# Some Stylized Facts on the Firm Size Distribution in Korea

```Japan- Korea Workshop on Productivity Analysis
Some Stylized Facts
on the Firm Size Distribution in Korea
Jeong-Dong Lee and Junho Na
Seoul National University
2012. 12. 7.
Contents
I. Background
II. Some Stylized Facts on the Firm Size Distribution in Korea
III. Change of Firm Size Distribution over time
I. Background
Implications of
Firm Size Distribution (FSD)
Heterogeneities of firm population
Vs.
Density
Density
Comparison of Firm Size Distributions
Log (Sales, Employee, Assets….)
Log (Sales, Employee, Assets….)
(d)
Vs.
Log (Sales, Employee, Assets….)
Density
Density
(c)
Log (Sales, Employee, Assets….)
3/23
Log-Normal vs. Pareto
Better fit of Pareto distribution to the right-skewed,
fat-tailed population
Linear
Log-Normal
Log
Heavy & Long TailLog
Straight Line
in Log-Log Axes
Concave
to Origin
Stable
Mean
Unstable
Finite
Variance
Potentially Infinite
Clearly defined
Confidence
Interval
often hard-to-tell
Vanishing Tail
Tail
Long & Fat Tail
Average Value
Important
Variables
Extreme Value
Independence,
Homogeneity
Assumption
Interdependence,
Heterogeneity
Relevant
Concepts
Mutually Causal,
Emergent Nonlinear
Dynamics,
Coevolution, Diversity
Linearity,
Randomness,
Equilibrium
Log
* Source : Andriani & McKelvey (2009)
Pareto
4/23
Interpretation of
Pareto Distribution
Pareto CDF
Interpretation
Pareto Distribution can be drawn in the form of
PDF or CDF
Log Probability Density
∶ () = Pr( = ) =  −
∶ () = Pr( ≥ ) =  −
OLS or MLE fitting
α : exponent of power law
= slope of power law line
α of PDF = a of CDF + 1
Usu. 2 < α < 3 in nature,
a little broader range in economics
Decrease of α = Existence of bigger firms
Log Size
Xmin : smallest value of range where power law
is valid = MES (Minimum Efficient Scale)
5/23
Stylized Facts on FSD
from Previous Researches
Statically asymmetric and dynamically robust
Asymmetric & Right-skewed
* Figure Source : Cabral & Mata (2003)
Robust over time
6/23
II. Firm Size Distributions in Korea
FSD of KOSPI firms
Power law holds… but weakly
KOSPI firms
(2009, non-financial, Size=Sales, N=651)*
CDF (Log-log form)
Median
= 3.30
Mean
= 3.38
S.D
= 0.75
Skewness = 0.26
Kurtosis = 3.55
Log (Sales)
* Data Source : KIS Value
Log Pr (X ≥ x)
(Cumulative Prabability Density)
Probability Density
DF (Log-real form)
α
= 1.84
Xmin = 10^11.9
Log (Sales)
8/23
FSD of the whole
enterprise population
Power law holds… and strongly
Nationwide enterprise survey
(2009, over 10 employees, Size=Employees, N=236,269)*
Median
= 18.00
Mean
= 39.51
Skewness = 155.28
Kurtosis
= 8294.33
Log (Employees)
* Data Source : Korea Statistics Bureau.
CDF (Log-log form)
Log Pr (X ≥ x)
(Cumulative Prabability Density)
Probability Density
DF (Log-real form)
α
= 2.46
Xmin = 64
Log (Employees)
9/23
FSD of
manufacturing plants (1)
Pareto distribution across diverse proxy measures on size
Korea mining & manufacturing survey
(2009, over 10 employees, N=47,059)
Size = Production
생산액
2009
0
Size = Employees
0
10
10
Value
10
Pr(X ≥ x)
-2
10
확률밀도
Total
fixed
assets
-1
-1
10
-3
10
-4
-2
N
-3
Median
-4
Mean
10
10
10
10
S.D.
-5
-5
10
-2
10
0
10
2
10
규 모(단 위 : 억 원 )
4
10
10
6
10
1
10
Size = Total fixed assets
2
3
4
47,059
47,059
47,059
47,059
29.4
19.0
20.0
11.8
201.6
43.7
768.7
66.4
275.0 14,856.2
1,123.6
3,316.7
5
10
10
10
Employees (2009, 명)
10
Skewness
Kurtosis
58.4
56.8
45.6
64.2
4,224.6
4,242.8
2,450.0
5,086.6
1.95
2.50
1.67
2.01
0
0
10
-1
10
10
10
Pr(X ≥ x)
-2
Pr(X≥x)
10
-3
10
-1
α
-2
Xmin
69.59
80.00
110.82
44.65
Proportion
of higher
20% groups
88.8%
66.0%
97.0%
87.2%
10
-3
10
-4
-4
10
10
-5
-5
10
Produc- Employtion
ees
-2
10
0
10
2
10
Total Asset (2009,
4
10
억 원)
6
10
3
10
10
VA (2009,
4
10
억원)
* Measuring units of production, fixed assets, value added = 100 million won
5
10
6
10
10/23
FSD of
manufacturing plants (2)
Pareto distribution of resource, capability, and performance
Korea mining & manufacturing survey
(2009, over 10 employees, N=47,059)
Resource
Capability
Performance
Total Fixed Assets
Labor Productivity
0
10
-1
10
-1
-1
10
10
Pr(X≥x)
α
= 1.67
Xmin= 110.82
-3
10
X
Pr(X ≥ x)
-2
-2
10
-3
10
α
= 2,86
Xmin= 89.07
⇒
-4
-4
10
0
10
2
10
Total Asset (2009,
4
10
억 원)
Highly
heterogeneous
6
10
10
-3
10
α
= 2.01
Xmin= 44.65
10
-5
-2
10
-2
10
-4
10
-5
Pr(X ≥ x)
10
10
10
0
0
10
-5
1
10
노동생산성(억원)
2
10
Relatively
homogenous
3
10
10
3
10
VA (2009,
4
10
억원)
5
10
6
10
Relatively
heterogeneous
11/23
FSD by Sectors (1)
Different slopes for different sectors
Korea mining & manufacturing survey
(2009, over 10 employees, Size = Production)
Food
(ind.code=10)
Textile
(ind.code=13)
식 료 품제 조 업 (산 업 코 드 =10)
0
10
1
2
3
10
10
생 산 액(2009, 억 원 )
10
4
10
제 지 업(산 업 코 드 =17)
3
5
10
-2
10
-3
10 -1
10
1
인 쇄 업(산 업 코 드 =18)
0
3
10 -1
10
5
10
10
생산액(2009, 억원)
10
Chemical
(ind.code=20)
10
10
-2
10
-4
1
10
10
생산액(2009, 억원)
Print & Publishing
(ind.code=18)
Paper
(ind.code=17)
0
-2
10
-3
-4
-1
10
10
10
10 -1
10
-4
-1
-1
Pr(X ≥ x)
-2
10
-3
10
-2
10
일 반 화 학 제 품제 조 업 (산 업 코 드 =20)
0
-3
0
10
1
2
3
10 -1
10
4
10
10
10
생산액(2009, 억원)
10
Pharmaceutical
(ind.code=21)
3
10
0
고 무,플 라 스 틱 제 조 업 (산 업 코 드 =22)
-1
-1
10
10
-1
-2
10
1
10
2
3
10
10
생산액(2009,억원)
4
10
10 0
10
-2
-3
-3
10
-4
0
10
-2
10
10
-3
10
-3
Pr(X≥x)
-2
10
Pr(X ≥ x)
-2
10
10
Pr(X≥x)
Pr(X ≥ x)
10
Pr(X ≥ x)
2
10
-1
10
1
10
10
생산액(2009,억원)
(ind.code=22)
제 약 업(산 업 코 드 =21)
0
0
10
Rubber, Plalstic
10
10
-1
10
목 재 제 조 업 (산 업 코 드 =16)
0
10
10
10
-3
10
가 죽,신 발 제 조 업 (산 업 코 드 =15)
0
10
10
Pr(X ≥ x)
Pr(X ≥ x)
-2
Wood
(ind.code=18)
-1
10
10
의 류 제 조 업 (산 업 코 드 =14)
0
10
-1
-1
10
Pr(X>=x)
섬 유 제 조 업 (산 업 코 드 =13)
0
10
10
Leather & Shoes
(ind.code=15)
Pr(X ≥ x)
0
Clothes
(ind.code=14)
10
-3
-4
1
10
2
3
10
10
생산액(2009,억원)
4
10
10
-1
10
0
10
1
10
2
3
10
10
생 산 액(2009, 억 원 )
4
10
5
10
10 -1
10
-4
0
10
1
2
3
10
10
10
생산액(2009,억원)
4
10
10
-2
10
0
10
2
10
생 산 액(2009,억 원 )
4
10
6
10
12/23
FSD by Sectors (1)
Korea mining & manufacturing survey
(2009, over 10 employees, Size = Production)
비철 제조업 (산업코드=23)
0
0
10
10
-1
-2
10
-3
-3
10
-4
10 -1
10
3
10 -1
10
5
10
전 기 장 비(산 업 코 드 =28)
0
자 동 차, 트 레 일 러 (산 업 코 드 =30)
0
10
-4
-4
0
2
10
10
생산액(2009,억원)
4
10
10
-2
10
0
2
10
10
생 산 액(2009,억 원 )
4
10 -1
10
6
10
10
Ship, Airplane
(ind.code=31)
기 계,장 비 (산 업 코 드 =29)
0
1
3
5
10
10
생산액(2009,억원)
10
Furnitures
(ind.code=32)
기 타 운 송 장 비 (조 선 등 , 산 업 코 드 =31)
가 구 제 조 업 (산 업 코 드 =32)
0
10
10
-1
10
-2
10
-3
-3
10
-1
-1
10
-2
10
Machinery
(ind.code=29)
10
10
10
10
-4
10
Automobiles
(ind.code=30)
Electric Instruments
(ind.code=28)
0
-2
10 -2
10
5
3
1
10
10
생산액(2009,억원)
-1
-1
10
의 료,정 밀 기 기 (산 업 코 드 =27)
0
10
10
10
-4
1
10
10
생산액(2009,억원)
(ind.code=27)
전 자 산 업 (산 업 코 드 =26)
0
-3
10
Medical, Precision
10
10
Pr(X ≥ x)
Pr(X ≥ x)
-2
금 속 제 조 업 (산 업 코 드 =25)
-1
10
10
Electronics
(ind.code=26)
10
-1
10
Pr(X ≥ x)
차 금 속 제 조 업(산 업 코 드 =24)
1
0
Metal Assembly
(ind.code=25)
Pr(X ≥ x)
Metals
(ind.code=24)
Pr(X≥x)
Non-ferro metal
(ind.code=23)
-1
10
10
-1
-3
0
10
2
10
생 산 액(2009, 억 원 )
4
10
6
10
10 -2
10
2
4
10
10
생산액(2009,억원)
6
10
10 -2
10
Pr(X ≥ x)
-2
10
10
-4
0
10
-2
10
-3
10
-4
-4
-2
10
-3
10
-3
10
10
-2
10
-2
10
Pr(X ≥ x)
-2
10
Pr(X ≥ x)
Pr(X ≥ x)
Pr(X≥x)
10
-3
-4
0
2
10
10
생산액(2009,억원)
4
10
10 -2
10
0
10
2
4
10
10
생산액(2009, 억원)
6
10
10 -1
10
0
10
1
2
3
10
10
10
생산액(2009,억원)
4
10
13/23
FSD by Sectors (2)
Alternative measures
alpha, Xmin (MES), Gini
ind.code
Sector
10
13
14
15
16
17
18
20
21
22
23
24
25
26
27
28
29
30
31
32
Foods
Textiles
Clothes
Leather & Shoes
Woods
Pulps & Papers
Printing & Publishing
Chemicals
Pharmaceuticals
Rubber & Plastic
Non-ferro Metals
Metals
Metal Assembly
Electronics
Medical, Precision Instruments
Electric Instruments
Machineries
Automobiles
Shipbuilding & Airplanes
Furniture
Total
No. of
observation
2,803
2,434
2,132
469
480
999
1,030
1,807
217
4,265
1,428
1,912
7,296
3,115
1,167
2,932
7,254
2,587
1,147
950
46,424
α
1.86
2.67
1.96
1.89
3.20
1.92
2.71
1.69
3.00
2.18
2.24
1.98
2.26
1.77
2.32
1.91
2.20
2.01
1.57
1.98
Gini
Xmin
(MES) Coefficient
75.8
0.78
111.1
0.64
71.2
0.84
19.4
0.72
88.9
0.58
28.5
0.73
42.3
0.53
54.5
0.87
634.0
0.67
58.1
0.67
84.8
0.75
360.5
0.87
74.8
0.70
178.8
0.94
97.2
0.71
46.1
0.80
99.3
0.72
200.9
0.89
178.7
0.95
14.5
0.72
HHI
0.003
0.008
0.016
0.017
0.010
0.012
0.005
0.014
0.019
0.004
0.055
0.042
0.014
0.047
0.028
0.017
0.006
0.069
0.122
0.014
14/23
FSD by Sectors (3)
Strong correlation between alpha and other measures
Correlations between Gini efficient and Pareto coefficient (α)
3.50
3.00
α
2.50
Corr. = - 0.80
2.00
1.50
1.00
-
0.20
0.40
0.60
0.80
1.00
1.20
Gini Coefficient
15/23
FSD by Sectors (4)
Weak correlation between alpha and MES
Correlations between MES and Pareto efficient (α) *
3.50
3.00
α
2.50
2.00
1.50
1.00
-
200.0
400.0
600.0
800.0
MES (100 million won)
* MES are measured by Xmin in Pareto CDF
16/23
FSD by Firm Characteristics
(1) R&D vs. No-R&D
Effort for innovation makes the population more
heterogeneous
Firm Activity Survey
(2009, all industries, over 50 employees, N=10,319)
no-R&D
0
그룹
10
No-R&D
Groups
-1
NoR&D
Groups
Pr(X ≥ x)
10
R&D
Groups
-2
10
-3
10
-4
10 -2
10
0
10
6
4
2
R&D
4,681
5,638
Median
183.4
313.6
Mean
756.9
2,172.2
4,214.7
17,384.2
10
10
10
매출액(2009,억원)
0
N
S.D
그룹
10
Skewness
31.1
30.8
-1
R&D
Groups
Pr(X ≥ x)
10
Kurtosis
1,396.2
1,351.3
-2
10
-3
α
10
-4
10
0
10
2
4
10
10
매출액(2009,억원)
6
10
Xmin
2.14
1,059.3
1.89
1,101.4
17/23
FSD by Firm Characteristics
(2) Export vs. Domestic
Export market makes the population more heterogeneous
Firm Activity Survey
(2009, all industries, over 50 employees, N=10,319)
내수기업 그룹
0
Domestic-only
Groups
10
Export
Groups
-1
Domestic
-only
Groups
Pr(X ≥ x)
10
-2
N
5,411
4,191
-3
Median
179.3
395.7
Mean
719.2
2,715.9
3,765.7
20,113.1
22.6
26.8
714.5
1,013.5
2.07
1.88
636.4
1,094.3
10
10
-4
10 -2
10
0
10
2
4
10
10
매출액(2009,억원)
6
10
S.D
수출 기업 그룹
0
10
Skewness
-1
Export
Groups
Pr(X ≥ x)
10
Kurtosis
-2
10
α
-3
10
-4
10
0
10
2
4
10
10
매출액(2009,억원)
6
Xmin
10
18/23
FSD by Firm Characteristics
(3) Affiliated or Independent
Affiliation makes the population more heterogeneous
Firm Activity Survey
(2009, all industries, over 50 employees, N=10,319)
독립 기업 그룹
0
10
Independent
Groups
-1
Independent
Groups
Pr(X ≥ x)
10
Affiliated
Groups
-2
10
N
5,145
5,174
Median
144.8
491.5
Mean
305.3
2,748.2
S.D
974.1
18,505.7
-3
10
-4
10 -2
10
0
10
2
10
매출액(2009,억원)
4
10
계열사 그룹
0
10
Skewness
-1
Affiliated
Groups
Pr(X ≥ x)
10
Kurtosis
28.5
1,224.0
28.1
1,146.6
-2
10
-3
10
α
-4
10 -2
10
0
10
2
4
10
10
매출액(2009,억원)
6
10
Xmin(=MES)
2.48
392.5
1.95
2,501.4
19/23
III. FSD over time
FSD by cohorts over time
Cohort effect for older firms
Korea mining & manufacturing survey
(1993~2003, over 5 employees, Size = Employees)
21/23
Becomes more homogeneous over time in terms of number
of employees
FSD over time
Changes of α values over time (1995~2009)
2.7
α
α
α
Asset α
2.5
2.3
2.1
1.9
1.7
Employee
Production
Fixed asset
1.5
1995 1997 1999 2001 2003 2005 2007 2009
1995
1996
1997
1998
1999
2000
2001
2002
2003
2004
2005
2006
2007
2008
2009
2.28
2.29
2.30
2.41
2.42
2.47
2.45
2.53
2.53
2.46
2.48
2.53
2.57
2.56
2.52
2.12
2.06
2.11
2.16
1.96
1.97
1.97
1.99
1.99
1.97
1.98
1.98
1.99
1.96
1.95
1.97
1.98
2.07
1.98
1.98
2.01
2.00
2.02
2.05
2.02
2.03
2.04
2.05
2.01
2.01
1.99
1.95
1.95
1.93
1.92
1.93
1.92
1.97
1.99
1.99
2.00
2.02
2.05
2.00
2.00
22/23
Thank you !
23/23
```

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