Arthur,

I can't find your file attachment, but your first example is clear enough with it. I take your point. The expenditure would be the same after A had fallen in price and B had increased in price so shouldn't we favour the formula that shows no change in price (Carli) over the one that shows a decrease (Jevons) If that was all there was to it, I would agree with you. But, as we both know, the RPI is a chain index with links every January. So the test that really matters is how well do the two formulas behave when they are chained. The answer is that the Jevons formula behaves very well and the Carli formula doesn't behave well at all.

Below I have shown what happens if the prices return to their level in period 0 in period 2. The link relative for the Jevons index for period 2 is just the reciprocal of its link relative for period 1, which you find unreasonable, so the chain index for period 2 with a period 0 base equals 1.0, as one would expect since the prices of A and B are the same in both periods.

The link relative for the Carli index for period 2 is 1.333. The situation going from period 1 to period 2 (identical to period 0) is comparable to the one going from period 0 to period 1 in the sense that the buyer can buy a unit of A and a unit of B for £2 in both periods. But whereas going from period 0 to period 1 the Carli index showed the nice link value of 1.0, so intuitively plausible, going from period 1 to period 2 it takes a value of 1.333. The difference is easily explained. Implicitly, the Carli links are what you would expect a Laspeyres price index to be if the quantities were inversely related to the prices. In the case of the period 0 to period 1 comparison the two items have the same price so they are given the same weight. In the case of the period 1 to period 2 comparison, item A's price is only a third that of item B's, so it is weighted much more heavily by the Carli index than B, leading to the big increase in the Carli link.

So although prices are the same in period 2 as in period 0, the chain Carli index shows that prices have risen by a third. This is why one national statistical agency after another has moved away from the Carli formula.

Possibly in certain special circumstances, its use could be defended. Arguably, it might be a suitable formula if the Carli index is the byproduct of pps samplng based on expenditures, and a new sample is drawn every year or every time the elementary aggregates are linked. You have convinced me that perhaps the total ban on the Carli formula by Eurostat is excessive. But is there anyplace in the existing RPI or CPI where the situation that I have just described holds. I rather doubt that there is.

| Prices | | |

| 0 | 1 | 2 |

A | 1 | 0.5 | 1 |

B | 1 | 1.5 | 1 |

| Link Relatives | |

| 0 | 1 | 2 |

A | 1.000 | 0.500 | 2.000 |

B | 1.000 | 1.500 | 0.667 |

Carli | 1.000 | 1.000 | 1.333 |

Jevons | 1.000 | 0.866 | 1.155 |

| Chain Index | |

Carli | 1.000 | 1.000 | 1.333 |

Jevons | 1.000 | 0.866 | 1.000 |

Original Message:

Sent: 03-08-2013 11:34

From: Arthur Barnett

Subject: The need for an uprating index

Jill,

My view on using the "same aggregation formulae" for indices differs from yours and possibly Gareth's.

I am reluctant to support statistics that potentially hide uncertainty and give the impression that statistical methodology is better than it is.

The formulae conventionally used all have their problems and therefore provide one of the dimensions of the uncertainty associated with price indices.

I am a little wary to offer simple illustrative examples as they can be over interpreted but hopefully the attached spreadsheet might be useful.

The examples are simplistic and far removed from the real world but hopefully illustrate a couple of points.

What the examples do not do, and are not intended to do, is to suggest Carli is superior to Jevons or vice versa. The intention is just to offer another illustration from a layman's perspective that both produce questionable results in different circumstances.

Basically the examples I have chosen show what can happen to arithmetic and geometric means of price relatives with symmetric additive changes in prices. I would argue that a layman would reasonably consider that these symmetric changes in price to cancel each other out and that "inflation" is unchanged because the aggregate cost of the two items remains the same in both time periods.

The simplest example is the top left hand one.

In this case there are 2 items A and B each with a price of £1 at time T1. The price of A decreases by 50p and B increases by 50p at time T2. Therefore the cost of the two items remains at £2 in total at time T2 the same as at T1.

The arithmetic mean of the price relatives is 1 which indicates no price change whereas the geometric mean is 0.87 indicating a fall in price which is arguably counter intuitive.

The other two examples on the left hand side vary the price of one of the items at T1 to £2. These examples show that the two averages can both imply consistent increases or decreases in "inflation". Again I would argue that the layman would again expect no change in "inflation" with these examples.

The first 3 examples illustrate how both means can give undesirable results. (As an aside the numbers can easily be changed so that the geometric mean returns a mean price relative of 1 - change the initial value of B to 50p in the third example.)

Now it can be argued that the price changes of 50p are untypically large as they equate to things like sale prices or two for one offers. The calculations to the right show what happens when the price change is reduced. The patterns remain the same but differences reduce. By the time the price change reduces to 1p there is very little difference in practice between all the calculations of averages.

These additional calculations illustrate that although there may be theoretical problems with both methodologies they may not always represent problems in practice depending upon the data. (As a further aside not all possible simple aggregation formulae will be produce undesirable results with these examples but such formulae are likely to demonstrate problems in other circumstances.)

And finally this brings me to a more general point. One of my concerns with price index methodology is the mathematics and the way it is used. With my use of simple examples I am potentially guilty of generalising from the particular but I hope that I have made it clear that that was not my intention. In general it seems to me that proponents of different methodologies are prone to selecting particular isolated mathematical results to support their case. In their defence it may be because there is no comprehensive body of supporting mathematics to describe how different functions will behave in different circumstances, or the mathematics exists but in practice is too difficult to access.

This is not to imply that mathematics has the answers which it clearly does not because the judgement of various disciplines is likely to dominate in describing such complex phenomena. However those judgements need to be based on a sound mathematical foundation.

Arthur

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Original Message:

Sent: 21-06-2013 10:51

From: Jill Leyland

Subject: The need for an uprating index

An issue I have been thinking about for a while is the distinction I believe exists between a consumer price index designed for uprating purposes and one for macroeconomic purposes. I attach a paper on the subject. This is intended as a think piece that will, I hope, stimulate debate, not as anything definitive. Comments are welcome!