If Laspeyre’s price index is $324$and Paasche’s price index is $144$ then Fisher’s ideal index is

1. Index numbers:

Index numbers are barometers of an economy. It is a specialised average designed to measure the changes in a group of variables over time.

2. Types of Index Numbers:

(i) Price Index Number:

Price index is a ‘Special type’ of average which studies net relative change in the prices of commodities, expressed in different units.

(ii) Quantity Index Number:

Quantity index numbers measures changes in volume of goods produced, the comparison is made in respect of quantity or volume.

(iii) Value Index Number.

Value index numbers study the changes in the total value of a certain period with the total value of the base period.

3. Types of Un-weighted index numbers:

(i) Simple aggregative:

$P_{01}=\frac{\sum p_1}{\sum p_0}\times 100$

(ii) Simple average of price relative method:

(a) Using arithmetic mean : $P_{01}=\frac{\Sigma \frac{{\rho }_1}{p_0}\times 100}{ \mathrm{N}}$

(b) Using geometric mean :$P_{01}=\mathrm{antilog}\frac{\sum \log \left(\frac{p_1}{p_0}\times 100\right)}{\mathrm{N}}$

4. Methods of Weighted aggregate index numbers:

(i) Laspeyeres index : $P_{01}^L=\frac{\sum p_1{q}_0}{\sum p_0{q}_0}\times 100$

(ii) Paasches index: $P_{01}^p=\frac{\sum p_1{q}_1}{\sum p_0{q}_1}\times 100$

(iii) Dorbish and Bowley index : $P_{01}^{DB}=\frac{P_{01}^L+P_{01}^P}{2}\times 100$

(iv) Fisher’s ideal index: $P^{DB}{}_{01}=\sqrt{P^L{}_{01}\times {P^P}_{01}}$

(v) Marshall-Edgeworth index :$P_{01}^{ME}=\frac{\sum p_1\left(q_0+q_1\right)}{\sum p_0\left(q_0+q_1\right)}\times 100$

(vi) Kelly’s index : $P_{01}^K=\frac{\sum p_{1}q}{\sum p_{0}q}\times 100, q=\frac{q_0+q_1}{2}$

5. Weighted average of price relatives:

(i) Weighted average price of relatives using arithmetic mean :

$P_{01}=\frac{\sum wp}{\sum w}\times 100$

(ii) Weighted average price of relatives using geometric mean :

$P_{01}=\text{ antilog }\left(\frac{\sum w\log p}{\sum w}\right)$

6. Base year:

The quantity index number measures the changes in the level of quantities of items consumed with reference to another year known as the base year.

7. Test satisfying index numbers:

(i) Time reversal test

(ii) Factor reversal

(iii) Circular test