The purchasing power parity (PPP) puzzle refers to the inability to reconcile the high short-run volatility of exchange rates with the glacial speed at which deviations from parity seem to damp out. Despite this, there is strong evidence of the long-run relationship between exchange rates and international price differentials. More recently, the alternative notion of non-linear mean reversion dynamics of rates is attracting substantial attention: as a consequence of various adjustment costs, there is a sizeable buffer region within which exchange rates can move independently of prices, generating persistent deviations from parity. This study seeks to integrate the idea of this “inaction band” with trade costs. When deviations fall within the band, movements in rates are close to a random walk; but when the threshold of the band is crossed, arbitrage causes movements back toward the band. To test this approach, we use a large number of disaggregated and highly tradable agricultural products over time in a large number of countries. We find some evidence in favour of the approach.