I want to model following situation:
1) There is a number of representatives of social groups (e. g. political parties).
2) Each of them devises an economic plan for the next year (N+1, N being current year).
3) The economy of the country consists of A...M industries. Their outputs and input-output matrix (in order to produce 1 unit of output of industry A, it requires X units of output of industry B, Y units of industry C etc.) for year N are known.
3) The basis for an economic plan is a "goal vector", in which for each industry there is a planned change (in percentage of output of that industry in year N).
For example, if there are industries A, B and C in the country, and some social group wants to increase the output of industry A by 10 %, keep the output of industry B at the level of year N, and reduce the output of industry C by 20 %, the "goal vector" would be (1.1, 1, 0.8).
4) All goal vectors are checked for feasibility and consistency.
5) From the feasible and consistent goal vectors, exactly one is chosen in scope of a voting.
6) Based on that goal vector, a system of rewards and punishments for the individual industries needs to be devised.
That system should ensure that the industries produce the amounts of final output, which is specified in the selected goal vector (as close as possible).
In reality those rewards and punishments may be
- tax load on the companies in the respective industries,
- interest rates on credits for the respective industries and
- tax load on the consumption of the output of the industries.
For modelling purposes, one number per industry is sufficient, for example
- > 1, if the industry is rewarded for producing more output in year N+1 than in year N,
- 0, if there are no rewards for producing more output and
- < 1, if the industry is punished for producing more output.
Let's call this number U.
The desired result is a vector u, with the number U for each industry.
Question: How can I calculate this vector for year N+1, when
- input-output matrix for year N,
- output of each industry for year N and
- desired final output of each industry for year N+1
are given? What approaches are there?