Parkinson's Law - a correction

 

From Parkinson's Law: Growth in numbers of administrative officials in the Royal Navy at a time when the actual administrative work to be done was decreasing.

 

 

 

In 1955 The Economist published "Parkinson's Law", also known as "The Rising Pyramid".  The author C. Northcote Parkinson (introduced as Raffles Professor of History at the University of Singapore) claimed to have discovered a scientific law for the growth in the number of administrative officials in government departments over time.

The model is simple: each bureaucrat employs two subordinates every doubling time.  The number has to be at least two since as a single subordinate makes a supervisor appear redundant.  And although the size of the organization doubles whilst the actual work (if any) does not, the work created by the additional channels of communication keeps everyone busy.

Parkinson provides compelling evidence for the theory by showing that two different departments - the Royal Navy and the Colonial Office - both showed the same pattern of an approximately 5% yearly increase, despite the fact that, over the periods studied, the responsibilities of each department were shrinking.  (Strangely, the same phenomenon can often be seen in private industry notwithstanding the supposed efficiency of the market - see David Graeber's book for details.)

Annoyingly, Parkinson let's himself down at the end of the paper when he attempts to derive a formula:

"In any public administrative department not actually at war a staff increase may be expected to follow this formula:

$x = \frac{2k^m + p}{n}$

where $k$ is the number of staff seeking promotion through the appointment of subordinates; $p$ represents the difference between the ages of appointment and retirement; $m$ is the number of man hours devoted to answering minutes within the department; and $n$ is the number of effective units being administered. Then $x$ will be the number of new staff required each year.

"Mathematicians will, of course, realize that to find the percentage increase they must multiply $x$ by 100 and divide by the total of the previous year (y), thus:

$\frac{100(2k^m + p)}{yn}$

"

A quick dimensional analysis identifies this as nonsense, which is such a shame given that the preceding work is so well researched and the qualitative conclusion so completely indisputable.  So let's see if we can put the theory on a sounder mathematical footing.

The exponential nature of the increase tells us

$\frac{dk}{dt} = \alpha k$

for some unknown constant $\alpha$.  If we let $N$ be the total number of generations in an organization then one generation corresponds to $p/N$ years.  But one generation in the Rising Pyramid is also the doubling time, and so

$e^{\alpha\frac{p}{N}} = 2$

Solving this for $\alpha$ gives us

$\alpha = \frac{N\ ln(2)}{p}$

And, noting that $x$ is in fact $dk/dt$ we find the correct formula to be

$x = \frac{N\ k\ ln(2)}{p}$

The second formula (for the percentage annual increase) is then just obtained by dividing by $k$ and multiplying by 100

$\frac{100\ N\ ln(2)}{p}$

Parkinson shows that this figure takes a value of about 5% over a wide range of organizations, and from that we can derive that a typical generation (or doubling time) of about 14 years.

It is hoped that this correction will be of service to operational researchers such as those at the Harvard Business School.