Solow Residual: Definition, Example, vs. TFP

What Is the Solow Residual?

The Solow residual is based on the work of Nobel prize-winning economist Robert Solow, whose growth model defined productivity growth as rising output with constant capital and labor. It tells you whether an economy is growing because of increases in capital or labor, or because those inputs are being used more efficiently. Solow found that only one-eighth of the increase in labor productivity in the United States between 1909 and 1949 could be attributed to increased capital. America, in other words, became great because of American innovation and know-how.

The Solow residual is the portion of an economy’s output growth that cannot be attributed to the accumulation of capital and labor, the factors of production. The Solow residual represents output growth that happens beyond the simple growth of inputs. As such, the Solow residual is often described as a measure of productivity growth due to technological innovation. The Solow residual is also referred to as total factor productivity (TFP).

Understanding the Solow Residual

The Solow residual is affected by a huge variety of technological, economic, and cultural factors. Innovation, investment in more productive sectors, and economic policies aimed at liberalization and competition all boost total factor productivity. Conversely, the Solow residual can be lowered by restrictive labor practices, excessive regulations, underdeveloped financial markets that fail to allocate capital efficiently, or anything else that affects the aggregate productivity of the economy. However, total factor productivity is often used as a proxy for technological progress and innovation. Differences in countries’ TFP levels are sometimes used to explain differences in economic development.

It is important to note that Solow did not use the term total factor productivity and did not consider his growth model or the residual bearing his name as having any kind of predictive function. Solow merely pointed out that growth was unaccounted for in a standard model and that the growth was likely attributable to innovations that spurred extra productivity. The Solow residual spurred improvements to economic models and measures, resulting in a better understanding of the importance of innovation—and investment in innovation—in improving a nation's economic performance.

Solow Residual Formula

The total factor productivity formula may take different forms depending on what is being calculated and more importantly what type of analysis is being performed. However, the general formula often the takes form of the following:

TFP = Y / (K^α * (L * H)^(1-α))

• Y represents the total output or GDP.
• K represents the capital input.
• L represents the labor input.
• H represents the human capital input.
• α (alpha) represents the output elasticity of capital, which measures the share of output attributed to capital.

At a high-level, the formula compares the expected production of capital (K), labor (L), and human capital (H) with the actual output (Y). The residual is the percentage of production increase that cannot be fully explained by changes in labor and capital inputs. It includes the impact of productivity-affecting technical advancements, efficiency gains, and other unnoticed factors.

Limitations of the Solow Residual

Though a valuable concept, there are limitations to the Solow residual worth mentioning. First, data on production, labor, and capital are often aggregated over the entire economy to calculate the Solow residual at the macroeconomic level. Aggregation may ignore heterogeneity and changes in productivity levels across various industries, geographies, or businesses, and this could cause measurement errors.

Estimating and measuring a number of variables, such as output, labor, and capital inputs, are necessary for calculating the Solow residual. The accuracy of the final TFP measurement can be impacted by guessing mistakes in any of these components. In addition, it might be difficult to assess capital input accurately.

The Solow residual tries to capture advancements in technology and efficiency, as these are often frequently considered to be residual factors. It can be challenging to pinpoint and attribute increases in productivity entirely to technical advancement because other elements, such as adjustments to managerial procedures or organizational efficiency, can also affect production. On a related note, it also does not distinguish between numerous elements like innovation, education, infrastructure, or regulatory policies that affect productivity changes.

Real-World Uses for the Solow Residual

As mentioned, the Solow residual has often been used as an explanation for the changing economic fortunes of national economies. For example, slowing growth in China has often been explained as an underlying productivity problem. In this interpretation, China's growth 'miracle' was the result of rapid capital accumulation and shifting underutilized labor into a modern capitalist economy.

According to World Bank, China's TFP had been consistently negative during the latter portion of the 2010's. World Bank argued it had wasted huge amounts of financial resources on inefficient state-owned enterprises in industries like steel, coal, and cement, and excess infrastructure.

Viewed through the lens of total factor productivity, China has managed to become an economic superpower through its sheer size rather than through gains in productivity. This lack of productivity is, however, becoming more of an issue as China has seemingly reached the limits of its appetite for market reforms. The 2020 research paper by World Bank states that should China work to reverse the "larger share of credit and investment to infrastructure and housing that led to lower returns to capital, a rapid buildup in debt, and higher risks to growth" in inefficient economies.

China may also see less access to potentially important technologies as the EU and U.S. have taken a firmer stance on sharing important intellectual property—something that can impact its Solow residual. Though it experienced double-digit growth each year over four decades, China’s labor force is contracting due to its decades-long “one-child” policy. Thus, China’s economic growth rate is susceptible to faltering.

The Bottom Line

Total factor productivity (TFP), also known as Solow's residual, is a measure of the share of production growth that cannot be explained by increases in inputs like capital and labor. It captures the impact of technical development, increases in production, and other unnoticed elements. The residual is the "unexplained" component of economic growth and sheds light on an economy's overall effectiveness and efficiency as well as the role that factors other than conventional inputs played in generating output.