Stability of money: phase transitions in an Ising economy

The stability of money value is an important requisite for a functioning economy, yet it critically depends on the actions of participants in the market themselves. Here we model the value of money as a dynamical variable that results from trading between agents. The basic trading scenario can be recast into an Ising-type spin model and is studied on the hierarchical network structure of a Cayley tree. We solve this model analytically and observe a phase transition between a one-state phase, always allowing for a stable money value, and a two-state phase, where an unstable (inflationary) phase occurs. The onset of inflation is discontinuous and follows a first-order phase transition. The stable phase provides a parameter region where money value is robust and can be stabilized without fine tuning.