What is the 'fixed point problem'?

Question

I understand what the fixed point is, but don't understand what the fixed point 'problem'. Is it resolved by 'fixed point iteration'? I am reading a paper, and the paper mentions that the default initialization is such that the code reaches the fixed point for the baseline economy after one iteration. Why does the fixed point matter in dynamic optimization?

Answer 1

In mathematics a fixed point is, in general, in a mapping of a space in itself, a point that corresponds to itself.

In particular, fixed points for functions $f(x)$ are value of the variables of the function such that

$$f(x)=x.$$ One can speak of fixed point for function from $\mathbb{R}^N$ to $\mathbb{R}^N$ (or, in general, from a metric space in itself), or for functions between topological spaces.

Fixed point theorems are a family of theorems about the existence of fixed points of a function. Well known theorems are the Contraction theorem, for functions from a metric space in itself , and Brouwer fixed point theorem in topology.

In discrete dynamic systems, which from a mathematical point of view are sequences by recurrence, we speak also of fixed points. They are points of equilibrium, in the sense that the dynamic system, once has arrived at that point, remains here, it is a 'state of rest' of the system.

And it can be seen that they are fixed point of a function, reconnecting to the definition above.

Actually, a dynamic discret system, given a function $f$ from $\mathbb{R}$ to $\mathbb{R}$$^{(1)}$, is recursively described as follows:

$$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;x_n=f(x_{n-1}) \;\,\; \;\,\;n=1, 2, ......\; \;\,\;\; \;\,\;(1)$$ $$x_0=p \;\;\;\;\;p\in \mathbb{R}$$

where $p$ is the initial condition.

A fixed point of the system, relative to $p$, is a fixed point of $f$, that is a point $x$ where

$$x_n=x_{n-1}=f(x_{n-1}).$$

A fixed poind is also called a stationary point.

And it is a point of equilibrium of the system, in the sense that the system, once has reached the fixed point $x_n$, gives always the same value $x_n$, it remains here forever (of course, if there isn't an exogenous disturbing factor that leads it away from equilibrium).

And, similarly, numerical analysis speaks of 'fixed point iteration', as a method to find a fixed point of a function. The concept is analogous to that of discrete dinamical systems.

I quote from Wikipedia "Fixed points":

"In numerical analysis, fixed-point iteration is a method of computing fixed points of a function. Specifically, given a function $f$ with the same domain and codomain, a point $x_{0}$ in the domain of $f$, the fixed point iteration is $$x_n=x_{n-1}=f(x_{n-1});\,\; \;\,\;\,\; \;\,\;$$ which gives rise to sequence $x_ 0, x_ 1, x_ 2, …$ of iterated function applications $x_ 0 , f (x_0),$ $f (f( x_0 )) , … $ which can to converge to a point x. Points that come back to the same value after a finite number of iterations of the function are called periodic points. A fixed point is a periodic point with period equal to one.

Unfortunately, I couldn't look at the paper you linked, because it is not of free access.

But you said that

the default initialization is such that the code reaches the fixed point for the baseline economy after one iteration.

so I suppose that concept is the same as in numerical analysis.

But apart from computation in numerical analysis, the key conceptual idea is that in a discrete dynamical system a fixed point is a point of equilibrium, in the sense described above.

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(1) Of course, there could be also dynamic systems in several variables.

Answer 2

At the end of the day, you are looking for an equilibrium. An equilibrium is essentially a fixed point. In equilibrium all agents maximize their payoffs, e.g., optimize (dynamically or statically) their strategies. Namely, an intersection of those optimal strategies is an equilibrium, or a fixed point.