Question
The quantity consumed and total expenditure of four different items are as given below. Find Paasche’s and Fisher’s index number for the year $2015$ with respect to the year $2013.$
Item Base year $2013$ Current year $2015$
  Total expenditure
(Rs.)
Consumption (Quantity) Total expenditure
(Rs.)
Consumption (Quantity)
$A$ $360$ $60 \ kg$ $375$ $25 \ kg$
$B$ $160$ $10$ litre $416$ $30$ litre
$C$ $480$ $15 \ kg$ $613.2$ $6 \ kg$
$D$ $336$ $3 \ kg$ $400$ $2.5 \ kg$

Answer

Here, quantity (consumption) of items and their total expenditure are given. We obtain the price per unit of item using the formula, Price per unit $=\frac{\text { Total expenditure }}{\text { Quantity }}$
We take $p_0=$ Price in $2013,$
$p_1 =$ Price in $2015,$
$q =$ Quantity in $2013$ and
$q_1=$ Quantity in $2015.$
 The table for calculation is prepared as follows :
Image
Paasche's index number:
$ I_p=\frac{\Sigma p_1 q_1}{\Sigma p_0 q_1}$
$=\frac{1804.2}{942}$
$=1.53$
Fisher's index number:
$\mathrm{I}_{\mathrm{F}}=\sqrt{\frac{\Sigma p_1 q_0}{\Sigma p_0 q_0} \times \frac{\Sigma p_1 q_1}{\Sigma p_0 q_1}} \times 100$
$=\sqrt{\frac{3121}{1336} \times \frac{1804.2}{942}} \times 100$
$=\sqrt{\frac{5630908.2}{1258512}} \times 100$
$=\sqrt{4.4742} \times 100$
$=2.1152 \times 100$
$=211.52$

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