Country and Firm Productivity Decomposition Using Cobb-Douglas
By Dr Staffan Canback, Tellusant
One of the strengths of my productivity framework is that it works for both countries and companies. Because I use the same underlying definitions, I can directly compare national productivity and firm productivity.
The basic production function is:
\[Y = A K^{\alpha} L^{1-\alpha}\]where:
- $(Y) = \textit{output (GDP, value added, gross profit, or revenue)}$
- $(A) = \textit{total factor productivity (TFP) = Solow residual}$
- $(K) = \text{capital input}$
- $(L) = \text{labor input}$
- $(\alpha) = \text{capital share}$
- $(1-\alpha) = \text{labor share}$
Growth Accounting
Taking growth rates:
\[g_Y = g_A + \alpha g_K + (1-\alpha)g_L\]where:
- $(g_Y) = \text{output growth}$
- $(g_A) = \text{TFP\ growth}$
- $(g_K) = \text{capital\ growth}$
- $(g_L) = \text{labor\ growth}$
Rearranging:
\[g_A = g_Y - \alpha g_K - (1-\alpha)g_L\]This is the standard TFP calculation already implemented in the model.
Labor Productivity
Labor productivity is:
\[\frac{Y}{L}\]Substituting the Cobb-Douglas production function:
\[\frac{Y}{L} = A\left(\frac{K}{L}\right)^{\alpha}\]Taking growth rates:
\[g_{Y/L} = g_A + \alpha(g_K-g_L)\]This yields an intuitive decomposition:
\[\text{Labor Productivity Growth} = \text{TFP Growth} + \text{Capital Deepening}\]where:
\[\text{Capital Deepening Contribution} = \alpha(g_K-g_L)\]Capital Productivity
Capital productivity is:
\[\frac{Y}{K}\]Substituting the Cobb-Douglas production function:
\[\frac{Y}{K} = A\left(\frac{L}{K}\right)^{1-\alpha}\]Taking growth rates:
\[g_{Y/K} = g_A + (1-\alpha)(g_L-g_K)\]This yields:
\[\text{Capital Productivity Growth} = \text{TFP Growth} + \text{Labor Deepening}\]where:
\[\text{Labor Deepening Contribution} = (1-\alpha)(g_L-g_K)\]Interpretation
Three productivity measures can now be reported:
| Measure | Formula |
|---|---|
| TFP Growth | $(g_A)$ |
| Labor Productivity Growth | $(g_A+\alpha(g_K-g_L))$ |
| Capital Productivity Growth | $(g_A+(1-\alpha)(g_L-g_K))$ |
The decomposition separates:
- Pure efficiency improvement (TFP)
- Capital deepening
- Labor deepening
Why This Matters
For countries:
- Capital and labor growth tend to be relatively smooth.
- Labor productivity and TFP often move together.
For firms:
- Capital growth and labor growth can diverge dramatically.
- Labor productivity may rise because of capital deepening rather than efficiency gains.
- TFP helps identify genuine operational improvement.
Examples:
- Amazon: strong capital accumulation and labor growth, but weak measured TFP.
- Microsoft: both capital deepening and positive TFP.
- Coca-Cola: labor reduction can boost labor productivity even when TFP is modest.
As a result, the decomposition is often more informative at the firm level than at the country level.
Reporting Framework
| Metric | Growth |
|---|---|
| Output Growth | x.x% |
| TFP Growth | x.x% |
| Labor Productivity Growth | x.x% |
| Capital Deepening Contribution | x.x% |
| Capital Productivity Growth | x.x% |
| Labor Deepening Contribution | x.x% |
Identities:
\[\text{Labor Productivity Growth} = \text{TFP Growth} + \text{Capital Deepening Contribution}\]and
\[\text{Capital Productivity Growth} = \text{TFP Growth} + \text{Labor Deepening Contribution}\]This provides a consistent framework for analyzing both countries and companies.