A fundamental assumption in most of economic modeling is that people maximize their utility subject to a budget constraint. This, as well as many other economic problems, mathematically translate into problems of maximization with constraints. A powerful and widely used method to tackle some of these problems is the method of Lagrange multipliers. Yet, the exposition of such method in standard textbooks is rather formal and utilitarian. In this paper we try to present it emphasising the fundamental intuitions behind the method.