Economic Planning: How Should an Economy Grow

Constraints

The following constraints are used in this example:

  • bounds on variables

  • for $i \in \text {INDUSTRIES}$,

    \[ \Variable{StaticProduction[i]} = \Argument{demand[i]} + \sum _{j \in \text {INDUSTRIES}} \Argument{production\_ coeff[i,j]} \cdot \Variable{StaticProduction[j]} \]

  • for $i \in \text {INDUSTRIES}$ and $\text {year} \in \{ 1, \dots , \Argument{num\_ years}+1\} $,

    \[ \Variable{ProductiveCapacity[i,year]} = \Argument{init\_ productive\_ capacity[i]} + \sum _{y=2}^\text {year} \Variable{ExtraCapacity[i,y]} \]

  • for $i \in \text {INDUSTRIES}$ and $\text {year} \in \text {YEARS0}$,

    \begin{align*} & \quad \Variable{Stock[i,year]} + \Variable{Production[i,year]} \\ & = (\text {if year}\ \in \text {YEARS then}\ \Argument{demand[i]}\ \text {else}\ 0) \\ & + \sum _{j \in \text {INDUSTRIES}} \Argument{production\_ coeff[i,j]} \cdot \Variable{Production[j,year}+1\Variable{]} \\ & + \sum _{j \in \text {INDUSTRIES}} \Argument{productive\_ capacity\_ coeff[i,j]} \cdot \Variable{ExtraCapacity[j,year}+2\Variable{]} \\ & + \Variable{Stock[i,year}+1\Variable{]} \end{align*}

  • for $\text {year} \in \{ 1, \dots , \Argument{num\_ years}+1\} $,

    \begin{align*} & \quad \sum _{j \in \text {INDUSTRIES}} \Argument{production\_ coeff[`manpower',j]} \cdot \Variable{Production[j,year]} \\ & + \sum _{j \in \text {INDUSTRIES}} \Argument{productive\_ capacity\_ coeff[`manpower',j]} \cdot \Variable{ExtraCapacity[j,year}+1\Variable{]} \\ & \le \Argument{manpower\_ capacity} \end{align*}

  • for $i \in \text {INDUSTRIES}$ and $\text {year} \in \{ 1, \dots , \Argument{num\_ years}+1\} $,

    \[ \Variable{Production[i,year]} \le \Variable{ProductiveCapacity[i,year]} \]