## Working Paper Ircres-CNR 06/2017 |
DOWNLOAD (pdf file) |

### A mathematical toy model of R&D process. How this model may be useful in studying territorial development

Angelo Bonomi ^{a}

^{a} Research Associate CNR-IRCRES, National Research Council, Research Institute on Sustainable Economic Growth,

via Real Collegio 30, Moncalieri (TO) – Italy

mail: This email address is being protected from spambots. You need JavaScript enabled to view it.

#### Abstract

This work describes a mathematical application of a technological model of the R&D process, presented in a previous work, with the objective to contribute to a better knowledge of relation between R&D investments and growth. The model considers R&D as an organized flux of knowledge and capitals generating new technologies and a general knowledge exploitable for further R&D activities. The mathematical model makes an oversimplification of the R&D activity considering R&D investments related to number of R&D projects carried out, and economic growth, stagnation or decline, related to the number of new technologies entering in use. The model considers the circulating knowledge in a territory in term of number of information packages generated by R&D projects and external contributions in term of scientific, technical or other knowledge. A combinatory process with all available packages gives the total number of potential innovative ideas, part of them generating R&D project proposals. The ratio between the number of R&D proposals and the total number of potential innovative ideas may be considered related to the *innovative system efficiency* of the territory. Proposals are selected forming the number of R&D projects effectively carried out following the adopted strategies for financing R&D projects. The number of new technologies entering in use depends on a selection rate of all R&D projects carried out, and the number of new successful technologies with high rates of return of investment depends on a selection rate of all new technologies entering in use. The study considers an application of the model consisting in the introduction of a variable number of initial R&D projects in a territory with various degrees of innovative efficiency resulting or not, after a certain time, in entering in use of new technologies and possible successful technologies. Calculations show that dependence curves, in term of number of carried out R&D projects as a function of the innovative efficiency of the territory, and following dependence of formation or not of new or successful technologies, delimit three specific areas in the diagram corresponding to development, stagnation and decline of the technological asset of the territory. The results of calculations of the model show how complex is the relation between R&D investments and economic growth, characterized by absence or weak growth at level of R&D investments under a critical value, and exponential growth above due to the autocatalytic effect of R&D. This discontinuity resulting by the model calculations is in contrast with assumed continuity of dependence of growth by R&D investments often considered in econometric models.

#### Keywords

Technology innovation, Research & development, R&D model, R&D management, Socio-economic growth, Territorial development.

#### JEL codes:

C6, O31, O32

DOI: 10.23760/2421-7158.2017.006