Power Laws in Economics

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Power Laws in Economics: An Introduction

Xavier Gabaix

August 14, 2014

Abstract

This paper is an elementary expository piece for the Journal of Economic Perspec-

tives. It presents a survey of various power laws for firms, cities, trade and finance,

as well as explanations that have been proposed for them. Then, it discusses to what

extent power laws may explain aggregate fluctuations. Its also given pointers to powerlaws outside of economics, and to open questions.

Samuelson (1969) was asked by a physicist for a law in economics that was both non-

trivial and true. This is a difficult challenge, as many (roughly) true results are in the

end rather trivial (e.g. demand curves slope downward), while many non-trivial results in

economics in fact require too much sophistication of the agents to be really true.1 Samuelson

answered the law of comparative advantage. The story does not say whether the physicist

was satisfied, as the law of comparative advantage is a qualitative law, not a quantitative

one. Since 1969, economics has found many qualitative insights, but many fewer reliablequantitative laws.

This article will make the case that, if asked now, Samuelson might mention various

power laws as non-trivial and true laws in economics.2

I start by providing several illustrations of power laws, for cities, firms, and the stock

market. I summarize some of the explanations that have been proposed. I then go on to

suggest that power laws may explain much, including aggregate fluctuations. I conclude with

comments on what is so special about power laws.

Prepared for the Journal of Economic Perspective. Comments most welcome: please email me if youfind that a major mechanism, power law, or reference is missing subject to space constraints. I thankJerome Williams for excellent research assistance.

1 See Gabaix (2014) for an exploration of this in basic microeconomics: many non-trivial predictionsof micro (e.g., Slutsky symmetry) fail when agents are boundedly rational they rely on too subtle anoptimization by the agents.

2 Other quantitative (approximate) laws are (i) the Black-Scholes formula, and (ii) some sophisticatedmechanisms in implementation theory.

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Figure 1: Rank-Size plots for all US cities with size over 250,000 in 2010. Source: Statisticalabstract of the US (2012). The line is the power law fit. The slope is103, suggestive ofZipfs law.

1 Some Empirical Power Laws

Let us look at some pictures of power laws.

1.1 City sizes

Let us look at the data on US cities of size (i.e., population) 250,000 or greater.3 We rank

cities by size, number 1 being the top city: #1 is New York, #2 Los Angeles, etc. We regress

log rank on log size, and find the following:

ln Rank= 788 103ln Size. (1)

This is close to a straight line (2 is 098), and the slope very close to 1 (the standard

deviation of the slope is 0.01).4 This means that the rank of a city is proportional to the

inverse of its size. This value of 1 has been found repeatedly across many cities andcountries, at least after the Middle Ages (Dittmar 2012). There is no artifactual reason to

find a power law, and even less for the slope to be 1.

To think about this type of regularity, it is useful to be a bit more abstract, and see the

3 This comprises all Metropolitan Statistical Areas (i.e., agglomerations) provided in the StatisticalAbstract of the US (2012).

4 Actually, the standard error returned by OLS is incorrect. The correct standard error is |slope|p

2=011, where= 184 is the number of cities in the sample (Gabaix and Ibragimov 2011).

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cities as coming from a underlying distribution: the probability that the size of a randomly

drawn city is greater than is proportional to 1 with' 1. More generally

(Size ) =

(2)

at least for above a cutoff(here, the 250,000 inhabitants cutoffgiven by the StatisticalAbstract of the US). A regularity of the type (2) is called a power law. In a given finitely-sized

sample, it generates an approximate relation of type (1): lnRank= lnSize.The interesting part is the coefficient , which is called the power law exponent of the

distribution (or Pareto exponent). A Zipfs law is a power law with an exponent of 1.

A lower means a higher degree of inequality in the distribution: it means a greater

probability offinding very large cities or (in another context) very high incomes. Indeed, the

moments of order greater than are infinite, while moments of order less than are finite.

For example, if= 103, the expected size is finite, but the variance is formally infinite.

In addition, the exponent is independent of the units (inhabitants or thousands of in-

habitants, say). This makes it at least conceivable that we might find an a priori constant

simple value (such as an integer) in various datasets.

What if we look at cities with size less than 250,000? Does Zipfs law still hold?5 Figure

2 shows the distribution of city sizes for the UK (where the data is particularly good). Here

we see the appearance of a straight line for cities of about size 500 and above. Zipfs law

holds very well too.

Why do we care? As Krugman (1996) put it, after having written his work on economic

geography: The failure of existing models to explain a striking empirical regularity (oneof the most overwhelming empirical regularities in economics!) indicates that despite con-

siderable recent progress in the modeling of urban systems, we are still missing something

extremely important. Suggestions are welcome. We shall see that since then, we have im-

proved our understanding of the origin of the Zipfs law, which has forced a great rethinking

about the origins of cities and firms, as we shall now see.

1.2 Firm sizes

We now look at the firm size distribution. Axtell (2001) presents it, from the US census data.

He bins firms according to their size (no. of employees), and plots the log of the number of

5 Because it is delicate to construct agglomerations (rather than fairly arbitrary legal entities), Rozenfeldet al. (2011) use a new algorithm constructing cities from fine-grained geographical data.

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Figure 2: Cumulative distribution of agglomerations in in the UK. We see a pretty goodpower law fit starting at about 500 inhabitants. The exponent is actually statistically non-different from 1 for size 12 000 inhabitants. Source: Rozenfeld at al. (2011).

firms within a bin. We see a straight line: this is a power law. Here we can even run the

regression in density, i.e. plot the number of cities of size approximately equal to . If

(2) holds, then the density of the firm size distribution is () = +1, so the slope in a

log-log plot should be (+ 1) (as ln () = (+ 1) ln +constant). Impressively, theexponent that Axtell finds is= 1059. This demonstrates a Zipfs law for firms.

Again, this has forced a rethinking offirms: most static theories (e.g. based on elasticityof demand, fixed cost, economies of scope, etc.) would not predict a Zipfs law. Some other

type of theory is needed, as we shall soon discuss.

1.3 Stock market movements

It is well-known that stock market returns are fat-tailed i.e. the probability of finding

extreme values is larger than for a Gaussian distribution of the same mean and standard

deviation. An energetic movement of physicists (econophysicists, a term coined after

geophysicists and biophysicists) has quantified a host of power laws in the stock market.For instance, the daily stock market movements are represented in Figure 4. They are

consistent with: (|| ) = with = 3, the cubic law of stock market returns.

The left panel of Figure 4 plots the distribution for four different sizes of stocks. The right

panel plots the distribution of normalized stock returns, i.e. of stock returns divided by

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Figure 3: Log frequency ln ()vs log size ln of U.S. firm sizes (by number of employees)for 1997. OLS fit gives a slope of 2.06 (s.e.= 0.054; 2 = 099). This corresponds to afrequency () 2059, i.e. a power law distribution with exponent = 1059. This isvery close to Zipfs law, which says that = 1. Source: Axtell (2001).

their standard deviation: after this normalization, the four different distributions collapse

onto the same curve. This is a type of universality a term much used in the power law

literature (and in physics) which means that different systems behave in the same way, after

some rescaling. This cubic law appears to hold for a variety of other international stock

markets too (Gopikrishnan et al. 1999).

Likewise, lots of other quantities are power law distributed (Plerou et al. 2005, Kyle and

Obizhaeva 2013, Bouchaud, Farmer and Lillo 2009, Geerolf 2014). For instance, the number

of trades per day is power law distributed with exponent of 3, while the number of sharestraded per time interval has an exponent of 1.5, and the price impact is proportional to the

volume to the power of 0.5.

Why do we care? One implication of the cubic law is that there are many more extreme

events than would occur if the distribution were Gaussian. More precisely, a 10 standard

deviation event and a 20 standard deviation event are, respectively, 53 = 125and103 = 1000

times less likely than a 2 standard deviation event, whereas if the distribution of returns

was Gaussian, the ratios would be 1022 and 1087, respectively. Indeed, in a stock market

comprising about 1000 stocks, a 10 standard deviation event happens in practice about every

day.

More deeply, the explanation for those regularities force us to rethink the functioning

of stock markets we shall discuss later theories that exactly explain these exponents in a

unified way.

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Figure 4: Cumultative distribution of daily stock market returns. The left panel shows thedistributions for 4 different sizes of stocks. The right panel shows the returns, normalized byvolatility. The slopes are close to 3, reflecting the cubic law of stock market fluctuations:(|| ) 3. The horizontal axis displays returns as high as 100 standard deviations.Source: Plerou et al. (1999).

1.4 Other power laws

Income and wealth also follow roughly power law distributions, as we have known since Pareto

(1896), who first documented power laws not only in economics, but indeed anywhere (to

the best of my knowledge). The distribution of wealth is more unequal than the distribution

of income: this makes sense, as differences in growth rate of wealth across individuals (due to

differences in returns or frugality) pile up and add an extra source of inequality. Typically,the exponent is around 1.5 for wealth, and between 1.5 and 3 for income. Given the recent

interest in income inequality, power laws and random growth processes are a central tool to

analyze those (Piketty and Zucman 2014, Atkinson, Piketty and Saez 2011, Benhabib, Bisin

and Zhu 2011, Lucas and Moll 2014).

2 What Causes Power Laws?

2.1 Random growth

The basic mechanism for generating power laws is proportional random growth (Champer-

nowne 1953, Simon 1955). Suppose that the we start with an initial distribution offirms, and

they grow and shrink randomly with independent shocks, and they satisfy Gibrats law:

all firms have the same expected growth rate and the same standard deviation of growth

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rate. This model does not even have a steady state distribution, as the distribution becomes

a lognormal with larger and larger variance. However, things change altogether if we add to

the model some friction that guarantees the existence of a steady state distribution. Suppose

for instance that there is also a lower bound on size, so that a firm size cannot go below a

given threshold. Then, the model yields a steady distribution, and it is a power law, with

some exponentthat depends on the details of the growth process.

Now, this mechanism generates a power law, but with an exponent that is generally not

1. Why would we have exponent of 1? This was Krugmans question earlier.

One explanation is given in Gabaix (1999) (see also Gabaix (2009) for a thorough review).

Suppose that the size of the friction (the lower bound) becomes very small, and that we

have a given exogenous population size to allocate in the system (between the different cities

or firms). Then, the exponentbecomes 1, rather than any other value.6 This is why we

observe lots of Zipfs laws: proportional random growth, with a small friction, and some

adding-up constraint for the total size of the system.Of course, this is the mechanical part of the explanation. An economist would like

to know why we have random growth in the first place, or in other words, why Gibrats

law holds. The simplest microfoundation is that cities and firms are basically constant

returns to scale, perhaps with small deviations from that benchmark, and lots of randomness.

Indeed, many fully economic models for the random growth of cities and firms have been

proposed since the 2000s (e.g., Rossi-Hansberg and Wright (2007) and Luttmer (2007)).

Likewise, for the income distribution, the details of the underlying mechanism (say, luck vs

thrift, responsive to incentives or not) are very important for a variety of questions, and

microfounded models are important. Still, to write them sensibly one needs to keep in mind

the core mechanics of these models here, random growth with power laws.

2.2 Matching and economics of superstars

Another manifestation of power laws is in the extremely high earnings of top earners in

areas of arts, sports and business. Rosen (1981) gives a qualitative explanation for this with

the economics of superstars. Gabaix and Landier (2008) give a tractable, calibratable

model of this phenomenon, along the following lines. Suppose that lots offirms, of different

sizes, compete to hire the talents of CEOs. The talent of a CEO is given by how much

6 One intuition is as follows: the exponent cannot be below 1, because then the distribution would haveinfinite mean. Indeed, an exponent just above 1 is the smallest consistent with a finite total population. Asthe friction becomes very small, the exponent becomes the fattest consistent with a finite population.

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(in percentages) she is expected to increase the firms profits. Competition implements the

efficient outcome, which is that the largest firm will be matched with the best CEO in the

economy, the second largest firm with the second best CEOs, etc.

One might think it hopeless to derive a quantitative theory, as the distribution of talent

is very hard to observe. However, we can draw on extreme value theory (which is a branch

of probability) to obtain some properties of the tail of the distribution of talent, without

knowing the distribution itself. One of them is that, given adjacent CEOs in the ordering of

talent, the approximate difference in talent between these two CEOs varies like a power law

of their rank. The exponent depends on the distribution, but the power law functional form

holds for essentially any reasonable distribution (in a way that can be made precise). Given

this, Gabaix and Landier work out the pay of CEO number , who manages a firm of size

(). We denote by() the size of a reference firm, i.e. the size of the median firm in

the S&P 500. The pay of CEO number is:

() = () ()1() (3)

where one calibrates = 13.7 This is a dual scaling relation, because it has two power

laws. We see how matching creates a dual scaling equation (38), or double power laws,

which has three implications:

(a)Cross-sectional prediction. In a given year, the compensation of a CEO is proportional

to the size of his firm to the power of13,()13.

(b) Time-series prediction. When the size of all largefirms is multiplied by (perhaps

over a decade), the compensation at all large firms is multiplied by . In particular, the payat the reference firm is proportional to the average market cap of a large firm.

(c)Cross-country prediction. Suppose that CEO labor markets are national rather than

integrated. For a given firm size , CEO compensation varies across countries with the

market capitalization of the reference firm,()23, using the same rank of the reference

firm across countries.

It turns out that all three predictions seem to hold empirically since the 1970s. This

explains why CEO pay has increased a lot firm size has increased.

Here, power laws and extreme value theory are the natural language for the economicsof superstars. In addition, very small differences in talent give rise to very large differences

in pay: in the Gabaix-Landier calibration, talent has finite support, but differences of talent

are unboundedly large. This is what happens when very large firms compete to hire the

7 Empirically, the exponent tends to be in the[03 04]range.

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services of CEOs.

The same logic should apply to other superstars market: apartments with a large view

on Central Park, but also sport and the price of works of art. When the wealth becomes

stronger and more unequal, the same should happen to say the price of works of art. As far

as I know, a systematic quantitative exploration of those issues still has to be done. 8

This line of thinking leads to a fresh way of thinking about pay-performance sensitiv-

ity, going beyond the classic result of Jensen and Murphy (1990). They define the Pay-

Performance Sensitivity (PPS) as how many dollars the CEO compensation (or wealth)

changes for a given dollar change in firm value. They find that the PPS is very small: CEOs

earn only $3 extra when their firm increases by $1000 in value: they conclude that corpo-

rate governance may not work well. However, Edmans, Gabaix and Landier (2009) propose

a different way to think of the benchmark incentives, resting on scaling arguments. Suppose

that to motivate the CEO, it is percent/percent incentives, not dollar/dollar incentives that

matter: namely, for a 1% of increase in firm value, the CEOs wealth should increase by%, where does not scale with size (this comes from preferences that are multiplicative in

effort and consumption). Then, if (3) holds, the PPS of Jensen-Murphy should decrease as

(Firm size)23.9 This is actually true empirically, as illustrated in Figure 5. Hence, think-

ing in terms of scaling leads to new predictions for pay-performance sensitivity, and actually

leads to propose new ones.

2.3 Optimization and Transfer of power laws

Optimization gives a good way to obtain power laws. For instance, the Allais-Baumol-Tobinrule for the demand for money (which scales as 12, where is the interest rate), is a

power law. The first scaling relation in economics (and, not coincidentally, the first non-

trivial empirical success in economics) may be Humes thought experiment that doubling

the money supply should lead (after a while) to a doubling of the price level a basic theory

that has stood the test of time.

Power laws have very good aggregation properties: taking the sum of two (independent)

power law distributions gives another power law distribution. Likewise, multiplying two

8 Behrens, Duranton and Robert-Nicoud (2013) propose a theory of Zipfs law based on matching.9 The percent/percent incentive is constant ( ln= , where is CEO pay, and the firm return),

so the Jensen-Murphy pay-performance sensitivity measure is,

:=

=

=

ln

=

0

= 001

using = 0 from (3, and for firm-independent constants 0 00

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2 0 2 4 61

0.5

0

0.5

1

1.5

2

ln Firm size

lnCE

Oc

ompensation

2 0 2 4 63.5

3

2.5

2

1.5

1

0.5

0

0.5

1

ln Firm size

lnPayperformancesensitivity

Figure 5: CEO pay and CEO pay-performance sensitivity vs firm size. Left panel. TheCEO compensation is the ex ante one, including Black-Scholes value of options granted.The slope is about 13, a reflection of Roberts law: PaySize with ' 13. Right panel.The pay-performance sensitivity is the Jensen-Murphy measure: by how many dollars doesthe CEO wealth change, for a given dollar change in firm value. The slope is about23,so that Size1 with ' 13. The scalings are predicted by the Edmans, Gabaix,Landier (2009) model. Source: The data and methodology is from Edmans et al. (2009), forthe years 1994-2008. Years are lumped together by reporting vs ln

andln

vsln

,

where indicate the median value of in year .

power laws, taking their max or their min, or a power, etc. gives again a power law distribu-

tion. This partly explains the prevalence of power laws: they survive many transformations

and the addition of noise.

3 Granularity: Aggregate fluctuations from microeco-

nomic shocks

I now turn to an application of power laws: getting a better sense of the origins of aggregate

fluctuations in GDP, exports and the stock market.

3.1 Basic ideas

Where do aggregate fluctuations come from? Gabaix (2011) proposes that idiosyncraticshocks to firms (or narrowly defined industries) can generate aggregate fluctuations. A

priori, an economist would say that this is not quantitatively possible: there are millions of

firms, so by an analogy similar to the central limit theorem, their total fluctuations should be

very small (technically, if there are firms, the total fluctuations should decay as 1

).

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However, when the firm size distribution is fat-tailed, things change completely: the central

limit theorem doesnt apply any more.10 Instead, GDP fluctuation decays in1 ln .

Empirically, the distribution of economic activity is indeed very concentrated amongst

firms. For instance, di Giovanni and Levchenko (2012) find that In Korea, the 10 biggest

business groups account for 54% of GDP and 51% of total exports.[...] The largest one,

Samsung, is responsible for 23% of exports and 14% of GDP. Hence, it is plausible that

idiosyncratic shocks to firms would affect GDP activity.

In that view, economic activity is not made of a smooth continuum of firms, but it is

made of incompressible grains of activities the firms whose fluctuations do not wash

out in the aggregate. The plain reason is that firms are big (and initial shocks are intensified

by a variety of generic amplification mechanisms, e.g. endogenous changes to hours worked).

Is this granular hypothesis relevant empirically? Gabaix (2011) finds that the idiosyn-

cratic shocks to large firms explain about 1/3 of GDP fluctuations in the US. Di Giovanni,

Levchenko and Mejean (2014) find that they explain over half the fluctuations in France.Further support is given in Foerster, Sarte and Watson (2011) for industrial production, and

di Giovanni and Levchenko (2012) for exports. The exploration continues.

There are two payoffs from this analysis: first, we may better understand the origins of

aggregatefluctuations. Second, these large idiosyncratic shocks may give a series of useful

instruments for macro. For instance, Amiti and Weinstein (2013) start form the fact that

banking is very concentrated, so idiosyncratic bank shocks may have strong ripple effects in

the economy, which allows them to quantify banking channels.11

Another implication of granularity is the importance of networks (Acemoglu et al. 2012).

Those large firm-level shocks propagate through networks, which create an interesting am-

plification mechanism, and a way to quantitatively see the propagation effects. Networks

are a particular case of granularity, rather than an alternative to it. They offer a way to

visualize the propagation of idiosyncratic firm shocks.

3.2 The great moderation: A granular post-mortem

This granular perspective offers a way to understand the time variations in volatility. For

instance, Carvalho and Gabaix (call granular or fundamental volatility the volatility that

10 A power-law variant holds, the Lvy central limit theorem.11 Incidentally, they find that idiosyncratic bank shock explain 40% of aggregate loan and investment

fluctuations.

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1960 1965 1970 1975 1980 1985 1990 1995 2000 2005

6

4

2

0

2

4

6

8

10x 10

3

year

CyclicalVolatilit

y(demeaned)

Figure 6: Fundamental Volatility and GDP Volatility. The squared line gives the granular /fundamental volatility (45, demeaned), i.e. the volatility that would arise if there were only

idiosyncratic sectoral shocks, and no aggregate shock. The solid and circle lines are annualized(and demeaned) estimates of GDP volatility, using respectively a rolling-window estimate and an

Hodrick-Prescott trend of instantaneous volatility. Source: Carvalho and Gabaix (2013).

would come only from idiosyncratic sectoral or firm-level shocks.12 Figure 6 reports their

findings. The fundamental volatility is quite correlated with actual volatility.

This gives an extra narrative for the great moderation and its undoing. i) The long

and large decline of fundamental volatility from the 1960s to the early 1990s is due to the

smaller size of a handful of heavy-manufacturing sectors: this created a great moderation

of volatility not due to monetary factors and the like, but simply because the economybecame more diversified. ii) The increased importance of the energy sector (which itself can

be traced to the rise of oil prices) accounts for the burst of volatility in the mid 1970s. iii)

The increase in the size of the financial sector is an important determinant of the increase

in fundamental volatility and of actual volatility in the 2000s. Similar findings hold for

the other developed countries considered.

Likewise, moving to explaining firm-level volatility, Kelly, Lustig and van Nieuwerburgh

(2013) find that taking account of the network can help us understand firm-level and ag-

gregate level stock market volatility: for instance, firms with few, and volatile, customers or

suppliers will have large volatility.

12 To operationalize this idea, they consider the fundamental volatility:=

rP

=1

GD P

22where

is the gross (notnet) output of sector , and is the standard deviation of the total factor productivity(TFP) in the sector.

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3.3 Origins of stock market crashes?

Gabaix et al. (2003, 2006) develop the hypothesis that stock market crashes are due to large

financial institutions selling under pressure in illiquid markets (see also Levy and Solomon

1996 and Solomon and Richmond 2001). This may account not only for large crashes, but

also for the whole distribution of mini-crashes described by the power law. Large institutionsare roughly Zipf distributed, but trade less than proportionally to their size to moderate price

impact; the price impact is itself proportional to the square root of the volume traded all of

these relationships arise endogenously. Still, when large institutions sell under time pressure,

they make the market fall and (if they are large enough, and under enough pressure) crash.

One can speculate that this type of mechanism might have been at work in a variety of

well-known events: the crash of Long Term Capital Management in the summer of 1998, the

rapid unwinding of very large stock positions by Socit Gnrale after the Kerviel rogue

trader scandal (which led stock markets to fall in 21-23 January 2008), and the flash crash

of May 2010. See the discussion in Gabaix et al. (2006) and Kyle and Obizhaeva (2013),

which also argue that the 1929 crash is an example of that phenomenon. This research

potential brings us closer to understanding the origins of stock market movements. 13

3.4 Volatility offirms and countries

Granularity can also explain the following fact: firms and countries have identical, non-

trivial, scaling of growth rates. Stanley et al. (1996) study how the volatility of the growth

rate offirms changes with size. To do this, they calculate the standard deviation ()of the

growth rate offirms sales,, and regress its log against log size. They find an approximately

affine relationship, displayed in Figure 7: ln firms () = ln + . This means that a firmof sizehas volatility proportional to with = 015.14

Lee et al. (1998) conduct the same analysis for country growth rates and find that

countries with a GDP of size also have a volatility proportional to 0

, with0 = 015

a form of diversification (Koren and Tenreyro 2013). The two graphs are plotted in Figure

7. The slopes are indeed very similar. This may be a type of universality. This may

be explained by granularity: if aggregate fluctuations come from microeconomic shocks, and

firm sizes follow Zipfs law, then the identical scaling of Figure 7 should hold true.

13 Barro and Jin (2011) document a power law distribution of macroeconomic disasters, which may explainmany puzzles in finance (Gabaix 2012).

14 Large firms have a smaller proportional standard deviation than small firms. However, this diversificationeffect is weaker than if a firm of size were composed ofindependent units of size 1, which would predict= 12. See Riccaboni et al. (2008).

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Figure 7: Standard deviation of the distribution of annual growth rates (log-log axes). Notethat ()decays with sizewith the same exponent for both countries and firms: () , with ' 16. The size is measured in sales for the companies (top axis) and in GDPfor the countries (bottom axis). The firm data are taken from the Compustat for the years

1974-1993, the GDP data from Summers and Heston (1991) for the years 1950-1992. Source:Lee et al. (1998).

4 Power laws and Universality outside of economics

Mathematics.The idea of universality comes from mathematics. The paradigm is the cen-

tral theorem: when forming an average of demeaned random variables, the distribution of the

average (suitably normalized, here multiplied by

)converges to the Gaussian distribution

a universal distribution independent of the details of the original distribution. This

phenomenon happens for other things, e.g. the distribution of extremes of distributions, andthe properties of matrices with random entries.

Physics. The physics of critical phenomena is full of universal laws, in the sense

that different materials behave in the exact same way, after normalization, around the phase

transition. There is a sophisticated machinery of the renormalization group to understand

this (see Sornette 2004). Lots of other phenomena, e.g. forest fires and rivers, have a

power law structure (partially explained by the theory of self-organized criticality). Perhaps

surprisingly, these notions of percolation and self-organized criticality havent (yet) found

widespread use in economics (see, however, Bak et al. 1993, Nirei 2006).

Networks. Networks are full of power laws: for instance (probably because of random

growth), the popularity of web sites (number of sites linking to it) is a power law (Barabasi

and Albert 1999).

Biology. Biology is replete with intriguing, universal relations of the power law type. For

instance, the energy that an animal of mass requires to function is proportional to34

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rather thanas a simple constant return to scale model would predict (Figure 11). West

et al. (1997) have proposed the following explanation: If one wants to design an optimal

network system to send nutrients to the animal, one designs a fractal system; the resulting

efficiency generates the34 law. There is an interesting lesson: lots of things could a priori

matter for energy consumption (e.g. climate, predator or prey status, thickness of the fur),

and they probably do matter, a little bit. However, in its essence, an animal is best viewed as

a network in which nutrients circulate at maximum efficiency. Understanding the power laws

forces the researcher to forget, in the first pass, about the details. Likewise, this research

shows similar laws for a host of variables, including life expectancy (which scales at 14).

Here the interpretation is that the animal is constructed optimally, given engineering

constraints given by biology. Perhaps surprisingly, this type of mechanism doesnt seem to

have been much studied in economics. For instance, the economy resembles a network with

power law distributed firms: does it come from optimality, as opposed to randomness? It

would be nice to know. Likewise, Zipfs law holds for the usage frequency of words. Thesimplest explanation is via random growth (as the popularity of words follow a random

growth process). However, perhaps that might reflect an optimal organization of mental

categories, perhaps in some tree-like structure? Again, one would like to know.

Figure 8: Metabolic rate (i.e., energy requirement per day) a function of mass across animals.

The slope of this log-log graph is 34 : the metabolic rate of an animal of mass isproportional to 34 (Kleibers law). Source: West, Brown and Enquist (2000).

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4.1 Conclusion

Power laws are everywhere (when datasets contains enough variation in some size-like

factor, such as income or number of employees) so all economists should know about

power laws, and the basic mechanisms that generate them.

Power laws can guide the researcher to the essence of the phenomenon. For instance,take city sizes: a priori, lots of things may be important for cities: specialization, transporta-

tion cost, elasticity of congestion vs positive externalities in human capital... The power law

approach concludes that while those things may exist, they are not the essence: the essence is

random growth with a small friction. Now, to generate the random growth, a judicious mix

of the traditional ingredients may be useful, but to orient the understanding, one should

first think about the essence, and only then about the economic underpinning.

Some open questions about power laws Now is a good time to answer questions

on power laws, as many new data are available.

Networks and granularity. How big is the volatility generated from idiosyncratic shocks,

propagated and amplified in networks?

The gravity equation. The trade flows between two countries is proportional to the GDP

of the two countries (which is trivial, and come from a simple CRS model), and declines

with distance (which is intuitive), as the inverse of distance to the power 1 the latter

scaling being very non-trivial. What explains this? Chaney (2013) proposes an ingenious

model linking this distance to the probability of forming a link in a random growth (for

power laws in trade, see also Melitz, Helpman and Yeaple (2004) and Eaton, Kortum andKramarz (2011)). Under Zipfs law, that generates the coefficient of 1. Similar scaling holds

for migration, see Levy (2013).

Why is the aggregate production (roughly) Cobb-Douglas with a capital share about

13? Jones (2005) generates the functional form, but not the particular exponent. Perhaps

generating it will suggest a deeper understand of the causes of technical progress.

Is random growth the canonical origin of power laws? or is it something else, like the

economics of superstars, or optimization? Does Gibrats law really hold?15

As big data makes lots of new datasets available, it will be important to order them.

Scaling questions are a natural way to do that, and have met with great success in the

15 Gibrats law seems to roughly hold (Ioannides and Overman 2003, Eeckhout 2004), though the issue isnot settled as the literature hasnt fully differentiated between permanent shocks and transitory ones andmeasurement error.

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natural sciences. Given the new availability of great datasets, the future of power laws seem

bright.

Recommendations for future readings A reader seeking a gentle introduction to

power law techniques might start with Gabaix (1999), and then move on to more systematicexposition in Gabaix (2009), which contains many other pointers. Mantegna and Stanley

(1999) contains an accessible introduction to the field of econophysics, while Sornette (1999)

contains many interesting physics mechanisms generating scaling. For extreme value theory,

Embrechts, Kluppelberg and Mikosch (1997) is very pedagogical. For networks, Jackson

(2010) and Newman (2010) are now classic references.

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Power Laws in Economics