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112 CHAPTER 4. THE LUCAS MODELcomplete insurance of idiosyncratic risk is for each individual to ownhalf of the domestic Þrm and half of the foreign Þrm 3ω xt = ω ∗ xt = ω yt = ω ∗ yt = 1 2 . (4.21)We call this a ‘pooling’ equilibrium because the implicit insurancescheme at work is that agents agree in advance that they will pooltheir risk by sharing the realized output equally.The solution under constant relative-risk aversion utility. Let’s adopta particular functional form for the utility function to get explicit solutions.We’ll let the period utility function be constant relative-riskaversion in C t = c θ xtcyt1−θ , a Cobb-Douglas index of the two goodsThenu(c x ,c y )= C1−γ t1 − γ . (4.22)u 1 (c xt ,c yt )= θC1−γ t,c xt(1 − θ)C1−γ tu 2 (c xt ,c yt )= .c ytand the Euler equations (4.6)—(4.13) becomeq t = 1 − θ x t, (4.23)θ y t" µCt+1 e (1−γ)Ãt= βE t 1+ e !#t+1, (4.24)x t x t+1C te ∗ " µCt+1t= βE tq t y t C t (1−γ)Ã!#1+ e∗ t+1. (4.25)q t+1 y t+1(79)⇒From (4.23) the real exchange rate q t is determined by relative outputlevels. (4.24) and (4.25) are stochastic difference equations in the ‘pricedividend’ratios e t /x t and e ∗ t /(q t y t ). If you iterate forward on them as3 Actually, Cole and Obstfeld [31]) showed that trade in goods alone are sufficientto achieve efficient risk sharing in the present model. These issues are dealt with inthe end-of-chapter problems.