Putting the Cycle back into Business Cycle Analysis
Paul Beaudry∗, Dana Galizia†and Franck Portier‡§
July 2016
Version 7.3 (incomplete)
Abstract
There is a long tradition in macroeconomics suggesting that market economies endogenously go through periods of booms and busts, with booms sowing the seeds of
the subsequent busts. One way such an idea can be captured mathematically is as a
limit cycle. For several reasons, limit cycles play almost no role in modern quantitative
business cycle analysis. In this paper we argue that limit cycles, or more precisely
stochastic limit cycles, offer an empirically relevant way of understanding post-war US
business fluctuations. To make the case, we proceed in three steps. First we re-examine
the spectral properties of several cyclically sensitive variables such as hours worked,
unemployment and capacity utilization. For all these series we document the presence
of an important peak in the spectral density at a periodicity of approximately 40 quarters. We then present a general class of models which has the potential of explaining
such observations. In particular this class of models, where complementarities play a
key role, embeds both the possibility of explaining such observations using traditional
mechanism or alternatively resulting from limit cycle behavior. Finally, we present and
estimate a New Keynesian type model which incorporates the elements presented in
our general structure and which is aimed at capturing elements of an accumulationliquidation cycle. The result of this estimation suggests that stochastic limit cycles
may offer a simple unified explanation to key cyclical properties of macroeconomic
data especially hours worked, interest rates and interest spreads.
Key Words: Business Cycle, Unemployment, Limit Cycle; JEL Class.: E3, E32,
E24
∗
Vancouver School of Economics, University of British Columbia and NBER
Department of Economics, Carleton University
‡
Toulouse School of Economics, Université Toulouse 1 Capitole and CEPR
§
The authors thank Jess Benhabib and Kiminori Matsuyama for helpful discussions. The authors would
also like to thank seminar participants at Banque de France, University of Manchester, Pompeu FabraToulouse “skiminar”, NBER Summer Institute, UCL, UCLA, UCSD, the University of Pennsylvania and
Northwestern University for comments. This is a heavily revisied version of a paper previously circulated
under the heading ”Reviving the Limit Cycle View of Macroeconomic Fluctuations”
†
1
Introduction
In most modern business cycle models, the underlying economic system is very stable. In
particular, in the absence of shocks, the variables in these systems tend to converge either
to a steady state or to a balanced growth path. In such frameworks, business cycles are
viewed as emerging from shocks—which could be either fundamental or non-fundamental—
that disturb an otherwise stable system. However, it is well known that this is not the only
framework that could give rise to business cycles. In particular, it may well be that economic
forces naturally produce cyclical phenomena; that is, in the absence of any shocks (including
shocks to agents’ beliefs) economic forces by themselves may favor recurrent periods of high
economic activity followed by periods of low economic activity. This sort of outcome will
arise, for example, if the underlying economic system embeds a limit cycle.1 In such a
framework, irregular business cycles can emerge from these underlying regular forces when
combined with shocks that move the system away from an attracting orbit.
The idea of self-sustaining trade cycles, to use the terminology of the early years of
macroeconomics, can be found in the dynamic version of the Keynesian theory proposed by
Kalecki [1937], and more formally later on in the nonlinear versions of Samuelson’s [1939]
accelerator proposed by Kaldor [1940], Hicks [1950] and Goodwin [1951].2 In the 1970s and
1980s, a larger literature emerged that examined the conditions under which qualitatively and
quantitatively reasonable economic fluctuations might occur in purely deterministic settings
(see, e.g., Benhabib and Nishimura [1979] and [1985], Day [1982] and [1983], Grandmont
[1985], Boldrin and Montrucchio [1986], Day and Shafer [1987]; for surveys of the literature,
see Boldrin and Woodford [1990] and Scheinkman [1990]). By the early 1990s, however, the
interest in such models for understanding business cycle fluctuations greatly diminished and
became quite removed from the mainstream research agenda.3
1
Informally, limit cycles are indefinitely repeating fluctuations that are capable of emerging in some
dynamic systems.
2
An earlier mention of self-sustaining cycles as a model for economic fluctuations is found in Le Corbeiller
[1933] in the first volume of Econometrica.
3
There are at least two strands of the macroeconomic literature that has productively continued to pursue
ideas related to limit cycles: a literature on innovation cycles and growth (see, for, example Shleifer [1986]
and Matsuyama [1999]), and a literature on endogenous credit cycles in an OLG setting (see, for example,
Azariadis and Smith [1998], Matsuyama [2007] and [2013], Myerson [2012] and Gu, Mattesini, Monnet, and
Wright [2013]).
1
There appear to be several key reasons why interest in limit cycles may have waned.
First, the earlier literature on deterministic fluctuations can be broadly sub-divided into two
categories: models with and without optimizing, forward-looking agents.4 The latter category, which were generally more capable of producing reasonable deterministic fluctuations
than the former, likely fell out of favor as macro in general moved toward more micro-founded
models.
Second, in the category of models featuring optimizing forward-looking agents, the primary focus was on models with a neoclassical, competitive-equilibrium structure.5 Such
models were often found to require relatively extreme parameter values in order to generate deterministic fluctuations. For example, the Turnpike Theorem of Scheinkman [1976]
establishes that, under certain basic conditions met by these models and holding all other
parameters constant, for a sufficiently high discount factor—i.e., for agents that are “forwardlooking” enough—the steady state of the model is globally attractive, so that persistent
deterministic fluctuations cannot appear.6 While in principle this does not rule out deterministic fluctuations completely, in practice the size of the discount factor needed to generate
them was often implausibly low. For example, in a survey of such models by Boldrin and
Woodford [1990], discount factors for several of the models they discuss were on the order of
0.3 or less.7 As the present paper illustrates, however, if one departs from the assumptions
of a neoclassical, competitive-equilibrium environment—for example, if there is a demand
externality—then a discount factor arbitrarily close to one can relatively easily support de4
The first category includes, for example, Benhabib and Nishimura [1979] and [1985], and Boldrin and
Montrucchio [1986], while the latter includes, for example, the pre-1970s examples cited above, as well as
Day [1982] and [1983].
5
While there are some exceptions, they are comparatively rare. Perhaps the clearest example is Hammour [1989], chapter 1, which is focused on deterministic fluctuations in an environment of increasing returns.
Other exceptions include models in the search literature that are capable of generating deterministic fluctuations, such as Diamond and Fudenberg [1989], Boldrin, Kiyotaki, and Wright [1993], Coles and Wright
[1998], Lagos and Wright [2003], Rocheteau and Wright [2013] and Gu, Mattesini, Monnet, and Wright
[2013]. Note however that these search papers were mainly concerned with characterizing the set of possible
equilibria for a particular model (which for some parameterizations included limit cycles), rather than being
focused on cycles directly.
6
See the discussion in section ?? for further details.
7
It is possible in principle to rationalize such low discount factors by choosing a longer period length for
the model. However, if households discount the future with a quarterly discount factor of 0.99 or greater—as
is frequently the case in the business-cycle literature—a factor of 0.3 would be associated with a period
length of 120+ quarters (30+ years). Since the minimum period length of a cycle is two periods, this would
generate cycles on the order of 60+ years, well outside of what is normally thought of as the business cycle.
2
terministic fluctuations in equilibrium.
Third, models producing periodic cycles—that is, cycles which exactly repeat themselves
every k periods—are clearly at odds with the data, where such consistently regular cycles
cannot be found.8 This can be observed by looking at the spectrum of data generated by
such models, which will generally feature one or more large spikes at frequencies associated
with k-period cycles. Spectra estimated on macroeconomic data generally lack such extreme
spikes,9 which suggests much less regularity in real-world cycles. To address this issue, papers from the earlier literature frequently sought to establish conditions under which such
irregular cycles could emerge in a purely deterministic setting via chaotic dynamics.10 While
in a number of cases this was found to be possible, the conditions are often more restrictive
than those required to generate simple periodic cycles. In contrast, rather than restricting
attention to a purely deterministic setting, this paper suggests embedding limit cycle mechanisms into a stochastic environment in which irregular fluctuations emerge as the interplay
between exogenous shocks and endogenous cycles.
This paper motivated with a desire to question the current business cycle theory consensus
which emphasizes the role of a pletora of shocks to explain fluctuations, with limited reliance
of systematic and recurrent endogenous forces. To this end, the main aim of this paper is
to explore whether the concept of a stochastic limit cycle offers a plausible and empirical
relevant alternative way of understanding US business cycle phenomena. Current popular
models of the business cycles, such as the Smets-Wouter model, explain business cycles
mainly as resulting from many different shocks. Such an approach can be considered as
primarily an episodic narrative of the business cycle, with each cycle being driven by a
possibly different shocks and where there is limited linkage from one cycle to the next. The
stochastic limit cycle perspective has the potential of offering a different, more systematic,
interpretation of macro-fluctuations whereby cyclical behavior is seen as part of ongoing
endogenous mechanism reflecting local instability and where shocks play the more limited
8
In Boldrin and Woodford [1990], the authors mention private communication with Sir John Hicks in
which Hicks indicated that he lost interest in endogenous cycle models because actual business cycles were
far from being regular periodic motions.
9
See Figure ??.
10
Roughly, chaotic fluctuations appear in systems where the orbits emanating from two different initial
points typically cannot be made arbitrarily close by choosing those initial points sufficiently close together.
Chaotic systems “appear” random, despite being entirely deterministic.
3
role of only emphasizing, attenuating and possibly postponing the natural tendencies of the
economy. The shocks themselves are not the primary and only cause of the cycle.
To explore the idea of stochastic limit cycles, we proceed by addressing three types
of issues. First, we document features of the US macroeconomic data which motivates our
analysis. In particular, we re-examine the spectral properties of several macroeconomic series
and show that these series exhibit important peaks in their spectral densities at a frequency of
around 40 quarters. We take these observations as highlighting important recurrent cyclical
features that need to be explained and where limit cycle may be one such explanation. It is
important to note that these patterns emergence most clearly when we use high resolution
techniques for the estimation of spectral densities. Second, we present a general class of model
which can explain through different mechanisms this type of spectrum observations, including
though standard impulses-and-propagation mechanisms, or through the more rarely studied
feature of stochastic limit cycles. This section builds on the work of Cooper and John (1988)
by illustrating in a systematic fashion the implications of strategic complementarities for
dynamic environments. As we shall emphasize, when complementarities are not too extreme,
this class of model does not generate multiple equilibrium. Instead, this class of model will
be shown to produce limit cycle behavior with a unique-equilibrium saddle-path structure.
In this sense, we will argue that this class of models offers a rich framework for exploring
a class of explanation to business cycles which is distinct from the idea of indeterminacy.
Finally, in the last section, we present a simple New Keynesian model with financial frictions
and durable good accumulation which fits into our general structure. The model is setup in
a way to be flexible enough to allow for the possibility of limit cycles but it does not impose
them. The type of cyclical behavior the model aims to capture is that generally associated
with what we like to call a accumulation-liquidation-cycle but other may prefer to call it a
credit cycle. In our setup, agents make their consumption decisions to satisfy their Euler
equation, where the interest they face reflecting both the policy rate set by the central bank
and a risk premium. When the stock of durable are sufficiently low (which may include
housing), agents will want to make new purchases and this will stimulate economic activity
and employment. As the economy starts booming, two forces come into play. One the one
hand the policy interest rate may increase as the result of central bank action but on the
4
other hand the risk premium on borrowing may decrease. The net effect of these two forces
drives the dynamics of the system, with the possibility of limit cycles arising. We use this
model as a lens to interpret post war macroeconomic data and especially to see if, when
estimated, it favors a choice of parameters which interprets the data as embedding limit
cycle forces. The model will be estimated using observations on hours work, interest rates
and interest spreads.
The remaining sections of the paper are as follows. In Section 1 we highlight a set of
spectral properties of US data on hours worked, unemployment, capacity utilization, gdp and
TFP which motivate our analysis and that later will be partly used in estimation. In passage,
we will discuss the potential value and tradeoffs of using higher resolution techniques when
estimating spectral densities. In Section 2 we present a simple reduced-form dynamic set-up
where agents both accumulate goods and interact with one another. These interaction can
take the form of either strategic substitute forces or strategic complementarities. In this
framework we will highlight the type of interactions between agents which tends to give rise
to limit cycles. In particular, we show how and when demand complementarities can cause
the steady state of the system to become locally unstable and for a limit cycle to appear
around it as part of a Hopf bifurcation.11,12 We further establish a simple condition under
which this limit cycle will be attractive. In this section, we downplay the role of expectation
of future actions as to make the analysis as simple as possible. We also use this section to
introduce the notion of a stochastic limit cycles and to illustrate the effects of shocks in such
a setup. In Section 2.6 we extend the setup to include forward looking elements which allows
us to discuss condition under which equilibrium behavior can generate limit cycles as the
unique saddle-path stable equilibrium. In Section 3.1, we extend a standard three equation
New Keynesien model in a manner that allows for the features highlight with our reduced
11
Since our model is formulated in discrete time, the bifurcation we consider is more appropriately referred
to as a Neimark-Sacker (rather than Hopf) bifurcation. Nonetheless, we will typically follow convention in
applying the term “Hopf bifurcation” to both continuous and discrete environments.
12
Informally, a Hopf bifurcation—which may occur in both continuous and discrete formulations—is characterized by a loss of stability in which the resulting limit cycle involves rotation around the steady state
in two-dimensional phase space. Discrete (but not continuous) systems may also produce limit cycles in a
one-dimensional setting via a “flip” bifurcation, in which case the system “jumps” back and forth over the
steady state. As discussed further in Section ??, for several reasons we will largely focus our discussion
around the dynamics associated with Hopf bifurcations.
5
form model. For example, the model is extended to an environment where consumption
services come for both the purchase of new goods as well from the services of accumulated
durable goods. We introduce complementarities in the set up by allowing for an interest
rate spread on household borrowing this is counter-cyclical. Section ?? takes the model to
the data to see whether estimation will reveal the presence of a limit cycle or whether the
data is better explained through more traditional channels. Since our estimation framework
allows the limit cycle to compete with exogenous disturbances in explaining the data, we
will be able to assess the extent to which it can reduce the reliance on persistent exogenous
disturbances in explaining business cycle fluctuations. Finally, in the last section we offer
concluding comments.
1
Motivating Observations
In this section we reexamine the cyclical properties of a set of quarterly U.S. macroeconomic variables covering the period 1947:1-2015:2
13
. One potential way of describing the
cyclical properties of stationary data is to focus on their spectrum. The spectrum depicts
the importance of cycles of different frequencies in explaining the data. As is well known,
if the spectrum of a time series displays a substantial peak at a given frequency, this is
an indication of recurrent cyclical phenomena at that frequency
14
. The difficulty in using
spectral analysis with macroeconomic data is that macroeconomic data are generally non
stationary and therefore some detrending procedure is needed to make the series stationary
before looking at its spectrum, This problem is especially acute for quantity variables such
as GDP. However, for labor market variables this problem is less sever as these data can be
considered close to stationary. Accordingly, we begin our exploration of cyclical properties
by focusing on U.S. nonfarm business hours worked per capita. The series is plotted in Panel
(a) Figure 1.
From this figure we can see many movements, but over the entire period, there is very
limited evidence of a trend. For this reason, it may be reasonable to look directly at the
spectrum of this series without any prior transformation. In Panel (b) Figure 1 we present
13
14
The source of these data are discussed in the appendix.
See for example Sargent [1987].
6
Figure 1: Properties of hours worked per capital
(a) Data
(b) Spectral Density
35
-7.85
Level
Various Bandpass
30
-7.9
25
-7.95
Logs
20
-8
15
-8.05
10
5
-8.1
0
4
-8.15
1950
1960
1970
1980
1990
2000
6
24
32
40
50 60
80
Log of Period
2010
Notes:
the spectrum for this series, over the range of periodicities from 4 to 80 quarters. The dark
line on the figure represents the spectrum of the level series , – i.e. without any attempt to
remove low frequency movements which reflect non-stationarities.15 .
Since we do not want to claim this this data is necessarily stationary, on the figure we
also plot the spectrum for different pre-transformations of this series where we removed very
low frequency movements using a low pass filter. In particular, the greys lines on the figure
represent the spectrum of the series after in it has first been detrended using a lowpass filter
aimed at removes frequencies greater than 1/100 quarters. In fact, we plot a whole sequence
of such series where we sequentially use the band pass filter to remove periodicities from 100
to 200 quarters. The first feature we want to emphasize from this figure is that, if one is
interested in the spectral properties of this series at periodicities below 60 quarters, these
properties are very robust to whether or not we remove very low frequency movements from
15
We obtain nonparametric power spectral density estimates by computing the discrete Fourier transform
(DFT) of the series (DFT) of X using a fast Fourier transform algorithm, and then smoothing it with a
Hamming kernel. One key element in this estimation is the number of point of the DFT, that determines
the frequency resolution. In order to be able to observe spectral density for frequencies corresponding to
periodicity 32 to 50 quarters, we choose a frequency resolution of 125.59 pHz (see the appendix for more
details).
7
the series.16
What does Panel (b) Figure 1 reveal about the cyclical properties of hours worked? To
us, the dominant features that emerges from Figure 2 is the distinct peak in the spectral
density at around 40 quarters. With this peak being much more pronounced that anything
arising at periodicities lower that 32 quarters. This suggest that a substantial fraction of
movements in hours worked reflect some type cyclical force with a periodicity of about 10
years. It is interesting to notice that this spectral hump is mainly contained in the range
of 32-50 quarters, and therefore just slightly out of the traditional range of business cycle
fluctuations that focuses on fluctuations between 6 and 32 quarters. Note also that this
hump, while at a slightly lower frequency than traditional business analysis, is not capturing
the type of medium run phenomena emphasized in work by Comin and Gertler [2006] who
focus indistinctly on cycles between 32 and 200 quarters. In our opinion, this hump should be
considered as part of business cycle phenomena and it suggests that the traditional definition
of the business cycle may be slightly too narrow. To make this case, in Panel (a) Figure 2
we plot the hours worked series after having removed fluctuations above 60 quarters using a
band pass filter. On the figure, we also emphasize NBER recession periods in grey. As can be
seen, the movements in hours including these slight lower frequencies match extremely well
with the standard narrative of the business cycle as opposed to reflecting lower frequency
movement distinct from the regular dating of business cycles.
To further support the notion of a peak in the spectrum of work input around the
periodicity of 40 quarter, in Panel (b) Figure 2 and Panel (c) Figure 2 we redo the same
exercise for two other measures of work activity. To be more precise, in Figure ?? we report
the spectrum for total hours worked per capita and in Panel (c) Figure 2 we report the
spectrum for the unemployment rate. In addition, in Panel (d) Figure 2 we report the
spectrum of a capital utilization measure. In all three cases, we plot both the spectrum
for the untransformed data, as well as plotting a set of spectrums were we first remove low
frequency movements. We highlight in dark grey the band of frequencies corresponding to
periodicity 32 to 50 quarters, and cut the spectrum at periodicity 60 quarters. For higher
16
If one is interested at looking a frequencies below 60 quarters for a series, but one is worried about
non-stationary properties of the data, these figures suggest that first detrending the series using a low pass
filter that removes frequencies above 100 may be an appropriate way to proceed.
8
Figure 2: Cyclical properties
(a) Filtered NF hours
(b) Spectrum total hours
25
Level
Various Bandpass
6
20
4
2
15
%
0
10
-2
-4
5
-6
0
4
-8
1950
1960
1970
1980
1990
2000
6
24
32
40
50
60
Log of Period
2010
(c) Spectrum of Unemployment
(d) Spectrum of Capacity utilization
40
4.5
Level
Various Bandpass
4
Level
Various Bandpass
35
3.5
30
3
25
2.5
20
2
15
1.5
10
1
5
0.5
0
0
4
6
24
32
40
50
4
60
6
24
Log of Period
Log of Period
Notes:
9
32
40
50
60
periods, as we have seen in Panel (b) Figure 1, lowpass filtering is likely to affect the spectrum
estimate. As can be seen, in all these figures we see a distinct peaks in the spectra around
40 quarters, and these peaks are apparent regardless of whether or not we first remove very
low frequency movements. Together Panel (b) to (d) in Figure 2 indicate that the aggregate
utilization of workers and capital by firms in the US exhibit important recurrent cyclical
phenomena at approximately ten year intervals. To explain such phenomena, one needs
to rely on a theory where substantial cyclical forces are present. While there are different
mechanisms which can explain such observations, we will explore in subsequent section the
plausibility and relevance stochastic limit cycle forces as a potential candidate.
Our observations of a distinct peak in the spectrum of a set of macroeconomic variables
may appear somewhat at odds with conventional wisdom. In particular, it is well known, at
least since Granger [1966], that several macroeconomic variables do not exhibit such peaks,
and for this reason the business cycle is often defined in terms of co-movement between
variables instead of reflecting distinct cyclical timing. This view of business cycle dynamics
depends on the variables one chooses to examine. We have focused on variables, we which
we like to call cyclically sensitive variables, where the size of business cycle fluctuations are
large in relation to trend movements. For such variables, the breakdown between trend and
cycle is less problematic and one can more easily detect spectrum peaks. In contrast, if one
focuses on quantity variables, for example GDP, one does to easily detect any such peaks.
To see this, in Figure 3 we report the same information we reported before regarding data
spectrum but in this case the series is real per capita GDP. Here we see that spectrum of the
un-treated data and the filtered data have very little in common. Since it does not make much
sense to report the spectrum of non-detrended GDP, because it is clearly non-stationary, in
Panel (a) Figure ?? we focus on the spectrum of GDP after we have removed movements
at periodicities greater than 100 quarters. The spectra in Panel (a) Figure ?? are in line
with conventional wisdom: even when we have removed very low frequency movements, we
still do not detect any substantial peak in the spectrum of GDP around 40 quarters. How
can this be? What explains the different spectral properties of output versus hours worked?
The main explanation to this puzzle lies in the behavior of TFP. In Panel (b) Figure ?? we
plot the spectrum of TFP after having removed low frequency movements in the same what
10
we have done for GDP and other variables. What is noticeable about the spectral density
of TFP in Panel (b) Figure ?? is the quick pick up it exhibits just above periodicities of 40
quarters. As with GDP, we do not see any marked peaks in the spectral density of TFP. The
interesting aspect to note is that if we add the spectral density of hours worked to that of
TFP, we almost exactly get that of GDP. This suggests that looking at the spectrum of GDP
may be a much less informative way to understand business cycle phenomena that looking
at the behavior of cyclical sensitive variables such as hours worked and capacity utilization.
Instead, GDP likely captures two distinct processes: the business cycle process associated
with the usage of factors and a lower frequency process associated with movement in TFP.
For this reason, we believe that business cycle analysis may gain by focusing more closely
on explaining the behavior of cyclically sensitive variables such as employment, which is less
likely contaminated by lower frequency movement in TFP, and this is precisely what we aim
to do.
Figure 3: Spectral Density of U.S. GDP per capita, Levels and Various Bandpass Filters
400
Level
Various Bandpass
350
300
250
200
150
100
50
0
4
6
24
Log of Period
Notes:
11
32
40
50
60
Figure 4: Spectral density for GDP and TFP
(a) Filtered GDP
(b) Filtered TFP
15
35
Various Bandpass
Various Bandpass
30
25
10
20
15
5
10
5
0
0
4
6
24
32
40
50
60
4
Log of Period
6
24
32
40
50
60
Log of Period
Notes:
2
Demand Complementarities as a source Limit Cycles: A simple Reduced-Form Model
In this section we present a simple reduced-form dynamic model aimed at illustrating how
and when limit cycles can emerge in an environment with demand complementarities and
how this can create a peak in the spectral density of a variable. As we show, the key
mechanism that allows the model to generate limit cycles is the interplay between demand
complementarities and the dynamic accumulation. We begin with a reduced-form setup so
as to highlight the generality of this mechanism, regardless of the source of the precise microfoundations that drive agents’ interactions. In the next section, we will provide a structural
model which fits into the class.
In addition to establishing that the model can, under fairly general conditions, produce a
Hopf bifurcation associated with an attractive limit cycle, an important goal of the analysis
in this section is to show that this happens even when we restrict the strength of the demand
complementarities to be too weak to create static multiple equilibrium. In fact, as we make
clear, the strength of the demand complementarities necessary for a limit cycle to appear in
this environment is always less than that needed to generate multiple equilibrium.
12
2.1
The Environment
Consider an environment with a large number N of agents indexed by i, where each agent
can accumulate a good Xit , which can be either productive capital or a durable consumption
good. The accumulation equation is given by
Xit+1 = (1 − δ)Xit + Iit ,
0 < δ < 1,
(1)
where Iit is agents i’s investment in the good. Suppose initially that there are no interactions
between agents and that the decision rule for agent i’s investment is given simply by
Iit = α0 − α1 Xit + α2 Iit−1 ,
(2)
where all parameters are strictly positive and 0 < α1 < 1, 0 < α2 < 1. In this decision rule,
the effect of Xit on investment is assumed to be negative so as to reflect some underlying
decreasing returns to capital accumulation, while the effect of past investment is positive so
as to reflect sluggish response that may be due, for example, to adjustment costs.17
When all agents behave symmetrically, the aggregate dynamics of the economy are given
by the linear system:
It
Xt
α0
It−1
α2 − α1 −α1 (1 − δ)
.
+
=
0
Xt−1
1
1−δ
{z
}
|
(3)
ML
The stability of this system is established in the following proposition.
Proposition 1. Both eigenvalues of the matrix ML lie strictly inside the unit circle. Therefore, system (3) is stable.
All proofs are given in the appendix. According to Proposition 1, the dynamics are
extreme simple, with the system converging to its steady state for any starting values of
Xit = Xt and Iit−1 = It−1 . We now add agent interactions to the model and study how is
the dynamics are affected.
17
We focus here on a system with two state variables. We do this since we need a system with at least two
state variables for the possibility of a Hopf bifurcation to arise. It is our conjecture that limit cycles that
emerge from Hopf bifurcations are more likely to be relevant for business cycle analysis than those arising
from flip bifurcations (which can arise in one-dimensional systems).
13
2.2
Adding Interactions Between Agents
To generalize the previous setup in order to allow for interactions between individuals, we
modify the investment decision rule to
P
Iit = α0 − α1 Xit + α2 Iit−1 + Et [F
Ijt
],
N
(4)
while keeping the law of motion for X (equation (1)) unchanged. Hence, in this setup agent
actions depend their expectation of other agents actions, where Et [] is the expecatition
operator. We assume that the function F (·) is continuous and differentiable at least three
times and that F (0) = 0. The function F (·) captures how the actions of others, summarized
P
by the average level of investment It ≡
Ijt /N , affect agent i’s investment decision Iit . For
example, the function F (·) can capture implicitly the price increase induced by the demand
of others if F 0 (·) < 0, or can capture demand complementarities if F 0 (·) > 0. As we will
focus on symmetric equilibrium and since there are no stochastic driving forces, we can drop
the expectation operator and treat the system as deterministic. In this formulation, we are
assuming that agents take the average actions in the economy as given, so that (4) can be
interpreted as agent i’s best-response rule to the average action.
Figure 5 illustrates this best-response rule for two different values of the intercept. Note
P
j
that in this diagram the intercept is given by αt = α0 −α1 ∞
j=0 (1−δ) Iit−1−j +α2 Iit−1 , which
is a function of the history of the economy prior to date t. Thus, past investment decisions
determine the location of the best-response rule, which in turn determines the current level
of investment, which then feeds into the determination of the next period’s intercept, and so
on. In order to rule out static multiple equilibria—that is, multiple solutions for It for given
values of Iit−1 and Xit —we assume that F 0 (It ) < 1.18 Thus, we are restricting attention
to cases where demand complementarities, if they are present, are never strong enough to
produce static multiple equilibrium.
In what follows we consider only symmetric equilibria, so that we may henceforth drop
the subscript i. We make the additional assumption that α2 < α1 /δ so that a steady state
necessarily exists and is unique, and let I s and X s denote the steady state values of I and
Under the condition F 0 (It ) < 1, the static equilibrium depicted in Figure 5 is generally viewed as stable
under a tâtonnement-type adjustment process. This stability property is not the focus of the current paper.
Instead we are interested in the explicit dynamics induced by the system (1) and (4).
18
14
Figure 5: Best-Response Rule for Two Different Histories
Iit
Iit = It
Iit = αt0 + F (It )
Iit = αt + F (It )
αt0
αt
It
Notes:
plots the best-response rule (equation (4)), Iit = αt + F (It ), with αt ≡ α0 −
P This figure
τI
0
α1 ∞
(1
−
δ)
it−1−j + α2 Iit−1 . The intercepts αt and αt correspond to two different histories of
j=0
the model.
15
X. Note that the condition α2 < α1 /δ will always be satisfied when δ is sufficiently small,
since both α1 and α2 are strictly positive.
Our goal is to examine how the dynamics of the system (1) and (4) are affected by the
properties of the interaction effects, and especially what conditions on F (·) will give rise to
a Hopf bifurcation that is associated with the emergence of an attractive limit cycle.
2.3
The Local Dynamics of the Model with Interactions Between
Agents
We now consider the dynamics implied by the bivariate system (1) and (4). In order to
understand those dynamics, it is useful to first look at local dynamics in the neighborhood
of the steady state. The first-order approximation of this dynamic system is given by
h
i
!
!
α1 (1−δ)
α1 (1−δ)
α2 −α1
α2 −α1
s
s
−
It
I
1
−
I
+
X
t−1
1−F 0 (I s )
1−F 0 (I s )
1−F 0 (I s )
= 1−F 0 (I s )
+
.
Xt
Xt−1
1
1−δ
0
|
{z
}
(5)
M
The eigenvalues of the matrix M are the solutions to the quadratic equation
Q(λ) = λ2 − T λ + D = 0,
where T is the trace of M (and also the sum of its eigenvalues) and D is the determinant of
M (and also the product of its eigenvalues). The two eigenvalues are thus given by
s 2
T
T
λ, λ = ±
−D
2
2
(6)
where
T ≡
α2 − α1
+1−δ
1 − F 0 (I s )
(7)
α2 (1 − δ)
.
1 − F 0 (I s )
(8)
and
D≡
When F 0 (I s ) = 0, the model dynamics are locally the same as in the model without
demand complementarities, and in particular, as noted in Proposition 1, the roots of the
system lie inside the unit circle in this case. More generally, we may (locally) parameterize
F by F 0 (I s ) and ask what happens as F 0 varies.
16
Geometric analysis:
It is informative to consider a geometric analysis of the location
of the two eigenvalues of the linearized system (5). This is done in Figure 6, which presents
the plane (T, D). A point in this plane is a pair (Trace, Determinant) of matrix M that
corresponds to a particular configuration of the model parameters (including F 0 (I s )). We
have drawn three lines and a parabola in that plane. The line BC corresponds to D = 1,
the line AB to the equation Q(−1) = 0 (⇔ D = −T − 1) and the line AC to the equation
[ at least one eigenvalue has a
Q(1) = 0 (⇔ D = T − 1). On the perimeter of triangle ABC,
modulus of one. We review in the appendix why both eigenvalues of the system are inside
[ while at least one eigenvalue is outside the unit
the unit circle when (T, D) is inside ABC,
[ Whether the eigenvalues are real or complex depends
circle when (T, D) is outside ABC.
2
on the sign of the discriminant of the equation Q(λ) = 0, which is given by ∆ ≡ T2 − D.
The parabola in Figure 6 corresponds to the equation ∆ = 0 (⇔ D = T 2 /4). Above the
parabola, the eigenvalues are complex and conjugate, while on or below the parabola they
are real. The possible configurations of the local dynamics can then be easily characterized
by considering the location of (T, D) within this diagram.
Without any restrictions on F 0 (I s ), the steady state can be locally stable, unstable, or a
saddle, with either real or complex eigenvalues. Proposition 1 proves that when F 0 (I s ) = 0,
the steady state values correspond to points E or E 0 (depending on the parameters) that
[ Furthermore, it is easy to verify that D is always greater than zero, so that
are inside ABC.
[ As an example, in Figure 6 we have put
the steady state must lie in the top part of ABC.
E and E 0 in the region of stability with complex eigenvalues.
As F 0 (I s ) varies, the eigenvalues of the system will vary, implying changes to the dynamic
behavior of I and X. From equations (7) and (8), assuming α1 6= α2 ,19 we may obtain the
following relationship between the trace and determinant of matrix M :
D=
α2 (1 − δ)
α2 (1 − δ)2
T−
.
α2 − α1
α2 − α1
(9)
Therefore, when F 0 (I s ) varies, T and D move along the line (9) in (T, D)-space, which allows
for an easy characterization of the impact of F 0 (I s ) on the location of the eigenvalues, and
therefore also of the stability of the steady state. We need to systematically consider the two
19
The case where α1 = α2 is addressed in the appendix.
17
cases α2 > α1 and α2 < α1 , as the line (9) slopes positively in the former case and negatively
in the latter.
Let us consider first the strategic substitutability case where F 0 (I s ) is negative; that is,
where the investment decisions of others have a negative effect on one’s own decision. In
that case, it is clear from equations (7) and (8) that
lim
F 0 (I s )→−∞
T =1−δ
and
lim
F 0 (I s )→−∞
D=0
We will denote the point (1 − δ, 0) by E1 on Figure 6. Note that E1 lies inside the “stability”
[
triangle ABC.
Let’s assume that the steady state without strategic interactions is E and α2 > α1 . When
F 0 (I s ) goes from 0 to −∞, the point (T, D) moves from E to E1 along the line given by
equation (9). This movement corresponds to the half-line denoted (a) on Figure 6. As E
[ and because the interior of ABC
[ is a convex
and E1 both belong to the interior of ABC,
[ The
set, any point of the segment [E, E1 ] also belongs to the interior of triangle ABC.
same argument applies if parameters are such that the steady state corresponds to E 0 and
α2 < α1 , with the relevant half-line being (a0 ). Thus, the following proposition holds:
Proposition 2. As F 0 (I s ) varies from 0 to −∞, the eigenvalues of M always stay within
the complex unit circle, and therefore the system remains locally stable.
Proposition 2 indicates that, when the actions of others play the role of strategic substitutes with one’s own action, the system always maintains stability. Since Walrasian settings
are typically characterized by strategic substitutability, this is one reason why dynamic Walrasian environments are generally stable.
We now turn to exploring how the presence of strategic complementarities (i.e., when
F 0 (I s ) > 0) affects the dynamics of the system. In Figure 6, a rise in F 0 (I s ) beginning from
F 0 (I s ) = 0 corresponds to movement along the line given by equation (9), starting from
the point E or E 0 and in the opposite direction to E1 . This movement is denoted by the
half-lines (b), (b’) or (b”) in Figure 6. It can also be easily verified that, as F 0 (I s ) gets
18
Figure 6: Local Stability when F 0 (I s ) ∈ (−∞, 1)
D
B
(b’)
(b)
1
E0
E
(b”)
(a’)
-2
C
-1
E1 (a)
1-δ
1
2
T
A -1
Saddle, real eigenvalues
Stable, real eigenvalues
Unstable, complex eigenvalues
Stable, complex eigenvalues
Unstable, real eigenvalues
Notes: This figure shows the plane (T, D), where T is the trace and D the determinant of matrix
M . The points E and E 0 correspond to two possible configurations of the model without demand
\
externalities. According to Proposition 1, those points are inside the triangle of stability ABC.
They are arbitrarily placed in the zone where the two eigenvalues are complex. E and arrows (a)
and (b) correspond to the case where α2 > α1 ; E 0 and arrows (a’), (b’) and (b”) to the case where
α2 < α1 . The case α1 = α2 is investigated in the appendix.
19
[ thereby causing
closer to one, (T, D) will necessarily cross the perimeter of triangle ABC,
the steady state to change from being locally stable to being unstable. The location of the
crossing depends on parameters.
Consider first the case where α2 > α1 , and assume that the steady state corresponds to
E on Figure 6. Under our earlier assumption that α2 < α1 /δ (which helped guarantee a
unique steady state), it can be verified that the half line (b) will never cross line segment
AC, which is the line segment associated with the largest of two real eigenvalues being equal
to one. Thus, when α2 > α1 , (b) must cross line segment BC; that is, the point at which
the system loses stability as F 0 (I s ) increases must be associated with complex eigenvalues.
Now consider the case where α2 < α1 , and assume that the steady state corresponds to
[ under two
E on Figure 6. In this case, (T, D) can cross the perimeter of triangle ABC
different possible configurations. If the slope of line (9) is sufficiently negative, it will cross
line segment BC, at which point the eigenvalues will be complex. This is the case drawn
in the figure as half line (b’). On the other hand, if the slope of line (9) is negative but
sufficiently flat, the point F 0 (I s ) = 0 will be associated with a point in (T, D)-space like E 0
rather that in E. In that case, as shown by half-line (b”) in Figure 6, as F 0 (I s ) increases
(T, D) will cross line segment AB, which is the line segment associated with the smaller of
two real eigenvalues being equal to negative one.
2.4
Bifurcations and the Occurrence of Limit Cycles
Our graphical analysis of the previous subsection shows that the presence of demand complementarities can change the qualitative dynamics of the system. In particular, if the
complementarities become strong enough the system will transition from being locally stable to being locally unstable. In the theory of dynamical systems, a change in local stability
when a parameter varies is referred to as a bifurcation. Bifurcations are of interest since
there is a close relationship between the nature of a bifurcation and the emergence of limit
cycles. We formalize the nature of possible bifurcations for our bivariate system (1) and (4)
in the following proposition.
Proposition 3. As F 0 (I s ) increases from zero towards one, the dynamic system given by
(1) and (4) will eventually become locally unstable. In particular:
20
− If α2 > α1 /(2 − δ)2 , then a Hopf (Neimark-Sacker) bifurcation will occur.
− If α2 < α1 /(2 − δ)2 , then a flip bifurcation will occur.
A flip bifurcation occurs with the appearance of an eigenvalue equal to negative one, and
a Hopf bifurcation with the appearance of two complex conjugate eigenvalues of modulus
one. The proof of Proposition 3 is given in the appendix, and involves establishing conditions
under which line (9) crosses line segment BC (for the Hopf bifurcation) or line segment AB
(for the flip bifurcation) as F 0 (I s ) increases from zero. In either case, the bifurcation that
occurs as F 0 (I s ) increases will be associated with the emergence of a limit cycle. In the
case of a flip bifurcation, the limit cycle that emerges close to the bifurcation point will be
of period two, i.e., will involve jumps back and forth over the steady state every period.
Such extreme fluctuations are unlikely to be very relevant for business cycle analysis.We
therefore henceforth focus on the more interesting case from our point of view, which is the
case where the system experiences a Hopf bifurcation. In this case, close to the bifurcation
point there will emerge around the steady state (in (X, I)-space) a unique isolated closed
invariant curve.20 Beginning from any point on this closed curve, the system will remain
on it thereafter, neither converging to a single point nor diverging to infinity, but instead
rotating around the steady state along that curve indefinitely. Further, in contrast to the
flip bifurcation, the cycle that emerges near a Hopf bifurcation may be of any period length
and hence the resulting dynamics appear more promising for understanding business cycles.
The condition on α2 under which a Hopf bifurcation (rather than a flip bifurcation) will
arise according to Proposition 3 may, at first pass, look rather restrictive. In fact, as δ → 1
this condition approaches α2 ≥ α1 , which necessitates a fairly large amount of sluggishness
in investment. However, if δ is small, the condition becomes significantly less restrictive.
For example, as δ → 0 the condition becomes α2 > α1 /4, a simple lower bound on the
degree of sluggishness. Given these considerations, one could loosely re-state Proposition 2
20
We have to this point side-stepped a minor technical issue, which is that, in discrete time, the bounded
non-convergent orbits appearing near a Hopf bifurcation will possess the basic qualitative features of a limit
cycle, but may never exactly repeat themselves and thus may not strictly speaking be limit cycles. For
example, in a bivariate discrete-time system characterized by rotation along the unit circle by θ radians per
period, if θ/π is irrational then the system will never return to the same point twice, despite the clear sense
in which the dynamics are cyclical. We will ignore this uninteresting technicality and apply the term “limit
cycle” broadly to also include the entire isolated closed invariant curve (e.g., the entire unit circle in the
example of this footnote) that emerges.
21
as indicating that if depreciation is not too fast, and sluggishness not too small, then the
system will experience a Hopf bifurcation as F 0 (I s ) increases from 0 towards 1.21
Having established conditions under which a Hopf bifurcation emerges, we turn now to
the question of whether such a limit cycle is attractive; that is, whether the economy would
be expected to converge towards such an orbit given an arbitrary starting point. To use
language from the theory of dynamical systems, a bifurcation may be either supercritical or
subcritical. In a supercritical bifurcation, the limit cycle emerges on the “unstable” side of
the bifurcation and attracts nearby orbits, while in a subcritical bifurcation the limit cycle
emerges on the “stable” side of the bifurcation and repels nearby orbits.
The emergence of a limit cycle is mainly of interest to us if it is attractive, so that
departures from the steady state will approach the limit cycle over time. The conditions
governing whether a Hopf bifurcation is supercritical or subcritical are often hard to state.
However, in our setup, a simple sufficient condition can be given to ensure that the Hopf
bifurcation is supercritical. This is stated in Proposition 4, where we make use of the Wan
[1978] theorem.
Proposition 4. If F 000 (I s ) is sufficiently negative, then the Hopf bifurcation noted in Proposition 3 will be supercritical.
The economics for why increasing F 0 (I s ) will cause the system to become unstable is
rather intuitive. A high value of F 0 (I s ) implies that an individual agent has a large incentive to accumulate more capital at times when other agents increase their accumulation.
This leads to a feedback effect whereby any initial individual desire to have high current
investment—due to some combination of a low current capital stock (the decreasing returns
channel) and a high level of investment in the previous period (the sluggishness channel)—
becomes amplified in equilibrium through a multiplier-type mechanism. When this feedback
effect is strong enough, it will cause small initial deviations from the steady state to grow
over time, pushing the system away from the fixed point. As a result, the economy will tend
to go through repeated episodes of periods of high accumulation followed by periods of low
accumulation, even in the absence of any exogenous shocks. Such behavior contrasts sharply
21
Note also that it is precisely when δ is small that our earlier condition on α2 guaranteeing a unique
steady state (i.e., α2 < α1 /δ) is also likely to be satisfied.
22
with the steady flow of I over time that would be the natural point of rest of the system in
the absence of complementarities.
The requirement that F 000 (I s ) be sufficiently negative for the emergence of an attractive limit cycle can also be related to economic forces. If the best-response function, F , is
positively sloped near the steady state and F 000 (I s ) is negative, then it will take an S-shaped
form.22 Note that Figure 5 was drawn with these features. The intuition for why an S-shaped
best-response function favors the emergence of an attractive limit cycle can be understood as
follows. As noted above, when the system is locally unstable, the demand complementarities
are strong enough near the steady state that any perturbation from that point will tend
to induce outward “explosive” forces. If the best-response function is S-shaped, however,
then as the system moves away from the steady state the demand complementarities will
eventually fade out (i.e., F 0 (I) eventually falls), so that the explosive forces that are in play
near the steady state are gradually replaced with inward “stabilizing” ones. As long as
F 000 (I s ) is sufficiently negative, these stabilizing forces will emerge quickly enough, and an
attractive limit cycle will appear at the boundary between the inner explosiveness region
and the outer stability region. Such a configuration for F is the one we have assumed when
drawing Figure 5.
If instead the best-response function has F 000 (I s ) > 0, then instead of dying out, the
demand complementarities would tend to grow in strength as the system moves away from
the steady state,23 so that inward stabilizing forces do not appear. In this case, when F 0 (I s )
is large enough for the system to become unstable, the Wan [1978] theorem implies the
presence of a subcritical Hopf bifurcation in which a repulsive limit cycle appears just before
the system becomes unstable.24
A parametric example of such an S-shaped function is the sigmoid function F (x) = 1+e1−x for x on the
real line.
23
A parametric example of such a function is the logit function, which is the reciprocal of the sigmoid,
y
and takes the form g(y) = log 1−y
for y ∈ (0, 1).
24
Note that subcriticality of a bifurcation does not necessarily imply global explosiveness on the unstable
side of the bifurcation, nor does it rule out the emergence of an attractive limit cycle in that region. Rather,
the results of this section (including those based on Wan [1978]) are inherently about the local behavior of the
system, where “local” in this case means “to a third-order Taylor approximation on some sufficiently small
neighborhood of the steady state”. Conclusions about the global behavior of the system cannot in general
be inferred from these local results. In particular, subcriticality only implies that if an attractive limit cycle
does emerge on the unstable side, then it must involve terms higher than third order. For example, if we
impose the additional assumptions that limI→0 F (I) = ∞ and limI→I¯ F (I) = −∞ for some I¯ ∈ (0, ∞), then
22
23
The general insight we take away from the Wan [1978] theorem regarding Hopf bifurcations is that attractive limit cycles are likely to emerge in our setting if demand complemenatrities are strong and create instability near the steady state, but tend to die out as one
moves away from the steady state. We may refer to such a setting as one with local demand
complementarities. In an economic environment, it is quite reasonable to expect that positive demand externalities are likely to die out if activity gets very large. For example, if
investment demand becomes sufficient large, some resource constraints are likely to become
binding, causing strategic substitutability to emerge in place of complementarities. Similarly,
physical constraints, such as a non-negativity restrictions on investment and capital or Inada
conditions implying that the marginal productivity of capital tends to infinity at zero are
reasonable considerations in economic environments that will limit systems from diverging
to zero or to negative activity. Such forces will in general favor the emergence of attractive
limit cycles in the presence of demand complementarities.
2.5
Stochastic Limit Cycles
As we have seen, in dynamic environments with accumulation, strategic complementarities
between agents’ action can readily create limit cycles in aggregate outcomes. We showed
this could arise even when individual level behavior favors stability in the sense that it would
converge to stability in the absence of agent interactions. Moreover, in our environment the
complementarities can be consider modest as they imply elasticities less than 1. However, if
the behavior of all agents is deterministic, then the resulting cyclical dynamics will not mimic
that of macroeconomic data as the the patterns will be far too regular. For example, in Figure
7 we report an example of the limit cycle dynamics implied by a simple parameterization
of Equation 4.25 While the dynamics are not those of a perfect sine or cosine wave, the
pattern is nonetheless very regular. For the concept of limit cycles to potentially have a
chance of helping explain macroeconomic fluctuations it is necessary to include exogenous
stochastic driving forces.26 For example, suppose we extend our previous representation of
agents behavior as follows
the system never becomes explosive, even if the Hopf bifurcation of Proposition 4 is subcritical.
25
Give details
26
An alternative root, which de not pursue here, would be to focus on chaotic dynamics.
24
P
Iit = α0 − α1 Xit + α2 Iit−1 + Et [F
Ijt
] + µt ,
N
(10)
where µt is a stationary stochastic process representing an exogenous force affecting
agents’ decisions to invest.27 How does the addition of this exogenous stochastic force affect
equilibrium dynamics? In this case, we can no longer say the the dynamic system exhibits
a limit cycle, instead we need to refer to the notion of stochastic limit cycles. What is
important to understand is that the addition of the stochastic term µt does not simply put
noise around an otherwise deterministic cycle. A potentially wrong interpretation of the
effects of shocks in a limit cycle environment is in depicted in Panel (a) of Figure 8. Instead,
the exogenous force continuously perturbs the limit cycle by creating both phase shifts and
amplitude shifts. To illustrate the effects of the stochastic force on dynamics, we return to
the parameterization used in Figure 7 and but add a shocks.28 . The resulting dynamic system
are presented in Panel (b) of Figure 8. This model would still have a deterministic cycle if no
shocks were present, but now instead of creating a smooth cycle, the model generates data
that look more like that observed for macroeconomic variables. These shocks transform an
otherwise smooth cycle into an outcome path where boom and bust have stochastic durations
and stochastic amplitude. As we will later show, the addition of such stochastic elements
also has the effect of changing the spectrum of a series. In the absence of stochastic elements,
the spectrum implied by a limit cycle with tend to have very extreme peaks. In contrast,
the presence of stochastic elements will tend to smooth out the spectrum. The presence of
shocks will not generally remove a spectral peak, but will make it less pronounced.
2.6
Adding forward looking elements
In the previous section we illustrated how agent interaction can create limit cycles when
agents are accumulating goods and when there is inertia in their behavior. However, the
framework we explored purposefully omitted any forward looking elements on the part of
agents. This omission had the advantage of allowing us to highlight key forces that can give
27
We are restricting attention to the introduction of stochastic elements which enter in a additive form.
Looking at the implications of allowing for stochastic elements which enter in a multiplicative form would
be interesting but is left for future work. As we will focus on symmetric equilibrium and since the stochastic
driving forces is common across agents, we can again drop the expectation operator.
28
Give details
25
Figure 7: Deterministic simulation
2
I
1.5
1
0.5
0
0
20
40
60
80
100
120
period
Notes:
Figure 8: Stochastic Limit Cycles
(a) Wrong Interpretation
(b) Effects of shocks
2
2
1.5
1.5
1
I
xt
0.5
0
1
-0.5
0.5
-1
-1.5
0
-2
0
50
100
150
0
200
20
40
60
period
t
Notes:
26
80
100
120
rise to limit cycles without needing to simultaneously address issues regarding an extra layer
of multiple equilibrium which may arise when agents are forward looking. In this section we
extend the results of the previous section to the case where individual agent’s decisions take
the form
N
X
Iit
Iit = α0 − α1 Xit + α2 It + α3 Et Iit+1 + Et [F (
)]
N
i=1
(11)
Here the determination of Xt still obeys Xit+1 = (1 − δ)Xit + Iit , and the function F (·)
controls the complementarity. As before, we focus on symmetric equilibrium and assume that
0 ≤ F 0 (·) < 1 so that the complementarity forces are never sufficiently strong to create static
P Iit
29
multiple equilibrium when agents take N
For now we return to a formulation
i=1 N as given.
where we omit any exogenous stochastic driving forces. Note that when we consider an
environment with forward looking agents, the model most often will include a transversality
condition. For this reason, let us assume that model also inlcudes a transversality condition
of the form limt−>∞ β t Xt = 0 (0 < β < 1).
With respect to our previous setup, the only difference in 11 is the inclusion of expectation
term Et Iit+1 . We again want to restrict attention to situations where the parameters are
such that there is a unique steady. The condition α1 > δ(α3 + α2 ) guarantees the absence of
multiple steady states when 0 < F 0 < 1. Note that as δ goes to zero, this constraints reduces
simply to a non-negativity constraint on α1 . We also want to assume that the parameters
are such that in the absence of any agent interaction (F 0 (I) = 0 for all I), the behavior of Iit
is determinate and converges to the the steady state, that is, that the system be saddle path
stable. Having α1 , α2 ∈]0, 1[ and α3 ∈]0, 1] is almost sufficient for this30 and accordingly
when illustrating properties of the system we will focus on situations where α1 , α2 ∈]0, 1[
and α3 are such that in the absence of complementarities the system is saddle path stable.
31
As before, let’s us denote the steady state of the system by I s and let complementarities
We may also want to assume that F 000 < and that F 0 is largest at the steady state.
2
2
We need the additional restriction that if (1 − δ)α2 > α3 then α1 > (1 − δ) + (α2 − α3 )[1 + α
α3 (1 − δ) ] −
α2
(1
−
δ).
α3
31
We are assuming that α3 is no greater than 1. However results from this section extend to the case
where α3 is also allowed to be ∈]0, 2[.
29
30
27
be governed by varying F 0 (I s ) between from 0 to 1. When F 0 (I s ) = 0, we starting from a
situation where the two roots of the dynamic system are within the unit circle and the third
root is greater than 1. This type of dynamic system can potentially give rise to the following
four types of bifurcations as F 0 (I s ) goes from zero to 1.32
1. The root that is great than one enters the unit circle
2. The two roots within the unit circle are or become complex and simultaneously leave
the unit circle (Hopf).
3. One of the roots within the unit circle become greater than 1 (Flip)
4. One of the roots within the unit circle become smaller than -1 (Fold)
The first bifurcation possibility corresponds to the case where the equilibrium paths of
the system becomes locally indeterminate as complementarities increase, in the sense of generating a continuum of multiple equilibrium outcomes. We will call this an indeterminancyinducing bifurcation. This is the type of bifurcation emphasized in the sunspot literature,
for example see Benhabib and Farmer (1993). The second bifurcation corresponds to a Hopf
bifurcation. The third possibility is a Flip bifurcation and the fourth is a Fold bifurcation.
The question that interests us most is under what condition a Hopf bifurcation arises as this
gives rise (locally) to limit cycles. However, before discussing this case, it is useful to first
recognize that in our setup case 1 and 4, while conceptually possible, never arise as indicated
by Proposition 5
Proposition 5. As 0 < F 0 (I s ) < 1 goes from 0 to 1, the dynamic system governed by 11
and 1 will not produce an indeterminancy-inducing bifurcation nor a fold bifurcation.
Proposition 5 is interesting as it implies that, although we have extended our framework
to include forward looking behavior in the presence of complementarities across agents,
this does not give rise to (self-fulfilling) multiple equilibrium in our setup. This may seem
surprising given that the literature on indeterminate dynamic equilibrium often seems to
draw on complementarities to generate multiple equilibrium. There are two reasons for this.
32
We focus here only on the first bifurcations of the system, that is, bifurcation that arise starting from
a situation where the system is initially has two roots inside the unit circle and one root outside.
28
In our set up, agents are forward looking but they also exhibit a form of sluggish behavior
captured by the term lit−1 . Such sluggishness make it harder for the equilibrium outcome to
exhibit indeterminacy as the system cannot react as aggressively to change in expectations.
Second, we are restricting attention to the case where α1 > δ(α2 + α3 ), as this eliminates
the possibility of multiple steady states. If however we were to remove this condition, then
the emergence of multiple dynamic equilibrium paths can emerge. The parameter set where
this could arise will nevertheless be quite small as long as δ is small. For our purpose, it
seems reasonable to maintain the assumption of that α1 > δ(α2 + α3 ) given that it ensures
a unique steady state and it is not very restrictive when δ is small. Proposition 5 further
implies that logical possibility of Fold type bifurcations is also ruled out in our setup (this
echoes our finding in the previous section). This is less surprising as such bifurcation are
generally associated with situations with multiple steady states, which we are eliminating
by assumption.
Proposition 5 leaves us with only two types of possible bifurcation arising as complementarity forces are increased: a Hopf bifurcation and a Flip bifurcation. Note that these were
also the only two types of bifurcation that could arise in our analysis of the previous section
where we omitted forward looking behavior. As previously noted, Flip bifurcations are unlikely to be very relevant in explaining macroeconomic phenomena as they induce cycles of
periodicity 2. In the current setup, such bifurcation can again only arise if α2 is sufficiently
small, as indicated by Proposition 6.
Proposition 6. As F 0 (I s ) goes from 0 to 1, the dynamic system governed by 11 and 1 will
lead to a Flip bifurcation only if α2 <
α1
2−δ
− α3
Since Flips bifurcation are not our focus, for the remaining discussion of this section we
will assume that α2 >
α1
2−δ
− α3 , or in other words, we will assume that individual level
behavior is characterized by sufficient sluggishness to rule out Flip bifurcations. Under this
added conditions, Proposition 5 and 6 imply that the only potential type of bifurcation in
the system governed by 11 and 1 is a Hopf bifurcation.
Let us emphasizes that the nature of the Hopf bifurcation in the presence of forward
looking elements is somewhat different from the case where such a force was non present. In
29
particular, the limit cycles that arises due to a Hopf bifurcation in the presence of forward
looking elements inherits in our setup a saddle path structure. Even if the limit cycle is
attractive, it will only be attractive on a two dimensional manifold. The transversality
condition will then force lt to jump –for any given initial values of Xt and lt−1 – on to the
stable manifold. Figure 9 illustrates the nature of phase diagram governing dynamics of an
attractive saddle-path stable limit cycle. As can be seen in the Figure, a two dimensional
manifold contains the attractive limit cycle, while the remaining unstable root makes all
point off this manifold lead to explosion.33
Figure 9: A Saddle Limit Cycle
Notes: the grey lines are two paths that converge to the limit cycle. The grey zone is the saddle-stable
manifold (which is here locally drawn as a linear plane, but which is not). The black lines are two paths for
which the jump variable has not jumped on the stable manifold, and which are therefore explosive.
We are now in a position to examine under what conditions an increase in complementary
33
The dynamic system we are considering is a three dimensional system. This type of system produced
a saddle path stable Hopf bifurcation which is distinct from the notion from the notion of the emergence
of indeterminacy. In contrast, if we were considering a two dimensional system, a Hopf bifurcation and
indeterminancy always arises together. This later case has appeared often in macroeconomic literature. In
contrast, the type of configuration implied by Figure9 is less common.
30
forces will lead to a Hopf bifurcation in our forward looking system. Proposition 7 gives a
necessary condition and Proposition 8 gives sufficient condition.
Proposition 7. As F 0 (I s ) goes from 0 to 1, the dynamic system governed by 11 and 1 can
lead to a Hopf bifurcation only if (1 − δ)α2 > α3
Proposition 8. As F 0 (I s ) goes from 0 to 1, the dynamic system governed by 11 and 1 will
α2
experience a Hopf bifurcation if 0 < (1 − δ)2 α32 − ((1 − δ)2 − 1)α2 − α3 − α1 < (1 − δ)( αα23 − 1)
Proposition 7 indicates that complementarities may not be sufficient to create Hopf bifurcation when agents are forward looking. This is in contrast to the case with no forward
looking elements where complementarities could always create local instability In effect, the
proposition indicates that the amount of sluggishness in the system needs to be sufficient
important (as governed by α2 ) relative to the forward looking behavior (as governed by α1 )
for complementarities to potentially create a Hopf bifurcation. For example, if α2 < α3 then
a Hopf bifurcation cannot arise. This condition will can be helpful when exploring whether
or not a particular of model may allow for limit cycles.
Proposition 8 provides a sufficient condition for the emergence of a Hopf bifurcation due
to complementarities in the presence of forward looking agents. This condition is not easily
interpretable. In order to get an idea on their meaning, in Figure 10 we illustrate the set of
points in the parameter space which can gives rise to a Hopf bifurcation as F 0 (I s ) goes from
0 to 1. The parameter box in the Figure corresponds to the set of α1 , α2 and α3 ∈ [0, 1]. The
light grey balls in the parameter box corresponds to a set of points where a Hopf bifurcation
will arise as F 0 (I s ) goes from 0 to 1.34 This figure is drawn for the case where δ = .01.
Similar figures arise for alternative values of δ. As can be seen on the figure, most of the
point satisfying the necessary condition (1 − δ)α2 > α3 actual support the emergence of a
limit cycle as complementarities forces are increased toward 1. In this sense, the condition
(1 − δ)α2 > α3 is can be seen as almost necessary and sufficient for the emergence of a Hopf
bifurcation, even though Proposition 8 indicates that the actual sufficient condition is more
subtle.
34
The silver balls in the figure represent parameter combinations where Flip bifurcation arise and the black
ball correspond to indeterminate bifurcation. Both these case are in areas ruled out by our assumptions.
31
Figure 10: Nature of Bifurcation (δ = .01)
Notes:
32
3
A New Keynesien Model with the Potential of Hidden Limit Cycles
3.1
The Model
In this section, we extend the workhorse three equation New Keynesien model to include
a set of elements which will allow it to capture forces included in our reduce form model.
This implies extending this simple model to include a form of capital accumulation, some
sluggishness in behavior and most importantly some form of complementarity. In the next
section we will bring this extended model to data, estimate it by simulated method of
moments (SMM) and then see if the resulting parameter estimates suggest or not the presence
of limit cycles forces in driving a set of business cycle observations.
Before we present the equations of the model, it is helpful to first briefly describe the
phenomena we are trying to embody with it. The model is aimed at capturing elements
generally associated with what we like to call the accumulation-liquidation cycle, but other
may prefer to call it a credit cycle.35 The narrative for such a cycle has many precedents in
the literature, whereby our formulation is only one possible interpretation. The main source
of complementarity in our model will run through the determination of the risk premium on
borrowing faced by households, and the main capital variable captures the accumulation of
household capital in the form of durable goods and/or houses. The interest rates faced by
households reflect both the behavior of the central bank and a risk premium. A limit cycle
can potentially arise in the setup when the process of accumulation interacts sufficiently
strongly with the determination of the risk premium. For example, when the economy is
perturbed from its steady state, instead of going back to its steady state, it may converge –
under certain parametrizations– to a boom bust cycle where consumption expenditures are
high for several periods, and the accumulation of goods is rapid when the risk premium is
low; the boom episode is eventually followed by a bust period where expenditures are low
due to the large accumulation of durable goods (or houses) and the simultaneous higher risk
premium. It is important to recognize that we are not developing a model that imposed the
presence of such boom bust cycles, but we are instead focusing on an environment which
35
WE are reluctant to call it a credit cycle as we adopt a representative agent setup where not credit is
actually extended in equilibrium.
33
is flexible enough to possibly capture such forces if they are present in the data. For most
parameter values the model will not produce limit cycles, it will only be for particular values
that limit cycles will arise. In the development of the model, we choose to keep all aspects
as simple as possible to keep a close connection with the reduced form model of previous
section. A more thorough derivation of a similar structure, where we are more explicit about
the source of market imperfections which cause complementarities, can be found in a earlier
version of this paper.36
The formal environment we consider is one with a representation household who can
purchase consumption services from the market. The preferences of the representative agent
P
i are given by j β j U (Cit+j −γCt+j−1 ) with U 0 (·) > 0, U 00 (·) < 0, 0 ≤ γ < 1. Cit represent the
consumption services purchased by household i in period t. This preference structure assumes
the presence of external habit which will be the source of sluggish in agent behavior.37 The
household’s problem takes the form
max
X
Et β j U (Cit+j − γCt+j−1 )
j
subject to
Ct+i + (1 + rt−1 )Bt + Ptx Xt+1 = yt + Bt+1 + rtx Xt + Pt Xt
where Bt+1 represents the borrowing of the household at time t to be repaid in period
t + 1, rt represent the nominal interest rate faced by the household at the time of borrowing,
yt is labor income and Xt represents the stock of of durable goods (or houses) held by the
household. Households are assume to buy all their consumption services, including those
flowing from durable goods, from the market. Even though households may hold durable
goods, they do not consume them directly. Instead they rent them our to firms at a rental
rate rtx per period. The firms in turn use the rented stock of durables, as well as labor, to
produce the consumption services. Under these conditions, the households Euler equation
takes the familiar form:38
36
See Beaudry, Galizia and Portier (2015).
We assume that habit is external to the individual only for simplicity. Extending the analysis to have
internal habit is not difficult.
1+r
38
We have here used the approximation 1+Π
= 1 + r + Π.
37
34
U 0 (Ct − γCt−1 ) = βt Et U (Ct+1 − γCt )(1 + rt − Πt+1 )
(12)
where Πt is the inflation rate and β is a discount factor which we allow to be stochastic.
In contrast to the baseline New Keynesian model, we will not assume that the interest rate
faced by households is equal to the policy rate set by the central bank. Instead we will
assume that the interest rate faced by households is equal to the policy interest rate (which
we denote it ) plus a risk premium rtp as given below:
(1 + rt ) = (1 + it + rtp )
(13)
The environment we want to consider in one where the risk premium rtp potentially
changes with the state of the economy. In fact, it is through this risk premium that complementarities between agents will arise. In particular, we want to allow for the risk premium
on loans to possibly decrease when the labor market is tight and it is less risky to lend to
households. To keep the model as simple as possible, we directly allow for a counter-cyclical
risk premium by positing it to be a non-increasing function of level of employment in the
economy, where the level employment is denoted by Lt . We could alternative present an
explicit mechanism to justify this mechanism, but such an addition complicates presentation
and we believe would not add much insight to the properties we will focus upon.39
rtp = R(lt )
R0 (Lt ) ≤ 0
(14)
In terms of central bank behavior, we assume that the central bank follows a slightly
modified Taylor type rule where interest rule reacts to expected inflation and to expected
market tightness as given by
it = Φ1 Et Πt+1 + Φ2 Et G(Lt+1 ) Φ1 > 0 G0 (L) > 0 and Φ2 > 0
(15)
By choosing to have the central bank react to expected developments to this economy, as
opposed to reacting only to contemporary variables, this will allow us to simplify the model
39
Note that the level of employment in this model can also be considered a measure of the output gap.
Hence, whenever we are referring to employment we could alternatively refer to the output gap.
35
considerably without loosing the flavor of standard New Keynesian models. In particular,
since our aim is to use this model to explain the dynamics of a set of real variables, and
since we would like to say away from the general debate of whether or a not a Phillips
curve properly calibrated to micro-economic data can explain inflation movements; we will
assume that the central bank sets Φ1 = 1. This parameterization implies that the central
bank varies its policy rate such that the expected real interest increases with expected labor
market activity. The attractiveness of this assumption is that it creates a model with a block
recursive structure where the real side of the economy can be determined independently of
the inflation rate. The inflation rate remains a functions of all the real variables but is not
affect them. The degree of price stickiness then governs the extent to which inflation varies
with economic activity. By varying the degree of price stickiness, inflation in the model
can become arbitrarily volatile or arbitrarily stable, but this margin does not affect other
variables. This convenient assumption will allow us to focus on the implications of the model
for real variables while allowing us to sidestep debates about inflation and the degree of price
stickiness necessary to match the data. All we need to assume regarding price setting for
our purposes is that there is some stickiness so that produced consumption services remain
demand determined.
We could at this point close the model by specifying that the production of Ct comes
exclusively from currently produced goods. However, if such were the case, the model would
not use Xt and there would be no role for accumulation forces. Since we want to allow for
accumulation forces, and more precisely we want to capture dynamics that may arise due to
the accumulation and liquidation of household capital (which could represent either houses
or durable goods), we will allow consumption services to come from either the production of
new goods or from the service flow of existing durables as follows:
Ct = sXt + F (Lt θt )
(16)
where Xt is the stock of durable goods, sXt is the service flow from durables and F (lt θt )
is the current production of goods. Without loss of generality we can set s = 1. The term
θt in the production of new goods represents productivity. The accumulation of durables is
in turn given
36
Xt+1 = (1 − δ)Xt + ψF (Lt θt ),
where we assume that a fraction
ψ
1−δ
(17)
< 1 of newly produced goods are durables.40 The
firms, who produce consumption services, are subject to nominal rigidities. However, as
we have assume that Φ1 = 1, we can bypass the need to explicit about the firms’ pricing
problem, and its implication for inflation, as the realizations of inflation do not feed back on
the determination of the real variables of interest to us.41
The model can be reduced to the accumulation 17 plus the Euler equation which we can
rewrite as
U 0 (Xt + F (Lt θt ) − γ(Xt−1 + F (Lt−1 θt−1 ))) = βEt U (Xt+1 + F (Lt+1 θt+1 ) − γ(Xt + F (Lt θt )))
(18)
−σ(1 + Φ2 G(Lt+1 ) + R(Lt ))
The attractive feature of Equation 18 is that it is in a form very similar to Equation
11. To see this even more clearly, let us posit the following simple functional forms. Let us
z
assume that the utility function is U (z) = − exp− Ω
0 < Ω, the production of new goods
is given by F (Lθ) = ln(Lθ), the policy function G(Lt ) = ln lt and θ is constant, then under
these conditions 18 can be rewritten as42
lt =
1
ψ
((1 − ΩΦ2 )Et lt+1 + γ(1 −
)lt−1 − κXt − Ω(R̃(lt ) − βt ))
1+γ−ψ
1−δ
where κ = (δ −
δγψ
)
1−δ
(19)
> 0 and R̃(lt ) = R(explt ). Here we have also added a stochastic
driving force to the model through shocks to ln β.
Equation 19 now has a form identical to that of Equation 11 as long as ΩΦ2 < 1. In
effect, one can interpret this Euler Equation as giving a structural interpretation to behavior
40
Our modeling structure implies that agents do not decide independently on the speed of accumulation
of durables and the purchase of new goods; instead these are in fixed proportions. This keeps the model
very tractable and gives it a structure almost identical to that we presented in previous sections. Allowing
for a more general treatment which decouples these two decisions is not difficult but makes the model more
complex and less directly comparable to the previous section. We have chosen the simpler root as to favor
more transparency regarding the functioning of the model.
41
Add some details?
42
In this derivation we used the approximation E ln z = ln Ez and that ln(1 + z) = z.
37
explored in the previous section. Since this equation is of the form we previously analyzed,
we know that it can potentially generate endogenous cyclical behavior in the form of limit
cycles. For this arise, we would need– among other conditions– that R̃0 (lt ) be sufficient
negative near the steady state. In other words, for limit cycles to arise we need the cyclical
sensitivity of the risk premium to be sufficient strong and negative. If this is the case, then
this model can in principle exhibit a type of accumulation-liquidation cycle. In such a case
the model’s steady state would be unstable and the forward looking agents in the model
would find it optimal to bunch their purchasing decisions. During booms, agents would
consume and accumulate durables. This would create a strong labor market which implies
that lending to household is less risky, which rationalizes a low risk premium on loans and
high purchases by households. Eventually the accumulation of goods would be sufficiently
important that new purchases would start to decline, which creates a bust in employment and
the risk premium on loans increases. The bust period acts as a liquidation period induced by
the high accumulation during the boom combined with high risk premium on loans during
the bust. The bust then comes to an end when the stock of durables is sufficiently depleted
and the purchasing of more new goods becomes desirable once again. While this narrative
has a similar flavor to that often associated with indeterminancy and multiple equilibrium,
it is important to recognize that in our framework such boom bust cycles do not require
expectation effects to be strong enough as to create self-fulfilling expectations.
Equation 19 is specified in terms of employment. Since it is more common in New
Keynesian models to focus output gaps , it is worth noticing that this equation can be
readily be interpreted as an equation for the (log) output gap. We prefer to specify it terms
of employment as the mapping to data is direct.
From Equation 19 we see that employment depends positively on both future and past
employment, as both these affect future and past consumption. These effects reflects those
commonly found in consumption Euler equations. In addition to these effects, Equation 19
includes the effect of the state variable Xt , whereby higher levels on Xt decrease employment since for the same consumption level, less employment is required. It is through this
channel that the model captures liquidation effects. Finally, employment decision exhibit
a complementarity structure through the risk premium. Increased employment lowers the
38
risk premium on borrowing, which favors more consumption and the purchase of new goods
which in turn stimulates employment. As long as −σ R̃0 is smaller than one, such a feedback
effect will not create static indeterminacy but, as we have seen, it may create limit cycles.
Our extended New Keynesian model jointly determines employment, the policy rate and
the the risk premium. One of the implication of this model is that interest rates and interest
rate spreads should share some cyclical properties to that of employment. For example, if
dynamics of employment are determined by this model and these dynamics induce a peak in
the spectrum of employment, then the policy rate set by the central bank and the interest
rates faced by private agents should also exhibit similar peaks in their spectrum. Hence,
before bringing this model to the data in a formal manner, it is desirable to first examine
the spectral properties of interest rates and the risk premium to see if they exhibit spectral
peaks somewhat similar to those observed for employment.
3.2
Spectral properties of the interest rates and interest spreads
In Panel (a) and (b) of Figure 11, we report an estimated of the spectral density of the real
return of US Federal fund43 and that of spreed between the Federal fund rate and BBA bonds,
which we will take as a our measure of the risk premium on agent borrowing.44 In each case,
we report the spectrum for the level of the series, as well as spectra for transformed series
where we have removed very low frequency movements using a low pass filter. As before,
the low pass filters is aimed to remove sequentially movements with periodicities below 100
to 200 quarters. The interesting aspect to note in both these figures is that these series
appear to exhibit a spectral peak in the vicinity of 40 quarters – regardless again on whether
or not we try to remove very low frequency movements. The peak around 40 quarters is
most noticeable in our spread series, but is also visible in the behavior of the policy rate.
These similarities are important since the simplicity of our extended New Keynesian model
implies that all three series should have somewhat similar spectral properties. The model
also implies that the policy interest rates should co-move positively with employment while
the interest spread should co-move negatively, and this is what we observe. While these
43
This corresponds to the interest on three month federal funds minus realized inflation over the holding
period.
44
In this footnote gives details about these data.
39
observation certainly do not validate the model, they represent preconditions for such a
model to be potentially relevant.
Figure 11: Spectral densities of the policy rate and the risk premium
(a) Policy rate
(b) Risk Premium
9
12
Level
Various Bandpass
Level
Various Bandpass
8
10
7
6
8
5
6
4
3
4
2
2
1
0
0
4
6
24
32
40
50
4
60
6
24
32
40
50
60
Log of Period
Log of Period
Notes:
3.3
Estimation
To summarize, our extended New Keynesian model can be reduced the following three
equations which simultaneously determine (log) employment lt , the real policy rate irt =
it − Et Πt+1 and a risk premium rtp .
lt =
X
sψ
1
(Et lt+1 + γ(1 −
)lt−1 − κ
ψ(1 − δ)j lt−j − σ(irt + rtp − βt ))
1 + γ − sψ
1−δ
j
(20)
irt = φ2 lt+1
rtp = R̃(lt )
There is one exogenous driving process in this model and this is captured by variations
in βt . We assume that βt follows a stationary AR(1) process βt = ρβt−1 + t , where t is a
white noise process with variance σ 2 .
40
To bring this model to data, the only remaining elements that needs to be specified
is the form of the functions R̃(·). Since we have no clear priors about the form of this
relationship, we approximate it by a third order polynomial. It is quite extreme to limit
the non-linearities of this model to working through only this one channel. Nonetheless we
choose this simplification as it keeps a structure that directly relates to the reduced form
models of the previous sections.
We estimate this model by simulated method of moments, where for each parameter set
the model is solved by a third order perturbation method.45 The estimation is somewhat
involved as it allows for the possibility of limit cycles in a model with forward looking agents.
To our knowledge, such a estimation is novel. The parameters of the model are chosen so
as to minimize its distance to a set of features of the data we have already emphasized. We
focus on three set of observations. The first set corresponds to the spectrum of hours worked
per capita depicted in Figure xx. We aim to fit the point estimates of this spectrum (using
the non pre-transformed data) between periodicities of 2 to 50 quarters. The second set of
observations targeted correspond to the spectrum of the interest rate spread. Here we also
aim to fit this spectrum on the interval between 2 and 50 quarters. Finally, we also target a
set of moments. These moments correspond the co-movement between employment and the
spread, as well as third and fourth order moments associated with these two variables.46 Our
estimation does not directly aim to fit the observation on the policy interest rate. Instead we
will use this dimension to provide a non-targeted dimension on which to judge the model’s
overall fit.
Our estimation procedure estimates 8 parameters. These are γ, Ω, Φ2 , σ, ρ and three
polynomial terms for the function R̃(·) denoted R1 , R2 and R3 . The two parameters δ and φ
are not estimated but are instead calibrate based on the share of consumption expenditures
on both durable goods and housing, and on their joint depreciate rate which imply that ψ
is set to .4 and δ is set to .05. The weighting matrix used in estimation is a simple identity
matrix.
45
Details in an appendix.
Each of these last moments are based on observations that are first filtered using a Band Pass filter
2-50. This is in line with our objective of using the current model to explain macro-economic fluctuations
arising at periodicities of 2-50 quarters.
46
41
The comparison between data targets and model counterparts are presented in Figures
12 and in Panel (a) of Figure 13. The models implication for the non-targeted policy interest
rate in given in Panel (b) of Figure 13. As can be seem from the figures, the models is capable
to replicating many of the salient features of all three variables of the model, albeit with
some exception. For example, the model does not capture the peak in the spectrum of the
interest rate spread observed around 20 quarters. The model also over predicts the variance
of the policy rate. Otherwise, it provides a quite reasonable fit on many dimensions. Note
that we are using only 8 parameters to fit a large set of features of the data. It is especially
interesting to see how well this parsimonious model fit the spectrum of hours work on most
of the interval between 2 and 50 quarters. It does not mimic all bumps and wiggles, but
it nicely captures the overall pattern and especially the marked peak around 40 quarters.
In Figure 14 we report the fit of the model with respect to set of moment targeted by the
estimation. As can be seen in the Figure, the model also does well at capturing this addition
properties of the data the exception of the third moment of risk premium.
The parameter estimates for the model are presented in Table 1 and the implied eigenvalues evaluated at the steady state are given in Table 2. The parameters estimates imply a
habit parameter of .56, which is with the values commonly found in the literature. We also
find that monetary authorities react reasonably aggressively to (expected) economic activity,
with an elasticity of the policy rate to employment of .187. Interesting, the model implies
that the first order effect (as captured by R1 on economic activity on the risk premium is
of the same order of magnitude – but of the opposite sign– to that on the policy rate. The
most interesting finding regarding these parameters estimates is that the shock process is
estimated to have an auto-correlation extremely close to zero. In fact, our estimate is slightly
negative. This implies that the model’s dynamics are almost complete due to endogenous
forces. The one parameter that may be considered somewhat large relative to the literature
is our estimate of Ω, which gives the elasticity of consumption decision to interest rates. Our
estimates suggest this elasticity is close to 5, indicating a very high response of consumption
expenditures to the interest rates faced by consumers. This finding is related to the fact
that, in our setup, agents’ consumption decisions are not only reacting to the policy interest
rate but are also reacting to both the policy rate and a risk premium. Although the sum
42
of these two parts is only moving mildly with activity, the net effect is still important given
that Ω is quite high.
Figure 12: Fit of Hours Spectrum
25
Data
Model
20
15
10
5
0
4
6
24
32
40
50
60
Period
Notes:
To illustrate the deterministic mechanisms implied by the parameter estimates, we report
results obtained when shutting down the estimated shock (i.e., setting σ = 0).47 Panel (a)
Figure 15 plots a simulated 212-quarter sample48 of log-hours generated from the deterministic version of the model. Two key properties should be noted. First, estimation of the
model has yielded a set of parameters that generate endogenous cyclical behavior, and these
cycles are of a reasonable length. This pattern is consistent with Table 2 which indicates
that the steady state is unstable with complex valued eigenvalues. The difficulty that some
earlier models had in generating cycles of quantitatively reasonable lengths may have been
one of the factors leading to limited interest in using a limit cycle framework for understanding business cycles. However, as this exercise demonstrates, reasonable-length endogenous
cycles can be generated in our framework relatively easily precisely because it possesses the
47
We first obtained the parameters from the full stochastic model. The simulation results for the deterministic model were then generated using these coefficients, but feeding in a constant value βt = 0 for the
exogenous process. In other words, agents in the deterministic model implicitly behave as though they live
in the stochastic world. As a result, any differences between the deterministic and the stochastic results in
this section are due exclusively to differences in the realized sequence of shocks, rather than differences in,
say, agents’ beliefs about the underlying data-generating process.
48
This is equal to the length of the sample period of the data.
43
Figure 13: Fit of Risk Premium and Policy rate
(a) Risk Premium
(b) Policy rate
9
12
Data
Model
8
Data
Model
10
7
6
8
5
6
4
3
4
2
2
1
0
0
4
6
24
32
40
50
60
4
6
24
Period
32
40
50
60
Period
Notes:
three key features we highlighted: diminishing returns to capital accumulation, sluggishness,
and local demand complementarities. However, notwithstanding the reasonable cycle length,
it is clear when comparing the simulated data in Panel (a) Figure 15 to the actual data that
the fluctuations in the deterministic version of the model are far too regular when compared
to the data.
Figure 14: Fit of Moments
Erp4
Eℓ4
Erp3
Eℓ3
Data Eℓr
p
Model
-1
-0.5
0
0.5
1
1.5
2
2.5
3
Notes:
These properties of the deterministic version of our estimated model—i.e., with a highly
44
Table 1: Parameter Values
γ
φ2
Ω
ρ
σ
R1
R3
R2
0.56285
0.18709
4.8205
-0.079408
0.73146
0.16286
-0.073629
-0.054307
Notes:
Table 2: Eigenvalues
R(λ)
1.0788
1.0788
2.6426
Im(λ)
|λ|
0.17329 1.0926
-0.17329 1.0926
0
2.6426
Notes:
Figure 15: Estimated model imposing no shocks
(a) Sample path
(b) Spectral Density
60
2
Data
Deterministic Simulation
50
1
0
40
Hours (%)
-1
30
-2
-3
20
-4
10
-5
0
-6
0
50
100
150
200
250
4
300
6
24
Period
Period
Notes:
45
32
40
50
60
Figure 16: Sample Path for Hours Generate by Model with Shocks
4
2
Hours (%)
0
-2
-4
-6
-8
50
100
150
200
250
Period
Notes:
regular cycle—can also be seen clearly in the frequency domain. Panel (b) Figure 15 plots
the spectrum for the deterministic version (dashed line), along with the spectrum for the
data (solid line) for comparison.49 This spectrum exhibits an extremely large peak at around
40 quarters. In contrast, the spectrum estimated from the data is much flatter and has a
peak at a lower frequency.
Re-introducing the estimated shocks into the model, we see a markedly different picture
in both the time and frequency domains. Figure 16 plots a 212-quarter arbitrary sample
of log-hours generated from the stochastic model. While clear cyclical patterns are evident
in the figure, it is immediately obvious that the inclusion of shocks results in fluctuations
that are significantly less regular than those generated in the deterministic model, appearing
qualitatively quite similar to the fluctuations found in actual data. This is confirmed by
the spectrum, which is previously presented in Figure alongside the data spectrum. The
stochastic model clearly matches the data quite well in this dimension, including possessing
a peak near 40 quarters and, as compared to the deterministic model, lacking any large spike.
It should be emphasized that the exogenous shock process in this model primarily accel49
Note that the model was not re-estimated after shutting down the TFP shock. As such, there may be
alternative parameterizations of the deterministic model that are better able to match the spectrum in the
data.
46
erates and decelerates the endogenous cyclical dynamics, causing significant random fluctuations in the length of the cycle while only modestly affecting its amplitude. For example, in
the deterministic version of the model the variance of log-hours is 6.65, while in the stochastic model it is 7.03, implying that almost 95% of the standard deviation of hours is due
to endogenous mechanisms. Moreover, it worth emphasizing once again that the estimated
shocks in the model are close to being iid, indicating that macroeconomic fluctuations can
be explained without a need to rely on persistent exogenous disturbances.
The role of complementarities in the model is extremely important. To see this, in
Figure 17 we plot the spectrum for hours worked implied by the model when we shut down
the endogenous risk premium (ie, we simulate the model using the estimated parameters
with the exception of setting ri = 0 for i = 1, 2, 3). As can be seen, in such a case the
model no longer mimic the properties of the data. In fact, the spectrum is almost flat and it
is very far from the data spectrum. This reflects the fact that, without complementarities,
the small amount of quasi-iid distrubances we estimates makes the outcomes of the model
exhibit very little variance.
Figure 17: Spectrum of model with complementarities turned off
25
Data
Model
20
15
10
5
0
4
6
24
Period
Notes:
47
32
40
50
60
Figure 18: The interest faced by agents implied by the model
4
3
Real Rate (% per annum)
2
1
0
-1
-2
-3
-4
-6
-4
-2
0
2
4
Hours (%)
Notes:
3.4
The relevance of linearity
In our estimation of model we have allowed the complementarities between agents, as captured by the function R̃(lt ), to be non-linear. Allowing for such non-linearity is essential for
the model to potentially produce limit cycles as it is this feature that permits the model to
be locally unstable while not being explosive. Note that we did not impose any restrictions
on these non-linearities besides that they be approximated by the a third order polynomial.
These non-linearities could in principle push the estimation to find the steady state to be
more or less stable than if linearity was imposed. Interestingly we find the system to be
locally unstable as implied by the implied eigenvalues near the steady state as presented
in the Table. If we choose instead to impose linearity, that is if we re-estimate the model
forcing R̃(lt ) to be linear, we find that the models’ steady state appears stable. Two of the
eigenvalues of the system in this case remain complex but the modulus drops from above
one to around .95.50 Finding that the estimation favor stability when estimated imposing
linearity is not surprising given that the data do not suggest explosive behavior. Moreover,
such a linear version of the model is capable to capturing some of the important features
of the spectral densities used in estimation but where it fits – by construction – less well is
50
The third eigenvalues maintains a modulus above 1.
48
with respect to certain higher moments. In particular, the non-linear version of the model
is capable to capturing very well the Kurtosis we observed in the empirical distribution of
hours. The data suggest that hour works exhibit a Kurtosis which implies thinner tails than
that implied by a normal distribution. Having a thin tailed distribution is precisely what
is implied by a limit cycle model as the non-linearities constrain the model’s capacity to
produce outcomes that are too far the steady state. In order to get a sense of the nonlinearity in the model, in Figure 18 we plot the function Φ2 lt + R̃(lt ) over the data relevant
range for lt . This relations captures the extent to which the interest rate in the model faced
by individuals, which reflects both the policy rate plus the risk premium, varies over the
business cycle. The figures is such at the steady this rate is normalized to zero. AS can
be seen, the interest rate faces by agents is close to unresponsive to activity close to the
steady state, and this is what is making the system unstable. In contrast, when far from
the steady state, this interest rate reacts more to economic condition allow the economy to
remain globally stable. In particular, this amount of non-linearity is sufficient to allow the
model to possess an unstable steady and thereby favor a limit cycle interpretation of US
macroeconomic fluctuations.
Conclusion
Why do market economies repeated go through periods of boom and bust? There are at
least two broad classes of explanations. On the one hand, it could be that the economy is
inherently stable and that the only reason for booms and busts is because of the adjustment
process to outside disturbances. On the other hand, is could be that the economy is locally
unstable in that it does not contain forces which push it towards a stable resting position.
Instead the economy internal forces keep it in constant motion but do not lead to explode
nor implode. In such a case, there still are outside disturbances, but these are not the
sole nor primarily driver of the business cycle. Shock in this environment interact with the
endogenous forces as to make fluctuations irregular and hard to predict.
In this paper our aim has been to examine the plausibility and relevance of the this
second, less mainstream, view of macroeconomic fluctuations. To this end, we have made
three contributions. First, we have documented a set of data patterns that should most
49
likely be present if the economy embeds limit cycle forces. In particular, we has shown
that several macroeconomic variables exhibit distinct peaks in the spectrum. These peak
arise around the periodicity of 40 quarters. While such periodicity is somewhat long relative
to the standard definition of the business cycle, we argue that they should be thought as
an integral part of business cycle phenomena as opposed to reflecting some more medium
run behavior. It is important to note that such observations do not imply the presence
of limit cycles, but instead can be taken as evidence of the possibility of their presence.
Second, we present a class of models which can produce limit cycles. Our goal in this section
has been to clarify how readily limit cycles can arise in environments commonly studied
by macroeconomist. The class of models we consider can be considered as the dynamic
extension of the class of models studied by Cooper and John (1988) in their seminal article
which emphasize how forces of strategic complementarities may help explain macroeconomic
behavior. Throughout this section we have aimed to clarified how the concept of limit cycles
is distinct from the concept of indeterminancy and multiple equilibrium (even though the two
can sometime coincide). Finally, in the last section of the paper, we extend a three equation
New Keynesian model in a manner that incorporates the elements emphasized in the second
section. The model is aimed at capturing a simple narrative of endogenous boom-bust cycles
where accumulation behavior can interact with financial conditions to sustain fluctuations.
When we explore the model empirically, we find that the data favor parameter estimates
which support the presence of limit cycle forces, that is, the data favor being explained
through limit cycles forces instead of through more conventional mechanisms. We also find
that shocks remain important in explaining macroeconomic fluctuations, but interestingly
the model only requires close to iid shocks to explain the data instead of the highly persistent
shocks often required in more mainstream models. It needs to be recognized that model we
present in the last section is highly stylized and it is certainly not a full description of
macroeconomic behavior. Nonetheless, it is interesting to see how such a parsimonious and
simple model can capture many features of macroeconomic data once limit cycles are allowed
to be part of the endogenous mechanisms. This, we believe, nicely illustrates the potential
empirical relevance of such a framework.
50
References
Azariadis, C., and B. Smith (1998): “Financial Intermediation and Regime Switching
in Business Cycles,” American Economic Review, 88(3), 516–536.
Benhabib, J., and K. Nishimura (1979): “The Hopf Bifurcation and the Existence and
Stability of Closed Orbits in Multisector Models of Optimal Economic Growth,” Journal
of Economic Theory, 21(3), 421–444.
(1985): “Competitive Equilibrium Cycles,” Journal of Economic Theory, 35(2),
284–306.
Boldrin, M., N. Kiyotaki, and R. Wright (1993): “A dynamic equilibrium model of
search, production, and exchange,” Journal of Economic Dynamics and Control, 17(5),
723–758.
Boldrin, M., and L. Montrucchio (1986): “On the Indeterminacy of Capital Accumulation Paths,” Journal of Economic Theory, 40(1), 26–39.
Boldrin, M., and M. Woodford (1990): “Equilibrium models displaying endogenous
fluctuations and chaos,” Journal of Monetary Economics, 25(2), 189–222.
Coles, M. G., and R. Wright (1998): “A Dynamic Equilibrium Model of Search, Bargaining, and Money,” Journal of Economic Theory, 78(1), 32–54.
Comin, D., and M. Gertler (2006): “Medium-Term Business Cycles,” American Economic Review, 96(3), 523–551.
Day, R. H. (1982): “Irregular Growth Cycles,” The American Economic Review, 72(3),
406–414.
(1983): “The Emergence of Chaos from Classical Economic Growth,” The Quarterly
Journal of Economics, 98(2), 201–213.
Day, R. H., and W. Shafer (1987): “Ergodic Fluctuations in Deterministic Economic
Models,” Journal of Economic Behavior and Organization, 8(3), 339–361.
Diamond, P., and D. Fudenberg (1989): “Rational expectations business cycles in search
equilibrium,” Journal of Political Economy, 97(3), 606–619.
51
Goodwin, R. (1951): “The Nonlinear Accelerator and the Persistence of Bisiness Cycles,”
Econometrica, 19(1), 1–17.
Grandmont, J.-M. (1985): “On Endogenous Competitive Business Cycles,” Econometrica, 53(5), 995–1045.
Granger, C. W. J. (1966): “The Typical Spectral Shape of an Economic Variable,”
Econometrica, 34(1), 150–161.
Gu, C., F. Mattesini, C. Monnet, and R. Wright (2013): “Endogenous Credit
Cycles,” Journal of Political Economy, 121(5), 940 – 965.
Hammour, M. L. (1989): “Endogenous Business Cycles: Some Theory and Evidence,”
PhD Thesis, Massachusetts Institute of Technology.
Hicks, J. (1950): A Contribution to the Theory of the Trade Cycle. Clarendon Press, Oxford.
Kaldor, N. (1940): “A Model of the Trade Cycle,” The Economic Journal, 50(197), 78–92.
Kalecki, M. (1937): “A Theory of the Business Cycle,” The Review of Economic Studies,
4(2), 77–97.
Kuznetsov, Y. A. (1998): Elements of Applied Bifurcation Theory, vol. 112 of Applied
Mathematical Sciences. Springer, New-York, 2 edn.
Lagos, R., and R. Wright (2003): “Dynamics, cycles, and sunspot equilibria in ’genuinely dynamic, fundamentally disaggregative’ models of money,” Journal of Economic
Theory, 109(2), 156–171.
Le Corbeiller, P. (1933): “Les systemes autoentretenus et les oscillations de relaxation,”
Econometrica, 1(3), 328–332.
Matsuyama, K. (1999): “Growing Through Cycles,” Econometrica, 67(2), 335–347.
(2007): “Credit Traps and Credit Cycles,” American Economic Review, 97(1),
503–516.
(2013): “The Good, the Bad, and the Ugly: An inquiry into the causes and nature
of credit cycles,” Theoretical Economics, 8(3), 623–651.
Myerson, R. B. (2012): “A Model of Moral-Hazard Credit Cycles,” Journal of Political
Economy, 120(5), 847–878.
52
Rocheteau, G., and R. Wright (2013): “Liquidity and asset-market dynamics,” Journal
of Monetary Economics, 60(2), 275–294.
Samuelson, P. (1939): “Interaction Between the Multiplier Analysis and the Principle of
Acceleration,” The Review of Economic Studies, 21(2), 75–78.
Sargent, T. J. (1987): Macroeconomic Theory. Emerald Group Publishing Limited, second
edn.
Scheinkman, J. (1976): “On Optimal Steady States of n-Sector Growth Models when
Utility is Discounted,” Journal of Economic Theory, 12(1), 11–30.
(1990): “Nonlinearities in Economic Dynamics,” The Economic Journal, 100(400),
33–48, Conference Papers.
Shleifer, A. (1986): “Implementation Cycles,” Journal of Political Economy, 94(6), 1163–
1190.
Wan, Y.-H. (1978): “Computations of the stability condition for the Hopf bifurcation of
diffeomorphisms on R2 ,” SIAM Journal of Applied Mathematics, 34(167-175).
Wikan, A. (2013): Discrete Dynamical Systems with an Introduction to Discrete Optimization Problems. Bookboon.com, London, U.K.
53
Appendix
A
A.1
Proofs of Propositions
Proposition 1
The two eigenvalues of matrix ML are the solution the equation
Q(λ) = λ2 − T λ + D = 0,
(A.1)
where T is the trace of the ML matrix (and also the sum of its eigenvalues) and D is the determinant
of the ML matrix (and also the product of its eigenvalues). The two eigenvalues are therefore given
by
s T
T 2
λ, λ = ±
−D
(A.2)
2
2
where
T = α2 − α1 + (1 − δ)
(A.3)
D = α2 (1 − δ).
(A.4)
and
From (A.4), we have that λλ ∈ (0, 1). Therefore, if the eigenvalues are complex, then their modulus
is between zero and one, and thus they are both inside the unit circle. If the two eigenvalues are
real, then they have the same sign and at least one of them is less than one in absolute value. From
(A.3), we have that λ + λ ∈ (−1, 2). Therefore, if the eigenvalues are both negative, then they are
both inside the unit circle. If they are both positive, let λ be the larger eigenvalue, and suppose
λ ≥ 1. Given that λλ < 1, we have λ < λ1 and thus λ + λ = T < 2 implies that λ + λ1 < 2, which
in turn implies (1 − λ)2 < 0. This is not possible and hence we must have λ < 1. Since λ is the
largest of two real positive eigenvalues, both eigenvalues must lie inside the unit circle.
A.2
Proposition 2
With demand complementarities, the trace and determinant of matrix M are given by
T =
α2 − α1
+ (1 − δ)
1 − F 0 (I s )
(A.5)
α2 (1 − δ)
.
1 − F 0 (I s )
(A.6)
and
D=
From equations (A.5) and (A.6), we have the following relationship between T and D:
D=
α2 (1 − δ)2
α2 (1 − δ)
T−
.
α2 − α1
α2 − α1
(A.7)
Therefore, when F 0 (I s ) varies, T and D move along the line (A.7) in the plane (T, D). We have
\ meaning that both eigenvalues of
shown that when F 0 (I s ) = 0, (T, D) belongs to the triangle ABC,
M are inside the unit circle. This corresponds to point E or point E 0 (depending of the configuration
of parameters) in Figure 6.
54
When F 0 (I s ) → −∞, we have D → 0 and T → 1−δ, which corresponds to point E1 in Figure 6.
\ both eigenvalues are inside the unit circle. When F 0 (I s )
As this point is inside the triangle ABC,
goes from 0 to −∞, (T, D) moves along the segment [E, E1 ] or [E 0 , E1 ]. Because both belong to
\ and because the interior of triangle ABC
\ is a convex set, both eigenvalues of matrix M stay
ABC
0
s
inside the unit circle when F (I ) goes from 0 to −∞.
A.3
Proposition 3
A flip bifurcation occurs with the appearance of an eigenvalue equal to -1, and a Hopf bifurcation
with the appearance of two complex conjugate eigenvalues of modulus 1. From (A.6) and (A.5),
we see that when F 0 (I s ) tends to 1 from below, D tends to +∞ and T tends to ±∞ depending on
the sign of α2 − α1 . Therefore, starting either from point E or E 0 (for which F 0 (I s ) = 0), (T, D)
\ At the point where the half-line along (A.7) starting from
will eventually exit the triangle ABC.
\ at least on eigenvalue will have a modulus one.
E (or E 0 ) crosses ABC,
Consider first the case α2 < α1 . In this case, the line (A.7) has a negative slope, and will cross
either segment AB ((a) in the figure) or segment BC ((b) in the figure). In case (a), we will have
a flip bifurcation since the eigenvalues will be real and one of them equal to -1 when crossing the
\ In case (b), we will have a Hopf bifurcation since the eigenvalues will be complex
triangle ABC.
\ We will be in case (b) when D = 1 and T > −2.
and will both have modulus 1 when crossing ABC.
0
s
D = 1 implies F (I ) = 1 − α2 (1 − δ). Plugging this into the expression for T , the condition T > −2
α1
−α1
α1
becomes 1 − δ + αα22(1−δ)
> −2 which can be simplified to α2 > (2−δ)
2 . Therefore, if α2 < (2−δ)2 , we
α1
have a flip bifurcation, while if α1 > α2 > (2−δ)2 , we have a Hopf bifurcation.
Consider next the case α2 > α1 . In this case, the line (A.7) has a positive slope, and could
potentially cross either segment AC or segment BC . If it crosses the segment BC, the eigenvalues
\ so that we will have a Hopf bifurcation. We
will be complex with modulus 1 when crossing ABC,
will be in this case when D = 1 and T < 2. D = 1 implies F 0 (I s ) = 1 − α2 (1 − δ). Plugging
−α1
this into the expression of T , the condition T < 2 becomes 1 − δ + αα22(1−δ)
< 2 which can be
α1
α1
simplified to α2 < δ2 . Therefore, if α1 < α2 < δ2 , we will have a Hopf bifurcation. If α2 > αδ21 ,
then as we increase F 0 we would cross the segment AC. However, this possibility is ruled out by
our assumption that α2 > αδ1 , which was imposed to guarantee a unique steady state.
Finally, in the case α1 = α2 , we always have T = 1 − δ, so that D increases with F 0 (I s ) along a
vertical line that necessarily crosses the segment BC, so that we have a Hopf bifurcation. Putting
all of these results together gives the conditions stated in Proposition 3.
A.4
Proposition 4
For this proposition, we make use of Wan’s [1978] theorem and of the formulation given by Wikan
[2013] (See Kuznetsov [1998] for a comprehensive exposition of bifurcation theory). For symmetric
allocations, our non-linear dynamical system is given by
It − F (It )
α2 − α1 −α1 (1 − δ)
It−1
α0
=
+
.
(A.8)
Xt
1
1−δ
Xt−1
0
To study the stability of the limit cycle in case this system goes through a Hopf bifurcation, we
need to write the system in the following “standard form”
y1t
cos θ − sin θ
y1t−1
f (y2t−1 , y2t−1 )
=
+
,
(A.9)
y2t
sin θ cos θ
y2t−1
g(y1t−1 , y2t−1 )
55
where y1 and y2 are (invertible) functions of I and X. Let µ be the bifurcation parameter (µ =
F 0 (I s ) in our case) and µ0 the value for which the Hopf bifurcation occurs. Define
d=
and
2
a = −Re
(1 − 2λ)λ
ξ11 ξ20
1−λ
d|λ(µ0 )|
dµ
!
1
− |ξ11 |2 − |ξ02 |2 + Re(λξ21 ),
2
where
1
[(f11 − f22 + 2g12 ) + i(g11 − g22 − 2f12 )] ,
8
1
≡ [(f11 + f22 ) + i(g11 + g22 )] ,
4
1
≡ [(f11 − f22 − 2g12 ) + i(g11 − g22 + 2f12 )] ,
8
1
≡
[(f111 + f122 + g112 + g222 ) + i(g111 + g122 − f112 − f222 )] .
16
ξ20 ≡
ξ11
ξ02
ξ21
According to Wan [1978], the Hopf bifurcation is supercritical if d > 0 and a < 0.
We first write (A.8) in the standard form (A.9). Denoting it = It − I s and xt = Xt − X s and
b
F (it ) = F (it + I s ), and recalling that F (I s ) = Fb(0) = 0, we can rewrite (A.8) as
it−1
α2 − α1 −α1 (1 − δ)
it − Fb(it )
.
(A.10)
=
1
1−δ
xt−1
xt
Define H(it ) = it − Fb(it ). Under the restriction F 0 (·) < 1, H is a strictly increasing function, and
is therefore invertible. Denote G(·) ≡ H −1 (·). Adding and subtracting to the right-hand side of
the first equation of (A.10) a first order approximation of G around zero, we obtain
!
α1 (1−δ)
α2 −α1
it−1
m(it−1 , xt−1 )
−
it − Fb(it )
0
s
0
s
1−F
(I
)
1−F
(I
)
=
+
,
(A.11)
xt−1
0
xt
1
1−δ
|
{z
}
M
with
α1 (1 − δ)
α2 − α1
m(it−1 , xt−1 ) ≡ G α1 (1 − δ)xt−1 + (α2 − α1 )it−1 +
xt−1 −
it−1 .
0
s
1 − F (I )
1 − F 0 (I s )
The eigenvalues of M are the solution of the equation
Q(λ) = λ2 − T λ + D = 0,
where T is the trace of the M matrix and D its determinant, with T =
α2 −α1
1−F 0 (I s )
+ (1 − δ) and
α2 (1−δ)
1−F 0 (I s ) .
At the Hopf bifurcation, D = 1 and the two eigenvalues are λ = cos θ ± i sin θ,
p
where θ is the angle between the vector T /2, D − (T /2)2 and the positive x-axis. Let λ be the
D =
eigenvalue with positive imaginary part and λ its conjugate, and let Λ and C be the two matrices
λ 0
Λ≡
0 λ
56
and
cos θ − sin θ
C≡
.
sin θ cos θ
By construction, λ and λ are the eigenvalues of C. We introduce matrices
sin θ
sin θ
VC ≡
−i sin θ i sin θ
and
VM
λ+δ−1 λ+δ−1
≡
1
1
whose columns are eigenvectors of C and M , respectively, and are thus such that C = VC ΛVC−1
−1
−1
and M = VM ΛVM
. We therefore have C = VC ΛVC−1 = VC VM
M VM VC−1 = BM B −1 with
0
sin θ
−1
B ≡ VC VM
=
.
−1 cos θ − (1 − δ)
Let us make the change of variables (y1t , y2t )0 = B × (it , xt )0 to obtain the “standard form” of (A.8)
y1t
y1t−1
cos θ − sin θ
f (y2t−1 , y2t−1 )
,
(A.9)
=
+
y2t
sin θ cos θ
y2t−1
g(y1t−1 , y2t−1 )
with
f (y2t−1 , y2t−1 ) = 0,
g(y1t−1 , y2t−1 ) = −G
γ
γ
1
1
1
y1t−1 − γ2 y2t−1 +
y
−
γ
y
1t−1
2 2t−1
sin θ
1 − F 0 (I s ) sin θ
and γ1 ≡ −α2 (1 − δ) + (α2 − α1 ) cos θ, γ2 ≡ α2 − α1 .
We can now check the conditions for the Hopf bifurcation to be supercritical, namely
d=
and
2
a = −Re
(1 − 2λ)λ
ξ11 ξ20
1−λ
d|λ(µ0 )|
>0
dµ
!
1
− |ξ11 |2 − |ξ02 |2 + Re(λξ21 ) < 0.
2
With µ ≡ F 0 (I s ) as the bifurcation parameter, we have |λ| = det(M ) =
d=
α2 (1−δ)
1−µ ,
so that
d|λ|
(1 − µ) + α2 (1 − δ)
=
> 0,
dµ
(1 − µ)2
as µ = F 0 (I s ) < 1.
Consider now the expression for a. As G(I) is the reciprocal function of I − F (I), we have
G000 =
F 000 (1 − F 0 )2 + 2F 002 (1 − F 0 )
,
(1 − F 0 )4
57
with F 0 < 1. This shows that G000 is an increasing function of F 000 . When F 000 becomes large
000
in absolute
terms and
negative, so does G . In the expression for a, the first three terms,
−Re
(1−2λ)λ
1−λ
2
ξ11 ξ20
− 21 |ξ11 |2 − |ξ02 |2 , are not functions of F 000 , while the last term is
α2 (1 − δ)
Re(λξ21 ) =
16
000
= κG ,
γ12
2
+ γ2 G000
sin2 θ
with κ > 0. If F 000 is sufficiently negative, then so will be G000 , and therefore Re(λξ21 ) and a.
Therefore, d > 0 and under the condition F 000 << 0, we have a < 0, in which case by Wan’s
[1978] theorem the limit cycle is supercritical.
A.5
Proposition 5
Since we are considering the properties of a deterministic system, we can replace Et lt+1 by lt+1 ,
and rewite Equation 11 and 1 as .
(1 − ρ)lt = −α1 Xt + α2 lt−1 + α3 lt+1
Xt+1 = (1 − δ)Xt + It
where 0 ≤ ρ < 1 stands in for F 0 (I s ) and we have omitted constant terms for simplicity.
Now rewrite the l equation as
51
lt+1 = a1 lt + a2 lt−1 + a3 Xt
α2
α1
where a1 = 1−ρ
α3 , a2 = − α3 , a3 = α3
The characteristic equation associated with this system is
λ3 − (1 − δ + a1 )λ2 + (a1 (1 − δ) − a2 − a3 )λ + (1 − δ)a2 = 0
Since this is a cubic equation, it can be factored into the form
(x − d)(x2 − τ x + c) = 0
q
q
2
2
where the roots of (x2 − τ x + c) , are τ2 + τ4 − c and τ2 − τ4 − c. If c >
modulus of the complex roots.
Equating coefficients we have
51
τ2
4 ,
then c is the
τ + d = 1 − δ + a1
(A.12)
c + τ d = a1 (1 − δ) − a2 − a3
(A.13)
cd = −(1 − δ)a2
(A.14)
Note that is formulation embeds the case where Xt+1 = (1 − δ)Xt + ψIt if we redefine α3 = α3 ψ
58
These three equations map the as into the roots of the system. From these equations we can
derives a set of necessary conditions for the types of bifurcation taking place at a first crossing
(were we are starting from an initial saddle situation with the unstable root is d > 1)
A bifurcation involving d = 1 implies
τ + 1 = (1 − δ) + a1
−(1 − δ)a2 + τ = a1 (1 − δ) − a2 − a3
or
δ(a2 − 1) = −δa1 − a3
this can arise with a1 > 0 only if
a3 < δ(1 − a2 )
or
α1 < δ(α3 + α2 )
which is rule out to ensure the absence of multiple steady states. Since both a Fold bifurcation
and an indeterminancy bifurcation involve d = 1, these are ruled out.
A.6
Proposition 6
For a Flip bifurcation we need to have d = −1, which implies that c = (1 − δ)a2 . Replacing this
into Equations A.12 and A.13, we get
τ − 1 = (1 − δ) + a1
(1 − δ)a2 − τ = a1 (1 − δ) − a2 − a3
or
(1 − δ)a2 − 2 + δ − a1 = a1 (1 − δ) − a2 − a3
This last equation will hold with a1 > 0 only if (2 − δ)(a2 − 1) > −a3 or α2 >
A.7
α1
2−δ
− α3
Proposition 7
A Hopf bifurcation implies arise with c = 1. From A.14 this implies that d = −(1 − δ)a2 , and we
know that d > 1 at this bifurcation, therefore −(1 − δ)a2 > 1 or (1 − δ)α2 > α3
59
A.8
Proposition 8
A Hopf bifurcation will arise if the exist at 0 < ρ < 1, such that at c = 1 we have d = −(1 − δ)a2 .
Equations A.12 and A.13 imply that at such a bifurcation we have
τ − (1 − δ)a2 = 1 − δ + a1
1 − (1 − δ)a2 τ = a1 (1 − δ) − a2 − a3
Using these equation to replace τ we get
1 − (1 − δ)2 a2 − (1 − δ)a1 a2 − (1 − δ)2 a22 = a1 (1 − δ) − a1 − a3
expressing this condition in terms of αs and ρ we get
α2
(1 − ρ) =
(1 − δ)2 α32 − ((1 − δ)2 − 1)α2 − α3 − α1
(1 − δ) αα23 − (1 − δ)
Hence, a Hopf bifurcation will arise if the condition in the proposition is met as this guarantees
the existence of a 0 < ρ < 1 which induces the bifurcation.
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