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Introduction to Modern Economic Growth problems, this is often made easier by using differential calculus The difficulty in using differential calculus with (6.1) is that the right hand side of this expression includes the value function V , which is endogenously determined We can only use differential calculus when we know from more primitive arguments that this value function is indeed differentiable The next theorem ensures that this is the case and also provides an expression for the derivative of the value function, which corresponds to a version of the familiar Envelope Theorem Recall that IntX denotes the interior of the set X and ∇x f denotes the gradient of the function f with respect to the vector x (see Mathematical Appendix) Theorem 6.6 (Differentiability of the Value Function) Suppose that Assumptions 6.1, 6.2, 6.3 and 6.5 hold Let π be the policy function defined above and assume that x0 ∈IntX and π (x0 ) ∈IntG (x0 ), then V (x) is continuously differentiable at x0 , with derivative given by (6.4) ∇V (x0 ) = ∇x U (x0 , π (x0 )) These results will enable us to use dynamic programming techniques in a wide variety of dynamic optimization problems Before doing so, we discuss how these results are proved The next section introduces a number of mathematical tools from basic functional analysis necessary for proving some of these theorems and Section 6.4 provides the proofs of all the results stated in this section 6.3 The Contraction Mapping Theorem and Applications* In this section, we present a number of mathematical results that are necessary for making progress with the dynamic programming formulation In this sense, the current section is a “digression” from the main story line and the material in this section, like that in the next section, can be skipped without interfering with the study of the rest of the book Nevertheless, the material in this and the next section are useful for a good understanding of foundations of dynamic programming and should enable the reader to achieve a better understanding of these methods Recall from the Mathematical Appendix that (S, d) is a metric space, if S is a space and d is a metric defined over this space with the usual properties The metric is referred to as “d” since it loosely corresponds to the “distance” between 266