# What is the result of the Bellman Equation

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

• Typos and misleading notations in the question. – Michael May 31 '20 at 1:20

## 1 Answer

...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, $$v(\cdot) = \nu(\cdot)$$.

• Thanks Michael , one last thing to make it 100% clear, what you are saying is that the result of Bellman eq is a function which is the value function for the SP. As both are the same if I value themsame point they are the same number. – Martin Mendina May 31 '20 at 20:19
• Yes, "the solution of Bellman eqn is a function which is the value function for the SP", in economics. To be more precise, the value function must necessarily satisfy the Bellman eqn, and conversely, if a solution of the Bellman eqn satisfies the tranversality condition, then it is the value function. But it's common in economics to ignore the 2nd statement, i.e. solve the Bellman eqn and just assume transversality condition holds for the found solution. – Michael May 31 '20 at 22:23