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


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.

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  1. There is also earlier work by Bhattacharya and Bunzel (2003a) and Bhattacharya and Bunzel (2003b), who study global dynamics in a search and matching framework but impose parametric restrictions and only consider the social-planner solution of the model. They establish the potential for n-period cycles in the model, but the modeling restrictions have been criticized by Shimer (2004). Our paper can be seen as contribution that unifies and clarifies these previous results.

  2. Another paper that studies global dynamics in a search and matching setting in labor and capital markets is Ernst and Semmler (2010). Their model has multiple steady states, one of which is a local attractor while another is saddle-path stable. Their analysis is fully numerical based on value-function iteration, whereas we solve the nonlinear equilibrium conditions that emerge from the first-order conditions.

  3. For expositional convenience, we present the problem of a representative firm only. We abstract from indexing the individual variables.

  4. Labor productivity \(A_{t}\) is generally assumed to be the main driver of business cycle fluctuations in the search and matching model (see Shimer 2005). We assume that \(A_{t}=A\) is constant throughout the exposition since we are abstracting from stochastic fluctuations.

  5. Note that newly matched workers who are separated from their job within the period reenter the matching pool immediately.

  6. We thus assume income pooling between employed and unemployed households and abstract from potential incentive problems concerning labor market search. This allows us to treat the labor market separate from the consumption choice. See Merz (1995) and Andolfatto (1996) for a discussion of these issues.

  7. This is a standard assumption in the literature. Shimer (2005) provides further discussion.

  8. They show that the interaction of the Fisher equation, that is, the relationship between nominal and real interest rates and expected inflation, with an ad hoc policy rule results in the existence of two steady states, one stable and one that is unstable globally. The key finding is that the globally unstable steady state is locally saddle-path stable and is actually the one that is imposed in linearized analyses. Benhabib et al. (2001) therefore argue that policy recommendations based on local analysis can be perilous in the global context (see Wolman and Couper 2003, for further discussion, and also Aruoba et al. 2018, for developing the empirical implications).

  9. Under risk aversion, the dynamics depend on the time path of output \(y_{t}\). Output is a function of employment \(n_{t}\), which evolves based on the law of motion (3). Since this feeds back onto the JCC via the definition of \(\theta _{t}=v_{t}/u_{t}\), it results in an interconnected two-equation system that cannot be solved analytically.

  10. The relationship between the backward and forward dynamics of nonlinear systems is an active area of research (see, for example, Kennedy and Stockman 2008 and references thereof). This distinction is immaterial for the study of linear systems since they are always invertible in this sense. That is, the properties of the forward dynamics are the ‘inverse’ of the properties of the backward map. If, on the other hand, one of the dynamic maps is a correspondence, this equivalence fails.

  11. It is straightforward to show that \(z_{0}\) is unique over \([0,\infty )\). Since \(f(0)=d>0\) and f is increasing on \([0,z_{\mathrm{max}})\), then \(f(z_{\mathrm{max}})>0\). Given that f is decreasing on \([z_{\mathrm{max}},\infty )\), the intersection point \(z_{0}\) such that \(f(z_{0})=0\) is unique. Note that the point \(z_{0}\) corresponds to the point q in the general notation used in Appendix A.3.

  12. In a linear model, an unstable equilibrium implies that \(\theta _{t}\) and \( v_{t}\) do grow without bound. While this is a possibility mathematically, it cannot be a rational expectations equilibrium since the resources \(\kappa \) needed to support increased vacancy postings would eventually exhaust finite production since the total size of the labor force is limited to one, see Eq. (2).

  13. We discuss this aspect in more detail in Sect. 4, as it bears more relevance for the quantitative implications of the model.

  14. The condition in (26) is expressed in implicit form since \(z_{\mathrm{SS}}\) depends on d, as well as ac,  and \(\xi \). We show explicit analytical results in terms of the structural model parameters for the benchmark case \( \xi =0.5\) in the next section.

  15. This parameterization is only one example of emergence of periodic doubling and chaos. These values are, however, economically plausible, as \(a=0.891\) is obtained by setting the discount rate \(\beta =0.99\) and the job separation rate \(\rho =0.1\). Under the Hosios condition \(\eta =\xi =0.5\), the value of \(c=0.4\) implies that m is around 0.9. For values of d between 5 and 5.5, the implied steady-state unemployment rate \(u_{\mathrm{SS}}\) ranges between 0.167 and 0.174, which is well within economically plausible bounds.

  16. Theorem 13 in Appendix presents the above proof for a general family of maps classified by Medio and Raines (2007) as Type B maps.

  17. The idea is to capture both measured unemployment in terms of recipients of unemployment benefits and potential job searchers that are only marginally attached to the labor force but are open to job search. Since we do not model labor force participation decisions, this is a shortcut to capturing effective labor market search. This approach has been taken by Cooley and Quadrini (1999) and Trigari (2009).

  18. In the figure, we extend the range of \(c=\beta \left( 1-\rho \right) m\eta \) beyond what is economically permissible since the map (37) is in principle not restricted in such manner.

  19. We also want to highlight one additional case under this benchmark parameterization. An alternative way to establish chaotic behavior is to use the logistic map \(r(\mu )=\mu r(1-r)\), which in the literature is a canonical example for demonstrating chaos in one-dimensional maps, as, for example, in Elaydi (2007). For the quadratic case \(\xi =0.5\), it is straightforward to pin down the values of parameters acd that establish qualitative, or topological, equivalence of the dynamic behavior of the iterates of the map f to those of the logistic map.

  20. In their benchmark study, Petrongolo and Pissarides (2001) find a value \(\xi \) of 0.7, while Hall and Schulhofer-Wohl (2015) report estimates that range between 0.28 and 0.7 from a wide variety of studies, data, and empirical approaches. Nevertheless, \(\xi =0.2\) would be considered below the plausible empirical range.

  21. For completeness, we also have that at \(0>f^{\prime }\left( z_{\mathrm{SS}}\right) >-1 \):

    $$\begin{aligned} \frac{1}{1+\frac{\beta ^{-1}+\left( 1-\rho \right) }{\rho }\frac{\xi }{\eta } }<u_{\mathrm{SS}}<\frac{1}{1+\frac{1-\rho }{\rho }\frac{\xi }{\eta }}, \end{aligned}$$

    and at \(-1>f^{\prime }\left( z_{\mathrm{SS}}\right) \):

    $$\begin{aligned} u_{\mathrm{SS}}<\frac{1}{1+\frac{\beta ^{-1}+\left( 1-\rho \right) }{\rho }\frac{\xi }{\eta }}. \end{aligned}$$
  22. Except for the case shown in m-\(\rho \) space of Fig. 8, a is fixed at 0.891, which is obtained by setting \(\beta =0.99\) and \( \rho =0.1\). We also impose the Hosios condition that sets \(\xi =\eta \).

  23. Log-linearizing this equation around the steady state would result in the same dynamic properties.

  24. An alternative way of seeing this is by inverting the linearized JCC. This implies the backward-looking representation \({\widehat{\theta }}_{t+1}=\left[ \beta (1-\rho )\left( 1-\frac{\eta }{\xi }m\theta _{\mathrm{SS}}^{1-\xi }\right) \right] ^{-1}{\widehat{\theta }}_{t}\). The root of this representation is the inverse of the root of the forward equation. Forward stability therefore implies backward instability, and vice versa. Given the parametric restrictions established in the theorem, the JCC would have explosive dynamics if expressed backward. Consequently, the only solution to be consistent with local stability is \({\widehat{\theta }}_{t}=0\). One important insight is that in the linear case the roots of the forward and the backward representation of the difference equation in question are the inverse of each other. Local analysis can therefore rely on either representation. This is, in general, not the case for global dynamics.

  25. This aspect and the existence of sunspot equilibria is treated more formally in Lubik and Schorfheide (2003) and Farmer et al. (2015).

  26. The figure also shows regions where the equilibrium does not exist. But for the purposes of this paper we rule these out on account of Lemma 2, which restricts the match efficiency to be less than one.

  27. The result is taken from Elaydi (2007).


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The authors wish to thank the editor Nicholas Yannelis and an anonymous referee for useful comments that improved the paper. We are also grateful to Andreas Hornstein, Hassan Sedaghat and Alex Wolman for many discussions. We would like to thank participants at the 25th Annual Symposium of the Society for Nonlinear Dynamics and Econometrics in Paris for constructive comments. The views expressed in this paper are those of the authors and should not be interpreted as those of the Federal Reserve Bank of Richmond or the Federal Reserve System.

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Correspondence to Thomas A. Lubik.



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:

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

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:

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

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)Footnote 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 JK of I with at most one common point such that

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

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:

  1. (i)

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

  2. (ii)

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

  3. (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

  1. (i)

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

  2. (ii)

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

  3. (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.


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

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

or for \(x\ne 0\)

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

The derivative of h(x) is given by:

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

which can be rewritten as:

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

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:

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

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
figure 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 \(z_{\mathrm{SS}}\).


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

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

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.


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:

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

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.

  1. (i)

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

  2. (ii)

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

  3. (iii)

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


  1. (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:

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

    and the result follows by Lemma 5.

  2. (ii)

    If \(f(d)>m_{+}\), then:

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

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

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

    and the result follows again by Lemma 5.

  3. (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.

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Lazaryan, N., Lubik, T.A. Global dynamics in a search and matching model of the labor market. Econ Theory 68, 461–497 (2019).

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  • Indeterminacy
  • Bifurcation
  • Chaos
  • Backward map
  • Forward map

JEL Classification

  • C62
  • C65
  • E24
  • J64