Question

For the given data, Obtain the Compute Index number using Fisher’s formula and show it satisfies time reversal test and factor reversal test

commodity	Base year		current year
commodity	Prices	Quantity	Price	Quantity
A	$10$	$12$	$12$	$15$
B	$7$	$15$	$5$	$20$
C	$5$	$24$	$9$	$20$
D	$16$	$5$	$14$	$5$

Accepted Answer

Given that,

commodity Base year Current year

Prices Quantity Price Quantity

A $10$ $12$ $12$ $15$

B $7$ $15$ $5$ $20$

C $5$ $24$ $9$ $20$

D $16$ $5$ $14$ $5$

Let $p_{0}, p_{1}, q_{0}, q_{1}$ are the prices and quantities of commodities.

Tabular form:

commodity Base year Current year

Prices $(p_{0})$ Quantity $(q_{0})$ Price $(p_{1})$ Quantity $q_{1}$ $p_{0} q_{0}$ $p_{1} q_{0}$ $p_{0} q_{1}$ $p_{1} q_{1}$

A $10$ $12$ $12$ $15$ $120$ $144$ $150$ $180$

B $7$ $15$ $5$ $20$ $105$ $75$ $140$ $100$

C $5$ $24$ $9$ $20$ $120$ $216$ $100$ $180$

D $16$ $5$ $14$ $5$ $80$ $70$ $80$ $70$

$Σ p_{0} q_{0} = 425$ $Σ p_{1} q_{0} = 505$ $Σ p_{0} q_{1} = 470$ $Σ p_{1} q_{1} = 530$

We know that, Laspeyre’s quantity index formula $Q_{01}^{L} = \frac{\sum p_{0} q_{1}}{\sum p_{0} q_{0}} \times 100$ , and Paasche’s quantity index formula $Q_{01}^{P} = \frac{\sum p_{1} q_{1}}{\sum p_{1} q_{0}} \times 100$ .
We know that, Fisher's quantity index number $Q_{01}^{F} = \sqrt{Q_{01}^{L} \times Q_{01}^{P}}$

$\Rightarrow \sqrt{110.58 \times 104.9} = 107.73$

We know that, Factor Reversal test $P_{01} \times Q_{01} = \frac{\sum p_{1} q_{1}}{\sum p_{0} q_{0}}$ , where $P_{01} = \sqrt{\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}} \times \frac{\sum p_{1} q_{1}}{\sum p_{0} q_{1}},} Q_{01} = \sqrt{\frac{\sum q_{1} p_{0}}{\sum q_{0} p_{0}} \times \frac{\sum q_{1} p_{1}}{\sum q_{0} p_{1}}}$

$\Rightarrow P_{01} \times Q_{01} = \sqrt{\frac{505}{425} \times \frac{530}{470} \times \frac{470}{425} \times \frac{530}{505}} = \frac{530}{425} = \frac{Σ p_{1} q_{1}}{Σ p_{0} q_{0}}$

We know that, Time reversal test $P_{01} \times P_{10} = 1$ , where $P_{10} = \sqrt{\frac{\sum p_{0} q_{1}}{\sum p_{1} q_{1}} \times \frac{\sum p_{0} q_{0}}{\sum p_{1} q_{0}}}, P_{01} = \sqrt{\frac{\sum p_{1} q_{0}}{\sum p_{0} q_{0}} \times \frac{\sum p_{1} q_{1}}{\sum p_{0} q_{1}}}$

$\Rightarrow \sqrt{\frac{505}{425} \times \frac{530}{470} \times \frac{470}{530} \times \frac{425}{505}} = 1$ .

$∴$ It satisfies time reversal test and factor reversal test.

Item	Price Rs in $2014$	Price Rs in $2015$
A	$6$	$10$
B	$2$	$2$
C	$4$	$6$
D	$10$	$12$
E	$8$	$12$

Items	year $2014$		year $2015$
Items	Price	Quantity	Price	Quantity
A	$6$	$50$	$10$	$56$
B	$2$	$100$	$2$	$120$
C	$4$	$60$	$6$	$60$
D	$10$	$30$	$12$	$24$
E	$8$	$40$	$12$	$36$

Item	Price Rs in $2014$	Price Rs in $2015$
A	$6$	$10$
B	$2$	$2$
C	$4$	$6$
D	$10$	$12$
E	$8$	$12$

Commodity	Year $2016$		Year $2017$
Commodity	Price	Quantity	Price	Quantity
A	$4$	$58500$	$6$	$62000$
B	$3.5$	$15630$	$5.5$	$13050$
C	$3$	$26230$	$5$	$25000$
D	$2.5$	$11360$	$4$	$10000$
E	$2$	$30000$	$3$	$31500$

Items	year $2014$		year $2015$
Items	Price	Quantity	Price	Quantity
A	$6$	$50$	$10$	$56$
B	$2$	$100$	$2$	$120$
C	$4$	$60$	$6$	$60$
D	$10$	$30$	$12$	$24$
E	$8$	$40$	$12$	$36$

For the given data, Obtain the Compute Index number using Fisher’s formula and show it satisfies time reversal test and factor reversal test

commodity Base year current year

Prices Quantity Price Quantity

A $10$ $12$ $12$ $15$

B $7$ $15$ $5$ $20$

C $5$ $24$ $9$ $20$

D $16$ $5$ $14$ $5$

Important Questions on Index Numbers

Learn and Prepare for any exam you want !

Commodity	$2010$		$2011$
Commodity	Price in Rs	Quantity	Price in Rs	Quantity
A	$15$	$15$	$22$	$12$
B	$20$	$5$	$27$	$4$
C	$4$	$10$	$7$	$5$

Item	Price Rs in $2006$	Price Rs in $2007$	quantity
A	$2$	$2.5$	$40$
B	$3$	$3.25$	$20$
C	$1.5$	$1.75$	10

Items	Jan $2015$		Jan $2018$
Items	Prices Rs Per kg	Quantity sold kg	Price Rs Per kg	Quantity sold kg
Veg oil	$26$	$40$	$31$	$45$
Sugar	$28$	$90$	$32$	$100$
Rice	$16$	$120$	$19$	$20$
Wheat	$15$	$110$	$18$	$130$

Commodity	$2000$		$2001$
Commodity	Price	Quantity	Price	Quantity
A	$2$	$7$	$3$	$5$
B	$5$	$11$	$6$	$10$
C	$3$	$14$	$5$	$11$
D	$4$	$16$	$4$	$18$

For the given data, Obtain the Compute Index number using Fisher’s formula and show it satisfies time reversal test and factor reversal test commodity Base year current year Prices Quantity Price Quantity A 10 12 12 15 B 7 15 5 20 C 5 24 9 20 D 16 5 14 5

Important Questions on Index Numbers

Learn and Prepare for any exam you want !

For the given data, Obtain the Compute Index number using Fisher’s formula and show it satisfies time reversal test and factor reversal test

commodity Base year current year

Prices Quantity Price Quantity

A $10$ $12$ $12$ $15$

B $7$ $15$ $5$ $20$

C $5$ $24$ $9$ $20$

D $16$ $5$ $14$ $5$