Interacting Agents Produce Endogenous Inequality

Introduction
The Model
Results
Conclusion

Endogeneous Inequality

Stephen Kinsella, Edward J. Nell, Matthias Greiﬀ

Annual Meeting of the Eastern Economic Association
February/March 2009, New York City

February 21, 2009

Stephen Kinsella, Edward J. Nell, Matthias Greiﬀ Endogeneous Inequality

Introduction
The Model Distributions of Income
Results Econophysics
Conclusion

Income Distributions and Econophysics
1
Distributions of Income
Econophysics

The Model
2
Behavioral Rules
Structure of the Model

Results
3
Model without bank
Introducing debt
Mobility

Conclusion
4


Introduction
Conclusion

Income Distribution

Vilfredo Pareto, “Ecrits sur la courbe de la repartition de la
richesse”, ch. 2, Librairie Droz, Geneve, (1896) 1965.
David G. Champernowne, “A model of income distribution”,
EJ, 1953.
Benoit Mandelbrot, “The Pareto-Levy Law and the
Distribution of Income”, IER, 1960.
John Angle (1983-. . . ), The Inequality Process.


Introduction
Conclusion

Conservation Principle

Conservation Law
Idea from physics: conservation of energy.
In econophysics: conservation of money.
We cannot keep track of all goods consumed.
A simple econophysics model
n agents, each agent has m Dollars
total amount of money M = n × m ¯
each period two agents are drawn and a random amount of
money is transferred from one agent to the other
nonnegativity constraint, mi ≥ 0


Introduction
Conclusion

Distribution of Money

distribution of money converges to a Boltzmann-Gibbs
exponential distribution (entropy increases)
thermodynamic equilibrium P(m) = c × e −m/m
¯

m: money temparature, c: normalizing constant
¯


nential Boltzmann- the Web page [35].Introduction
The final distribution is universal despiteof Income rules
The Model Distributions different
for ∆m. To amplify this point further, Ref. [25] also con-
Conclusion
. (7) sidered a toy model, where ∆m was taken to be a ran-
Distribution of Money dom fraction of the average amount of money of the two
d Tm is the “money
agents: ∆m = ν(mi + mj )/2. This model produced the
average amount of
where M is the total
ts. 5 5
N=500, M=5*10 , time=4*10 .
5] performed agent- 18
y transfers between 16
n the same amount !mquot;, T
of agents (i, j) was 14
3
as transferred from 12
Probability, P(m)

was repeated many

log P(m)
2
ility distribution of 10
animation videos at 1
8
itory period, money
0
6
nary form shown in 0 1000 2000 3000
Money, m
n is very well fitted 4

2
e considered in Ref.
amount was fixed 0
0 1000 2000 3000 4000 5000 6000
cally, it means that Money, m
s for the same price
shows that the ini- FIG. 1 Histogram and points: Stationary probability distri-
dens toFigure: Boltzmann-Gibbs P (m) obtained in agent-based computer sim-
bution of money exponential distribution for money
a symmet- (Source:
ulations. Solid curves: Fits to the Boltzmann-Gibbs law (7).
r a diffusion process.
p around the m = 0 2008). lines: The initial distribution of money. (Reproduced
Yakovenko Vertical
from Ref. [25])
e boundary, because

Introduction
Conclusion

Critique & Modiﬁcations

Critique
Model is attractive in its simplicity but represents a rather
primitive picture of the market.
Agents are characterized by their amount of money.
Modiﬁcations
Heterogeneous agents (in terms of money, abilities,
opportunities, and savings rates).
Ability changes as agents spend money on education.


Introduction
The Model Behavioral Rules
Results Structure of the Model
Conclusion

The Question

How is inequality of incomes and wealth generated?

Simple four sector model.
Conservation law should be fulﬁlled.
Model should produce exponential and power-law distributions
of income.
Model should be more realistic but not too complex.


Introduction
Conclusion

Characteristics of the Model

no representative agent
no utility function
no production function
no rational expectations
large number of heterogeneous agents
individual behavior is unpredictable
individuals follow simple rules
indeterminacy at the micro level (random selection from a
given distribution)


Introduction
Conclusion

Four Sectors

In the simplest version of our model we have three sectors.
Workers...
search for work.
work for a wage or get dole.
spend money on consumption (demand).
Firms...
hire workers.
pay wages.
receive revenue from selling output.
Government: collects taxes and provides dole.
Add banking sector later.


Introduction
Conclusion

Wage Bargaining

Each agents’ reservation wage is given by:
w (m, a, o) : R3 → R+
Hiring rule:
Every unemployed worker is matched with a randomly chosen
firm.
Provided that the firm has enough money to pay the wage,
wF W
they sign a wage contract with probability p = min w W , w F .
w

Firing rule: If a firm has not enough money to pay all its
employees, layoff workers until the firm can pay the wagebill
for the remaining workers.
Wage payment rule: Employees get their wages.
Dole payment rule: Unemployed workers get a dole-income
which is a fraction of their reservation wage.

Introduction
Conclusion

Demand

Each agent (workers and firms) spends money on average
once a month.
Agents save a fraction of their money sm, (s ∈ [0, 1]).
The agent (=buyer) spends a random fraction u (u ∈ U [0, 1])
of his remaining money (1 − s)m on consumption,
∆m = u(1 − s)m.
A fraction t∆m goes to the government as tax income (t =
tax rate).
The remaining part (1 − t)∆m is transferred to seller.
The seller is a firm. The probability that a particular firm is
choosen is proportional to the sum of its employees’ abilities.


Introduction
Conclusion

Ability & Education

Modelled in a rather crude way.
Assume that a fraction of money is spend on education.
Ability increases by η = U [0, o] ∆m.
But more sophisticated versions are possible...


Introduction
Conclusion

Government

Government is passive (no government spending besides dole).
Spend money on dole.
Collect taxes on consumption.
Increase tax rate if government deﬁcit, decrease if surplus.


Introduction
Conclusion

Structure of the Model

Each month the following happens:
Wage bargaining.
Demand.
At the end of each year we collect data on income distribution
(and other data).


Introduction
Model without bank
The Model
Introducing debt
Results
Mobility
Conclusion

Income Distribution

0.14

0.12

0.10

0.08

0.06

0.04

0.02

5 10 15 20 25 30

Figure: Probability density function for 95% of incomes and exponential
distribution.


Introduction
Model without bank
The Model
Introducing debt
Results
Mobility
Conclusion

Banks: zero interest rate

debt is permitted (negative money)
unlimited borrowing has to be precluded
total amount of debt is limited by minimum reserve
requirement for banks, M = M0 r
maximal debt of any agent is limited by, mi > −md ∀i
¯
debt: increase in money temparature
Money supply ’increases’ (money multiplier) but conservation
law is still fulﬁlled!


Introduction
Model without bank
The Model
Introducing debt
Results
Mobility
Conclusion

Bond Market

Introduce a market for one-year bonds.
Agents can save (buy bonds) or get a loan (sell bonds).
Higher interest rate r increases supply and reduces demand.
Trading at disequilibrium.
Interest rate r adjusts.
r increases if excess demand for bonds.
r decreases if excess supply for bonds.


Introduction
Model without bank
The Model
Introducing debt
Results
Mobility
Conclusion

Interest Rate Adjustment

r
r

supply
supply

r2
r1
r1

demand
demand

t t+1

Figure: Excess demand in the bond market.


Introduction
Model without bank
The Model
Introducing debt
Results
Mobility
Conclusion

Measuring Mobility

N
1 0 − log mi1 |
i=1 | log mi
Mb = N

Two time period framework.
Money at time t: m0 = (m1 , m2 , . . . , mN ) .
0 0 0

Money at time t + 10: m1 = (m1 , m2 , . . . , mN ) .
1 1 1

Source: G.S. Fields & E.A. Ok, “Measuring Movement of Income”,
Economica (1999).


Introduction
Model without bank
The Model
Introducing debt
Results
Mobility
Conclusion

Mobility
mobility
1.5

1.4

1.3

1.2

1.1

1.0

0.9

spending
0.2 0.4 0.6 0.8

Figure: Absolute mobility and spending.

Higher savings → lower mobility.
Higher mobility if debt is allowed for.

Introduction
The Model
Results
Conclusion

Conclusion

summarize main results from our model


Introduction
The Model
Results
Conclusion

Problems

Workers income is now wage income plus interest earned /
paid (on bonds).
Income can get negative if interest payment > wage income.
A possible solution: Restrict borrowing and introduce a
minimum wage such that income from minimum wage is
suﬃciently high to pay interest.
Or: Allow for agents to get bankrupt. (Interest rate on
borrowing > interest rate on lending.)


Introduction
The Model
Results
Conclusion

Further research and possible extensions

Further research:
Allow for more than one bank.
Look at ﬁrm size distribution.
Fit model to actual data.
Possible Extensions:
Add production technology and growth.
Introduce central bank.
Look at the eﬀects of policy.


Introduction
The Model
Results
Conclusion

Conclusion

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http://matthiasgreiﬀ.wordpress.com
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Interacting Agents Produce Endogenous Inequality

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