Monotonic Saddle-Path Dynamics

Alan G. Isaac

Department of Economics, American University, Washington, DC 20016

Abstract

The paper provides necessary and sufficient conditions for the existence of a monotonic saddle path in second-order difference equations. The conditions are illustrated with three popular exchange-rate models. Keywords: saddle path, difference equations JEL classification: C0, E0, F3

Requests

The paper is forthcoming in Economics Letters, and I am happy to provide it as correspondence. Please contact me at aisaac@american.edu. A synopsis of the paper follows.

Monotonic Saddle Paths

Standard discussions of difference equations concentrate on stability conditions, which are therefore well-known (e.g., Sargent 1979). However, economists are often more interested in dynamic systems displaying “saddle-path stability”. Consider the following linear, second-order difference equation with forcing function f_{t} and constant real coefficients b and c.

(F^{2} + bF + c)x_{t} = f_{t}  (1)

Claim:

The difference equation (1) has a monotonic saddle path iff either c > 0 and 0 > 1 + b + c, or c < 0, 1 + b + c > 0, and 1 − b + c < 0.

Proof: The proof is contained in the paper, which is available from the author.

Applications to Exchange Rates

Consider the partial adjustment approach of Woo (1985), the sticky-price approach of Dornbusch (1976) as implemented in Wickens (1985), and the portfolio effects approach as implemented by Driskill, Mark, and Sheffrin (1992). These models produce the following characteristic polynomials.

Characteristic Polynomials for Three Studies

Woo (1985)  F^{2} − (1 + \frac{1}{β} ) F + \frac{α}{β}
Wickens (1985)  F^{2} − (1 + μ + \frac{θ}{λ}) F + μ
Driskill, Mark, and Sheffrin (1992)  F^{2} − (2 + \frac{α}{η}) F + 1

All parameters are as defined in the original studies, and theory assigns them positive values. As a result, it is immediate that the Wickens model and the Driskill, Mark, and Sheffrin model are saddle-path stable. In the Woo model, this result follows from the magnitude of α: it is a partial adjustment parameter, so α ∈ (0, 1). This means Woo (p.4) did not need to assume the existence of a saddle path in his model: the model structure implies it.

Conclusion

If natural restrictions are sufficient to the needs of the analyst, this can