Price index theory

4.15 The remainder of Price index theory focuses on the comparison of prices over time (temporal indexes). Expenditure on an individual item is the product of price and quantity, that is:

\(e_t = p_tq_t \space \space \space \space \space \space (4.1)\)

where \(e\) is expenditure, \(p\) is price, \(q\) is quantity and the subscript \(t\) refers to the time periods at which the observations are made.

4.16 Consider the expenditures on the same commodity in two different times periods. Changes in these expenditures can reflect changes in the price, changes in the quantity, or a combination of both price and quantity changes. For example, suppose the price of Granny Smith apples at a particular market is $2.00 per kg in period one, and it rises to $2.50 per kg in period two. The change in the price of apples between these two periods 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 a consumer bought exactly the same quantity of apples in the two periods, the expenditure on Granny Smith apples would rise by 25%. However, if the amount purchased in the first period was 10 kg, and the amount purchased in the second period was 12 kg, the quantity would also have risen by a factor of 12/10 = 1.20 or 20%. In these circumstances, the total expenditure on apples increases from $20 in the first period (10 kg at $2.00 per kg), to $30 in the second period (12 at $2.50 per kg), an increase in expenditure of $10 or 50%. The ratio of the current expenditure to the previous expenditure is the product of the change in price and the change in quantity (1.25 x 1.20 = 1.50).

4.17 The ratio between the price in the current period and the price in the reference period is called a price relative. A price relative shows the change in price for one item only (e.g. the pricing of Granny Smith apples at one particular fruit market).

In terms of the formula in equation 4.1: 

\(e_1\) (expenditure in period 1) = \(p_1\) ($2.00) x \(q_1\) (10kg) = $20, and
\(e_2\) (expenditure in period 2) = \(p_2\) ($2.50) x \(q_2\) (12kg) = $30
where: \(p_1\) is the price per kg in period 1; \(q_1\) is the quantity in period 1;
\(p_2\) is the price per kg in period 2 and \(q_2\) is the quantity in period 2.

The ratio between the prices in the two periods, \(p_2\) and \(p_1\) ($2.50/$2.00 = 1.25) is the price relative.³

4.18 It is only necessary to have observations on two of the three components of equation 4.1 to analyse contributions to change in the expenditure. Using the apple example, suppose observations were only available on expenditure and price. The expenditure could be divided by the price to estimate the quantity (or the movements in expenditure and price could be used). Alternatively, if only expenditure and quantity data were available, expenditure could be divided by quantity to derive what's known as the 'unit (price) value'.

4.19 Now consider the case of price and quantity (and expenditure) observations on many commodities. The quantity measurements can have many dimensions, such as kilograms, tonnes, or even units (e.g. number of motor cars), and the quantities and prices of items are likely to show different movements between periods. Answers are required to questions such as these: what is the change over time in the quantity of commodities, and what is the contribution of price changes to changes in the expenditure on the bundle of commodities over time? Answering these questions 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.

4.20 In essence, an index number is an average of either prices or quantities compared with the corresponding average in a base period. The problem is how to calculate the average.

4.21 More formally, the price index problem is how to derive an index of price \((I^P)\) and an index of quantity \((I^Q)\) such that the product of the two is the change in the total value of the items between the base period \((0)\) and any other period \((t)\), that is

\({I{^P_t}I{^Q_t}={V_t/V_0}} \space \space \space \space \space (4.2)\)

where \(V_t\) is the value of all items in period \(t\) and \(V_0\) is their value in period \(0\) (base period). Based on equation (4.1), can be represented as:

\({V_t} = {\sum {v_{it}}} = {\sum {p_{it}{q_{it}}}} \space \space \space \space \space (4.3)\)

that is, the sum of the product of prices and quantities of each item denoted by subscript \(i\). The summation range \((i =1..N)\) is not shown in order to make the formula more readable.

4.22 In presenting index number formulas, a simple starting point is to compare two sets of prices (sometimes called bilateral indexes). Consider price movements between two periods, where the first period is denoted as period \(0\) and the second period as period \(t\) (period \(0\) occurs before period \(t\)). To calculate the price index, the quantities need to be fixed at the same period in time. The initial question is what period should be used to determine the basket (or quantities). There are several possibilities.

(i) The quantities of the first (or 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 basket of goods and services that was purchased in the first period. Estimating the cost of the basket 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 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 \(I_{Lt}\). It may be represented, with a base of 100.0, as:

\({I_{Lt}}=\frac{\sum p_{it}q_{i0}}{\sum p_{i0}q_{i0}} \times 100 \space \space \space \space \space (4.4)\)

(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 basket that was purchased in the second period. Estimating the cost of purchasing the second period's basket 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 total price of the basket in the second period compared to the total price of the 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 \(I_{Pt}\). It may be represented, with a base of 100.0, as:

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

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

This approach tries to overcome some of the inherent difficulties of using a basket fixed at either time period. 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,⁴ which is the geometric mean of the Laspeyres and Paasche indexes:

\(I_{Ft} = {(I_{Lt}I_{Pt}) ^ \frac {1}{2}} \space \space \space \space \space (4.6)\)

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

\(I_{Tt} = \prod \limits_i (\frac {p_{it}}{p_{i0}})^{s_{i}} \space \space \space \space \space (4.7)\)

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

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

4.23 The Laspeyres and Paasche formulas are expressed above in terms of quantities and prices. However, in practice, quantities might not be observable or meaningful (e.g. consider the quantity dimension of legal services, public transport, and education). Thus in practice, the Laspeyres formula is typically estimated using expenditure shares to weight price relatives - this is numerically equivalent to the formula (4.4) above.

4.24 To derive the price relatives form of the Laspeyres index, multiply the numerator of equation 4.4 by \(\frac {p_{i0}}{p_{i0}}\)and rearrange to obtain:

\({I_{t}}= \sum \frac {p_{it}}{P_{i0}} (\frac{p_{i0}q_{i0}}{\sum p_{i0}q_{i0}}) \times 100 \space \space \space \space \space (4.8)\)

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

\(w_{i0} = \frac{p_{i0}q_{i0}}{\sum p_{i0}q_{i0}} = \frac{e_{i0}}{\sum {e_{i0}}} \space \space \space \space \space (4.9)\)

then the Laspeyres formula may be expressed as:

\({I_{it}}= \sum w_{i0} (\frac{p_{it}}{p_{i0}}) \times 100 \space \space \space \space \space (4.10)\)

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

4.25 In a similar manner, the Paasche index may be constructed using expenditure weights. In equation 4.5, multiply the denominator by \(\frac {P_{it}}{P_{it}}\)and rearrange terms to obtain:

\(I_{Pt} = \frac{\sum p_{it}q_{it}}{\sum p_{it}q_{it} \frac{p_{i0}}{p_{it}}} = \frac{1}{\sum \frac{p_{i0}}{p_{it}}} (\frac {\sum p_{it}q_{it}}{p_{it}q{_{it}}}) \times 100 \space \space \space \space \space (4.11)\)

which may be expressed as:

\(I_{Pt} = \frac{1}{\sum w_{it} \frac{p_{i0}}{p_{it}}} \times 100 \space \space \space \space \space (4.12)\)

which is the inverse of a ‘backward’ Laspeyres index (i.e. a Laspeyres index going from period t to period \(0\) using period \(t\) expenditure weights).⁵

4.26 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 must be quantities.

4.27 An example of creating index numbers using the above formulas is presented in Table 4.1. For the purposes of this exercise, a limited range of the types of commodities households might purchase is used. The quantities that these items would typically be measured in may vary. There are likely to be differences in price behaviour of the commodities over time. Further, the quantities of these items households purchase may vary over time in response to changes in prices (of both the item and other items) and household incomes.

4.28 Differences that might arise in price changes (and, by implication expenditure patterns) are illustrated by the following:

• prices of high labour content items, such as services like a haircut, will tend to show steady trends over time relative to other items;
• prices of high technology goods, such as tablets, tend to decline over time, either absolutely or relative to other items, reflecting productivity and technological advances;
• prices of some items, such as fresh fruit, are affected by climatic and seasonal influences and so have volatile price movements; and
• prices of some items might at times be influenced by changes in taxation rates (e.g. tobacco).

4.29 Price changes influence, to varying degrees, the quantities of items households purchase. For some items, such as basic food stuffs, the quantities purchased may show little change in response to price changes. For other items, the quantities households purchase may change by a smaller or greater proportionate amount than the price change.⁶

4.30 The examples in Table 4.1 reflect some of these possibilities.

4.31 In Table 4.1 the different index formulas 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 in periods after the base period, with the Fisher Ideal and the Törnqvist of similar magnitude falling between the index numbers produced by the other two formulas. In other words the Laspeyres index will generally produce a higher (lower) measure of price increase (decrease) than the other formulas and the Paasche index a lower (higher) measure of price increase (decrease) in periods after the base period.⁷

4.32 With the recent ability of National Statistical Offices to access transactions (scanner) data for use in their CPIs, new index construction methods have been developed to make use of the available price and quantity data in each period. These methods borrow heavily from existing methods in the production of spatial price indexes, in particular multilateral indexes. These multilateral index methods are being adapted for the purpose of producing temporal price indexes, which can be used in the production of the CPI. The use of transactions data and multilateral index methods in the Australian CPI are discussed further in Use of transactions data in the Australian CPI.

4.35 The following illustrate the index number calculations:

Laspeyres

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

Paasche

\(= 1 / [(0.4220 / 1.0345) + (0.1424 / 0.8182) + (0.0768 / 1.0500) + (0.2321 / 0.9167) + (0.1266 / 1.0909)] \times 100 \\ = 97.62\)

Fisher

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

Törnqvist is best calculated by first taking the logs of the index formula

\(= \frac{1}{2} \times (0.3932 + 0.4220) \times ln (1.0345) \\ + \frac{1}{2} \times (0.1864 + 0.1424) \times ln (0.8182) \\ + \frac{1}{2} \times (0.1085 + 0.0768) \times ln (1.0500) \\ + \frac{1}{2} \times (0.1627 + 0.2321) \times ln (0.9167) \\ + \frac{1}{2} \times (0.1492 + 0.1266) \times ln (1.0909) \\ = -0.0199\)

and then taking the exponent multiplied by 100

\(= e^{-0.0199} \times 100 \\ =98.04\)

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

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

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

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

And the Fisher Chain Index for period 2 is calculated as follows:

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

OR

\((128.9 \times 127.8)^{\frac12} \\ = 128.3\)

4.37 An index formula is said to be 'transitive' if the index number derived directly is identical to the number derived by chaining. In general, no weighted index formula will be transitive because period-to-period calculation of the index involves changing the weights for each calculation. This can be seen in Table 4.2 where in period 2 the direct Laspeyres (130.2) is different to the chain Laspeyres (128.9) due to the different quantities. The index formulas in Table 4.2 will only result in transitivity if there is no change in the quantity of each item in each period or if all prices show the same movement. In both these unlikely cases, all the formulas (Laspeyres, Paasche and Fisher) will produce the same result.

4.38 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 are fixed at some earlier base period. On the other hand, the direct Paasche formula requires both current period price observations and current period weights before the index can be calculated.

4.39 The households’ expenditures on all consumer goods and services in the Consumer Price Index (CPI) basket is mainly sourced from information derived from the Household Expenditure Survey (HES). However, the results from the HES are not available until approximately 12 months after the end of the survey. The Laspeyres index requires either quantities or expenditure in the base period which would mean the CPI would be unable to be calculated on these expenditures until approximately 16 months after the HES is completed.

4.40 The CPI is a quarterly survey which means the ABS must continue to calculate the CPI on the old expenditures until the new expenditures are available. When the new expenditures are available, a statistical office can then recalculate the CPI based on the new weights. However, this will lead to revisions to previously published CPI estimates which is not desirable for any contract indexation. The alternative is to use a class of price indexes called a Lowe index which defines the index as the percentage change, between the periods compared, in the total cost of purchasing a fixed basket of quantities. Most statistical offices make use of some kind of Lowe index in practice.

4.41 To calculate a price index, any set of quantities could be used. These do not have to be restricted to quantities or expenditures purchased in one period and could be arithmetic or geometric averages of the quantities of multiple periods. For the Australian CPI, the quarterly percentage change from the December quarter 2017 onwards is mainly based on the HES which was collected in respect of the financial year 2015-16. Prior to this, the CPI from the September quarter 2011 was based on the HES which was collected in respect of the financial year 2009-10. For a complete listing of the historical CPI weighting patterns see Consumer Price Index: Historical Weighting Patterns, 1948-2017 (cat. no. 6431.0).

4.42 The period whose quantities are actually used in a CPI is described as the weight reference period. In the 17th series this generally refers to the HES which is 2015-16 and it will be denoted as period \(b\). With the CPI being annually re-weighted, period \(b\) will be updated each year to the second most recent financial year (e.g. for 2018, period b is 2016-17). Period \(0\) is the price reference period which is the most recent September quarter. The Lowe index using the quantities of period \(b\) can be written as follows:

\(P_{L0} = \frac{\sum ^n _{i=1} p^t _iq^b_i}{\sum ^n_{i=1}p^0_i q^b_i} = {\sum^n_{i=1}} (\frac {p^t_i}{p^0_i})s^{0b}_i \)

where: \(s^{0b}_i = \frac{p^0 _iq^b_i}{\sum ^n_{i=1}p^0_i q^b_i} \space \space \space \space \space (4.13)\)

4.43 Similar to the Laspeyres index described earlier, the Lowe index can be calculated as either the ratio of prices and quantities, or as an arithmetic weighted average of the price relatives. The expenditures refer to quantities in period \(b\) (e.g. 2016-17) and prices in period \(0\) (e.g. September quarter 2018). Lowe indexes are widely used for CPI purposes.

4.44 The Laspeyres and Paasche indexes are two special cases of the Lowe price index. When the quantities are those of the price reference period, that is when \(b=0\), the Laspeyres index is obtained. When quantities are those of the other period, that is when \(b=t\), the Paasche index is obtained.

4.45 In some situations, it is not possible or meaningful to derive weights in either quantity or expenditure terms for each price observation. This is typically so for a narrowly defined commodity grouping in which there might be many sellers (or producers). Information might not be available on the total volume of sales of the item 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 in the CPI in many countries that the price indexes at the lowest level (where prices enter the index) are calculated using an equally weighted formula, such as an arithmetic mean or a geometric mean.

4.46 Suppose there are price observations for \(N\) items in period \(0\) and period \(t\). Then three approaches⁸ for constructing an equally weighted index are as follows.

(i) Calculate the arithmetic mean of prices in both periods and obtain the relative of the current period’s average to the base period’s average (i.e. divide the current period’s average by the base period’s average). This is the relative of the arithmetic mean of prices (RAP) approach, also referred to as the Dutot formula:

\(I_D = \frac {\frac{1}{N} \sum p_{it}} {\frac{1}{N} \sum p_{i0}} \space \space \space \space \space (4.14)\)

(ii) For each item, calculate its price relative (i.e. divide the price in the current period by the price in the base period) and then take the arithmetic average of these relatives. This is the arithmetic mean of price relatives (APR) approach, also referred to as the Carli formula:

\(I_C = {\frac{1}{N} \sum} {\frac{p_{it}}{p_{i0}}} \space \space \space \space \space (4.15)\)

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

\(I_G = \prod ({\frac{p_{it}}{p_{i0}}})^{\frac{1}{N}} \space \space \space \space \space (4.16)\)

4.47 Although these formulas apply equal weights, the implicit basis of the weights differs. The geometric mean applies weights such that the expenditure shares of each observation are the same in each period. In other words, it is assumed that as an item becomes more (less) expensive relative to other items in the sample the quantity declines (increases) with the percentage change in the quantity offsetting the percentage change in the price. The RAP formula assumes equal quantities in both periods. That is, the RAP assumes there is no change in the quantity of an item purchased regardless of either its price movement or that of other items in the sample. The APR assumes equal expenditures in the base period with quantities being inversely proportional to base period prices.

4.48 The following are calculations of the equal weight indexes using the data in Table 4.2. Setting period \(0\) as the base with a value of 100.0, the following index numbers are obtained in period \(t\):

RAP formula: \(113.5 = \frac {\frac{1}{3} (12+13+17)} {\frac{1}{3} (10+12+15)} \times 100\)

ARP formula: \(113.9 = {\frac{1}{3} ({\frac {12}{10}}+{\frac {13}{12}}+{\frac {17}{15}})} \times 100\)

GM formula: \(113.9 = 3 {\sqrt {{\frac {12}{10}}\times {\frac {13}{12}}\times {\frac {17}{15}}}}\times 100\)

4.49 Theory suggests that the APR formula will produce the largest estimate of price change, the GM the least and the RAP a little larger but close to the GM.¹⁰ Empirical examples generally support this proposition,¹¹ although with a small sample as in the example above, substantially different rankings for the RAP formula are possible depending on the prices.

4.50 The behaviour of these formulas under chaining and direct estimation is shown in Table 4.3 using the price data from Table 4.2. The RAP and GM formulas are transitive, but not the APR.

4.54 It is appropriate at this point to re-examine the decomposition of an expenditure aggregate into price and quantity components introduced in equation 4.1. It is important to know the form of the quantity index when a particular form of the price index is used (and vice versa) to ensure the accurate decomposition of the value change.

4.55 A value is the product of a price and a quantity (in its simplest form, the price of a single item multiplied by 1 is the value of the item). It follows that changes in the value of expenditure on an item from period to period are the result of changes in the prices or quantities or both. If any two of the value, price or quantity are known, the third can be derived (i.e. \(E = {P\times Q}\), where \(E\) = expenditure, \(P\) = price and \(Q\) = quantity), e.g. \(Q={E/P}\). The calculation is straightforward when a single item is involved. However, in the case of an expenditure total that is the sum of several items, breaking up that expenditure into its price and quantity components becomes more complicated.

4.56 Price indexes provide a means of removing the effects of price changes from changes in expenditure so that the underlying changes in quantity can be identified. In the Australian National Accounts, price indexes are widely used in the process of estimating changes in volumes of expenditure, production etc. The process of using price indexes in this way is known as price deflation, with the index termed a deflator. The form of price index (current or fixed weighted) will determine the resulting index of quantity change.

4.57 The change in an expenditure or value aggregate between period \(0\) and \(t\) may be expressed as:

\({\frac {E_t}{E_0}} = \frac {\sum p_{it}q_{it}}{\sum p_{i0}q_{i0}}\space \space \space \space \space (4.17)\)

4.58 Multiplying the right-hand side of equation (4.17) by \(\frac {\sum p_{it}q_{i0}}{\sum p_{it}q_{i0}}\)allows the equation to be expressed as:

\({\frac {E_t}{E_0}} = \frac {\sum p_{it}q_{i0}}{\sum p_{i0}q_{i0}}\times \frac {\sum p_{it}q_{it}}{\sum p_{it}q_{i0}} \space \space \space \space \space (4.18)\)

where the first term on the right-hand side of the equals sign is a Laspeyres price index and the second is a Paasche volume index.¹³ This is referred to as the Laspeyres decomposition. In other words, if an index of value change is deflated by a base-period-weighted price index, then the index of quantity change is a current-period-weighted quantity index.

4.59 An alternative decomposition of the change in the expenditure aggregate is obtained by multiplying the right-hand side of (4.17) by \(\frac {\sum p_{i0}q_{it}}{\sum p_{i0}q_{it}}\)which produces:

\({\frac {E_t}{E_0}} = \frac {\sum p_{it}q_{it}}{\sum p_{i0}q_{it}}\times \frac {\sum p_{i0}q_{it}}{\sum p_{i0}q_{i0}} \space \space \space \space \space (4.19)\)

where the first term on the right-hand side of the equals sign is a Paasche price index and the second is a Laspeyres volume index. This is referred to as the Paasche decomposition. In other words, if an index of value change is deflated by a current-period-weighted price index, then the index of quantity change is a base-period-weighted quantity index.

4.60 A similar decomposition can also be undertaken for the Fisher Ideal index. By taking the geometric average of the alternative Laspeyres and Paasche decomposition of value change (right-hand sides of equations (4.18) and (4.19)) it can be shown that value change is the product of Fisher Ideal price and quantity indexes.

4.61 All the index formulas discussed above require observations on the same items in each period. In some situations it may be necessary to change the items 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 respondent goes out of business;
• the sample needs to be updated to reflect changes in the market shares of respondents;
• to introduce a new respondent; or
• to include a new item.

4.62 It is important that changes in price samples are introduced without distorting the level of the index for the price sample. This usually involves a process commonly referred to as splicing. Splicing is similar to chaining except that it is carried out at the level of the price sample. An example of handling a sample change is shown in Table 4.4, for equally weighted indexes assuming a new respondent is introduced in period \(t\). A price is also observed for the new respondent in the previous period \(t-1\). The inclusion of the new respondent causes the geometric mean to fall from $5.94 to $5.83. The index should capture the effect of respondent 4's price movement between period \(t-1\) and t without capturing this recorded price change due to the inclusion of a new respondent.

4.63 In the case of the APR and GM formulas, the process involves:

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

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

4.65 In the case of the 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 base period and period \(t\) , the movement in the average price was 1.333 (6.00/4.50) without the new respondent;
• between period \(t-1\) and \(t\), the movement in the average price was 1.063 (6.25/5.88) including the new respondent in both periods; and
• thus the index for period \(t\) is 141.7 (1.333 x 1.063 x 100).