Question

Using the following data, construct Fisher’s Ideal index and show how it satisfies Factor Reversal Test and Time Reversal Test?

Commodity	Price in Rupees per unit		Number of units
	Basic year	Current year	Base year	Current year
A	6	10	50	56
B	2	2	100	120
C	4	6	60	60
D	10	12	50	24
E	8	12	40	36

Answer 1

Commodity	Base year		Current year		p0q0	p0q1	p1q0	p1q1
	p0	q0	p1	q1
A	6	10	50	56	300	336	500	560
B	2	2	100	120	200	240	200	240
C	4	6	60	60	240	240	360	360
D	10	12	50	24	500	240	600	288
E	8	12	40	36	320	288	480	432
Total					`sum"p"_0"q"_0` = 1560	`sum"p"_0"q"_1` = 1344	`sum"p"_1"q"_0` = 2140	`sum"p"_1"q"_1` = 1880

Fisher’s Price Index Number

`"P"_01^"F" = sqrt((sum"p"_1"q"_0)/(sum"p"_0"q"_0) xx (sum"p"_1"q"_1)/(sum"p"_0"q"_1)) xx 100`

= `sqrt(2140/1560 xx 1880/1344) xx 100`

= `sqrt((40,23,200)/(20,96,640)) xx 100`

= `sqrt(1.92) xx 100`

= `1.385 xx 100`

= 138.5

Time Reversal Test: To prove P₀₁ × P₁₀ = 1

P₀₁ × P₁₀ = `sqrt((sum"p"_1"q"_0 xx sum"p"_1"q"_1)/(sum"p"_0"q"_0 xx sum"p"_0"q"_1)) xx sqrt((sum"p"_0"q"_1 xx sum"p"_0"q"_0)/(sum"p"_1"q"_1 xx sum"p"_1"q"_0))`

= `sqrt(2140/1560 xx 1880/1344 xx 1344/1880 xx 1560/2140)`

P₀₁ × P₁₀ = 1

Time reversal test is satisfied.

Factor Reversal Test: To prove P₀₁ × Q₀₁ = `(sum"p"_1"q"_1)/(sum"p"_0"q"_0)`

= `sqrt((sum"p"_1"q"_0 xx sum"p"_1"q"_1)/(sum"p"_0"q"_0 xx sum"p"_0"q"_1)) xx sqrt((sum"q"_1"P"_0 xx sum"q"_1"P"_1)/(sum"q"_0"p"_0 xx sum"q"_0"p"_1))`

= `sqrt(2140 /1560 xx 1880/1344 xx 1344/1560 xx 188/2140)`

= `sqrt((1880 xx 1880)/(1560 xx 1560)`

= `1880/1560`

⇒ `"P"_01 xx "Q"_01 = (sum"p"_1"q"_1)/(sum"p"_0"q"_0)`

Factor reversal test is satisfied.