Economics of Small Business

Eighth Week

Gibrat’s Law

Distribution and Inequality

• Vilfredo Pareto discovered that the skewed distribution of income in different countries follows the same mathematical form.

• This is called the “Pareto Distribution.” • In other applications it is also called the Zipf,

Mandelbrot, or Bradford distribution. • Robert Gibrat (1904–1980) investigated the

distribution of the sizes of firms.

Random Growth

• Gibrat, writing in 1931, argued that the growth rate of a firm is a random variable, independent of the size of the firm.

• That does not mean that the distribution of firm sizes does not change.

• A firm that (randomly) grows faster in one period will have a larger base, and so grow more on the average, in the next period.

Lognormal

• This will approach a “lognormal” distribution

• Where Z is a normal distribution, that is

• Like the other distributions we have discussed, the lognormal is highly skewed.

Example from the Reading

Lognormal Distribution

Here is a lognormal distribution of firm sizes out of 10,000 firms, with ¾ no-employee firms. (This is zero-inflated; that is, X is the size of the firm plus one – since the lognormal would never show a value of zero.)

Test

• Observations of the distribution of firm sizes showed that it does indeed approximate a lognormal distribution pretty closely.

• This was taken as evidence in favor of “Gibrat’s Law.”

• That seemed to settle the matter for several decades.

Pareto Distribution

Here is a Pareto distribution constructed in the same way.

Negative Binomial Distribution

Here is a negative binomial distribution constructed in the same way (except that it is not zero-inflated – the negative binomial can take a zero value.)

Not Conclusive

• We see that the lognormal distribution, the Pareto and negative binomial distributions look very similar.

• Thus, evidence that the distribution of firm sizes is approximately lognormal does not give very strong evidence for “Gibrat’s Law.”

• Which distribution are we observing?

Revisit

• In the later twentieth century, “Gibrat’s law” was revisited, with mixed results.

• It is true that a large component of the rate of growth of firms is random.

• However, in more recent studies, with better data, there is a good deal of evidence that “Gibrat’s Law” is not true in general.

Deviations 1• With some exceptions studies of various industries

in various countries, and internationally, tend to show that1. Smaller firms tend to grow faster than larger (but

recall the regression fallacy.)

2. Younger firms tend to grow faster than older

3. There is some autocorrelation – firms that grow faster in one year are likely also to grow faster in the following year.

4. Growth is more predictable – with a smaller standard deviation – in larger firms.

Deviations 2

• Each of these points contradicts “Gibrat’s Law.”• However, for the most part, the deviations are

small. • “Gibrat’s Law” may hold, to a very good

approximation, for large, mature, well-financed firms.

• Most early studies focused mostly on firms of that kind, and this may explain some of the difference.

Survivors

• Here is another issue: some firms cease to exist in any given period. How should they be treated? 1. Count them with a growth rate of -1?

2. Apply “Gibrat’s Law” only to survivors?

• The first option would present some statistical difficulties!

• However, the second implies some bias, since the evidence clearly shows that the probability of exit decreases with firm size and age.

Entry and Survival

• The survival, growth and performance of new entries is an important issue in this literature.

• Results are mixed here. – Younger small firms are less likely to survive– Some studies find that firms that are larger at

their startup are more likely to survive, some not.

Differences by Industry

• Some studies find differences by industry, both in the long term and especially for the survival of start-ups.

• Some unsurprising results: in recent years – firms have been more likely to survive, and grow

rapidly, if they are innovative, capital intensive, and in a growing industry;

– more likely to grow rapidly, but less likely to survive, if there are economies of scale.

Interim Conclusion

• Scanning many studies, it now seems pretty clear that 1. To the extent that “Gibrat’s Law” is

approximately true, it applies to larger, mature firms that survive from one period to the next.

2. For SMEs, there are important deviations.a. Smaller firms grow faster

b. Younger firms grow faster

Further Studies 1

• A recent study finds also that 1. There are some predictable differences between

“high-tech” and other firms.

2. Cash flow and internal finance have positive effects on growth and survival, especially for young firms.

3. Debt has positive, but the flow of interest negative, impacts on small firm growth, especially for young firms.

• These results suggest that asset constraint is important for young and small firms.

Further Studies 2

• The same study finds also that labor productivity and R&D have somewhat complicated impacts.Labor productivity has a greater impact on the growth

of older firms. R&D is positive in general, but a combination of R&D

and a debt burden can be a negative influence on growth, especially in high-tech firms.

• This study was on Portugese data. Other studies of Netherlands, British, and worldwide pharmaceutical industries find some similar deviations.

Common Sense and Theory

• Much of this seems pretty common-sense. • It also fits with some basic economic theory –

with one important twist:Start-up small firms that survive grow steadily to the

minimum efficient size. However, a barrier to this growth is that they may face

limited access to capital to support the growth.Once they attain minimum efficient scale, “Gibrat’s

Law” may be approximately correct.

The Reading

• Your reading for this week reinforces this view, again using data from Portugal.

• It contrasts data on a group of larger, more mature firms (that are legally required to report financial data) with data for all employers, including the smaller.

• For the mature firms, the distribution appears lognormal; but for the bigger set it deviates from the lognormal, as we saw before.

Example, Again

Age 1

• They do not have a measure of firm age.

• However, they have the tenure of the longest-employed employee!

• This sets a lower bound to the age of the firm.

• They use this to group firms in approximate age categories.

Age 2

• Using this, they offer further evidence that the age of the firm makes a difference.

Age 2

• This also applies to a group of survivor firms that entered in the same year, as they grow older.

Details

• These curves are computed from the data on the number of firms near each size measure, smoothed in a pretty simple way.

• Since the second diagram only looked at survivor firms, we can be sure that the changing shape of the distribution is not a consequence of firms dropping out, but of different changes in the size of the surviving firms.

Heterogeneity 1

• The paper borrows part of a model due to Nobel Laureate Robert Lucas, that assumes that business founders are not all alike, but instead are heterogeneous.

• Each individual is characterized by a measure of her or his managerial efficiency, .

• Lucas tells us that if is less than some lower limit, the person will not start a business but instead look for a job.

Heterogeneity 2

• The Cabral and Mata paper uses this in a slightly different way.

• They assume that the “optimal” firm size for a founder depends on her or his efficiency .

• Thus it is written as s*().• Probably more efficient managers prefer larger

firms, ceteris paribus. • The distribution of optimum firm sizes is

presumably lognormal.

Wealth

• The size of a new firm will be limited by the founder’s wealth (both as direct investment and collateral for loans)

• The wealth of a potential business founder is represented by w(z,).

• The vector z represents the observable characteristics of the founder, and is a random variable.

Minimum

• Thus, at the time of founding, the size of the firm is the minimum of s*(and the limit set by the founder’s wealth, w(z,).

• As time goes on, however, fewer and fewer of the surviving firms are limited by the proprietor’s assets.

• Thus the long-run distribution is the distribution of optimal sizes – approximately longnormal

Wealth, Again

• Unfortunately, they did not have data on wealth. • They considered age and education as proxies for

wealth. • Their statistical analysis suggested age is better:

consistently with their model, founders’ age seemed to impact younger firms but not older.

• (Are they stacking the debt a little, here?)

Wealth and Founder’s Age

• Older founders may have more wealth because they have had more time to save.

• We also have evidence from another source that persons who have received cash gifts or inheritances are more likely to start businesses than others.

• Inheritance, at least, is likely to be correlated with age.

Modeling

• To model this, they assume1. that the wealth limit increases with the square of the

founder’s age,

2. and is lognormally distributed, for a given age;

3. that the scale after seven years is the optimal scale;

4. That the probability of asset constraint is inversely proportional to the founder’s age.

• The years for this study are 1984 and 1991.

Wealth Limit 1

• Thus the wealth limit is written as

• where ai is the founder’s age, i is normally disbributed and is an arbitrary constant.

Wealth Limit 2

• Taking logarithms, we have

• So we see that the wealth limit is distributed lognormally, given the age of the founder.

• The standard deviation of i is , another arbitrary constant.

Probability

• The probability of an asset constraint is modeled as

• Where is an arbitrary constant.• The idea is that as the founder gets older, the

probability increases that he can raise enough investment capital for an efficient scale of the business.

Calibration

• We want to choose values of and so that the model comes close to the facts.

• This is called calibration of the model.

• In econometrics, we use statistical procedures to do that.

• However, the authors instead use a trial-and-error approach.

Simulation

• For a computer simulation of a partly random process, we rely on a kind of program called a pseudorandom number generator.

• Such a program gives a series of numbers distributed like a particular distribution – normal, for example.

Simulated Entry 1

1. Choose trial values for , and .

2. Using a pseudorandom number generator with standard deviation , choose a trial value of i.

3. Using a pseudorandom number generator and some algebra, choose a trial value of ai corresponding to the actual distribution of ages.

4. Compute

Simulated Entry 2

5. Again using a pseudorandom number generator choose a trial value of s* that corresponds to the actual distribution in the later year.

6. Yet again using a pseudorandom number generator, replace with s* with probability

7. This gives us a starting scale for a hypothetical firm with those constants.

Simulated Entry 3

8. Wash, rinse, repeat!• The authors used that loop to generate 515

simulations for each of 2000 sets of arbitrary constants .

• They chose constants that gave distributions that best matched the distribution of their 515 observations from 1984 and 1991, statistically speaking.

The Fit

Double-Check

• To measure the explanatory power of their model, the authors compare it to another model that makes the 1984 and 1991 sizes proportional.

• The calculate a measure of improved fit somewhat like R2, which would be the proportion of the observed variation that is explained by the model.

• On that basis, they say that the model explains between 70% and 75% of unexplained variation.

• In the appendix, they present some other data to argue that Portugal is not a special case.

Reflections

• How much confidence can we put in the conclusions of this reading?

• If it were the only evidence we have, it would be persuasive, but not conclusive.

• It is a clever piece of work, but that in itself is a worry!

• It is one stone in the balance. • But, as we have seen, we have a considerable

number of other stones in the same dish.

Overall Conclusions

• On the evidence to date, we can be reasonably sure that 1. “Gibrat’s Law” may be applicable to large,

mature firms, but not to small firms in general nor, especially, to startups.

2. Asset constraint and limited access to finance is one important reason for this.