Solving a Saddlepath Unstable Model with Complex-Valued Eigenvalues
Ric D. Herbert
This paper investigates the success of the well-known reverse-shooting and forward-shooting algorithms in finding stable solutions for linear macroeconomic models that both possess the particular property known as saddle-path instabiity and also have highly cyclic dynamic properties. It is anticipated that assessing how well these algorithms cope with solving highly cyclic models will also provide insights into how well they are likely to cope with solving non-linear models. In this paper, a perfect foresight version of the well-known Cagan (1956) model of the monetary dynamics of hyperinflation is augmented with a labour market. Additional eignevalues are then generated through sluggish adjustment mechanisms for wages and for the supply of labour. This process provides the simplest model with stable complex-valued eigenvalues and a saddlepath: a model with two stable complex-valued eigenvalues and one unstable real-valued eigenvalue. Using this model, it is possible to define an indexing parameter that, when varied, determines a range of values for the stable eignvalues: from (i) real-valued, to (ii) complex-valued with small absolute imaginary part, to (iii) complex-valued with large absolute imaginary part. This leads to corresponding stable time-paths for the model, which are (i) either humped or monotonic, (ii) cyclic but with infrequent cycles, and (iii) cyclic but with frequent cycles. This paper then compares the properties of solutions derived using the reverse-shooting and forward-shooting approaches as the magnitude of the indexing parameter (and hence of the cycles) is allowed to vary. In the highly oscillatory case, we show that the success of both approaches is crucially dependent on the choice of ODE solver and of parameters.
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