[latexpage]The purpose of this post is to explain the different ways of solving intertemporal optimization problems. We consider the example of the model of Hansen that belongs to the class of real business cycle models (RBC). A brief explanation of the model’s structure is necessary before attacking directly the maximization problem under constraints.
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,