What are saddles?

In chapter 4 we discussed a variety of trajectories in the phase plane. But the one trajectory we did not discuss was the one associated with a saddle-path solution. Saddle-path solutions have entered economics in a major way. They became most popular when rational expectations became a major specification of expectations formation. In earlier chapters we have considered adaptive expectations. The simplest one of all was that individuals expected the price in the current period to be equal to the price last period. Even the more complex expectations that base current prices on the trend in prices are backward-looking. In other words, the current expected price depends in some way on past prices. Not only are they backward-looking, but they also take no account of the modelling of the variable in question. Rational expectations theory takes quite a different approach. For the moment we shall simply say that in a rational expectations model, expected prices take account of all information, including the model under investigation, and based on that information attempts to minimise any errors. If successful in doing this, it is like having perfect foresight. In this chapter we shall use the assumption of perfect foresight as our means of analysing rational expectations.

Rational expectations modelling is more complex than the models we have discussed so far. So in this chapter we shall do two things only. First we will consider some simple examples so we can see how to derive saddle-point solutions and investigate their stability/instability.