Price index theory

The value of an individual product is the product of price and quantity, that is:

\(v^t=p^tq^t \space \space (3.1)\)

where \(v\) is value, \(p\) is price, \(q\) is quantity and the superscript \(t\) refers to the periods at which the observations are made. For an output index, the value of concern is revenue. For an input index, the value of concern is expenditure.

Decomposition of a change in a value can be illustrated using equation (3.1), as in the following example: 

Suppose the price of tinned apples from a particular producer is $2.00 per 440g tin at a particular time. Suppose further that the price rises to $2.50 per 440g tin at a later time. The movement in the price of apples from the first to the later period is obtained from the ratio of the price in the second period to the price in the first period, that is $2.50/$2.00 = 1.25 or an increase of 25% in the price.

If the producer sold exactly the same quantity of tinned apples in the two periods, the revenue from the sale (the value of the sale) would rise by 25%.

However, if the amount sold in the first period was 1,000 tins, and the amount sold in the second period was 1,200 tins, the quantity would also have risen, by 1,200/1,000 = 1.20 or 20%. In these circumstances, the total revenue from sale of tinned apples increases from $2,000 in the first period (1,000 tins at $2.00 per tin), to $3,000 in the second period (1,200 tins at $2.50 per tin), an overall increase in revenue (value) of $1,000, or 50%. The overall increase in value is the product of the ratios of the change in price and the change in quantity (1.25 x 1.20 = 1.50).

For an individual product, the ratio of the price in one period and the price in an earlier period is called a price relative. A price relative shows the change in price for one product only (e.g. the price of a tin of apples from one particular producer).

In terms of the formula in equation 3.1:

The ratio of the prices in the two periods, \(p^2\) and \(p^1\)

($2.50/$2.00 = 1.25) is the price relative \(\big(\frac{p^2}{p^1}\big)\)

Now consider the case of price and quantity (and value) observations on many products. The quantity measurements can have many dimensions, such as number of units (e.g. tins), kilograms, metres, litres or even time (for services). Further, the quantities and prices of products are likely to show different movements between periods. Determining the respective movements in price and quantity between periods is the task of index numbers; to summarise the information on sets of prices and quantities into single measures to assist in understanding and analysing changes.

In essence, an index number is an average of either prices or quantities. The problem is how the average should be calculated.

More formally, the price index problem is how to derive numbers \(I^t_{PRICE}\) (an index of price) and \(I^t_{QUANTITY}\) (an index of quantity) such that the product of the two is the change in the total value of the products between the reference period \((0)\) and any other period \((t)\), that is:

\(I^t_{PRICE}=\frac{P^t}{P^0}\), and

\(I^t_{QUANTITY} = \frac{Q^t}{Q^0}\), then

\(I^t_{PRICE} \times I^t_{QUANTITY}= \frac{P^t}{P^0}\times\frac{Q^t}{Q^0}\)

                                  \(=\frac{P^tQ^t}{P^0Q^0}\)

                                  \(=\frac{V^t}{V^0} \space \space \space (3.2)\)

where \(P^t\), \(Q^t\) and \(V^t\) are respectively, price, quantity and value of all products in period \(t\) and \(P^0\), \(Q^0\) and \(V^0\) are respectively, their prices, quantities and values in period \(0\) (the reference period). Based on equation (3.1), \(V^t\) can be represented as:

\(V^t=\sum^N_\limits {i=1}v^t_i\)

     \(=\sum^N_\limits {i=1}p^t_iq^t_i \space \space \space (3.3)\)

that is, the sum of the product of prices and quantities of each product denoted by subscript \(i\).

As stated earlier, one way to measure the price component of the change in value is by holding the quantities constant. 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 basket (or quantities).

The options are to use:

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

Estimating the cost of the basket in the second period’s prices simply requires multiplying the quantities of products 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 basket to the total price of the basket 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

\(I^t_{Laspeyres}=\frac{\sum^N_\limits {i=1}p^t_iq^0_i}{\sum^N_\limits {i-1}p^0_iq^0_i} \times 100 \space \space \space (3.4)\)

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

Estimating the cost of purchasing the second period’s basket in the first period simply requires multiplying the quantities of products purchased in the second period by the prices prevailing in the first period. A price index is obtained from the ratio of the total price of the basket in the second period to the total price of the same basket valued at 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:

\(I^t_{Paasche}=\frac{\sum^N_\limits {i=1}p^tq^t}{\sum^N_\limits {i=1}p^0q^t}\times 100 \space \space \space (3.5)\)

(iii) A combination (or average) of quantities in both periods³.

In the absence of any firm indication that either period is the better to use as the reference, then a combination of the two is a sensible compromise. In practice this approach is most frequent in:

a) the Fisher Ideal price index⁴, which is the geometric mean of the Laspeyres and Paasche indexes:

\(I^t_{Fisher}=(I^t_{Laspeyres}\times I^t_{Paasche})^{\frac{1}{2}}\)

\(= \sqrt {I^t_{Laspeyres} \times I^t_{Paasche}} \space \space \space (3.6)\)

and

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

\(I^t_{Törnqvist}=\prod^n_\limits {i=1}\big(\frac{p^t_i}{p^0_1}\big)^{s_i}\times 100 \space \space \space (3.7)\)

where

\(s_i = \frac {1}{2}\bigg(\frac{v^0_i}{\sum^n_\limits{i=1}v^0_i}+\frac{v^t_i}{\sum^n_\limits{i=1}v^t_i}\bigg)\)

is the average of the value shares for the \(i\)th product in the two periods.

The Fisher Ideal and Törnqvist indexes are often described as symmetrically weighted indexes in that they treat the weights from the two periods equally⁵.

The Laspeyres and Paasche formulae are expressed above in terms of quantities and prices. In practice quantities might not be observable or meaningful (for example, how would the quantities of legal services, public transport and education be measured?). Thus in practice the Laspeyres formula is typically estimated using value shares to weight together price relatives; this is numerically equivalent to the formula (3.4) above.

To derive the price relatives form of the Laspeyres index, multiply the numerator of equation (3.4) by  \(\frac{p^t_i}{p^0_i}\) and rearrange to obtain:

\(I^t_{Laspeyres}=\sum^n_\limits{i=1}\bigg(\frac{p^t_i}{p^0_i}\bigg)\frac{P^0_1q^0_1}{\sum^n_\limits{i=1}p^0_iq_i^0}\)

               \(=\sum^n_\limits{i=1} \bigg(\frac{p^t_i}{p^0_i}\bigg) s^0_i \space \space \space (3.8)\)

where \(s^0_i\) represents the value share of product i in the reference period.

\(s^0_i=\frac{p^0_iq^0_i}{\sum^n_\limits{i=1}v^0_ip^0_iq^0_i} \space \space \space (3.9)\)

3.15 To derive the price relatives form of the Paasche index, multiply the denominator (3.5) by \(\frac{p^t_i}{p^0_i}\) and rearrange to obtain:

\(I^t_{Paasche} = \Bigg(\frac{\sum^n_\limits{i=1}p^t_iq^t_i}{\sum^n_\limits{i=1}p^0_iq^t_i\frac{p^t_i}{p_i^t}}\Bigg)\times 100 = \frac{1}{\sum^n_\limits{i=1}\frac{p^0_i}{p^t_i}}\bigg(\frac{\sum^n_\limits{i=1}p^t_iq^t_i}{p^t_iq^t_i}\bigg)\times100 \space \space \space (3.10)\)

Which may be expressed as:

\(I^t_{Paasche}=\frac{1}{\sum^n_\limits{i=1}\big(\frac{p^0_i}{p^t_i}\big)\times S^t_i} \times 100 \space \space \space (3.11)\)

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

An example of creating index numbers using the above formulae is presented in Table 3.1 below.

The following illustrate the index number calculations:

(a) Laspeyres

\(= (0.3932 \times 1.0345) + (0.1864 \times 0.8182) + (0.1085 \times 1.0500) + (0.1627 \times 0.9167) \\+ (0.1492 \times 1.0909) \times 100 \\= 98.51\)

(b)Paasche

\(= 1/((0.4220 \times 1.0345) + (0.1424 \times 0.8182) + (0.0768 \times 1.0500) + (0.2321 \times 0.9167)\\ + (0.1266 / 1.0909)) \times 100 \\ = 97.61\)

Fisher

\(= (98.51 \times 97.62)^{1/2} \\= 98.06\)

(c) Törnqvist best calculated by first taking the logs of the index formula
\(= 1/2 \times (0.3932 + 0.4220) \times ln(1.0345)\)
\(+ 1/2 \times (0.1864 + 0.1424) \times ln(0.8182)\)
\(+ 1/2 \times (0.1085 + 0.0768) \times ln(1.0500)\)
\(+ 1/2 \times (0.1627 + 0.2321) \times ln(0.9167)\)
\(+ 1/2 \times (0.1492 + 0.1266) \times ln(1.0909)\)
\(= -0.0199\)

and then taking the exponent multiplied by 100
\(= e^{-0.0199} *100\)
\(= 98.03\)

In Table 3.1 the different index formulae produce different index numbers and thus different estimates of the price movements. Typically the Laspeyres formula will produce a higher index number than the Paasche formula, with the Fisher Ideal and the Törnqvist of similar magnitude falling between the index numbers produced by the other two formulae. In other words the Laspeyres index will generally show a higher (lower) price rise (fall) than the other formulae and the Paasche index a lower (higher price rise (fall))⁶.

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

\((114.2/100) * (112.9/100) * 100 \\= 128.9\)

The Paasche Chain Index for period 2 is calculated as follows:

\((113.8/100) * (112.3/100) * 100 \\= 127.8\)

The Fisher Chain Index for period 2 is calculated as follows:

\((114/100) * (112.6/100) * 100\\ = 128.3 \)

OR

\((128.9 * 127.8)^{1/2} \\= 128.3 \)

The direct Laspeyres formula has the advantage that the index can be extended to include another period’s price observations when available, as the weights (quantities or value shares) are held fixed at some earlier period. On the other hand, the direct Paasche formula requires both current period price observations and current period weights before the index can be extended.

In some situations it is not possible or meaningful to derive weights for each price observation. This is typically so for a narrowly defined product grouping in which there might be many sellers (or producers). Information might not be available on the overall volume of sales of the product or for the individual sellers or producers from whom the sample of price observations is taken. In these cases it seems appropriate not to weight, or more correctly to assign an equal weight, to each price observation. It is a common practice that the price indexes at the lowest level (where prices enter the index) are calculated using an equal-weights formula, based on arithmetic means or a geometric mean.

Suppose there are price observations for \(n\) products in period \(0\) and \(t\). Then three approaches for constructing an equal weights index are⁷ ⁸:

1. calculate the arithmetic mean of prices in both periods and obtain the relative of the second period’s average with respect to the first period’s average (i.e. divide the second period’s average by the first period’s average). This is the Dutot formula also referred to as the relative of the arithmetic mean of prices (RAP) approach:

\(I^t_{Dutot}=\frac{\frac{1}{n}\sum^n_\limits{i=1}p^t_i}{\frac{1}{n}\sum^n_\limits{i=1}p^0_t} \space \space \space (4.12)\)

2. for each product, calculate its price relative (i.e. divide price in the second period by the price in the first period) and then take the arithmetic average of these relatives. This is the Carli formula, also referred to as the arithmetic mean of price relatives (APR) approach:

\(I^t_{Carli}=\frac 1{n}\sum^n_\limits{i=1}\frac {p^t_i}{p^0_i} \space \space \space (4.13)\)

3. for each product calculate its price relative and then take the geometric mean⁹( of the relatives. This is the Jevons formula, also referred to as the geometric mean (GM) approach:

\(I^t_{Jevons}=\prod^n_\limits{i=1}\bigg(\frac{p^t_i}{p^0_i}\bigg)^{\frac{1}{n}}\)

\(= \frac{\big(\prod^n_\limits{i=1}p^t_i\big)^{ \frac 1{n}}}{\big(\prod^n_\limits{i=1}p^o_i)^{\frac{1}{n}}} \space \space \space (4.14)\)

The following are calculations of the equal weight indexes using the data in Table 3.2. Setting period \(0\) as the reference with a value of 100.0, the following index numbers are obtained in period \(t\):

Dutot (RAP) formula: \(113.5 = \frac{\frac1{3}(12+13+17)}{\frac1{3}(10+12+15)}\times 100\)

Carli (APR) formula: \(113.9 = \frac1{3}(\frac{12}{10}+\frac{13}{12}+\frac{17}{15})\times 100\)

Jevons (GM) formula: \(113.8= \space^3\sqrt{\frac{12}{10}\times\frac{13}{12}\times\frac{17}{15}}\times 100\)

Theory suggests that the APR formula will generally show the largest estimate of price change, the GM¹⁰  the least and the RAP a little larger but close to the GM formula. Real life examples generally support this proposition¹¹, although with a small sample, as in the above example, different rankings for the RAP formula are possible depending on the prices.

The behaviour of these formulae under chain linking and direct estimation is shown in Table 3.3 using the price data from Table 3.2. It is noted that the RAP and GM formulae are transitive (the index number derived by the direct method is identical to that derived by the chain link method), but not the APR.

One practical issue with price index construction when using the Laspeyres approach is that it may not always be possible to obtain values at the desired reference period. For example, the values may only be available from an earlier period. In this situation, a value is price updated so that it is composed of quantities in period b (some period prior to period 0) valued at the price level of period 0. The Laspeyres index in this form is referred to as a Lowe Index. The Lowe index is used by most National Statistical Offices to compile official price indexes. The Lowe index is expressed as follows:

\(\frac{\sum^n_\limits{i=1}p^t_iq^b_i}{\sum^n_\limits{i=1}p^0_iq^b_i} \times 100 \space \space \space (4.15)\)

All the index formulae discussed above require observations on the same products in each period. In some situations it may be necessary to change the products or outlets included in the price sample or, if weights are used, to re-weight the price observations. Examples of changes in a price sample include: a data provider goes out of business; or the sample needs to be updated to reflect changes in the market shares of providers; to introduce a new provider; or to include a new product.

It is important that changes in price samples are introduced without distorting the level of the index for the price sample. This is usually done by a process commonly called 'splicing'. Splicing is similar to chain linking except that it is carried out at the price sample level. An example of handling a sample change is shown in table 3.5, for equal weighted indexes assuming a new provider is introduced in period \(t\). A price is also observed for the new provider in period \(t-1\) . The inclusion of the new provider causes the geometric mean to fall from $5.94 to $5.83. We do not want this price change to be reflected in the index but we do want to capture the effect of provider 4’s price movement between period \(t-1\) and \(t\).

In the case of the arithmetic mean of price relatives and geometric mean formulae, this is done by:

• setting the previous period price relative for period \(t\) for the new provider (4) equal to the average of the price relatives of the three providers included in period \(t-1\) (1.326)
• applying the movement in provider 4’s price between period \(t-1\) and \(t\) to derive a price relative for period \(t \space (6.00/5.50\times 1.326=1.447)\).

For these two formulae, the average of the price relatives is effectively the index number, so the geometric mean index for period \(t-1\) is 132.6 and for period t is 141.6.

In the case of the relative of the arithmetic mean of prices formula (RAP) formula, the method is similar but prices are used instead of price relatives. The RAP formula uses the arithmetic mean of prices (not the arithmetic mean of the price relatives). The index for RAP can be calculated from the period to period price movements:

• between the reference period and period \(t-1\) , the movement in the average price was 1.333 (6.00/4.50) without the new provider
• between period \(t-1\) and \(t\), the movement in the average price was 1.063 (6.25/5.88) including the new provider in both periods

Thus the index for period \( t\) is 141.7 (1.333 1.063 100).