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#### Question 1:

Construct the Index Number for 2011 with 2015 as base from the following prices of commodities by simple (Unweighted) aggregative method.

 Commodities : A B C D E Prices in ₹(2015) : 50 40 10 5 2 Prices in  ₹(2011) : 80 60 20 10 6
[Index Number = 164.48]

 Commodity Prices in 2015 (p0) Prices in 2011 (p1) A B C D E 50 40 10   5   2 80 60 20 10   6 ∑ p0 = 107 ∑ p1 = 176

${p}_{01}=\frac{\mathrm{\Sigma }{p}_{1}}{\mathrm{\Sigma }{p}_{0}}×100=\frac{176}{107}=164.48$

Thus, price index is 164.48.

#### Question 2:

Using the following data and 2008 as the base period, compute simple aggregative price indices for the two fuels.

 Item Producer's Price (in ₹) 2008 2009 2010 Coal ( ₹) 5 3 4 Crude oil ( ₹) 2 3 4

 Item Prices in 2008 (p0) Prices in 2009 (p1) Prices in 2010 (p2) Coal Crude oil 5 2 3 3 4 4 ∑ p0 = 7 ∑ p1 = 6 ∑ p2 = 8

${p}_{01}=\frac{\mathrm{\Sigma }{p}_{1}}{\mathrm{\Sigma }{p}_{0}}×100=\frac{6}{7}×100=85.71\phantom{\rule{0ex}{0ex}}{p}_{02}=\frac{\mathrm{\Sigma }{p}_{2}}{\mathrm{\Sigma }{p}_{0}}×100=\frac{8}{7}×100=114.28$
Thus, index  number for 2009 is 85.71 and index number for 2010 is 114.28

#### Question 3:

Calculate the index number for 2014 with 2013 as base from the following prices of the commodities by simple (unweighted) aggregative method.

 Commodity and unit Price ( ₹) (2013) Price ( ₹) (2014) Butter per kg 20.00 22.00 Milk per litre 3.00 4.50 Cheese per Tin 18.00 19.80 Bread per Kg 2.00 3.80 Eggs per Dozen 4.00 4.50

 Commodity Prices in 2013 (p0) Prices in 2014 (p1) Butter Milk Cheese Bread Eggs 20 3 18 2 4 22 4.5 19.80 3.80 4.50 ∑ p0 = 47 ∑ p1 = 54.6

${p}_{01}=\frac{\mathrm{\Sigma }{p}_{1}}{\mathrm{\Sigma }{p}_{0}}×100=\frac{54.6}{47}×100=\frac{5460}{47}=116.17$
Thus, index number is 116.17.

#### Question 4:

Calculate Quantity Index Numbers from the following data by simple aggregative method taking quantity of 2011 as base.

 Commodity Quantity (in tons) 2011 2012 2013 2014 2015 A 0.30 0.33 0.36 0.36 0.39 B 0.25 0.24 0.30 0.32 0.30 C 0.20 0.25 0.28 0.32 0.30 D 2.00 2.40 2.50 2.50 2.60

[Quantity Index No. : 2012 = 117.1; 2013 = 125.1; 2014 = 127.3 ; 2015 = 130.5]

 Commodity Quantity in 2011 (q0) Quantity in  2012 (q1) Quantity in  2013 (q2) Quantity in 2014 (q3) Quantity in 2015 (q4) A B C D 0.30 0.25 0.20 2.00 0.33 0.24 0.25 2.4 0.36 0.30 0.28 2.50 0.36 0.32 0.32 2.50 0.39 0.30 0.30 2.60 ∑ q0 = 2.75 ∑ q1 = 3.22 ∑ q2 = 3.44 ∑ q3 = 3.5 ∑ q4 = 3.59

${q}_{01}=\frac{\mathrm{\Sigma }{q}_{1}}{\mathrm{\Sigma }{q}_{0}}×100=\frac{3.22}{2.75}×100=117.1\phantom{\rule{0ex}{0ex}}$
Thus, quantity index for 2012 is 117.1.
${q}_{02}=\frac{\mathrm{\Sigma }{q}_{2}}{\mathrm{\Sigma }{q}_{0}}×100=\frac{3.44}{2.75}×100=125.1\phantom{\rule{0ex}{0ex}}$
Thus, quantity index for 2013 is 125.1.
${q}_{03}=\frac{\mathrm{\Sigma }{q}_{3}}{\mathrm{\Sigma }{q}_{0}}×100=\frac{3.5}{2.75}×100=127.3\phantom{\rule{0ex}{0ex}}$
Thus, quantity index for 2014 is 127.3.
${q}_{04}=\frac{\mathrm{\Sigma }{q}_{4}}{\mathrm{\Sigma }{q}_{0}}×100=\frac{3.59}{2.75}×100=\frac{359}{2.75}=130.5\phantom{\rule{0ex}{0ex}}$
Thus, quantity index for 2015 is 130.5.

#### Question 5:

Calcualte index number for 2017 on the base prices for 2013 from the following by average of price relative method.

 Items : Bricks Timber Plaster Board Sand Cement Price in  ₹ (2013) : 10 20 5 2 7 Price in  ₹ (2017) : 16 21 6 3 14
[Index No. = 147]

 Item Prices in 2013 (p0) Prices in 2017 (p1) Price relatives of 2017 in relation to 2013 $\left(\frac{{p}_{1}}{{p}_{0}}×100\right)$ Brick 10 16 $\frac{16}{10}×100=160$ Timber 20 21 $\frac{21}{20}×100=105$ Plaster Board 5 6 $\frac{6}{5}×100=120$ Sand 2 3 $\frac{3}{2}×100=150$ Cement 7 14 $\frac{14}{7}×100=200$ $\mathrm{\Sigma }\left(\frac{{p}_{1}}{{p}_{0}}×100\right)=735$

${p}_{01}=\frac{\mathrm{\Sigma }\left(\frac{{p}_{1}}{{p}_{0}}×100\right)}{N}=\frac{735}{5}=147$
Thus, index number is 147.

#### Question 6:

Construct the index number for 2016 taking 2006 as base by price relative method using arithmetic mean.

 Commodities : A B C D Price in  ₹ (2006) : 10 20 30 40 Price in  ₹ (2016) : 13 17 60 70
[Index No. = 147.5]

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Curriculum

1 - CBSE

13 - XI-Commerce
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16 - Economics
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7 - NM Shah (2018)
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9 - Introduction to Index Numbers
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363
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Question

Calcualte index number for 2017 on the base prices for 2013 from the following by average of price relative method.

 Items : Bricks Timber Plaster Board Sand Cement Price in  ₹ (2013) : 10 20 5 2 7 Price in  ₹ (2017) : 16 21 6 3 14

[Index No. = 147]

Solution

Comment

Submit
 Commodities Prices in 2006 (p0) Prices in 2016 (p1) $\left(\frac{{\mathbit{p}}_{\mathbf{1}}}{{\mathbit{p}}_{\mathbf{0}}}\mathbf{×}\mathbf{100}\right)$ A 10 13 $\frac{13}{10}×100=130$ B 20 17 $\frac{17}{20}×100=85$ C 30 60 $\frac{60}{30}×100=200$ D 40 70 $\frac{70}{40}×100=175$ $\mathrm{\Sigma }\left(\frac{{p}_{1}}{{p}_{0}}×100\right)=590$

${p}_{01}=\frac{\mathrm{\Sigma }\left(\frac{{p}_{1}}{{p}_{0}}×100\right)}{N}=\frac{590}{4}=147.5$
Thus, index number is 147.5

#### Question 7:

Construct Index Number for each year from the following average annual wholesale prices of cotton with 2001 as base .

 Year Wholesale Prices (in ₹) Year Wholesale Prices (in ₹) 2001 75 2006 70 2002 50 2007 69 2003 65 2008 75 2004 60 2009 84 2005 72 2010 80

 Prices in 2001 (p0) Prices in 2002 (p1) Prices in 2003 (p2) Prices in 2004 (p3) Prices in 2005  (p4) Prices in 2006  (p5) Prices in 2007 (p6) Prices in 2008 (p7) Prices in 2009 (p8) Prices in 2010 (p9) 75 50 65 60 72 70 69 75 84 80

${p}_{01}=\frac{\mathrm{\Sigma }{p}_{1}}{\mathrm{\Sigma }{p}_{0}}×100=\frac{50}{75}×100=66.67\phantom{\rule{0ex}{0ex}}{p}_{02}=\frac{\mathrm{\Sigma }{p}_{2}}{\mathrm{\Sigma }{p}_{0}}×100=\frac{65}{75}×100=86.67\phantom{\rule{0ex}{0ex}}{p}_{03}=\frac{\mathrm{\Sigma }{p}_{3}}{\mathrm{\Sigma }{p}_{0}}×100=\frac{60}{75}×100=80\phantom{\rule{0ex}{0ex}}{p}_{04}=\frac{\mathrm{\Sigma }{p}_{4}}{\mathrm{\Sigma }{p}_{0}}×100=\frac{72}{75}×100=96\phantom{\rule{0ex}{0ex}}{p}_{05}=\frac{\mathrm{\Sigma }{p}_{5}}{\mathrm{\Sigma }{p}_{0}}×100=\frac{70}{75}×100=93.33\phantom{\rule{0ex}{0ex}}{p}_{06}=\frac{\mathrm{\Sigma }{p}_{6}}{\mathrm{\Sigma }{p}_{0}}×100=\frac{69}{75}×100=92\phantom{\rule{0ex}{0ex}}{p}_{07}=\frac{\mathrm{\Sigma }{p}_{7}}{\mathrm{\Sigma }{p}_{0}}×100=\frac{75}{75}×100=100\phantom{\rule{0ex}{0ex}}{p}_{08}=\frac{\mathrm{\Sigma }{p}_{8}}{\mathrm{\Sigma }{p}_{0}}×100=\frac{84}{75}×100=112\phantom{\rule{0ex}{0ex}}{p}_{09}=\frac{\mathrm{\Sigma }{p}_{9}}{\mathrm{\Sigma }{p}_{0}}×100=\frac{80}{75}×100=106.67$

#### Question 8:

The group indices of prices of commodities for second week of Sept. 2017 and the group weights are given below. Compute the index number by family budget method.

 Group Weights Index Food Article 31 473.6 Manufactures 30 390.2 Industrial Raw Material 18 510.2 Semi-Manufactures 17 403.3 Miscellaneous 4 624.4

 Group Weights (w) Index (x) wx Food Article Manufactures Industrial Raw Material Semi-Manufactures Miscellaneous 31 30 18 17 4 473.6 390.2 510.2 403.3 624.4 14681.6 11706 9183.6 6856.1 2497.6 ∑w = 100 ∑wx = 44924.9

$\mathrm{WPI}=\frac{\Sigma wx}{\Sigma w}=\frac{44924.9}{100}=449.249$
Thus, the index of wholesale prices is 449.29.

#### Question 9:

Calculate price index number for 2016 of following data by weighted aggregative method using (a) Laspeyre's method, (b)  Paasche's method, (c) Fisher's method.

 Commodity Price (2012) Quantity (2012) Price (2016) Quantity (2016) A 4 20 6 10 B 3 15 5 23 C 2 25 3 15 D 5 10 4 15

 Commodity Base Year Current Year Prices p0 Quantity q0 Prices p1 Quantity q1 p0q0 p0q1 p1q0 p1q1 A B C D 4 3 2 5 20 15 25 10 6 5 3 4 10 23 15 40 80 45 50 50 40 69 30 200 120 75 75 40 60 115 45 160 ∑p0q0 = 225 ∑p0q1 = 339 ∑p1q0 = 310 ∑p1q1 = 380

(a)

(b)

(c)

Note: As per the textbook, the price index using Paasche's method is 158.99 and Fisher's method is 148.1. However, as per the above solution the price index using Paasche's method should be 112.09 and Fisher's method should be 124.26.

#### Question 10:

From the data given below , construct Laspeyre's , Paasche's and Fisher's price index and quantity index numbers with base 2015 and interpret.​

 Commodity 2015 2016 Price (₹) Quantity (Kg) Price (₹) Quantity (Kg) A B C 4 3 8 2 5 2 6 2 4 3 1 6

 Commodity Base Year Current Year Price p0 Quantity q0 Price p1 Quantity q1 p0q0 p0q1 p1q0 p1q1 A B C 4 3 8 2 5 2 6 2 4 3 1 6 8 15 16 12 3 48 12 10 8 18 2 24 ∑p0q0 = 39 ∑p0q1 = 63 ∑p1q0 = 30 ∑p1q1 = 44

Laspeyre's Price index =  $\frac{\mathrm{\Sigma }{p}_{1}{q}_{0}}{\mathrm{\Sigma }{p}_{0}{q}_{0}}×100=\frac{30}{39}×100=76.92$
Laspeyre's Quantity index = $\frac{\mathrm{\Sigma }{p}_{0}{q}_{1}}{\mathrm{\Sigma }{p}_{0}{q}_{0}}×100=\frac{63}{39}×100=161.53$

Paasche's Price index = $\frac{\mathrm{\Sigma }{p}_{1}{q}_{1}}{\mathrm{\Sigma }{p}_{0}{q}_{1}}×100=\frac{44}{63}×100=69.84$

Paasche's Quantity index = $\frac{\mathrm{\Sigma }{p}_{1}{q}_{1}}{\mathrm{\Sigma }{p}_{1}{q}_{0}}×100=\frac{44}{30}×100=146.67$

Fisher's Price index = $\sqrt{\frac{\mathrm{\Sigma }{p}_{1}{q}_{1}}{\mathrm{\Sigma }{p}_{0}{q}_{1}}×\frac{\mathrm{\Sigma }{p}_{1}{q}_{0}}{\mathrm{\Sigma }{p}_{0}{q}_{0}}}×100=\sqrt{\frac{44}{63}×\frac{30}{39}}×100=73.29\phantom{\rule{0ex}{0ex}}$

Fisher's Quantity index =$\sqrt{\frac{\mathrm{\Sigma }{p}_{1}{q}_{1}}{\mathrm{\Sigma }{p}_{1}{q}_{0}}×\frac{\mathrm{\Sigma }{p}_{0}{q}_{1}}{\mathrm{\Sigma }{p}_{0}{q}_{0}}}×100=\sqrt{\frac{44}{30}×\frac{63}{39}}×100=153.92\phantom{\rule{0ex}{0ex}}$
Note: As per the textbook, the quantity indices using Laspeyre's and Paasche's methods are 143.18 and 130. However, as per the above solution the quantity indices using Laspeyre's and Paasche's methods should be 161.53 and 146.67.

#### Question 11:

Calculate weighted aggregative of actual  price index number and quantity index number from the following data using (i) Laspeyre's Method , and (ii) Paasche's Method and (iii) Fisher's Method and interpret them.

 Commodity Base year Current Year Quantity lbs. Price  per lb. Quantity lbs. Price  ​per lb. Bread 6 40 paise 7 30 paise Meat 4 45 paise 5 50 paise Tea 0.5 90 paise 1.5 40 pAise

 Base Year Current Year Quantity q0 Price p0 Quantity q1 Price p1 p0q0 p1q1 p1q0 p0q1 Bread Meat Tea 6 4 0.5 40 45 90 7 5 1.5 30 50 40 240 180 45 210 250 60 180 200 20 280 225 135 ∑p0q0 = 465 ∑p1q1 = 520 ∑p1q0 = 400 ∑p0q1 = 640

(i) Price Index number
(a) Laspeyre's ${p}_{01}=\frac{\mathrm{\Sigma }{p}_{1}{q}_{0}}{\mathrm{\Sigma }{p}_{0}{q}_{0}}×100=\frac{400}{465}×100=86.02$

(b) Paasche's ${p}_{01}=\frac{\mathrm{\Sigma }{p}_{1}{q}_{1}}{\mathrm{\Sigma }{p}_{0}{q}_{1}}×100=\frac{520}{640}×100=81.25$
(c) Fisher's ${p}_{01}=\sqrt{\frac{\mathrm{\Sigma }{p}_{1}{q}_{1}}{\mathrm{\Sigma }{p}_{0}{q}_{1}}×\frac{\mathrm{\Sigma }{p}_{1}{q}_{0}}{\mathrm{\Sigma }{p}_{0}{q}_{0}}}×100=\sqrt{\frac{520}{640}×\frac{400}{465}}×100=83.60\phantom{\rule{0ex}{0ex}}$

(ii) Quantity Index number
(a) Laspeyre's ${q}_{01}=\frac{\mathrm{\Sigma }{q}_{1}{p}_{0}}{\mathrm{\Sigma }{q}_{0}{p}_{0}}×100=\frac{640}{465}×100=137.63$

(b) Paasche's ${q}_{01}=\frac{\mathrm{\Sigma }{q}_{1}{p}_{1}}{\mathrm{\Sigma }{q}_{0}{p}_{1}}×100=\frac{520}{400}×100=130$

(c) Fisher's ${q}_{01}=\sqrt{\frac{\mathrm{\Sigma }{p}_{0}{q}_{1}}{\mathrm{\Sigma }{p}_{0}{q}_{0}}×\frac{\mathrm{\Sigma }{p}_{1}{q}_{0}}{\mathrm{\Sigma }{p}_{1}{q}_{0}}}×100=\sqrt{\frac{520}{400}×\frac{640}{465}}×100=133.76\phantom{\rule{0ex}{0ex}}$

The price index number for current year is 86.02 and 81.25 as per Laspeyres's method and Paasche's method respectively. This implies that there is net decrease in the prices in current year by 13.98% and 18.75% from the base year as per Laspeyres's method and Paasche's method respectively.

The quantity index number for current year is 137.63 and 130 as per Laspeyres's method and Paasche's method respectively. This implies that there is net increase in the quantity demanded by 37.63% and 30% from the base year as per Laspeyres's method and Paasche's method respectively.

#### Question 12:

Calculate price index number by weighted average of price relative method.

 Commodity Price Base Year (in ₹) Price current year ( in ₹) Quantity Base year  (in Kg) A 6.0 8.0 40 B 3.0 3.2 80 C 2.0 3.0 2

 Commodity Price in Base Year (p0) Price in Quantity Year (p1) Quantity in Base Year (q0) $\mathbit{R}\mathbf{=}\left(\frac{{p}_{\mathit{1}}}{{p}_{\mathit{0}}}\mathbf{×}\mathbf{100}\right)$ V (p0q0) RV A B C 6 3 2 8 3.2 3 40 80 20 133.33 106.67 150 240 240 40 31999.2 25600.8 6000 ∑V = 520 ∑RV = 63600

Weighted Average of Price Relative Method

Thus, price index number is 122.30.

#### Question 13:

Prepare consumer price index numbers from the following data for 2014 and 2013  taking 2012 as base.

 Group Price (in ₹) 2012 2013 2014 A 20.00 24.00 21.00 B 1.25 1.50 1.00 C 5.00 8.00 8.00 D 2.00 2.25 2.12

Give weights to four groups as 4, 3, 2 and 1 respectively.

Price Index for 2013

 Group p0 (2012) p1 (2013) W $\mathbit{R}\mathbit{=}\left(\frac{{\mathbit{p}}_{\mathbf{1}}}{{\mathbit{p}}_{\mathbf{0}}}\mathbf{×}\mathbf{100}\right)$ WR A B C D 20 1.25 5 2 24 1.5 8 2.25 4 3 2 1 120 120 160 112.5 480 360 320 112.5 ∑W= 10 ∑WR=1272.5

Thus, consumer price index for 2013 is 127.25.

Price Index number for 2014
 Group p0 (2012) p1 (2014) W $\mathbit{R}\mathbit{=}\left(\frac{{\mathbit{p}}_{\mathbf{1}}}{{\mathbit{p}}_{\mathbf{0}}}\mathbf{×}\mathbf{100}\right)$ WR A B C D 20 1.25 5 2 21 1 8 2.12 4 3 2 1 105 80 160 106 420 240 320 106 ∑W= 10 ∑WR=1086

Thus, consumer price index for 2014 is 108.6.

Note: As per textbook, consumer index for 2014 is 109.78. However, as per the above solution, consumer index for 2014 should be 108.6.

#### Question 14:

From the data given below construct the consumer price index number:

 Commodity Price Relatives Weights Food 250 45 Rent 150 15 Clothing 320 20 Fuel and Lighting 190 5 Miscellaneous 300 15