How much economic activity happens? (Reprise)

The question came up once before on the blog. One way of answering it is to say that supply and demand will be equalised, so that the quantity of goods supplied is such that buyers and sellers agree on a common price. Equivalently, we could look at pricing in terms of marginal costs and marginal revenues. Or we could draw a supply and demand diagram, which would give more information because it shows the way in which adjustment to the equilibrium occurs.

None of the stated answers describes the full dynamics of the situation. A supply and demand diagram for example, together with knowledge about the presence of a market mechanism, does not tell us how long the adjustment will take. The set of supply and demand equations would have to be supplemented by an equation giving the market adjustment process over time when out of equilibrium, and the evolution of the equilibrium over time.

Compressing the set of equations into a compact form may yield analytical benefits. Representation in terms of maximisation or minimisation of an integral term seems promising, as it puts Lagrangian theory at our disposal.

Demonstrating the equivalence of the various forms of supply and demand interaction requires some algebraic microeconomic theory, but doesn’t appear intractable. To show the equivalence of integral minimisation to a supply and demand equalisation, it may also be possible to borrow a proof from physical science. The integral minimisation associated with energy conservation can be shown to be equivalent to the equalisation of motion forces and gravitational forces at each point in time, so the proof should translate.