G. Vermandel Ph.D. Candidate in Economics

Dynamic optimization : an example via Hansen’s Model

The purpose of this post is to explain the different ways to solve problems of intertemporal optimization. We take here the example of the famous model of Hansen that belongs to the class of real business cycle models. A brief explanation is necessary before attacking directly the maximization under constraints.
The agent that maximizes utility:

(1)  

with the consumption and the leisure, so with the labor. The agent divides his time between work and leisure. Empirically, we set the working time to eight hours per week, or 1 / 3 of a day.
La fonction d'utilité spécifique utilisée est :

with , it was linearized by log.
Moreover, the production function is Cobb-Douglas technology with stochastic form,

(2)  

with is the stochastic variable of technology following the autoregressive random process :

for to make process stable.
Capital accumulation is also a constraint for the household, it is:

(3)  

And the feasibility constraint is such that the application can not exceed supply:

(4)  

In fact, the impossibility of stock equivalent to writing .
The point of indifference between factors of production involves a payment to the marginal productivity of factors. For capital, its performance is:

(5)  

and wage,

(6)  

Factors of production are bound by,

et used to simplify the first-order conditions. The household can now maximize its utility (1) under three constraints (2), (3) and (4),

The household optimise his quantity of work and capital in order to get the maximum utility. However, the capital needs one period to take effect, for this reason why optimization deals with instead of . Once maximized and the constraints satisfied, we obtain the first-order conditions,

These two equations are used to define the optimal path that will maximize the household utility. Economists commonly use three tools to obtain these two equations,

• Partial derivatives,
• Dynamic Lagrangian,
• Value function.