Index NumbersClass 11 Statistics For Economics NCERT Solutions
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Q1EXERCISES
An index number which accounts for the relative importance of the items is known as
(i)
weighted index
(ii)
simple aggregative index
(iii)
simple average of relatives
Solution
(i) weighted index
Explanation: A weighted index, as described in the chapter, is one where the relative importance of different items is taken into account by assigning them specific weights. This is in contrast to simple indices where all items are treated as having equal importance.
Q2EXERCISES
In most of the weighted index numbers the weight pertains to
(i)
base year
(ii)
current year
(iii)
both base and current year
Solution
(i) base year
Explanation: The chapter explains that in general, the base period weight is preferred to the current period weight. This is because calculating the weight every year is inconvenient, and using a fixed base year allows for more consistent and comparable measurements over time. Laspeyre's price index, for example, uses base period quantities as weights.
Q3EXERCISES
The impact of change in the price of a commodity with little weight in the index will be
(i)
small
(ii)
large
(iii)
uncertain
Solution
(i) small
Explanation: In a weighted index number, the weight assigned to a commodity reflects its relative importance. Therefore, a commodity with a small weight will have a minimal impact on the overall index value, even if its price changes significantly.
Q4EXERCISES
A consumer price index measures changes in
(i)
retail prices
(ii)
wholesale prices
(iii)
producers prices
Solution
(i) retail prices
Explanation: The chapter states that the Consumer Price Index (CPI), also known as the cost of living index, measures the average change in retail prices. It reflects the prices that consumers pay for a specific basket of goods and services.
Q5EXERCISES
The item having the highest weight in consumer price index for industrial workers is
(i)
Food
(ii)
Housing
(iii)
Clothing
Solution
(i) Food
Explanation: According to the table of weights for the All-India Combined Consumer Price Index provided in the chapter, the 'Food and beverages' group has the highest weight at 45.86%. This indicates that food items constitute the largest portion of the expenditure for the consumer groups measured by this index.
Q6EXERCISES
In general, inflation is calculated by using
(i)
wholesale price index
(ii)
consumer price index
(iii)
producers' price index
Solution
(i) wholesale price index
Explanation: The chapter explicitly states that the Wholesale Price Index (WPI) is widely used to measure the rate of inflation. The change in the 'All Commodities' WPI is often referred to as 'Headline Inflation'.
Q7EXERCISES
Why do we need an index number?
Solution
We need index numbers because they are statistical devices that simplify complex data and measure changes over time. They serve several important purposes:
- Summarizing Change: An index number provides a single summary figure that measures the overall change in a group of related variables, such as the prices of many commodities or the output of various industries. This is much easier to understand than looking at individual changes.
- Policy Making: Index numbers like the Consumer Price Index (CPI) and Wholesale Price Index (WPI) are crucial for economic policy formulation. They are used in wage negotiations, income policy, taxation, and for measuring inflation.
- Measuring Purchasing Power: CPI is used to calculate the purchasing power of money and to determine real wages, which helps in understanding the actual standard of living.
- Economic Barometer: Indices like the Index of Industrial Production (IIP) and Sensex act as barometers for the health of the industrial sector and the stock market, respectively, guiding policymakers and investors.
Q8EXERCISES
What are the desirable properties of the base period?
Solution
The desirable properties of a base period for an index number are:
- Normality: The base year should be a normal period, free from unusual events or extreme fluctuations. Years affected by events like wars, famines, droughts, or major economic crises should be avoided as they would not provide a representative benchmark for comparison.
- Recency: The base period should not be too far in the past. Economic and consumption patterns change over time; new products are introduced and old ones disappear. A recent base year ensures that the comparisons being made are relevant and meaningful.
Q9EXERCISES
Why is it essential to have different CPI for different categories of consumers?
Solution
It is essential to have different Consumer Price Indices (CPI) for different categories of consumers because consumption patterns and the relative importance of items vary significantly across these groups. For example:
- Different Consumption Baskets: The basket of goods and services consumed by an industrial worker is very different from that of an agricultural labourer or an urban non-manual employee. Food might constitute a larger portion of an agricultural labourer's budget, while housing and transport might be more significant for an urban employee.
- Varying Importance (Weights): The relative importance (weight) of each item differs. As the chapter notes, a rise in petrol prices may not directly impact the living conditions of poor agricultural labourers as much as it would affect an urban commuter.
By constructing separate CPIs for specific groups (like Industrial Workers, Agricultural Labourers, etc.), policymakers can get a more accurate picture of how price changes affect the cost of living for each particular segment of the population, allowing for more targeted and effective policies.
Q10EXERCISES
What does a consumer price index for industrial workers measure?
Solution
A Consumer Price Index (CPI) for industrial workers measures the average change over time in the retail prices of a fixed basket of goods and services consumed by a typical industrial worker's household. It essentially measures the change in the cost of living for this specific group. For instance, if the CPI for industrial workers (with base year 2001=100) is 277 in a later year, it means that an industrial worker would need Rs 277 to purchase the same basket of goods and services that cost Rs 100 in 2001.
Q11EXERCISES
What is the difference between a price index and a quantity index?
Solution
The main difference between a price index and a quantity index lies in what they measure:
- Price Index: A price index measures the average change in the prices of a group of commodities over a period. Its purpose is to show the general trend of price levels. Examples include the Consumer Price Index (CPI) and the Wholesale Price Index (WPI).
- Quantity Index: A quantity index measures the change in the physical volume of a group of items. It tracks changes in production, construction, or employment, keeping the price factor constant. An example is the Index of Industrial Production (IIP), which measures the change in the level of output in the industrial sector.
Q12EXERCISES
Is the change in any price reflected in a price index number?
Solution
No, the change in the price of any and every commodity is not reflected in a price index number. A price index is constructed based on a selected basket of representative items. Therefore:
- Inclusion in the Basket: If the price of a commodity that is not included in the selected basket changes, it will not be reflected in the index.
- Weightage: For items that are included, the impact of a price change depends on the weight assigned to that item. A significant price change in an item with a very low weight will have only a small impact on the overall index.
Q13EXERCISES
Can the CPI for urban non-manual employees represent the changes in the cost of living of the President of India?
Solution
No, the CPI for urban non-manual employees cannot represent the changes in the cost of living of the President of India. The reason is that the consumption basket and expenditure pattern of the President would be vastly different from that of a typical urban non-manual employee. The goods and services consumed, and their relative importance (weights), would not align. A CPI is designed for a specific, homogeneous group of consumers, and the President's lifestyle and expenditure do not fit into this category. Therefore, using this specific CPI would give a completely inaccurate and unrepresentative measure of changes in the President's cost of living.
Q14EXERCISES
The monthly per capita expenditure incurred by workers for an industrial centre during 1980 and 2005 on the following items are given below. The weights of these items are 75,10,5,6 and 4 respectively. Prepare a weighted index number for cost of living for 2005 with 1980 as the base.
Items Price in 1980 Price in 2005 Food 100 200 Clothing 20 25 Fuel & lighting 15 20 House rent 30 40 Misc 35 65
Solution
To prepare the weighted index number for the cost of living for 2005 with 1980 as the base, we will use the weighted average of price relatives method. The formula is:
Cost of Living Index = (ΣWR) / ΣW
Where:
- W = Weight of the item
- R = Price Relative = (Price in Current Year / Price in Base Year) × 100 = (P₁/P₀) × 100
First, we calculate the Price Relative (R) for each item.
- Food: R = (200 / 100) × 100 = 200
- Clothing: R = (25 / 20) × 100 = 125
- Fuel & lighting: R = (20 / 15) × 100 = 133.33
- House rent: R = (40 / 30) × 100 = 133.33
- Misc: R = (65 / 35) × 100 = 185.71
Next, we create a table to calculate WR for each item.
| Items | Weight (W) | Price Relative (R) | WR |
|---|---|---|---|
| Food | 75 | 200.00 | 15000.00 |
| Clothing | 10 | 125.00 | 1250.00 |
| Fuel & lighting | 5 | 133.33 | 666.65 |
| House rent | 6 | 133.33 | 799.98 |
| Misc | 4 | 185.71 | 742.84 |
| Total | ΣW = 100 | ΣWR = 18459.47 |
Now, we apply the formula:
Cost of Living Index = ΣWR / ΣW = 18459.47 / 100 = 184.59
The weighted index number for the cost of living for 2005 is 184.59. This indicates that the cost of living for the workers increased by 84.59% from 1980 to 2005.
Q15EXERCISES
Read the following table carefully and give your comments.
INDEX OF INDUSTRIAL PRODUCTION BASE 1993-94
Industry Weight in % 1996-97 2003-2004 General index 100 130.8 189.0 Mining and quarrying 10.73 118.2 146.9 Manufacturing 79.58 133.6 196.6 Electricity 10.69 122.0 172.6
Solution
Based on the provided table for the Index of Industrial Production (IIP), the following comments can be made:
-
Overall Industrial Growth: The General Index, which represents the overall industrial production, increased from 130.8 in 1996-97 to 189.0 in 2003-04. This indicates a robust growth in the industrial sector of the economy during this period.
-
Dominance of Manufacturing Sector: The Manufacturing sector holds the highest weight (79.58%) in the IIP. It also shows the most significant growth, with its index value rising from 133.6 to 196.6. This implies that the manufacturing sector was the primary driver of the overall industrial growth.
-
Performance of Other Sectors:
- The Mining and quarrying sector, with a weight of 10.73%, grew from an index of 118.2 to 146.9. Its growth was positive but slower compared to the manufacturing sector.
- The Electricity sector, with a weight of 10.69%, also expanded, with its index moving from 122.0 to 172.6. Its growth was stronger than mining but less than manufacturing.
In conclusion, the data shows a healthy expansion of the industrial economy between 1996-97 and 2003-04, led overwhelmingly by strong performance in the manufacturing sector.
Q16EXERCISES
Try to list the important items of consumption in your family.
Solution
This is an activity-based question that requires a personalized answer. The important items of consumption in a typical family can be categorized as follows:
- Food:
- Cereals (Wheat, Rice)
- Pulses (Dal)
- Vegetables and Fruits
- Milk and Dairy Products (Butter, Ghee, Cheese)
- Edible Oils
- Sugar, Spices
- Meat, Fish, Eggs
- Housing:
- House Rent or Home Loan EMI
- Electricity Bill
- Water Bill
- Clothing and Footwear:
- Apparel for family members
- Shoes, Sandals
- Fuel and Light:
- LPG Cylinder
- Petrol/Diesel for vehicles
- Miscellaneous:
- Education (School fees, books, stationery)
- Healthcare (Medicines, doctor's fees)
- Transportation (Public transport fares, vehicle maintenance)
- Communication (Mobile phone/internet bills)
- Personal care items (Soap, toothpaste)
- Recreation and Entertainment (Movies, travel)
Q17EXERCISES
If the salary of a person in the base year is Rs 4,000 per annum and the current year salary is Rs 6,000, by how much should his salary be raised to maintain the same standard of living if the CPI is 400?
Solution
To maintain the same standard of living, the person's salary must increase in proportion to the increase in the Consumer Price Index (CPI).
- Base Year Salary: Rs 4,000
- Base Year CPI: 100 (The CPI of the base year is always 100)
- Current Year CPI: 400
First, we calculate the salary required in the current year to maintain the same standard of living as the base year.
Required Current Salary = Base Year Salary × (Current Year CPI / Base Year CPI)
Required Current Salary = Rs 4,000 × (400 / 100)
Required Current Salary = Rs 4,000 × 4
Required Current Salary = Rs 16,000
This means the person needs a salary of Rs 16,000 to have the same purchasing power they had in the base year.
Now, we find out how much of a raise is needed.
- Current Salary: Rs 6,000
- Required Salary: Rs 16,000
Required Raise = Required Salary - Current Salary
Required Raise = Rs 16,000 - Rs 6,000
Required Raise = Rs 10,000
Therefore, his salary should be raised by Rs 10,000 to maintain the same standard of living.
Q18EXERCISES
The consumer price index for June, 2005 was 125. The food index was 120 and that of other items 135. What is the percentage of the total weight given to food?
Solution
We can solve this using the formula for a weighted index.
Let:
- The weight of food be W₁
- The weight of other items be W₂
- The total weight is 100, so W₁ + W₂ = 100, which means W₂ = 100 - W₁
Given:
- Overall CPI = 125
- Food Index (I₁) = 120
- Other Items Index (I₂) = 135
The formula for the overall CPI is:
Overall CPI = (W₁I₁ + W₂I₂) / (W₁ + W₂)
Substituting the values:
125 = (W₁ × 120 + (100 - W₁) × 135) / 100
Now, we solve for W₁:
125 × 100 = 120W₁ + 13500 - 135W₁
12500 = 13500 - 15W₁
15W₁ = 13500 - 12500
15W₁ = 1000
W₁ = 1000 / 15
W₁ = 66.67
Therefore, the percentage of the total weight given to food is 66.67%.
Q19EXERCISES
An enquiry into the budgets of the middle class families in a certain city gave the following information;
Expenses on items Food 35% Fuel 10% Clothing 20% Rent 15% Misc. 20% Price (in Rs) in 2004 1500 250 750 300 400 Price (in Rs) in 1995 1400 200 500 200 250
Solution
To find the cost of living index for 2004 with 1995 as the base year, we will use the weighted average of price relatives method.
Formula: Cost of Living Index = ΣWR / ΣW
Where:
- W = Weight (given as a percentage)
- R = Price Relative = (Price in 2004 / Price in 1995) × 100
Step 1: Calculate the Price Relative (R) for each item.
- Food: R = (1500 / 1400) × 100 = 107.14
- Fuel: R = (250 / 200) × 100 = 125.00
- Clothing: R = (750 / 500) × 100 = 150.00
- Rent: R = (300 / 200) × 100 = 150.00
- Misc.: R = (400 / 250) × 100 = 160.00
Step 2: Calculate WR for each item and sum them up.
The total weight (ΣW) = 35 + 10 + 20 + 15 + 20 = 100.
| Expenses on Items | Weight (W) | Price Relative (R) | WR |
|---|---|---|---|
| Food | 35 | 107.14 | 3749.90 |
| Fuel | 10 | 125.00 | 1250.00 |
| Clothing | 20 | 150.00 | 3000.00 |
| Rent | 15 | 150.00 | 2250.00 |
| Misc. | 20 | 160.00 | 3200.00 |
| Total | ΣW = 100 | ΣWR = 13449.90 |
Step 3: Calculate the Cost of Living Index.
Cost of Living Index = ΣWR / ΣW = 13449.90 / 100 = 134.50
The cost of living index for the year 2004 as compared with 1995 is 134.50. This indicates an increase of 34.50% in the cost of living for middle-class families in that city over this period.
Q20EXERCISES
Record the daily expenditure, quantities bought and prices paid per unit of the daily purchases of your family for two weeks. How has the price change affected your family?
Solution
This is an activity-based question requiring personal data collection and analysis. Here is a guide on how to approach it:
Step 1: Data Collection (for two weeks)
Create a daily log for two consecutive weeks. For each day, record:
- Item: Name of the item purchased (e.g., Milk, Bread, Petrol, Vegetables).
- Quantity: The amount bought (e.g., 1 litre, 1 loaf, 5 litres, 1 kg).
- Price per Unit: The cost for one unit (e.g., Rs 50/litre, Rs 40/loaf).
- Total Expenditure: Quantity × Price per Unit.
Step 2: Analysis
- Week 1 vs. Week 2: Compare the prices of the same items between the first and second week. Note which items became more expensive, cheaper, or stayed the same.
- Construct a Simple Price Index: Choose the first week as the base period. For a few key items, calculate a simple price index for the second week using the formula:
Index = (Price in Week 2 / Price in Week 1) × 100. - Impact on Family Budget: Analyze how the price changes affected your family's expenditure.
- Did the total weekly expenditure increase or decrease?
- If prices of essential items (like milk or vegetables) increased, did your family have to cut back on other expenses (like entertainment or non-essential items)?
- Did your family buy less of an item because its price went up? This shows the impact on consumption patterns.
Example of how to state the effect:
"During the two weeks, the price of tomatoes increased by 15%. As a result, our family's total expenditure on vegetables went up by Rs 50 in the second week. To manage the budget, we had to reduce our spending on snacks. This shows that even a small price change in an essential commodity directly affects our family's spending choices and purchasing power."
Q21EXERCISES
Given the following data-
Year CPI of industrial workers (1982=100) CPI of agricultural labourers (1986-87 = 100) WPI (1993-94=100) 1995-96 313 234 121.6 1996-97 342 256 127.2 1997-98 366 264 132.8 1998-99 414 293 140.7 1999-00 428 306 145.3 2000-01 444 306 155.7 2001-02 463 309 161.3 2002-03 482 319 166.8 2003-04 500 331 175.9
Source: Economic Survey, 2004-2005, Government of India
(i)
Comment on the relative values of the index numbers.
(ii)
Are they comparable?
Solution
(i)
Comment on the relative values of the index numbers:
- Upward Trend: All three indices—CPI for Industrial Workers (CPI-IW), CPI for Agricultural Labourers (CPI-AL), and Wholesale Price Index (WPI)—show a consistent and continuous increase from 1995-96 to 2003-04. This indicates a general trend of rising prices and inflation across different sectors of the economy during this period.
- Magnitude of Values: The absolute values of CPI-IW are the highest, followed by CPI-AL, and then WPI. This difference is primarily due to their different base years. CPI-IW has the oldest base year (1982), so it has accumulated price increases over a longer period, resulting in a higher index value. Similarly, CPI-AL (base 1986-87) has a higher value than WPI (base 1993-94).
- Rate of Increase: The cost of living for both industrial and agricultural workers, as indicated by their respective CPIs, appears to have risen substantially. For example, the CPI-IW increased from 313 to 500, a significant jump that reflects a sharp decline in the purchasing power of money for this group.
(ii)
Are they comparable?
No, the index numbers are not directly comparable in their absolute forms. The reasons are:
- Different Base Years: Each index has a different base year (1982, 1986-87, and 1993-94). An index value of 313 for CPI-IW means prices are 213% higher than in 1982, while an index of 121.6 for WPI means prices are 21.6% higher than in 1993-94. Comparing 313 with 121.6 is meaningless because they are measured from different starting points.
- Different Baskets of Goods: The three indices measure price changes for different sets of commodities and services. WPI tracks wholesale prices of goods only (no services), while the CPIs track retail prices of a basket of goods and services consumed by specific population groups.
However, while their absolute values cannot be compared, the rate of change (e.g., annual percentage increase) of each index can be calculated and compared. This would help in analyzing whether the inflation rate was higher at the wholesale level or for specific consumer groups in a particular year.
Q22EXERCISES
The monthly expenditure (Rs.) of a family on some important items and the Goods and Services Tax (GST) rates applicable to these items is as follows:
Item Monthly Expense(Rs) GST Rate % Cereals 1500 0 Eggs 250 0 Fish, Meat 250 0 Medicines 50 5 Biogas 50 5 Transport 100 5 Butter 50 12 Babool 10 12 Tomato Ketchup 40 12 Biscuits 75 18 Cakes, Pastries 25 18 Branded Garments 100 18 Vacuum Cleaner, Car 1000 28
Calculate the average tax rate as far as this family is concerned.
Solution
To calculate the average GST rate for this family, we need to use the formula for a weighted average. Here, the expenditure on each item acts as the weight (w), and the GST rate is the variable (x).
The average tax rate is calculated as: Average GST Rate = (Σwx) / (Σw)
First, we group the items by their GST rate and calculate the total expenditure for each category.
- Category 1 (0% GST):
- Expenditure = 1500 (Cereals) + 250 (Eggs) + 250 (Fish, Meat) = Rs 2000
- Category 2 (5% GST):
- Expenditure = 50 (Medicines) + 50 (Biogas) + 100 (Transport) = Rs 200
- Category 3 (12% GST):
- Expenditure = 50 (Butter) + 10 (Babool) + 40 (Tomato Ketchup) = Rs 100
- Category 4 (18% GST):
- Expenditure = 75 (Biscuits) + 25 (Cakes, Pastries) + 100 (Branded Garments) = Rs 200
- Category 5 (28% GST):
- Expenditure = Rs 1000 (Vacuum Cleaner, Car)
Now, we create a table to calculate the total tax paid (wx) and total expenditure (Σw).
| Category (GST Rate) | Expenditure (w) | GST Rate (x) | Total Tax Paid (wx) |
|---|---|---|---|
| 1 (0%) | 2000 | 0.00 | 0 |
| 2 (5%) | 200 | 0.05 | 10 |
| 3 (12%) | 100 | 0.12 | 12 |
| 4 (18%) | 200 | 0.18 | 36 |
| 5 (28%) | 1000 | 0.28 | 280 |
| Total | Σw = 3500 | Σwx = 338 |
Finally, we calculate the average GST rate:
Average GST Rate = Σwx / Σw = 338 / 3500 = 0.09657
To express this as a percentage, we multiply by 100:
Average GST Rate = 0.09657 × 100 = 9.66%
The average tax rate as far as this family is concerned is 9.66%.