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Methods and Materials of Demography  - Incorrect Use Of The Exponential Method

External Links:

Wikipedia - Logarithm article
Wikipedia - Common logarithm (base 10) article
Wikipedia - Natural logarithm (base e) article
Wikipedia  e article
Wikipedia Exponential Function article (includes exponential laws/rules)

OECD - Glossary of statistical terms - Exponential Growth
Geometric Growth

World Bank
- About Data - Data & Statistics - Growth Rates

 

The Exponential Method

 "If you understand exponentials, the key to many of the secrets of the Universe is in your hand." Carl Sagan's Billions And Billions (1997) p.23

Introduction

I think Carl Sagan has a point, and so this article is a modest attempt to show how easy it is to relax our grip on exponentials. They're slippery little suckers, as you'll see.

In this case, it is the exponential method itself that is being examined.

Standing on the shoulders of giants

According to Trefil (2002), we have Sir Isaac Newton's calculus to thank for our understanding of the exponential method:

When the net increase in a population is proportional to the number of individuals, the population will grow exponentially.

...Now, if dN is the net number of individuals added to the population in a time dt, and if there are N individuals all told in a population, then the conditions for exponential growth will be met if

dN = rNdt

The letter r (the constant of proportionality) in the above differential equation is effectively the rate of compound interest. Trefil continues:

Ever since Newton invented the Calculus in the 17th century, we have known how to solve this equation for N, the number of people in the population at any given time....The solution is:

N=N0 e rt

where N0 represents the starting population. 

Thomas Robert Malthus (Footnote No. 5 to Book II Chapter IX of the 6th edition of his Essay On the Principle of Population) used a different exponential method, based on logarithms (base 10). I've re-produced Malthus' example here, with the addition of step numbers, and a few minor changes for readability):

A general formula for estimating the population of a country at any distance from a certain period, under given circumstances of births and mortality, may be found in Bridge's Elements of Algebra, p. 225:

Log. A = log. P + n log. 1 + (m-b) / mb

A representing the required population at the end of any number of years; n the number of years; P the actual population at the given period; 1/m the proportion of yearly deaths to the population; or ratio of mortality; 1/b the proportion of yearly births to the population, or ratio of births. In the present case:

[1] P = 9,287,000; n = 10; m = 47; b = 29 1/2.

[2] (m-b) / mb = 1/79

[3] (m-b) / mb = 80/79

[4] The log. of 80/79 = 0.00546   

[5] \   n log. 1 + (m-b) / mb) = 0.05460

[6] Log. P. = 6.96787 

[7] 6.96787 (from [6])  added to 0.05460 (from [5]) = 7.02247 

[8] 7.02247 is the log of A, the number answering to which is 10,531,000.

The US Census Bureau use an exponential method similar to the one Malthus used, though theirs is based on Natural Logarithms:

[1] r = 100 * ln [ P(t+n) / P(t) ] / n 

where:

The US Census Bureau has got it wrong, as they should state:

[2] ln(r) = 100 * ln [ P(t+n) / P(t) ] / n 

[2a] Thus r = e ln(r)

[2b] But r is the Exponential Factor, or growth ratio. To get the growth rate (let's call it R):

R = r -1

Even though the US Census Bureau method [1]  is commonly used by demographers worldwide, their approach is also flawed, as they incorrectly represent the Natural Logarithm of the annual growth ratio (r) as a real number - see US Census Bureau  - Incorrect Use Of The Exponential Method for more. Essentially, unlike Malthus' use of Bridge's Elements of Algebra ([8]), they forgot to convert a logarithm back into a real number as would be obvious from the corrected US Census Bureau method [2] .

Exponential Growth and Geometric Growth 

Curiously, both  the OECD and the World Bank seem to agree with the fiction that the exponential growth rate can be obtained by the same exponential method as that used by the US Census Bureau:

[1]  r = 100 * ln [ Pn / P1 ] / n 

where:

Both  the OECD and the World Bank have got it wrong, as they should state:

[2]  ln(r) = 100 * ln [ Pn / P1 ] / n 

[2a] Thus r = e ln(r)

[2b] But r is the Exponential Factor, or growth ratio. To get the growth rate (let's call it R):

R = r -1

In fact, all the OECD and the World Bank method [1]  derives is the Natural Logarithm of the exponential factor or growth ratio. In other words, r is a Natural Logarithm and not a real number as per the corrected OECD and the World Bank method [2] .

Both world bodies then go on to state that the geometric growth rate (...applicable to compound growth over discrete periods...) can be derived as follows:

[3] r = exp(100 * ln [ Pn / P1 ] / n) -1

This is fascinating, as in fact the exponential factor of growth is derived as follows:

[4] exponential factor = exp( 100 * ln [ Pn / P1 ] / n)

and the actual exponential growth rate is derived as follows:

[5] r = exp( 100 * ln [ Pn / P1 ] / n) -1 

[6] the previous step can thus be simplified to:

r = exponential factor - 1

Clearly, [3] and [5] are exactly the same algebraic equation. The only difference is one of interpretation. The OECD and the World Bank stipulate that [1] produces the actual exponential growth rate, when in fact it only produces the Natural Logarithm of the exponential factor (or growth ratio). Both world bodies have failed to follow the standard exponential rule that you must convert a Natural Logarithm to a real number precisely as I have done at [4]. 

This result does not have a proper mathematical name, so I've dubbed it the exponential factor of growth. This represent the multiplying factor by which the starting population has grown. For positive growth rates, the exponential factor will be greater than 1. For negative growth rates, the exponential factor will be less than 1. 

[5] (or [6]) is the actual exponential growth rate, so there is no need for something called the geometric growth rate.

I note that both the OECD and World Bank refer to a compound growth model (based on geometric growth rates) being more appropriate than an exponential growth model for economic phenomena. Yet there is no such separate phenomena as geometric growth rates, at least, not as described by these two world bodies. However, I do believe that a growth model based on variable rate compound interest is the key to all growth (for populations, or in the world of finance) - I call this model the Couttsian Growth Model (see Growth Models for more). 

Conclusion

Sometimes humans are clever. They stand on the shoulders of giants and learn from the past. Malthus was one such individual, as witnessed by his excellent use of the exponential method. Sometimes humans are not so clever, and they fall off those shoulders. I hope my article allows us to climb back up again.

Bibliography

Cassell's Laws Of Nature - James Trefil (2002)

Malthus, Thomas Robert, An Essay on the Principle of Population. J. Johnson. 1798. (1st edition) Library of Economics and Liberty.  

Malthus, Thomas Robert, An Essay on the Principle of Population. John Murray. 1826. (6th edition) Library of Economics and Liberty.