The modified golden rule of a multi-sector economy

We consider an n-good model of optimal accumulation determined by a technology, a utility function, and a discount factor. A technology is δ-productive if it contains an input-output pair such that the discounted output vector strictly dominates the input vector. We show that a δ-productive technology has a non-trivial modified golden rule. We also report a counter- example of David Starrett showing that the modified golden rule need not have turnpike properties, and that there may exist non-trivial periodic optimal consumption plans even when there is no non-trivial modified golden rule.