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Outline: Output Validation From Firm Empirics to General Principles. Firm data highly regular (universe of all firms) Power law firm sizes, by various measures What is a typical firm? Conceptual/mathematical challenges Heavy-tailed firm growth rates

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slide1

Outline: Output ValidationFrom Firm Empirics to General Principles

  • Firm data highly regular (universe of all firms)
    • Power law firm sizes, by various measures
      • What is a typical firm?
      • Conceptual/mathematical challenges
    • Heavy-tailed firm growth rates
      • Why doesn’t the central limit theorem work?
    • Wage-firm size effects
  • Agent models are multi-level:
    • Validation at distinct levels
summary from yesterday
Summary from Yesterday
  • Interacting agent model of firm formation
  • Features of agent computing:
    • Agents seek utility gains; perpetualadaptation emerges
    • Intrinsically multi-level
    • Full distributional information available
  • Potentially costly:
    • Sensitivity analysis
    • Calibration/estimation
u s firm sizes are zipf distributed rl axtell science 293 sept 7 2001 pp 1818 20
“U.S. Firm Sizes are Zipf Distributed,”RL Axtell, Science, 293 (Sept 7, 2001), pp. 1818-20

For empirical PDF, slope ~ -2.06,

thus tail CDF has slope ~ -1.06

Pr[S≥si] = 1-F(si) = si-a

slide4

“U.S. Firm Sizes are Zipf Distributed,”RL Axtell, Science, 293 (Sept 7, 2001), pp. 1818-20

For empirical PDF, slope ~ -2.06,

thus tail CDF has slope ~ -1.06

Average firm size ~ 20

Median ~ 3-4

Mode = 1

Pr[S≥si] = 1-F(si) = si-a

alternative notions of firm size
Alternative Notions of Firm Size
  • Simon: Skewness not sensitive to how firm size is defined
  • For Compustat, size distributions are robust to variations including revenue, market capitalization and earnings
  • For Census, receipts are also Zipf-distributed
slide6

Alternative Notions of Firm Size

  • Simon: Skewness not sensitive to how firm size is defined
  • For Compustat, size distributions are robust to variations including revenue, market capitalization and earnings
  • For Census, receipts are also Zipf-distributed

Firm size in $106

slide7

Alternative Notions of Firm Size

  • Simon: Skewness not sensitive to how firm size is defined
  • For Compustat, size distributions are robust to variations including revenue, market capitalization and earnings
  • For Census, receipts are also Zipf-distributed

DeVany on the distribution of movie receipts:

a ~ 1.25 => the ‘know nothing’ principle

Firm size in $106

self employment
Self-Employment
  • 15.5 million businesses with receipts but no employees:
    • Full-time self-employed
    • Farms
    • Other (e.g., part-time secondary employment)
slide9

Self-Employment

  • 15.5 million businesses with receipts but no employees:
    • Full-time self-employed
    • Farms
    • Other (e.g., part-time secondary employment)
slide10

What Size is a Typical Firm?

Existence of moments depends on a

  • First moment doesn’t exist if a ≤ 1: a ~ 1.06
    • Alternative measures of location:
      • Geometric mean: s0 a exp(1/a) ~ 2.57(for U.S. firms)
      • Harmonic mean (E[S-1]-1): s0 a (1+1/a) ~ 1.94(for U.S. firms)
      • Median: s0 a 21/a ~ 1.92 (for U.S. firms)
  • Second moment doesn’t exist since a ≤ 2

Moments exist for finite

samples

Non-existence means

non-convergence

history i gibrat
History I: Gibrat
  • Informal sample of French firms in the 1920s
  • Found firms sizes approximately lognormally distributed
  • Described ‘law of proportional growth’ process to explain the data
  • Important problems with this ‘law’
  • Early empirical data censored with respect to small firms
history ii simon and co authors
History II: Simon and co-authors
  • Described entry and exit of firms via Yule process (discrete valued random variables
  • Characterized size distribution for publicly-traded (largest) companies in U.S. and Britain
    • Pareto tail (large sizes)
  • Explored serial correlation in growth rates
  • Famous debate with Mandelbrot
  • Caustically critiqued conventional theory of the firm
history iii industrial organization
History III: Industrial Organization
  • Quandt [1966] studied a variety of industries and found no functional form that fit well across all industries
  • Schmalansee [1988] recapitulated Quandt
  • 1990s: All discussion of firm size distribution disappears from modern IO texts
  • Sutton (1990s): game theoretic models leading to ‘bounds of size’ approach to intra-industry size distributions
history iv stanley et al 1995
History IV: Stanley et al. [1995]
  • Using Compustat data over several years found the lognormal to best fit the data in manufacturing
  • 11,000+ publicly traded firms
  • More than 2000 firms report no employees! Ostensibly holding companies
  • Beginning of Econophysics!
sba census vs compustat data
SBA/Census vs Compustat Data
  • Qualitative structure: increasing numbers of progressively smaller firms
  • Comparison: 5.5 million U.S. firms
slide16

What is the Origin of the Zipf?

  • Hypothesis 1: Zipf in all industries => Zipf overall
slide17

What is the Origin of the Zipf?

  • Hypothesis 1: Zipf in all industries => Zipf overall Refuted by Quandt [1966]
slide18

What is the Origin of the Zipf?

  • Hypothesis 1: Zipf in all industries => Zipf overall Refuted by Quandt [1966]
  • Hypothesis 2: Zipf distribution of industry sizes => Zipf overall
slide19

What is the Origin of the Zipf?

  • Hypothesis 1: Zipf in all industries => Zipf overall Refuted by Quandt [1966]
  • Hypothesis 2: Zipf distribution of industry sizes => Zipf overall No!
slide20

What is the Origin of the Zipf?

  • Hypothesis 1: Zipf in all industries => Zipf overall Refuted by Quandt [1966]
  • Hypothesis 2: Zipf distribution of industry sizes => Zipf overall No!
  • Hypothesis 3: Zipf dist. of market sizes
slide21

What is the Origin of the Zipf?

  • Hypothesis 1: Zipf in all industries => Zipf overall Refuted by Quandt [1966]
  • Hypothesis 2: Zipf distribution of industry sizes => Zipf overall No!
  • Hypothesis 3: Zipf dist. of market sizes No!
slide22

What is the Origin of the Zipf?

  • Hypothesis 1: Zipf in all industries => Zipf overall Refuted by Quandt [1966]
  • Hypothesis 2: Zipf distribution of industry sizes => Zipf overall No!
  • Hypothesis 3: Zipf dist. of market sizes No!
  • Hypothesis 4: Exponential distribution of firms in each industry and exponential distribution of inverse average firm size
slide23

Origin of the Zipf, hypothesis 4

Sutton [1998] gives as a bound an exponential distribution

of firm sizes by industry

slide24

Origin of the Zipf, hypothesis 4

Sutton [1998] gives as a bound an exponential distribution

of firm sizes by industry

Exponential distribution of firm sizes by industry: p exp(-ps)

Exponential distribution of reciprocal firm means: q exp(-qp)

slide25

Origin of the Zipf, hypothesis 4

Sutton [1998] gives as a bound an exponential distribution

of firm sizes by industry

Exponential distribution of firm sizes by industry: p exp(-ps)

Exponential distribution of reciprocal firm means: q exp(-qp)

slide29

Origin of the Zipf: Sutton

Frequency

Average firm size across industries

slide30

Firm Growth Rates areLaplace Distributed: Publicly-Traded

Stanley, Amaral, Buldyrev, Havlin,

Leschhorn, Maass,, Salinger and Stanley,

Nature, 379 (1996): 804-6

properties of subbotin distribution
Properties of Subbotin distribution
  • Laplace (double exponential) and normal as special cases
  • Heavy tailed vis-à-vis the normal
  • Recent work by S Kotz and co-authors characterizes the Laplace as the limit distribution of normalized sums of arbitrarily-distributed random variables having a random number of summands (terms)
slide34

Variance in Firm Growth RatesScales Inversely (Declines) with Size

s ~ r0-b

b ≈ 0.15 ± 0.03 (sales)

b ≈ 0.16 ± 0.03 (employees)

Stanley, Amaral, Buldyrev, Havlin, Leschhorn, Maass, Salinger and Stanley, Nature, 379 (1996): 804-6

anomalous scaling
Anomalous Scaling…
  • Consider a firm made up of divisions:
    • If the divisions were independent then s would scale like s-1/2
    • If the divisions were completely correlated then s would be independent of size (scale like s0)
    • Reality is interior between these extremes
  • Stanley et al. get this by coupling divisions
  • Sutton postulates that division size is a random partition of the overall firm size
  • Wyart and Bouchaud specify a Pareto distribution of firm sizes
slide36

More Firm Facts

  • Wage rates increase in firm size (Brown and Medoff):
    • Log(wages)  Log(size)
  • Constant returns to scale at aggregate level (Basu)
  • More variance in job destruction time series than in job creation (Davis and Haltiwanger)
  • ‘Stylized’ facts:
    • Growth rate variance falls with age
    • Probability of exit falls with age
slide37

Requirements of an Empirically Accurate ‘Theory of the Firm’

  • Produces a power law distribution of firm sizes
  • Generates Laplace (double exponential) distribution of growth rates
  • Yields variance in growth rates that decreases with size according to a power law
  • Wage-size effect obtains
  • Constant returns to scale
  • Methodologically individualist (i.e., written at the agent level)
  • No microeconomic/game theoretic explanation for any of these
slide38

Firm Size Distribution

Firm sizes are Pareto distributed, fs(1+a)

a ≈ -1.09

slide39

Productivity: Output vs. Size

Constant returns at the aggregate level despite

increasing returns at the local level

slide40

Firm Growth Rate Distribution

Growth rates Laplace distributed by K-S test

Stanley et al [1996]: Growth rates Laplace distributed

slide41

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)

slide42

Wages as a Function of Firm Size:Search Networks Based on Firms

Brown and Medoff [1992]: wages  size 0.10

slide43

Wages as a Function of Firm Size:Search Networks Based on Firms

Brown and Medoff [1992]: wages  size 0.10

slide44

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

slide45

Summary:An Empirically-Oriented Theory

  • Produces a right-skewed distribution of firm sizes (near Pareto law)
  • Generates heavy-tailed distribution of growth rates
  • Yields variance in growth rates that decreases with size according to a power law
  • Wage-size effect emerges
  • Constant returns to scale at aggregate level
  • Methodologically individualist
background
Background
  • Agent models are multi-level systems
  • Empirical relevance can be achieved at different levels
  • Observation: For most of what we do, 2 levels are active

Macro-dynamics

g: Rm

Rm

y(t+1)

y(t)

a: Rn

Rm

m < n

x(t)

x(t+1)

f: Rn

Rn

Micro-dynamics

update to understanding our creations sfi bulletin 1994
Update to“Understanding Our Creations…, ”SFI Bulletin, 1994
  • Multiple levels of empirical relevance:
    • Level 0: Micro-level, qualitative agreement
    • Level 1: Macro-level, qualitative agreement
    • Level 2: Macro-level, quantitative agreement
    • Level 3: Micro-level, quantitative agreement
  • Then, few examples beyond level 0
distinct classes of abms
Distinct Classes of ABMs

Qualitative

Quantitative

Macro

Micro

distinct classes of abms1
Distinct Classes of ABMs

Qualitative

Quantitative

Macro

Micro

distinct classes of abms2
Distinct Classes of ABMs

Qualitative

Quantitative

Macro

Micro

distinct classes of abms3
Distinct Classes of ABMs

Qualitative

Quantitative

Macro

Micro

natural development cycle
Natural Development Cycle

Qualitative

Quantitative

Macro

Micro

terminology
Terminology

Qualitative

Quantitative

Macro

VALIDATION

Micro

terminology1
Terminology

Qualitative

Quantitative

Macro

VALIDATION

CALIBRATION

Micro

terminology2
Terminology

Qualitative

Quantitative

Macro

VALIDATION

CALIBRATION

ESTIMATION

Micro

examples
Examples

Qualitative

Quantitative

Macro

Micro

examples1
Examples

Qualitative

Quantitative

Macro

Retirement

Micro

examples2
Examples

Qualitative

Quantitative

Anasazi

Macro

Retirement

Micro

examples3
Examples

Qualitative

Quantitative

Anasazi

Macro

Retirement

Micro

examples4
Examples

Qualitative

Quantitative

Anasazi

Macro

Retirement

Firms

Micro

examples5
Examples

Qualitative

Quantitative

Anasazi

Macro

Retirement

Firms

Smoking

Micro

examples6
Examples

Qualitative

Quantitative

Easter Island

Anasazi

Macro

Retirement

Firms

Smoking

Micro

examples7
Examples

Qualitative

Quantitative

Easter Island

Anasazi

Macro

Retirement

Firms

Smoking

Micro

models demo d
Models Demo’d
  • ZI traders (Level 1)
  • Retirement (Level 1)
  • Smoking (Level 3)
  • Firms (Level 2)
  • Anasazi (Level 2)
  • Commons (Level 1)
  • Easter Island (Level 1)
easter island
Easter Island
  • Small Pacific Island 2500 miles West of Chile
  • Initially settled by Polynesians
  • Initially a paradise, with virgin palm stands, easy fishing, available fresh water
  • Notable for giant stone statues
  • Over-exploitation of environment led to societal collapse
  • Today, a paradigm of unsustainability
easter island abm motivations
Easter Island ABM: Motivations
  • Papers by Brander and Taylor in AER on bioeconomic ODE models of Easter Island
  • No agency in these models (no statues!)
  • Population dynamics basis for empirics
  • Agent models as generalizations of systems dynamics models
  • Scale comparable to Anasazi
easter island abm execution
Easter Island ABM: Execution
  • Island biogeography coded
  • Fishing is primary source of nutrition
  • ‘Excess’ labor expended on statue creation
  • Over-exploitation leads to declining welfare, brutish society (deaths due to conflict)
  • Loss of trees eliminates large fish from diet
  • Heterogeneous agent model much richer than ODE model
conclusion
Conclusion
  • Empirical ambitions of agent models constrained by data
  • Agent models amenable, even desirous of micro-data
  • There is a natural agent model development cycle toward fine resolution models
  • Today, micro-data availability is main impediment to high resolution models