# Box: Basic Formulas for Calculating Chain-Type Real Output and Price Measures

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_ = {∑ p_t-1_q_t_ ∑ p_t-1_q_t-1_} × {∑ p_t_q_t_ ∑ p_t_q_t-1_},

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

$Q_t_^L^ = \left\{\sum p_t-1_q_t_\sum p_t-1_q_t-1_\right\},$

and the second term is a Paasche quantity index ($Q_t_^P^$), or

Q_t_^P^ = {∑ p_t_q_t_ ∑ p_t_q_t-1_},

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

$Q_t_^F^ =Q_t_^L^ × Q_t_^P^$.

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

$100\left(Q_t_^F^ - 1.0\right).$

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

$P_t_^F^ =\sum p_t_q_t-1_\sum p_t-1_q_t-1_\right\} × \sum p_t_q_t_\sum p_t-1_q_t_\right\}$, which is the geometric mean of a Laspeyres price index ($P_t_^L^$) and a Paasche price index ($P_t_^P^$), or

$P_t_^F^ =P_t_^L^ × P_t_^P^.$

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

$I_t_^F^ = I_t-1_^F^ × Q_t_^F^,$

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/

$\left\{\sum p_t_q_t_\sum p_t-1_q_t-1_\right\} =\left\{\sum p_t_q_t-1_\sum p_t-1_q_t-1_\right\} × \left\{\sum p_t_q_t_\sum p_t-1_q_t\right\}×\left\{\sum p_t-1_q_t_\sum p_t-1_q_t-1_\right\} × \left\{\sum p_t_q_t_\sum p_t_q_t-1_\right\}.$

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,

$CD_t_^F^ = \sum p_b_q_b_ × I_t_^F^ / 100.$

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:

$IPD_t_^F^ = \left\{\sum p_t_q_t_CD_t_^F^\right\} × 100.$

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.