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What Is Gibrat’s Law?

Growth patterns in nature, economics, and social systems often follow surprisingly predictable paths. One of the most important principles governing these patterns is Gibrat’s Law, which explains why so many phenomena end up following lognormal distributions rather than the familiar bell curve of normal distributions.

Gibrat’s Law states that growth rates don’t depend on current size. Whether something is large or small right now has no bearing on how fast it grows next. This appears in many different phenomena, from city populations and company revenues to biological organisms and word frequencies in languages. Robert Gibrat, a French statistician, formalized this concept in the 1930s while studying economic data. The underlying principle shows up throughout natural and social systems.

The Mathematical Foundation

When growth rates are independent of current size, we can express this mathematically. If we have a variable X at time t and it grows by a random proportion each period, we can write:

where ε represents the random growth rate, which doesn’t depend on the current value of X(t). Taking the logarithm of both sides gives us:

This reveals why lognormal distributions emerge. When you multiply many independent random variables together (like repeated growth factors), their product tends toward a lognormal distribution. This happens because multiplying the original values is equivalent to adding their logarithms.

If log(X) is normally distributed, then X itself equals e^(log(X)), which is the exponential of that normal distribution. This exponential transformation stretches the symmetric bell curve into the characteristic right-skewed lognormal shape, bunched up on the left with a long tail extending right.

A Concrete Example: Following One Company Over Time

Let’s track a hypothetical company starting with $100,000 in revenue. Each year, it grows by a random percentage. For example, sometimes 10%, sometimes 5% or 15%, and sometimes -3%. Here’s what happens:

• Year 1: \$100,000 × 1.10 = \$110,000 (10% growth)
• Year 2: \$110,000 × 1.05 = \$115,500 (5% growth)
• Year 3: \$115,500 × 1.15 = \$132,825 (15% growth)
• Year 4: \$132,825 × 0.97 = \$128,840 (-3% decline)

Now, if we track the logarithm of revenue instead (using natural logs, base e):

• log($100,000) = 11.51
• log($110,000) = 11.61 (added 0.10)
• log($115,500) = 11.66 (added 0.05)
• log($132,825) = 11.80 (added 0.14)

The logarithms grow by adding small increments. Instead of multiplying revenue by growth factors, we’re now adding growth increments to the logarithms. Specifically, we’re adding log(1+ε) values each period. When the growth rates ε are small, log(1+ε) ≈ ε, so we’re approximately adding the growth rates themselves.

According to the Central Limit Theorem, when we add many independent random variables with finite variance, their sum approaches a normal distribution. Since log(revenue) is the sum of many log(1+ε) terms, log(revenue) becomes normally distributed over time. When we convert back to actual dollar amounts using the exponential function, we get the characteristic right-skewed lognormal shape.

Now imagine we have 1,000 companies that all started with different amounts and grew according to this same process. After many years, the distribution of their final sizes will be lognormal. Most companies will be relatively small, but a few will have grown to be much larger due to a series of good growth years.

Where Gibrat’s Law Shows Up

Gibrat’s Law can be observed in many scenarios:

City Populations: Large cities don’t grow faster than small cities in percentage terms. A city of 100,000 might grow by 2% in a year, while a city of 1 million might also grow by 2%. This proportional growth pattern helps explain why city size distributions are often close to lognormal.

Company Sizes: Research shows that company growth rates are independent of current company size. Small startups and large corporations can both experience high or low growth rates, leading to a lognormal distribution of company revenues and employee counts across industries.

Biological Systems: Cell division, population growth, and protein concentrations in cells often follow patterns consistent with Gibrat’s Law, resulting in lognormal distributions of biological measurements.

Income Distributions: While income distributions are complex and often follow power-law patterns in the upper tail, the middle portions exhibit lognormal characteristics due to proportional growth effects in career advancement and investment returns.

Practical Implications

Recognizing when Gibrat’s Law applies has practical consequences. If you’re analyzing data that results from proportional growth processes, assuming a normal distribution can lead to incorrect conclusions about probabilities, risks, and typical values.

For business planning, understanding that company growth follows Gibrat’s Law suggests that current size alone doesn’t predict future growth rates. This insight influences investment strategies and market competition analysis. In policy making, recognizing lognormal patterns in urban growth helps city planners understand why some cities become much larger than others, even when starting from similar conditions.

Gibrat’s Law explains why lognormal distributions appear across different fields. Proportional growth processes naturally create these skewed distributions, giving us insight into everything from economic inequality to biological diversity. When you encounter highly skewed data with a long right tail, the underlying process might involve proportional growth governed by Gibrat’s Law.