The objective function of a representative household is to maximize the sum of the expected utility,

\begin{equation}\label{eq:MaxUtility}

\max \sum\nolimits_{0}^{\infty }\beta ^{t}u\left( c_{t},l_{t}\right)

\end{equation}

where $c_{t}$ is the consumption, $l_{t}$ the leisure, so $l_{t}=1-h_{t}$ with $h_{t}$ the hours worked. The agent divides his time (normalized to 1) between work and leisure. Empirically, we set the working time to eight hours per week, or 1 / 3 of a day. Moreover, $\beta ^{t}$ is the discount factor so that expected utlity in $t+2$ is lower than in $t+1$.

The functional form of the utility function is thus given by,

\[

u(c_{t},1-h_{t})=\ln c_{t}+A\ln (1-h_{t})

\] with $A>0$ a rescale parameter.

Moreover, the production function is Cobb-Douglas technology with stochastic form,

\begin{equation}\label{eq:Contrainte1}

f\left( \lambda _{t},k_{t},h_{t}\right) =\lambda _{t}\left( k_{t}\right)

^{\theta }\left( h_{t}\right) ^{1-\theta }

\end{equation}

where $\lambda _{t}$ is the stochastic variable of technology following the autoregressive random process :

\[

\lambda _{t+1}=\gamma \lambda _{t}+\varepsilon _{t+1}

\] for $0<\gamma <1$ to make process stable. Capital accumulation is also a constraint for the household, it is: \begin{equation}\label{eq:Contrainte2} k_{t+1}=\left( 1-\delta \right) k_{t}+i_{t} \end{equation} And the feasibility constraint is such that the application can not exceed supply: \begin{equation}\label{eq:Contrainte3} f\left( \lambda _{t},k_{t},h_{t}\right) \geq c_{t}+i_{t} \end{equation} In fact, the impossibility of stock equivalent to writing $f\left( \lambda _{t},k_{t},h_{t}\right) =c_{t}+i_{t}$. The point of indifference between factors of production involves a payment to the marginal productivity of factors. For capital, its performance is: \begin{eqnarray} r_{t} &=&\frac{\partial f\left( \lambda _{t},k_{t},h_{t}\right) }{\partial k_{t}} \\ &=&\theta \lambda _{t}\left( \frac{k_{t}}{h_{t}}\right) ^{\theta -1} \end{eqnarray} and wage, \begin{eqnarray} w_{t} &=&\frac{\partial f\left( \lambda _{t},k_{t},h_{t}\right) }{\partial h_{t}} \\ &=&\left( 1-\theta \right) \lambda _{t}\left( \frac{k_{t}}{h_{t}}\right) ^{\theta } \end{eqnarray} Factors of production are bound by, \[ \theta w_{t}h_{t}=\left( 1-\theta \right) k_{t}r_{t} \] $r_{t}$ et $w_{t}$ used to simplify the first-order conditions. The household can now maximize its utility (\ref{eq:MaxUtility}) under three constraints (\ref{eq:Contrainte1}), (\ref{eq:Contrainte2}) and (\ref{eq:Contrainte3}), \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 household optimizes his quantity of work $h_ {t}$ and capital $k_ {t+1}$ in order to get the maximum utility. However, the capital needs one period to take effect, for this reason why optimization deals with $k_ {t +1}$ instead of $k_ {t}$. Once maximized and the constraints satisfied, we obtain the 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*} These two equations are used to define the optimal path that will maximize the household utility. Economists commonly use three mathematic tools to obtain these two previous equations,