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Multiplicador lagrangiano
max∑tβt[ct−1/2(1−xt)2],max∑tβt[ct−1/2(1−xt)2],\begin{equation} \max \sum_{t}\beta^{t}[c_{t}-1/2(1-x_{t})^{2}], \end{equation} s.t.ct+qtbt+1≤(1−τt)(1−xt)+bt, s.t.ct+qtbt+1≤(1−τt)(1−xt)+bt,\begin{equation} \ s.t. c_{t}+q_{t}b_{t+1} \leq (1-\tau_{t})(1-x_{t})+b_{t}, \end{equation}ττ\tau L=∑tβt[ct−1/2(1−xt)2]+λ[ct+qtbt+1−bt−(1−τt)(1−xt)] L=∑tβt[ct−1/2(1−xt)2]+λ[ct+qtbt+1−bt−(1−τt)(1−xt)]\begin{equation} \ L=\sum_{t}\beta^{t}[c_{t}-1/2(1-x_{t})^{2}]+\lambda[c^{t}+q_{t}b_{t+1}-b_{t}-(1-\tau_{t})(1-x_{t})] \end{equation} Condições de primeira ordem w.r.t.ct:βt−λ=0w.r.t.ct:βt−λ=0\begin{equation} w.r.t. c_{t}: \beta^{t}-\lambda=0 \end{equation} w.r.t.xt:−βt(1−xt)(−1)−λ(1−τt)(−1)=0w.r.t.xt:−βt(1−xt)(−1)−λ(1−τt)(−1)=0\begin{equation} w.r.t. x_{t}: -\beta^{t}(1-x_{t})(-1)-\lambda(1-\tau_{t})(-1)=0 \end{equation} w.r.t.bt:?w.r.t.bt:?\begin{equation} w.r.t. b_{t}: ? \end{equation}