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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.
5 References
Acemoglu, Daron, Vasco M. Carvalho, Asuman Ozdaglar, and Alireza Tahbaz-Salehi. 2012.
The network origins of aggregate fluctuations. Econometrica, 80(5): 19772016.
Amiti, Mary and David Weinstein. 2013. How Much do Bank Shocks Affect Investment?
Evidence from Matched Bank-Firm Data, NBER Working Paper 18890.
Atkinson, Anthony, Emmanuel Saez and Thomas Piketty. 2011. Top Incomes in the
Long Run of History. Journal of Economic Literature, 49(1): 371
Axtell, Robert L. 2001. Zipf distribution of US firm sizes. Science, 293(5536): 1818
1820.
Bak, Per, Kan Chen, Jos Scheinkman, and Michael Woodford. Aggregate fluctuations
from independent sectoral shocks: self-organized criticality in a model of production and
inventory dynamics. Ricerche Economiche, 47(1): 330.
Barabsi, Albert-Lszl, and Rka Albert. 1999. Emergence of scaling in random
networks. Science, 286(5439): 509512.
Barro, Robert J., and Tao Jin. 2011. On the size distribution of macroeconomic disas-
ters. Econometrica 79(5): 1567-1589.
Behrens, Kristian, Gilles Duranton and Frdric Robert-Nicoud. 2013. Productive
Cities: Sorting, Selection, and Agglomeration.Journal of Political Economy, 122(3): 507553.
Benhabib, Jess, Alberto Bisin, and Shenghao Zhu. 2011. The distribution of wealth
and fiscal policy in economies with finitely lived agents. Econometrica, 79(1): 123157.
17
8/11/2019 Power Laws in Economics
18/21
Bouchaud, Jean-Philippe, J. Doyne Farmer, and Fabrizio Lillo. 2009. How markets
slowly digest changes in supply and demand. Handbook of Financial Markets: Dynamics
and Evolution, pp. 57160.
Champernowne, D. G. 1953. A model of income distribution. The Economic Journal,
63 (250): 31851.
Chaney, Thomas. 2013. The Gravity Equation in International Trade: an Explanation.
Working Paper. Toulouse.
Di Giovanni, Julian and Andrei Levchenko. 2012. Country Size, International Trade,
and Aggregate Fluctuations in Granular Economies. Journal of Political Economy, 120(6):
10831132.
Di Giovanni, Julian and Andrei Levchenko. 2014. Firms, Destinations, and Aggregate
Fluctuations. Econometrica, 82(4): 13031340.
Eaton, Jonathan, Samuel Kortum, and Francis Kramarz. 2011. An Anatomy of Inter-
national Trade: Evidence from French Firms. Econometrica, 79(5): 1453-1498.Edmans, Alex, Xavier Gabaix, and Augustin Landier. 2009. A multiplicative model
of optimal CEO incentives in market equilibrium. Review of Financial Studies, 22(12):
48814917.
Eeckhout, Jan. 2004. Gibrats law for (all) cities. American Economic Review, 94(5):
142951.
Embrechts, Paul, Claudia Klppelberg, and Thomas Mikosch. 1997. Modelling Extremal
Events for Insurance and Finance. New York: Springer.
Foerster, Andrew, Pierre-Daniel Sarte and Mark Watson. 2011. Sectoral versus Aggre-
gate Shocks: A Structural Factor Analysis of Industrial Production. Journal of Political
Economy, 119(1): 138.
Fu, Dongfeng, Fabio Pammolli, Sergey V. Buldyrev, Massimo Riccaboni, Kaushik Matia,
Kazuko Yamasaki, and H. Eugene Stanley. 2005. The growth of business firms: theoret-
ical framework and empirical evidence. Proceedings of the National Academy of Sciences
102(52): 188016.
Gabaix, Xavier. 1999. Zipfs law for cities: An explanation. Quarterly Journal of
Economics, 114(3): 73967.
Gabaix, Xavier. 2011. The granular origins of aggregate fluctuations. Econometrica,79(3): 733772.
Gabaix, Xavier. 2012. Variable rare disasters: A tractable framework for ten puzzles in
macro-finance. Quarterly Journal of Economics, 127(2): 645700.
Gabaix, Xavier. A Sparsity-Based Model of Bounded Rationality. Quarterly Journal
18
8/11/2019 Power Laws in Economics
19/21
of Economics, forthcoming.
Gabaix, Xavier, Parameswaran Gopikrishnan, Vasiliki Plerou, and H. Eugene Stanley.
2003. A theory of power-law distributions in financial market fluctuations. Nature,
423(6937): 267270.
Gabaix, Xavier, Parameswaran Gopikrishnan, Vasiliki Plerou, and H. Eugene Stanley.
2006. Institutional investors and stock market volatility. Quarterly Journal of Economics,
121(2): 461504.
Gabaix, Xavier and Rustam Ibragimov. 2011. Rank-1/2: A simple way to improve the
OLS estimation of tail exponents. Journal of Business Economics and Statistics, 29(1):
2439.
Gabaix, Xavier and Augustin Landier. 2008. Why has CEO pay increased so much?
Quarterly Journal of Economics, 123(1): 49100.
Geerolf, Francois. 2014. A Theory of Power Law Distributions for the Returns to
Capital and of the Credit Spread Puzzle. Working Paper, UCLA.Gopikrishnan, Parameswaran, Vasiliki Plerou, Luis A. Nunes Amaral, Martin Meyer,
and H. Eugene Stanley. 1999. Scaling of the distribution offluctuations offinancial market
indices. Physical Review E 60(5): 53055316.
Melitz, Marc, Elhanan Helpman, and Stephen Yeaple. 2004. Export versus FDI with
heterogeneous firms.American Economic Review, 94(1): 300316.
Ioannides, Yannis M., and Henry G. Overman. 2003. Zipfs law for cities: an empirical
examination.Regional Science and Urban Economics, 33(2): 127137.
Jackson, Matthew O. 2010. Social and Economic Networks. Princeton University Press.
Jensen, Michael C., and Kevin J. Murphy. 1990. Performance pay and top-management
incentives. Journal of Political Economy, 98(2): 225264.
Jones, Charles I. 2005. The shape of production function and the direction of technical
change. Quarterly Journal of Economics, 120(2): 517549.
Kelly, Bryan, Hanno Lustig and Stijn Van Nieuwerburgh. 2013. Firm Volatility in
Granular Networks. Working paper.
Koren, Mikls, and Silvana Tenreyro. 2013. Technological diversification. American
Economic Review 103(1): 378-414.
Krugman, Paul. 1996. Confronting the Mystery of Urban Hierarchy. Journal of theJapanese and International economies, 10(4): 399418.
Kyle, Albert, and Anna Obizhaeva. 2013. Market microstructure invariance: Theory
and empirical tests. Working Paper.
Kyle, Albert, and Anna Obizhaeva. 2013. Large bets and stock market crashes.
19
8/11/2019 Power Laws in Economics
20/21
Working paper.
Lee, Youngki, Lus A. Nunes Amaral, Martin Meyer, David Canning, and H. Eugene
Stanley. 1998. Universal Features in the Growth Dynamics of Complex Organizations.
Physical Review Letters81(15): 32753278.
Levy, Moshe. 2010. Scale-free human migration and the geography of social networks.
Physica A: Statistical Mechanics and its Applications, 389(21): 49134917.
Levy, Moshe, and Sorin Solomon. 1996. Power laws are logarithmic Boltzmann laws.
International Journal of Modern Physics C 7(4): 595601.
Lillo, Fabrizio, and Rosario N. Mantegna. 2003. Power-law relaxation in a complex
system: Omori law after a financial market crash. Physical Review E 68(1): 01611924.
Luttmer, Erzo GJ. 2007. Selection, growth, and the size distribution offirms. Quarterly
Journal of Economics, 122(3): 11031144.
Lucas, Robert and Ben Moll. 2014. Knowledge Growth and the Allocation of Time.
Journal of Political Economy, 122(1): 151.Mantegna, Rosario Nunzio, and Harry Eugene Stanley. 2000. An Introduction to Econo-
physics: Correlations and Complexity in Finance. Cambridge University Press.
Newman, Mark. 2010. Networks: An introduction. Oxford University Press.
Nirei, Makoto. 2006. Threshold behavior and aggregate fluctuation. Journal of Eco-
nomic Theory, 127(1): 309322.
Pareto, Vilfredo. 1896. Cours dconomie politique. Librairie Droz.
Piketty, Thomas, and Gabriel Zucman. 2014. Capital is back: Wealth-income ratios in
rich countries, 17002010. Quarterly Journal of Economics, forthcoming.
Plerou, Vasiliki, Parameswaran Gopikrishnan, Luis A. Nunes Amaral, Martin Meyer,
and H. Eugene Stanley. 1999. Scaling of the distribution of price fluctuations of individual
companies.Physical Review E, 60(6): 651929.
Plerou, Vasiliki, Parameswaran Gopikrishnan, and H. Eugene Stanley. 2005. Quanti-
fying fluctuations in market liquidity: Analysis of the bid-ask spread. Physical Review E,
71(4): 046131.
Riccaboni, Massimo, Fabio Pammolli, Sergey V. Buldyrev, Linda Ponta, and H. Eugene
Stanley. 2008. The size variance relationship of business firm growth rates. Proceedings
of the National Academy of Sciences, 105(50): 1959519600.Simon H. 1955. On a class of skew distribution functions. Biometrika. 44:425-40
Solomon, Sorin, and Peter Richmond. 2001. Power laws of wealth, market order volumes
and market returns.Physica A: Statistical Mechanics and its Applications, 299(1): 188197.
Rosen, Sherwin. 1981. The Economics of Superstars. American Economic Review,
20
8/11/2019 Power Laws in Economics
21/21
71(5): 845858.
Rossi-Hansberg, Esteban, and Mark LJ Wright. 2007. Urban structure and growth.
Review of Economic Studies, 74(2): 597624.
Rozenfeld, Hernn D., Diego Rybski, Xavier Gabaix, and Hernn A. Makse. 2009. The
Area and Population of Cities: New Insights from a Different Perspective on Cities. Amer-
ican Economic Review, 101(5): 2205-2025.
Samuelson, Paul A. 1969. The way of an economist. International economic relations:
Proceedings of the third congress of the international economic association 111. London:
MacMillan.
Sornette, Didier. 2006. Critical phenomena in natural sciences: chaos, fractals, selfor-
ganization and disorder: concepts and tools.Springer Science & Business.
Stanley, Michael HR, Luis AN Amaral, Sergey V. Buldyrev, Shlomo Havlin, Heiko
Leschhorn, Philipp Maass, Michael A. Salinger, and H. Eugene Stanley. 1996. Scaling
behaviour in the growth of companies. Nature, 379(6568): 804806.United States Census. 2012. Statistical Abstract of the US. http://www.census.gov/compendia/s
accessed July 26 2014.
West, Geoffrey B., James H. Brown, and Brian J. Enquist. 1997. A general model for
the origin of allometric scaling laws in biology. Science, 276(5309): 122126.
West, Geoffrey B., James H. Brown, and Brian J. Enquist. 2000. The origin of universal
scaling laws in biology. In Scaling in Biology,eds. JH Brown, GB West. Oxford: Oxford
Unive. Press
Zipf, George Kingsley. 1949. Human behavior and the principle of least effort. Cambidge:
Addison-Wesley.
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