这是一份stanford斯坦福大学ECON 242作业代写的成功案
The consumer faces the following period-by-period budget constraint:
$$
A_{t+1}=\left(1+r_{t}\right)\left(A_{t}+y_{t}+w_{t} n_{t}-c_{t}\right)
$$
The Bellman equation for the problem is:
$$
V\left(A_{t}\right)=\max u\left(c_{t}, n_{t}\right)+\beta V\left(\left(1+r_{t}\right)\left(A_{t}+y_{t}+w_{t} n_{t}-c_{t}\right)\right)
$$
The $\mathrm{FOCs}$ for the problem read:
$$
\begin{aligned}
\lambda_{t} &=\beta\left(1+r_{t}\right) V_{t+1}^{\prime}\left(A_{t+1}\right) \
u_{n}\left(c_{t}, n_{t}\right) &=w \lambda_{t}
\end{aligned}
$$
By envelope $\lambda_{t}=V^{\prime}\left(A_{t}\right)$. Therefore, the conditions become:
$$
\begin{aligned}
\lambda_{t} &=\beta\left(1+r_{t}\right) V_{t+1}^{\prime}\left(\lambda_{t+1}\right) \
u_{n}\left(c_{t}, n_{t}\right) &=\lambda_{t} w_{t}
\end{aligned}
$$
ECON 242COURSE NOTES ：
The Lagrangian for the problem reads:
$$
L=[u(z-T(z))+\lambda T(z)] h(z)
$$
Where $\lambda$ is constant across individuals and measures the value of government revenues in equilibrium. The optimal choice of $T(z)$ delivers the following first order condition:
$$
\frac{\partial L}{\partial T(z)}=\left[-u^{\prime}(z-T(z))+\lambda\right] h(z)=0
$$
Rearranging:
$$
u^{\prime}(z-T(z))=\lambda
$$