Outline: Output Validation From Firm Empirics to General Principles Firm data highly regular (universe of all firms) –Power law firm sizes, by various

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

• Interacting agent model of firm formation• Features of agent computing:

– Agents seek utility gains; perpetual adaptation 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



For empirical PDF, slope ~ -2.06,thus tail CDF has slope ~ -1.06

Pr[S≥si] = 1-F(si) = si





For empirical PDF, slope ~ -2.06,thus tail CDF has slope ~ -1.06

Average firm size ~ 20Median ~ 3-4

Mode = 1

Pr[S≥si] = 1-F(si) = si

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






Firm size in $106

DeVany on the distribution of movie receipts: ~ 1.25 => the ‘know nothing’ principle

Self-EmploymentSelf-Employment

• 15.5 million businesses with receipts but no employees:– Full-time self-employed

– Farms

– Other (e.g., part-time secondary employment)

Self-EmploymentSelf-Employment

• 15.5 million businesses with receipts but no employees:– Full-time self-employed

– Farms

– Other (e.g., part-time secondary employment)

Pr S ≥si[ ] =s0

si +1

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

α

What Size is a Typical Firm?

What Size is a Typical Firm?

Existence of moments depends on – First moment doesn’t exist if ≤ 1: ~ 1.06

• Alternative measures of location:– Geometric mean: s0

exp(1/) ~ 2.57 (for U.S. firms)

– Harmonic mean (E[S-1]-1): s0 (1+1/) ~ 1.94 (for U.S. firms)

– Median: s0 21/ ~ 1.92 (for U.S. firms)

– Second moment doesn’t exist since ≤ 2Moments exist for finite

samples

Non-existence means

non-convergence

History I: GibratHistory 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

• 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 II: Simon and co-authors

History II: Simon and co-authors

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

Size class Census/SBA Compustat0 719,978 2576

0 - 4 3,358,048 26995 - 9 1,006,897 149

10 - 19 593,696 25120 - 99 487,491 1287

100 - 499 79,707 2123500+ 16,079 4267

What is the Origin of the Zipf?


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



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




• Hypothesis 2: Zipf distribution of industry sizes => Zipf overall




• Hypothesis 2: Zipf distribution of industry sizes => Zipf overall No!





• Hypothesis 3: Zipf dist. of market sizes





• Hypothesis 3: Zipf dist. of market sizes 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

Origin of the Zipf, hypothesis 4


Sutton [1998] gives as a bound an exponential distributionof firm sizes by industry



Exponential distribution of firm sizes by industry: p exp(-ps)Exponential distribution of reciprocal firm means: q exp(-qp)




qexp−qp( )pexp−ps( )dp0∞∫ =

qq+s

Exponential distribution of firm sizes by industry: p exp(-ps)Exponential distribution of reciprocal firm means: q exp(-qp)


Origin of the Zipf: SuttonOrigin of the Zipf: Sutton

€

ψ s; s( ) =1sexp −

ss

⎛

⎝⎜

⎞

⎠⎟


€

ψ s; s( ) =1sexp −

ss

⎛

⎝⎜

⎞

⎠⎟

€

a s ; λ, β( ) =λβ

Γ β( ) s1+βexp −

λs

⎛

⎝⎜

⎞

⎠⎟


€

ψ s; s( ) =1sexp −

ss

⎛

⎝⎜

⎞

⎠⎟

€

a s ; λ, β( ) =λβ

Γ β( ) s1+βexp −

λs

⎛

⎝⎜

⎞

⎠⎟

€

f s; λ, β( ) = a s ; λ, β( )ψ s; s( )d s0∞∫ =βpβ 1

λ + s

⎛

⎝⎜

⎞

⎠⎟1+β


Average firm size across industries

Frequency

Firm Growth Rates areLaplace Distributed: Publicly-

Traded

Firm Growth Rates areLaplace Distributed: Publicly-

Traded

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

rt ≡lnSt+1

St

p(r)=12σ

exp−2r−r σ

⎛

⎝ ⎜

⎞

⎠ ⎟

Firm Growth Rates areSubbotin Distributed:

Universe

Firm Growth Rates areLaplace Distributed: Over

Time

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)

Variance in Firm Growth Rates

Scales Inversely (Declines) with Size

Variance in Firm Growth Rates

Scales Inversely (Declines) with Size

~ r0β

β ≈ 0.15 ± 0.03 (sales)β ≈ 0.16 ± 0.03 (employees)

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

Anomalous Scaling…

• Consider a firm made up of divisions:– If the divisions were independent then would scale

like s-1/2

– If the divisions were completely correlated then 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

• 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

More Firm FactsMore Firm Facts

Requirements of an Empirically Accurate ‘Theory

of the Firm’

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

Firm Size DistributionFirm Size Distribution

Firm sizes are Pareto distributed, f s1+

≈ -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

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

Summary:An Empirically-Oriented

Theory

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

ThreeDistinct Kinds

ofEmpirically-RelevantAgent-Based Models

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

x(t) x(t+1)f: Rn Rn

y(t) y(t+1)g: Rm Rm

a: Rn Rm

m < n

Micro-dynamics

Macro-dynamics

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

Level 0

Qualitative Quantitative

Micro

Macro


Level 1

Level 0


Micro

Macro


Level 1 Level 2

Level 0


Micro

Macro


Level 1 Level 2

Level 0 Level 3


Micro

Macro

Natural Development Cycle

Level 1 Level 2

Level 0 Level 3


Micro

Macro

Terminology

Level 1 Level 2

Level 0 Level 3


Micro

MacroVALIDATION

Terminology

Level 1 Level 2

Level 0 Level 3


Micro

MacroVALIDATION

CALIBRATION

Terminology

Level 1 Level 2

Level 0 Level 3


Micro

MacroVALIDATION

CALIBRATION

ESTIMATION

Examples

Level 1 Level 2

Level 0 Level 3


Micro

Macro

Examples

Level 1 Level 2

Level 0 Level 3


Micro

Macro

Retirement

Examples

Level 1 Level 2

Level 0 Level 3


Micro

Macro

Retirement

Anasazi

Examples

Level 1 FINANCE

Level 0 Level 3


Micro

Macro

Retirement

Anasazi

Examples

Level 1 FINANCE

Level 0 Level 3


Micro

Macro

Retirement

Anasazi

Firms

Examples

Level 1 FINANCE

Level 0 Level 3


Micro

Macro

Retirement

Anasazi

Firms

Smoking

Examples

Level 1 FINANCE

Level 0 Level 3


Micro

Macro

Retirement

Anasazi

Firms

Smoking

Easter Island

Examples

Level 1 FINANCE

Level 0 Level 3


Micro

Macro

Retirement

Anasazi

Firms

Smoking

Easter Island

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)

Model Types

ModelMacro

Data?Quality

Micro

Data?Quality

Dynamic

Data?

Retirement yes good no N/A yes

Smoking

Firms

Anasazi

Easter Island

Model Types

ModelMacro

Data?Quality

Micro

Data?Quality

Dynamic

Data?


Smoking yes good yes good no

Firms

Anasazi

Easter Island

Model Types

ModelMacro

Data?Quality

Micro

Data?Quality

Dynamic

Data?



Firms yes good partial good no

Anasazi

Easter Island

Model Types

ModelMacro

Data?Quality

Micro

Data?Quality

Dynamic

Data?




Anasazi yes good yes OK yes

Easter Island

Model Types

ModelMacro

Data?Quality

Micro

Data?Quality

Dynamic

Data?




Anasazi yes good yes OK yes

Easter Island

yes poor no N/A yes

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

• 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

• 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

• 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

Documents

Outline: Output Validation From Firm Empirics to General Principles Firm data highly regular (universe of all firms) –Power law firm sizes, by various