(i) weighted index
Reason: Weighted index numbers assign weights to items based on their relative importance (e.g., higher weight for food than salt), unlike simple indices which treat all items equally.

(i) base year
Reason: Methods like Laspeyres’ Index, which is widely used, utilize base year quantities ($q_0$) as weights because base year data is fixed and readily available.

(i) small
Reason: In a weighted index, the impact of a price change is proportional to the weight of the item. An item with low importance (weight) will not significantly affect the overall index value.

(i) retail prices
Reason: CPI tracks the cost of a basket of goods and services purchased by the average consumer, which are always bought at retail prices.

(i) Food
Reason: For industrial workers (and lower-income groups in general), a major portion of income is spent on food; hence, it carries the highest weight in the CPI basket.

(i) wholesale price index
Reason: Traditionally in India, the Wholesale Price Index (WPI) was the primary measure for calculating headline inflation, although the focus has shifted towards CPI in recent policy frameworks. In the context of standard textbook definitions, WPI is often cited as the general measure.

Index numbers are indispensable tools for economic analysis because:

• Measure of Change: They quantify changes in variables like prices or production over time.
• Barometer of Economy: They act as economic barometers (e.g., CPI measures inflation).
• Comparability: They allow comparison of data expressed in different units.
• Deflating: They are used to adjust wages and income for inflation (calculating real income).

The base period (year) should have the following properties:

• Normal Year: It should be a year of economic stability, free from wars, floods, or famines.
• Recent Period: It should not be too distant in the past, as consumer tastes and technology change over time.
• Data Availability: Reliable statistical data for that year must be available.

Different categories of consumers (e.g., Industrial Workers, Agricultural Labourers) have different consumption patterns.
For example, an industrial worker may spend more on rent and transport, while an agricultural labourer may spend more on food. A single index cannot represent the cost of living for all groups accurately; hence, separate CPIs are needed.

It measures the average change over time in the prices of a fixed basket of goods and services consumed specifically by industrial workers. It is often used to determine the Dearness Allowance (DA) for employees.

• Price Index: Measures changes in the price level of goods over time (e.g., CPI, WPI). It ignores changes in quantity.
• Quantity Index: Measures changes in the volume or quantity of goods produced or consumed over time (e.g., Index of Industrial Production – IIP). It ignores changes in price.

No. An index number is based on a sample basket of representative goods and services. Changes in the prices of items not included in this basket (e.g., a specific luxury car or a rare niche product) will not be reflected in the index.

No. The President of India belongs to a high-income bracket with a consumption pattern vastly different from that of “urban non-manual employees” (middle class). The basket of goods, quality of items, and weights assigned would be incomparable; therefore, that CPI would not accurately reflect the President’s cost of living.

Since Quantities are not given but Weights ($W$) are given directly, we use the Weighted Average of Price Relatives Method.
$$ \text{Formula: } I = \frac{\sum RW}{\sum W} $$
Where Price Relative $R = \frac{P_1}{P_0} \times 100$

Items	Weight ($W$)	Price 1980 ($P_0$)	Price 2005 ($P_1$)	Relative ($R$)	Product ($RW$)
Food	75	100	200	$\frac{200}{100} \times 100 = 200$	15,000
Clothing	10	20	25	$\frac{25}{20} \times 100 = 125$	1,250
Fuel	5	15	20	$\frac{20}{15} \times 100 = 133.3$	666.65
Rent	6	30	40	$\frac{40}{30} \times 100 = 133.3$	799.98
Misc	4	35	65	$\frac{65}{35} \times 100 = 185.7$	742.85
Total	$\sum W = 100$				$\sum RW = 18459.48$

Calculation:
$$ \text{Cost of Living Index} = \frac{18459.48}{100} = \mathbf{184.59} $$
Interpretation: The cost of living has increased by approx 84.6% in 2005 compared to 1980.

Analysis of the Data:

• Overall Growth: The General Index of industrial production has risen significantly from 130.8 in 1996-97 to 189.0 in 2003-04, indicating robust industrial growth.
• Sectoral Dominance: The Manufacturing sector holds the highest weight (79.36%), meaning it drives the general index. It also shows the highest growth (index rose to 196.6).
• Electricity: This sector also showed healthy growth, rising from 122.0 to 172.6.
• Lagging Sector: Mining and Quarrying has the lowest index value (146.9) in 2003-04 compared to other sectors, indicating it grew at a slower pace than Manufacturing and Electricity.

(Student Activity. A sample list is provided below:)

• Food: Wheat, Rice, Milk, Vegetables, Fruits, Oil, Spices.
• Housing: Rent or Maintenance, Electricity Bill.
• Clothing: Daily wear, School uniform.
• Education: School fees, Books, Stationery.
• Transport: Petrol/Diesel, Bus fare.
• Communication: Mobile recharge, Internet bill.
• Medical: Doctor fees, Medicines.

Given: Base Salary ($S_0$) = Rs 4,000. Current Salary ($S_1$) = Rs 6,000. CPI = 400.
Standard Base CPI is assumed to be 100.

Calculation:
Salary required to maintain the same standard of living:
$$ \text{Required Salary} = \frac{\text{CPI} \times \text{Base Salary}}{100} $$
$$ = \frac{400 \times 4000}{100} = \text{Rs } 16,000 $$
Raise Needed:
Current Salary = Rs 6,000.
Raise = Required Salary – Current Salary
Raise = $16,000 – 6,000$ = Rs 10,000.

Given: General Index ($I$) = 125. Food Index ($I_F$) = 120. Other Index ($I_O$) = 135.
Let the weight of Food be $x$.
Since total weight is percentage based, Weight of Other items = $(100 – x)$.

Formula: $I = \frac{I_F \cdot x + I_O \cdot (100-x)}{100}$
$125 = \frac{120x + 135(100 – x)}{100}$
$12500 = 120x + 13500 – 135x$
$12500 – 13500 = 120x – 135x$
$-1000 = -15x$
$x = \frac{1000}{15} = 66.67$

Result: Percentage weight given to food is 66.67%.

Using the Family Budget Method (Weighted Average of Price Relatives):
$$ \text{Index} = \frac{\sum RW}{\sum W} $$
Where Price Relative $R = (P_1 / P_0) \times 100$.

Items	Weight ($W$)	Price 1995 ($P_0$)	Price 2004 ($P_1$)	$R = \frac{P_1}{P_0} \times 100$	Product ($RW$)
Food	35	1400	1500	107.14	3749.9
Fuel	10	200	250	125.00	1250.0
Clothing	20	500	750	150.00	3000.0
Rent	15	200	300	150.00	2250.0
Misc	20	250	400	160.00	3200.0
Total	100				13449.9

Calculation:
$$ \text{Index} = \frac{13449.9}{100} = \mathbf{134.5} $$
Interpretation: The cost of living rose by 34.5% in 2004 compared to 1995.

(Student Activity)
Guidance: Create a table with columns for Date, Item, Quantity, Price Paid.
Compare the prices with those from a few months ago. If prices have risen (Inflation), your family might have:

• Reduced consumption of non-essential items.
• Shifted to cheaper brands.
• Experienced a decrease in savings.

(i) Comment on Relative Values:

• Trend: All three indices (CPI-IW, CPI-AL, WPI) show a consistent upward trend, indicating inflation over the decade.
• Magnitude: CPI for Industrial Workers (313 to 500) and Agricultural Labourers (234 to 331) have much higher absolute values than WPI (121.6 to 175.9).

(ii) Are they comparable?
No, strictly speaking, they are not directly comparable because:

• Different Base Years: CPI-IW is based on 1982=100, CPI-AL on 1986-87=100, and WPI on 1993-94=100.
• Different Baskets: The basket of goods for a worker differs from that of a wholesaler.
• To compare them effectively, they must be shifted to a common base year (Splicing).

Step 1: Classify items and calculate Total Expenditure per Tax Slab.

Category (Tax Rate $X$)	Items Included	Total Exp ($W$)	Product ($WX$)
0%	Cereals (1500) + Eggs (250) + Fish/Meat (250)	2000	$2000 \times 0 = 0$
5%	Medicines (50) + Biogas (50) + Transport (100)	200	$200 \times 5 = 1000$
12%	Butter (50) + Babool (10) + Ketchup (40)	100	$100 \times 12 = 1200$
18%	Biscuits (75) + Cakes (25) + Garments (100)	200	$200 \times 18 = 3600$
28%	Vacuum Cleaner, Car (1000)	1000	$1000 \times 28 = 28000$
Total		$\sum W = 3500$	$\sum WX = 33800$

Step 2: Calculate Weighted Average Tax Rate.
$$ \text{Average Rate} = \frac{\sum WX}{\sum W} = \frac{33800}{3500} $$
$$ = 9.657… \approx \mathbf{9.66\%} $$
(Note: The textbook solution calculates using decimal weights [e.g., 0.05 for 5%], leading to 0.966. If using percentage as whole numbers like above, the result is 9.66, which is the same as 9.66%).