# Post-COVID Inflation &
the Monetary Policy Dilemma:
An Agent-Based Scenario Analysis

Max Sina Knicker1, Karl Naumann-Woleske1, Jean-Philippe Bouchaud1,2,3,
Francesco Zamponi4

(  1 Chair of EconophysiX and Complex Systems, LadHyX, UMR CNRS 7646, Ecole
Polytechnique Paris
2 Capital Fund Management, 75007 Paris, France
3 Académie des Sciences, Quai de Conti, 75006 Paris, France
4 Laboratoire de Physique de l’Ecole Normale Supérieure, ENS, Université PSL,
CNRS, Sorbonne Université, Université de Paris, F-75005 Paris, France
)

###### Abstract

The economic shocks that followed the COVID-19 pandemic have brought to light
the difficulty, both for academics and policy makers, of describing and
predicting the dynamics of inflation. This paper offers an alternative
modelling approach. We study the 2020-2023 period within the well-studied
Mark-0 Agent-Based Model, in which economic agents act and react according to
plausible behavioural rules. We include in particular a mechanism through
which trust of economic agents in the Central Bank can de-anchor.

We investigate the influence of regulatory policies on inflationary dynamics
resulting from three exogenous shocks, calibrated on those that followed the
COVID-19 pandemic: a production/consumption shock due to COVID-related
lockdowns, a supply-chain shock, and an energy price shock exacerbated by the
Russian invasion of Ukraine. By exploring the impact of these shocks under
different assumptions about monetary policy efficacy and transmission
channels, we review various explanations for the resurgence of inflation in
the United States, including demand-pull, cost-push, and profit-driven
factors.

Our main results are four-fold: (i) without appropriate policy, the shocked
economy can take years to recover, or even tip over into a deep recession;
(ii) the response to policy is non-monotonic, leading to a narrow window of
“optimal” policy responses due to the trade-off between inflation and
unemployment; (iii) the success of monetary policy in curbing inflation is
primarily due to expectation anchoring, rather than to direct impact of
interest rate hikes; (iv) the two most sensitive model parameters are those
describing wage and price indexation.

Our analysis contributes to the ongoing debate on the appropriate policy
responses to post-COVID inflation. We offer a flexible framework allowing one
to assess different policy options and accommodate various behavioural
idiosyncrasies. The model can generate a variety of plausible scenarios,
including runaway hyperinflation. The results of our study have implications
for Central Bank decision-making, and offers an easy-to-use tool that may help
anticipate the consequences of different monetary and fiscal policies.

###### Contents

  1. 1 Introduction
  2. 2 The Mark-0 Model
    1. 2.1 Model Overview
    2. 2.2 Inflation Expectations and Monetary Policy
  3. 3 Policy Channels in Mark-0
    1. 3.1 The Interest Rate Channel
    2. 3.2 The Expectation Channel
    3. 3.3 Credit Regulatory Policy
    4. 3.4 Helicopter Money
    5. 3.5 Windfall Tax
  4. 4 Stationary Dynamics: the Role of the Central Bank
    1. 4.1 The Economic Equilibrium Without Monetary Policy
    2. 4.2 Stabilizing Inflation through Monetary Policy
  5. 5 The COVID Shock and its Aftermath in the Absence of Monetary Policy
    1. 5.1 Modelling the Three COVID Shocks
    2. 5.2 The Effect of Shocks Without Stabilisation Policies
    3. 5.3 Sensitivity to Shock Magnitude and the Role of Easy-Credit Policy
  6. 6 Monetary Policy Response to Inflationary Shocks
    1. 6.1 An Inactive Central Bank
    2. 6.2 Monetary Policy with Anchored Trust
    3. 6.3 Monetary Policy with Floating Trust
    4. 6.4 Sensitivity to the Strength of the Monetary Policy
    5. 6.5 Sensitivity to Transmission Channels
  7. 7 Fiscal Stimulus
    1. 7.1 The Effects of Helicopter Money
    2. 7.2 The Effects of a Windfall Tax
  8. 8 Model Sensitivity: The Dangers of a Wage-Price Spiral
    1. 8.1 “Sloppiness” Analysis
    2. 8.2 The Risk of a Hyperinflation Spiral
  9. 9 Summary & Conclusions
  10. A Appendices
    1. A.1 Table of Parameters
    2. A.2 counterfactuals
    3. A.3 Monetary Policy
    4. A.4 Monetary Policy with Stronger Central Bank Reaction
    5. A.5 Sensitivity of Monetary Policy to $\alpha_{c}$ and $\alpha_{\Gamma}$
    6. A.6 Helicopter Money
    7. A.7 Windfall Tax
    8. A.8 Sloppiness
    9. A.9 Sensitivity to the Indexation Parameters $g_{p}$ and $g_{w}$

## 1 Introduction

Inflation has captured global attention since the onset of the COVID-19
pandemic (from now on, simply “COVID”) in 2020. In the United States annual
inflation reached 4.8% in 2021 and peaked at 9.1% in June 2022,111see e.g.
U.S. Bureau of Labor Statistics, Consumer Price Index for All Urban Consumers:
All Items in U.S. City Average [CPIAUCSL], retrieved from FRED, Federal
Reserve Bank of St. Louis; May 24, 2023. while Europe experienced highs of
11.5% in October 2022.222see e.g. Eurostat EuroIndicators, report n. 31/2023,
https://ec.europa.eu/eurostat/documents/2995521/16310161/2-17032023-AP-EN.pdf
Competing narratives have emerged to explain the mechanisms driving this
inflationary surge, which has persisted longer than expected. Policymakers
have been blindsided by inadequate models, with the Bank of England admitting
it has “big lessons to learn” from failure to forecast inflation using
existing models.333see for example the Financial Times,
https://www.ft.com/content/b972f5e3-4f03-4986-890d-5443878424ac  This paper
aims to explore the influence of fiscal and monetary policies on prevailing
inflationary dynamics within a complex macroeconomic environment, modeled
using the Mark-0 Agent-based Model that is based on alternative modeling
foundations.

During the period of high inflation in 2021 and 2022, several theories emerged
to explain the underlying causes of persisting inflationary trends. These
interpretations led to differing views on appropriate policy responses,
ranging from monetary policy interventions such as interest rate hikes to more
targeted fiscal policies and price controls. Our analysis contributes to the
debate on the appropriate policy responses to post-COVID inflation by
providing a flexible framework to assess different policy options in the
context of various inflation drivers, including demand-pull, cost-push, and
profit-driven inflation. Our framework can accommodate varying behavioral
foundations within a complex economy, including agents’ trust in the Central
Bank’s clout and the anchoring of inflation expectations. Our main conclusions
are that (i) the economic recovery after the shocks, especially in absence of
mitigating policies, can be much more sluggish than expected – or even fall
into deep recessions beyond dangerous tipping points or “dark
corners”(Blanchard, 2014); (ii) the policy response (both from the government,
the Central Bank, and other public authorities) has to navigate a narrow path,
facing the trade-off between high inflation (with the risk of a runaway
scenario) and high unemployment (with the risk of an economic collapse); (iii)
the success of monetary policy in curbing inflation is primarily due to
expectation anchoring, rather than to direct impact of interest rate hikes;
(iv) the two most sensitive model parameters (in terms of the model output)
are those describing wage and price indexation, or in other words the
bargaining power of workers and the market power of firms.

An initially dominant view of the post-COVID inflation was based on the “too
much money chasing too few goods” theory of inflation (see e.g. de Soyres et
al., 2022b, a), also known as demand-pull inflation. This narrative focuses on
the large amount of fiscal stimulus, such as the Coronavirus Aid, Relief and
Economy Security Act (the “CARES Act”) or the American Rescue Plan in the
U.S., that have led to excess demand due to increases in disposable income
while supply has not adjusted, thus pushing up prices. Some, such as Ferguson
& Storm (2023) agree with the demand-pull inflation analysis but contend that
“the final cause of the inflationary surge in the U.S., therefore, was in
large measure the unequal (wealth) effects of ultra-loose monetary policy
during 2020-2021”.

In contrast to the excess demand interpretation, scholars such as Stiglitz &
Regmi (2022) conclude that “today’s inflation is largely driven by supply
shocks and sectoral demand shifts, not by excess aggregate demand”. In
particular, the energy and food price shocks following COVID and the Russian
invasion of Ukraine in 2022 were major causes of inflation, but this time from
a cost-push inflation perspective wherein firms pass on increases in costs to
consumers through prices. The strength of the inflationary surge was in large
part due to the systemically important nature of the sectors where inflation
occurred: energy and food (Weber et al., 2022). Combined with supply chain
bottlenecks that have also made headlines after COVID, these observations
suggest an alternative cost-push scenario that is outside the realm of Central
Bank’s policy toolkit. This is also the conclusion of Bernanke and Blanchard
(2023), where a detailed decomposition of inflation over different factors is
proposed.444We became aware of this paper, dated May 23, 2023, in the last
week before finalizing our own work. Similarly to our own approach, Bernanke
and Blanchard (2023) include wage bargaining, labor tightness and trust
anchoring in their framework. Among the most important differences, however,
are the absence of (i) supply-demand imbalances in the price setting mechanism
and (ii) possible price gouging effects. Furthermore, the various “tipping
points” that are found within our model do not exist in their simplified
specification.

Finally, a recent debate has emerged around profit-driven inflation, wherein
sellers have increased prices beyond the increase in costs they face, thus
expanding their profit margins. This is based on the observation that profits
have increased sharply in the current inflationary climate, as compared to the
observed amount of cost increases, even when including concerns of a wage-
price spiral, and the excess demand cited by a monetarist perspective (Glover
et al., 2023; Stiglitz & Regmi, 2022). Weber & Wasner (2023) analysed firms’
earnings calls and posit an interpretation based on imperfect competition and
market power. Specifically, in an environment of cost increases and supply
bottlenecks, firms’ market power temporarily increases. This may be enhanced
by consumer uncertainty, because when inflation is high, uncertainty about
prices also becomes high, such that consumers may be prone to accepting
unreasonably high prices (an effect sometimes called “consumer
discombobulation”), thus allowing companies to enlarge their margins.

All of these different interpretations lead to different guidelines for how
policy should respond, and whether it should at all. From a standard
monetarist perspective, the Central Bank should raise interest rates in order
to push down demand, thus solving the excess demand scenario. On the other
hand, in a cost-push scenario, the Central Bank’s rate has no effect on the
external cost increases and may actually harm the situation if firms choose to
pass on the increased costs of debt to consumers. On this, Stiglitz & Regmi
(2022) suggest “monetary policy, then, is too blunt an instrument because it
will greatly reduce inflation only at the cost of unnecessarily high
unemployment, with severe adverse distributive consequences”, as in their view
inflation is not due to excess demand. In reality, the U.S. Federal Reserve
has hiked rates at the fastest rate since Paul Volcker was chairman, in line
with a monetarist view. Simultaneously, in 2023 inflation has begun easing,
which begs the question of whether the reduction in inflation throughout 2022
is due to monetary policy or external factors such as the easing of energy
prices as with the oil crises of the 1970s (Blinder, 1982; Blinder and Rudd,
2013). In recent years, we have observed that given the Central Bank’s mandate
to keep inflation low and stable, they are expected to act, generally through
interest rate mechanisms and communication.555There is an ongoing impression
that monetary policy is the “only game in town” when it comes to inflation
response. The question remains, to what effect? Are dips in inflation due to
external factors or to Central Bank policy? What are the consequences of
raising rates in this environment? Are the consequences of not acting on
inflation with monetary policy greater than those when one does? After all,
Bruno and Easterly (1998) found that “countries can manage to live with
relatively high – around 15-30 percent – inflation for long periods”. It is
these questions upon which we aim to shed some light by considering various
scenarios generated through an Agent-based Model, that allows us to run
various counterfactual scenarios.

To conduct our analysis, we use the Mark-0 Agent-based Model (ABM) originally
proposed by Gualdi et al. (2015), and extended in Gualdi et al. (2017);
Bouchaud et al. (2018) to study monetary policy and inflation, with an early
application to the effects of the COVID pandemic by Sharma et al. (2020) and a
model-exploration study by Naumann-Woleske et al. (2023). Agent-based Models
are computational scenario-generators in which a large number of individual
agents, in our case firms, interact based on a set of behavioral rules (see
Haldane & Turrell, 2019; Dosi & Roventini, 2019, for more detailed
introductions).666Dawid and Delli Gatti (2018) review several major
Macroeconomic Agent-based Models. These models allow us to study (possibly)
out-of-equilibrium systems by generating emergent macroeconomic dynamics, many
of which have been successfully replicated (see Dosi et al., 2017, for
instance). Agent-Based Models also allow one, with minimal cost, to study the
impact of several effects, such as the (de-)anchoring of trust in the Central
Bank policies, which could be highly relevant in practice but are very hard to
take into account within standard rational equilibrium models.

The Mark-0 ABM is a simplified model of a closed macroeconomy that nonetheless
generates a wide variety of phenomena, from stable low unemployment and
inflation, to endogenous crises that may oscillate regularly or punctuate long
periods of recovery, or even to runaway inflation. The philosophy behind the
model is to generate qualitatively plausible scenarios. In this context,
Gualdi et al. (2017) studied the efficacy of monetary policy in maintaining
low unemployment and inflation, and “find that provided the economy is far
from phase boundaries (or ‘dark corners’ (Blanchard, 2014)) such policies can
be successful, whereas too aggressive policies may in fact, unwillingly, drive
the economy to an unstable state, where large swings of inflation and
unemployment occur.” An analysis using the Mark-0 model predicted, as early as
June 2020, that the post-COVID recovery could be more sluggish than expected
and lead to a period of sustained inflation (Sharma et al., 2020).

In this paper we present a more detailed study of the post-COVID recovery by
considering three distinct shocks occurring in the Mark-0 model: (1) a COVID-
shock that negatively impacts firms productivity and consumer demand for the
period of lockdowns, (2) a supply-chain shock on firms’ productivity that
mimics the after-effects of COVID on global value chains, and (3) an energy
price shock that is exacerbated by the Russian invasion of Ukraine. Each of
these shocks is calibrated to macroeconomic time-series for the United States,
such that the magnitude and duration match the observed data. We then study
the effects of these shocks under different assumptions about the activity and
efficacy of monetary policy, including the strength of monetary policy
reactions to inflation, the ability of the Central Bank to influence firms’
expectations, and the structure of firm decision-making on prices and wages.

Our first results confirm that in the presence of properly calibrated shocks,
the economic recovery in absence of any mitigating policy is extremely
sluggish, taking least several years (if not much more) for the economy to
return to the pre-crisis equilibrium. We note that this happens with shocks
that are calibrated on the observed post-COVID macroeconomic time series,
which already incorporate the effect of actually implemented policies; we
expect that in absence of any policies the impact would have been even
greater. A second set of results concerns the impact of such policies. We find
that the model properly accounts for the inflation-unemployment trade-off,
which leads to a narrow window in which policy can be efficient. For example,
a disproportionate response of the Central Bank to inflation leads to
unnecessary high unemployment, and an oversized injection of Helicopter Money
leads to an unnecessary high inflation. Finally, we find that the anchoring
(or de-anchoring) of economic agents’ trust in the Central Bank strongly
influences the dynamics of inflation, which confirms the necessity for the
Central Bank to maintain its credibility. We conclude that each policy must be
carefully tuned to achieve the desired result, and we believe that our
modelling tool can guide policy makers in this respect.

Our primary objective is to provide possible scenarios and counterfactuals. As
discussed at length in our previous papers (Gualdi et al., 2015; Bouchaud et
al., 2018; Sharma et al., 2020), our ambition is not to provide precise
predictions based on a fully calibrated model, but rather a tool for decision
makers to help them apprehend different possible outcomes and anticipate
unintended consequences and potential counter-intuitive impacts of their
policies. We hope that Mark-0 can be usefully added to the policymakers
toolbox and help them navigate a radically complex world (see e.g. King & Kay
(2020); Bouchaud (2021)).

The remainder of the paper is structured as follows: Section 2 gives an
overview of the Mark-0 model and the adaptations made for this paper. Pursuing
this, Section 3 then outlines the policy channels that the model contains:
interest rate, expectation management, access to credit, Helicopter Money and
Windfall Tax. Section 4 discusses the model’s dynamics in absence of exogenous
shocks. In Section 5 we calibrate our three shocks, and show how they affect
the macroeconomic dynamics with or without an easy credit policy, but without
any monetary policy interventions by the Central Bank. In Section 6 we
introduce and discuss monetary policy through interest rates and expectations
management, and highlight its effects. In Section 7 we add fiscal policy by
examining two distinct kind of stimuli, Helicopter Money and a Windfall Tax.
Section 8 presents a discussion of the robustness of our model to variations
of its parameters, and highlights the risk of hyperinflation. Finally, Section
9 summarizes our results and lays out some perspectives and ideas for future
work.

## 2 The Mark-0 Model

The Mark-0 Model is a stylized macroeconomic agent-based model that is
qualitatively very rich, as many plausible – and sometimes un-intuitive –
economic phenomena can emerge at the macro scale. The model was created as a
minimalist reduction of the Mark family of Agent-Based Models originally
developed by Gaffeo et al. (2008) and Gatti et al. (2011). Mark-0 was proposed
by Gualdi et al. (2015), and expanded to monetary policy questions by Gualdi
et al. (2017) and Bouchaud et al. (2018). More recently, an analysis of
possible economic responses to the COVID-pandemic were studied using Mark-0 by
Sharma et al. (2020). Unless stated otherwise, we use the model description
and formalism of Sharma et al. (2020). In this section, we outline the
relevant equations for the model and explain the economic intuition behind all
of them.

### 2.1 Model Overview

The model comprises a large set of firms producing a homogeneous consumption
good that is purchased by a representative household sector. The household
sector goes to the market with a consumption budget $C_{B}$ computed as a
fraction of savings, $S(t)$, wages $W(t)$, payouts from the energy sector
$\delta_{e}\mathcal{E}_{e}(t)$, and interest on deposits $\rho_{d}(t)S(t)$,

$C_{B}(t)=c(t)\left[S(t)+W(t)+\delta_{e}\mathcal{E}_{e}(t)+\rho_{d}(t)S(t)\right],$
(1)

where $c(t)\in(0,1)$ is the consumption propensity out of wealth and income,
defined as

$c(t)=c_{0}\left[1+\alpha_{c}\left(\hat{\pi}(t)-\rho_{d}(t)\right)\right],\qquad\alpha_{c}\geq
0,$ (2)

where $\hat{\pi}(t)$, specified by Eq. (20) below, is the expected future
inflation. Eq. (2) mimics the behaviour of the classical Euler equation, as
the household consumes more when real interest rates,
$\rho_{d}(t)-\hat{\pi}(t)$, are lower.

Households choose to split their consumption budget between firms based on an
intensity of choice model

$D_{i}(t)=\frac{C_{B}(t)}{p_{i}(t)}\frac{\exp(-\beta
p_{i}(t))}{\sum_{j}\exp(-\beta
p_{j}(t))},\quad\text{s.t.}\quad\sum_{i}p_{i}(t)D_{i}(t)\equiv C_{B}(t),$ (3)

with a price sensitivity $\beta$.777Good differentiation could easily be
included at this stage by replacing $p_{i}$ by $p_{i}/\psi_{i}$, where
$\psi_{i}$ is the preference for good $i$.

Because the model considers out-of-equilibrium situations, demand $D_{i}$ for
good $i$ and production $Y_{i}$ may not match, leading to a realized
consumption $c_{i}^{R}$ given by $\min(D_{i},Y_{i})$. Meanwhile, firms produce
consumption goods based on a linear production function,

$Y_{i}(t)=\zeta(t)N_{i}(t),$ (4)

dependent only on the firm’s employed labour force $N_{i}$ and time-dependent
labour-productivity $\zeta$. At each time step, corresponding to one month
throughout this paper, firms can adapt to the current economic environment by
choosing three firm-specific variables: the target production $Y_{i}(t)$,
price $p_{i}(t)$, and wage offered to their employees, $w_{i}(t)$. We describe
these in turn, but note that the time step of one month and the different
update parameters $\gamma,g$ are assumed to be small such that production,
prices and wages evolve slowly between $t$ and $t+1$, barring the role of
external shocks of the COVID type, as discussed in section 5 below.

  1. 1.

Production update. Production adapts to the observed gap between supply and
demand at the previous time step as follows:

$\displaystyle\begin{split}\text{If }Y_{i}(t)<D_{i}(t)&\hskip
10.0pt\Rightarrow\hskip
10.0ptY_{i}(t+1)=Y_{i}(t)+\min\\{\eta^{+}_{i}(D_{i}(t)-Y_{i}(t)),\zeta
u^{\star}_{i}(t)\\}\\\ \text{If }Y_{i}(t)>D_{i}(t)&\hskip
10.0pt\Rightarrow\hskip
10.0ptY_{i}(t+1)=Y_{i}(t)-\eta^{-}_{i}[Y_{i}(t)-D_{i}(t)]\\\ \end{split}$ (5)

where $u^{\star}_{i}(t)$ is the maximum number of unemployed workers available
to the firm $i$ at time $t$, which depends on the wage the firm pays, see
Sharma et al. (2020) for details. The speed at which firms hire and fire
workers depends on their level of financial fragility $\Phi_{i}$, defined as
the debt-to-sales ratio, where debt is $\mathcal{D}_{i}(t)$:888Note that this
definition of fragility $\Phi$ slightly differs from that used in our previous
publication, in particular the normalisation by the bankruptcy threshold
$\Theta$.

$\Phi_{i}(t)=\frac{1}{\Theta}\,\frac{\mathcal{D}_{i}(t)}{\min{(p_{i}(t)D_{i}(t),p_{i}(t)Y_{i}(t))}}.$
(6)

Non-indebted firms have zero fragility and $\Theta$ is the maximum debt-to-
sales ratio allowed by firms creditors, beyond which firms are declared
bankrupt.999For the detailed bankruptcy settlement, see Gualdi et al. (2017);
Bouchaud et al. (2018) Firms that are close to bankruptcy (i.e. $\Phi\approx
1$) are arguably faster to fire and slower to hire, and vice-versa for healthy
firms. The coefficients of the hiring and firing rates $\eta^{\pm}_{i}$ for
firm $i$ (belonging to $[0,1]$) are given by:

$\eta^{\pm}_{i}=[\\![{\eta_{0}^{\pm}\,(1\mp\Gamma(t)\Phi_{i}(t))}]\\!],$ (7)

where $\eta_{0}^{\pm}$ are fixed coefficients, identical for all firms, and
$[\\![{x}]\\!]=x$ when $x\in(0,1)$, $[\\![{x}]\\!]=1$ for $x\geq 1$ and
$[\\![{x}]\\!]=0$ when $x\leq 0$. The factor $\Gamma>0$ measures how the
financial fragility of firms influences their hiring/firing policy, since a
larger value of $\Phi_{i}$ then leads to a faster downward adjustment of the
workforce when the firm is over-producing, and a slower (more cautious) upward
adjustment when the firm is under-producing. $\Gamma$ itself depends on the
inflation-adjusted interest rate and takes the following form:

$\displaystyle\Gamma(t)=\max\Big{[}\alpha_{\Gamma}\left(\rho_{\ell}(t)-\hat{\pi}(t)\right),\Gamma_{0}\Big{]},\qquad\Gamma_{0},\alpha_{\Gamma}\geq
0,$ (8)

where $\rho_{\ell}(t)$ is the rate at which firms can borrow, $\Gamma_{0}$ and
$\alpha_{\Gamma}$ are free, non negative parameters, the latter being similar
to $\alpha_{c}$ that captures the influence of the real interest rate on the
hiring/firing policy of firms.

  2. 2.

Price update. Compared to the previous versions of the Mark-0 model, in this
paper the firms’ price update, $\Delta
p_{i}(t+1)=({p_{i}(t+1)-p_{i}(t)})/{p_{i}(t)}$, explicitly takes into account
the demand-output gap, and a contribution for the change in exogenous (e.g.
energy) price $\Delta p_{e}$ (in %), to with:

$\displaystyle\Delta p_{i}(t+1)$
$\displaystyle=\gamma\xi_{i}(t)\cdot\frac{D_{i}(t)}{Y_{i}(t)}+g_{p}\hat{\pi}(t)+g_{e}\Delta
p_{e}(t)\quad$ $\displaystyle\mathrm{if}\quad$
$\displaystyle\begin{cases}D_{i}(t)>Y_{i}(t)\\\
p_{i}(t)<\overline{p}(t)\end{cases}$ (9) $\displaystyle\Delta p_{i}(t+1)$
$\displaystyle=-\gamma\xi_{i}(t)\cdot\frac{Y_{i}(t)}{D_{i}(t)}+g_{p}\hat{\pi}(t)+g_{e}\Delta
p_{e}(t)\quad$ $\displaystyle\mathrm{if}\quad$
$\displaystyle\begin{cases}D_{i}(t)<Y_{i}(t)\\\
p_{i}(t)>\overline{p}(t)\end{cases}$ (10) $\displaystyle\Delta p_{i}(t+1)$
$\displaystyle=g_{p}\hat{\pi}(t)+g_{e}\Delta p_{e}(t)\quad$
$\displaystyle\mathrm{otherwise},$ (11)

where

     * •

$\xi_{i}(t)$ are independent uniform $U[0,1]$ random variables scaled up by
the actual demand-output ratio ${D_{i}(t)}/{Y_{i}(t)}$ (or its inverse), in
order to mimic an increased pressure on prices in case of supply (or demand)
gluts. When ${D_{i}(t)}\approx{Y_{i}(t)}$, this reduces to the rule used in
previous papers, see Gualdi et al. (2015);

     * •

$\hat{\pi}(t)$ is the expected next-period inflation in % per month, that
firms take into account – or may even amplify – in their price setting
mechanism;

     * •

$\Delta p_{e}(t)$ are exogenous price variations that are partially or fully
passed on to final customers.

The magnitudes of the corresponding price adjustments are determined by
parameters: $\gamma$ for supply-demand imbalance, $g_{p}$ for inflation
expectations, and $g_{e}$ for changes in the exogenous price. In this regard,
$g_{e}$ can be thought of as the effective energy-share of production that
firms want to pass on to customers, whereas $g_{p}>1$ would correspond to the
much discussed concept of “greedflation”.

Note that $\overline{p}$ is the consumer price index, defined as

$\overline{p}=\frac{\sum_{i}c^{R}_{i}p_{i}}{\sum_{i}c^{R}_{i}}\ ,\qquad
c^{R}_{i}=\min(D_{i},Y_{i})\ ,$ (12)

with $c^{R}_{i}$ the realized consumption of product $i$.101010Here again, our
present definition of $\overline{p}$ slightly differs from that used in our
previous papers, where we used a production weighted index instead of a
consumption based index. We conform in this paper with the more standard
definition of inflation based on the Consumer Price Index (CPI).

  3. 3.

Wage update. In similar fashion, firms update wages as

$\displaystyle\Delta w_{i}(t+1)$
$\displaystyle=\gamma(1-u(t))\left(1-\Gamma(t)\Phi_{i}(t)\right)\,\xi^{\prime}_{i}(t)+g_{w}\hat{\pi}(t)\quad$
$\displaystyle\mathrm{if}\quad\begin{cases}D_{i}(t)>Y_{i}(t)\\\
\mathcal{P}_{i}(t)>0\end{cases}$ (13) $\displaystyle\Delta w_{i}(t+1)$
$\displaystyle=-\gamma
u(t)\left(1+\Gamma(t)\Phi_{i}(t)\right)\,\xi^{\prime}_{i}(t)+g_{w}\hat{\pi}(t)\quad$
$\displaystyle\mathrm{if}\quad\begin{cases}D_{i}(t)<Y_{i}(t)\\\
\mathcal{P}_{i}(t)<0\end{cases}$ (14) $\displaystyle\Delta w_{i}(t+1)$
$\displaystyle=g_{w}\hat{\pi}(t)$ $\displaystyle\mathrm{otherwise}$ (15)

with $\Delta w_{i}(t+1)=({w_{i}(t+1)-w_{i}(t)})/w_{i}(t)$, and where $u(t)$
represents the unemployment rate, $\xi^{\prime}_{i}(t)$ is a $U[0,1]$ random
realisation, and $g_{w}$ is the wage sensitivity to expected inflation, which
could be seen as worker’s bargaining power. Firm profits $\mathcal{P}_{i}$
include the cost of debt $\mathcal{D}_{i}$, the revenue on cash
$\mathcal{E}_{i}$ and the cost of energy $g_{e}p_{e}Y_{i}$:

$\displaystyle\mathcal{P}_{i}=p_{i}(t)\min\\{Y_{i}(t),D_{i}(t)\\}-w_{i}(t)Y_{i}(t)+\rho_{d}\mathcal{E}_{i}(t)-\rho_{\ell}\mathcal{D}_{i}(t)-g_{e}p_{e}(t)Y_{i}(t)\
.$ (16)

The conditions of wage change therefore depend on the demand-supply imbalance,
current firm profit $\mathcal{P}_{i}(t)$ and firm financial health, and also
on the tension on the labour market, since lower unemployment leads to higher
wage increase. This allows the model to reproduce Phillips curve effects.

  4. 4.

Energy Sector. In this paper, we consider an exogenously varying energy price,
$p_{e}(t)$. Before the explicit shock introduced in Section 5, we consider the
real energy price to be constant and equal to the average price of goods, i.e.
$p_{e}(t)=\overline{p}(t)$. In each period, firms pay a total amount
$g_{e}p_{e}(t)\sum_{i}Y_{i}(t)$ to the energy sector. Subsequently, the energy
sector pays a fraction of its accumulated profits to households at a rate
$\delta_{e}$. Therefore the cash balance $\mathcal{E}_{e}$ of the energy
sector writes as

$\mathcal{E}_{e}(t+1)=\mathcal{E}_{e}(t)+g_{e}p_{e}(t)\sum_{i}Y_{i}(t)-\delta_{e}\mathcal{E}_{e}(t)$
(17)

with the fraction $\delta_{e}$ of the energy profits that is paid out to
households as income. Here, $\delta_{e}$ can be understood as dividends, share
sales, or other channels through which energy sector profits circulate back
into the economy. The way we introduce the energy sector in this paper as an
accounting identity with exogenous prices is arguably simplistic, and can
certainly be improved in a future version of the model.

### 2.2 Inflation Expectations and Monetary Policy

The equations above depend on inflation rate expectations $\hat{\pi}(t)$, that
we now define. We consider that the measure of realized inflation is given by
the change in consumption-weighted average price,

$\pi(t)=\frac{\overline{p}(t)-\overline{p}(t-1)}{\overline{p}(t-1)},$ (18)

where $\overline{p}$ is defined in Eq. (12).

In the model, agents form expectations of future inflation partly on the basis
of past realisations and partly on the basis of their trust in the Central
Bank ability to enforce its inflation target. More precisely, they use an
exponentially weighted moving average over realised inflation,

$\pi_{\text{ema}}(t)=(1-\omega)\pi_{\text{ema}}(t-1)+\omega\pi(t),$ (19)

where $\omega$ sets a memory time over which agents perceive realized
inflation, equal to $-[\log(1-\omega)]^{-1}\approx\omega^{-1}$ for small
$\omega$. Together with the Central Bank’s communicated inflation target,
$\pi^{\star}$, all agents form the same inflation expectation, given by a
weighted average of $\pi_{\text{ema}}(t)$ and $\pi^{\star}$:

$\hat{\pi}(t)=(1-\tau^{T}(t))\pi_{\text{ema}}(t)+\tau^{T}(t)\pi^{\star},$ (20)

where $\tau^{T}(t)$ is the degree to which expectations are anchored around
the Central Bank’s target.111111For empirical work on the question of trust,
see e.g. Christelis et al. (2020). Consistently with the definition of
$\pi_{\text{ema}}$ and the long-term nature of the inflation target, we
interpret $\hat{\pi}(t)$ as the expectation of long-term inflation.121212This
is in contrast to the common formulations in DSGE models that operate with
one-period-ahead expectations ($t\to t+1$)

As far as monetary policy is concerned, we assume that the Central Bank sets
the baseline interest rate, $\rho_{0}$, via a classical Taylor rule based on
observed (realized) inflation $\pi_{\text{ema}}$:

$\rho_{0}=\rho^{\star}+\phi_{\pi}\left(\pi_{\text{ema}}(t)-\pi^{\star}\right),$
(21)

with reaction strength $\phi_{\pi}$. The baseline rate $\rho_{0}$ is then
translated into a time-dependent rate on loans, $\rho_{\ell}(t)$, and
deposits, $\rho_{d}(t)$, adjusting for the cost of bankruptcies (see Gualdi et
al., 2017; Sharma et al., 2020, for more details). In this paper, we do not
consider a double mandate for the Central Bank as in Gualdi et al. (2017),
instead focusing only on inflation. We will comment about this below, as
strict inflation control may turn out to be highly detrimental to
unemployment.

Finally, we allow inflation expectation anchoring to evolve dynamically
via131313Note that in our previous paper (Bouchaud et al. (2018)), the
anchoring parameter $\tau^{T}$ was assumed to be time independent, a case we
will call “Anchored Trust” henceforth.

$\tau^{T}(t+1)=(1-\omega)\tau^{T}(t)+\omega\exp{\left[-\alpha_{I}\frac{|\pi(t)-\pi^{\star}|}{\pi^{\star}\phi_{\pi}}\right]}.$
(22)

This equation aims at capturing the fact that the degree of expectation
anchoring depends on how closely the realised inflation actually matches the
Central Bank target. This is factored into an exponentially weighted moving
average with memory time $\approx\omega^{-1}$: realized and target inflation
must differ significantly, and for sufficiently long times, for agents to lose
trust in the Central Bank. A larger $\alpha_{I}$ means economic agents lose
trust in the Central Bank more abruptly as the gap between realized inflation
and inflation target becomes significant. We have included the factor
$\phi_{\pi}$ in the denominator to emphasize the fact that stronger commitment
of the Central Bank should decrease the sensitivity of anchoring on realized
inflation, but of course this extra factor can be reabsorbed into
$\alpha_{I}$.

## 3 Policy Channels in Mark-0

In the augmented Mark-0 model, the central authorities can influence
macroeconomic dynamics through (i) the manipulation of interest rates, (ii)
anchoring inflation expectations, and (iii) regulating the amount of debt
firms can accumulate. Additionally, there is the possibility of fiscal policy
in the form of direct injection of cash in the economy (“Helicopter Money”)
and Windfall Tax on the energy sector. These manipulations take effect through
firms’ financial fragility, which directly determines the probability of
bankruptcies and indirectly changes the propensity of firms to hire and fire
workers, as well as the wage setting behavior. Interest rates and cash
injection also have effects on the willingness of consumers to spend. In
addition, there is an expectations channel whereby firms and households may or
may not trust the Central Bank’s communicated inflation target depending on
how close actual inflation is to the target. We now consider these mechanics
in turn, as these will turn out to be important in response to the COVID
related shocks that we will introduce in the next section.

### 3.1 The Interest Rate Channel

The Central Bank’s baseline interest rate $\rho_{0}$ additively influences
both deposit ($\rho_{d}$) and lending ($\rho_{\ell}$) rates. Through this
channel, the interest rate affects firms’ wages and hiring strategies, as well
as households’ spending.

The impact on households is straightforward and parallels that of the standard
Euler equation of intertemporal substitution: higher interest rates, all else
being equal, reduce the propensity to consume out of income and wealth through
parameter $\alpha_{c}$ (Eq. (2)), effectively decreasing current demand in
favor of later consumption. This decrease leads firms to reduce or maintain
prices due to excess supply (see Eq. (10)), thereby lowering inflation.

The impact of interest rates on firm behavior occurs through the coefficient
$\alpha_{\Gamma}$, defined in Eq. (8), which influences firms’ production
target and wage adjustment. In all cases $\alpha_{\Gamma}$ affects firm
behavior in combination with the firm’s fragility $\Phi$ (where $\Phi=1$
implies bankruptcy, see Eq. (6)). An increase in the baseline rate will
increase $\Gamma$, which implies that firms have stronger reactions to excess
demand or excess supply. Unhealthy firms ($\Phi>0$) will therefore more
cautiously expand production when needed ($\eta^{+}$), but more abruptly fire
staff in the case of excess supply ($\eta^{-}$). Hence, an increase of the
baseline rate $\rho_{0}$ will tend to decrease production and increase
unemployment as fragile firms feel the brunt of the cost of debt. For similar
reasons, an increase of $\rho_{0}$ implies a stronger downward pressure on
wages.

Therefore, in Mark-0 an increase of the baseline rate ceteris paribus induces
a reduction of consumption and an increase of unemployment. The success of a
rate hike in curbing inflation depends on the strength of the response of
households and of firms to interest rates, and hence on the value of the
parameters $\alpha_{c}$ and $\alpha_{\Gamma}$. If, as some authors argue
(Reis, 2022), the reaction of households and of firms to interest rates are
subdued, large hikes would be necessary for monetary policy to have a
significant effect on inflation, in part by bringing about a recession
(Stiglitz & Regmi, 2022).

### 3.2 The Expectation Channel

With rising global inflation, the debate about the importance of inflation
expectations for macroeconomic dynamics has been rekindled. Some, such as Rudd
(2022), argue that short-term inflation expectations play no role in
macroeconomic dynamics, while others, like Reis (2021), maintain their
importance but question whether they are “de-anchored” from Central Bank’s
targets. In the present version of the Mark-0, long-term inflation
expectations $\hat{\pi}$ play a role in price and wage settings: formally, the
factor $g_{p}$ in Eqs. (9) and (10) determines the fraction of expected
inflation reflected in firms’ next-period prices. In a similar style, the
factor $g_{w}$ in Eqs. (13) and (14) controls the rate of wage adaptation to
inflation in response to firms’ expected inflation, and is thus adjacent to
the idea of labour-bargaining power.

The long term inflation expectation, $\hat{\pi}$, is determined as a mixture
of past realized inflation and the Central Bank target, with the Central
Bank’s effectiveness in reaching its target dictating the level of “anchoring”
$\tau^{T}$ (see Eq. (20) and (22)). With this formulation, economic agents
lose confidence in Central Bank actions when inflation significantly and
persistently deviates from the Central Bank’s inflation target. The parameter
$\alpha_{I}$ represents the sensitivity of trust to the Central Bank’s ability
to control inflation.

De-anchored beliefs imply firms’ inflation expectation is based solely on past
observation, running a risk of a wage-price spiral or a hyper-inflation
collapse, depending on the imbalance between the parameters $g_{p}$ and
$g_{w}$. The simplest case is equality in bargaining and market power,
$g_{w}=g_{p}$, which implies that on average both prices and wages are raised
in the same proportion to expected inflation, thus having no real effects on
the economy. One then faces either stable inflation when $g_{w}=g_{p}<1$ or
possible hyper-inflation when $g_{w}=g_{p}>1$, see section 8.

A more interesting case is $g_{p}\neq g_{w}$, implying a differentiation in
power. For $g_{p}>g_{w}$, labour bargaining power is lower than firms pricing
power. In this case, firms raise prices by more than wages, eroding the
purchasing power of workers, which results in lower demand that makes price
hikes less likely until firms’ excess demand becomes excess supply, reversing
the cycle. For $g_{w}>g_{p}$, on the other hand, an unstable wage spiral may
set in, with higher wages driving demand up, leading to further increases of
prices as firms face excess demand. The situation $g_{p}>1$, at least for
firms with a healthy balance sheet, would correspond to “greedflation”, i.e.
firms trying to use inflation to hide increases in their profits.

### 3.3 Credit Regulatory Policy

The last channel for the monetary authority to affect the economy within the
framework of Mark-0 is by means of the default threshold, $\Theta$. Earlier
work on the Mark-0 model found that the default threshold is a key parameter
causing a phase transition between a regime of high unemployment and/or
endogenous crises (low $\Theta$), and of stable full employment (higher
$\Theta$ conditional on low interest rates) (see Gualdi et al., 2015). This is
because higher thresholds give firms more time to adjust their production and
price strategy to return to a profitable state, assuming interest payments are
low. Empirically, both in the CARES act and the CAA there were relaxations of
the bankruptcy laws in the United States.141414CARES refers to the Coronavirus
Aid, Relief, and Economic Security Act, and the CAA refers to the Consolidated
Appropriations Act of 2021. Direct state aid to firms may also prevent
bankruptcies from occurring. In the realm of our model these policies are
mimicked by increasing the bankruptcy threshold $\Theta$ of firms during the
shock and reducing it again after the shock. The “Easy-Credit policy” proposed
in Sharma et al. (2020) amounts to set the value of $\Theta$ such that it
remains commensurate to the current average firm fragility, to wit

$\Theta(t)=\max(\mu\langle\phi\rangle(t),\Theta_{0}),$ (23)

where $\Theta_{0}$ is the bankruptcy threshold before the shock, $\mu$ a
multiplier, and $\langle\phi\rangle(t)$ the firm-wide average debt-to-sales
ratio at time $t$, meaning that only the firms whose debt-to-sales ratio
exceeds $\mu\langle\phi\rangle$ must file for bankruptcy. Throughout the
following we will use $\mu=1.3$.

This implies that with easier credit access, firms can accumulate debt during
the shock without the fear of bankruptcy. This, in turn, affects the firing
and hiring policies of firms as they can continue with their business as
usual, leading to a hiring and firing as if they were not fragile (see Eq.
(6)). Sharma et al. (2020) have shown that such policy is effective in
speeding up the recovery to pre-shock levels after severe COVID-like shocks,
without such policy long-lasting recessions would otherwise ensue.

### 3.4 Helicopter Money

When interest rates are low and access to credit is loose, but shocks
nonetheless persist, central authorities may turn to less conventional
policies such as “Helicopter Money” to address economic downturn. Introduced
by Friedman (1969), this implies an injection of money directly into the hands
of economic agents to increase spending and thus economic output.151515By
contrast, quantitative easing implies that money created by the Central Bank
is used to purchase government bonds and distributed through the government,
whereas Helicopter Money is directly distributed in the economy (Ugai et al.,
2007). While some view Helicopter Money as a potential cause of hyperinflation
and currency devaluation, it can be an effective response to financial crises
and pandemics by stimulating real economic activity (Reis & Tenreyro, 2022).
In this paper we consider Helicopter Money as introduced by Friedman: a
distribution of newly printed money by the Central Bank directly to
households, resulting in a decrease in the Central Bank’s net worth and an
increase in the net worth of households. This injection of money is modeled as
a one-time increase of savings of households by a factor $\kappa_{H}>1$, such
that $S\rightarrow\kappa_{H}S$ instantaneously after all shocks. The increased
savings of households boost demand through an increased consumption budget
$C_{B}$ (see Eq. (3)) and finally end up on the balance sheet of firms where
it is then redistributed to households as wages and dividends.

### 3.5 Windfall Tax

Whereas Helicopter Money stimulates consumption by injecting external funds
into households savings, a tax on “windfall” profits, which refer to large
unforeseen profits (Chennells, 1997), achieves a similar outcome by
redistributing money through taxation. Following the Russian invasion of
Ukraine, there has been a significant surge in fossil fuel prices, leading to
unexpectedly high returns for utilities and fossil fuel producing companies
(Weber, 2022), and profit-driven price markups for other firms (Weber &
Wasner, 2023), leading to strong debate amongst policymakers about
implementing a Windfall Tax. The specific design of the Windfall Tax as a
rent-sharing fiscal policy may vary depending on the particular circumstances
(Baunsgaard and Vernon, 2022).

In this paper, we implement a “Windfall Tax” as a temporary increase of the
fraction of energy sector cash balance that is re-injected to the savings of
households, i.e. $\delta_{e}\rightarrow\delta_{e}+\Delta\delta_{e}$. This tax
has a duration of two years, commencing one year prior to the end of the price
shock. The primary objective of increasing $\delta_{e}$ is to boost
consumption by enhancing households’ savings through the redistribution of the
increased cash balance in the energy sector that follow the energy price
increases.

## 4 Stationary Dynamics: the Role of the Central Bank

### 4.1 The Economic Equilibrium Without Monetary Policy

Figure 1: Economic dashboard for the three Central Bank scenarios described in
sections 4.1, 4.2 in absence of any policy and shock: Blue lines: Inactive
Central Bank scenario ($\pi^{\star}=\phi_{\pi}=0,\;\rho^{\star}=1.2\%$ p.a.).
Orange lines: Reactive Central Bank with Anchored Trust scenario
($\pi^{\star}=2.4\%\,\,\text{p.a.},\;\phi_{\pi}=1,\;\tau^{T}=0.95$). Green
lines: Reactive Central Bank with Floating Trust scenario. The complete
parameter set can be seen in table 1. The two insets correspond to the
demand/output ratio and the bankruptcy rate.

To begin our analysis, we consider the economy without shocks. Our first
choice of parameters corresponds to a Central Bank that does not react to
inflation (i.e. $\pi^{\star}=\phi_{\pi}=0$) while maintaining low baseline
interest rates ($\rho^{\star}=1.2\%$ p.a.). By consequence, agents form
inflation expectations based only on their own observations ($\tau^{T}=0$, see
Eq. (20)). In this scenario, called “Inactive CB”, and for the choice of
parameters detailed in Table 1, the economy spontaneously evolves into a
steady state of full capacity, as evidenced by the low, oscillating
unemployment rate around $0.5\%$ (see Figure 1). There are small endogenous
output and unemployment fluctuations of roughly two years in duration.
Simultaneously, there are stronger oscillations in the demand and supply for
goods, and in real wages. These fluctuations cause firms to adjust both their
production and prices, thus generating a small “business cycle” with periodic
imbalance between demand and output due to price adjustments. But in the
absence of an active Central Bank target, the amplitude of the resulting
inflation oscillations is found to be substantial, ranging between $2\%$ and
$8\%$ p.a., suggesting a role for monetary policy.

Note however that depending on the choice of parameters, the economy may
settle into a much less favourable state. In particular, when the hire-to-fire
ratio is too small or when the baseline interest rate is too high, the economy
collapses, see Gualdi et al. (2015); Bouchaud et al. (2018).

### 4.2 Stabilizing Inflation through Monetary Policy

At this point, we introduce an active Central Bank and study its impact on
inflation and unemployment. Specifically, we set the Central Bank’s inflation
target to $\pi^{\star}=2.4\%$ p.a. and its reaction to inflation variation to
$\phi_{\pi}=1$. In this configuration, we assume that the Central Bank works
with a well-anchored inflation expectation fixed at $\tau^{T}=0.95$. In the
pre-COVID period core inflation was quite stable as a result of an arguably
successful monetary policy (Miles et al., 2017). Simultaneously, the Central
Banks had been acting in a transparent and measurable way despite the
historically low interest rates (Reis, 2021), suggesting a high degree of
trust. We call this constant $\tau^{T}$ scenario “Reactive CB, Anchored Trust”
in the following. In this case, Figure 1 shows that inflation is reduced to an
average of $\langle\pi_{AT}\rangle\approx 2.7\%$, close to the $2.4\%$ target,
while nearly completely suppressing the business cycle with a very low stable
unemployment. On the other hand, there is a slight reduction in real wages and
higher interest rates.

Finally, we introduce a third scenario, referred to as “Reactive CB, Floating
Trust”, where we allow agents’ trust in the Central Bank to vary depending on
the perceived success of the CB to bring inflation to roost, see $\tau^{T}(t)$
defined in Eq. (22). We choose as parameter values $\omega=0.2$ (corresponding
to a memory time of 5 months) and a sensitivity to off-target inflation of
$\alpha_{I}=0.4$. Again, all other parameters are taken from Table 1. In the
absence of external shocks, this configuration leads to a similarly high
degree of trust in the Central Bank, and is otherwise identical to the case
where trust is anchored – simply because inflation is on target.

## 5 The COVID Shock and its Aftermath in the Absence of Monetary Policy

Figure 2: Empirical Shocks: Fitting of shock scenarios to macro data from the
US. The empirical series (dashed lines) are indexed to January 2020. (Left
Panel) Empirical series for the Personal Consumption Expenditure (orange,
dashed line) and Industrial Production Index (blue, dashed line). Solid lines
represent our model for the shocks to consumption propensity $c$ (orange) and
productivity $\zeta$ (blue). Dark grey area corresponds to the COVID Shock and
the grey area to the Supply Chain Shock in the shock scenarios that we apply
to Mark-0. (Right Panel) WTI Crude oil spot price during the periods of shock
(green, dashed line). Modelling of the artificial price series (green, solid)
in the Mark-0 model for a period of two years (light grey area). Data
retrieved from FRED (2023b, a, c).

### 5.1 Modelling the Three COVID Shocks

In order to assess the effects of recovery and monetary policies in response
to the events of 2020-2022, we introduce in Mark-0 three shocks calibrated on
U.S. data: (1) a COVID shock, (2) a supply-chain shock, and (3) an energy
price shock. The data was retrieved from the Federal Reserve Bank of St.
Louis, and are shown in Figure 2 where we have indexed values to January 2020.
We now discuss these shocks in turn:

  1. 1.

COVID: The first major COVID outbreak emerged in the US in February 2020,
prompting the US government to declare a public health emergency, followed by
the implementation of stay-at-home orders in March of the same year. The COVID
outbreak led to a significant reduction in personal consumption expenditure by
households, falling by around $15\%$ in February 2020 (Figure 2, dashed orange
line), which was of the same order of magnitude as the decline in the
Industrial Production Index during the same month (dashed blue line). It took
about 5 months for aggregate Personal Consumption Expenditure to recover to
its pre-pandemic level. In accordance with this, we consider a shock to the
consumption propensity $c(t)$ of households by $15\%$, which appears in Eq.
(2), recovering linearly over 5 periods (solid orange line in Figure 2).

  2. 2.

Supply Chains: During the initial COVID outbreak, firms in the US laid off a
large number of workers, reducing their production to a similar degree as the
personal consumption expenditure. However, while personal consumption
recovered within 5 months, it took an additional 10-15 months for industrial
production to return to its pre-pandemic state, as can be seen in the
industrial production index (dashed blue line, Figure 2). This was due to a
plethora of idiosyncratic supply chain disruptions, such as logistics and
transportation difficulties, semiconductor shortages, pandemic-related
restrictions on economic activity, and labor shortages that led to the slower
recovery of production in the industrial sector (Attinasi et al., 2022). In
the context of this paper, we model this by a shock to firm productivity
$\zeta(t)$ (defined in Eq. (4)) of an initial magnitude of 15%, with a
recovery of 15 months, see solid blue line in Figure 2.

  3. 3.

Energy Prices: Finally, we consider the energy price shock. The reduction of
demand and production throughout the pandemic led to a supply glut in energy
markets, which led to a steep decrease in energy prices, such as oil, by up to
70% for immediate delivery of West Texas Intermediate crude oil (Figure 2,
dashed green line). As the recovery period began, external factors such as
extreme weather conditions in various parts of the world and maintenance work
that had been postponed during the pandemic caused a surge in demand, and the
energy prices thus rebounded quickly (Zakeri et al., 2022). Unfortunately,
with the Russian invasion of Ukraine in February 2022 this rebound was further
exacerbated to a global energy crisis, due to Russia’s position as a major
global exporter of natural gas and oil. By June 2022, the WTI crude oil spot
price had peaked, rising nearly 100% compared to pre-pandemic levels.
Following this, the global recovery and adjustment of energy markets has led
to a sharp easing of energy prices. For Mark-0, these processes are external
to the model, such that we introduce here an exogenous price shock to firms’
price update Eqs. (9)-(11). Specifically, firms’ prices change by an
additional exogenous factor $g_{e}\Delta p_{e}(t)$, where $g_{e}$ is a
constant factor akin to the energy-share in production, and $\Delta p_{e}(t)$
is the time-dependent monthly percentage change in energy prices. Our
artificial energy price series is shown in Figure 2 as the solid green line,
and is based on the WTI Spot Price.161616Prior to the shock, we set
$p_{e}(t)=\overline{p}(t)$. Alternative calibrations with the energy component
of the U.S. Consumer Price Index or the Henry Hub Natural Gas price lead to
similar shapes and magnitudes. In the following we will choose $g_{e}=3.15\%$,
i.e. half of the energy share of the GDP in the US, where the factor 2
accounts for inventories and partial substitutability. We also posit that the
profits of the energy sector are transferred back to households at an
effective rate of $\delta_{e}=4\%$ per time step, unless stated otherwise.

### 5.2 The Effect of Shocks Without Stabilisation Policies

Figure 3: Economic dashboard for the three shocks in the Inactive Central Bank
scenario and in the absence of any policies: The dynamics for the three shock
scenarios, COVID only (blue), COVID and Supply Chain shock (orange) and all
shocks (green) when the Central Bank is inactive and no policy is applied. The
areas shaded in grey indicate the duration of the three shocks: the COVID
shock lasting until the end of the dark grey area, the supply chain shock
until the end of the grey area, and the energy price shock until the end of
the light grey area. The dashed black line represents pre-shock averages. In
the first two cases, the economy is able to recover on its own (although with
significant fluctuations that last for a few years), but gradually collapses
when the energy shock is turned on.

To develop some intuition, we now build three counterfactual cases where (i)
only the COVID shock, (ii) COVID + supply chain shocks and (iii) COVID +
supply chain + energy price shocks hit the Mark-0 economy, without any
stabilisation policy put in place. It should be noted that this exercise is
somewhat optimistic, as our shocks have been calibrated on an economy where
emergency policies were actually implemented. This is discussed further below.

  1. Case (i)

After the impact of only the COVID shock, the economy remains operational with
full capacity but experiences a persistent and high inflation that peaks at
15.5% and only starts to recede by the end of 2023, with significant
fluctuations that persist for a few years (Figure 3, blue). All other
observables return to their pre-shock levels by the end of 2023, with small
oscillations around the steady state in accordance with the Inactive Central
Bank scenario without shocks.

  2. Case (ii)

When adding the supply chain shock (Figure 3, orange), the dynamics remain
very similar: the economy returns to its steady state without fiscal or
monetary policy. Remarkably, inflationary dynamics are almost identical,
albeit with a slightly higher peak value (16.0%) reached slightly earlier in
2023. The biggest difference is the prolongation of the recovery period, which
leads to a steeper real wage dip.

  3. Case (iii)

Taking now the energy price shock into consideration, the situation abruptly
changes. Initially, we observe a short and strong deflation period due to the
steep drop in energy prices, increasing real wages, supply-demand imbalance
and the fragility of firms due to the decreased prices.171717 We consider here
a direct price-shock transmission. In practice, energy is purchased in various
forms and often with inventories and financial contracts to insure against
price volatility. In this respect, it might make more sense to define the
effective price of energy $p_{e}(t)$ in the production process as an
exponential moving average of the WTI spot price. This would smooth out the
initial dip and lead to more realistic inflation time series. We leave this
for a later study. Immediately following this, we observe an explosion in
inflation with a peak of 22.7%. Firms fragility increases due to the price
shock which causes bankruptcies (peaking at 10 % rate), unemployment and a
decrease of wages. In the long run, this disruption is so strong that wages
cannot recover. This starts a feedback cycle of low demand that leads to a
decrease of output, which causes a reduction in savings and therefore demand
drops. This feedback cycle consistently increases unemployment, and decreases
savings and wages, which eventually ends in a collapse of the economy.

In the realm of the Mark-0 model, the COVID and supply chain shocks would thus
have had a minor long term impact on output, but would have led to substantial
medium term inflation due to excess savings, as predicted using the Mark-0
model as early as June 2020 in Sharma et al. (2020). The energy price shock,
however, is the last straw on the camel’s back and, in the absence of monetary
and fiscal policies, has a strongly detrimental impact on the economy, causing
it to collapse. This is a consequence of the existence of discontinuous
transition boundaries (a.k.a. tipping points) in parameter space, as
emphasized in Gualdi et al. (2015) and, within the specific context of the
COVID shock, in Sharma et al. (2020). As a scenario-generating tool for
policymakers, our model demonstrates that in the absence of mitigating
policies, the full sequence of COVID, supply chain and energy shocks can
trigger a negative feedback cycle, manifested as a downward wage spiral, that
results in a collapse of demand and a full blown crisis. However, we see that
the first two shocks, as modelled above, are of sufficiently mild amplitude
not to trigger a complete collapse of the economy. In the following section,
we study counterfactuals with larger “bare” (unmitigated) shock amplitudes,
where mitigating policy measures are indeed needed to prevent a catastrophic
collapse of output beyond a certain tipping amplitude of the initial COVID
shock alone.

### 5.3 Sensitivity to Shock Magnitude and the Role of Easy-Credit Policy

Figure 4: Counterfactual COVID shock in the Inactive Central Bank scenario and
in the absence of any policies: Unemployment (left) and inflation (right) in
the Inactive Central Bank scenario without any policy, with an amplified COVID
shock by a factor $\kappa$ and Supply Chain shock. $\kappa=1$ (purple line)
corresponds to the COVID shock considered in the previous section, with a very
low level of unemployment. As soon as $\kappa\geq 1.6$, the economy is unable
to recover spontaneously, and output collapses as demonstrated in the
$\kappa=1.6$ case (green line).  Figure 5: Economic dashboard for a stronger
COVID shock in the Inactive Central Bank scenario, with and without Easy-
Credit policy: The dynamics of the Inactive CB scenario in case (ii) (COVID
and supply chain shock but no energy price shock), for a COVID shock strength
of $\kappa=1.6$ without Easy-Credit policy (blue) and $\kappa=1.6$ with Easy-
Credit policy, $\mu=1.3$ (orange). We see that policy is effective in
preventing the economic collapse in the case of the stronger COVID shock.

Our model is based on data that already incorporates policy measures, which
were indeed in place in 2020-21 to mitigate shocks. As such, it is possible
that the unmitigated shocks were actually more severe than what we observe
retrospectively in data. Here, we investigate counterfactual (ii)-scenarios
with different shock magnitudes, where we shut down the final energy shock but
scale the amplitude of the observed COVID shock by a factor $\kappa$ (see
Figure 2, left panel dark grey area). This stronger shock could have taken
place if for example the CARES act and the CAA where not authorized.181818The
supply chain shock is left unchanged in the present exercise, but we observe
almost the same phenomenon without such additional shock. As $\kappa$
increases, firms go progressively bankrupt due to their high financial
fragility $\Phi$, leading to a higher unemployment rate during the shock.
Households save money during the shock, resulting in higher demand, leading to
upward pressure on prices and somewhat higher inflation. Therefore in the long
run, the scenarios with intensified shocks $\kappa\lesssim 1.6$ very slightly
increases inflation and long-term unemployment. However, for larger $\kappa$,
a tipping point is reached, as in Sharma et al. (2020): more initial
bankruptcies and firms’ hesitation to increase wages after the shock leads to
a collapse of output and deflation, see Figure 4 (for a full dashboard see
Figure 10 in Appendix A.2).

In this case, allowing firms to accumulate more debt by easing bankruptcy
rules is highly effective to keep the economy on an even keel. In fact, as
shown in Figure 5, the “Easy-Credit” policy defined by Eq. (23) with $\mu=1.3$
for the bankruptcy threshold $\Theta$ manages to substantially reduce the
impact of strong COVID-related shocks, even with an increased initial COVID
shock ($\kappa=1.6$) that would collapse the economy on its own. With Easy-
Credit policy, firms can maintain wages and do not need to fire employees to
remain solvent, allowing the economy to recover. However, unemployment still
peaks above 10% (with a second hump mid 2024) and inflation reaches 16.5% at
the end of 2022. As we will discuss in section 8 below, the main issue with
such a period of high inflation is the risk of de-anchoring inflation
expectations, opening the path to possible hyper-inflation.

In order to control inflation, one needs to consider the effect of monetary
policy. We thus now turn to the study of the same sequence of three non-
amplified shocks, with Easy-Credit policy and with a fully active Central
Bank, distinguishing between the case of Anchored Trust and the case of
Floating Trust in its ability to curb inflation.

## 6 Monetary Policy Response to Inflationary Shocks

In the previous section, we observed that the loosening of regulatory
bankruptcy policy during severe shocks prevents the Mark-0 economy from
collapsing, but does not address the issue of high inflation rates (Section
5.3). To study inflation mitigation policies, we now introduce a Taylor-rule
based monetary policy into the mix, i.e. we combine the situation explored in
Section 5 (all shocks and Easy-Credit policy without Central Bank) with the
different monetary policy scenarios discussed in Section 4. All numerical
experiments henceforth are run with the Easy-Credit policy described in Eq.
(23) with $\mu=1.3$, as in Section 5. As the economy returns to its steady
state, the Easy-Credit policy becomes equivalent to the fixed default
threshold, such that its duration depends on the recovery period.191919Eq.
(23) indeed implies $\Theta(t)=\Theta_{0}$ in the steady state when the
average fragility is low, which is the case for the choice of parameters made
in this paper. Note that in what follows, we consider the three shocks
together, keeping the amplitude of the COVID shock to the one calibrated on
the observed data (i.e. $\kappa=1$ in the language of Section 5). As already
discussed, such calibration may underestimate the severity of the unmitigated
shock, since the US government immediately implemented Easy-Credit and
Helicopter Money measures to alleviate the COVID shock. Note also that
unemployment is not an explicit target of our Central Bank, in the sense that
the Taylor-rule is only responsive to realized inflation and not to
unemployment.

### 6.1 An Inactive Central Bank

We begin our study by considering again the situation with an Inactive Central
Bank as in Section 4.1, but now with all shocks together and with the Easy-
Credit policy being implemented (Figure 6, full blue line; see Figure 14 in
Appendix A.3 for all macroeconomic timeseries). In this case, over the first
few months, the combination of shocks causes a spike in unemployment that
peaks around 28% during a deflationary phase, but the economy quickly recovers
to full employment, yet at the price of a sustained inflationary period that
peaks above 20% before slowly reverting to equilibrium, consistently with the
discussion of Section 5: the Easy-Credit policy can avoid full collapse, but
only at the price of high inflation.

### 6.2 Monetary Policy with Anchored Trust

We next consider a monetary policy experiment with a responsive Central Bank
in the (optimistic) “Anchored Trust” scenario. In the Mark-0 model, as in real
life, the Central Bank’s aim is to keep inflation on target. Here, following
Section 4.2, the inflation target is chosen to be $\pi^{\star}=2.4\%$ p.a. and
the Taylor rule strength is $\phi_{\pi}=1$. Moreover, we optimistically assume
that the Central Bank has successfully communicated to all economic actors
that it has inflation under control, thereby convincing them to believe that
long-term inflation will be close to the Central Bank’s inflation target, with
a fixed anchor weight $\tau^{T}=0.95$.

In the case of Anchored Trust and variable interest rates with all three
shocks (Figure 6, dashed orange line), the Central Bank is successful in
taming inflation: peak inflation is lower than in the case without an active
central bank (peak at 9.7% vs 24.1% in the Inactive Central Bank scenario) and
quickly tapers off. Although the inflation rate is not in line with the
Central Bank’s target throughout the crisis, it appears that monetary policy
is able to reign in inflation. The reduction in inflation is not primarily due
to the impact of interest rate policy but rather to the strong anchoring of
expectations, which significantly dampens expected (and thus realized)
inflation, as upward pressure on both prices and wages are reduced. Such
anchoring moderates the price increases after the initial COVID shock, but
does not dampen the deflationary dynamics towards the end of the energy price
shock, as this is driven by a strongly increasing unemployment that puts
downward pressure on wages (see Figure 14 in Appendix A.3 for all
macroeconomic timeseries).

The initial price shock in the first few months causes bankruptcies that are
less numerous with a reactive Central Bank, hence also reducing the initial
unemployment spike (peak at 28.5% in the Inactive Central Bank scenario and
5.2% with Active Central Bank with Anchored Trust). However, at the end of the
shock the Central Bank effort to control inflation results in a peak
unemployment of 14.6 % in Feb 2024, 8% higher than in the Inactive Central
Bank scenario. We see that in the Mark-0 model, the Central Bank is stuck
between a rock and a hard place with its inflation-unemployment trade-off,
meaning that in the context of these shocks, monetary policy is not a panacea:
some form of Keynesian stimulus or “Helicopter Money” is needed in conjunction
with monetary policy to restore the economy to its pre-crisis steady-state –
see section 7.1 below. In this regard, a Central Bank with an explicit dual-
mandate would potentially navigate the trade-off better.

Figure 6: Unemployment and Inflation with all shocks, Easy-Credit policy, and
three distinct monetary policy scenarios. (Blue lines) Inactive Central Bank
scenario. (Orange lines) Reactive Central Bank with Anchored Trust, with
anchor parameter $\tau^{T}=0.95$. A Taylor rule policy successfully decreases
peak inflation but increases peak unemployment. (Green lines) Reactive Central
Bank with Floating Trust. Here, trust is eroded during the high inflation
period. Monetary policy then fails at reducing inflation, with the risk of
hyper-inflation lurking (see section 8); unemployment remains lower because of
the higher inflation.

### 6.3 Monetary Policy with Floating Trust

The scenario of high and Anchored Trust in monetary policy for long-term
inflation expectations might be a good approximation during long periods of
stability, but in periods of shocks, the anchoring $\tau^{T}$ may decrease
when inflation deviates strongly from the Central Bank’s target (Reis, 2021).
We model such a Floating Trust effect through Eq. (22), where the anchoring
parameter decreases when the spread between observed and target inflation
increases. We set $\alpha_{I}=0.4$ and a memory time of $\sim 5$ months
($\omega=0.2$). These values imply that when inflation reaches $4$ times the
Central Bank’s target $\pi^{\star}$, trust in the Central Bank falls to
approximately a third of its initial value after approximately one year.
Larger values of $\alpha_{I}$ would lead to an even steeper de-anchoring of
inflation expectations. Such a loss of faith in monetary authorities further
increases realized inflation as economic agents expect higher inflation in the
future and take this into account when setting prices and wages. This may lead
to a self-fulfilling feedback loop between expected inflation and real
inflation through a wage-price spiral (see section 8).

Compared to the Anchored Trust case, the surge of inflation after COVID leads
to a loss in trust that almost vanishes (i.e. $\tau^{T}\to 0$) during the
energy price shock. The consequence of this is a higher realized inflation
rate, that reaches a peak value only slightly below the Inactive Central Bank
scenario (Figure 6, right).202020For the full economic dashboard in the case
of dynamic expectations, see Appendix A.3, Fig. 13, and compare with Fig. 12.
Unemployment, on the other hand, turns out to be higher than with an Inactive
Central Bank (see Figure 6, left). Expectation de-anchoring can lead to a dire
situation where high interest rates fail to rein in high inflation while
driving the economy into a deep recession.

A strong factor of the loss in trust is the exogenous nature of the energy
price movements. Despite the strong hike up to a real interest rate of 3.4%
per year (18.4% nominal target rate), inflation remains far above target. Once
these shocks have passed, the Central Bank is arguably quite successful in
keeping inflation closer to its steady state value. However, similarly to the
Anchored Trust case, the economy now contends with a persistent, relatively
high unemployment rate. The unemployment here is driven by the energy price
shock that causes a strong surge in inflation, which erodes household savings
and reduces real wages by 4% compared to its steady state value, thus leading
to a drop in demand (see Figure 13 in Appendix A.3). In contrast to the
Anchored Trust scenario, unemployment in the Floating Trust scenario recovers
to full employment faster. This is driven by the comparatively higher real
wages, which allows for a faster recovery of the consumption budget and
consequently output. This implies that a loss of trust may be beneficial in
terms of unemployment by promoting a faster adjustment to exogenous price
levels as they occur.

As with both the Inactive Central Bank and Anchored Trust scenario, strong
exogenous inflation outside of the Central Bank’s sphere of influence leads to
a severe contraction of the economy, which recovers only slowly. The
convergence to pre-crisis equilibrium takes more than 15 years in the Anchored
Trust scenario and 6 years in the Floating Trust scenario. This suggests that
fiscal policy to shore up consumer’s budget or reduce energy price effects on
firms and households can be a more effective tool than monetary policy, in
face of exogenous price shocks.

### 6.4 Sensitivity to the Strength of the Monetary Policy

In circumstances of crises, as with our shocks, Central Banks should possibly
react more strongly to maintain inflation and keep expectations anchored. We
thus consider a stronger Central Bank Taylor reactivity parameter
$\phi_{\pi}=2.0$ (see Appendix A.4, Figure 15 and 16 for details).

In the case of Anchored Trust, an increase in $\phi_{\pi}$ hardly changes
inflation, but stronger interest rate hikes result in significantly higher
unemployment rates. Here, stronger monetary policy is detrimental. By
contrast, in the Floating Trust scenario with dynamic trust, on the other
hand, an increase in $\phi_{\pi}$ signals a stronger commitment of the Central
Bank to control inflation. Therefore, all else being the same, inflation
expectations are lower which, in a self-fulfilling manner, cuts realized
inflation from 18.43% to 12.3% and keeps trust better anchored. However, as in
the Anchored Trust case, the sharper interest rate hikes by the Central Bank
lead to a significant increase in the unemployment rate (from 9.3% to 15.2%)
and a much longer recovery time for the economy as a whole. Again, our model
confirms and quantifies the observation made by Stiglitz & Regmi (2022),
quoted in the introduction: monetary policy interest tools tend to be too
blunt, curbing inflation at the cost of unnecessarily high unemployment.

### 6.5 Sensitivity to Transmission Channels

Within our modelling framework, the impact of monetary policy relies on the
efficiency of three transmission channels: (a) expectation anchoring,
discussed above; (b) sensitivity of consumption on real interest rate, through
parameter $\alpha_{c}$; (c) sensitivity of firms’ hiring and wage policies on
real interest rate, through parameter $\alpha_{\Gamma}$. We have run some
simulations to check the dependence of the state of the economy on these two
parameters in the Anchored Trust case, see Appendix A.5. The conclusion of
this exercise is that increasing $\alpha_{c}$ and $\alpha_{\Gamma}$ has a
minor direct effect on inflation but a significant effect on unemployment,
depending on the sign of the real interest rate $\rho-\hat{\pi}$. As expected,
a raise of interest rates degrades economic activity, all the more so when the
sensitivity of firms and households to the real rate is higher. As already
stated, the main transmission channel through which monetary policy impacts
the economy is expectation anchoring, rather than cost of loans or income on
savings.

## 7 Fiscal Stimulus

Section 6 showed that in the presence of exogenous inflation drivers, the
Central Bank can reduce inflation only at the cost of high unemployment. The
mechanisms responsible for unemployment are the increased burden of debt that
weighs on wages and increases the speed at which firms lay off workers. This
in turn lowers demand, an effect amplified by the erosion of household’s
purchasing power by the lingering inflation. One way to tackle this issue is
to ensure that demand does not drop as severely by increasing it through
fiscal stimulus (see Section 3.4). We test this policy device here by
considering different kinds of fiscal stimuli. Unless otherwise indicated, the
fiscal stimuli are applied on the scenario of Section 6.3, i.e. with all
shocks, Easy-Credit policy, and an Active Central Bank with Floating Trust,
which we believe to be the scenario closest to reality.

### 7.1 The Effects of Helicopter Money

Figure 7: Helicopter Money in the Reactive Central Bank with Floating Trust
scenario, Easy-Credit policy, and all shocks. Unemployment (left) and
inflation (right) for a helicopter drop of size $\kappa_{H}S$ one month after
the price shock. Already with $\kappa_{H}\geq 0.2$ unemployment is reduced
almost to zero. A further increase of Helicopter Money only increases the
duration of the high-inflation period, without further reducing unemployment.

Following Sharma et al. (2020), we begin by considering a one-time increase of
the households’ budget by a factor $\kappa_{H}\in[0\%,60\%]$ nine months after
the end of the energy price shock (July 2023). The goal is to find an optimal
multiplier $\kappa_{H}$ that reduces unemployment while keeping excess
inflation to a minimum in amplitude and duration. Here we find that this
minimum is reached for $\kappa_{H}\approx 30\%$ (see Figure 7 for the Reactive
Central Bank with Floating Trust case). 212121See Appendix A.6, Figures 19, 20
for the other two monetary policy scenarios, and Figures 21, 22 and 23 for the
complete economic dashboards of all scenarios. This choice of $\kappa_{H}$
matches the magnitude of the “Emergency Money for the People Act”, which
entailed the US government providing direct payments of 2000 USD per month for
a maximum of 12 months to support individuals during the COVID pandemic.

In all considered scenarios, the fiscal stimulus package leads to a quick
recovery of production to its pre-shock levels and unemployment at near 0%,
thus successfully eliminating the steep recession and long recovery following
the energy price shock, as shown in Figure 7. However, policymakers face an
inflation-unemployment trade-off, as the stimulus generates a spurt of
inflation. In the absence of monetary authority, inflation due to the
injection of money rises to 16.2%. However, this is an endogenous inflation,
which means that in the Floating Trust case, the Central Bank can raise rates
to curb consumption propensity and keep inflation below 10% p.a. (Figure 7).
Increasing the stimulus leads to a higher inflation peak and a longer duration
of high inflation.

The inflationary period after the stimulus is in part due to a short-term de-
anchoring of expectations as the demand stimulus leads to higher prices in the
context of a tight labour market preventing production increases. Yet, here
one can take the position that the economy is in good shape from a
macroeconomic perspective, with near-zero unemployment despite an above-target
inflation.222222Because Mark-0 treats households at the aggregate level, we do
not analyze the distributional consequences of inflation here, though
inflation is always and everywhere differential in its effects. Naturally, the
Anchored Trust case would be much more favorable as the high unemployment
problem is resolved with only a minimal rise in inflation, around 5%
annualized over 8 months, as soon as $\kappa_{H}\gtrsim 0.3$.

### 7.2 The Effects of a Windfall Tax

Figure 8: Windfall Tax in the Reactive Central Bank with Floating Trust
scenario, Easy-Credit policy, and all shocks. Unemployment (left) and
inflation (right) for a Windfall Tax of $\delta_{e}+\Delta\delta_{e}$ one year
before the end of the price shock with a duration of two years. With
$\Delta\delta_{e}\approx 3\%$ unemployment is reduced strongly. A further
increase of tax does only increase unemployment again. For even larger
$\Delta\delta_{e}$, inflation increases because of increased demand due to
increased savings.

Another stimulus that central authorities can use to alleviate the impact of
the price shock is a Windfall Tax. This policy redistributes the excess
profits generated by the energy sector as a result of rising energy prices to
household savings with the aim to stimulate consumption and consequently
reduce unemployment. We test the effectiveness of the policy, which in our
modelling consists in an increase of the energy sector payout rate
$\delta_{e}\rightarrow\delta_{e}+\Delta\delta_{e}$ one year before the end of
the price shock with a duration of two years, with
$\Delta\delta_{e}\in[0\%,16\%]$.232323We remind here that $\delta_{e}$ is the
fraction of the energy sector cash balance redistributed to households at each
time step, see Eq. (17).

Our aim is to design the fiscal stimulus in a manner that minimizes both
unemployment and inflation. We find that for $\Delta\delta_{e}=4\%$ the excess
profits of the energy sector are redistributed such that unemployment is
significantly reduced at all times while inflation remains under control.
Further increasing $\Delta\delta_{e}$ reduces the unemployment in 2023 to
almost zero, but at the cost of increasing unemployment in 2024 after the
Windfall Tax has ended (see Figure 8).242424For more detail, see Appendix A.7,
Figures 24, 25. As long as the tax is active, the increase in $\delta_{e}$
results in higher savings, leading to an increased consumption budget and
therefore higher demand. This has the effect of reducing unemployment as firms
expand production, as well as reducing firm fragility due to increased
profitability. However, as the Windfall Tax increases in the Floating Trust
scenario, inflation also increases and expectations remain de-anchored for
longer, compelling the Central Bank to increase interest rates. As the tax
ends, the household’s consumption budget has been reduced to below the case
without a Windfall Tax, and the real interest rate remains high. This leads to
a negative spiral of lower demand and output, higher unemployment, leading to
lower income, until the system equilibrates again.

The inflationary effects of the increase in savings due to the Windfall Tax
remain minor for small values of $\Delta\delta_{e}$. However, once the
Windfall Tax exceeds $\Delta\delta_{e}\sim 12\%$, a longer period of high
inflation after the shock is observed (see Figure 8 and Appendix A.7, Figure
28) as inflation expectations remain de-anchored for a longer period.

The Anchored Trust scenario presents a different challenge. Again, up to a
threshold, the Windfall Tax reduces the initial and long-term unemployment
spike, with $\Delta\delta_{e}=4\%$ still remaining a robust choice (see
Appendix A.7, Figures 25, 27). However, this time at the cost of inflation (or
less deflation) on the way back to the pre-crisis steady state. In this
regard, policymakers might opt instead to minimize the volatility of inflation
to let it smoothly return to the Central Bank’s target rate, which actually
suggests a higher Windfall Tax of $\Delta\delta_{e}=8\%$ for instance.

We conclude that when implementing a Windfall Tax as a fiscal stimulus,
excessive tax levels can have the unintended consequence of exacerbating long-
term unemployment, while a measured approach can both reduce unemployment and
smooth inflation dynamics.

## 8 Model Sensitivity: The Dangers of a Wage-Price Spiral

### 8.1 “Sloppiness” Analysis

To develop a deeper understanding of the reaction to shocks of our model
economy, we explore the parameter sensitivity of this configuration of the
Mark-0 model using the “sloppy model” methodology put forth by Sethna and
collaborators (see Transtrum et al. (2015) for an introduction) and applied to
macroeconomic models by Naumann-Woleske et al. (2023). This method identifies
the stiffest parameter combinations, along which one observes the largest
change in outcomes for small perturbations. In particular, we consider the
parameter sensitivity of the dynamics of the inflation rate and of the
unemployment rate (for methodological details, see Appendix A.8 and Naumann-
Woleske et al. (2023)). Overall, this approach provides a more comprehensive
understanding of parameter sensitivity by considering the curvature of the
model manifold, as compared to one-at-a-time parameter sensitivity analysis.

With respect to both inflation and unemployment, the balance of wage
indexation (bargaining power) $g_{w}$ to price indexation (market power)
$g_{p}$, has by far the strongest influence on the outcome dynamics for the
inactive CB and Floating Trust scenarios (see detailed results in Appendix A.8
and A.9). Beyond inflation itself, eigenvectors are combinations of various
pricing parameters including $\gamma$, $g_{p}$ and $g_{e}$ for both
unemployment and inflation. Central Bank parameters are present but not
significantly. This aligns with our findings that the Mark-0 economy always
maintains a tight connection between unemployment and inflation, indicating
that it may not be possible to minimize one without affecting the other (see
Appendix A.8, Figures 31, 32).

Figure 9: Hyperinflation tipping points. Left: Stable inflation vs.
Hyperinflation in the plane ($g_{p}=g_{w}$, $\tau^{T}$) in the Anchored Trust
scenario and in absence of exogenous shocks. Note that inflation can be stable
even when $g_{p}=g_{w}>1$ for strong enough anchoring. Middle: Stable
inflation vs. Hyperinflation in the plane ($g_{p}=g_{w}$, $\alpha_{I}$) in the
Floating Trust case and in absence of exogenous shocks. Right: inflation rate
(top) and unemployment (bottom) as a function of time, in the Floating Trust
case, $\alpha_{I}=0.3$ and different values of $g_{p}=g_{w}$, after the three
COVID shocks. When indexation is too strong, an hyperinflation regime sets in
at the end of the energy price shock.

### 8.2 The Risk of a Hyperinflation Spiral

We can interpret the sensitivity to $g_{w}$ and $g_{p}$ in light of the
dangers of a hyperinflation episode resulting from a wage-price spiral.
Alvarez et al. (2022) cite a concern that hyperinflation may occur if firms
increase wages in response to higher inflation, leading to an increase in
purchasing power and ultimately feeding into a wage-price spiral in the
current macroeconomic environment. This feedback loop is influenced by the
indexation of prices and wages to firms’ inflation expectations (Holland,
1988), i.e. by the value of parameters $g_{p}$ and $g_{w}$ in our model. In
the simplest case where bargaining power and market power are equal
($g_{p}=g_{w}$), the economy reaches a stable inflationary state, marked by
cyclical fluctuations due to mismatches in supply and demand (see Figure 1 in
Section 2, for which $g_{w}=g_{p}=0.8$). When indexation is weak
($g_{p}=g_{w}<1$), neither wages nor prices fully incorporate inflation
expectations, and fluctuations in the inflation rate due to mismatches in
demand and supply are dampened. Conversely, for strong indexation
($g_{p}=g_{w}>1$), the economy may enter a state of hyperinflation, though as
wages and prices increase equally fast, there are no real effects (omitting,
of course, the cost of inflation itself due to “menu costs”). Only when
$g_{p}\neq g_{w}$ does the economy collapse.

Introducing monetary policy implies that hyperinflation can be staved off due
to the anchoring of inflation expectations. In the case of Anchored Trust,
inflation remains stable until a critical point $g_{p}=g_{w}<g^{\star}$ where
$g^{\star}>1$ depends on the commitment of the Central Bank as well as the
strength of expectation anchoring to the its target (see Figure 9, left
panel). The same holds true with Floating Trust (middle panel). However, in
this case an interesting range of parameters where inflation remains stable
for $g^{\star}>g_{w}=g_{p}>1$, until a strong enough shock occurs, which
triggers a loss of trust in the Central Bank, tipping the economy into a
hyperinflation phase (as illustrated in Figure 9, right panel). This is
precisely the scenario that Central Bankers want to avoid by doing “whatever
it takes”.

If bargaining and market power differ ($g_{p}\neq g_{w}$), the effect on
unemployment is significant (see Appendix A.9, Figure 34). In the Anchored
Trust case, when market power is greater than bargaining power – a.k.a.
“greedflation” – ($g_{p}>g_{w}$), unemployment increases after the shock and
only slowly decreases as wages adjust at a slower pace to pre-crisis levels,
leading to sluggish demand recovery. Consequently, firms do not see the need
to increase their production faster. In contrast, if bargaining power exceeds
market power ($g_{w}>g_{p}$), wages recover more quickly, resulting in an
earlier recovery of pre-crisis purchasing power, which helps to reduce
unemployment. This conclusion is in line with the result of section 7.1 on the
impact of Helicopter Money or, more generally, of Keynesian stimulus:
increasing the consumption budget of households is quite efficient at reviving
an ailing economy, but only if hyperinflation can be avoided.

## 9 Summary & Conclusions

In this paper, we have expanded the Mark-0 Agent-based Model to assess the
inflationary dynamics following the COVID pandemic and energy crisis of
2020-2023. Our results highlight the narrow path of monetary policy to balance
unemployment and inflation, as well as the benefits of joint monetary and
fiscal policy packages.

We extended the Mark-0 model used in Sharma et al. (2020) in two directions
(Section 2): First, a dynamically evolving trust in the Central Bank’s ability
to control inflation, such that expected inflation remains anchored to the
Central Bank’s target if trust is high, but persistent off-target inflation
realisations lead to a loss of trust in the Central Bank. Second, a simple
“exogenous” energy sector from which firms buy energy that allows us to
introduce an energy price, and slowly re-inject the energy profits into the
economy as dividend payouts. This slightly expanded Mark-0 model allows us to
investigate several policy channels (Section 3): (a) monetary policy via the
management of interest rates and expectations, (b) an Easy-Credit regulatory
policy, and fiscal policies such as (c) Helicopter Money and (d) a Windfall
Tax.

The Mark-0 steady state economy is then perturbed by three calibrated shocks
(Section 5): a consumption propensity drop (representing lockdowns), a
productivity drop (representing supply chains disruptions), and an energy
price shock (first a drop, representing reduced demand during lockdowns,
followed by a rise due do demand recovery exacerbated by the Russian invasion
of Ukraine). We show in Section 5 that the Easy-Credit policy, which was one
of the first emergency responses to the COVID pandemic, is able to alleviate
the impact of the shocks, but at the price of high and sustained inflation, as
observed in the data and predicted as early as June 2020 by Sharma et al.
(2020) within the Mark-0 framework.

We investigated whether monetary policy alone can control a surge of
inflation, and at what cost to unemployment (Section 6). Our results show that
if agents’ expectations remain fully anchored, inflation remains closer to the
Central Bank target, but at the price of a strong recession leading to a wave
of unemployment. Meanwhile, if agents’ expectations evolve dynamically,
inflation rises well above target during the shock leading to de-anchored
expectations. Monetary policy is then essentially ineffective in controlling
inflation while causing unemployment to increase in comparison to a baseline
case with no monetary policy at all. This is robust to variations in
parameters related to the channels of monetary policy. Our framework thus
supports the view that the efficacy of monetary policy comes at the cost of
unnecessarily high unemployment (Stiglitz & Regmi, 2022).

Within Mark-0, trust anchoring is thus a crucial determinant of the success of
the Central Bank inflation mitigation policy, far more important than the
direct economic impact of higher interest rates (noting that we do not model
in detail a financial sector). This resonates with Bernanke’s statement:
“Expectations matter so much that a Central Bank may be able to help make
policy more effective by working to shape those expectations”,252525see
Bernanke, B. (2013), “Communication and Monetary Policy”, speech at “National
Economists Club Annual Dinner”.
(https://www.federalreserve.gov/newsevents/speech/bernanke20131119a.pdf) and
points to the importance of narratives in shaping expectations (Shiller,
2020). In turn, fine-tuned fiscal policy combined with monetary policy can be
successful in controlling inflation while keeping unemployment around
acceptable levels (Section 7). However, this requires a high degree of
precision to be effective: too weak a fiscal stimulus is ineffective, and too
large a stimulus leads to further high inflation. Finally, in Section 8 we
show that the pricing power of firms and the bargaining power of workers play
a crucial role. Depending on their relation, the economy can experience a
runaway hyperinflation wage-price spiral. This points to an additional
difficulty in properly calibrating monetary and fiscal policy due to a variety
of possible tipping points or “dark corners” lurking around.

Overall, this study of the Mark-0 economy shows that (i) the economic recovery
can be very sluggish due to self-fulfilling expectations and other non-linear
feedback loops, (ii) there is a tension between inflation and unemployment
that is robustly captured by the model, (iii) in the Mark-0 economy, monetary
policy is effective at controlling inflation provided trust is anchored (but
not because of real economic effects of higher interest rates), and (iv)
fiscal policy can alleviate some of the negative unemployment effects of
inflation-focused monetary policy. While our study did not exhaust all
possible cases and parameters, we believe that it illustrates both the realism
and the richness of the Mark-0 economy. It offers a versatile tool with which
policy makers can easily play in order to forge their intuition about what may
happen if they turn this knob or add that policy measure. Indeed, almost all
possible narratives that emerged during the recent debate about post-COVID
inflation can be captured and reproduced by our model with proper choices of
parameters and shock specifications. Furthermore, the procedure proposed in
Section 8 can be used to rigorously establish which predictions of the model
are robust to the choice of parameters, most of which are difficult to
properly calibrate on data.

Our model is obviously incomplete and improvable on many counts. In
particular, (i) considering a dual mandate central bank as in Bouchaud et al.
(2018); Gualdi et al. (2017) to address negative unemployment side-effects,
(ii) a disaggregated household sector (e.g. distinguishing wage and rent
earners, and by accumulated wealth) to assess distributional implications of
policy, (iii) production networks and energy supply chains with differentiated
products and commodities (as in Dessertaine et al., 2022, for instance), which
introduces firm distribution effects and systemic risk, (iv) a financial
sector to more closely model interest rate pass-through, lending choices and
financial systemic risk, and (v) more detailed labour market structures to
account for the change in occupational structure post-covid and demographic
trends. However, we do believe, as we already argued in Gualdi et al. (2015),
that the Mark-0 model should already be part of the toolkit of Central Banks,
if only as an inspiring scenario generator, or “telescope for the mind”,
especially in times of great modelling uncertainty during which it is crucial
to be at least “roughly right” and avoid being blindsided by spurious Black
Swans.262626See for example Mark Buchanan, “This Economy Does Not Compute”,
New York Times, October 2008, Gualdi et al. (2015); King & Kay (2020);
Bouchaud (2021), or the recent Financial Times piece cited in the
introduction, https://www.ft.com/content/b972f5e3-4f03-4986-890d-5443878424ac

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##  Appendix A Appendices

### A.1 Table of Parameters

Table 1: Inactive Central Bank parameter set for Mark-0 Model (see Section 4.1) . Parameters, which change for the Reactive Central Bank and Anchored/Floating Trust scenarios, can be taken from sections 6.2 and 6.3 respectively and are indicated in the table with a star. All rates are given in monthly time scales. | Parameter description | Value
---|---|---
$R_{0}$ | Ratio of hiring-firing rate ($\eta^{+}/\eta^{-}$): | 2.0
$\Theta$ | Maximum credit supply available to firms : | 4.0
$\Gamma_{0}$ | Financial Fragility sensitivity: | 0.0
$\rho_{0}$ | Baseline interest rate: | 0.001*
$\alpha_{c}$ | Influence of deposit rates on consumption | 12
$\phi_{\pi}$ | Intensity of interest rate policy of Central Bank: | 0.0*
$\pi^{*}$ | Cental Bank inflation target: | 0.0*
$\tau^{T}$ | Inflation target parameter: | 0.0*
$g_{w}$ | Factor to adjust wages to inflation expectations: | 0.8
$g_{p}$ | Factor to adjust prices to inflation expectations: | 0.8
$y_{0}$ | Initial production: | 0.5
$\gamma_{p}$ | Parameter to set adjustment of prices: | 0.01
$\eta_{0}^{-}$ | Firing propensity: | 0.2
$\alpha_{\Gamma}$ | Influence of loans interest rate on hiring-firing policy: | 450
$G_{0}$ | Fraction of savings in consumption budget: | 0.5
$\phi$ | Revival probability per unit time: | 0.1
$\omega$ | Moving average parameter: | 0.2
$\delta$ | Dividend rate: | 0.02
$\delta_{e}$ | Fraction of energy sector’s equity redistributed: | 0.04
$g_{e}$ | Share of Energy Price share in GDP: | 0.0325
$\mu$ | Easy-Credit policy multiplier: | 1.3

### A.2 counterfactuals

Figure 10: Economic dashboard for the Inactive Central Bank scenario without
any policy and with a stronger COVID shock: The dynamics for a COVID shock
strength of $\kappa=1.0$ (blue), $\kappa=1.3$ (orange) and $\kappa=1.6$
(green) in the Inactive Central Bank scenario without any policy. Once the
COVID shock gets too strong, the economy collapses in the long run.

### A.3 Monetary Policy

Figure 11: Dynamics for the three shocks for the Inactive Central Bank
scenario: The dynamics for the three shocks, COVID only (blue), COVID and
Supply Chain shock (orange) and all shocks (green) for the Inactive Central
Bank scenario with Easy-Credit policy. The areas shaded in grey indicate the
duration of the three shocks: the COVID shock lasting until the end of the
dark grey area, the supply chain shock until the end of the grey area, and the
price shock until the end of the light grey area. Figure 12: Dynamics for the
three shocks for the Reactive Central Bank with Anchored Trust scenario: The
dynamics for the three shocks, COVID only (blue), COVID and Supply Chain shock
(orange) and all shocks (green) to the scenario with reactive Central Bank and
Anchored Trust of economic agents with an Easy-Credit policy. The areas shaded
in grey indicate the duration of the three shocks: the COVID shock lasting
until the end of the dark grey area, the supply chain shock until the end of
the grey area, and the price shock until the end of the light grey area.
Figure 13: Dynamics for the three shocks for the Reactive Central Bank with
Floating Trust scenario: The dynamics for the three shocks, COVID only (blue),
COVID and Supply Chain shock (orange) and all shocks (green) to the scenario
with reactive Central Bank and Floating Trust of economic agents with an Easy-
Credit policy. The areas shaded in grey indicate the duration of the three
shocks: the COVID shock lasting until the end of the dark grey area, the
supply chain shock until the end of the grey area, and the price shock until
the end of the light grey area. Figure 14: Dynamics for all scenarios and all
shocks: The full dashboard for Figure 6 All dynamics are with Easy-Credit
policy in the Inactive Central Bank scenario (blue), Reactive Central Bank
with Anchored trust scenario (orange) and Reactive Central Bank with Floating
Trust scenario (green). The areas shaded in grey indicate the duration of the
three shocks: the COVID shock lasting until the end of the dark grey area, the
supply chain shock until the end of the grey area, and the price shock until
the end of the light grey area.

### A.4 Monetary Policy with Stronger Central Bank Reaction

Figure 15: Economic dashboard for the stronger Reactive Central Bank with
Anchored Trust scenario and all shocks. The dynamics for a Central Bank
strength of $\phi_{\pi}=1.0$ (blue) and $\phi_{\pi}=2.0$ (orange). A stronger
central bank is not able to decrease inflation due to an external price shock,
but with stronger policies, the consumption of households reduces, which leads
to an increase in unemployment. Figure 16: Economic dashboard for the stronger
Reactive Central Bank with Floating Trust scenario and all shocks. The
dynamics for a Central Bank strength of $\phi_{\pi}=1.0$ (blue) and
$\phi_{\pi}=2.0$ (orange) with Floating Trust of economic agents. A stronger
Central bank can reduce excess inflation at the cost of increased
unemployment. The central bank must maintain a balance between falling
inflation and rising unemployment.

### A.5 Sensitivity of Monetary Policy to $\alpha_{c}$ and $\alpha_{\Gamma}$

Figure 17: Sensitivity of $\alpha_{c}$ for the Reactive Central Bank with
Anchored Trust scenario and all shocks. As already described in Gualdi et al.
(2017), larger $\alpha_{c}$ lead to a greater magnification of price trends
(cet. par.). This amplification, in turn, causes the fluctuations in
unemployment and inflation. Furthermore, larger $\alpha_{c}$ increases
consumption which reduces unemployment due to higher demand.  Figure 18:
Sensitivity of $\alpha_{\Gamma}$ for the Reactive Central Bank with Anchored
Trust scenario and all shocks. Larger values of $\alpha_{\Gamma}$ increases
the influence of financial fragility on the hiring/firing policy of firms
(cet. par.). This in turn leads to a larger downward adjustment of the
workforce which increases unemployment.

### A.6 Helicopter Money

Figure 19: Helicopter Money in the Inactive Central Bank scenario and all
shocks. Unemployment (left) and inflation (right) for a helicopter drop of
size $\kappa_{H}S$ one month after the price shock. Already with
$\kappa_{H}\geq 0.2$ unemployment is reduced almost to zero. A further
increase of Helicopter Money only increases inflation without reducing
unemployment more. Figure 20: Helicopter Money in the Reactive Central Bank
with Anchored Trust scenario. Unemployment (left) and inflation (right) for a
helicopter drop of size $\kappa_{H}S$ one month after the price shock. With
$\kappa_{H}\geq 0.3$ unemployment is reduced almost to zero. A further
increase of Helicopter Money only increases inflation without reducing
unemployment further. However, the Central Bank policy manages to keep
inflation under control. Figure 21: Economic dashboard for Helicopter Money in
the Inactive Central Bank scenario and all shocks. Full dynamics for scenario
without Helicopter Money (blue), for Helicopter Money with $\kappa_{H}=0.2$
(orange) and $\kappa_{H}=0.6$ (green) one month after the price shock and with
Easy-Credit policy. Figure 22: Economic dashboard for Helicopter Money in the
Reactive Central Bank with Anchored Trust scenario and all shocks. Full
dynamics for scenario without Helicopter Money (blue), for Helicopter Money
with $\kappa_{H}=0.3$ (orange) and $\kappa_{H}=0.6$ (green) one month after
the price shock and with Easy-Credit policy. Figure 23: Economic dashboard for
Helicopter Money in the Reactive Central Bank with Floating Trust and all
shocks. Full dynamics for scenario without Helicopter Money (blue), for
Helicopter Money with $\kappa_{H}=0.2$ (orange) and $\kappa_{H}=0.6$ (green)
one month after the price shock and with Easy-Credit policy.

### A.7 Windfall Tax

Figure 24: Windfall Tax in the Inactive Central Bank scenario. Unemployment
(left) and inflation (right) for Windfall Tax of $\delta_{e}+\Delta\delta_{e}$
one year before the end of the price shock with a duration of two years. With
$\Delta\delta_{e}\approx 2\%$ unemployment is reduced strongly. A further
increase of tax does only increase unemployment again. For larger dividends,
inflation increases because increased demand due to increased savings. Figure
25: Windfall Tax in the Reactive CB with Anchored Trust scenario. Unemployment
(left) and inflation (right) for Windfall Tax of $\Delta\delta_{e}$ one year
before the end of the price shock with a duration of two years. With
$\Delta\delta_{e}\approx 6\%$ unemployment is reduced strongly. A further
increase of $\Delta\delta_{e}$ leads to a significant resurgence of
unemployment.  Figure 26: Economic dashboard for Windfall Tax in the Inactive
Central Bank scenario, all shocks. Full dynamics for the Inactive Central Bank
scenario without Windfall Tax (blue), for tax of $\Delta\delta_{e}=2\%$
(orange) and $\Delta\delta_{e}=16\%$ (green) one year before the end of the
price shock with a duration of two years. Figure 27: Economic dashboard for
Windfall Tax in the Reactive Central Bank with Anchored Trust scenario and all
shocks. Full dynamics for scenario without Windfall Tax (blue), for tax of
$\Delta\delta_{e}=6\%$ (orange) and $\Delta\delta_{e}=16\%$ (green) one year
before the end of the price shock with a duration of two years. Figure 28:
Economic dashboard for Windfall Tax in the Reactive Central Bank with Floating
Trust scenario and all shocks. Full dynamics for scenario without Windfall Tax
(blue), for tax of $\Delta\delta_{e}=3\%$ (orange) and $\Delta\delta_{e}=16\%$
(green) one year before the end of the price shock with a duration of two
years.

### A.8 Sloppiness

For the sloppiness analysis we follow the same approach as in Naumann-Woleske
et al. (2023). We define a mean-square loss error which is dependent on the
set of parameters $\Phi$ and a small deviation $\delta$. The loss function can
be written as

$\mathcal{L}(\Phi,\delta)=\frac{1}{2SKT}\sum_{s}\sum_{k}\sum_{t}\left(\frac{y_{s,k,t}(\Phi+\delta)-y_{s,k,t}(\Phi)}{\left\lVert
y_{s,k}(\Phi)\right\rVert}\right)^{2},$ (24)

where $y_{s,k,t}(\Phi)$ is the realisation of output variable
$k\in\\{1,\dots,K\\}$ at time $t\in\\{1,\dots,T\\}$ for random realisation
$s\in\\{1,\dots,S\\}$. In this case, we use the output variables inflation and
unemployment with 50 random realisations and we average over a period of 11
years that stars 1 year before the COVID shock in February 2020.

The Hessian matrix for the mean-squared loss at point $\Phi$ is then defined
as

$H_{i,j}^{\mathcal{L}}(\Phi):=\left.\frac{d^{2}\mathcal{L}(\Phi,\delta)}{d\log\Phi_{i}d\log\Phi_{j}}\right|_{\delta=0},$
(25)

$H_{i,j}^{\mathcal{L}}(\Phi)=\frac{1}{SKT}\sum_{s}\sum_{k}\sum_{t}\frac{1}{\left\lVert
y_{s,k,t}(\Phi)\right\rVert^{2}}\frac{dy_{s,k,t}(\Phi)}{d\log\Phi_{i}}\frac{dy_{s,k,t}(\Phi)}{d\log\Phi_{j}},$
(26)

Since the parameters in general have different order of magnitudes, we take
the derivative with respect to the log parameters to only consider relative
parameter changes. Note, that in Eq. 26 the second derivative term vanishes as
we evaluate $H_{i,j}^{\mathcal{L}}(\Phi)$ at $\delta=0$.

The Hessian matrix can be decomposed into its eigenvalues and eigenvectors
where the eigenvalues indicates the relative importance of the corresponding
eigenvectors that represent the linear combination of parameters in parameter
space. We are interested in the stiff directions, so the eigenvectors with
largest eigenvalues that bring the biggest change to the dynamics of the
system.

The eigenvalue spectrum spans several decades (Figure 29), which indicates the
sloppiness of the model at that set of parameters. Based on the spectrum, we
select the four largest eigenvectors to consider for our analysis (see Figures
30, 31, 32).

Figure 29: Eigenvalue Spectrum for three scenarios and all shocks. Sloppiness
parameter $\epsilon=0.01$, $n_{seeds}=50$ target variable inflation (red)
unemployment (blue), (a,b) Inactive Central Bank, (c,d) Reactive Central Bank
with Anchored Trust, (e,f) Reactive Central Bank with Floating Trust. Figure
30: Eigenvectors of the Hessian matrix for the Inactive Central Bank Scenario
and all shocks. The length of the eigenvectors is normed to 1 and the colors
indicate the target variables, unemployment (blue) and inflation (red).
Figure 31: Eigenvectors of the Hessian matrix for the Reactive Central Bank
with Anchored Trust scenario and all shocks. The length of the eigenvectors is
normed to 1 and the colors indicate the target variables, unemployment (blue)
and inflation (red).  Figure 32: Eigenvectors of the Hessian matrix for the
Reactive Central Bank with Floating Trust scenario and all shocks. The length
of the eigenvectors is normed to 1 and the colors indicate the target
variables, unemployment (blue) and inflation (red).

### A.9 Sensitivity to the Indexation Parameters $g_{p}$ and $g_{w}$

Figure 33: Sensitivity of $g_{p}=g_{w}=const$ for the Reactive Central Bank
with Anchored Trust scenario and all shocks. Increasing $g_{p}=g_{w}$
increases slightly inflation (cet. par.), causing slightly higher interest
rates which leads to a reduction of consumption budget that reduces demand and
therefore increases unemployment. Figure 34: Sensitivity of
$\frac{g_{p}+g_{w}}{2}=const$ for the reactive Central Bank with Anchored
Trust scenario and all shocks. (purple) When $g_{w}>g_{w}$, there is a higher
bargaining power of worker which increases wages (cet. par.), therefore there
is a higher consumption budget and higher demand which decreases unemployment.
(green) For larger market power ($g_{p}>g_{w}$), there is the reverse effect
of decreased wages causing a decrease in consumption budget and demand which
therefore increases unemployment.