Time reversal test is not satisfied by: A. Laspeyres’ price index num…

The correct answer is Option (2) → A, C only

An index number satisfies the time reversal test when \(P_{01} \times P_{10} = 1\) (indices written as ratios, without the factor 100). Reversing time swaps the suffixes 0 and 1.

Simple aggregative index:

    \( \displaystyle P_{01} \times P_{10} = \frac{\sum p_1}{\sum p_0} \times \frac{\sum p_0}{\sum p_1} = 1 \)

Laspeyres’ index (base-year quantities as weights):

    \( \displaystyle P_{01} \times P_{10} = \frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_0 q_1}{\sum p_1 q_1} \)

which is not 1 in general, because the weights change from \(q_0\) to \(q_1\) when time is reversed.

Paasche index (current-year quantities as weights):

    \( \displaystyle P_{01} \times P_{10} = \frac{\sum p_1 q_1}{\sum p_0 q_1} \times \frac{\sum p_0 q_0}{\sum p_1 q_0} \)

which is not 1 in general, for the same reason.

Fisher’s index is the square root of the product of the Laspeyres’ and Paasche indices, so

    \( \displaystyle P_{01} \times P_{10} = \sqrt{\frac{\sum p_1 q_0}{\sum p_0 q_0} \times \frac{\sum p_1 q_1}{\sum p_0 q_1} \times \frac{\sum p_0 q_1}{\sum p_1 q_1} \times \frac{\sum p_0 q_0}{\sum p_1 q_0}} \)

    \( = \sqrt{1} = 1 \)

So the test holds for Fisher’s and the simple aggregative index, and fails for Laspeyres’ and Paasche indices.

Checking the statements

(A) Laspeyres’ price index number — Correct. It fails the test: its weights change from \(q_0\) to \(q_1\) when time is reversed.

(B) Fisher’s price index number — Incorrect. All four factors under the root cancel, so \(P_{01} \times P_{10} = 1\).

(C) Paasche price index number — Correct. It fails the test: its weights change from \(q_1\) to \(q_0\) when time is reversed.

(D) Simple aggregative price index number — Incorrect. \(\frac{\sum p_1}{\sum p_0} \times \frac{\sum p_0}{\sum p_1} = 1\), so it satisfies the test.

Note: This topic was not in the CUET 2026 syllabus.