Consumer's problem \begin{equation} \max \sum_{t}\beta^{t}[c_{t}-1/2(1-x_{t})^{2}], \end{equation} \begin{equation} \ s.t. c_{t}+q_{t}b_{t+1} \leq (1-\tau_{t})(1-x_{t})+b_{t}, \end{equation} where c=consumption, x=leisure(1-x=labor), $\tau$=labor income tax, b_{t+1}=the gov's bond, which is sold in period t at price q_{t}. Time endowment is normalized to 1.
\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} First order conditions \begin{equation} w.r.t. c_{t}: \beta^{t}-\lambda=0 \end{equation} \begin{equation} w.r.t. x_{t}: -\beta^{t}(1-x_{t})(-1)-\lambda(1-\tau_{t})(-1)=0 \end{equation} \begin{equation} w.r.t. b_{t}: ? \end{equation}