On the specification of the asset evolution equation in consumption models

Dynamic programming is often used by researchers to find the first-order Euler relation when the discounted utility of an infinite stream of consumption is maximized, subject to one or more constraints. Although the consumption model is the same in most studies, the specification of the constraint, that is, the asset evolution equation, differs from author to author. In fact there are two general formulations of this equation. The purpose of this article is to check whether these two different formulations alter the first-order conditions. The surprising but reassuring result is that they do not. This result holds for both discrete and continuous-time variables. This article also solves the Euler relations for two customary functional forms of the utility.