Banerjee’s Factorial Index Numbers

The underlying idea of Banerjee’s so-called factorial approach to the construction of index numbers1 appears to be that if commodity aggregates, like single commodities, had a price measure and a volume measure of their own, then these measures (which we shall denote by symbols with an asterisk) should be such that P j * Q k * = Σ P j i Q k i ( j , k = 0 , 1 ) $$P_j^*Q_k^* = \Sigma P_j^iQ_k^i\quad \left( {j,k = 0,1} \right)$$ and the index numbers of price and volume based on them should be P = P 1 * / P 0 * a n d Q = Q 1 * / Q 0 * $$P = P_1^*/P_0^*\;and\;Q = Q_1^*/Q_0^*$$ Forming (as in factorial design analysis) the products [ P 1 * + P 0 * ] [ Q 1 * + Q 0 * ] = a [ P 1 * − P 0 * ] [ Q 1 * − Q 0 * ] = b [ P 1 * + P 0 * ] [ Q 1 * + Q 0 * ] = c [ P 1 * − P 0 * ] [ Q 1 * − Q 0 * ] = d $$\begin{gathered} \left[ {P_1^* + P_0^*} \right]\left[ {Q_1^* + Q_0^*} \right] = a \hfill \\ \left[ {P_1^* - P_0^*} \right]\left[ {Q_1^* - Q_0^*} \right] = b \hfill \\ \left[ {P_1^* + P_0^*} \right]\left[ {Q_1^* + Q_0^*} \right] = c \hfill \\ \left[ {P_1^* - P_0^*} \right]\left[ {Q_1^* - Q_0^*} \right] = d \hfill \\ \end{gathered}$$ and dividing them by V 0 = P 0 * Q 0 * = Σ P 0 i Q 0 i $${V_0} = P_0^*Q_0^* = \Sigma P_0^iQ_0^i$$ Banerjee then concludes that the aggregate price and volume indices should be such that P j * Q k * = Σ P j i Q k i ( j , k = 0 , 1 ) $$P_{j}^{*}Q_{k}^{*}=\Sigma P_{j}^{i}Q_{k}^{i}\quad \left( {j,k=0,1} \right)$$ It is at this point that it becomes clear that the underlying idea of the factorial approach is erroneous, because there exist no P, Q pair that can satisfy simultaneously all four of these relations.