Chapter 4 - Price index theory

4.9 In presenting index number formulas a simple starting point is to compare two sets of prices. Consider price movements between two time periods, where the first period is denoted as period 0 and the second period as period t (period 0 occurs before period t). In order to calculate the price index, the quantities need to be held fixed at some point in time. The initial question is what period should be used to determine the quantities. The options are to use:

(i) The quantities of the first (i.e. earlier) period. 

This approach answers the question ‘how much would it cost in the second period, relative to the first period, to purchase the same quantities of items as purchased in the first period’. Estimating the cost in the second period’s prices simply requires multiplying the quantities of items purchased in the first period by the prices that prevailed in the second period. A price index is obtained from the ratio of the revalued cost using the second period prices to the cost in the first period. This approach was proposed by Laspeyres in 1871 and is referred to as a Laspeyres price index. It may be represented, with a base of 100.0, as:

\((4.1) \space I_{Lt} =\frac{\sum{p_{it}}q_{i0}}{\sum{p_{i0}}q_{i0}} \times 100\)

where:
\(p = price \\ q = quantity \\ i = item. \)

(ii) The quantities of the second (or more recent) period. 

This approach answers the question ‘how much would it have cost in the first period, relative to the second period, to purchase the same quantities of items as purchased in the second period'. Estimating the cost in the first period simply requires multiplying the quantities of items purchased in the second period by the prices prevailing in the first period. A price index is obtained from the ratio of the cost in the second period to the revalued cost using the first period’s prices. This approach was proposed by Paasche in 1874 and is referred to as a Paasche price index. It may be represented, with a base of 100.0, as: 

\((4.2) \space I_{Pt} =\frac{\sum{p_{it}}q_{it}}{\sum{p_{i0}}q_{it}} \times 100\)

(iii) A combination (or average) of quantities in both periods. 

This approach tries to overcome some of the inherent difficulties of using quantities of items fixed at either point in time. In the absence of any firm indication that either period is the better to use as the base or reference, then a combination of the two is a sensible compromise. In practice this approach is most frequent in:

a) the Fisher Ideal price index, which is the geometric mean of the Laspeyres and Paasche indexes: 

\((4.3) \space I_{Pt} =(I_{Lt}I_{Pt}) ^{\frac{1}{2}}\)

b) the Törnqvist price index, which is a weighted geometric mean of the price relatives where the weights are the average shares of total values in the two periods, that is: 

\((4.4) \space I_{Tt}=\prod_\limits{i}\big(\frac{p_{it}}{p_{i0}}\big)^{s_{i}}\)

where \(s_{i}=\frac{1}{2}\big(e_{i0}/\sum e_{i0}+e_{i1}/\sum e_{i1}\big)\) is the average of the expenditure shares for the \(i^{th}\) item in the two periods.

4.10 The Fisher Ideal and Törnqvist indexes are often described as ‘symmetrically weighted indexes’ because they treat the weights from the two periods equally.

4.11 All price indexes in the ABS use the Laspeyres price index formula. This is largely because the weights used are held fixed at some earlier base period. On the other hand the Paasche, Fisher Ideal and Törnqvist formulas all require current period weights. Timely information of this nature is often very difficult to obtain.

4.12 It should be noted that the different index formulas produce different index numbers and thus different estimates of price movement. Typically the Laspeyres formula will produce a higher index number than the Paasche formula, with the Fisher Ideal and the Törnqvist index numbers falling between the two. In other words the Laspeyres index will generally produce a higher measure of price increase (or lower decrease) than the other formulas and the Paasche index a lower measure of price increase (or larger decrease).

4.13 Only the Laspeyres price index formula will be considered from this point. Further information regarding price index theory can be found in Consumer Price Index: Concepts, Sources and Methods, 2011 (cat. no. 6461.0).

4.14 The Laspeyres formula above is expressed in terms of quantities and prices. In practice, quantities might not be observable or meaningful. Thus in practice the Laspeyres formula is typically estimated using expenditure shares to weight price relatives - this is numerically equivalent to the formula (4.1) above.

4.15 To derive the price relatives form of the Laspeyres index, multiply the numerator of equation (4.1) by \(\frac{p_{i0}}{p_{i0}}\)and rearrange to obtain:

\((4.5) \space I_{t} = \sum \frac{p_{it}}{p_{io}}\big( \frac{p_{i0}q_{i0}}{\sum p_{i0}q_{i0}}\big)\times 100\)

where the term in parentheses represents the expenditure share of item \(i\) in the reference (or, more commonly labelled, base) period. Let:

\((4.6) \space w_{i0} = \frac{p_{i0}q_{i0}}{\sum p_{i0}q_{i0}}\)

then the Laspeyres formula may be expressed as:

\((4.7) \space I_{it}=\sum w_{i0} \big( \frac{p_{it}}{p_{i0}}\big)\times 100\)

where \(\frac {p_{it}}{p_{i0}}\)is the price relative for the \(i^{th}\) item

4.16 The important point to note here is that if price relatives are used then value (or expenditure) weights must also be used. On the other hand, if prices are used directly rather than in their relative form, then the weights used must be quantities.

The following illustrates the Laspeyres index number calculation: 

\(Laspeyres = 100.0 \times [(0.4310 \times 1.1000) + (0.1293 \times 1.3333) + (0.0776 \times 1.4444) \space + \)

\( (0.2586 \times 0.6667) + (0.1035 \times 1.1000)]. \)
 

In this example, the Laspeyres Chain Index for period 2 is calculated as follows:

\(Index_{t=2} = (104.5/100) \times (117.0/100) \times 100 \\= 122.3.\)