# Production system and representation of economic sectors - C3IAM

${\displaystyle Y_{i,r}=CES(X_{j,r};\rho )=A_{i}\centerdot \sum _{j}^{\centerdot }({\boldsymbol {\alpha }}_{j,r}\centerdot {X_{j},r}^{\rho })^{\frac {1}{\rho }}}$ (1)
Where  ${\displaystyle Y_{i,r}}$refer to the output of sector ${\displaystyle i}$ in region ${\displaystyle r}$, ${\displaystyle X_{i,r}}$is the input of sector ${\displaystyle i}$ in region ${\displaystyle r}$, ${\displaystyle A_{i}}$is shift parameter, ${\displaystyle {\boldsymbol {\alpha }}}$is the share parameter,  ${\displaystyle \rho ={\frac {1}{1-\sigma }}}$is the substitution parameter, ${\displaystyle \sigma }$is the elasticity of substitution.