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Worked Examples · Example 8
Q.

Calculate Fisher's price index number for the given data and verify that it satisfies the time-reversal test.

CommodityPrice 2008 (p0p_0)Price 2012 (p1p_1)Quantity 2008 (Q0Q_0)Quantity 2012 (Q1Q_1)
Rice101346
Wheat151878
Rent252959
Fuel1114810
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Build the four aggregate sums, then Fisher =L×P×100≈121.24=\sqrt{L\times P}\times100\approx121.24; since P01×P10=1P_{01}\times P_{10}=1 the time-reversal test holds.

P01F=∑p1q0∑p0q0×∑p1q1∑p0q1×100P_{01}^{F}=\sqrt{\frac{\sum p_1q_0}{\sum p_0q_0}\times\frac{\sum p_1q_1}{\sum p_0q_1}}\times100

Time-reversal test: P01×P10=1P_{01}\times P_{10}=1 (indices as ratios, without the ×100\times100), where P10=∑p0q1∑p1q1×∑p0q0∑p1q0P_{10}=\sqrt{\dfrac{\sum p_0q_1}{\sum p_1q_1}\times\dfrac{\sum p_0q_0}{\sum p_1q_0}}.

  1. Build the working table of p1q0, p0q0, p1q1, p0q1p_1q_0,\ p_0q_0,\ p_1q_1,\ p_0q_1.
Commodityp0p_0p1p_1q0q_0q1q_1p1q0p_1q_0p0q0p_0q_0p1q1p_1q_1p0q1p_0q_1
Rice10134652407860
Wheat151878126105144120
Rent252959145125261225
Fuel111481011288140110
Total435\mathbf{435}358\mathbf{358}623\mathbf{623}515\mathbf{515}
  1. Aggregate sums: ∑p1q0=435, ∑p0q0=358, ∑p1q1=623, ∑p0q1=515.\sum p_1q_0=435,\ \sum p_0q_0=358,\ \sum p_1q_1=623,\ \sum p_0q_1=515.

  2. Laspeyres L=∑p1q0∑p0q0×100=435358×100=121.51.L=\dfrac{\sum p_1q_0}{\sum p_0q_0}\times100=\dfrac{435}{358}\times100=121.51.

  3. Paasche P=∑p1q1∑p0q1×100=623515×100=120.97.P=\dfrac{\sum p_1q_1}{\sum p_0q_1}\times100=\dfrac{623}{515}\times100=120.97. …

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