Forecasting errors and bounded rationality: An example

We consider the problem of a decision maker (a 'firm') which has to try to 'match' a deterministic, infinite price sequence - produce one unit when the price is high and produce nothing when the price is low. The firm cannot observe the current price but can be interpreted as being fully informed about the 'model' which generates the price sequence. The firm bears a complexity cost of computing the price sequence. We show that for any complexity cost measure which satisfies some standard axioms, there exists a (computable) deterministic price sequence which forces the firm to take the wrong decision infinitely often. We also show that 'continuity at the limit' holds in our model: as the relative importance of the complexity cost declines, the firm's decision rule will converge (pointwise) to the rule which matches the price sequence in every period.