Directed technical change and environmental quality

Abstract

We propose a directed technical change model with two sectors, clean and dirty, to analyze the impact of the degree of substitutability between sectors and the degree of scale effects on the environmental quality. The technological knowledge is biased towards the clean sector; i.e., the environmental quality is improved whenever the elasticity of substitution between inputs in both sectors increases and, along with that, the economy: (i) is rich in renewable capital, (ii) has higher relative supply of clean labor under scale effects, and (iii) enjoys higher relative R&D productivity in the clean sector. The improvement in the environmental quality benefits the welfare. Moreover, the growth rate is higher in the presence of scale effects.

Appendices

Appendix 1. Final good prices in the two sectors

From the maximization problem max π = Y − P_DY_D − P_CY_C, with $Y={\left({\varsigma}_D{Y}_D^{\frac{\varepsilon -1}{\varepsilon }}+{\varsigma}_C{Y}_C^{\frac{\varepsilon -1}{\varepsilon }}\right)}^{\frac{1}{\varepsilon -1}}$, the first-order condition with respect to Y_D

$$ \frac{\partial \pi }{\partial {Y}_D}=\frac{\varepsilon }{\varepsilon -1}{\left({\varsigma}_D{Y}_D^{\frac{\varepsilon -1}{\varepsilon }}+{\varsigma}_C{Y}_C^{\frac{\varepsilon -1}{\varepsilon }}\right)}^{\frac{1}{\varepsilon -1}}\frac{\varepsilon -1}{\varepsilon }{\varsigma}_D{Y}_D^{-\frac{1}{\varepsilon }}-{P}_D=0 $$

reveals that the price of the final good in the D-Sector is,

$$ {P}_D={\left({\varsigma}_D{Y}_D^{\frac{\varepsilon -1}{\varepsilon }}+{\varsigma}_C{Y}_C^{\frac{\varepsilon -1}{\varepsilon }}\right)}^{\frac{1}{\varepsilon -1}}{\varsigma}_D{Y}_D^{-\frac{1}{\varepsilon }}. $$

Similarly for the C-sector, from the first-order condition with respect to Y_C, we obtain

$$ {P}_C={\left({\varsigma}_D{Y}_D^{\frac{\varepsilon -1}{\varepsilon }}+{\varsigma}_C{Y}_C^{\frac{\varepsilon -1}{\varepsilon }}\right)}^{\frac{1}{\varepsilon -1}}{\varsigma}_C{Y}_C^{-\frac{1}{\varepsilon }}. $$

Finally, Eq. (4) is easily obtained

$$ \frac{P_C}{P_D}=\frac{\varsigma_C}{\varsigma_D}{\left(\frac{Y_C}{Y_D}\right)}^{-\frac{1}{\varepsilon }}. $$

Appendix 2. Inverse demand for labor derivation

Considering $\mathit{\max}\ {\pi}_D={P}_D{Y}_D-{w}_D{L}_D-{\int}_0^{A_D}{q}_D(j){x}_D(j) dj$, where ${Y}_D=\frac{L_D^{\alpha}\cdotp {R}^{\beta }}{1-\alpha -\beta }{\int}_0^{A_D}{x}_D{(j)}^{1-\alpha -\beta } dj$ (5), the first-order conditions yields

$$ {\displaystyle \begin{array}{c}\frac{\partial {\pi}_D}{\partial {L}_D}=\alpha {P}_D\frac{L_D^{\alpha -1}{R}^{\beta }}{1-\alpha -\beta }{\int}_0^{A_D}{x}_D{(j)}^{1-\alpha -\beta } dj-{w}_D\\ {}=\alpha \frac{P_D}{L_D}{Y}_D-{w}_D=0,\end{array}} $$

and thus ${w}_D=\alpha \frac{P_D}{L_D}{Y}_D$;

$$ \frac{\partial {\pi}_D}{\partial {x}_D}={P}_D{L}_D^{\alpha }{R}^{\beta }{x}_D{(j)}^{-\alpha -\beta }-{q}_D(j)=0 $$

and thus ${x}_D(j)={\left(\frac{P_D}{q_D(j)}{L}_D^{\alpha }{R}^{\beta}\right)}^{\frac{1}{\alpha +\beta }}$. Plugging the latter into the former, it results that

$$ {\displaystyle \begin{array}{c}{w}_D=\alpha \frac{P_D}{L_D}\frac{L_D^{\alpha }{R}^{\beta }}{1-\alpha -\beta }{\int}_0^{A_D}{x}_D{(j)}^{1-\alpha -\beta } dj\\ {}=\alpha \frac{P_D}{L_D}\frac{L_D^{\alpha }{R}^{\beta }}{1-\alpha -\beta }{\int}_0^{A_D}{P}_D^{\frac{1-\alpha -\beta }{\alpha +\beta }}{q}_D{(j)}^{-\frac{1-\alpha -\beta }{\alpha +\beta }}{L}_D^{\frac{\alpha \left(1-\alpha -\beta \right)}{\alpha +\beta }}{R}^{\frac{ta\left(1-\alpha -\beta \right)}{\alpha +\beta }} dj\\ {}=\frac{\alpha }{1-\alpha -\beta }{P}_D^{\frac{1}{\alpha +\beta }}{q}_D{(i)}^{-\frac{1-\alpha -\beta }{\alpha +\beta }}{\left(\frac{L_D}{R}\right)}^{-\frac{\beta }{\alpha +\beta }}.\end{array}} $$

Appendix 3. Relative price of the C-factor deduction

From (4) $\frac{P_C}{P_D}=\frac{\varsigma_C}{\varsigma_D}{\left(\frac{Y_C}{Y_D}\right)}^{-\frac{1}{\varepsilon }}$, and bearing in mind (11) and (12), we obtain

$$ \frac{P_C}{P_D}=\frac{\varsigma_C}{\varsigma_D}{\left[\frac{\frac{A_C}{1-\alpha -\beta }{\left({P}_C^{\left(1-\alpha -\beta \right)}{L}_C^{\alpha }{K}^{\beta}\right)}^{\frac{1}{\alpha +\beta }}}{\frac{A_D}{1-\alpha -\beta }{\left({P}_D^{\left(1-\alpha -\beta \right)}{L}_D^{\alpha }{R}^{\beta}\right)}^{\frac{1}{\alpha +\beta }}}\right]}^{-\frac{1}{\varepsilon }}. $$

$$ {\left(\frac{P_C}{P_D}\right)}^{\frac{\left(1-\alpha -\beta \right)+\varepsilon \left(\alpha +\beta \right)}{\alpha +\beta }}={\varsigma}^{\varepsilon }{\left(\frac{A_C}{A_D}\right)}^{-1}{\left(\frac{L_C}{L_D}\right)}^{-\frac{\alpha }{\alpha +\beta }}{\left(\frac{K}{R}\right)}^{-\frac{\beta }{\alpha +\beta }} $$

and, therefore, Eq. (17) is recovered

$$ \frac{P_C}{P_D}={\varsigma}^{\frac{\varepsilon \left(\alpha +\beta \right)}{\sigma }}{\left[{\left(\frac{A_C}{A_D}\right)}^{\alpha +\beta }{\left(\frac{L_C}{L_D}\right)}^{\alpha }{\left(\frac{K}{R}\right)}^{\beta}\right]}^{-\frac{1}{\sigma }}. $$

Appendix 4. Relative return on clean-skilled capital deduction

Combining Eqs. (13), (14), and (17) we get

$$ {\displaystyle \begin{array}{c}W=\frac{w_C}{w_D}=\frac{\frac{\alpha }{1-\alpha -\beta }{P}_C^{\frac{1}{\alpha +\beta }}{L}_C^{\frac{-\beta }{\alpha +\beta }}{K}^{\frac{\beta }{\alpha +\beta }}}{\frac{\alpha }{1-\alpha -\beta }{P}_D^{\frac{1}{\alpha +\beta }}{L}_D^{\frac{-\beta }{\alpha +\beta }}R\frac{\beta }{\alpha +\beta }}\\ {}={\left(\frac{P_C}{P_D}\right)}^{\frac{1}{\alpha +\beta }}{\left(\frac{L_C}{L_D}\right)}^{\frac{-\beta }{\alpha +\beta }}{\left(\frac{K}{R}\right)}^{\frac{\beta }{\alpha +\beta }}\\ {}={\left[{\varsigma}^{-\varepsilon}\left(\frac{A_C}{A_D}\right){\left(\frac{L_C}{L_D}\right)}^{\frac{\alpha +\beta \sigma}{\alpha +\beta }}{\left(\frac{K}{R}\right)}^{\frac{\beta \left(1-\sigma \right)}{\alpha +\beta }}\right]}^{-\frac{1}{\sigma }}.\end{array}} $$

Appendix 5. Relative profitability of innovators in the C-sector deduction

Dividing Eq. (16) by (15) and combining with (17) we have

$$ {\displaystyle \begin{array}{c}\frac{\pi_C}{\pi_D}=\frac{\left(\alpha +\beta \right){P}_C^{\frac{1}{\alpha +\beta }}{\left(\frac{L_C}{K}\right)}^{\frac{\alpha }{\alpha +\beta }}}{\left(\alpha +\beta \right){P}_D^{\frac{1}{\alpha +\beta }}{\left(\frac{L_D}{R}\right)}^{\frac{\alpha }{\alpha +\beta }}}\\ {}={\varsigma}^{\frac{\varepsilon }{\sigma }}{\left(\frac{A_C}{A_D}\right)}^{\frac{\alpha +\beta }{\alpha +\beta}\left(-\frac{1}{\sigma}\right)}{\left(\frac{L_C}{L_D}\right)}^{\frac{\alpha \left(\sigma -1\right)}{\alpha +\beta}\frac{1}{\sigma }}{\left(\frac{K}{R}\right)}^{\frac{\beta \left(\sigma -1\right)}{\alpha +\beta}\frac{1}{\sigma }}\\ {}={\varsigma}^{\frac{\varepsilon }{\sigma }}{\left(\frac{A_C}{A_D}\right)}^{-\frac{1}{\sigma }}{\left[{\left(\frac{L_C}{L_D}\right)}^{\alpha }{\left(\frac{K}{R}\right)}^{\beta}\right]}^{\frac{\sigma -1}{\left(\alpha +\beta \right)\sigma }}.\end{array}} $$

Equation (19) is thus recovered.

Appendix 6. Directed technical change result

Solving Eq. (19) with respect to $\frac{A_C}{A_D}$, yelds

$$ \frac{A_C}{A_D}={\varsigma}^{\varepsilon }{\left(\frac{\lambda_C}{\lambda_D}\frac{L_C^{-{\gamma}_{L_C}}}{L_D^{-{\gamma}_{L_D}}}\right)}^{\sigma }{\left[{\left(\frac{L_C}{L_D}\right)}^{\alpha }{\left(\frac{K}{R}\right)}^{\beta}\right]}^{\frac{\sigma -1}{\alpha +\beta }}, $$

which meets Eq. (22).

Appendix 7. Skill premium deduction with directed technical change

Substituting (22) into (18), we obtain

$$ {\displaystyle \begin{array}{c}W=\frac{w_C}{w_D}={\left[{\varsigma}^{-\varepsilon}\left(\frac{A_C}{A_D}\right){\left(\frac{L_C}{L_D}\right)}^{\frac{\alpha +\beta \sigma}{\alpha +\beta }}{\left(\frac{K}{R}\right)}^{\frac{\beta \left(1-\sigma \right)}{\alpha +\beta }}\right]}^{-\frac{1}{\sigma }}\\ {}={\left[{\left(\frac{\lambda_C}{\lambda_D}\right)}^{\sigma }{\left(\frac{L_C^{-{\gamma}_{L_C}}}{L_D^{-{\gamma}_{L_D}}}\right)}^{\sigma }{\left(\frac{L_C}{L_D}\right)}^{\frac{\alpha \left(\sigma -1\right)}{\alpha +\beta }}{\left(\frac{L_C}{L_D}\right)}^{\frac{+\beta \sigma}{\alpha +\beta }}{\left(\frac{K}{R}\right)}^{\frac{\beta \left(\sigma -1\right)}{\alpha +\beta }}{\left(\frac{K}{R}\right)}^{\frac{\beta \left(1-\sigma \right)}{\alpha +\beta }}\right]}^{-\frac{1}{\sigma }}\\ {}={\left[\left(\frac{\lambda_C}{\lambda_D}\right)\left(\frac{L_C^{-{\gamma}_{L_C}}}{L_D^{-{\gamma}_{L_D}}}\right)\left(\frac{L_C}{L_D}\right)\right]}^{-1}.\end{array}} $$

Equation (23) is thus deduced.