A Static Theory of Pareto DistributionsBank of France / Sciences Po Workshop on ”the Granularity of

Macroeconomic Fluctuations: where do we stand?”

Francois GeerolfUCLA

June 24, 2016

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Pareto distributions

§ 958

c

A

LA COURBE DES REVENUS 305

Fig. 47.

q

\

B

poser en ligne droite 1. Disons immédiatement que nous allons retrouver cette tendance dans les nombreux exemples que nous aurons encore à examiner.

Un autre fait, tout aussi, et même plus remarquable, c'est que les courbes de la réparti-tion des revenus, en Angleterre

Schedule D - Année 1893-94.

x N

f GREAT BRITAIN -- .

150 400 6iS '17 7-17 200 234 '185 9 3f"l5 ::lOO '121 996 4 592 400 74 041

l !li: 1

500 54 419 600 42 072 1 428 700 St 269 1 104 800 29311 940 900 2.') 033 771

1000 22896 684 2000 9880 271

6069 142 4, '161 88

1 5000 3081 68 1

10000 1 104 22

et en Irlande, présentent un parallélisme à peu près complet. Ce fait est à rapprocher d'un autre, que nous allons bientôt constater: les inclinaisons des lignes mm, pq obtenues pour dif-

(958) 1 C'est-à-dire que la courbe réelle est interpolée par une droite dont l'équation est (1) log N = log A - ",log X.

L'équation générale de la courbe est peut-être

(2) log = log A - ", log (a + x) - ; mais ce n'est que dans un seul cas (Oldenbourg) que nous avons trouvé une valeur appréciable pour f3. Il est donc fort probable que f3 est, en gé-néral, négligeable, et qu'on a simplement

(3) log N = log A = ", log (a += x). Quallli il s'agit du revenu total, a est aussi, en général, fort petit et le

plus souvent, de l'ordre des erreurs d'observation. Nous sommes donc ainsi ramené à l'équation (1).

§ 958

c

A

LA COURBE DES REVENUS 305

Fig. 47.

q

\

B

poser en ligne droite 1. Disons immédiatement que nous allons retrouver cette tendance dans les nombreux exemples que nous aurons encore à examiner.

Un autre fait, tout aussi, et même plus remarquable, c'est que les courbes de la réparti-tion des revenus, en Angleterre

Schedule D - Année 1893-94.

x N

f GREAT BRITAIN -- .

150 400 6iS '17 7-17 200 234 '185 9 3f"l5 ::lOO '121 996 4 592 400 74 041

l !li: 1

500 54 419 600 42 072 1 428 700 St 269 1 104 800 29311 940 900 2.') 033 771

1000 22896 684 2000 9880 271

6069 142 4, '161 88

1 5000 3081 68 1

10000 1 104 22

et en Irlande, présentent un parallélisme à peu près complet. Ce fait est à rapprocher d'un autre, que nous allons bientôt constater: les inclinaisons des lignes mm, pq obtenues pour dif-

(958) 1 C'est-à-dire que la courbe réelle est interpolée par une droite dont l'équation est (1) log N = log A - ",log X.

L'équation générale de la courbe est peut-être

(2) log = log A - ", log (a + x) - ; mais ce n'est que dans un seul cas (Oldenbourg) que nous avons trouvé une valeur appréciable pour f3. Il est donc fort probable que f3 est, en gé-néral, négligeable, et qu'on a simplement

(3) log N = log A = ", log (a += x). Quallli il s'agit du revenu total, a est aussi, en général, fort petit et le

plus souvent, de l'ordre des erreurs d'observation. Nous sommes donc ainsi ramené à l'équation (1).

I 1890s, tax tabulations: Pareto plots N of peoplewith incomes ≥ x :

logN = logA− α log x .

I Same α: England, Ireland, Prussia, Saxe, andPeru.

I Pareto 6= Bell-Shaped curve. Few empiricalregularities in economics.

I With Pareto:I No scale. US: y50 = $51, 939 < yav = $72, 641.I Long tails. Top 1% gets ≈ 20% of pre-tax income.

1 / 28

Pareto distribution for US Labor Incomes, 2008.Source: Statistics of Income (Public Use)

Year: 2008 −− Slope: −1.94

−6

−5.

5−

5−

4.5

−4

−3.

5−

3−

2.5

−2

−1.

5−

1−

.50

Log 1

0 S

urvi

vor

2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 7.5Log10 Labor Income ($)

Labor Income ($) Fitted values

2 / 28

Zipf’s Law for Firm SizesDistribution of US Firm Sizes. Source: Axtell (2001).

Slope: 2.059 (density) ⇒ Tail coeff: 1.059 . ”Zipf’s law”.R E P O R T S

10-

10-»

10-10

10 102 103 10"Firm size (employees)

105 106

19.0 (21.8 for firms larger than 0). Clearly,the COMPUSTAT data are heavily censoredwith respect to small firms. Such firms playimportant roles in the economy (75, 16).

For further analysis, I used a tabulation fromCensus in which successive bins are of increas-ing size in powers of three. The modal firm sizeis 1, whereas the median is 3 (4 if size 0 firmsare not counted) These data are approximatelyZipf-distributed {a = 1.059), as determined byordinary least squares (OLS) regression in log-log coordinates (Fig. 1). There are too few verysmall and very large firms with respect to theZipf fit, presumably due to finite size effects,yet the power law distribution well describesthe data over nearly six decades of firm size(from 10° to 10* employees). This result sug-gests both that a common mechanism of firmgrowth operates on firms of all sizes, and thatthe fundamental unit of analysis is the individ-ual employee.

But firms having a single employee arenot the smallest economic entities in the U.S.economy. Although there were some 5.5 mil-lion firms that had at least one employee atsome time during 1997, there were another15.4 million business entities in that yearwith no employees. These are predominantlyself-employed individuals and partnerships,and are called "nonemployer" firms by Cen-sus. These smallest of firms account for near-ly $600 billion in receipts in 1997. Yet, ifthese firms are included in the overall firmsize distribution, the Zipf distribution still fitsthe data well. To see this, Eq. 1 must bemodified to accommodate firms having noemployees

= ( ; ^ ) , ^ ¡ ^ 0 , a > 0 (2)

Table 2. Power law exponent for U.S. firms in1992, firms with employees and all firms. Resultsusing OLS regression on Census data, with stan-dard errors in parentheses.

Type Estimated a Adjusted

Firms with employees 0.994 (0.043) 0.995All businesses 0.995 (0.031) 0.994

Fig. 1. Histogram of U.S. firm sizes,by employees. Data are for 1997from the U.S. Census Bureau, tab-ulated in bins having width in-creasing in powers of three (30).The solid line is the OLS regressionline through the data, and it has aslope of 2.059 (SE = 0.054; adjust-ed R = 0.992), meaning that a =1.059; maximum likelihood andnonparametric methods yield sim-ilar results. The data are slightlyconcave to the origin in log-logcoordinates, reflecting finite sizecutoffs at the limits of very smalland very large firms.

Here, OLS yields an estimate of a = 1.098(SE = 0.064), and the adjusted R^ = 0.977.Including self-employment drives the aver-age firm size down to 5.0 employees/firm,and makes the median number of employees0.

An interesting property of firm size distri-butions noted in previous studies of largefirms is that the qualitative character of suchdistributions is independent of how size isdefined (7). Although the position of individ-ual firms in a size distribution does depend onthe definifion of size, the shape of the distri-bution does not. This also holds for the Cen-sus data. Basing firm size on receipts, a Zipfdistribution describes the data (a = 0.994)(Fig. 2). Here, modal and median firm reve-nues are each less than $100,000, and theaverage is $173,000/firm.

As a further test on the robustness of theseresults, I repeated these analyses for Censusdata from 1992. Average firm size was slight-ly smaller then, at 20.9 employees/firm (ex-cluding size 0 firms). But overall, the Zipfdistribudon is as strong (Table 2).

Virtually all U.S. firms experienced sig-nificant changes in revenue and work forcefrom 1992 to 1997. Thus, individual firmsmigrated up and down the Zipf distribution,but economic forces seem to have renderedany systematic deviations from it short-lived.Even the substantial merger and acquisitionactivity of this period seemed to have little

1

10-1

i 10-3

2o.10-5

10-«10" 10^ 108

Receipts (1997$)1010

Fig. 2. Tail cumulative distribution function ofU.S. firm sizes, by receipts in dollars. Data arefor 1997 from the U.S. Census Bureau, tabulat-ed in bins whose width increases in powers of10. The solid line is the OLS regression linethrough the data and has slope of 0.994 (SE =0.064; adjusted R = 0.976).

effect on the overall firm size distribution.There are a variety of stochastic growth

processes that converge to Pareto and Zipfdistribufions (7, 5, 77,18). Empirically, thereis support for Gibrat-like processes in whichaverage growth rates are independent of size{19, 20) and growth rate variance declineswith size (27, 22). Consider a variation of theGibrat process known as the Kesten process{23-25), in which sizes are bounded frombelow; i.e.,

s,{t + I) = max[so,y{t)sM] (3)where 7 is a random growth rate. For nearlyany growth rate distribution, this processyields Pareto distributions that have the ex-ponent Oi defined implicitly by {26)

N =a - 1

(4)

where N is the total number of firms and A isthenumberof employees. For A' = 5.5 X 10''and ^ = 105 X 10^ as in 1997 (excludingself-employment), SQ = 1 implies a = 0.997,a value close to my empirical finding. Similarresults are obtained for each year backthrough 1988 (Table 3).

Table 3. Theoretical power law exponents for U.S. firms over a 10-year period. Note that even thoughthe number of firms and total employees each increased over this period, as did the average firm size, thevalue of a was approximately unchanged.

Year

1997199619951994199319921991199019891988

Firms

5,541,9185,478,0475,369,0685,276,9645,193,6425,095,3565,051,0255,073,7955,021,3154,954,645

Employees

105,299,123102,187,297100,314,94696,721,59494,773,91392,825,79792,307,55993,469,27591,626,09487,844,303

Mean firm size

19.0018.6518.6818.3318.2518.2218.2818.4218.2517.73

a, from (4)

0.99660.99860.99831.00041.00081.00091.00040.99951.00061.0039

Viiww.sciencemagorg SCIENCE VOL 293 7 SEPTEMBER 2001 1819

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Theories of Pareto Distributions in EconomicsWhy Pareto? May reflect some fundamental economic principle:

1. Pareto distributed primitives. Explain one Pareto withanother Pareto.

I Lucas (1978), Chaney (2008), Gabaix, Landier (2008), etc.

2. Paretos from random growth models.

I Champernowne (1953), Simon, Bonini (1958), Kesten (1973),Gabaix (1999), Gabaix, Lasry, Lions, Moll (2016), etc.

3. New from this paper: Paretos from ProductionFunctions. Assignment models with positive sorting, with aspecial form of production function.

I Presentation: Garicano (2000) model.I Property of the production function, not of specific

microfoundations.I Another example: Geerolf (2015).

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This paper

I A particular version of Garicano (2000). Under limitedassumptions on the skill distribution:

I L layers of hierarchy = weak law of Pareto for span ofcontrol with coefficient:

αL = 1 +1

L− 1. α+∞ = 1 .

⇒ a new theory of Zipf’s law for firm sizes.

I Pareto for labor incomes, with βL ∈ [1,+∞], when top skillsare scarce enough.

I Data supports these predictions:

I Existing US data.

I French matched employer-employee.

5 / 28

”Big Picture” Takeaways

When the production function has a certain property:

1. Pareto-like heterogeneity from infinitesimal ex-antedifferences.⇒ New intuition for why firm sizes and labor incomes are soheterogenous despite small observable differences.

2. Pareto is a local result:

I Deviations from Pareto to measure heterogeneity?

I Alternative to:

I Optimal taxation: Pareto distributed skills.

I Trade: Pareto distributed firm productivities.

I Misallocation: Pareto distributed manager/firm productivities.

6 / 28

Literature

I Pareto Distributions. Pareto (1895, 1896).

I Literature in Physics. ”Change of Variable near the origin”.Sornette (2002), Newman (2005), Sornette (2006).

I Random Growth. Zipf (1949), Champernowne (1953), Simon andBonini (1958), Kesten (1973), Sutton (1997), Gabaix (1999), Axtell(2001), Luttmer (2007), Gabaix, Lasry, Lions, Moll (2016).

I Competitive Assignment Models. Roy (1950), Rosen (1981),Sattinger (1975), Kremer (1993), Tervio (2008), Gabaix, Landier(2008).

I Span of Control. Lucas (1978), Rosen (1981), Rosen (1982),Rossi-Hansberg, Wright (2007).

I Organizational Structure. Calvo, Wellisz (1978,1979), Garicano(2000), Garicano, Rossi-Hansberg (2004, 2006), Antras, Garicano,Rossi-Hansberg (2006), Caliendo, Monte, Rossi-Hansberg (2015).

7 / 28

Overview

Environment

Span of control with 2 layers

Span of control with L layers - Zipf’s Law

Empirics

Labor Income Distribution

Conclusion

8 / 28

Environment

Span of control with 2 layers

Span of control with L layers - Zipf’s Law

Empirics

Labor Income Distribution

Conclusion

8 / 28

A Garicano (2000) Economy 1/2I Agents: continuum, measure 1. 1 unit of time.

I 1 good. 1 unit of time → 1 good.

I Agents: different exogenous skills. Agent with skill x cansolve problems in [0, x ].

I Distribution of skills x : c.d.f. F (.), density f (.) on [1−∆, 1].

∆: Heterogeneity in Skills.

F (.): Skill Distribution.

I Workers encounter problems in production. Draw a unitcontinuum of different problems on [0, 1] in c.d.f. G (.),uniform w.l.o.g. :

I When they know the solution: produce 1 unit of the good.

I When they don’t: can ask someone else for a solution.h < 1: manager’s time cost to listen to one problem.

h: Helping Time.9 / 28

A Garicano (2000) Economy 2/2

I Assumption 1: x unknown.

I Assumption 2: h low enough: always hierarchies.

I Assumption 3: one manager with time 1 at the top.

10 / 28

Environment

Span of control with 2 layers

Span of control with L layers - Zipf’s Law

Empirics

Labor Income Distribution

Conclusion

10 / 28

Imposing 2 layersI Occupational cutoff: z2 splits managers and workers.

Workers

1-Δ 1 skill

z1 z2 z3

y = m(x)

y

Managers

x

x = e(y)

I Span of control of y and output of (x , y)

n(y) =1

h(1− x)Q(x , y) =

y

h(1− x).

I Planner’s problem. Positive sorting (∂2/∂x∂y > 0).y = m(x). Workers x fail to solve 1− x problems:

f (y)dy = h(1− x)f (x)dx ⇒ f (m(x))m′(x) = h(1− x)f (x).

I z2, m(.) unknowns. One initial value problem, one equation:

m(1−∆) = z2, m(z2) = 1.11 / 28

Uniform Distribution

Proposition

With f (x) = 1/∆ on [1−∆, 1],

(a) Distribution of the span of control of managers is a truncatedPareto (2):

P[N ≥ n] =n2

1− (n/n)2

(1

n2− 1

n2

)with n =

1

h∆and n =

1√1 + h2∆2 − 1

∼∆→02

h2∆2.

(b) When ∆→ 0, n→∞ and n/n→∞, the distribution of spanof control is a full Pareto (2).

Intuition: size biased distribution n(m(x)) is truncated Pareto (1),thus span of control distribution n(y) is truncated Pareto (2):

n(y) = n(m(x)) =1

h(1− x).

12 / 28

Pareto PlotExample: h = 70%, f (.) uniform with ∆ = 30%, 10%, 2%.

Δ=30%

Δ=10%

Δ=2%

1 2 3 4 5 6 7

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Log10Firm Size

Log 10Survivor

13 / 28

Non-uniform distributionI Because of the ”blowing up” of the denominator, the result

holds true even if density is not uniform.

I If f (1) 6= 0, then the previous results hold in the upper tailonly. (weak law of Pareto)

I Example with an increasing distribution.

Uniform

f(.)

11-Δ

Δ

1/Δ

x

Increasing

f(.)

11-Δ

Δ

2/Δ

x

14 / 28

Non-uniform distribution: Weak law of ParetoExample: h = 70%, f (.) increasing with ∆ = 30%, 10%, 2%.

Δ=30%

Δ=10%

Δ=2%

1 2 3 4 5 6 7

-3.5

-3.0

-2.5

-2.0

-1.5

-1.0

-0.5

0.0

Log10Firm Size

Log 10Survivor

I Quantitatively, for slowly varying densities, ”upper tail”corresponds to the bulk of the distribution.

15 / 28

Relaxing f (1) > 0

I If f (1) = 0. Illustration: Polynomial Functions:

f (x) =ρ+ 1

∆ρ+1(1− x)ρ if x ∈ [1−∆, 1]

I ⇒ Much smaller firm sizes. Pareto (2) dominates in uppertail.

16 / 28

Environment

Span of control with 2 layers

Span of control with L layers - Zipf’s Law

Empirics

Labor Income Distribution

Conclusion

16 / 28

Occupational Choice, given L

I In equilibrium, agents split into L types according to their skillsin [1−∆, 1] = [z1, zL+1], and form a hierarchical organization:

Workers

1-Δ 1 skill

z1 z2 z3

x2 = m(x1)

x2

Managersof type 2

zL+1

Managersof type L

zL xL. . . xL-1

xL-1 = e(xL)

Managersof type L-1

x1zL-1

17 / 28

Firm with L = 3 layers

Workers Manager of type 2

Manager of type 3Firm 1 (Most Productive)

Skill

(CEOs)Firm 2Firm 3 (Least Productive)

1-Δ

1

x1 x2

x3

I Positive Sorting.

I Span of control of manager of type 2 x2 (same):

n2→1(x2) =1

h(1− x1)⇒ f (x2)dx2 = h(1− x1)f (x1)dx1.

I Intermediary Span of control of manager of type 3 x3:

n3→2(x3) =1

h1− x2

1− x1

⇒ f (x3)dx3 = h1− x2

1− x1f (x2)dx2.

17 / 28

Any L

I First layer always special with:

m′(x1)f (m(x1)) = h(1− x1)f (x1).

I Subsequent layers l ∈ [2, ..., L− 1] with conditional probability:

m′(xl)f (m(xl)) = h1− xl

1−m−1(xl)f (xl).

I Matching the more skilled and less skilled:

I L− 1 initial conditions.I L− 1 equations for occupational cutoffs.

I Equilibrium number of layers: fixed cost, or indivisibility witha discrete number N of agents:

L = maxL

{L s.t. 1− zL ≥

1

N

}.

18 / 28

Zipf’s Law for Firm Sizes

I Total Span of control n(xL) ≡ nL→1(xL) is given by:

n(xL) = nL→L−1(xL) ∗ nL−1→L−2(xL−1) ∗ ... ∗ n2→1(x2)

n(xL) =L−1∏l=1

nl+1→l(xl).

I Again, generalizing α2 = 2 and α3 = 3/2 by iteration, the tailexponent for n(xL) is:

αL = 1 +1

L− 1.

I When L→∞, Zipf’s law for firm sizes:

α+∞ = 1 .

19 / 28

Environment

Span of control with 2 layers

Span of control with L layers - Zipf’s Law

Empirics

Labor Income Distribution

Conclusion

19 / 28

French DADS - Plants per Firms

I Pareto on most of the range consistent with the model: theuniform distribution = better approximation locally.

20 / 28

Distribution of US Firms and Establishments. Source:Census Bureau.

I Equivalent for the US? Establishment Level.

-5-4

-3-2

-10

Log 10

Sur

vivo

r

0 1 2 3 4Log10 Size

Establishments Firms

Firms: 1.01 Establishments: 1.33 21 / 28

Environment

Span of control with 2 layers

Span of control with L layers - Zipf’s Law

Empirics

Labor Income Distribution

Conclusion

21 / 28

Labor Income DistributionI Decentralizing skill prices w(.) , 2 layers:

w(y) = maxx

y − w(x)

h(1− x).

I Envelope condition:

w ′(y) =1

h(1− x)= n(y) ⇒ dw(y(n))

dn︸ ︷︷ ︸W ′(n)

= n(y(n))︸ ︷︷ ︸≡S(n)

dy(n)

dn︸ ︷︷ ︸T ′(n)

.

I Compare with Gabaix, Landier (2008). 1 to 1 matching ofCEOs with T (n) to firms with S(n) ∼ Zipf:

maxn

AS(m)γT (n)−W (n) ⇒n=m W ′(n) = AS(n)γT ′(n).

Here S(n) ∼ Zipf endogenously. Microfoundation forTalent ∗ Span of Control, and γ = 1.

I Slight difference: Zipf’s law is truncated ⇒ integratedtruncated Pareto distribution. (hypergeometric function)

22 / 28

Reduced form VS full modelIs a reduced form approach sufficient then ? Not always:

I Calculate all wages. ”Trickle-down” effects.

I Largest firms 6= largest incomes. ρ = 0 ⇒ very small laborincome heterogeneity. Managers compete. Ambiguouscorrelation between wages and size:

I Conditional on ρ, positive correlation between wages and size.

I When ρ↗, wages ↗ size ↘.

I Relate change in firm sizes to deep parameters. Here hand ∆ shift the distribution out.

I But: truncation is key for comparative statics of the Paretodistribution:

I Gabaix and Landier (2008) attribute the 5x increase in CEOcompensation to a 5-fold in the scale. h or ∆.

I Difficulty: α = −3 in 1970s to α = −1.8 now. In Gabaix andLandier (2008), only the scale changes, not the tail index.

23 / 28

Labor Income Distribution: effect of ∆

Δ=90%

Δ=60%

Δ=30%

0.0 0.5 1.0 1.5 2.0-6

-5

-4

-3

-2

-1

0

Log10Labor Income

Log 10Survivor

24 / 28

Labor Income Distribution: effect of h

h =1%

h =5%

h =10%

0.0 0.5 1.0 1.5 2.0-6

-5

-4

-3

-2

-1

0

Log10Wages

Log 10Survivor

25 / 28

Environment

Span of control with 2 layers

Span of control with L layers - Zipf’s Law

Empirics

Labor Income Distribution

Conclusion

25 / 28

Conclusion: coming back to Axtell (2001)What does not matter for heterogeneity under a power lawproduction function

26 / 28

Conclusion: coming back to Axtell (2001)What matters for heterogeneity under a power law productionfunction

27 / 28

Conclusion

I Main takeaways:

I Stylized model can account for Paretian firm size and laborincome distribution, regardless of the ability distribution.

I New intuition for the disconnect between outcomes andprimitives: ”power law change of variable near the origin”.Endogenous ”economics of superstars”.

I Pareto optimal Pareto distributions.

I Future work:

I Other microfoundations for power-law production functions.

I Implications for heterogenous firms in trade, development.Example: Hsieh-Klenow facts = higher ∆?

I Firm / worker fixed effects in AKM-type regressions?

I Optimal taxation with effort choice.

28 / 28