Growth Accounting and Capital Deepening: Panel Evidence

1. Introduction and Data Description

This study uses a panel dataset of country-level macroeconomic variables commonly employed in growth accounting and productivity analysis. The dataset we used is the Penn World Table, which includes annual observations on real output, capital stock, employment, hours worked, and human capital. Real variables are measured at constant 2017 prices and expressed in purchasing power parity (PPP) units.

The core variables used in the empirical analysis are described below, including their variable codes and definitions as provided in the dataset.

• countrycode: Three-letter ISO country code identifying each country.

• year: Calendar year of observation.

• rgdpo: Output-side real GDP at chained PPPs (millions of 2017 US dollars).

• emp: Number of people engaged (in millions).

• avh: Average annual hours worked by people engaged.

• rkna: Capital services at constant 2017 national prices (index, 2017=1).

• hc: Human capital index based on years of schooling and returns to education.

• rtfpna: Total factor productivity at constant national prices (index, 2017=1).

From these variables, output per worker and output per hour are constructed as follows:

• Output per worker:

y_pw = rgdpo / emp

• Output per hour:

y_ph = rgdpo / (emp * avh)

Capital per worker and capital per hour are defined analogously using capital services (rkna). Growth rates are computed as first differences of natural logarithms, which approximate percentage changes.

2. Theoretical Framework

We begin with a standard Cobb–Douglas production function for country i at time t:

Dividing both sides by labor Lᵢₜ yields output per worker:

Taking logarithms and first differences gives the growth accounting decomposition:

This expression decomposes output growth into capital deepening, human capital accumulation, and total factor productivity (TFP) growth.

Where:

• y=Y/L(output per worker or per hour)

• k=K/L

• H= human capital index

• A= total factor productivity (TFP)

This gives the growth accounting decomposition:

3. Growth Accounting Decomposition

Using α = 0.33, average annual output per worker growth equals 2.54%. The decomposition shows that TFP growth accounts for approximately 1.9%, while capital deepening and human capital contributes 0.31% and 0.06%, respectively.

For output per hour, average growth equals 2.83%. TFP contributes 2.17%, while capital deepening contributes −0.23% and human capital 0.06%.

Thus, long-run growth in the sample is primarily driven by TFP growth rather than factor accumulation.

4. Econometric Specification

To validate the decomposition econometrically, the following panel regression with country and year fixed effects is estimated:

Where μᵢ denotes country fixed effects and γₜ denotes year fixed effects. Standard errors are clustered at the country level.

5. Regression Results

For output per worker growth, the estimated coefficient on capital deepening is 0.92 and highly statistically significant. The coefficient on human capital growth is statistically indistinguishable from zero.

Key findings:

• Capital elasticity ≈ 0.92

• Highly statistically significant (t = 37.5)

• Human capital coefficient ≈ 0 (insignificant)

For output per hour growth, the capital coefficient equals 0.896, again highly significant, while human capital remains insignificant.

Same pattern:

• Capital coefficient ≈ 0.90

• Human capital insignificant

Very consistent across specifications.

The magnitude of the capital coefficient substantially exceeds the theoretical capital share α = 0.33 implied by the Cobb–Douglas framework.

6. Graphical Evidence: TFP and Labor Productivity

The figures plot (i) output per worker against TFP levels and (ii) output per hour against TFP levels. Both figures are intended as descriptive diagnostics illustrating the cross-sectional and within-sample dispersion in productivity outcomes across countries and years.

Figure 1. Total Factor Productivity as the Output per Worker Driver

Figure 1 plots output per worker against TFP at constant national prices (2017 = 1). The fitted curve summarizes the average relationship between productivity and output per worker across the sample.

Discussion:
Consistent with the growth-accounting identity in Section 3, higher TFP levels are associated with substantially higher output per worker. The wide vertical dispersion at lower TFP values suggests considerable heterogeneity in output per worker among low- and middle-productivity observations, potentially reflecting differences in factor accumulation, sectoral composition, measurement error, or transitional dynamics. At higher TFP values, the cloud becomes sparser, indicating fewer country-year observations at very high productivity levels in the sample.

Figure 2. Total Factor Productivity as the Output per Hour Driver

Figure 2 plots output per hour against TFP at constant national prices (2017 = 1). The figure illustrates how productivity differences translate into differences in output per unit of labor input measured in hours.

Discussion:
The positive association between TFP and output per hour mirrors the relationship observed for output per worker. Because output per hour adjusts for cross-country differences in annual hours worked, Figure 2 provides a labor-intensity-adjusted view of productivity. The concentration of observations around TFP values near 1 (normalized around 2017) is expected given the construction of the TFP index, while the dispersion in output per hour indicates meaningful differences in effective production efficiency and/or complementary inputs across observations.

Link to the econometric results:

Taken together, Figures 1 and 2 provide descriptive support for the central role of productivity in explaining variation in labor productivity outcomes. This is consistent with the growth-accounting decomposition, in which the TFP residual accounts for the largest share of average output growth, even though the panel regressions show a strong contemporaneous association between capital deepening and output growth (likely reflecting endogeneity and joint responses to productivity shocks).

7. Interpretation

The discrepancy between the accounting parameter α and the regression coefficient likely reflects endogeneity. Productivity shocks may simultaneously increase output growth and induce capital accumulation, leading to upward bias in OLS estimates.

While the accounting decomposition indicates that TFP growth is the primary driver of long-run growth, the regression results reveal strong contemporaneous comovement between capital deepening and output growth.

Overall, the results suggest that productivity dynamics play a central role in long-run development, while capital accumulation responds strongly to short-run growth fluctuations.

8. Summary

Using a Cobb–Douglas production function framework, we decompose output growth into capital deepening, human capital accumulation, and TFP growth:

With α = 0.33, average output per worker growth of 2.5% per year is primarily driven by TFP growth (≈2.0%), while capital deepening and human capital contribute modestly.

However, panel fixed-effects regressions reveal a strong contemporaneous relationship between capital growth and output growth, with an estimated elasticity near 0.9. This likely reflects endogeneity and simultaneity between investment and productivity shocks rather than a true production elasticity.

8. Conclusion: Implications for Private Equity and M&A Country-Level Diligence

This analysis provides a macroeconomic lens through which Private Equity and M&A investors can evaluate country-level operating environments when assessing acquisitions of productive firms. By decomposing output growth into capital deepening, human capital accumulation, and total factor productivity (TFP), we isolate the structural drivers of economic performance rather than relying solely on headline GDP growth.

The growth-accounting framework,

demonstrates that, across the sample, long-run growth is predominantly driven by TFP rather than factor accumulation. While capital deepening and human capital contribute modestly on average, TFP growth explains the majority of sustained increases in output per worker and output per hour.

For investors, this distinction is critical.

GDP growth driven primarily by capital accumulation is a good signal and explains an important part of economic growth, but it may reflect cyclical investment waves, credit expansion, or policy stimulus. In contrast, TFP-driven growth signals improvements in efficiency, technology adoption, institutional quality, infrastructure, managerial practices, and competitive dynamics. These structural factors are far more relevant for long-term value creation and exit multiples.

The panel regression results further indicate a strong contemporaneous relationship between capital growth and output growth, with estimated elasticities near unity. From an investor perspective, this suggests that capital formation and productivity shocks move jointly in the short to medium run. In practice, this means that high-growth environments often exhibit both rapid investment and rising productivity—but the regression evidence also cautions against interpreting capital accumulation alone as the causal driver of performance. Productivity improvements likely underpin both.

For country-level diligence in Private Equity and strategic M&A, several actionable implications emerge:

1. Prioritize Productivity Regimes Over Growth Rates

Countries exhibiting sustained TFP growth provide more robust platforms for operational scaling, margin expansion, and EBITDA multiple stability. Productivity-enhancing environments typically correlate with stronger institutions, better contract enforcement, technological diffusion, and competitive market structures.

2. Differentiate Structural Growth from Credit-Driven Expansion

Capital-deepening episodes may temporarily boost output but can reverse if financing conditions tighten. Investors should assess whether growth is efficiency-based (TFP) or leverage-driven.

3. Assess Labor Productivity Dispersion

The graphical analysis shows wide variation in output per worker at lower productivity levels. This dispersion implies potential operational upside for investors capable of importing managerial practices, governance improvements, or technological upgrades into underperforming markets.

4. Evaluate Human Capital Dynamics Carefully

While within-country variation in the human capital index is limited in the regression framework, cross-country differences in workforce quality remain economically meaningful for long-horizon investors, particularly in sectors dependent on skilled labor and innovation.

5. Macro–Micro Alignment

Acquiring a productive firm in a high-TFP-growth environment increases the probability that firm-level improvements compound with macro tailwinds. Conversely, in low-productivity regimes, value creation may rely more heavily on firm-specific transformation rather than macro support.

In summary, this analysis reinforces a core principle of value investing at the country level: sustainable returns are more likely in economies where productivity growth—not merely capital accumulation—drives expansion. For Private Equity and M&A investors conducting cross-border diligence, incorporating growth-accounting diagnostics into macro screening enhances the ability to distinguish structurally attractive jurisdictions from those experiencing temporary, factor-driven growth cycles.

Ultimately, productivity is the macro analogue of operational excellence at the firm level. Countries that systematically improve efficiency create environments in which productive firms can scale, generate resilient cash flows, and deliver superior risk-adjusted returns over the investment horizon.

9. Sources & References

Feenstra, Robert C., Robert Inklaar and Marcel P. Timmer (2015), "The Next Generation of the Penn World Table" American Economic Review, 105(10), 3150-3182, available for download at www.ggdc.net/pwt