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5.1. CALIBRATING THE ONE-SECTOR GROWTH MODEL 145where a 0 = U c g 1 + βU c g 2 =0,a 1 = βU cc g 1 g 2 ,a 2 = U cc g1 2 + βU cc g2 2 + βU c g 22 ,a 3 = U cc g 1 g 2 ,a 4 = βU c g 32 + βU cc g 2 g 3 ,a 5 = U cc g 1 g 3 .The derivatives are evaluated at steady state values.A second but equivalent option is to take a second—order Taylorapproximation to the objective function around the steady state and tosolve the resulting quadratic optimization problem. The second optionis equivalent to the Þrst because it yields linear Þrst—order conditionsaround the steady state. To pursue the second option, recall that λ t =(k t+1 ,k t ,A t ) 0 . Write the period utility function in the unconstrainedoptimization problem asR(λ t )=U[g(λ t )]. (5.20)Let R j = ∂R(λ t )/∂λ jt be the partial derivative of R(λ t ) with respectto the j−th element of λ t and R ij = ∂ 2 R(λ t )/(∂λ it ∂λ jt ) be the secondcross-partial derivative. Since R ij = R ji the relevant derivatives are,R 1 = U c g 1 ,R 2 = U c g 2 ,R 3 = U c g 3 ,R 11 = U cc g 2 1,R 22 = U cc g 2 2, +U c g 22R 33 = U cc g 2 3,R 12 = U cc g 1 g 2 ,R 13 = U cc g 1 g 3 ,R 23 = U cc g 2 g 3 + U c g 23 .The second-order Taylor expansion of the period utility function isR(λ t ) = R(λ)+R 1 (k t+1 − k)+R 2 (k t − k)+R 3 (A t − A)+ 1 2 R 11(k t+1 − k) 2