Odd period cycles and ergodic properties in price dynamics for an exchange economy

First, we clarify what we mean by a Li-Yorke chaos, a turbulence, and a topological chaos. (There are several definitions of a chaos in the literature.) The following definitions are taken from [Ruette, 2017, Def. 5.1] and [Block and Coppel, 1992, Chap. \@slowromancapii@]. Let $g$ be a continuous map of a closed interval $I$ into itself:

It is known that a map $g$ is topologically chaotic if and only if $g$ has a periodic point whose period is not a power of $2$, see [Block and Coppel, 1992, Chap. \@slowromancapii@]. This implies that a map $g$ is topologically chaotic if and only if the topological entropy of $g$ is positive, see [Block and Coppel, 1992, Chap. \@slowromancapviii@]. In the first part of this paper, we focus on a topological chaos (or a positive topological entropy) in the context of price dynamics. See [Ruette, 2017] for more characterisations and (subtle) mutual relations of various kinds of chaos.

Here, we explain a key result to obtain Theorem 1.2. Along the way, we give definitions for all the unexplained notation (such as $\mathfrak{G}$ and $E$) in Theorem 1.2. First, we recall the following mathematical result characterising the existence of a topological chaos for a unimodal interval map [Deng et al., 2022, Cor. 3]. Theorem 1.2 is a (highly non-trivial as seen in Section 3) consequence (or a special case) of [Deng et al., 2022, Cor. 3]. Let $\mathfrak{G}$ be the set of continuous maps from a closed interval $[a,b]$ to itself so that an arbitrary element $g\in\mathfrak{G}$ satisfies the following two properties:

1. 1.

   there exists $m\in(a,b)$ with the map $g$ strictly decreasing on $[a,m]$ and strictly increasing on $[m,b]$.

2. 2.

   $g(a)>a$, $g(b)\leq b$, and $g(x)<x$ for all $x\in[m,b)$.

For $g\in\mathfrak{G}$, let $\Pi:=\{x\in[a,m]\mid g(x)\in[a,m]\textup{ and }g^{2}(x)=x\}$. Now we are ready to state [Deng et al., 2022, Cor. 3]:

We keep the same notation $f$, $\bar{x}$, $\bar{y}$, $\alpha$, $\beta$, and $\lambda$ from Equation (1.1). We write ${\left.\kern-1.2ptf\right|_{E}}$ for the restriction of $f$ to the closed interval $E:=[f(\sqrt{\bar{y}\lambda\beta}),f^{2}(\sqrt{\bar{y}\lambda\beta})+\sqrt{\bar{y}\lambda\beta}]$. (We will explain the significance of the number $\sqrt{\bar{y}\lambda\beta}$ and the reason for the choice of $E$ in the next section.) Note that in the next section we will show that $f(\sqrt{\bar{y}\lambda\beta})<f^{2}(\sqrt{\bar{y}\lambda\beta})+\sqrt{\bar{y}\lambda\beta})$ (so, $E$ is a non-degenerate closed interval) and that ${\left.\kern-1.2ptf\right|_{E}}$ is a map to $E$.

Now, we give several comments/interpretations on Theorem 1.2. First, in the next section, we will show that the condition $\frac{\bar{y}\beta}{{\bar{x}}^{2}(1-\alpha)^{2}}<\lambda<\frac{4\bar{y}\beta}{{\bar{x}}^{2}(1-\alpha)^{2}}$ in Theorem 1.2 is the weakest condition for $f$ to be economically meaningful (to keep $p>0$) and also for ${\left.\kern-1.2ptf\right|_{E}}$ to be in $\mathfrak{G}$. As stated in Introduction, the upper bound for $\lambda$ is not really interesting and the real meat is in the lower bounds in parts 2,3, and 4.

Second, it is well-know that the vertical stretch the graph of $f$ controls the existence of a chaos: if we stretch the graph further, we are likely to obtain a chaos. Now, looking at Equation (1.1), we see that if we make $\alpha$, $\beta$, $\bar{y}$ small, or $\bar{x}$ large, the ”valley” of the graph of $f$ goes deep down. Also, it is clear that a small $\lambda$ makes the graph of $f$ ”flat”. Our lower bound $L:=\frac{25\bar{y}\beta}{9{\bar{x}}^{2}(1-\alpha)^{2}}$ agrees with these observations: $L$ is an increasing function of $\alpha$, $\beta$, $\bar{y}$ and a decreasing function of $\bar{x}$. In other words, it is easy to generate a chaos (small $\lambda$ gives a chaos) if $\alpha$, $\beta$, or $\bar{y}$ is small or $\bar{x}$ is large. We give an economic explanation for this. By looking at the form of the excess demand function $z(p)=\bar{y}\beta/p-\bar{x}(1-\alpha)$ (or the demand function of each consumer for good $x$, that is, $x=\alpha\bar{x}$ for consumer 1 and $x=\frac{\beta\bar{y}}{p}$ for consumer 2 respectively), we see that $p$ goes up very sharply (almost irrespective of the parameter values) after $p$ gets close to $0$. To generate a chaos, $p$ needs to drop sharply afterwords, that is possible (or at least easy) if $\alpha$, $\beta$, or $\bar{y}$ is small or $\bar{x}$ is large for the following reasons: 1. if $\alpha$ or $\beta$ is small, the demand for good $1$ is weak (thus $p$ drops sharply), 2. if $\bar{y}$ is small, then the demand of consumer 2 for good $x$ (that is excessively strong when $p$ is close to $0$) drops sharply (since the budget for consumer 2 is tight), 3. if $\bar{x}$ is large, when $p$ is very high, a large excess supply happens.

Third, part 4 of Theorem 1.2 shows that if $\lambda$ is sufficiently large, for any initial $p>0$, our price dynamics eventually trap $p$ inside a (Li-Yorke) chaotic region $E$. This is interesting since parts 1, 2, and 3 say nothing about what is going on outside of $E$. This sort of analysis is not done in [Deng et al., 2022].

To end this section, we give an application of Theorem 1.2. By the Sharkovsky order in [Sharkovsky, 1964], we know that if the map $f$ has a cycle of period three, then it also has a cycle of any odd order. Thus, it is natural to guess that if $\lambda$ is close to the lower bound for $\lambda$ in Theorem 1.2 (but still above the lower bound), the map $f$ has an odd period cycle but no period three cycle. We give one example where this is actually the case. Using our concrete characterisation of the existence of an odd period cycle, we obtain:

Note that in this case, by Theorem 1.2 a necessary and sufficient condition for the existence of an odd period cycle is $\frac{25\bar{y}\beta}{9{\bar{x}}^{2}(1-\alpha)^{2}}=2.77<\lambda<4=\frac{4\bar{y}\beta}{{\bar{x}}^{2}(1-\alpha)^{2}}$. (We use this condition in Section 6.) Our numerical computation (see Section 4 for details) shows that Proposition 2.4 is almost an if and only if statement: for $\lambda\geq 3.01$, we get a period three cycle. Although many examples of this sort can be obtained by the same method, a complete characterisation for the existence of a period three cycle for a unimodal map is not known. (Thus it is not possible to obtain the precise $\lambda$ value where a bifurcation happens.) We leave it for a future work.

Let $I$ be a compact interval on $\mathbb{R}$. Let $\mathcal{B}$ denote the Borel $\sigma$-algebra of $I$. Let $\zeta:\mathcal{B}\rightarrow[0,\infty]$ be a measure on $I$. A measurable map $g:I\rightarrow I$ is called ergodic with respect to $\zeta$ if whenever $g^{-1}(A)=A$ for $A\in\mathcal{B}$, then either $\zeta(A)=0$ or $\zeta(I\backslash A)=0$. We say that a measure $\zeta$ is g-invariant if $\zeta(g^{-1}(A))=\zeta(A)$ for any $A\in\mathcal{B}$. We write $\mu$ for the Lebesgue measure on $I$. A measure $\zeta$ is called absolutely continuous with respect to $\mu$ if whenever $\mu(A)=0$ for $A\in\mathcal{B}$ then $\zeta(A)=0$. We write an ”acim” for a measure that is absolutely continuous with respect to $\mu$ and $g$-invariant (if $g$ is clear from the context, we just say ”invariant”). Remember that if $\zeta$ is an acim then there exists a ($\mu$-integrable) density function (Radon-Nikodym derivative) $\xi:I\rightarrow\mathbb{R}$ with $d\zeta=\xi d\mu$, see [Collet and Eckmann, 1980, \@slowromancap[email protected]]. It is well-known that if $g$ is ergodic with respect to an acim $\zeta$, then we can give an estimate of $\xi$ (hence an estimate of $\zeta$) using some iterates of $g$, see [Day, 1998, 8.5.2 and 8.5.3] (we use this argument below in Theorem 5.11).

Here, we recall (a special case of) the famous Birkhoff’s ergodic theorem [Day, 1998, Thm. 8.2], which states that under certain conditions the time average of $g$ is equal to its space average:

In particular, Proposition 5.1 says that (if the conditions are met) the time average of $g$ converges to a constant (for almost all $x$), in other words, we can ”predict” the future on average. We call the integral on the right-hand side (or sometimes the sum inside the limit on the left-hand side) of Equation (5.1), that is $\int_{I}g\;d\zeta$ (or sometimes $\frac{1}{n}\sum_{k=0}^{n-1}g^{k}(x)$), the ergodic sum of $g$ (which we are referring to would be clear from the context). In the following, we try to apply Proposition 5.1 to our price dynamics defined by ${\left.\kern-1.2ptf\right|_{E}}$ (with the condition $\frac{\bar{y}\beta}{{\bar{x}}^{2}(1-\alpha)^{2}}<\lambda<\frac{4\bar{y}\beta}{{\bar{x}}^{2}(1-\alpha)^{2}}$ to force that $p>0$ and ${\left.\kern-1.2ptf\right|_{E}}\in\mathfrak{G}$). In the following, we write $f$ for ${\left.\kern-1.2ptf\right|_{E}}$ to ease the notation. So we need to show that $f$ has an absolutely continuous invariant measure $\zeta$, $f$ is ergodic with respect to $\zeta$, and $f$ is $\zeta$-integrable. (The last condition is clear since $f$ is continuous and bounded on the compact interval $E$.)

In general, it is pretty difficult to prove the existence of an acim for a measurable transformation $g$ except for some special cases such as when $g$ is ”expansive” or ”iteratively expansive” as studied in a classical paper of Lasota and Yorke [Lasota and Yorke, 1973]. Recall that $g$ is called expansive if $g$ is piecewise $C^{2}$ and $|g^{\prime}(x)|>1$ for $\mu$-almost all $x$. A typical example of an expansive map is a well-studied ”tent map” [Day, 1998, 8.5.4 and 8.5.5]. Further recall that a slightly more general ”iteratively expansive” map, that is a piecewise $C^{2}$ map with $|g^{\prime}(x)|^{n}>1$ for some positive integer $n>1$ for $\mu$-almost all $x$. From [Lasota and Yorke, 1973], we know that a density function $\xi$ with $d\zeta=\xi d\mu$ is a fixed point of the Perron-Frobenius operator $P$ from the set of measurable function on $I$ to itself, and that a uniform expansion of a map $g$ helps a lot to make $P$ a ”nice” operator. As a result, it is not too hard to prove the existence of an acim in these (iteratively) expansive cases. See [W. de Melo, 1993, Chap. 5], [Shen and van Strien, 2014, Sec. 4], and [Lyubich, 2012] for an overview of this problem and also see [Sato and Yano, 2012] and [Sato and Yano, 2013] for applications of an acim for an iteratively expansive map in economics.

In this paper, our function $f$ is not (even iteratively) expansive since it has a critical point $s$. (So this is a hard case.) Thus, to establish the existence of an acim, we need some deep analytical results to investigate the counter play between the contraction of $f$ near the critical point $s$ and the expansion of $f$ at $f(s)$ (that is far from $s$). Now, we restrict the class of functions we consider to so-called $S$-unimodal maps (due to Singer [Singer, 1978]), whose ergodic properties are well studied, see [Collet and Eckmann, 1980, Part \@slowromancapii@], [W. de Melo, 1993, Chap. 5], [Avila and Moreira, 2005], [Avila et al., 2003] for example. (Our function $f$ is actually $S$-unimodal as we will show below.) Let $g$ be a measurable transformation defined on a compact interval $I=[a,b]$ of $\mathbb{R}$.

Note that a well-known ”logistic map” ($g(x)=rx(1-x)$ for $r\in(0,4]$) is $S$-unimodal. The first key result in this section is