Lecture 2: Mathematical preliminaries
This lecture lays down the mathematical foundations for solving dynamic programming problems. First, we clarify that value function iteration is best understood as convergence of functions that requires appropriate spaces and appropriate operators. As convergence of functions is to be measured as distance between functions, we need a metric vector space--a space the elements of which are points (somehow understood) and the distance between points is measured with metrics. Such points may have various interpretations. To show this, we quickly get from a simple two-dimension Euclidean space to higher-dimension Euclidean spaces of points and, finally, to the space of infinite-dimension points in which points are understood as functions. At this stage we arrive at the space of continuous functions, so the distance between points is simply the distiance between functions--the convergence of functions is thus no more than a process in which the distance between two points is decreasing. Hence the convergence of functions can be understood and measured very intuitively. Then, having all this in hand, we use the Contraction Mapping Theorem to ensure convergence in value function iteration. We also use the Theorem of the Maximum to introduce such basic concepts as correspondence (to be used later as the feasible set for optimum choices) or continuity understood as lower and upper hemi-continuity of a correspondence. As an example, at the end of the lecture we examine the famous Cass-Koopmans growth model to see these concepts in operation. Title page:(0:00) Value function iteration:(0:11) On the convergence of functions:(7:06) The convergence of functions, demonstrated:(9:54) How to measure convergence?:(17:51) More on how to measure convergence:(24:44) How to ensure convergence?:(39:41) The theorem of the maximum:(51:16) The Cass-Koopmans growth model:(56:34)
This lecture lays down the mathematical foundations for solving dynamic programming problems. First, we clarify that value function iteration is best understood as convergence of functions that requires appropriate spaces and appropriate operators. As convergence of functions is to be measured as distance between functions, we need a metric vector space--a space the elements of which are points (somehow understood) and the distance between points is measured with metrics. Such points may have various interpretations. To show this, we quickly get from a simple two-dimension Euclidean space to higher-dimension Euclidean spaces of points and, finally, to the space of infinite-dimension points in which points are understood as functions. At this stage we arrive at the space of continuous functions, so the distance between points is simply the distiance between functions--the convergence of functions is thus no more than a process in which the distance between two points is decreasing. Hence the convergence of functions can be understood and measured very intuitively. Then, having all this in hand, we use the Contraction Mapping Theorem to ensure convergence in value function iteration. We also use the Theorem of the Maximum to introduce such basic concepts as correspondence (to be used later as the feasible set for optimum choices) or continuity understood as lower and upper hemi-continuity of a correspondence. As an example, at the end of the lecture we examine the famous Cass-Koopmans growth model to see these concepts in operation. Title page:(0:00) Value function iteration:(0:11) On the convergence of functions:(7:06) The convergence of functions, demonstrated:(9:54) How to measure convergence?:(17:51) More on how to measure convergence:(24:44) How to ensure convergence?:(39:41) The theorem of the maximum:(51:16) The Cass-Koopmans growth model:(56:34)