Ideal Indexes

Abstract

Among many index numbers, the two most favoured because of algebraic simplicity and ease of computation are those advocated by E. Laspeyres in 1864 and by H. Paasche in 1874. There are n commodities, indexed from 1 to n. At time point t, the price vector is p_t = {p_1t,…, p_nt} and the quantity vector q_t = {q_1t,…, q_nt}. p_sq_t denotes ${\sum}_{i=1}^n{p}_{is}{q}_{it}$. Let P_st and Q_st be the price and quantity indexes from time s to t. Then, these two indexes are

$$ {\displaystyle \begin{array}{cc}\hfill \mathrm{Laspeyres}{P}_{st}^L={p}_t{q}_s/{p}_s{q}_s,\hfill & \hfill {Q}_{st}^L={p}_s{q}_t/{p}_s{q}_s\hfill \\ {}\hfill \mathrm{Paasche}{P}_{st}^P={p}_t{q}_t/{p}_s{q}_t,\hfill & \hfill {Q}_{st}^P={p}_t{q}_t/{p}_t{q}_s\hfill \end{array}} $$