On optimal regimes of knowledge exchange: a model of recombinant growth and firm networks

[T]he ultimate limits to growth lie not so much in our ability to generate new ideas as in our ability to process an abundance of potentially new ideas into usable form

(M. Weitzman)

Abstract

The literature has documented two patterns of knowledge exchange: free sharing of knowledge and barter exchange. The former has been coined as collective invention, while the latter is observed in the form of R&D alliance. This study, for the first time, compares these two modes of cooperation in creating and diffusing new knowledge. Doing so, we take seriously the network character of knowledge and the skewed distribution of innovation size by proposing a novel model. In this model, knowledge is represented by distinct letters and words constructed thereof and accumulated by agents over time. Discovering new words agents recombine available knowledge pieces not randomly but following certain ideas, semi-definite structures on what words can be further constructed. We proceed by allocating agents in a network and allowing them to cooperate over direct ties either in a regime of collective invention or bilateral R&D alliances. We find networks with skewed degree distribution as most productive under R&D alliances and perfect IPR since they best concentrate scarce resources in discovering different knowledge combinations. In contrast, under collective invention and imperfect IPR, clustered networks better diffuse valuable ideas and knowledge resulting in the overall superior performance. Furthermore, under imperfect IPR, collective invention raises the inequality in payoffs among agents in networks with skewed degree distribution but reduces it for clustered topologies. The latter brings a novel explanation on why industries in the past have experienced a shift in the dominant pattern of knowledge exchange.

Appendices

Appendix

The four stylized network topologies have been implemented as follows:

• The regular (periodic) lattice represents a network in which agents are connected with few nearest neighbors (the number must be even, and is equal to four in our case). The resulting network has highest clustering (i.e., partners of my partners tend to be partners as well), no degree asymmetry (everyone has the same number of edges) and relatively high mean path length (average number of edges to connect any two agents in the network).

• The small world network is constructed out of regular lattice following Watts and Strogatz (1998) by sequentially considering each edge and rewiring it to a different randomly chosen node with a small probability p (typically between 1% and 10%, we use 5%). The distinct property of small world network is that for the small p the average clustering remains high, while mean path lengths drops dramatically. The degree distribution becomes (slightly) skewed.

• The random network is constructed by applying the Watts–Strogatz algorithm but with a 100% probability of rewiring each link.²⁷ The resulting network becomes completely disordered with negligible clustering, short average path length and higher degree asymmetry.

• Finally, to construct the scale-free network, the algorithm proposed by Barabási and Albert (1999) is used. Its formation intuition is the preferential attachment mechanism implying that more connected nodes are more likely to receive new links. As a result, one obtains a graph with a high asymmetry in degree distribution following a power law (we use a power exponent $\gamma$ equaling 3), shortest average path length among the considered networks and low clustering.

A graphical representation of the four networks together with the estimates of their structural properties is given in Fig. 14. All four network topologies are implemented to produce a connected graph, i.e., it is possible to get from every node in the graph to every other node in the graph through a series of edges, called a path.

Table 1 Parameters used

	Regular	Small world	Random	Scale free
Clustering	0.50	0.38	0.04	0.11
	(0.00)	(0.03)	(0.01)	(0.03)
Mean path length	12.88	5.11	3.45	3.08
	(0.00)	(0.44)	(0.07)	(0.07)
Degree asymmetry	0.00	0.15	0.47	0.86
	(0.00)	(0.02)	(0.03)	(0.08)

Fig. 14

Four stylized networks considered. Note: The networks illustrated have a regular, b small world, c random and d scale-free topology. The parameters are chosen according to the baseline scenario specified in Table 1. Degree asymmetry is measured by the coefficient of variation. Values are reported in averages over 100 replications (with standard deviations in parentheses)

Fig. 15

Letters’ frequency in the corpus of English words

On generating new ideas. A word citing a discovered word is chosen to generate a new idea probabilistically more often if the number of extra letters it contains compared to the cited word is small. More formally, denote the number of extra letters as l. Then, the given word is selected to generate a new idea if

$$\begin{aligned} u \times e^{l-3}<\epsilon , \ \text {where} \ u \in U(0,1) \end{aligned}$$

(1)

Thus, while words containing one or two extra letters are chosen with a high likelihood (72% and 27%, respectively), words with more than five new letters have very low chance to be selected (see Fig. 16).

Fig. 16

Probability to select a word for idea generation

Additional results

See Figs. 17 and 18.

Fig. 17

Number and value of inventions in the perfect IPR regime under alternative letter sampling (see Fig. 15)

Fig. 18

Evolution of Gini coefficient for words generated by agents under perfect IPR. Note: Results for collective invention regime are plotted on the left chart, while for R&D alliances—on the right chart. Average Gini coefficient ± one standard deviation over 20 restarts are plotted