lower than *MEY* or *MSY*, depending
on the intertemporal preferences of society concerning resource use.

Fishing effort investment decisions are related to the expectation that the fishing unit (i.e.
vessel + gears) assures positive net revenues throughout its lifetime. An approach to the
incorporating problem of the dimension and importance of time as a key factor in investment
and development of fisheries, is to consider the *preferences* in the consumption of a certain
good in different periods.

For example, consider an individual that has the alternative of consuming goods in the present
or in subsequent years. This person will not necessarily be indifferent to the choice of spending
now or in the future, even if prices remain constant. Indeed, consumption in a period
constitutes a different good to consume in another period. Each society member has *temporal
preferences* concerning the consumption of a good in different time periods. This is measured
by the *marginal time preference rate (MTPR)*. If an individual is indifferent between consuming
an additional 1 US$ in a year or 1.10 US$ in the following, he has a *MTPR* of 10% per year.
The term*marginal* is used because a *MTPR* measures the individual preference between small
increments in consumption through time. This presupposes that the individual has dissimilar
expectations about the amount of a good that will consume in different periods (see Sudgen &
Williams, 1978 for a detailed discussion on the subject).

Preference analysis in the use of a fishery resource could not be static, for two reasons: 1) its
renewable nature implies variability in availability and uncertainty in its magnitude through time;
2) a different temporal marginal preference of resource use will exist according to the type of
fishery considered. For example, open access fisheries are generally characterized by a high
*MTPR*, because of the inherent characteristics of fish stocks developed in Chapter 1. Thus,
there will be incentives to increase fishing effort levels (and thus yields and profits) in the short-
run, having little or no concern for the future. In mechanized fisheries, the investment carried
out in planning and developing fishing activities is not immediately paid. The lifetime of the
fishing unit should be taken into account to evaluate the investment magnitude, as well as
present and future costs, and the probable revenues derived from fishing. In these cases, it is
probable that, under precautionary management schemes (e.g. limited entry), a low *MTPR*
occurs, in order to favor investments and to sustain the resource in the long run.

Consider a fisher A who has to decide on how to distribute his consumption activities in two
time periods *t _{1}* and

**Figure 2.8.** Intertemporal neutral preference (adapted from Randall, 1981).

Now consider a fisher B and his respective utility function (indifference curve *l _{b}*) in two-time
periods

**Figure 2.9.** Intertemporal positive preference (adapted from Randall, 1981).

Fisher C (Fig. 2.10) has a *negative time preference* or a low *MTPR*. He will transfer part of his
consumption in the current period to the subsequent one, in such a way that his consumption
will be Q^{*}_{1c} + Q^{*}_{2c}. Thus, his optimal intertemporal consumption is achieved at the tangent
between *WW* and the indifference curve l^{*}_{c}.

**Figure 2.10.** Intertemporal negative preference (adapted from Randall, 1981).

The *present value* of flow of future revenues allows for comparisons of money during
different time periods. The *discount rate (d)* is used for this purpose. Individuals with different
time preferences adjust their intertemporal consumption profile so as to be indifferent between
now and later. For example, suppose that an individual has a *MTPR* of *d* per time period, i.e.,
he is indifferent between an extra consumption of 1 unit in period 0 and 1 + *d* units in period 1.
Analogously, he would be indifferent to the alternative of consuming (1 + d)^{2} extra units in
period 2. Thus, an extra unit consumed in period 1 has a *present value* of (1/1+*d*) units in
period 0, and an extra unit consumed in period 2 has a *present value* of 1/(1+*d*)^{2} units in
period 0, and so on for *t* periods (1/ (1+ *d*)^{t}). The *d* rate to which future revenues are
discounted at present values is the *discount rate*. A higher discount rate would lead to a lower
present value, and vice versa. The discount rate differs from *MTPR* in that it does not imply an
interpretation of the rate to which it refers, but rather it is simply a *number*, generally constant,
used in arithmetic manipulations.

Related concepts, such as the *discount and compensation factors*, could be used for
calculations. The *discount factor v _{td}* is the present value of an accumulated unit in the period t

The *compensation factor a _{d}* is the present value of a flow of revenues based on the
accumulation of one unit in each of the

where: t=1,2,...,n. In terms of the discount factor:

The compensation factor is useful when the flow of costs and benefits through time is *the
same*. This is rarely the case in fisheries.

Although the equilibrium estimators *MSY,MEY* and *BE* are useful benchmarks as *RPs* in the
bioeconomic analysis of fisheries, their static nature diminish their reliability as appropriate
management tools. Considering it extremely unlikely that the fishery system reflects equilibrium
states, the dynamic fitting of the Schaefer-Gordon model should be preferred to its static
counterpart, as it takes into account the intertemporal flow of costs and benefits from different
fishing effort levels and dynamic biomass fluctuations. Thus, a fishery should be managed by
maximizing a dynamic exploitation pattern more than by setting a specific (static) sustainable
yield level. Indeed, a fishery will be economically efficient *if it maximizes the net present value
of catches*. Since these are autocorrelated in time, the bioeconomic static analysis loses
significance as an appropriate fisheries management tool.

The net present value of a flow of benefits and costs through time could be expressed as:

where *PV*π is the present value of the net revenues π(*t*).

Clark (1985) develops a bioeconomic dynamic model based on the concept of a sole owner
attemoting to maximize his profits from a fishery. This concept does not mean a monoply, in
which prices could be fixed as a result of market control. Indeed, the dynamic Gordan-Schaefer
model developed by clark assumes thar the industry is a price taker (i.e.,prices are constant
through time). The model considers the discount rate d in its *continuos* form. The discount
factor defined in (2.56) is expressed in its exponential form as:

In this expression,δ is the annual continous discount rate (Clark, 1985). According to (2.60), δ is related with d in the following form:

δ=ln(*1+d*) **(2.61)**

The present value of the revenues π(ι) in a time interval (O,T) will be:

In the long run (Τ = + ∞), a single fishery owner will tend to maximize the present value of *π(t)*.
Thus, substituting *π(t)* in (2.62):

The above is subjected to the differential equation that defines the classic surplus production model:

where f(ι)>0 and that the initial biomass B_{o} is known. Solving for f(ι) in (2.64),
substituting in (2.63) and integrating by parts, Clark (1985) showed that the optimum biomass
level (B_{opt} for a given discount rate is given by:

where B_{BE} is defined as *c/pq* (see eq. 2.12). Optimum biomass *B _{OPT}* decreases as δ increases,
and consequently will approach the biomass at bioeconomic equilibrium

The optimal sustainable yield (OSY) and optimal effort (f_{opt}levels for a given price of time δ
are obtained by:

Sustainable exploitation of a fishery resource requires that the sum of the present value of net
revenues be maximized. Setting sustainable yield levels for this purpose will depend on: (a) the
biological balance between recruitment, somatic growth and mortality rates; (b) dynamic
fluctuations in costs and prices in a regional and international context, probably reflected in the
interest rate; and (c) socio-economic and political conditions. Expectation of changes in costs,
prices and stock magnitude, should be included in the bioeconomic analysis of a fishery
through a weighed analysis of the probability distribution of alternative management actions,
based on a dynamic stochastic approach. In this context, the selection of a specific *discount
rate* value will be critical in setting an adequate exploitation strategy, and will depend on the
expected variability in the bio-socio-economic variables above mentioned.

A high rate of discounting (δ → +α) will threaten the viability of the resource. In this case, the
dynamic *MEY* will tend to BE. On the contrary, when resource characteristics support a long-term
exploitation strategy, there is a certain stability in prices and costs, socio-economic
conditions encourage investments, and the future is not discounted. Thus as δ → 0, the
dynamic and static *MEY's* will coincide. In general, the dynamic *MEY* will fluctuate between
these two extreme situations (Anderson, 1986).

According to clark (1973; 1985), high δ values increase the risk of stock overexploitation.
Hannesson (1986; 1987) expresses that this view *ignores the implications derived from a high
interest rate in the cost of capital, an effect previously* established by Farzin (1984) for non-renewable
resources. According to Hannesson, a high *d* value will increase harvesting costs,
since sudden exploitation will require short-term investments in gear, equipment, etc.

Hannesson (1986) discusses the ambiguity of δ: on the one hand, it expresses the return rate
required for achieving short-term profits, in such a way that high *d* values will imply high
exploitation rates and a decrease in stock availability through time. On the other hand, δ
expresses the *opportunity cost of capital* to be invested in the fishing unit (*e.g.* vessel, gears);
thus a high δ value will imply high operation costs and therefore optimal exploitation rates at
lower levels than in the previous case, promoting an increase in stock availability.