Editing EconMult (section)

==Catch production==
A ''fishery'' is in EconMult defined as a unique fleet/targeted species combination. The total catch within each fishery may include all biomass units defined in the model. Each biomass unit vessel catch is computed by [[Cobb–Douglas]] [[production function]], applying two input variables: ''number of fishing days '''d''')'' and ''biomass unit '''x''')''. In the matrix below each column represents a fleet group and each row a targeted species so each element in the matrix is a fishery and gives the vessel catch of the biomass units represented. The biomass units represent all the targeted species. Each catch is represented a Cobb–Douglas production equation as shown in the vessel yield matrix (Y):

<math>
Y_{V} = \begin{pmatrix} \begin{pmatrix} 
q_{1,1,1} d_{1,1} v_{1,1}^{\alpha_{1,1,1}-1} x_{1}^{\beta_{1,1,1}} \\ \vdots \\ q_{1,1,i} d_{1,1} v_{1,1}^{\alpha_{1,1,i}-1} x_{i}^{\beta_{1,1,i}} 
\end{pmatrix} & \cdots & \begin{pmatrix}
q_{1,n,1} d_{1,n} v_{1,n}^{\alpha_{1,n,1}-1} x_{1}^{\beta_{1,n,1}} \\ \vdots \\ q_{1,n,i} d_{1,n} v_{1,n}^{\alpha_{1,n,i}-1} x_{i}^{\beta_{1,n,i}} 
\end{pmatrix} \\ \vdots & \ddots & \vdots \\ \begin{pmatrix} 
q_{j,1,1} d_{j,1} v_{j,1}^{\alpha_{j,1,1}-1} x_{1}^{\beta_{j,1,1}} \\ \vdots \\ q_{j,1,i} d_{j,1} v_{j,1}^{\alpha_{j,1,i}-1} x_{i}^{\beta_{j,1,i}} 
\end{pmatrix} & \cdots & \begin{pmatrix}
q_{j,n,1} d_{j,n} v_{j,n}^{\alpha_{j,n,1}-1} x_{1}^{\beta_{j,n,1}} \\ \vdots \\ q_{j,n,i} d_{j,n} v_{j,n}^{\alpha_{j,n,i}-1} x_{i}^{\beta_{j,n,i}} 
\end{pmatrix} \end{pmatrix}
</math>

The corresponding fleet catches are

<math>
Y = \begin{pmatrix} \begin{pmatrix} 
q_{1,1,1} d_{1,1} v_{1,1}^{\alpha_{1,1,1}} x_{1}^{\beta_{1,1,1}} \\ \vdots \\ q_{1,1,i} d_{1,1} v_{1,1}^{\alpha_{1,1,i}} x_{i}^{\beta_{1,1,i}} 
\end{pmatrix} & \cdots & \begin{pmatrix}
q_{1,n,1} d_{1,n} v_{1,n}^{\alpha_{1,n,1}} x_{1}^{\beta_{1,n,1}} \\ \vdots \\ q_{1,n,i} d_{1,n} v_{1,n}^{\alpha_{1,n,i}} x_{i}^{\beta_{1,n,i}} 
\end{pmatrix} \\ \vdots & \ddots & \vdots \\ \begin{pmatrix} 
q_{j,1,1} d_{j,1} v_{j,1}^{\alpha_{j,1,1}} x_{1}^{\beta_{j,1,1}} \\ \vdots \\ q_{j,1,i} d_{j,1} v_{j,1}^{\alpha_{j,1,i}} x_{i}^{\beta_{j,1,i}} 
\end{pmatrix} & \cdots & \begin{pmatrix}
q_{j,n,1} d_{j,n} v_{j,n}^{\alpha_{j,n,1}} x_{1}^{\beta_{j,n,1}} \\ \vdots \\ q_{j,n,i} d_{j,n} v_{j,n}^{\alpha_{j,n,i}} x_{i}^{\beta_{j,n,i}} 
\end{pmatrix} \end{pmatrix}
</math>

''α'' ''β'' and ''q'' are parameters, the first two known as output elasticities of effort and biomass respectively, ''q'' is often referred to as ''the catchability coefficient''. All the three parameters have the same dimension as the matrix above, e.g. the catchability coefficient:

<math>Q = \begin{pmatrix} \begin{pmatrix} 
q_{1,1,1} \\ \vdots \\ q_{1,1,i} 
\end{pmatrix} & \cdots & \begin{pmatrix}
q_{1,n,1} \\ \vdots \\ q_{1,n,i} 
\end{pmatrix} \\ \vdots & \ddots & \vdots \\ \begin{pmatrix} 
q_{j,1,1} \\ \vdots \\ q_{j,1,i} 
\end{pmatrix} & \cdots & \begin{pmatrix}
q_{j,n,1} \\ \vdots \\ q_{j,n,i} 
\end{pmatrix} \end{pmatrix}
</math>