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  p_o, 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 p_o and p₁ 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,t-1/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_tq_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_tLP_t = FP_t², 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_tFP_t:

(4b)

or:

(4c)

by consolidating the sums of the values in current dollars:

(5)

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