3.1 Growth and Development Accounting

Empirically evaluating growth models presents a significant challenge. One of the most direct methods for connecting the Solow model to real-world data is through growth accounting. This methodology provides a framework for decomposing observed economic growth into the contributions from its constituent parts: the growth of the capital stock, the growth of the labor force, and a residual component attributed to technological progress.

Starting with the aggregate production function and assuming competitive markets where factors are paid their marginal products, we can derive the fundamental growth accounting equation in continuous time: g = αK * gK + αL * gL + x Here, g, gK, and gL are the growth rates of output, capital, and labor, respectively. The weights αK and αL are the income shares of capital and labor in the national economy. The final term, x, is the residual component that cannot be explained by the growth in measured inputs. This residual is commonly known as Total Factor Productivity (TFP) growth, or the “Solow residual”. It is our empirical measure of technological progress.

While theoretically straightforward, the TFP residual has been a subject of intense debate. The economist Moses Abramovitz famously described it as “the measure of our ignorance,” because it captures everything that affects output growth other than the change in measured inputs. Consequently, the Solow residual can be inflated by various sources of mismeasurement. For instance, if the quality of labor improves due to better education and training (human capital), but we only measure hours worked, this quality improvement will be incorrectly attributed to TFP. Similarly, unmeasured improvements in the quality of capital goods can also lead to an overestimation of the contribution of technology.

The same accounting framework can be adapted to analyze differences across countries at a single point in time, a methodology known as development accounting. Instead of decomposing growth over time, this approach decomposes cross-country income differences into components attributable to differences in:

1. Physical capital per worker (k)
2. Human capital per worker (h)
3. TFP levels (A)

The typical approach, used by economists like Hall and Jones, is to assume a specific functional form for production, usually a Cobb-Douglas production function: Yj = Kj^(α) (AjHj)^(1-α) Using data on output per worker (yj), physical capital per worker (kj), and human capital per worker (hj) for each country j, one can then calculate the relative TFP level (Aj) as the residual that makes the equation hold true. This calibration exercise yields two main findings:

1. Differences in physical and human capital are indeed very important and are highly correlated with cross-country income differences. Rich countries have vastly more physical and human capital per worker than poor countries.
2. However, even after accounting for these capital inputs, there remain very large residual TFP differences across countries. These TFP gaps account for a significant portion of the observed variation in income per capita, highlighting the critical importance of understanding differences in technology and production efficiency.

While these accounting exercises are illuminating, they are essentially static comparisons. A more dynamic test of the Solow model involves analyzing its predictions about convergence, which requires the use of regression analysis.

3.2 Regression Analysis and Conditional Convergence

The Solow model’s theory of transitional dynamics provides a powerful and testable prediction: countries that are farther from their own steady state should grow faster. This convergence hypothesis can be directly tested using cross-country regression analysis. By taking a linear approximation of the Solow model’s dynamics around the steady state, we can derive a basic convergence regression equation: gi,t,t-1 = b0 + b1 * log yi,t-1 + εi,t In this equation, gi,t,t-1 is the average growth rate of country i, log yi,t-1 is its initial log income per capita, and εi,t is an error term. The model predicts that the coefficient b1 should be negative, capturing the convergence effect.

However, when this “unconditional” convergence regression is applied to a broad sample of countries, the prediction often fails; the estimated b1 is typically not significantly negative. The theoretical reason for this failure is that the simple regression assumes all countries are converging to the same steady state (y*), which is implicitly captured in the constant term b0. In reality, countries have very different steady states due to differences in saving rates, population growth, and other structural factors. A poor country might not grow quickly if its own steady state is also very low.

This leads to the more nuanced concept of “conditional convergence”. To test this, the regression equation is augmented with variables that control for the determinants of each country’s steady state: gi,t,t-1 = b0 + b1 * log yi,t-1 + β’Xi,t + εi,t Here, the vector Xi,t includes variables such as the investment rates for physical and human capital (sK, sH) and the rate of population growth (n). Empirical work by Mankiw, Romer, and Weil (MRW) demonstrated that once these control variables are included, the coefficient b1 becomes robustly negative, providing strong evidence in favor of conditional convergence.

Despite its influence, the MRW approach faces two significant challenges that cast doubt on its conclusions about the relative importance of capital versus technology:

1. The Orthogonality Assumption: The regression implicitly assumes that the unobserved technology levels (Aj) are uncorrelated with the included control variables, like investment rates. This assumption is highly implausible. It is much more likely that countries with higher productivity (high Aj) will also have higher investment rates. This correlation would create an omitted variable bias, causing the regression to overstate the explanatory power of capital accumulation and understate the importance of TFP differences.
2. Implausible Parameter Estimates: The estimated coefficient on human capital investment implies a return to schooling that is far larger than the estimates derived from microeconometric studies of individual wages (i.e., Mincerian regressions). This discrepancy again suggests an omitted variable bias. The high estimated return to human capital is likely capturing the effect of omitted technology, which is positively correlated with both human capital investment and income.

In summary, the Solow model provides a powerful initial framework for thinking about growth. However, empirical tests of the model, whether through accounting exercises or regression analysis, consistently point to the overwhelming importance of a factor that the model itself leaves unexplained: cross-country differences in technology and productivity. This major shortcoming motivates the development of micro-founded models and theories of endogenous technological change, which seek to open up this “black box.”