A pitchfork bifurcation in the tatonnement process

This note extends the example of Gale (1963) by considering the continuous time tatonnement process for a class of two agent, two commodity exchange economies, parametrized by a number µ∈(0,1). We demonstrate that as the parameter passes a threshold value µ* the unique, globally stable competitive equilibrium loses local stability while two new locally stable equilibria appear. Intuitively, as µ increases the income effect become increasingly more important relative to substitution effect, and eventually overwhelms the latter. As the parameter µ approaches 1, the economy tends to the example considered by Gale, as does the limiting behavior of the tatonnement.