Take logs of the production function and differentiate with respect to time. The multiplicative form becomes additive:
gY = gA + α · gK + (1 − α) · gLOutput growth is the sum of technology growth, capital growth weighted by capital's income share, and labour growth weighted by labour's. Rearranged, it defines the residual:
gA = gY − α · gK − (1 − α) · gLThat rearrangement is the whole of growth accounting, and Robert Solow published it in 1957, a year after the growth model itself. Three inputs, one weight, one subtraction.
How are the inputs measured?
The mechanics that decide whether your answer means anything are all in the inputs. gY is real output growth, chain-linked at constant prices, from the national accounts. gL is hours worked rather than headcount wherever hours are published. A country whose average working week shortens by 5% over a decade has added labour input more slowly than its employment count suggests.
And gK requires a capital stock, which almost no country publishes in the form you want, so you build your own.
The standard construction is the perpetual inventory method. Start from an initial stock, then roll it forward:
K[t+1] = (1 − δ) · K[t] + I[t]where I[t] is real gross fixed capital formation in year t and δ is the depreciation rate. Two inputs to that recursion are yours to choose, the initial stock K[0] and δ, and both move the answer.
Shifting India's assumed FY2014 stock by 10% either way moves gK from 8.39% to 9.24% or 7.64%, which moves the residual from 1.480 to 1.139 or 1.777. Moving δ between 4% and 6% moves the residual from 1.235 to 1.719. Publish the point estimate alone and you have published one number from a range you never showed anyone.
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How do you check the decomposition?
The verification that makes a decomposition admissible is the per-worker identity. Subtract gL from both sides of the growth equation and the same numbers must reappear in a different arrangement:
gY − gL = α · (gK − gL) + gAThe left side is growth in output per worker. The right side splits it into capital deepening, meaning capital growing faster than labour, and the residual.
If your two routes do not agree to machine precision you have an arithmetic error, not a modelling disagreement. Worked examples 1 and 2 run both routes and close to zero.