Solution to the Bellman equation is a fixed point

Question

I have recently started studying dynamic optimization. I cannot quite wrap my head around the fact that the value function of the Bellman equation is a fixed point of a contraction mapping. As far my understanding is rather naive: if the problem is finite, say: $$\sum_{t=0}^T \beta^tu(c_t)$$ we construct the Bellman equation from the end, as if we knew the maximum possible value of the sequence in advance. Starting from the last period $T$, we just repeat the maximization by adding an optimal term reflecting current period utility $u(c_t)$, until we arrive to the period $0$. From here I can clearly see how contraction mapping works. But the infinite case is not so easy for me to comprehend: I can only suppose, that, by iteration of the Bellman operator $(Bv)(x)$, we perform a "calibration" of the policy function until we find the value function (i.e. the maximum possible utility given our transversality conditions) $(Bv)(x)=v(x)$. Am I, at least, thinking in the right direction, or this idea should be understood in a different way? Thank you in advance. (Also, this is my first question on .stackexchange ever, and if there are any issues with presentation of my question, please, let me know)

Answer 1

I am by no means an expert on this, but maybe this helps. Here is a simple example for a bellman equation

$V(y) = \max_x u(x,y) + \beta V(y')$

$s.t. \, y' = f(x,y)$

This is a functional equation in an unknown function V. A solution to this problem is a function V that satisfies the equation above. If you look at the equation, it's pretty clear that the solution has to be a fixed point of the operator on the RHS of the bellman equation: if you take the correct V and an arbitrary y and calculate

$\max_x u(x,y) + \beta V(y')$

$s.t. \, y' = f(x,y)$

you will get $V(y)$. The operator that is the RHS of the Bellman equation operates on functions, and the solution is a fixed point in some space of functions.

It's a different question whether this fixed point exists and how to find it. Here, you appeal to the contraction mapping theorem: under typical assumptions on u and provided $\beta<1$, the maximization step above is a contraction mapping for any guess of V. This means that there exists a unique fixed point V, and you can find it by successive iteration.