UNIT 4 : INDEX NUMBER & TIME SERIES ( 20 Marks)
	INDEX NUMBERS
	DEFN OF INDEX NUMBER
	TERMINOLOGY
	BASE PERIOD/YEAR
	CURRENT PERIOD / YEAR
	INDEX NUMBER OF BASE YEAR IS ASSUMED TO BE 100
	Base year : Reference Period . It is denoted by suffix '0'
	Current year : The year for Comparison : It is denoted by suffix '1'
	po : Price of a commodity in the base year
	p1 : Price of a commodity in the current year
	qo : Quantity of the commodity consumed /purchased during the base year
	q1 : Quantity of the commodity consumed /purchased during the current year
	w : Weights assigned to a commodity
	i = Simple index number or price relative = p1/po x 100
	A : SIMPLE (UNWEIGHTED) INDEX NUMBER
	(i) Simple Aggregative Method
	I = ∑ p1/  ∑ p0 *100
	(ii) Simple Average of Price Relative
	I= ∑ i/n
	i= p1/po x 100
	n is number of commodities			(div d)
	Examples
	Eg 1 : Calculate index number using the formula of Simple Aggregative and Simple Average of Price Relative
	Commodity	Price
		Base Year (2005)	Current Year (2010)
	A	20	25
	B	25	35
	C	40	50
	D	4	8
	Solution
	Commodity	Price
		Base Year (po)	Current Year (p1)	Price Relative = p1/po *100
	A	20	25	125
	B	25	35	140
	C	40	50	125
	D	4	8	200
	Total	89	118	590
	(i) Simple Aggregative Method
	I =  ∑ p1/  ∑ p0 *100		=	118/89*100	=	132.5842697
	(ii) Simple Average of Price Relative
	I=  ∑ i/n	=	590/4	=	147.5
	(div c)
	Eg 2 : Calculate index number using the formula of Simple Aggregative and Simple Average of Price Relative
	Commodity	Price
		2020	2010
	A	35	30
	B	50	40
	C	120	100
	D	50	20
	E	70	100
	Solution
	Commodity	Price
		2020 (p1)	2010 (po)	Price Relative = i= p1/po *100
	A	35	30	116.67
	B	50	40	125
	C	120	100	120
	D	50	20	250
	E	70	100	70.00
	Total	325	290	681.67
	(i) Simple Aggregative Method
	I =  ∑ p1/  ∑ p0 *100		=	325/290*100	=	112.07
	(ii) Simple Average of Price Relative
	I=  ∑i/n		=	681.67/5	=	136.334
	(div A)
	Eg 3 : Calculate index number using the formula of Simple Aggregative and Simple Average of Price Relative
	Commodity	Price
		Base Year (2010)	Current Year (2015)
	Rice	30	50
	Wheat	25	65
	Salt	10	15
	Sugar	20	30
	Solution
	Commodity	Price
		Base Year (po)	Current Year (p1)	Price Relative = i=p1/po *100
	Rice	30	50	166.67
	Wheat	25	65	260.00
	Salt	10	15	150.00
	Sugar	20	30	150.00
	Total	85	160	726.67
	(i) Simple Aggregative Method
	I =  ∑ p1/  ∑ p0 *100		=	160/85 *100	=	188.2352941
	(ii) Simple Average of Price Relative
	I=  ∑ i/n	=	726.67/4	=	181.67
	Eg 3 : Calculate index number using the formula of Simple Aggregative and Simple Average of Price Relative
	Commodity	Price
		2010	2000
	Rice	70	50
	Wheat	40	35
	Salt	8	10
	Sugar	30	20
	Oil	100	80
	Solution
	Commodity	Price
		2010 (p1)	2000 (po)	Price Relative = i= p1/po *100
	Rice	70	50	140.00
	Wheat	40	35	114.29
	Salt	8	10	80.00
	Sugar	30	20	150.00
	Oil	100	80	125.00
	Total	248	195	609.29
	(i) Simple Aggregative Method
	I =  ∑ p1/  ∑ p0 *100 =		248/195 *100 =	127.18
	(ii) Simple Average of Price Relative
	I=  ∑ i/n =	609.29/5 =	121.858
	B : COMPOSITE INDEX NUMBER
	(I) FIVE IMPORTANT FORMULAES (BASED ON QUANTITIES)								keywords for learning purpose
	1) Laspeyer's Index Numbers = IL =			∑p1q0/∑poqo *100					1000			(only qo is considered in the formula)
	2) Paasche's Index Numbers = Ip =			∑p1q1/∑poq1 *100					1101			(only q1 is considered in the formula)
	3) Fisher's Index Number = IF =			SQRT (IL * Ip)	=	√ (IL * Ip)			Geometric mean of IL & Ip
	4) Drobish Bowley's Index Number = IDB =			(IL+Ip)/2					Arithmetic mean of IL & Ip
	5) Marshal Edgeworth Index Number =			(∑p1q0 +∑p1q1)/(∑poqo +∑poq1) *100					Numerator values of IL & Ip added And Divided by added Denominator value of IL & IP
	(II) Weighted Index Number (WEIGHTS GIVEN INSTEAD OF QUANTITIES)								(multiply by w in the simple index number formula)
	(i) Weighted Aggregative Method =			I =  ∑ p1w/  ∑ p0w *100
	(ii) Weighted Average of Price Relative =			I=  ∑iw/∑w
		(family budget method)
				i=p1/po *100
	Weighted Average of Price Relative is also called as COST OF LIVING NUMBER BY FAMILY BUDGET METHOD
	Eg 1: Calculate Index Number using Laspeyre's, Paasche's , Fisher's, Bowley's and Edgeworth Formula
	COMMODITY	2001		2011
		PRICE	QUANTITY	PRICE	QUANTITY
	A	25	4	30	5
	B	20	3	30	7
	C	10	9	12	8
	D	15	10	20	12
	Solution
	COMMODITY	2001		2011
		PRICE (po)	QUANTITY (qo)	PRICE (p1)	QUANTITY (q1)	poqo	poq1	p1qo	p1q1
	A	25	4	30	5	100	125	120	150							100	125	120	150
	B	20	3	30	7	60	140	90	210							60	140	90	210
	C	10	9	12	8	90	80	108	96							90	80	108	96
	D	15	10	20	12	150	180	200	240							150	180	200	240
					Total	400	525	518	696							400	525	518	696
	1) Laspeyer's Index Numbers = IL =			∑p1q0/∑poqo *100 =		518/400 *100 =		129.50
	2) Paasche's Index Numbers = Ip =			∑p1q1/∑poq1 *100 =		696/525 *100 =		132.57
	3) Fisher's Index Number = IF =			SQRT (IL * Ip)	=	√ (IL * Ip) =		√ 129.57*132.57=		131.03
	4) Drobish Bowley's Index Number = IDB =			(IL+Ip)/2=	131.04
	5) Marshal Edgeworth Index Number =			(∑p1q0 +∑p1q1)/(∑poqo +∑poq1) *100 =			(518+696)/(400+525)*100 =			131.243
	Eg 2: Calculate Index Number using Laspeyre's, Paasche's , Fisher's, Drobish's and Marshal's Formula
	COMMODITY	PRICE		QUANTITY
		2010	2015	2010	2015
	A	30	40	8	7
	B	5	8	7	10
	C	20	25	10	9
	D	10	12	5	10
	Solution
	COMMODITY	PRICE		QUANTITY		poqo	poq1	p1qo	p1q1
		2010 (po)	2015(p1)	2010 (qo)	2015 (q1)
	A	30	40	8	7	240	210	320	280
	B	5	8	7	10	35	50	56	80
	C	20	25	10	9	200	180	250	225
	D	10	12	5	10	50	100	60	120
					Total	525	540	686	705
	1) Laspeyer's Index Numbers = IL =			∑p1q0/∑poqo *100 =		686/525*100 =		130.67
	2) Paasche's Index Numbers = Ip =			∑p1q1/∑poq1 *100 =		705/540*100 =		130.56
	3) Fisher's Index Number = IF =			SQRT (IL * Ip)	=	√ (IL * Ip) =		130.61
	4) Drobish Bowley's Index Number = IDB =			(IL+Ip)/2=	(130.67+130.56)/2			130.615
	5) Marshal Edgeworth Index Number =			(∑p1q0 +∑p1q1)/(∑poqo +∑poq1) *100 =			(686+705)/(525+540) *100 =			130.61033
	Eg 3: Calculate Index Number using Laspeyre's, Paasche's , Fisher's Formula
	COMMODITY	PRICE		QUANTITY
		2010	2000	2010	2000
	Wheat	50	40	6	5
	Rice	70	60	10	8
	Oil	100	80	5	4
	Sugar	30	20	4	5
	Salt	10	8	2	2
	Solution
	COMMODITY	PRICE		QUANTITY
		2010 (p1)	2000 (po)	2010 (q1)	2000 (qo)	poqo	poq1	p1qo	p1q1
	Wheat	50	40	6	5	200	240	250	300
	Rice	70	60	10	8	480	600	560	700
	Oil	100	80	5	4	320	400	400	500
	Sugar	30	20	4	5	100	80	150	120
	Salt	10	8	2	2	16	16	20	20
					Total	1116	1336	1380	1640
	1) Laspeyer's Index Numbers = IL =			∑p1q0/∑poqo *100 =		1380/1116*100=		123.66
	2) Paasche's Index Numbers = Ip =			∑p1q1/∑poq1 *100 =		1640/1336*100 =		122.75
	3) Fisher's Index Number = IF =			SQRT (IL * Ip)	=	√ (IL * Ip) =		123.20
	4) Drobish Bowley's Index Number = IDB =			(IL+Ip)/2=	(123.66 + 122.75)/2			123.21
	5) Marshal Edgeworth Index Number =			(∑p1q0 +∑p1q1)/(∑poqo +∑poq1) *100 =			(1380+1640)/(1116+1336) *100 =			130.61
	Eg 4: Calculate Index Number using Laspeyre's, Paasche's , Fisher's & Bowley's Formula
	COMMODITY	PRICE		QUANTITY
		2010	2020	2010	2020
	Wheat	20	40	4	5
	Rice	40	60	5	4
	Oil	75	100	2	3
	Sugar	25	30	2	3
	Salt	10	10	1	2
	COMMODITY	PRICE		QUANTITY
		2010 (po)	2020 (p1)	2010 (qo)	2020 (q1)	poqo	poq1	p1qo	p1q1
	Wheat	20	40	4	5	80	100	160	200
	Rice	40	60	5	4	200	160	300	240
	Oil	75	100	2	3	150	225	200	300
	Sugar	25	30	2	3	50	75	60	90
	Salt	10	10	1	2	10	20	10	20
					Total	490	580	730	850
	1) Laspeyer's Index Numbers = IL =			∑p1q0/∑poqo *100 =		148.98
	2) Paasche's Index Numbers = Ip =			∑p1q1/∑poq1 *100 =		146.55
	3) Fisher's Index Number = IF =			SQRT (IL * Ip)	=	√ (IL * Ip) =	147.76
	4) Drobish Bowley's Index Number = IDB =			(IL+Ip)/2=	147.77
	(II) WEIGHTED INDEX NUMBERS (WEIGHTS GIVEN INSTEAD OF QUANTITIES)
	Eg 1 : Calculate  Index Number by  Weighted Aggregated Method
	COMMODITY	PRICE
		2010	2020	Weights
	Wheat	20	40	25
	Rice	40	60	30
	Oil	75	100	5
	Sugar	25	30	10
	Salt	10	10	2
	Solution
	COMMODITY	PRICE
		2010 (po)	2020 (p1)	Weights  (w)	p1w	pow
	Wheat	20	40	25	1000	500
	Rice	40	60	30	1800	1200
	Oil	75	100	5	500	375
	Sugar	25	30	10	300	250
	Salt	10	10	2	20	20
					3620	2345
	(i) Weighted Aggregative Method =			I =  ∑ p1w/  ∑ pow *100 =		154.37
	Eg 2 : Calculate  Index Number by  Weighted Aggregated Method
	COMMODITY	PRICE
		2000	1990	Weights
	A	10	8	25
	B	50	30	30
	C	45	40	5
	D	20	15	10
	E	10	7	2
	Solution
	COMMODITY	PRICE
		2000 (p1)	1990 (po)	Weights  (w)	p1w	pow
	A	10	8	25	250	200
	B	50	30	30	1500	900
	C	45	40	5	225	200
	D	20	15	10	200	150
	E	10	7	5	50	35
					2225	1485
	(i) Weighted Aggregative Method =			I =  ∑ p1w/  ∑ pow *100 =		149.83
	Eg 3 : Calculate  Index Number by  Weighted Average OF Price Relative Method
	COMMODITY	PRICE
		2010	2020	Weights
	Wheat	20	40	25
	Rice	40	60	30
	Oil	75	100	5
	Sugar	25	30	10
	Salt	10	10	2
	Solution
	COMMODITY	PRICE
		2010 (po)	2020 (p1)	Weights  (w)	I =p1/po*100	I*w =Iw
	Wheat	20	40	25	200	5000
	Rice	40	60	30	150	4500
	Oil	75	100	5	133.33	666.65
	Sugar	25	30	10	120.00	1200
	Salt	10	10	2	100	200
				72		11566.65
	(ii) Weighted Average of Price Relative =			∑Iw/∑w =	160.65
				I=p1/po *100
	Eg 4 : Calculate  Index Number by  Weighted Average OF Price Relative Method
	COMMODITY	PRICE
		2010	2020	Weights
	A	15	18	20
	B	25	30	10
	C	30	35	5
	D	10	15	15
	E	5	10	25
	Solution
	COMMODITY	PRICE
		2010 (po)	2020 (p1)	Weights  (w)	I =p1/po*100	I*w =Iw
	A	15	18	20	120	240
	B	25	30	10	120	1200
	C	30	35	5	116.67	583.35
	D	10	15	15	150	2250
	E	5	10	25	200	5000
				75		9273.35
	(i) Weighted Average of Price Relative =			I=  ∑iw/∑w =	123.64
				i=p1/po *100
	Weighted Average of Price Relative formulae is same as COST OF LIVING NUMBER BY FAMILY BUDGET METHOD
	Eg 4 : Calculate  Index Number by  Weighted Average OF Price Relative Method
	Group	Group Index	Weights
	A	150	20
	B	120	10
	C	130	5
	D	140	15
	E	110	25
	Solution
	Group	Group Index (I)	Weights  (w)	Iw
	A	150	20	3000
	B	120	10	1200
	C	130	5	650
	D	140	15	2100
	E	110	25	2750
			75	9700
	(i) Weighted Average of Price Relative =			I=  ∑iw/∑w =	129.33
	C :Cost of Living Index Number
	(i) By Aggregate Expenditure method =			∑p1q0/∑poqo *100 =
	(LASPEYER'S index No.)
	(ii) By Family Budget Method			=  ∑Iw/∑w
	More Imp
	( Same formula as Weighted Average Prize Relative Method)
	(If I is not given then I is calculated using this formulae)			I=p1/po *100
	Eg 1 : Calculate Cost of Living Index Number by  Aggregate Expenditure Method
	COMMODITY	PRICE		QUANTITY
		2010	2020	2010
	Wheat	20	40	4
	Rice	40	60	5
	Oil	75	100	2
	Sugar	25	30	2
	Salt	10	10	1
	Solution
	COMMODITY	PRICE		QUANTITY
		2010 (po)	2020 (p1)	2010 (qo)	poqo	p1qo
	Wheat	20	40	4	80	160
	Rice	40	60	5	200	300
	Oil	75	100	2	150	200
	Sugar	25	30	2	50	60
	Salt	10	10	1	10	10
				Total	490	730
	(i) Cost of Living by Aggregate Expenditure method =			∑p1q0/∑poqo *100 =		148.98
		(same formulae as Laspeyers formula)
	Eg 2 : Calculate Cost of Living Index Number by Family Budget Method
	Group	Group Index	Weights
	A	150	20
	B	220	10
	C	160	25
	D	170	30
	E	200	15
	SOLUTION
	Group	Group Index (I)	Weights (W)	IW
	A	150	20	3000
	B	220	10	2200
	C	160	25	4000
	D	170	30	5100
	E	200	15	3000
			100	17300
	Cost of living Index Number By Family Budget Method =			=  ∑Iw/∑w =	173
	(It is the Same formula as Weighted Average of Prize Relative Method)
	Eg 3 : Construct Cost of Living Index Number by Family Budget Method
		Price
	Group	Base Year	Current Year	Weights
	Food	25	40	20
	Clothing	15	30	25
	Fuel	10	12	10
	Rent	3	4	25
	Miscellaneous	10	15	15
	Solution
		Price
	Group	po	p1	w	I=p1/po*100	Iw
	Food	25	40	20	160	3200
	Clothing	15	30	25	200	5000
	Fuel	10	12	10	120	1200
	Rent	3	4	25	133.333	3333.25
	Miscellaneous	10	15	15	150	2250
				95		14983.25
	Cost of living Index Number By Family Budget Method =			=  ∑Iw/∑w =	157.7184211
	D: REAL INCOME OR WAGES
	REAL INCOME =	CURRENT INCOME/ INDEX OF CURRENT YEAR *100
	EG1 : CALCULATE REAL INCOME FOR THE FOLOWING DATA
	YEAR	INCOME (Rs)	INDEX NO.
	2000	14000	100
	2001	15000	120
	2002	16000	140
	2004	17000	150
	2005	18000	200
	SOLUTION
	YEAR	INCOME (RS)	INDEX NO.	REAL INCOME
	2000	14000	100	14000
	2001	15000	120	12500
	2002	16000	140	11428.57
	2004	17000	150	11333.33
	2005	18000	200	9000
	REAL INCOME =	CURRENT INCOME/ INDEX OF CURRENT YEAR *100
	EG1 : Construct REAL INCOME FOR THE GIVEN DATA
	YEAR	INCOME (RS)	INDEX NO.
	2000	10000	100
	2001	15000	120
	2002	20000	140
	2004	25000	150
	2005	30000	200
	SOLUTION
	YEAR	INCOME (Rs)	INDEX NO.	REAL INCOME
	2000	10000	100	10000
	2001	15000	120	12500
	2002	20000	140	14285.71
	2004	25000	150	16666.67
	2005	30000	200	15000.00
	STEPS IN CONSTRUCTION OF INDEX NUMBER : THEORY (READ IT)