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Growth accounting
Assigning observed growth to its drivers — and reading honestly the residual that remains.
gᵧ = α·g_K + (1−α)·g_L + g_AProblem / motivation
A country sees its GDP grow by 4% a year. How much comes from the factories it added, how much from the hands it hired, and how much from simply producing BETTER with the same ones? The method that answers is called growth accounting — and the first time it was applied, its answer redirected the whole discipline.
It starts from one way of writing output: Y = A·Kᵅ·L¹⁻ᵅ, where K is capital (machines, buildings, infrastructure), L is labour (hours, hands), and A an overall efficiency — how well the two are put to use. This form is called a Cobb-Douglas production function, and it is not picked at random: it is what will make the exponent α MEASURABLE, as we shall see.
This course does three things. It PROVES the two rules that take you from levels to growth rates — most treatments take them as given, referring you to logarithms; here they are proved in three lines of arithmetic, and we shall put a number on what the shortcut costs. It then decomposes an observed growth rate into three contributions. And it ends on the question that matters: what is the STATUS of the third term? ⚠️ It is not a forecast, and this method is not a model: it predicts nothing, it attributes after the fact. That is what separates it from the course “The Solow model”, which uses the same algebra to predict.
Assumptions
In your view, which assumptions are needed for this model to hold? Jot down your ideas — no lead is wrong, this is your worksheet.
Your worksheet is still empty. Go for it: propose at least one idea.
0 idea(s) proposedFormalization
Let us write output as a PRODUCT of three things: Y = A·Kᵅ·L¹⁻ᵅ. Capital K and labour L are the factors; A is the efficiency with which they are combined — what economists call total factor productivity, TFP. The exponent α, between 0 and 1, sets the weight of capital. Why this form rather than another? Because it has a property no more general notation offers: α is a FIXED number in it, and under perfect competition that number is exactly capital's share of national income (assumption 3) — about one third in the data. So the model can be calibrated without ever estimating the production function: it is enough to read the national accounts. It is a convenient choice, not a law of nature (limits 2 and 4).
We want to get from levels to growth rates. First brick: when two quantities multiply, their rates add up. Proof, without logarithms: if one is multiplied by (1 + a) and the other by (1 + b), the product is multiplied by (1 + a)(1 + b) = 1 + a + b + **ab**. Everything is there: we get back the sum a + b, plus a cross term ab. Let us put a number on it in our case — capital +6%, labour +1%: 1.06 × 1.01 = 1.0706, that is +7.06% when the sum announces 7%. The cross term costs six hundredths of a point. This is the same bridge as in the course “The causes of inflation” (where M·V = P·Y becomes g(M) + g(V) ≈ π + g(Y)): one rule, three courses on this site — and it is where the “≈” we shall see appear comes from.
Second brick, the one that is always taken as given: if K grows by g, by how much does Kᵅ grow? Take the case that concerns us, α = 1/3 — so we are dealing with a cube root. Let us try the answer 1 + g/3 and check it by cubing: (1 + g/3)³ = 1 + g + g²/3 + g³/27. The last two terms are tiny for small g, so (1 + g/3)³ ≈ 1 + g — in other words **(1 + g)^(1/3) ≈ 1 + g/3**, which is what had to be proved. Numerical check at g = 6%: our approximation gives 1.061208 instead of 1.06, and the exact cube root of 1.06 is 1.0196 — that is +1.96% where the rule announces 6/3 = 2%. Four hundredths of a point apart. The reasoning holds for any α (with a power 1/α in place of the cube): no logarithm needed any more, and above all we now KNOW what the rule neglects.
Let us assemble. Y is the product of A, of Kᵅ and of L¹⁻ᵅ: by the first rule, its growth rate is the sum of the three rates; by the second, that of Kᵅ is α·g_K and that of L¹⁻ᵅ is (1 − α)·g_L. Hence **gᵧ = α·g_K + (1 − α)·g_L + g_A**. Read it left to right: observed growth is the SUM of three contributions — capital, labour, efficiency. Then turn it over: g_A = gᵧ − α·g_K − (1 − α)·g_L. ⚠️ And take the measure of what this second writing implies: g_A appears nowhere in the data, it is DEFINED as what is left over. This is what we call the Solow residual, and it is the whole subject of the end of the course. Click each term:
Solving / calculation
Let us decompose an observed growth rate, then check that the two rules proved above have not lied to us — a check almost no treatment carries out. The data: gᵧ = 4%, g_K = 6%, g_L = 1%, and α = 1/3 read from the national accounts.
- 1. What we observe, and what we assume
measured: gᵧ = 4% · g_K = 6% · g_L = 1% ‖ calibrated: α = 1/3three measurements and one parameter — the residual is not on the list - 2. The contribution of capital
α · g_K = (1/3) × 6= 2.00 points - 3. The contribution of labour
(1 − α) · g_L = (2/3) × 1≈ 0.67 point - ⭐ 4. The residual, by SUBTRACTION (and by nothing else)
g_A = gᵧ − α·g_K − (1 − α)·g_L = 4 − 2.00 − 0.67g_A ≈ 1.33 point - 5. What TFP weighs in growth
1.33 / 4 × 100≈ 33% — a third attributed to a factor no machine contains - ⭐ 6. Did the two rules lie? Check in EXACT arithmetic
factors: 1.06^(1/3) × 1.01^(2/3) = 1.026399 ‖ residual: 1.04 / 1.026399 − 1contribution of factors 2.64% against 2.67 by the rules, and residual 1.33% against 1.33: the shortcut costs three hundredths of a point - 7. The same decomposition PER WORKER, by two routes
gᵧ − g_L = 4 − 1 ‖ α·(g_K − g_L) + g_A = (1/3) × 5 + 1.333.00% on both sides — growth per head comes from capital deepening and from the residual - ⚠️ 8. What the decomposition does not say
why is g_A 1.33 rather than 0?the method says nothing about it — it attributes, it does not explain
Three things to take away. (1) The formula did not fall from the sky: it comes out of two arithmetic rules provable on scrap paper, and the exact check in step 6 shows that the shortcut costs only three hundredths of a point at these orders of magnitude. (2) Out of 4 points of growth, 2 are attributed to capital, 0.67 to labour, and 1.33 are left over — but “left over” is the right phrase: this third number is a balance, obtained by removing from observed growth everything we knew how to count. (3) That is why reading it calls for caution: a residual that grows can signal an acceleration of innovation… or a measurement of the factors that has become too generous. The interpretation settles what can, and only what can, be drawn from it.
g_A = gᵧ − α·g_K − (1 − α)·g_LThis simulator imitates the method exactly: you set what a statistician MEASURES — observed growth and that of the two factors — and the residual follows by subtraction. ⚠️ A word on α, which is set here in percentage points: at 33% capital weighs 1.98 points and the residual 1.35, where the solving step — which assumes exactly one third — gives 2.00 and 1.33. Two hundredths apart, and they come from the slider's rounding, not from a computation error. Three things to try. One: raise g_K alone and watch the residual melt away by just as much, without any innovation having disappeared. Two: reproduce Solow 1957 by pushing TFP's share beyond 80%. Three: keep comparing the last line — the gap between the rules and the exact computation — and watch it grow with the rates.
- Capital 1,98
- Labour 0,67
- Residual (TFP) 1,35
- observed gᵧ 4 %
Economic interpretation
The decomposition is arithmetically impeccable; it is its INTERPRETATION that calls for discipline. Four readings, two of them warnings that are nearly always skipped.
Moses Abramovitz's phrase has stuck because it is accurate: g_A captures genuine technical progress, but also the measurement errors in capital and labour, organisational gains, unmodelled scale effects, changes in the composition of the workforce, and even the imperfections of the price index used to compute gᵧ (course “Nominal vs real GDP”). Everything not counted elsewhere is in there. That is why a residual of 87% is no proof that technical progress explains 87% of growth: it is the measure of what capital and labour, AS WE HAVE COUNTED THEM, fail to explain.
Accumulating capital runs into diminishing returns: it raises the LEVEL of long-run income without durably installing a higher growth rate. Only a TFP that grows continuously sustains permanent growth in GDP per capita. That result is not proved here — it is proved in the course “The Solow model”, which shows that accumulation alone runs out of steam and that long-run growth reduces to that of A. Growth accounting, for its part, supplies the empirical measure: it is what showed that this A carried most of the weight.
Here is the point that separates rigorous use from abuse. Since g_A is DEFINED as gᵧ − α·g_K − (1 − α)·g_L, the equation is true by construction, whatever the data: it cannot be contradicted, so it predicts nothing. What can be tested are the assumptions that make α measurable and the measurements that feed the first three terms. This is exactly the grid of the course “The quantity theory of money”, where M·V = P·Y is an irrefutable identity that becomes a theory only by betting on the stability of V: the formula measures, the assumptions qualify. Hold on to the practical consequence: “growth accounting shows that…” is a phrase that must always be followed by “provided capital is correctly measured”.
This decomposition circulates elsewhere under other names, and setting them side by side lights up all three. In “Okun's law”, the “moving walkway” — potential growth as the sum of productivity gains and of new workers arriving — is this same accounting in aggregate form: what Okun calls productivity brings together what this course separates, TFP and capital deepening. In “The Solow model”, the residual becomes the A that the model posits from OUTSIDE, without explaining it — and it is that finding, born of measurement, that launched endogenous growth theories. Here, finally, it is neither a potential nor a parameter: it is a balance computed on observed data. One algebra, three statuses.
Limits / critiques
Exercises
α = 0.3; g_K = 5%; g_L = 2%; g_A = 1.5%. Compute GDP growth gᵧ (in %).
α = 1/3 and capital grows by 9%. What contribution of capital does the rule of the course announce (in points)?
In a country's national accounts, capital income amounts to 300 €bn out of a national income of 1,200 €bn. Under the assumptions of the course, what value of α should be taken?
A statistical institute discovers that it was UNDER-estimating the growth of capital (intangible capital left out). Once the measurement is corrected, the recomputed Solow residual will be…
Capital is multiplied by 1.06 and labour by 1.02. By how much does their PRODUCT increase exactly (in %, two decimals)?
A country grows by 3% (gᵧ). With α = 1/3, g_K = 3% and g_L = 3%, compute the Solow residual g_A (in %).
α = 1/3; capital grows by 4%, labour by 1%, TFP by 1%. By how much does GDP PER WORKER grow (in %)?
True or false: if the decomposition did not fit a country's data, we would conclude that the growth accounting equation is false.