Learning-by-doing in R&D, knowledge threshold, and technological divide

Abstract

This paper presents a simple R&D-based growth model of the “technological divide,” in which learning-by-doing (investing) in R&D and a threshold level of technological knowledge jointly determine the pattern of economic growth. Specifically, the model generates differences in the growth pattern primarily by modifying the underlying parameters that govern the evolution of economy-wide technological competence or dynamic R&D productivity. The technological divide arises at the threshold level of technological knowledge, which is largely affected by the quality of socio-technological infrastructure. Government policies aimed at enhancing the quality of socio-technological infrastructure can help countries escape from the “technology divide” trap by lowering the knowledge threshold. While the model preserves the spirit of the R&D-based endogenous growth model in the sense of its policy effects and the endogenous evolution of technological competence, the model does not need to reach the scale effect directly, where an increase in the size of an economy generates more rapid growth.

Appendix

Derivation of Eq. 3

The first-order conditions for firm i with respect to R&D expenditure (r) and price (p) are respectively as follows:

$$ m+(p-c)m_p =0\,\,( \mbox{from}\,\;\prod\nolimits_{p} =0)\,\;\mbox{and} $$

(15)

$$ (p-c)Qm_r -(mQc_r +1)=0 \;(\mbox{from}\,\;\prod\nolimits_{r} =0), $$

(16)

where m _p is the marginal market share effect of price, m _r is the marginal market share effect of R&D expenditure, and c _r is the marginal production-cost effect of (product) R&D. Applying the price elasticity of market share $(\varepsilon^{pm}=-(p/m)m_{p})$ to Eq. 15 yields (p − c)/p = 1/ε ^pm. Similarly, by defining (r/m)m _r as the R&D elasticity of market share (ε ^rm) and applying it to Eq. 16, we get r = (1/Φ)(p − c) mQ ε ^rm, where Φ denotes the total marginal cost of R&D (i.e., Φ = 1 + mQc _r ). Here, for simplicity, we assume that the production-cost effect of product R&D is absent (c _r  = 0 and thus Φ = 1). Then, combining these results yields a preliminary version of firm i’s profit-maximizing R&D expenditure and R&D intensity:

$$ r=({{\varepsilon ^{rm}} / \varepsilon ^{pm}})S\,\;\mbox{and thus}\,\;\Omega \equiv r / S=({{\varepsilon ^{rm}} / {\varepsilon ^{pm}}}), $$

(17)

where S denotes firm sales (S = pmQ).

From the definitions of the elasticities of market share with respect to R&D expenditure and price, we get $\varepsilon^{rm} = (r/m)m_{r} = (r/m)m_{U}U_{T}T_{r}$ and $\varepsilon^{pm}=-(p/m)m_{p} = -(p/m)m_{U}U_{p}$. Using the elasticity of market share with respect to consumer utility, $\varepsilon^{Um} = (U/m)m_{U}$, the quality elasticity of consumer utility, $\delta^{T} = (T/U)U_{T}$, the price elasticity of consumer utility, $\delta^{P}=-(p/U)U_{p}$, and technological competence, σ = (r/T)T _r , the elasticities of market share with respect to R&D expenditure and price can be $\varepsilon^{rm} = (r/m)m_{r} = (r/m)m_{U}U_{T}T_{r} = \varepsilon^{Um} \delta^{T} \sigma$ and $\varepsilon^{pm}=-(p/m)m_{p}=-(p/m)m_{U}U_{p}= \varepsilon^{Um} \delta^{p}$. Incorporating these results into Eq. 17 yields Eq. 3.

Derivation of Eq. 4

From Eq. 3, we get $\sum\nolimits_{i=1}^N {r_i} =\delta \sum\nolimits_{i=1}^N {(\sigma _i S_i )}$. Dividing both sides by industry sales, $\sum\nolimits_{i=1}^N {S_i}$, yields Eq. 4. Note that ${\sum\nolimits_{i=1}^N {(\sigma _i S_i )} } \big/{\sum\nolimits_{i=1}^N {S_i}}=\sum\nolimits_{i=1}^N {(\sigma _i s_i)}$, where $s_i ={S_i} \big/{\sum\nolimits_{i=1}^N {S_i}}$.

Derivation of Eq. 5

From the assumption that each industry has its own level of technological competence (i.e., all firms in each industry have the same level of technological competence and thus σ _i  = σ), Eq. 4 can be reduced to

$$ \Omega ^I\equiv {r^I} /{S^I}={\sum\nolimits_{i=1}^N {r_i } } \Big/ {\sum\nolimits_{i=1}^N {S_i } }=\delta \sigma ,\,i=\mbox{1, 2, 3,...,}N. $$

(18)

Here, note that the sum of all the market shares is unity (i.e., Σ _i s _i  = 1, i = 1, 2, 3, ..., N). From Eq. 18, industry R&D expenditure is r ^I = δσ δ ^I. Then, the R&D outlay at the economy level is obtained by summing up industry R&D expenditures:

$$ r^E\equiv \sum\nolimits_{k=1}^Z {r_k } =\delta \sum\nolimits_{k=1}^Z ( \sigma _k S_k ),\,\;k=1, 2, 3,...,Z. $$

(19)

By dividing both sides of Eq. 19 by the aggregate output for the economy $(S^E\equiv \sum\nolimits_k {S_k} ,\,k=1,2,3,...,Z)$, we get the R&D intensity for the economy:

$$ \Omega ^E\equiv {r^E} /{S^E}=\delta \sum\nolimits_{k=1}^Z {(\sigma _k s_k )} ,\,\;k=1, 2, 3,...,Z, $$

(20)

where $s_{k}=S_{k}/S^{E}$. Note that Eq. 20 is the same as Eq. 5.