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7.3. LONG-RUN DETERMINANTS OF THE REAL EXCHANGE RATE213What About the Long-Run?Since the international law-of-one price and purchasing-power parityhas Þrmly been shown to break down in the short run, the next stepmight be to ask whether purchasing-power parity holds in the long run.Recent work on this issue proceeds by testing for a unit root in the logreal exchange rate. The null hypothesis in popular unit-root tests isthat the series being examined contains a unit root. But before we jumpin we should ask whether these tests are interesting from an economicperspective. In order for unit-root tests on the real exchange rate tobe interesting, the null hypothesis (that the real exchange rate has aunit root) should have a Þrm theoretical foundation. Otherwise, if wedo not reject the unit root, we learn only that the test has insufficientpower to reject a null hypothesis that we know to be false, and if wedo reject the unit root, we have only conÞrmed what we believed to betrue in the Þrst place.The next section covers the Balassa-Samuelson model which providesa theoretical justiÞcationforPPPtobeviolatedeveninthelongrun.7.3 Long-Run Determinants of the RealExchange RateWe study a two-sector small open economy. The sectors are a tradablegoodssector and a nontradable-goods sector. The terms of trade (therelative price of exports in terms of imports) are given by world conditionsand are assumed to be Þxed. Before formally developing themodel, it will be useful to consider the following sectoral decompositionof the real exchange rate.Sectoral Real Exchange Rate DecompositionLet P T be the price of the tradable-good and P N be the price of the ⇐(129)nontradable-good, and let the general price level be given by the Cobb-Douglas formP =(P T ) θ (P N ) 1−θ , (7.3)