In Part I of this book we have shown how the motion of an economy is described by a differential equation whose solution depends in an essential way on three kinds of hypotheses: the structure of the production process, which includes some form of technical progress, the saving and investment decisions made by society, and the growth of the labour force. In particular, to any savings-investment hypothesis corresponds a particular trajectory of income per person.

The natural question to ask now is the following: among all possible growth paths that would result from society's savings-investment decisions, is there one that would be optimal? The answer to this question requires that we first define an optimality criterion. Once this is done, we need to develop the tools necessary to solve such problems. This is the object of the second part of this book. We will provide an introduction to the calculus of variations and the Pontryagin maximum principle. Care will be taken to give economic interpretations of each, i.e. intuitive ways of obtaining their fundamental equations.