Optimal R&D investment with learning-by-doing: Multiple steady states and thresholds

Alfred Greiner*, Anton Bondarev

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

6 Citations (Scopus)


In this paper, we present an intertemporal optimization problem of a representative R&D firm that simultaneously invests in horizontal and vertical innovations. We posit that learning-by-doing makes the process of quality improvements a positive function of the number of existing technologies with the function displaying a convex-concave form. We show that multiple steady states can arise with 2 being saddle point stable and 1 unstable with complex conjugate eigenvalues. Thus, a threshold with respect to the variety of technologies exists that separates the 2 basins of attractions. From an economic point of view, this implies that a lock-in effect can occur such that it is optimal for the firm to produce only few technologies at a low quality when the initial number of technologies falls short of the threshold. Hence, history matters as concerns the state of development implying that past investments and innovations determine whether the firm produces a large or a small variety of high- or low-quality technologies, respectively.

Original languageEnglish
Pages (from-to)956-962
Number of pages7
JournalOptimal Control Applications and Methods
Issue number6
Publication statusPublished - 1 Nov 2017
Externally publishedYes


  • horizontal and vertical innovations
  • lock-in
  • multiple steady states
  • optimal control
  • thresholds


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