# Capital Demand Driven Business Cycles: Mechanism and Effects

Karl Naumann-Woleske<EMAIL_ADDRESS>Michael Benzaquen
<EMAIL_ADDRESS>Maxim Gusev<EMAIL_ADDRESS>Dimitri
Kroujiline<EMAIL_ADDRESS>Chair of Econophysics and Complex
Systems, Ecole Polytechnique, 91128 Palaiseau Cedex, France  LadHyX, UMR CNRS
7646, Ecole Polytechnique, 91128 Palaiseau Cedex, France  New Approaches to
Economic Challenges Unit (NAEC), OECD, 75016 Paris, France  Capital Fund
Management, 75007 Paris, France  LGT Capital Partners, 8808 Pfäffikon,
Switzerland

###### Abstract

We develop a tractable macroeconomic model that captures dynamic behaviors
across multiple timescales, including business cycles. The model is anchored
in a dynamic capital demand framework reflecting an interactions-based process
whereby firms determine capital needs and make investment decisions on a micro
level. We derive equations for aggregate demand from this micro setting and
embed them in the Solow growth economy. As a result, we obtain a closed-form
dynamical system with which we study economic fluctuations and their impact on
long-term growth. For realistic parameters, the model has two attracting
equilibria: one at which the economy contracts and one at which it expands.
This bi-stable configuration gives rise to quasiperiodic fluctuations,
characterized by the economy’s prolonged entrapment in either a contraction or
expansion mode punctuated by rapid alternations between them. We identify the
underlying endogenous mechanism as a coherence resonance phenomenon. In
addition, the model admits a stochastic limit cycle likewise capable of
generating quasiperiodic fluctuations; however, we show that these
fluctuations cannot be realized as they induce unrealistic growth dynamics. We
further find that while the fluctuations powered by coherence resonance can
cause substantial excursions from the equilibrium growth path, such deviations
vanish in the long run as supply and demand converge.

###### keywords:

Business Cycles , Economic Growth , Interactions-Based Models , Dynamical
Systems , Limit Cycle , Coherence Resonance

##  1 Introduction

> My view was, and still is, that the most urgent current analytical need was
> for a way of fitting together short-run macroeconomics, when the main action
> consists of variations in aggregate demand, with the long run factors
> represented by the neoclassical growth model, when the main action is on the
> supply side. Another way of saying this is that short-run and long-run
> models of macroeconomic behavior need a way to merge in a practical
> macroeconomics of the medium run (Solow, 2005).

The field of economics has long been aware of a conceptual dichotomy between
studies of short-term dynamics and models of long-term growth. An early
distinction was made between the Hicks IS-LM model (1937) and the Solow growth
model (1956). The developments in both approaches have captured important
dynamics at their respective timescales, such as short-term demand effects and
endogenous drivers of long-term growth (e.g. Aghion and Howitt, 1992). Yet it
is not well understood how the dynamics at different timescales are
interlinked and how medium-term disequilibrium dynamics impact the long-term
growth trend of the economy.

Since the World War II, the United States of America alone has faced twelve
recessions. While the severe short-term consequences of these crises are
appreciated, understanding of the long-lasting impact on growth remains
underdeveloped. The pervasive recurrence of booms and busts has thus sparked
research into the linkages between economic volatility and growth (Cooley and
Prescott, 1995; Aghion and Howitt, 2006; Priesmeier and Stähler, 2011; Bakas
et al., 2019). Theoretical as well as empirical investigations have turned out
to be inconclusive, as authors disagree on both the sign and magnitude of the
ultimate effect of volatility on growth.111 We suggest Bakas et al. (2019) for
a comprehensive review of this literature.  Theoretical literature is divided
into two dominant strands that stem from either Schumpeterian notions, in
which volatility is good for growth (based on Schumpeter, 1939, 1942), or the
learning-by-doing concept (based on Arrow, 1962), where volatility is
detrimental to growth. The conflicting theoretical frameworks and ambiguous
empirical findings indicate that new, alternative approaches may be needed to
decipher the genuine nature of the relationship between volatility and growth.
Current literature does not generally consider the impact of the interactions
among economic agents and their collective dynamics on long-term growth. It is
this impact and its underlying mechanisms that we seek to capture and explain.

We are motivated by the micro-to-macro approach of agent-based modeling
(LeBaron and Tesfatsion, 2008; Dawid and Delli Gatti, 2018; Hommes and
LeBaron, 2018) and, especially, the Keynes-meets-Schumpeter class of models
(Dosi et al., 2010, 2015) that study the linkages between endogenous growth
and demand policy. While agent-based models successfully capture many complex
phenomena, they are generally analytically intractable, making the analysis of
the precise mechanics linking volatility and growth difficult. Our approach
remains distinct as we aim to derive a tractable system of equations for the
aggregate dynamics from micro-level interactions.

This paper’s objective is to develop a model of capital demand driven economic
fluctuations, in which interactions among agents to coordinate on economic
outcomes lead to periods of booms and busts, and apply it to examine how
fluctuations affect the economy across different timescales and possibly shape
its long-term growth. Inspired by Keynes (1936), our focus on capital demand
is motivated by the observation that firms’ investment is both pro-cyclical
and volatile (Stock and Watson, 1999), suggesting investment decisions play a
key role in business cycles. We treat investment decision-making as an
interactions-based process whereby firm managers exchange views and affect
each other’s opinions. In other words, we emphasize strategic complementarity
and peer influence that cause managers to coalign their individual
expectations at the micro level. We use the framework developed in Gusev et
al. (2015) and Kroujiline et al. (2016) to describe this interaction process
mathematically and derive the macroscopic equations governing the dynamics of
aggregate capital demand. To close the economy while highlighting the demand-
driven effects, we attach these equations to a simple supply side component
represented by the Solow growth model (1956).

As a result, we obtain a closed-form dynamical system, hereafter the Dynamic
Solow model, which enables us to study a broad range of economic behaviors.
The model’s primary contribution is the identification of a new mechanism of
business cycles that captures their quasiperiodic nature characterized by one
or several peaks in a wide distribution of cycle lengths.

We show that, for economically realistic parameters, the Dynamic Solow model
admits two attracting equilibria that entrap the economy in either a
contraction or expansion.222The 2008 crisis gave new impetus to revisiting the
single equilibrium framework; e.g. Vines and Wills (2020) recently made the
case for moving towards a multi-equilibrium paradigm.  The equilibria are
indeterminate (Benhabib and Farmer, 1999) as both the path to and the choice
of equilibrium depend on the beliefs of the agents themselves. The entrapment
is asymmetric because technological progress, introduced externally, causes
the economy to stay on average longer in expansion than contraction,
contributing to long-term growth. The flow of exogenous news continually
perturbs the economy stochastically and prevents it from settling at either
equilibrium. Over time, the economy tends to drift slowly towards the boundary
between the contraction and expansion regions, making it easier for a news
shock to instigate a regime transition in line with the “small shock, large
business cycle” effect (Bernanke et al., 1996). This endogenous mechanism
generates quasiperiodic fluctuations as it involves both deterministic
dynamics and stochastic forcing.

Such a mechanism, whereby noise applied to a dynamical system leads to a
quasiperiodic response, is known as coherence resonance (Pikovsky and Kurths,
1997). It occurs in situations where the system has long unclosed trajectories
such that even small amounts of noise can effectively reconnect them and thus
create a quasiperiodic limit cycle. Coherence resonance emerges naturally in
bi-stable systems333In dynamical systems, bistability means the system has two
stable equilibrium states., including our model.

The coherence resonance mechanism differentiates the Dynamic Solow model from
preceding research that has often considered limit cycles as the endogenous
source of economic fluctuations.444 The early literature comprises Hicks
(1937), Kaldor (1940) and Goodwin (1951). Later reviews include Boldrin and
Woodford (1990), Scheinkman (1990), Lorenz (1993) and Gandolfo (2009). We also
note Beaudry et al. (2020) as an influential recent investigation.  In
particular, Beaudry et al. (2020) propose an extended Dynamic Stochastic
General Equilibrium model, in which the quasiperiodic character of
fluctuations comes from noise acting directly on a periodic limit cycle. We
argue, however, that coherence resonance may be the preferred route to
generating business cycles as it requires noise only as a catalyst, thus
relying much less on random shocks to reproduce regime variability.
Furthermore, we show that the fluctuations produced by a noise-perturbed limit
cycle, which is as well recovered in a certain parameter range in our model,
dampen long-term growth and unrealistically cause capital demand to diverge
from supply in the long run.

We note that the Dynamic Solow model nests two limiting cases that match those
of previous literature. In the case where capital demand is persistently
higher than supply, the model recovers the exponential equilibrium growth of
the classic Solow model. In the opposite case, where capital demand is
persistently lower than supply, the model exhibits quasiperiodic fluctuations
driven by a coherence resonance mechanism similar to that in Kroujiline et al.
(2019).

We explore the Dynamic Solow model numerically across multiple timescales,
from months to centuries, and identify business cycles as quasiperiodic
fluctuations that most frequently last 40-70 years. These fluctuations may be
associated with Kondratieff cycles if interpreted as investment
driven.555Kondratieff himself attributed these cycles to capital investment
dynamics. This interpretation was further advanced by a number of papers in
the 1980s. Kondratieff cycles are, however, more commonly linked to
technological innovation. There have also been attempts to combine investment
and innovation explanations. For a review see Korotayev and Tsirel (2010).
Korotayev and Tsirel (2010) employ spectral analysis to suggest the existence
of long-term business cycles. However, the academic community remains divided
on this issue and the research has been focused primarily on the fluctuations
in the 8-12 year range. These shorter-term cycles cannot emerge in our model
because it does not include accelerators such as the financial sector or
household debt.

Currently, many macroeconomic models describe an economy in or near
equilibrium. Most prominent is the Dynamic Stochastic General Equilibrium
class of models (see Christiano et al., 2018; Kaplan and Violante, 2018, for
recent reviews). While behavioral limitations and various frictions have been
considered, these models operate in an adiabatic regime where equilibrium is
reached more quickly than the environment changes. In other words, there is
some form of perfect coordination (e.g. market clearing where supply and
demand equate) among all agents at each point in time. Over long timescales
this treatment may be justified, but in the near term coordination failures
are inevitable, leading to pronounced fluctuations and persistent spells of
disequilibrium.

The Dynamic Solow model enables us to study both the disequilibrium
fluctuations and the equilibrium growth. We examine the impact of fluctuations
on growth and show that fluctuations can affect economic expansion over
extended time intervals. However, the deviations from the balanced growth path
disappear with time as demand and supply converge asymptotically in the long
run.

The remainder of this paper is structured as follows. In Section 2 we
introduce and explain the mechanics of dynamic capital demand and the Solow
growth framework within which it rests. Section 3 considers two limiting
cases: first, we obtain the equilibrium growth path when capital demand
exceeds supply; and second, we investigate the demand dynamics and highlight
the mechanism underlying fluctuations when capital supply exceeds demand.
Section 4 formulates and studies the general case of the Dynamic Solow model,
focusing on the analysis of mid-term fluctuations and long-term growth.
Finally, Section 5 concludes by reflecting on the work done and suggests
further avenues of research.

##  2 The Dynamic Solow Model

This section develops the Dynamic Solow model.666M. Gusev and D. Kroujiline
formulated and presented the basic ideas behind the model at the Macroeconomic
Instability seminars of the ”Rebuilding Macroeconomics” project
(www.rebuildingmacroeconomics.ac.uk) run by the National Institute of Economic
and Social Research in London (Gusev and Kroujiline, 2020). The modeling
framework is set out in Section 2.1 and the equations of the model components
are derived in Sections 2.2-2.5.777The code for this model is written in
Python and is available at github.com/KarlNaumann/DynamicSolowModel.

###  2.1 Model Structure

The Dynamic Solow model is illustrated in Figure 1. It consists of a dynamic
demand framework that we propose to describe how firms determine capital needs
and make investment decisions (right loop), to which we attach the familiar
circular income flow of the Solow growth economy (left loop).888 We choose
this supply-side framework for the following reasons: (i) the capital supply
dynamics are less important on the timescales where we expect to find
fluctuations and thus can be modeled approximately; (ii) the assumption that
households save a constant fraction of income is an appropriate leading-order
approximation since it is the first term in the Taylor series expansion of
savings as a general function of income; and (iii) the Solow model is a
parsimonious representation of economic growth, sharing the basics with many
macroeconomic models, which may be helpful to extending our approach to more
sophisticated settings.

Figure 1: A conceptual flowchart of the Dynamic Solow model. Each individual
circle depicts an entity or agent. The circles’ color indicates whether they
belong to the same entity (notably, firm managers and firm factories are both
parts of the firm). Labeled black arrows define the flows between the
respective entities or agents. The orange loops highlight (i) the Solow
economy (left loop) and (ii) the dynamic demand decision-making process
introduced in this section (right loop).

In the Solow economy, households supply capital through savings and firms
convert capital into output for household consumption. Both households and
firms are static decision-makers. Households save a fixed share of income and
firms convert all supplied capital into production. In contrast, we aim to
describe how firms develop a strategic business outlook based on their reading
of the current economic situation and accordingly determine their capital
needs so as to adjust production capacity. Firms thus become active decision-
makers, which results in a dynamically evolving capital demand.

Organizational decision-making is a complex process with competing goals and
targets, often based on industry-standard business planning and operating
procedures (Cyert and March, 1992; Miller and Cardinal, 1994). Without needing
to make firm goals explicit, we posit that corporate decision-making can be
viewed as a composite of two distinct processes occurring on different
timescales. First, there is information gathering and analysis, characterized
by the frequency with which exogenous information such as ad-hoc company news,
monthly statistics releases or quarterly earnings reports becomes available.
Second, there is the formation of firms’ expectations about the future based
on the analysis of collected information, which is then translated into
investment decisions. Note that the strategic aspect of investment decision-
making implies longer timescales than those of information gathering and
analysis.

We model this two-tier decision-making on the microscale by introducing two
classes of agents: analysts who collect and analyze relevant information and
managers who use this analysis to develop a business outlook and make
investment decisions. There are industries where these two classes of agents
actually exist (e.g. analysts and investors in finance), whereas in other
situations this division serves as a metaphor for the different actions
completed by an individual participant. Our objective is to derive the macro-
level equations for aggregate demand from this micro setting.

External information enters the decision-making process at the analyst level.
Initially, we may neglect the cost side and focus solely on revenue
generation, elevating in relevance the expectation of future consumption.
Motivated by recent work on extrapolative beliefs in finance (Greenwood and
Shleifer, 2014; Kuchler and Zafar, 2019; Da et al., 2021), we assume that
analysts base their expectations primarily on the current state of the economy
by extrapolating the consumption growth into the future. As such, we wish to
carve out consumption growth as the most relevant information stream and model
all other news as exogenous noise (treating news shocks similarly to Angeletos
and La’O, 2013; Angeletos et al., 2020; Beaudry and Portier, 2014). We can
further simplify by replacing consumption with production since consumption is
approximated as a constant fraction of production in the model. The resulting
system acquires a feedback mechanism as higher output growth leads to
increasing expectations that cause greater investment, inducing further
increases in output growth and starting the process anew.

On the manager level, we emphasize the impact of the opinions and actions of
competitors on decision-making, following the growing body of research on peer
influence in business (Griskevicius et al., 2008) and strategic
complementarity (Cooper and John, 1988; Beaudry et al., 2020). More
specifically, we assume that managers exchange views within their peer network
with the purpose of coaligning their expectations about the economy.

The Dynamic Solow model employs, as discussed, two different processes for
capital demand and supply: firms determine capital needs dynamically via
individual interactions and economic feedback while households supply capital
in proportion to income. Thus, demand and supply do not necessarily match at
each point in time, which brings us to the discussion of capital market
clearing on different timescales. The dynamic demand discovery process occurs
on timescales much shorter than the timescale of technological growth. At
these short and intermediate timescales – relevant to information gathering,
investment decision-making and production adjustment – prices are rigid and we
expect demand and supply to behave inelastically. However, over long time
horizons in which the economy is advancing along the equilibrium growth path,
prices become flexible and the capital market clears via price adjustment.
Therefore, we expect that demand and supply converge in the long run.

As such, the conceptual framework behind the model is now complete. The
remainder of Section 2 is as follows. Section 2.2 extends the usual equation
for aggregate economic production to include the (shorter) timescales at which
production capacity adjusts. Section 2.3 briefly introduces a representative
household and the capital motion equation. Section 2.4 derives the equations
for aggregate capital demand from the micro-level agent-based formulation
outlined above. Finally, Section 2.5 sets out conditions for capital market
clearing.

###  2.2 Production

We represent aggregate output by a Cobb-Douglas production function that takes
invested capital as an input,999 For simplicity, we normalize the initial
technology level to unity and do not consider labor, which is equivalent to
normalizing the labor component to unity or taking the variables in per capita
terms.  generically written as

$Y=e^{\varepsilon t}K^{\rho},$ (1)

with output $Y$, invested capital $K$, capital share in production $\rho$ and
technology growth rate $\varepsilon$. Equation (1) implies that output adjusts
immediately to any change in capital. In other words, it is only valid on
timescales longer than the time it takes to adjust the production capacity
(e.g. the construction of a new factory or installation of new machinery).
Since we are also concerned with decision-making processes that occur at much
shorter timescales than production adjustment, we introduce a dynamic form of
production

$\tau_{y}\dot{Y}=-Y+e^{\varepsilon t}K^{\rho},$ (2)

where the dot denotes the derivative with respect to time and $1\ll\tau_{y}\ll
1/\varepsilon$ is the characteristic timescale of production capacity
adjustment.101010 In reality, the adjustment periods are asymmetric because it
is easier to reduce capacity than to increase it. For simplicity, we treat
capacity increases and decreases as if they were symmetric.  In the short run,
this equation describes the dynamic adjustment of output to new capital
levels. In the long run, we recover the Cobb-Douglas production form (1) as
$\tau_{y}\dot{Y}$ becomes negligibly small for $t\gg\tau_{y}$.

Finally, we rewrite equation (2) with log variables $k=\ln K$ and $y=\ln Y$ as

$\tau_{y}\dot{y}=e^{\rho k+\varepsilon t-y}-1.$ (3)

###  2.3 Households and Capital Supply

We consider a single representative household that is the owner of the firm
and thus receives $Y$ as income. A fixed proportion of income, expressed as
$\lambda Y$, is saved and the remainder is consumed. This is a convenient
simplification that allows us to focus on the effects of dynamic capital
demand. A constant savings rate can also be viewed as a leading-order Taylor
expansion of household savings as a general function of income, making it a
sensible first approximation.

The total savings are available to firms to invest. We denote them as capital
supply $K_{s}$. The working capital used in production, $K$, suffers
depreciation at a rate $\delta$. As households are the owners of the capital,
the loss $\delta K$ is attributed to the capital supply. Consequently, the
supply dynamics take the form

$\dot{K}_{s}=\lambda Y-\delta K.$ (4)

Setting $k_{s}=\ln K_{s}$, we reformulate equation (4) using log variables as

$\dot{k}_{s}=\lambda e^{y-k_{s}}-\delta e^{k-k_{s}}.$ (5)

###  2.4 Dynamic Capital Demand

In this section, we derive the equations for aggregate capital demand. As set
out in Section 2.1, this derivation is based on a micro-level framework that
divides the firms’ investment planning into two processes occurring at
different speeds: fast-paced information gathering and analysis; and slow-
paced decision-making. We model these processes with two classes of
interacting agents: analysts who collect and analyze relevant information; and
managers who use this analysis to develop their strategic business outlook and
make investment decisions.111111This modeling approach adapts the investor-
analyst interaction framework, developed for the stock market in Gusev et al.
(2015) and Kroujiline et al. (2016), to the macroeconomic context.

In mathematical terms, we consider two large groups of agents: analysts
$i\in\\{1,\dots,N_{h}\\}$ and managers $j\in\\{1,\ldots,N_{s}\\}$, where
$N_{h}\gg 1$ and $N_{s}\gg 1$. Each analyst and manager has a positive or
negative expectation about the future path of production, respectively
$h_{i}=\pm 1$ and $s_{j}=\pm 1$. The agents interact by exchanging opinions.
As a result, the agents influence each other’s expectations and tend to
coalign them. To stay general, we assume analysts and managers interact among
themselves and with each other. These individual interactions drive the
evolution of the macroscopic variables: average analyst expectation $h$
(information) and average manager expectation $s$ (sentiment).

At each moment of time $t$, sentiment $s$ is given by

$s(t)=n_{+}(t)-n_{-}(t),$ (6)

where $n_{+}={N_{s}^{+}}/{N_{s}}$ and $n_{-}={N_{s}^{-}}/{N_{s}}$, with
$N_{s}^{+}$ and $N_{s}^{-}$ representing the respective number of optimists
$(s_{j}=1)$ and pessimists ($s_{j}=-1$). By construction, $s$ varies between
$-1$ and $1$. At the leading order, we treat interaction as though each
$s_{j}$ is affected by the collective opinions $s$ and $h$ (similarly
constructed), each forcing $s_{j}$ in their respective directions.121212 This
treatment, known as the mean-field approach, is the leading-order
approximation for a general interaction topology.  As a result of this
simplification, we can introduce the total force of peer influence $F_{s}$
acting on each manager as

$F_{s}(s,h)=\beta_{1}s(t)+\beta_{2}h(t)+E_{s}(t),$ (7)

where $\beta_{1}>0$ and $\beta_{2}>0$ are the sensitivities and $E_{s}$
denotes general exogenous influences (to be specified later). Equation (7)
implies that as the collective expectations of managers and analysts grow more
optimistic, the stronger the force exerted on a pessimistic manager to reverse
her views (and vice versa).

In addition, managers may be affected by a multitude of idiosyncratic factors
causing them to occasionally change opinions irrespective of other
participants. We treat them as random disturbances and, accordingly, introduce
the transition rates $p^{-+}$ as the probability per unit time for a manager
to switch from a negative to positive opinion and $p^{+-}$ as the probability
per unit time of the opposite change. We can express the changes in $n_{+}$
and $n_{-}$ over a time interval $\Delta t$ as

$\displaystyle n_{+}(t+\Delta t)$ $\displaystyle=$ $\displaystyle
n_{+}(t)+\Delta t\left(n_{-}(t)p^{-+}(t)-n_{+}(t)p^{+-}(t)\right),$ (8)
$\displaystyle n_{-}(t+\Delta t)$ $\displaystyle=$ $\displaystyle
n_{-}(t)+\Delta t\left(n_{+}(t)p^{+-}(t)-n_{-}(t)p^{-+}(t)\right).$ (9)

Noting that $n_{+}=(1+s)/2$ and $n_{-}=(1-s)/2$, we subtract (9) from (8) to
obtain in the limit $\Delta t\rightarrow 0$

$\dot{s}=(1-s)p^{-+}-(1+s)p^{+-}.$ (10)

To complete the derivation, we must find out how the transition rates depend
on peer influence: $p^{-+}=p^{-+}(F_{s})$ and $p^{+-}=p^{+-}(F_{s})$. It
follows from (8) that in the state of equilibrium, when $n_{\pm}(t+\Delta
t)=n_{\pm}(t)$, the condition
${p^{-+}}/{p^{+-}}={n_{+}}/{n_{-}}={N_{s}^{+}}/{N_{s}^{-}}$ holds. Thus
${p^{-+}}/{p^{+-}}$ can be interpreted as the ratio of optimists to
pessimists. We can assume this ratio changes proportionally to a change in
$F_{s}$, that is
$d{\left(N_{s}^{+}/N_{s}^{-}\right)}/\left(N_{s}^{+}/N_{s}^{-}\right)=\alpha
dF_{s}$ where $\alpha$ is a positive constant. This interpretation allows us
to write $d{\left(p^{-+}/p^{+-}\right)}/\left(p^{-+}/p^{+-}\right)=\alpha
dF_{s}$, which leads to

$\frac{p^{-+}}{p^{+-}}=e^{\alpha F_{s}}.$ (11)

Condition (11) implies correctly that $p^{-+}>p^{+-}$ for $F_{s}>0$,
$p^{-+}=p^{+-}$ for $F_{s}=0$ and $p^{-+}<p^{+-}$ for $F_{s}<0$. To obtain the
final condition required to determine $p^{-+}$ and $p^{+-}$ uniquely, we
introduce the characteristic time $\tau_{s}$ over which individual
expectations change due to random disturbances. Since $p^{-+}$ and $p^{+-}$
are per unit time, $\tau_{s}$ represents the characteristic time over which
the total probability for a manager to reverse her expectation is unity:131313
Consider the impact of a random disturbance. At its end, the agent’s state
will remain or change to its opposite. Introduce $p^{{}^{\prime}++}$ and
$p^{{}^{\prime}+-}$ as the probabilities for the agent, in state $+1$, to end
up in state $+1$ or $-1$, respectively, and $p^{{}^{\prime}--}$ and
$p^{{}^{\prime}-+}$ for the agent, in state $-1$, to end up in state $-1$ or
$+1$, respectively. Thus, $p^{{}^{\prime}++}+p^{{}^{\prime}+-}=1$ and
$p^{{}^{\prime}--}+p^{{}^{\prime}-+}=1$. Assuming that the ending state does
not depend on the initial state, i.e. $p^{{}^{\prime}++}=p^{{}^{\prime}-+}$
and $p^{{}^{\prime}--}=p^{{}^{\prime}+-}$, we obtain
$p^{{}^{\prime}-+}+p^{{}^{\prime}+-}=1$. If the disturbances are frequent on
the timescale of interest, we can transform the discrete probabilities into
continuous transition rates via $p^{-+}={p^{{}^{\prime}-+}}/{\tau_{s}}$ and
$p^{+-}={p^{{}^{\prime}+-}}/{\tau_{s}}$ to recover equation (12).

$(p^{-+}+p^{+-})\tau_{s}=1.$ (12)

Together conditions (11) and (12) imply the transition rates:

$\displaystyle p^{-+}$ $\displaystyle=\frac{1}{\tau_{s}\left(1+e^{-\alpha
F_{s}}\right)}\;,$ $\displaystyle p^{+-}$
$\displaystyle=\frac{1}{\tau_{s}\left(1+e^{\alpha F_{s}}\right)}\;.$ (13)

Equations (13) allow us to rewrite (10) as

$\tau_{s}\dot{s}=-s+\tanh\left(F_{s}\right)=-s+\tanh\left(\beta_{1}s+\beta_{2}h+E_{s}\right),$
(14)

where ${\alpha}/{2}$ is absorbed into $\beta_{1}$ and $\beta_{2}$ without loss
of generality. Note that $\tau_{s}$ acquires a dual meaning: at the micro
level, $\tau_{s}$ is akin to the manager’s average memory timespan; at the
macro level, $\tau_{s}$ is the characteristic time of variation in the
aggregate expectation of managers.

Applying this approach to model the dynamics of analyst expectations yields
the same form of the evolution equation for information $h$:

$\tau_{h}\dot{h}=-h+\tanh\left(F_{h}\right)=-h+\tanh\left(\beta_{3}s+\beta_{4}h+E_{h}\right),$
(15)

where $\tau_{h}$ represents the analyst’s average memory timespan on the micro
level and the characteristic time of the variation in the aggregate
expectation of analysts on the macro level. Similarly, $F_{h}$ is the peer
influence acting on the analysts’ expectations, which is linear in $s$ and $h$
with sensitivities $\beta_{3}$ and $\beta_{4}$, and $E_{h}$ denotes general
exogenous influences.

Equations (14) and (15) describe a generalized interactions-based process of
decision-making. We now make several assumptions to adapt it to the capital
demand discovery mechanism of the Dynamic Solow model (Figure 1).

First, we assume managers receive information only via analysts and
accordingly set $E_{s}=0$. Second, we assume analysts are affected, first and
foremost, by the news about economic development and only thereafter by all
other news. More specifically, we assume the average analyst projects the
output trend forward in time (extrapolative beliefs) and we treat all other
relevant news as exogenous noise. Thus we set

$E_{h}=\gamma\dot{y}+\xi_{t},$ (16)

with sensitivity $\gamma$ and news noise $\xi_{t}$ acting on the timescale
$\tau_{\xi}\ll\tau_{h}$. The latter implies that changes to expectations are
impacted by short-term shocks with no relation to economic fundamentals (as
suggested, for example, by Angeletos et al. (2020)).

Third, we establish separate timescales for information processing and
expectation formation. That is, we assume information is received and
processed much faster than it takes managers to adapt their long-term outlook
and form investment decisions. Therefore: $\tau_{h}\ll\tau_{s}$. Fourth, as
$\tau_{h}$ is much shorter than $\tau_{s}$, we assume direct interactions are
less important for analysts than for managers and we take
$\beta_{3}=\beta_{4}=0$ for simplicity.

The final step is to model the link between sentiment and capital demand.
Consider a firm whose managers have just decided on capital allocation in line
with their collective sentiment. The following day, all else being equal, the
managers will not revisit this decision unless their sentiment changes.
Therefore, in the short run where $t\ll\tau_{s}$ (that is, over time horizons
where the memory of past sentiment persists), capital demand must be driven by
change in sentiment. Conversely, over longer horizons where $t\gg\tau_{s}$,
the connection between previous decisions and sentiment becomes weaker and,
therefore, investment decisions must be based on the level of sentiment itself
in the long run. For lack of simpler alternatives, we superpose these two
asymptotic regimes, $\dot{k}_{d}\sim\dot{s}$ for $t\ll\tau_{s}$ and
$\dot{k}_{d}\sim s$ for $t\gg\tau_{s}$, and, as a result, arrive at a complete
system of equations for capital demand:

$\displaystyle\dot{k}_{d}$ $\displaystyle=$ $\displaystyle
c_{1}\dot{s}+c_{2}s,$ (17) $\displaystyle\tau_{s}\dot{s}$ $\displaystyle=$
$\displaystyle-s+\tanh\left(\beta_{1}s+\beta_{2}h\right),$ (18)
$\displaystyle\tau_{h}\dot{h}$ $\displaystyle=$
$\displaystyle-h+\tanh\left(\gamma\dot{y}+\xi_{t}\right),$ (19)

where $c_{1}>0$ and $c_{2}>0$ represent the capital demand sensitivity to a
change in sentiment $\dot{s}$ and the level of sentiment $s$, respectively;
and $\gamma>0$ represents the sensitivity of information $h$ to the state of
the economy or, in other words, the strength of economic feedback.141414Gusev
et al. (2015) derived equations (18) and (19) in the context of their
generalized Ising model. The derivation presented here provides a more
intuitive, albeit less rigorous, treatment. Equation (18), with $h$ as an
exogenous variable, was originally obtained by Suzuki and Kubo (1968) for the
classic Ising (1925) model in statistical mechanics. Following Haag and
Weidlich (1983), equation (18) (often in its stationary form) has frequently
appeared in the socioeconomic context. For reviews of the interactions-based
approaches reliant on the statistical mechanics methods and, particularly, the
applications of the Ising model and its variants to the opinion dynamics
problems in economics and finance, we recommend Brock and Durlauf (2001),
Bouchaud (2013) and Slanina (2013). We also note that, unlike equations (18)
and (19) that are derived from the micro level, equation (17) is obtained as a
phenomenological relation between business sentiment and capital demand.
Equation (17) was initially suggested as a link between investor sentiment and
stock market returns in Gusev et al. (2015).

###  2.5 Capital Market Clearing

At the relatively short time horizons relevant to information gathering,
investment decision-making and production adjustment, prices are not flexible
enough to efficiently match capital demand $k_{d}$ and supply $k_{s}$, which
are determined independently from each other. Accordingly, we introduce an
inelastic market clearing condition for log invested capital $k$ as

$k=\min\left(k_{s},k_{d}\right),$ (20)

to be satisfied at each moment in time. In contrast to the classic framework,
in which all household savings are used in production, this condition implies
that only a portion of savings will be invested should demand fall short of
supply (with the remainder retained in household savings accounts).

Equation (20) is a local clearing condition that reflects the short-term price
rigidity. Therefore, as was discussed in Section 2.1, this equation cannot
remain valid over long-term horizons during which prices become sufficiently
flexible to match demand and supply. As such, we supplement (20) with an
asymptotic clearing condition that holds in the timescale of long-term
economic growth:151515The asymptotic relation (21) means that the relative
error between $k_{s}$ and $k_{d}$ goes to zero as $t$ goes to infinity.

$k_{s}\sim k_{d}\quad\text{for}\quad t\geq O\left(1/\varepsilon\right)\gg 1.$
(21)

Together, equations (20) and (21) interlink the supply and demand components
and close the Dynamic Solow model.

At this point, it may be useful to discuss the characteristic timescales in
the model. The timescales we have encountered are differentiated in length
such that $\tau_{\xi}\ll\tau_{h}\ll\tau_{s}\ll\tau_{y}\ll 1/\varepsilon$.
Economically, information gathering occurs on a relatively short timescale,
$\tau_{h}$ (with the publication of, for example, monthly and quarterly
corporate reports and industry data releases); investment decisions require
more time, $\tau_{s}$ (as processed through, for example, annual board
meetings); and the implementation of changes to production levels takes much
longer, $\tau_{y}$ (the time needed for material adjustments such as
infrastructure development). We set $\tau_{h}=25$, $\tau_{s}=250$ and
$\tau_{y}=1000$ in units of business days (250 business days = 1 year). We
further assume the timespan of exogenous news events to be on average one week
and set $\tau_{\xi}=5$ as the news noise decorrelation timescale (see C).
Finally, we take technology growth rate $\varepsilon=2.5\times 10^{-5}$, which
implies the timescale of 160 years.161616 In equation (1), the term
$\exp(\varepsilon t)$, commonly referred to as total factor productivity,
yields a growth rate of about 0.6% p.a. for $\varepsilon=2.5\times 10^{-5}$,
which is not far from the estimates based on the 2005-2016 period using
Fernald (2016).

##  3 Two Limiting Cases

In this section, we inspect two cases that follow from the market clearing
condition (20): first, the supply-driven case, $k_{d}>k_{s}$ such that
$k=k_{s}$, which recovers a Solow-type growth economy; and, second, the
demand-driven case, $k_{d}<k_{s}$ such that $k=k_{d}$, in which the economic
fluctuations emerge.

###  3.1 Supply-Driven Case $k_{d}>k_{s}$

In the supply-driven case, the market clearing condition yields $K=K_{s}$
(firms use all available capital for production)171717It is convenient to use
the non-logarithmic variables for this analysis.. Consequently, the Dynamic
Solow model is reduced to equations (2) and (4) which can be expressed as a
single second-order differential equation:

$\tau_{Y}\ddot{K}+(1+\tau_{Y}\delta)\dot{K}+\delta K=\lambda
K^{\rho}e^{\varepsilon t}.$ (22)

For $t\sim 1/\varepsilon$ and longer time intervals, the derivative terms in
equation (22) become negligibly small and we recover the equilibrium growth
path. On shorter timescales, $t\sim\tau_{y}$, equation (22) describes
adjustment towards the equilibrium growth path. These two effects can be
observed simultaneously by deriving an approximate solution to equation (22)
for $t\geq O(\tau_{y})$ (see B). The resulting production path is given by

$Y=\left(\frac{\lambda}{\delta}\right)^{\frac{\rho}{1-\rho}}\left(\left(Be^{-\left(\frac{1-\rho}{\tau_{y}}\right)t}+1\right)^{\frac{1}{1-\rho}}+e^{\left(\frac{\varepsilon}{1-\rho}\right)t}-1\right),$
(23)

where $B$ is the constant of integration.181818 This derivation is valid for
the parameter values provided in A, subject to the simplifying assumption
$\tau_{y}\delta\gg 1$ which allows us to obtain the solution in a compact
form.  Equation (23) explains the output dynamics between intermediate and
long-term timescales, capturing both the long-term growth of the classic Solow
model (given by the second exponent) and the intermediate relaxation towards
the same (given by the first exponent). The approximate analytic solution (23)
and the exact numerical solution to equation (22) are compared in Figure 2.

Figure 2: Output $Y(t)$ in the supply-driven case represented by the numerical
solution of equation (22) (dashed blue line) and the approximate solution (23)
(solid red line). The precision of the approximate solution is improved with a
greater timescale separation $\tau_{y}\ll 1/\varepsilon$. Parameters:
$\rho=1/3$, $\tau_{y}=1000$, $\lambda=0.15$, $\varepsilon=10^{-5}$ and
$\delta=0.02$ with an integration constant of $B=1.5$. The inset box
highlights the intermediate adjustment of output from an arbitrary initial
value to the equilibrium growth path.

###  3.2 Demand-Driven Case $k_{d}<k_{s}$

In the demand-driven case, the market clearing condition yields $k=k_{d}$. The
Dynamic Solow model is specified at this limit by equations (3) and (17)-(19)
(in this case, equation 5 decouples and no longer affects production). To
facilitate our analysis, we introduce the variable $z=\rho k_{d}+\varepsilon
t-y$, which makes the model solutions bounded in the $(s,h,z)$-space (see D).
Economically, $z$ represents the direction and strength of economic growth.
This follows from rewriting equation (3) as $\tau_{y}\dot{y}=e^{z}-1$, noting
that for $z>0$ production expands, for $z<0$ it contracts and $z=0$ is a
production fixed point. Using $z$, we re-express the model as a three-
dimensional dynamical system that is bounded and autonomous in the absence of
exogenous noise $\xi_{t}$:

$\displaystyle\dot{z}$ $\displaystyle=\rho c_{1}\dot{s}+\rho
c_{2}s-\omega_{Y}\left(e^{z}-1\right)+\varepsilon$ (24a)
$\displaystyle\tau_{s}\dot{s}$
$\displaystyle=-s+\tanh\left(\beta_{1}s+\beta_{2}h\right)$ (24b)
$\displaystyle\tau_{h}\dot{h}$
$\displaystyle=-h+\tanh\left(\gamma\omega_{y}\left(e^{z}-1\right)+\xi_{t}\right),$
(24c)

where, for convenience, $\omega_{y}=1/\tau_{y}$.

This dynamical system is parametrized and examined in detail in C. For the
relevant range of parameters it has three equilibria: a stable focus where
sentiment is positive ($s>0$) and the economy is expanding ($z>0$), a stable
focus where sentiment is negative ($s<0$) and the economy is contracting
($z<0$) and an unstable saddle point in between.191919For convenience, we
classify 3D equilibrium points using more familiar 2D terminology. As such:
(i) the stable (unstable) node has three negative (positive) real eigenvalues;
(ii) the focus has one real and two complex eigenvalues and is stable if the
real eigenvalue and the real parts of complex eigenvalues are all negative and
unstable otherwise; and (iii) the saddle is always unstable as it has three
real eigenvalues that do not all have the same sign. In the figures, the
stable points are green and unstable points are red, while the nodes are
marked by triangles, foci by squares and saddles by circles. The location,
basin of attraction and stability of the equilibria are primarily affected by
the parameters $c_{2}$ (sensitivity to sentiment levels) and $\gamma$
(sensitivity to economic feedback). In particular, an increasing $c_{2}$
strengthens convergence towards the equilibria, so the system acquires greater
stability.

(a) Coherence resonance for $c_{2}=7\times 10^{-4}$ and $\gamma=2000$. This
subcritical regime presents a bi-stable configuration of equilibria: green
squares denote the two stable foci and the red circle an unstable saddle. Red
trajectories terminate at the $s<0$ focus in which the economy contracts and
blue trajectories terminate at the $s>0$ focus in which the economy expands.
The long trajectories passing near one focus and ending at the other are of a
particular interest as they provide the pathway for the economy’s regime
transitions.

(b) Limit cycle for $c_{2}=1\times 10^{-4}$ and $\gamma=4000$. In this
supercritical regime, only the positive ($s>0$) equilibrium point survives,
having bifurcated into an unstable focus, and a large stable limit cycle
emerges that propels the economy between contraction and expansion with a
constant frequency.

Figure 3: 3D phase portraits ($\xi_{t}=0$) in the $(s,h,z)$-space: (a) the
coherence resonance regime and (b) the limit cycle regime. The long
trajectories in (a) can be viewed as segments of the limit cycle in (b), which
remain unconnected for $\xi_{t}=0$. Parameters other than $c_{2}$ and $\gamma$
are from the base case in A. Figure 4: Left: The coherence resonance phase
portrait in Figure 3(a) projected on the $s,z$-plane. The approximate boundary
between the contraction and expansion regions is indicated by a dashed black
line. When the economy moves along the long trajectories, it traverses quickly
the distance in $s$ between the foci. However, the movement speed declines
sharply with the proximity to equilibrium. As a result, the economy slowly
ascends along the $z$-axis towards the left focus in the contraction mode and
slowly descends to the right focus in the expansion mode. As the economy is
undergoing this gradual drift, the distance to the boundary separating
contraction and expansion diminishes. Right: The limit cycle phase portrait in
Figure 3(b) projected on the $s,z$-plane.

If $c_{2}$ is below a certain critical value, equations (24) generate a
periodic limit cycle. The idea that limit cycles provide a mechanism of
economic fluctuations dates back to Kalecki (1937), Kaldor (1940), Hicks
(1950) and Goodwin (1951). The empirical irrelevance of periodic limit cycles
led to a diminished interest in this research direction202020 Nevertheless, a
similar line of research has been pursued in overlapping generations models
and innovation cycles. See Hommes (2013) and Beaudry et al. (2020) for
references. ; however, Beaudry et al. (2020) have reinitiated the discussion
by proposing that cyclicality can arise from stochastic limit cycles “wherein
the system is buffeted by exogenous shocks, but where the deterministic part
of the system admits a limit cycle”. In our system, exogenous news noise
$\xi_{t}$ similarly detunes limit cycle periodicity. This mechanism, however,
cannot explain the “small shock, large crisis” effect or reproduce the general
variability present in the real-world economy. At the other extreme, our
system generates noise-prevailing behaviors with weak cyclicality. Neither
extreme accurately reflects empirical observations and thus we seek a sensible
balance between these features in a parameter regime that produces significant
dynamic effects but precedes the limit cycle formation (C).

Figure 5: A simulated economy’s path for nonzero $\xi_{t}$ in the coherence
resonance case, smoothed by a Fourier filter to remove harmonics with periods
less than 500 business days for a better visualization in the 3D phase space.
The regions where the trajectory is dense indicate the contraction and
expansion attractors, around which the economy spends most of its time. The
relatively straight path segments between the attractors correspond to the
economic regime transitions that occur on a relatively rapid timescale.
Parameters are from the base case in A.

To this end, we consider a subcritical regime with $c_{2}$ above but close to
its critical value at which the limit cycle emerges. In this situation,
henceforth referred to as the coherence resonance regime, the foci are always
stable, thus acting as attractors entrapping the economy. In Figures 3 and 4,
we compare the phase portraits (i.e. the system trajectories for $\xi_{t}=0$)
of the coherence resonance and limit cycle regimes. In the coherence resonance
case, we take note of the unclosed largescale trajectories that pass near one
attractor and converge to the other. These trajectories, which can be viewed
as segments of a limit cycle, are the pathways along which the economy moves
between contraction and expansion.

Figure 6: Sentiment $s(t)$ (left) and output $y(t)$ (right) on a simulated
economy’s path for nonzero $\xi_{t}$ in the coherence resonance regime. Output
undergoes fluctuations within a general growth trend as sentiment evolution
exhibits a distinct bi-stable pattern. Parameters are from the base case in A.

The dynamics of business cycles for nonzero $\xi_{t}$ are visualized in Figure
5. The economy’s trajectory displays distinctly bi-stable behavior as it
spends most of its time near each focus and transits swiftly between them.
When captive to an attractor, the trajectory follows an orbit around the
corresponding focus, buffeted by exogenous noise $\xi_{t}$, preventing it from
settling. Simultaneously, the economy drifts slowly towards the boundary
between attracting regions (Figure 4(left)), making it easier for a random
news shock to thrust it across the boundary to be caught by the other
attractor. The news shocks $\xi_{t}$ thus fulfill a dual purpose: they perturb
the economy from equilibrium and provide a trigger that alternates the
economic regime between expansions and recessions.

This mechanism can be classified as coherence resonance, a phenomenon whereby
noise applied to a dynamical system leads to a quasiperiodic response
(Pikovsky and Kurths, 1997). Coherence resonance normally occurs in bi-stable
systems that are stochastically forced and in which key variables evolve on
different timescales. The Dynamic Solow model satisfies these requirements:
(i) news shocks provide a stochastic force; (ii) two stable equilibria emerge
in the relevant parameter range; and (iii) the separation of characteristic
timescales follows from the dynamics of corporate decision-making processes.
The three-dimensionality of equations (24) introduces an important novel
feature into the classic two-dimensional case of coherence resonance: the
above-mentioned slow drift of the economy’s trajectory, which gradually
increases the probability of regime transition.212121 This drift imposes a
slow timescale $t\geq O(\tau_{y})$ on the frequency of the economy’s
fluctuations by modulating the probability of regime transition. This effect
was first observed and explained for a similar dynamical system in Kroujiline
et al. (2019).  This novel feature nonetheless leaves the basic mechanism
unchanged: exogenous noise forces the economy across the boundary separating
the regions of different dynamics, effectively reconnecting the trajectories
between attractors. As a result, the economy undergoes quasiperiodic
fluctuations consisting of alternating periods of expansion and recession
punctuated by sharp transitions (as in Figure 6).

While both coherence resonance and limit cycle occur in the economically
realistic range of parameters (see C), we will show that only coherence
resonance is compatible with the long-term economic growth when studying the
asymptotic convergence of capital supply and demand in the presence of
fluctuations driven by the coherence resonance and limit cycle mechanisms in
Section 4.2.

##  4 Business Cycles and Long-Term Growth in the General Case

While the supply- and demand-driven cases have been instructive for
highlighting the mechanisms underlying economic dynamics, their applicability
as standalone models is limited as supply and demand converge in the long run
(equation (21)). As such, our primary focus is on the general case in which
supply and demand coevolve, potentially leading to an interplay of supply- and
demand-driven dynamics. We formulate the general case in Section 4.1, study
long-term growth rates in Section 4.2 and examine economic fluctuations in
Section 4.3.

###  4.1 Formulation of the General Case

In the general case, invested capital $k$ can alternate between $k_{d}$
(demand-driven regime) and $k_{s}$ (supply-driven regime) in accordance with
the market clearing condition (20). As discussed in Section 2.1, firms’
decision-making processes are influenced by feedback from the economy.
However, the supply-driven regime represents a special situation in which
firms’ investment decisions do not affect economic output as production is
determined in this case solely by capital availability. In other words, the
supply-driven regime implies a Solow-type growth economy propelled by
expectations of future consumption so high as to induce firms to utilize all
capital supplied by households in production. Therefore, $\dot{y}$, which is
positive in this regime, holds no additional information for managers, who are
already overwhelmingly bullish about the economy. The idiosyncratic news
$\xi_{t}$ remains the only source of nontrivial information, thereby becoming
the focus of managers and analysts alike. Thus, economic feedback
$\gamma\dot{y}$ vanishes as a decision factor in the supply-driven regime.

Following this argument, we account for regime-dependent variation in feedback
strength by introducing a regime-specific factor $H(k_{d},k_{s})$ that
regulates the impact of feedback in equation (19):

$\displaystyle\tau_{h}\dot{h}$ $\displaystyle=$
$\displaystyle-h+\tanh\left(\gamma\dot{y}H(k_{s},k_{d})+\xi_{t}\right),$ (25)

where

$H(k_{s},k_{d})=\begin{cases}1&\text{if }k_{d}\leq k_{s}\\\ 0&\text{if
}k_{d}>k_{s}\end{cases}.$ (26)

The Dynamic Solow model is then represented in the general case by the
following system of equations:

$\displaystyle\tau_{Y}\dot{y}$ $\displaystyle=e^{\rho k+\varepsilon t-y}-1,$
(27) $\displaystyle\dot{k}_{s}$ $\displaystyle=\lambda e^{y-k_{s}}-\delta
e^{k-k_{s}},$ (28) $\displaystyle\dot{k}_{d}$
$\displaystyle=c_{1}\dot{s}+c_{2}s,$ (29) $\displaystyle\tau_{s}\dot{s}$
$\displaystyle=-s+\tanh\left(\beta_{1}s+\beta_{2}h\right),$ (30)
$\displaystyle\tau_{h}\dot{h}$
$\displaystyle=-h+\tanh\left(\gamma\dot{y}H(k_{s},k_{d})+\xi_{t}\right),$ (31)
$\displaystyle k$ $\displaystyle=\min(k_{d},k_{s}),$ (32) $\displaystyle
k_{s}$ $\displaystyle\sim k_{d}\quad\text{for}\quad t\geq O(1/\varepsilon)\gg
1,$ (33)

where (27) is the dynamic equation governing production; (28) describes the
motion of capital supply; (29)-(31) govern the feedback-driven dynamics that
link information $h$, sentiment $s$ and capital demand $k_{d}$; (32) defines
the locally-inelastic market clearing condition; and (33) represents long-term
market clearing that takes the form of an asymptotic boundary condition at
large $t$.

###  4.2 Growth and Convergence in the Long Run

The Dynamic Solow model (27)-(33) covers two regimes with different dynamics:
a demand-driven regime with endogenous fluctuations and a supply-driven regime
without them. Both regimes are expected to participate in the model’s general
case, owing to the convergence of supply and demand in the long run under
equation (33).

Equation (33) is central to our present analysis. Based on the regime
definitions, this equation is satisfied when supply grows faster than demand
in the supply-driven regime and, conversely, when demand grows faster than
supply in the demand-driven regime. Under the demand-driven regime, the two
possible mechanisms of fluctuations – limit cycle and coherence resonance –
may entail different growth rates, validating the mechanism if demand grows
fast enough to satisfy (33) and invalidating it otherwise.

This section aims to determine (i) the impact of fluctuations on growth; (ii)
the mechanism of fluctuations compatible with equation (33); and (iii) the
actual growth dynamics realized in the model. We first consider separately the
supply- and demand-driven regimes (Sections 4.2.1 and 4.2.2) and then tackle
the general case (Section 4.2.3). D provides the derivations of the equations
herein.

####  4.2.1 Asymptotic Growth in the Supply-Driven Case ($k_{d}>k_{s}$)

We show in D that the economy’s long-term growth in the supply-driven case is
given by

$\displaystyle y_{0}$ $\displaystyle=k_{s0}=\frac{\varepsilon}{1-\rho}\equiv
R,$ (34) $\displaystyle k_{d0}$ $\displaystyle=0,$ (35)

where $y_{0}$, $k_{s0}$ and $k_{d0}$ represent, respectively, the log output,
log supply and log demand growth rates; $\rho=1/3$ is the capital share in
production; and $R$ denotes the classic Solow growth rate. As expected, the
growth rate is not influenced by demand dynamics and matches $R$. These
estimates are verified by numerical simulations (see Figure 7). Note that
supply always catches up with demand as $k_{s0}>k_{d0}$ in this case.

Figure 7: A long-term simulation that captures asymptotic growth in the
enforced supply-driven regime ($k=k_{s}$). Output $y$ and supply $k_{s}$ grow
steadily at the Solow rate $R$ while demand $k_{d}$ stagnates. As such,
$k_{s}$ will eventually reach and exceed $k_{d}$, at which point the supply-
driven regime will be succeeded by the demand-driven regime. Parameters are
from the base case in A.

####  4.2.2 Asymptotic Growth in the Demand-Driven Case ($k_{d}<k_{s}$)

We show in D that the economy’s long-term growth in the demand-driven case
satisfies

$y_{0}=k_{s0}=R+\rho\left(k_{d0}-R\right).$ (36)

According to this equation, output becomes dependent on demand, which itself
undergoes endogenous fluctuations, and thus demand dynamics affect the long-
term behavior of the economy. The magnitude of $k_{d0}$ determines whether the
economy expands faster or slower than the classic Solow economy: $y_{0}>R$ if
$k_{d0}>R$ and $y_{0}<R$ if $k_{d0}<R$ (the latter condition including an
important case when $k_{d0}=0$ that yields an especially slow growth rate
$y_{0}=k_{s0}=\varepsilon$). Next, we estimate $k_{d0}$ numerically under the
effect of limit cycle and coherence resonance.

Figure 8 depicts the growth dynamics driven by a periodic limit cycle
($\xi_{t}=0$). We observe that $k_{d0}$ stays close to zero and $y_{0}$ and
$k_{s0}$ match $\varepsilon$ closely in accordance with (36), meaning the
economy grows only through improvements in production efficiency. Figure 9
displays similar dynamics for the limit cycle perturbed by exogenous noise
$\xi_{t}$. It follows that limit cycles, whether periodic or stochastic, lead
to a growth rate of less than $R$.

The above result can be explained by noting that an economy on a limit cycle
trajectory spends roughly an equal amount of time in expansion ($s>0$) as in
contraction ($s<0$) and, consequently, $s$ exhibits on this trajectory a long-
term average value of zero. In D, we find that $k_{d0}$ is proportional to the
long-term average of $s$, implying $k_{d0}$ tends to zero as well; therefore,
demand can never catch up with supply due to the difference in their growth
rates. In sum, the fluctuations generated by a limit cycle detract from long-
term growth and fail to satisfy equation (33).

Figure 8: A long-term simulation that captures asymptotic growth in the
enforced demand-driven regime ($k=k_{d}$) powered by a periodic limit cycle
($\xi_{t}=0$) with $\gamma=1000$, $c_{2}=2\times 10^{-5}$ and all other
parameters from the base case in A. Left: Production $y$ grows at a rate lower
than the Solow rate $R$ while demand $k_{d}$ stagnates (and, in fact, appears
to gradually decrease, which could be attributed to the slight asymmetry of
the limit cycle with respect to $s$). Since $k_{s}$ and $y$ grow at the same
rate (equation (36)), $k_{d}$ cannot catch up with $k_{s}$. Right: Sentiment
$s(t)$ demonstrates the limit cycle’s periodicity. Figure 9: A long-term
simulation that captures asymptotic growth in the enforced demand-driven
regime ($k=k_{d}$) powered by a stochastic limit cycle ($\xi_{t}\neq 0$) with
the same parameters as in Figure 8. Left: Production $y$ grows at a rate lower
than the Solow rate $R$ while capital demand $k_{d}$ stagnates (exogenous
noise evidently erasing the limit cycle’s asymmetry visible in Figure 8).
Since $k_{s}$ and $y$ grow at the same rate (equation (36)), $k_{d}$ cannot
catch up with $k_{s}$. Right: Sentiment $s(t)$ is no longer periodic due to
the impact of $\xi_{t}$.

Coherence resonance induces a drastically different long-term dynamic despite
the visually similar fluctuations (see Figure 10). Demand grows asymptotically
at $k_{d0}>R$, leading to accelerated economic growth of $y_{0}=k_{s0}\equiv
R^{\star}>R$ in accordance with equation (36). This fast-paced growth, made
possible by excess capital, $k_{s}>k_{d}$, available for investment in the
demand-driven regime, is explained by technological progress
($\varepsilon>0$), causing the economy to spend on average more time in
expansion than contraction. We further observe that $k_{d0}>R^{\star}$; that
is, demand grows faster than both supply and output.222222This result is
consistent with equation (36), which can be rewritten as
$k_{d0}-R^{\star}=((1-\rho)/\rho)(R^{\star}-R)$, so that $k_{d0}>R^{\star}$
since $R^{\star}>R$ and $\rho<1$. Therefore, demand powered by coherence
resonance always catches up with supply.

Figure 10: A long-term simulation that captures asymptotic growth in the
enforced demand-driven regime ($k=k_{d}$) powered by coherence resonance in
the base case parameter regime (A). Left: Output $y$ and supply $k_{s}$ expand
at the rate $R^{\star}>R$ and demand $k_{d}$ grows faster than both $y$ and
$k_{s}$. As such, $k_{d}$ eventually reaches and exceeds $k_{s}$, at which
point the demand-driven regime is succeeded by the supply-driven regime.
Right: Sentiment $s(t)$ exhibits a bi-stable behavior typical of coherence
resonance.

####  4.2.3 Asymptotic Growth in the General Case

We have shown that fluctuations affect growth in the demand-driven regime of
the Dynamic Solow model. In particular, limit cycles generate fluctuations
that contribute negatively to growth, thus failing to satisfy the asymptotic
boundary condition (33). Therefore, such fluctuations cannot be realized,
which rules out limit cycles as the mechanism responsible for business cycles.

Figure 11: A long-term simulation that captures asymptotic growth in the
general case. Left: Output $y$, supply $k_{s}$ and demand $k_{d}$ grow at the
Solow rate $R$, demonstrating the asymptotic convergence on the economy’s
trajectory. Right: The interplay of supply- and demand-driven regimes on a
subsection of the same trajectory. Shaded segments correspond to periods when
$k_{d}>k_{s}$. Parameters are from the base case in A.

By contrast, coherence resonance produces fluctuations that contribute
positively to growth, so that demand always catches up with supply. As this
occurs, the system transits into the supply-driven regime in which supply
grows faster than demand. Once supply has exceeded demand, the system switches
back into the demand-driven dynamics. The regime cycle has thus come full
circle, ensuring (33) is satisfied in the long run. As such, the economy’s
path realized in the general case is forged by a regime interplay where the
supply-driven equilibrium dynamics and the demand-driven fluctuations, powered
by coherence resonance, continuously succeed one another.

Our simulations show the economy grows asymptotically at the Solow rate $R$.
This result is not entirely unexpected. As capital supply and demand converge
over the long run, capital invested into production during the supply- and
demand-driven regime segments of the economy’s trajectory must also match
asymptotically, as follows from (32). Consequently, the economy’s average
growth rate across supply-driven segments is equal to the average growth rate
across demand-driven segments. As the economy expands at $R$ in the supply-
driven regime, the same growth rate is achieved, on average, across the
demand-driven segments,232323The demand-driven economy cannot reach the
asymptotic growth rate $R^{\star}>R$ as a result of the segments’ finite
duration and an adverse growth bias due to the typical alignment of demand-
driven segments with recessionary periods. meaning $R$ is also the overall
rate of expansion. Figure 11 displays a simulation capturing the realized
asymptotic growth path in the general case and highlights the interplay of the
supply- and demand-driven dynamics.

To sum up, the asymptotic growth rates in the demand-driven regime depend on
the mechanism underlying economic fluctuations. Fluctuations driven by a limit
cycle cannot be realized since they do not satisfy the convergence between
supply and demand in the long run. The economy’s trajectory realized in the
general case of the Dynamic Solow model consists of a chain of supply-driven
regimes in which the economy experiences the equilibrium growth and demand-
driven regimes in which fluctuations emerge via coherence resonance. Overall,
the economy grows asymptotically at the classic Solow rate $R$. Although
fluctuations can cause large excursions from this equilibrium growth path, the
deviations disappear on the large timescales relevant for the convergence of
supply and demand.

###  4.3 Business Cycle Dynamics

Our analysis of asymptotic growth in the preceding section has led us to
conclude that coherence resonance is the relevant endogenous mechanism
underlying economic dynamics as it enables the convergence of capital demand
and supply over the long run. In this section, we focus on the intermediate
timescale to examine endogenous fluctuations produced by the Dynamic Solow
model (27)-(33) in the coherence resonance regime.

Figure 12: A simulation of the economy’s trajectory over the medium term in
the model’s general case, which highlights a sequence of supply- and demand-
driven regimes. Shaded segments correspond to periods during which
$k_{d}>k_{s}$. Parameters are from the base case in A

.

Figure 12 depicts a typical realization of the economy’s trajectory over the
medium term. The economy undergoes a sequence of supply- and demand-driven
dynamic behaviors, as indicated, respectively, by shaded and unshaded
segments. In the demand-driven case, in which demand is below supply,
sentiment (lower panel) exhibits distinctively bi-stable behavior, staying for
long periods near the positive (expansion) and negative (contraction)
equilibria and traversing quickly the distance between them during economic
regime transitions. This sentiment behavior leads to fluctuations in demand
(middle panel) that, in turn, induce business cycles around the long-term
growth trend (upper panel). Conversely, during periods when supply is the
limiting factor, sentiment follows a random walk due to the absence of
economic feedback and the supply-driven economy exhibits the equilibrium
growth dynamics.

The long-term simulations demonstrate that demand stays below supply on
average $\sim 70\%$ of the time. This can be interpreted as the firms’
decision to hold excess capital (as, for example, noted in Fair, 2020) as the
entire capital supply is made available to firms, implying a capital
utilization rate below 100% over extended periods.242424The utilization rate
measures current production as a proportion of potential production. In our
context, the analogous measure is the ratio of capital in production in the
demand-driven regime $(k=k_{d})$ to capital available $(k_{s})$. For empirical
evidence of firms holding excess capital, see the FRED series “Capacity
Utilization: Total Index” (CAPUTLB50001SQ) or the Census Bureau’s National
Emergency Utilization Rate. See also Murphy (2017) for a study of the
persistence of excess capital.

Figure 13: Histogram of the duration of simulated business cycles. The cycles
are based on the detrended production, $y-Rt$, with respect to the best fit
straight line which coincides with the equilibrium growth path, given by the
Solow growth rate $R$ (Section 4.2.3). Duration is calculated as the time
interval between two successive zero crossings in the same direction by
detrended production. The histogram, based on the 5 year bins, is truncated at
10 years to eliminate noise artifacts and at 150 years to highlight its peak.
The parameters are from the base case in A.  Figure 14: Histogram of the
duration of simulated sentiment cycles. The cycles are defined as the
roundtrip of sentiment between the positive and negative equilibria. Duration
is calculated as the time interval between two successive zero crossings in
the same direction by sentiment $s$ in the enforced demand-driven case. The
histogram, based on the 5 year bins, is truncated at 10 years to eliminate
noise artifacts and at 150 years to highlight its peak. The parameters are
from the base case in A.

Figure 13 is a histogram of business cycle periods simulated by the model. It
displays a wide distribution with a peak in the 40-70 year interval (with over
50% of the periods falling into this range), indicating the presence of
quasiperiodic fluctuations. To confirm the source of these fluctuations, we
inspect the distribution of the lengths of sentiment cycles, defined as the
roundtrip of sentiment between the positive and negative equilibria (such as
those depicted in the lower panel in Figure 12). This distribution, shown in
Figure 14, also peaks at 40-70 years. It follows that business cycles are, as
expected, linked to sentiment transitions from one equilibrium to the other
driven by coherence resonance. Therefore, we affirm coherence resonance is the
relevant mechanism forming the quasiperiodic fluctuations in output captured
in Figure 13.

In C, we show that parameter $c_{2}$, which defines the sensitivity of capital
demand to sentiment, is key to the business cycle duration: the lower $c_{2}$,
the shorter the average duration of business cycles. We also show there that
the model admits coherence resonance only if $c_{2}$ is above a certain
critical value and tune the model to be in a regime with $c_{2}$ close to this
value. It follows that coherence resonance – as a mechanism of business cycles
driven by firms’ investment – imposes a natural minimum duration threshold,
ruling out fluctuations with a characteristic timespan shorter than the
Kondratieff-like 40-70 years.

In current literature, business cycles are typically estimated to last 8-12
years. However, a direct comparison of the duration would be misleading as our
model, centered on capital demand dynamics, does not include links to the
faster-paced processes, such as credit or equity market dynamics, that can
accelerate business cycles through further interactions with the real economy.
In other words, our model captures capital demand driven cycles, which are
arguably just one of a number of fluctuation modes that reinforce or otherwise
affect each other to produce the business cycles observed in the real world.

On that point, we take note of Kroujiline et al. (2019) that studies combined
effects in a coupled macroeconomic system, attaching the interactions-based
stock market model of Gusev et al. (2015) (capable of producing relatively
short-term endogenous cycles) to the simple phenomenological model of the
economy of Blanchard (1981) (within which output follows slow relaxation
dynamics) to obtain quasiperiodic fluctuations with the same frequency as
observed business cycles. A natural next step would be to investigate whether
a more advanced coupled system, where both the financial sector and the real
economy experience nonlinear endogenous dynamics at different frequencies can
replicate and explain observed macroeconomic behaviors in greater
detail.252525Such as the stock market model of Gusev et al. (2015) and the
present Dynamic Solow model of the economy, which share a similar framework
for micro-level interactions.

##  5 Conclusion

In this paper we have developed the Dynamic Solow model, a tractable
macroeconomic model that captures dynamic behaviors across multiple
timescales, and applied it to study economic fluctuations and their impact on
long-term growth.

Our model consists of a dynamic capital demand component, representing an
interactions-based process whereby firms determine capital needs and make
investment decisions, and a simple capital supply component in the form of a
Solow growth economy. These components are interlinked via a capital market,
which comprises a local inelastic market clearing condition reflecting short-
term price rigidity and an asymptotic market clearing condition valid for
timescales in which prices are sufficiently flexible to match supply and
demand. Starting from the micro-level interactions among firms, we derived the
macroscopic equations for capital demand that constitute the dynamic core of
the model and attached them to a capital motion equation for a static
representative household (providing a dynamic version of the Cobb-Douglas
production equation to capture short-term processes) and the aforementioned
market clearing conditions. As a result, we have obtained a closed-form
nonlinear dynamical system that allows for the examination of a broad range of
economic behaviors.

The Dynamic Solow model admits two characteristic regimes, depending on
whether capital demand exceeds supply or not. When demand exceeds supply,
supply drives output and the dynamic demand component decouples from the rest
of the economy, placing the economy on the familiar equilibrium growth path.
Otherwise, demand drives output and the model is shown, for economically
realistic parameters, to possess two attracting equilibria, one where the
economy contracts and the other where it expands. This bi-stable geometry
gives rise to business cycles manifested as endogenous fluctuations, wherein
the economy’s long entrapment in recessions and expansions is punctuated by
rapid alternations between them. We show that, in our model, the economy’s
realized trajectory is forged by an interplay of these regimes such that the
supply-driven equilibrium dynamics and demand-driven fluctuations continuously
succeed one another. We further show that the economy spends around 70% of its
time in the demand-driven regime, indicating fluctuations represent a
prevalent economic behavior.

We identify a coherence resonance phenomenon, whereby noise applied to a
dynamical system leads to a quasiperiodic response, to be the mechanism behind
demand-driven fluctuations. In our model, exogenous noise (representing news
received by analysts) instigates the economy’s transition from one equilibrium
to the other, resulting in recurrent booms and busts. As such, news shocks act
as a catalyst, which is compatible with the “small shocks, large cycle” effect
observed in the real-world economy. In addition, under a different range of
parameter values, we obtain a stochastic limit cycle (i.e. a limit cycle
perturbed by exogenous noise) likewise capable of generating endogenous
fluctuations. We show, however, that this type of fluctuations cannot be
realized as the growth dynamics induced by it do not allow supply and demand
to converge in the long run. While both limit cycle and coherence resonance
mechanisms are hardwired in our model, in the sense that the parameter ranges
must be appropriately selected, we conjecture that in reality the economy
self-regulates towards the coherence resonance parameter ranges via long-term
price adjustment responsible for the convergence of supply and demand in the
long run.

The distribution of the business cycle periods simulated by our model displays
a peak in the Kondratieff range of 40-70 years, demonstrating the
quasiperiodic character of demand-driven fluctuations. We further find
coherence resonance imposes a minimum duration threshold that rules out
fluctuations peaking at shorter lengths. This result seems sensible because
our model, centered on capital demand dynamics, has no links to faster-paced
processes (such as credit or equity market dynamics) that can accelerate
fluctuations to be in line with the observed business cycles. A natural
extension would be to develop a coupled system, within which both the
financial sector representing such faster-paced processes and the real economy
experience nonlinear endogenous dynamics at different characteristic
frequencies.

Our simulations show that although demand-driven fluctuations occasionally
cause large excursions from the equilibrium growth path, the deviations vanish
in the long run as supply and demand converge. In our model, the equilibrium
growth path is defined by the Solow growth rate in which technology growth
appears, simplistically, as a fixed exogenous parameter. From this
perspective, endogenizing technological progress in the model may help better
understand the long-term impact of fluctuations on growth, presenting an
intriguing topic for future research.

##  6 Acknowledgments

We deeply thank J.-P. Bouchaud who contributed to the early stages of this
work and Dmitry Ushanov for contributing ideas on the numerical analysis of
the model. We also thank Mikhail Romanov for helpful comments and suggestions.
This research was conducted within LGT Capital Partners and the Econophysics &
Complex Systems Research Chair, the latter under the aegis of the Fondation du
Risque, the Fondation de l’Ecole polytechnique, the Ecole polytechnique and
Capital Fund Management. Maxim Gusev and Dimitri Kroujiline are grateful to
LGT Capital Partners for financial support. Karl Naumann-Woleske also
acknowledges the support from the New Approaches to Economic Challenges Unit
at the Organization for Economic Cooperation and Development (OECD).

## References

  * Aghion and Howitt (1992) Aghion, P., Howitt, P., 1992\.  A Model of Growth Through Creative Destruction.  Econometrica 60, 323–351.  doi:10.2307/2951599.
  * Aghion and Howitt (2006) Aghion, P., Howitt, P., 2006\.  Joseph Schumpeter Lecture Appropriate Growth Policy: A Unifying Framework.  Journal of the European Economic Association 4, 269–314.  doi:10.1162/jeea.2006.4.2-3.269.
  * Angeletos et al. (2020) Angeletos, G.M., Collard, F., Dellas, H., 2020.  Business-Cycle Anatomy.  American Economic Review 110, 3030–3070.  doi:10.1257/aer.20181174.
  * Angeletos and La’O (2013) Angeletos, G.M., La’O, J., 2013\.  Sentiments.  Econometrica 81, 739–779.  doi:10.3982/ECTA10008.
  * Arrow (1962) Arrow, K.J., 1962.  The Economic Implications of Learning by Doing.  The Review of Economic Studies 29, 155–173.  doi:10.2307/2295952.
  * Bakas et al. (2019) Bakas, D., Chortareas, G., Magkonis, G., 2019.  Volatility and growth: A not so straightforward relationship.  Oxford Economic Papers 71, 874–907.  doi:10.1093/oep/gpy065.
  * Beaudry et al. (2020) Beaudry, P., Galizia, D., Portier, F., 2020.  Putting the Cycle Back into Business Cycle Analysis.  American Economic Review 110, 1–47.  doi:10.1257/aer.20190789.
  * Beaudry and Portier (2014) Beaudry, P., Portier, F., 2014\.  News-Driven Business Cycles: Insights and Challenges.  Journal of Economic Literature 52, 993–1074.
  * Benhabib and Farmer (1999) Benhabib, J., Farmer, R.E.A., 1999\.  Chapter 6 Indeterminacy and sunspots in macroeconomics, in: Handbook of Macroeconomics. Elsevier. volume 1, pp. 387–448.  doi:10.1016/S1574-0048(99)01009-5.
  * Bernanke et al. (1996) Bernanke, B., Gertler, M., Gilchrist, S., 1996.  The Financial Accelerator and the Flight to Quality.  The Review of Economics and Statistics 78, 1–15.
  * Blanchard (1981) Blanchard, O.J., 1981.  Output, the stock market, and interest rates.  The American Economic Review 71, 132–143.
  * Boldrin and Woodford (1990) Boldrin, M., Woodford, M., 1990\.  Equilibrium models displaying endogenous fluctuations and chaos: A survey.  Journal of Monetary Economics 25, 189–222.  doi:10.1016/0304-3932(90)90013-T.
  * Bouchaud (2013) Bouchaud, J.P., 2013.  Crises and Collective Socio-Economic Phenomena: Simple Models and Challenges.  Journal of Statistical Physics 151, 567–606.
  * Brock and Durlauf (2001) Brock, W., Durlauf, S., 2001\.  Interactions-based models, in: Handbook of Econometrics. Elsevier, pp. 3297–3380.
  * Christiano et al. (2018) Christiano, L.J., Eichenbaum, M.S., Trabandt, M., 2018.  On DSGE Models.  Journal of Economic Perspectives 32, 113–140.  doi:10.1257/jep.32.3.113.
  * Cooley and Prescott (1995) Cooley, T., Prescott, E., 1995\.  Economic growth and business cycle, in: Frontiers of Business Cycle Research. Princeton University Press, Princeton, N.J.
  * Cooper and John (1988) Cooper, R., John, A., 1988.  Coordinating Coordination Failures in Keynesian Models.  The Quarterly Journal of Economics 103, 441–463.  doi:10.2307/1885539.
  * Cyert and March (1992) Cyert, R.M., March, J.G., 1992\.  Behavioral Theory of the Firm.  2nd edition ed., Wiley-Blackwell, Cambridge, Mass., USA.
  * Da et al. (2021) Da, Z., Huang, X., Jin, L.J., 2021.  Extrapolative beliefs in the cross-section: What can we learn from the crowds?  Journal of Financial Economics 140, 175–196.  doi:10.1016/j.jfineco.2020.10.003.
  * Dawid and Delli Gatti (2018) Dawid, H., Delli Gatti, D., 2018\.  Agent-Based Macroeconomics, in: Handbook of Computational Economics. Elsevier. volume 4, pp. 63–156.  doi:10.1016/bs.hescom.2018.02.003.
  * Dosi et al. (2015) Dosi, G., Fagiolo, G., Napoletano, M., Roventini, A., Treibich, T., 2015.  Fiscal and monetary policies in complex evolving economies.  Journal of Economic Dynamics and Control 52, 166–189.  doi:10.1016/j.jedc.2014.11.014.
  * Dosi et al. (2010) Dosi, G., Fagiolo, G., Roventini, A., 2010.  Schumpeter meeting Keynes: A policy-friendly model of endogenous growth and business cycles.  Journal of Economic Dynamics and Control 34, 1748–1767.  doi:10.1016/j.jedc.2010.06.018.
  * Fair (2020) Fair, R.C., 2020.  Some important macro points.  Oxford Review of Economic Policy 36, 556–578.  doi:10.1093/oxrep/graa011.
  * Fernald (2016) Fernald, J.G., 2016.  Reassessing Longer-Run U.S. Growth: How Low?  Federal Reserve Bank of San Francisco, Working Paper Series , 01–25doi:10.24148/wp2016-18.
  * Gandolfo (2009) Gandolfo, G., 2009.  Economic Dynamics.  Fourth ed., Springer-Verlag, Berlin Heidelberg.
  * Goodwin (1951) Goodwin, R., 1951.  The Nonlinear Accelerator and the Persistence of Business Cycles.  Econometrica 19, 1–17.  doi:10.2307/1907905.
  * Greenwood and Shleifer (2014) Greenwood, R., Shleifer, A., 2014\.  Expectations of Returns and Expected Returns.  The Review of Financial Studies 27, 714–746.  doi:10.1093/rfs/hht082.
  * Griskevicius et al. (2008) Griskevicius, V., Cialdini, R., Goldstein, N., 2008.  Applying (and Resisiting) Peer Influence.  MIT Sloan Management Review 49.
  * Gusev and Kroujiline (2020) Gusev, M., Kroujiline, D., 2020\.  A Simple Economic Model with Interactions.  SSRN Scholarly Paper ID 3552539\. Social Science Research Network. Rochester, NY.  doi:10.2139/ssrn.3552539.
  * Gusev et al. (2015) Gusev, M., Kroujiline, D., Govorkov, B., Sharov, S.V., Ushanov, D., Zhilyaev, M., 2015\.  Predictable markets? A news-driven model of the stock market.  Algorithmic Finance 4, 5–51.  doi:10.3233/AF-150042.
  * Haag and Weidlich (1983) Haag, G., Weidlich, W., 1983\.  Concepts and Models of a Quantitative Sociology: The Dynamics of Interacting Populations.  First ed., Springer.
  * Hicks (1937) Hicks, J.R., 1937.  Mr. Keynes and the "Classics"; A Suggested Interpretation.  Econometrica 5, 147–159.  doi:10.2307/1907242.
  * Hicks (1950) Hicks, J.R., 1950.  A Contribution to the Theory of the Trade Cycle.  Oxford University Press, Oxford.
  * Hommes (2013) Hommes, C., 2013.  Behavioral Rationality and Heterogeneous Expectations in Complex Economic Systems.  Cambridge University Press, Cambridge.  doi:10.1017/CBO9781139094276.
  * Hommes and LeBaron (2018) Hommes, C.H., LeBaron, B.D., 2018\.  Handbook of Computational Economics. Volume 4, Heterogeneous Agent Modeling. volume 4 of Handbook of Computational Economics.  North Holland, Amsterdam.
  * Ising (1925) Ising, E., 1925.  Beitrag zur Theorie des Ferromagnetismus.  Zeitschrift für Physik 31, 253–258.  doi:10.1007/BF02980577.
  * Kaldor (1940) Kaldor, N., 1940.  A Model of the Trade Cycle.  The Economic Journal 50, 78–92.  doi:10.2307/2225740.
  * Kalecki (1937) Kalecki, M., 1937.  A Theory of the Business Cycle.  The Review of Economic Studies 4, 77–97.  doi:10.2307/2967606.
  * Kaplan and Violante (2018) Kaplan, G., Violante, G.L., 2018\.  Microeconomic Heterogeneity and Macroeconomic Shocks.  Journal of Economic Perspectives 32, 167–194.  doi:10.1257/jep.32.3.167.
  * Keynes (1936) Keynes, J.M., 1936.  The General Theory of Employment, Interest, and Money.  Macmillan, London.
  * Korotayev and Tsirel (2010) Korotayev, A.V., Tsirel, S.V., 2010\.  A Spectral Analysis of World GDP Dynamics: Kondratieff Waves, Kuznets Swings, Juglar and Kitchin Cycles in Global Economic Development, and the 2008–2009 Economic Crisis.  Structure and Dynamics 4.
  * Kroujiline et al. (2016) Kroujiline, D., Gusev, M., Ushanov, D., Sharov, S.V., Govorkov, B., 2016.  Forecasting stock market returns over multiple time horizons.  Quantitative Finance 16, 1695–1712.  doi:10.1080/14697688.2016.1176241.
  * Kroujiline et al. (2019) Kroujiline, D., Gusev, M., Ushanov, D., Sharov, S.V., Govorkov, B., 2019.  An endogenous mechanism of business cycles.  Algorithmic Finance , 1–22doi:10.3233/AF-190292.
  * Kuchler and Zafar (2019) Kuchler, T., Zafar, B., 2019\.  Personal Experiences and Expectations about Aggregate Outcomes.  The Journal of Finance 74, 2491–2542.  doi:10.1111/jofi.12819.
  * LeBaron and Tesfatsion (2008) LeBaron, B., Tesfatsion, L., 2008\.  Modeling Macroeconomies as Open-Ended Dynamic Systems of Interacting Agents.  American Economic Review 98, 246–250.  doi:10.1257/aer.98.2.246.
  * Lorenz (1993) Lorenz, H.W., 1993.  Nonlinear Dynamical Economics and Chaotic Motion.  Second ed., Springer-Verlag, Berlin Heidelberg.  doi:10.1007/978-3-642-78324-1.
  * Miller and Cardinal (1994) Miller, C.C., Cardinal, L.B., 1994\.  Strategic Planning and Firm Performance: A Synthesis of More than Two Decades of Research.  The Academy of Management Journal 37, 1649–1665.  doi:10.2307/256804.
  * Murphy (2017) Murphy, D., 2017.  Excess capacity in a fixed-cost economy.  European Economic Review 91, 245–260.  doi:10.1016/j.euroecorev.2016.11.002.
  * Pikovsky and Kurths (1997) Pikovsky, A., Kurths, J., 1997\.  Coherence resonance in a noise-driven excitable system.  Physical Review Letters 78, 775–778.
  * Priesmeier and Stähler (2011) Priesmeier, C., Stähler, N., 2011\.  Long dark shadows or innovative spirits? the effects of (smoothing) business cycles on economic growth: A survey of the literature.  Journal of Economic Surveys 25, 898–912.  doi:10.1111/j.1467-6419.2011.00694.x.
  * Scheinkman (1990) Scheinkman, J.A., 1990.  Nonlinearities in Economic Dynamics.  The Economic Journal 100, 33–48.  doi:10.2307/2234182.
  * Schumpeter (1939) Schumpeter, J.A., 1939.  Business Cycles: A Theoretical, Historical, and Statistical Analysis of the Capitalist Process.  McGraw Hill, New York.
  * Schumpeter (1942) Schumpeter, J.A., 1942.  Capitalism, Socialism, and Democracy.  Harper and Brothers, New York.
  * Slanina (2013) Slanina, F., 2013.  Essentials of Econophysics Modelling.  Oxford University Press, Oxford.  doi:10.1093/acprof:oso/9780199299683.001.0001.
  * Solow (1956) Solow, R.M., 1956.  A Contribution to the Theory of Economic Growth.  The Quarterly Journal of Economics 70, 65.  doi:10.2307/1884513.
  * Solow (2005) Solow, R.M., 2005.  Reflections on Growth Theory, in: Handbook of Economic Growth. Elsevier. volume 1, Part A, pp. 3–10.
  * Stock and Watson (1999) Stock, J.H., Watson, M.W., 1999\.  Business Cycle Fluctuations in U.S. Macroeconomic Time Series, in: Taylor, J.B., Woodford, M. (Eds.), Handbook of Macroeconomics. Elsevier, New York. volume 1A, pp. 3–64.  doi:10.3386/w6528.
  * Suzuki and Kubo (1968) Suzuki, M., Kubo, R., 1968.  Dynamics of the Ising Model near the Critical Point. I.  Journal of the Physical Society of Japan 24, 51–60.  doi:10.1143/JPSJ.24.51.
  * Vines and Wills (2020) Vines, D., Wills, S., 2020.  The rebuilding macroeconomic theory project part II: Multiple equilibria, toy models, and policy models in a new macroeconomic paradigm.  Oxford Review of Economic Policy 36, 427–497.  doi:10.1093/oxrep/graa066.

##  Appendix A Model Variables and Parameters

Table 1: Table of Notation | Definition | Type | Base Case Value
---|---|---|---
Production (Section 2.2)
$Y$ | Production | Variable | -
$K$ | Capital in production | Variable | -
$y$ | Log production | Variable | -
$k$ | Log capital in production | Variable | -
$\tau_{Y}$ | Production characteristic timescale (business days) | Constant | $1000$
$A_{0}$ | Initial technology level | Constant | 1
$\varepsilon$ | Daily technology growth rate | Constant | $2.5\times 10^{-5}$
$\rho$ | Capital share in production | Constant | $1/3$
Capital Supply (Section 2.3)
$K_{s}$ | Capital supply | Variable | -
$k_{s}$ | Log capital supply | Variable | -
$\lambda$ | Proportion of income saved | Constant | $0.15$
$\delta$ | Daily depreciation rate | Constant | $2\times 10^{-4}$
Capital Demand (Section 2.4)
$K_{d}$ | Capital demand | Variable | -
$k_{d}$ | Log capital demand | Variable | -
$s$ | Sentiment level | Variable | -
$h$ | Information level | Variable | -
$c_{1}$ | Capital demand sensitivity to $\dot{s}$ | Constant | $3$
$c_{2}$ | Capital demand sensitivity to $s$ | Constant | $7\times 10^{-4}$
$\beta_{1}$ | Sentiment herding factor | Constant | $1.1$
$\beta_{2}$ | Sentiment sensitivity to $h$ | Constant | $1.0$
$\gamma$ | Feedback strength factor | Constant | $2000$
$\xi_{t}$ | News shocks | Exogenous noise | -
$\tau_{s}$ | Sentiment characteristic timescale (business days) | Constant | $250$
$\tau_{h}$ | Information characteristic timescale (business days) | Constant | $25$
$\tau_{\xi}$ | News shocks characteristic timescale (business days) | Constant | $5$
Limiting Cases (Section 3)
$z$ | Production growth indicator | Variable | -
$\omega_{y}$ | $\omega_{y}=\tau_{y}^{-1}$ | Constant | 0.001
General Case (Section 4)
$y_{0},k_{s0},k_{d0}$ | Asymptotic growth rates of log variables | Variable | -
$R$ | Classic Solow growth rate $R=\varepsilon/(1-\rho)$ (daily) | Constant | $3.75\times 10^{-5}$
$1/\varepsilon$ | Technology growth timescale (business days) | Constant | $4\times 10^{4}$

##  Appendix B Approximate Solution in the Supply-Driven Regime

In this appendix, we solve equation (22) approximately through use of the
boundary layer technique and obtain the economy’s path in analytic form in the
intermediate and long run under the supply-driven regime ($K=K_{s}$).

The starting point of our derivation is equation (22), for convenience
repeated here:

$\tau_{Y}\ddot{K}+(1+\tau_{Y}\delta)\dot{K}+\delta K=\lambda
K^{\rho}e^{\varepsilon t}.$ (37)

Recall that $1\ll\tau_{Y}\ll 1/\varepsilon$, where $\tau_{Y}$ is the timescale
in which output adjusts to changes in the level of capital and $1/\varepsilon$
is the timescale of output growth in the long run. We aim to capture the
dynamics on these two timescales by solving equation (37) on the interval
$t\geq O(\tau_{y})$. For simplicity, we assume that $\tau_{Y}\delta\gg 1$,
which implies that $\tau_{Y}\delta\dot{K}$ is much larger than $\dot{K}$ and
$\tau_{Y}\ddot{K}$ on the interval $t\geq O(\tau_{y})$, allowing us to derive
a more compact solution.

First, we consider equation (37) for $t\gg\tau_{y}$. In this outer region,
$\tau_{y}\dot{K}\ll K$ and we can approximate the solution to (37) by the
solution to equation:

$\delta K_{o}=\lambda K_{o}^{\rho}e^{\varepsilon t},$ (38)

which is given by

$K_{o}=\left(\frac{\lambda}{\delta}e^{\varepsilon
t}\right)^{\frac{1}{1-\rho}}.$ (39)

Next, we consider equation (37) on the interval $O(\tau_{y})\leq t\ll
1/\varepsilon$, where $e^{\varepsilon t}\rightarrow 1$ and $\tau_{y}\dot{K}$
is not necessarily substantively smaller than $K$. In this inner region, we
can approximate the solution to (37) by the solution to

$\tau_{y}\dot{K}_{i}+K_{i}=\frac{\lambda}{\delta}K^{\rho}_{i}.$ (40)

This is the Bernoulli equation and its solution is given by

$K_{i}=\left(Be^{-(1-\rho)\frac{t}{\tau_{y}}}+\frac{\lambda}{\delta}\right)^{\frac{1}{1-\rho}},$
(41)

where $B$ is the constant of integration.

Solutions $K_{o}$ and $K_{i}$ must match in the overlapping interval
$\tau_{Y}\ll t\ll 1/\varepsilon$. This is satisfied for any value of $B$ since
$K_{o}\rightarrow\left(\frac{\lambda}{\delta}\right)^{\frac{1}{1-\rho}}$ and
$K_{i}\rightarrow\left(\frac{\lambda}{\delta}\right)^{\frac{1}{1-\rho}}$, as
follows from (39) and (41). Thus, we approximate the solution to equation (37)
in this region by

$K_{m}=\left(\frac{\lambda}{\delta}\right)^{\frac{1}{1-\rho}}.$ (42)

The approximate solution to equation (37) that is uniformly valid for all
$t\geq O(\tau_{y})$ is given by

$K=K_{i}+K_{o}-K_{m}=\left(\frac{\lambda}{\delta}\right)^{\frac{1}{1-\rho}}\left(\left(Be^{-\left(\frac{1-\rho}{\tau_{y}}\right)t}+1\right)^{\frac{1}{1-\rho}}+e^{\left(\frac{\varepsilon}{1-\rho}\right)t}-1\right),$
(43)

where $B$ has been rescaled for convenience.

As a final step, we obtain the solution for output $Y$ by inverting the
equation of capital motion (4):

$Y=\frac{1}{\lambda}\left(\dot{K}+\delta K\right).$ (44)

Note that $\dot{K}\ll\delta K$ on the interval $t\geq O(\tau_{Y})$ due to the
simplifying assumption $\tau_{Y}\delta\gg 1$. Therefore, the corresponding
uniform approximation for output $Y$, valid for all $t\geq O(\tau_{Y})$, is
given by

$Y=\left(\frac{\lambda}{\delta}\right)^{\frac{\rho}{1-\rho}}\left(\left(Be^{-\left(\frac{1-\rho}{\tau_{y}}\right)t}+1\right)^{\frac{1}{1-\rho}}+e^{\left(\frac{\varepsilon}{1-\rho}\right)t}-1\right).$
(45)

##  Appendix C Model Parameterization

In this appendix, we examine the model’s parameters and discuss how they
affect the behavior of the dynamical system (24) in the phase space.

We begin with equation (24b) that describes sentiment dynamics. Parameter
$\beta_{1}$ defines the relative importance of the herding and random
behaviors of firms. In an unforced situation ($\beta_{2}=0$), the number of
stable equilibrium points, to which the firms’ sentiment $s$ converges,
doubles at $\beta_{1}=1$ from one to two. For $\beta_{1}<1$, random behavior
prevails since there is a single equilibrium at $s=0$, meaning firms fail to
reach a consensus opinion. Conversely, for $\beta_{1}>1$, herding behavior
rules as equation (24b) generates a polarized, bi-stable environment with one
pessimistic ($s<0$) and one optimistic ($s>0$) equilibrium states. It is
sensible to assume $\beta_{1}\sim 1$, otherwise firms would unrealistically
behave either randomly or in perfect synchronicity. We set $\beta_{1}=1.1$,
implying a slight prevalence of herding over randomness. In addition, we set
$\beta_{2}=1$ to ensure that analysts’ influence on firms’ managers likewise
appears in the leading order.

We now consider the information dynamics in (24c). The terms under the
hyperbolic tangent describe the impacts of economic growth and exogenous news
on the collective opinion of analysts $h$. We assume these two sources of
information are of equal importance. Thus, we expect that
$\gamma\omega_{y}=O(1)$ in the feedback term and we model $\xi_{t}$ as an
Ornstein-Uhlenbeck process with an $O(1)$ standard deviation and short
decorrelation timescale $\tau_{\xi}$. Note that $\omega_{y}\ll 1$ and
accordingly $\gamma\gg 1$.

Finally, we inspect the economic dynamics in (24a). In this equation,
different terms determine leading behaviors on separate timescales. We show in
D that the last three terms (with technology growth rate $\varepsilon$
estimated on the basis of observed total factor productivity) are in balance
in the long run. However, if we consider short timescales, the change in
sentiment $\dot{s}$ becomes dominant. Thus, equation (24a) can be approximated
in the short run as $\dot{z}\sim\rho c_{1}\dot{s}$ and we set $\rho c_{1}=1$.
We also note that by construction $c_{2}\ll c_{1}$ to ensure that the term
$c_{2}s$ does not contribute to capital demand dynamics on short timescales.
Hence we expect $c_{2}\ll 1$.

As highlighted in Section 2, there is a segregation of characteristic
timescales that emerges naturally from the types of decisions faced by the
different agents in the model: $\tau_{\xi}\ll\tau_{h}\ll\tau_{s}\ll\tau_{y}\ll
1/\varepsilon$. This segregation facilitates the transfer of the impact of
instantaneous news shocks $\xi_{t}$ across multiple timescales. The estimates
for the timescales are discussed in Section 2.5.

The parameters $c_{2}$ and $\gamma$ are central to the system’s behavior in
the phase space. Increasing $c_{2}$ stabilizes the system, strengthening
convergence towards the stable equilibria and creating a higher barrier
between attracting regions. The role of $\gamma$ is twofold. As $\gamma$ grows
from zero, its immediate effect is to destabilize the system due to growing
economic feedback. However, as $\gamma$ continues to increase, it exerts a
stabilizing effect similar to that of $c_{2}$ because of the term $\gamma
c_{2}$ in the equilibrium condition:

$\textrm{arctanh}(s)-\beta_{1}s=\beta_{2}\tanh\left(\gamma
c_{2}s+\gamma\varepsilon\right),$ (46)

which follows from equations (24) for $\dot{h}=\dot{s}=\dot{z}=\xi_{t}=0$.
Consequently, the potential to generate autonomous economic instability is
limited. In particular, there exists a critical value262626 Subject to the
values set for the other parameters.  of $c_{2}\sim 10^{-4}$ below which
feedback may generate a limit cycle and above which it does not. Figure 15
depicts the formation and subsequent destruction, for $c_{2}=10^{-4}$, of the
limit cycle as $\gamma$ increases.

Figure 15: Development of a stable limit cycle with increasing $\gamma$ for
$\xi_{t}=0$, $c_{2}=1\times 10^{-4}$ and all other parameter values from the
base case (A). The left panels show the phase portraits projected on the
($s,z$)-plane and the right panels plot $s(t)$. Classification of equilibrium
points is provided in footnote 19 in Section 3.2. As $\gamma$ increases, a
large stable limit cycle emerges and then vanishes, demonstrating the
destabilizing effect of $\gamma$ at low values and its stabilizing effect at
high values. (i) Stable dynamics for $\gamma=350$: red trajectories converge
to the left focus and blue trajectories converge to the right focus. (ii) The
equilibria become unstable and a large stable limit cycle emerges for
$\gamma=1000$. (iii) The left node and the saddle point vanish while the limit
cycle persists at $\gamma=4000$. (iv) The dynamics are again stable at
$\gamma=15000$: trajectories converge to the stable focus and the limit cycle
disappears.

In this paper, we argue that realistic economic behaviors cannot be explained
by a stochastic limit cycle. Therefore, we proceed to study the system for
$c_{2}\gtrsim 10^{-4}$, which ensures a bi-stable configuration without a
limit cycle. Figure 16 illustrates that as $c_{2}$ increases, the barrier
between attracting regions grows stronger, resulting in less frequent
crossings from one region to the other (i.e. cycle duration increases). We
seek $c_{2}$ at the lower end of this range to reduce cycle duration.

Figure 16: The effect of $c_{2}$ on the dynamics of sentiment $s(t)$ for
$\xi_{t}\neq 0$. As $c_{2}$ increases from the base case value $c_{2}=7\times
10^{-4}$ (left) to $c_{2}=9.5\times 10^{-4}$ (right), the barrier separating
the two attracting regions grows stronger. The system spends more time captive
to the attractors, reducing the frequency of the crossings between them and
lengthening the duration of fluctuations. Note that the system tends to stay
longer at the expansion attractor (where $s>0$) owing to the asymmetry induced
by technological growth $\varepsilon>0$. All other parameters are from the
base case (A).

Similarly, the barrier between attracting regions grows stronger as $\gamma$
increases, resulting in infrequent transitions between the attractors.
Relatively small values of $\gamma$, however, dampen feedback, leading to weak
dynamics and stochastic-like behavior. Accordingly, we focus on values of
$\gamma$ between these two extremes. Figure 17 depicts the dynamics under
different values of $\gamma$, with balanced dynamic behaviors for a reasonably
wide range thereof.

Figure 17: The effect of $\gamma$ on the dynamics of sentiment $s(t)$ for
$\xi_{t}\neq 0$. At low $\gamma$, the system’s behavior is dominated by noise
as the barrier between the two attracting regions is weak. As $\gamma$
increases, the barrier grows stronger and the system becomes extremely bi-
stable. Reasonably balanced dynamics emerge in the range from $\gamma=1500$ to
$\gamma=2500$. Note the asymmetry caused by technological growth becomes
exacerbated as $\gamma$ increases in accordance with equation (46). All other
parameters are from the base case (A).

We select $c_{2}=7\times 10^{-4}$ and $\gamma=2000$ for the base case studied
in Sections 3.2 and 4. Note that $c_{2}\ll 1$, $\gamma\gg 1$ and
$\gamma\omega_{y}=O(1)$, as expected. Figure 18 provides the base case phase
portrait ($\xi_{t}=0$) projected on the $(s,h)-$plane, showing attracting
regions around the two stable equilibria as well as long trajectories passing
near each attractor and ending at the opposite equilibrium. These
trajectories, which emerge due to strong feedback ($\gamma\gg 1$), allow the
economy to transition quickly between expansions and contractions.

Figure 18: System dynamics in the base case (A). Left: A phase portrait
($\xi_{t}=0$) projected on the ($s,h$)-plane. The portrait depicts stable
foci, separated by a saddle point, and the large trajectories relevant for
regime transitions. Right: A trajectory ($\xi_{t}\neq 0$) projected on the
($s,h$)-plane. The stable foci are at the center of the two attracting
regions, within which the trajectory is dense. The transit of the economy
between these regions corresponds to regime transitions between contractions
and expansions, occurring at much shorter intervals than the periods during
which the economy is captive to an attractor. The trajectory was smoothed by a
Fourier filter to remove harmonics with periods less than 500 business days
for clean visualization. Figure 19: The effect of $\varepsilon$ on the
dynamics of sentiment $s(t)$ for $\xi_{t}\neq 0$. As $\varepsilon$ increases
from the base case value $\varepsilon=2.5\times 10^{-5}$ (left) to
$\varepsilon=7.5\times 10^{-5}$ (right), the system behavior begins to exhibit
a stronger asymmetry between the contraction and expansion attractors. All
other parameters are from the base case (A).

The attractors are not connected and the economy cannot cross the boundary
separating them in the absence of exogenous news shocks $\xi_{t}$. It takes a
random news event to force the economy, entrapped by one attractor, across the
boundary. Once it crosses the boundary, the economy finds itself on the long
trajectory that takes it swiftly to the other attractor, where the economy
remains captive until another news event instigates the next regime transition
by again forcing the economy across the boundary. At this point the economy is
carried back to the entrapment region where it started. This is the coherence
resonance mechanism that is at the heart of the economic fluctuations in our
model.

As a final comment, we note that if $\varepsilon=0$, the equilibrium condition
(46) is symmetric to $s\to-s$. Technology growth, $\varepsilon>0$, causes an
asymmetry272727Note that $\gamma$ amplifies the asymmetry via the term
$\gamma\varepsilon$ in equation (46). wherein the equilibrium at $s>0$ becomes
stronger than that at $s<0$ (to the extent that the latter vanishes above a
certain threshold). As a result, the system tends to stay longer in the region
where economic sentiment is positive, accelerating the economy’s long-term
growth. The asymmetry, however, vanishes in the limit cycle regime, whether
periodic or stochastic (Section 4.2). Figure 19 illustrates this asymmetric
behavior.

##  Appendix D Asymptotic Analysis of Long-Term Growth

In this appendix, we study the behavior of the Dynamic Solow model in the long
run by seeking $y\sim y_{0}t$, $k_{d}\sim k_{d0}t$ and $k_{s}\sim k_{s0}t$ in
equations (27)-(33) at large values of $t$.

###  D.1 Asymptotic Behavior in the Supply-Driven Regime ($k_{d}>k_{s}$)

We first consider the situation where capital demand exceeds supply, which
entails $k=k_{s}$ under the market clearing condition (32), and obtain the
resulting growth rates.

For $t\gg 1$, the production equation (27) becomes

$e^{(\rho k_{s0}+\varepsilon-y_{0})t}-1=\tau_{y}y_{0}.$ (47)

Consequently, $(\rho k_{s0}+\varepsilon-y_{0})t$ must be constant, which in
turn implies that

$y_{0}=\rho k_{s0}+\varepsilon,$ (48)

with a precision of up to $O(1/t)$. Similarly, capital supply equation (28)
yields

$k_{s0}=\lambda e^{(y_{0}-k_{s0})t}-\delta,$ (49)

so that $(y_{0}-k_{s0})t$ is constant and, therefore, with a precision of up
to $O(1/t)$:

$k_{s0}=y_{0}.$ (50)

It follows from equations 48 and 50 that

$y_{0}=k_{s0}=\frac{\varepsilon}{1-\rho}\equiv R,$ (51)

where $R$ denotes the classic Solow growth rate.282828This same result also
follows from equation (23) for $t{\geq}O(1/\varepsilon)$.

To determine the growth rate of capital demand $k_{d0}$, we average equation
(29) with respect to time, noting that $\bar{\dot{s}}=0$ since $s$ is bounded:

$k_{d0}=c_{2}\bar{s},$ (52)

where the bar denotes the time average.

Then we average equation (31) while noting that $\bar{\dot{h}}=0$ since $h$ is
bounded and that $H(k_{s},k_{d})=0$ from (26) (no feedback) to obtain

$\bar{h}=\overline{\tanh\left(\xi_{t}\right)}=\tanh\left(\overline{\xi_{t}}\right)=0,$
(53)

where we have assumed that fluctuations are small to allow us to take averages
under the hyperbolic tangent292929This simplifying assumption does not
severely restrict applicability as $\tanh{x}$ is approximated reasonably well
by a linear function for $-1{\leq}x{\leq}1$ and the noise amplitude is $O(1)$
in (53).. Similarly averaging equation (30) leads to

$\bar{s}=\overline{\tanh\left(\beta_{1}s+\beta_{2}h\right)}=\tanh\left(\beta_{1}\bar{s}\right).$
(54)

Equation (54) has three solutions for $\beta_{1}>1$: $s=0$, $s_{-}<0$, and
$s_{+}>0$, where $s_{-}=-s_{+}$. Our focus is on $s_{-}$ and $s_{+}$ as they
correspond to the stable equilibrium points303030For the base case value
$\beta_{1}=1.1$, we have $s_{\pm}\approx\pm 0.5$ from (54).. The system spends
most of its time in the attracting regions that surround each of these two
equilibria and transits rapidly between them when forced by exogenous noise.
In the long run, the time spent in transit is negligible relative to the
length of time during which the system is entrapped by the attractors. The
attractors have the same strength and are located symmetrically in $s$, thus
the system tends to spend an equal amount of time at each of them at large
$t$. Therefore, its average position with respect to sentiment $s$ must be
zero. More formally, taking $s_{-}$ and $s_{+}$ as the attractors’ proxies, we
estimate the long-term average sentiment as

$\displaystyle\bar{s}=\frac{1}{2}\left(s_{-}+s_{+}\right)=0.$ (55)

Hence equation (52) yields

$k_{d0}=0.$ (56)

This result is intuitively clear: the growth of demand is driven in the long
run by average sentiment, which converges to zero because its dynamics are
symmetric in the absence of feedback. We conclude that in the supply-driven
regime the economy’s growth is, as expected, independent of capital demand and
matches the classic Solow growth, $y_{0}=k_{s0}=R$, while capital demand is
stagnating ($k_{d0}=0$). We verify these results via numerical simulations in
Section 4.2.

###  D.2 Asymptotic Behavior in the Demand-Driven Regime ($k_{d}<k_{s}$)

In the demand-driven regime, the market clearing condition (32) yields
$k=k_{d}$, so that equation (27) becomes

$e^{(\rho k_{d0}+\varepsilon-y_{0})t}-1=\tau_{y}y_{0}.$ (57)

Consequently,

$y_{0}=\rho k_{d0}+\varepsilon,$ (58)

with a precision of up to $O(1/t)$. Similarly, equation (28) takes the form:

$k_{s0}=\lambda e^{(y_{0}-k_{s0})t}-\delta e^{k_{d}-k_{s}}.$ (59)

The term $\delta e^{k_{d}-k_{s}}$ can be neglected as it is exponentially
small for $k_{d}<k_{s}$; therefore, with a precision of up to $O(1/t)$:

$y_{0}=k_{s0}.$ (60)

And finally, averaging equation (29) leads to

$k_{d0}=c_{2}\bar{s}.$ (61)

We can rewrite equations (58) and (60) as

$y_{0}=k_{s0}=R+\rho\left(k_{d0}-R\right).$ (62)

It follows that if $\bar{s}>\frac{R}{c_{2}}$, then the economy’s long-term
growth exceeds the classic Solow growth rate $R$. For the base case values of
$c_{2}$, $\varepsilon$ and $\rho$ in our model, we find $\bar{s}>0.05$.

To estimate $\bar{s}$, we must consider three types of characteristic behavior
possible in the demand-driven regime: noise-driven, limit cycle and coherence
resonance behavior. Noise-driven behavior prevails when feedback is weak. This
situation is, in its limit, equivalent to that of the supply-driven regime in
which sentiment behaves symmetrically with respect to the origin. Therefore,
$\bar{s}\rightarrow 0$. Thus, the noise-driven mode generates growth
$y_{0}\rightarrow\varepsilon$, which is lower than $R$.

The growth in the two other modes is studied numerically in Section 4.2. For
completeness, we briefly note, first, limit cycles (periodic or stochastic)
lead to $\bar{s}\rightarrow 0$ and $y_{0}\rightarrow\varepsilon$ (as the
economy tends to spend a half of its time in the region where $s>0$ and the
other half where $s<0$) and, second, coherence resonance yields $\bar{s}>0.05$
and $y_{0}>R$, owing to the attractors’ asymmetry caused by technological
growth ($\varepsilon>0$) in the presence of economic feedback ($\gamma>0$).

As a final remark, it follows from (58) that, asymptotically, $z\sim
z_{0}t\sim(\rho k_{d0}+\varepsilon-y_{0})t\sim O(1)$. The system’s motion is
therefore bounded in $z$. Its motion is likewise bounded in $s$ and $h$, which
vary between -1 and 1, as, at the boundaries, $\dot{s}$ and $\dot{h}$ are
directed into the domain of motion as follows, respectively, from equations
(30) and (31). Thus, the system’s phase trajectories are bounded in the
$(s,h,z)$-space.