Education and training in a model of endogenous growth with creative wear-and-tear

Abstract

How does the rate at which firms adopt new technologies affect the level of education and training of a country’s workforce? What is then the mix of general education and technology-specific training that maximises the growth rate of an economy? We try to answer these questions by developing an endogenous growth model which focuses on privately financed general education and firm financed technology specific training in a setting where creative destruction renders technologies gradually obsolete. We reproduce some stylized facts regarding the technology-education-training relationship and we show how the optimum amount of time devoted to education and training is affected by the rate of technical change itself. In particular, we find that a faster arrival of new technologies shifts the private knowledge portfolio towards general human capital, less prone to creative destruction. We also find that households tend to under-invest in education, thus leading to lower growth rates than technically feasible, and higher training costs than absolutely necessary. This suggests that there is room for education policy reducing private education fees.

Appendix: The core equations of the model and the nature of the steady state

This Appendix provides the simultaneous equation system (implemented in Mathematica 9 (© Wolfram)) that we used to obtain the results presented above. The core-equations of the simultaneous system are given by:

$$H_{R} + H_{P} = H \cdot \left( {1 - \varphi - \psi } \right)$$

(35)

$$\varphi = \gamma_{1} \cdot (\zeta 1 - \zeta 2)/(1 + \gamma 1 \cdot \zeta 1)$$

(36)

$$\psi = \gamma 1 \cdot \zeta 2/(1 + \gamma 1 \cdot \zeta 1)$$

(37)

$$\hat{A} = \delta \cdot \varsigma 0 \cdot \varphi^{\varsigma 1 - \varsigma 2} \cdot \psi^{\varsigma 2} \cdot H_{R}$$

(38)

$${\text{J}} = H_{P} /\left( {1 + \hat{A} \cdot {\text{s}}} \right)$$

(39)

$$T = H_{P} \cdot \hat{A} \cdot {\text{s}}/\left( {1 + \hat{A} \cdot {\text{s}}} \right)$$

(40)

$$\begin{aligned} ( - \exp \;(s \cdot \rho ) \cdot \alpha \cdot (\hat{A} \cdot s \cdot (\alpha - 1) + \alpha - \alpha \cdot \gamma 1) \cdot (\hat{A} \cdot (\alpha + \theta - \alpha \cdot \theta ) + \rho ) + \hfill \\ \,exp\;(\hat{A} \cdot s \cdot (\alpha - 1) \cdot (\theta - 1)) \cdot (\hat{A} \cdot (1 - 2\alpha + (\alpha - 1) \cdot \theta ) - \rho ) \cdot \hfill \\ (\hat{A} \cdot s \cdot \alpha^{2} \cdot (\theta - 1) + \alpha \cdot (\gamma 1 - 1 + s \cdot (\hat{A} \cdot (1 - 2\theta ) - \rho )))/ \hfill \\ (\alpha \cdot ( - exp\;(s \cdot \rho ) \cdot \alpha \cdot (\hat{A} \cdot (\alpha + \theta - \alpha \cdot \theta ) + \rho ) + \hfill \\ exp(\hat{A} \cdot s \cdot (\alpha - 1) \cdot (\theta - 1)) \cdot (\hat{A} \cdot (2\alpha - 1 + \theta - \alpha \theta ) + \rho ))) = 0 \hfill \\ \end{aligned}$$

(41)

$$\begin{array}{*{20}c} {H_{P} = - \left( {\exp \left( s \cdot \left( {r + \hat{A} \cdot \alpha } \right) \right)\cdot \left( {1 + \hat{A} \cdot s} \right) \cdot \alpha \cdot \left( {r^{2} - r \cdot \hat{A} - \alpha \cdot \hat{A}^{2} + 2 \cdot r \cdot \alpha \cdot \hat{A} + \left( {\alpha \cdot \hat{A}} \right)^{2} } \right) \cdot \varphi ^{{\zeta 2 - \zeta 1}} \cdot \psi ^{{ - \zeta 2}} } \right)/} \\ {\left( {\exp \left( {\hat{A} \cdot s} \right) \cdot \left( {\hat{A} - r - 2 \cdot \hat{A} \cdot \alpha } \right) + \exp \left( {\left( {r + \hat{A} \cdot \alpha } \right) \cdot s} \right) \cdot \left( {r \cdot \alpha + \hat{A} \cdot \alpha ^{2} } \right) \cdot \delta \cdot \zeta 0} \right)} \\ \end{array}$$

(42)

$$r = \hat{A} \cdot \left( {1 - \alpha } \right) \cdot \theta + \rho $$

(43)

The variables for which this system is solved are $H_{R} ,H_{P} ,J,T,\varphi ,\psi ,\hat{A},s,r$. Solving the system consisting of just Eqs. (35)–(42) for a given value of r gives rise to (an implicit) growth-supply side relationship between $\hat{A}$ and the interest rate r, as suggested by Eq. (34). (43) reproduces the growth-demand side of the model. The final solution of the model can be thought of as being obtained by confronting the growth supply side with the growth demand side of the model.

A final remark concerns the nature of the steady state solution obtained by the system above. Contrary to the Love of Variety growth models by Romer (1990) and Jones (1995), for example, our model does not account for physical capital accumulation. That allows all final output to be consumed, and there is no direct real trade-off between consumption now and consumption in the future because the capital stock as a stock of ‘cumulative consumption foregone’ (cf. Romer (1990)) does not exist. Hence, the hump-shaped capital growth locus one usually finds in growth models with fixed capital formation is missing here. The intertemporal trade-off between current and future consumption that does play a role in our model is due to the allocation of labor time over its various uses. Since we made the assumption of the existence of a steady state in order to arrive at the simple expressions for the allocation of labor time as given by (30.A) and (30.B), part of the more intricate dynamic relationship between the allocation of labor and the actual/transitional growth rate has effectively been disregarded for now. However, as the model generates a priori sensible outcomes, it seems to be worthwhile to look into the matter of (the stability of) transitional growth more closely in the future.