3  Chained indices

In addition to choosing an index number formula, a choice needs to be made whether to construct fixed-weight or chained indices. A fixed-weight quantity index compares quantities in period relative to some fixed base period (which is why fixed-weight indices are also known as fixed-base indices). Information on price movements and therefore weighting changes in the intervening periods are ignored. In contrast, a chained index compares quantities between two periods taking into account information on weighting changes in the intervening period or periods. In other words, a chained index uses price information that is more representative of that faced by economic agents in each period than does a fixed-weight index.

When relative prices change, relative quantities also usually change. For example, if the price of a particular good rises relative to all other goods in an economy because of an increase in demand, then price taking firms will tend to produce more of this good relative to other goods. Alternatively, consumers will tend to substitute away from goods that have become relatively more expensive to less expensive goods. Using a fixed-weight index to measure quantity changes in the presence of relative price changes will introduce substitution bias into the quantity index because information on relative price changes is not taken into account when measuring quantity changes. Moreover, the substitution bias usually becomes larger over time, as the fixed weights become more unrepresentative of those faced by agents when measuring quantity changes in more recent periods. Chaining fixed-weight indices helps to alleviate the substitution bias.

The following numerical example illustrates how the use of a fixed-weight index may give a distorted measure of quantity movements, owing to substitution bias, and how chaining helps to alleviate this problem. Construction of the Laspeyres output quantity index, using nominal output shares in period as weights for the quantity ratios, was shown in Table 3 (and is also reproduced in panel I of Table 5). This Laspeyres quantity index indicates that aggregate output in the two good economy increased by 18% between period and period and 23% between period and period . Because the price of is increasing relatively more than the price of , firms will produce relatively more of to reap higher profits. However, because weights are fixed in period , too little weight is given to and too much weight to when constructing the aggregate output quantity index beyond period . In other words, in this economy with price taking firms, the Laspeyres output quantity index is biased downwards, tending to understate aggregate quantity movements.

Table 5 – Chaining using Laspeyres output indices

Panel I Fixed-weight Laspeyres index (Base weights period 0 )					Panel II Fixed-weight Laspeyres index (Base weights period 1)				Panel III Chained Laspeyres index
				Percentage change				Percentage change		Percentage change
t=0	0.545	0.455	1.000			0.400	0.379	0.779	1.000
t=1	0.636	0.545	1.182	18.2%	0.467	0.533	1.000	28.4%	1.182	18.2%
t=2	0.727	0.727	1.455	23.1%	0.533	0.800	1.333	33.3%	1.576	33.3%

The downward bias in aggregate output movements using the Laspeyres quantity index with nominal output shares fixed in period 0 is shown by constructing the Laspeyres output quantity index using nominal output shares from period 1 as the fixed weights. This is shown in panel II of Table 5. The increase in aggregate output between period 1 and period 2 is 33%, compared to an increase of 23% when the Laspeyres output quantity index is calculated using nominal output shares from period 0 as the fixed weights. The larger increase in aggregate output when weights are used from period 1 is owing to greater weight being given to increases in and lesser weight being given to increases in than when period 0 weights are used.

The chained output quantity index is formed by linking fixed weight quantity indices. In the above example, the percentage increase in aggregate output between period 0 and 1 is derived from the fixed-weight Laspeyres index constructed using fixed weights from period 0 (see panel III of Table 5). The percentage increase in the chained Laspeyres quantity index between period 1 and 2 is derived from the fixed-weight Laspeyres quantity index that was constructed using fixed weights from period 1.

More generally, a chained index is constructed as follows:

(7)    

where denotes the chained index between period 0 and t and the direct index between period t - 1 and t. Chaining can be applied to any of the index number formulae outlined in equations (2) to (5). Using the example presented in Table 5, the chained Laspeyres index in period 2 is calculated as follows:

(8)