Public Policy Sources #37:
Growth accounting: determining sources of economic growth

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To motivate an understanding of why productivity growth is so important for a sustained increase in real per-capita incomes, I will briefly discuss how economists determine the sources of economic growth. This is an essential exercise, for a thorough understanding of the concept of productivity growth requires us to define rigorously what productivity is and how it contributes to economic growth. Although this approach--called growth accounting--is not without flaws, it is instructive to start here to illustrate some basic concepts that are essential to the economics of growth.

Central to our analysis is the reality that in an economy, firms take factor inputs such as labour, capital, and raw materials and convert these inputs into desirable outputs--goods and services. The relationship between factor inputs and outputs can be summarized mathematically by what economists call a production function. A production function is a purely technological relation that tells us the maximum output that an economy could produce using a given amount of inputs. Formally, a production function could be written as follows:

Y(t) = A(t)F [K(t), L(t)] (1)

where Y(t) is output at time t, K(t) is the stock of capital (i.e. machines and equipment) at time t, and L(t) is labour (i.e. the total number of people working or the total time worked) at time t. Hence, output in period t is produced by combining capital and labour in the function F with some object A(t) (which we will define shortly). It is useful to think of F as the technology for putting capital and labour together. Economists generally assume that the function F is increasing in both K and L. That is, the more capital or labour employed, the more output produced. If workers work harder or the economy accumulates more capital (i.e. firms accumulate more machines and equipment), total output rises.

The variable A(t) is defined as total factor productivity. The higher the value of A(t), the more output a given amount of capital and labour can produce. Therefore, as A(t) increases, the economy is able to produce more and more output from a given stock of labour and capital inputs. When economists speak of productivity growth, this is essentially what they mean--the growth rate is the economy's ability to produce output from a given stock of input. In ordinary parlance, one can think of productivity growth as "working smarter" rather than "working harder." While it is true (from the discussion above) that working harder (i.e. increasing the stock of labour or capital) will increase total output, the total factor productivity of an economy only increases if people "work smarter" and learn to obtain more output from a given supply of inputs. "Working smarter" therefore corresponds to an increase in A(t).

It is important to note that in popular discussions, different definitions of the term "productivity" are employed. For instance, journalists for the business press often speak of productivity in terms of average labour productivity (i.e. output per worker, or Y/L). While average labour productivity is an important indicator for measuring the economic well-being of the average worker, it provides an incomplete measure of the productivity of the economy as a whole, since, according to our framework, increases in both capital and labour input will increase total output. A rising ratio of output per worker may not necessarily indicate growing productivity throughout the economy, since growth in output per worker could be entirely due to an increase in the available stock of machinery and equipment (i.e. an increase in the capital stock) rather than an increase in A(t). In other words, average labour productivity is a poor indicator of economy-wide productivity since increases in the ratio of output per worker could be entirely due to agents in the economy "working harder" (i.e. accumulating more capital) rather than "working smarter."

What causes total factor productivity A(t) to rise over time? Improvements in technology--the invention of the internal combustion engine, the introduction of electricity, of semiconductors--clearly increase total factor productivity. With better machines, tools, and equipment, it is possible to use labour and capital more efficiently. Improvements in the education and skill of the labour force also cause A(t) to rise since more skilled workers may work more efficiently with capital inputs than less skilled workers. Organizational change may also induce growth in total factor productivity. More efficient ways of motivating workers to perform productive tasks, improvements in teamwork and so on will result in increases in total factor productivity. Additionally, the institutional environment plays a critical role in determining the overall productivity of the economy. Countries where corruption is rampant or where basic infrastructure is inadequate are likely to have much slower (possibly negative) growth rates in total factor productivity. Conversely, countries with a stable legal and political framework that rewards investment and innovation are likely to have faster rates of total factor productivity. This list of factors that influence the growth rate of total factor productivity A(t) is certainly not exhaustive but it does point out a few of the channels through which public policy may affect productivity performance.

Productivity is the cornerstone of economic growth. We are wealthier than our ancestors and our Third-World neighbours largely because we are more productive than they are. Productivity also affects our position relative to other developed countries in the world: the more productive we are in Canada, the higher our per-capita GDP is in the world rankings. In other words, productivity is the reason why the developed world enjoys a standard of living higher than that of the Third World today or that of the developed world one hundred years ago (Mokyr 1990; Lipsey 1996).

To illustrate why this is so, it is first necessary to develop a theoretical framework to determine the sources of economic growth. The relationship between the economy's inputs and outputs can be represented by a production function, which tells us that we can get more output for three reasons: (i) because people are working harder or more people are working (higher L); (ii) because workers have more tools and equipment to work with (higher K); or (iii) because labour and capital are combined more efficiently (higher A). My objective is to determine how much each of these factors contributes to total output growth. That is, I want to decompose the growth rate of total output into components due to each of these three elements. This is what economists call growth accounting.1

To do this requires that one assume a particular functional form for the production function. As is standard in this literature, I will assume that the production function takes the following algebraic form:

Y(t) = A(t)K(t)xL(t)1-x (2)

where 0 < x < 1. A production function of this form is called a Cobb-Douglas production function. Cobb-Douglas production functions have a number of convenient economic properties. For our purposes, it is sufficient to note that a Cobb-Douglas production function embodies the assumption of constant returns to scale (i.e. if we double our inputs of labour and capital, we double total output). Furthermore, it can be shown that for a Cobb-Douglas production function, the exponent x corresponds to capital's share of total output. For most industrialized countries like Canada or the United States, x is approximately 30 percent, or 0.3. Conversely, labour's share of total output is 70 percent, or 0.7 (Hall, Taylor, and Rudin 1995).

To perform growth accounting, it is necessary to do a bit of tedious algebra. Since my goal is to provide a non-technical overview of the issues, I will not formally derive the results here. Interested readers can see Appendix 1 for the algebraic details. After doing the algebra, however, it is possible to obtain the following expression:

D Y(t)/Y(t) = D A(t)/A(t) + x[ D K(t)/K(t)] + (3)
(1-x)[ D L(t)/L(t)]

For any variable Z(t), D Z(t)/Z(t) is the growth rate (i.e. the percentage increase) of Z between periods t and t-1. Thus, D Y(t)/Y(t) denotes the growth rate of real output, D A(t)/A(t) denotes growth rate of total factor productivity, D K(t)/K(t) is the growth rate of the capital stock, and D L(t)/L(t) is the growth rate of the stock of labour. Hence, equation (3) says that the growth rate of output is equal to the growth rate of total factor productivity plus a weighted sum of the growth rate of the capital stock and the growth rate of the stock of labour. This is the fundamental growth accounting equation for it tells how the growth rate of real output can be partitioned into the growth rates of its component parts.

In principle, the growth rates of output ( D Y/Y), the capital stock ( D K/K), and the stock of labour ( D L/L) are measurable over any interval of time. Moreover, as noted earlier, the variable x is capital's share of total output. For Canada, the value of x is roughly 0.3. Hence, the only variable that is not directly measurable is D A/A, the growth rate of total factor productivity. Note, however, that knowledge of all the other variables enables us to estimate total factor productivity's growth rate ( D A/A). In particular, by solving equation (3) for D A/A it is possible to find the growth rate in total factor productivity as a residual of the other growth rates. That is, I can compute the growth rate of total factor productivity by using the following expression:

D A(t)/A(t) = D Y(t)/Y(t) - x[ D K(t)/K(t)] - (4)
(1-x)[ D L(t)/L(t)]

All empirical estimates of total factor productivity growth are obtained by using these means. Total factor productivity growth is not a variable that can be measured directly. However, through growth accounting, it is possible to measure the total factor productivity growth indirectly as a residual of our measurable growth rates. Solow (1957) was one of the first economists to use this procedure; hence, estimates of total factor productivity growth are sometimes called estimates of the Solow Residual.

Qualifications

A few qualifications regarding this estimation process should be noted. The first is that, since total factor productivity growth is a residual, measurement error in any of the observable growth rates will affect estimates of total factor productivity growth. For instance, if the growth rate of the labour force or the capital stock is underestimated, then measured total factor productivity growth will be biased upwards (i.e. overestimated). If output growth is underestimated, then total factor productivity growth will be biased downward (i.e. underestimated). Hence, empirical estimates of productivity growth are only as good as the underlying data.

Another important fact that should be noted is that, since total factor productivity growth is not directly measurable, it is an inherently vague magnitude. In a sense, total factor productivity growth is a catch-all expression for everything that affects growth but cannot be observed. Many economists believe that estimates that find a large value for total factor productivity growth reveal our ignorance about the sources of economic growth (Griliches 1994; Solow 1994). Knowing that total factor productivity growth is important may not tell us very much, since it is not quite clear exactly what "total factor productivity" is. As noted earlier, there are many factors that could influence total factor productivity growth. Simply measuring the magnitude of this variable does not tell us what underlies it.

One should also bear in mind that the enigmatic nature of total factor productivity growth is the source of some confusion in discussions about the sources of economic growth. For instance, an improvement in the quality of labour (due, say, to an increase in the average skill level of workers) is not directly measurable. In addition, improvements in the quality of the existing stock of capital are not easily measured. It is not obvious whether such improvements in input quality should be attributed to productivity growth or simply input growth (should we count more skilled workers as simply more workers, i.e. larger L, or productivity improvement, i.e. larger A?). In practice, most researchers attempt to incorporate quality change into our measures of inputs but nevertheless this ambiguity should be emphasized.

A final, technical, digression should be made before we move on. Often one is interested in accounting for the growth rate in per-capita output rather than the growth rate of total output. This is motivated by the fact that improvements in material living standards are much better approximated by using output per capita than total output. With little change to the underlying growth-accounting framework, it is possible to account for growth rate in per-capita output. Dividing the Cobb-Douglas production function by L(t), the stock of labour, yields an expression for output in per-capita terms:

Y(t)/L(t) = A(t)[K(t)/L(t)]x (5)

Define y(t) = Y(t)/L(t) and k(t) = K(t)/L(t). Following a procedure similar to that outlined in Appendix 1, I can decompose the growth rate of output per worker ( D y(t)/y(t)) as follows:

D y(t)/y(t) = D A(t)/A(t) + x[ D k(t)/k(t)] (6)

Equation (6) states that output per worker can rise for two reasons: (i) total factor productivity A(t) is rising, and (ii) the amount of capital per worker (k(t)) is increasing. Notice, however, that the impact of the growth rate of capital per worker on the growth rate of output per worker is scaled down by x, capital's share of total output. Hence, a one-percentage increase in total factor productivity has a larger impact on the growth rate of output per worker than a one-percentage increase in the ratio of capital per worker.

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