Exchange rates, tradables and non-tradables

I am reading a recent review of evidence on exchange rate fluctuations and the links to prices of internationally tradable goods and non-tradable goods. The evidence shows that exchange rate changes are closely linked to the variations in both tradable and non-tradable good prices domestically. Earlier theorists proposed that only tradable good prices would be closely linked with exchange rate fluctuations, since exchange rates are determined in part by purchases of currencies in order to buy tradable goods, and evidently not purchases of non-tradable goods.

Not having reached the end of the review yet, I haven't read the author's proposed theoretical resolution of the difficulty, but it does seem a priori that the proposition of no link between non-tradable prices and exchange rates is theoretically clearly flawed. One would expect the prices of tradable goods and non-tradable goods to operate within bands relative to each other that are reasonably similar even in entirely separate economies if the production functions and inputs for the goods are similar. So in economies where the exchange rates are determined by tradable prices, they will also be linked to non-tradable prices.

The review may propose this mechanism too. I'll find out today. In theoretical work, the effort is often in the exact mathematical modelling rather than the basic ideas.