Skill-Biased Innovation, Growth, and Inequality

Abstract

Parallel to the widespread inequality phenomena in the advanced countries, there has been a growing literature about skill-biased technology and inequality. Among them, having analyzed the dynamics of technical unemployment in the framework of the induced factor bias of innovation, Stiglitz (2014) argued that the formulation of induced skill-biased innovation is one of the promising researches for analyzing the various inequalities prevailing in the OECD countries. One of the implications of the induced innovation framework in line with Kennedy (1964) and Samuelson (1966) is that relatively increasing of the factor share can induce firms to introduce its own factor-augmenting technical progress in the maximization of the instantaneous cost reduction rate of change on the concavity of the innovation frontier.

Appendix

1.1 Aggregate Elasticity of Substitution

We can derive the endogenous elasticity of substitution as follows. w / ris given by the following equation:

$$\begin{aligned} w/r= & {} \{\textit{uw}_{1}+(1-u)w_{2}\} /\{f(l_{1},l_{2})-f_{1}(l_{1},l_{2})l_{1}-f_{2}(l_{1},l_{2})l_{2}\}\nonumber \\= & {} v (l_{1}, l_{2}, u) \end{aligned}$$

(3.60)

In this case, $l_{1}= u/k$ and $l_{2}=(1-u)$ where $k\equiv K/L$. Denoting $\theta _{l1k}\equiv (k/l_{1})(dl_{1}/dk), \theta _{l2k}\equiv (k/l_{2})(dl_{2}/dk)$, we have the following equations:

$$\begin{aligned} \theta _{l1k}=-1, \theta _{l2k}=- 1 \end{aligned}$$

(3.61)

Moreover, defining $\eta _{1}\equiv (l_{1}/v)\partial v/\partial l_{1}$ and $\eta _{2}\equiv (l_{2}/v)\partial v/\partial l_{2}$, the aggregate elasticity of substitution $\sigma ^{A}$ is then written as

$$\begin{aligned} \sigma ^{A}\equiv & {} [(w/r)/k]dk/d(w/r)\nonumber \\= & {} \{(k/v)dv/dk\}^{-1}\nonumber \\= & {} (\eta _{1}\theta _{l1k} + \eta _{2}\theta _{l2k})^{-1}\nonumber \\= & {} (-\eta _{1}-\eta _{2})^{-1} \end{aligned}$$

(3.62)

Calculating each $\eta _{1}\equiv (l_{1}/v)\partial v/\partial l_{1}, \eta _{2}\equiv (l_{2}/v) \partial v/\partial l_{2}$, and specifying these in our two-level CES production function yield

$$\begin{aligned} \eta _{1}= (b\sigma _{2}^{-1}-\sigma _{1}^{-1})a/(a+ b\kappa ), \eta _{2}=-\sigma _{2}^{-1}b/(a+b)<0 \end{aligned}$$

(3.63)

Thus, the aggregate elasticity of substitution is given by

$$\begin{aligned} \sigma ^{A}= & {} (-\eta _{1 }-\eta _{2})^{-1}\nonumber \\= & {} (a+b\kappa )/(a\sigma _{1}^{-1}+ b\kappa \sigma _{2}^{-1}) \end{aligned}$$

(3.64)