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IntroductionThe Model

ResultsConclusion

Interacting Heterogeneous Agents

Stephen Kinsella, Edward J. Nell, Matthias Greiff

February 27, 2009

Stephen Kinsella, Edward J. Nell, Matthias Greiff Interacting Heterogeneous Agents


ResultsConclusion

Econophysics

1 Income Distributions and EconophysicsEconophysics

2 The ModelLabor MarketEducationProductionDemandBanksStructure of the Model

3 ResultsMobilityIncome Distribution

4 Conclusion



ResultsConclusion

Econophysics

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 m̄ Dollars initiallytotal amount of money M = n × m̄each period two agents are drawn and a random amount ofmoney is transferred from one agent to the othernonnegativity constraint, mi ≥ 0



ResultsConclusion

Econophysics

Distribution of Money

distribution of money converges to a Boltzmann-Gibbsexponential distribution (entropy increases)

thermodynamic equilibrium P(m) = c × e−m/m̄

m̄: money temparature, c : normalizing constant



ResultsConclusion

Econophysics

Distribution of Money

5

III.D. As a starting point, Ref. [25] first considered sim-ple models, where debt is not permitted. This means thatmoney balances of agents cannot go below zero: mi ! 0for all i. Transaction (5) takes place only when an agenthas enough money to pay the price: mi ! !m, otherwisethe transaction does not take place. If an agent spendsall money, the balance drops to zero mi = 0, so the agentcannot buy any goods from other agents. However, thisagent can still produce goods or services, sell them toother agents, and receive money for that. In real life,cash balance dropping to zero is not at all unusual forpeople who live from paycheck to paycheck.

The conservation law is the key feature for the success-ful functioning of money. If the agents were permittedto “manufacture” money, they would be printing moneyand buying all goods for nothing, which would be a dis-aster. The physical medium of money is not essential, aslong as the conservation law is enforced. Money may bein the form of paper cash, but today it is more often rep-resented by digits in computerized bank accounts. Theconservation law is the fundamental principle of account-ing, whether in the single-entry or the double-entry form.More discussion of banks and debt is given in Sec. III.D.

C. The Boltzmann-Gibbs distribution of money

Having recognized the principle of money conserva-tion, Ref. [25] argued that the stationary distribution ofmoney should be given by the exponential Boltzmann-Gibbs function analogous to Eq. (1)

P (m) = c e!m/Tm . (7)

Here c is a normalizing constant, and Tm is the “moneytemperature”, which is equal to the average amount ofmoney per agent: T = "m# = M/N , where M is the totalmoney, and N is the number of agents.

To verify this conjecture, Ref. [25] performed agent-based computer simulations of money transfers betweenagents. Initially all agents were given the same amountof money, say, $1000. Then, a pair of agents (i, j) wasrandomly selected, the amount !m was transferred fromone agent to another, and the process was repeated manytimes. Time evolution of the probability distribution ofmoney P (m) can be seen in computer animation videos atthe Web pages [35, 36]. After a transitory period, moneydistribution converges to the stationary form shown inFig. 1. As expected, the distribution is very well fittedby the exponential function (7).

Several di"erent rules for !m were considered in Ref.[25]. In one model, the transferred amount was fixedto a constant !m = $1. Economically, it means thatall agents were selling their products for the same price!m = $1. Computer animation [35] shows that the ini-tial distribution of money first broadens to a symmet-ric, Gaussian curve, characteristic for a di"usion process.Then, the distribution starts to pile up around the m = 0state, which acts as the impenetrable boundary, because

of the imposed condition m ! 0. As a result, P (m) be-comes skewed (asymmetric) and eventually reaches thestationary exponential shape, as shown in Fig. 1. Theboundary at m = 0 is analogous to the ground state en-ergy in statistical physics. Without this boundary con-dition, the probability distribution of money would notreach a stationary state. Computer animation [35, 36]also shows how the entropy of money distribution, de-fined as S/N = $

!

k P (mk) lnP (mk), grows from theinitial value S = 0, when all agents have the same money,to the maximal value at the statistical equilibrium.

While the model with !m = 1 is very simple and in-structive, it is not very realistic, because all prices aretaken to be the same. In another model considered inRef. [25], !m in each transaction is taken to be a ran-dom fraction of the average amount of money per agent,i.e., !m = !(M/N), where ! is a uniformly distributedrandom number between 0 and 1. The random distri-bution of !m is supposed to represent the wide varietyof prices for di"erent products in the real economy. Itreflects the fact that agents buy and consume many dif-ferent types of products, some of them simple and cheap,some sophisticated and expensive. Moreover, di"erentagents like to consume these products in di"erent quan-tities, so there is variation of paid amounts !m, eventhough the unit price of the same product is constant.Computer simulation of this model produces exactly thesame stationary distribution (7), as in the first model.Computer animation for this model is also available onthe Web page [35].

The final distribution is universal despite di"erent rulesfor !m. To amplify this point further, Ref. [25] also con-sidered a toy model, where !m was taken to be a ran-dom fraction of the average amount of money of the twoagents: !m = !(mi + mj)/2. This model produced the

0 1000 2000 3000 4000 5000 60000

2

4

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10

12

14

16

18

Money, m

Pro

ba

bili

ty,

P(m

)

N=500, M=5*105, time=4*10

5.

!m", T

0 1000 2000 30000

1

2

3

Money, m

log

P(m

)

FIG. 1 Histogram and points: Stationary probability distri-bution of money P (m) obtained in agent-based computer sim-ulations. Solid curves: Fits to the Boltzmann-Gibbs law (7).Vertical lines: The initial distribution of money. (Reproducedfrom Ref. [25])

Figure: Boltzmann-Gibbs exponential distribution for money (Source:Yakovenko 2008).



ResultsConclusion

Econophysics

Critique & Modifications

Critique

Model is attractive in its simplicity but represents a ratherprimitive picture of the market.Agents are characterized only by their amount of money.Data on wealth is rarely available, but data on income is.

Modifications

Heterogeneous agents (in terms of money, abilities,opportunities, and savings rates).Ability changes as agents spend money on education.



ResultsConclusion

Labor MarketEducationProductionDemandBanksStructure of the Model

The Question

How is inequality of incomes generated?

Simple four sector model.

Conservation law should be fulfilled.

Model should produce exponential (or gamma) and power-lawdistributions of income.

Inequality of income between and within classes should beexplained.



ResultsConclusion


Characteristics of the Model

no representative agent

no utility 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 agiven distribution)



ResultsConclusion


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.spend money on education.

Firms...

hire workers.pay wages.receive revenue from selling output.

Government: collects taxes and provides dole.

Add banking sector later.



ResultsConclusion


Wage Bargaining

Hiring rule:

Each agents’ reservation wage is given by:w(m, θ, o) : R3 → R+.Every unemployed worker is matched with a randomly chosenfirm.If the firm’s res. wage exceeds the worker’s res. wage, theysign a wage contract.

If a firm has not enough money to pay all its employees, layoffworkers until the firm can pay the wagebill for the remainingworkers.

Unemployed workers get a dole-income which is a fraction oftheir reservation wage.



ResultsConclusion


Ability & Education

Workers can be of five types.

no degreecollege degreeBA degreeMA degreePhD

Workers are born with innate abilities which they can augmentby further training and education. The workers skills can besummed up in a measure of the workers productivity, θw

i .



ResultsConclusion


Education Levels

a0

innate ability,productivity

a0'

me

a0 = innate abilityme = money spend on education

BA

College

MA

PhD

θt+1 = f (θt ,me , o)



ResultsConclusion


Production & Capacity Utilization

Think of the production sector as a vertically integrated linearproduction model (neo-Ricardian).

In each market there will be winners and loosers, the higherearnings of the successful are exactly balanced by the lowerearnings of the less successful.

If θw < θf worker performs inadequately (accidents,slowdowns).

A firm produces its highest potential output if θw ≥ θf for allemployees.

output=∑

min[θw , θf

]Stephen Kinsella, Edward J. Nell, Matthias Greiff Interacting Heterogeneous Agents


ResultsConclusion


Demand

Each agent (workers and firms) spends money on averageonce 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 ischoosen is proportional to its output.



ResultsConclusion


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.



ResultsConclusion


Banks

Debt is permitted (negative money).

Unlimited borrowing has to be precluded.

Total amount of debt is limited by minimum reserverequirement for banks, M = M0

rr .Maximum debt of any agent is limited by, mi > −m̄d∀i .

Debt: increase in money temparature.

Money supply ’increases’ (money multiplier) but conservationlaw is still fulfilled!



ResultsConclusion


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.



ResultsConclusion


Interest Rate Adjustment

supply

demand

r

r1

supply

demand

r

r1r2

t t+1

Figure: Excess demand in the bond market.



ResultsConclusion


Structure of the Model

At the beginning of the year agents buy or sell one-year bonds.

Workers die and get born.

Each month the following happens:

Wage bargaining, hiring and firing.Effective Demand.Education.

At the end of each year we collect data on income distribution(and other data).



ResultsConclusion

MobilityIncome Distribution

Measuring Mobility

Mb = 1N

∑Ni=1 | log m0

i − log m1i |

Two time period framework.

Money at time t: m0 = (m01,m

02, . . . ,m

0N)′.

Money at time t + 10: m1 = (m11,m

12, . . . ,m

1N)′.

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



ResultsConclusion


Mobility

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æ

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æ

æ

æ

à

à

àà

à

à

à

à

à

ì

ì

ìì

ì

ìì

ì ì

0.2 0.4 0.6 0.8spending

0.9

1.0

1.1

1.2

1.3

1.4

1.5

mobility

Figure: Absolute mobility and spending.

Higher savings → lower mobility.

Higher mobility if debt is allowed for.

Positive interest rate reduces mobility.



ResultsConclusion


Income Distribution by Education

5 10 15 20 25

0.05

0.10

0.15

Figure: Income distribution for different levels of education.



ResultsConclusion

Conclusion

Workers are heterogeneous with respect to wealth, ability, andopportunities.

Almost no restrictions on agents behavior.

Markets generate surpluses that go to the successful, gains arecarried forward through time.

Labor MarketEducation

Differences in wealth, ability, and opportunity translate intoincome inequality within and between classes.



ResultsConclusion

Problems

Workers income is wage income plus interest earned / paid(on bonds).

Income can get negative if interest payment > wage income.

A possible solution: Restrict borrowing and introduce aminimum wage such that income from minimum wage issufficiently high to pay interest.

Or: Allow for agents to go bankrupt. (Interest rate onborrowing > interest rate on lending.)



ResultsConclusion

Further research and possible extensions

Further research:

Allow for more than one bank.

Look at firm size distribution.

Fit model to actual data (Irland 2000-2006).

Possible Extensions:

Introduce central bank.

Look at the effects of policy.



ResultsConclusion

Stephen Kinsella

[email protected]://www.stephenkinsella.net

Edward J. Nell

[email protected]

Matthias Greiff

[email protected]://matthiasgreiff.wordpress.com
