Solow Residual, The

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Solow Residual, The


A growth accounting exercise is used to break down the growth of output into the growth of the factors of productioncapital and laborand the growth of the efficiency in the utilization of these factors. The measure of this efficiency is usually referred to as total factor productivity (TFP). For policy purposes, it may matter whether output growth stems from factor accumulation or from increases in TFP.

The American economist Robert M. Solow set up the grounds for growth accounting in a 1957 article. Solow considered a neoclassical production function,

where Yt is aggregate output, Kt is the stock of physical capital, Lt is the labor force, and At represents TFP, which appears in a Hicks neutral way. After some simple transformations, this equation can be written in terms of the growth rates of these variables. For simplicity, consider a Cobb-Douglas production function with 0 < α < 1. Then, taking natural logarithms and differentiating both sides of (1) with respect to time t, the growth rate of aggregate output can be expressed as,

(For a variable E = Y, A, K, L, the term E̊ stands for the derivative of E with respect to time t, and so E̊ /E stands for the growth rate.) Note that the growth rates of physical capital and labor are weighted by α and (1-α ). As is well known, these weights correspond to the respective shares of rental payments for capital and labor in total income. With available data on α and the growth rates for output, physical capital, and labor, TFP growth can be computed from (2) as the residual. Accordingly, TFP growth is the so-called Solow residual.

Solow carried out this exercise for the U.S. economy for the 19091949 period, during which output per man-hour approximately doubled. According to Solows estimates, about one-eighth of the increment in labor productivity could be attributed to increased capital per man-hour, and the remaining seven-eighths to the residual. But the residual seemed too large.

To be sure, TFP is shaped by a broad range of influencesa variety of technological, economic, and cultural factors. These factors include innovations in technology, the shift of underemployed labor from agriculture to more productive sectors, economic policies aimed at liberalization and competition, and changes in shopping habits from tiny shops to department stores. Usually, these changes will increase TFP, although TFP may fall for other reasons, such as trade union restrictions, environmental regulations, and safety measures that limit the use of production factors. (For example, suppose that to conduct weight-lifting exercises a gym requires a spotter; then, two people are needed for a single task, so this rule would decrease TFP.) Other factors that may influence TFP are frictions in financial markets, physical and human capital externalities, public expenditures, or any other element that affects the aggregate productivity of the economy.

Measurement is crucial for comprehending the Solow residual. First, observe that aggregate output is roughly the value of market goods and services produced in a society, but for most purposes this measure is too narrow because it leaves out many basic activities that enhance welfare. For instance, from the weight-lifting example we can see that a safety measure will usually decrease output for the benefit of protecting human lives, and it should be clear that the beneficial effects of this rule will not affect output directly. Moreover, output and other aggregate variables may be measured with error; indeed, many Internet activities are not satisfactorily treated in the national accounts. Second, there is the problem of quality adjustment. Various goods and services (e.g., cars, cellular phones) did not exist in the past or are now of much better quality, but these quality improvements are not well recorded in the statistics. Third, there are lags in the processes of innovation, learning, and implementation of technologies. Some current investments will see most of their payoffs far in the future and cannot be evaluated according to todays productivity. From 1973 to 1989, the United States and some other Western economies experienced a slowdown in TFP growth. Presumably, this productivity slowdown happened because these advanced economies were adjusting to the era of information and communication technologies; in the meantime, productivityas shown in the statisticswas low.

In spite of these measurement problems, various studies have analyzed the determinants of the Solow residual (Denison 1962; Jorgenson and Griliches 1967) with emphasis on embodied and disembodied technological progress. Advances in technology may be embodied in the latest vintages of capital. Thus, new capital is better than old capital, not just because old capital has suffered wear and tear, but because of the quality improvement that comes with new capital. Therefore, a part of technological progress is embodied in Kt and failure to allow for this rise in quality may overstate the growth assigned to TFP. Similar considerations apply to labor: New generations entering the labor force are better educated and by all counts are more productive. In contrast, disembodied technological progress, included in TFP, will be associated with new modes of organization and operation of inputs, as well as other improvements not incorporated into the quality of factors of production. In practice, it has proved difficult to offer reliable estimates for the importance of embodied and disembodied technological progress. Sizable estimates have been reported for the contribution of embodied technological progress in physical capital to growth, but it is puzzling that many cross-country studies (Pritchett 2001) have found that the estimates for human capital to growth are insignificant or do not have the desired sign.

With the availability of broad sets of data in recent years, it has been possible to make cross-country comparisons of Solow residuals. These exercises offer new possibilities to test theories of economic growth. For a broad collection of countries, some studies contend that the growth process can be explained by factor accumulation. For instance, this suggests that the observed high growth rates for output in some fast-growing countries in Southeast Asia may not be long lasting, since there may be decreasing returns in the accumulation of these factors and further investments may become less productive. These studies have been criticized on the grounds of poor measurement of human capital, high physical capital shares, and biased estimates from endogeneity in the variables (Klenow and Rodríguez-Clare 1997; Easterly and Levine 2001). Therefore, the prevailing view is that to a great extent cross-country differences in output levels and growth rates should be attributed to the Solow residual.

In summary, the Solow residual is that part of output growth that cannot be attributed to the accumulation of capital and labor. There is a variety of factors that may contribute to output growth and hence the residual may be sizable. Quantifying the main determinants of the Solow residual may be instrumental in comparisons of growth experiences across countries and in testing theories of economic growth.

SEE ALSO Accumulation of Capital; Change, Technological; Neoclassical Growth Model; Production Function; Solow, Robert M.


Denison, Edward F. 1962. The Sources of Economic Growth in the United States and the Alternatives Before Us. Supplementary Paper no. 13. New York: Committee for Economic Development.

Easterly, William, and Ross Levine. 2001. Its Not Factor Accumulation. World Bank Economic Review 15 (2): 177219.

Jorgenson, Dale W., and Zvi Griliches. 1967. The Explanation of Productivity Change. Review of Economic Studies 34 (2): 249280.

Klenow, Peter J., and Andrés Rodríguez-Clare. 1997. The Neoclassical Revival in Growth Economics: Has It Gone Too Far? NBER Macroeconomics Annual 12: 13103.

Pritchett, Lant. 2001. Where Has All the Education Gone? World Bank Economic Review 15 (3): 367391.

Solow, Robert M. 1957. Technical Change and the Aggregate Production Function. Review of Economics and Statistics 39: 312320.

Fernando García-Belenguer

Manuel S. Santos

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Solow Residual, The

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