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Outline: Output Validation From Firm Empirics to General Principles

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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; 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

For empirical PDF, slope ~ -2.06,

thus tail CDF has slope ~ -1.06

Pr[S≥si] = 1-F(si) = si-a

“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

- 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

- 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

- 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

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

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

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

- 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

- 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

- 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]

- 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

- Qualitative structure: increasing numbers of progressively smaller firms
- Comparison: 5.5 million U.S. firms

What is the Origin of the Zipf?

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

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?

- 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?

- 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?

- 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?

- 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?

- 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

Sutton [1998] gives as a bound an exponential distribution

of firm sizes by industry

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)

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)

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

- 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 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…

- 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

- 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

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

Constant returns at the aggregate level despite

increasing returns at the local level

Growth rates Laplace distributed by K-S test

Stanley et al [1996]: Growth rates Laplace distributed

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)

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

Brown and Medoff [1992]: wages size 0.10

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

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

- 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

- 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

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

- 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

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