Beyond Econ 101: Modelling Supply and Demand Cycles as Coupled, Damped, Driven Harmonic Oscillators

The supply and demand curve (x) taught in introductory economics represents the aggregate state of the supply and demand for a particular good or service at a moment in time.

In the real world, supply and demand do not rise or fall with gentle de-accelerations or accelerations.  Rather, with demand held constant, as supply passes a certain price threshold, suppliers will stop producing the good or service, and begin producing a substitute that can be supplied at lower cost.  Similarly, with supply held constant, when demand reaches a certain floor of zero gains in value from increased quantity, demand will begin to rise in the market for a substitute as consumers begin to substitute lower cost substitutes.  Thus aggregate supply and demand will oscillate in time rather than remaining static.   Schumpeter was the first to observe  this effect when he observed how innovation in a market would lead price to decline as consumers switched from an old to a new good or service, even as supply for the old remained steady.

Schumpeter gives us a theory that explains how price will fluctuate in time.  If we next add in transactions costs (resistance), liquidity (inductance), and external money supply (capacitance) we have enough variables and enough understanding of their interaction to develop a fairly rich description of the observable behavior of price over time in various markets.

For example, supply and demand can seldom be approximated as decoupled from one another (and therefore constant with respect to one another).  The coupling of real supply and demand cycles is better approximated as two coupled, damped, driven harmonic oscillators.  The strength of the coupling will be a function of the "stray impedance" of the market, which arises from any coupling between the independent elements of the "circuits."  Usually, however, transactions costs are the mechanism for coupling.  Beyond a certain threshold, transactions costs will grow nonlinearly, and introduce the effects of non-linear coupling into the system.  When the coupling is weak, fluctuations in supply and demand will merely add or subtract linearly to produce a relatively steady price -- i.e., a steady-state market price that is observed in most markets, most of the time.

The mathematical model is simple.  Undergraduate physics majors could work out the dynamics on paper.  The static (x) model drops out by measuring each cycle within a window in time.

Here are a few predictions from the model.  When the coupling between the oscillators is weak, they'll simply superpose linearly -- i.e., one frequency will simply constructively and destructively interfere with the other.  I.e., the supply cycle will run smoothly regardless of whether demand is blowing up or stagnant; and vice-versa.  When the coupling is strong, however, non-linear effects emerge.  The frequency of the two cycles will "beat" and to produce "envelope frequency" = supply frequency + demand frequency, modulated at "beat frequency" = supply frequency - demand frequency.