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5.2. CALIBRATING A TWO-COUNTRY MODEL 153Next, transform the constrained optimization problem into an unconstrainedproblem by substituting (5.34) and (5.35) into (5.27). Theproblem is now to maximize³ωE t u[g(λt )] + βU[g(λ t+1 )] + β 2 U[g(λ t+2 )] + ···´(5.36)³+(1 − ω)E t u[h(λt )] + βU[h(λ t+1 )] + β 2 U[h(λ t+2 )] + ···´.At date t, the choice variables available to the planner are k t+1 ,k ∗ t+1,and c ∗ t . Differentiating (5.36) with respect to these variables and rearrangingresults in the Euler equationsγU c (c t ) = βE t U c (c t+1 )[g 3 (λ t+1 )], (5.37)γU c (c t ) = βE t U c (c t+1 )[g 4 (λ t+1 )], (5.38)U c (c t ) = [(1− ω)/ω]U c (c ∗ t ). (5.39)(5.39) is the Pareto—Optimal risk sharing rule which sets home marginalutility proportional to foreign marginal utility. Under log utility, homeand foreign per capita consumption are perfectly correlated,c t =[ω/(1 − ω)]c ∗ t .The Two-Country Steady StateFrom (5.37) and (5.38) we obtain y/k = y ∗ /k ∗ =(γ/β+δ−1)/α. We’vealready determined that c =[ω/(1 − ω)]c ∗ = ωc w where c w = c + c ∗is world consumption. From the production functions (5.28)—(5.29) weget k =(y/k) 1/(α−1) and k ∗ =(y ∗ /k ∗ ) 1/(α−1) . From (5.30)—(5.31) weget i = i ∗ =(γ + δ − 1)k. It follows that c = ωc w = ω[y + y ∗ − (i + i ∗ )]=2ω[y − i].Thus y − c − i =(1− 2ω)(y − i) and unless ω =1/2, the currentaccount will not be balanced in the steady state. If ω > 1/2 thehomecountry spends in excess of GDP and runs a current account deÞcit.How can this be? In the market (competitive equilibrium) interpretation,the excess absorption is Þnanced by interest income earned on pastlending to the foreign country. Foreigners need to produce in excess oftheir consumption and investment to service the debt. In a sense, theyhave ‘over-invested’ in physical capital.In the planning problem, the social planner simply takes away someof the foreign output and gives it to domestic agents. Due to the