# Economic Crises in a Model with Capital Scarcity and Self-Reflexive
Confidence

Federico Guglielmo Morelli  Both authors contributed equally to this work.
LPTMC, UMR CNRS 7600, Sorbonne Université, 75005 Paris, France  LadHyX, UMR
CNRS 7646, Ecole Polytechnique, 91128 Palaiseau, France  Chair of Econophysics
& Complex Systems, Ecole polytechnique, 91128 Palaiseau, France  Karl Naumann-
Woleske  Both authors contributed equally to this work. LadHyX, UMR CNRS 7646,
Ecole Polytechnique, 91128 Palaiseau, France  Chair of Econophysics & Complex
Systems, Ecole polytechnique, 91128 Palaiseau, France  New Approaches to
Economic Challenges Unit (NAEC), OECD, 75016 Paris, France  Michael Benzaquen
LadHyX, UMR CNRS 7646, Ecole Polytechnique, 91128 Palaiseau, France  Chair of
Econophysics & Complex Systems, Ecole polytechnique, 91128 Palaiseau, France
Capital Fund Management, 23-25, Rue de l’Université 75007 Paris, France  Marco
Tarzia  LPTMC, UMR CNRS 7600, Sorbonne Université, 75005 Paris, France
Institut Universitaire de France, 1 rue Descartes, 75005 Paris, France  Jean-
Philippe Bouchaud  Chair of Econophysics & Complex Systems, Ecole
polytechnique, 91128 Palaiseau, France  Capital Fund Management, 23-25, Rue de
l’Université 75007 Paris, France  Académie des Sciences, Quai de Conti, 75006
Paris, France

###### Abstract

In the General Theory, Keynes remarked that the economy’s state depends on
expectations, and that these expectations can be subject to sudden swings. In
this work, we develop a multiple equilibria behavioural business cycle model
that can account for demand or supply collapses due to abrupt drops in
consumer confidence, which affect both consumption propensity and investment.
We show that, depending on the model parameters, four qualitatively different
outcomes can emerge, characterised by the frequency of capital scarcity and/or
demand crises. In the absence of policy measures, the duration of such crises
can increase by orders of magnitude when parameters are varied, as a result of
the “paradox of thrift”. Our model suggests policy recommendations that
prevent the economy from getting trapped in extended stretches of low output,
low investment and high unemployment.

###### Contents

  1. I Introduction
  2. II A Behavioural Business Cycle Model
    1. II.1 The Household
    2. II.2 The Firm
    3. II.3 The Leontief Limit
      1. II.3.1 Abundant Capital
      2. II.3.2 Scarce Capital
      3. II.3.3 Discussion
    4. II.4 The Risk of Investment
    5. II.5 Spending and Investing
      1. II.5.1 The Consumption Rate
      2. II.5.2 The Investment Allocation
    6. II.6 Summary & Orders of Magnitude
  3. III Crises & Phase Diagrams
    1. III.1 Crises Indicators
    2. III.2 Prosperous Stability
    3. III.3 Prevalent Capital Scarcity
    4. III.4 Prevalent Consumption Crisis
    5. III.5 Capital and Consumption Crises
    6. III.6 Summary
    7. III.7 Investment and Crisis Recovery
  4. IV Discussion and Conclusions

The theory can be summed up by saying that, given the psychology of the
public, the level of output and employment as a whole depends on the amount of
investment. I put it in this way, not because this is the only factor on which
aggregate output depends, but because it is usual in a complex system to
regard as the “causa causans” that factor which is most prone to sudden and
wide fluctuation. More comprehensively, aggregate output depends on the
propensity to hoard, on the policy of the monetary authority as it affects the
quantity of money, on the state of confidence concerning the prospective yield
of capital assets, on the propensity to spend and on the social factors which
influence the level of the money wage.  Keynes (Keynes1936, )

##  I Introduction

The years following 2008 were marked by the great financial crisis, and with
it a crisis for economic theory (Kirman2010, ). As for the great depression of
the 1930s, there was a failure to predict the crisis amongst economic
orthodoxy.111See the extensive discussions in (Buiter2009, ; Blanchard2018, ;
DosiRoventini2019, ; Korinek2017, ; Romer2016, ; Stiglitz2018, ), with a
recent review in Fair2020  Despite its failures in predicting the recession
(ChristianoEtAl2018, ) or the sluggish recovery (LindeEtAl2016, ), the
mainstream Dynamic Stochastic General Equilibrium (DSGE) class of models have
remained the core macroeconomic framework and workhorse tool of policy
(KaplanViolante2018, ). While calls to reform these models have been made
Stiglitz2018 ; VinesWills2018 ; VinesWills2020 , the basic framework with a
single rational representative agent often remains a baseline assumption when
studying business cycles, although heterogeneous agents new Keynesian models
(HANK) have recently been considered as well.222For a non-exhaustive list of
prominent examples see (KaplanEtAl2018, ; KaplanViolante2014, ; McKayReis2016,
). The decisions of such a representative agent, which include capital
investment decisions, determine the trajectory of the economy and are based on
the optimisation of a utility function with static parameters, with no space
for “animal spirits” or confidence effects (although see BarskySims2012  and
the discussion below). In such models, “dark corners” are absent and crises
can only be the result of major exogenous shocks.333“Dark corners” refers to a
particularly insightful piece by O. Blanchard in 2014 entitled “Where Danger
Lurks”, see Blanchard2014

This must be contrasted with Keynes’ intuition, which, as rephrased by Minsky
(Minsky1976, ), was that the subjective evaluation of prospects over a time
horizon is the major proximate basis for investment and portfolio decisions,
and these subjective estimates are changeable. Expectations can indeed be
subject to rapid changes, disagreement and irrationality, as reflected in the
high volatility of investment (StockWatson1999, ), and the abrupt nature of
expansions and recessions. The investor behaviour behind these swings are
indeed often referred to as animal spirits or irrational exuberance
(Shiller2005, ; AkerlofShiller2010, ). There is now a rich literature on
irrational behaviour across economics (see (Hommes2021, ) for a recent
review). However, it has not been fully dovetailed into more traditional
business cycle models. One can find some boundedly rational components in DSGE
models,444 See (FrankeWesterhoff2017, ) for an early review of animal spirits
in macroeconomic models.  such as (CorneaEtAl2019, ; HommesLustenhouwer2019, ;
Ozden2021, ) that focus on learning in expectations formation in a single-
actor model, as well as (JumpLevine2019, ; Gabaix2020, ) that use various
different utility specifications in DSGE. But apart from Refs. BarskySims2012
; DenizAslanoglu2014 ; Brenneisen2020 , there is surprisingly little work
attempting to factor confidence or sentiment into the DSGE framework as an
explicit variable. This is despite some empirical work suggesting that
consumer confidence contains important information for forecasting personal
spending and consumption Blanchard1993 ; CarrollEtAl1994 ;
MatsusakaSbordone1995 .

Whereas the adapted New Keynesian model of Barsky and Sims BarskySims2012 ;
Brenneisen2020  conflates confidence with forecasting with imperfect signals,
or private news about future technological states, we rather want to focus
here on Keynes’ animal spirit, self-reflexive facet of confidence, which can
be subject lead to abrupt shifts like in 1929 or 2008.555More recently
AngeletosEtAl2018 ; AngeletosLaO2013  consider sentiments as uncertainty about
the beliefs of others. For another strand of the literature on sudden
breakdown of confidence, see Bouchaud2013 ; GamaBatistaEtAl2015  and
references therein. As a first step to incorporate such effects and assess
their impact on the economy, some of us (MorelliEtAl2020, ) recently proposed
a generalisation of a simple monetary model in which the household’s
propensity to consume depends on the prior state of the economy, which
generates either optimism or anxiety. This feedback can amplify productivity
shocks, and lead to the appearance of a second equilibrium characterised by
low consumption and high unemployment, and crises resulting from self-induced
confidence collapse. The existence of two very different macroeconomic
equilibria has also been recently suggested in another context in
(CarlinSoskice2018, ).

In the present paper, our aim is to significantly extend the work of
(MorelliEtAl2020, ) by including capital investment as a factor determining
the trajectory of the economy. We assume that capital and labour are
essentially non-substitutable, and posit a behavioural rule for investment
that accounts for both consumer confidence and for the quality of the returns
generated by risky capital investment. This expanded framework allows us to
investigate the joint dynamics of confidence, capital availability and output.
In a nutshell, our model attempts to capture many of the ideas so clearly
expressed by Keynes in the opening quote above, while keeping part of the
scaffolding of standard business cycle models.

Figure 1: Trajectories of the OECD confidence index, changes in consumption
[‰], and investment [%] in the United States over a period from the beginning
of 2000 to the present. The data were taken from FRED and OECD.

In fact, one of our motivations for building such behavioural business cycle
models stems from the Great Recession, sparked by Lehman’s bankruptcy that led
to a sudden collapse in the confidence of both households and investors. This
was followed by an almost immediate downfall of both investment and
consumption. These stylized facts can be observed in Figure 1. It took six
years to recover to prior confidence levels, even as investment and
consumption grew in the medium-term. It is difficult to believe that the Great
Recession was the result of a major exogenous shock.666 Lo2012  notes that
there is no consensus narrative of the causes for the crisis. In addition,
from a DSGE model perspective, AngeletosEtAl2020  recently ruled out many of
the common exogenous DSGE shocks as explanatory candidates for business cycles
Rather, Keynes’ story assigning the abruptness of the crisis to a shift in
investment decisions is much more plausible. Indeed, anecdotal evidence
reported by prominent actors at the time strongly suggests that confidence
collapse played an essential role in the unfolding of the crisis – see the
riveting account of Ben Bernanke et al. in BernankeEtAl2019 .

The consumption and investment trajectories generated by our model can be
grouped into four distinct categories differentiated by the prevalence of
crises in consumption and the scarcity of capital for production. We recover
the bi-stable behaviour obtained in Ref. MorelliEtAl2020 , alternating between
enduring spells of high and low confidence. In addition, the household’s
investment behaviour can lead to capital scarcity, i.e. periods where capital
is the limiting factor to output. During these instances there is an increased
risk of a confidence collapse and an ensuing low-consumption depression where
the household consumes a small fraction of disposable income and invests
cautiously.

Furthermore, when compared with the results of our previous version of the
model where capital is absent (or rather, assumed to be so abundant that
keeping track of it is unnecessary), we find that low output periods can last
orders of magnitude longer. This is because, in the absence of suitable policy
measures, investment remains low and capital scarcity prevents the economy
from recovering.

This multiple equilibria scenario is an attempt to move away from the over-
simplified, but still dominant single equilibrium paradigm, following recent
calls to that effect VinesWills2020 ; Greene2021 . Note that the coexistence
of different equilibria is also the hallmark of recent agent based models, see
e.g. GualdiEtAl2015 ; SharmaEtAl2020  and refs. therein.

The qualitative results of our behavioural business cycle model suggests
various policy measures, in terms of narratives Shiller2019  that may change
the perception of the future of the economy and the attractiveness of
investment in productive capital. In particular, we emphasise the crucial need
to maintain capital investment at a sufficiently high level throughout crisis
periods, in order to allow for a quick recovery when the economic conditions
improve.

The manuscript is organised as follows. In Section II we build up our business
cycle model based on (MorelliEtAl2020, ), and outline our two novel additions.
We then show the various dynamics the model can generate an reveal its phase
diagram in Section III. We discuss in particular how capital scarcity
increases the probability of consumption crises, and lead to a multifold
increase of the recovery time (see Section III.7). Section IV concludes by
discussing the policy implications of our findings, and the avenues for
possible extensions of the model.

##  II A Behavioural Business Cycle Model

The framework presented here hybridises some standard assumptions used in the
New Keynesian Dynamic Stochastic General Equilibrium (DSGE) model (see
(Gali2015, )) with plausible behavioural assumptions about consumption
propensity and investment strategies. The environment is based on two blocks:
the representative consumer and the representative firm. At this point, we
neglect inter-temporal effects and do not attempt to model inflation dynamics
and monetary policy, although these features could be included at a later
stage. Nonetheless, the phenomenology of our model is already quite rich and
needs to be streamlined before exploring further the dynamics of prices. All
variables and notations are reported in Table 1.

###  II.1 The Household

The household sector derives utility from a composite consumption good and
provides labour services to the firm. At each time $t$, the representative
household maximises its instantaneous utility,

$U_{t}(c_{t},n_{t}):=G_{t}\cdot\log c_{t}-\gamma\cdot n^{2}_{t}\ ,$ (1)

where $c_{t}$ and $n_{t}$ denote respectively the level of aggregate
consumption and the aggregate amount of working hours the household provides
to the firm, $G_{t}$ is the (time dependent) propensity to consume out of
income, and $\gamma$ is the disutility of labour (which we fix to 1 for the
numerical analysis).

Each period, the household faces a budget constraint given by its real income
$\mathcal{I}_{t}$,

$\mathcal{I}_{t}:=w_{t}\cdot
n_{t}+\frac{b_{t-1}}{1+\pi_{t}}+q_{t-1}\cdot\frac{k_{t-1}}{1+\pi_{t}}\ ,$ (2)

which is funded by three sources: (i) the real wage rate $w_{t}$ paid by the
firm for a unit of labour $n_{t}$, (ii) the real value of the maturating
single-period bonds $b_{t-1}$, purchased at time $t-1$ at the price
$(1+r_{t-1})^{-1}$ and paying $(1+\pi_{t})^{-1}$ at time $t$, where $r_{t}$ is
the interest rate and $\pi_{t}$ is the inflation rate, and (iii) the realised
yield $q_{t-1}$ per unit of real capital $k_{t}$ that the firm pays to the
household in return for investment. We henceforth assume a constant interest
rate $r=0.15\%$ and inflation $\pi=0.1\%$, keeping in mind a unit time scale
corresponding to a month or quarter.

Figure 2: A schematic representation illustrating the division of income, i.e.
the budget constraint, by the household.

Total spending, correspondingly, consists in good consumption (with the price
of good set to unity), purchases of new bonds and topping up the firm’s
capital. Maximisation of the household’s utility (Eq. (1)) leads to the
familiar state equation

$n_{t}\cdot c_{t}-\frac{G_{t}\cdot w_{t}}{2\gamma}=0\ ,$ (3)

describing the trade-off between consumption and labour in the current period
$t$.

Interestingly, Eq. (3) can also be interpreted in a way that lends itself to a
natural generalisation for investment decisions. Suppose one starts with a
time independent utility function, Eq. (1) with $G_{t}\equiv 1$, which is now
optimised under the constraint that the total budget devoted to consumption is
a fixed fraction $G_{t}\in[0,1]$ of the income $\mathcal{I}_{t}$, i.e.

$c_{t}=G_{t}\cdot\mathcal{I}_{t}.$ (4)

It is easy to show that the very same equation Eq. (3) immediately follows. We
posit that the remaining fraction $1-G_{t}$ of income is invested in bonds and
capital, i.e.

$\displaystyle i_{t}=(1-G_{t})\cdot\mathcal{I}_{t}\ ,$ (5)

where a fraction $F_{t}\cdot i_{t}$ (with $F_{t}\in[0,1]$) is allocated to
productive capital, and the remainder $(1-F_{t})\cdot i_{t}$ is invested in
bonds – see Fig. 2 for a pie chart summarising the household spending and
investment decision.

The capital level available to the firm thus evolves as

$k_{t}=(1-\delta)\cdot k_{t-1}+F_{t}\cdot(1-G_{t})\cdot\mathcal{I}_{t}\ ,$ (6)

where $\delta$ is the capital depreciation rate. The remaining investment is
allocated to bonds at price $(1+r)^{-1}$, so

$\displaystyle\frac{b_{t}}{1+r}$ $\displaystyle=$
$\displaystyle(1-F_{t})\cdot(1-G_{t})\cdot\mathcal{I}_{t}$ (7)

The quantities $G_{t}$ and $F_{t}$ aim to capture confidence effects and the
attractiveness of risky capital investment, respectively, and are specified in
section II.5 below.

###  II.2 The Firm

The economy’s productive sector is made up of a single representative firm,
which transforms labour $n_{t}$ and capital $k_{t}$ into a composite good
$y_{t}$ consumed by the representative household. The firm’s production
technology is given by a Constant Elasticity of Substitution (CES) function
with constant returns to scale,777In full generality, the CES function should
be written as $\left(\alpha\cdot k_{t}^{-\rho}+(1-\alpha)\cdot(\kappa
n_{t})^{-\rho}\right)^{-\frac{1}{\rho}}$, where $\kappa$ is another parameter.
However one can always set $\kappa=1$ at the expense of rescaling the
disutility of labour parameter according to $\gamma\to\kappa\gamma$.

$y_{t}=z_{t}\cdot\left(\alpha\cdot k_{t}^{-\rho}+(1-\alpha)\cdot
n_{t}^{-\rho}\right)^{-\frac{1}{\rho}}\ ,$ (8)

where $\alpha=1/3$ is the capital share in production, $1/(1+\rho)$ is the
elasticity of substitution between capital and labour with $\rho>0$, and
$z_{t}>0$ is a stationary exogenous technological process. It is given by
$z_{t}=z_{0}e^{\mathfrak{z}_{t}}$, where $\mathfrak{z}_{t}$ follows an AR(1)
process:

$\mathfrak{z}_{t}=\eta\cdot\mathfrak{z}_{t-1}+\sqrt{1-\eta^{2}}\cdot\mathcal{N}(0,\sigma^{2})\
,$ (9)

with first-order autocorrelation $\eta$, which affects the correlation time of
the technology shocks. (In the following we will fix $\eta=0.5$, corresponding
to a correlation time of a few months). The base level $z_{0}$ corresponds to
the most probable value of productivity. Note, importantly, that $z_{0}$ has
units of [Time]-1, i.e. the amount of goods that can be produced per unit time
for a given level of capital and labour. As our focus is on economic
fluctuations, we abstract from production growth in the present model, i.e.
the secular dependence of $z_{0}$ on time.

The CES production function nests two important limits that affect economic
dynamics. As $\rho\to 0^{+}$, the production function becomes perfectly
elastic and recovers the Cobb-Douglas form
($y^{CD}_{t}=z_{t}n_{t}^{1-\alpha}k_{t}^{\alpha}$), whereas in the limit
$\rho\to+\infty$ the firm produces via an inelastic Leontief function
($y^{L}_{t}=z_{t}\min(n_{t},k_{t})$).888Keeping the parameter $\kappa$ free
(see previous footnote), the Leontief function would read
$y^{L}_{t}=z_{t}\min(\kappa n_{t},k_{t})$, i.e. $\kappa^{-1}$ measures the
amount of labour equivalent to one unit of capital. Throughout the following,
we choose $\rho=7$, corresponding to a near Leontief limit, i.e. a very small
amount of substitutability between capital and labour. We will briefly comment
in section III.7 the impact of higher substitutability.

The firm maximises its target profit $\mathcal{P}_{t}$

$\mathcal{P}_{t}=p_{t}\cdot y_{t}-w_{t}\cdot n_{t}-{q}^{*}_{t}\cdot k_{t}\
,\quad(p_{t}\equiv 1),$ (10)

with respect to the labour supply $n_{t}$ and the capital level $k_{t}$, where
$p_{t}$ is set to unity and correspondingly $w_{t}$ is the real wage and
${q}^{*}_{t}$ is the real rent on capital. Under the assumption that the
market clears, i.e.

$y_{t}=c_{t}\ ,$ (11)

one finds

$\displaystyle\tilde{w}_{t}$ $\displaystyle=$
$\displaystyle(1-\alpha)\,\left(\frac{\tilde{c}_{t}}{n_{t}}\right)^{1+\rho}$
(12) $\displaystyle\tilde{q}^{*}_{t}$ $\displaystyle=$
$\displaystyle\alpha\,\left(\frac{\tilde{c}_{t}}{k_{t}}\right)^{1+\rho}$ (13)

where, generically, $\tilde{x}:=x/z$. Note that, as it should be, wages,
consumption and rent on capital are all in units of $z$, i.e. unit time scale
(e.g. a month or a quarter).

Combining the household’s state equation, Eq. (3) and the equation for the
real wage, Eq. (12), the consumption level $c_{t}$ must satisfy

$\tilde{c}_{t}^{2}=\frac{G_{t}}{2\gamma}(1-\alpha)^{\frac{2}{\rho}}\left[1-\alpha\left(\frac{\tilde{c}_{t}}{k_{t}}\right)^{\rho}\right]^{1+\frac{2}{\rho}}\
,$ (14)

As both sides of Eq. (14) are monotonous, this ensures a unique solution for
any given level of capital $k_{t}$ and consumption rate $G_{t}$. As expected,
the consumption at time $t$ increases if the capital $k_{t}$ is increased
and/or the consumption rate $G_{t}$ is increased.

###  II.3 The Leontief Limit

In this section we discuss in detail the Leontief limit of the equations
derived in the previous section. Such an analysis will greatly help
understanding the dynamics of the model that will be described below.

Figure 3: The figure shows the behaviour of rescaled consumption
$\tilde{c}_{t}$, wages $\tilde{w}_{t}$, and rent on capital
$\tilde{q}^{*}_{t}$ as a function of $k_{t}$, in the Leontief limit, i.e.
$\rho\to+\infty$. Note the qualitative change of behaviour between
$k_{t}>\sqrt{G_{t}/2\gamma}$ and $k_{t}<\sqrt{G_{t}/2\gamma}$.

####  II.3.1 Abundant Capital

Assume first that $\tilde{c}_{t}<k_{t}$ and $\rho\to\infty$. Then
$(\tilde{c}_{t}/k_{t})^{\rho}\to 0$ and one finds

$\tilde{c}_{t}\approx\sqrt{\frac{G_{t}}{2\gamma}}$ (15)

This is only consistent with our working hypothesis when

$\frac{G_{t}}{2\gamma}<k_{t}^{2}.$ (16)

In this regime one finds, using Equation (3):

$n_{t}=\tilde{c}_{t}\tilde{w}_{t},$ (17)

which once plugged back in Equation (12) leads to

$\tilde{w}_{t}=(1-\alpha)^{1/(2+\rho)}\approx 1.$ (18)

Since $\tilde{c}_{t}<k_{t}$, one concludes from Eq. (13) that the rent on
capital $\tilde{q}_{t}^{*}$ is exponentially small. Intuitively, as labour is
the limiting factor, consumption is directly proportional to how much the
household chooses to work and the productivity at that time, while capital has
no distinct effects on the economy.

####  II.3.2 Scarce Capital

Now let us look at the regime

$\frac{G_{t}}{2\gamma}>k_{t}^{2}.$ (19)

We introduce the notation $\beta_{t}:=2k_{t}^{2}\gamma/G_{t}$ for further use.
We hypothesise that the solution for $\tilde{c}_{t}$ in this regime is of the
form

$\tilde{c}_{t}=k_{t}e^{-x_{t}/\rho}$ (20)

where $x_{t}=O(1)$ is to be determined. Plugging in Eq. (14), we find:

$k_{t}^{2}e^{-2x_{t}/\rho}=\frac{G_{t}}{2\gamma}(1-\alpha)^{-2/\rho}\left(1-\alpha
e^{-x_{t}}\right)^{1+2/\rho},$ (21)

or, for $\rho\to+\infty$,

$e^{-x_{t}}=\frac{1-\beta_{t}}{\alpha}.$ (22)

The equation for $\tilde{q}_{t}^{*}$ then leads to

$\tilde{q}_{t}^{*}=\alpha e^{-x_{t}}=1-\beta_{t},$ (23)

which indeed vanishes when $\beta_{t}=1$, correctly matching the regime where
capital is plentiful, whereas $\tilde{q}_{t}^{*}$ tends to unity when
$\beta_{t}\to 0$, i.e. where $k_{t}\to 0$.

With $\tilde{c}_{t}=k_{t}e^{-x_{t}/\rho}$, one finds from Equation (3)

$n_{t}=\frac{G_{t}w_{t}}{2\gamma k_{t}}e^{x_{t}/\rho}.$ (24)

Finally, plugging into the equation for wages,

$\tilde{w}_{t}^{2+\rho}=(1-\alpha)e^{-x_{t}}\beta_{t}^{1+\rho},$ (25)

or in the limit $\rho\to\infty$,

$\tilde{w}_{t}=\beta_{t}+O(\rho^{-1}),$ (26)

and hence $n_{t}=k_{t}+O(\rho^{-1})$. Again, this solution matches with the
result $\tilde{w}_{t}=1$ obtained for $\beta_{t}\geq 1$.

####  II.3.3 Discussion

A summary sketch of the above results is provided in Fig. 3, as available
capital $k_{t}$ is varied. Part of the dynamical properties of our model can
be inferred from this figure: when capital is lush, return on capital is small
and investment decreases (i.e. $F_{t}$ decreases). If investment falls below
the level of capital depreciation $\delta$, then $k_{t}$ will fall until the
level $\sqrt{G_{t}/2\gamma}$ is reached. At this point, return on capital
$q^{*}_{t}$ increases, promoting investment. When consumption propensity
$G_{t}$ increases, $k_{t}$ may fall behind, again leading to an increase of
$q^{*}_{t}$. Hence we expect a regime where the economy stabilises close to
the point where $k_{t}\approx\sqrt{G_{t}/2\gamma}$, where capital and labour
are tracking each other, and interest on capital and wages are neither very
small, nor saturated to their maximum value $w_{\max}=q^{*}_{\max}=z$.

###  II.4 The Risk of Investment

Rigidity and costs to capital usage are typically introduced through
adjustment costs to capital utilisation (e.g. see (SmetsWouters2007, )). In
this paper we take a different route. Rather than empowering the household to
choose the firm’s utilisation rate, we suppose the household invests in
capital and gives operational control of capital to the firm. In exchange it
is promised a return ${q}^{*}$ per unit capital and per unit time scale (month
or quarter). However, as the volatility of the stock-market attests to, such a
return is not assured. Hence, we introduce an intrinsic state-dependent risk
to the returns on capital999 More sophisticated distributions can be
considered. We use this simple form to keep the number of parameters of the
model as small as possible.  $\xi\in[0,1]$ as a modifier, such that the rate
actually paid by the firm is:

$q_{t}={q}^{*}_{t}\cdot\xi_{t}\leq{q}^{*}_{t},$ (27)

where $\xi$ is distributed as

$p(\xi)=a\cdot\xi^{a-1},$ (28)

where parameter $a$ controls the intensity of the risk. Note indeed that
$\mathbb{E}[\xi]=a/(1+a)$ and $\mathbb{V}[\xi]=a/(2+a)(1+a)^{2}$. Hence the
larger the value of $a$, the more $p(\xi)$ is concentrated around $\xi=1$
(full payment). This formulation of risk implies that the representative firm
pays out at most the marginal productivity of capital, but more likely only
pays a fraction of this, corresponding to an effective description of
financial distress and bankruptcy within a representative firm setup. In most
simulations we set $a=15$, such that the return is on average $93.75\%$ of the
promised return. In an extended version of the model, the parameter $a$ could
itself be a function of the state of the economy (in particular of the
availability of capital), but we will not consider this possibility here.

###  II.5 Spending and Investing

The model laid out in Sections II.1-II.4 contains two dynamic variables, the
consumption rate $G_{t}$ in Eq. (4) and the investment allocation rate $F_{t}$
in Eq. (5), which have not been specified yet. These two variables are
responsible for the feedback mechanisms which are at the core of the dynamical
evolution of our model economy. Here we elaborate on these mechanisms and
provide the economic intuition behind them.

####  II.5.1 The Consumption Rate

As in Refs. MorelliEtAl2020 ; DenizAslanoglu2014 , we postulate that the
consumption rate $G_{t}$ (or propensity, see section II.1) is a function of
the consumer confidence index  $\mathcal{C}_{t}$, that we model as a real
variable $\in[-1,1]$ and, possibly, on the difference between the expected
inflation rate $\widehat{\pi}_{t}:=\mathbb{E}_{t}[\pi]$ and the bond rate
$r_{t}$:

$G_{t}:=G_{t}(\mathcal{C}_{t},\widehat{\pi}_{t}-r_{t},\ldots),$ (29)

where the dependence on the second variable is a way to effectively encode the
content of the standard Euler equation without explicitly introducing an
inter-temporal optimisation of utility, and where the $\ldots$ leaves room to
possible additional variables. But since in the present paper we assume both
inflation and interest rates to be constant, the second variable will be
dropped altogether. As far as the first variable is concerned, we follow our
previous work in MorelliEtAl2020 , where we postulated that confidence of a
given household is impacted by the level of consumption of other households in
the previous time step. In a mean-field limit, this self-reflexive mechanism
writes

$\displaystyle\mathcal{C}_{t}=\tanh\left(\theta_{c}\cdot(c_{t-1}-c_{0})\right),$
(30)

The parameter $c_{0}$ is a “confidence threshold” where the concavity of
$\mathcal{C}(c)$ changes (if $c_{t-1}<c_{0}$, $\mathcal{C}$ is closer to $1$
while if $c_{t-1}>c_{0}$, $\mathcal{C}(c)$ is closer to $-1$). Parameter
$\theta_{c}>0$ sets the width of the consumption interval over which the
transition from low confidence to high confidence takes place. One could
introduce, as in e.g. BeaudryPortier2014 ; KroujilineEtAl2019 ;
BeaudryEtAl2020 , the impact of macroeconomic news as an extra contribution to
the argument of the $\tanh$ function. This would describe how the consumer
confidence index is further modulated by some exogenous shocks, but we leave
such an extension for future work.

Back to the consumption rate $G_{t}$, we write

$\displaystyle G_{t}$ $\displaystyle=$
$\displaystyle\frac{1}{2}\big{[}G_{\min}+G_{\max}+(G_{\max}-G_{\min})\cdot\mathcal{C}_{t}\big{]}\,$
(31)

where $0\leq G_{\min}<G_{\max}\leq 1$ are the minimum and maximum proportions
of income the household will consume. We fix $G_{\min}=0.05$ to ensure the
household will consume whenever its income is positive (necessary
consumption). Similarly, we set $G_{\max}=0.95$ to account for a minimal form
of precautionary savings in response to some uncertainty regarding the future.

The intuition behind Eqs. (30) and (31) is that when consumption is above the
threshold, $c>c_{0}$, there is high confidence in the future of the economy,
thereby a large fraction of income, $G_{t}\to G_{\max}$, is consumed. High
confidence represents the belief that future income will be sufficient to
maintain high consumption with a minimal amount of savings to sustain capital
levels. Conversely, when consumption is below the threshold, $c<c_{0}$, the
consumption rate collapses, $G_{t}\to G_{\min}$. Following a shock or
deterioration in consumption to below the confidence threshold, there is
uncertainty about the future economy and whether future consumption is
assured. This induces the household to save more for the future, effectively
reducing the current demand. Economically, $c_{0}$ is thus similar to minimal
consumption: it defines the threshold beyond which there is a panic where the
household’s “survival” is in question.

The parameter $\theta_{c}$ modulates the households reaction to a breach of
necessary consumption, and can be described as the household’s panic polarity.
For high $\theta_{c}$, the household requires only a relatively small shock
below $c_{0}$ to reduce the consumption rate to its minimum. This leads to a
bi-stable savings behaviour with sharp transitions. Conversely, as
$\theta_{c}\to 0$ the household becomes unresponsive to the state of the
economy, consuming half its income regardless of high or low preceding
consumption. The intermediate levels of $\theta_{c}$ describe the smoothness
of the adjustment to consumption shocks.

According to MorelliEtAl2020  there are four distinct “phases”, i.e. regions
of qualitatively comparable dynamics, that are distinguished by the bi-
stability of $G_{t}$. We can observe in particular a phase of high persistent
consumption with no crises, high consumption with short downward spikes, or a
phase with alternating periods of high consumption (booms) and low consumption
(busts).

####  II.5.2 The Investment Allocation

In each period, the household must allocate its savings between one-period
bonds and capital. It does so through an allocation decision $F_{t}$ based on
the household’s observation of the economy, and its beliefs about future risk
and return. The novelty of our model lies in the behavioural foundation that
determines the proportion of new investment dedicated to bonds, $F_{t}$.

Investment beliefs are shaped by two factors: (i) an estimate of the expected
risk-adjusted excess returns to capital investment, given by a Sharpe ratio
$\mathcal{S}_{t}$ Sharpe1966 , and (ii) the current confidence level
$\mathcal{C}_{t}$ about the future state of the economy.

The Sharpe ratio $\mathcal{S}_{t}$ is an estimate of the risk-adjusted real
return, $q_{t}-\delta$, of investing capital in the firm versus holding risk-
free bonds ($b_{t}$) paying $r$. It increases as the returns to capital
increase or become less volatile. We assume that estimates of the future
Sharpe ratio are only based on exponential moving averages of past
(observable) realised returns, which is a form of extrapolative
beliefs.101010See DaEtAl2021 ; KuchlerZafar2019  for recent empirical work on
extrapolative beliefs. In other words, i.e. the mean $\mu^{q}$ and standard
deviation $\sigma^{q}$ of the return stream are computed as

$\displaystyle\mu^{q}_{t}$ $\displaystyle=$
$\displaystyle\lambda\cdot\mu^{q}_{t-1}+(1-\lambda)\cdot q_{t}$ (32)
$\displaystyle(\sigma_{t}^{q})^{2}$ $\displaystyle=$
$\displaystyle\lambda\cdot(\sigma^{q}_{t-1})^{2}+(1-\lambda)\cdot(q_{t}-\mu^{q}_{t})^{2}$
(33) $\displaystyle\mathcal{S}_{t}$ $\displaystyle:=$
$\displaystyle\mathcal{N}\cdot\frac{\mu^{q}_{t}-r_{t}-\delta}{\sigma_{t}^{q}}$
(34)

with an exponential moving average defined by a gain parameter
$\lambda\in(0,1)$, corresponding to a memory time scale equal to
$\mathcal{T}_{\lambda}:=1/|\log\lambda|$: a larger $\lambda$ implies that a
higher weight is given to recent observations. The factor $\mathcal{N}\approx
1/4$ is quite arbitrary, but chosen such that, when compared to the confidence
in Eq. (35) below, the two terms are of similar magnitude. (Note that this
choice is in fact immaterial, since changing $\mathcal{N}$ is equivalent to
rescaling the parameter $\nu$ defined in Eq. (35) below.)

The interpretation of the Sharpe ratio is as follows: a positive signal
$\mathcal{S}_{t}>0$ suggests that the expected real return to capital
investment exceeds the returns to risk-free bonds. The magnitude of
$\mathcal{S}_{t}$ is inversely proportional to the risk of capital investment,
as measured by the estimated volatility $\sigma_{t}^{q}$. Thus in a high-
volatility environment the signal might be positive but weak.

The second indicator potentially influencing the household investment decision
is the confidence index, $\mathcal{C}_{t}$, as previously defined. In periods
where the household has low confidence, there is a reduced impetus to invest
in risky assets because households wish to guarantee next-period income. These
are often periods of crisis with a higher volatility in returns. Since bonds
are risk-free, this leads to a higher allocation of funds to bonds, ceteris
paribus. Conversely, higher confidence about the future means more appetite
for risk, and hence a higher fraction of the savings invested in the capital
of firms and a lower fraction invested in bonds.

We postulate that the propensity, $F_{t}$, to make risky bets is a function of
the overall sentiment $\Sigma_{t}$, computed as a linear combination of the
Sharpe ratio and of the confidence:

$\Sigma_{t}=\nu\cdot\mathcal{S}_{t}+(1-\nu)\cdot\mathcal{C}_{t},$ (35)

where $\nu\in[0,1]$ is the weight the household gives to its estimates of
risk-adjusted return $\mathcal{S}_{t}$ and its confidence level
$\mathcal{C}_{t}$. When $\nu=1$, the household’s confidence plays no role in
the investment rule. For positive Sharpe ratio and confidence indicator, the
sentiment is positive, $\Sigma_{t}>0$, indicating a willingness to invest in
risky capital. But if $\nu<1$ sentiment can turn negative even when the Sharpe
ratio is high, because of a high level of anxiety about the future state of
the economy, encoded as a negative value of $\mathcal{C}_{t}$.

Finally, the unbounded sentiment $\Sigma_{t}$ is transformed into a portfolio
allocation to capital $F_{t}\in[0,1]$ via,

$F_{t}=\frac{1}{2}\left[F_{\max}+F_{\min}+(F_{\max}-F_{\min})\cdot\tanh(\theta_{k}\cdot\Sigma_{t})\right],$
(36)

where $F_{\max}$ and $F_{\min}$ represent the maximum and the minimum
proportion of total investment $i_{t}$ invested into capital. In the
following, the allocation decision is bounded between $F_{\min}=0$ and
$F_{\max}=1$, which precludes any divestment (or short-selling) of capital or
bonds.111111 One could allow for divestment by $F_{\min}<0$, however, this
would require a more elaborate form for Eq. (36). The parameter $\theta_{k}$
represents the sensitivity of the portfolio allocation to the agent’s
sentiment and sets the width of the sentiment interval over which the capital
allocation goes from $F_{\min}$ to $F_{\max}$, that is how polar the
investment decision is. For $\theta_{k}\to\infty$, the allocation becomes
binary, leading to either $F_{t}=F_{\min}$ when sentiment is negative or
$F_{t}=F_{\max}$ when sentiment is positive.

In the following part, we fix the sensitivities to a rather high value
$\theta_{k}=\theta_{c}=15$, such that the transitions between different
regimes are sharp.

###  II.6 Summary & Orders of Magnitude

In this section we have set up a business cycle model incorporating two
behavioural mechanisms: a self-reflexive consumption rate decision, already
advocated in Ref. MorelliEtAl2020 , and an investment allocation decision. The
novelty of this paper lies in the behavioural foundation that determines the
proportion of new risky investment $F_{t}$, which depends directly on three
key parameters, $\lambda,\nu,\theta_{k}$, describing the “sentiment” of the
household, i.e. its risk aversion. $F_{t}$ also indirectly depends on the risk
intensity parameter $a$ and the capital depreciation rate $\delta$. The
consumption decision depends on two parameters $c_{0}$ and $\theta_{c}$ that
define the household confidence about its future welfare.

In the following we discuss how the parameters of these two feedback
mechanisms strongly affect the model’s dynamics. Note that a very important
parameter of the model is the baseline productivity $z_{0}$, which fixes the
scale of the consumption, wages and rent on capital (all per unit time scale).
In the following, we choose $z_{0}=0.05$, corresponding to an annual
productivity of capital of $20\%$ if the unit time step is a quarter and
$60\%$ if it is a month.121212 To estimate an appropriate order of magnitude
for $z_{0}$, we considered the gross value added by non-financial corporations
in the U.S. divided by the current cost net stock of fixed assets together
with total wages (as a proxy for labour), which shows a downward trend to
approximately 28% p.a.

Among all the parameters of the model, three have an interpretation in terms
of time scales:

  * •

$\eta$, which appears in the dynamics of the productivity shocks, that we have
fixed to $0.5$ thoughout this study, corresponding to a time scale
$\mathcal{T}_{\eta}=1/|\log\eta|$ of a few months ;

  * •

$\lambda$, which is the gain parameter used by investors to estimate the
Sharpe ratio of risky investments, corresponding to a time scale
$\mathcal{T}_{\lambda}=1/|\log\lambda|$. Our default value will be
$\lambda=0.95$, corresponding to $\mathcal{T}_{\lambda}\approx 20$ or $5$
years if the unit time is a quarter or a month respectively;

  * •

$\delta$, the capital depreciation rate, which we choose in the range $0.001$
– $0.02$, corresponding to a typical replacement time of capital
$\mathcal{T}_{\delta}=1/|\log(1-\delta)|\approx 12$ – $250$ years when the
unit time is a quarter, and three times less if it is a month. Hence
$\delta=0.001$ means essentially no depreciation of capital.

The role and the effect of varying these timescales is studied in detail in
Section III.7. An important remark, at this stage is that, while our choice of
one quarter as the unit time step is quite arbitrary, a combination of
parameters that is crucial for the properties of the model is the
dimensionless product $z_{0}\cdot\mathcal{T}_{\delta}\approx z_{0}/\delta$,
i.e. how much goods can be produced (per unit capital) over the life-cycle of
capital.

Figure 4: This figure shows different sections of the phase space. The three
rightmost upper panels display the $\log_{10}$ probability of capital shortage
$\Xi_{k}$ as a function of the Sharpe ratio weight $\nu$ and of the confidence
threshold $c_{0}$. The bottom row display the $\log_{10}$ probability of
consumption crises $\Xi_{c}$ as a function of the same parameters. The dark
red zones correspond to the regions where $\Xi_{k/c}>0.99$, where crises are
permanent. Each column corresponds to a different choice of the capital
depreciation, from $\delta=0.001$ to $\delta=0.02$. We set $a=15$ and
$\sigma_{z}=0.15$. As $\delta$ increases, the
$(\mathcal{H}_{k},\mathcal{H}_{c})$ regimes becomes widespread. In each panel
we have marked the points chosen to illustrate the different phases of the
model, together with their label (see text). The dynamics trajectories and the
corresponding histograms are reported in Fig. 5. The two leftmost panels show
another section of the phase space, varying $\log\ a$ and $\sigma_{z}$, with
$c_{0}=0.017$, $\nu=1$ and $\delta=0.005$ fixed (the two solid dots there
correspond to the same solid dots of the middle panels, in the
$\mathcal{H}_{k}\mathcal{H}_{c}$ phase).

##  III Crises & Phase Diagrams

In this section we first investigate numerically the phase diagram of our
self-reflexive business cycle model and highlight the different dynamical
features that the model can generate. We choose as control parameters those
which govern the behaviour of our two feedback mechanisms: the consumption
propensity $G_{t}$ and the risky investment decision $F_{t}$. In order to
navigate through the following paragraphs, let us explain in a nutshell what
is expected to happen in the model.

If a productivity shock causes confidence to drop, consumption propensity
$G_{t}$ and consumption both drop as well, whereas the saving rate $1-G_{t}$
increases. Because consumption drops, unemployment rises and capital becomes
superfluous, leading to a decrease of the rent on capital $q^{*}$. Because the
fraction of savings invested in capital $F_{t}$ depends both on $q^{*}$
(through the Sharpe ratio) and on the level of confidence (with a weight
$1-\nu$), the amount invested in risky capital, given by $(1-G_{t})\cdot
F_{t}\cdot\mathcal{I}_{t}$ can either increase (if the factor $1-G_{t}$
dominates) or decrease (if the factor $F_{t}$ dominates), depending on
parameters and conditions. In the second situation, and if capital
depreciation is fast, one may face a situation where consumption is impaired
and capital becomes scarce at the same time, making recovery more difficult
and leading to long periods where the economy is trapped in a low output
state.

###  III.1 Crises Indicators

We focus on two distinct phenomena exhibited by our model: consumption crises
and capital scarcity.

  * •

Consumption crises occur in periods where the household’s consumption,
$c_{t}$, falls below its threshold, $c_{0}$. In other words, we have a low
demand for consumption which leads the economy into a stagnating low-output
state. The severity of such consumption crises is measured as

$\displaystyle\Xi_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{T}\sum_{t=0}^{T}\left(1-\frac{c_{t}}{c_{0}}\right)\,\Theta(c_{0}-c_{t}),$
(37)

where $\Theta(x\geq 0)=1$ and $\Theta(x<0)=0$ and $T$ is the total simulation
time. This indicator counts the fraction of time consumption $c_{t}$ is low,
weighted by the relative distance between $c_{t}$ and $c_{0}$.

  * •

Since we are considering an economy defined by low substitutability between
capital and labour (i.e. $\rho\gg 1$ in the CES production function), we
define capital scarcity as the periods where production is determined by
capital levels, i.e. $k_{t}\leq n_{t}$. The severity of capital crises is
similarly measured as

$\displaystyle\Xi_{k}$ $\displaystyle=$
$\displaystyle\frac{1}{T}\sum_{t=0}^{T}\left(1-\frac{k_{t}}{n_{t}}\right)\,\Theta(n_{t}-k_{t}).$
(38)

In a sense, one can consider these two phenomena as demand and supply crises.

  * •

In the consumption crisis state the household does not wish to spend on
consumption, hence we see a low aggregate demand. Provided capital
depreciation is low, this is also a state of excess capital ($k>n$) and low
returns on capital.

  * •

In the capital scarcity state, the firm is bound in its production by the
supply of capital, hence it can be viewed as a form of supply crisis.

Both phenomena can be more or less frequent, and at first glance unrelated,
but closer scrutiny reveals that in some regions of parameters, these two
types of crises interact with one another. To differentiate between
characteristic behaviours we distinguish between four different phases in the
space of the parameters defined by the values that the indicators $\Xi_{k}$
and $\Xi_{c}$ take: $(\mathcal{L}_{k},\mathcal{L}_{c})$,
$(\mathcal{L}_{k},\mathcal{H}_{c})$, $(\mathcal{H}_{k},\mathcal{L}_{c})$,
$(\mathcal{H}_{k},\mathcal{H}_{c})$, where $\mathcal{L}$ and $\mathcal{H}$
represent the “low prevalence” and “high prevalence” of each phenomena $c$ or
$k$, respectively. There is however no strict definition of the boundary
between high and low prevalence regimes. As a convention, we consider that the
crisis prevalence is high when $\Xi\gtrsim 10^{-2}$.

Given this setup, we first focus on the effects of three key parameters: the
depreciation rate $\delta$, the weight $\nu$ of the Sharpe ratio in the
investment decision, and the consumption threshold $c_{0}$. Other parameters
are fixed to $z_{0}=0.05$, $\lambda=0.95$ (i.e. $\mathcal{T}_{\lambda}=20$),
$a=15$ and $\sigma_{z}=0.15$.

Figure 4 presents heat-maps of the severity of capital crises
$\log_{10}\Xi_{k}$ (top) and consumption crises $\log_{10}\Xi_{c}$ (bottom)
across parameter combinations, where red indicates high prevalence. We show
two representative sections of the parameter space: the planes $(c_{0},\nu)$
(left) and $(\sigma_{z},a)$ (right). Since we present the logarithms of
$\Xi_{k,c}$, the crossovers between high and low prevalence of different
phases are rapid but smooth, i.e. there are no sharp phase transitions that
characterise the system’s behaviour. From each phase we study a point of the
line $\nu=1$ and different values of $c_{0}$ (marked by points in Figure 4),
with all other parameters fixed and plot the dynamics of consumption $c_{t}$,
labour $n_{t}$, capital $k_{t}$, and the measured Sharpe ratio
$\mathcal{S}_{t}$ in Figure 5. Note that changing the value of parameters
(including $\nu$) while staying in the same phase leads to qualitatively
similar trajectories.

Figure 5: The upper panels show sample dynamics trajectories for each phase
presented in Fig. 4 (marked by circles). In the topmost panels the solid black
line corresponds to the consumption $c_{t}$, while the dashed red and blue
horizontal lines show, respectively, the confidence threshold $c_{0}$ and the
average consumption in the $\delta=0$ scenario. Grey (resp. pink) background
indicate capital scarcity (resp. consumption crisis). The middle row presents
the dynamics of capital $k_{t}$ (solid orange), labour $n_{t}$ (solid green)
and Sharpe ratio $\mathcal{S}_{t}$ (solid grey, with levels shown on the right
y-axis). The lower panels show the histograms of consumption (black dots) and
labour (green crosses) in a $\log-\log$ scale. The green (resp. grey) dashed
curve corresponds to the $\delta=0$ baseline value for labour (resp.
consumption), with $c_{0}$ indicated as a vertical red line. In the
$\mathcal{H}_{c}$ phase, the histograms of consumption and labour become bi-
modal, corresponding to high output and low output regimes. For all
simulations $\nu=1$, $a=15$, $\sigma_{z}=0.15$. Specific parameters are
$\mathcal{L}_{k}\mathcal{L}_{c}$: $\delta=0.001$, $c_{0}=0.001$,
$\mathcal{L}_{k}\mathcal{H}_{c}$: $\delta=0.001$, $c_{0}=0.019$,
$\mathcal{H}_{k}\mathcal{L}_{c}$: $\delta=0.02$, $c_{0}=0.001$,
$\mathcal{H}_{k}\mathcal{H}_{c}$: $\delta=0.005$, $c_{0}=0.017$.

###  III.2 Prosperous Stability

As shown in Figure 5, leftmost column, the $\mathcal{L}_{k}\mathcal{L}_{c}$
phase is characterised by a stable capital surplus, low interest on capital
and rare consumption crises. The depreciation of capital $\delta$ is so small
that even with a puny level of investment, capital is always in excess and
labour is the limiting factor. The stable capital surplus, in combination with
a low confidence threshold $c_{0}$, means that productivity shocks $z_{t}$
hardly ever reach the required magnitude to trigger a consumption crisis, and
if it does, recovery is almost immediate.

The corresponding bottom panel of Figure 5, shows that the consumption level
has normal fluctuations, entirely due to exogenous productivity shocks
$z_{t}$, around a single high-consumption equilibrium. A corollary of the
large capital excess is that the labour supply is nearly constant (extremely
narrow-distribution in the $\mathcal{L}_{k}\mathcal{L}_{c}$ panel of Figure
5).

As the depreciation rate $\delta$ increases, the average excess of capital
supply over labour shrinks, increasing the prevalence of capital scarcity.
Accordingly, the phase $\mathcal{L}_{k}$ quickly disappears upon increasing
$\delta$, leading to a pervasive $\mathcal{H}_{k}\mathcal{L}_{c}$ phase (see
e.g. Fig. 4, third column, which shows that capital is always scarce when
$\delta=0.02$). As $\delta$ is further increased, the $\mathcal{L}_{c}$ phase
is more and more confined to small values of $c_{0}$, i.e. when confidence is
intrinsically robust.

###  III.3 Prevalent Capital Scarcity

The $\mathcal{H}_{k}\mathcal{L}_{c}$ phase is characterised by persistent
capital scarcity with rare consumption crises, and is confined within a low
$c_{0}$ “band” in the $(c_{0},\nu)$ plane when $\delta$ is large enough. Since
$c_{0}$ is low, confidence of is generally high and therefore the household
systematically consumes a large proportion $G_{t}$ of its income, leaving only
a small share for investment. Because of capital depreciation, the economy
settles in a regime where $k_{t}<\sqrt{G_{t}/2\gamma}$, meaning that
production is limited by capital, wages are low and rent on capital is high
(i.e. the left region in Fig. 3). Hence, the average consumption level is
lower than the maximal consumption level reached in the
$\mathcal{L}_{k}\mathcal{L}_{c}$ phase – see Figure 5, second column.

But since $c_{t}$ is now closer to $c_{0}$, consumption crises are lurking
around and the economy can flip into the $\mathcal{H}_{k}\mathcal{H}_{c}$ if
$c_{0}$ increases and/or if productivity shocks are stronger (higher
$\sigma_{z}$). This is clearly confirmed by the phase diagram of Fig. 4. In
fact, comparing the phase diagrams for $\delta=0.005$ and $\delta=0.02$, we
see that faster depreciation of capital converts large swaths of
$\mathcal{H}_{k}\mathcal{L}_{c}$ phase into $\mathcal{H}_{k}\mathcal{H}_{c}$.
Hence, in this case, investment crises (i.e. the supply side) do trigger
consumption crises (i.e. the demand side) by reducing the difference between
$c_{t}$ and $c_{0}$ – see also the discussion in section III.5.

###  III.4 Prevalent Consumption Crisis

When the depreciation rate is sufficiently small but the confidence threshold
increases, capital remains abundant but self-reflexive confidence crises can
hurl the system into a low consumption, low employment regime as a result of
random productivity shocks. This is the $\mathcal{L}_{k}\mathcal{H}_{c}$
phase. Since capital is high, its level does not impact the level of
production, and interest on capital is small. Hence the model becomes
completely equivalent, in this regime, to the one studied in MorelliEtAl2020 ,
where the dynamics is characterised entirely by the consumption propensity
$G_{t}$ and is dominated by frequent consumption crises, induced by breakdown
of collective confidence.

As argued in MorelliEtAl2020  and shown in Figure 5, third column, consumption
then displays bi-stable dynamics, where high and low consumption regimes
alternate. Correspondingly, the distributions of consumption and labour reveal
a secondary peak centred around the low-consumption equilibrium.

Note that during consumption crises (i.e. $G_{t}\searrow$) capital becomes
even more abundant relative to labour (recall that one needs to compare
$k_{t}$ with $\sqrt{G_{t}/2\gamma}$) and therefore return on capital and
Sharpe ratio both fall, as can be seen in the pink shaded region of Fig. 5,
third column. If we are in a region where consumption crises are short enough
compared to both the time $\mathcal{T}_{\lambda}$ over which the Sharpe ratio
is estimated and the capital depreciation time $\mathcal{T}_{\delta}$, then
one can avoid a capital crisis when confidence comes back. Otherwise, the
economy enters a turbulent $\mathcal{H}_{k}\mathcal{H}_{c}$ phase with both
capital and consumption crises.

###  III.5 Capital and Consumption Crises

This final $\mathcal{H}_{k}\mathcal{H}_{c}$ phase has both persistent capital
scarcity and consumption crises. As anticipated above, capital crises can
trigger consumption crises, because capital scarcity drives consumption closer
to the confidence threshold $c_{0}$, below which consumption drops and
precautionary savings increase. One can then enter a doom loop (similar to
Keynes’ famous paradox of thrift) where now capital is too high and leads to a
reduction of incentive to invest away from bonds. Hence, as shown in the
fourth column of Fig. 5, capital and labour fluctuate around low levels, with
intertwined periods of capital scarcity (grey regions) and high unemployment
(pink regions). The Sharpe ratio gyrates rather strongly between negative
values and values close to unity, with a significant negative skewness. The
economy is unstable and always far from its optimal state.

Recall that we have fixed the interest rate on bonds to a constant value. But
with a massive demand for bonds, as expected in the
$\mathcal{H}_{k}\mathcal{H}_{c}$ phase, one should expect the government to
borrow at low rates and prop up the economy with public investment, a feature
not modelled in the current framework, but certainly worth accounting for in a
later version of the model.

###  III.6 Summary

To summarise, we have identified four qualitatively different phases of the
dynamics. Possibly the most interesting (and novel) one is
$\mathcal{H}_{k}\mathcal{H}_{c}$, where capital scarcity is persistent,
thereby triggering consumption crises. In this phase the economy is unstable,
as capital becomes scarce the likelihood of a consumption crisis increases,
and vice versa, low consumption drives rent on capital down and increases the
risk aversion of investors.

We have underlined the role of the capital depreciation rate $\delta$ in
determining the fate of our model economy. In fact, when capital and
infrastructure are sufficiently long-lived such that
$z_{0}\cdot\mathcal{T}_{\delta}$ is large, the economy reaches a stable and
prosperous state $\mathcal{L}_{k}\mathcal{L}_{c}$, provided self-induced
confidence crises are rare enough (i.e. $c_{0}$ small). Conversely, when
$z_{0}\cdot\mathcal{T}_{\delta}$ is low, capital depreciates too quickly and
this dents the rents that can be expected by investors. The economy quickly
becomes under-capitalised and inefficient, especially because the dearth of
capital makes confidence crises more probable, paving the way for the
existence of a dysfunctional $\mathcal{H}_{k}\mathcal{H}_{c}$ region in the
phase diagram.

###  III.7 Investment and Crisis Recovery

Figure 6:  The upper (resp. lower) left heat-map shows $\log_{10}\Xi_{c}$
(resp. average Sharpe ratio) as a function of the memory time scale
$\mathcal{T}_{\lambda}$ and the weight parameter $\nu$. The dark red zones
correspond to the regions where $\Xi_{c}>0.99$, where crises are permanent.
The right set of panels shows snapshots of the dynamics of $c_{t}$, $k_{t}$
and $n_{t}$, corresponding to recovery (left) and crisis formation (right),
both for $\mathcal{T}_{\lambda}=20$ and $\nu=0.75$. The two bottom rows show
the histograms of Sharpe ratio $\mathcal{S}$ and crisis duration
$\mathcal{T}_{<}$ for four values of $\mathcal{T}_{\lambda}$: 2, 10, 20 and
50, all for the same value of $\nu$ (shown as symbols in the two heat-maps on
the left). The faded grey lines show the same histograms in the benchmark case
$\nu=1$ that correspond to the $\mathcal{H}_{k}\mathcal{H}_{c}$ point in Fig.
4. Other parameters used are: $\delta=0.005$, $c_{0}=0.017$, $\sigma_{z}=0.15$
and $a=15$.

In the previous section, we have explained how capital depreciation can cause
instabilities, and the appearance of a $\mathcal{H}_{k}\mathcal{H}_{c}$ phase
where both capital and consumption undergo regular crises. In this section, we
want to explore the influence of the memory timescale $\mathcal{T}_{\lambda}$,
which is the history span over which investors assess the Sharpe ratio of
capital investment, and of the sentiment parameter $\nu$ on the time needed
for recovery when in a crisis period. We focus on this turbulent phase of the
economy. We will fix the other two timescales
$\mathcal{T}_{\delta},\mathcal{T}_{\eta}$ defined in section II.6 to,
respectively, $200$ and $2$.

Our benchmark will thus be the $\mathcal{H}_{k}\mathcal{H}_{c}$ point in Fig.
4, corresponding to $\lambda=0.95$, $\delta=0.005$, $\nu=1$ and $c_{0}=0.017$
(with $a$ and $\sigma_{z}$ also fixed at their baseline values). Looking at
the statistics of the high consumption periods and of the low consumption
periods, we conclude that the prosperous periods last a time $\mathcal{T}_{>}$
of the order of $\mathcal{T}_{\delta}=200$ (data not shown), whereas crises
are rather short, of the order of $\mathcal{T}_{<}\approx 10$, see Fig. 6,
plain vertical lines in the third graph of the bottom row, which corresponds
to $\mathcal{T}_{\lambda}=20$, i.e. $\lambda=0.95$. The full distribution of
$\mathcal{T}_{<}$ and of the Sharpe ratio $\mathcal{S}$ for $\nu=1$ are shown
in light grey, and reveals that whereas its time averaged value of
$\mathcal{S}$ is clearly positive and equal to
$\langle\mathcal{S}\rangle_{t}\approx 0.71$, its full distribution is uni-
modal but quite broad and negatively skewed.

Fig. 6, left graphs shows the consumption crisis prevalence $\Xi_{c}$ and the
average Sharpe ratio $\mathcal{S}$ as $1-\nu$ (weighing confidence in the
investment allocation decision) and $\mathcal{T}_{\lambda}$ are varied. One
sees that decreasing $\nu$ or increasing $\mathcal{T}_{\lambda}$ leads to an
increase of $\Xi_{c}$, at least in the range shown,
$\mathcal{T}_{\lambda}\lesssim 300$. The evolution of the average Sharpe ratio
is more complex, reflecting the non-trivial shape of its distribution function
(bi-modal and skewed, see below). But certainly as agents pay less attention
to the actual return on capital and are more affected by the level of
confidence, the average Sharpe becomes strongly negative (red region of the
diagram) and the economy gets trapped forever in a low consumption, low
investment regime where $\theta_{k}\Sigma$ is negative (see Eq. (36)).

Now, let us look at a cut along the direction $\nu=0.75$, corresponding to a
$25\%$ weight given to confidence in the allocation decision, as
$\mathcal{T}_{\lambda}$ is varied. For this particular value of $\nu$, the
average Sharpe ratio is close to zero and only weakly depends on
$\mathcal{T}_{\lambda}$ (Fig. 6, bottom left graph). But from the bottom row
of Fig. 6, we see that when $\mathcal{T}_{\lambda}\gtrsim 10$, the
distribution of Sharpe ratios becomes bi-modal and with a skewness that
decreases as $\mathcal{T}_{\lambda}$ increases. This can be rationalised as
follows:

  * •

The peak corresponding to positive Sharpe ratios comes from prosperous
periods, where consumption is high and capital relatively scarce, leading to a
positive return on capital $q^{*}$: see Fig. 6, top right graphs: in the high
consumption phase, the orange line (capital) is below the green line (labour).

  * •

The peak corresponding to zero Sharpe comes from crises periods, where capital
is in slight excess of labour, leading to a small return on capital (see again
Fig. 6, top right graphs, and section II.3).

  * •

The fat left tail corresponding to negative Sharpes comes from the transitory
periods between high confidence and low confidence, when consumption and
labour collapse but capital depreciates much more slowly. In this case, return
on capital plummets and the Sharpe ratio becomes negative.

  * •

As $\mathcal{T}_{\lambda}$ increases, the weight of these transitory regimes
in the estimate of the Sharpe ratio becomes small, and the fat left tail
disappears, as crises become less frequent and much longer.131313For very
large $\mathcal{T}_{\lambda}$, the situation changes again, see below.

Whereas the length of the prosperous periods $\mathcal{T}_{>}$ is unchanged
compared to the benchmark $\nu=1$ case for all values of
$\mathcal{T}_{\lambda}\gtrsim 10$, the length of the crisis periods
$\mathcal{T}_{<}$ increases by more than a 100 times as $\nu$ is decreased
from $1$ to $0.75$ (compare the grey line and the coloured points in bottom
row in Fig. 6). The first observation is due to the fact that the Sharpe ratio
estimated when in a high output period is clearly in positive territory and
quite insensitive to $\mathcal{T}_{\lambda}$ (see the histograms in Fig. 6).
This means that capital supply is also independent of $\mathcal{T}_{\lambda}$
and that the confidence collapse mechanism must be identical to the one
described in our previous work MorelliEtAl2020  and not triggered by a lack of
investment.141414 This is not to say that the crisis frequency is not related
to capital abundance. As already noted in section III.5, as capital
depreciation $\delta$ increases, available capital decreases, which leads to a
lowering of output $c_{t}$. Since the distance between $c_{t}$ and the
threshold $c_{0}$ is a crucial determinant of the probability of a confidence
crisis, the region $\mathcal{H}_{k}\mathcal{H}_{c}$ becomes pervasive as
$\delta$ increases, see again Fig. 4.

On the contrary, the mechanism by which confidence is restored is strongly
impacted by the value of the memory time $\mathcal{T}_{\lambda}$. When the
economy is in a consumption crisis, the returns on capital are very small.
Thus, averaged over a sufficiently long time period, the Sharpe ratio is well
defined and also small (see the narrow peaks in the Sharpe ratio distribution
in Figure 6). Combined with the low confidence dampener on sentiment
$(1-\nu)\cdot\mathcal{C}_{t}$, this leads to a negligible investment flow. So
whereas positive productivity shocks should put the economy back on an even
keel, the level of capital is lagging, which creates a ceiling that prevents
consumption (and hence confidence) from increasing substantially and returning
to the high consumption case.

Interestingly, the dependence of $\mathcal{T}_{<}$ on $\mathcal{T}_{\lambda}$
is in fact non-monotonic. For very large $\mathcal{T}_{\lambda}\gtrsim 1000$,
the memory of prosperous periods persists even during the crises, so that the
Sharpe ratio and investment always remain high. In such cases,
$\mathcal{T}_{<}$ abruptly drops back to small values $\lesssim 5$ (data not
shown). With extremely small probability, however, the system remains trapped
in a crisis forever.

In the opposite case of a small enough $\mathcal{T}_{\lambda}$, the short
periods where consumption increases due to productivity shocks allow
sufficiently rapid increases in capital rent to encourage immediate
investment. This is enough to prop up capital and allows confidence to be
fully restored as labour and consumption will grow with the limiting factor
$k_{t}$. For an example of these positive spikes of consumption, see top
centre panel in Fig. 6. The same effect takes place if $\nu$ is increased back
to $1$, where only realised Sharpe affects investment. In this case, the drag
on capital due to low confidence levels is absent, and the system is able to
pull itself out of the rut much more efficiently, leading to shorter crisis
periods. But for lower values of $\nu$ (higher impact of household confidence
on the investment propensity), the dearth of capital in crisis periods is such
that the economy is unable to ever recover, i.e. $\mathcal{T}_{<}=+\infty$ for
all purposes.

From a policy point of view, reducing interest rates has the direct effect of
increasing the Sharpe ratio and reducing the return to bonds, thus promoting
investment and making the transition back to the high consumption state
easier. However, this may require the central bank to set interest rates $r$
to negative values, as $r$ which might already be close to zero due to prior
crises. Besides monetary policy, other measures that improve confidence (e.g.
central bank messaging) and/or promote investment into productive capital
would have a similar impact (for instance if the government decides on strong
fiscal measures that include investment into productive capital, such as
through mission-oriented policies or infrastructure spending).

Finally, let us mention that while the existence of consumption crisis is
independent of the substitutability parameter $\rho$, the duration of the low
investment, low consumption periods is also highly sensitive to
substitutability effects. We have indeed found that when $\rho$ is
sufficiently small, i.e. for production functions closer to Cobb-Douglas than
to Leontief, recovery is much faster (data not shown). This could have been
expected: lack of capital can now be compensated by labour, expediting the
transition back to a prosperous state of affairs.

##  IV Discussion and Conclusions

We have constructed a behavioural real business cycle model in which labour
and capital are nearly unsubstitutable. In the model, consumption and
investment are controlled by (a) the confidence of households, which is self-
reflexive (i.e. agents take cues from the consumption of other agents to
determine their consumption budget) and (b) the quality of the excess real
return to capital, as measured by the Sharpe ratio. As we have shown in our
previous work MorelliEtAl2020 , the self-referential nature of confidence
amplifies the effect of productivity shocks on output, and can lead to crises
where consumption abruptly jumps from a high equilibrium level to a low
equilibrium level. Depending on the parameters of the model, these crises can
be more or less frequent, and the low consumption periods can be of various
duration: short spikes (“V-shaped crises”) or long drawn-out phases (“L-shaped
crises”).

In the present study we investigate how the introduction of capital affects
these dynamical patterns. In our model, capital can either be abundant (in
which case labour is the limiting factor to production) or scarce. The main
factors determining the quantity of working capital are the depreciation rate
and the propensity of the households to save and invest, which itself depends
on the return on capital. The resulting phenomenology of the model is quite
rich. Our analysis reveals the following main takeaways:

  1. 1.

Higher capital depreciation rates, ceteris paribus, lead to capital scarcity
and limit production. This makes the economy more prone to confidence crises,
increasing their prevalence;

  2. 2.

Increasing the influence of the level of confidence in capital allocation
decisions creates a feedback loop similar to Keynes’ paradox of thrift,
destabilising and trapping the economy into a non-optimal low consumption
state;

  3. 3.

The time during which the economy remains in a low output state is highly
sensitive to the time span over which investors compute the Sharpe ratio.
Increasing this memory timescale leads to sluggish adjustments of
investment.151515Note however, as reported in the previous section, that
extremely long memory timescales allow the Sharpe ratio to stick to high
values. Consequently, instantaneous increase of capital returns due to
productivity upticks are not sufficient to boost the investment propensity.
This leads to a persistence of capital scarcity, and prevents the economy from
escaping the low output trap.

Our findings have different policy implications. As already emphasised in our
previous paper MorelliEtAl2020 , if self-reflexive feedback loops exist, then
governments and monetary authorities should not only manage inflation
expectations but more broadly confidence in the future prospects of the
economy. Although confidence indices are routinely measured by polling
institutes (see e.g. Fig. 1 in Section I), the inclusion of such indices in
macroeconomic DSGE models and the importance of narratives Shiller2019  have
never really been considered seriously beyond the impact of news shocks on
productivity.161616 e.g. the work of BeaudryEtAl2020  reflecting interactions
and complementarity. Also AngeletosEtAl2020  showed that single news shocks
(confidence shocks) are sufficient to fit empirically the effects of business
cycles. See also DenizAslanoglu2014 .

Beyond communication and narratives, our model suggests that monetary
authorities should also directly promote investment, in particular during
recessions. This is needed to prevent the economy being trapped in a low
output, low confidence environment. Although this conclusion looks perfectly
intuitive, our model reveals that a lack of capital can prolong crisis periods
by orders of magnitude, and convert V-shaped crises into L-shaped crises.
Boosting investment in working capital can be done through traditional
channels, by lowering the risk-free interest rate (possibly making it
negative) or by direct Keynesian investments in infrastructure and in
innovation, which have the double effect of increasing the productivity of
capital and propping up household confidence.

There are of course many directions in which our model should be extended and
improved. The first obvious one is to allow interest rates and inflation to be
dynamical variables, and to introduce an explicit monetary policy with the
central bank monitoring inflation and confidence. A fully developed DSGE model
building upon the framework proposed here would be welcome. Other relevant
extensions could be to include a feedback mechanism between confidence and the
time scale $\mathcal{T}_{\lambda}$ or the sentiment parameter $\nu$. This
would allow potentially relevant panic effects to set in the model, and
capture what happened in 2008, for example.

Another possible extension is to allow the parameter $a$ which describes the
default risk on capital to depend on the state of the economy, since
bankruptcies are more frequent when the economy is in a low output, low
investment regime.

Last, but not least, we have assumed that confidence is only a function of
past realised output, but other factors should obviously be taken into account
to model the dynamics of the confidence index, in particular financial news
(like in 2008) or geopolitical news. Our framework would lead to scenarios
where a shock like Lehman’s bankruptcy simultaneously affects both consumption
and investment, leading to a deep and prolonged recession, even in the absence
of any “true” productivity shock. Conversely, good news about the future (e.g.
technology shocks) could help recovering faster from the low output trap.

It would also be interesting to look for a “grand unification” between the
type of behavioural business cycle/DSGE models considered in this paper and
heterogeneous agent based models studied in the recent literature, which
generically give rise to similar crises and bi-stable dynamics between high
output and low output regimes of the economy (see e.g. GualdiEtAl2015 ;
SharmaEtAl2020 ).

##### Acknowledgements

We deeply thank Davide Luzzati who contributed to the early stages of this
work, and Francesco Zamponi for many insightful discussions. This research was
conducted within the Econophysics & Complex Systems Research Chair, under the
aegis of the Fondation du Risque, the Fondation de l’Ecole polytechnique, the
Ecole polytechnique and Capital Fund Management. 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

  * [1] John Maynard Keynes.  The General Theory of Employment, Interest, and Money.  Macmillan, London, 1936.
  * [2] Alan Kirman.  The Economic Crisis is a Crisis for Economic Theory.  CESifo Economic Studies, 56(4):498–535, 2010.
  * [3] Willem Buiter.  The unfortunate uselessness of most ’state of the art’ academic monetary economics.  VOX EU, June 2009.
  * [4] Olivier Blanchard.  On the future of macroeconomic models.  Oxford Review of Economic Policy, 34(1-2):43–54, January 2018.
  * [5] Giovanni Dosi and Andrea Roventini.  More is different … and complex! the case for agent-based macroeconomics.  Journal of Evolutionary Economics, 29(1):1–37, March 2019.
  * [6] A Korinek.  Thoughts on DSGE macroeconomics: Matching the moment, but missing the point?  In Economic Theory and Public Policies: Joseph Stiglitz and the Teaching of Economics. Columbia University Press, 2017.
  * [7] Paul Romer.  The Trouble With Macroeconomics.  2016\.
  * [8] Joseph E Stiglitz.  Where modern macroeconomics went wrong.  Oxford Review of Economic Policy, 34(1-2):70–106, 2018.
  * [9] Ray C Fair.  Some important macro points.  Oxford Review of Economic Policy, 36(3):556–578, December 2020\.
  * [10] Lawrence J. Christiano, Martin S. Eichenbaum, and Mathias Trabandt.  On DSGE Models.  Journal of Economic Perspectives, 32(3):113–140, August 2018.
  * [11] J. Lindé, F. Smets, and R. Wouters.  Challenges for Central Banks’ Macro Models.  In Handbook of Macroeconomics, volume 2, pages 2185–2262. Elsevier, 2016.
  * [12] Greg Kaplan and Giovanni L. Violante.  Microeconomic Heterogeneity and Macroeconomic Shocks.  Journal of Economic Perspectives, 32(3):167–194, August 2018.
  * [13] David Vines and Samuel Wills.  The rebuilding macroeconomic theory project: An analytical assessment.  Oxford Review of Economic Policy, 34(1-2):1–42, January 2018.
  * [14] David Vines and Samuel Wills.  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(3):427–497, December 2020\.
  * [15] Greg Kaplan, Benjamin Moll, and Giovanni L. Violante.  Monetary Policy According to HANK.  American Economic Review, 108(3):697–743, March 2018.
  * [16] Greg Kaplan and Giovanni L. Violante.  A model of the consumption response to fiscal stimulus payments.  Econometrica, 82(4):1199–1239, 2014.
  * [17] Alisdair McKay and Ricardo Reis.  The Role of Automatic Stabilizers in the U.S. Business Cycle.  Econometrica, 84(1):141–194, 2016.
  * [18] Robert B Barsky and Eric R Sims.  Information, Animal Spirits, and the Meaning of Innovations in Consumer Confidence.  American Economic Review, 102(4):1343–1377, June 2012.
  * [19] Olivier Blanchard.  Where Danger Lurks.  Finance and Development (IMF), 2014.
  * [20] Hyman P. Minsky.  John Maynard Keynes.  Springer, 1976.
  * [21] James H. Stock and Mark W. Watson.  Business Cycle Fluctuations in U.S. Macroeconomic Time Series.  In John B Taylor and Michael Woodford, editors, Handbook of Macroeconomics, volume 1A, pages 3–64. Elsevier, New York, 1999.
  * [22] Robert J. Shiller.  Irrational Exuberance.  Currency/Doubleday, New York, 2nd ed edition, 2005.
  * [23] George A. Akerlof and Robert J. Shiller.  Animal Spirits.  Princeton University Press, 2010.
  * [24] Cars Hommes.  Behavioral and Experimental Macroeconomics and Policy Analysis: A Complex Systems Approach.  Journal of Economic Literature, 59(1):149–219, March 2021.
  * [25] Reiner Franke and Frank Westerhoff.  Taking Stock: A Rigorous Modelling of Animal Spirits in Macroeconomics.  Journal of Economic Surveys, 31(5):1152–1182, 2017.
  * [26] A Cornea, Cars H Hommes, and Domenico Massaro.  Behavioral heterogeneity in U.S. inflation dynamics.  Journal of Business & Economic Statistics, 37(2):288–300, 2019\.
  * [27] Cars Hommes and Joep Lustenhouwer.  Managing Unanchored, Heterogeneous Expectations and Liquidity Traps.  Journal of Economic Dynamics and Control, 101:1–16, April 2019\.
  * [28] Tolga Ozden.  Heterogeneous Expectations and the Business Cycle at the Effective Lower Bound.  2021\.
  * [29] Robert Calvert Jump and Paul Levine.  Behavioural New Keynesian models.  Journal of Macroeconomics, 59:59–77, March 2019.
  * [30] Xavier Gabaix.  A Behavioral New Keynesian Model.  American Economic Review, 110(8):2271–2327, August 2020.
  * [31] Pinar Deniz and Erhan Aslanoglu.  Consumer Confidence in a DSGE Model for Turkey.  In Allied Social Science Associations Annual Meeting, Philadelphia, PA, 2014. Allied Social Science Associations.
  * [32] Jan-Niklas Brenneisen.  Monetary policy under imperfect information and consumer confidence.  2020\.
  * [33] Olivier Blanchard.  Consumption and the Recession of 1990-1991.  The American Economic Review, 83(2):270–274, 1993.
  * [34] Christopher D. Carroll, Jeffrey C. Fuhrer, and David W. Wilcox.  Does Consumer Sentiment Forecast Household Spending? If So, Why?  The American Economic Review, 84(5):1397–1408, 1994.
  * [35] John Matsusaka and Argia Sbordone.  Consumer Confidence and Economic Fluctuations.  Economic Inquiry, 33(2):296–318, 1995.
  * [36] George-Marios Angeletos, Fabrice Collard, and Harris Dellas.  Quantifying Confidence.  Econometrica, 86(5):1689–1726, 2018.
  * [37] George-Marios Angeletos and Jennifer La’O.  Sentiments.  Econometrica, 81(2):739–779, 2013.
  * [38] Jean-Philippe Bouchaud.  Crises and Collective Socio-Economic Phenomena: Simple Models and Challenges.  Journal of Statistical Physics, 151(3-4):567–606, 2013.
  * [39] Joao Gama Batista, Jean-Philippe Bouchaud, and Damien Challet.  Sudden Trust Collapse in Networked Societies.  Eur. Phys. J. B, 88(3), 2015.
  * [40] Federico Morelli, Michael Benzaquen, Marco Tarzia, and Jean-Philippe Bouchaud.  Confidence Collapse in a Multi-Household, Self-Reflexive DSGE Model.  PNAS, 2020.
  * [41] Wendy Carlin and David Soskice.  Stagnant productivity and low unemployment: Stuck in a Keynesian equilibrium.  Oxford Review of Economic Policy, 34(1-2):169–194, January 2018\.
  * [42] Andrew W. Lo.  Reading about the Financial Crisis: A Twenty-One-Book Review.  Journal of Economic Literature, 50(1):151–178, March 2012.
  * [43] George-Marios Angeletos, Fabrice Collard, and Harris Dellas.  Business-Cycle Anatomy.  American Economic Review, 110(10):3030–3070, October 2020.
  * [44] Ben S. Bernanke, Timothy F. Geithner, and Henry M. Paulson Jr.  Firefighting: The Financial Crisis and Its Lessons.  Penguin Books, New York, 2019.
  * [45] Megan Greene.  Joe Biden’s experiment could revolutionise economic thinking.  Financial Times, May 2021.
  * [46] Stanislao Gualdi, Marco Tarzia, Francesco Zamponi, and Jean-Philippe Bouchaud.  Tipping points in macroeconomic agent-based models.  Journal of Economic Dynamics and Control, 50(C):29–61, 2015.
  * [47] Dhruv Sharma, Jean-Philippe Bouchaud, Stanislao Gualdi, Marco Tarzia, and Francesco Zamponi.  V-, U-, L-, or W-shaped recovery after COVID? Insights from an Agent Based Model.  PLoS One, 16(3), July 2020.
  * [48] Robert J. Shiller.  Narrative Economics.  2019\.
  * [49] Jordi Galí.  Monetary Policy, Inflation, and the Business Cycle: An Introduction to the New Keynesian Framework.  Princeton University Press, Princeton, N.J, second edition, 2015.
  * [50] Frank Smets and Rafael Wouters.  Shocks and frictions in US business cycles: A Bayesian DSGE approach.  American economic review, 97(3):586–606, 2007.
  * [51] Paul Beaudry and Franck Portier.  News-Driven Business Cycles: Insights and Challenges.  Journal of Economic Literature, 52(4):993–1074, 2014.
  * [52] Dimitri Kroujiline, Maxim Gusev, Dmitry Ushanov, Sergey V. Sharov, and Boris Govorkov.  An endogenous mechanism of business cycles.  Algorithmic Finance, pages 1–22, October 2019.
  * [53] Paul Beaudry, Dana Galizia, and Franck Portier.  Putting the Cycle Back into Business Cycle Analysis.  American Economic Review, 110(1):1–47, January 2020.
  * [54] William F. Sharpe.  Mutual Fund Performance.  The Journal of Business, 39(1):119–138, 1966.
  * [55] Zhi Da, Xing Huang, and Lawrence J. Jin.  Extrapolative beliefs in the cross-section: What can we learn from the crowds?  Journal of Financial Economics, 140(1):175–196, April 2021.
  * [56] Theresa Kuchler and Basit Zafar.  Personal Experiences and Expectations about Aggregate Outcomes.  The Journal of Finance, 74(5):2491–2542, 2019.

Table 1: Notation used in the model | Description | Definition | Value(s)
---|---|---|---
Variables
$c_{t}$ | Real consumption | Sec. II.1 Eq. (1) |
$n_{t}$ | Labour | Sec. II.1 Eq. (1) |
${G}_{t}$ | Propensity to consume from Income | Sec. II.1 Eq. (31) | $[G_{\min},G_{\max}]$
$\mathcal{I}_{t}$ | Income in period $t$ | Sec. II.1 Eq. (2) |
$w_{t}$ | Real wage | Sec. II.1 Eq. (12) |
$b_{t}$ | Real income from one-period bond purchased in $t$ | Sec. II.1 Eq. (7) |
$k_{t}$ | Real capital | Sec. II.1 Eq. (6) |
$q_{t}$ | Realised yield on capital | Sec. II.1 Eq. (6) | $[0,{q}^{\star}_{t}]$
$i_{t}$ | Real value of total investment | Sec. II.1 Eq. (5) | $(1-G_{t})\mathcal{I}_{t}$
$F_{t}$ | Allocation of investment to capital | Sec. II.1 Eq. (36) | $[F_{\min},F_{\max}]$
$z_{t}$ | Total factor productivity (TFP) | Sec. II.2 Eq. (9) | $[0,\infty)$
${q}^{\star}_{t}$ | Ideal capital returns | Sec. II.2 Eq. (13) |
$\xi$ | Investment risk process | Sec. II.4 Eq. (27) | $[0,1]$
$\mu_{t}^{q}$ | Estimated expected return | Sec. II.5 Eq. (32) |
$\sigma_{t}^{q}$ | Estimated investment volatility | Sec. II.5 Eq. (33) |
$\mathcal{S}_{t}$ | Estimated Sharpe ratio | Sec. II.5 Eq. (34) |
$\mathcal{C}_{t}$ | Consumption confidence | Sec. II.5 Eq. (30) | [-1, 1]
$\Sigma_{t}$ | Unbounded sentiment | Sec. II.5 Eq. (35) |
Studied Parameters
$\delta$ | Depreciation rate | Sec. II.1 Eq. (6) | $0.001,0.005,0.02$
a | Investment risk multiplier | Sec. II.4 Eq. (28) | $15$
$c_{0}$ | Consumption confidence threshold | Sec. II.5 Eq. (30) | $[0,0.025]$
$\lambda$ | Memory kernel for the Sharpe ratio | Sec. II.5 Eq. (32) | $0.607,0.905,0.951,0.98$
$\nu$ | Interpolation between $\mathcal{S}_{t}$ and $\mathcal{C}_{t}$ | Sec. II.5 Eq. (35) | $0.75,1.0$
Fixed Parameters
$\gamma$ | Disutility of labour | Sec. II.1 Eq. (1) | 1
$\pi_{t}$ | Inflation rate | Sec. II.1 Eq. (2) | 0.1%
$r_{t}$ | Real interest rate | Sec. II.1 Eq. (2) | 0.15%
$p_{t}$ | Price level | Sec. II.1 Eq. (2) | 1
$\alpha$ | Capital share in production | Sec. II.2 Eq. (8) | $1/3$
$1/(1+\rho)$ | Elasticity of substitution $k$ vs. $n$ | Sec. II.2 Eq. (8) | -7
$z_{0}$ | Baseline value of the total factor productivity | Sec. II.2 Eq. (9) | 0.05
$\eta$ | Autocorrelation of total factor productivity | Sec. II.2 Eq. (9) | 0.5
$\theta_{c}$ | Consumption rate transition width | Sec. II.5 Eq. (30) | 300
$G_{\min}$ | Minimum consumption rate | Sec. II.5 Eq. (31) | 0.05
$G_{\max}$ | Maximum consumption rate | Sec. II.5 Eq. (31) | 0.95
$\mathcal{N}$ | Scaling factor for the Sharpe ratio $\mathcal{S}_{t}$ | Sec. II.5 Eq. (34) | 1/4
$F_{\min}$ | Minimum capital allocation | Sec. II.5 Eq. (36) | 0.0
$F_{\max}$ | Maximum capital allocation | Sec. II.5 Eq. (36) | 1.0
$\theta_{k}$ | Allocation transition width | Sec. II.5 Eq. (36) | 15
Additional Parameters
$\Xi_{k}$ | Weighted proportion of time when $k_{t}<n_{t}$ | Sec. III Eq. (38) | $[0,1]$
$\Xi_{c}$ | Weighted proportion of time when $c_{t}<c_{0}$ | Sec. III Eq. (37) | $[0,1]$
$[\mathcal{L}_{k},\mathcal{H}_{k}]$ | Low/High frequency $\Xi_{k}$ | Sec. III |
$[\mathcal{L}_{c},\mathcal{H}_{c}]$ | Low/High frequency $\Xi_{c}$ | Sec. III |
$\mathcal{T}_{\lambda}$ | Timescale of allocation $S_{t}$ | Sec. II.6 | $1/|\log(\lambda)|$
$\mathcal{T}_{\eta}$ | Timescale of TFP shock $z_{t}$ | Sec. II.6 | $1/|\log(\eta)|$
$\mathcal{T}_{\delta}$ | Timescale of capital $k_{t}$ | Sec. II.6 | $1/|\log(1-\delta)|$
$\mathcal{T}_{<}$ | Average duration of consumption crises | Sec. III.7 |
$\mathcal{T}_{>}$ | Average duration of high output periods | Sec. III.7 |