Blackwell's contraction mapping theorem

Proof of the Theorem

For all ${\displaystyle u}$ and ${\displaystyle v\in X}$, ${\displaystyle u\le v+||v-u||}$. Properties 1. and 2. imply that ${\displaystyle T(u)\le T(v+||v-u||)\le T(v)+\beta ||v-u||}$, hence, ${\displaystyle T(u)-T(v)\le \beta ||v-u||}$. The symmetric follows from a similar argument and we prove that ${\displaystyle T}$ is a contraction mapping.

Applications

The cake eating problem

An agent has access to only one cake for its entire, infinite, life. It has to decide the optimal way to consume it. It evaluates a consumption plan, ${\displaystyle c_t}$, by using a separable utility function, ${\displaystyle \sum _{t=0}^{\mathrm{\infty }}{\beta }^t\frac{c_t^{1-\sigma }}{1-\sigma }}$, with discounting factor ${\displaystyle \beta \in (0,1)}$. Its problem can be summarized as

${\displaystyle \underset{c_t}{max}\sum _{t=0}^{\mathrm{\infty }}{\beta }^t\frac{c_t^{1-\sigma }}{1-\sigma }\text{ subject to }{x}_t=x_{t-1}-c_t\text{, }{x}_{-1}=0\text{, }{c}_t\ge 0\text{ and }{x}_t\ge 0\mathrm{\forall }t\in {\mathbb{Z}}_{+}}$. ()

Applying Bellman's principle of optimality we find (1)'s corresponding Bellman equation

${\displaystyle V(c)=\underset{c^{\prime }}{max}\frac{c^{1-\sigma }}{1-\sigma }+\beta V(c^{\prime })}$. ()

It can be proven that the solution to this functional equation, if it exists, is equivalent to the solution of (1). To prove its existence we can resort to Blackwell's sufficient conditions.

Define the operator ${\displaystyle T(V(c))=\underset{c^{\prime }}{max}\frac{c^{1-\sigma }}{1-\sigma }+\beta V(c^{\prime })}$. A solution to (2) is equivalent to finding a fixed-point for our operator. If we prove that this operator is a contraction mapping then we can use Banach's fixed-point theorem, and conclude that there is indeed a solution to (1).

First note that ${\displaystyle T}$ is defined over the space of bounded functions since for all feasible consumption plans, ${\displaystyle \sum _{t=0}^{\mathrm{\infty }}{\beta }^t\frac{c_t^{1-\sigma }}{1-\sigma }\le \sum _{t=0}^{\mathrm{\infty }}{\beta }^t\frac{1^{1-\sigma }}{1-\sigma }=\frac{1}{(1-\beta )(1-\sigma )}<\mathrm{\infty }}$. Endowing it with the sup-norm we conclude that the domain and co-domain are ordered normed vector spaces. We are just left with verifying that the conditions for Blackwell's theorem are respected:

1. (monotonicity) ${\displaystyle \text{if }V(c)\ge U(c)\mathrm{\forall }c\in [0,1]}$ then ${\displaystyle T(V(c))=\underset{c^{\prime }}{max}\frac{c^{1-\sigma }}{1-\sigma }+\beta V(c^{\prime })\ge \underset{c^{\prime }}{max}\frac{c^{1-\sigma }}{1-\sigma }+\beta U(c^{\prime })=T(U(c))}$
2. (discounting) ${\displaystyle T(V(c)+a)=\underset{c^{\prime }}{max}\frac{c^{1-\sigma }}{1-\sigma }+\beta (V(c^{\prime })+a)\le \underset{c^{\prime }}{max}\frac{c^{1-\sigma }}{1-\sigma }+\beta V(c^{\prime })+\beta a=T(V(c))+\beta a}$ where ${\displaystyle a}$ is a constant function.