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7.3. LONG-RUN DETERMINANTS OF THE REAL EXCHANGE RATE215economy takes the price of traded goods as given. We’ll set P T =1. Ris the rental rate on capital, W is the wage rate, and P N is the price ofnontraded goods, all stated in terms of the traded good.Competitive Þrms take factor and output prices as given and chooseK and L to maximize proÞts. The intersectoral mobility of labor andcapital equalizes factor prices paid in the tradable and nontradablesectors. The tradable-good Þrm chooses K T and L T to maximize proÞtsA T L (1−α T )T K α TT − (WL T + RK T ). (7.8)The nontradable-good Þrm’s problem is to choose K N and L N to maximizeP N A N L (1−α N )N K α NN − (WL N + RK N ). (7.9)Let k ≡ (K/L) denote the capital—labor ratio. It follows from theÞrst order conditionsR = A T α T (k T ) α T −1 , (7.10)R = P N A N α N (k N ) α N −1 , (7.11)W = A T (1 − α T )(k T ) α T, (7.12)W = P N A N (1 − α N )(k N ) α N. (7.13)The international mobility of capital combined with the small countryassumption implies that R is exogeneously given by the world rentalrate on capital. (7.10)-(7.13) form four equations in the four unknowns(P N ,W,k T ,k N ).To solve the model, Þrst obtain the traded-goods sector capital-laborratio from (7.10)¸ 1·αT A T(1−α T )k T =. (7.14)RNext, substitute (7.14) into (7.12) to get the wage rateW =(1− α T )(A T )1(1−α T )Substituting (7.15) into (7.13), you getk N =⎛⎜⎝(1 − α T ) A(1 − α N )1(1−α T )T¸ ·αTαT1−α T. (7.15)R³ αTR´ αT1−α TP N A N⎞⎟⎠1α N. (7.16)