The Relation Between P S and P F

Törnqvist (1958) pointed out in a comment on the new index numbers that, while they have the property P L − Q L = P S − Q S , P S Q S = V $${P_L} - {Q_L} = {P_S} - {Q_S},{P_S}{Q_S} = V$$ the Fisher index numbers have the property P L / Q L = P F / Q F , P F Q F = V $${P_L}/{Q_L} = {P_F}/{Q_F},{P_F}{Q_F} = V$$ This interesting observation led him to suspect that in practice P S will be almost equal to P F . In order to verify that this indeed would normally be the case, he developed a formula for the difference between P S and P F . This formula should have been P S − P F = ( P L − Q L ) ( P F − P P ) P S + Q L − ( P L − P F ) $${P_S} - {P_F} = \frac{{\left( {{P_L} - {Q_L}} \right)\left( {{P_F} - {P_P}} \right)}}{{{P_S} + {Q_L} - \left( {{P_L} - {P_F}} \right)}}$$ but inadvertently he replaced the (P F − P P ) in the numerator by (P L − P F ). Using his formula he found that P S − P F = 0.0053 for P L − P F = 0.1, P L − Q L = 0.1 and P S + Q L = 2. (The correct formula gives the same result for P F − P P = 0.1, P L − Q L = 0.1 and (P S + Q L ) − (P L + P F ) = Q S + P F = 1.9.) On the basis of this finding Törnqvist concluded: ‘A difference of the order of size of 10 per cent between the price index of Laspeyres and Fisher occurs only in rare practical cases. Stuvel’s index will thus rarely differ from Fisher’s ideal index by an amount large enough to be seen in the figures used for presenting the results.’