Global dynamics in a search and matching model of the labor market

Abstract

We study global and local dynamics of a simple search and matching model of the labor market. We show that the model exhibits chaotic and periodic dynamics for empirically plausible parameter values both in backward and forward time. In contrast to the global results, we show that the model can be locally indeterminate or have no equilibrium at all, but only for parameterizations that are empirically unreasonable. In contrast to earlier work, we establish these results analytically without placing numerical restrictions on the parameters.

Appendix

In this appendix, we list definitions and results necessary for the establishment of periodic and chaotic solutions in the search and matching model, as discussed in the paper.

1.1 Preliminaries

Let $f:\mathbb {R}\rightarrow \mathbb {R}$ be a map and consider the first-order difference equation given by:

$$\begin{aligned} x_{t+1}=f(x_{t}). \end{aligned}$$

(50)

Definition 1

(Invariance) The interval $I\subset \mathbb {R}$ is invariant under f if $f(I)\subseteq I$. For the first-order equation in (50), the above definition implies that if the initial value $x_{0}\in I$, then $x_{t}\in I$ for $t>0$.

Definition 2

(Periodic points) Let p be a nonnegative integer and let $f^{p}=f\circ f\circ \cdots \circ f$ be the composition of the map f with itself p times. The point $s\in \mathbb {R}$ is a p-periodic point of the map f if $f^{p}(s)=s$. The first-order equation in (50) has a periodic solution of period p if the map f has a p-periodic point. In this case, we say that the equation in (50) has a periodic solution of period p (or a p-cycle), i.e., $x_{t+p}=x_{t}$ for all $t\ge 0$.

The following result in Block and Coppel (1986) establishes sufficient conditions for existence of periodic points of odd periods.

Lemma 5

Let $f:\mathbb {R}\rightarrow \mathbb {R}$ be a continuous map. If for some odd integer $p>1$ there exists a point x such that:

$$\begin{aligned} f^{p}(x)\le x<f(x)\quad \mathrm{or}\quad f(x)<x\le f^{p}(x), \end{aligned}$$

(51)

then f has a periodic point of period p.

We next list Sharkovski’s ordering of positive integers defined as follows (see Elaydi 2007, for more):

$$\begin{aligned} 3&\vartriangleleft 5\vartriangleleft 7\vartriangleleft \cdots \\ 2\cdot 3&\vartriangleleft 2\cdot 5 \vartriangleleft 2 \cdot 7 \vartriangleleft \cdots \\ 2^2\cdot 3&\vartriangleleft 2^2\cdot 5 \vartriangleleft 2^2 \cdot 7 \vartriangleleft \cdots \\ \cdots&\nonumber \\ 2^n\cdot 3&\vartriangleleft 2^n\cdot 5 \vartriangleleft 2^n \cdot 7 \vartriangleleft \cdots \\ \cdots&\\ 2^n&\vartriangleleft 2^{n-1} \vartriangleleft \cdots \vartriangleleft 2^2\vartriangleleft 2 \vartriangleleft 1 \end{aligned}$$

Now the theorem.

Theorem 5

(Sharkovski)²⁷ Let $f:I\rightarrow I$ be a continuous map on the interval I, where I may be finite, infinite, or the whole real line. If f has a periodic point of period k, then it has a periodic point of period r for all r with $k\vartriangleleft r$.

Given Sharkovski’s ordering, the above theorem states that if a function f has a periodic point of period 3, then it has periodic points of all periods, which is stated as a theorem below.

Theorem 6

(Li and Yorke 1975) Let $f:I\rightarrow I$ be a continuous map on an interval $I\subseteq \mathbb {R}$. If f has a periodic point in I of period 3, then f has a periodic point of every integer period $k\ge 1$.

There are several, not necessarily equivalent, definitions of chaos in mathematical literature. The more commonly used ones are those in the sense of Li and Yorke, Devaney, and Block and Coppel (see Aulbach and Kieninger 2001 for more details). For the purpose of this paper, below we list the definition of chaos in the sense of Block and Coppel and refer to the result in Aulbach and Kieninger (2001) that establishes equivalence between chaos in the sense of Block and Coppel to that of Devaney.

Definition 3

A map $f:I\rightarrow I$ is called turbulent if there exist compact subintervals J, K of I with at most one common point such that

$$\begin{aligned} J\cup K\subseteq f(J)\cap f(K). \end{aligned}$$

(52)

If J and K are disjoint, then f is said to be strictly turbulent.

Theorem 7

(Chaos in the sense of Block and Coppel) A continuous map $f:I\rightarrow I$ on a nontrivial compact interval I is chaotic in the sense of Block and Coppel if and only if one of the following equivalent conditions is satisfied:

• (i) $f^m$ is turbulent for some $m\in \mathbb {N}$.

• (ii) $f^m$ is strictly turbulent for some $m\in \mathbb {N}$.

• (iii) f has a periodic point whose period is not a power of 2.

Theorem 8

(Aulbach and Kieninger 2001) A continuous map $f:I\rightarrow I$ on an interval I is chaotic in Devaney sense if and only if it is chaotic in the Block and Coppel sense.

Our next theorem is from Elaydi (2007) and lists conditions under which period-doubling bifurcations occur.

Theorem 9

(Period-Doubling Bifurcation) Let a one-parameter family $F_{\mu }(x)$ be written as a map of two variables, i.e., $H(\mu ,x): \mathbb {R}\times \mathbb {R}\rightarrow \mathbb {R}$ and let $x^{*}$ be the fixed point of $F_{m}u$. Suppose that

• (i) $H_{\mu }(x^*)=x^*$ for all $\mu$ in an interval around a threshold point $\mu ^*$.

• (ii) $H^{\prime }_{\mu ^*}(x^*)=-1$.

• (iii) $\frac{\partial ^{2} H^2}{\partial \mu \partial x^*}(\mu ^*, x^*)\ne 0$.

where $H^2(\mu , x)=H(H(\mu , x))$. Then there exists an interval I about $x^*$ and a function $p:I\rightarrow \mathbb {R}$ such that $H_{p(x)}(x)\ne x$, but $H^2_{p(x)}(x)= x$.

Finally, we state the result of Kennedy and Stockman (2008) that relates the solutions of a map iterated backward in time to those of forward representation. For establishment of periodic solutions in the forward map, existence of periodic solutions in the backward map is sufficient. Using the same notation as in Kennedy and Stockman (2008), the map $f^{-1}$ is defined for the map f on a metric space X with $f:X\rightarrow X$, regardless whether f is multi-valued or not. Their main result states:

Theorem 10

Let $f:X\rightarrow X$ be continuous on a metric space X. Then f is chaotic on X in the sense of Devaney if and only if $f^{-1}$ is chaotic on X.

The above theorem is an important result showing that models with backward dynamics are chaotic going forward in time if and only if they are chaotic going backward in time. Hence, establishment of chaotic solutions in backward dynamics is sufficient for existence of chaotic forward dynamics.

1.2 Fixed points of the g-map

Theorem 11

The map $g(x)=(ax^{\xi }-cx+d)^{\frac{1}{\xi }}$ can have two positive fixed points.

Proof

The fixed points of the map g must satisfy the expression:

$$\begin{aligned} x=(ax^{\xi }-cx+d)^{\frac{1}{\xi }}, \end{aligned}$$

(53)

or for $x\ne 0$

$$\begin{aligned} h(x):=\frac{(ax^{\xi }-cx+d)^{\frac{1}{\xi }}}{x}=1 \end{aligned}$$

(54)

The derivative of h(x) is given by:

$$\begin{aligned} h^{\prime }(x)=\frac{1}{\xi x^{2}}\left[ (ax^{\xi }-cx+d)^{\frac{1}{\xi } -1}(\xi ax^{\xi -1})x-(ax^{\xi }-cx+d)^{\frac{1}{\xi }}\right] , \end{aligned}$$

(55)

which can be rewritten as:

$$\begin{aligned} h^{\prime }(x)=\frac{1}{\xi x^{2}}\left[ (ax^{\xi }-cx+d)^{\frac{1-\xi }{\xi } }\right] \left[ (\xi -1)ax^{\xi }-d\right] . \end{aligned}$$

(56)

Next, we determine the behavior of h(x) via the sign of its derivative $h^{\prime }(x)$. First, note that $\displaystyle \lim _{x\rightarrow 0^{+}}h(x)=\infty$. Since $0<\xi <1$, then $(\xi -1)ax^{\xi }-d<0$ for all $x\ge 0$. Also, we let:

$$\begin{aligned} \phi (x)=ax^{\xi }-cx+d\quad \text {with}\quad \phi (0)=d>0,\quad \phi (d/a)^{\frac{ 1}{\xi }})=-c(d/a)^{\frac{1}{\xi }}<0,\quad \end{aligned}$$

(57)

hence there exists a point $x^{*}\in \left( 0,\left( \frac{d}{a}\right) ^{\frac{1}{\xi }}\right)$ such that $\phi (x^{*})=0$. Moreover, $\phi (x)>0$ for $x\in (0,x^{*})$, $\phi (x)<0$ for $x>x^{*}$, and $h(x^{*})=g(x^{*})=0$.

Now, if $\frac{1}{\xi }=2k$ for some positive integer $k\ge 1$, then $\frac{ 1}{\xi }-1$ is odd, which means that $(\phi (x))^{1/\xi -1}$ is positive on $(0,x^{*})$ and negative on $(x^{*},\infty )$. This means h(x) is decreasing on $(0,x^{*})$ and increasing on $(x^{*},\infty )$ and is exactly 0 at $x^{*}$. Therefore, there exist precisely, two points $x^{\prime }$ and $x^{\prime \prime }$, at which $h(x^{\prime })=h(x^{\prime \prime })=1$, hence $x^{\prime }$ and $x^{\prime \prime }$ are the two fixed points of g(x), which proves the above claim.

If, on the other hand, $\frac{1}{\xi }=2k+1$, then $(\phi (x))^{1/\xi -1}>0$ for $x>0$, hence h is decreasing on $(0,\infty )$ and is equal to one at precisely one point, and in this case, the positive fixed point of the map g is unique. $\square$

See an example of the map g at $a=0.8,c=0.7$, and $d=2$ with $\xi =0.5$ in Fig. 11. The two steady states are clearly discernible at the intersection points of the map g with the identity line.

Fig. 11

Multiple steady states of map g at $a=0.8$, $c=0.7$, $d=2$ and $\xi =0.5$

1.3 Steady state of the f-map

Lemma 6

Equation (17) has a unique positive steady state $z_{\mathrm{SS}}$.

Proof

The steady state(s) of (17) must be the fixed point(s) of the map f in (18). The fixed point(s) ${\bar{z}}$ of f must satisfy the equation

$$\begin{aligned} a{\bar{z}}-c{\bar{z}}^{1/\xi }+d={\bar{z}}. \end{aligned}$$

(58)

To show that such a point ${\bar{z}}$ exists, we define a function $\phi (z)=(1-a)z+cz^{1/\xi }-d$. Since $\phi (0)=-d<0$ and $\phi \left( \frac{d}{ 1-a}\right) =c\left( \frac{d}{1-a}\right) ^{1/\xi }>0$, there exists a point ${\bar{z}}\in \left( 0,\frac{d}{1-a}\right)$ such that $\phi ({\bar{z}})=0$. To show that ${\bar{z}}$ is unique, we note that the derivative $\phi ^{\prime }(z)=(1-a)+\frac{c}{\xi }z^{\frac{1-\xi }{\xi }}>0$ for $z>0$. Therefore, $\phi$ is increasing on $(0,\infty )$, and hence its zero is unique. It follows that ${\bar{z}}=z_{\mathrm{SS}}$ is the unique positive fixed point of the map f and the equation in (17) has a unique positive steady state. $\square$

1.4 Chaos in type-B maps

Suppose $f:\mathbb {R}\rightarrow \mathbb {R}$ is a function with a critical point $m>0$ such that f is increasing on [0, m) and decreasing on $(m, \infty )$ and $f(0)=d>0$. Under appropriate scaling, this type of a map has been characterized by Medio and Raines (2007) as a Type-B map. We establish sufficient conditions for existence of periodic and chaotic solutions for a general class of such maps.

Given that f is decreasing on $(m, \infty )$ and $f(m)>0$, there exists a real number $q>m$, such that $f(q)=0$ (i.e., q is the preimage of 0). This gives us the following result.

Theorem 12

If $f(m)\le q$, then the interval [0, q] is invariant under f.

Proof

Let $x\in [0, q]$. Then $f(x)\le f(m)\le q$ for all $x\ge 0$. Further, if $0\le x\le m$, then $f(x)\ge f(0)=d>0$ since f is increasing on [0, m), and if $m\le x\le q$, then $f(x)\ge f(q)=0$ since f is decreasing on [0, q]. $\square$

Now, for the Type-B map defined above, for any point $y\in [0,q]$, there exists a pair of real numbers $y_{-}$ and $y_{+}$ such that $f(y_{-})=f(y_{+})=y$, i.e., $y_{-}$ and $y_{+}$ are preimages of y. Moreover, if $z<y$, then:

$$\begin{aligned} z_{-}<y_{-}<m<y_{+}<z_{+}. \end{aligned}$$

(59)

We use this to establish sufficient conditions for existence of odd periodic points in Type-B maps.

Theorem 13

Let f be a Type-B map defined above.

• (i) If $m>d$ and $f(m)=q$, then f has a periodic point of period 3 in [0, q].

• (ii) If $f(d)\ge m_{+}$ and $f(m)=q$, then f has a periodic point of period 5 in [0, q].

• (iii) If $f(d)=q$, then f has a periodic point of period 3 in [0, q].

Proof

• (i) If $m>d$, then $f(m)=q\ge m$, $f^{2}(m)=f(q)=0$ and $f^{3}(m)=f(0)=d<m$, hence:

  $$\begin{aligned} d=f^{3}(m)<m\le f(m)=q, \end{aligned}$$

  (60)

  and the result follows by Lemma 5.

• (ii) If $f(d)>m_{+}$, then:

  $$\begin{aligned} m_{+}\rightarrow m\rightarrow q\rightarrow 0\rightarrow d\rightarrow f(d), \end{aligned}$$

  (61)

  i.e., $f(d)=f^{5}(m_{+})$ and:

  $$\begin{aligned} m=f(m_{+})<m_{+}\le f^{5}(m_{+}), \end{aligned}$$

  (62)

  and the result follows again by Lemma 5.

• (iii) Setting $f(d)=q$ pins down exactly the cycle $\{\ldots 0, d, q, 0, d, q, \ldots \}$ as $f(0)=d$, $f(d)=q$, $f(q)=0$. $\square$

As a corollary, we also have the following result.

Corollary 1

If any of the hypotheses (i), (ii), or (iii) in Theorem 13 hold, then f has periodic points of every periods in [0, q] (except for 3 in case of (ii)) and is chaotic in the sense of Block and Coppel, and Devaney.