The Solow Residual

A growth accounting exercise is intended to break down the growth of output into the growth of the factors of production--capital and labor-- and 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 henceforth). For policy purposes it may matter whether output growth stems from factor accumulation or from increases in TFP.

Robert M. Solow (1957) set up the grounds for growth accounting. He considered a neoclassical production function

Yt  At F(K t , Lt ) (1)

where Yt is aggregate output, K t 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

 1 F(K t , Lt )  K t Lt with 0    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

Y˙ /Y  A˙ / A  (K˙ / K)  (1)(L˙ / L) (2)

(For a variable E  Y, A, K, L the term E˙ stands for the derivative of E with respect to time t , and so E˙ / E stands for the growth rate.) Note that the growth rates of physical capital and labor are weighted by  and (1) . 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  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 US economy for the period 1909-49 where output per man hour approximately doubled. According to his 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. The residual seemed too big!

To be sure, TFP is conformed by a broad range of influences—a variety of technological, economic and cultural factors. Think of technological innovations, underemployed labor shifting 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.

Notwithstanding, TFP may go down for some other reasons such as trade unions restrictions, environmental regulations and safety measures that limit the use of production factors. (By way of example, suppose that for some weight-lifting exercises your gym requires a spotter; then, two people are needed for a single task and 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 affect the aggregate productivity of the economy.

Measurement is also 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 as it leaves out many basic activities that enhance welfare. For instance, from the preceding example we can see that a safety measure will usually decrease output in the benefit of protecting human lives, and it should be clear that the beneficial effects of this rule will not affect output directly. Also, 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 have most of their payoffs in a far future, and cannot be evaluated according to today’s productivity. For a period ranging from 1973 to 1989 the US and some other Western economies experienced a slowdown in TFP growth. Presumably, this productivity slowdown happened because these advanced economies were getting transformed to the era of information and communication technologies, and in the meantime productivity --as shown in the statistics -- was quite low.

In spite of these measurement problems, various works have analyzed the determinants of the Solow residual [e.g., Edward F. Denison (1962) and Dale W. Jorgenson and Zvi

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 also because of the quality improvement that comes with new capital. Therefore, a

part of technological progress is embodied in K t 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. Sizeable estimates have been reported for the contribution of embodied technological progress in physical capital to growth, but it seems puzzling that many cross-country studies [e.g., Lant Pritchett (2001)] have found that the estimates for human capital to growth are insignificant or do not have the desired sign.

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 quite difficult to offer reliable estimates for the importance of embodied and disembodied technological progress.

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 that includes the fast growing countries of East Asia, some studies contend that the growth process can be explained by factor accumulation. This suggests that the observed high growth rates for output 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 works have been criticized on the grounds of poor measurement of human capital, high physical capital shares and biased estimates from endogeneity in the variables [see Peter J. Klenow and Andrés Rodríguez-Clare (1997) and William

Easterly and Ross Levine (2001)]. Therefore, the prevailing view is that to a great extent cross-country differences in output 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 quite sizable. Quantifying the main determinants of the Solow residual may be instrumental in comparisons of growth experiences across countries and to test theories of economic growth.

References

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. "It's Not Factor Accumulation" The World

Bank Economic Review 15(2): 177-219.

Klenow Peter J. and Andrés Rodríguez-Clare 1997. “The Neoclassical Revival in

Growth Economics: Has It Gone Too Far?” NBER Macroeconomics Annual 1997, 12:

13-103.

Jorgenson, Dale W. and Zvi Griliches. 1967. “The Explanation of Productivity

Change.” The Review of Economic Studies 34 (2): 249-280.

Pritchett, Lant 2001. "Where Has All the Education Gone?" The World Bank Economic

Review 15(3): 367-391.

Solow, Robert M.. 1957. “Technical Change and the Aggregate Production Function.”

Review of Economics and Statisticsl 39: 312-320.