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5.1. CALIBRATING THE ONE-SECTOR GROWTH MODEL 143Letting g j = ∂c t /∂λ jt be the partial derivative of g(λ t ) with respectto the j−th element of λ t and g ij = ∂ 2 c t /(∂λ it ∂λ jt ) be the secondcross-partial derivative, for future reference you haveg 1 = −γ,g 2 = f k (A, k)+(1− δ),g 3 = y/A,g 11 = g 12 = g 21 = g 13 = g 31 = g 33 =0,g 22 = f kk (A, k),g 23 = g 32 = αk α−1 .Now substitute (5.11) into (5.6) to transform the constrained optimizationproblem into an unconstrained problem. You want to maximizeE t ∞ Xj=0β j u[g(λ t+j )], (5.12)where g(λ t ) is given in (5.11). At date t, k t is pre-determined and theonly choice variable is i t and choosing i t is equivalent to choosing k t+1 .The Þrst-order conditions for all t are−γu c (c t )+βE t u c (c t+1 )[f k (A t+1 ,k t+1 )+(1− δ)] = 0. (5.13)Notice that c t must obey the consolidated budget constraint (5.11). Itfollows that (5.13) is a nonlinear stochastic difference equation in k t .Analytic solutions to such equations are not easy to obtain so we resortto approximation methods.The Steady StateWe will compute the approximate solution around the model’s steadystate. InordertodothatweneedÞrst to Þnd the steady state. Denotesteady state values of output, consumption, investment and capitaly, c, i, k without the time subscript and let the steady state value ofA =1.Since f k = αk α−1 = α(y/k), (5.13) becomes γ = β[α(y/k) +(1 − δ)] from which we obtain the steady state output to capital ratio