Optimization with partial derivatives

The method of resolution via partial derivative solves only simple maximization of intertemporal optimization programs. The Hansen's program is,

$\begin{equation*} \left\{ \begin{array}{c} \max_{k_{t+1},h_{t}}\sum\nolimits_{0}^{\infty }\beta ^{t}\left( \ln c_{t}+A\ln \left( 1-h_{t}\right) \right) \\ sc:f\left( \lambda _{t},k_{t},h_{t}\right) =\lambda _{t}\left( k_{t}\right) ^{\theta }\left( h_{t}\right) ^{1-\theta } \\ sc:k_{t+1}=\left( 1-\delta \right) k_{t}+i_{t} \\ sc:f\left( \lambda _{t},k_{t},h_{t}\right) \geq c_{t}+i_{t}% \end{array}% \right. \end{equation*}$

The constraints' satisfaction allows us to rewrite, $\omega$ , the objective function,

$\begin{eqnarray*} \omega &=&\sum\beta ^{t}\left( \ln c_{t}+A\ln \left( 1-h_{t}\right) \right) \\ &=&\sum\beta ^{t}\left( \ln \left( f\left( \lambda _{t},k_{t},h_{t}\right) -i_{t}\right) +A\ln \left( 1-h_{t}\right) \right) \\ &=&\sum\beta ^{t}\left( \ln \left( \lambda _{t} k_{t} ^{\theta }h_{t} ^{1-\theta }-k_{t+1}+\left( 1-\delta \right) k_{t}\right) +A\ln \left( 1-h_{t}\right) \right) \end{eqnarray*}$

Thus, the optimization involves the quantities $k_{t+1}$ et $h_{t}$ ,

$\begin{eqnarray*} FOC &:&\frac{\partial \omega }{\partial k_{t+1}}=0 \\ &\Leftrightarrow &\frac{-\beta ^{t+1}}{\lambda _{t}k_{t}^{\theta }h_{t}^{1-\theta }-k_{t+1}+\left( 1-\delta \right) k_{t}}+E_{t}\left\{ \frac{\beta ^{t+2}\left( \theta \lambda _{t+1}k_{t+1}^{\theta -1}h_{t+1}^{1-\theta }+\left( 1-\delta \right) \right) }{\lambda _{t+1}k_{t+1}^{\theta }h_{t+1}^{1-\theta }-k_{t+2}+\left( 1-\delta \right) k_{t+1}}\right\} =0 \\ &\Leftrightarrow &\beta E_{t}\left\{ \frac{\theta \lambda _{t+1}k_{t+1}^{\theta -1}h_{t+1}^{1-\theta }+\left( 1-\delta \right) }{% \lambda _{t+1}k_{t+1}^{\theta }h_{t+1}^{1-\theta }-k_{t+2}+\left( 1-\delta \right) k_{t+1}}\right\} =\frac{1}{\lambda _{t}k_{t}^{\theta }h_{t}^{1-\theta }-k_{t+1}+\left( 1-\delta \right) k_{t}} \\ &\Leftrightarrow &\beta E_{t}\left\{ \frac{r_{t+1}+\left( 1-\delta \right) }{% c_{t+1}}\right\} =\frac{1}{c_{t}} \end{eqnarray*}$

Concerning $h_{t}$ ,

$\begin{eqnarray*} FOC &:&\frac{\partial \omega }{\partial h_{t}}=0 \\ &\Leftrightarrow &\beta ^{t}\left( \frac{\left( 1-\theta \right) \lambda _{t}k_{t}^{\theta }h_{t}^{-\theta }}{\lambda _{t}k_{t}^{\theta }h_{t}^{1-\theta }-k_{t+1}+\left( 1-\delta \right) k_{t}}+\frac{-A}{1-h_{t}}% \right) =0 \\ &\Leftrightarrow &\left( 1-\theta \right) \lambda _{t}\left( \frac{k_{t}}{% h_{t}}\right) ^{\theta }\left( 1-h_{t}\right) =A\left( \lambda _{t}k_{t}^{\theta }h_{t}^{1-\theta }-k_{t+1}+\left( 1-\delta \right) k_{t}\right) \\ &\Leftrightarrow &w_{t}\left( 1-h_{t}\right) =Ac_{t} \end{eqnarray*}$

We find out our first-order conditions,

$\begin{equation*} \left\{ \begin{array}{c} \beta E_{t}\left\{ \frac{r_{t+1}+\left( 1-\delta \right) }{c_{t+1}}\right\} =% \frac{1}{c_{t}} \\ w_{t}\left( 1-h_{t}\right) =Ac_{t}% \end{array}% \right. \end{equation*}$

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G. Vermandel Ph.D. Candidate in Economics

Optimization with partial derivatives

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