Annex I: An outline of the model

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Annex I: An outline of the model

It is possible to gain some sense of the scale of potential costs and benefits arising from open archiving federally funded R&D articles output by using a modified Solow-Swan model (Houghton et al. 2006; Houghton and Sheehan 2009; Houghton and Oppenheim et al. 2009). This section presents an outline of the basis and development of the model used.

Returns to R&D in a simple Solow-Swan model

In the basic Solow-Swan model, the key elements are a production function:

where A is an index of technology, K is the capital stock and L is the supply of labor, with both K and L are taken to be fully employed by virtue of the competitive markets assumption, and an accumulation equation:  

where  is the net investment or the change in the net capital stock, equal to gross investment less depreciation, and d is a constant depreciation rate. Substituting (1) into (2) gives

From (3) it is possible to determine the conditions for steady state growth in the capital stock.
Re-arranging, taking logarithms, differentiating with respect to time and imposing the condition that for steady state growth:


where  is the single constant steady state rate of growth of capital stock, consumption and output, respectively.

The main features of the Solow-Swan model are apparent from equation (4). Firstly, if technology and labor supply are fixed, the steady state growth rate is zero. That is, there is no endogenous growth in the model, growth being driven in the steady state by change in the exogenous variables. Secondly, if one of technology and population show positive growth then the steady state growth rate of the economy is proportional to the growth rate in that variable; if both rates are positive the economy’s growth rate is a weighted average of the two. Thirdly, the steady state growth rate does not depend on either the level of savings or of investment in the economy. An economy that continuously saves and invests 20% of national income will have a higher level of output than one investing 5%, but it will not have a higher steady state growth rate. Thus the broad economic message of the Solow-Swan model is that steady growth is possible in a purely competitive world, provided that there is growth in either population or technology, or both.

Contributions to growth and total factor productivity

Solow (1957) further developed this model in a way that provided the foundations for subsequent ‘growth accounting’. Starting with total differentiation of the production function (1), and substituting for the partial derivatives of Y from (1) with respect to each of its arguments, yields:

Equation (5) can then be used in two main ways in the empirical study of growth.

Given that in the competitive model capital and labor are paid their marginal products and assuming constant returns to scale, β and α can be estimated from the relative shares of capital and labor. A variant of (5) with those weights can then be used to estimate the relative contribution of capital, labor, technology and other factors to growth. Solow made pioneering estimates in 1957, the results of which he later described as “startling” (Solow 1987), and these have been much refined and amplified by Denison (1985) and others. Solow found that 7/8th of the growth in real output per worker in the US economy between 1909 and 1949 was due to “technical change in the broadest sense” and only 1/8th to capital formation. Denison’s 1985 estimates covered the US economy for the period 1929 to 1982. Of the growth in real business output of 3.1% per annum over that period, he found that the increase in labor input with constant educational qualifications accounted for about 25% and capital input for 12%. Most of the remainder is accounted for by technological progress and by the increased human capital of the workforce. What was “startling” about these results was the relatively minor contribution to output growth arising from the increase in the traditional factors of production, capital and labor.

The other related use of equation (5) is to estimate the “Solow residual”, or total factor productivity. This is defined as the difference between output growth and the weighted sum of the growth rates of factor inputs (K and L), using constant return to scale weights. That is, total factor productivity growth (TFP) is given by:

where β = 1 - α and β and α are derived from the shares of capital and labor in total income.

Total factor productivity is thus the growth in output not accounted for, on these assumptions, by the growth in capital and labor inputs. This method is now used very widely around the world in measuring productivity. This recent use has confirmed the broad Solow-Denison findings, in that for most modern economies total factor productivity growth is significantly more important than expansion of inputs in explaining total output growth. However, it must be remembered that the method rests on the assumptions embedded in the Solow model and that, as a consequence, the finding that the larger proportion of growth is to be explained by an exogenous “technical change in the broadest sense” constitutes something of an admission of defeat for economic analysis.

Estimating the rate of return to R&D

While there are recognized limitations to the traditional growth model approach, this basic framework has been widely used in estimating the rate of return to R&D. The standard approach to estimating returns to R&D is to divide the technology variable A in (1) into two components, a stock of R&D knowledge variable R and a variable Z that represents a matrix of other factors affecting productivity growth. The production function then becomes:

and the counterpart of equation (5) becomes:


That is, the rate of growth of the R&D knowledge stock (i.e. accumulated R&D expenditure or R&D capital) contributes to output growth as a factor of production, with elasticity γ. The rate of return to knowledge (γy/γR) is that continuing average per cent increment in output resulting from a one per cent increase in the knowledge stock. This can be readily derived from the elasticity γ by

The normal approach to creating a measure of the stock of R&D knowledge, for a given industry or for the economy as a whole, is to use the perpetual inventory method to create the knowledge stock from the flows of R&D, using the relationship:

where d is the rate of obsolescence of the knowledge stock. This method also requires some starting estimates (R0) of the knowledge stock, and estimates can be sensitive to that assumption.
Then the capital stock at time t is given by:

Given a series for R and for the variables Z, it is then possible to estimate γ by either of the two methods noted above: estimate equation (8) with the parameters α .. η unconstrained, or obtain estimates of the parameters α and β (constrained to be equal to one) from the factor shares of capital and labor, calculate TFP by a variant of (7) and regress R and Z on TFP to obtain γ.

Incorporating the efficiency of research and accessibility of knowledge

This standard approach makes some key simplifying assumptions. Here we note three in particular. It is assumed that:

  • All R&D generates knowledge that is useful in economic or social terms efficiency of R&D;
  • All knowledge is equally accessible to all entities that could make productive use of it (accessibility of knowledge); and
  • All types of knowledge are equally substitutable across firms and uses substitutability.

A good deal of work has been done to address the fact that the substitutability assumption is not realistic, as particular types of knowledge are often specialized to particular industries and applications. Much less has been done on the other two assumptions, which are our focus.

We define an accessibility parameter e as the proportion of the R&D knowledge stock that is accessible to those who could use it productively, and an ‘efficiency’ of R&D parameter f as the proportion of R&D spending that generates useful knowledge. Then starting with a given stock of useful knowledge R*0 at the start of period zero, useful knowledge at the start of period 1 will be given by:

where the contribution of R&D in period zero to the knowledge stock is reduced by the parameter φ to allow for unproductive R&D. This means that the stock of useful knowledge at period t is given by:

If the period over which knowledge is accumulated is long, so that (1 - δ)t R*0 is small relative to R*t, then R*t can be approximated by φR. However, only a proportion of useful knowledge may be accessible, so that accessible useful knowledge at period t is εR*t, and hence approximately φεRt, where Rt is the stock of knowledge as calculated under the standard methods.

Using this approximation and noting that it is accessible useful knowledge that is the correct factor in the production function, (6) becomes:

If φ and ε are independent functions of time, then the results of estimating a linearized version of (14) that excludes them will be misleading. However, if we assume that these parameters reflect institutional structures for research and research commercialization in a given country, and can hence be taken as fixed (and as less than or equal to one), then the standard results stand, but need to be reinterpreted. Again using R as the stock of knowledge calculated by the standard method (which assumes φ = ε = 1) and Ras the corresponding accessible stock of useful knowledge, then R = R*/φε, and the rate of return to useful and accessible knowledge becomes:

Thus, if f and/or e are less than one, the rate of return to Ris greater than that to R by the factor 1/fe. This does not imply that the measured rate of return to R is biased, because R* = feR.

Assume now that there is a one-off increase in the value of f and e, from the constant values of f0and e0 to new values of (1 + df)f0 and (1 + de)e0, respectively. Then the rate of return to R*, that is:

is fixed, but the return to R will increase: 

It follows from (17) that, because the increase in efficiency and accessibility leads to a higher value of R* for a given level of R, the rate of return to R will increase by the compound rate of increase of the percentage changes in f and e.

The basic result of the foregoing is that, if accessibility and efficiency are constant over the estimation period, but then show a one-off increase, then, to a close approximation, the return to R&D will increase by the same percentage increase as that in the accessibility and efficiency parameters.

Some methodological notes on the model

While this model specification follows an established literature on the estimation of returns to R&D, there are a number of conceptual difficulties that need to be considered in applying this methodology to estimating the returns to knowledge generated by scholarly publications (i.e. journal articles). The first is that the measure of R&D used in the model is expenditure on R&D. This includes many activities that are broader than the creation of the stock of knowledge arising from the writing and publication of scholarly journal articles, which is the focus of this study.

Martin and Tang (2007) explored seven mechanism or channels through which the benefits of publicly funded research may flow through to the economy or to society more generally, namely: (i) an increase in the stock of useful knowledge; (ii) the supply of skilled graduates and researchers; (iii) the creation of new scientific instrumentation and methodologies; (iv) the development of networks and stimulation of social interaction; (v) the enhancement of problem solving capacity; (vi) the creation of new firms; and (vii) the provision of social knowledge.[4] Hence, it is not sufficient to simply treat R&D expenditure as adding to the total stock of codified knowledge. Other channels, such as the enhancement of problem solving capacity, reflect increases in tacit knowledge. The training of skilled graduates in basic research is an important part of the R&D function as is their use of tacit knowledge to find and interpret specialized knowledge to solve problems as part of the innovation process.

The second and related issue is the complexity of the innovation process itself. The production function form of the returns to R&D equation proposed herein suggests a simple linear (science push) model of innovation, in which R&D is simply another factor of production. However, it is widely acknowledged that this fails to capture the complex feedback loops of the process, as suggested by the Kline and Rosenberg (1986) chain link model, which at least conceptually captures this complexity (Figure AI.1). It suggests that, in addition to the creation of new ideas and designs from research and their conversion into commercially available technologies, successful innovation depends on feedback from a myriad of actors in the innovation system, including customers, marketing departments, suppliers, etc.

Figure AI.1: Chain Link model of commercial innovation

Source: DEST (2003) Mapping Australian Science and Innovation, Canberra, p114. Based on Kline and Rosenberg (1986).

These two factors, the multiple mechanisms through which research impacts on innovation and the complexity of the innovation process itself, make it difficult to ascribe the results of research published in journal articles to particular innovations. This ‘attribution problem’ has resulted in some estimates of returns to R&D being upwardly biased because of the failure to properly match streams of research costs to streams of outputs (Alston and Pardy 2001). Various approaches have been adopted to deal with this problem. One approach is to introduce control variables for non-research factors into the equations used to estimate returns to R&D (Alston and Pardy 2001). Another is to selectively identify the influence of the stock of knowledge by substituting measures of the stock of publications for broader measures of R&D, such as expenditure, in the returns to R&D equation (Adams 1990, Verspagen 2004).

This suggests that where broader approaches to measuring the returns to R&D are used, such as in this study, some care is required to properly attribute the general returns to R&D to the development of the stock of knowledge represented by scholarly publications. For a single country, such as the US, this requires not only consideration of the extent to which scholarly publications relate to the returns to R&D from federally funded research in the US, but also how spillovers occur between countries (Jaffe 1989).

In this study, our approach is to: (i) estimate the proportion of total R&D activity devoted to the production and use of journal articles in terms of researcher time spent reading and writing journal articles; and (ii) estimate the extent of international spillovers from the localization of returns to R&D reported by economic studies, national versus international article and patent citation patterns, and national versus international downloads from existing open access archives (See Annex II for details).

[4] Although one could argue that enhanced access is important in all of these (arguably, with the exception of the third). More open access would effectively increase the stock of useful knowledge that is accessible to would-be users; contribute through impacts on education to enhancing the supply and skills of researchers; enable the development of networks on the basis of a shared, common and complete set of information; enhance problem solving capacity by providing necessary supporting information; enable the provision of a range of social knowledge (e.g. in health care); and provide opportunities for the emergence of new firms and new industries (e.g. text and data mining, metadata and discovery tools).