10 Time Series & Index Numbers · 10.1 Time series and index numbers
10.1.3 Base shifting, splicing and deflating index numbers; cost of living index and limitations
Base shifting re-expresses an index on a new base year, splicing joins an old and a new series into one continuous series, deflating turns money values into real values by dividing by a price index, and the cost of living (consumer price) index measures the change in the cost of a fixed basket bought by a particular class of people.
- APPSC Paper I
- Advanced level
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Core facts
- Base shifting: new index = old index of the year ÷ old index of the new base year × 100.
- Splicing needs an overlapping year that appears in both series.
- Forward (old series continued on its old base): new-series index × old-series index of the overlap year ÷ 100.
- Backward (old series brought onto the new base): old-series index × 100 ÷ old-series index of the overlap year.
- Deflating: real value = money value ÷ price index × 100. Real wage = money wage ÷ CPI × 100.
- Purchasing power of money = 100 ÷ price index (in rupees of the base year): it moves opposite to the price index.
- Cost of living index (CLI) = consumer price index: measures the change in the cost of buying the same goods and services for a specified class (industrial workers, farm labourers, urban employees). It measures changes in prices, not in the standard of living.
- Based on a family budget enquiry; major groups: food, clothing, fuel and light, house rent, miscellaneous; uses retail prices.
- Two methods:
- Aggregate expenditure method: CLI = Σp₁q₀ ÷ Σp₀q₀ × 100 (Laspeyres).
- Family budget method: CLI = ΣRW ÷ ΣW, R = p₁/p₀ × 100, W = p₀q₀ (base-year expenditure). Gives the same result as method 1.
- Uses of the CLI: fixing dearness allowance and wages, deflating incomes, measuring purchasing power, price policy.
- India (2026): CPI (Rural, Urban, Combined) by NSO, MoSPI, new series base 2024 from February 2026; CPI-IW by the Labour Bureau, base 2016 = 100, used for DA of central government employees.
Example 1. An index (2018 = 100) is 150 in 2022 and 180 in 2024. Shift the base to 2022 and give 2024. Solution. 180 ÷ 150 × 100. Answer: 120
Example 2. The CPI rises from 100 to 125. What happens to the purchasing power of a rupee? Solution. 100 ÷ 125 = 0.80. Answer: it falls to 80 paise of base-year value (a 20% fall)
Explained
1. Base shifting. Needed when the base year is too old, or to compare two series with different bases.
Example 3. Shift the base of this index from 2018 to 2022.
| Year | Index (2018 = 100) | Index (2022 = 100) |
|---|---|---|
| 2018 | 100 | 66.67 |
| 2019 | 110 | 73.33 |
| 2020 | 120 | 80.00 |
| 2021 | 135 | 90.00 |
| 2022 | 150 | 100.00 |
| 2023 | 165 | 110.00 |
| 2024 | 180 | 120.00 |
Solution. Divide each old index by 150 (the old index of 2022) and multiply by 100, e.g. 2019: 110 ÷ 150 × 100 = 73.33. Answer: as in the last column
Strictly, the correct way is to recompute the index with new-base prices and weights; dividing by the base-year index is an approximation that is exact only for indices that pass the circular test (such as fixed-weight aggregative indices).
2. Splicing.
Example 4. An old series (2015 = 100) ends with 160 in 2020. A new series (2020 = 100) gives 2021: 105, 2022: 112, 2023: 120. Splice the new series to the old base. Solution. Multiply each new index by 160 ÷ 100 = 1.6. 2021: 105 × 1.6 = 168.0; 2022: 112 × 1.6 = 179.2; 2023: 120 × 1.6 = 192.0. Answer: 168.0, 179.2, 192.0 (base 2015)
Example 5. Splice the old series backward onto the 2020 base: 2016: 110, 2017: 118, 2018: 130, 2019: 144, 2020: 160 (2015 = 100). Solution. Multiply each by 100 ÷ 160. 2016: 68.75; 2017: 73.75; 2018: 81.25; 2019: 90.00; 2020: 100. Answer: 68.75, 73.75, 81.25, 90, 100 (base 2020)
Official series use the same idea, called a linking factor, when a CPI or WPI moves to a new base.
3. Deflating.
Example 6. Find the real wages at base-year prices.
| Year | Money wage (Rs) | CPI | Real wage (Rs) |
|---|---|---|---|
| 2021 | 24,000 | 125 | 19,200 |
| 2022 | 26,400 | 132 | 20,000 |
| 2023 | 28,800 | 144 | 20,000 |
| 2024 | 31,500 | 150 | 21,000 |
Solution. Real wage = money wage ÷ CPI × 100, e.g. 2021: 24,000 ÷ 125 × 100 = 19,200. Money wages rose 31.25% (24,000 to 31,500) but real wages rose only 9.375% (19,200 to 21,000); in 2023 the money rise was eaten up by prices. Answer: as in the last column
4. Cost of living index: family budget method.
Example 7. Find the CLI.
| Group | Weight (% of spending) | Price relative |
|---|---|---|
| Food | 40 | 150 |
| Clothing | 10 | 125 |
| Fuel and light | 10 | 140 |
| House rent | 20 | 120 |
| Miscellaneous | 20 | 130 |
Solution. ΣRW = 40 × 150 + 10 × 125 + 10 × 140 + 20 × 120 + 20 × 130 = 6,000 + 1,250 + 1,400 + 2,400 + 2,600 = 13,650. ΣW = 100. CLI = 13,650 ÷ 100. Answer: 136.5
Example 8. With the CLI of Example 7, a worker earned Rs 20,000 in the base year. What income keeps his standard of living? What is the real value of Rs 28,000 now? Solution. Needed income = 20,000 × 136.5 ÷ 100 = 27,300 (so a DA of Rs 7,300). Real value = 28,000 ÷ 136.5 × 100 = 20,512.82. Answer: Rs 27,300; about Rs 20,513 at base prices
5. Cost of living index: aggregate expenditure method.
Example 9. A family's base-year basket: rice 100 kg, dal 10 kg, oil 5 litres, sugar 8 kg. Base prices (Rs) 40, 100, 120, 40; current prices 52, 130, 150, 45. Find the CLI. Solution. Σp₀q₀ = 4,000 + 1,000 + 600 + 320 = 5,920. Σp₁q₀ = 5,200 + 1,300 + 750 + 360 = 7,610. CLI = 7,610 ÷ 5,920 × 100. Answer: 128.55 (about)
6. Steps in constructing a cost of living index. Decide the class of people and the area → conduct a family budget enquiry to find what and how much they buy → choose items in each group → collect retail prices from local markets → fix weights (expenditure shares) → compute by either method.
Deeper
1. Limitations of index numbers.
- Only approximate indicators: based on samples of items, markets and families.
- Quality changes and new products are hard to handle; a fixed basket goes out of date, so the base must be revised.
- Different formulas give different results from the same data (Examples in STA-TI-02).
- Each index suits only its purpose: a CPI for industrial workers does not measure the cost of living of farmers or of the rich.
- A CLI shows changes in the cost of a fixed basket, not changes in living standards; consumers substitute cheaper goods, which a Laspeyres basket ignores (upward bias).
- International and long-period comparisons are weak because habits, goods and weights differ.
- Can be manipulated by choosing a favourable base year, items or weights.
2. Why base years are revised in India. Consumption shifts away from food towards services and transport, so old weights overstate food. The CPI of NSO moved from base 2012 to base 2024 in February 2026, using weights from the Household Consumption Expenditure Survey 2023-24 with a lower food weight; the WPI moved from 2011-12 to base 2022-23 in June 2026 alongside a new Producer Price Index (see ECO-IN-04). Linking factors splice the old and new series.
3. Deflating national income. Real GDP = nominal GDP ÷ GDP deflator × 100; growth in real terms is roughly nominal growth minus inflation. Using the wrong index (e.g. a CPI for a production series) gives misleading real values.
Example 10. Nominal income is Rs 1,20,000 when the price index is 160 (base 100). What is the real income? If real income is to stay at Rs 80,000, what nominal income is needed at index 160? Solution. 1,20,000 ÷ 160 × 100 = 75,000. Needed: 80,000 × 160 ÷ 100 = 1,28,000. Answer: Rs 75,000; Rs 1,28,000
AP angle
- The Directorate of Economics and Statistics (DES), Andhra Pradesh, collects retail and wholesale prices of essential commodities from market centres across the State through its district and mandal field staff.
- DES uses price indices and deflators to convert the State's GSDP at current prices into GSDP at constant prices, and links old and new series when the national base year changes.
Common traps
- ✗ "To shift the base, subtract the new base's index from each value." ✓ Divide by the new base year's old index and multiply by 100.
- ✗ "Splicing needs no common year." ✓ It needs an overlapping year present in both series.
- ✗ "Real wage = money wage × price index ÷ 100." ✓ Divide: money wage ÷ price index × 100.
- ✗ "CLI measures changes in the standard of living." ✓ It measures changes in the cost of a fixed basket.
- ✗ "The family budget method and the aggregate expenditure method give different answers." ✓ With base-year weights they give the same answer.
- ✗ "CLI uses wholesale prices." ✓ It uses retail prices.
- ✗ "Price index up from 100 to 125 means purchasing power falls 25%." ✓ It falls to 100/125 = 0.80, a 20% fall.
Quick revision
- Base shift: old index ÷ old index of new base × 100.
- Splicing: needs an overlap year; forward × (old overlap ÷ 100), backward × (100 ÷ old overlap).
- Real value = money value ÷ price index × 100.
- Purchasing power of money = 100 ÷ price index.
- CLI = consumer price index for a class of people; retail prices; family budget enquiry.
- Aggregate expenditure: Σp₁q₀/Σp₀q₀ × 100.
- Family budget: ΣRW/ΣW with W = p₀q₀ (same answer).
- Income to keep living standard = base income × CLI ÷ 100.
- Limits: approximate, quality changes, outdated basket, purpose-specific, formula-dependent.
- CPI base 2024 (Feb 2026, NSO); CPI-IW base 2016 (Labour Bureau, DA).
Practise this topic. Questions on Index numbers: construction, tests & cost of living, with answers explained.
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