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154 CHAPTER 5. INTERNATIONAL REAL BUSINESS CYCLESconcavity of the production function, optimality requires that the worldcapital stock be split up between the two countries so as to equate themarginal product of capital at home and abroad. Since technology isidentical in the 2 countries, this implies equalization of national capitalstocks, k = k ∗ , and income levels y = y ∗ , even if consumption differs,c 6= c ∗ .Quadratic ApproximationYou can solve the model by taking the quadratic approximation of theunconstrained objective function about the steady state. Let R be theperiod weighted average of home and foreign utilityR(λ t )=ωU[g(λ t )] + (1 − ω)U[h(λ t )].Let R j = ωU c (c)g j +(1− ω)U c (c ∗ )h j , j =1,...,7betheÞrst partialderivative of R with respect to the j−the element of λ t . Denote thesecond partial derivative of R byR jk = ∂R(λ) = ω[U c (c)g jk +U cc g j g k ]+(1−ω)[U c (c ∗ )h jk +U cc (c ∗ )h j h k ].∂λ j ∂λ k(5.40)Let q =(R 1 ,...,R 7 ) 0 be the gradient vector, Q be the Hessian matrixof second partial derivatives whose j, k−th element is Q jk =(1/2)R j,k .Then the second-order Taylor approximation to the period utility functionisR(λ t )=[q +(λ t − λ) 0 Q](λ t − λ),and you can rewrite (5.36) asX ∞max E t β j [q +(λ t+j − λ) 0 Q](λ t+j − λ). (5.41)j=0Let Q j• be the j−th row of the matrix Q. TheÞrst-order conditionsare(k t+1 ): 0=R 1 + βR 3 + Q 1• (λ t − λ)+βQ 3• (λ t+1 − λ), (5.42)(kt+1) ∗ : 0=R 2 + βR 4 + Q 2• (λ t − λ)+βQ 4• (λ t+1 λ), (5.43)(c ∗ t ): 0=R 7 + Q 7• (λ t − λ). (5.44)