Question

Find the following data calculate fisher's ideal inderx and proves that it satisfies TRT and FRT

baseyear current year

Commodity price Qty price Qty

A 4 60 6 60

B 10 30 12 24

C 6 50 10 56

D 2 100 2 120

Answer 1

Fisher's ideal Index is

$=\sqrt{\dfrac{\sum P_1Q_o}{\sum P_oQ_o}\times \dfrac{\sum P_1Q_1}{\sum P_oQ_1}}\times 100\\[9pt]=\sqrt{\dfrac{1420}{1040}\times \dfrac{1148}{1056}}\times 100=1.3679\times 100=136.79$

$TRT\implies P_{o1}\times P_{1o}=1$

LHS-

$=\sqrt{\dfrac{\sum P_1Q_o}{\sum P_oQ_o}\times \dfrac{\sum P_1Q_1}{\sum P_oQ_1}}\times \sqrt{\dfrac{\sum P_oQ_1}{\sum P_1Q_1}\times \dfrac{\sum P_o Q_o}{P_1 Q_o}}\\[9pt]=\sqrt{1}=1=RHS$

TRT satisfied.

$FRT\implies P_{o1}\times Q_{o1}=\dfrac{\sum P_1Q_1}{\sum P_o Q_o}$

So LHS $=\sqrt{\dfrac{\sum P_1Q_o}{\sum P_oQ_o}\times \dfrac{\sum P_1Q_1}{\sum P_oQ_1}}\times \sqrt{\dfrac{\sum Q_1P_o}{\sum P_oQ_o}\times \dfrac{\sum Q_1P_1}{\sum Q_oP_1}}$

$=\sqrt{\dfrac{( 1448)^2}{(1040)^2}}=\dfrac{1148}{1040}=\dfrac{\sum P_1Q_1}{\sum P_oQ_o}$ =RHs

FRT Satisfied.