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Chain Fisher volume index - Methodology

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Contribution to growth formula

The axiomatic approach suggested by Yuri Dikhanov (1997) and Christian Ehemann (1997) of the Bureau of Economic Analysis is based on the fact that a Fisher volume index is an average of a Laspeyres index, which evaluates the variation in a quantity for a component i at price  po, and a Paasche index, which evaluates the variation in volume for the same component at price p1. An additive decomposition of a Fisher index using a weighted average for prices po and p1 to evaluate the variation in volume for component i is therefore justified. In general, such a decomposition can take the following form:

Equation 1

where G(t) represents some kind of index. The solution for λ can be deducted as follows:

Equation 1a
(1b) =>

Equation 1b

(1c) =>

Equation 1c

(1d) =>

Equation 1d


Equation 2

.Where G(t) is given by a Laspeyres index, we would have:

Equation 2a

which leads immediately to λ=0.  

Thus, for a Laspeyres index, the contribution %Δ i,t-1/t is given by (by subtracting 1 and multiplying formula (1) by 100):

Equation 3

which takes us back to a Laspeyres formula as described by the Equation (3) of the text. This result is understandable in light of the additive consistency of the Laspeyres index.

 Where G(t) is given by a Fisher index, we have:

Equation 2b

where FVt is the Fisher volume index at time t. By dividing the numerator and denominator by Σptqt , we obtain:

Equation 2c


Equation 2d

where PPt is the Paasche price index at time t, and FPt is the Fisher price index at time t. If we multiply (2d) by FPt at the numerator and denominator, we obtain:


Equation 2e

which, multiplied by PPt at the numerator and denominator, becomes:


Equation 2f

by reducing and using the equivalence PPtLPt = FPt2, we obtain:


Equation 2g

Equation (1) then becomes:


Equation 4

To obtain the contribution of a single component to the percentage growth of the aggregate, we take FVt-1 and multiply by 100:


Equation 4a

This formula is sometimes difficult to operationalize given that it is expressed in terms of price and quantity. For a version expressed in terms of current dollar values and prices (as used by the IEAD), we can multiply the numerator and denominator by FVtFPt:


Equation 4b



by consolidating the sums of the values in current dollars:


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