We want to analyse specialization matters in a given network (N,g).
Nodes represent individuals that can produce goods and services (just like in our usual economy) and that can be consumers too.
This problem studies optimal production allocation:
- We suppose that individual need at most $\alpha$ kind of good or service per time (t) (in futher problem, we will analyse the case in which $\alpha$ is stochastic).
- Goods are evaluated in terms of quantities, and services in terms of hours (hours allocated for their production). So $\alpha \in \mathbb{R}^+$.
- At each time t, individual $j$ ask for a quantity $q_i^j$ for a specific kind of good or services $i$ with probability $p_j$.
The question is : how many kind of good or service each individual (node) has to product, and in which quantity, in order to give satisfaction to all the network at each moment t?
Moreover, we suppose that each individual can product at least $E(\frac{\alpha}{N})$ kind of goods/services each time (t).