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228 CHAPTER 7. THE REAL EXCHANGE RATEProblems1. (Heterogeneous commodity baskets). Suppose there are two goods,both of which are internationally traded and for whom the law of oneprice holds,p 1t = s t + p ∗ 1t, p 2t = s t + p ∗ 2t,where p i is the home currency price of good i, p ∗ i is the foreign currencyprice, and s is the nominal exchange rate, all in logarithms. Assumefurther that the nominal exchange rate follows a unit-root process,s t = s t−1 + v t where v t is a stationary process, and that foreign pricesare driven by a common stochastic trend, zt∗p ∗ 1t = z ∗ t + ² ∗ 1t p ∗ 2t = z ∗ t + ² ∗ 2t.where z ∗ t = z ∗ t−1 + u t, ² ∗ it , (i = 1, 2) are stationary processes, and u t isiid with E(u t )=0,E(u 2 t )=σ 2 u. Show that even if the price levels areconstructed as,p t = φp 1t +(1 − φ)p 2t , p ∗ t = φ ∗ p ∗ 1t +(1 − φ ∗ )p ∗ 2t,with φ 6= φ ∗ ,thatp t − (s t + p ∗ t ) is a stationary process.