The classical dichotomy predicts that all of the time series variance in the aggregate real exchange rate is accounted for by non-traded goods in the CPI basket because traded goods obey the Law of One Price. In stark contrast, Engel (1999) found that traded goods had comparable volatility to the aggregate real exchange rate. Our work reconciles these two views by successfully applying the classical dichotomy at the level of intermediate inputs into the production of final goods using highly disaggregated retail price data. Since the typical good found in the CPI basket is about equal parts traded and non-traded inputs, we conclude that the classical dichotomy applied to intermediate inputs restores its conceptual value.