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Bioeconomic Module Mathematical Formulation

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Ricardo Oliveros Ramos edited this page Jun 11, 2026 · 2 revisions

This page presents the mathematical formulation of the Economic Module. It includes the equations used to represent accessible biomass, size-dependent harvest, fishing costs, profit, consumer demand, prices, social optimum and calibration relationships.

For the conceptual interpretation of these equations, see Bioeconomic Module Concepts and Assumptions. For the list of parameters, see Bioeconomic Module Parameters. For data needs and estimation workflow, see Bioeconomic Module Data Requirements and Bioeconomic Module Calibration.

Costs

Based on TQV18, QSK+18 and LDDS19.

Fishing costs depend on the availability of fish and the amount of fish harvested. We assume perfect selectivity with respect to species, but imperfect selectivity with respect to the size of species. Depending on the gear, larger or smaller fish of a species can be retained by the gear. The part of the population that could potentially be harvested is the accessible biomass. For example, with a trawl-like fishing net the probability of retainment increases with the size of the fish by a sigmoid pattern:

$$ q_{i,s}(\sigma_{i,t})=\frac{1}{1+e^{-\omega_i \ (l_{i,s}-\sigma_{i,t})}} $$

where $l_s$ is the length of fish in size class $s$, mesh size is $\sigma_t$ for year $t$ and the selectivity accuracy $\omega$ is assumed to be constant.

The total available (or accessible) biomass of a species $i$ is therefore:

$$ B_{i,t}=\sum_{s} q_{i,s}(\sigma_{i,t})\ w_{i,s}\ N_{i,s,t} $$

where $w_{i,s}$ is the weight of fish species $i$ in size class $s$ and $N_{i,s,t}$ is the number of fish in that size class in a given year.

When harvesting a total biomass of $H_{i,t}$, this is distributed to the size classes according to the retainment, therefore

$$ h_{i,s,t}=q_{i,s}(\sigma_{i,t}) \ w_{i,s} \ N_{i,s,t}\frac{H_{i,t}}{B_{i,t}} $$

is the biomass of fish harvested from each size class in a given year.

The costs of harvesting fish increases with harvest and decreases when more fish are accessible.

$$ C_{i,t}=c_{i,t}\ \frac{H_{i,t}}{B_{i,t}^{\chi_i}}, $$

Here, $c_{i,t}$ are baseline harvesting costs that may increase (with increasing fuel prices) or decrease (with technological advancement) over time, such that $c_{i,t}=c_{i,0} e^{\tau_i(t-t_0)}$ with trend $\tau_i$. The parameter $\chi_i$ is the stock elasticity. When $\chi_i<1$ the stock is hyperstable, meaning that at constant effort an increase in available biomass does not necessarily result in a higher harvest.

Fishers' profit is the revenue from selling fish minus costs of harvesting:

$$ \Pi_{i,t} = \sum_{s} p_{i,s,t} h_{i,s,t} - C_{i,t} $$

The profit margin is the fraction of profits with respect to revenue:

$$ \pi_{i,t} = \frac{\sum_s p_{i,s,t} h_{is,t,} - C_{i,t}}{\sum_s p_{i,s,t} h_{is,t,}} $$

Demand

Based on QR13, QRS+16, QSK+18, GQ16, TQV18 and LDDS19.

Consumers have preferences for certain species but also size classes of fish. Yet, when preferred species are rare, and thus expensive, consumers can shift their consumption towards more abundant species, depending on the elasticity of demand.

The more elastic demand is, the easier it is to substitute. The same holds for different size classes of fish, where demand is likely to be even more elastic; you would not pay an extremely higher price for one big fish than for the same weight of small fish.

The utility of fish consumption is:

$$ \nu_t = \left( \sum_{i=1}^I \alpha_{i} \left( \sum_{s=1}^S \beta_{i,s,t} , h_{i,s,t}^{\frac{\mu_i-1}{\mu_i}} \right) ^{\frac{\mu_i}{\mu_i-1} \frac{\sigma-1}{\sigma}} \right) ^{\frac{\sigma}{\sigma-1}}. $$

If the species has only one size class, the consumer cannot substitute between size classes but can substitute between species, thus utility of fish consumption, for s = 1, is as follows:

$$ \nu_t = \left( \sum_{i=1}^I \alpha_{i} h_{i,s,t} ^ {\frac{\sigma-1}{\sigma}} \right) ^{\frac{\sigma}{\sigma-1}} $$

where $\mu_i$ is the elasticity of substitution for different size classes of fish species $i$, $\sigma$ is the elasticity of substitution between different species, $\beta_{i,s}$ are the constant consumer preferences for size classes and $\alpha_i$ are consumer preferences for species.

Consumers pay for fish consumption, therefore total utility is:

$$ u(\nu_t) = \begin{cases} \gamma \ln(\nu_t) - \sum_i \sum_s p_{i,s,t} , h_{i,s,t}& \text{ if } \eta=1 \gamma \frac{\eta}{\eta-1} , \nu_t^{\frac{\eta-1}{\eta}} - \sum_i \sum_s p_{i,s,t} , h_{i,s,t} & \text{else} \end{cases} $$

where $\gamma$ is the total expenditure on fish when $\eta=1$.

Consumers maximise utility over the consumption of fish $h_{i,s,t}$ resulting in the inverse demand function:

$$ p_{i,s,t}= \gamma \left( \beta_{i,s} h_{i,s,t}^{-\frac{1}{\mu_i}} \right) \alpha_i \left( \sum_k \beta_{i,k} h_{i,k,t} ^{\frac{\mu_i - 1}{\mu_i}}\right)^{\frac{\mu_i}{\mu_i-1}\frac{\sigma-1}{\sigma}-1} \nu^{\frac{1}{\sigma}-\frac{1}{\eta}}. $$

and for s = 1:

$$ p_{i,s,t}= \gamma h_{i,s,t}^{-\frac{1}{\mu_i}} \alpha_i \nu^{\frac{1}{\sigma}-\frac{1}{\eta}}. $$

Social optimum

In total, the utility derived from fish consumption is the benefit obtained from fish consumption of different species and size classes minus the costs needed for their extraction. Since individuals discount future utility (we prefer to have something today rather than tomorrow), the total net present value is:

$$ \begin{align*} NPV & = \sum_{t=t_0}^{t_0+T} \delta^t \left( u(\nu_t) + \sum_i \Pi_{i,t} \right) & = \sum_{t=t_0}^{t_0+T} \delta^t \left( \left(\gamma , \frac{\eta}{\eta-1} , \nu_t^{\frac{\eta-1}{\eta}} - \sum_i \sum_s p_{i,s,t} , h_{i,s,t} \right) + \left( \sum_i \sum_s p_{i,s,t} , h_{i,s,t} - \sum_i C_{i,t}\right) \right) & = \sum_{t=t_0}^{t_0+T} \delta^t \left(\gamma , \frac{\eta}{\eta-1} , \nu_t^{\frac{\eta-1}{\eta}} - \sum_i C_{i,t}\right) \end{align*}$$

where $\delta$ is a discount factor. The fist term is the consumers' utility and the last term is fishermen's profit.

A social planner, that accounts for consumer and producer surplus, would maximise this net present value over future control variables in the fishery, here future harvests and mesh sizes.

Calibration relationships

We estimate parameters of the cost function given catch, accessible biomass and profitability data. Species and size class preferences are estimated using the inverse demand function assuming market equilibrium.

The full calibration workflow is described in Bioeconomic Module Calibration. The equations below are included here because they define the mathematical relationships used for parameter estimation.

Cost parameter estimation

Rearranging the definition of the profit margin we get:

$$ \ln(1-\pi_{i,t}) + \ln \left(\frac{\sum_s , p_{i,s,t} h_{i,s,t}}{H_{i,t}} \right)= \ln(c_{i0}) -\chi_i * \ln(B_{it}) + \tau_i , t, $$

and we can estimate baseline costs $c_{i,0}$, stock elasticity $\chi_{i}$ and the time trend on fishing costs $\tau_{i}$.

Demand parameter estimation

Considering each species as a monospecific model, we rearrange the demand function from SZQ23 to get:

$$ p_{i,s, t} h_{i,s,t}^{\frac{1}{\mu_i}} =\beta_{i,s} \sum_j p_{i,j,t} h_{i,j,t}^ {\frac{1}{\mu_i}} $$

Rearranging the inverse demand function we get:

$$ P_{i,t} H_{i,t}^{\frac 1 \sigma} = \alpha_i \sum_j P_{j,t} H_{j,t}^{\frac 1 \sigma} $$

where $P_{i,t}:=\sum_k \frac{p_{i,k,t} , h_{i,k,t}^{\frac 1 \mu_i}}{\beta_{i,k}}$ and $H_{i,t} :=\left(\sum_k \beta_{i,k} , h_{i,k,t}^{\frac{\mu_i -1}{\mu_i}} \right)^{\frac{\mu_i-\sigma}{\mu_i-1}}$. We can first estimate size preferences $\beta_{i,s}$ for each species and then species preferences $\alpha_i$.

and for s = 1:

$$ p_{i,s,t} h_{i,s,t}^{\frac{1}{\sigma}} =\alpha_{i} \sum_i p_{i,j,t} h_{i,j,t}^ {\frac{1}{\sigma}} $$

The elasticities of substitution between size classes $\mu_{i}$ and between species $\sigma$ are either taken from the literature or assumed.

Furthermore, given $\nu_t$ as defined before and using:

$$ \ln \left( \sum_i \sum_s p_{i,s,t} , h_{i,s,t} \right) = \ln(\gamma) + \frac{\eta-1}{\eta} \ln(\nu_t) $$

we can estimate $\gamma$ and $\eta$.

TQV18: Olli Tahvonen, Martin F. Quaas, and Rüdiger Voss. Harvesting selectivity and stochastic recruitment in economic models of age-structured fisheries. Journal of Environmental Economics and Management, 92:659–676, 2018.

QSK+18: Martin F. Quaas, Max T. Stoeven, Bernd Klauer, Thomas Petersen, and Johannes Schiller. Windows of opportunity for sustainable fisheries management: the case of Eastern Baltic cod. Environmental and Resource Economics, 70(2):323–341, 2018.

LDDS19: Kira Lancker, A.L. Deppenmeier, T. Demissie, and J.O. Schmidt. Climate change adaptation and the role of fuel subsidies: an empirical bio-economic modeling study for an artisanal open-access fishery. PLoS ONE, 14(8):e0220433, 2019.

QR13: Martin F. Quaas and Till Requate. Sushi or fish fingers? Seafood diversity, collapsing fish stocks and multi-species fishery management. Scandinavian Journal of Economics, 115(2):381–422, 2013.

QRS+16: Martin F. Quaas, Thorsten B. H. Reusch, Jörn O. Schmidt, Olli Tahvonen, and Rudi Voss. It is the econ- omy, stupid! Projecting the fate of fish populations using ecological-economic modeling. Global Change Biology, 22(1):264–270, 2016.

GQ16: Rolf A. Groeneveld and Martin F. Quaas. Promoting selective fisheries through certification? an analysis of the PNA unassociated-sets purse seine fishery. Fisheries Research, 182:69 – 78, 2016.

SZQ23: Hanna Schenk, Fabian Zimmermann and Martin F. Quaas. The economics of reversing fisheries-induced evolution. Nature Sustainability, 6:706 – 711, 2023.

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