Economic Module

Economic Module

In this section, information on the Economic Module OSMOSE-EC is provided.

Introduction

The economic model takes into account supply and demand dynamics. Supply depends mainly on fishing costst that are stock-dependent and technology-dependent. Demand is driven by the price that consumers are willing to pay and consumer preferences for different species and different size classes. Since evolutionary change impacts demographic changes within the fish population it is particularly interesting to study economic impacts given consumer preferences for size classes.

Costs

Based on TQV18, QSK+18, 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

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

with length of fish $l_s$ in size class $s$, mesh sieze $\sigma_t$ in year $t$ and selectivity accuracy $\omega$ 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 over time (with increasing fuel prices) or decrease (with technological advancement), sucht 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.

Fishermen's 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 revenues

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

Demand

Based on QR13, QRS+16, QSK+18, GQ16, TQV18, 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 high price for a bigger fish than for the same mass of small fishes.

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 is

for s = 1

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

Here $\mu_i$ 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

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

with discount factor $\delta$. 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

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 and assuming market equilibrium.

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

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

with $P_{i,t}:=\sum_k \frac{p_{i,k,t} , h_{i,k,t}^{\frac 1 \mu_i}}{\beta_{i,k}}$ 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 estimate $\gamma$ and $\eta$.

Data requirements

For the parameterisation we need

• profit margins $\pi_{it}$. If this is not available and the fishery has been operating at or close to open access conditions the profit margin is zero.
• prices per kg $p_{i,s,t}$
• the weight of individuals $w_{i,s}$ in kg
• the catchability $q_{i,s,t}$, the proportion of individuals in a size class that are retained in the net. Needs to be between 0 and 1. Alternatively one can fit the catchability to fishing mortalities $F_{i,s,t} = F_{i,t}^{max} , q_{i,s,t}$.
• population numbers $N_{i,s,t}$
• instead of catchability and numbers it is possible to provide instead the harvest $h_{i,s,t}$ and the accessible biomass $B_{i,t}$

for species (i), size class (s) and year (t) as indicated.

Parameters

Short explanation of parameters.

Estimated parameters:

Parameter	Description
$c_{i,0}$	baseline costs of harvesting in the year $t_0$ (usually 2020)
$\tau_{i}$	exponential time trend (positive or negative) on costs of harvesting
$\chi_{i}$	stock elasticity. If >1 the stock is hypersensitive: fishing costs decline fast with accessible biomass. If <1 the stock is hyperstable and fishing costs remain relatively constant with the available biomass
$\beta_{i,s}$	consumer preferences for different sizes (s) of each species (i). $sum_s \beta_{i,s}=1$
$\alpha_{i}$	consumer preferences for different species (i). $sum_i \alpha_{i}=1$
$\eta$	elasticity of demand for fish
$\gamma$	weight of fish consumption in total utility. When $\eta=1$ it is the total expenditure on fish

Parameters from the litterature:

Parameter	Description
$\mu_i$	elasticity of substitution between different sizes of a species (i). Higher values indicate easier substitutability
$\sigma$	elasticity of substitution between species of fish. Should be lower than $\mu_i$ since sizes are better substitutes than species.

Output

• Total profit
• Total revenue
• Total costs
• Future prices

Scenarios: change harvest ($H_{i, t}$), change distribution of harvest over size classes, delete time trend ($\tau_{i}=0$), see what happens when consumers prefer larger or smaller fish (change $\beta_{i,s}$)...

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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.