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6.2. RATIONAL RISK PREMIA 173The intertemporal marginal rate of substitution will play a key role.In aggregate asset-pricing applications, it is common to work with percapita consumption data. One way to justify using such data in theutility function in Lucas’s two-country model is to assume that the periodutility function is homothetic and that the relative price betweenthe home good and the foreign good (the real exchange rate) is constant.This allows you to write the representative agent’s intertemporalmarginal rate of substitution between t and t +1asµ t+1 = βu0 (C t+1 ), (6.10)u 0 (C t )where u 0 (C t ) is the representative agent’s marginal utility evaluated atequilibrium consumption. 3Let P t be the domestic price level and let β isthesubjectivediscountfactor. A speculative position in a forward contract requires noinvestment at time t. If the agent is behaving optimally, the expectedmarginal utility from the real payoff from buying the foreign currencyforward is E t [u 0 (c t+1 )(F t − S t+1 )/P t+1 ] = 0. To express the Euler equationin terms of stationary random variables so that their unconditionalvariances and unconditional covariances between random variables exist,multiply both sides by β and divide by u 0 (c t )toget"#F t − S t+1E t µ t+1 . =0, (6.11)P t+1(6.11) is key to understanding the demand for forward foreign exchangerisk-premia in the intertemporal asset pricing framework. Keep in mindthat the intertemporal marginal rate of substitution varies inverselywith consumption growth so that when the agent experiences the goodstate, consumption growth is high and the intertemporal marginal rateof substitution is low.3 If the period utility function in Lucas’s two-good model isu(C t )= C1−γ twith C1−γ t = Cxt θ C1−θ yt the intertemporal marginal rate of substitutionis β(C t+1 /C t ) 1−γ (C xt /C xt+1 ). But if the relative price between X and Y isconstant, the growth rate of consumption of X is the same as the growth rate ofthe consumption index and the intertemporal marginal rate of substitution becomesthat in (6.10)