# Robust-less-fragile: Tackling Systemic Risk and Financial Contagion in a Macro Agent-Based Model

Gianluca Pallante, Mattia Guerini, Mauro Napoletano, Andrea Roventini

<figure><img src="image_1.png" /></figure>

## To cite this version:

Gianluca Pallante, Mattia Guerini, Mauro Napoletano, Andrea Roventini. Robust-less-fragile: Tackling Systemic Risk and Financial Contagion in a Macro Agent-Based Model. 2024. hal-04576530

HAL Id: hal-04576530

https://sciencespo.hal.science/hal-04576530

Preprint submitted on 15 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>

Distributed under a Creative Commons Attribution - ShareAlike 4.0 International License
# Robust-less-fragile: Tackling Systemic Risk and Financial Contagion in a Macro Agent-Based Model

Gianluca Pallante, Mattia Guerini, Mauro Napoletano, Andrea Roventini

<figure><img src="image_1.png" /></figure>

## To cite this version:

Gianluca Pallante, Mattia Guerini, Mauro Napoletano, Andrea Roventini. Robust-less-fragile: Tackling Systemic Risk and Financial Contagion in a Macro Agent-Based Model. 2024. hal-04576530

HAL Id: hal-04576530

https://sciencespo.hal.science/hal-04576530v1

Preprint submitted on 15 May 2024

HAL is a multi-disciplinary open access
archive for the deposit and dissemination of sci-
entific research documents, whether they are pub-
lished or not. The documents may come from
teaching and research institutions in France or
abroad, or from public or private research centers.

L'archive ouverte pluridisciplinaire HAL, est
destinée au dépôt et à la diffusion de documents
scientifiques de niveau recherche, publiés ou non,
émanant des établissements d'enseignement et de
recherche français ou étrangers, des laboratoires
publics ou privés.

<figure><img src="image_2.png" /></figure>
ROBUST-LESS-FRAGILE: TACKLING
SYSTEMIC RISK AND FINANCIAL CONTAGION
IN A MACRO AGENT-BASED MODEL

Gianluca Pallente

Mattia Guerini

Mauro Napoletano

Andrea Roventini
ROBUST-LESS-FRAGILE: TACKLING
SYSTEMIC RISK AND FINANCIAL CONTAGION
IN A MACRO AGENT-BASED MODEL

Gianluca Pallente

Mattia Guerini

Mauro Napoletano

Andrea Roventini
EDITORIAL BOARD

Chair: **Xavier Ragot** (Sciences Po, OFCE)

**Members:** **Jérôme Creel** (Sciences Po, OFCE), **Eric Heyer** (Sciences Po, OFCE), **Sarah Guillou** (Sciences Po, OFCE), **Xavier Timbeau** (Sciences Po, OFCE)

CONTACT US

OFCE

10 place de Catalogne | 75014 Paris | France
Tél. +33 1 44 18 54 24
www.ofce.fr

WORKING PAPER CITATION

This Working Paper:

Gianluca Pallante, Mattia Guerini, Mauro Napoletano and Andrea Roventini,

Robust-less-fragile: Tackling Systemic Risk and Financial Contagion in a Macro Agent-Based Model
*Sciences Po OFCE Working Paper*, n° 06/2024.

Downloaded from URL: [www.ofce.sciences-po.fr/pdf/dtravail/WP2024-06.pdf](www.ofce.sciences-po.fr/pdf/dtravail/WP2024-06.pdf)

DOI - ISSN
EDITORIAL BOARD

Chair: **Xavier Ragot** (Sciences Po, OFCE)

**Members:** **Jérôme Creel** (Sciences Po, OFCE), **Eric Heyer** (Sciences Po, OFCE), **Sarah Guillou** (Sciences Po, OFCE), **Xavier Timbeau** (Sciences Po, OFCE)

CONTACT US

OFCE

10 place de Catalogne | 75014 Paris | France
Tél. +33 1 44 18 54 24
www.ofce.fr

WORKING PAPER CITATION

This Working Paper:

Gianluca Pallante, Mattia Guerini, Mauro Napoletano and Andrea Roventini,

Robust-less-fragile: Tackling Systemic Risk and Financial Contagion in a Macro Agent-Based Model
*Sciences Po OFCE Working Paper*, n° 06/2024.

Downloaded from URL: [www.ofce.sciences-po.fr/pdf/dtravail/WP2024-06.pdf](www.ofce.sciences-po.fr/pdf/dtravail/WP2024-06.pdf)

DOI - ISSN
## ABOUT THE AUTHOR

Gianluca Pallante, Institute of Economics and l'EmbeDS, Scuola Superiore Sant'Anna, Pisa, Italy,

Email Address: gianluca.pallante@santannapisa.it

Mattia Guerini, Université Côte d'Azur, CNRS, GREDEG, France

Email Address: mattia.guerini@unibs.it

Mauro Napoletano, Université Côte d'Azur, CNRS, GREDEG, and OFCE-Sciences Po, France

Email Address: mauro.napoletano@sciencespo.fr

Andrea Roventini, Institute of Economics and l'EmbeDS, Scuola Superiore Sant'Anna, Pisa, Italy,

Email Address: andrea.roventini@santannapisa.it

## ABSTRACT

We extend the Schumpeter meeting Keynes (K+S; see Dosi et al., 2010, 2013, 2015) to model the emergence and the dynamics of an interbank network in the money market. The extended model allows banks to directly exchange funds, while evaluating their interbank positions using a network-based clearing mechanism (NEVA, see Barucca et al., 2020). These novel adds on, allow us to better measure financial contagion and systemic risk events in the model and to study the possible interactions between micro-prudential and macro-prudential policies. We find that the model can replicate new stylized facts concerning the topology of the interbank network, as well as the dynamics of individual banks' balance sheets. Policy results suggest that the economic system at large can benefit from the introduction of a micro-prudential regulation that takes into account the inter-bank network relationships. Such a policy decreases the incidence of systemic risk events and the bankruptcies of financial institutions. Moreover, a trade-off between financial stability and macroeconomic performance does not emerge in a two-pillar regulatory framework grounded on i) a Basel III macro-prudential regulation and ii) a NEVA-based micro-prudential policy. Indeed, the NEVA allows the economic system to achieve financial stability without overly stringent capital requirements.

## KEYWORDS

Financial contagion, Systemic risk, Micro-prudential policy, Macro-prudential policy, Macroeconomic stability, Agent-based computational economics.

## JEL

C63, E32, E42, E58, G18.

## RESUME

Nous étendons le modèle multi-agents (ABM) Schumpeter Keynes (K+S ; voir Dosi et al., 2010, 2013, 2015) pour étudier l'émergence et la dynamique d'un réseau interbancaire sur le marché monétaire. Dans le modèle les banques peuvent échanger directement des fonds sur le marché interbancaire, tout en évaluant leurs positions interbancaires à l'aide d'un mécanisme d'évaluation des actifs au sein d'un réseau financier (NEVA, voir Barucca et al., 2020). Ces nouveaux ajouts nous permettent de mieux mesurer la contagion financière et les événements de risque systémique dans le modèle et d'étudier les interactions possibles entre les politiques micro-prudentielles et macro-prudentielles. Nous constatons que le modèle peut reproduire de nouveaux faits stylisés concernant la topologie du réseau interbancaire, ainsi que la dynamique des bilans des banques individuelles. Les résultats sur les effets des politiques micro- et macro-prudentielles suggèrent que le système économique dans son ensemble peut bénéficier de l'introduction d'une réglementation microprudentielle qui prend en compte les relations dans le réseau interbancaire. Une telle politique diminue l'incidence des événements liés au risque systémique et des faillites des institutions financières. De plus, un arbitrage entre stabilité financière et performance macroéconomique n'apparaît pas dans un cadre réglementaire à deux piliers fondé sur i) une réglementation macroprudentielle de Bâle III et ii) une politique microprudentielle basée sur la NEVA. En effet, la NEVA permet au système économique d'atteindre la stabilité financière sans introduire des limites trop strictes en termes d'adéquation des fonds propres des banques.

## MOTS CLES

Contagion financière, Risque systémique, Politique microprudentielle, Politique macroprudentielle, Stabilité macroéconomique, Modélisation multi-agents (ABM).
## ABOUT THE AUTHOR

Gianluca Pallante, Institute of Economics and l'EmbeDS, Scuola Superiore Sant'Anna, Pisa, Italy,

Email Address: gianluca.pallante@santannapisa.it

Mattia Guerini, Université Côte d'Azur, CNRS, GREDEG, France

Email Address: mattia.guerini@unibs.it

Mauro Napoletano, Université Côte d'Azur, CNRS, GREDEG, and OFCE-Sciences Po, France

Email Address: mauro.napoletano@sciencespo.fr

Andrea Roventini, Institute of Economics and l'EmbeDS, Scuola Superiore Sant'Anna, Pisa, Italy,

Email Address: andrea.roventini@santannapisa.it

## ABSTRACT

We extend the Schumpeter meeting Keynes (K+S; see Dosi et al., 2010, 2013, 2015) to model the emergence and the dynamics of an interbank network in the money market. The extended model allows banks to directly exchange funds, while evaluating their interbank positions using a network-based clearing mechanism (NEVA, see Barucca et al., 2020). These novel adds on, allow us to better measure financial contagion and systemic risk events in the model and to study the possible interactions between micro-prudential and macro-prudential policies. We find that the model can replicate new stylized facts concerning the topology of the interbank network, as well as the dynamics of individual banks' balance sheets. Policy results suggest that the economic system at large can benefit from the introduction of a micro-prudential regulation that takes into account the inter-bank network relationships. Such a policy decreases the incidence of systemic risk events and the bankruptcies of financial institutions. Moreover, a trade-off between financial stability and macroeconomic performance does not emerge in a two-pillar regulatory framework grounded on i) a Basel III macro-prudential regulation and ii) a NEVA-based micro-prudential policy. Indeed, the NEVA allows the economic system to achieve financial stability without overly stringent capital requirements.

## KEYWORDS

Financial contagion, Systemic risk, Micro-prudential policy, Macro-prudential policy, Macroeconomic stability, Agent-based computational economics.

## JEL

C63, E32, E42, E58, G18.

## RESUME

Nous étendons le modèle multi-agents (ABM) Schumpeter Keynes (K+S ; voir Dosi et al., 2010, 2013, 2015) pour étudier l'émergence et la dynamique d'un réseau interbancaire sur le marché monétaire. Dans le modèle les banques peuvent échanger directement des fonds sur le marché interbancaire, tout en évaluant leurs positions interbancaires à l'aide d'un mécanisme d'évaluation des actifs au sein d'un réseau financier (NEVA, voir Barucca et al., 2020). Ces nouveaux ajouts nous permettent de mieux mesurer la contagion financière et les événements de risque systémique dans le modèle et d'étudier les interactions possibles entre les politiques micro-prudentielles et macro-prudentielles. Nous constatons que le modèle peut reproduire de nouveaux faits stylisés concernant la topologie du réseau interbancaire, ainsi que la dynamique des bilans des banques individuelles. Les résultats sur les effets des politiques micro- et macro-prudentielles suggèrent que le système économique dans son ensemble peut bénéficier de l'introduction d'une réglementation microprudentielle qui prend en compte les relations dans le réseau interbancaire. Une telle politique diminue l'incidence des événements liés au risque systémique et des faillites des institutions financières. De plus, un arbitrage entre stabilité financière et performance macroéconomique n'apparaît pas dans un cadre réglementaire à deux piliers fondé sur i) une réglementation macroprudentielle de Bâle III et ii) une politique microprudentielle basée sur la NEVA. En effet, la NEVA permet au système économique d'atteindre la stabilité financière sans introduire des limites trop strictes en termes d'adéquation des fonds propres des banques.

## MOTS CLES

Contagion financière, Risque systémique, Politique microprudentielle, Politique macroprudentielle, Stabilité macroéconomique, Modélisation multi-agents (ABM).
<figure><img src="image_3.png" /></figure>
<figure><img src="image_3.png" /></figure>
# Robust-less-fragile: Tackling Systemic Risk and Financial Contagion in a Macro Agent-Based Model

Gianluca Pallante*<fnref n="1" />, Mattia Guerini<fnref n="3" />,<fnref n="4" />,<fnref n="5" />,<fnref n="2" />, Mauro Napoletano<fnref n="5" />,<fnref n="6" />,<fnref n="2" />, and Andrea Roventini<fnref n="1" />

## Abstract

We extend the *Schumpeter meeting Keynes* (K+S; see Dosi et al., 2010, 2013, 2015) to model the emergence and the dynamics of an interbank network in the money market. The extended model allows banks to directly exchange funds, while evaluating their interbank positions using a network-based clearing mechanism (NEVA, see Barucca et al., 2020). These novel adds on, allow us to better measure financial contagion and systemic risk events in the model and to study the possible interactions between micro-prudential and macro-prudential policies. We find that the model can replicate new stylized facts concerning the topology of the interbank network, as well as the dynamics of individual banks' balance sheets. Policy results suggest that the economic system at large can benefit from the introduction of a micro-prudential regulation that takes into account the interbank network relationships. Such a policy decreases the incidence of systemic risk events and the bankruptcies of financial institutions. Moreover, a trade-off between financial stability and macroeconomic performance does not emerge in a two-pillar regulatory framework grounded on i) a Basel III macro-prudential regulation and ii) a NEVA-based micro-prudential policy. Indeed, the NEVA allows the economic system to achieve financial stability without overly stringent capital requirements.

**Keywords:** Financial contagion, Systemic risk, Micro-prudential policy, Macro-prudential policy, Macroeconomic stability, Agent-based computational economics

**JEL Codes:** C63, E32, E42, E58, G18

<footnote n="1">¹Institute of Economics and l'EmbeDS, Scuola Superiore Sant'Anna, Pisa, Italy</footnote>

<footnote n="2">²Institute of Economics, Scuola Superiore Sant'Anna, Pisa, Italy</footnote>

<footnote n="3">³Department of Economics and Management, University of Brescia, Italy</footnote>

<footnote n="4">⁴Fondazione ENI Enrico Mattei, Milano, Italy</footnote>

<footnote n="5">⁵Université Côte d'Azur, CNRS, GREDEG, France</footnote>

<footnote n="6">⁶Sciences Po, OFCE, France</footnote>

<footnote>*Corresponding author: gianluca.pallante@santannapisa.it - Institute of Economics and l'EMbeDS, Scuola Superiore Sant'Anna, piazza Martiri della Libertà 33, 56127, Pisa (IT).</footnote>

<footnote>**Acknowledgements:** We thank, among others, Paolo Barucca, Alessandro Caiani, Matteo Coronese, Domenico delli Gatti, Giovanni Dosi, Francesco Lamperti, Antoine Mandel, Mattia Montagna, Lilit Popoyan, Isabelle Salle, Willi Semmler, Tania Treibich and Arzu Uluc. We are also grateful to participants of the 2022 ABM4Policy Workshop, WEHIA 2021 and EEA 2020 conferences. The authors acknowledge the support by the European Union's Horizon 2020 research and innovation program under grant agreement No. 822781 - GROWINPRO.</footnote>
# Robust-less-fragile: Tackling Systemic Risk and Financial Contagion in a Macro Agent-Based Model

Gianluca Pallante*¹, Mattia Guerini³,⁴,⁵,², Mauro Napoletano⁵,⁶,², and Andrea Roventini¹

¹Institute of Economics and l'EmbeDS, Scuola Superiore Sant'Anna, Pisa, Italy

²Institute of Economics, Scuola Superiore Sant'Anna, Pisa, Italy

³Department of Economics and Management, University of Brescia, Italy

⁴Fondazione ENI Enrico Mattei, Milano, Italy

⁵Université Côte d'Azur, CNRS, GREDEG, France

⁶Sciences Po, OFCE, France

## Abstract

We extend the *Schumpeter meeting Keynes* (K+S; see Dosi et al., 2010, 2013, 2015) to model the emergence and the dynamics of an interbank network in the money market. The extended model allows banks to directly exchange funds, while evaluating their interbank positions using a network-based clearing mechanism (NEVA, see Barucca et al., 2020). These novel adds on, allow us to better measure financial contagion and systemic risk events in the model and to study the possible interactions between micro-prudential and macro-prudential policies. We find that the model can replicate new stylized facts concerning the topology of the interbank network, as well as the dynamics of individual banks' balance sheets. Policy results suggest that the economic system at large can benefit from the introduction of a micro-prudential regulation that takes into account the interbank network relationships. Such a policy decreases the incidence of systemic risk events and the bankruptcies of financial institutions. Moreover, a trade-off between financial stability and macroeconomic performance does not emerge in a two-pillar regulatory framework grounded on i) a Basel III macro-prudential regulation and ii) a NEVA-based micro-prudential policy. Indeed, the NEVA allows the economic system to achieve financial stability without overly stringent capital requirements.

**Keywords:** Financial contagion, Systemic risk, Micro-prudential policy, Macro-prudential policy, Macroeconomic stability, Agent-based computational economics

**JEL Codes:** C63, E32, E42, E58, G18

<footnote>*Corresponding author: gianluca.pallante@santannapisa.it - Institute of Economics and l'EMbeDS, Scuola Superiore Sant'Anna, piazza Martiri della Libertà 33, 56127, Pisa (IT).</footnote>

<footnote>**Acknowledgements:** We thank, among others, Paolo Barucca, Alessandro Caiani, Matteo Coronese, Domenico delli Gatti, Giovanni Dosi, Francesco Lamperti, Antoine Mandel, Mattia Montagna, Lilit Popoyan, Isabelle Salle, Willi Semmler, Tania Treibich and Arzu Uluc. We are also grateful to participants of the 2022 ABM4Policy Workshop, WEHIA 2021 and EEA 2020 conferences. The authors acknowledge the support by the European Union's Horizon 2020 research and innovation program under grant agreement No. 822781 - GROWINPRO.</footnote>
# 1 Introduction

In this paper, we extend the *Schumpeter meeting Keynes* (K+S) agent-based model (Dosi et al., 2010, 2013, 2015) to account for the endogenous formation of an interbank market and to study the emergence of systemic risk, financial crises, and the possible interactions between micro-prudential and macro-prudential policies.

The complex interactions in financial networks among economic agents have a fundamental role in the building up of systemic risk and the emergence of financial crises. Indeed, *network interconnectedness* – which was substantially underestimated before 2008 – has amplified the negative effects of the sub-prime mortgage bubble, paving the way to the Great Recession (Haldane, 2012; Battiston et al., 2016; Dosi and Roventini, 2019a; Chen, 2022). Understanding the relationship between interconnectedness, financial system stability, and macroeconomic resilience remains a paramount area of interest in the research agenda. However, a consensus has not yet been consolidated regarding the mechanisms, as well as the policy measures required to address the prevailing challenges arising from asymmetric information, moral hazard, and neglected network externalities inherent in interbank markets (Altinoglu and Stiglitz, 2023).

Relatedly, the interplay between different micro-prudential and macro-prudential policies in limiting the emergence of financial crises and taming their negative effects is still not clear. An increasingly connected financial system may favor risk diversification and so resilience to micro shocks, but it can also turn out to be more exposed to shocks’ diffusion (Battiston et al., 2012; Glasserman and Young, 2016). According to this view, micro- and macro-prudential policies can affect this trade-off over three different channels.

The first one concerns shocks arising from and propagating in financial markets. A typical case is the so-called balance sheet contagion, which arises when a shock to one bank's balance sheet negatively affects those of all the other financial institutions that hold the assets of such a bank. This might even trigger an avalanche of losses among interbank counterparts (Kiyotaki and Moore, 2002; Gai et al., 2011; Battiston et al., 2012; Luu et al., 2021). The second channel concerns shocks that originate in the financial sector, but in turn affect the real one. Indeed, shocks to the leverage of a commercial bank affect its lending decisions and increase the probability for non-financial corporations to become credit-rationed (Adrian and Shin, 2008; Brunnermeier and Pedersen, 2009; Laeven and Valencia, 2012; Gross et al., 2018). The third channel concerns shocks born in the real economy that hit the financial industry. This might occur when competitive pressures or demand shocks induce the default of a large firm to which a bank is heavily exposed. The default can lead to non-performing loans and ultimately erode the lenders' equity, which in turn become riskier for all its counterparts over the financial system (Kiyotaki and Moore, 1997; Boissay, 2006; Luu and Lux, 2019; Popoyan et al., 2020).

Empirical works have tried to estimate the relevance of these effects. However, a full identification of each of those forces is extremely challenging because they are often connected in complex and non-linear ways. Agent-based models can be employed to study these issues, as they allow one to study and model direct interaction between heterogeneous agents (Farmer and Foley, 2009; Fagiolo and Roventini, 2017; Dawid and Delli Gatti, 2018; Dosi and Roventini, 2019b). An increasing number of small-scale agent-based models have studied the interaction between contagion effects and systemic risk (Georg, 2013; Montagna and Kok, 2016; Reale, 2024). However, the interbank networks' properties have often been studied in isolation, either empirically (e.g., Liu et al., 2020) or by means
# 1 Introduction

In this paper, we extend the *Schumpeter meeting Keynes* (K+S) agent-based model (Dosi et al., 2010, 2013, 2015) to account for the endogenous formation of an interbank market and to study the emergence of systemic risk, financial crises, and the possible interactions between micro-prudential and macro-prudential policies.

The complex interactions in financial networks among economic agents have a fundamental role in the building up of systemic risk and the emergence of financial crises. Indeed, *network interconnectedness* – which was substantially underestimated before 2008 – has amplified the negative effects of the sub-prime mortgage bubble, paving the way to the Great Recession (Haldane, 2012; Battiston et al., 2016; Dosi and Roventini, 2019a; Chen, 2022). Understanding the relationship between interconnectedness, financial system stability, and macroeconomic resilience remains a paramount area of interest in the research agenda. However, a consensus has not yet been consolidated regarding the mechanisms, as well as the policy measures required to address the prevailing challenges arising from asymmetric information, moral hazard, and neglected network externalities inherent in interbank markets (Altinoglu and Stiglitz, 2023).

Relatedly, the interplay between different micro-prudential and macro-prudential policies in limiting the emergence of financial crises and taming their negative effects is still not clear. An increasingly connected financial system may favor risk diversification and so resilience to micro shocks, but it can also turn out to be more exposed to shocks’ diffusion (Battiston et al., 2012; Glasserman and Young, 2016). According to this view, micro- and macro-prudential policies can affect this trade-off over three different channels.

The first one concerns shocks arising from and propagating in financial markets. A typical case is the so-called balance sheet contagion, which arises when a shock to one bank's balance sheet negatively affects those of all the other financial institutions that hold the assets of such a bank. This might even trigger an avalanche of losses among interbank counterparts (Kiyotaki and Moore, 2002; Gai et al., 2011; Battiston et al., 2012; Luu et al., 2021). The second channel concerns shocks that originate in the financial sector, but in turn affect the real one. Indeed, shocks to the leverage of a commercial bank affect its lending decisions and increase the probability for non-financial corporations to become credit-rationed (Adrian and Shin, 2008; Brunnermeier and Pedersen, 2009; Laeven and Valencia, 2012; Gross et al., 2018). The third channel concerns shocks born in the real economy that hit the financial industry. This might occur when competitive pressures or demand shocks induce the default of a large firm to which a bank is heavily exposed. The default can lead to non-performing loans and ultimately erode the lenders' equity, which in turn become riskier for all its counterparts over the financial system (Kiyotaki and Moore, 1997; Boissay, 2006; Luu and Lux, 2019; Popoyan et al., 2020).

Empirical works have tried to estimate the relevance of these effects. However, a full identification of each of those forces is extremely challenging because they are often connected in complex and non-linear ways. Agent-based models can be employed to study these issues, as they allow one to study and model direct interaction between heterogeneous agents (Farmer and Foley, 2009; Fagiolo and Roventini, 2017; Dawid and Delli Gatti, 2018; Dosi and Roventini, 2019b). An increasing number of small-scale agent-based models have studied the interaction between contagion effects and systemic risk (Georg, 2013; Montagna and Kok, 2016; Reale, 2024). However, the interbank networks' properties have often been studied in isolation, either empirically (e.g., Liu et al., 2020) or by means
of simple models of statistical mechanics (e.g., Lux, 2015). On the other side, most of the agent-based
macroeconomic models abstract from an interbank market. Notable exceptions are the agent-based
models by Delli Gatti et al. (2010), Tedeschi et al. (2012), Popoyan et al. (2017), Gurgone et al. (2018),
Popoyan et al. (2020), Catullo et al. (2021) in which interbank networks are nested into larger agent-
based models to draw and analyze the macroeconomics consequences of systemic risk. At the same
time, these models typically focus on business cycles frequencies and they lack endogenous dynamics
of innovation and technology diffusion, which would allows one to study also the possible long-run
consequences of financial crises.

We contribute to these streams of literature by including an interbank network in the medium-
scale K+S agent-based model (Dosi et al., 2010, 2013, 2015) which can be used to jointly analyze short-
and long-run macroeconomic dynamics as it generates both endogenous growth and business cycles
punctuated by financial crises as emergent properties. In particular, to account for the contagion
effects and systemic risk, we exploit the network of connections between banks embedded in the K+S
model assuming that the payment flows among different firms must be managed by the commercial
banks. This allows us to create a multilayered network where the shocks can originate either in the
real or the financial sector, and they can propagate both within and between networks. Furthermore,
we endow banks in the model with a common network-based micro-prudential tool called Network
Valuation (NEVA, see Barucca et al., 2020). Generalizing the Eisenberg and Noe (2001) framework,
the NEVA accounts for financial interconnectedness and it ensures systemic risk minimization with
minimal assumptions on banks' behavior and rationality.

To the best of our knowledge, this is the first medium-scale macroeconomic agent-based model
attempting to study the macro-financial effects stemming from the common adoption of a micro-
prudential tool (NEVA) that evaluate banks' interbank exposures in order to minimize systemic risk.
Moreover, the extended K+S model enables one to analyze, in a unified framework, the interactions
between micro- and macro-prudential policies and their joint impact on financial stability and eco-
nomic dynamics. This is particularly relevant because, even if these two policies are often modeled as
separate tools, they inevitably share objectives, transmission mechanisms (Altunbas et al., 2018) and
possible complementarities (Osinski et al., 2013), which can be exploited by policy-makers.

Simulation results show that the model can account for a rich list of stylized fact concerning the
network structure of the interbank market (sizeable volume of interbank likanges, dissassortativity,
vertex centrality and bank size relationship) and the co-movements of macro-financial variables.
Moreover, we find that the economic system at large can benefit from the introduction of a micro-
prudential regulation that takes into account the interbank network relationships. Indeed, the use
of the NEVA framework decreases the incidence of systemic risk events and of bankruptcy episodes
of financial institutions, notwithstanding the tightness of mandatory capital requirements. In addi-
tion, a trade-off between financial stability and macroeconomic performance does not emerge in a
two-pillar regulatory framework grounded on a Basel III macro-prudential regulation and a NEVA
micro-prudential policy. Indeed, the NEVA lessens the pressure on credit supply and in turns on
macroeconomic performance resulting from a too stringent macro-prudential regulation. In other
words, the NEVA allows the economic system to achieve financial and macroeconomic stability with-
out relying on overly stringent capital requirements.

The rest of the paper is structured as follows. Section 2 describes the model and it focuses on the
of simple models of statistical mechanics (e.g., Lux, 2015). On the other side, most of the agent-based
macroeconomic models abstract from an interbank market. Notable exceptions are the agent-based
models by Delli Gatti et al. (2010), Tedeschi et al. (2012), Popoyan et al. (2017), Gurgone et al. (2018),
Popoyan et al. (2020), Catullo et al. (2021) in which interbank networks are nested into larger agent-
based models to draw and analyze the macroeconomics consequences of systemic risk. At the same
time, these models typically focus on business cycles frequencies and they lack endogenous dynamics
of innovation and technology diffusion, which would allows one to study also the possible long-run
consequences of financial crises.

We contribute to these streams of literature by including an interbank network in the medium-
scale K+S agent-based model (Dosi et al., 2010, 2013, 2015) which can be used to jointly analyze short-
and long-run macroeconomic dynamics as it generates both endogenous growth and business cycles
punctuated by financial crises as emergent properties. In particular, to account for the contagion
effects and systemic risk, we exploit the network of connections between banks embedded in the K+S
model assuming that the payment flows among different firms must be managed by the commercial
banks. This allows us to create a multilayered network where the shocks can originate either in the
real or the financial sector, and they can propagate both within and between networks. Furthermore,
we endow banks in the model with a common network-based micro-prudential tool called Network
Valuation (NEVA, see Barucca et al., 2020). Generalizing the Eisenberg and Noe (2001) framework,
the NEVA accounts for financial interconnectedness and it ensures systemic risk minimization with
minimal assumptions on banks' behavior and rationality.

To the best of our knowledge, this is the first medium-scale macroeconomic agent-based model
attempting to study the macro-financial effects stemming from the common adoption of a micro-
prudential tool (NEVA) that evaluate banks' interbank exposures in order to minimize systemic risk.
Moreover, the extended K+S model enables one to analyze, in a unified framework, the interactions
between micro- and macro-prudential policies and their joint impact on financial stability and eco-
nomic dynamics. This is particularly relevant because, even if these two policies are often modeled as
separate tools, they inevitably share objectives, transmission mechanisms (Altunbas et al., 2018) and
possible complementarities (Osinski et al., 2013), which can be exploited by policy-makers.

Simulation results show that the model can account for a rich list of stylized fact concerning the
network structure of the interbank market (sizeable volume of interbank likanges, dissassortativity,
vertex centrality and bank size relationship) and the co-movements of macro-financial variables.
Moreover, we find that the economic system at large can benefit from the introduction of a micro-
prudential regulation that takes into account the interbank network relationships. Indeed, the use
of the NEVA framework decreases the incidence of systemic risk events and of bankruptcy episodes
of financial institutions, notwithstanding the tightness of mandatory capital requirements. In addi-
tion, a trade-off between financial stability and macroeconomic performance does not emerge in a
two-pillar regulatory framework grounded on a Basel III macro-prudential regulation and a NEVA
micro-prudential policy. Indeed, the NEVA lessens the pressure on credit supply and in turns on
macroeconomic performance resulting from a too stringent macro-prudential regulation. In other
words, the NEVA allows the economic system to achieve financial and macroeconomic stability with-
out relying on overly stringent capital requirements.

The rest of the paper is structured as follows. Section 2 describes the model and it focuses on the
mechanisms bringing to the formation and the evolution of interbank linkages and to the functioning
of the NEVA micro-prudential tool. Section 3 presents the results of our simulations experiments.
Lastly, Section 4 concludes.

## 2 The K+S model with an interbank market

The model belongs to the *Schumpeter meeting Keynes* family of models (K+S; see Dosi et al., 2010, 2013, 2015; Lamperti et al., 2018). It features vertically differentiated enterprises producing either a final consumption good or a capital good. Capital-good firms (indexed by $i = 1, ..., F_K$), produce vintages of machines, whose technology is defined by heterogeneous levels of labour productivity and energy efficiency.<fnref n="1" /> These three dimensions analogously define the technology behind the production process of capital-good firms. The technology of both production processes and machines results from a process of endogenous technical change, the Schumpeterian growth engine of the model, primarily fueled by R&D expenditures, financed by a share of $F_K$ firms’ sales.

Machines are eventually sold as an input for production to downstream consumption-good firms (indexed by $j = 1, ..., F_C$). The Keynesian source of endogenous business cycles of the model is driven by the dynamics of investment in machines by consumption-good firms. This investment depends on consumption-good firms’ expected final demand. In case internal funds are not sufficient to meet their desired levels of investments, consumption-good firms rely on external source of finance, applying for loans from the banking sector.

The model also incorporates a household sector that consumes the final good out of wage income. The public sector collects taxes on incomes, i.e., on firm profits and wages, and when unemployment rises consumers are paid a subsidy, proportional to the current wage level. Finally, given the inflation and the unemployment rate target, the central bank implements its monetary policy by setting the main interest rate following a Taylor-type rule. All agents are heterogeneous in their state variables (e.g., number of clients, technology in use, productivity, R&D intensity, market shares, profits, net-worth, etc.). Moreover, agents are boundedly rational and use simple heuristics to make their decisions and they do not have complete knowledge of the network of economic relationships emerging in each market.

We extend the K+S model by introducing an interbank market where banks can directly interact by exchanging funds. This allows us to obtain a more realistic representation of the determination of credit supply, and to better account for counterparty risk and systemic risk. Indeed, the supply of credit of each bank does not depend only on its balance sheet and the creditworthiness of its clients, but also on the overall risk perception (possibly related to the business cycle phase), as well as on the possibility of exchanging liquidity with other financial institutions over the money market.

We model direct banks' interactions by exploiting the relationships (i) between consumption-good firms (clients) and capital-good firms (suppliers) in the market for machines; and (ii) between the banks and both types of firms (see Figure 1). This approach is grounded on the fact that a key role of the banking sector is to handle the system of payments among firms.<fnref n="2" />

<footnote n="1">¹Our model nests into an agent-based integrated assessment model (Lamperti et al., 2018, 2019, 2021), where the economy features a stylized energy sector and can interact with a climate box. In this version of the model, we keep the modeling of the energy sector as simple as possible (see Section A.3 in Appendix A) and we do not allow the climate box to interact with the economic system.</footnote>

<footnote n="2">²In line with Galbiati and Soramäki (2011), we model payments through a decentralized mechanism.</footnote>
mechanisms bringing to the formation and the evolution of interbank linkages and to the functioning
of the NEVA micro-prudential tool. Section 3 presents the results of our simulations experiments.
Lastly, Section 4 concludes.

## 2 The K+S model with an interbank market

The model belongs to the *Schumpeter meeting Keynes* family of models (K+S; see Dosi et al., 2010, 2013, 2015; Lamperti et al., 2018). It features vertically differentiated enterprises producing either a final consumption good or a capital good. Capital-good firms (indexed by $i = 1, \dots, F_K$), produce vintages of machines, whose technology is defined by heterogeneous levels of labour productivity and energy efficiency.<fnref n="1" /> These three dimensions analogously define the technology behind the production process of capital-good firms. The technology of both production processes and machines results from a process of endogenous technical change, the Schumpeterian growth engine of the model, primarily fueled by R&D expenditures, financed by a share of $F_K$ firms’ sales.

Machines are eventually sold as an input for production to downstream consumption-good firms (indexed by $j = 1, \dots, F_C$). The Keynesian source of endogenous business cycles of the model is driven by the dynamics of investment in machines by consumption-good firms. This investment depends on consumption-good firms’ expected final demand. In case internal funds are not sufficient to meet their desired levels of investments, consumption-good firms rely on external source of finance, applying for loans from the banking sector.

The model also incorporates a household sector that consumes the final good out of wage income. The public sector collects taxes on incomes, i.e., on firm profits and wages, and when unemployment rises consumers are paid a subsidy, proportional to the current wage level. Finally, given the inflation and the unemployment rate target, the central bank implements its monetary policy by setting the main interest rate following a Taylor-type rule. All agents are heterogeneous in their state variables (e.g., number of clients, technology in use, productivity, R&D intensity, market shares, profits, net-worth, etc.). Moreover, agents are boundedly rational and use simple heuristics to make their decisions and they do not have complete knowledge of the network of economic relationships emerging in each market.

We extend the K+S model by introducing an interbank market where banks can directly interact by exchanging funds. This allows us to obtain a more realistic representation of the determination of credit supply, and to better account for counterparty risk and systemic risk. Indeed, the supply of credit of each bank does not depend only on its balance sheet and the creditworthiness of its clients, but also on the overall risk perception (possibly related to the business cycle phase), as well as on the possibility of exchanging liquidity with other financial institutions over the money market.

We model direct banks' interactions by exploiting the relationships (i) between consumption-good firms (clients) and capital-good firms (suppliers) in the market for machines; and (ii) between the banks and both types of firms (see Figure 1). This approach is grounded on the fact that a key role of the banking sector is to handle the system of payments among firms.<fnref n="2" />

<footnote n="1">¹Our model nests into an agent-based integrated assessment model (Lamperti et al., 2018, 2019, 2021), where the economy features a stylized energy sector and can interact with a climate box. In this version of the model, we keep the modeling of the energy sector as simple as possible (see Section A.3 in Appendix A) and we do not allow the climate box to interact with the economic system.</footnote>

<footnote n="2">²In line with Galbiati and Soramäki (2011), we model payments through a decentralized mechanism.</footnote>
<figure><img src="image_4.png" /><figcaption>**Figure 1:** Multi-layered network structure in the model. The solid arrow displays a representative transaction that occurs in the interbank market from Bank 2 to Bank 1. capital-good firm 1 (a client of Bank 1) supplies a machine to consumption-good firm 3 (a client of Bank 2) and must receive deposits in exchange. As banks are responsible for arranging payments between the firms, our reduced form representation of the interbank market shows a mechanism for the formation of a direct payment link from Bank 2 (which takes deposits from the bank account of consumption-good firm 3) to Bank 1 (which use the received deposits and place them in the bank account of capital-good firm 1).</figcaption></figure>

In the model, whenever a consumption-good firm buys a new machine, it transfers the corre-
sponding money value of the transaction to a capital-good firm. We rule out the possibility of a direct
cash transfer between firms, and we assume that the payment occurs through a deposit transfer from
the bank where the consumption-good firm holds its bank account to the one where the capital-good
firm holds its own account. This payment operation puts the two banks in direct connection. In
Section 2.3, we shall also describe in detail how the payment system endogenously generates an
endogenous network of interbank linkages. In this network, each bank represents a node/vertex,
and a payment represents a link/edge. All links are directed and weighted. The direction of the link
depends on the buyer-seller relationship between the two banks' clients. The weights of the links are
instead defined by the amount of the payment between the two parties.

Given the focus of this paper, in the following sections, we provide a detailed description of
the functioning of the banking sector. Specifically, we describe how credit supply is determined,
the generation and evolution of the interbank network, and the characteristics of micro- and macro-
prudential policies. The other elements of the model, such as technical change, investment decisions
of the firms, households' consumption, and government's behavior, are detailed in Appendix A.

**2.1 Banks' lending and macro-prudential regulation**

Modern banking institutions do not just intermediate funds between savers and borrowers. They can create money *ex-nihilo* (McLeay et al., 2014) and play a central role in the system of payments in the economy. Today most of the money in circulation is endogenously created by private commercial banks in the form of loans granted to the private sector (see Godley and Lavoie, 2006; Lavoie, 2014; Ihrig et al., 2021, among the others). This happens because every time a bank lends out new funds (such as loans to firms) a deposit of the same amount is automatically created on the liability side of the bank's balance sheet to match the new asset. This deposit corresponds to the funds that the borrower has obtained from the bank and which have been contingently deposited at the bank itself. Most deposits are therefore liabilities created by banks themselves rather than already existing funds
<figure><img src="image_4.png" /><figcaption>**Figure 1:** Multi-layered network structure in the model. The solid arrow displays a representative transaction that occurs in the interbank market from Bank 2 to Bank 1. capital-good firm 1 (a client of Bank 1) supplies a machine to consumption-good firm 3 (a client of Bank 2) and must receive deposits in exchange. As banks are responsible for arranging payments between the firms, our reduced form representation of the interbank market shows a mechanism for the formation of a direct payment link from Bank 2 (which takes deposits from the bank account of consumption-good firm 3) to Bank 1 (which use the received deposits and place them in the bank account of capital-good firm 1).</figcaption></figure>

In the model, whenever a consumption-good firm buys a new machine, it transfers the corre-
sponding money value of the transaction to a capital-good firm. We rule out the possibility of a direct
cash transfer between firms, and we assume that the payment occurs through a deposit transfer from
the bank where the consumption-good firm holds its bank account to the one where the capital-good
firm holds its own account. This payment operation puts the two banks in direct connection. In
Section 2.3, we shall also describe in detail how the payment system endogenously generates an
endogenous network of interbank linkages. In this network, each bank represents a node/vertex,
and a payment represents a link/edge. All links are directed and weighted. The direction of the link
depends on the buyer-seller relationship between the two banks' clients. The weights of the links are
instead defined by the amount of the payment between the two parties.

Given the focus of this paper, in the following sections, we provide a detailed description of
the functioning of the banking sector. Specifically, we describe how credit supply is determined,
the generation and evolution of the interbank network, and the characteristics of micro- and macro-
prudential policies. The other elements of the model, such as technical change, investment decisions
of the firms, households' consumption, and government's behavior, are detailed in Appendix A.

2.1 Banks’ lending and macro-prudential regulation

Modern banking institutions do not just intermediate funds between savers and borrowers. They can
create money *ex-nihilo* (McLeay et al., 2014) and play a central role in the system of payments in the
economy. Today most of the money in circulation is endogenously created by private commercial
banks in the form of loans granted to the private sector (see Godley and Lavoie, 2006; Lavoie, 2014;
Ihrig et al., 2021, among the others). This happens because every time a bank lends out new funds
(such as loans to firms) a deposit of the same amount is automatically created on the liability side
of the bank's balance sheet to match the new asset. This deposit corresponds to the funds that the
borrower has obtained from the bank and which have been contingently deposited at the bank itself.
Most deposits are therefore liabilities created by banks themselves rather than already existing funds
provided by saving agents and ready to be lent out to firms.<fnref n="3" />

Our model comprises *B* heterogeneous commercial banks that can create money endogenously along the lines outlined above. Moreover, in line with the empirical evidence (Berger et al., 1995) the size of each bank, as measured by the number of clients, is drawn from a Pareto distribution – under the constraint that the total number of clients of all banks adds up to the total number of firms in the system.

As in Dosi et al. (2015), banks provide loans only to consumption-good firms and are subject to macro-prudential regulation. We assume that for a generic bank *k*, the total credit supply $TC_{k,t}$ depends on three main factors: (i) the bank's past equity, $E_{k,t-1}$; (ii) the bank's past ratio of non-performing loans to total assets, $Bda_{k,t-1}$; and (iii) a macro-prudential counter-cyclical capital buffer $CCB_t$. Formally, a bank's credit supply is defined as:

$$TC_{k,t} = \frac{E_{k,t-1}}{\tau^B(1 + CCB_t + \beta Bda_{k,t-1})} \quad (1)$$

The macro-prudential setting in the model is in line with the Basel III regulatory framework. In particular, we assume that the authority fixes two types of constraints, which are both present in Equation 1: a fixed *capital requirement* captured by the parameter $\tau^B \in (0, 1]$ that we set equal to 4.5% as within the Basel III framework; a time-varying *counter-cyclical capital buffer* captured by the variable $CCB_t$. Similarly to Popoyan et al. (2017), the counter-cyclical buffer depends non-linearly on the aggregate credit-to-GDP gap $CG_t$, measured as the deviation of the credit-to-GDP ratio from its 5-year moving average trend. More specifically the buffer writes:

$$CCB_t = \begin{cases} 0, & \text{if } CG_t < J \\ \frac{(CG_t-J)}{(H-J)}0.025, & \text{if } J \le CG_t \le H \\ 0.025, & \text{if } CG_t > H \end{cases} \quad (2)$$

where $J = 2\%$ and $H = 10\%$ measure the adjustment factors based on historical evidence about banking crises (see BIS, 2010, pag.13). Overall, a tighter macro-prudential regulation, as measured by a higher value of $\tau^B$, decreases the credit supply for all banks. However, the aggregate effects on the macroeconomic and financial systems are non-trivial. On the one hand, the lower credit supply might limit growth possibilities for firms. On the other hand, the regulation can also prevent the formation of *boom-and-bust credit cycles* thus stabilizing the economy (Schularick and Taylor, 2012).

Bank equity $E_{k,t-1}$ also plays a prominent role in determining the credit supply, as specified in Equation (1). In banks' balance sheets, equity is a residual claim measuring the difference between assets and liabilities. In our model, it will consequently depend on the value of each bank's interbank claims. In Section 2.5 we shall discuss how the valuation of the interbank claims affects the equity value of the bank and indirectly the supply of credit in the economy.

<footnote n="3">³See McLeay et al. (2014) and Ihrig et al. (2021) for more details about the implications of the theory of endogenous money for the functioning of the banking sector.</footnote>
provided by saving agents and ready to be lent out to firms.<fnref n="3" />

Our model comprises *B* heterogeneous commercial banks that can create money endogenously along the lines outlined above. Moreover, in line with the empirical evidence (Berger et al., 1995) the size of each bank, as measured by the number of clients, is drawn from a Pareto distribution – under the constraint that the total number of clients of all banks adds up to the total number of firms in the system.

As in Dosi et al. (2015), banks provide loans only to consumption-good firms and are subject to macro-prudential regulation. We assume that for a generic bank *k*, the total credit supply $TC_{k,t}$ depends on three main factors: (i) the bank's past equity, $E_{k,t-1}$; (ii) the bank's past ratio of non-performing loans to total assets, $Bda_{k,t-1}$; and (iii) a macro-prudential counter-cyclical capital buffer $CCB_t$. Formally, a bank's credit supply is defined as:

$$TC_{k,t} = \frac{E_{k,t-1}}{\tau^B(1 + CCB_t + \beta Bda_{k,t-1})} \quad (1)$$

The macro-prudential setting in the model is in line with the Basel III regulatory framework. In particular, we assume that the authority fixes two types of constraints, which are both present in Equation 1: a fixed *capital requirement* captured by the parameter $\tau^B \in (0, 1]$ that we set equal to 4.5% as within the Basel III framework; a time-varying *counter-cyclical capital buffer* captured by the variable $CCB_t$. Similarly to Popoyan et al. (2017), the counter-cyclical buffer depends non-linearly on the aggregate credit-to-GDP gap $CG_t$, measured as the deviation of the credit-to-GDP ratio from its 5-year moving average trend. More specifically the buffer writes:

$$CCB_t = \begin{cases} 0, & \text{if } CG_t < J \\ \frac{(CG_t-J)}{(H-J)}0.025, & \text{if } J \le CG_t \le H \\ 0.025, & \text{if } CG_t > H \end{cases} \quad (2)$$

where $J = 2\%$ and $H = 10\%$ measure the adjustment factors based on historical evidence about banking crises (see BIS, 2010, pag.13). Overall, a tighter macro-prudential regulation, as measured by a higher value of $\tau^B$, decreases the credit supply for all banks. However, the aggregate effects on the macroeconomic and financial systems are non-trivial. On the one hand, the lower credit supply might limit growth possibilities for firms. On the other hand, the regulation can also prevent the formation of *boom-and-bust credit cycles* thus stabilizing the economy (Schularick and Taylor, 2012).

Bank equity $E_{k,t-1}$ also plays a prominent role in determining the credit supply, as specified in Equation (1). In banks' balance sheets, equity is a residual claim measuring the difference between assets and liabilities. In our model, it will consequently depend on the value of each bank's interbank claims. In Section 2.5 we shall discuss how the valuation of the interbank claims affects the equity value of the bank and indirectly the supply of credit in the economy.

<footnote n="3">³See McLeay et al. (2014) and Ihrig et al. (2021) for more details about the implications of the theory of endogenous money for the functioning of the banking sector.</footnote>
## 2.2 The interests rate structure

The model features several interest rates because of the various activities banks perform. Let us begin by determining the interest rate charged by a bank to its client asking for a new loan ($r_t^{deb}$). This rate depends on two components: a mark-up common to all banks, ($\mu^{deb}$) applied on the Central Bank interest rate ($r_t^{cb}$); a client-specific risk-premium that is associated with the fragility of the borrowing client cl. The latter is constructed by classifying clients into four credit classes that correspond to the four quartiles (i.e., $q^{cl} = \{1, 2, 3, 4\}$) of the distribution of firm financial fragility (see Dosi et al., 2015, for a similar approach). For each client cl belonging to the quartile $q^{cl}$ at time t the bank k applies a risk-premium that is defined as follows:

$$r_{k,cl,t}^{deb} = r_t^{cb}(1 + \mu^{deb})[1 + (q^{cl} - 1)k_{const}], \quad (3)$$

where $k_{const} > 0$ is a scaling parameter.

Furthermore, interbank assets (see Section 2.3 for a description of their determination) yield an interest rate $r_t^{IB}$ equal to:

$$r_t^{IB} = (1 - md^{IB})r_t^{cb} \quad (4)$$

where $md^{IB}$ is a mark-down on $r_t^{cb}$ in line with the empirical evidence showing that in most economies, the money market rate is often highly correlated to the main refinancing rate, but slightly lower.

The deposit rate $r_t^D$ that banks pay on the deposits from their clients is determined similarly, i.e. by applying a mark-down $md^{dep}$ on the central bank main refinancing rate. The same approach is used to set the interest rate on government bonds $r_t^{bonds}$ and the rate $r_t^{res}$ at which bank reserves at the central bank are rewarded (see Table 5 for the values of the mark-down applied to set these interest rates in the model).

Finally, the main refinancing operation rate is set by the central bank according to a dual-mandate Taylor rule:

$$r_t^{cb} = r^T + \gamma_{\pi}(\pi_t - \pi^T) + \gamma_{U}(U_t - U^T) \quad \text{with} \quad \gamma_{\pi} > 1, \quad \gamma_{U} \ge 0 \quad (5)$$

where the terms in parentheses represent inflation rate and unemployment rate gaps, while the parameters ($\gamma_{\pi}, \gamma_{U}$) measure the aggressiveness of the central bank with respect to each objective.

Overall, the interest rate structure in the model is such that the following inequalities hold (see also Table 5):

$$r_t^D \le r_t^{res} \le r_t^{IB} < r_t^{cb} \le r_t^{bonds} \le r_t^{deb}. \quad (6)$$

## 2.3 Link creation in the interbank network

A key working assumption of our model is that banks manage payments between consumption-good and capital-good firms. This generates an endogenous network of interbank claims that evolves as time goes by. In this section, we describe the creation of new links in the foregoing interbank network. The next section describes instead how links can be destroyed.

To explain the process of link formation in the model, let us consider the case where a capital-good firm $f_k$ sells a new machinery to a consumption-good firm $f_c$. The two firms are, respectively, clients of the banks $b_k$ and $b_c$. Let us also consider a situation in which the $f_c$ firm does not have enough internal funds to buy the new machine and asks for a loan from its associated bank, which
## 2.2 The interests rate structure

The model features several interest rates because of the various activities banks perform. Let us begin by determining the interest rate charged by a bank to its client asking for a new loan ($r_t^{deb}$). This rate depends on two components: a mark-up common to all banks, ($\mu^{deb}$) applied on the Central Bank interest rate ($r_t^{cb}$); a client-specific risk-premium that is associated with the fragility of the borrowing client cl. The latter is constructed by classifying clients into four credit classes that correspond to the four quartiles (i.e., $q^{cl} = \{1, 2, 3, 4\}$) of the distribution of firm financial fragility (see Dosi et al., 2015, for a similar approach). For each client cl belonging to the quartile $q^{cl}$ at time t the bank k applies a risk-premium that is defined as follows:

$$r_{k,cl,t}^{deb} = r_t^{cb}(1 + \mu^{deb})[1 + (q^{cl} - 1)k_{const}], \quad (3)$$

where $k_{const} > 0$ is a scaling parameter.

Furthermore, interbank assets (see Section 2.3 for a description of their determination) yield an interest rate $r_t^{IB}$ equal to:

$$r_t^{IB} = (1 - md^{IB})r_t^{cb} \quad (4)$$

where $md^{IB}$ is a mark-down on $r_t^{cb}$ in line with the empirical evidence showing that in most economies, the money market rate is often highly correlated to the main refinancing rate, but slightly lower.

The deposit rate $r_t^D$ that banks pay on the deposits from their clients is determined similarly, i.e. by applying a mark-down $md^{dep}$ on the central bank main refinancing rate. The same approach is used to set the interest rate on government bonds $r_t^{bonds}$ and the rate $r_t^{res}$ at which bank reserves at the central bank are rewarded (see Table 5 for the values of the mark-down applied to set these interest rates in the model).

Finally, the main refinancing operation rate is set by the central bank according to a dual-mandate Taylor rule:

$$r_t^{cb} = r^T + \gamma_{\pi}(\pi_t - \pi^T) + \gamma_{U}(U_t - U^T) \quad \text{with} \quad \gamma_{\pi} > 1, \quad \gamma_{U} \ge 0 \quad (5)$$

where the terms in parentheses represent inflation rate and unemployment rate gaps, while the parameters ($\gamma_{\pi}, \gamma_{U}$) measure the aggressiveness of the central bank with respect to each objective.

Overall, the interest rate structure in the model is such that the following inequalities hold (see also Table 5):

$$r_t^D \le r_t^{res} \le r_t^{IB} < r_t^{cb} \le r_t^{bonds} \le r_t^{deb}. \quad (6)$$

## 2.3 Link creation in the interbank network

A key working assumption of our model is that banks manage payments between consumption-good and capital-good firms. This generates an endogenous network of interbank claims that evolves as time goes by. In this section, we describe the creation of new links in the foregoing interbank network. The next section describes instead how links can be destroyed.

To explain the process of link formation in the model, let us consider the case where a capital-good firm $f_k$ sells a new machinery to a consumption-good firm $f_c$. The two firms are, respectively, clients of the banks $b_k$ and $b_c$. Let us also consider a situation in which the $f_c$ firm does not have enough internal funds to buy the new machine and asks for a loan from its associated bank, which
endogenously creates new money in the system (Figure 2 Panel A). The new loan is an asset for $b_c$ and a liability for $f_c$. Given the assumption that all payments are managed by banks, the payment between the firms originates a transfer of deposits and an exchange of interbank funds between the two banks $b_c$ and $b_k$. The transaction, therefore, implies a change in the level of deposits and the creation of an interbank relationship between the two banks.

<figure><img src="image_5.png" /><figcaption>**Figure 2:** Stylized examples of interbank network: link formation (**Panel A**), pulsing weights (**Panel B**) and link persistence (**Panel C**). *Notes:* The figure shows the T-account plot for representative balance sheets of consumption- and capital-good firms ($f_c$ and $f_k$, respectively) and the banks to which they are connected ($b_c$ and $b_k$). The figure depicts the evolution in the balance sheet composition and highlights how interbank linkages emerges within the system of payments embedded in the model. See text, Section 2.3 - 2.4, for a more detailed description.</figcaption></figure>

At this stage, two alternative possibilities can emerge in our model according to the ability of the firm $f_c$ to collect enough funds to repay the debt or, instead, to default (partially or totally) on its outstanding debt. Let us start from the first case (Figure 2 Panel B). In such a situation, the consumption-good firm $f_c$ has collected enough funds from its sales and, at the end of the period, it has a sufficient amount of liquidity to repay the loan. Thus, the firm $f_c$ deposits its profits at its bank $b_c$ and, consequently, this bank can immediately close the interbank liability position towards the other bank $b_k$. Hence, the interbank link is immediately closed and our interbank network is characterized only by *pulsing weights*. If this occurs for all firms, then all payments are settled and all the interbank claims are cleared. The interbank network would be represented by a null adjacency matrix. When firm $f_c$ defaults instead, the bank $b_c$ experiences a loss on the loan previously granted to $f_c$ — i.e., a bad debt (Figure 2 Panel C). In this case, a fraction of the credit loss is transmitted to the payment that the bank $b_c$ would have done to the other bank $b_k$ to close its interbank liability, if the
endogenously creates new money in the system (Figure 2 Panel A). The new loan is an asset for $b_c$ and a liability for $f_c$. Given the assumption that all payments are managed by banks, the payment between the firms originates a transfer of deposits and an exchange of interbank funds between the two banks $b_c$ and $b_k$. The transaction, therefore, implies a change in the level of deposits and the creation of an interbank relationship between the two banks.

**A Link formation**

<figure><img src="image_5.png" /></figure>

**B Pulsing weights**

<figure><img src="image_6.png" /></figure>

**C Link persistence**

<figure><img src="image_7.png" /><figcaption>**Figure 2:** Stylized examples of interbank network: link formation (**Panel A**), pulsing weights (**Panel B**) and link persistence (**Panel C**). *Notes:* The figure shows the T-account plot for representative balance sheets of consumption- and capital-good firms ($f_c$ and $f_k$, respectively) and the banks to which they are connected ($b_c$ and $b_k$). The figure depicts the evolution in the balance sheet composition and highlights how interbank linkages emerges within the system of payments embedded in the model. See text, Section 2.3 - 2.4, for a more detailed description.</figcaption></figure>

At this stage, two alternative possibilities can emerge in our model according to the ability of the firm $f_c$ to collect enough funds to repay the debt or, instead, to default (partially or totally) on its outstanding debt. Let us start from the first case (Figure 2 Panel B). In such a situation, the consumption-good firm $f_c$ has collected enough funds from its sales and, at the end of the period, it has a sufficient amount of liquidity to repay the loan. Thus, the firm $f_c$ deposits its profits at its bank $b_c$ and, consequently, this bank can immediately close the interbank liability position towards the other bank $b_k$. Hence, the interbank link is immediately closed and our interbank network is characterized only by *pulsing weights*. If this occurs for all firms, then all payments are settled and all the interbank claims are cleared. The interbank network would be represented by a null adjacency matrix. When firm $f_c$ defaults instead, the bank $b_c$ experiences a loss on the loan previously granted to $f_c$ – i.e., a bad debt (Figure 2 Panel C). In this case, a fraction of the credit loss is transmitted to the payment that the bank $b_c$ would have done to the other bank $b_k$ to close its interbank liability, if the
firm $f_c$ had not defaulted. The remaining fraction of the loss is absorbed by a reduction in the equity of $b_c$. Hence, in this situation, the link between the two banks survives over time since the interbank payment did not occur yet.

This simple mechanism for the creation of connections among banks is sufficient to characterize the interbank system by means of an adjacency matrix of the form:

$$L = \begin{bmatrix} 0 & \ell_{1,2} & \cdots & \ell_{1,B} \\ \ell_{2,1} & 0 & & \vdots \\ \vdots & \ddots & \ddots & \vdots \\ \ell_{B,1} & \cdots & \cdots & 0 \end{bmatrix} = A' \quad (7)$$

where the matrix $L$ is the interbank liability matrix.<fnref n="4" /> Each entry $\ell_{k,b}$ measures the nominal value of the interbank debt that a generic bank $k$ has towards another generic bank $b$ and that is not regulated within a period. The stock of interbank liabilities of a bank $k$ is $IB_k^L = \sum_{b=1}^B \ell_{k,b}$ (i.e., the sum of the $k^{th}$ row of the matrix $L$); the total amount of interbank assets of a bank $k$ is instead $IB_k^A = \sum_{b=1}^B \ell_{b,k}$ (i.e., the sum of the $k^{th}$ column of the matrix $L$).

## 2.4 Link destruction in the interbank network

After an interbank link between two banks is created, one bank becomes a creditor of another bank – in our previous example, the bank $b_k$ is the creditor of bank $b_c$. A creditor bank is the holder of an interbank asset and it can demand the reimbursement of the deposit from the debtor bank at any time. We assume that the creditor bank decides to close (or not to close) the interbank position according to a relatively simple heuristic that involves the opportunity cost of holding the asset for another period. More in detail, we assume that a bank holding the interbank asset has two options. The first one consists in the rollover of the interbank credit for another period, which is a risky asset and pays the interbank rate $r_t^{IB}$, with a probability $(1 - p_{b,t})$ – that is, the probability that the debtor bank will not default. The second one is a risk-free alternative according to which the bank immediately closes the interbank position, obtains the liquidity, and deposits it at the central bank. This alternative pays the risk-free rate $r_t^{res}$. In a nutshell, the choice corresponds to an investment decision between a risky asset and a risk-free one. Formally, the decision by the creditor bank $k$ to destroy the interbank with bank $b$ at time $t$ reads:

$$Pr(CLOSE_{k,b,t} = 1) = Pr(r_t^{res} - r_t^{IB}(1 - p_t^b) > \xi_{k,b,t}) \quad (8)$$

$$\xi_{k,b,t} = \begin{cases} 1 & \text{if } IB_k^A / (\sum_k IB_k^A) \le \Upsilon \\ 0 & \text{otherwise} \end{cases} \quad (9)$$

with $\Upsilon$ being a random variable with a uniform distribution $U[m_t, n_t]$. The support of this distribution is denoted by $m_t$ and $n_t$. These are, respectively, the minimum and maximum interbank exposures as a share of total interbank exposure at time $t$. Comparative static of Equation (8) suggests that, if the default probability of the borrower ($p_{b,t}$) is too large to compensate the spread between the returns on the interbank market ($r_t^{IB}$) and on the central bank deposits ($r_t^{res}$), then it is preferable for bank $k$

<footnote n="4">⁴$L = A'$ implies that $A$ is the corresponding interbank asset matrix.</footnote>
firm $f_c$ had not defaulted. The remaining fraction of the loss is absorbed by a reduction in the equity of $b_c$. Hence, in this situation, the link between the two banks survives over time since the interbank payment did not occur yet.

This simple mechanism for the creation of connections among banks is sufficient to characterize the interbank system by means of an adjacency matrix of the form:

$$L = \begin{bmatrix} 0 & l_{1,2} & \cdots & l_{1,B} \\ l_{2,1} & 0 & & \vdots \\ \vdots & & \ddots & \vdots \\ l_{B,1} & \cdots & \cdots & 0 \end{bmatrix} = A' \quad (7)$$

where the matrix $L$ is the interbank liability matrix.<fnref n="4" /> Each entry $l_{k,b}$ measures the nominal value of the interbank debt that a generic bank $k$ has towards another generic bank $b$ and that is not regulated within a period. The stock of interbank liabilities of a bank $k$ is $IB_k^L = \sum_{b=1}^B l_{k,b}$ (i.e., the sum of the $k^{th}$ row of the matrix $L$); the total amount of interbank assets of a bank $k$ is instead $IB_k^A = \sum_{b=1}^B l_{b,k}$ (i.e., the sum of the $k^{th}$ column of the matrix $L$).

## 2.4 Link destruction in the interbank network

After an interbank link between two banks is created, one bank becomes a creditor of another bank – in our previous example, the bank $b_k$ is the creditor of bank $b_c$. A creditor bank is the holder of an interbank asset and it can demand the reimbursement of the deposit from the debtor bank at any time. We assume that the creditor bank decides to close (or not to close) the interbank position according to a relatively simple heuristic that involves the opportunity cost of holding the asset for another period. More in detail, we assume that a bank holding the interbank asset has two options. The first one consists in the rollover of the interbank credit for another period, which is a risky asset and pays the interbank rate $r_t^{IB}$, with a probability $(1 - p_{b,t})$ – that is, the probability that the debtor bank will not default. The second one is a risk-free alternative according to which the bank immediately closes the interbank position, obtains the liquidity, and deposits it at the central bank. This alternative pays the risk-free rate $r_t^{res}$. In a nutshell, the choice corresponds to an investment decision between a risky asset and a risk-free one. Formally, the decision by the creditor bank $k$ to destroy the interbank with bank $b$ at time $t$ reads:

$$Pr(CLOSE_{k,b,t} = 1) = Pr(r_t^{res} - r_t^{IB}(1 - p_t^b) > \xi_{k,b,t}) \quad (8)$$

$$\xi_{k,b,t} = \begin{cases} 1 & \text{if } IB_k^A / (\sum_k IB_k^A) \le \Upsilon \\ 0 & \text{otherwise} \end{cases} \quad (9)$$

with $\Upsilon$ being a random variable with a uniform distribution $U[m_t, n_t]$. The support of this distribution is denoted by $m_t$ and $n_t$. These are, respectively, the minimum and maximum interbank exposures as a share of total interbank exposure at time $t$. Comparative static of Equation (8) suggests that, if the default probability of the borrower ($p_{b,t}$) is too large to compensate the spread between the returns on the interbank market ($r_t^{IB}$) and on the central bank deposits ($r_t^{res}$), then it is preferable for bank $k$

<footnote n="4">⁴$L = A'$ implies that $A$ is the corresponding interbank asset matrix.</footnote>
to close the risky position towards bank b. However, if $p_{b,t} = 1$, bank k can still decide to keep the interbank position open. This event occurs whenever the variable $\xi_{k,b,t} = 1$, namely when the share of interbank assets that bank k holds, is not higher than the random realization of $\Upsilon$.<fnref n="5" />

Following Gai et al. (2011), in our baseline specification the probability of default $p_{b,t}$ of bank b is fully exogenous and drawn from a uniformly distributed random variable $U(0, 2h_0)$ whose expected value $h_0 \in (0, 0.5]$ can be interpreted as the haircut rate, which is usually defined as the percentage deduction from the value of the collateral required to obtain financing.<fnref n="6" /> In this setting, realizations of $U(0, 2h_0)$ reflect idiosyncratic shocks to the perceived underlying risk of the interbank debtor. In our equation, therefore, a smaller haircut maps into a reduction in the perceived risk associated to the bankruptcy of the counterpart and to a lower probability of closing the interbank position. Therefore the counterparty's probability of default is an important trigger for the banks' actions and can have relevant consequences on the profitability and the credit supply of each single bank. At the aggregate level, instead, it can affect the overall interbank network topology and systemic risk.

## 2.5 Banks' equity

The credit supply of a bank does not only depend upon the macro-prudential regulation, but also on banks' equity (see Section 2.1). We now introduce the law of motion of a bank's equity and we carefully describe how micro-prudential regulation can impact on equity and, in turn, on credit supply and macroeconomic fundamentals.

At the end of each period, the profit of a generic bank k is determined as:

$$ \Pi_{k,t} = \left( \sum_{cl=1}^{Cl_k} r_{k,cl,t}^{deb} Loans_{k,cl,t} \right) + r_t^{res} Res_{k,t} + r_t^{bonds} Bonds_{k,t} - r_t^D Depo_{k,t} - BadDebt_{k,t} + r_t^{IB} NetIB_{k,t} \quad (10) $$

where $Cl_k$ indicates the number of clients of the bank k, and the $BadDebt_{k,t}$ is strictly positive only in those periods in which the bank experiences some losses on its outstanding loans due to the bankruptcy of some of its clients. Banks earn profits on the outstanding loans to consumption-good firms ($Loans_{k,t}$), on the stock of reserves at the central bank ($Res_{k,t}$), on the yields provided by sovereign bonds ($Bonds_{k,t}$), and on the interbank assets ($IB_{k,t}^A$). At the same time, banks pay an interest rate on firms' deposits ($Depo_{k,t}$) and on the interbank liabilities ($IB_{k,t}^L$).<fnref n="7" />

At the end of the period, the after-tax profits of banks – i.e. $Net\Pi_{k,t} = (1 - tr)\Pi_{k,t}$ – are stockpiled to the bank's net worth. Thus, the equity of the generic bank k evolves accordingly to the following law of motion:

$$ E_{k,t} = Loans_{k,t} + Res_{k,t} + Bonds_{k,t} + IB_{k,t}^A - IB_{k,t}^L - Depo_{k,t} + Net\Pi_{k,t} \quad (11) $$

If the equity of the bank is negative the bank goes bankrupt, and the interconnected banks in the interbank network L may only partially recover their interbank claims (i.e., $l_{b,k}$) up to a fraction $\rho^{IB}$ of

<footnote n="5">⁵In the simplest case in which $\Upsilon \sim U[\bar{m}, \bar{n}]$, and the share of bank's k interbank asset $IB_k^A/(\sum_k IB_k^A) = (\bar{n} + \bar{m})/2$, then $\xi_{k,b,t} = 1$ with 50% probability. However, since the support of $\Upsilon$ changes over time according to the minimum and maximum share of interbank exposure of the system ($m_t$ and $n_t$, respectively), the probability of keeping the position open anyway regardless the estimated counterparty's probability of default $p_{b,t}$ changes accordingly. Hence, a bank with a relatively lower interbank exposure will tend to keep the position open towards the debtor bank b.</footnote>

<footnote n="6">⁶See Section 2.6 for scenarios in which $p_{b,t}$ is endogenously determined.</footnote>

<footnote n="7">⁷For sake of brevity, in Equation (10) we have directly reported the net interbank position $NetIB_{k,t}$.</footnote>
to close the risky position towards bank b. However, if $p_{b,t} = 1$, bank k can still decide to keep the interbank position open. This event occurs whenever the variable $\xi_{k,b,t} = 1$, namely when the share of interbank assets that bank k holds, is not higher than the random realization of $\Upsilon$.<fnref n="5" />

Following Gai et al. (2011), in our baseline specification the probability of default $p_{b,t}$ of bank b is fully exogenous and drawn from a uniformly distributed random variable $U(0, 2h_0)$ whose expected value $h_0 \in (0, 0.5]$ can be interpreted as the haircut rate, which is usually defined as the percentage deduction from the value of the collateral required to obtain financing.<fnref n="6" /> In this setting, realizations of $U(0, 2h_0)$ reflect idiosyncratic shocks to the perceived underlying risk of the interbank debtor. In our equation, therefore, a smaller haircut maps into a reduction in the perceived risk associated to the bankruptcy of the counterpart and to a lower probability of closing the interbank position. Therefore the counterparty's probability of default is an important trigger for the banks' actions and can have relevant consequences on the profitability and the credit supply of each single bank. At the aggregate level, instead, it can affect the overall interbank network topology and systemic risk.

## 2.5 Banks' equity

The credit supply of a bank does not only depend upon the macro-prudential regulation, but also on banks' equity (see Section 2.1). We now introduce the law of motion of a bank's equity and we carefully describe how micro-prudential regulation can impact on equity and, in turn, on credit supply and macroeconomic fundamentals.

At the end of each period, the profit of a generic bank k is determined as:

$$ \Pi_{k,t} = \left( \sum_{cl=1}^{Cl_k} r_{k,cl,t}^{deb} Loans_{k,cl,t} \right) + r_t^{res} Res_{k,t} + r_t^{bonds} Bonds_{k,t} - r_t^D Depo_{k,t} - BadDebt_{k,t} + r_t^{IB} NetIB_{k,t} \quad (10) $$

where $Cl_k$ indicates the number of clients of the bank k, and the $BadDebt_{k,t}$ is strictly positive only in those periods in which the bank experiences some losses on its outstanding loans due to the bankruptcy of some of its clients. Banks earn profits on the outstanding loans to consumption-good firms ($Loans_{k,t}$), on the stock of reserves at the central bank ($Res_{k,t}$), on the yields provided by sovereign bonds ($Bonds_{k,t}$), and on the interbank assets ($IB_{k,t}^A$). At the same time, banks pay an interest rate on firms' deposits ($Depo_{k,t}$) and on the interbank liabilities ($IB_{k,t}^L$).<fnref n="7" />

At the end of the period, the after-tax profits of banks – i.e. $Net\Pi_{k,t} = (1 - tr)\Pi_{k,t}$ – are stockpiled to the bank's net worth. Thus, the equity of the generic bank k evolves accordingly to the following law of motion:

$$ E_{k,t} = Loans_{k,t} + Res_{k,t} + Bonds_{k,t} + IB_{k,t}^A - IB_{k,t}^L - Depo_{k,t} + Net\Pi_{k,t} \quad (11) $$

If the equity of the bank is negative the bank goes bankrupt, and the interconnected banks in the interbank network L may only partially recover their interbank claims (i.e., $l_{b,k}$) up to a fraction $\rho^{IB}$ of

<footnote n="5">⁵In the simplest case in which $\Upsilon \sim U[\bar{m}, \bar{n}]$, and the share of bank's k interbank asset $IB_k^A/(\sum_k IB_k^A) = (\bar{n} + \bar{m})/2$, then $\xi_{k,b,t} = 1$ with 50% probability. However, since the support of $\Upsilon$ changes over time according to the minimum and maximum share of interbank exposure of the system ($m_t$ and $n_t$, respectively), the probability of keeping the position open anyway regardless the estimated counterparty's probability of default $p_{b,t}$ changes accordingly. Hence, a bank with a relatively lower interbank exposure will tend to keep the position open towards the debtor bank b.</footnote>

<footnote n="6">⁶See Section 2.6 for scenarios in which $p_{b,t}$ is endogenously determined.</footnote>

<footnote n="7">⁷For sake of brevity, in Equation (10) we have directly reported the net interbank position $NetIB_{k,t}$.</footnote>
their original claims. In our baseline scenario, we set $\rho^{IB} = 0$. Moreover, to keep the number of banks constant within a simulation, we assume that when a bank goes bankrupt the Government steps in by providing fresh capital to the defaulted bank. The equity of the bank after the government bailout is a fraction $\vartheta \sim U(0.1, 0.9)$ of the smallest incumbent banks equity, provided it meets the regulatory capital requirements.

In our baseline scenario equity is evaluated at its book value, as calculated in Equation (11). However, such a formulation does not take into account the fact that all the other banks with whom the generic bank $k$ has direct or indirect interconnections in interbank network can default. Therefore, the book value provides only an imperfect representation of the market value of the bank.

A well-designed micro-prudential regulation provides banks with the tools to better evaluate their interbank claims. If appropriately evaluated, the equity value of banks can systematically vary from the book value. This difference might give rise to sizable impact on the credit supply of each bank, limiting their exposure to counterparty and systemic risks. In particular, in this model we investigate the properties of the NEVA mechanism (Barucca et al., 2020), which can be employed as a behavioural and micro-prudential device aimed at protecting the banks from the counterparty default risk, as well as from credit risk. The next section describes the implementation of such a mechanism in our model.

## 2.6 The NEVA clearing mechanism and micro-prudential regulation

Persistent interbank links between two banks endogenously emerge in the model when a consumption-good firm defaults on its debt. As time goes by, the owner of an interbank claim that decides not to close the position remains the holder of an interbank asset which is valued at its historical nominal value. However, this value could be misleading as it does not take into account two relevant risk factors: the counterparty risk and the credit risk. If one accounts for these two sources of risk, the market value of an interbank asset may change over time also impacting on the final value of a bank's equity, as in Equation (11). By introducing in the model a clearing mechanism that allows all banks to constantly update the market evaluation of their interbank claims, we thus allow firms to constantly update also the value of their equity.

An example of the above interbank clearing mechanism is the proposed by Eisenberg and Noe (2001). This mechanism provides (under some regularity conditions) a unique evaluation vector of the outstanding interbank claims based upon the probability of default of all banks. However, the Eisenberg and Noe (2001) evaluation model relies on two strong assumptions. First, it assumes that the evaluation of the interbank claims by each bank is carried out only at the maturity (i.e., ex-post) rather than at each period (i.e., ex-ante). Second, it assumes that all banks have a complete knowledge of the underlying network structure and the financial conditions of all the other banks. These two issues are solved by Barucca et al. (2020) with the development of the NEVA (Network Valuation), which generalizes the Eisenberg and Noe (2001) framework. More specifically, the NEVA model allows banks to evaluate their claims at any time (not only at maturity) with a local knowledge of the network. In particular, all banks are assumed to have information about their own financial situation and about those of their first neighbors, i.e. the banks with whom they have a direct interbank link.

Following Barucca et al. (2020), we thus aggregate all assets and liabilities outside the interbank market (i.e., loans, bonds, reserves and deposits) into a unique balance sheet item which we label the
their original claims. In our baseline scenario, we set $\rho^{IB} = 0$. Moreover, to keep the number of banks constant within a simulation, we assume that when a bank goes bankrupt the Government steps in by providing fresh capital to the defaulted bank. The equity of the bank after the government bailout is a fraction $\vartheta \sim U(0.1, 0.9)$ of the smallest incumbent banks equity, provided it meets the regulatory capital requirements.

In our baseline scenario equity is evaluated at its book value, as calculated in Equation (11). However, such a formulation does not take into account the fact that all the other banks with whom the generic bank $k$ has direct or indirect interconnections in interbank network can default. Therefore, the book value provides only an imperfect representation of the market value of the bank.

A well-designed micro-prudential regulation provides banks with the tools to better evaluate their interbank claims. If appropriately evaluated, the equity value of banks can systematically vary from the book value. This difference might give rise to sizable impact on the credit supply of each bank, limiting their exposure to counterparty and systemic risks. In particular, in this model we investigate the properties of the NEVA mechanism (Barucca et al., 2020), which can be employed as a behavioural and micro-prudential device aimed at protecting the banks from the counterparty default risk, as well as from credit risk. The next section describes the implementation of such a mechanism in our model.

## 2.6 The NEVA clearing mechanism and micro-prudential regulation

Persistent interbank links between two banks endogenously emerge in the model when a consumption-good firm defaults on its debt. As time goes by, the owner of an interbank claim that decides not to close the position remains the holder of an interbank asset which is valued at its historical nominal value. However, this value could be misleading as it does not take into account two relevant risk factors: the counterparty risk and the credit risk. If one accounts for these two sources of risk, the market value of an interbank asset may change over time also impacting on the final value of a bank's equity, as in Equation (11). By introducing in the model a clearing mechanism that allows all banks to constantly update the market evaluation of their interbank claims, we thus allow firms to constantly update also the value of their equity.

An example of the above interbank clearing mechanism is the proposed by Eisenberg and Noe (2001). This mechanism provides (under some regularity conditions) a unique evaluation vector of the outstanding interbank claims based upon the probability of default of all banks. However, the Eisenberg and Noe (2001) evaluation model relies on two strong assumptions. First, it assumes that the evaluation of the interbank claims by each bank is carried out only at the maturity (i.e., ex-post) rather than at each period (i.e., ex-ante). Second, it assumes that all banks have a complete knowledge of the underlying network structure and the financial conditions of all the other banks. These two issues are solved by Barucca et al. (2020) with the development of the NEVA (Network Valuation), which generalizes the Eisenberg and Noe (2001) framework. More specifically, the NEVA model allows banks to evaluate their claims at any time (not only at maturity) with a local knowledge of the network. In particular, all banks are assumed to have information about their own financial situation and about those of their first neighbors, i.e. the banks with whom they have a direct interbank link.

Following Barucca et al. (2020), we thus aggregate all assets and liabilities outside the interbank market (i.e., loans, bonds, reserves and deposits) into a unique balance sheet item which we label the
"Net External Asset" and we denote it by $\Theta_{k,t}$. The book value of the generic bank's equity presented in Equation (11) can be reformulated as:

$$E_{k,t} = \Theta_{k,t} + \sum_{b=1}^{B} \ell_{k,b,t} - \sum_{b=1}^{B} \ell_{b,k,t} = \Theta_{k,t} + IB_{k,t}^{A} - IB_{k,t}^{L} \quad (12)$$

When the NEVA framework is implemented at each time step, a bank's interbank claims are evaluated in a way that potential losses associated with the bankruptcy of other banks are taken into account.

This above evaluation mechanism replaces the historical book value of equity in our model. To account for this new element, the equity Equation (12) becomes:

$$E_{k,t}^{NEVA} = \Theta_{k,t} + \sum_{b=1}^{B} \ell_{k,b,t} V_{k,b,t}^{NEVA} - \sum_{b=1}^{B} \ell_{b,k,t} = \Theta_{k,t} + IB_{k,t}^{A,NEVA} - IB_{k,t}^{L} \quad (13)$$

where $V_{k,b,t}^{NEVA}$ is the evaluation vector that bank k associates to all the direct links with other b banks in the interbank market at time t, and it is computed using the clearing mechanism described in Barucca et al. (2020). In short, this corresponds to the expected value of a valuation function that depends, on the one side, upon the expected value of the counterparty b's equity, and, on the other side, upon the expected value of b's assets under the circumstance that b defaults. Following Barucca et al. (2020), the vector $V_{k,b,t}^{NEVA}$ can be expressed as:

$$V_{k,b,t}^{NEVA} = 1 - p_{b,t}^{NEVA} - \rho_{b,t}^{NEVA}, \quad \forall k. \quad (14)$$

The term $p_{b,t}^{NEVA}$ measures the default probability of a bank b and measures the counterparty risk. The term $\rho_{b,t}^{NEVA}$, instead, is an endogenously-determined recovery rate on the assets of b, in the case b shall default. The two variables are determined as follows:

$$p_{b,t} \approx \mathbb{E}[1_{\Delta\Theta_{b,t} < E_{b,t} | \Theta_{b,t}}] \quad (15)$$

$$\rho_{b,t} \approx \mathbb{E}\left[\left(\frac{E_{b,t} + \Delta\Theta_{b,k} + IB_{b,t}^{L}}{IB_{b,t}^{L}}\right)1_{-IB_{b,t}^{L} - E_{b,t} \le \Delta\Theta_{b,t} < -E_{b,t} | \Theta_{b,t}}\right] \quad (16)$$

where $\Delta\Theta_{b,t}$ is an unexpected change in net external assets that in our model refers to the increase in the bad debt, which might unexpectedly occur at time t.<fnref n="8" />

In our model, the NEVA can be used for two different purposes, which will characterize two different scenarios. First, it can be used by each bank as a device to evaluate counterparty risk. Using the NEVA to evaluate the probability of default of its counterparts, a bank might be more careful in its decision to close or not to close a persistent interbank link, see Equation (8). Second, the NEVA can be used as a micro-prudential tool by a regulatory authority that tries to limit the credit exposure of the banking industry. We assume that the regulatory authority asks to all the banks that have an interbank claim to another bank b to reduce their leverage whenever the evaluation of the bank b is below unity – i.e., if $V_{b,t}^{NEVA} < 1$. In particular, we assume that under this micro-prudential regulatory

<footnote n="8">⁸Notice that the book value evaluation, can also be seen as a very particular case of the NEVA, in which $V_{k,b,t} = 1$ for all the interbank positions and for all periods. This will simply lead to Equation (12). In addition under some regularity conditions (see Barucca et al., 2020, pag. 1189-1195) the solution for the vector of the default probabilities of the NEVA algorithm is unique and equivalent to the one by Eisenberg and Noe (2001).</footnote>
"Net External Asset" and we denote it by $\Theta_{k,t}$. The book value of the generic bank's equity presented in Equation (11) can be reformulated as:

$$E_{k,t} = \Theta_{k,t} + \sum_{b=1}^{B} \ell_{k,b,t} - \sum_{b=1}^{B} \ell_{b,k,t} = \Theta_{k,t} + IB_{k,t}^{A} - IB_{k,t}^{L} \quad (12)$$

When the NEVA framework is implemented at each time step, a bank's interbank claims are evaluated in a way that potential losses associated with the bankruptcy of other banks are taken into account.

This above evaluation mechanism replaces the historical book value of equity in our model. To account for this new element, the equity Equation (12) becomes:

$$E_{k,t}^{NEVA} = \Theta_{k,t} + \sum_{b=1}^{B} \ell_{k,b,t} V_{k,b,t}^{NEVA} - \sum_{b=1}^{B} \ell_{b,k,t} = \Theta_{k,t} + IB_{k,t}^{A,NEVA} - IB_{k,t}^{L} \quad (13)$$

where $V_{k,b,t}^{NEVA}$ is the evaluation vector that bank k associates to all the direct links with other b banks in the interbank market at time t, and it is computed using the clearing mechanism described in Barucca et al. (2020). In short, this corresponds to the expected value of a valuation function that depends, on the one side, upon the expected value of the counterparty b's equity, and, on the other side, upon the expected value of b's assets under the circumstance that b defaults. Following Barucca et al. (2020), the vector $V_{k,b,t}^{NEVA}$ can be expressed as:

$$V_{k,b,t}^{NEVA} = 1 - p_{b,t}^{NEVA} - \rho_{b,t}^{NEVA}, \quad \forall k. \quad (14)$$

The term $p_{b,t}^{NEVA}$ measures the default probability of a bank b and measures the counterparty risk. The term $\rho_{b,t}^{NEVA}$, instead, is an endogenously-determined recovery rate on the assets of b, in the case b shall default. The two variables are determined as follows:

$$p_{b,t} \approx \mathbb{E}[1_{\Delta\Theta_{b,t} < E_{b,t} | \Theta_{b,t}}] \quad (15)$$

$$\rho_{b,t} \approx \mathbb{E}\left[\left(\frac{E_{b,t} + \Delta\Theta_{b,k} + IB_{b,t}^{L}}{IB_{b,t}^{L}}\right)1_{-IB_{b,t}^{L} - E_{b,t} \le \Delta\Theta_{b,t} < -E_{b,t} | \Theta_{b,t}}\right] \quad (16)$$

where $\Delta\Theta_{b,t}$ is an unexpected change in net external assets that in our model refers to the increase in the bad debt, which might unexpectedly occur at time t.<fnref n="8" />

In our model, the NEVA can be used for two different purposes, which will characterize two different scenarios. First, it can be used by each bank as a device to evaluate counterparty risk. Using the NEVA to evaluate the probability of default of its counterparts, a bank might be more careful in its decision to close or not to close a persistent interbank link, see Equation (8). Second, the NEVA can be used as a micro-prudential tool by a regulatory authority that tries to limit the credit exposure of the banking industry. We assume that the regulatory authority asks to all the banks that have an interbank claim to another bank b to reduce their leverage whenever the evaluation of the bank b is below unity – i.e., if $V_{b,t}^{NEVA} < 1$. In particular, we assume that under this micro-prudential regulatory

<footnote n="8">⁸Notice that the book value evaluation, can also be seen as a very particular case of the NEVA, in which $V_{k,b,t} = 1$ for all the interbank positions and for all periods. This will simply lead to Equation (12). In addition under some regularity conditions (see Barucca et al., 2020, pag. 1189-1195) the solution for the vector of the default probabilities of the NEVA algorithm is unique and equivalent to the one by Eisenberg and Noe (2001).</footnote>
settings, the banks should always take into account a fall in the evaluation of $V_{.,b}^{NEVA}$ in their measure of financial fragility, which is captured by the variable $Bda_{k,t}$. More specifically, for all banks holding a credit with $b$ the prudential buffer writes:

$$ Bda_{k,t} = \frac{\gamma_{BD}BadDebt_{k,t}}{\Theta_{k,t} + \sum_{b=1}^{B} \ell_{k,b,t} V_{k,b,t}^{NEVA}} \quad (17) $$

If equity were evaluated at its book value, then the supply of credit (see Equation 1) would be completely inelastic to the evolution of risk in the interbank market. The introduction of the NEVA relaxes this feature.

## 2.7 Risk Evaluation Scenarios

The decision of a creditor bank to close an interbank position in the model depends upon the probability of default of the borrower bank, measured by $p_{b,t}$, as well as on the relative interbank exposure of the lender, captured by $\xi_{k,b,t}$, (see Equation (8) and Section 2.4). We can use these two variables to define four alternative interbank risk evaluation scenarios. The details of these scenarios are summarized in Table 1

**Baseline scenario (BASE).** In our baseline specification, the NEVA is switched off. In this scenario, $p_{b,t}$ is exogenous and drawn from a Uniform distribution $U(0, 2h_0)$, where the expected value $h_0$ can be interpreted as the average of idiosyncratic shocks to the haircut on the collateral that the creditor can obtain in case of default of a debtor bank.

**Semi-endogenous (SEMI).** This scenario is similar to the baseline, except that now the creditor banks can close the riskiest positions in the interbank market. More specifically, banks apply the previous decision rule based on the haircut and a draw from a uniform distribution, only if they have an open interbank claim toward the riskiest bank in the system, as measured by the firm with the largest share of non-performing loans.

**NEVA.** In this scenario, a creditor bank in the interbank market decides to close a position according to Equation (13) and hence it employs the NEVA clearing vector for better evaluating the counterpart risk.

**NEVA².** This last scenario is characterized by the presence of NEVA, employed both to guide the banks' decision about closing or not interbank claims according to Equation (13), as well as to fix the supply of credit for the next period, by evaluating interbank exposures according to Equation (17).

<table><caption>**Table 1:** Interbank market scenarios</caption><thead><tr><th>Scenario</th><th>Probability of default p<sub>b,t</sub></th><th>Valuation of Interbank Claims</th></tr></thead><tbody><tr><td>BASE</td><td>U(0, 2h<sub>0</sub>)</td><td>X</td></tr><tr><td>SEMI</td><td>U(0, 2h<sub>0</sub>), for Bank b: arg max<sub>b∈B</sub> Bda<sub>b,t</sub></td><td>X</td></tr><tr><td>NEVA</td><td>p<sub>b,t</sub><sup>NEVA</sup></td><td>X</td></tr><tr><td>NEVA<sup>2</sup></td><td>p<sub>b,t</sub><sup>NEVA</sup></td><td>Σ<sub>b=1</sub><sup>B</sup> ℓ<sub>k,b,t</sub> V<sub>k,b,t</sub><sup>NEVA</sup></td></tr></tbody></table>
settings, the banks should always take into account a fall in the evaluation of $V_{.,b}^{NEVA}$ in their measure of financial fragility, which is captured by the variable $Bda_{k,t}$. More specifically, for all banks holding a credit with b the prudential buffer writes:

$$Bda_{k,t} = \frac{\gamma_{BD}BadDebt_{k,t}}{\Theta_{k,t} + \sum_{b=1}^{B} \ell_{k,b,t} V_{k,b,t}^{NEVA}} \quad (17)$$

If equity were evaluated at its book value, then the supply of credit (see Equation 1) would be completely inelastic to the evolution of risk in the interbank market. The introduction of the NEVA relaxes this feature.

## 2.7 Risk Evaluation Scenarios

The decision of a creditor bank to close an interbank position in the model depends upon the probability of default of the borrower bank, measured by $p_{b,t}$, as well as on the relative interbank exposure of the lender, captured by $\xi_{k,b,t}$, (see Equation (8) and Section 2.4). We can use these two variables to define four alternative interbank risk evaluation scenarios. The details of these scenarios are summarized in Table 1

**Baseline scenario (BASE).** In our baseline specification, the NEVA is switched off. In this scenario, $p_{b,t}$ is exogenous and drawn from a Uniform distribution $U(0, 2h_0)$, where the expected value $h_0$ can be interpreted as the average of idiosyncratic shocks to the haircut on the collateral that the creditor can obtain in case of default of a debtor bank.

**Semi-endogenous (SEMI).** This scenario is similar to the baseline, except that now the creditor banks can close the riskiest positions in the interbank market. More specifically, banks apply the previous decision rule based on the haircut and a draw from a uniform distribution, only if they have an open interbank claim toward the riskiest bank in the system, as measured by the firm with the largest share of non-performing loans.

**NEVA.** In this scenario, a creditor bank in the interbank market decides to close a position according to Equation (13) and hence it employs the NEVA clearing vector for better evaluating the counterpart risk.

**NEVA².** This last scenario is characterized by the presence of NEVA, employed both to guide the banks' decision about closing or not interbank claims according to Equation (13), as well as to fix the supply of credit for the next period, by evaluating interbank exposures according to Equation (17).

<table><caption>**Table 1:** Interbank market scenarios</caption><thead><tr><th>Scenario</th><th>Probability of default p<sub>b,t</sub></th><th>Valuation of Interbank Claims</th></tr></thead><tbody><tr><td>BASE</td><td>U(0, 2h<sub>0</sub>)</td><td>X</td></tr><tr><td>SEMI</td><td>U(0, 2h<sub>0</sub>), for Bank b: arg max<sub>b∈B</sub> Bda<sub>b,t</sub></td><td>X</td></tr><tr><td>NEVA</td><td>p<sub>b,t</sub><sup>NEVA</sup></td><td>X</td></tr><tr><td>NEVA<sup>2</sup></td><td>p<sub>b,t</sub><sup>NEVA</sup></td><td>Σ<sub>b=1</sub><sup>B</sup> ℓ<sub>k,b,t</sub> V<sub>k,b,t</sub><sup>NEVA</sup></td></tr></tbody></table>
## 2.8 Timeline of events

At each time step $t$ of our simulations, agents take the following decisions:

1. Policy variables (e.g., capital requirement, tax rate, central bank interest rate, etc.) are fixed.

2. Machines ordered by consumption-good firms in the previous period are delivered and will become part of the capital stock.

3. Capital and consumption-good firms calculate costs and set their prices.

4. The maximal amount of credit supplied by each bank to consumption-good firms is determined according to Equation (1).

5. Capital-good firms signal the discovery of new machines, if any, to consumption-good firms.

6. Consumption-good firms fix their desired level of production and investment and eventually their demand for bank credit.

7. Banks rank firms' request and provide credit. Credit rationing may possibly arise.

8. Consumption-good firms receive the "brochures" of the brand new machines produced by capital-good firms and buy the most convenient.

9. Banks settle the payments between consumption-good and capital-good firms.

10. Both firms calculate the labour input necessary for production.

11. Production plans are undertaken.

12. Consumption-good firms' market shares are allocated based on their competitiveness.

13. Firms in both sectors compute profits. If profits are positive, consumption-good firms pay back their loans to their bank and deposit their net savings, if any. If some consumption-good firms are unable to pay back their loans, banks involved in the payment system open an interbank position.

14. Banks evaluate their interbank exposures and decide whether to close their interbank claim according to Equation (8). Firms' and banks' profits are taxed.

15. Entry and exit of consumption-good and capital-good firms takes place. In both sectors firms with market shares smaller than a minimum threshold and firms with negative net liquid assets exit from the market and are replaced by new entrants.

16. Bank profits are calculated. Banks pay taxes and dividends.

17. Government bails out banks with realized negative equity.

18. Government calculate its budget and, if negative, deficits are financed by banks.

19. Aggregate macroeconomic and financial variables are calculated.

20. Capital-good firms perform R&D, trying to discover new products and more efficient production techniques and to imitate the technologies and the products of their competitors.
## 2.8 Timeline of events

At each time step $t$ of our simulations, agents take the following decisions:

1. Policy variables (e.g., capital requirement, tax rate, central bank interest rate, etc.) are fixed.

2. Machines ordered by consumption-good firms in the previous period are delivered and will become part of the capital stock.

3. Capital and consumption-good firms calculate costs and set their prices.

4. The maximal amount of credit supplied by each bank to consumption-good firms is determined according to Equation (1).

5. Capital-good firms signal the discovery of new machines, if any, to consumption-good firms.

6. Consumption-good firms fix their desired level of production and investment and eventually their demand for bank credit.

7. Banks rank firms' request and provide credit. Credit rationing may possibly arise.

8. Consumption-good firms receive the "brochures" of the brand new machines produced by capital-good firms and buy the most convenient.

9. Banks settle the payments between consumption-good and capital-good firms.

10. Both firms calculate the labour input necessary for production.

11. Production plans are undertaken.

12. Consumption-good firms' market shares are allocated based on their competitiveness.

13. Firms in both sectors compute profits. If profits are positive, consumption-good firms pay back their loans to their bank and deposit their net savings, if any. If some consumption-good firms are unable to pay back their loans, banks involved in the payment system open an interbank position.

14. Banks evaluate their interbank exposures and decide whether to close their interbank claim according to Equation (8). Firms' and banks' profits are taxed.

15. Entry and exit of consumption-good and capital-good firms takes place. In both sectors firms with market shares smaller than a minimum threshold and firms with negative net liquid assets exit from the market and are replaced by new entrants.

16. Bank profits are calculated. Banks pay taxes and dividends.

17. Government bails out banks with realized negative equity.

18. Government calculate its budget and, if negative, deficits are financed by banks.

19. Aggregate macroeconomic and financial variables are calculated.

20. Capital-good firms perform R&D, trying to discover new products and more efficient production techniques and to imitate the technologies and the products of their competitors.
## 3 Simulation results

In this section, we present the main results obtained from our extensive Monte Carlo simulations of our model in the baseline scenario. For this baseline simulation setting, we run 1000 Monte Carlo simulations in which only the pseudo-Random Number Generator has been modified. This allows us to obtain a sufficient set of observations to test the statistical significance of our claims. In addition, we discard the first 300 time observations for each batch run to ensure that the statistics are washed away by the noise due to initial conditions.

To broadly validate the model, we follow the indirect inference approach (see Windrum et al., 2007; Fagiolo and Roventini, 2017) also employed by all the previous versions of the *Schumpeter meeting Keynes* family of models. In particular, we evaluate the ability of our model to replicate a wide range of stylized facts at the micro-, meso- and macroeconomic levels. Specifically focusing on this model, the indirect calibration process aims to qualitatively replicate observed data concerning macro-financial factors and several statistical characteristics of the interbank network involving borrowing and lending connections (see Fagiolo et al., 2019). Finally, we also verify the continued validity of the micro- and macro- properties reproduced by earlier versions of the K+S models within our extended framework.

<table><thead><tr><th></th><th>μ̄</th><th>μ<sub>0.5</sub></th><th>σ</th><th></th><th>μ̄</th><th>μ<sub>0.5</sub></th><th>σ</th></tr></thead><tbody><tr><td>GDP</td><td>0.029</td><td>0.030</td><td>0.003</td><td>Deposits</td><td>0.059</td><td>0.062</td><td>0.010</td></tr><tr><td>Consumption</td><td>0.029</td><td>0.030</td><td>0.003</td><td>Bad Debt</td><td>0.059</td><td>0.060</td><td>0.050</td></tr><tr><td>Investment</td><td>0.028</td><td>0.029</td><td>0.005</td><td>Cash Reserves</td><td>0.057</td><td>0.061</td><td>0.013</td></tr><tr><td>Unempl. rate</td><td>0.069</td><td>0.037</td><td>0.111</td><td>Interbank Exposure<sup>c</sup></td><td>0.060</td><td>0.062</td><td>0.018</td></tr><tr><td>Inflation</td><td>0.031</td><td>0.032</td><td>0.005</td><td>Total Credit</td><td>0.059</td><td>0.061</td><td>0.009</td></tr><tr><td>Debt/GDP</td><td>0.847</td><td>0.116</td><td>3.226</td><td>Losses due to contagion</td><td>0.031</td><td>0.000</td><td>0.104</td></tr><tr><td>Bank defaults<sup>a</sup></td><td>28.000</td><td>3.000</td><td>71.000</td><td>Bank Fragility<sup>d</sup></td><td>0.083</td><td>0.070</td><td>0.066</td></tr><tr><td>Fin. constraints<sup>b</sup></td><td>0.250</td><td>0.223</td><td>0.095</td><td>Bank Equity</td><td>0.057</td><td>0.061</td><td>0.013</td></tr></tbody></table>

<sup>a</sup>: Number of bank defaults

<sup>b</sup>: Share of credit rationed firms

<sup>c</sup>: Interbank losses over interbank claims

<sup>d</sup>: Average financial fragility index *Bda* as in Equation (1)

<caption>**Table 2:** Summary statistics of the main aggregate macroeconomic and financial variables growth rates (unless otherwise stated). The average $\bar{\mu}$, median $\mu_{0.5}$ and standard deviation $\sigma$ are computed across 1000 Monte Carlo experiments in the baseline configuration.</caption>

Table 2 presents a collection of key Monte Carlo statistics produced by our model in the baseline scenario. The results of the table show that our model can generate the following outcomes: (i) realistic growth rates for GDP, consumption, and investment, arranged in an accurate hierarchy of volatility; (ii) low unemployment rates; and (iii) consistent inflation stability. This indicates that in the baseline scenario, the model effectively emulates a healthy economic system characterized by endogenous growth and business cycles. Moreover, financial variables, which are significantly impacted by the extensions described in Section 2 exhibit, on average, higher rates of growth compared to real variables, but they are also much more volatile. This finding is compatible with the empirical evidence brought about by Zoltan and Kumhof (2015).

In the next Section we will discuss in detail the ability of the model to account for a set of stylized facts related to the co-movements of financial aggregates and fundamental features inherent to interbank networks. Notably, K+S models (Dosi et al., 2010, 2013, 2015; Lamperti et al., 2018; Guerini et al., 2022) can account, jointly, for a wide ensemble of empirical regularities, making it
## 3 Simulation results

In this section, we present the main results obtained from our extensive Monte Carlo simulations of our model in the baseline scenario. For this baseline simulation setting, we run 1000 Monte Carlo simulations in which only the pseudo-Random Number Generator has been modified. This allows us to obtain a sufficient set of observations to test the statistical significance of our claims. In addition, we discard the first 300 time observations for each batch run to ensure that the statistics are washed away by the noise due to initial conditions.

To broadly validate the model, we follow the indirect inference approach (see Windrum et al., 2007; Fagiolo and Roventini, 2017) also employed by all the previous versions of the *Schumpeter meeting Keynes* family of models. In particular, we evaluate the ability of our model to replicate a wide range of stylized facts at the micro-, meso- and macroeconomic levels. Specifically focusing on this model, the indirect calibration process aims to qualitatively replicate observed data concerning macro-financial factors and several statistical characteristics of the interbank network involving borrowing and lending connections (see Fagiolo et al., 2019). Finally, we also verify the continued validity of the micro- and macro- properties reproduced by earlier versions of the K+S models within our extended framework.

<table><thead><tr><th></th><th>μ̄</th><th>μ<sub>0.5</sub></th><th>σ</th><th></th><th>μ̄</th><th>μ<sub>0.5</sub></th><th>σ</th></tr></thead><tbody><tr><td>GDP</td><td>0.029</td><td>0.030</td><td>0.003</td><td>Deposits</td><td>0.059</td><td>0.062</td><td>0.010</td></tr><tr><td>Consumption</td><td>0.029</td><td>0.030</td><td>0.003</td><td>Bad Debt</td><td>0.059</td><td>0.060</td><td>0.050</td></tr><tr><td>Investment</td><td>0.028</td><td>0.029</td><td>0.005</td><td>Cash Reserves</td><td>0.057</td><td>0.061</td><td>0.013</td></tr><tr><td>Unempl. rate</td><td>0.069</td><td>0.037</td><td>0.111</td><td>Interbank Exposure<sup>c</sup></td><td>0.060</td><td>0.062</td><td>0.018</td></tr><tr><td>Inflation</td><td>0.031</td><td>0.032</td><td>0.005</td><td>Total Credit</td><td>0.059</td><td>0.061</td><td>0.009</td></tr><tr><td>Debt/GDP</td><td>0.847</td><td>0.116</td><td>3.226</td><td>Losses due to contagion</td><td>0.031</td><td>0.000</td><td>0.104</td></tr><tr><td>Bank defaults<sup>a</sup></td><td>28.000</td><td>3.000</td><td>71.000</td><td>Bank Fragility<sup>d</sup></td><td>0.083</td><td>0.070</td><td>0.066</td></tr><tr><td>Fin. constraints<sup>b</sup></td><td>0.250</td><td>0.223</td><td>0.095</td><td>Bank Equity</td><td>0.057</td><td>0.061</td><td>0.013</td></tr></tbody></table>

<sup>a</sup>: Number of bank defaults

<sup>b</sup>: Share of credit rationed firms

<sup>c</sup>: Interbank losses over interbank claims

<sup>d</sup>: Average financial fragility index *Bda* as in Equation (1)

<caption>**Table 2:** Summary statistics of the main aggregate macroeconomic and financial variables growth rates (unless otherwise stated). The average $\bar{\mu}$, median $\mu_{0.5}$ and standard deviation $\sigma$ are computed across 1000 Monte Carlo experiments in the baseline configuration.</caption>

Table 2 presents a collection of key Monte Carlo statistics produced by our model in the baseline scenario. The results of the table show that our model can generate the following outcomes: (i) realistic growth rates for GDP, consumption, and investment, arranged in an accurate hierarchy of volatility; (ii) low unemployment rates; and (iii) consistent inflation stability. This indicates that in the baseline scenario, the model effectively emulates a healthy economic system characterized by endogenous growth and business cycles. Moreover, financial variables, which are significantly impacted by the extensions described in Section 2 exhibit, on average, higher rates of growth compared to real variables, but they are also much more volatile. This finding is compatible with the empirical evidence brought about by Zoltan and Kumhof (2015).

In the next Section we will discuss in detail the ability of the model to account for a set of stylized facts related to the co-movements of financial aggregates and fundamental features inherent to interbank networks. Notably, K+S models (Dosi et al., 2010, 2013, 2015; Lamperti et al., 2018; Guerini et al., 2022) can account, jointly, for a wide ensemble of empirical regularities, making it
suitable to deliver credible insights into policy scenarios (Dawid and Delli Gatti, 2018; Dosi and Roventini, 2019b). For the full list of stylized facts replicated by the model, we refer to Table 4 in Appendix B.

## 3.1 Empirical predictions of the model

Our model incorporates a process of firm financing that involves money creation by banks (Palley, 1996; Godley and Lavoie, 2006; Jakab and Kumhof, 2014; Caverzasi and Godin, 2015; Jakab and Kumhof, 2018) and wherein the lending decisions depend on leverage and capital requirements (see Equation 1). This is in line with the working of modern financial markets (McLeay et al., 2014; Ihrig et al., 2021). In this framework, whenever a bank extends fresh funds through lending, these funds are promptly deposited by the borrowers. This straightforward mechanism results in a nearly one-to-one adjustment of the banks' debt to a variation in assets. On the other hand, equity remains relatively unresponsive to changes in assets. Indeed, equity will increase only if banks generate profits, which is not immediately related to assets' variation.

This distinction in the sensitivity of debt and equity to fluctuations in assets has also been empirically recorded by both Adrian et al. (2013) and Zoltan and Kumhof (2015). Our agent-based model replicates this empirical finding as an emergent property of the system. Panel A of Figure 3 presents the relationship between the log-changes in banks' equity (red) and debt (blue) as a function of the log-changes in total assets. It is immediate to observe that the elasticity of debt to assets is high and slightly larger than one. Conversely, the response of equity to changes in assets is rather flat.

Panel B of Figure 3 shows instead the distribution of the two elasticity parameters, estimated with simulated bank-level data. It is easy to observe that the estimate for the Debt-to-Assets elasticity is centered around 1, while the Equity-to-Assets elasticity is centered around 0. Models utilizing the loanable funds approach (examples are, but not limited to Bernanke et al., 1999; Gertler and Kiyotaki, 2010; Adrian and Boyarchenko, 2012), in which banks solely serve as intermediaries for private sector savings, often fail to replicate such empirical patterns. Our results suggest that a financing mechanism driven by endogenous money creation offers a more credible description of banking industry behavior.

Next, we focus on the composition of the asset and liability sides of individual banks' balance sheets over time in a typical Monte Carlo simulation (cf., Figure 4). First, coherently with Jakab and Kumhof (2018), our model predicts that banks' balance sheets experience frequent oscillations in the level of their interbank assets and liabilities. Second, the asset side (Panel A of the figure) shows that the loans granted to the consumption-good firms and reserves constitute the most relevant components of the asset side of the balance sheet. Pooling all bank-level observations, these two components, on average, represent 58% and 33% of the whole amount of assets. Loans to consumption-good firms and interbank positions are the components determining banks' heterogeneity in the assets' and liabilities composition.<fnref n="9" />

To analyze how the creation and destruction of interbank claims affect balance sheet adjustments, let us consider the example of a representative bank (bank 5 in the simulation) in the last interval of a Monte Carlo iteration (few time steps before period 450). Looking at its liabilities (cf., Panel B

<footnote n="9">⁹Similarly to Dosi et al. (2015), the government bonds are bought in proportion to the bank's size and play a marginal role in our model.</footnote>
suitable to deliver credible insights into policy scenarios (Dawid and Delli Gatti, 2018; Dosi and Roventini, 2019b). For the full list of stylized facts replicated by the model, we refer to Table 4 in Appendix B.

## 3.1 Empirical predictions of the model

Our model incorporates a process of firm financing that involves money creation by banks (Palley, 1996; Godley and Lavoie, 2006; Jakab and Kumhof, 2014; Caverzasi and Godin, 2015; Jakab and Kumhof, 2018) and wherein the lending decisions depend on leverage and capital requirements (see Equation 1). This is in line with the working of modern financial markets (McLeay et al., 2014; Ihrig et al., 2021). In this framework, whenever a bank extends fresh funds through lending, these funds are promptly deposited by the borrowers. This straightforward mechanism results in a nearly one-to-one adjustment of the banks' debt to a variation in assets. On the other hand, equity remains relatively unresponsive to changes in assets. Indeed, equity will increase only if banks generate profits, which is not immediately related to assets' variation.

This distinction in the sensitivity of debt and equity to fluctuations in assets has also been empirically recorded by both Adrian et al. (2013) and Zoltan and Kumhof (2015). Our agent-based model replicates this empirical finding as an emergent property of the system. Panel A of Figure 3 presents the relationship between the log-changes in banks' equity (red) and debt (blue) as a function of the log-changes in total assets. It is immediate to observe that the elasticity of debt to assets is high and slightly larger than one. Conversely, the response of equity to changes in assets is rather flat.

Panel B of Figure 3 shows instead the distribution of the two elasticity parameters, estimated with simulated bank-level data. It is easy to observe that the estimate for the Debt-to-Assets elasticity is centered around 1, while the Equity-to-Assets elasticity is centered around 0. Models utilizing the loanable funds approach (examples are, but not limited to Bernanke et al., 1999; Gertler and Kiyotaki, 2010; Adrian and Boyarchenko, 2012), in which banks solely serve as intermediaries for private sector savings, often fail to replicate such empirical patterns. Our results suggest that a financing mechanism driven by endogenous money creation offers a more credible description of banking industry behavior.

Next, we focus on the composition of the asset and liability sides of individual banks' balance sheets over time in a typical Monte Carlo simulation (cf., Figure 4). First, coherently with Jakab and Kumhof (2018), our model predicts that banks' balance sheets experience frequent oscillations in the level of their interbank assets and liabilities. Second, the asset side (Panel A of the figure) shows that the loans granted to the consumption-good firms and reserves constitute the most relevant components of the asset side of the balance sheet. Pooling all bank-level observations, these two components, on average, represent 58% and 33% of the whole amount of assets. Loans to consumption-good firms and interbank positions are the components determining banks' heterogeneity in the assets' and liabilities composition.<fnref n="9" />

To analyze how the creation and destruction of interbank claims affect balance sheet adjustments, let us consider the example of a representative bank (bank 5 in the simulation) in the last interval of a Monte Carlo iteration (few time steps before period 450). Looking at its liabilities (cf., Panel B

<footnote n="9">⁹Similarly to Dosi et al. (2015), the government bonds are bought in proportion to the bank's size and play a marginal role in our model.</footnote>
<figure><img src="image_6.png" /><figcaption>**Figure 3: Panel A.** A 2-dimensional density plot (log) changes in bank Debt/Equity (y-axis) vs Bank Assets (x-axis), along with a 45 degrees line (dotted). **Panel B.** Univariate distribution of the bank-level elasticities of bank Debt to bank Assets (top) are centered around one (dotted vertical line). Univariate distribution of the bank-level elasticities of bank Equity to bank Assets (bottom) are centered around zero (dashed vertical line). All observations are at bank-level, pooled across all time periods for all Monte Carlo iterations.</figcaption></figure>

of Figure 4), we observe that the bank records a sudden increase in the share of its interbank debt, which emerges due to the default of at least one of its consumption-good firm clients. Part of this loan corresponds to the payment made in advance to buy the machinery of capital-good firms, which are in turn connected to banks that would experience sudden increases in the share of interbank assets (in the example, the increase is visible in the asset side of bank 1 and 2).

Up to now our discussion focused on the model’s predictions about the macroeconomic properties of the banking sector and on the microeconomic properties of individual banks’ balance sheets. Let us now examine the model’s predictions concerning the statistical properties of the interbank network. More specifically, we examine network assortativity and network density.<fnref n="10" /> In addition, we study node-specific features of the network, like the centrality and exposure of the single banks, split by their size class.

Empirical studies (see Soramäki et al., 2007; Cocco et al., 2009, among the others) show that the network of interbank credit-debit relationship is disassortative and has a low density, recorded to be about 0.3% on average. Moreover, the network typically features a high-density core (i.e., a group of few densely connected banks) and a low-density periphery (i.e., banks that have few connections. See Alves et al., 2013; Aldasoro and Alves, 2018).

In particular, for the banking industry case, disassortativity implies that small banks, whose size is measured by their number of clients, will tend to be connected to banks with large sizes.

Our mechanism of interbank link formation, despite its simplicity, reproduces all above-mentioned stylized facts about the structure of an interbank network. Indeed, our model generates a network with a density that remains quite stable over time and averages 15% for the overall network (see Panel A of Figure 5). In addition, consistent with empirical evidence, we find can identify a core of the network with high density (around 35%) and a periphery with lower density (slightly below 10%). Panel B of Figure 5 presents instead the Monte Carlo average of the assortativity index over

<footnote n="10">¹⁰A network is said to be assortative (disassortative) if there is a higher likelihood that a network’s node is connected to other nodes with similar (different) characteristics. (For the definition of assortativity indexes see Section 3.E of Newman, 2003).</footnote>
<figure><img src="image_8.png" /><figcaption>**Figure 3: Panel A.** A 2-dimensional density plot (log) changes in bank Debt/Equity (y-axis) vs Bank Assets (x-axis), along with a 45 degrees line (dotted). **Panel B.** Univariate distribution of the bank-level elasticities of bank Debt to bank Assets (top) are centered around one (dotted vertical line). Univariate distribution of the bank-level elasticities of bank Equity to bank Assets (bottom) are centered around zero (dashed vertical line). All observations are at bank-level, pooled across all time periods for all Monte Carlo iterations.</figcaption></figure>

of Figure 4), we observe that the bank records a sudden increase in the share of its interbank debt, which emerges due to the default of at least one of its consumption-good firm clients. Part of this loan corresponds to the payment made in advance to buy the machinery of capital-good firms, which are in turn connected to banks that would experience sudden increases in the share of interbank assets (in the example, the increase is visible in the asset side of bank 1 and 2).

Up to now our discussion focused on the model’s predictions about the macroeconomic properties of the banking sector and on the microeconomic properties of individual banks’ balance sheets. Let us now examine the model’s predictions concerning the statistical properties of the interbank network. More specifically, we examine network assortativity and network density.<fnref n="10" /> In addition, we study node-specific features of the network, like the centrality and exposure of the single banks, split by their size class.

Empirical studies (see Soramäki et al., 2007; Cocco et al., 2009, among the others) show that the network of interbank credit-debit relationship is disassortative and has a low density, recorded to be about 0.3% on average. Moreover, the network typically features a high-density core (i.e., a group of few densely connected banks) and a low-density periphery (i.e., banks that have few connections. See Alves et al., 2013; Aldasoro and Alves, 2018).

In particular, for the banking industry case, disassortativity implies that small banks, whose size is measured by their number of clients, will tend to be connected to banks with large sizes.

Our mechanism of interbank link formation, despite its simplicity, reproduces all above-mentioned stylized facts about the structure of an interbank network. Indeed, our model generates a network with a density that remains quite stable over time and averages 15% for the overall network (see Panel A of Figure 5). In addition, consistent with empirical evidence, we find can identify a core of the network with high density (around 35%) and a periphery with lower density (slightly below 10%). Panel B of Figure 5 presents instead the Monte Carlo average of the assortativity index over

<footnote n="10">¹⁰A network is said to be assortative (disassortative) if there is a higher likelihood that a network’s node is connected to other nodes with similar (different) characteristics. (For the definition of assortativity indexes see Section 3.E of Newman, 2003).</footnote>
<figure><img src="image_7.png" /></figure>

<figure><img src="image_8.png" /><figcaption>**Figure 4:** Composition of the assets (Panel A) and liability (Panel B) sides of individual banks' balance sheets in a representative Monte Carlo simulation.</figcaption></figure>
<figure><img src="image_9.png" /></figure>

<figure><img src="image_10.png" /><figcaption>**Figure 4:** Composition of the assets (Panel A) and liability (Panel B) sides of individual banks' balance sheets in a representative Monte Carlo simulation.</figcaption></figure>
A Interbank Network Density

<figure><img src="image_9.png" /></figure>

B Assortativity index

<figure><img src="image_10.png" /><figcaption>**Figure 5:** Assortativity (A) and density (B) indexes of the emergent interbank network. The indexes are computed as averages of the 1000 Monte Carlo simulations at each time step along with the 5-th and 95-th percentiles confidence bands (light blue). We also report a 90% confidence interval for the sample mean (dark blue).</figcaption></figure>

time. This is always negative, thus confirming that the interbank network generated by our model is
disassortative.

A Average Vertex Centrality

<figure><img src="image_11.png" /></figure>

B Net Interbank position

<figure><img src="image_12.png" /><figcaption>**Figure 6:** Panel A shows the Monte Carlo average number of links by type and bank size. Panel B plots the evolution of the Monte Carlo average value of net interbank assets as a percentage of total interbank exposures (breakdown by size). Confidence bands denote the 5% and 95% percentiles and the 90% confidence interval for the sample mean (lighter and darker colours, respectively).</figcaption></figure>

Let us now consider node-specific network statistics. In this respect, a major role is played by
measures of centrality which capture the relative importance of each node in the network.<fnref n="11" /> The
results are also disaggregated by bank size classes (see Figure 6). In particular, for each size class, we
report the average centrality of the banks belonging to that specific class.<fnref n="12" /> A standard result in the

<footnote n="11">¹¹There is a plethora of centrality measures that can be employed. In what follows we focus on of the most popular measures: the overall centrality accounting for incoming and outgoing links; the in-degree centrality accounting only for the incoming ones; and the out-degree centrality taking into account only the outgoing links. (See Newman, 2018).</footnote>

<footnote n="12">¹²Taking into account that the distribution of banks' clients is Pareto, the size-class categories are constructed as follows: big banks are those in top 20% of the distribution, medium-sized banks are between the 30-th and 80-th percentiles, small</footnote>
<figure><img src="image_11.png" /><figcaption>**Figure 5:** Assortativity (A) and density (B) indexes of the emergent interbank network. The indexes are computed as averages of the 1000 Monte Carlo simulations at each time step along with the 5-th and 95-th percentiles confidence bands (light blue). We also report a 90% confidence interval for the sample mean (dark blue).</figcaption></figure>

time. This is always negative, thus confirming that the interbank network generated by our model is disassortative.

<figure><img src="image_12.png" /><figcaption>**Figure 6:** Panel A shows the Monte Carlo average number of links by type and bank size. Panel B plots the evolution of the Monte Carlo average value of net interbank assets as a percentage of total interbank exposures (breakdown by size). Confidence bands denote the 5% and 95% percentiles and the 90% confidence interval for the sample mean (lighter and darker colours, respectively).</figcaption></figure>

Let us now consider node-specific network statistics. In this respect, a major role is played by measures of centrality which capture the relative importance of each node in the network.<fnref n="11" /> The results are also disaggregated by bank size classes (see Figure 6). In particular, for each size class, we report the average centrality of the banks belonging to that specific class.<fnref n="12" /> A standard result in the

<footnote n="11">¹¹There is a plethora of centrality measures that can be employed. In what follows we focus on of the most popular measures: the overall centrality accounting for incoming and outgoing links; the in-degree centrality accounting only for the incoming ones; and the out-degree centrality taking into account only the outgoing links. (See Newman, 2018).</footnote>

<footnote n="12">¹²Taking into account that the distribution of banks' clients is Pareto, the size-class categories are constructed as follows: big banks are those in top 20% of the distribution, medium-sized banks are between the 30-th and 80-th percentiles, small</footnote>
empirical interbank network literature is that big banks are significantly more central, especially when the outgoing links are considered (Craig and von Peter, 2014; Fricke and Lux, 2015). Furthermore, empirical evidence suggests that larger banks tend to borrow cash reserves in the interbank market (see Müller, 2006, for the Swiss interbank market). The results from our simulations are consistent with this evidence (see Figure 6). The result is explained by the fact that big banks have many more consumption-good firm clients who can default on their debts. And when such clients default, these banks become borrowers in the interbank market.

Finally, the simulation results indicate that our model generates a banking structure wherein small banks are liquidity providers; medium-sized banks are liquidity neutral; and large banks are liquidity borrowers (see the right panel of Figure 6).<sup>13</sup> This is also a prominent feature observed in real interbank networks (see Liu et al., 2020).

## 3.2 The effects of a network-based micro-prudential regulation

Given the good empirical performance of the model, in this section we present the results of policy evaluation exercises where we compare the performance of the economy and the banking sector in the four different scenarios described in Section 2.7. These exercises aim to test whether the NEVA clearing mechanism can dampen systemic risk, as measured by the total number of banks’ defaults and by the losses experienced by non-defaulting banks because of the contagion from defaulted ones.

As explained in Section 2.3, the NEVA algorithm endows banks with an adaptive local-information criterion that helps them to better and promptly assess the current value of their interbank claims, also taking into account the possibility that their counterparts will default. Therefore, the two scenarios in which it is employed can be considered scenarios wherein banks are endowed with a new micro-prudential tool. Finding that the NEVA has a beneficial impact on the financial soundness of banks and systemic risk reduction in these scenarios would be good news for the financial regulator, as this mechanism is much less demanding in terms of information requirements about the interbank network. Indeed, in contrast to the Eisenberg and Noe (2001) alternative, which requires global knowledge of the whole network structure, the NEVA only requires that each bank has information about the financial soundness of its interbank counterparts.

Table 3 compares the simulation results about the variables capturing systemic risk (first horizontal block), the banking sector (second block), and overall macroeconomic performance (third block) in the semi-endogenous, NEVA and NEVA<sup>2</sup> scenario with respect to the baseline one.

First, none of the above alternative scenarios involves a significant variation in credit supply, as well as on variables measuring macroeconomic performance (like GDP growth, the unemployment rate, investment growth, etc.). In addition, also average bank performance (captured by bank profits, bank equity, bad debt and bank fragility) is not significantly affected.

The significant effects mostly concern the systemic risk in the interbank market. In particular, the semi-endogenous scenario (SEMI) does not yield significant improvements in performance in terms of systemic risk mitigation. In contrast, when the NEVA framework is applied *solely* to compute the probability of default of banks’ counterparties (NEVA scenario), we find that the number of bank

<footnote>banks are below the 30-th percentile.</footnote>

<footnote n="13"><sup>13</sup>The net interbank positions are measured for each bank *k* as the difference between the bank's total interbank assets $IB_k^A$ and its total interbank liabilities $IB_k^L$. At each period *t*, we then average the net interbank position of each bank in the specific size class and we compute the share, as a percentage of the total interbank exposure of the system.</footnote>
empirical interbank network literature is that big banks are significantly more central, especially when the outgoing links are considered (Craig and von Peter, 2014; Fricke and Lux, 2015). Furthermore, empirical evidence suggests that larger banks tend to borrow cash reserves in the interbank market (see Müller, 2006, for the Swiss interbank market). The results from our simulations are consistent with this evidence (see Figure 6). The result is explained by the fact that big banks have many more consumption-good firm clients who can default on their debts. And when such clients default, these banks become borrowers in the interbank market.

Finally, the simulation results indicate that our model generates a banking structure wherein small banks are liquidity providers; medium-sized banks are liquidity neutral; and large banks are liquidity borrowers (see the right panel of Figure 6).<sup>13</sup> This is also a prominent feature observed in real interbank networks (see Liu et al., 2020).

## 3.2 The effects of a network-based micro-prudential regulation

Given the good empirical performance of the model, in this section we present the results of policy evaluation exercises where we compare the performance of the economy and the banking sector in the four different scenarios described in Section 2.7. These exercises aim to test whether the NEVA clearing mechanism can dampen systemic risk, as measured by the total number of banks’ defaults and by the losses experienced by non-defaulting banks because of the contagion from defaulted ones.

As explained in Section 2.3, the NEVA algorithm endows banks with an adaptive local-information criterion that helps them to better and promptly assess the current value of their interbank claims, also taking into account the possibility that their counterparts will default. Therefore, the two scenarios in which it is employed can be considered scenarios wherein banks are endowed with a new micro-prudential tool. Finding that the NEVA has a beneficial impact on the financial soundness of banks and systemic risk reduction in these scenarios would be good news for the financial regulator, as this mechanism is much less demanding in terms of information requirements about the interbank network. Indeed, in contrast to the Eisenberg and Noe (2001) alternative, which requires global knowledge of the whole network structure, the NEVA only requires that each bank has information about the financial soundness of its interbank counterparts.

Table 3 compares the simulation results about the variables capturing systemic risk (first horizontal block), the banking sector (second block), and overall macroeconomic performance (third block) in the semi-endogenous, NEVA and NEVA<sup>2</sup> scenario with respect to the baseline one.

First, none of the above alternative scenarios involves a significant variation in credit supply, as well as on variables measuring macroeconomic performance (like GDP growth, the unemployment rate, investment growth, etc.). In addition, also average bank performance (captured by bank profits, bank equity, bad debt and bank fragility) is not significantly affected.

The significant effects mostly concern the systemic risk in the interbank market. In particular, the semi-endogenous scenario (SEMI) does not yield significant improvements in performance in terms of systemic risk mitigation. In contrast, when the NEVA framework is applied *solely* to compute the probability of default of banks’ counterparties (NEVA scenario), we find that the number of bank

<footnote>banks are below the 30-th percentile.</footnote>

<footnote n="13"><sup>13</sup>The net interbank positions are measured for each bank *k* as the difference between the bank's total interbank assets $IB_k^A$ and its total interbank liabilities $IB_k^L$. At each period *t*, we then average the net interbank position of each bank in the specific size class and we compute the share, as a percentage of the total interbank exposure of the system.</footnote>
<table><thead><tr><th>Scenario</th><th>Bank defaults</th><th>Loss due to contagion</th><th>Interbank exposure</th><th>Bailout costs</th><th>Credit Supply</th></tr></thead><tbody><tr><td>SEMI</td><td>0.866</td><td>0.832*</td><td>1</td><td>0.849</td><td>1.011</td></tr><tr><td>NEVA</td><td>0.508***</td><td>0.269***</td><td>0.992</td><td>0.572***</td><td>1</td></tr><tr><td>NEVA<sup>2</sup></td><td>0.483***</td><td>0.263***</td><td>0.992</td><td>0.542***</td><td>1.006</td></tr><tr><td></td><th>Total Loans</th><th>Bank Profits</th><th>Bank Equity</th><th>Bad Debt</th><th>Bank Fragility (Bda)</th></tr><tr><td>SEMI</td><td>1.005</td><td>1.042</td><td>1.01</td><td>0.952</td><td>0.976</td></tr><tr><td>NEVA</td><td>0.996</td><td>0.992</td><td>0.999</td><td>0.986</td><td>0.999</td></tr><tr><td>NEVA<sup>2</sup></td><td>0.996</td><td>1.012</td><td>1.005</td><td>0.972</td><td>1.018</td></tr><tr><td></td><th>GDP growth</th><th>Unempl. rate</th><th>Deb/GDP ratio</th><th>Investment growth</th><th>Financial constraints</th></tr><tr><td>SEMI</td><td>1.002</td><td>0.937</td><td>0.806</td><td>1.009</td><td>0.988</td></tr><tr><td>NEVA</td><td>1.004</td><td>0.893</td><td>0.806</td><td>1.002</td><td>0.99</td></tr><tr><td>NEVA<sup>2</sup></td><td>1.003</td><td>0.897</td><td>0.815</td><td>1.002</td><td>0.99</td></tr></tbody></table>

Notes. The table reports the ratio of the average values of economic indicators of a particular scenario relative to the baseline scenario. Values higher than one implies that the average value of the economic variable is higher than in the baseline scenario. The null hypothesis is that there is no statistical difference in the means calculated in the two scenarios.

*p<0.1; **p<0.05; ***p<0.01.

<caption>**Table 3:** Comparison of scenarios to the baseline. Main indicators for systemic risk, financial and macroeconomic performances.</caption>

defaults decrease by 48.2%. In addition, losses due to contagion effects contract by 73.1%, and bailout costs are lowered by 42.8%, without significantly affecting exposure in interbank markets.

Moreover, if NEVA's micro-prudential potential is fully exploited and it is used *also* for an internal assessment for bank credit provision (NEVA² scenario), we observe a further reduction in the number of bank defaults, losses due to contagion and bailout costs by 51.7%, 73.7% and 45.8%, respectively. Hence, the NEVA is extremely effective in taming the perilous effects of systemic risk.

The results discussed indicate that the use of the NEVA has significant effects in terms of systemic risk mitigation while not having any negative impact either on banks or macroeconomic performance. Accordingly, they suggest that the use of such a micro-prudential tool is not characterized by a trade-off between financial stability and macroeconomic performance. Encouraged by this finding, in the following section we explore whether this trade-off shows up instead for banks of different sizes, thus offering insights on the possible distributional impacts of NEVA. Furthermore, we investigate the possible interactions between a micro-prudential tool, like the NEVA framework, and macro-prudential regulatory conditions, represented by the Basel III macro-prudential regulation.

## 3.3 Heterogeneity analysis

The results discussed in the previous section may hide the presence of significant heterogeneity across banks, generated by the distributional effects of the NEVA within the banking system.

To detect the possible heterogeneous impact of the NEVA, we study the effects brought about by the introduction of the NEVA on different categories of banks. In particular, for each category of bank size (big, medium, small), we compute the instances of bank defaults, the losses due to
<table><thead><tr><th>Scenario</th><th>Bank defaults</th><th>Loss due to contagion</th><th>Interbank exposure</th><th>Bailout costs</th><th>Credit Supply</th></tr></thead><tbody><tr><td>SEMI</td><td>0.866</td><td>0.832*</td><td>1</td><td>0.849</td><td>1.011</td></tr><tr><td>NEVA</td><td>0.508***</td><td>0.269***</td><td>0.992</td><td>0.572***</td><td>1</td></tr><tr><td>NEVA<sup>2</sup></td><td>0.483***</td><td>0.263***</td><td>0.992</td><td>0.542***</td><td>1.006</td></tr><tr><td></td><th>Total Loans</th><th>Bank Profits</th><th>Bank Equity</th><th>Bad Debt</th><th>Bank Fragility (Bda)</th></tr><tr><td>SEMI</td><td>1.005</td><td>1.042</td><td>1.01</td><td>0.952</td><td>0.976</td></tr><tr><td>NEVA</td><td>0.996</td><td>0.992</td><td>0.999</td><td>0.986</td><td>0.999</td></tr><tr><td>NEVA<sup>2</sup></td><td>0.996</td><td>1.012</td><td>1.005</td><td>0.972</td><td>1.018</td></tr><tr><td></td><th>GDP growth</th><th>Unempl. rate</th><th>Deb/GDP ratio</th><th>Investment growth</th><th>Financial constraints</th></tr><tr><td>SEMI</td><td>1.002</td><td>0.937</td><td>0.806</td><td>1.009</td><td>0.988</td></tr><tr><td>NEVA</td><td>1.004</td><td>0.893</td><td>0.806</td><td>1.002</td><td>0.99</td></tr><tr><td>NEVA<sup>2</sup></td><td>1.003</td><td>0.897</td><td>0.815</td><td>1.002</td><td>0.99</td></tr></tbody></table>

Notes. The table reports the ratio of the average values of economic indicators of a particular scenario relative to the baseline scenario. Values higher than one implies that the average value of the economic variable is higher than in the baseline scenario. The null hypothesis is that there is no statistical difference in the means calculated in the two scenarios.

<caption>**Table 3:** Comparison of scenarios to the baseline. Main indicators for systemic risk, financial and macroeconomic performances.</caption>

defaults decrease by 48.2%. In addition, losses due to contagion effects contract by 73.1%, and bailout costs are lowered by 42.8%, without significantly affecting exposure in interbank markets.

Moreover, if NEVA's micro-prudential potential is fully exploited and it is used *also* for an internal assessment for bank credit provision (NEVA² scenario), we observe a further reduction in the number of bank defaults, losses due to contagion and bailout costs by 51.7%, 73.7% and 45.8%, respectively. Hence, the NEVA is extremely effective in taming the perilous effects of systemic risk.

The results discussed indicate that the use of the NEVA has significant effects in terms of systemic risk mitigation while not having any negative impact either on banks or macroeconomic performance. Accordingly, they suggest that the use of such a micro-prudential tool is not characterized by a trade-off between financial stability and macroeconomic performance. Encouraged by this finding, in the following section we explore whether this trade-off shows up instead for banks of different sizes, thus offering insights on the possible distributional impacts of NEVA. Furthermore, we investigate the possible interactions between a micro-prudential tool, like the NEVA framework, and macro-prudential regulatory conditions, represented by the Basel III macro-prudential regulation.

## 3.3 Heterogeneity analysis

The results discussed in the previous section may hide the presence of significant heterogeneity across banks, generated by the distributional effects of the NEVA within the banking system.

To detect the possible heterogeneous impact of the NEVA, we study the effects brought about by the introduction of the NEVA on different categories of banks. In particular, for each category of bank size (big, medium, small), we compute the instances of bank defaults, the losses due to

<footnote>*p<0.1; **p<0.05; ***p<0.01.</footnote>
contagion, the growth rate of the total loans to consumption-good firms and the growth rate of firms'
investments. This exercise allows us to understand whether the NEVA affects some banks and their
credit transmission more than others. As usual, we run t-tests to assess whether there are, on average,
statistically significant differences across the scenarios.

Figure 7 illustrates the findings of our study, which reveals the heterogeneity underlying the aggregate patterns discussed in the previous section. We first focus on systemic risk indicators (first and second row of Figure 7). Our results suggest that the use of both NEVA and NEVA² yields advantages to all banks regardless of their size. However, larger banks get a more pronounced benefit. This is because large banks are also more connected in the interbank network (see Section 3.1). Accordingly, a better evaluation of their counterparties’ assets yields larger reductions in contagion-related losses for these banks and as a consequence, in their default rate.

The last two rows of Figure 7 illustrate the distributional impact of the NEVA on the real sector by
showing, respectively, the variation in the loans to firms of banks on different size classes (third row)
and the variation in the investment of firms connected to banks of different sizes (fourth row). The
two panels indicate that the more prudential behavior implied by the NEVA does not bring significant
reductions either in the amount of loans that banks of different sizes grant to consumption-good firms
or in the investment of the firms connected to them (cf., the NEVA and NEVA² scenarios with the
BASE one). Accordingly, the absence of a significant impact of the NEVA on the real sector that we
highlighted in the previous section is not generated by a composition effect.

**3.4 Micro-prudential and macro-prudential interactions**

In the previous sections we highlighted that the introduction of the NEVA allows a significant
mitigation of systemic risk in interbank markets while not generating credit-crunch effects and/or a
worse macroeconomic performance.

However, how does the NEVA micro-prudential tool interact with the Basel III macro-prudential
regulation? In particular, we want to study whether a trade-off between financial stability and
macroeconomic performance emerges when these two different sets of regulatory instruments inter-
act. Indeed, as micro- and macro-prudential policies inherently share objectives and transmission
mechanisms (Altunbas et al., 2018; Osinski et al., 2013), their simultaneous adoption could trigger
non-trivial effects on systemic-risk-related variables and the macro-financial outlook.

For this purpose, we exploit the versatility of our agent-based model to perform Monte Carlo
simulation experiments where we interact the NEVA and NEVA² micro-prudential tools with more
or less stringent macro-prudential policy regimes identified by mandatory capital requirements,
captured in our model by the parameter τ^B in Equation (1). More specifically, we let the macro-
prudential parameter τ^B to vary from 2% to 7% (higher values implies a more stringent macro-
prudential regulation) and for each value of τ^B, we simulate the micro-prudential policy scenarios
described in Section 2.7. We take as benchmark for this experiment the baseline scenario (BASE) with
τ^B = 4.5% (cf., Tables 2 and 3) to ease the comparability of the results.

The results of the above experiment are presented in Figures 8 and 9 for all the analyzed com-
binations of the macro-prudential parameter (τ<sup>B</sup>) and of the micro-prudential settings. Each bar
of the plots in the two figures show the difference between the value of a variable of interest in a
given scenario and the value of the same variable in the benchmark scenario, along with a 90%-level
contagion, the growth rate of the total loans to consumption-good firms and the growth rate of firms'
investments. This exercise allows us to understand whether the NEVA affects some banks and their
credit transmission more than others. As usual, we run t-tests to assess whether there are, on average,
statistically significant differences across the scenarios.

Figure 7 illustrates the findings of our study, which reveals the heterogeneity underlying the aggregate patterns discussed in the previous section. We first focus on systemic risk indicators (first and second row of Figure 7). Our results suggest that the use of both NEVA and NEVA² yields advantages to all banks regardless of their size. However, larger banks get a more pronounced benefit. This is because large banks are also more connected in the interbank network (see Section 3.1). Accordingly, a better evaluation of their counterparties’ assets yields larger reductions in contagion-related losses for these banks and as a consequence, in their default rate.

The last two rows of Figure 7 illustrate the distributional impact of the NEVA on the real sector by
showing, respectively, the variation in the loans to firms of banks on different size classes (third row)
and the variation in the investment of firms connected to banks of different sizes (fourth row). The
two panels indicate that the more prudential behavior implied by the NEVA does not bring significant
reductions either in the amount of loans that banks of different sizes grant to consumption-good firms
or in the investment of the firms connected to them (cf., the NEVA and NEVA² scenarios with the
BASE one). Accordingly, the absence of a significant impact of the NEVA on the real sector that we
highlighted in the previous section is not generated by a composition effect.

3.4 Micro-prudential and macro-prudential interactions

In the previous sections we highlighted that the introduction of the NEVA allows a significant
mitigation of systemic risk in interbank markets while not generating credit-crunch effects and/or a
worse macroeconomic performance.

However, how does the NEVA micro-prudential tool interact with the Basel III macro-prudential
regulation? In particular, we want to study whether a trade-off between financial stability and
macroeconomic performance emerges when these two different sets of regulatory instruments inter-
act. Indeed, as micro- and macro-prudential policies inherently share objectives and transmission
mechanisms (Altunbas et al., 2018; Osinski et al., 2013), their simultaneous adoption could trigger
non-trivial effects on systemic-risk-related variables and the macro-financial outlook.

For this purpose, we exploit the versatility of our agent-based model to perform Monte Carlo
simulation experiments where we interact the NEVA and NEVA² micro-prudential tools with more
or less stringent macro-prudential policy regimes identified by mandatory capital requirements,
captured in our model by the parameter τ^B in Equation (1). More specifically, we let the macro-
prudential parameter τ^B to vary from 2% to 7% (higher values implies a more stringent macro-
prudential regulation) and for each value of τ^B, we simulate the micro-prudential policy scenarios
described in Section 2.7. We take as benchmark for this experiment the baseline scenario (BASE) with
τ^B = 4.5% (cf., Tables 2 and 3) to ease the comparability of the results.

The results of the above experiment are presented in Figures 8 and 9 for all the analyzed com-
binations of the macro-prudential parameter (τ<sup>B</sup>) and of the micro-prudential settings. Each bar
of the plots in the two figures show the difference between the value of a variable of interest in a
given scenario and the value of the same variable in the benchmark scenario, along with a 90%-level
<figure><img src="image_13.png" /><figcaption>**Figure 7:** The effects of the NEVA implementation on different categories of banks. *Notes:* The figure shows, for each category of bank and firm size, the distribution of the 1000 Monte Carlo percentage difference between the value of statistic X under the scenario of interest *s* and a baseline scenario *base*, $(X_s - X_{base})/X_{base}$, with a 90% confidence level.</figcaption></figure>
<figure><img src="image_13.png" /><figcaption>**Figure 7:** The effects of the NEVA implementation on different categories of banks. *Notes:* The figure shows, for each category of bank and firm size, the distribution of the 1000 Monte Carlo percentage difference between the value of statistic X under the scenario of interest *s* and a baseline scenario *base*, $(X_s - X_{base})/X_{base}$, with a 90% confidence level.</figcaption></figure>
confidence interval. The value of the variable in the benchmark scenario (BASE with $\tau^B = 4.5\%$) is
centered on zero by construction.

The analysis of the bar plots in Figures 8 and 9 confirms that the implementation of the NEVA
brings a significant reduction in the values of the systemic risk-related variables (i.e., number of
bank defaults, losses due to contagion and bailout costs approximately fall by 30%, 75% and 50%,
respectively) *notwithstanding* the strength of macro-prudential policy (Figure 8, top 3 panels). Such
results are robust even in the NEVA² scenario, i.e. when the tool is implemented also for determining
lending decisions. The NEVA thus appears to be invariant and orthogonal to the macro-prudential
regime in terms of systemic risk mitigation.

However, two outcomes emerge when we examine the impact on credit and macroeconomic
variables. First, we observe stronger credit crunch effects as the macro-prudential policy gets tighter
(Figure 8, bottom 3 panels). The mechanism is straightforward: an increase in capital requirements
(higher $\tau^B$) leads to a contraction in the supply of credit - see Equation (1) - given the loans demand.
This increases the share of credit-constrained firms (see Figure 9, first panel from the top) that need
to revise downward their investment and production plans, thus hampering economic growth.

Second, we observe *non-linear* interaction effects between micro- and macro-prudential policy
tools. For instance, Figures 8 and 9 show that for values of $\tau^B$ lower than 4.5%, none of the macroe-
conomic and financial variables are significantly affected by the introduction of the NEVA. This is
because, in presence of looser capital requirements, the NEVA regulatory tool contributes to keep
the credit supply steady, thus taming the risk associated to the formation of credit booms. In such
a stable macro-financial environment, firms are generally not credit rationed (see also Figure 9, top
panel). They can therefore pursue their desired investment plans, which in turns sustain the pace of
economic growth (cf., Figure 9).

On the contrary, the credit crunch triggered by tighter macro-prudential requirements ($\tau^B > 4.5\%$) is reinforced by the presence of the NEVA micro-prudential tool. In this case, the total loans granted to firms decrease from -2.3% to -6.3% when $\tau^B$ is equal to 5% and 7%, respectively (see Figure 8); in the more stringent macro-prudential scenario, as Figure 9 highlights, financial constraints increase by 12.4%, while investment growth and GDP growth decrease by -4.8% and -2.4%, respectively.<fnref n="14" /> Nonetheless, even in presence of tighter capital requirements, the introduction of the NEVA brings significant improvements in the mitigation of systemic risk (see Figure 8 top 2 panels).

Overall, our results suggest that there are strong complementarities between the NEVA micro-
prudential tool and the BASEL III-like macro-prudential tools. More precisely, the introduction
of the NEVA could allow policy makers to relax (or at least not to exacerbate) mandatory capital
requirements. In particular, a mix represented by the adoption of the NEVA micro-prudential tool
and a less stringent macro-prudential regulation successfully tames systemic risk without constraining
the flow of bank credit to firms in the real economy.

# 4 Conclusion

This work extends the *Schumpeter meeting Keynes* model (Dosi et al., 2010, 2013, 2015) introducing an
explicit payment system between the economic agents that leads to the endogenous formation of an

<footnote n="14">¹⁴For detailed magnitudes of this effects see Table 6 in Appendix B.</footnote>
confidence interval. The value of the variable in the benchmark scenario (BASE with $\tau^B = 4.5\%$) is
centered on zero by construction.

The analysis of the bar plots in Figures 8 and 9 confirms that the implementation of the NEVA
brings a significant reduction in the values of the systemic risk-related variables (i.e., number of
bank defaults, losses due to contagion and bailout costs approximately fall by 30%, 75% and 50%,
respectively) *notwithstanding* the strength of macro-prudential policy (Figure 8, top 3 panels). Such
results are robust even in the NEVA² scenario, i.e. when the tool is implemented also for determining
lending decisions. The NEVA thus appears to be invariant and orthogonal to the macro-prudential
regime in terms of systemic risk mitigation.

However, two outcomes emerge when we examine the impact on credit and macroeconomic
variables. First, we observe stronger credit crunch effects as the macro-prudential policy gets tighter
(Figure 8, bottom 3 panels). The mechanism is straightforward: an increase in capital requirements
(higher $\tau^B$) leads to a contraction in the supply of credit - see Equation (1) - given the loans demand.
This increases the share of credit-constrained firms (see Figure 9, first panel from the top) that need
to revise downward their investment and production plans, thus hampering economic growth.

Second, we observe *non-linear* interaction effects between micro- and macro-prudential policy
tools. For instance, Figures 8 and 9 show that for values of $\tau^B$ lower than 4.5%, none of the macroe-
conomic and financial variables are significantly affected by the introduction of the NEVA. This is
because, in presence of looser capital requirements, the NEVA regulatory tool contributes to keep
the credit supply steady, thus taming the risk associated to the formation of credit booms. In such
a stable macro-financial environment, firms are generally not credit rationed (see also Figure 9, top
panel). They can therefore pursue their desired investment plans, which in turns sustain the pace of
economic growth (cf., Figure 9).

On the contrary, the credit crunch triggered by tighter macro-prudential requirements ($\tau^B > 4.5\%$) is reinforced by the presence of the NEVA micro-prudential tool. In this case, the total loans granted to firms decrease from -2.3% to -6.3% when $\tau^B$ is equal to 5% and 7%, respectively (see Figure 8); in the more stringent macro-prudential scenario, as Figure 9 highlights, financial constraints increase by 12.4%, while investment growth and GDP growth decrease by -4.8% and -2.4%, respectively.<fnref n="14" /> Nonetheless, even in presence of tighter capital requirements, the introduction of the NEVA brings significant improvements in the mitigation of systemic risk (see Figure 8 top 2 panels).

Overall, our results suggest that there are strong complementarities between the NEVA micro-
prudential tool and the BASEL III-like macro-prudential tools. More precisely, the introduction
of the NEVA could allow policy makers to relax (or at least not to exacerbate) mandatory capital
requirements. In particular, a mix represented by the adoption of the NEVA micro-prudential tool
and a less stringent macro-prudential regulation successfully tames systemic risk without constraining
the flow of bank credit to firms in the real economy.

# 4 Conclusion

This work extends the *Schumpeter meeting Keynes* model (Dosi et al., 2010, 2013, 2015) introducing an
explicit payment system between the economic agents that leads to the endogenous formation of an

<footnote n="14">¹⁴For detailed magnitudes of this effects see Table 6 in Appendix B.</footnote>
<figure><img src="image_14.png" /><figcaption>**Figure 8:** Systemic-risk related and macrofinancial effects of NEVA implementation under different capital requirement regimes. *Notes*: The figure shows, for a set of financial indicators, the distribution of the 1000 Monte Carlo percentage difference between the value of statistic X under the scenario of interest *s* and a baseline scenario *base*, $(X_s - X_{base})/X_{base}$, with a 90% confidence level. In order to have statistics comparable across different capital requirement regimes, a reference-baseline scenario is fixed at $\tau^B = 0.045$, i.e., the baseline parametrization in Section 3, centered around 0 by construction (darker purple bars).</figcaption></figure>
<figure><img src="image_14.png" /><figcaption>**Figure 8:** Systemic-risk related and macrofinancial effects of NEVA implementation under different capital requirement regimes. *Notes*: The figure shows, for a set of financial indicators, the distribution of the 1000 Monte Carlo percentage difference between the value of statistic X under the scenario of interest *s* and a baseline scenario *base*, ($X_s - X_{base})/X_{base}$, with a 90% confidence level. In order to have statistics comparable across different capital requirement regimes, a reference-baseline scenario is fixed at $\tau^B = 0.045$, i.e., the baseline parametrization in Section 3, centered around 0 by construction (darker purple bars).</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Figure 9:** Real effects of NEVA implementation under different capital requirement regimes. *Notes*: The figure shows, for a set of macroeconomic indicators, the distribution of the 1000 Monte Carlo percentage difference between the value of statistic X under the scenario of interest *s* and a baseline scenario *base*, ($X_s - X_{base})/X_{base}$, with a 90% confidence level. In order to have statistics comparable across different capital requirement regimes, a reference-baseline scenario is fixed at $\tau^B = 0.045$, i.e., the baseline parametrization in Section 3, centered around 0 by construction (darker purple bars).</figcaption></figure>
<figure><img src="image_15.png" /><figcaption>**Figure 9:** Real effects of NEVA implementation under different capital requirement regimes. *Notes*: The figure shows, for a set of macroeconomic indicators, the distribution of the 1000 Monte Carlo percentage difference between the value of statistic X under the scenario of interest *s* and a baseline scenario *base*, ($X_s - X_{base})/X_{base}$, with a 90% confidence level. In order to have statistics comparable across different capital requirement regimes, a reference-baseline scenario is fixed at $\tau^B = 0.045$, i.e., the baseline parametrization in Section 3, centered around 0 by construction (darker purple bars).</figcaption></figure>
interbank network. In the model, payments among firms for capital goods are mediated by deposits exchanges between their banks which are closed within a period. Indeed, under normal conditions, a bank advances resources for a consumption-good firm at the moment of the purchase of machines from a capital-good firm and recovers them at the end of the same period if the consumption-good firm is solvent, thereby closing the position in the interbank market. However, if the consumption-good firm is insolvent, the advances made by the buyer's bank to the seller's one results in an interbank position that cannot be immediately closed and thus persists over time. This simple mechanisms generates endogenously a sizeable volume of interbank assets and liabilities without imposing any restrictive assumptions on banks' behaviour, and results in an endogenous interbank network whose features are consistent with recent empirical evidence.

By employing extensive Monte Carlo simulations, we show that our model is able to jointly replicate a wide ensemble of stylized facts concerning growth and business cycles, the properties of firm size distributions (as in Dosi et al., 2010, 2013, 2015; Guerini et al., 2022), and key features of empirically-observed interbank markets. It is therefore suitable to be used as a policy laboratory to test the financial and real impact of alternative regulatory mechanisms impacting on banks' behavior. In this respect, we study whether the NEVA clearing mechanism (Barucca et al., 2020) can be employed as a *micro-prudential* instrument to mitigate financial instability. NEVA equips banks with a tool for the endogenous evaluation of interbank claims. Such an evaluation considers the interconnectedness within the interbank network, effectively protecting banks from increasing systemic risk. We study the effectiveness of this tool in different scenarios: in the NEVA scenario, banks evaluate the counterparty probability of default in order to decide whether to keep open (or not) the interbank claims; in NEVA² scenario, banks adopt the tool also to adjust their credit supply in a more prudent fashion when dealing with relatively risky interbank clients.

Simulation results show that NEVA helps mitigating the harmful effects of systemic risk, measured by the number of bank defaults and losses due to contagion effects. In addition, the foregoing mitigation effects are stronger whenever the NEVA is used both for evaluating the interbank counterparty's probability of default and for making lending decisions based on a prudent evaluation of interbank exposures. Moreover, we do not detect any distributional effects of the adoption of the NEVA across bank size. In other words, the benefits from systemic risk reduction are shared among all banks regardless of their size. Furthermore, we do not observe the emergence of a trade-off between financial stability and macroeconomic performance in our simulation, which suggests that the more cautious approach towards interbank interconnectedness implied by the NEVA does not result in a lower levels of credit provision to the real economy and therefore in worse short- and long-run macroeconomic performances.

We then further explore the interplay between micro- and macro-prudential policies. We find non-linear interactions between the NEVA and the Basel III tools. More specifically, in presence of looser capital requirement, the NEVA micro-prudential tool mitigates systemic risk, while not constraining the flow of credit to firms. On the contrary, when macroprudential regulation becomes tighter, the NEVA still prevents the build-up of systemic risk but it exacerbates to fall of credit to the real side of the economy. The non-linear interactions between the NEVA and Basel III framework has two relevant policy implications. First, the NEVA is always able to mitigate systemic risk in more or less stringent Basel III framework. Second, the implementation of NEVA micro-prudential
interbank network. In the model, payments among firms for capital goods are mediated by deposits exchanges between their banks which are closed within a period. Indeed, under normal conditions, a bank advances resources for a consumption-good firm at the moment of the purchase of machines from a capital-good firm and recovers them at the end of the same period if the consumption-good firm is solvent, thereby closing the position in the interbank market. However, if the consumption-good firm is insolvent, the advances made by the buyer's bank to the seller's one results in an interbank position that cannot be immediately closed and thus persists over time. This simple mechanisms generates endogenously a sizeable volume of interbank assets and liabilities without imposing any restrictive assumptions on banks' behaviour, and results in an endogenous interbank network whose features are consistent with recent empirical evidence.

By employing extensive Monte Carlo simulations, we show that our model is able to jointly replicate a wide ensemble of stylized facts concerning growth and business cycles, the properties of firm size distributions (as in Dosi et al., 2010, 2013, 2015; Guerini et al., 2022), and key features of empirically-observed interbank markets. It is therefore suitable to be used as a policy laboratory to test the financial and real impact of alternative regulatory mechanisms impacting on banks' behavior. In this respect, we study whether the NEVA clearing mechanism (Barucca et al., 2020) can be employed as a *micro-prudential* instrument to mitigate financial instability. NEVA equips banks with a tool for the endogenous evaluation of interbank claims. Such an evaluation considers the interconnectedness within the interbank network, effectively protecting banks from increasing systemic risk. We study the effectiveness of this tool in different scenarios: in the NEVA scenario, banks evaluate the counterparty probability of default in order to decide whether to keep open (or not) the interbank claims; in NEVA² scenario, banks adopt the tool also to adjust their credit supply in a more prudent fashion when dealing with relatively risky interbank clients.

Simulation results show that NEVA helps mitigating the harmful effects of systemic risk, measured by the number of bank defaults and losses due to contagion effects. In addition, the foregoing mitigation effects are stronger whenever the NEVA is used both for evaluating the interbank counterparty's probability of default and for making lending decisions based on a prudent evaluation of interbank exposures. Moreover, we do not detect any distributional effects of the adoption of the NEVA across bank size. In other words, the benefits from systemic risk reduction are shared among all banks regardless of their size. Furthermore, we do not observe the emergence of a trade-off between financial stability and macroeconomic performance in our simulation, which suggests that the more cautious approach towards interbank interconnectedness implied by the NEVA does not result in a lower levels of credit provision to the real economy and therefore in worse short- and long-run macroeconomic performances.

We then further explore the interplay between micro- and macro-prudential policies. We find non-linear interactions between the NEVA and the Basel III tools. More specifically, in presence of looser capital requirement, the NEVA micro-prudential tool mitigates systemic risk, while not constraining the flow of credit to firms. On the contrary, when macroprudential regulation becomes tighter, the NEVA still prevents the build-up of systemic risk but it exacerbates to fall of credit to the real side of the economy. The non-linear interactions between the NEVA and Basel III framework has two relevant policy implications. First, the NEVA is always able to mitigate systemic risk in more or less stringent Basel III framework. Second, the implementation of NEVA micro-prudential
regulation allow to pursue financial stability with less stringent mandatory capital requirements thus
supporting a more abundant flow of bank credit to finance firms' investment and production plans.

The model could be extended in different ways. First, similarly to Guerini et al. (2022), one could enrich the dynamics of the interbank market by letting banks trade other financial instruments, e.g., government bonds, thus determining the creation of secured/unsecured transactions. Second, since our model is nested into an agent-based integrated assessment model (Lamperti et al., 2018), it could be employed to evaluate the financial risks related to the green energy transition (Lamperti et al., 2019, 2021). This would allow one to study the rising contagion and systemic risks related to the formation of stranded assets and the policies to be implemented in order to tame their destabilizing effects. Third, similarly to Popoyan et al. (2017, 2020), one could introduce in the model an explicit market for liquidity which is now automatically deposited at central bank facilities in the form of cash reserves. This would also introduce another source of financial instability in the model and allow the study of the impact of unconventional monetary policies.
regulation allow to pursue financial stability with less stringent mandatory capital requirements thus
supporting a more abundant flow of bank credit to finance firms' investment and production plans.

The model could be extended in different ways. First, similarly to Guerini et al. (2022), one could enrich the dynamics of the interbank market by letting banks trade other financial instruments, e.g., government bonds, thus determining the creation of secured/unsecured transactions. Second, since our model is nested into an agent-based integrated assessment model (Lamperti et al., 2018), it could be employed to evaluate the financial risks related to the green energy transition (Lamperti et al., 2019, 2021). This would allow one to study the rising contagion and systemic risks related to the formation of stranded assets and the policies to be implemented in order to tame their destabilizing effects. Third, similarly to Popoyan et al. (2017, 2020), one could introduce in the model an explicit market for liquidity which is now automatically deposited at central bank facilities in the form of cash reserves. This would also introduce another source of financial instability in the model and allow the study of the impact of unconventional monetary policies.
# A Appendix: The model

This appendix contains the full formal structure of the model as originally developed by Dosi et al. (2010, 2013, 2015), Lamperti et al. (2018). The description of the original model and those parts that have not been modified, heavily draws on the latter references.

We begin with the description of the technological search processes and the determination of production and prices in the *capital-good sector* and the equations related to the determination of production, investment, prices and profits in the *consumption-good sector*, as well as those related to the *public sector*.

## A.1 The capital-good sector

The economy is characterized by two vertically-integrated sectors, a upstream capital-intensive sector that sells machines to a downstream sector which produces an homogeneous bundle of goods.

In this version of the model, upstream firms use labor and energy as an input of production. Innovation and imitation activities are undertaken to boost productivity, and to cut production costs; moreover, they are carried on by firms' investments in R&D, which are ultimately a share of past revenues.

The technology of the machines of vintage $\tau$ is captured by their labor productivity (LP) and energy efficiency (EE) it is represented by a set of coefficients ($A_{i,\tau}^l, B_{i,\tau}^l$), where $l = \{LP, EE\}$.

The coefficient $A_{i,\tau}^{LP}$ represents the productivity of the machinery in the consumption-good industry; $B_{i,\tau}^{LP}$ is the productivity of the process leading to the manufacturing of the capital good. Similarly, $A_{i,\tau}^{EE}$ and $B_{i,\tau}^{EE}$ characterize the level of energy efficiency in the production processes of both type of goods.

Upstream firms, subject to market selection forces, need to improve their technology in order to increase their productivity and to gain market power. They can do it by means of innovation and imitation, which are both costly. Following Dosi et al. (2010), both innovation and imitation are modeled in two steps. In the first, the dynamics of technical change randomly determines the success of both innovation and imitation processes: this is modeled by realization of Bernoulli-distributed random variables in which the level of R&D investments positively determine the probability that the innovation is successful. In a second step, the size of the technological improvement is stochastically determined:

$$A_{i,\tau+1}^l = A_{i,\tau}(1 + \chi_{A,i}^l) \quad l = \{LP, EE\} \quad (18)$$

$$B_{i,\tau+1}^l = B_{i,\tau}(1 + \chi_{B,i}^l) \quad l = \{LP, EE\} \quad (19)$$

where $\chi_{A,i}$ and $\chi_{B,i}$ are i.i.d. realization of $\text{Beta}(\alpha, \beta)$ random variable with support given by the interval $[\underline{x}^l, \bar{x}^l]$, which characterize the technological opportunity space (Dosi, 1988). If an upstream firm successfully innovate, close competitors can increase the chances of being imitators.
# A Appendix: The model

This appendix contains the full formal structure of the model as originally developed by Dosi et al. (2010, 2013, 2015), Lamperti et al. (2018). The description of the original model and those parts that have not been modified, heavily draws on the latter references.

We begin with the description of the technological search processes and the determination of production and prices in the *capital-good sector* and the equations related to the determination of production, investment, prices and profits in the *consumption-good sector*, as well as those related to the *public sector*.

## A.1 The capital-good sector

The economy is characterized by two vertically-integrated sectors, a upstream capital-intensive sector that sells machines to a downstream sector which produces an homogeneous bundle of goods.

In this version of the model, upstream firms use labor and energy as an input of production. Innovation and imitation activities are undertaken to boost productivity, and to cut production costs; moreover, they are carried on by firms' investments in R&D, which are ultimately a share of past revenues.

The technology of the machines of vintage $\tau$ is captured by their labor productivity (LP) and energy efficiency (EE) it is represented by a set of coefficients ($A_{i,\tau}^l, B_{i,\tau}^l$), where $l = \{LP, EE\}$.

The coefficient $A_{i,\tau}^{LP}$ represents the productivity of the machinery in the consumption-good industry; $B_{i,\tau}^{LP}$ is the productivity of the process leading to the manufacturing of the capital good. Similarly, $A_{i,\tau}^{EE}$ and $B_{i,\tau}^{EE}$ characterize the level of energy efficiency in the production processes of both type of goods.

Upstream firms, subject to market selection forces, need to improve their technology in order to increase their productivity and to gain market power. They can do it by means of innovation and imitation, which are both costly. Following Dosi et al. (2010), both innovation and imitation are modeled in two steps. In the first, the dynamics of technical change randomly determines the success of both innovation and imitation processes: this is modeled by realization of Bernoulli-distributed random variables in which the level of R&D investments positively determine the probability that the innovation is successful. In a second step, the size of the technological improvement is stochastically determined:

$$A_{i,\tau+1}^l = A_{i,\tau}(1 + \chi_{A,i}^l) \quad l = \{LP, EE\} \quad (18)$$

$$B_{i,\tau+1}^l = B_{i,\tau}(1 + \chi_{B,i}^l) \quad l = \{LP, EE\} \quad (19)$$

where $\chi_{A,i}$ and $\chi_{B,i}$ are i.i.d. realization of $\text{Beta}(\alpha, \beta)$ random variable with support given by the interval $[\underline{x}^l, \bar{x}^l]$, which characterize the technological opportunity space (Dosi, 1988). If an upstream firm successfully innovate, close competitors can increase the chances of being imitators.
## A.2 The consumption-good sector

Downstream firms manufacture a homogeneous bundle of goods by using machineries bought from upstream firms, with constant returns to scale. Workers consumption determines the level of demand to be satisfied and accordingly, firms adaptively update their production plans $Q_j^d$ (also considering desired inventories ($N_j^d$) and the actual stock ($N_j$)) according to the expected level of demand $D_j^e = f[D_{j,t-1}, D_{j,t-2}, \dots, D_{j,t-h}]$:

$$Q_{j,t}^d = D_{j,t}^e + N_{j,t}^d - N_{j,t}, \qquad (20)$$

where $N_j(t) = \iota D_j^e(t)$, $\iota \in [0, 1]$.

The production levels of downstream firms are constrained by the level of their capital stock ($K^d$). Accordingly, if production plans require more capital, firms undertake expansionary investments, namely that increase their production capacity.

$$E I_{j,t}^d = K_{j,t}^d - K_{j,t}. \qquad (21)$$

Firms also undertake investments aimed at replacing machineries that are become technologically obsolete in terms of productivity performances that is, for a given set of capital goods $\Xi_i(t)$, the vintage $\tau$ is substituted with a more productive one if

$$\frac{p^{new}}{c_{j,t}^{con} - c^{new}} = \frac{p^{new}}{\frac{w_t}{A_{i,\tau}^{LP}} + \frac{c_t^{en}}{A_{i,\tau}^{EE}} - c_j^{new}} \le b \qquad (22)$$

with $p^{new}$ and $c^{new}$ being the price of the machinery and its unitary cost of production, respectively. The parameter $b$ discounts firms' "patience" on the rate of return on investments.

The choice of the upstream supplier is determined by a price/productivity ratio of those vintages that the downstream firm can observe. Being characterized by the presence of systematic information asymmetries, the choice of the consumption-good firm canis restricted to a subset of upstream producers. Since the production of machineries requires some time, consumption-good firms first order the capital good that delivered at the end of the period. The price of each machine-tool has a mark-up on its cost.

When comes at how to finance their investments, consumption-good firms operate in imperfect credit markets (in the spirit of (Stiglitz and Weiss, 1981)). Internal finance has priority: if are not able to fully cover production and investment costs, they will rely on external finance, by borrowing funds from a bank in the form of a credit line. Given the total credit (exposure) of a bank, the latter lends out money to firms on a pecking-order, determined by the ratio between equity and sales (see Dosi et al., 2013). If the credit demanded by consumption-good firms exceeds its supply, firms are credit- rationed. Also in the downstream sector, firms charge a markup over the unit cost of production according to the following rule:

$$p_{j,t}^{con} = c_{j,t}^{con}[1 + \mu_{j,t}]. \qquad (23)$$

The choice of the markup is determined by selection processes of the markets in which firms operate. In particular, it depends on the evolution firms' market share, $f_j$:
## A.2 The consumption-good sector

Downstream firms manufacture a homogeneous bundle of goods by using machineries bought from upstream firms, with constant returns to scale. Workers consumption determines the level of demand to be satisfied and accordingly, firms adaptively update their production plans $Q_j^d$ (also considering desired inventories ($N_j^d$) and the actual stock ($N_j$)) according to the expected level of demand $D_j^e = f[D_{j,t-1}, D_{j,t-2}, \dots, D_{j,t-h}]$:

$$Q_{j,t}^d = D_{j,t}^e + N_{j,t}^d - N_{j,t}, \qquad (20)$$

where $N_j(t) = \iota D_j^e(t)$, $\iota \in [0, 1]$.

The production levels of downstream firms are constrained by the level of their capital stock ($K^d$). Accordingly, if production plans require more capital, firms undertake expansionary investments, namely that increase their production capacity.

$$EI_{j,t}^d = K_{j,t}^d - K_{j,t}. \qquad (21)$$

Firms also undertake investments aimed at replacing machineries that are become technologically obsolete in terms of productivity performances that is, for a given set of capital goods $\Xi_i(t)$, the vintage $\tau$ is substituted with a more productive one if

$$\frac{p^{new}}{c_{j,t}^{con} - c^{new}} = \frac{p^{new}}{\frac{w_t}{A_{i,\tau}^{LP}} + \frac{c_t^{en}}{A_{i,\tau}^{EE}} - c_j^{new}} \le b \qquad (22)$$

with $p^{new}$ and $c^{new}$ being the price of the machinery and its unitary cost of production, respectively. The parameter $b$ discounts firms' "patience" on the rate of return on investments.

The choice of the upstream supplier is determined by a price/productivity ratio of those vintages that the downstream firm can observe. Being characterized by the presence of systematic information asymmetries, the choice of the consumption-good firm canis restricted to a subset of upstream producers. Since the production of machineries requires some time, consumption-good firms first order the capital good that delivered at the end of the period. The price of each machine-tool has a mark-up on its cost.

When comes at how to finance their investments, consumption-good firms operate in imperfect credit markets (in the spirit of (Stiglitz and Weiss, 1981)). Internal finance has priority: if are not able to fully cover production and investment costs, they will rely on external finance, by borrowing funds from a bank in the form of a credit line. Given the total credit (exposure) of a bank, the latter lends out money to firms on a pecking-order, determined by the ratio between equity and sales (see Dosi et al., 2013). If the credit demanded by consumption-good firms exceeds its supply, firms are credit- rationed. Also in the downstream sector, firms charge a markup over the unit cost of production according to the following rule:

$$p_{j,t}^{con} = c_{j,t}^{con}[1 + \mu_{j,t}]. \qquad (23)$$

The choice of the markup is determined by selection processes of the markets in which firms operate. In particular, it depends on the evolution firms' market share, $f_j$:
$$ \mu_{j,t} = \mu_{j,t-1} \left[ 1 + \nu \frac{f_{j,t-1} - f_{j,t-2}}{f_{j,t-2}} \right] \quad (24) $$

with $0 \le \nu \le 1$.

Moreover, market shares evolution is governed by a “quasi replicator” mechanism: less competitive firms are driven out from the market as the level of their competitiveness decreases.

At the end of every period, all firms' profits (net of taxes) are computed and the level of cash reserves is updated. If the latter is negative or market share goes to zero, a firm exits the market and it is replaced by a new entrant.

### A.3 The Energy Industry

The model that we are extending, relies on an energy sector (Lamperti et al., 2018). Upstream and downstream sectors produce their goods also using energy as input, which is produced by a vertically-integrated monopolist owning power plants. In this model interactions and feedbacks stemming from the energy sector that performs R&D for green and dirty technologies are not relevant for the macro-financial dynamics. The only feedback mechanism that we embed is on the dynamics of unit cost of production of the two competitive sectors, as the latter use energy for manufacturing their products. For a more detailed discussion of the functioning of this sector we refer to section 2.2.1 of Lamperti et al. (2018)

### A.4 The public sector

The public sector collects taxes on incomes (firm profits and wages) and when unemployment rises consumers are paid a subsidy, proportional to the current level of market wage. As in other versions of the model, wages are determined by institutional, market-related and macroeconomic factors. Accordingly, they ultimately depend on the inflation gap, average productivity, and unemployment rate, as follows:

$$ \frac{\Delta w_t}{w_{t-1}} = \pi^T + \psi_1(\pi_{t-1} - \pi^T) + \psi_2 \frac{\Delta \overline{AB}_t}{\overline{AB}_{t-1}} - \psi_3 \frac{\Delta U_t}{U_{t-1}}, \quad (25) $$

where $\overline{AB}$ stands for the economy-wide average productivity and $\psi_1, \psi_2, \psi_3 > 0$. The sum of all unemployment subsidies adds up to the level of Government expenditures $G_t = w_t^u(L^S - L_t^D)$. Since workers consume all their income, aggregate consumption is determined by the sum of all incomes, both from employed and unemployed: $C_t = w_t L_t^D + G_t$.

The tax rate is fixed at $tr$. Public expenditures also comprises the bank bailout costs. Public deficit is calculated accordingly and it set to be equal to $Def_t = Debt_t^{cost} + G_t^{bailout} + G_t - Tax_t$. Whenever the deficit is positive, the Government issues bonds that are acquired by banks according to their size.

All the aggregate variables are then the result of microeconomic behavior and interaction. Since there are not intermediate goods, aggregate production is the sum of firms' value added. National accounting entities are also met: the value of total production corresponds to the sum of aggregate consumption, investment and change in inventories ($\Delta N_t$):
$$ \mu_{j,t} = \mu_{j,t-1} \left[ 1 + \nu \frac{f_{j,t-1} - f_{j,t-2}}{f_{j,t-2}} \right] \qquad (24) $$

with $0 \le \nu \le 1$.

Moreover, market shares evolution is governed by a “quasi replicator” mechanism: less competitive firms are driven out from the market as the level of their competitiveness decreases.

At the end of every period, all firms' profits (net of taxes) are computed and the level of cash reserves is updated. If the latter is negative or market share goes to zero, a firm exits the market and it is replaced by a new entrant.

### A.3 The Energy Industry

The model that we are extending, relies on an energy sector (Lamperti et al., 2018). Upstream and downstream sectors produce their goods also using energy as input, which is produced by a vertically-integrated monopolist owning power plants. In this model interactions and feedbacks stemming from the energy sector that performs R&D for green and dirty technologies are not relevant for the macro-financial dynamics. The only feedback mechanism that we embed is on the dynamics of unit cost of production of the two competitive sectors, as the latter use energy for manufacturing their products. For a more detailed discussion of the functioning of this sector we refer to section 2.2.1 of Lamperti et al. (2018)

### A.4 The public sector

The public sector collects taxes on incomes (firm profits and wages) and when unemployment rises consumers are paid a subsidy, proportional to the current level of market wage. As in other versions of the model, wages are determined by institutional, market-related and macroeconomic factors. Accordingly, they ultimately depend on the inflation gap, average productivity, and unemployment rate, as follows:

$$ \frac{\Delta w_t}{w_{t-1}} = \pi^T + \psi_1(\pi_{t-1} - \pi^T) + \psi_2 \frac{\Delta \overline{AB}_t}{\overline{AB}_{t-1}} - \psi_3 \frac{\Delta U_t}{U_{t-1}}, \qquad (25) $$

where $\overline{AB}$ stands for the economy-wide average productivity and $\psi_1, \psi_2, \psi_3 > 0$. The sum of all unemployment subsidies adds up to the level of Government expenditures $G_t = w_t^u(L^S - L_t^D)$. Since workers consume all their income, aggregate consumption is determined by the sum of all incomes, both from employed and unemployed: $C_t = w_t L_t^D + G_t$.

The tax rate is fixed at $tr$. Public expenditures also comprises the bank bailout costs. Public deficit is calculated accordingly and it set to be equal to $Def_t = Debt_t^{cost} + G_t^{bailout} + G_t - Tax_t$. Whenever the deficit is positive, the Government issues bonds that are acquired by banks according to their size.

All the aggregate variables are then the result of microeconomic behavior and interaction. Since there are not intermediate goods, aggregate production is the sum of firms' value added. National accounting entities are also met: the value of total production corresponds to the sum of aggregate consumption, investment and change in inventories ($\Delta N_t$):
$$ \sum_{i=1}^{F_1} Q_{i,t} + \sum_{j=1}^{F_2} Q_{j,t} = Y_t = C_t + I_t + \Delta N_t $$
$$ \sum_{i=1}^{F_1} Q_{i,t} + \sum_{j=1}^{F_2} Q_{j,t} = Y_t = C_t + I_t + \Delta N_t $$
B Appendix: Tables and Figures

<table><thead><tr><th>Stylized facts</th><th>Empirical studies (among others)</th></tr></thead><tbody><tr><td><strong>Macroeconomic stylized facts</strong><br> SF1 Endogenous self-sustained growth with persistent fluctuations<br> SF2 Fat-tailed GDP growth-rate distribution<br> SF3 Recession duration exponentially distributed<br> SF4 Relative volatility of GDP, consumption, investments and debt<br> SF5 Cross-correlations of macro variables<br> SF6 Pro-cyclical aggregate R&amp;D investment<br> SF7 Cross-correlations of credit-related variables<br> SF8 Cross-correlation between firm debt and loan losses<br> SF9 Cross-correlation financial aggregates</td><td>Burns and Mitchell (1946); Kuznets and Murphy (1966)<br> Zarnowitz (1985); Stock and Watson (1999)<br> Fagiolo et al. (2008); Castaldi and Dosi (2009)<br> Ausloos et al. (2004); Wright (2005)<br> Stock and Watson (1999); Napoletano et al. (2006)<br> Stock and Watson (1999); Napoletano et al. (2006)<br> Wälde and Woitek (2004)<br> Lown and Morgan (2006); Leary (2009)<br> Foos et al. (2010)<br> Adrian et al. (2013); Zoltan and Kumhof (2015)</td></tr><tr><td><strong>Interbank network stylized facts</strong><br> SF10 Disassortativity<br> SF11 Centrality-bank size relation<br> SF12 Heterogeneous interbank network density<br> SF13 Large (small) banks are net borrowers (lenders)</td><td>Soramäki et al. (2007); Cocco et al. (2009)<br> Craig and von Peter (2014); Fricke and Lux (2015)<br> Alves et al. (2013), Aldasoro and Alves (2018)<br> Müller (2006); Liu et al. (2020)</td></tr><tr><td><strong>Microeconomic stylized facts</strong><br> SF14 Firm (log) size distribution is right-skewed<br> SF15 Fat-tailed firm growth-rate distribution<br> SF16 Productivity heterogeneity across firms<br> SF17 Persistent productivity differential across firms<br> SF18 Lumpy investment rates at firm-level</td><td>Dosi (2005)<br> Bottazzi and Secchi (2003, 2006)<br> Bartelsman and Doms (2000); Dosi (2005)<br> Bartelsman and Doms (2000); Dosi (2005)<br> Doms et al. (1998)</td></tr></tbody></table>

Notes: The table reports the full list of stylized facts that our model is able to replicate. In *italics*, we highlight those that are relative to our version.

<caption>**Table 4:** Main empirical stylized facts replicated by the model.</caption>