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108 CHAPTER 4. THE LUCAS MODELsubject to (4.3).You can transform the constrained optimum problem into an unconstrainedoptimum problem by substituting c xt from (4.3) into (4.4).Theobjectivefunctionbecomesu(ω xt−1 (x t + e t )+ω yt−1 (q t y t + e ∗ t ) − e t ω xt − e ∗ t ω yt − q t c yt ,c yt )+E t [βu(ω xt (x t+1 + e t+1 )+ω yt (q t+1 y t+1 + e ∗ t+1)(4.5)−e t+1 ω xt+1 − e ∗ t+1ω yt+1 − q t+1 c yt+1 ,c yt+1 )] + ···(77)⇒Let u 1 (c xt ,c yt )=∂u(c xt ,c yt )/∂c xt be the marginal utility of x-consumptionand u 2 (c xt ,c yt )=∂u(c xt ,c yt )/∂c yt be the marginal utility of y-consumption.Differentiating (4.5) with respect to c yt , ω xt ,andω yt , setting the resultto zero and rearranging yields the Euler equationsc yt : q t u 1 (c xt ,c yt )=u 2 (c xt ,c yt ), (4.6)ω xt : e t u 1 (c xt ,c yt )=βE t [u 1 (c xt+1 ,c yt+1 )(x t+1 + e t+1 )], (4.7)ω yt : e ∗ t u 1 (c xt ,c yt )=βE t [u 1 (c xt+1 ,c yt+1 )(q t+1 y t+1 + e ∗ t+1)]. (4.8)These equations must hold if the agent is behaving optimally. (4.6)is the standard intratemporal optimality condition that equates therelative price between x and y to their marginal rate of substitution.Reallocating consumption by adding a unit of c y increases utility byu 2 (·). This is Þnanced by giving up q t units of c x , each unit of whichcosts u 1 (·) units of utility for a total utility cost of q t u 1 (·). If the individualis behaving optimally, no such reallocations of the consumptionplan yields a net gain in utility.(4.7) is the intertemporal Euler equation for purchases of the domesticequity. The left side is the utility cost of the marginal purchaseof domestic equity. To buy incremental shares of the domestic Þrm, itcosts the individual e t units of c x , each unit of which lowers utility byu 1 (c xt ,c yt ). The right hand side of (4.7) is the utility expected to bederived from the payoff of the marginal investment. If the individualis behaving optimally, no such reallocations between consumption andsaving can yield a net increase in utility. An analogous interpretationholds for intertemporal reallocations of consumption and purchases ofthe foreign equity in (4.8).