Annex: What are the empirical studies of R&D measuring?
This annex provides a more mathematical description of what the empirical studies analysing R&D actually measure. The rate of return to R&D (ρ), and the elasticity of output with respect to the knowledge R&D stock (β) are defined mathematically as:[19]
where 
is output (value-added), 
is the knowledge R&D stock, and 
is a time index. The change in the knowledge R&D stock, 
, is simply the flow of R&D in a given period.
Note that the rate of return is closely related to the “internal rate of return” often used for financial calculations,[20] and the elasticity is a standard measure.[21] Also note that there is a one-to-one relationship between β and ρ, for a given knowledge R&D stock to output ratio.
Studies that estimate the rate of return (ρ) generally use an equation that takes the form[22]
,
where 
, and μ is a time trend. That is, the studies regress the change in MFP against the flow of R&D (as a proportion of value-added).
Studies that estimate the elasticity of output with respect to the knowledge R&D stock (β) generally use an equation that takes the form[23]
,
where A is a constant. That is, the studies regress the level of MFP against the knowledge R&D stock. Note that in this form, the parameter β can be interpreted as the elasticity of MFP with respect to the knowledge R&D stock.
Studies of the returns to R&D can be illustrated graphically as shown below. The horizontal axis measures R&D activity and the vertical axis measures costs and benefits. The flat lines indicate the marginal social costs (MSC) and marginal private costs (MPC), assuming that both are constant and identical before any subsidy. The curves show the marginal social benefits (MSB) and marginal private benefits (MPB), assuming diminishing marginal benefits and social benefits that exceed private benefits. R0 is the level of R&D chosen in the absence of a subsidy. The level of subsidy shown in the figure induces firms to increase their R&D hours from R0 to the socially optimal level R1 where MSB = MSC.
Figure 2 – Private and social marginal costs and benefits to R&D
Note that empirical studies that estimate the private and social rates of return effectively give the bracketed vertical distance on the diagram above (i.e. the gap between MPB and MSB at R0). The studies do not show the horizontal distance between R0 and R1 – to know this we need an estimate of the shape of the functions. However, the fact that the vertical gap as measured in many empirical studies is relatively large suggests that we may be some way from the social optimum of R&D performed.
Notes
- [19]Start with a Cobb-Douglas production function of the form , where Y is output (value-added), A is a constant, D is the knowledge stock, K is capital, L is labour, and μ is a time trend. Take the partial derivative of Y with respect to D and solve for β.
- [20]The internal rate of return calculation solves the following equation for i,
- [21]This interpretation is seen more easily by rewriting the equation as .
- [22]Start with the expression for MFP implied by the production function in footnote 19: . Take logs and derivatives and substitute the expression for β.
- [23]Start with the expression for MFP described in footnote 22. Take logs.
