Tests of Adequacy — Why Fisher's Index Is Called "Ideal"

A price index formula can be judged against certain purely ARITHMETIC consistency checks, independent of which actual numbers go into it. Two of these, both satisfied by Fisher's Ideal Index but NOT by Laspeyres or Paasche alone, are why it earns the name "ideal" in the Maharashtra HSC (MSBSHSE) Std XII Economics syllabus.

Time Reversal Test. If the base and current years are SWAPPED, the index computed "backward" ($P_{10}$) should be the exact reciprocal of the index computed "forward" ($P_{01}$), so that:

Fisher's Ideal Index satisfies this exactly, because $F_{01}=\sqrt{L_{01}P_{01}}$ and $F_{10}=\sqrt{L_{10}P_{10}}$, and it can be shown algebraically that $L_{01}=1/P_{10}$ and $P_{01}=1/L_{10}$ — so the two square roots are automatically exact reciprocals of each other. Neither Laspeyres alone nor Paasche alone passes this test on its own.

Factor Reversal Test. Multiplying a PRICE index and the CORRESPONDING quantity index (the same formula, with $p$ and $q$ interchanged) should reproduce exactly the VALUE index, $V_{01}=(\sum p_1q_1/\sum p_0q_0)\times100$:

$P_{01}\times Q_{01} = V_{01} \quad(\text{as pure ratios})$ …