# Box: Calculation of Component Contributions to the Change in GDP

### Changes from preceding period

The contributions to the change in real GDP provide a measure of the composition of GDP growth that is not affected by the nonadditivity of the GDP components. Two formulas for the contributions of components to the percent change in real GDP are used—one for years following the base year, and the other for the base year and for years preceding the base year.

For years following the base year, the contribution to the percent change in real GDP in year t attributable to the quantity change in component i ($C%\Delta _i,t_$) is

$C%\Delta _i,t_ = 100 × \left\{\left(p_i,t-1_ P_t_^P^ + p_i,t_\right) × \left(q_i,t_ - q_i,t-1_\right)2 × IPD_t_^F^ × CD_t-1_^F^\right\},$

where $CD_t-1_^F^$ is the chained-dollar GDP in year t-1;

$IPD_t_^F^$ is the implicit price deflator for GDP in year t;

$p_i,t_$ is the price of component i in year t;

$P_t_^P^$ is the Paasche price index for GDP in year t;

and

$q_i,t_$ is the quantity of component i in year t.

For the base year and years preceding the base year, $C%\Delta _i,t_$ is $C%\Delta _i,t_ = 100 × \left\{&left;\left( \left\{p_i,t_P_t_^L^\right\} + p_i,t-1_&right;\right) × \left(q_i,t_ - q_i,t-1_\right)2 × IPD_t-1_^F^ × CD_t-1_^F^\right\} ,$

where $P_t_^L^$ is the Laspeyres price index for GDP in year t.

Because these contributions to the percent change are additive, they can be used to calculate the contributions to subaggregates as well as the contributions of the detailed components.

The formula used for the contributions for years after the base period can also be used for the most recent quarters (the "tail" period), except the quarterly results are expressed at annual rates./1/

### Changes over other periods

Users can also prepare close approximations of contributions to real GDP growth or to the growth of other aggregates using chain-type annual-weighted indexes. In effect, users compute a chained-dollar series for a particular period using the percent changes in the chain-type annual-weighted indexes to compute chained-dollar series indexed to the current dollars of the base period appropriate for the analysis. (For references to additional information on these calculations, see footnote 9 in the text.)

Footnotes:

1. The formulas must be modified for other quarters and for the most recent year because for these periods, chained output is calculated in a slightly different manner.

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