Proposition 2 The exponential moving average in eqs. (7)-(8) can be rewritten as
Proof. See the Appendix for a proof. Substitution of eq. (17) into eq. (6) yields
which can be approximated by
if we assume that ω > 1, which means that the long-period moving average more strongly depends on recent exchange rates than when 0 < ω< 1 holds. The approximation in eq. (19) makes the model more tractable to analyze. (The approximation can be motivated by the fact that it is a truncated Taylor series, and that the mathematics of catastrophe theory (i.e., the theory that deals with the dynamics when at least one of the variables make discontinuous changes, like the money supply in this letter encourages us to work with cubic Taylor series without bothering with the remainder.)
Proposition 3 When the expectations of the chartists is described by eq. (19), the model is characterized by saddle-path stability.
By Mikael Bask
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