This box illustrates the basic calculations for chain-type real GDP output
and price measures. The formula used to calculate the annual change in real GDP
is a "Fisher Ideal" formula ($Q^F^\_t\_$) that uses weights for 2 adjacent
years (years *t-1* and *t*)./1/ The formula for
the change in real GDP in year *t* relative to its value in year *t-1*
is

Q^F^_t_ =

where the *p*'s and *q*'s represent prices and quantities in the
2 years.

Because the first term in the Fisher Ideal formula is a Laspeyres quantity index ($Q\_t\_^L^$), or

$Q\_t\_^L^\; =\; \{\sum \; p\_t-1\_q\_t\_\sum \; p\_t-1\_q\_t-1\_\},$

and the second term is a Paasche quantity index ($Q\_t\_^P^$), or

Q_t_^P^ = {∑ p_t_q_t_

the Fisher formula can also be expressed for year *t* as the geometric
mean of these indexes as follows:

$Q\_t\_^F^\; =Q\_t\_^L^\; \times \; Q\_t\_^P^$.

The percent change in real GDP (or a GDP component) from year *t-1* to
year *t* is calculated as

$100(Q\_t\_^F^\; -\; 1.0).$

Similarly, price indexes are calculated using the Fisher Ideal formula,

$P\_t\_^F^\; =\sum \; p\_t\_q\_t-1\_\sum \; p\_t-1\_q\_t-1\_\}\; \times \; \sum \; p\_t\_q\_t\_\sum \; p\_t-1\_q\_t\_\}$, which is the geometric mean of a Laspeyres price index ($P\_t\_^L^$) and a Paasche price index ($P\_t\_^P^$), or

$P\_t\_^F^\; =P\_t\_^L^\; \times \; P\_t\_^P^.$

The chain-type real output and price indexes are presented with the base
year (*b*) equal to 100; that is, $I\_b\_\; =\; 100$. In general, the quantity index
value for period *t* is

$I\_t\_^F^\; =\; I\_t-1\_^F^\; \times \; Q\_t\_^F^,$

and the price index is calculated analogously.

The current-dollar change from year *t-1* to year *t* expressed
as a ratio is $\sum \; p\_t\_q\_t\_\; /\; \sum \; p\_t-1\_q\_t-1\_$. It is equal to the
product of the Fisher Ideal price and quantity indexes:/2/

$\{\sum \; p\_t\_q\_t\_\sum \; p\_t-1\_q\_t-1\_\}\; =\{\sum \; p\_t\_q\_t-1\_\sum \; p\_t-1\_q\_t-1\_\}\; \times \; \{\sum \; p\_t\_q\_t\_\sum \; p\_t-1\_q\_t\}\times \{\sum \; p\_t-1\_q\_t\_\sum \; p\_t-1\_q\_t-1\_\}\; \times \; \{\sum \; p\_t\_q\_t\_\sum \; p\_t\_q\_t-1\_\}.$

The chained-dollar value ($CD\_t\_^F^$) is calculated by multiplying the index
value by the base-period current-dollar value ($\sum \; p\_b\_q\_b\_$) and dividing
by 100. For period *t*,

$CD\_t\_^F^\; =\; \sum \; p\_b\_q\_b\_\; \times \; I\_t\_^F^\; /\; 100.$

The implicit price deflator ($IPD\_t\_^F^$) for period *t* is calculated
as the ratio of the current-dollar value to the corresponding chained-dollar
value multiplied by 100 as follows:

$IPD\_t\_^F^\; =\; \{\sum \; p\_t\_q\_t\_CD\_t\_^F^\}\; \times \; 100.$

1. Only annual weights are used in the calculations, including those for monthly and quarterly changes in real measures. The formulas shown apply only to annual estimates. Modified formulas are used for calculating quarterly estimates (see text).

2. See text footnote 9.