Verify that Fisher's Ideal Index Number satisfies the Factor Reversal Test, for the same data.

Exercises · Q7

Q. Verify that Fisher's Ideal Index Number satisfies the Factor Reversal Test, for the same data.

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✓ Free question

Step 1 — Compute Fisher's quantity index $Q_{01}$ (swap $p\leftrightarrow q$ in the Fisher formula).

Q_{01}^{F}=\sqrt{\dfrac{\Sigma q_1p_0}{\Sigma q_0p_0}\times\dfrac{\Sigma q_1p_1}{\Sigma q_0p_1}}\times100=\sqrt{\dfrac{400}{300}\times\dfrac{461}{347}}\times100=\sqrt{1.7714}\times100\approx133.09

Step 2 — Multiply the price index ratio and the quantity index ratio.

P_{01}^{F}\times Q_{01}^{F}=1.1546\times1.3309\approx1.5367

Step 3 — Compute the true value ratio directly, as an independent dual-check.

\dfrac{\Sigma p_1q_1}{\Sigma p_0q_0}=\dfrac{461}{300}=1.5367

Both routes agree at 1.5367.

✓Final answer

P_{01}^{F}\times Q_{01}^{F}\approx1.5367=\dfrac{\Sigma p_1q_1}{\Sigma p_0q_0}

Q.Verify that Fisher's Ideal Index Number satisfies the Factor Reversal Test, for the same data.