This is “Growth Accounting”, section 16.17 from the book Theory and Applications of Macroeconomics (v. 1.0).
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Growth accounting is a tool that tells us how changes in real gross domestic product (real GDP) in an economy are due to changes in available capital, labor, human capital, and technology. Economists have shown that, under reasonably general circumstances, the change in output in an economy can be written as follows:output growth rate = a × capital stock growth rate + [(1 − a) × labor hours growth rate]+ [(1 − a) × human capital growth rate] + technology growth rate.
In this equation, a is just a number. For example, if a = 1/3, the growth in output is as follows:output growth rate = (1/3 × capital stock growth rate) + (2/3 × labor hours growth rate)+ (2/3 × human capital growth rate) + technology growth rate.
Growth rates can be positive or negative, so we can use this equation to analyze decreases in GDP as well as increases. This expression for the growth rate of output, by the way, is obtained by applying the rules of growth rates (discussed in Section 16.11 "Growth Rates") to the Cobb-Douglas aggregate production function (discussed in Section 16.15 "The Aggregate Production Function").
What can we measure in this expression? We can measure the growth in output, the growth in the capital stock, and the growth in labor hours. Human capital is more difficult to measure, but we can use information on schooling, literacy rates, and so forth. We cannot, however, measure the growth rate of technology. So we use the growth accounting equation to infer the growth in technology from the things we can measure. Rearranging the growth accounting equation,technology growth rate = output growth rate − (a × capital stock growth rate)− [(1 − a) × labor hours growth rate] − [(1 − a) × human capital growth rate].
So if we know the number a, we are done—we can use measures of the growth in output, labor, capital stock, and human capital to solve for the technology growth rate. In fact, we do have a way of measuring a. The technical details are not important here, but a good measure of (1 − a) is simply the total payments to labor in the economy (that is, the total of wages and other compensation) as a fraction of overall GDP. For most economies, a is in the range of about 1/3 to 1/2.