Multi-Agent Firms Rob Axtell. Multi-Agent Firms: How does this fit in to what we have done? Graduate microeconomics: –Markets –Games –Firms Common criticism

Multi-Agent Firms

Rob Axtell

Multi-Agent Firms:How does this fit in to what we have

done?• Graduate microeconomics:

– Markets

– Games

– Firms

• Common criticism of general equilibrium theory: it is not strategic (e.g., Bob Anderson’s 201A, 201B,…201)

• More fundamental criticism of the theory of the firm: not even methodological individualist [Winter 1993]!

Not Today’s Outline• Goal: Reproduce empirical data on U.S. firms

– Firm sizes

– Firm growth rates

– Wage-size effects

– …

• Formulate using game theory– Conventional ‘solution concepts’ not useful

– Constant adaptation at the agent level

– Against the ‘Nash program’ of game theory

• From firms to cities to countries...

Outline of Lecture 11MAS Model of Firm Formation and

Dynamics(Paper + model:

www.brookings.edu/dynamics/papers/firms)

• Goal: Represent a firm with multiple agents

• Start with a population of agents:– What economic environment induces firm

formation? We want to grow firms.– Equilibrium? Stability? Dynamics?– What relevance to empirical data?

• Next: Empirical validation of model output

• Later: From firms to cities to countries...

Many Theories of the Firm

Many Theories of the Firm

• Textbook orthodoxy: Firms as black boxes– Production function specifies technology

– Profit maximization specifies behavior

– Winter’s critique: Not even methodologically individualist

• Coase and Williamson (‘New Institutionalism’):– “Transaction cost” approach

• Principal-agent (game theoretic) approaches:– Firm as nexus of contracts (incomplete contracts)

• Firm as information processing network (e.g., Radner)• Evolutionary economics (Nelson and Winter):

– Purposive instead of maximizing behavior

• Industrial organization– Modern game theoretic orientation has little connection to data

Some (Old) Empirical DataFortune 1000 c. 1970s

from Ijiri and Simon [1977]

Critique of the Neoclassical(U-Shaped) Cost Function

Critique of the Neoclassical(U-Shaped) Cost Function

“[T]heory says nothing about whether the same cost curves are supposed to prevail for all of the firms in an industry, or whether, on the contrary, each firm has its own cost curve and its own optimum scale. If the former then all firms in the industry should be the same size. A prediction could hardly be more completely falsified by the facts than this one is. Virtually every industry that has been examined exhibits...a highly skewed distribution of sizes with very large firms existing side by side with others of modest size. If each firm, on the other hand, has its own peculiar optimum, then the theory says nothing about what the resulting distribution of these optima for the industry should be. Thus, the theory either predicts the facts incorrectly or it makes no prediction at all.”

More Critique...More Critique...

“All these factors make static cost theory both irrelevant for understanding the size distribution of firms in the real world and empirically vacuous.”

“Economics is not a discipline in which hypotheses that follow from classical assumptions, or that are necessary for classical conclusions, are quickly abandoned in the face of hostile evidence”

More Critique...More Critique...

“All these factors make static cost theory both irrelevant for understanding the size distribution of firms in the real world and empirically vacuous.”

“Economics is not a discipline in which hypotheses that follow from classical assumptions, or that are necessary for classical conclusions, are quickly abandoned in the face of hostile evidence”

Herbert Simon [1958]

• Heterogeneous agents: replace representative agent, focus on distribution of behavior instead of average behavior; endogenous heterogeneity

• Bounded rationality: essentially impossible to give agents full rationality in non-trivial environments

• Local/social interactions: agent-agent interactions mediated by inhomogeneous topology (e.g., graph, social network, space)

• Focus on dynamics: no assumption of equilibrium; paths to equilibrium and non-equilibrium adjustments

• Each realization a sufficiency theorem

Features of Agent Computation

Features of Agent Computation

Synopsis of EndogenousFirm Formation Model

Synopsis of EndogenousFirm Formation Model

• Heterogeneous population of agents• Situated in an environment of increasing

returns (team production)• Agents are boundedly rational (locally

purposive not hyper-rational)• Rules for dividing team output (compensation

systems)• Agents have social networks from which they

learn about job opportunities

An Analytical Modelof Firm Formation

An Analytical Modelof Firm Formation

Set-Up: Consider a group of N agents, each of whom supplies input (‘effort’) ei [0,1] Total effort level: E = i{1..N} ei

Total output: O(E) = aE + bE, a, b≥ 0 b = 0 means constant returns, b > 0 is increasing returns Agents receive equal shares of output:

S(E) = O(E)/N Agents have Cobb-Douglas preferences for income (output shares) and leisure,

Ui(ei) = S(ei,E~i)i (1-ei)

1-i

EquilibriumEquilibrium

Proposition 1: Nash equilibrium exists and is unique


ei

* i

, E~i

max 0,a 2b E

~i i a2

4abi

2 1 E~ i

4b2i

2 1 E~i

2

2b 1 i



ei

* i

, E~i

max 0,a 2b E

~i i a2

4abi

2 1 E~ i

4b2i

2 1 E~i

2

2b 1 i

5 10 15 20

0.2

0.4

0.6

0.8

1

q = 0.95

q = 0.90

q = 0.80

q = 0.50

ei*

E~i


Moral Hazard in Team Production


Proposition 2: Agents under-supply input at Nash equilibrium



e2

e1

Consider a 2 agent team:

Proposition 2: Agents under-supply input at Nash equilibrium

Homogeneous TeamsHomogeneous Teams

5 10 15 20 25Size0.51

1.52

2.53

3.54

IndividualUtility

q=0.9

q=0.8

q=0.7

q=0.5

Utility as a function of team size and agent type


5 10 15 20 25Size0.51

1.52

2.53

3.54

IndividualUtility

q=0.9

q=0.8

q=0.7

q=0.5


0 0.2 0.4 0.6 0.8 1 q1

2

5

10

20

50

100200

OptimalSize

Optimal team size as a functionof agent type

Stability, IStability, I

‘Best reply’ effort adjustment: Agents know last period’s output

ei t+1( ) =max0,−a−2b E~i t( )−θi( ) + a2 +4abθi

2 1+E~i t( )( )+4b2θi2 1+E~i t( )( )

2

2b 1+θi( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥



J ij ≡∂ei

∂ej

=

−1+θi2 a+2b1+E~i

*( )

a2 +4abθi2 1+E~i

*( ) +4b2θi2 1+E~i

*( )2

1+θi


2 1+E~i t( )( )+4b2θi2 1+E~i t( )( )

2

2b 1+θi( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥



J ij ≡∂ei

∂ej

=

−1+θi2 a+2b1+E~i

*( )

a2 +4abθi2 1+E~i

*( ) +4b2θi2 1+E~i

*( )2

1+θi


2 1+E~i t( )( )+4b2θi2 1+E~i t( )( )

2

2b 1+θi( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

For a « b or E~i:

J ij ≈θi −1θi +1

∈[−1,0]



J =

0 k1 L k1

k2 0 L k2

M O M

kN L kN 0

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

J ij ≡∂ei

∂ej

=

−1+θi2 a+2b1+E~i

*( )

a2 +4abθi2 1+E~i

*( ) +4b2θi2 1+E~i

*( )2

1+θi


2 1+E~i t( )( )+4b2θi2 1+E~i t( )( )

2

2b 1+θi( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

For a « b or E~i:


∈[−1,0]



mini

ri ≤λ0 ≤maxi

ri

mini

ci ≤λ0 ≤maxi

ciJ =

0 k1 L k1

k2 0 L k2

M O M

kN L kN 0

⎡

⎣

⎢ ⎢ ⎢ ⎢ ⎢

⎤

⎦

⎥ ⎥ ⎥ ⎥ ⎥

J ij ≡∂ei

∂ej

=

−1+θi2 a+2b1+E~i

*( )

a2 +4abθi2 1+E~i

*( ) +4b2θi2 1+E~i

*( )2

1+θi


2 1+E~i t( )( )+4b2θi2 1+E~i t( )( )

2

2b 1+θi( )

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥

For a « b or E~i:


∈[−1,0]

Stability, IIStability, II

Proposition 3: There is an upper bound on stable group size



N−1( )mini

ki ≤λ0 ≤ N−1( )maxi

kiUsing row sums:



N−1( )mini


ki

Nmax≤1

mini

ki

+1⎢

⎣ ⎢ ⎢

⎥

⎦ ⎥ ⎥ ≈

21−max

iθi

⎢

⎣ ⎢ ⎢

⎥

⎦ ⎥ ⎥

Using row sums:



N−1( )mini


ki

Nmax≤1

mini

ki

+1⎢

⎣ ⎢ ⎢

⎥

⎦ ⎥ ⎥ ≈

21−max

iθi

⎢

⎣ ⎢ ⎢

⎥

⎦ ⎥ ⎥

Using row sums:

Thus, the agent who most prefers income determines maximum size



N−1( )mini


ki

kii−1

N

∑ −maxi

ki ≤λ0 ≤ kii−1

N

∑ −mini

ki

Nmax≤1

mini

ki

+1⎢

⎣ ⎢ ⎢

⎥

⎦ ⎥ ⎥ ≈

21−max

iθi

⎢

⎣ ⎢ ⎢

⎥

⎦ ⎥ ⎥

Using row sums:


Using column sums:



N−1( )mini


ki

kii−1

N

∑ −maxi

ki ≤λ0 ≤ kii−1

N

∑ −mini

ki

Nmax≤1

mini

ki

+1⎢

⎣ ⎢ ⎢

⎥

⎦ ⎥ ⎥ ≈

21−max

iθi

⎢

⎣ ⎢ ⎢

⎥

⎦ ⎥ ⎥

Using row sums:


Using column sums:

Nmax≤1+max

iki

k

⎢

⎣ ⎢ ⎢

⎥

⎦ ⎥ ⎥ ≈

1+mini

θi

f θ ( )

⎢

⎣ ⎢ ⎢

⎥

⎦ ⎥ ⎥

For homogeneous groups:

Stability, IIIStability, III

Stability boundary is closeto size at which individualand group utilities aremaximized


5 10 15 20 25Size0.51

1.52

2.53

3.54

IndividualUtility

q=0.9

q=0.8

q=0.7

q=0.5


0 0.2 0.4 0.6 0.8 1 q1

2

5

10

20

50

100200

OptimalSize

Optimal team size as a functionof agent type




0 0.2 0.4 0.6 0.8 1 q1

2

5

10

20

50

100200

OptimalSize




Optimal firms live on the edge of chaos!


For heterogeneous groups:

Agent with largest preference for income determines maximum stable group size



Optimal firms live on the edge of chaos!

Nmax=2

1−maxi

θi

Motivations for aComputational Model


• Representative agent/representative group formulation

• Exclusive focus on equilibria, which provide no information since they are unstable

• Unstable equilibria not explosive

• Analogy with financial markets, turbulence

• Perfectly-informed, perfectly rational agents

• Synchronous updating of model with equations

Deficiencies of the analytical model:



• Representative agent/representative group formulation

• Exclusive focus on equilibria, which provide no information since they are unstable

• Perfectly-informed, perfectly rational agents

• Synchronous updating of model with equations

Deficiencies of the analytical model:

Agent-based computational modeling perfectly suited to by-pass these problems

• Preference parameter, , distributed uniformly on (0,1)

• Firm output: O(E) = E + E, ≥ 1

• Agents are randomly activated

• Each computes its optimal effort level, e*, for:

• staying a member of its present firm;

• moving to a different firm (random graph);

• starting a new firm;

• The option that yields the greatest utility is selected

The Computational Model

with Agents

The Computational Model

with Agents

<Run Firms code>

Firm Size DistributionFirm Size Distribution

Firm sizes are Pareto distributed, f s

≈ -1.09

Productivity: Output vs. Size

Productivity: Output vs. Size

Constant returns at the aggregate level despiteincreasing returns at the local level

Firm Growth Rate Distribution

Firm Growth Rate Distribution

Growth rates Laplace distributed by K-S test

0.2 0.5 1 2 5 10Growth Rate1.¥10- 6

0.00001

0.0001

0.001

0.01

0.1

1

Frequency

Stanley et al [1996]: Growth rates Laplace distributed

Variance in Growth Rates

as a Function of Firm Size

Variance in Growth Rates

as a Function of Firm Size

1 5 10 50 100 500Size

0.15

0.2

0.3

0.5

0.7

1

sr

slope = -0.174 ± 0.004

Stanley et al. [1996]: Slope ≈ -0.16 ± 0.03 (dubbed 1/6 law)

Wages as a Function of Firm Size:

Search Networks Based on Firms



Brown and Medoff [1992]: wages size 0.10





Brown and Medoff [1992]: wages size 0.10

Firm Lifetime Distribution

Firm Lifetime Distribution

1 10 100 1000 10000 100000.Rank

100

200

300

400

500Lifetime

Data on firm lifetimes is complicated by effects of mergers, acquisitions, bankruptcies, buy-outs, and so onOver the past 25 years, ~10% of 5000 largest firms disappear each year

• Importance of (locally) purposive behavior

• Vary a, b, and : Greater increasing returns means larger firms

• Alternative specifications of preferences

• Role of social networks

• Agent ‘loyalty’ is a stabilizing force in large firms

• Bounded rationality: groping for better effort levels

• Alternative compensation schemes

• Firm founder sets hiring standards

• Firm founder acts as residual claimant

Effect of Model ParametrizationEffect of Model Parametrization

• Importance of (locally) purposive behavior

• Vary a, b, and : Greater increasing returns means larger firms

• Alternative specifications of preferences

• Role of social networks

• Agent ‘loyalty’ is a stabilizing force in large firms

• Bounded rationality: groping for better effort levels

• Alternative compensation schemes

• Firm founder sets hiring standards

• Firm founder acts as residual claimant

Effect of Model ParametrizationEffect of Model Parametrization

Sensitivity to CompensationSensitivity to

Compensation

Compensation proportional to input: Si(ei,E) = eiO(E)/E

All firms now stable

Mixed CompensationMixed Compensation

Linear combination of compensation policies:

Si(ei,E) = (ei/E+(1-)/N)O(E)

Mixed CompensationMixed Compensation

Linear combination of compensation policies:

Now firms are again unstable

Si(ei,E) = (ei/E+(1-)/N)O(E)

SummarySummary

• Heterogeneous agents who ‘best reply’ locally and out-of-equilibrium in an economic environment of increasing returns with free agent entry and exit are sufficient to generate firms

• Highly non-stationary (turbulent) micro-data, stationary macro-data

• Constant returns at the aggregate level

• A microeconomic explanation of the empirical data

• Successful firms are those that can attract and maintain high productivity workers; profit maximization, to the extent it exists, is a by-product

• Analytically difficult model tractable with agents

Documents

Multi-Agent Firms Rob Axtell. Multi-Agent Firms: How does this fit in to what we have done? Graduate microeconomics: –Markets –Games –Firms Common criticism