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Practice Exercise · Q9
Q.

Based on the given data, check whether i) Paasche's formula and, ii) Fisher's formula will satisfy the time reversal test:

CommodityBase Year PriceBase Year QuantityCurrent Year PriceCurrent Year Quantity
P410615
Q615420
R85104
Sikkim CbseNCERTSubjective· 5mImportance★★★★★est
68% · 27/40 Questions
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Using the four sums ∑p1q0=170, ∑p0q0=170, ∑p1q1=210, ∑p0q1=212\sum p_1q_0=170,\ \sum p_0q_0=170,\ \sum p_1q_1=210,\ \sum p_0q_1=212: Paasche's index gives P01×P10=0.9906≠1P_{01}\times P_{10}=0.9906\neq1 (fails), while Fisher's gives P01×P10=1P_{01}\times P_{10}=1 (satisfies).

Time-reversal test (indices as ratios, i.e. omit the ×100\times100): P01×P10=1P_{01}\times P_{10}=1.

P01P=∑p1q1∑p0q1,P10P=∑p0q0∑p1q0P^{P}_{01}=\dfrac{\sum p_1q_1}{\sum p_0q_1},\quad P^{P}_{10}=\dfrac{\sum p_0q_0}{\sum p_1q_0}

P01F=∑p1q0∑p0q0⋅∑p1q1∑p0q1,P10F=∑p0q1∑p1q1⋅∑p0q0∑p1q0P^{F}_{01}=\sqrt{\dfrac{\sum p_1q_0}{\sum p_0q_0}\cdot\dfrac{\sum p_1q_1}{\sum p_0q_1}},\quad P^{F}_{10}=\sqrt{\dfrac{\sum p_0q_1}{\sum p_1q_1}\cdot\dfrac{\sum p_0q_0}{\sum p_1q_0}}

Working table

Commodityp0p_0q0q_0p1p_1q1q_1p1q0p_1q_0p0q0p_0q_0p1q1p_1q_1p0q1p_0q_1
P41061560409060
Q615420609080120
R8510450404032
Total170170210212
  1. Column sums: ∑p1q0=60+60+50=170\sum p_1q_0=60+60+50=170; ∑p0q0=40+90+40=170\sum p_0q_0=40+90+40=170; ∑p1q1=90+80+40=210\sum p_1q_1=90+80+40=210; ∑p0q1=60+120+32=212\sum p_0q_1=60+120+32=212.
  2. Paasche forward: P01P=∑p1q1∑p0q1=210212=0.99057P^{P}_{01}=\dfrac{\sum p_1q_1}{\sum p_0q_1}=\dfrac{210}{212}=0.99057.
  3. Paasche reversed (swap the roles of the years): P10P=∑p0q0∑p1q0=170170=1P^{P}_{10}=\dfrac{\sum p_0q_0}{\sum p_1q_0}=\dfrac{170}{170}=1.
  4. Product P01P×P10P=0.99057×1=0.99057≠1P^{P}_{01}\times P^{P}_{10}=0.99057\times1=0.99057\neq1 ⇒\Rightarrow Paasche does NOT satisfy the time-reversal test. …

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