The ‘principle of invariance’ in currency systems: a comment on Caianiello et al.

Abstract

In an often-cited article in this journal, Caianiello et al. (1982. International Journal of General Systems, 8 (2), 81–92) formulate a ‘principle of invariance’ for currency systems. They build on this principle to explain the distribution of the coins and banknotes in circulation over the different denominations, and analyse how a currency system adjusts to inflation. This paper shows that Caianiello et al.'s distribution law is flawed because the principle of invariance is false.

Keywords:

• currency denominations
• principle of invariance
• hierarchical self-organizing systems
• refining transformations

Acknowledgements

We are grateful to Steven Vanduffel and the three referees for helpful comments.

Notes

 1. Email: [email protected]

 2. Email: [email protected]

 3. Note that Caianiello et al. use the term ‘monetary systems’, which we think is too broad.

 4. For the case where overpayment and the return of change are allowed, see Van Hove and Heyndels (1996) and Van Hove (2001).

 5. The value of the largest denomination is fixed in order to compare denominational structures over identical intervals.

 6. Note that Hentsch does not specify the factor of proportionality.

 7. As an aside, while many of the circulation figures per denomination are – when transferred into logs – to a greater or less extent aligned along a line with a slope of 0.5, Caianiello et al. fail to check whether the intercept is also in accordance with the distribution law that they have derived. Indeed, given Equation (2), , so that the intercept should equal .

 8. Caianiello et al. assume that the velocities of circulation of all tokens equal 1, which is a particular case of ours. Indeed, they seem to overlook that there is a difference between the number of tokens exchanged in a given period (the flow) and the number of tokens in circulation at a given point in time (the stock). As a matter of fact, in Section 2 of their article (where they derive formulas for the flows), they define N as ‘the minimum number of coins which are necessary to obtain all the integers between zero and ’ (op. cit. p. 84). In Section 3 (where they reason in terms of stocks), they use the same symbol to designate ‘the total number of elements’ in the system; that is, in circulation (o.c., p. 85). [As will be shown below, this confusion lies at the heart of the fatal error in their proof.] The gap between their Sections 2 and 3 can only be closed by assuming that the velocity of circulation of all tokens is 1; that is, that all tokens are used once and only once in the period considered.

 9. Note that Caianiello et al. do not demonstrate that equals ; they simply state it.

10. Note that this is where Caianiello et al. go wrong. In Equation (1), on p. 86, they (implicitly) make the jump from N as designating the number of tokens exchanged to N as shorthand for the number of tokens in circulation; see footnote 8 on this. The same equation is also the starting point of their proof that the mean value remains unchanged under a p-refinement. Unfortunately, they use the same N in their expressions for both and and thus in effect only demonstrate that the value of the coins and notes exchanged remains the same – which is self-evident; see the main text.

Additional information

Notes on contributors

Y. Bouhdaoui

1

D. Bounie

2