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Knowledge growth, complexity and the returns to R&D

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Abstract

The introduction of a production function of technology embodying laws of returns to research and development (R&D) is now standard practice in growth theory. This paper offers a critical evaluation, in the light of a generalized N–K model, of some recent contributions suggesting foundations for the existence of laws of returns to R&D. It is argued that such contributions fail to analyze the way in which research and development activity in the technological and scientific domains affect the dimension, the hierarchic structure and the complexity of knowledge search spaces. In the attempt at moving some analytical steps in this direction, this paper considers the possibility that modularity effectively counters the rise in complexity which would follow from idea growth and the increasing number of potential interactions between component ideas. It is argued that the force of the modularity argument finds its limits in the face of radical innovations that are general purpose, but entail a deconstruction and reconstruction of the hierarchy of technological interactions. It is also suggested that niche creation and knowledge spillovers elicit the early development and subsequent diffusion of such radical innovations.

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Notes

  1. Caballero and Jaffe (1993), Jones (2002), Porter and Stern (2000), and many others, discuss some of the issues involved in the choice of different specifications for the technology of the R&D sector. Admittedly, their discussion is only tentative, reflecting the poor understanding of the grounds for bringing knowledge growth under the discipline of law-like regularities.

  2. See also Eicher (2003).

  3. To corroborate this view, Reiter (1992) draws upon studies on creativity such as Koestler (1964) and Perkins (1981). Classic references in this respect are Poincaré (1908) and Hadamard (1949), which are rational reconstructions of mathematical creation. Weitzman (1996) also refers to anecdotal evidence on Edison’s invention of the electric candle, and to Usher (1929) and Schumpeter (1934). The notion of useful-knowledge production through recombination had been already hinted at by Adam Smith, when he observed that “... philosophers, or men of speculation, ... are often capable of combining together the powers of the most distant and dissimilar objects” (Smith 1776, p. 8, italics added).

  4. More formally, H(t)=C 2 (A(t))−C 2 (A(t−1)) where C 2 (A) is the set of pair-wise combination of the elements in A.

  5. Reiter (1992) agrees on the point that ideas production through recombination is subject to increasing returns, but, somewhat arbitrarily, the process there is formalized in terms of exponential, rather than combinatorial, growth.

  6. Interestingly enough, these authors try to embed in their formalization not only the economic, but also the social and institutional constraints on the pace of technological progress along a given trajectory.

  7. In economics and organization theory the N–K model has been mainly used to study the relative evolutionary effectiveness of different search procedures: local versus global (Frenken et al. 1999; Marengo et al. 2000); centralized versus decentralized (Kauffman and Macready 1995; Frenken and Valente 2003; Ethiraj and Levinthal 2004); and cognitive versus experiential (Gavetti and Levinthal 2000). The N-K model has been also used to discuss the relation between complexity, modularity and vertical (dis)integration (Marengo 2000). A common theme in this literature is the interdependence between the design of a search landscape, in particular, the number and location of the points that will count as local optima, and the design of a search procedure on the landscape (McKelvey 1999; Levinthal and Warglien 1999, Auerswald et al. 2000; Hovhannisian 2003).

  8. In the lines of enquiry considered in the previous footnote, an original problem is posed, such as the exploration of a technological landscape of exogenously given complexity. The influence of bounded rationality and computing costs on the evolution of search heuristics is then investigated. It is finally shown how different organizational choices concerning the search procedure deform the search landscape corresponding to the original problem in different directions.

  9. This echoes Nelson’s (2000) distinction between knowledge as a ‘body of practice’ and as a ‘body of understanding’.

  10. For the time being, we abstract from the otherwise important distinction between information and knowledge and the fact that techniques, like routines, may contain elements of tacit knowledge. These notions are introduced in Sections 5 and 6 below.

  11. T={0, 1}N; so there are 2N possible designs of T.

  12. \(V = 1 \mathord{\left/ {\vphantom {1 N}} \right. \kern-\nulldelimiterspace} N{\sum\limits_{j{\text{ = 1}}}^{\text{N}} {V_{j} } }\). For the sake of simplicity, we assume that K is constant across the components of T. Our definition of a complementary input does not correspond to the more restrictive definition used by Milgrom and Roberts (1990), or Topkis (1998), both of which are based on the mathematical notion of supermodularity.

  13. When complementarity is maximal (K=N-1) the fitness landscape is random, in the sense that the fitness values of neighbouring states are totally uncorrelated. The cases K=0 and K=N−1 lend themselves to formal analysis (Macken et al. 1991; Kauffman 1993). Here, we stress the sample properties of the large family of correlated N–K landscapes, which spans the parameter space between the single-peaked (K=0) and the random (K=N−1) landscape.

  14. Frenken et al. 1999, p. 152.

  15. The step-size-m problem has a solution that is strictly lower than N if, but not only if, a complete decomposition exists containing proper subspaces of the original problem space. (see Frenken et al. 1999 ; Marengo 2000). The solution to the step-size-m problem is related to the mathematical notion of cover size (Page 1996).

  16. The problem is NP-complete (Rivkin 2000, pp. 832–833).

  17. Frenken et al. (1999) and Marengo (2000) build upon Simon (1962, 1973) to suggest that an ɛ-satisficing solution (a solution not more than ɛ far from optimal fitness) to the search problem will be available if the problem structure is nearly decomposable. This is the case if there exists a decomposition of the search space into subspaces with a hierarchic structure such that components belonging to the same subspace receive sufficiently weak (in their fitness effects) epistatic links from components belonging to different subspaces at the same or at higher levels in the hierarchy. For a negligibly small ɛ, the formal conditions for near-decomposability are strong. Moreover, finding the optimal near-decomposition corresponding to a preassigned ɛ is itself a complex task. Quoting Chapman et al. (2001): “System design is an NP-complete problem”.

  18. Frenken et al. (1999) simulate a selection environment where subpopulations of agents are characterized by different decompositions of a given problem and are faced by constraints on their computing resources. They show that subpopulations with suboptimal decompositions tend to invade the population.

  19. Kauffman’s fitter dynamics follows from the decomposition of the N-bit search space into N subspaces, each of size 1 bit: to generate one new configuration from a current configuration, one of the N subspaces is randomly selected and its configuration mutated; if the fitness value of the new design thus obtained is higher than the fitness value of the current design, the idea moves to the new configuration. The procedure is then iterated until a local maximum of V( ) is reached. The fitter dynamics are non-deterministic, but a local peak of the landscape is a stationary state of these dynamics.

  20. It is worth stressing that the accepted mutations and the rest points of a decentralized search procedure would not generally correspond to those of a centralized procedure.

  21. If the interdependent subspaces of an incorrect decomposition are decentralized to parallel, rather than sequential, search procedures (Ethiraj and Levinthal 2004), simultaneity will accelerate search on the subspaces, but the outcome will hardly improve, because co-evolutionary disorder increases.

  22. Kauffman’s requirement that K increases linearly with N is unnecessary.

  23. \(V = \frac{1}{F}{\sum\limits_{f = 1}^F {V_{f} } }\).

  24. The qualification is added to relate the following discussion to the notion of near-decomposability.

  25. Cf. Altenberg (1995), p. 231. The number of components that affect the performance of function f in a non-negligible way is the polygeny of f. M has F polygeny column vectors and N pleiotropy row vectors. The application of this extended framework to the domain of technological evolution is not new. Frenken and Nuvolari (2004) partly draw upon Altenberg’s work to reconstruct the early development of the steam engine.

  26. The grouping of the subset of string components serving the same subset of functions within a relatively isolated ‘gene net’ “... means that genetic change can occur in one of these gene nets without influencing the others, thereby much increasing its chance of being viable. The grouping leads to a limiting of pleiotropy and provides a way in which complex developing organisms can change in evolution.” (Bonner 1988, p. 175, emphasis added.) This implies that genetic change can proceed rather smoothly along the subset of dimensions participating in the regulation of a functional complex occasionally conveying adaptive advantages, while preserving the functionality of the whole.

  27. Cf. Baldwin and Clark (2000), Buenstorf (2005), Langlois (2002), Brusoni and Prencipe (2001). These studies do not address the relation between selection for modularity and paradigmatic changes in knowledge discussed in this paper.

  28. Again, the case studies in Windrum and Frenken (2003) provide telling examples.

  29. The example shows also that if information codification-and-transmission cost declines, through technical change, visible information may increase, making flexible coordination between activities possible without loss of efficiency.

  30. Decentralization responds to a large number of incentives, some of which are non-cognitive and are discussed in the well-known property-right, principal–agent and transaction-costs literature. It is well beyond the scope of this paper to provide a coherent integration between the cognitive and non-cognitive incentives. For a recent attempt, see Dosi et al. (2001).

  31. The division of innovative labour between specialized suppliers and users is shown to increase with the number n of users, or application sectors and with the fixed cost k born by an application sector to develop its local technology. Division of innovative labour will instead decrease with the size q of each application sector and with the additional production cost d, if any, incurred by the application sector, when it uses the general-purpose technology developed by the specialized supplier, instead of the home-made, more targeted, local technology. Advances in science and technology, resulting from the R&D investments of the specialized knowledge suppliers, “often create technological links among formerly distant industries ... inducing a convergence in (their) technical bases”(p. 156). Correspondingly, the number n of users is increased. The same investments tend to lower the additional cost d (p. 157). cf. Arora et al. 2001, chapter 6).

  32. The process decreases the additional cost d defined in the previous footnote.

  33. Pavitt’s (1998) argument that radical innovations are not necessarily disruptive of corporate competencies does not contradict the proposition in the text. The reason is that a possibly small but crucial part of the in-house corporation competencies may extend across its disciplinary subfields of activity and across its organization modules.

  34. See also Fleming and Sorensen (2001b).

  35. Gavetti and Levinthal (2000) discuss at some length the nature of the distinction.

  36. This marks a difference with the biological-niche literature quoted in Levinthal (1998).

  37. Our focus is precisely on the role of niche creation and knowledge spillovers in the development and diffusion of radical general-purpose innovation entailing discontinuous structural change. This marks a difference, not only with Levinthal (1998), but also with the reconstruction made in Frenken and Nuvolari (2004) of the early development of the steam engine in the light of complexity theory.

  38. Cf. Hughes (1983), chapter II.

  39. The expression is from Mokyr (2002, p. 19).

  40. Cf. Hughes (1983), chapter V.

  41. Cf. David (1992), p. 157.

  42. The proposition is referred to as Fisher’s principle; (cf. Metcalfe 1994).

  43. Many examples are provided by ideas evolution in physics. For instance, direct recombination of ideas from relativity theory and quantum mechanics is forbidden by the inconsistencies between the original formulations of these theories. Recently, the prospect of useful recombination between the two sets of ideas has been offered by the expansion of the knowledge set brought by the new and still-incomplete quantum theory of gravity (Smolin 2001).

  44. Namely, ϕ(0, J)=ϕ(S, 0)=0 for any J and S>0.

  45. A(t+1)=A(t)+B(t).

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Acknowledgment

I wish to thank Carlo Zappia, Stefano Vannucci and an anonymous referee for helpful comments and constructive criticism on a previous version of the paper. Financial support from Siena University and the Italian MIUR: COFIN 2002 (prot. 2002131335_008) is gratefully acknowledged.

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    Mauro Caminati

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Caminati, M. Knowledge growth, complexity and the returns to R&D. J Evol Econ 16, 207–229 (2006). https://doi.org/10.1007/s00191-005-0007-z

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