Economic growth and economic development 280

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
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