# Solution to the Bellman equation is a fixed point

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)

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.

• Thanks, after reading your post I see that it it indeed a fixed point by definition. Yes, I just assumed, that utility function is bounded, etc., by default. I have found an article that explains the point in more detail – dpd Jan 26 '17 at 0:51
• @due_to_revision if you're happy with the answer, you should consider accepting it, so the site doesn't keep your question in the 'Unanswered' queue. – Theoretical Economist Jan 28 '17 at 1:33