# Technical appendix to ‘How much have UK academics lost…’

This is a technical appendix to my earlier post ‘How much have UK academics lost in real terms?’.

[**UPDATE:** See this commentary on Jevons versus Carli in the Royal Statistical Society’s “Response to Department for Work and Pensions consultation on Security and Sustainability in Defined Benefit Pension Schemes”: ‘While more robust than the Carli it [Jevons] can underestimate in certain circumstances, albeit its underestimation tends to be much less severe than Carli’s overestimation.’]

Here’s a link to the expert advice from Erwin Diewert on which the UK Statistics Authority drew in reaching their conclusion that RPI should no longer be recognised as a national statistic and that a new index — RPIJ — should be published, which merits such recognition.

These are the most relevant excerpts from Diewert’s note:

2. How Can a Jevons Index be Justified if a COGI [Cost of Goods Index] is the Target Index?

…Our reason for taking the geometric mean … is that taking the geometric mean leads to an index that satisfies the importanttime reversal testas applied to elementary indexes…. [T]he Jevons index [which takes the geometric mean] has an additional important property that none of the other elementary index formulae in common use possess: namely it satisfies the followingtransitivityorcircularitytest:[1]

(10) P(p⁰,p¹)P(p¹,p²) = P(p⁰,p²) for all strictly positive p⁰, p¹ and p².

…An important implication of an index that satisfies the circularity test is this: if the index is chained over time (which is the case for both the RPI and CPI) and a base period price vector is repeated at some future period, then the chained index will correctly indicate that no price change has occurred…. [T]he Jevons index possesses more desirable properties (i.e., satisfies more desirable tests) than competing elementary indexes[2]and hence seems to be a reasonable choice as an elementary index.

3. Why is the Failure of the Carli Index to Pass the Time Reversal Test so Important?

Satisfaction of thetime reversal testfor an elementary index, P(p⁰,p¹), can be written as follows:

(11) P(p⁰,p¹) P(p¹,p⁰) = 1 for all strictly positive p⁰ and p¹.

The equation (11) has the following interpretation. Use the elementary index to compute (one plus) the rate of price change P(p⁰,p¹) that results from the period 0 price vector p⁰ changing to the period 1 price vector p¹. Now suppose that the period 2 price vector reverts back to the period 0 price vector p⁰ and compute (one plus) the rate of price change P(p¹,p⁰) that results from this change. The product of the two price changes, P(p⁰,p¹) P(p¹,p⁰), should equal 1 to indicate that no overall price change has taken place between periods 0 and 1. Here is the problem with the Carli formula: not only does not satisfy (11) but itfails(11) with the following definite inequality:

(12) Pc(p⁰,p¹)Pc(p¹,p⁰) > 1

unless the price vector p¹ is proportional to p⁰ (so that p¹ =λp⁰ for some scalarλ> 0), in which case, (11) will hold. The main implication of the inequality (12) is that the use ofthe Carli index will tend to give higher measured rates of inflationthan a formula which satisfies the time reversal test (using the same data set and the same weighting). There are numerous empirical examples of this upward bias in the Carli formula, that start with the numerical results in Fisher (1922).[3]The upward bias can be substantial if monthly chained Carli indexes are used.[4]

Fisher (1922; 66 and 383) was the first to establish the upward bias of the Carli index[5]and he made the following observations on its use by statistical agencies:

“In fields other than index numbers it is often the best form of average to use. But we shall see that the simple arithmetic average produces one of the very worst of index numbers. And if this book has no other effect than to lead to the total abandonment of the simple arithmetic type of index number, it will have served a useful purpose.” Irving Fisher (1922; 29–30).

It took 70 to 80 years before Fisher’s advice was followed by major statistical agencies around the world.[6]

[1]The term circularity test is due to Fisher (1922; 413).

[2]See Diewert (1995) (2012; 36–39) on the test approach to elementary indexes.

[3]In more recent times, the empirical results in Szulc (1983) (1987) were extremely influential, leading Statistics Canada to abandon the use of the Carli formula in 1978 for the Dutot and then later for the Jevons formula. Other statistical agencies eventually followed this example.

[4]See the examples in the ONS (2012b) and the example in Appendix D in particular which deals with the price bouncing behavior that was first emphasized by Szulc (1983) (1987).

[5]See also Szulc (1987; 12) and Dalén (1992; 139). Dalén (1994; 150–151) provides some nice intuitive explanations for the upward bias of the Carli index.

[6]Evans (2012) shows that the UK is the only European Union country that uses the Carli index in its national CPI. Of course, the Eurostat mandated HICP has ruled out the use of the Carli index from its inception.