Multi-Agent Firms

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Multi-Agent Firms

• Rob Axtell

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Multi-Agent FirmsHow 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 Andersons 201A,
  201B,201W)
• More fundamental criticism of the theory of the
  firm not even methodological individualist
  Winter 1993!

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Not Todays 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...

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Outline of Lecture 11MAS Model of Firm Formation
and Dynamics(Paper model www.brookings.edu/dyn
amics/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...

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Many Theories of the Firm

• Textbook orthodoxy Firms as black boxes
• Production function specifies technology
• Profit maximization specifies behavior
• Winters 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

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Some (Old) Empirical DataFortune 1000 c. 1970s
from Ijiri and Simon 1977
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Critique of the Neoclassical(U-Shaped) Cost
Function
Theory 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.
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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
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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
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Features of Agent Computation

• 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

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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

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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 gt 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,Ei)?i
(1-ei)1-?i
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Equilibrium
Proposition 1 Nash equilibrium exists and is
unique
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Equilibrium
Proposition 1 Nash equilibrium exists and is
unique
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Equilibrium
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Moral Hazard in Team Production
Proposition 2 Agents under-supply input at Nash
equilibrium
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Moral Hazard in Team Production
Proposition 2 Agents under-supply input at Nash
equilibrium
Consider a 2 agent team
e2
e1
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Homogeneous Teams
Utility as a function of team size and agent type
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Homogeneous Teams
Utility as a function of team size and agent type
Optimal team size as a function of agent type
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Stability, I
Best reply effort adjustment Agents know last
periods output
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Stability, I
Best reply effort adjustment Agents know last
periods output
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Stability, I
Best reply effort adjustment Agents know last
periods output
For a b or Ei
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Stability, I
Best reply effort adjustment Agents know last
periods output
For a b or Ei
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Stability, I
Best reply effort adjustment Agents know last
periods output
For a b or Ei
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Stability, II
Proposition 3 There is an upper bound on stable
group size
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Stability, II
Proposition 3 There is an upper bound on stable
group size
Using row sums
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Stability, II
Proposition 3 There is an upper bound on stable
group size
Using row sums
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Stability, II
Proposition 3 There is an upper bound on stable
group size
Using row sums
Thus, the agent who most prefers income
determines maximum size
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Stability, II
Proposition 3 There is an upper bound on stable
group size
Using row sums
Thus, the agent who most prefers income
determines maximum size
Using column sums
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Stability, II
Proposition 3 There is an upper bound on stable
group size
Using row sums
Thus, the agent who most prefers income
determines maximum size
Using column sums
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Stability, III
For homogeneous groups
Stability boundary is close to size at which
individual and group utilities are maximized
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Homogeneous Teams
Utility as a function of team size and agent type
Optimal team size as a function of agent type
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Stability, III
For homogeneous groups
Stability boundary is close to size at which
individual and group utilities are maximized
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Stability, III
For homogeneous groups
Stability boundary is close to size at which
individual and group utilities are maximized
Optimal firms live on the edge of chaos!
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Stability, III
For homogeneous groups For
heterogeneous groups Agent with
largest preference for income
determines maximum stable group size
Stability boundary is close to size at which
individual and group utilities are maximized
Optimal firms live on the edge of chaos!
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Motivations for aComputational Model
Deficiencies of the analytical 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

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Motivations for aComputational Model
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

? Agent-based computational modeling
perfectly suited to by-pass these problems
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The Computational Modelwith Agents

• 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

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ltRun Firms codegt
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Firm Size Distribution
Firm sizes are Pareto distributed, f ? s?(1a)
a -1.09
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Productivity Output vs. Size
Constant returns at the aggregate level
despite increasing returns at the local level
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Firm Growth Rate Distribution
Growth rates Laplace distributed by K-S test
Stanley et al 1996 Growth rates Laplace
distributed
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Variance in Growth Ratesas a Function of Firm
Size
slope -0.174 0.004
Stanley et al. 1996 Slope -0.16 0.03
(dubbed 1/6 law)
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Wages as a Function of Firm SizeSearch Networks
Based on Firms
Brown and Medoff 1992 wages ? size 0.10
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Wages as a Function of Firm SizeSearch Networks
Based on Firms
Brown and Medoff 1992 wages ? size 0.10
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Firm Lifetime Distribution
Data on firm lifetimes is complicated by effects
of mergers, acquisitions, bankruptcies,
buy-outs, and so on Over the past 25 years, 10
of 5000 largest firms disappear each year
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Effect 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

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Effect 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

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Sensitivity to Compensation
Compensation proportional to input Si(ei,E)
eiO(E)/E
All firms now stable
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Mixed Compensation
Linear combination of compensation policies
Si(ei,E) (aei/E(1-a)/N)O(E)
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Mixed Compensation
Linear combination of compensation policies
Si(ei,E) (aei/E(1-a)/N)O(E)
Now firms are again unstable
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Summary

• 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