Factor Reversal Test (FRT) 🔄

Proposed by Irving Fisher, this is the second most important test for index numbers.

The Concept 💡

An index number formula should permit the interchange of Prices (P) and Quantities (Q) without inconsistent results.

Wait, what does that mean? It means:

Price Index * Quantity Index = Value Index

Just like in real life: Price per unit * Quantity sold = Total Value

The Condition 📝

The product of the Price Index (P_01) and Quantity Index (Q_01) should equal the ratio of total value of the current period to the base period.

P_01 * Q_01 = (∑ p1 q1) / (∑ p0 q0) = V_01

• P01: Price Index formula (prices change, quantities are weights).
• Q01: Quantity Index formula (quantities change, prices are weights - calculated by swapping p and q in the P01 formula).

Use Case: Fisher's Index 🏆

Only Fisher’s Ideal Index satisfies this test.

Proof:

1. Fisher's Price Index (P_01):

   √[ (∑ p1 q0 / ∑ p0 q0) * (∑ p1 q1 / ∑ p0 q1) ]
   

2. Fisher's Quantity Index (Q_01): (Swap p and q in the above formula)

   √[ (∑ q1 p0 / ∑ q0 p0) * (∑ q1 p1 / ∑ q0 p1) ]
   

3. Multiply (P_01 * Q_01):

   √[ (∑ p1 q0 / ∑ p0 q0) * (∑ p1 q1 / ∑ p0 q1) * (∑ q1 p0 / ∑ q0 p0) * (∑ q1 p1 / ∑ q0 p1) ]
   

   (Rearranging terms...)

   • ∑ p1 q0 cancels ∑ q0 p1.
   • ∑ p0 q1 cancels ∑ q1 p0.
   • We are left with:

   √[ (∑ p1 q1 / ∑ p0 q0) * (∑ p1 q1 / ∑ p0 q0) ]
   = (∑ p1 q1 / ∑ p0 q0)
   

   (Which is Value Index)

Result: Passed! ✅

Summary

Note: Laspeyres, Paasche, and Marshall-Edgeworth DO NOT pass this test.

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