What is the result of the Bellman Equation

Question

I'm just starting with dynamic optimization and although I understant the proof's of the theorem I'm not able to fully understand whether the bellman equation is a function , a function valuated at some number (therefore a number) or both. I'm using Stokey Lucas and Prescot Recursive methods for economics dynamics. And they state in chapter 4.1 and 4.2: Given some assumptions about the feasibility set and the $F(x_t,x_{t+1})$ This problems are equivalent:

1) Sequential problem (SP) $sup_{x_{t+1}}(\Sigma_{t=0}^\infty\beta^tF(x_t,x_{t+1}))$ st $x_{t+1}\in\Gamma(x_{t+1})$ for all t

2) Functional Equation (FE)$\ v(x)= sup_{y\in\Gamma(x)}[F(x,y)+\beta v(y)]$

What is not clear to me is that: In 1 the result of applying sup operator is a NUMBER (Value function valuated at $x_0$ While in 2 as there is a functional equation, the result is a function. The authors seem to talk about a number (chapter 4.1) but then (in chapter 4.2) they state that applying the contraction mapping theorem to 2 we get the solution which is the unique fixed point in the set of continous bounded function, therefore the result is a function

So the solution is a number or a function?

Thanks in advance

Answer 1

...the result of applying sup operator is a NUMBER...

Read it carefully. The equation is

$$ v(x_0) = \sup_{ \{x_t \}_{t \geq 1}} \cdots \quad (1) $$ This defines a function $v$, called the value function, which is a type of indirect utility function. For given $x = x_0$, the value of $v(x)$ is defined to be the sup on the RHS, taken over feasible sequences $\{ x_t \}$.

...there is a functional equation, the result is a function...

Yes, the function $v(\cdot)$ defined by $(1)$ satisfies the function (Bellman) equation. Equality is in the sense of functions. The LHS is $v(\cdot)$ defined by $(1)$. The RHS is another function $$ \nu(x) = \sup_{y\in\Gamma(x)}[F(x,y)+\beta v(y)]. $$ The claim---the dynamic programming principle---is that they are equal, $v(\cdot) = \nu(\cdot)$.