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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 p_{o}, and a Paasche index, which evaluates the variation in volume for the same component at price p_{1}. An additive decomposition of a Fisher index using a weighted average for prices p_{o} and p_{1} to evaluate the variation in volume for component i is therefore justified. In general, such a decomposition can take the following form:
(1) 
where G(t) represents some kind of index. The solution for λ can be deducted as follows:
(1a) 
(1b)  => 

(1c)  => 

(1d)  => 

thus:
(2) 
.Where G(t) is given by a Laspeyres index, we would have:
(2a) 
which leads immediately to λ=0.
Thus, for a Laspeyres index, the contribution %Δ _{i,t1/t} is given by (by subtracting 1 and multiplying formula (1) by 100):
(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:
(2b) 
where FV_{t} is the Fisher volume index at time t. By dividing the numerator and denominator by Σp_{t}q_{t} , we obtain:
(2c) 
or:
(2d) 
where PP_{t} is the Paasche price index at time t, and FP_{t} is the Fisher price index at time t. If we multiply (2d) by FP_{t} at the numerator and denominator, we obtain:
(2e) 
which, multiplied by PP_{t} at the numerator and denominator, becomes:
(2f) 

by reducing and using the equivalence PP_{t}LP_{t} = FP_{t}^{2}, we obtain:
(2g) 

Equation (1) then becomes:
(4) 

To obtain the contribution of a single component to the percentage growth of the aggregate, we take FV_{t}1 and multiply by 100:
(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 FV_{t}FP_{t}:
(4b) 

or:
(4c) 

by consolidating the sums of the values in current dollars:
(5) 

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