- See Chapter 15 Basic Index Number Theory in IMF (2010) Producer Price Index Manual: Theory and Practice. Washington, DC: International Monetary Fund (IMF).

# Chapter 6 Price and volume measures

## Introduction

6.1 In the Australian economy, millions of economic transactions take place every day involving the production and sale of goods and services (products). The monetary value of each of these transactions is a product of the quantity produced or sold at a price per unit. In a particular period, the total value of all transactions taking place in an economy is simply the sum of the individual transaction values in that period. This is referred to as the current price value.

6.2 For many purposes, economists and other analysts wish to measure the volume growth of production and expenditures; that is, growth free of the effects of price change. The current price values are subject to the effects of changing prices and so they are unsatisfactory for these purposes. Consider the sale of beef and chicken in the following example:

In period 1, 20 kilos of beef are sold at $1.00 per kilo for a value of $20.00 and 10 kilos of chicken are sold at $2.00 per kilo for a value of $20.00. Total sales of meat are valued at $40.00.

In period 2, 18 kilos of beef are sold at $1.10 per kilo for a value of $19.80 and 12 kilos of chicken are sold at $2.00 per kilo for a value of $24.00. Total sales of meat are valued at $43.80.

- In this example, it can help to think of the kilos of beef/chicken as the ‘volume’ estimate, and the value as the current price, with the amount per kilo as the price. This exemplifies the key components in estimating volumes.

6.3 Total sales of meat have increased from $40.00 in period 1 to $43.80 in period 2, but what is the growth in volume terms? One way of answering this question is to hold prices constant in the two periods, at say period 1 prices. The total value of sales in period 2 at period 1 prices is $42.00 (18 kilos of beef @ $1.00 plus 12 kilos of chicken @ $2.00). At period 1 prices, the total value of meat sales has increased from $40.00 to $42.00, which is an increase of 5%. This can be expressed algebraically as:

\(\large{\frac{p_{beef}^1 q_{beef}^2+p_{chicken}^1 q_{chicken}^2}{p_{beef}^1 q_{beef}^1+p_{chicken}^1 q_{chicken}^1}=\frac{(1.00×18)+(2.00×12)}{(1.00×20)+(2.00×10) }=\frac{18.00+24.00}{20.00+20.00}=\frac{42.00}{40.00}=1.05}\)

where \(p\) represents the price and \(q\) represents the quantity.

6.4 This expression is called a Laspeyres volume index. The defining feature is that in calculating growth from one period to another, the prices of the earlier period are applied to both periods.

6.5 Another way of estimating the volume growth of meat sales is to hold prices constant at period 2 prices. The value of meat sales in period 1 at period 2 prices is $42.00 (20 kilos of beef @ $1.10 per kilo plus 10 kilos of chicken @ $2.00 per kilo). This gives volume growth of 4.3% between the two periods and can be written algebraically as:

\(\large{\frac{p_{beef}^2 q_{beef}^2+p_{chicken}^2 q_{chicken}^2}{p_{beef}^2 q_{beef}^1+p_{chicken}^2 q_{chicken}^1 }=\frac{(1.10×18)+(2.00×12)}{(1.10×20)+(2.00×10) }=\frac{19.80+24.00}{22.00+20.00}=\frac{43.80}{42.00}=1.043}\)

6.6 This expression is called a Paasche volume index. The defining feature is that in calculating growth from one period to another, the prices of the later period are applied to both periods.

6.7 Both the Laspeyres and Paasche indexes are equally valid for calculating the volume growth of meat sales between period 1 and period 2, yet they give different answers. This suggests that an average of the two may be a better estimate than either of them. Fisher’s Ideal Index hereafter referred to as the Fisher index is the geometric mean of the Laspeyres and Paasche and is considered to be a superior index³³.

6.8 Up until the beginning of the twenty first century, most OECD member countries derived volume estimates of aggregates by holding prices constant in a base year; that is, constant price estimates. In effect, constant price estimates are a sequence of Laspeyres indexes from the base year to the current period multiplied by the current price value in the base year. Over time, price relativities change and when estimating volume growth from one period to another it is best to use prices at or about the current period. Both the 1993 and 2008 SNAs recommend the abandonment of constant price estimates in favour of chain volume estimates. Chain volume estimates are derived by linking together period-to-period indexes, such as Laspeyres, Paasche or Fisher indexes.

6.9 While chain volume estimates are generally superior to constant price estimates in terms of deriving volume growth rates, their use raises a number of issues such as:

- which index formula should be used (Laspeyres, Paasche or Fisher)?
- how frequently should the fixed prices change - quarterly or annually?
- if annually, how should quarterly indexes be derived and how should they be linked together? and
- unlike constant price estimates, chain indexes are not generally additive; how should contributions to growth be derived?

6.10 Annex A to this chapter addresses these issues in detail whilst this chapter outlines how volume estimates are actually derived in the ASNA.

6.11 There are two principal steps in deriving volume estimates of national accounts aggregates:

- the derivation of elemental volume indexes at the most detailed level practicable; and
- the aggregation of the elemental volume indexes to the desired level, such as GDP.

6.12 The chapter addresses the second step first because it is best to consider the nature of the aggregate volume indexes before describing how the elemental indexes are derived.

## Terminology

6.13 Before proceeding to discuss the aggregation of volume estimates it is necessary to define some of the key terminology to be used to minimise the risk of confusion.

6.14 The base period for an elemental volume index is the period for which the prices are fixed. Hence a Laspeyres volume index from time 0 to time t can be written as:

\(\large{\frac{q^tp^0}{q^0p^0}}\)

and a constant price estimate can be written as: \(q^tp^0\)

6.15 The Laspeyres volume index is equal to the constant price value for period t divided by the current price value for period 0. When elemental volume indexes are aggregated, the current price values in the base period form the weights for combining the elemental volume indexes. The derivation of elemental volume indexes is discussed later in this chapter.

6.16 The reference period is the period for which an index series is set equal to 100. This is an arbitrary number used in order to compare prices or volumes over time. Where the index is instead represented as a volume measure, the reference period of the series is set equal to the current price value. This allows the volume series to be expressed in terms of currency units.

6.17 For constant price estimates the base period and the reference period coincide. For chain volume indexes there is only one reference period, but there are many base periods chained together.