# A Simple Nonlinear Dynamic Model for Unemployment: Explaining the Spanish Case

Authors: João Ricardo Faria; Miguel A. León-Ledesma
Journal: Discrete Dynamics in Nature and Society
 (2008)

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## Abstract

Spanish unemployment is characterized by three distinct regimes of low, medium, and high unemployment and by a fast transition between them. This paper presents a simple nonlinear dynamic model that is able to explain this behavior with multiple equilibria and jumps describing the transition between equilibria. The model has only a small number of parameters capturing the fundamentals of labor markets and macroeconomic and institutional factors. The model is capable of generating unemployment dynamics that encompass the “unique” natural rate hypothesis, the structuralist hypothesis, and the hysteresis hypothesis.


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## Body

## 1. Introduction

Spain's unemployment performance has been the focus of much theoretical and empirical literature. (On Spanish unemployment see, among others,
Bentolila and Blanchard [1], Bentolila and Dolado [2], Dolado and Jimeno
[3], Dolado et al. [4], Juselius and Ordonez [5], Romero-Ávila and Usabiaga [6].)
After a period of very low unemployment, during the 1980s and 1990s, Spain's
unemployment rose to the highest of the OECD group reaching levels above 20%.
In recent years, however, Spain has become the main source of employment creation in Europe,
which has led to a dramatic fall in the unemployment rate to levels of around
8%. What is remarkable about Spanish unemployment during this period is that it
appears to have jumped very quickly from a low to a high unemployment
equilibrium. As a result, any model of the Spanish unemployment should be able
to yield multiple equilibria and a fast transition between them.This paper puts forward a simple nonlinear dynamic model that is able to
generate these regime changes and their rapid transition with a small number of
parameters capturing the main factors of Spanish unemployment, including labor
market fundamentals and macroeconomic and institutional factors that may have
acted upon it. In the model, changes in the institutional characteristics of
the labor market or macroeconomic conditions can potentially lead to large
unemployment equilibrium jumps. The model is also able to encompass the three
main unemployment dynamics theories in the literature, namely, the natural rate
hypothesis, the structuralist hypothesis, and unemployment hysteresis.After briefly characterizing the Spanish unemployment in Section2, we
present the model in Section 3. Section 4 provides some links between the
empirical characterization and the theoretical model, while Section 5 concludes.


## 2. Characterizing Spanish Unemployment

As mentioned earlier, the evolution of
Spanish unemployment has been characterized by three main phases (see Figure1). The first one is between 1965 and the second half of the 1970s, where
unemployment is very low at levels averaging about 2% up to 1977. In the second
phase, between 1980 and the second half of the 1990s, unemployment increases
dramatically and stays at a very high level for almost 30 years. Indeed,
unemployment rates reach a peak in 1994 at 24%. The third phase is
characterized by a rapid decrease in unemployment between the second half of
the 1990s and 2007. The current unemployment rate in 2007 has recovered to
levels of about 8% (all
the data is quarterly and refers to OECD’s standardized unemployment rate).Figure 1
Spanish unemployment 1965:1-2007:2.These rapid changes in unemployment
rates have coincided with important events and reforms taking place in the
Spanish economy in the last 30 years. (For an overview of labor market reforms in Spain
see Ferreiro and Serrano [7].) The low Spanish unemployment during the 1960s and early 1970s
reflected a rapidly growing economy during the catch-up process, but also the
fact that migration absorbed large part of the excess labor force. With the
process of democratic transition, the government embarked on important reforms
of the labor market and taxation systems to mimic the European postwar welfare
state institutions. Important events were the approval of the Workers’ Statute (1977) and the tax
reform (1977-78). The first legally recognized trade unions and collective
bargaining and gave preference to permanent contracts, whereas the second
modernized the tax system with the introduction of personal income tax and
corporate tax to substitute the archaic tax system of the Francoist era. The
new system also extended unemployment benefits to job losers. These changes
coincided with a deep economic recession, deindustrialization,  and high
inflation. The high unemployment rates that ensued planted the seeds for
another labor market reform in 1984 to introduce more flexibility especially
for temporary employment. (This
period also saw several tax reforms and the introduction of the VAT as general
indirect tax in preparation for the Spanish entry in the European Community.) This reform also
coincided with an economic recovery which saw unemployment rates falling but
only to levels close to 16%. The ERM crisis in 1991 generated another deep
recession and unemployment rates increased dramatically in the following three
years. This led to another labor market reform in 1994 to give flexibility to
the wage setting process with preference for decentralized bargaining. Another
two reforms were introduced in 1997 and 2002 to increase the flexibility of the
labor market and reduce the duality generated by the increase in temporary
work. The reforms during the 1990s and 2000s also changed (reduced)
substantially unemployment benefits and eligibility criteria. These changes
also coincided with tax reforms to improve the efficiency of tax collection,
reduce the number of tax brackets, and, during the 2000s, reduce the high
marginal tax rates. All these reforms took place in a context of strong trade
liberalization, especially with the accession to the EU (then European
Economic Community) in 1986, deregulation of goods markets and
privatization.Against this background, the rapid and
significant unemployment changes can be characterized as a multiple equilibria
pattern. To illustrate this point, we can use a Markov switching in mean (MSM) model
as put forward by Hamilton [8], to characterize the dynamics of the Spanish unemployment rate. We estimated
the following MSM dynamic model for the unemployment rate: (2.1)[ut−μ(st)]=∑p=1Pϕp[ut−p−μ(st)]+εt, where ut is the unemployment rate, p is the lag augmentation, ϕp are autoregressive coefficients, εt is an i.i.d. N(0,σ) error term, and μ(st) is the mean of the unemployment rate that
depends on st which is the unobservable realization of M states that is governed by a
discrete-time, discrete-state Markov stochastic process. This process is
defined by the transition probabilities Pr(st+1=j∣st=i)=pij with ∑j=1Mpij=1,∀i.This is a simple
representation of a multiple equilibria process for unemployment (see León-Ledesma and
McAdam [9] for the case of Eastern European countries). In our case, we allow a maximum number
of 3 states (M=3), that is, the
variable is allowed to switch between three different regimes st=1,2,3. (The model allows for a
lag augmentation of 4.)Estimation of the model using the EM
algorithm yields three clear states of high, medium, and low unemployment. The
results in Table1 show the mean unemployment rates in regimes 1, 2, and 3, the
transition probabilities between regimes, and the timing of the three regimes.
The actual and mean unemployment, together with its estimated mean and the
smoothed transition probabilities are plotted in Figure 2. The results show a
dramatic increase in unemployment in 1978 from its previous 2.2% mean. For a
short period it moves into the second regime with a mean of 10.8% and then
rapidly moves in 1982 into the third high-unemployment regime (mean 19.5%). By
1999, unemployment exits the high-unemployment regime into the medium-unemployment
one. The three regimes are very persistent, especially regimes 1 and 3 (low and
high) as the probabilities of transition out of the current regime are very
low. Following Psaradakis and Spagnolo [10], we also report the Akaike information
criterion (AIC) to select the number of states which favors the choice of M=3.Table 1
Markov-switching model for Spain’s unemployment.
Regime meansMeanSt.
errorRegime 12.2160.316Regime 210.770.407Regime 319.520.294Transition
probabilitiesRegime
1Regime
2Regime
3Regime 10.98150.018531.27E-28Regime 23.09E-120.9780.022Regime 32.61E-150.01420.9858Regime
duration% Observ.DurationRegime 15353.96Regime 246.545.45Regime 370.570.42Regime datingRegime 11965:1–1978:1Regime 21978:2–1981:41999:4–2007:2Regime 31982:1–1999:3AICM=1 (linear)6.92M=25.57M=34.65Unemployment and regime probabilities for Spain.
(a)(b)(c)(d)The reported changes in equilibrium
unemployment appear to coincide with some of the major labor market and tax
reforms previously identified. They also coincide with major changes in
economic performance, although these events may not be independent as some of
the reforms took place after policy makers realized the need for reform and
achieved sufficient political consensus. Hence, the Spanish unemployment can be
characterized by multiple equilibria, where regime changes coincide with major economic
and institutional changes in the Spanish economy. For this reason, a model of
the Spanish unemployment dynamics has to be able to yield multiple equilibria
as a consequence of internal and external changes in the labor market, and
rapid transition between states.


## 3. The Model

The model describes the time evolution of Spanish unemployment,u•≡du/dt,
as a result of two different forces, which we call internal and external. The
internal forces are the labor market fundamentals, the ones that lie beneath
the labor market equilibrium, affecting labor supply and labor demand. These
internal forces include workers’ and trade unions’ preferences, outside options
(such as unemployment benefits), bargaining power, firms’ technology, and
market power. These variables are all arguments of the internal forces
function, denoted by F(u,X), where X is the vector of internal
forces.The external forces are external interventions aimed at reducing
unemployment, affecting labor market equilibrium besides labor market
fundamentals. Among these external forces are macroeconomic policies and
institutional changes related to fiscal and monetary policies and goods markets.
These forces are represented by the functionG(u,Y) where Y is
the vector of external forces.Unemployment evolves according to(3.1)u•≡F(u,X)−G(u,Y).It is clear that in the steady state we have thatu=u(X,Y),
which is the desired result of any model that aims at describing unemployment
as a function of labor market fundamentals and institutional and macroeconomic characteristics
of the economy.However, this standard model is not a priori equipped to deal with
multiple equilibria or to explain the transitions from one equilibrium
unemployment to another. If Spanish unemployment is characterized by several
distinct regimes, it has multiple equilibria. Moreover, the change from one
regime to another appears to have been very fast. As a consequence, a
reasonable model of Spanish unemployment has to generate at least two steady
state equilibria and be able to explain the factors that led to a fast
transition between them. The model in (3.1) can do this by assuming
very simple nonlinear dynamics.Starting with functionF(u,X), and in order to keep the model as
simple as possible, it is enough to assume that internal forces act to limit growth
towards an upper limit, that is, unemployment cannot affect the whole working
population, normalized to one. Therefore, F(u,X)=r(X)(1−u),
where the relative rate of change r is a function of X.In the same vein, let us assume that functionG(u,Y) captures all
external interventions aimed at reducing unemployment. For instance, it is
well-known that government’s taxes may affect unemployment (e.g., [11–14]), so
the government may be tempted to put forward tax schemes to reduce unemployment;
the same holds true for other macroeconomic policies and institutional changes
(e.g., [15–17]). The main
characteristic of our function G(u,Y) is that it has an upper limit to the rate of
unemployment reduction, that is, when unemployment is too high, these external forces
may not be effective in reducing unemployment due to the limit imposed by labor
market tightness. This means that there is an upper limit to the rate of reduction
of unemployment due to these external interventions in the labor market. This
is a reasonable assumption because all these forces are external to the
fundamentals of the labor market. A simple formulation of function G(u,Y) is the following: G(u,Y)=b(Y)(u/(a2+u2)),
where the upper limit b is a function of Y, and a is a
positive constant.SubstitutingF(u,X) and G(u,Y) into (3.1) yields (3.2)u•=r(X)(1−u)−b(Y)ua2+u2.This model has multiple equilibria, the equilibrium values of
unemploymentu must satisfy (3.3)u•=0⇒r(X)(1−u)=b(Y)ua2+u2.In order to make the analysis of the model simpler, we scale the equation
by makingλ=u/a,
and multiplying by a/b(Y),
yielding (3.4)ab(Y)r(X)(1−aλ)=λ1+λ2.The points where the left and right-hand sides of (3.4)
intersect are the equilibria for λ, and, equivalently, u. The two
sides of (3.4) are plotted in Figure 3. The left-hand side is a straight line with intercepts a(r(X)/b(Y)),
and 1/a. This last intercept is by definition fixed, while the first one
varies with the parameters of r(X) and b(Y). The right-hand side
of (3.2) is a peaked curve, which describes the external effort in reducing
unemployment. This curve crosses the origin and is asymptotic to the λ axis at high unemployment. The equilibria for
unemployment are defined where the straight line intersects the peaked
unemployment reducing curve. The number and location of these intersections
depends on the parameters of r(X) and b(Y).Figure 3
From low to high unemployment equilibrium.Figure3 describes the case of two equilibria (low and
high unemployment) and also shows that any given change in r(X) and b(Y),
that increases the intercept a(r(X)/b(Y)),
may trigger a jump in the equilibrium rates of unemployment from low to high.
As a consequence, this model is able to explain the evolution of the Spanish
unemployment from a low equilibrium to a high equilibrium in such a short
period of time given any small change in the parameters of r(X) and b(Y).An example of the workings of the
model is the following. The economy is initially at pointA and the
government changes the eligibility rules of unemployment benefits increasing
the pool of unemployed receiving benefits. This makes the function r(X) increase,
holding everything else constant, shifting the slope of the straight line to
the right which triggers a major change from a low to high unemployment
equilibrium as in point B. This is one of the hypotheses entertained by
Blanchard and Jimeno [18] when explaining differences in unemployment
performance in Spain and Portugal.
As mentioned earlier, the democratic transition period was characterized by the
passing of the workers statute and an
extension of unemployment benefits to displaced workers in the late 1970s and
early 1980s (see [19]). From the point of view of our model, this can
trigger a rapid and dramatic increase in unemployment equilibrium.The reverse is also
true, the model is able to generate a jump from high unemployment to low
unemployment, and this case is depicted in Figure4. The economy is initially
in a high unemployment equilibrium and for any slight change in the
parameters of r(X) and b(Y), triggering a fall in a(r(X)/b(Y)),
this leads to a jump from the high to a low unemployment equilibrium. As
an example, the economy is initially at point A and the government
decreases tax progressivity. For countries characterized by industrial
bargaining, such as Spain,
Brunello and Sonedda [20] found that this government action may decrease
unemployment. (See
also Raurich et al. [21] for a model that explains the European unemployment
hysteresis phenomenon in terms of fiscal policies.) Referring to Figure 4, a reduction in tax
progressivity increases b(Y), holding everything else constant, shifting
the straight line to the left, making the economy jump from the high to low
unemployment equilibrium, as in point B.Figure 4
From high to low unemployment equilibrium.The model can be
generalised to represent more complex internal forces, such as a logistic
growth (which implies multiplying the functionF(⋅) by u and
introducing an exogenous term representing the carrying capacity of the labour
market). In this case, the dynamics remain the same as long as a quadratic term
for u appears on the numerator of the external forces function. An
analogous case is analyzed by Ludwig et al. [22], who also discuss in depth
the cusp catastrophe that characterizes this type of phenomena. A cusp
catastrophe describes the case of a quickly adjusting variable and two
parameters. In the case of our model, this variable is unemployment and r and b are the parameters (the classic references
of catastrophe theory are Thom [23] and Zeeman [24]).These simple nonlinear
dynamics can also encompass the case of three equilibria, one of them is an unstable
equilibrium. Figure5 represents this case. The economy is initially at point A,
which is an unstable equilibrium. Any positive or negative shock can then move
the economy towards the stable equilibria, which are the high or low
unemployment equilibria (points B and C, resp.). All that is
necessary for this large unemployment change is a temporary shock. In this respect,
the model can generate dynamics similar to those emphasised by the unemployment
hysteresis literature (see
Blanchard and Summers [25] and Røed [26]). This is because temporary shocks can have
permanent effects on unemployment. This is an important feature of the model.Figure 5
Unstable equilibrium.The empirical literature
on unemployment dynamics has distinguished between three competing views,
namely, the “unique” natural rate Friedman-Phelps hypothesis, the structuralist
hypothesis represented by Phelps [15], and the hysteresis hypothesis (See,
amongst many others, Camarero et al. [27]). Although hysteresis is usually represented as a
unit root in unemployment, Jaeger and Parkinson [28] have shown that all that
is necessary for hysteresis is that temporary shocks change the equilibrium
unemployment. For the structuralist hypothesis, however, changes in equilibrium
unemployment stem from permanent shocks, which in our model are generated by
changes to the parameters governing external or internal labour market forces.
Hence our model is capable of generating unemployment dynamics that encompass
these two hypotheses (it
is also possible, e.g., that a very large temporary shock changes the
incentives that policy makers face, leading to a permanent change in, e.g.,
unemployment eligibility, the level of protection, and bargaining systems. This
seems to be a relevant hypothesis for the Spanish case). The Friedman-Phelps natural rate case
would simply arise in the particular case in which there are no changes in the
parameters governing functions r(X) and b(Y).


## 4. Reconciling Empirical Results with the Model

Our empirical results show that over the 1965–2007 time horizon,
the Spanish unemployment jumped from regime 1 to regime 2 to regime 3, and then
back to regime 2 (as in Figure2). How can these three regimens be reconciled
with the two stable equilibria of the topology envisaged by the formal model and
by Figures 3, 4, and 5?Evidence and theory appear to be at odds since the latter allows for
either a single high stable equilibrium unemployment, or for a single low
stable equilibrium unemployment, or for two stable (one high and one low)
equilibria and one intermediate unstable equilibrium. In other words there are
three regimens and at most two stable equilibria (we thank one of the referees and an associate
editor for this observation).
Here we try to reconcile the existence of three regimes for Spanish
unemployment with our theoretical arguments.In order to tackle this issue consider Figure6, which reproduces Figure 5 and includes a new straight line in which we assume that parameter a has changed exogenously to a′,a′>a, and the new straight line  has intercepts a′(r(X)/b(Y)),
and 1/a′. Notice that the intercept 1/a′ is to the left of the intercept 1/a. The figure has now three stable equilibria, including one stable
equilibrium for intermediary unemployment.Figure 6
Three stable equilibria.The reconciliation between our empirical results and the model runs along
the following lines. The economy is initially at low unemployment equilibrium,
L. There is an exogenous change in the parametera to a′, which leads the
economy to a new intermediary, and stable, unemployment equilibrium, I. In that
case, any given change in r(X) and b(Y)—that increases
the intercept a′(r(X)/b(Y))—may trigger a
jump in the equilibrium rates of unemployment from intermediary to high, H. The
economy may stay at the high unemployment equilibrium for a while and new
changes in r(X) and b(Y) that decrease the intercept a′(r(X)/b(Y)) may trigger another jump in the equilibrium
rates of unemployment from high to intermediary. Indeed, one can also think of
these dynamics as a result of several changes in parameter a alone as, for instance, a change from a to a’ and then back to a. This explanation shows that the
formal model can also encompass a case with 3 stable equilibria as long as we
allow for parameter a to change
exogenously.


## 5. Concluding Remarks

The Spanish unemployment over the last 40 years is characterized by three
regimes of low, medium, and high unemployment and by a fast transition between
them. This paper puts forward a simple nonlinear dynamic model that is able to generate
multiple unemployment equilibria and rapid transition from low to high
unemployment. Additionally, the model is able to represent, with a small number
of parameters, the main determinants of unemployment as represented by the
fundamentals of labor market and macroeconomic and institutional factors. The model is capable of generating
unemployment dynamics that encompass the “unique” natural rate hypothesis, the
structuralist hypothesis, and the hysteresis hypothesis.



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