Jay: Some comments in-line below:

On 01-Feb-2012 jay ginn wrote:
> Dear Radstatisticians,
>
> A mathematician colleague needs stats. advice but doesnt
> use email; hence my intervention. He serves on a committee
> concerned with improving the measure of inflation and has
> done a lot of work on the different ways that RPI and CPI
> are calculated. I believe at least one RSS member is on
> the committee, someone from ONS and some employer and
> TU reps.
>
> Apart from the different baskets of goods/services used
> in the two measures, a major difference is that RPI uses
> an arithmetic mean and CPI a geometric mean. The former
> takes the mean of quantities (sum/n), while the latter is
> sum/nth root.

The description "sum/n" is OK for the arithmetic mean, but
the geometric mean is calculated as "nth root of product".
Thus the geometric mean of {1,2,3,4,5,6} is

  (1*2*3*4*5*6) ^ (1/6) = 720 ^ (0.16666...) = 2.993795...

(i.e. 3.0 to 1 decimal place). The arithmetic mean is

  (1+2+3+4+5+6)/6 = 21/6 = 3.5

Note that the geometric mean is smaller than the artihmetic
mean, and that is a general result for the geometric mean
versus the arithmetic mean of numbers (which should be
non-negative) provided they are not all equal (if they are
all equal then the arithmetic mean equals the geometric mean).

This may be relevant to reasons for preferring the geometric
mean when calculating a price index! (I was not previously
aware that the RPI and the CPI used different definitions
for calculating the mean).

> As my colleague understands it, this means the CPI method
> assumes that changes in future purchasing behaviour (towards
> selecting cheaper goods) can be predicted by price rises
> whereas RPI reflects past purchasing behaviour (which has
> already shifted in response to price changes). He argues
> there is no justification for using the geometric mean since
> both measures take account of behavioural response.

I do not, myself, see any reason why using the geometric mean
should correspond to a predictive property while using the
arithmetic mean should rather reflect a summary of past
behaviour. Both are measures of price change (though calculated
differently), and each could be used for both purposes!

I have had a look at the Wikipedia entry:

http://en.wikipedia.org/wiki/Consumer_Price_Index_(United_Kingdom)

 "Note that unlike the RPI, the CPI takes the geometric mean
  of prices to aggregate items at the lowest levels, instead
  of the arithmetic mean. This means that the CPI will generally
  be lower than the RPI. The rationale is that this accounts
  for changes in consumer spending behaviour as a result of
  differences in price changes amongst products. According to
  the ONS, this difference is the largest contributing factor
  to the differences between the RPI and the CPI.

If we are going to look at differences between these two measures
in terms of what they are most sensitive to, then the story can
get a bit more complicated.

It seems that the RPI is a weighted arithmetic mean of prices
at a particular time, and is compared to its "baseline" value
at a given previous time, and the *percentage* change is
evaluated. If all items in the calculation change by the same
percentage (relative to their baseline values), then the
percentage change in the RPI will also be the same as this.

  (1.1*1 + 1.1*2 + 1.1*3 + 1.1*4 + 1.1*5 + 1.1*6)/6
= 1.1*(1 + 2 + 3 + 4 + 5 + 6)/6 = 1.1*21/6 = 1.1*3.5

so 10% increase.

Likewise, the CPI can be compared to a "baseline" value, and
the percentage change evaluated. And again, if all items change
in price by the same percentage, then the CPI will change by
that same percentage. Example (10% change across the board):

  (1.1*1 + 1.1*2 + 1.1*3 + 1.1*4 + 1.1*5 + 1.1*6)
= (1.1)*(1+2+3+4+5+6)/6 = 1.1*3.5, so 10* increase

  ((1.1*1)*(1.1*2)*(1.1*3)*(1.1*4)*(1.1*5)*(1.1*6))^(1/6)
= ((1.1^6)*(1*2*3*4*5*6))^(1/6)
= ((1.1^6)^(1/6))*((1*2*3*4*5*6)^(1/6))
= 1.1*((1*2*3*4*5*6)^(1/6)), so 10% increase.

However, the sensitivities of the RPI and the CPI to a
percentage change in low values are different from their
sensitivities to a percentage change in high values. Example:

RPI (low end):
  (1.1*1 + 2 + 3 + 4 + 5 + 6)/6 = 3.516667
a 0.476% change from the previous value of 3.5

RPI (high end):
  (1 + 2 + 3 + 4 + 5 + 1.1*6)/6 = 3.6
a 2.857% change from the previous value of 3.5

CPI (low end):
  (1.1*1*2*3*4*5*6)^(1/6) = 3.041731
= (1.1^(1/6))*((1*2*3*4*5*6)^(1/6))
= (101.6012)*((1*2*3*4*5*6)^(1/6))
i.e. a 1.6% increase over the previous value; *and* this
result is the same whether the "1" increases by 10% or
the "6" increases by 10% (or any other item; so we don't
need to do "CPI high end").

Thus the RPI is more sensitive to a given percentage increase
in price for higher-priced items than for lower-priced items,
while for the CPI the effect is the same whatever the price.

> I know that ONS are working on a better measure, especially
> in terms of including housing costs, but they intend to
> continue using the geometric mean and this is what my colleague
> is challenging. He would value some help from a statistician
> on this issue. Could someone either confirm or disagree with
> what he says; and if agreeing, provide a reference he could
> use to support what he says.on the committee?

Well, as I said above, I don't see what differences in the
values and sensitivities of these two methods of evaluation
(such as I exemplified above) may necessarily have to do
with whether either reflects past, or alternatively future,
behaviour by consumers in response to/in anticipation of
changes in prices. Both could be used for either purpose.

As far as consumers' perceptions of price change are concerned,
the more relevant aspect of the RPI/CPI difference probably
lies in their different sensitivities (see above). I would
be tempted to suspect that most consumers are likely to be
more influenced by a given percentage at the high end, since
larger expenditures would take a larger chunk out of their
expenditure margins.

If a Mars bar goes up by 10% from 55p to 60p, then a Mars
bar a day for a month will cost you £1.50 a month extra,
say 0.75% out of a person's monthly food expendture margin
of say £200. On the other hand, if a 75cl bottle of Veuve
Clicquot champagne (say 6 glasses) goes up by 10% from
£35 to £38.50, then a glass of champagne a day (5 bottles
per month) will take 5*£3.50 = £17.50 (8.75%) out of your
monthly margin. So you may cut back on the champagne (or cut
out the Mars bars to help pay for it ... ).

However, perhaps consumers tend to think in percentage
terms anyway, and might not particularly perceive the
difference between the effects on their budgets of Mars
bars or Veuve Clicquot.

On the other hand, if their mortgage repayment went up
by 10% from say £500 to £550, then this could be more
noticeable, and induce abstinence from champagne and
Mars bars altogether. (Though still unlikely to influence
someone who can afford a personal margin of £1000 for food).

However, I am no expert whatever on the psychology and
economics of consumer behaviour. But it would be interesting
to see studies taking account of the kind of effects
I have been looking at above. And maybe such studies have
indeed had an influence on the choice between RPI and CPI.

> best wishes
> Jay

And best wishes to you!
Ted.

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E-Mail: (Ted Harding) <[log in to unmask]>
Date: 01-Feb-2012 Time: 23:16:37
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