# Box: Computation of the Chain-Type Quantity Indexes for Double-Deflated Industries

For this comprehensive revision, BEA introduces annual chain-type quantity indexes as the measure of real gross output, intermediate inputs, and GPO for industries and industry groups. Each link in the chain-type quantity index is a Fisher quantity index for two adjacent years. Each annual Fisher quantity index, in turn, is the geometric mean of Laspeyeres and Paasche quantity indexes for the two adjacent years.

The formulas below summarize the computation of the Fisher chain-type quantity indexes of real gross output, intermediate inputs, and GPO for an industry or industry group. In the notation, L refers to the Laspeyeres quantity index; P refers to the Paasche quantity index; F refers to the Fisher quantity index; and C refers to the Fisher chain-type quantity index. The subscripts indicate time periods; $L_t-1,t_$ is the Laspeyeres quantity index for the two adjacent years, t-1 and t. The superscript GO refers to gross output; II refers to intermediate inputs; and GPO refers to gross product originating. Lowercase p and q refer to detailed prices and quantities, respectively.

Laspeyeres quantity indexes for gross output, intermediate inputs, and GPO, respectively, are

$L^GO^_t-1,t_ = \left\{\sum p^GO^_t-1_ q^GO^_t_\sum p^GO^_t-1_ q^GO^_t-1_\right\},$

$L^II^_t-1,t_ = \left\{\sum p^II^_t-1_ q^II^_t_\sum p^II^_t-1_ q^II^_t-1_\right\},$ and

$L^GPO^_t-1,t_ = \left\{\left(\sum p^GO^_t-1_ q^GO^_t_\right) - \left(\sum p^II^_t-1_ q^II^_t_\right)\left(\sum p^GO^_t-1_ q^GO^_t-1_\right) - \left(\sum p^II^_t-1_ q^II^_t-1_\right)\right\}.$

Paasche quantity indexes for gross output, intermediate inputs, and GPO are

$P^GO^_t-1,t_ = \left\{\sum p^GO^_t_ q^GO^_t_\sum p^GO^_t_ q^GO^_t-1_\right\},$

$P^II^_t-1,t_ = \left\{\sum p^II^_t_ q^II^_t_\sum p^II^_t_ q^II^_t-1_\right\},$ and

$P^GPO^_t-1,t_ = \left\{\left(\sum p^GO^_t_ q^GO^_t_\right) - \left(\sum p^II^_t_ q^II^_t_\right)\left(\sum p^GO^_t_ q^GO^_t-1_\right) - \left(\sum p^II^_t_ q^II^_t-1_\right)\right\}.$

Fisher quantity indexes for gross output, intermediate inputs, and GPO are

$F^GO^_t-1,t_ =L^GO^_t-1,t_ × P^GO^_t-1,t_,$

$F^II^_t-1,t_ =L^II^_t-1,t_ × P^II^_t-1,t_,$ and

$F^GPO^_t-1,t_ =L^GPO^_t-1,t_ × P^GPO^_t-1,t_.$

Fisher chain-type quantity indexes for gross output, intermediate inputs, and GPO for years following the base year are

$C^GO^_t_ = C^GO^_t-1_ × F^GO^_t-1,t_,$

$C^II^_t_ = C^II^_t-1_ × F^II^_t-1,t_,$and

$C^GPO^_t_ = C^GPO^_t-1_ × F^GPO^_t-1,t_$.

In the base year (1992 for this comprehensive revision),

$C^GO^_t_ = C^II^_t_ = C^GPO^_t_ = 100.$

The above formulas are applied to GPO industries, to industry groups such as durable goods manufacturing, and to aggregates such as private industries.

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