PaascheFormeln

PaascheFormeln refers to a set of index formulas used to measure changes in prices and quantities over time, named after the German economist Hermann Paasche. They are primarily applied in price statistics and national accounts to construct Paasche price and Paasche quantity indices, which are weighted by current-period data.

Paasche price index: P_P = (sum over goods i of p_i,t · q_i,t) / (sum over i of p_i,0 · q_i,t). Here prices in period t are multiplied by current-period quantities, which serve as weights. The index shows how much the total expenditure would change if prices move from the base period to the current period using the current basket of quantities.

Paasche quantity index: Q_P = (sum over i of p_i,t · q_i,t) / (sum over i of p_i,t · q_i,0). Weights are derived from current-period prices, and the numerator uses current prices with current quantities while the denominator uses current prices with base-period quantities.

For comparison, the Laspeyres counterparts use base-period quantities as weights: P_L = (sum p_i,t · q_i,0) / (sum p_i,0 · q_i,0) and Q_L = (sum p_i,0 · q_i,t) / (sum p_i,0 · q_i,0). Because of weighting, Paasche indices tend to reflect substitutions toward cheaper goods and often show smaller price changes than Laspeyres indices, while Laspeyres may overstate them.

In practice, PaascheFormeln are used alongside Laspeyres and other indices to bound true changes in price levels and real quantities. They are also employed in chain indices to update weights over time, providing a dynamic measure of inflation and real output. See also Fisher index, Laspeyres index, and chain indices.