How to Use This Page
1. Solve each exercise problem on your own first — attempt every step before looking here. 2. Compare your method with the step-by-step solution shown. 3. If your answer differs, rework from the step where you diverged. 4. Pay special attention to the Common Mistakes section — these are the errors that cost marks in exams. 5. Use the Exam Checklist before your test to ensure you have covered every topic.
Key Concepts (Quick Recap)
Index Numbers
- Price Index: Measures relative change in prices
- Quantity Index: Measures relative change in quantities
- Value Index: Measures relative change in total value (price × quantity)
- Simple Aggregate Index: (Σp₁/Σp₀) × 100
- Simple Average of Relatives: Average of (p₁/p₀ × 100) for all commodities
- Weighted Index Numbers use weights to reflect relative importance
Important Price Index Formulas
- Laspeyres Price Index (LPI): (Σp₁q₀/Σp₀q₀) × 100 — uses base year quantities as weights
- Paasche's Price Index (PPI): (Σp₁q₁/Σp₀q₁) × 100 — uses current year quantities as weights
- Fisher's Ideal Index: √(LPI × PPI) — satisfies time reversal and factor reversal tests
- Marshall-Edgeworth Index: (Σp₁(q₀+q₁))/(Σp₀(q₀+q₁)) × 100
- Consumer Price Index (CPI): (Σp₁q₀/Σp₀q₀) × 100 (Laspeyres form for consumer basket)
Tests for Index Numbers
- Time Reversal Test: P₀₁ × P₁₀ = 1 (should hold)
- Factor Reversal Test: P₀₁ × Q₀₁ = (Σp₁q₁)/(Σp₀q₀) (should hold)
- Circular Test: P₀₁ × P₁₂ × P₂₀ = 1 (should hold)
Time Series Components
- Trend (T): Long-term direction (upward or downward)
- Seasonal (S): Regular, predictable patterns within a year
- Cyclical (C): Medium-term business cycles (2-10 years)
- Irregular (I): Random, unpredictable fluctuations
Time Series Models
- Additive Model: Y = T + S + C + I
- Multiplicative Model: Y = T × S × C × I
Trend Measurement Methods
- Freehand method: Draw a trend line visually
- Semi-average method: Divide data into two equal halves, compute averages, join them
- Moving average method: Average of successive groups; smooths fluctuations
- Least squares method: Fit a line Y = a + bX using Y = a + bX where a = ΣY/n, b = ΣXY/ΣX²
Moving Average
- Odd period (3, 5, 7-year): Direct averaging
- Even period (4, 6, 8-year): Requires centering — take a 2-period moving average of successive moving averages
- Formula for 3-year: M₃ = (y₁ + y₂ + y₃)/3
- Formula for 4-year centered: First compute 4-year MA, then compute 2-year centered MA
I. Choose the Best Answer (MCQs)
1. Laspeyres price index uses: (a) Current year quantities as weights (b) Base year quantities as weights (c) Average quantities as weights (d) No weights
Answer(b) Base year quantities as weights Explanation: Laspeyres index LPI = (Σp₁q₀/Σp₀q₀) × 100, where q₀ are base year quantities used as weights.
2. Fisher's ideal index is the: (a) Arithmetic mean of LPI and PPI (b) Geometric mean of LPI and PPI (c) Harmonic mean of LPI and PPI (d) Sum of LPI and PPI
Answer(b) Geometric mean of LPI and PPI Explanation: Fisher's Index = √(LPI × PPI). It is the geometric mean of Laspeyres and Paasche's indices.
3. The base period index number is always: (a) 0 (b) 10 (c) 100 (d) 1
Answer(c) 100 Explanation: By convention, the base period index is set to 100, and all other periods are expressed relative to this base.
4. In a 5-year moving average, how many values are lost at each end? (a) 1 (b) 2 (c) 3 (d) 5
Answer(b) 2 Explanation: For a k-period moving average, k/2 values are lost at each end. For 5-year: 5/2 = 2.5, so 2 values are lost at each end (we round down for odd periods since the middle value is centred).
5. Which index number satisfies both the time reversal test and the factor reversal test? (a) Laspeyres (b) Paasche's (c) Fisher's (d) Marshall-Edgeworth
Answer(c) Fisher's Explanation: Fisher's ideal index satisfies both the time reversal test (P₀₁ × P₁₀ = 1) and the factor reversal test (P₀₁ × Q₀₁ = V₀₁).
6. In the multiplicative model of time series, Y equals: (a) T + S + C + I (b) T × S × C × I (c) T × S + C × I (d) T + S × C × I
Answer(b) T × S × C × I Explanation: The multiplicative model assumes Y = T × S × C × I, where the components multiply rather than add.
7. The cyclical component of a time series refers to: (a) Long-term trend (b) Seasonal patterns within a year (c) Business cycles of 2-10 years (d) Random fluctuations
Answer(c) Business cycles of 2-10 years Explanation: Cyclical variations are medium-term oscillations around the trend, typically lasting 2-10 years, associated with business cycles (boom, recession, depression, recovery).
8. If the simple aggregate price index is 125, it means: (a) Prices decreased by 25% (b) Prices increased by 25% (c) Prices increased by 125% (d) Prices are 125 times the base
Answer(b) Prices increased by 25% Explanation: Index of 125 means current prices are 125% of base prices, i.e., prices have increased by 25% from the base period.
II. Fill in the Blanks
1. The formula for Laspeyres price index is (Σp₁q₀/Σp₀q₀) × 100.
2. Fisher's ideal index is the geometric mean of Laspeyres and Paasche's indices.
3. The base period is assigned the index number 100.
4. In the additive model of time series, Y = T + S + C + I.
5. The time reversal test requires that P₀₁ × P₁₀ = 1.
6. For a 4-year moving average, centering is required because 4 is an even number.
7. The least squares method fits a trend line of the form Y = a + bX.
8. Seasonal variations are regular and predictable fluctuations that repeat every year.
III. True or False
1. Statement: Laspeyres index tends to overstate price changes.
True. Laspeyres uses base year quantities as weights, which means it doesn't account for consumers substituting away from goods that have become more expensive. This tends to overstate the true cost of living increase.
2. Statement: Paasche's index uses base year quantities as weights.
False. Correction: Paasche's index uses current year quantities as weights. PPI = (Σp₁q₁/Σp₀q₁) × 100.
3. Statement: Moving average method completely eliminates trend from the data.
False. Correction: The moving average method is used to isolate the trend by eliminating seasonal and irregular variations. It does not eliminate the trend — it reveals the trend.
4. Statement: In the least squares method, ΣY = na + bΣX when ΣX = 0.
True. When ΣX = 0, the normal equations simplify to: ΣY = na and ΣXY = bΣX². So a = ΣY/n and b = ΣXY/ΣX².
5. Statement: Consumer Price Index (CPI) is also called cost of living index.
True. CPI measures changes in the retail prices of goods and services consumed by a specific group, and it reflects the cost of living for that group.
6. Statement: An index number of 90 means prices have increased by 90%.
False. Correction: An index of 90 means current prices are 90% of base prices, i.e., prices have decreased by 10%.
IV. Match the Following
| Column A | Column B |
|---|
| 1. Laspeyres index | (a) √(LPI × PPI) |
| 2. Paasche's index | (b) Uses base year quantities |
| 3. Fisher's index | (c) Uses current year quantities |
| 4. Time series trend | (d) Long-term movement |
| 5. Seasonal variation | (e) Regular pattern within a year |
Answer1 → (b): Laspeyres uses base year quantities q₀. 2 → (c): Paasche's uses current year quantities q₁. 3 → (a): Fisher's = √(LPI × PPI). 4 → (d): Trend is the long-term upward or downward movement. 5 → (e): Seasonal variation is a regular pattern that repeats within each year.
V. Answer Briefly
1. What are the uses of index numbers?
Answer
Index numbers are used to: 1. Measure changes in prices, quantities, or values over time 2. Compare cost of living across different regions or periods 3. Formulate economic policies (inflation targeting, wage adjustments) 4. Deflate monetary values to real terms (e.g., real income = nominal income/CPI × 100) 5. Evaluate purchasing power of money 6. Forecast future trends
2. Explain the steps in constructing a price index number.
Answer
Steps to construct a price index number: 1. Define the purpose: Decide what the index will measure 2. Select the base period: Choose a normal, representative period 3. Select commodities: Choose representative items consumed by the group 4. Collect price data: Obtain prices for base and current periods 5. Assign weights: Determine the importance of each commodity 6. Choose the formula: Laspeyres, Paasche's, or Fisher's 7. Compute the index: Apply the formula and calculate
3. What are the components of a time series? Explain each briefly.
Answer
1. Trend (T): The long-term general direction of movement — upward, downward, or stationary. Reflects the underlying growth or decline. 2. Seasonal variation (S): Regular, predictable patterns that repeat within a year (e.g., higher sales during festivals). 3. Cyclical variation (C): Medium-term oscillations around the trend (typically 2-10 years) — boom, recession, depression, recovery. 4. Irregular variation (I): Random, unpredictable events like strikes, natural disasters, or sudden policy changes.
4. Distinguish between additive and multiplicative models.
Answer
| Feature | Additive Model | Multiplicative Model | |---|---|---| | Formula | Y = T + S + C + I | Y = T × S × C × I | | Assumption | Components are independent | Components interact | | Units | All in same units | T in units; S, C, I as ratios | | Use | When variations are roughly constant | When variations change proportionally | | Isolation | Subtract trend: Y − T = S + C + I | Divide by trend: Y/T = S × C × I |
5. What is the moving average method? When is centering required?
Answer
The moving average method computes successive averages of overlapping groups of k consecutive observations to smooth out fluctuations and reveal the trend. For a k-year moving average: - Odd k (3, 5, 7): The average is centred on the middle year directly. - Even k (4, 6, 8): The average falls between two years, so centering is required — take a 2-period moving average of successive moving averages to centre them on specific years.
6. Write the formula for computing the trend line using the method of least squares.
Answer
The trend line is Y = a + bX where: When ΣX = 0 (by choosing the origin at the middle): - a = ΣY/n (intercept) - b = ΣXY/ΣX² (slope) The X values are coded as ..., −3, −2, −1, 0, 1, 2, 3, ... for odd n, or ..., −5, −3, −1, 1, 3, 5, ... for even n (with step of 2).
VI. Answer in Detail
1. From the following data, calculate: (i) Laspeyres Price Index, (ii) Paasche's Price Index, (iii) Fisher's Ideal Index.
Answer
| Commodity | Base Year Price (p₀) | Base Year Qty (q₀) | Current Year Price (p₁) | Current Year Qty (q₁) |
|---|
| A | 10 | 20 | 12 | 15 |
| B | 15 | 10 | 18 | 12 |
| C | 20 | 5 | 25 | 4 |
Step 1: Set up the calculation table | Commodity | p₀q₀ | p₁q₀ | p₀q₁ | p₁q₁ | |---|---|---|---|---| | A | 10×20 = 200 | 12×20 = 240 | 10×15 = 150 | 12×15 = 180 | | B | 15×10 = 150 | 18×10 = 180 | 15×12 = 180 | 18×12 = 216 | | C | 20×5 = 100 | 25×5 = 125 | 20×4 = 80 | 25×4 = 100 | | Total | 450 | 545 | 410 | 496 | (i) Laspeyres Price Index LPI = (Σp₁q₀/Σp₀q₀) × 100 = (545/450) × 100 = 121.11 (ii) Paasche's Price Index PPI = (Σp₁q₁/Σp₀q₁) × 100 = (496/410) × 100 = 120.98 (iii) Fisher's Ideal Index FI = √(LPI × PPI) = √(121.11 × 120.98) = √14651.12 = 121.04
2. Calculate the 3-year and 5-year moving averages from the following data:
Answer
| Year | 2018 | 2019 | 2020 | 2021 | 2022 | 2023 | 2024 |
|---|
| Value | 20 | 25 | 30 | 28 | 35 | 40 | 38 |
3-Year Moving Average: M₃(2019) = (20 + 25 + 30)/3 = 75/3 = 25.00 M₃(2020) = (25 + 30 + 28)/3 = 83/3 = 27.67 M₃(2021) = (30 + 28 + 35)/3 = 93/3 = 31.00 M₃(2022) = (28 + 35 + 40)/3 = 103/3 = 34.33 M₃(2023) = (35 + 40 + 38)/3 = 113/3 = 37.67 2 values are lost (2018 and 2024). 5-Year Moving Average: M₅(2020) = (20 + 25 + 30 + 28 + 35)/5 = 138/5 = 27.60 M₅(2021) = (25 + 30 + 28 + 35 + 40)/5 = 158/5 = 31.60 M₅(2022) = (30 + 28 + 35 + 40 + 38)/5 = 171/5 = 34.20 4 values are lost (2018, 2019, 2023, 2024). The moving averages show an overall upward trend in the data.
3. Calculate the 4-year centred moving average from the following data:
Answer
| Year | Value |
|---|
| 2017 | 10 |
| 2018 | 15 |
| 2019 | 18 |
| 2020 | 22 |
| 2021 | 20 |
| 2022 | 25 |
| 2023 | 28 |
Step 1: 4-Year Moving Totals and Averages | Moving Total | Period | 4-Year MA | |---|---|---| | 10+15+18+22 = 65 | 2017-2020 | 65/4 = 16.25 | | 15+18+22+20 = 75 | 2018-2021 | 75/4 = 18.75 | | 18+22+20+25 = 85 | 2019-2022 | 85/4 = 21.25 | | 22+20+25+28 = 95 | 2020-2023 | 95/4 = 23.75 | Step 2: Centre the 4-Year Moving Averages (2-Year MA of successive 4-year MAs) | Centred at | Centred MA | |---|---| | 2019.5 | (16.25 + 18.75)/2 = 17.50 | | 2020.5 | (18.75 + 21.25)/2 = 20.00 | | 2021.5 | (21.25 + 23.75)/2 = 22.50 | The centred moving averages are placed at mid-years. The trend values show a steady increase.
4. Fit a straight line trend by the method of least squares and estimate the value for 2026.
Answer
| Year | 2019 | 2020 | 2021 | 2022 | 2023 |
|---|
| Sales (₹ lakh) | 10 | 12 | 14 | 16 | 18 |
Step 1: Code X values (odd n = 5) Middle year 2021 gets X = 0. X: −2, −1, 0, 1, 2 | Year | Y | X | XY | X² | |---|---|---|---|---| | 2019 | 10 | −2 | −20 | 4 | | 2020 | 12 | −1 | −12 | 1 | | 2021 | 14 | 0 | 0 | 0 | | 2022 | 16 | 1 | 16 | 1 | | 2023 | 18 | 2 | 36 | 4 | | Total | 70 | 0 | 20 | 10 | Step 2: Compute a and b a = ΣY/n = 70/5 = 14 b = ΣXY/ΣX² = 20/10 = 2 Step 3: Trend equation Y = 14 + 2X (where X = 0 at 2021, unit = 1 year) Step 4: Estimate for 2026 X for 2026 = 2026 − 2021 = 5 Y = 14 + 2(5) = 14 + 10 = ₹24 lakh
5. From the following data, calculate the Consumer Price Index (CPI) and determine the real wages.
Answer
| Group | Weight (w) | Current Index (I) |
|---|
| Food | 50 | 130 |
| Clothing | 20 | 120 |
| Housing | 15 | 110 |
| Fuel | 10 | 140 |
| Misc | 5 | 150 |
The money wage is ₹8,000.
Step 1: Calculate CPI CPI = (ΣwI)/(Σw) = (50×130 + 20×120 + 15×110 + 10×140 + 5×150)/(50+20+15+10+5) = (6500 + 2400 + 1650 + 1400 + 750)/100 = 12700/100 = 127 Step 2: Calculate Real Wage Real Wage = (Money Wage/CPI) × 100 = (8000/127) × 100 = 62.99 × 100 = ₹6,299 (approximately) Though the money wage is ₹8,000, the purchasing power is equivalent to ₹6,299 in base period terms — a 27% increase in prices has reduced the real value.
6. Verify that Fisher's ideal index satisfies the time reversal test for the following data.
Answer
| Commodity | p₀ | q₀ | p₁ | q₁ |
|---|
| A | 4 | 10 | 6 | 8 |
| B | 3 | 8 | 5 | 10 |
Step 1: Calculate LPI and PPI | Commodity | p₀q₀ | p₁q₀ | p₀q₁ | p₁q₁ | |---|---|---|---|---| | A | 40 | 60 | 32 | 48 | | B | 24 | 40 | 30 | 50 | | Total | 64 | 100 | 62 | 98 | LPI = (Σp₁q₀/Σp₀q₀) × 100 = (100/64) × 100 = 156.25 PPI = (Σp₁q₁/Σp₀q₁) × 100 = (98/62) × 100 = 158.06 Step 2: Calculate Fisher's Index P₀₁ P₀₁ = √(156.25 × 158.06) = √24696.875 ≈ 157.15 Step 3: Calculate P₁₀ (reverse the roles) P₁₀(Laspeyres) = (Σp₀q₁/Σp₁q₁) × 100 = (62/98) × 100 = 63.27 P₁₀(Paasche's) = (Σp₀q₀/Σp₁q₀) × 100 = (64/100) × 100 = 64.00 P₁₀(Fisher) = √(63.27 × 64.00) = √4049.28 ≈ 63.64 Step 4: Verify Time Reversal Test P₀₁ × P₁₀ = 157.15 × 63.64 / (100 × 100) = 9999.89/10000 ≈ 1.00 Since P₀₁ × P₁₀ ≈ 1, the time reversal test is satisfied. ✓
Common Mistakes to Avoid
1. Confusing base year and current year in formulas — In LPI, the denominator Σp₀q₀ uses base year prices AND base year quantities. In PPI, the denominator Σp₀q₁ uses base year prices but current year quantities. Always double-check which year's data goes where.
2. Forgetting to centre even-period moving averages — For 4-year, 6-year, etc. moving averages, you must take a 2-period moving average of successive MAs to centre them. Skipping this step gives trend values positioned between years.
3. Incorrect X-coding in least squares — For odd n, code as ..., −2, −1, 0, 1, 2, ... with origin at the middle year and step of 1. For even n, code as ..., −5, −3, −1, 1, 3, 5, ... with step of 2. This ensures ΣX = 0.
4. Misinterpreting index numbers — An index of 130 means prices are 130% of base (i.e., 30% increase), NOT 130% increase. An index of 80 means prices are 80% of base (i.e., 20% decrease).
5. Not converting money wages to real wages correctly — Real Wage = (Money Wage/CPI) × 100. Do NOT forget to multiply by 100 after division.
6. Confusing additive and multiplicative models — In additive model Y = T + S + C + I, to isolate seasonal, subtract trend. In multiplicative model Y = T × S × C × I, to isolate seasonal, divide by trend.
Exam Checklist
- [ ] I can compute Laspeyres, Paasche's, and Fisher's index numbers from given data
- [ ] I can verify time reversal and factor reversal tests
- [ ] I can calculate Consumer Price Index using weighted average of group indices
- [ ] I can compute moving averages (both odd and even periods, with centering)
- [ ] I can fit a trend line using least squares method and estimate future values
- [ ] I can distinguish between the four components of time series (T, S, C, I)
- [ ] I can convert money wages to real wages using CPI
- [ ] I can identify which index formula to use based on the problem context
