Analysis and decomposition of scope economies: R&D at US research universities

Abstract

This article uses a multi output production function to analyse economies of scope between patents and R&D in US research universities. It evaluates the tradeoffs and/or synergies that arise between traditional university research outputs (articles and doctorates) and academic patents. It also investigates the sources of economies of scope and the relative roles of complementarity, scale and convexity. Nonparametric Data Envelopment Analysis (DEA) estimates of scope economies using R&D input and output data from 92 research universities show significant economies of scope between articles and patents but only modest complementarities except in a few cases. The analysis shows how scale effects (for small universities) and convexity effects can contribute to economies of scope.

Acknowledgements

This work was supported by a grant from the National Science Foundation, SES-0424772. Seniority is equally shared. The authors would like to thank seminar participants at the University of British Colombia, the University of Lille, the University of Wisconsin, as well as Bronwyn Hall and Suzanne Scotchmer for helpful suggestions on an earlier draft of the paper. Any remaining errors are the sole responsibility of the authors.

Notes

¹ It also establishes linkages with the work on Panzar (1989), Milgrom and Roberts (1990), Topkis (1998) and others on the role of supermodularity and complementarity in scope economies.

² This is because, by construction in Fig. 1, the distance is equal to and the distance is equal to .

³ Note that other measures have been developed in the literature. They include the directional distance function D _g(z, g) discussed by Chambers et al. (1996) and Färe and Grosskopf (2000): D _g(z, g) ≡ max_β{β: (z + β g) ∈ F}. Since it satisfies σ(z, g) = −D _g(z, g), it should be clear that the analysis presented below could be presented equivalently using the directional distance function. Other measures include Shephard's (1970) output distance function D _O(z) ≡ min_θ{θ: (−x, y/θ) ∈ F}, and Shephard's input distance function D _I(z) ≡ max_θ {θ: (−x/θ, y) ∈ F}. The relationships between these functions and the shortage function have been analysed in the literature (Chambers et al., 1996; Färe and Grosskopf, 2000). However, by measuring input or output proportions, the Shephard's distance functions are not additive across firms. As such they do not provide attractive measurements for analysing economies of scope.

⁴ To establish the linkage with the measure of scope economies proposed by Baumol et al. consider the case where g = (g_x, 0). Then (π _a − π _s) can be written as (π _a − π _s) =  – p _x  · [x + σ(−x, y, g) g _x ]. This reduces to the cost-based measure of economies of scope proposed by Baumol et al. in the case where [x/K + σ(−x/K, y ^k , g) g _x ] and [x + σ(−x, y, g) g _x ] are cost-minimizing inputs for the corresponding firms.

⁵ Note that S in (2) measures the productivity benefits from output diversification, and is expressed in number of units of the reference bundle g. As such, S depends on the choice of g. This should be kept in mind in the interpretation of S.

⁶ While Proposition 1 assumes K = 2, Chavas and Kim (2007) present the general case where K ≥ 2.

⁷ When the nonzero elements of g include only outputs, note that Y(x/2) (and hence the production frontier HH′) remain constant in Fig. 1. Then, the distance between HH′ and point B, C, D or E (as measured by the shortage function) can be assessed directly from Fig. 1: it is the number of units of g that separate each point from the production frontier HH′. However, when the nonzero elements of the reference bundle g include inputs, evaluating the shortage function at point B, C, D or E involves a shift in the feasible set Y(x/2) and thus in the production frontier HH′. And in the case where the nonzero elements of g include only inputs, then the shortage function evaluated at some output point is the number of units of g that shift HH′ to this point.

⁸ Note that we did not take into consideration self-citations as ISI does not differentiate self-citations.

⁹ The analysis was also conducted without quality adjustments. This affected the quantitative estimates of scope economies, indicating that quality adjustments matter. However, the heterogeneity in scope estimates across universities was found to hold with or without quality adjustments.

¹⁰ It should be noted that this approach helps to resolve the time dependency of citation counts. It also reduces the truncation problem associated with relatively recent research outputs (which do not have enough time to generate citations) and with relatively old research outputs (which may receive few recent citations).

¹¹ Alternatively, we could have used econometric tools to estimate a parametric form of the shortage function (as a multi-product production function). However, while insisting on a flexible specification, collinearity problems become serious and make the econometric approach unattractive when dealing with the number of inputs and outputs (both current and lagged) we considered.

¹² The analysis was also conducted for alternative choices of the reference bundle g. As discussed above, the measure of scope economies S in (2) depends on g. Our choice of g (reflecting current and lagged faculty, along with proportions of current and lagged post-docs and doctorates in their final year of study) was made on a priori grounds. It allows for a convenient interpretation of our results (with S intuitively measured in ‘faculty input’).

¹³ Below, the linear programming problem (6) is solved using GAMS software.

¹⁴ Note that the nature of returns to scale can be assessed by making use of the CRTS-DEA representation of the technology: F ^c = {z: }. Under CRTS, Equation 6 becomes: min_γ,λ {γ: }, which has for solution γ ^c(z) and λ ^c(z). Then, assuming technical efficiency, we can establish that z is in a region of IRTS, CRTS or DRTS depending upon whether , respectively.

¹⁵ We define small and large universities by their size of total faculty salary. Among 92 universities in our dataset, the 46 universities whose total faculty salary is above the average are considered to be large universities.

¹⁶ This result is not surprising given that our estimation of scope economies involves evaluating the productivity effects of dividing each university into two smaller (and more specialized) universities.