Growth Accounting and TFP

A country grows at six percent a year for a decade. Its statistics office reports that the capital stock rose, employment rose, and the rest, about half of the total, was “total factor productivity”. Newspapers translate that as technological progress and the finance ministry cites it as evidence that reforms worked. Five years later a research team remeasures the same decade with hours instead of headcount and capital services instead of the capital stock, and the productivity half shrinks to a fifth. Nothing happened to the economy in between. What changed was how carefully the inputs were counted, and growth accounting is the exercise that makes the dependence explicit: it attributes growth to measured inputs and calls whatever is left over productivity, so every improvement in measurement moves the line between them.

The Arithmetic Is a Subtraction

Start from a production function in which output depends on capital, labor and a term for how efficiently they are combined.

Take logs and differentiate with respect to time. Growth in output becomes a weighted sum of growth in the inputs plus growth in the efficiency term.

Everything except \(\hat{A}\) can be measured, at least in principle: output growth from the national accounts, capital and labor growth from their own statistics, and \(\alpha\) from the share of national income paid to capital, on the assumption that factors are paid their marginal products. The productivity term is then obtained by rearranging.

That is the Solow residual, and the word residual is exact. It is not estimated, observed or modelled. It is what remains after the measured contributions have been subtracted from measured growth, which makes it the sum of technological progress and of everything the subtraction got wrong. Our article on the Cobb-Douglas production function derives the function and its properties; this article is about what happens when it is used as a ledger.

Everything Unmeasured Lands in the Residual

The residual absorbs error from each of the three terms subtracted from it, and the errors are not small. Labor is usually measured as employment, but people work different hours in booms and slumps, and a fall in hours that is not counted shows up as a fall in productivity. Labor is also treated as homogeneous, so a workforce that becomes more educated contributes more output per head, and if education is not measured the extra output is credited to the residual rather than to the workers. Our article on human capital describes the quantity being left out.

Capital is worse. The stock of machines is not the flow of services they provide, and utilisation moves sharply over the cycle: a factory running one shift in a recession and three in a boom has the same capital stock and triple the capital input. Growth accounting done with the stock attributes the boom to productivity and the recession to its collapse, which is one reason the residual is so strongly procyclical and one reason the productivity shocks in our article on the real business cycle model are contested. Capital is also not one thing. A dollar of computers delivers more services and depreciates faster than a dollar of buildings, so an economy shifting toward equipment gains capital input that a single deflated stock understates.

Then there is the weight. Setting \(\alpha\) equal to capital’s income share assumes competitive factor markets and constant returns to scale. Where firms have market power, the share of income going to profits exceeds capital’s marginal contribution, the weight is wrong, and the residual is wrong with it. Where returns to scale are increasing, a rise in all inputs raises output more than proportionally and the excess is booked as productivity. None of these is technological progress. All of them are in \(\hat{A}\).

Figure 1. The Same Decade of Growth, Measured Three Ways

Stylized illustration of one growth rate attributed under three input definitions. The shares are illustrative and are not estimates for any economy.

The Debate That Turned on How Inputs Were Counted

The clearest demonstration of how much the residual depends on measurement is the argument over East Asian growth in the 1990s. The four economies that grew fastest in the world for three decades were widely described as productivity miracles. Careful accounting with hours, educational attainment and disaggregated capital found that most of the growth was accumulation: rising participation, rising hours, rapidly rising schooling, and investment rates that reached forty percent of output. The productivity residual, after those adjustments, was ordinary. The conclusion drawn was that the growth was the result of mobilising inputs on an extraordinary scale, and since inputs cannot rise forever, that the rate could not be sustained.

The counter-argument accepted the arithmetic and questioned the weights. With a capital share taken from the income accounts, a fast-accumulating economy gets a large capital contribution. With a capital share inferred from the return on capital, which fell as the stock grew, the contribution is smaller and more of the growth returns to the residual. Both sides were doing growth accounting on the same countries and the same decades. They disagreed about the measured inputs and about \(\alpha\), and the residual moved by enough to change the story. That is not a failure of the method. It is the method working as designed, showing exactly where the interpretation depends on a measurement choice.

Table 1. What Ends Up in the Residual, and What Removes It

Source of the gap	Where it appears	What careful accounting does about it
Hours worked	Employment counted as if hours were constant	Use hours; the residual loses its cyclical swing
Labor quality	A more educated workforce credited to productivity	Weight workers by schooling and experience
Capital utilisation	Idle machines counted as active input	Proxy utilisation with energy use or shifts
Capital composition	Equipment and structures treated as one stock	Measure capital services with asset-specific rental prices
Market power	Profit share used as the capital weight	Estimate the weight from the return on capital, not the income share
Reallocation	Labor moving from farms to factories credited to technology	Account by sector; the aggregate gain is composition
Genuine technical change	What remains after all of the above	Nothing; this is the part the residual was meant to capture

Why the Residual Still Matters After All of This

It would be easy to conclude that the residual is too contaminated to mean anything, and that would be the wrong lesson. The reason growth accounting is worth doing is that the distinction it draws, between growing by using more and growing by using better, is the one that decides whether growth can continue. Accumulation runs into diminishing returns, which is the central result of our article on the Solow-Swan growth model: an economy that grows by investing more will see each additional unit of capital add less, and the growth rate must fall toward whatever the residual can sustain on its own. An economy that grows through the residual faces no such ceiling in the model, which is why the theories in our article on endogenous growth are theories of what generates the residual rather than of accumulation.

For a middle-income economy this is the whole question. A country that has grown for two decades by moving workers out of agriculture and raising its investment rate has been growing through the first two terms of the equation, and both have limits: the farm labor runs out and the return on capital falls. The transition to growing through the residual is what our article on the middle-income trap describes as the step most economies fail to make. Growth accounting cannot make the transition happen, but it can say whether it has started, and it is the only tool that can.

Cross-country comparisons of productivity levels, as opposed to growth rates, are the most fragile use of the method and the one to read with most care. Comparing the level of \(A\) between two economies requires comparable capital stocks, which means comparable prices for machines across countries, comparable depreciation, and comparable labor quality, and each comparison inherits every measurement problem above at once. The datasets that produce these comparisons state their methods carefully and revise them regularly, and a productivity gap between two countries can move by a third between editions of the same table. Our article on what productivity is covers the labor productivity measure that is far more reliably comparable, because it needs only output and hours.

Reading a Growth Decomposition

Five questions settle whether a decomposition can bear the weight put on it. Is labor measured in hours or heads, and is it quality-adjusted? Is capital a stock or a flow of services, and is utilisation controlled for? Where does the capital share come from, and would the conclusion survive a different value? Is the accounting done for the whole economy or by sector, since a large share of aggregate productivity growth in a developing economy is reallocation between sectors rather than progress within any? And over what horizon, because the residual is procyclical and a decomposition that starts in a recession and ends in a boom will find productivity growth that a longer window would not.

A decomposition that answers those questions is a statement about where growth came from, with a stated margin for what the inputs could not capture. One that does not is a statement about how the inputs were measured, dressed as a finding about technology. The difference between the two is the difference between the first and third bars of the figure, and it is often the whole of a policy conclusion.

MASEconomics Explains

Conclusion

Growth accounting is a subtraction, and the number it produces is defined by what was subtracted. Total factor productivity is growth that the measured inputs do not explain, which makes it equal parts technological progress and measurement shortfall, in proportions that no decomposition can state from its own output. The same decade can show productivity as half of growth or a fifth of it depending on whether workers are counted in heads or hours, whether capital is a stock or a flow of services, and where the capital weight was taken from. The East Asian debate was settled by none of the participants, because it was a debate about those choices.

The method survives its own weaknesses because the question it asks is the right one. Growth from using more inputs meets diminishing returns and stops; growth from using inputs better does not, in the models that matter, and whether an economy has begun to make that transition is the question on which its future rate depends. A decomposition that states how it measured labor, capital and the weight, and shows how the answer moves when those choices move, is evidence about that transition. One that reports a productivity share as though it were read off an instrument has reported its own assumptions.

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Thanks for reading! A residual is a confession of what the accounts could not count, and it should be read as one. Happy learning with MASEconomics