Total Factor Productivity and the Growth Accounting Equation

In growth accounting, TFP means Total Factor Productivity. It is the part of output growth that cannot be explained by measured growth in capital and labor. CFA Level III candidates use this framework to forecast trend GDP growth — which then anchors expected long-run equity returns in the capital market expectations (CME) process.

Symbols used in this article

Symbol	Meaning
$Y$	Output (GDP for a country, or production for a firm)
$A$	Total Factor Productivity — "the technology level"
$K$	Capital stock (machines, buildings, infrastructure, computers)
$L$	Labor input (workers or hours worked)
$\alpha$	Capital share of income (typically ~0.30 for developed economies)
$1 - \alpha$	Labor share of income (typically ~0.70)
$g_X$	Growth rate of variable $X$ (e.g., $g_Y$ = output growth)
$g_A$	TFP growth — the "Solow residual"

The Cobb-Douglas production function

The starting point is the Cobb-Douglas production function:

$$Y = A K^{\alpha} L^{1-\alpha}$$

Output $Y$ depends on three things: productivity $A$, capital $K$, and labor $L$. The exponents $\alpha$ and $1-\alpha$ are the capital and labor shares of income — empirically about 0.30 and 0.70 in developed economies.

The growth accounting equation

Take logs of both sides and differentiate with respect to time. You get:

$$g_Y = g_A + \alpha g_K + (1-\alpha) g_L$$

Rearrange to solve for the unexplained piece:

$$g_A = g_Y - \alpha g_K - (1-\alpha) g_L$$

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In plain English: output growth = productivity growth + capital contribution + labor contribution. Whatever output growth you cannot explain with measured capital and labor growth is assigned to TFP.

A worked example

Suppose you observe:

• $g_Y = 5\%$ (output growth)
• $g_K = 4\%$ (capital growth)
• $g_L = 2\%$ (labor growth)
• $\alpha = 0.3$ (capital share)

Plug into the growth-accounting equation:

$$g_A = 5\% - 0.3 \times 4\% - 0.7 \times 2\%$$ $$g_A = 5\% - 1.2\% - 1.4\% = 2.4\%$$

Output grew by 5%, but capital and labor only explain 2.6 percentage points of that growth. The remaining 2.4 percentage points are counted as TFP growth.

Why TFP is called "the measure of our ignorance"

TFP is not directly observed. You measure GDP, capital stock, and labor — then call the unexplained remainder "productivity." Whatever you cannot account for through inputs becomes the residual.

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TFP can reflect:

• Better technology (microchips, AI, biotechnology)
• Better management (lean manufacturing, just-in-time inventory)
• Improved institutions (rule of law, property rights, financial markets)
• Smarter logistics (containerization, e-commerce)
• Better education quality (skill upgrading not captured in hours worked)
• Economies of scale (urbanization, network effects)
• Measurement error (we mismeasured GDP, capital, or labor)

This is why economists sometimes call TFP "the measure of our ignorance" — it is what we cannot explain.

Capital deepening vs TFP growth

The growth-accounting decomposition reveals a critical distinction:

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Source of growth	Mechanism	Sustainable long-term?
Capital deepening	More machines per worker	NO — diminishing returns to $K$
TFP growth	Better technology, organization	YES — no upper limit

This is the fundamental insight of growth theory. A country that grows via capital deepening alone (building more factories, more roads, more machines) will eventually run into diminishing returns. A country whose growth comes from TFP can sustain that growth indefinitely. This is why economists worry when China growth becomes increasingly capital-deepening (declining marginal productivity of capital) while TFP growth slows.

Cross-country interpretation

Apply the same framework to compare two economies with identical 5% output growth:

Economy	$g_Y$	$g_K$	$g_L$	$\alpha$	TFP growth $g_A$
A (mature, high TFP)	5%	3%	1%	0.30	$5 - 0.9 - 0.7 = 3.4\%$
B (catching up, capital-driven)	5%	10%	1%	0.30	$5 - 3.0 - 0.7 = 1.3\%$

Both economies grow at 5%, but economy A is far more sustainable because most of its growth comes from genuine productivity gains. Economy B is bringing more capital to bear without much technology — it will eventually run into diminishing returns.

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Why CFA candidates care: anchoring CME

The CFA Level III curriculum uses growth accounting to anchor long-run real equity returns. The chain of reasoning:

1. Forecast trend GDP growth using $g_Y = g_A + \alpha g_K + (1-\alpha) g_L$
2. Trend GDP growth $\approx$ trend corporate-earnings growth (Gordon-style approximation)
3. Trend earnings growth + dividend yield $\approx$ long-run real equity return

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For an established economy like the US, plausible long-run inputs:

$$g_Y \approx \underbrace{1.5\%}_{g_A} + \underbrace{0.3 \times 2.0\%}_{\alpha g_K} + \underbrace{0.7 \times 0.5\%}_{(1-\alpha) g_L} = 1.5 + 0.6 + 0.35 \approx 2.4\%$$

The implication: long-run real US GDP grows at about 2.4% — and that anchors long-run real equity expectations.

The CFA-exam pattern

When the vignette gives you values for output growth, capital growth, labor growth, and capital share, you should be able to:

1. Compute TFP as the residual: $g_A = g_Y - \alpha g_K - (1-\alpha) g_L$
2. Interpret TFP vs capital deepening — which dominates the growth story?
3. Project long-run growth — strip out cyclical components, keep trend
4. Anchor real returns — use trend GDP growth as a proxy for real corporate-earnings growth

Practice these computations in our CFA Level III question bank.