An economic table can show component volume measures of 200 and 200 alongside an aggregate of 450. The apparent 50-unit gap need not be an arithmetic mistake or a missing industry. Separately chained series do not generally retain additivity across components.
This guide constructs that result from two goods and three periods. You will distinguish current values from volume change, calculate successive links and see why subtracting published components can create an imaginary residual. All figures are original teaching data; the model illustrates chain linking rather than reproducing a country’s national accounts.
THE GOLD STANDARD OF ECONOMICS
Choose the question you need to answer
Read the inputs · Chapters 1–2. Separate values, quantities and price weights.
Build and interpret · Chapters 3–5. Chain the aggregate, expose non-additivity and calculate a contribution.
Practise and continue · Chapters 6–8. Change prices and test tempting residuals.
Open the chapter contents · Try the transfer checks · The Gold Standard of Economics
Chapter contents
Read the inputs · Chapters 1–2
Build and interpret · Chapters 3–5
Practise and continue · Chapters 6–8
A current-price value adds prices multiplied by quantities in the same period. If more is spent, quantities may have increased, prices may have risen or both may have changed. A volume measure aims to represent change after accounting for price differences under its stated method.
Let p stand for a price and q for a quantity. In a two-good economy, current value in period t is Vₜ = p_Aₜq_Aₜ + p_Bₜq_Bₜ. For this learning model, goods have stable definitions and quality. There are no intermediate inputs, taxes or other national-accounting adjustments.
A fixed-price volume calculation values every period’s quantities at one period’s prices. Chain linking instead joins successive comparisons whose price weights can change. These are different constructions, even if both are presented with a common reference period.
The Office for National Statistics explains the distinction between weighting periods and presentation references, and warns that chained component series need not sum to their aggregate. Our calculation below makes that consequence visible without assuming every statistical agency uses this exact simplified construction.
In period 0, both goods cost 1 unit and each has quantity 100. Current value is 1 × 100 + 1 × 100 = 200. Use this as the reference value for the chained aggregate.
In period 1, A’s quantity rises to 200 while B stays at 100. A’s price remains 1 and B’s rises to 2. Current value is 200 + 200 = 400. This increase contains both quantity and price changes.
In period 2, A remains at quantity 200 and B rises to 200. Prices stay at 1 and 2. Current value is 200 + 400 = 600. All period-1-to-2 value growth happens to reflect quantity change because prices are unchanged over that link.
Keep the elementary data. They allow us to calculate comparable-price changes and verify every aggregate. If only the final table were available, a non-additivity gap would not by itself reveal which prices, quantities or index formula produced it.
| Period | A price | A quantity | B price | B quantity | Current value |
|---|---|---|---|---|---|
| 0 | 1 | 100 | 1 | 100 | 200 |
| 1 | 1 | 200 | 2 | 100 | 400 |
| 2 | 1 | 200 | 2 | 200 | 600 |
For each link use a Laspeyres quantity factor: value the new quantities at previous-period prices, then divide by the previous period’s current value. This keeps prices fixed within the link while allowing their weights to update between links.
From 0 to 1, new quantities at period-0 prices are 1 × 200 + 1 × 100 = 300. Divide by V₀ = 200 to obtain factor 1.5, a 50% volume increase. Starting from chained value 200 gives C₁ = 200 × 1.5 = 300.
From 1 to 2, new quantities at period-1 prices are 1 × 200 + 2 × 200 = 600. Divide by V₁ = 400 for factor 1.5. Link that to C₁: C₂ = 300 × 1.5 = 450.
The chained sequence is therefore 200, 300, 450. Expressed as an index with period 0 = 100, it is 100, 150, 225. Volume has risen 125% over the whole interval. Current value rose from 200 to 600, or 200%; those are different statements.
Notice the denominator in the second link: 400 is the value of period-1 quantities at period-1 prices. Using chained value 300 there would mix valuation bases. Calculate the link first on its consistent price basis, then apply it to the chained series.
For A alone, quantities double from 100 to 200 and then stay constant. Its separately linked volume series, referenced to its initial value 100, is 100, 200, 200.
For B alone, quantity stays at 100 and then doubles to 200. Its series is 100, 100, 200. For one stable-quality good, the quantity ratios telescope, leaving its initial price multiplied by each period’s quantity.
The component sums are 200, 300 and 400. The aggregate’s independently chained sequence is 200, 300 and 450. At period 2, the aggregate exceeds the sum by 50, even though there are only two goods and every input has been accounted for.
The aggregate used B’s higher period-1 price when weighting its later quantity increase. Adding the separately referenced components does not reconstruct that changing aggregate weighting. Aggregating and chaining are operations whose order matters.
The gap is not automatically a stock, a transaction or an unreported sector. Calling it “other production” would invent economic content. It is a discrepancy between these constructed volume measures, distinct from an actual statistical discrepancy that may also occur in real accounts.
| Period | A chained volume | B chained volume | Sum of components | Chained aggregate |
|---|---|---|---|---|
| 0 | 100 | 100 | 200 | 200 |
| 1 | 200 | 100 | 300 | 300 |
| 2 | 200 | 200 | 400 | 450 |
If you need a current-price share in period 2, B contributes 400 of the current value 600: two-thirds. Dividing B’s chained component 200 by aggregate chained value 450 gives about 44.44%, but it is not that current-price share.
For a contribution to the 1-to-2 volume growth in this toy model, work on the link’s period-1 price basis. A’s contribution is 1 × (200 − 200)/400 = 0. B’s is 2 × (200 − 100)/400 = 0.50. Together they explain the link’s 50% growth: 0 and 50 percentage points.
Do not obtain that contribution by subtracting separately chained component levels and dividing by the previous chained aggregate. B’s chained increase is 100; 100/300 gives 33.33%, which fails to explain the aggregate’s 50% link growth.
For real tables, use the agency’s published contribution series or its specified calculation, and keep the release vintage, units, reference and adjustment status consistent. Different index formulas, quarterly procedures and tail treatments can change which calculations are valid. The example demonstrates why a shortcut fails, not a universal replacement formula.
Transfer check: remove the price-weight change
Keep the same quantities but hold both prices at 1 in every period. The first link remains 300/200 = 1.5. The second becomes 400/300 = 4/3. The chained aggregate ends at 400, equal to the component sum. Chained series need not be non-additive in every case.
Transfer check: subtract a component
Using the original example, someone computes “B volume” as 450 − 200 = 250. The separately chained B series says 200. The residual 250 follows the aggregate-minus-A construction; it does not recover B’s independently chained measure. Label the operation rather than treating the two results as interchangeable.
Transfer check: relabel the aggregate’s reference
Set period 1 of the original aggregate to 100. The sequence becomes approximately 66.6667, 100, 150. Its link growth rates remain 50% and 50%. This rescales the existing aggregate; it does not replace its chain of weights with one fixed basket.
Practical evidence check
Given an unfamiliar economic table, identify current prices, fixed prices, chained indices and chained currency-referenced levels. Read the additivity notes before making a component sum, share or residual. Then test your chosen operation on the complete two-good dataset above.
Does a currency label make chained volumes ordinary money amounts?
No. A series can be scaled to a reference year’s currency value while preserving a sequence of volume comparisons. Its units do not make it a record of transactions at current prices.
Is a non-additivity gap just rounding?
Not necessarily. This example produces an exact gap of 50 before rounding. Rounding can introduce an additional, separate discrepancy in published data.
Can changing the reference restore every accounting identity?
Do not assume it. Reference choices and publication methods affect where additivity holds. Consult the series notes rather than extending a reference-period equality to all periods.
Is chain linking the same as price-index rebasing?
No. Re-referencing rescales an existing series. Chain linking joins comparisons with changing weighting periods. The earlier rebasing guide develops the scale-change question.
A useful economic explanation describes the measure before diagnosing a gap. It asks whether the table records transactions, fixed-price quantities or separately linked volume series, and chooses an operation that the construction supports.
The learning check is simple: reproduce 450 from the original prices and quantities, then explain why 200 + 200 still equals 400 without either calculation being wrong. Once that relationship is clear, a published warning about non-additivity becomes a practical reading instruction.
Subject and learning map
The Gold Standard of Economics
Education: learning that transfers into life
How X Works: connected subject hub
Economic measures and their interpretation
Price-index rebasing and consistent comparisons
National accounts, GDP and productivity literacy
Macroeconomics: interpreting the wider economy
Economists: evidence and clear policy trade-offs
Authoritative sources
Sources checked on 21 October 2026. Numerical cases and practice tasks are original teaching examples, not observations from a published experiment or official statistics.
Office for National Statistics: Chain-linking methods used within the UK National Accounts — Methodological reference explaining changing weights, re-referencing and non-additivity. The two-good calculation here is a simplified annual Laspeyres teaching model, not an implementation of every ONS procedure.