Weighted Index Numbers Dorbish-Bowley's index number Example-5

1. Find Laspeyre's index number, Paasche's index number, Dorbish-Bowley's index number
Item Price0 Quantity0 Price1 Quantity1
Rice 39 1 40 1.5
Milk 40 12 44 10
Bread 45 2 50 1.5
Banana 30 2 36 1.5

Solution:

Item	$p_0$	$q_0$	$p_1$	$q_1$	$p_1q_0$	$p_0q_0$	$p_1q_1$	$p_0q_1$
Rice	39	1	40	1.5	40	39	60	58.5
Milk	40	12	44	10	528	480	440	400
Bread	45	2	50	1.5	100	90	75	67.5
Banana	30	2	36	1.5	72	60	54	45
---	---	---	---	---	---	---	---	---
Total					$740$	$669$	$629$	$571$

1. Laspeyre's index number

$I_L=(sum p_1q_0)/(sum p_0q_0) xx 100$

$=(740)/(669) xx 100$

$=110.61$

Thus, there is a rise of

$(110.61-100)=10.61%$ in prices

2. Paasche's index number

$I_P=(sum p_1q_1)/(sum p_0q_1) xx 100$

$=(629)/(571) xx 100$

$=110.16$

Thus, there is a rise of

$(110.16-100)=10.16%$ in prices

3. Dorbish and Bowley's price index

$I_D=(I_L + I_P)/2$

$=(110.6129+110.1576)/2$

$=(220.7705)/2$

$=110.39$

Thus, there is a rise of

$(110.39-100)=10.39%$ in prices

2. Find Dorbish-Bowley's index number
Item Price0 Quantity0 Price1 Quantity1
A 10 20 12 22
B 8 16 8 18
C 5 10 6 11
D 4 7 4 8

Solution:

Item	$p_0$	$q_0$	$p_1$	$q_1$	$p_1q_0$	$p_0q_0$	$p_1q_1$	$p_0q_1$
A	10	20	12	22	240	200	264	220
B	8	16	8	18	128	128	144	144
C	5	10	6	11	60	50	66	55
D	4	7	4	8	28	28	32	32
---	---	---	---	---	---	---	---	---
Total					$456$	$406$	$506$	$451$

1. Dorbish and Bowley's price index

$I_D=((sum p_1q_0)/(sum p_0q_0) + (sum p_1q_1)/(sum p_0q_1))/2 xx 100$

$=((456)/(406) + (506)/(451))/2 xx 100$

$=(2.2451)/2 xx 100$

$=1.1226 xx 100$

$=112.26$

Thus, there is a rise of

$(112.26-100)=12.26%$ in prices

3. Find Dorbish-Bowley's index number
Item Price0 Quantity0 Price1 Quantity1
A 3 25 5 28
B 1 50 3 60
C 2 30 1 30
D 5 15 6 12

Solution:

Item	$p_0$	$q_0$	$p_1$	$q_1$	$p_1q_0$	$p_0q_0$	$p_1q_1$	$p_0q_1$
A	3	25	5	28	125	75	140	84
B	1	50	3	60	150	50	180	60
C	2	30	1	30	30	60	30	60
D	5	15	6	12	90	75	72	60
---	---	---	---	---	---	---	---	---
Total					$395$	$260$	$422$	$264$

1. Dorbish and Bowley's price index

$I_D=((sum p_1q_0)/(sum p_0q_0) + (sum p_1q_1)/(sum p_0q_1))/2 xx 100$

$=((395)/(260) + (422)/(264))/2 xx 100$

$=(3.1177)/2 xx 100$

$=1.5589 xx 100$

$=155.89$

Thus, there is a rise of

$(155.89-100)=55.89%$ in prices

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5. Dorbish-Bowley's index number Example-5