Outline: Output Validation From Firm Empirics to General Principles Firm data highly regular...
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Transcript of Outline: Output Validation From Firm Empirics to General Principles Firm data highly regular...
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
“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
“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
Average firm size ~ 20Median ~ 3-4
Mode = 1
Pr[S≥si] = 1-F(si) = si
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
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
Firm size in $106
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
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?
What is the Origin of the Zipf?
• Hypothesis 1: Zipf in all industries => Zipf overall
What is the Origin of the Zipf?
What is the Origin of the Zipf?
• Hypothesis 1: Zipf in all industries => Zipf overall Refuted by Quandt [1966]
What is the Origin of the Zipf?
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
What is the Origin of the Zipf?
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!
What is the Origin of the Zipf?
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
What is the Origin of the Zipf?
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!
What is the Origin of the Zipf?
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
Origin of the Zipf, hypothesis 4
Origin of the Zipf, hypothesis 4
Sutton [1998] gives as a bound an exponential distributionof firm sizes by industry
Origin of the Zipf, hypothesis 4
Origin of the Zipf, hypothesis 4
Exponential distribution of firm sizes by industry: p exp(-ps)Exponential distribution of reciprocal firm means: q exp(-qp)
Sutton [1998] gives as a bound an exponential distributionof firm sizes by industry
Origin of the Zipf, hypothesis 4
Origin of the Zipf, hypothesis 4
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)
Sutton [1998] gives as a bound an exponential distributionof firm sizes by industry
Origin of the Zipf: SuttonOrigin of the Zipf: Sutton
€
ψ s; s( ) =1sexp −
ss
⎛
⎝⎜
⎞
⎠⎟
Origin of the Zipf: SuttonOrigin of the Zipf: Sutton
€
ψ s; s( ) =1sexp −
ss
⎛
⎝⎜
⎞
⎠⎟
€
a s ; λ, β( ) =λβ
Γ β( ) s1+βexp −
λs
⎛
⎝⎜
⎞
⎠⎟
Origin of the Zipf: SuttonOrigin of the Zipf: Sutton
€
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
Origin of the Zipf: SuttonOrigin of the Zipf: Sutton
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
Wages as a Function of Firm Size:
Search Networks Based on Firms
Brown and Medoff [1992]: wages size 0.10
Wages as a Function of Firm Size:
Search Networks Based on Firms
Wages as a Function of Firm Size:
Search Networks Based on Firms
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
Distinct Classes of ABMs
Level 1
Level 0
Qualitative Quantitative
Micro
Macro
Distinct Classes of ABMs
Level 1 Level 2
Level 0
Qualitative Quantitative
Micro
Macro
Distinct Classes of ABMs
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Natural Development Cycle
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Terminology
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
MacroVALIDATION
Terminology
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
MacroVALIDATION
CALIBRATION
Terminology
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
MacroVALIDATION
CALIBRATION
ESTIMATION
Examples
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Examples
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Retirement
Examples
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Retirement
Anasazi
Examples
Level 1 FINANCE
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Retirement
Anasazi
Examples
Level 1 FINANCE
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Retirement
Anasazi
Firms
Examples
Level 1 FINANCE
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Retirement
Anasazi
Firms
Smoking
Examples
Level 1 FINANCE
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Retirement
Anasazi
Firms
Smoking
Easter Island
Examples
Level 1 FINANCE
Level 0 Level 3
Qualitative Quantitative
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?
Retirement yes good no N/A yes
Smoking yes good yes good no
Firms
Anasazi
Easter Island
Model Types
ModelMacro
Data?Quality
Micro
Data?Quality
Dynamic
Data?
Retirement yes good no N/A yes
Smoking yes good yes good no
Firms yes good partial good no
Anasazi
Easter Island
Model Types
ModelMacro
Data?Quality
Micro
Data?Quality
Dynamic
Data?
Retirement yes good no N/A yes
Smoking yes good yes good no
Firms yes good partial good no
Anasazi yes good yes OK yes
Easter Island
Model Types
ModelMacro
Data?Quality
Micro
Data?Quality
Dynamic
Data?
Retirement yes good no N/A yes
Smoking yes good yes good no
Firms yes good partial good no
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