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Transcript of Outline: Output Validation From Firm Empirics to General Principles Firm data highly regular...
![Page 1: Outline: Output Validation From Firm Empirics to General Principles Firm data highly regular (universe of all firms) –Power law firm sizes, by various.](https://reader036.fdocuments.us/reader036/viewer/2022062719/56649ec95503460f94bd77d0/html5/thumbnails/1.jpg)
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
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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
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“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
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“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
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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
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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
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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
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Self-EmploymentSelf-Employment
• 15.5 million businesses with receipts but no employees:– Full-time self-employed
– Farms
– Other (e.g., part-time secondary employment)
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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
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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
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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
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• 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
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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
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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!
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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
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What is the Origin of the Zipf?
What is the Origin of the Zipf?
• Hypothesis 1: Zipf in all industries => Zipf overall
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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]
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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
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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!
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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
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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!
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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
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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
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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
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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
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Origin of the Zipf: SuttonOrigin of the Zipf: Sutton
€
ψ s; s( ) =1sexp −
ss
⎛
⎝⎜
⎞
⎠⎟
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Origin of the Zipf: SuttonOrigin of the Zipf: Sutton
€
ψ s; s( ) =1sexp −
ss
⎛
⎝⎜
⎞
⎠⎟
€
a s ; λ, β( ) =λβ
Γ β( ) s1+βexp −
λs
⎛
⎝⎜
⎞
⎠⎟
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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
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Origin of the Zipf: SuttonOrigin of the Zipf: Sutton
Average firm size across industries
Frequency
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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
⎛
⎝ ⎜
⎞
⎠ ⎟
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Firm Growth Rates areSubbotin Distributed:
Universe
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Firm Growth Rates areLaplace Distributed: Over
Time
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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)
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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
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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
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• 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
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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
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Firm Size DistributionFirm Size Distribution
Firm sizes are Pareto distributed, f s1+
≈ -1.09
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Productivity: Output vs. Size
Productivity: Output vs. Size
Constant returns at the aggregate level despiteincreasing returns at the local level
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Firm Growth Rate Distribution
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 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)
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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
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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
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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
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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
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ThreeDistinct Kinds
ofEmpirically-RelevantAgent-Based Models
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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
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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
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Distinct Classes of ABMs
Level 0
Qualitative Quantitative
Micro
Macro
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Distinct Classes of ABMs
Level 1
Level 0
Qualitative Quantitative
Micro
Macro
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Distinct Classes of ABMs
Level 1 Level 2
Level 0
Qualitative Quantitative
Micro
Macro
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Distinct Classes of ABMs
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
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Natural Development Cycle
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
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Terminology
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
MacroVALIDATION
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Terminology
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
MacroVALIDATION
CALIBRATION
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Terminology
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
MacroVALIDATION
CALIBRATION
ESTIMATION
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Examples
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
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Examples
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Retirement
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Examples
Level 1 Level 2
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Retirement
Anasazi
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Examples
Level 1 FINANCE
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Retirement
Anasazi
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Examples
Level 1 FINANCE
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Retirement
Anasazi
Firms
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Examples
Level 1 FINANCE
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Retirement
Anasazi
Firms
Smoking
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Examples
Level 1 FINANCE
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Retirement
Anasazi
Firms
Smoking
Easter Island
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Examples
Level 1 FINANCE
Level 0 Level 3
Qualitative Quantitative
Micro
Macro
Retirement
Anasazi
Firms
Smoking
Easter Island
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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)
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Model Types
ModelMacro
Data?Quality
Micro
Data?Quality
Dynamic
Data?
Retirement yes good no N/A yes
Smoking
Firms
Anasazi
Easter Island
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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
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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
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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
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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
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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
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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
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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
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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