Macroeconomics 4

More on Bellman equation

$$v(x)=\max _{y \in \Gamma(x)}\{F(x, y)+\beta v(y)\}$$

we define the following operator T:

$$(T v)(x)=\max _{y \in \Gamma(x)}\{F(x, y)+\beta v(y)\}$$

A solution to the functional equation is then a fixed point of this operator

$$v^{*}=T v^{*}$$

• Complete Metric Spaces (domain and range of the operator T)
• Contraction Mapping Theorem (existence and uniqueness of the operator T)
• Blackwell's theorem (sufficient condition for an operator to be a contraction)

CMT

Definition 33 Let $(S, d)$ be a metric space and $T: S \rightarrow S$ be a function mapping $S$ into itself. The function $T$ is a contraction mapping if there exists a number $\beta \in(0,1)$ satisfying

$$d(T x, T y) \leq \beta d(x, y) \text { for all } x, y \in S$$

The number $\beta$ is called the modulus of the contraction mapping.

Theorem 35 ==CMT== Let $(S, d)$ be a complete metric space and suppose that $T: S \rightarrow S$ is a contraction mapping with modulus $\beta$. Then a) the operator $T$ has exactly one fixed point $v^{*} \in S$ and b) for any $v_{0} \in S$, and any $n \in \mathbf{N}$ we have

$$d\left(T^{n} v_{0}, v^{*}\right) \leq \beta^{n} d\left(v_{0}, v^{*}\right)$$

Blackwell's Theorem

Theorem 37 Let $X \subseteq \mathbf{R}^{L}$ and $B(X)$ be the space of bounded functions $f$ : $X \rightarrow \mathbf{R}$ with the $d$ being the sup-norm. Let $T: B(X) \rightarrow B(X)$ be an operator satisfying

1. Monotonicity: If $f, g \in B(X)$ are such that $f(x) \leq g(x)$ for all $x \in X$, then $(T f)(x) \leq(T g)(x)$ for all $x \in X$
2. Discounting: Let the function $f+a$, for $f \in B(X)$ and $a \in \mathbf{R}_{+}$be defined by $(f+a)(x)=f(x)+a$ (i.e. for all $x$ the number a is added to $f(x))$. There exists $\beta \in(0,1)$ such that for all $f \in B(X), a \geq 0$ and all $x \in X$
   $$[T(f+a)](x) \leq[T f](x)+\beta a$$
   If these two conditions are satisfied, then the operator $T$ is a contraction with modulus $\beta$.

tip

The neoclassical growth model with bounded utility satisfies the Sufficient conditions for a contraction and there is a unique fixed point to the functional equation that can be computed from any starting guess $v_0$ be repeated application of the T-operator.