# 6.2. 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.

## 6.2.1. 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 $$\tau_{i}$$.

## 6.2.2. Demand parameter estimation

Rearranging the inverse demand function we get

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

and

$\mathcal{P}_{i,t} \, \mathcal{H}_{i,t}^{\frac 1 \sigma} = \alpha_i \, \sum_j \mathcal{P}_{j,t} \, \mathcal{H}_{j,t}^{\frac 1 \sigma}$

with $$\mathcal{P}_{i,t}:=\sum_k \frac{p_{i,k,t} \, h_{i,k,t}^{\frac 1 \mu_i}}{\beta_{i,k}}$$ and $$\mathcal{H}_{i,t} :=\left(\sum_k \beta_{i,k} \, h_{i,k,t}^{\frac{\mu_i -1}{\mu_i}} \right)^{\frac{\mu_i-\sigma}{\mu_i-1}}$$ we can first estimate size preferences $$\beta_{i,s}$$ for each species and then species preferences $$\alpha_i$$.

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 $$\eta$$.

## 6.2.3. 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 $$F_{i,s,t}=F^{max}_{i,t} \, q_{i,s,t}$$.

• 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.