A Mathematical Framework for Evolutionary Economics

We suggest a mathematical description of economic ``variables'', such as commodities, actors, values (quantities of numéraire), prices, durations, etc., as ``evolutions'' depending on time, not only as static states ranging over vector spaces. The states of these variables (stocks) are ``related'' at each instant by ``inert relations'' and, together with their time derivatives (flows), by ``kinetic relations''. When inert and kinematic relations are given independently of each other, it may happen than some of some, all or none evolutions are ``viable'' in the sense they satisfy both inert and kinematic relations. Hence the task is to reduce the inert and kinematic relations for restoring the viability requirement: the reduced inert relation is called its ``viability kernel'' and the restricted kinematic relation is called its ``regulator''. This regulator governs the viable economic evolutions of these variables and their derivatives. Knowing it, it allows to describe one of them, the endowment in means of payment of an economy, for instance, in function of all remaining variables. This choice can be interpreted as a ``budgetary rule'' depending on both inert and kinetic relations. It may happen that \emph{a priori} budgetary rules designed independently of inert and kinematic relations, such as variants of the Walras law, do not fit this budgetary rule, and thus trigger the dysviability of the economy. They are indeed sufficient (but not necessary) conditions designed for proving the existence of an equilibrium, a state which does not evolve, not for governing viable evolutions.