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Gibrat distribution

From phenomenon to model

Robert Gibrat proposed a proportional-growth explanation for economic sizes in 1931. The law of Gibrat has been studied for firms, cities, and other units that grow multiplicatively.

What to look at

Its support starts at loc and extends to infinity. scale acts on the logarithmic scale in the implemented model; comparisons with other texts require checking conventions.

Useful relationships

It is essentially a lognormal parameterization with location. Normal describes accumulated log-growth; Pareto and Dagum often complement the upper tail of size and income distributions.

Names you may also encounter

It is also called the Gibrat distribution or Gibrat lognormal. The law of Gibrat is a growth hypothesis, not merely a density name.

When it makes sense

Gibrat gives a generative hypothesis: many small independent growth rates produce log-scale dispersion. If growth systematically depends on size, the interpretation fails even if the curve fits.

What not to assume

Do not infer proportional growth from an observed lognormal: the same shape can arise from other multiplicative mechanisms.

The growth law that became Lognormal

Robert Gibrat proposed that proportional growth of a firm could be independent of its size. If many growth factors multiply over time, logarithms add and final size may approach a Lognormal law. The Gibrat distribution is consequently a lognormal parameterization.

The law of proportionate effect is an economic hypothesis, not a universal mathematical identity. Small and large firms may have different growth rates, volatility, and exit probabilities. Selection among survivors also distorts the observed distribution. Fitting Gibrat describes sizes compatible with multiplicative growth, but testing the law requires longitudinal growth data rather than a single cross-sectional snapshot of sizes.

Decision guide

A good candidate when: a positive size is compatible with many multiplicative growth factors and the model’s fixed log-scale dispersion is plausible.

Compare it with: Lognormal with free shape and Pareto for the tail. A compatible cross-sectional distribution does not prove proportional growth; longitudinal data are required.

References

Gibrat Distribution: equations and calculator

Distribution defintion

X∼Gibrat(Loc,Sc)X\sim\mathrm{Gibrat}\left(\text{Loc},\text{Sc}\right)

Distribution domain

x∈(Loc,∞)x\in\left(\text{Loc},\infty\right)

Parameters domain and parameters constraints

Loc∈R,Sc∈R+\text{Loc}\in\mathbb{R},\text{Sc}\in\mathbb{R}^{+}

Cumulative distribution function

FX(x)=Φ(ln⁡x)=12(1+erf(ln⁡z(x)2))F_{X}\left(x\right)=\Phi\left(\ln x\right)=\frac{1}{2}\left(1+\mathrm{erf}\left(\frac{\ln z(x)}{\sqrt{2}}\right)\right)

Probability density function

fX(x)=1Sc1x2πexp⁡(−12(ln⁡z(x))2)f_{X}\left(x\right)=\frac{1}{\text{Sc}}\frac{1}{x\sqrt{2\pi}}\exp\left(-\frac{1}{2}\left(\ln z(x)\right)^{2}\right)

Percent point function/Sample

FX−1(u)=Loc+Sc×exp⁡(Φ−1(u))F^{-1}_{X}\left(u\right)=\text{Loc}+\text{Sc}\times \exp\left(\Phi^{-1}\left(u\right)\right)

Non-central parametric moments

μk′=E[Xk]=∫Loc∞xkfX(x)dx=exp⁡(k22)\mu'_{k}=E[X^k]=\int_{\text{Loc}}^{\infty}x^{k}f_{X}\left(x\right)dx=\exp\left(\frac{k^{2}}{2}\right)

Parametric mean

Mean(X)=Loc+Sc⋅μ~1′=Loc+Sc⋅e\mathrm{Mean}(X)=\text{Loc}+\text{Sc}\cdot\tilde{\mu}'_{1}=\text{Loc}+\text{Sc}\cdot\sqrt{e}

Parametric variance

Variance(X)=Sc2⋅(μ~2′−μ~1′2)=Sc2[e2−e]\mathrm{Variance}(X)=\text{Sc}^{2}\cdot(\tilde{\mu}'_{2}-\tilde{\mu}'^{2}_{1})=\text{Sc}^{2}\left[e{^2}-e\right]

Parametric skewness

Skewness(X)=μ3′−3μ2′μ1′+2μ1′3(μ2′−μ1′2)1.5=e−1(2+e)\mathrm{Skewness}(X)=\frac{\mu'_{3}-3\mu'_{2}\mu'_{1}+2\mu'^{3}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{1.5}}=\sqrt{e-1}\left(2+e\right)

Parametric kurtosis

Kurtosis(X)=μ4′−4μ1′μ3′+6μ1′2μ2′−3μ1′4(μ2′−μ1′2)2=e4+2e3+3e2−3\mathrm{Kurtosis}(X)=\frac{\mu'_{4}-4\mu'_{1}\mu'_{3}+6\mu'^{2}_{1}\mu'_{2}-3\mu'^{4}_{1}}{(\mu'_{2}-\mu'^{2}_{1})^{2}}=e^{4}+2e^{3}+3e^{2}-3

Parametric median

Median(X)=Loc+Sc×exp⁡(Φ−1(1/2))\mathrm{Median}(X)=\text{Loc}+\text{Sc}\times \exp\left(\Phi^{-1}\left(1/2\right)\right)

Parametric mode

Mode(X)=Loc+Sce\mathrm{Mode}(X)=\text{Loc}+\frac{\text{Sc}}{e}

Additional information and definitions

Loc:Location parameter\text{Loc}:\text{Location parameter}
Sc:Scale parameter\text{Sc}:\text{Scale parameter}
z(x)=(x−Loc)/Scz\left(x\right)=\left(x-\text{Loc}\right)/\text{Sc}
u:Uniform[0,1] random varibleu:\text{Uniform[0,1] random varible}
Φ(x):CDF normal standard distribution\Phi\left(x\right):\text{CDF normal standard distribution}
Φ−1(x):PPF normal standard distribution\Phi^{-1}\left(x\right):\text{PPF normal standard distribution}
erf(x):Error function\mathrm{erf}(x):\text{Error function}