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Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography

The Financial Instability Hypothesis: a Stochastic Microfoundation Framework

Carl Chiarella and Corrado Di Guilmi

School of Finance and Economics - University of Technology, Sydney

Computing in Economics and Finance

15th International Conference

Sydney 17/07/2009

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography

Aim

To consistently microfound the model of financial instability proposed by Minsky (1975) and Taylor and O’Connell (1985) in which investors’ expectations drive investments, according to the mechanism first described by Keynes.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography

Aim

To consistently microfound the model of financial instability proposed by Minsky (1975) and Taylor and O’Connell (1985) in which investors’ expectations drive investments, according to the mechanism first described by Keynes.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography

The issue: Heterogeneous and interacting agents

Minsky (1963): Financial Instability Hypothesis:

Three types of firms with respect to their financial soundness: hedge, speculative, Ponzi; “Shifts of firms among classes as the economy evolves in historical time underlie much of its cyclical behavior. This detail is rich and illuminating but beyond the reach of mere algebra” [Taylor and O’Connell, 1985].

↓ Two different methods for model solution:

1

the agent based model with numerical simulation;

2

the stochastic dynamic aggregation framework [Aoki and Yoshikawa, 2006].

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography

The issue: Heterogeneous and interacting agents

Minsky (1963): Financial Instability Hypothesis:

Three types of firms with respect to their financial soundness: hedge, speculative, Ponzi; “Shifts of firms among classes as the economy evolves in historical time underlie much of its cyclical behavior. This detail is rich and illuminating but beyond the reach of mere algebra” [Taylor and O’Connell, 1985].

↓ Two different methods for model solution:

1

the agent based model with numerical simulation;

2

the stochastic dynamic aggregation framework [Aoki and Yoshikawa, 2006].

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography

The issue: Heterogeneous and interacting agents

Minsky (1963): Financial Instability Hypothesis:

Three types of firms with respect to their financial soundness: hedge, speculative, Ponzi; “Shifts of firms among classes as the economy evolves in historical time underlie much of its cyclical behavior. This detail is rich and illuminating but beyond the reach of mere algebra” [Taylor and O’Connell, 1985].

↓ Two different methods for model solution:

1

the agent based model with numerical simulation;

2

the stochastic dynamic aggregation framework [Aoki and Yoshikawa, 2006].

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography

The issue: Heterogeneous and interacting agents

Minsky (1963): Financial Instability Hypothesis:

Three types of firms with respect to their financial soundness: hedge, speculative, Ponzi; “Shifts of firms among classes as the economy evolves in historical time underlie much of its cyclical behavior. This detail is rich and illuminating but beyond the reach of mere algebra” [Taylor and O’Connell, 1985].

↓ Two different methods for model solution:

1

the agent based model with numerical simulation;

2

the stochastic dynamic aggregation framework [Aoki and Yoshikawa, 2006].

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography

The issue: Heterogeneous and interacting agents

Minsky (1963): Financial Instability Hypothesis:

Three types of firms with respect to their financial soundness: hedge, speculative, Ponzi; “Shifts of firms among classes as the economy evolves in historical time underlie much of its cyclical behavior. This detail is rich and illuminating but beyond the reach of mere algebra” [Taylor and O’Connell, 1985].

↓ Two different methods for model solution:

1

the agent based model with numerical simulation;

2

the stochastic dynamic aggregation framework [Aoki and Yoshikawa, 2006].

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography

The issue: Heterogeneous and interacting agents

Minsky (1963): Financial Instability Hypothesis:

Three types of firms with respect to their financial soundness: hedge, speculative, Ponzi; “Shifts of firms among classes as the economy evolves in historical time underlie much of its cyclical behavior. This detail is rich and illuminating but beyond the reach of mere algebra” [Taylor and O’Connell, 1985].

↓ Two different methods for model solution:

1

the agent based model with numerical simulation;

2

the stochastic dynamic aggregation framework [Aoki and Yoshikawa, 2006].

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography

The issue: Heterogeneous and interacting agents

Minsky (1963): Financial Instability Hypothesis:

Three types of firms with respect to their financial soundness: hedge, speculative, Ponzi; “Shifts of firms among classes as the economy evolves in historical time underlie much of its cyclical behavior. This detail is rich and illuminating but beyond the reach of mere algebra” [Taylor and O’Connell, 1985].

↓ Two different methods for model solution:

1

the agent based model with numerical simulation;

2

the stochastic dynamic aggregation framework [Aoki and Yoshikawa, 2006].

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography

Outline

1

Introduction

2

Agent based model Hypotheses

3

Stochastic dynamics Set up Master equation Analytical solution

4

Simulations Results

5

Concluding remarks Results Future research

6

Bibliography

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography

Outline of the model

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Firms

A firm j decides on investment based on the shadow-price of capital Pj

k(t):

I j(t) = aPj

k(t)

(1) where

the shadow-price of capital is Pj

k(t) = (r(t) + ρj(t))P

i(t) (2) ρj is the expected difference of return to capital for the firm j with respect to the average level r; i is the interest rate, P is the final good price and a is a parameter.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Firms

A firm j decides on investment based on the shadow-price of capital Pj

k(t):

I j(t) = aPj

k(t)

(1) where

the shadow-price of capital is Pj

k(t) = (r(t) + ρj(t))P

i(t) (2) ρj is the expected difference of return to capital for the firm j with respect to the average level r; i is the interest rate, P is the final good price and a is a parameter.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Firms

A firm j decides on investment based on the shadow-price of capital Pj

k(t):

I j(t) = aPj

k(t)

(1) where

the shadow-price of capital is Pj

k(t) = (r(t) + ρj(t))P

i(t) (2) ρj is the expected difference of return to capital for the firm j with respect to the average level r; i is the interest rate, P is the final good price and a is a parameter.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Firms

A firm j decides on investment based on the shadow-price of capital Pj

k(t):

I j(t) = aPj

k(t)

(1) where

the shadow-price of capital is Pj

k(t) = (r(t) + ρj(t))P

i(t) (2) ρj is the expected difference of return to capital for the firm j with respect to the average level r; i is the interest rate, P is the final good price and a is a parameter.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Firms

A firm j decides on investment based on the shadow-price of capital Pj

k(t):

I j(t) = aPj

k(t)

(1) where

the shadow-price of capital is Pj

k(t) = (r(t) + ρj(t))P

i(t) (2) ρj is the expected difference of return to capital for the firm j with respect to the average level r; i is the interest rate, P is the final good price and a is a parameter.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Firms prefer to finance their investments:

first with retained earnings Aj and, then with new equities E j or debt Dj (in a proportion dependent on the level of interest rate)

Firms are classified into two groups according to their level of debt Dj:

state z = 1: speculative firms: Dj(t) > 0 state z = 2: hedge firms: Dj(t) = 0

Correspondingly, there are two types of shares in the market, with prices Pe,1 and Pe,2.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Firms prefer to finance their investments:

first with retained earnings Aj and, then with new equities E j or debt Dj (in a proportion dependent on the level of interest rate)

Firms are classified into two groups according to their level of debt Dj:

state z = 1: speculative firms: Dj(t) > 0 state z = 2: hedge firms: Dj(t) = 0

Correspondingly, there are two types of shares in the market, with prices Pe,1 and Pe,2.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Firms prefer to finance their investments:

first with retained earnings Aj and, then with new equities E j or debt Dj (in a proportion dependent on the level of interest rate)

Firms are classified into two groups according to their level of debt Dj:

state z = 1: speculative firms: Dj(t) > 0 state z = 2: hedge firms: Dj(t) = 0

Correspondingly, there are two types of shares in the market, with prices Pe,1 and Pe,2.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Firms prefer to finance their investments:

first with retained earnings Aj and, then with new equities E j or debt Dj (in a proportion dependent on the level of interest rate)

Firms are classified into two groups according to their level of debt Dj:

state z = 1: speculative firms: Dj(t) > 0 state z = 2: hedge firms: Dj(t) = 0

Correspondingly, there are two types of shares in the market, with prices Pe,1 and Pe,2.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Firms prefer to finance their investments:

first with retained earnings Aj and, then with new equities E j or debt Dj (in a proportion dependent on the level of interest rate)

Firms are classified into two groups according to their level of debt Dj:

state z = 1: speculative firms: Dj(t) > 0 state z = 2: hedge firms: Dj(t) = 0

Correspondingly, there are two types of shares in the market, with prices Pe,1 and Pe,2.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

A firm fails if Dj(t) > cK j(t), with c > 1. The probability of new firm entering the system is directly proportional to the variation in the aggregate output observed in the previous period.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

A firm fails if Dj(t) > cK j(t), with c > 1. The probability of new firm entering the system is directly proportional to the variation in the aggregate output observed in the previous period.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

A firm fails if Dj(t) > cK j(t), with c > 1. The probability of new firm entering the system is directly proportional to the variation in the aggregate output observed in the previous period.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Investors

Two possible types of investors: chartists (proportion nc) and fundamentalists (proportion 1 − nc); we assume that chartists favour the speculative firms: ρj

1(t) = nc(t)

˜ ̟j(t) , ρj

2(t) = 1 − nc(t)

˜ ̟j(t) ; where

˜ ̟j is an idiosyncratic random variable; the proportion of chartists in the market nc is randomly drawn from a uniform distribution.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Investors

Two possible types of investors: chartists (proportion nc) and fundamentalists (proportion 1 − nc); we assume that chartists favour the speculative firms: ρj

1(t) = nc(t)

˜ ̟j(t) , ρj

2(t) = 1 − nc(t)

˜ ̟j(t) ; where

˜ ̟j is an idiosyncratic random variable; the proportion of chartists in the market nc is randomly drawn from a uniform distribution.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Investors

Two possible types of investors: chartists (proportion nc) and fundamentalists (proportion 1 − nc); we assume that chartists favour the speculative firms: ρj

1(t) = nc(t)

˜ ̟j(t) , ρj

2(t) = 1 − nc(t)

˜ ̟j(t) ; where

˜ ̟j is an idiosyncratic random variable; the proportion of chartists in the market nc is randomly drawn from a uniform distribution.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Investors

Two possible types of investors: chartists (proportion nc) and fundamentalists (proportion 1 − nc); we assume that chartists favour the speculative firms: ρj

1(t) = nc(t)

˜ ̟j(t) , ρj

2(t) = 1 − nc(t)

˜ ̟j(t) ; where

˜ ̟j is an idiosyncratic random variable; the proportion of chartists in the market nc is randomly drawn from a uniform distribution.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Wealth allocation

using the mean field approximations ρ1 and ρ2 (a statistic of ρ1

z, ..., ρj z, ..., ρN z ), prices and allocations of the wealth W are

calculated according to            ǫ1(i, ρ1, ρ2, ψ)W = Pe,1E1 ǫ2(i, ρ1, ρ2, ψ)W = Pe,2E2 β(i, ρ1, ρ2, ψ)W = D Ψ(i, ρ1, ρ2, ψ)W = M W = Pe1E1 + Pe2E2 + D + M (3) where the parameter ψ reflects the availability of near money activities.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Wealth allocation

using the mean field approximations ρ1 and ρ2 (a statistic of ρ1

z, ..., ρj z, ..., ρN z ), prices and allocations of the wealth W are

calculated according to            ǫ1(i, ρ1, ρ2, ψ)W = Pe,1E1 ǫ2(i, ρ1, ρ2, ψ)W = Pe,2E2 β(i, ρ1, ρ2, ψ)W = D Ψ(i, ρ1, ρ2, ψ)W = M W = Pe1E1 + Pe2E2 + D + M (3) where the parameter ψ reflects the availability of near money activities.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Wealth allocation

using the mean field approximations ρ1 and ρ2 (a statistic of ρ1

z, ..., ρj z, ..., ρN z ), prices and allocations of the wealth W are

calculated according to            ǫ1(i, ρ1, ρ2, ψ)W = Pe,1E1 ǫ2(i, ρ1, ρ2, ψ)W = Pe,2E2 β(i, ρ1, ρ2, ψ)W = D Ψ(i, ρ1, ρ2, ψ)W = M W = Pe1E1 + Pe2E2 + D + M (3) where the parameter ψ reflects the availability of near money activities.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

Wealth allocation

using the mean field approximations ρ1 and ρ2 (a statistic of ρ1

z, ..., ρj z, ..., ρN z ), prices and allocations of the wealth W are

calculated according to            ǫ1(i, ρ1, ρ2, ψ)W = Pe,1E1 ǫ2(i, ρ1, ρ2, ψ)W = Pe,2E2 β(i, ρ1, ρ2, ψ)W = D Ψ(i, ρ1, ρ2, ψ)W = M W = Pe1E1 + Pe2E2 + D + M (3) where the parameter ψ reflects the availability of near money activities.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

The variable ρ

The key variable for the allocation of wealth is ρj. It influences:

the level of firms’ investment through the shadow price Pj

k(t) = (r(t)+ρj(t))P i(t)

; the prices of shares Pe,1 and Pe,2 in system (3), reflecting the investors’ expectations on the different firms.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

The variable ρ

The key variable for the allocation of wealth is ρj. It influences:

the level of firms’ investment through the shadow price Pj

k(t) = (r(t)+ρj(t))P i(t)

; the prices of shares Pe,1 and Pe,2 in system (3), reflecting the investors’ expectations on the different firms.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

The variable ρ

The key variable for the allocation of wealth is ρj. It influences:

the level of firms’ investment through the shadow price Pj

k(t) = (r(t)+ρj(t))P i(t)

; the prices of shares Pe,1 and Pe,2 in system (3), reflecting the investors’ expectations on the different firms.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Hypotheses

The variable ρ

The key variable for the allocation of wealth is ρj. It influences:

the level of firms’ investment through the shadow price Pj

k(t) = (r(t)+ρj(t))P i(t)

; the prices of shares Pe,1 and Pe,2 in system (3), reflecting the investors’ expectations on the different firms.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Set up

The two dynamics

Using the mean field approximations ρ1 and ρ2 it is possible to replicate the model for a representative hedge firm and for a representative speculative firm; thus the model is able to generate dynamics in two different ways:

an agent based approach with N different agents; a stochastic approximation, with 2 different firms: one “good” and one “stressed”.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Set up

The two dynamics

Using the mean field approximations ρ1 and ρ2 it is possible to replicate the model for a representative hedge firm and for a representative speculative firm; thus the model is able to generate dynamics in two different ways:

an agent based approach with N different agents; a stochastic approximation, with 2 different firms: one “good” and one “stressed”.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Set up

The two dynamics

Using the mean field approximations ρ1 and ρ2 it is possible to replicate the model for a representative hedge firm and for a representative speculative firm; thus the model is able to generate dynamics in two different ways:

an agent based approach with N different agents; a stochastic approximation, with 2 different firms: one “good” and one “stressed”.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Set up

The two dynamics

Using the mean field approximations ρ1 and ρ2 it is possible to replicate the model for a representative hedge firm and for a representative speculative firm; thus the model is able to generate dynamics in two different ways:

an agent based approach with N different agents; a stochastic approximation, with 2 different firms: one “good” and one “stressed”.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Master equation

Hypotheses:

the number of firms is fixed and equal to N; the evolution of the numbers of firms of the two types N1 and N2 follows a continuous time jump Markov process.

The stochastic dynamics of the the proportion of the two types of firms can be described by a master equation: dp(Nz, t) dt = λp(Nz −1, t)+µp(Nz +1, t)−{[(λ + µ)p(Nz, t)]} (4) where λ and µ are the transition rates.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Master equation

Hypotheses:

the number of firms is fixed and equal to N; the evolution of the numbers of firms of the two types N1 and N2 follows a continuous time jump Markov process.

The stochastic dynamics of the the proportion of the two types of firms can be described by a master equation: dp(Nz, t) dt = λp(Nz −1, t)+µp(Nz +1, t)−{[(λ + µ)p(Nz, t)]} (4) where λ and µ are the transition rates.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Analytical solution

Using the asymptotic solution [Di Guilmi, 2008], the dynamics of the economy can be represented by the following system:    dn1(t) = (λn1(t) − (λ + µ)[n1(t)]2)dt + σ dθ dK(t) = dI(t)dt = N {I1(t)n1(t) + I2(t)[1 − n1(t)]} dt (5) where

θ(s) = C exp

• − s2

2σ2

• :

σ2 = λµ (λ + µ)2 (6)

with n1 indicates the proportion of speculative firms; s representing the fluctuations component of the stochastic process for n1. s representing the fluctuations component of the stochastic process for n1.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Analytical solution

Using the asymptotic solution [Di Guilmi, 2008], the dynamics of the economy can be represented by the following system:    dn1(t) = (λn1(t) − (λ + µ)[n1(t)]2)dt + σ dθ dK(t) = dI(t)dt = N {I1(t)n1(t) + I2(t)[1 − n1(t)]} dt (5) where

θ(s) = C exp

• − s2

2σ2

• :

σ2 = λµ (λ + µ)2 (6)

with n1 indicates the proportion of speculative firms; s representing the fluctuations component of the stochastic process for n1. s representing the fluctuations component of the stochastic process for n1.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Analytical solution

Using the asymptotic solution [Di Guilmi, 2008], the dynamics of the economy can be represented by the following system:    dn1(t) = (λn1(t) − (λ + µ)[n1(t)]2)dt + σ dθ dK(t) = dI(t)dt = N {I1(t)n1(t) + I2(t)[1 − n1(t)]} dt (5) where

θ(s) = C exp

• − s2

2σ2

• :

σ2 = λµ (λ + µ)2 (6)

with n1 indicates the proportion of speculative firms; s representing the fluctuations component of the stochastic process for n1. s representing the fluctuations component of the stochastic process for n1.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Analytical solution

Using the asymptotic solution [Di Guilmi, 2008], the dynamics of the economy can be represented by the following system:    dn1(t) = (λn1(t) − (λ + µ)[n1(t)]2)dt + σ dθ dK(t) = dI(t)dt = N {I1(t)n1(t) + I2(t)[1 − n1(t)]} dt (5) where

θ(s) = C exp

• − s2

2σ2

• :

σ2 = λµ (λ + µ)2 (6)

with n1 indicates the proportion of speculative firms; s representing the fluctuations component of the stochastic process for n1. s representing the fluctuations component of the stochastic process for n1.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Results

Figure: Capital (upper panel) and share of speculative firms (lower panel).

Agent based model (black continuous line) and stochastic dynamics (red dashed line).

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Results

Figure: Rate of bankruptcy and aggregate capital (right axis). Simulation

• f agent based model.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Results

Figure: Distribution of debt (upper panel) and capital (lower panel) during debt cycle.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Results

Figure: Dynamics of aggregate capital for different values of ψ (Monte Carlo simulation - upper panel) and c (lower panel).

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Results

Figure: Distribution of capital across firms conditioned on a. Simulation

• f agent based model.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Results

a consistent microfoudation of Minsky’s investment framework; a model with heterogeneous agents for which an approximated analytical solution can be obtained:

results satisfactorily mimic the outcomes of an agent based model with a much higher degree of heterogeneity;

a tool to analyse the effect of instability in financial markets

• n the real sector of the economy.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Results

a consistent microfoudation of Minsky’s investment framework; a model with heterogeneous agents for which an approximated analytical solution can be obtained:

results satisfactorily mimic the outcomes of an agent based model with a much higher degree of heterogeneity;

a tool to analyse the effect of instability in financial markets

• n the real sector of the economy.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Results

a consistent microfoudation of Minsky’s investment framework; a model with heterogeneous agents for which an approximated analytical solution can be obtained:

results satisfactorily mimic the outcomes of an agent based model with a much higher degree of heterogeneity;

a tool to analyse the effect of instability in financial markets

• n the real sector of the economy.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Results

a consistent microfoudation of Minsky’s investment framework; a model with heterogeneous agents for which an approximated analytical solution can be obtained:

results satisfactorily mimic the outcomes of an agent based model with a much higher degree of heterogeneity;

a tool to analyse the effect of instability in financial markets

• n the real sector of the economy.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Results

a consistent microfoudation of Minsky’s investment framework; a model with heterogeneous agents for which an approximated analytical solution can be obtained:

results satisfactorily mimic the outcomes of an agent based model with a much higher degree of heterogeneity;

a tool to analyse the effect of instability in financial markets

• n the real sector of the economy.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Future research

the identification of the conditions under which the system generates speculative bubbles and how they burst; a more refined modelling of financial markets; the study of the effects of:

additional hypotheses on firms (e. g. the possibility of buying their own shares); high leverage; various forms of speculative behaviour; the shifting of debt: introduction of banking and public sector; government policies: fiscal, monetary; regulatory framework.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Future research

the identification of the conditions under which the system generates speculative bubbles and how they burst; a more refined modelling of financial markets; the study of the effects of:

additional hypotheses on firms (e. g. the possibility of buying their own shares); high leverage; various forms of speculative behaviour; the shifting of debt: introduction of banking and public sector; government policies: fiscal, monetary; regulatory framework.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Future research

the identification of the conditions under which the system generates speculative bubbles and how they burst; a more refined modelling of financial markets; the study of the effects of:

additional hypotheses on firms (e. g. the possibility of buying their own shares); high leverage; various forms of speculative behaviour; the shifting of debt: introduction of banking and public sector; government policies: fiscal, monetary; regulatory framework.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Future research

the identification of the conditions under which the system generates speculative bubbles and how they burst; a more refined modelling of financial markets; the study of the effects of:

additional hypotheses on firms (e. g. the possibility of buying their own shares); high leverage; various forms of speculative behaviour; the shifting of debt: introduction of banking and public sector; government policies: fiscal, monetary; regulatory framework.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Future research

the identification of the conditions under which the system generates speculative bubbles and how they burst; a more refined modelling of financial markets; the study of the effects of:

additional hypotheses on firms (e. g. the possibility of buying their own shares); high leverage; various forms of speculative behaviour; the shifting of debt: introduction of banking and public sector; government policies: fiscal, monetary; regulatory framework.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Future research

the identification of the conditions under which the system generates speculative bubbles and how they burst; a more refined modelling of financial markets; the study of the effects of:

additional hypotheses on firms (e. g. the possibility of buying their own shares); high leverage; various forms of speculative behaviour; the shifting of debt: introduction of banking and public sector; government policies: fiscal, monetary; regulatory framework.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Future research

the identification of the conditions under which the system generates speculative bubbles and how they burst; a more refined modelling of financial markets; the study of the effects of:

additional hypotheses on firms (e. g. the possibility of buying their own shares); high leverage; various forms of speculative behaviour; the shifting of debt: introduction of banking and public sector; government policies: fiscal, monetary; regulatory framework.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Future research

the identification of the conditions under which the system generates speculative bubbles and how they burst; a more refined modelling of financial markets; the study of the effects of:

additional hypotheses on firms (e. g. the possibility of buying their own shares); high leverage; various forms of speculative behaviour; the shifting of debt: introduction of banking and public sector; government policies: fiscal, monetary; regulatory framework.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography Future research

the identification of the conditions under which the system generates speculative bubbles and how they burst; a more refined modelling of financial markets; the study of the effects of:

additional hypotheses on firms (e. g. the possibility of buying their own shares); high leverage; various forms of speculative behaviour; the shifting of debt: introduction of banking and public sector; government policies: fiscal, monetary; regulatory framework.

Introduction Agent based model Stochastic dynamics Simulations Concluding remarks Bibliography

Aoki, Masanao and Yoshikawa, Hiroshi. Reconstructing Macroeconomics. Cambridge University Press, 2006. Di Guilmi, Corrado. The generation of business fluctuations: financial fragility and mean-field interaction. Peter Lang Publishing Group: Frankfurt/M., 2008. Minsky, Hyman. John Maynard Keynes. Columbia University Press: New York, 1975. Minsky, Hyman. Inflation, recession and economic policy. New York: ME Sharpe, 1982. Taylor, Lance and O’Connell, Stephen. A Minsky Crisis, The Quarterly Journal of Economics, 1985, 100(5): 871-85.