Seminar talk: A new approach to business fluctuations: heterogeneous interacting agents, scaling laws and financial fragility

1. Notes: A new approach to business
fluctuations: heterogeneous interacting
agents, scaling laws and financial fragility
(JEBO 2005)
Li Xihao
Polytechnic University of Marche,
Ancona, Italy
xihao.li@gmail.com
2013. 03. 25

2. Introduction: Main Points
◮
The paper: (Gatti D D, Guilmi C D, Gaffeo E, et al. 2005): A
new approach to business fluctuations: heterogeneous
interacting agents, scaling laws and financial fragility. Journal
of Economic behavior & organization, 2005, 56(4): 489-512.
◮
The idea: By agent-based modeling and simulation, the
interaction of heterogeneous financially fragile firms and a
banking sector replicates business fluctuation.

3. Introduction: Agent-based Methodology
◮
Model the economy from ‘bottom-up’.
◮
Agents are heterogeneous, interacting with each other.
◮
Networks to capture the interrelationship among agents.
◮
interaction among agents in ‘micro-foundation’ produces the
economic phenomenon in ‘macro-level’.

4. Empirical Observations
◮
(Axtell 2001) and (Gaffeo et al. 2003) have shown the
distribution of firms size follows a Zipf or power law.
◮
(Stanley et al. 1996) and (Amaral et al. 1997) have found the
growth rate of firms output follows a Laplace distribution.

5. Firms’ size: Power law Vs. Lognormal distribution

6. Model: the economy
◮
Economy with goods market and credit markets, in discrete
time periods t = 1, 2, . . ..
◮
Nt firms and one banking sector.

7. Firm i = 1, . . . , N: I
◮
Use capital Kit to produce goods, under a linear production
technology: Yit = φKit .
◮
Firm receives a selling price Pit fluctuating around the average
market price Pt , with Pit = uit Pt , E(uit ) = 1.
◮
Firm get loans Lit from the banking sector under the interest
rate rit . Firm’s financial cost is rit (Lit + Ait ) = rit Kit .
◮
Firm’s profit: πit = uit Yit − grit Kit = (uit φ − grit )Kit , g > 1.
Firm’s expected profit: E(πit ) = (φ − grit )Kit .
◮
Firm’s net worth: Ait = Ait−1 + πit .
◮
Firm goes bankrupt when Ait < 0, the threshold of firm’s
bankruptcy is:
1
Ait−1
uit ≡ (grit −
).
φ
Kit

8. Firm i = 1, . . . , N: II
◮
Firm’s net worth: Ait = Ait−1 + πit .
◮
Firm goes bankrupt when Ait < 0, the threshold of firm’s
bankruptcy is:
Ait−1
1
).
uit ≡ (grit −
φ
Kit
◮
2
Firm faces bankruptcy cost C f when uit < uit : C f = cYit .
◮
Firm maximizes its expected profit, with its objective function:
Γit
= E(πit ) − E(C f )
= (φ − grit )Kit −
◮
φc
2
(grit Kit − Ait−1 Kit ).
2
The optimal capital stock is:
d
Kit =
Ait−1
φ − grit
+
.
cφgrit
2grit

9. Firm i = 1, . . . , N: III
◮
◮
d
To keep optimal capital stock, firm invests Iit = Kit − Kit−1 .
Iit is either from profits or by loan in the credit market, i.e.
Iit = πit−1 + ∆Lit = πit−1 + (Lit − Lit−1 )
The demand for credit is:
Ld =
it
1 − 2grit
φ − grit
− πit−1 + (
)Ait−1 .
cφgrit
2grit

10. The Banking Sector: I
◮
◮
◮
The total credit supply from the banking sector is:
Ls = Et + Dt = Et−1 /v .
t
K
A
Credit for firm i is: Ls = λLs Kit−1 + (1 − λ)Ls Ait−1 , with
t t−1
t t−1
it
Kt−1 = i Kit−1 and At−1 = i Ait−1 .
K
Denote κit−1 = Kit−1 and αit−1 =
t−1
interest rate for firm i is:
rit =
Ait−1
At−1 ,
the equilibrium
2 + Ait−1
.
2cg ((1/φc) + πit−1 + Ait−1 ) + 2cgLs [λκit−1 + (1 − λ)αit−1 ]
t

11. The Banking Sector: II
◮
The banking sector’s profit is:
B
πt =
rit Ls − rt [(1 − ω)Dt−1 + Et−1 ],
it
i ∈Nt
where the return rate of the bank’s equity is rt , the average of
1
lending interest rate; and (1−ω) rt is the borrowing rate of the
deposits Dt−1 .
◮
When a firm goes bankrupt, the banking sector bears the bad
debt loss: Bit = Lit − Kit .
◮
The banking sector’s equity evolves:
B
Et = πt + Et−1 −
Bit−1 .
i ∈Ωt−1

12. Entry-Exit Mechanism
◮
Firm i goes bankrupt if Ait < 0.
◮
Number of new entry firms is:
entry
Nt
= NPr (entry ) =
N
,
1 + exp[d(rt−1 − e)]
with d, e, and N > 1 are constants.
◮
New entry firm j has initial capital stock Kj0 randomly drawn
from a uniform distribution with the mode of the size
distribution of surviving firms.
◮
New entry firm j endows with an equity ratio aj0 = Aj0 /Kj0
equal to the mode of the equity base distribution of surviving
firms.

13. Simulation: Setup
◮
Simulation with 10, 000 firms for 1, 000 periods.

14. Simulation Result: Aggregate Output

15. Simulation Result: Growth Rates of Aggregate Output

16. Simulation Result: Firm Size

17. Simulation Result: Growth Rate Distribution

18. Open Topics
◮
Micro foundation of Macroeconomic models.
◮
Representative Agents (RA) or Heterogenous Interactive
Agents (HIA).

19. References
◮
◮
◮
◮
◮
(Gatti et al. 2005): Gatti D D, Guilmi C D, Gaffeo E, et al. A
new approach to business fluctuations: heterogeneous
interacting agents, scaling laws and financial fragility. Journal
of Economic behavior & organization, 2005, 56(4): 489-512.
(Axtell 2001): Axtell, R., 2001. Zipf distribution of U.S. firm
sizes. Science 293, 1818C1820.
(Gaffeo et al. 2003): Gaffeo, E., Gallegati, M., Palestrini, A.,
2003. On the size distribution of firms. Additional evidence
from the G7 countries. Physica A 324, 117C123.
(Stanley et al. 1996): Stanley, M., Amaral, L., Buldyrev, S.,
Havlin, S., Leschorn, H., Maas, P., Salinger, M., Stanley, E.,
1996. Scaling behavior in the growth of companies. Nature
379, 804C806.
(Amaral et al. 1997): Amaral, L., Buldyrev, S., Havlin, S.,
Leschhorn, H., Maas, P., Salinger,M., Stanley, E., Stanley, M.,
1997. Scaling behavior in Economics: I. Empirical results for
company growth. Journal de Physique 7, 621C633.