Write output as a function of two inputs and a technology level:
Y = A · K^α · L^(1−α)Y is real output, K the real stock of productive capital, L labour input measured in workers or in hours, and A a technology or efficiency term that scales everything. The exponents are the elasticities of output with respect to each input: raise K by 1% holding everything else fixed and Y rises by α percent.
Constant returns to scale and diminishing returns
Two properties of this form do all the work and both are testable rather than decorative. Constant returns to scale means the exponents sum to one, so doubling both inputs doubles output; that is what makes the function a description of an economy rather than of a firm with a fixed factor. Diminishing marginal returns to each input follows from α lying strictly between zero and one, because the marginal product of capital, α · A · K^(α−1) · L^(1−α), falls as K rises. Give a worker a first lathe and output jumps. The tenth lathe, with the same technology and the same worker, barely moves anything.
One further property is why the Cobb-Douglas form survives in applied work. Under competitive factor markets each factor is paid its marginal product, so the share of output going to capital works out to exactly α. That converts an unobservable exponent into something you can read off a national accounts table.
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How do you find α from national accounts?
Reading it off is harder than it sounds, and most published decompositions go quietly wrong at exactly this step. The United States can get close to a clean answer because most income arrives as either employee compensation or corporate operating surplus. India cannot, because a very large share of value added is mixed income earned by own-account enterprises and unincorporated households, where the same person supplies both the labour and the capital and the accounts cannot separate them.
Take India's gross value added split into roughly 32% compensation of employees, 26% operating surplus and 42% mixed income, which is the shape the National Accounts show (check against the current National Accounts Statistics release before quoting it). The capital share you get depends entirely on how you split that 42%:
| Treatment of mixed income | Labour share | α | TFP residual | TFP share of growth |
|---|---|---|---|---|
| All to capital | 0.3200 | 0.6800 | −0.372 pp | −6.31% |
| Split half and half | 0.5300 | 0.4700 | 1.017 pp | 17.24% |
| Two thirds to labour | 0.6000 | 0.4000 | 1.480 pp | 25.09% |
| All to labour | 0.7400 | 0.2600 | 2.406 pp | 40.80% |
The residual runs from minus 0.372 points to plus 2.406 points, and the only thing that changed was an accounting convention about the income of shopkeepers and farmers. Publish an Indian TFP number without naming the mixed-income convention behind it and you have published a preference dressed as a measurement.
This node uses α = 0.40 for India, which corresponds to allocating two thirds of mixed income to labour, and α = 0.35 for the United States, which is the standard value and consistent with a labour share near 0.65. Both choices are stated every time they are used and both carry a sensitivity table.