The Environmental Tax: Effects on Inequality and Growth

Abstract

Within an R&D-driven growth model, this paper studies how an environmental tax and its cost both for firms and consumers affect individuals’ incentives for human capital accumulation, income inequality, and the per capita growth rate. The results show that when a low share of the environmental tax on consumption is levied, a tighter environmental tax results in an increase in individuals’ human capital accumulation and income inequality between both unskilled and skilled workers and among skilled workers and spurs the per capita growth rate. A numerical simulation for the U.S. economy illustrates the results and shows that the increase in income inequality is very modest compared to the large increase in the per capita output growth rate. Moreover, it can be seen that a no-carbon-pricing green policy with command and control instruments, for example, has a negative effect on both the incentive for human capital accumulation and the per capita growth rate.

Appendices

Appendix 1: Individuals’ and Firms’ Decision

• 1. The supply of the unskilled and skilled labor force is the same as in Dinopoulos and Segerstrom (1999). The supply of unskilled labor at time t, $L\left( t\right)$, equals the number of individuals in the population who decide to remain unskilled, i.e. $L\left( t\right) =\theta _{0}N\left( t\right)$. For the derivation of the skilled labor force at time t, note that $\left( 1-\theta _{0}\right) N\left( t\right)$ individuals either work as skilled workers or are training to become skilled workers. In this subpopulation the skilled workers are all individuals that were born between $\left( t-D\right)$ and $\left( t-T\right)$, $\int _{t-D}^{t-T}b\left( 1-\theta _{0}\right) N\left( s\right) ds=\left( 1-\theta _{0}\right) \psi N\left( t\right)$, where $\psi \equiv \left( e^{n\left( D-T\right) }\right) /\left( e^{nD}-1\right) <1$. The average skill level of workers who have finished training is $\left[ \left( \theta _{0}-\gamma \right) /2\right] +$ $\left[ \left( 1-\gamma \right) /2\right] =$ $\left( \theta _{0}+1-2\gamma \right) /2$. Therefore the supply of skilled labor at time t, measured in efficiency units, is given by $H\left( t\right) =\left( \theta _{0}+1-2\gamma \right) \left( 1-\theta _{0}\right) \frac{\psi }{2}N\left( t\right)$. □

• 2. In the case of non-drastic innovations and Bertrand competition along each variety ${\tilde{\omega }}\in \left[ 0,\beta \right]$, a limit pricing strategy by the top-quality producer is adopted. Due to free entry in producing the second best quality, profit flows should be zero in equilibrium. In light of the above, the price-quality ratio of the top quality leader producing quality $\left( j+1\right)$ of a variety is $\frac{ p_{{\tilde{\omega }},n}}{\lambda ^{j+1}}$, where $\lambda$ is the exogenous and constant quality jump in each variety, whereas the price-quality ratio of the follower producing quality j of the same variety is $\frac{\left[ \phi +\left( 1-\eta \right) \tau _{q}\right] }{\lambda ^{j}}$. The quality leader has the lowest price-quality ratio whenever $\frac{p_{{\tilde{\omega }} ,n}}{\lambda ^{j+1}}\le$ $\frac{\left[ \phi +\left( 1-\eta \right) \tau _{q}\right] }{\lambda ^{j}}$, i.e., whenever $p_{{\tilde{\omega }},n}\le$ $\lambda \left[ \phi +\left( 1-\eta \right) \tau _{q}\right]$, which implies $p_{{\tilde{\omega }},n}=\lambda \left[ \phi +\left( 1-\eta \right) \tau _{q} \right]$.

  The price charged to consumers gross of the tax burden is $p_{{\tilde{\omega }} }=p_{n,{\tilde{\omega }}}+\eta \tau _{q}$. When all the fiscal burden falls on consumers—i.e. $\eta \rightarrow 1$—the price is $p_{{\tilde{\omega }} }=\lambda \phi +\tau _{q}$. On the contrary, when all the fiscal burden falls on firms—i.e. $\eta \rightarrow 0$—the price is $p_{{\tilde{\omega }} }=\lambda \left( \phi +\tau _{q}\right)$.

  Profit flows of the patent holder along each variety ${\tilde{\omega }}$ solves the following maximization problem:

  $$\begin{aligned} \underset{q_{{\tilde{\omega }}}\left( t\right) }{Max}\left( p_{n,{\tilde{\omega }} }+\eta \tau _{q}\right) q_{{\tilde{\omega }}}\left( t\right) -\left[ \phi +\left( 1-\eta \right) \tau _{q}\right] q_{{\tilde{\omega }}}\left( t\right) -\eta \tau _{q}q_{{\tilde{\omega }}}\left( t\right) , \end{aligned}$$

  (33)

  where instantaneous profit flows net of the tax burden charged on consumers are considered. The maximization problem (33) reduces to:

  $$\begin{aligned} \underset{q_{{\tilde{\omega }}}\left( t\right) }{Max}p_{n,{\tilde{\omega }}}\left( q_{{\tilde{\omega }}}\right) q_{{\tilde{\omega }}}-\left[ \phi +\left( 1-\eta \right) \tau _{q}\right] q_{{\tilde{\omega }}}\left( t\right) . \end{aligned}$$

  (34)

  The solution to this maximization problem implies $p_{{\tilde{\omega }} ,n}=\lambda \left[ \phi +\left( 1-\eta \right) \tau _{q}\right]$, and profit flows net of the tax burden are:

  $$\begin{aligned} \pi ({\tilde{\omega }},t)=\left( \lambda -1\right) \left[ \phi +\left( 1-\eta \right) \tau _{q}\right] q_{{\tilde{\omega }}}\left( t\right) . \end{aligned}$$

  (35)

  The price gross of the environmental tax on consumers is $p_{{\tilde{\omega }} }=p_{n,{\tilde{\omega }}}+\eta \tau _{q}=\lambda \phi +\tau _{q}\left[ \lambda \left( 1-\eta \right) +\eta \right]$. □

• 3. Let $v\left( {\tilde{\omega }},t\right)$ denote the expected discounted profit flows of a successful firm in variety ${\tilde{\omega }}$ at time t. Since each leader is targeted by R&D firms that try to discover the next quality leader product, the shareholder suffers a loss $v\left( {\tilde{\omega }},t\right)$, with probability $I\left( {\tilde{\omega }},t\right) dt$. Thus, the event of no innovation occurs with probability $1-I\left( {\tilde{\omega }} ,t\right) dt$. Over a time interval dt, the shareholder of a stock issued by a successful R&D firm receives a dividend $\pi \left( {\tilde{\omega }} ,t\right) dt$, and the value of the firm appreciates by $dv\left( {\tilde{\omega }},t\right) ={\dot{v}}\left( {\tilde{\omega }},t\right) dt$. Since the stock market is assumed to be perfectly efficient, the expected rate of return of a stock issued by a successful R&D firm must be equal to the riskless rate of return r:

  $$\begin{aligned} rdt=\frac{{\dot{v}}\left( {\tilde{\omega }},t\right) }{v\left( {\tilde{\omega }} ,t\right) }\left[ 1-I\left( {\tilde{\omega }},t\right) dt\right] dt-\frac{ v\left( {\tilde{\omega }},t\right) -0}{v\left( {\tilde{\omega }},t\right) }I\left( {\tilde{\omega }},t\right) dt+\frac{\pi \left( {\tilde{\omega }},t\right) }{ v\left( {\tilde{\omega }},t\right) }dt. \end{aligned}$$

  (36)

  Dividing by dt, and taking the limits as $dt\rightarrow 0$, the following condition for the expected discounted value of the firm producing variety ${\tilde{\omega }}$ is obtained:

  $$\begin{aligned} v\left( {\tilde{\omega }},t\right) =\frac{\pi \left( {\tilde{\omega }},t\right) }{ r+I\left( {\tilde{\omega }},t\right) -\frac{\overset{.}{v}\left( {\tilde{\omega }} ,t\right) }{v\left( {\tilde{\omega }},t\right) }}, \end{aligned}$$

  (37)

  As in Dinopoulos and Segerstrom (1999), along the BGP the per-capita variables all grow at the same rate, it follows that $\frac{\overset{.}{v} \left( \omega ,t\right) }{v\left( \omega ,t\right) }=\frac{\overset{.}{\pi } \left( \omega ,t\right) }{\pi \left( \omega ,t\right) }=n$, and $\frac{ \overset{.}{v}\left( {\tilde{\omega }},t\right) }{v\left( {\tilde{\omega }} ,t\right) }=\frac{\overset{.}{\pi }\left( {\tilde{\omega }},t\right) }{\pi \left( {\tilde{\omega }},t\right) }=n$. Hence, the expected discounted value (37) boils down to:

  $$\begin{aligned} v\left( {\tilde{\omega }},t\right) =\frac{\pi \left( {\tilde{\omega }},t\right) }{ \rho +I\left( {\tilde{\omega }},t\right) -n}. \end{aligned}$$

  (38)

  where $r=\rho$ since the analysis refers to the BGP, and $\rho >n$. □

Appendix 2: Comparative Statics

In the following equations, variables refer to time t, and therefore the time index has been eliminated for the sake of simplicity, unless it is strictly necessary for comprehension of the text. The first part of the Appendix proves the existence of a unique steady state value for threshold ability parameter $\theta _{0}$, and the comparative static exercises are implemented to show the effects of a tighter tax $\tau _{q}$ and higher tax share $\eta$ on the incentive for human capital accumulation. The second part of the Appendix show the effects of a tighter tax $\tau _{q}$ and a higher tax share $\eta$ on profit flows of dirty and clean firms.

• 1. Consider Eq. (32) which can be written as:

  $$\begin{aligned} \left( \theta _{0}+1-2\gamma \right) \left( 1-\theta _{0}\right) \frac{\psi }{2}=ak\left( \beta I\left( {\tilde{\omega }}\right) +\left( 1-\beta \right) I\left( \omega \right) \right) . \end{aligned}$$

  (39)

  Considering both Eqs. (29) and (30), the skilled labor market clearing condition above can be written as:

  $$\begin{aligned} \begin{array}{l} \left( \theta _{0}+1-2\gamma \right) \left( 1-\theta _{0}\right) \frac{\psi }{2} =\theta _{0}\left( \theta _{0}-\gamma \right) \Omega -ak\left( \rho -n\right) , \end{array} \end{aligned}$$

  (40)

  where $\Omega \equiv \left( \lambda -1\right) \lambda \iota \frac{\beta [\phi +\left( 1-\eta \right) \tau _{q}]+\left( 1-\beta \right) \iota \left[ \lambda \left( \phi +\left( 1-\eta \right) \tau _{q}\right) +\eta \tau _{q}\right] }{\sigma B}$.

  The left hand side of Eq. (40) is a strictly concave quadratic polynomial in $\theta _{0}$ with roots $\left( 2\gamma -1\right)$ and 1. The right hand side of the same equation is a strictly convex quadratic polynomial in $\theta _{0}$ with two real roots, one negative and one positive, where the positive root is:

  $$\begin{aligned} \theta _{0}=\frac{\gamma \Omega +\sqrt{\left( \gamma \Omega \right) ^{2}+4\Omega \left( \rho -n\right) }}{2\Omega }\in \left( \gamma ,1\right) , \end{aligned}$$

  (41)

  if the stated parameter restrictions are satisfied. Therefore, one and only one real and positive steady state solution $\theta _{0}^{*}\in \left( \gamma ,1\right)$ exists.

  Let us introduce the function

  $$\begin{aligned} G\left( \theta _{0},\tau _{q}\right) =\theta _{0}\left( \theta _{0}-\gamma \right) \Omega -ak\left( \rho -n\right) -\left( \theta _{0}+1-2\gamma \right) \left( 1-\theta _{0}\right) \frac{\psi }{2}=0. \end{aligned}$$

  Using the Implicit Function Theorem for the function $G\left( \theta _{0},\tau _{q}\right)$, the following relationships are proven:

  $$\begin{aligned} \frac{\partial G}{\partial \theta _{0}}=\left( 2\theta _{0}-\gamma \right) \Omega +\left( \theta _{0}-\gamma \right) \psi >0, \end{aligned}$$

  (42)
  $$\begin{aligned} \frac{\partial G}{\partial \tau _{q}}&= \left[ \frac{\theta _{0}\left( \theta _{0}-\gamma \right) \left( \lambda -1\right) \lambda \iota }{\sigma }\right] \nonumber \\&\quad *\left[ \frac{\left[ \left( 1-\beta \right) \iota \left( \lambda \left( 1-\eta \right) +\eta \right) +\beta \left( 1-\eta \right) \right] B-\beta [\phi +\left( 1-\eta \right) \tau _{q}]\left( 1-\beta \right) \iota _{w_{L}}\left[ \lambda \left( 1-\eta \right) +\eta \right] }{B^{2}}\right] , \end{aligned}$$

  (43)

  which after some tedious calculations can be rewritten as:

  $$\begin{aligned} \frac{\partial G}{\partial \tau _{q}}&= \left[ \frac{\theta _{0}\left( \theta _{0}-\gamma \right) \left( \lambda -1\right) \lambda \iota }{\sigma }\right] \nonumber \\&\quad \left[ \frac{\left( 1-\beta \right) \beta \phi _{w_{L}}\lambda \iota ^{2}\left( \lambda \left( 1-\eta \right) +\eta \right) +\beta ^{2}\left( 1-\eta \right) \lambda \iota \phi _{w_{L}}-\eta \phi \beta \left( 1-\beta \right) \iota _{w_{L}}}{B^{2}}\right] . \end{aligned}$$

  (44)

  The following inequality is a sufficient condition to get $\frac{\partial G}{ \partial \tau _{q}}>0$:

  $$\begin{aligned} \eta <{\bar{\eta }}\equiv \frac{1}{1+\frac{1}{\lambda }\frac{1-\beta }{\beta } \frac{\iota _{w_{L}}}{\phi _{w_{L}}}\frac{\phi }{\iota }}\in \left( 0,1\right) . \end{aligned}$$

  (45)

  It is worth noting that inequality $\frac{\partial G}{\partial \tau _{q}}>0$ also holds when the parameter restriction $\frac{\phi \iota _{w_{L}}}{\iota ^{2}\phi _{w_{L}}}<\lambda$ is satisfied. If condition (45) is satisfied, using the Implicit Function Theorem for the function $G\left( \theta _{0},\tau _{q}\right)$, the following relationships is proven:

  $$\begin{aligned} \frac{\partial \theta _{0}}{\partial \tau _{q}}=-\frac{\frac{\partial G}{ \partial \tau _{q}}}{\frac{\partial G}{\partial \theta _{0}}}<0. \end{aligned}$$

  (46)

  Note that if no dirty varieties existed in the economy, i.e. $\beta =0$, the condition (45) would be $\eta <0$, which is never satisfied. On the contrary, if no clean varieties existed in the economy, i.e. $\beta =1$, the condition (45) would be $\eta <1$, which is always satisfied.

  Using the Implicit Function Theorem for the function $G\left( \theta _{0},\tau _{q}\right)$, the following relationship is proven:

  $$\begin{aligned} \frac{\partial G}{\partial \eta }&= \left[ \beta \tau _{q}\left( \lambda -1\right) \lambda \iota \right] \nonumber \\&\quad *\left[ \frac{-\lambda \iota \beta \phi _{w_{L}}-\left( 1-\beta \right) \left( \lambda -1\right) \lambda \iota ^{2}\phi _{w_{L}}-\left( 1-\beta \right) \iota _{w_{L}}\left( 1+\eta \right) }{\left( \sigma B\right) ^{2}} \right] <0. \end{aligned}$$

  (47)

  Therefore, using again the Implicit Function Theorem for the function $G\left( \theta _{0},\tau _{q}\right)$, the following relationship is proven:

  $$\begin{aligned} \frac{\partial \theta _{0}}{\partial \eta }=-\frac{\frac{\partial G}{ \partial \eta }}{\frac{\partial G}{\partial \theta _{0}}}>0. \end{aligned}$$

  (48)

  □

• 2. This second part of “Appendix 2” shows the effects of a tighter environmental tax $\tau _{q}$ and a higher tax share $\eta$ on profit flows of dirty and clean varieties.

Let a tighter environmental tax $\tau _{q}$ be levied. The marginal change in net profit flows in each variety ${\tilde{\omega }}$ is:

$$\begin{aligned} \frac{\partial \pi \left( {\tilde{\omega }}\right) }{\partial \tau _{q}}&= \frac{ N\left( \lambda -1\right) \lambda \iota }{B^{2}}[B\frac{\partial \theta _{0} }{\partial \tau _{q}}\left( \phi +\left( 1-\eta \right) \tau _{q}\right) +B\theta _{0}\left( 1-\eta \right) + \nonumber \\& \quad -\left( 1-\beta \right) \iota _{w_{L}}\theta _{0}\left( \phi +\left( 1-\eta \right) \tau _{q}\right) \left( \lambda \left( 1-\eta \right) +\eta \right) ]. \end{aligned}$$

(49)

After some tedious calculations, the condition (49) can be written as:

$$\begin{aligned} \frac{\partial \pi \left( {\tilde{\omega }}\right) }{\partial \tau _{q}}&= \left[ \frac{N\left( \lambda -1\right) \lambda \iota }{B^{2}}\right] \nonumber \\& \quad *\left[ B\frac{\partial \theta _{0}}{\partial \tau _{q}}\left( \phi +\left( 1-\eta \right) \tau _{q}\right) +\theta _{0}\left( \beta \lambda \iota \phi _{w_{L}}\left( 1-\eta \right) -\left( 1-\beta \right) \eta \phi \iota _{w_{L}}\right) \right] , \end{aligned}$$

(50)

When both inequalities $\eta >{\bar{\eta }}$ and $\frac{\phi \iota _{w_{L}}}{ \iota ^{2}\phi _{w_{L}}}<\lambda$ hold, a tighter tax implies lower profit flows of each dirty variety.

Next, it is proven that if condition (45) is satisfied, i.e., $\eta <{\bar{\eta }}$, a tighter tax $\tau _{q}$ implies higher profit flows of the dirty and clean varieties. The proof uses the cost function of dirty firms and therefore results hold for any degree of substitution between labor and dirty inputs. Moreover, according to the preferences as in Eq. (2), the proof uses weak substitutability between dirty and clean varieties. The proof is by contradiction and follows three steps.

• 1. This point proves that a tighter tax $\tau _{q}$ decreases the total cost of production of each dirty variety. Let $TC\left( {\tilde{\omega }} \right)$ denote the total cost of production of each dirty variety: $TC\left( {\tilde{\omega }}\right) =\left[ \phi +\left( 1-\eta \right) \tau _{q} \right] q_{{\tilde{\omega }}}\left( p_{{\tilde{\omega }}}\right)$. Then, the total cost of production of each dirty variety is a decreasing function in the environmental tax if the following inequality is satisfied: $\frac{ \partial TC\left( {\tilde{\omega }}\right) }{\partial \tau _{q}}=\left( 1-\eta \right) q_{{\tilde{\omega }}}\left( p_{{\tilde{\omega }}}\right) +\left[ \phi +\left( 1-\eta \right) \tau _{q}\right] \frac{\partial q_{{\tilde{\omega }} }\left( p_{{\tilde{\omega }}}\right) }{\partial p_{{\tilde{\omega }}}}\frac{ \partial p_{{\tilde{\omega }}}}{\partial \tau _{q}}<0$. After some algebraic manipulations such an inequality can be written as: $\frac{\partial TC\left( {\tilde{\omega }}\right) }{\partial \tau _{q}}=\left| \epsilon _{_{{\tilde{\omega }}}}\right| \phi \eta +\left( \left| \epsilon _{_{{\tilde{\omega }} }}\right| -1\right) \left( 1-\eta \right) \left( \lambda \phi +\lambda \left( 1-\eta \right) \tau _{q}+\eta \left( 1-\eta \right) \tau _{q}\right) >0$, that always holds because inequality $\left| \epsilon _{_{\tilde{ \omega }}}\right| \ge 1$ holds, where $\left| \epsilon _{_{{\tilde{\omega }}}}\right|$ denotes the (absolute value of the) price elasticity of demand for each dirty variety. Therefore, the total cost of production of each dirty variety $TC\left( {\tilde{\omega }}\right)$ is a decreasing function of the environmental tax $\tau _{q}$. 2. This point proves that since a tighter tax implies a reduction in the total cost of production $TC\left( {\tilde{\omega }}\right)$ of each dirty variety, when the (positive) price effect outweighs—in absolute value—the (negative) quantity effect on profits, the change in profit flows of each dirty variety outweighs—in absolute value—the change in profit flows of the same dirty variety obtained when the (negative) quantity effect outweighs—in absolute value—the (positive) price effect. Preferences as in Eq. (2) imply a unit constant elasticity of substitution between dirty and clean varieties. Due to the CES utility function, the relative change in the ratio between the quantity of any two varieties ${\tilde{\omega }}$ and $\omega$, i.e., $\left( q_{\omega }/q_{{\tilde{\omega }} }\right)$ have to be equal to the relative change in the respective price ratio $\left( p_{{\tilde{\omega }}}/p_{\omega }\right)$ to get the constant value of the elasticity of substitution along each indifference curve. A tighter tax implies a higher relative price of each dirty to clean variety $\left( p_{{\tilde{\omega }}}/p_{\omega }\right)$ and a lower demand for each dirty variety $q_{{\tilde{\omega }}}$. Therefore, when in each dirty variety ${\tilde{\omega }}$ the (positive) price effect outweighs—in absolute value—the (negative) quantity effect, a tighter tax $\tau _{q}$ together with the value of the CES imply a higher demand for each clean variety $q_{\omega }$, and then higher profit flows of each clean variety are obtained (bear in mind that both the price and the production cost of each clean variety are unchanged). Moreover, because the total cost of production $TC\left( {\tilde{\omega }}\right)$ of each dirty variety is a decreasing function of the environmental tax $\tau _{q}$ (step 1 above), when the (positive) price effect outweighs—in absolute value—the (negative) quantity effect of each dirty variety, a tighter tax implies higher profit flows of each dirty variety.

On the contrary, when in each dirty variety ${\tilde{\omega }}$ the (negative) quantity effect outweighs—in absolute value—the (positive) price effect, a tighter tax $\tau _{q}$ together with the value of the CES implies a lower demand for each clean variety $q_{\omega }$, and then lower profit flows of each clean variety are obtained. Moreover, because the total cost of production $TC\left( {\tilde{\omega }}\right)$ of each dirty variety is a decreasing function of the environmental tax $\tau _{q}$ (step 1 above), when the (negative) quantity effect outweighs—in absolute value—the (positive) price effect of each dirty variety, a tighter tax can imply either higher or lower profit flows of each dirty variety. In light of the above, because the reduction in the total cost of production $TC\left( {\tilde{\omega }}\right)$ of each dirty variety is the same, when the (positive) price effect outweighs—in absolute value—the (negative) quantity effect, the change in profit flows of each dirty variety outweighs—in absolute value—the change in profit flows of the same dirty variety obtained when the (negative) quantity effect outweighs—in absolute value—the (positive) price effect. 3. If condition (45) is satisfied, i.e. $\eta <{\bar{\eta }}$, then $\frac{\partial \theta _{0}}{\partial \tau _{q}}<0$ and a tighter tax determines a stronger incentive for human capital accumulation and a higher aggregate innovation rate. From step 2 above, when the (positive) price effect outweighs—in absolute value—the (negative) quantity effect of each dirty variety, a tighter tax implies higher profit flows of dirty and clean varieties that together determine the stronger incentive for human capital accumulation and the higher aggregate innovation rate. On the contrary, when the (negative) quantity effect outweighs—in absolute value—the (positive) price effect of each dirty variety, a tighter tax implies lower demand and profit flows of each clean variety. In such a case, the (same) higher incentive for human capital accumulation and the (same) higher aggregate innovation rate should both only stand on the shoulders of higher profit flows of dirty varieties, and a contradiction with step 2 arises. Indeed, step 2 shows that when the (positive) price effect outweighs—in absolute value—the (negative) quantity effect the increase of profit flows of each dirty variety outweighs—in absolute value—the change in profit flows of the same variety obtained when the (negative) quantity effect outweighs—in absolute value—the (positive) price effect.

In light of steps 1 to 3, if condition (45) is satisfied, i.e. $\eta <{\bar{\eta }}$, a tighter tax $\tau _{q}$ implies higher profit flows of dirty and clean varieties. Results hold for any degree of substitution between labor and dirty and clean inputs in the manufacturing sector, and even with weak substitutability between the demand for dirty and clean varieties. □

Finally, the effect of a higher tax share $\eta$ on profit flows of dirty and clean varieties is analyzed. Let a higher share $\eta$ of the environmental tax on final consumption be levied. The marginal change in net profit flows in each dirty and clean variety ${\tilde{\omega }}$ and $\omega$ respectively are:

$$\begin{aligned} \frac{\partial \pi \left( {\tilde{\omega }}\right) }{\partial \eta }&= \frac{ N\left( \lambda -1\right) \lambda \iota }{B^{2}}[B\frac{\partial \theta _{0} }{\partial \eta }\left( \phi +\left( 1-\eta \right) \tau _{q}\right) -\theta _{0}\lambda \iota \beta \phi _{w_{L}}\tau _{q}+ \nonumber \\& \quad -\left( 1-\beta \right) \theta _{0}\iota _{w_{L}}\tau _{q}\left[ \phi +\left( 1-\eta \right) \tau _{q}+\eta \tau _{q}\right] <0, \end{aligned}$$

(51)

and

$$\begin{aligned} \frac{\partial \pi \left( \omega \right) }{\partial \eta }=\left[ N\left( \lambda -1\right) \iota \right] *\frac{\frac{\partial \theta _{0}}{ \partial \eta }{\hat{B}}-\theta _{0}\left[ \frac{\beta \phi _{w_{L}}\tau _{q}\left( \lambda -1\right) }{\left[ \lambda \left( \phi +\left( 1-\eta \right) \tau _{q}\right) +\eta \tau _{q}\right] ^{2}}\right] }{{\hat{B}}^{2}} <0, \end{aligned}$$

(52)

where ${\hat{B}}=\frac{B}{\lambda \iota \left[ \lambda \left( \phi +\left( 1-\eta \right) \tau _{q}\right) +\eta \tau _{q}\right] }$. □

Appendix 3: Numerical Simulation

• 1. This Appendix explains how the parameter values are calibrated. In the baseline calibration analysis equations (39) and ( 40) are used to get the threshold ability parameter $\theta _{0}$ and the aggregate innovation rate I. To this aim, the variables $\sigma \equiv \frac{\left( 1-e^{-\rho D}\right) }{\left( e^{-\rho T}-e^{-\rho D}\right) }$, $\psi \equiv \frac{\left( e^{n\left( D-T\right) }\right) }{\left( e^{nD}-1\right) }$, $\gamma =\theta _{0}-\frac{ \sigma }{\left( w_{H}/w_{L}\right) }$, $\Omega \equiv \frac{\left( \lambda -1\right) \lambda \iota \beta [\phi +\left( 1-\eta \right) \tau _{q}]+\left( 1-\beta \right) \iota \left[ \lambda \left( \phi +\left( 1-\eta \right) \tau _{q}\right) +\eta \tau _{q}\right] }{\sigma B}$, and $B\equiv \beta \phi _{w_{L}}\lambda \iota +\left( 1-\beta \right) \iota _{w_{L}}\left[ \lambda \left( \phi +\left( 1-\eta \right) \tau _{q}\right) +\eta \tau _{q} \right]$ are used. Table 5 shows some key parameter values with the respective source used.

Table 5 Baseline calibration U.S

The population growth rate n is calibrated to allow for fertility differentials between less educated and skilled individuals. The details are in point 2 of this Appendix.

To calculate $\sigma \equiv \frac{\left( 1-e^{-\rho D}\right) }{\left( e^{-\rho T}-e^{-\rho D}\right) }$, and $\psi \equiv \frac{\left( e^{n\left( D-T\right) }\right) }{\left( e^{nD}-1\right) }$, the subjective discount rate $\rho$ is set at 0.045 to generate an interest rate of $4.5\%$. This value is the long-term real interest rate for the period 1990–2019 taken from the OECD Statistics dataset (annual percentage) and it also coincides with the estimated value of Neves et al. (2018). A different value of $\rho$ from that set here does not alter the calibration qualitative results. Moreover, it is assumed that the amount of time an individual spends at work D is 40 years, and the length of training T is 4 years. These are standard measures for a developed economy as in Dinopoulos and Segerstrom (1999). So we get $\sigma =1.246$ and $\psi =0.92$.

The size of innovation measures the gross mark-up enjoyed by innovators. Hall (2018) estimates the mark-up to be in the range of 1.04 and 1.85 between 1988 and 2015 for the U.S. economy with an average value of $\lambda =1.38$. The parameter $\beta$ represents the share of varieties producing polluting emissions in the economy. To get a consistent measure for the flow of polluting emissions the share $\beta$ is calibrated together with the variable representing the polluting emissions flow $e_{{\tilde{\omega }}}$. Therefore, data on all GHG emissions (Total GHG per unit of GDP in CO2 equivalents, units: Kilograms per 1000 U.S. dollars, Thousands) for the period 1990–2018 from the OECD Statistics dataset are utilized. All GHG emissions data from OECD statistics include: 1A1. Energy industries, 1A2. Manufacturing industries and construction; 1A3. Transport; 1A4. Residential and other sectors; 1A5. Other-Energy; 1B. Fugitive emissions from fuels; 1C. CO2 from transport and storage; 2. Industrial process and product use; 3. Agriculture; 5. Waste; 6. Others. Land use, land use change and forestry are only excluded. The average value for the period 1990–2018 is $\beta e_{ {\tilde{\omega }}}=0.54$. Considering other measures of polluting emissions given by all the available actual data from the WDI—namely CO2 emissions (kt) and Other greenhouse gas emissions, HFC, PFC and SF6 (thousand metric tons of CO2 equivalent, converted to kt) as well as CO2 emissions (kg per 2017 PPP $ss$ of GDP)—does not alter the calibration results.

To calibrate the tax on pollution, data on all environmentally related taxes (henceforth: ERT) as a percentage of the GDP from the OECD Statistics dataset for the period 1995–2016 for the U.S. economy are utilized. These data on ERT refer to energy products (including vehicle fuels); motor vehicles and transport services; measured or estimated emissions to air and water, ozone depleting substances, certain non-point sources of water pollution, waste management and noise, as well as management of water, land, soil, forests, biodiversity, wildlife and fish stocks. The data have been cross-validated and complemented with Revenue statistics from the OECD Tax statistics database and official national sources. The ERT average value for the period 1995–2016 for the U.S. is $\tau _{q}=0.87$. To calibrate the share $\eta$ of the environmental tax on consumption for the U.S. data on retail sales tax are needed. Indeed, the U.S. is the only OECD country that employs a retail sales tax rather than a value added tax (VAT) as the principal consumption tax. The retail sales tax in the U.S. is not a federal tax and is imposed at state and local government level. To fix ideas, compared to the other OECD countries, the U.S. has the lowest proportion of revenue from general consumption in its total tax revenue ($8.2\%$) in 2018, far below the OECD average ($21.2\%$) (Bradbury and Buydens 2019; Walczak 2021). Since $\eta$ denotes the share of ERT on final consumption, the ratio of total revenues from ERT on final consumption to total ERT revenues is used to calibrate $\eta$. Homogeneous available aggregate data at the federal government level for total revenues from ERT on final consumption and total revenues raised by ERT come from the OECD (2021) for the period 1994–2006 and refer to Gasohol tax. Therefore, the average value of the ratio of total revenues from ERT on final consumption to total ERT revenues from Gasohol tax for the period 1994–2006 is used to obtain $\eta \simeq 0.17$. Note that, since data on Gasohol tax are only available at federal level, the actual value of the tax share $\eta$ in the U.S. economy is higher, and the baseline calibration value $\eta \simeq 0.17$ is the less favorable for the validity of policy implications and results.

Due to the lack of physical capital in the model set-up, the total cost of dirty and clean firms, i.e., $\phi$ and $\iota$, are reduced by the average value (as percentage of GDP) of Gross Capital Formation (henceforth: GCF). The U.S. GCF average value (as a percentage of GDP) for the period 1990–2019 comes from the OECD (2021) data and is 0.213, so that $\phi =\iota =1-0.213=0.787$ are obtained. Consistently with the value of all the other variables, the parameters obtained with the Shephard’s lemma—i.e. $\phi _{w_{L}}$ and $\iota _{w_{L}}$—should represent the value of the labor employment as a percentage of the GDP. The U.S. labor share average value for the period 1990–2016 comes from the U.S. Bureau of Labor Statistics (2017) and is $\phi _{w_{L}}=\iota _{w_{L}}=0.6$. All these values are used to calculate $\Omega \equiv \frac{\left( \lambda -1\right) \lambda \iota \beta [\phi +\left( 1-\eta \right) \tau _{q}]+\left( 1-\beta \right) \iota \left[ \lambda \left( \phi +\left( 1-\eta \right) \tau _{q}\right) +\eta \tau _{q}\right] }{\sigma B}$ and $B\equiv \beta \phi _{w_{L}}\lambda \iota +\left( 1-\beta \right) \iota _{w_{L}}\left[ \lambda \left( \phi +\left( 1-\eta \right) \tau _{q}\right) +\eta \tau _{q}\right]$.

To calculate $\gamma =\theta _{0}-\frac{\sigma }{\left( w_{H}/w_{L}\right) }$ , the variable $\left( 1-\theta _{0}\right)$ represents the population fraction that becomes skilled. The value $\theta _{0}=0.729$ for the U.S. is obtained from Barro and Jong-Wha (2013) and refers to the average value of the educational attainment of the total population aged 25 and over that has completed tertiary level of education for the period 1990–2019. Consistently with this value, the measure of the skill premium ${\tilde{w}}_{H}=\left( w_{H}/w_{L}\right) =1.73$ comes from Goldin and Katz (2007) as documented in Fig. 1 and Table A1.8—also calculated in Neves et al. (2018)—and refers to the average skill premium for the period 1990–2005. The parameter $\gamma$ is then internally calibrated through the Eq. (6), and is $\gamma =\theta _{0}-\frac{\sigma }{\left( w_{H}/w_{L}\right) }=0.0088$.

The labor coefficient a in the R&D sector is set to 1 for the U.S. as a benchmark parameter. The R&D complexity parameter k for the PEG specification is calibrated internally at k $=0.816$. In the calibration procedure both the labor coefficient $a=1$ and the R&D complexity parameter $k=0.816$ as in Eq. (17) are calculated to render the simulated values of the threshold ability parameter $\theta _{0}$ and the innovation rate I consistent with actual data. In particular, in the baseline calibration the threshold ability value $\theta _{0}$ obtained through the calibration is compared to the average value of the educational attainment of the total population aged 25 and over that has completed tertiary level of education for the period 1990–2019 obtained from Barro and Jong-Wha (2013). The innovation rate obtained through the calibration is compared with average R&D expenditures (as a percentage of GDP) for the period 1990–2019, which is used as a proxy for the innovation rate as in Berman–Bound–Griliches (1994).

In the sensitivity tests represented in Figs. 1a, b, 2a, b, 3a, b and 4a, b, Eqs. (25), (26), (29), and (30) are used to calibrate both the profit flows and the innovation rate of the dirty and clean varieties. The aggregate innovation rate is obtained by weighting the innovation rate of the dirty and clean varieties by their respective share (as a percentage of GDP), $\beta =0.54$ and $\left( 1-\beta \right) =0.46$, in the economy.

To calibrate the centralized economy solution, the value of parameter A in the pollution externality function around 1 allows the calibration results to always hold and a sensitivity analysis for different values of such a parameter is conducted. The parameter values for the weight of polluting emissions $\nu$ and the rate at which pollution recycles itself in the environment $\delta$ are fixed respectively to $\nu =1$ and $\delta =0.1$. In the calibration analysis, the initial population size is set to $N_{0}=1$ . Perturbations of each of these values do not alter the calibration results. To calculate the social optimum values as in “Appendix 3” of the threshold ability parameter ${\hat{\theta }}_{0}$ and of the innovation rate ${\hat{I}}$ equations (73) and (63) are used, and to calculate social optimum values in its stronger version as in “Appendix 4” of the threshold ability parameter ${\hat{\theta }}_{0,s}$ and of the innovation rate ${\hat{I}}_{s}$, Eqs. (85) and (81) are used.

• 2. The population growth rate n is calibrated to allow for fertility differentials between less educated and skilled individuals. Since the model assumes an exogenous and constant population growth rate, endogeneity between education and fertility and mortality rates is not accounted for. To calculate the net fertility rate, the number of births and deaths by educational group in the U.S. is needed. The number of births by educational group is obtained from Hazan and Zaobi (2014). Considering the female population aged 15-49, the authors analyze the cross-sectional relationship between fertility and women’s education in the U.S. for the period 2001–2011 and estimate the total fertility rate (henceforth: TFR) by classifying women into five educational groups: no high school degree, a high school degree, some college, a college degree and an advanced degree. Moreover, to account for the drawback that women are assigned to an educational group according to their educational attainment at the time of the survey, which may differ a great deal from their completed schooling especially for young women who are still in their schooling period, Hazan and Zaobi (2014) estimate the “hybrid fertility rate” as in Shang and Weinberg (2013) combining children born at a specified age and current age-specific-fertility-rates from that age until the end of the fecundity period. Then, to go back to the number of births by educational group from the “hybrid fertility rate” the percentage of female population by educational groups is needed. The available data on the percentage of female population by educational group from the WDI allow the non-cumulative percentage of female population of three educational groups to be calculated for the period 2004–2011. Each woman is counted once and assigned to the educational group according to her highest attended degree. In particular, the percentage of the female population aged 25 and over that attained (not cumulative percentage) no high school diploma ($f_{nh}$), a high school diploma ($f_{h}$), college and more ($f_{c}$) are obtained with values $f_{nh}=0.13$, $f_{nh}=0.564$, $f_{c}=0.3036$ respectively. To be consistent, the percentages of the female population with no school diploma ($f_{nh}=0.13$) and with a high school diploma ($f_{h}=0.564$) are used as weight to obtain the number of births from the “hybrid fertility rate” respectively of the educational groups no high school degree and high school degree of Hazan and Zaobi (2014), and the percentage of female population with college and more ($f_{c}=0.3060$) is used as weight to obtain the number of births of the educational groups some college, college degree and advanced degree of Hazan and Zaobi (2014). At this point, using the standard technique indicated by the U.S. Census Bureau’s (2021), the number of births by educational group are obtained. Data on the number of deaths by the same educational groups (no high school degree, a high school degree, a college degree and more) come from Osario and Prisinzano (2020a, 2020b) and consider total population aged 44–89 for the period 2001–2011. Combining the births and deaths by educational group, the average net “hybrid fertility rate” ($n_{fh}$) allowing for differentials between less educated and skilled households for the period 2001–2011 is calculated and is $n_{fh}=0.0147$ which is used in the baseline calibration, i.e. $n=$ $n_{fh}=0.0147$ is used. Moreover, following the same procedure as above the net fertility rate ($n_{f}$) for the period 2001–2011 is also calculated and is $n_{f}=0.0154$. Using either $n_{f}=0.0154$ or $n_{fh}=0.0147$ does not alter the calibration results. Moreover, the calibration results hold even when the average population growth rate is used. In particular, the data for the population growth rate n comes from the WDI and is the average value for the period 1990–2019, $n=0.01$.

Appendix 4: Social Optimum Weak Sustainability

This Appendix considers the social optimum. To begin with, let describe both the environmental negative externality term in the utility function (2) and the dynamic equation governing the pollution first. As in Weitzman (2003), let us consider the disutility of being forced to live with a stock of pollution at each time $t\ge 0$, which depends on the overall stock of pollution $P\left( t\right)$. In particular, the disutility term in the utility function (2) is $E\left( t\right) =\ln A^{P\left( t\right) }$, with $A>1$. It is assumed that if the level of pollution exceeds an upper bound ${\hat{G}}>0$, the planet melts and all life ceases. We can turn this problem considering the relevant state variable of the economy to be the good of ‘environmental quality’ S rather than the bad of pollution so that $S\left( t\right) ={\hat{G}}-P\left( t\right)$, where $S\left( t\right)$ is the stock of global environmental quality at each time $t\ge 0$ that can be degraded by global polluting emissions. Then, the disutility term becomes $\ln A^{\left( {\hat{G}}-S\left( t\right) \right) }$. The relevant state variable for the social planner is the relative contribution of the country under analysis to the stock of global pollution. In this way, when making its choices, the social planner considers the following dynamic law of the stock of pollution:

$$\begin{aligned} {\dot{P}}\left( t\right) =\nu e_{{\tilde{\omega }}}Q_{{\tilde{\omega }}}\left( t\right) -\delta P\left( t\right) , \end{aligned}$$

(53)

where $e_{{\tilde{\omega }}}Q_{{\tilde{\omega }}}\left( t\right)$ is the aggregate quantity of goods and services that generate the flow polluting emissions $e_{{\tilde{\omega }}}$ per unit of product. The term $\delta$ is the rate at which pollution recycles itself in the environment. Recalling the definition of the environmental quality for the economy at hand, the contribution $S\left( t\right) ={\hat{G}}-P\left( t\right)$, the relevant state-variable differential equation is transformed from (53) to

$$\begin{aligned} {\dot{S}}\left( t\right) =-{\dot{P}}\left( t\right) =-\nu e_{{\tilde{\omega }}}Q_{ {\tilde{\omega }}}\left( t\right) +\delta \left( {\hat{G}}-S\left( t\right) \right) . \end{aligned}$$

(54)

Given this dynamic equation of the stock of environmental quality together with the dynamic equation governing the expected number of innovations, the social planner determines the optimal manufacturing production of both dirty and clean varieties, the optimal threshold ability to accumulate human capital $\theta _{0}$, the stock of sustainable pollution which pins down the negative environmental externality, and the optimal aggregate rate of innovation I. In particular, two alternative scenarios of the social planner are analyzed: the former is similar to the traditional approach of Weitzman (2003) and considers a gradual degradation of the environmental quality over time (this “Appendix 4”), the second one considers no (more) degradation of the environmental quality over time (“Appendix 5”).

The social planner takes as given the stock of both non-renewable and renewable energy because it is assumed that the economy only uses an infinitesimal share of these energy sources available worldwide. We first examine the problem of the static allocation of labor across manufacturing industries, taking the total number of workers as given. Dropping the time argument and taking as given the negative externality from pollution, the social planner considers the entire population at each time, i.e. all the households type $\theta$, and solves:

$$\begin{aligned} \begin{array}{l} Max\,\,\,\int _{0}^{1}u_{\theta }d\theta \\ s.t.\,\,\,N\theta _{0}=\left[ \int _{0}^{\beta }\phi _{w_{L}}q_{{\tilde{\omega }}}d {\tilde{\omega }}+\int _{\beta }^{1}\iota _{w_{L}}q_{\omega }d\omega \right] N, \end{array} \end{aligned}$$

(55)

where $u_{\theta }$ is given in (2). With $\vartheta$ denoting the Lagrange multiplier, the first order conditions for each dirty and clean varieties respectively are:

$$\begin{aligned} \frac{1}{q_{{\tilde{\omega }}}}=\vartheta \phi _{w_{L}}, \end{aligned}$$

(56)

and

$$\begin{aligned} \frac{1}{q_{\omega }}=\vartheta \iota _{w_{L}}. \end{aligned}$$

(57)

Due to the symmetry of both conditions (56) and (57) it follows that both dirty and clean varieties $q_{ {\tilde{\omega }}}$ and $q_{\omega }$ are symmetric for all ${\tilde{\omega }}$ and $\omega$ respectively. The ratio of condition (56) to (57) gives rise to $q_{\omega }=\frac{\phi _{w_{L}}}{ \iota _{w_{L}}}q_{{\tilde{\omega }}}$. Substituting this expression into the static constraint as in (55) gives rise to

$$\begin{aligned} q_{{\tilde{\omega }}}=\frac{\theta _{0}}{\phi _{w_{L}}}, \end{aligned}$$

(58)

and substituting this last condition for $q_{\omega }=\frac{\phi _{w_{L}}}{ \iota _{w_{L}}}q_{{\tilde{\omega }}}$ gives rise to

$$\begin{aligned} q_{\omega }=\frac{\theta _{0}}{\iota _{w_{L}}}. \end{aligned}$$

(59)

Let us turn now to the dynamic optimization problem of the social planner. Since the total employment in manufacturing is taken as given in the static problem, the social planner chooses the optimal threshold ability $\theta _{0}$ to accumulate human capital taking as exogenously given the constant birth rate b, the constant dying rate d, with $b-d=n>0$, and the duration of the training period. In this way, the social planner determines the optimal manufacturing production of both dirty and clean varieties and the stock of pollution which pins down the negative environmental externality. Moreover, because the social planner chooses the optimal threshold ability to accumulate human capital $\theta _{0}$, it determines the aggregate rate of innovation I. The expected number of innovations before time t of each dirty and clean variety equals $J_{{\tilde{\omega }} }\left( t\right) =\int _{0}^{t}I_{{\tilde{\omega }}}\left( s\right) ds$ and $J_{\omega }\left( t\right) =\int _{0}^{t}I_{\omega }\left( s\right) ds$, respectively. Let us define $J=\beta J_{{\tilde{\omega }}}+\left( 1-\beta \right) J_{\omega }$ to be the economy-wide expected number of innovations.

Finally, let us consider the disutility of being forced to live with a stock of pollution $P\left( t\right)$ at each time $t\ge 0$. The relevant state variable for the social planner under consideration is its relative contribution to the stock of global pollution. In this way, the stock of pollution of the social planner under consideration evolves according to the following dynamic law:

$$\begin{aligned} {\dot{P}}\left( t\right) =\nu e_{{\tilde{\omega }}}Q_{{\tilde{\omega }}}\left( t\right) -\delta P\left( t\right) , \end{aligned}$$

(60)

where $e_{{\tilde{\omega }}}Q_{{\tilde{\omega }}}\left( t\right) =e_{{\tilde{\omega }} }\beta q_{{\tilde{\omega }}}N\left( t\right)$ from Eq. (58) are the polluting emissions. The term $\delta$ is the rate at which pollution recycles itself in the environment. Recalling the definition of the environmental quality for the economy at hand $S\left( t\right) ={\hat{G}}-P\left( t\right)$, and dropping the time index, the relevant state-variable differential equation is transformed from (60) to

$$\begin{aligned} {\dot{S}}=-{\dot{P}}=-\nu \left( \frac{e_{{\tilde{\omega }}}\beta \theta _{0}}{\phi _{w_{L}}}\right) N+\delta {\hat{G}}-\delta S. \end{aligned}$$

(61)

These facts together with what we know to be the optimal allocation of resources across varieties, allows us to write the following problem

$$\begin{aligned} Max\int _{0}^{\infty }e^{-\left( \rho -n\right) t}\left[ \begin{array}{l} \left( \ln \lambda \right) J+\ln \theta _{0}+\beta \ln \phi _{w_{L}} \\ \quad +\left( 1-\beta \right) \ln \iota _{w_{L}}+S\ln A-{\hat{G}}\ln A \end{array} \right] dt. \end{aligned}$$

(62)

We maximize (62) subject to the non-negativity constraint $I\ge 0$, the resource constraint

$$\begin{aligned} \begin{array}{l} N\theta _{0}+\left( \theta _{0}+1-2\gamma \right) \left( 1-\theta _{0}\right) \frac{\psi }{2}N \\ \quad =\left( \beta \phi _{w_{L}}q_{{\tilde{\omega }}}+\left( 1-\beta \right) \iota _{w_{L}}q_{\omega }\right) N+a\left( \beta I_{{\tilde{\omega }}}X_{{\tilde{\omega }}}+\left( 1-\beta \right) I_{\omega }X_{\omega }\right) , \end{array} \end{aligned}$$

(63)

the dynamic equations

$$\begin{aligned} {\dot{J}}&= I_{,} \end{aligned}$$

(64)

$$\begin{aligned} {\dot{S}}&= -\nu \frac{e_{{\tilde{\omega }}}\beta \theta _{0}}{\phi _{w_{L}}} N+\delta {\hat{G}}-{\bar{\delta }}S. \end{aligned}$$

(65)

and the inequality constraint for the threshold ability parameter $\theta _{0}$

$$\begin{aligned} \gamma \le \theta _{0}\le 1. \end{aligned}$$

(66)

Considering Eqs. (58), (59) and $X_{{\tilde{\omega }}}=X_{\omega }=kN$, the resource constraint is:

$$\begin{aligned} \left( \theta _{0}+1-2\gamma \right) \left( 1-\theta _{0}\right) \frac{\psi }{2ak}=I. \end{aligned}$$

(67)

Dropping constant and given terms and substituting the resource constraint for $I=\beta I_{{\tilde{\omega }}}+\left( 1-\beta \right) I_{\omega }$ in (64), the current value Hamiltonian is

$$\begin{aligned} H_{cv}&= \left( \ln \lambda \right) J+\ln \theta _{0}+S_{i}\ln A \nonumber \\&\quad +\mu \left[ \left( \theta _{0}+1-2\gamma \right) \left( 1-\theta _{0}\right) \frac{\psi }{2ak}\right] \nonumber \\& \quad+\mu _{p}\left[ -\nu \frac{e_{{\tilde{\omega }}}\beta \theta _{0}}{\phi _{w_{L}} }N+\delta {\hat{G}}-\delta S\right] +\mu _{1}\left[ \theta _{0}-\gamma \right] +\mu _{2}\left[ 1-\theta _{0}\right] , \end{aligned}$$

(68)

where $\mu$ and $\mu _{p}$ are the costate variables, and $\mu _{1}$ and $\mu _{2}$ are the Lagrange multipliers for the inequality constraint (66). Using Pontryagin’s maximum principle we derive the necessary and sufficient conditions for a maximum that apply whenever the non-negativity constraint $I\ge 0$ does not bind:

$$\begin{aligned}&\frac{1}{\theta _{0}}+\mu \frac{\psi }{ak}\left( \gamma -\theta _{0}\right) -\mu _{p}\frac{e_{{\tilde{\omega }}}v\beta }{\phi _{w_{L}}}N+\mu _{1}-\mu _{2}=0, \end{aligned}$$

(69)

$$\begin{aligned}&\ln \lambda =\left( \rho -n\right) \mu -{\dot{\mu }}, \end{aligned}$$

(70)

$$\begin{aligned}&\ln A-\delta \mu _{p}=\left( \rho -n\right) \mu _{p}-{\dot{\mu }}_{p}, \end{aligned}$$

(71)

the transversality conditions $\lim _{t\rightarrow \infty } e^{-\left( \rho -n\right) t}\mu J=0$ and $\lim _{t\rightarrow \infty }e^{-\left( \rho -n\right) t}\mu _{p}S=0$, and the slackness conditions $\mu _{1}\left( \theta _{0}-\gamma \right) =0$, $\mu _{2}\left( 1-\theta _{0}\right) =0$. The solutions to (70) and (71) together with the appropriate transversality condition require that $\mu = \frac{\ln \lambda }{\rho -n}>0$ for all t, and $\mu _{p}=\frac{\ln A}{\rho -n+\delta }>0$ for all t, where $\rho >n$ is assumed. Let us consider the optimal condition (69). The term $\mu _{p}\frac{v\beta }{ \phi _{w_{L}}}N$ grows at the population growth rate n. To get a solution to the above problem let us start at time zero, $t=0$. Let us suppose that at the very beginning of the period $\gamma<\theta _{0}<1$, so that the optimal condition (69) is

$$\begin{aligned} \frac{1}{\theta _{0}}+\mu \frac{\psi }{ak}\left( \gamma -\theta _{0}\right) -\mu _{p}\frac{e_{{\tilde{\omega }}}v\beta }{\phi _{w_{L}}}N_{0}=0. \end{aligned}$$

(72)

Substituting the value of both $\mu$ and $\mu _{p}$ into (69) we find

$$\begin{aligned} \mu \frac{\psi }{ak}\theta _{0}^{2}+\left( \mu _{p}\frac{ve_{{\tilde{\omega }} }\beta N_{0}}{\phi _{w_{L}}}-\mu \frac{\psi \gamma }{ak}\right) \theta _{0}-1=0. \end{aligned}$$

(73)

Let us define $\Delta =\mu \frac{\psi \gamma }{ak}-\mu _{p}\frac{ve_{{\tilde{\omega }}}\beta N_{0}}{\phi _{w_{L}}}$, Eq. (73) is a strictly convex quadratic polynomial in $\theta _{0}$ with two real roots, one negative and one positive, where the positive root is:

$$\begin{aligned} {\hat{\theta }}_{0}=\frac{-\Delta +\sqrt{\Delta ^{2}+4\mu \frac{\psi }{ak}}}{ 2\mu \frac{\psi }{ak}}. \end{aligned}$$

(74)

Some tedious calculations shows that ${\hat{\theta }}_{0}>\gamma$ always holds, and ${\hat{\theta }}_{0}<1$ when $\frac{\mu \psi }{ak}\left( 1+\Delta \right) -1>0$ holds. Therefore, at time $t=0$ an optimal threshold ability parameter ${\hat{\theta }}_{0}\in \left( \gamma ,1\right)$ exists. When time goes on, and starting with the optimal threshold ability ${\hat{\theta }} _{0}\in \left( \gamma ,1\right)$, the optimal condition (69) can be written as

$$\begin{aligned} \frac{1}{\theta _{0}}+\mu \frac{\psi }{ak}\left( \gamma -\theta _{0}\right) =\mu _{p}\frac{ve_{{\tilde{\omega }}}\beta }{\phi _{w_{L}}}N_{0}e^{nt}. \end{aligned}$$

(75)

The term on the right hand side grows at the population growth rate, so that the right hand side also has to grow. This can only happen if the threshold ability $\theta _{0}$ decreases over time, indeed the left hand side of the optimal condition (75) strictly increases in $\theta _{0}$. Therefore, as time goes on the optimal threshold ability decreases until it reaches its greatest lower bound $\theta _{0}=\gamma$. When the optimal threshold ability reaches this value, it can not further decrease and the Lagrange multipliers are $\mu _{2}=0$ and $\mu _{1}=\mu _{p} \frac{ve_{{\tilde{\omega }}}\beta }{\phi _{w_{L}}}N_{0}e^{nt}-\frac{1}{\gamma } >0$, where $\mu _{1}$ represents the shadow price of the environmental negative externality the economy pays to get the constant threshold ability parameter $\theta _{0}=\gamma$ over time. □

Appendix 5: Social Optimum Strong Sustainability

The problem of the social planner can be rewritten considering a ‘stronger sustainability’ approach which consists in a constant level of the stock of pollution, i.e. ${\dot{P}}\left( t\right) =0$. Whenever the effective recycling term $\delta$ is positive, this sustainability approach does not imply zero emissions. Indeed, the rate at which pollution recycles itself in the environment $\delta$ allows the polluting emissions to be recycled at each time $t>0$. ${\dot{P}}\left( t\right) =0$, and dropping the time index, implies that the relevant state-variable differential equation is transformed from (60) to

$$\begin{aligned} {\dot{S}}_{i}=-{\dot{P}}_{i}=0\Rightarrow S_{i}=-\frac{\nu }{\delta } \left( \frac{e_{{\tilde{\omega }}}\beta \theta _{0}}{\phi _{w_{L}}}\right) N+ {\hat{G}}. \end{aligned}$$

(76)

These facts together with what we know to be the optimal allocation of resources across varieties allow us to write the following problem

$$\begin{aligned} Max\int _{0}^{\infty }e^{-\left( \rho -n\right) t}\left[ \begin{array}{l} \left( \ln \lambda \right) J+\ln \theta _{0}+\beta \ln \phi _{w_{L}} \\ \quad +\left( 1-\beta \right) \ln \iota _{w_{L}}+S\ln A-{\hat{G}}\ln A \end{array} \right] dt. \end{aligned}$$

(77)

We maximize (77) subject to the non-negativity constraint $I\ge 0$, the resource constraint

$$\begin{aligned} \begin{array}{l} N\theta _{0}+\left( \theta _{0}+1-2\gamma \right) \left( 1-\theta _{0}\right) \frac{\psi }{2}N \\ \quad =\left( \beta \phi _{w_{L}}q_{{\tilde{\omega }}}+\left( 1-\beta \right) \iota _{w_{L}}q_{\omega }\right) N+a\left( \beta I_{{\tilde{\omega }}}X_{{\tilde{\omega }}}+\left( 1-\beta \right) I_{\omega }X_{\omega }\right) , \end{array} \end{aligned}$$

(78)

and the dynamic equation

$$\begin{aligned} {\dot{J}}=I_{,} \end{aligned}$$

(79)

and the inequality constraint for the threshold ability parameter $\theta _{0}$

$$\begin{aligned} \gamma \le \theta _{0}\le 1. \end{aligned}$$

(80)

Considering Eqs. (58), (59) and $X_{{\tilde{\omega }}}=X_{\omega }=kN$, the resource constraint is:

$$\begin{aligned} \left( \theta _{0}+1-2\gamma \right) \left( 1-\theta _{0}\right) \frac{\psi }{2ak}=I. \end{aligned}$$

(81)

Dropping constant terms and substituting the resource constraint for $I=\beta I_{{\tilde{\omega }}}+\left( 1-\beta \right) I_{\omega }$ in (79), the current value Hamiltonian is

$$\begin{aligned} H_{cv}&= \left( \ln \lambda \right) J+\ln \theta _{0}+\left( -\frac{\nu }{ \delta }\frac{e_{{\tilde{\omega }}}\beta \theta _{0}}{\phi _{w_{L}}}N+{\hat{G}} \right) \ln A \nonumber \\& \quad +\mu _{s}\left[ \left( \theta _{0}+1-2\gamma \right) \left( 1-\theta _{0}\right) \frac{\psi }{2ak}\right] +\mu _{1,s}\left[ \theta _{0}-\gamma \right] +\mu _{2,s}\left[ 1-\theta _{0}\right] , \end{aligned}$$

(82)

where $\mu _{s}$ is the costate variables, and $\mu _{1,s}$ and $\mu _{2,s}$ are the Lagrange multipliers for the inequality constraint (80). Using Pontryagin’s maximum principle we derive the necessary and sufficient conditions for a maximum that apply whenever the non-negativity constraint $I\ge 0$ does not bind:

$$\begin{aligned}&\frac{1}{\theta _{0}}+\mu _{s}\frac{\psi }{ak}\left( \gamma -\theta _{0}\right) -\frac{\nu }{\delta }\frac{e_{{\tilde{\omega }}}\beta N}{\phi _{w_{L}}}\ln A=0, \end{aligned}$$

(83)

$$\begin{aligned}&\ln \lambda =\left( \rho -n\right) \mu _{s}-{\dot{\mu }}_{s}, \end{aligned}$$

(84)

and the transversality condition $\lim _{t\rightarrow \infty } e^{-\left( \rho -n\right) t}\mu _{s}\left( t\right) J\left( t\right) =0$. The solutions to (84) together with the transversality condition require that $\mu _{s}=\frac{\ln \lambda }{\rho -n}>0$ for all t . Repeating the same reasoning lines as the optimal social problem in “Appendix 3”, and substituting the value of $\mu _{s}$ in (83) we find:

$$\begin{aligned} \mu _{s}\frac{\psi }{ak}\theta _{0}^{2}+\left( \frac{\nu }{\delta }\frac{e_{ {\tilde{\omega }}}\beta N_{0}}{\phi _{w_{L}}}\ln A-\mu _{s}\frac{\psi \gamma }{ ak}\right) \theta _{0}-1=0. \end{aligned}$$

(85)

Let us define $\Delta _{s}=\mu _{s}\frac{\psi \gamma }{ak}-\frac{\nu }{ \delta }\frac{e_{{\tilde{\omega }}}\beta N_{0}}{\phi _{w_{L}}}\ln A$, Eq. (85) is a strictly convex quadratic polynomial in $\theta _{0}$ with two real roots, one negative and one positive, where the positive root is:

$$\begin{aligned} {\hat{\theta }}_{0,s}=\frac{-\Delta _{s}+\sqrt{\Delta _{s}^{2}+4\mu _{s}\frac{ \psi }{ak}}}{2\mu _{s}\frac{\psi }{ak}}. \end{aligned}$$

(86)

Some tedious calculations shows that ${\hat{\theta }}_{0,s}>\gamma$ always holds, and ${\hat{\theta }}_{0,s}<1$ when $\frac{\mu _{s}\psi }{ak}\left( 1+\Delta _{s}\right) -1>0$ holds.

Therefore, at time $t=0$ an optimal threshold ability parameter ${\hat{\theta }} _{0,s}\in \left( \gamma ,1\right)$ exists. When time goes on, and starting with the optimal threshold ability ${\hat{\theta }}_{0,s}\in \left( \gamma ,1\right)$, the optimal condition (83) can be written as

$$\begin{aligned} \frac{1}{\theta _{0}}+\mu _{s}\frac{\psi }{ak}\left( \gamma -\theta _{0}\right) =\frac{\nu }{\delta }\frac{e_{{\tilde{\omega }}}\beta }{\phi _{w_{L}}}\left( N_{0}e^{nt}\right) \ln A. \end{aligned}$$

(87)

The term on the right hand side grows at the population growth rate, so that the right hand side also has to grow. This can only happen if the threshold ability $\theta _{0}$ decreases over time, indeed the left hand side of the optimal condition (87) strictly increases in $\theta _{0}$. Therefore, as time goes on the optimal threshold ability decreases until it reaches its greatest lower bound $\theta _{0}=\gamma$. When the optimal threshold ability reaches this value, it can not further decrease and the Lagrange multipliers are $\mu _{2,s}=0$ and $\mu _{1,s}= \frac{\nu }{\delta }\frac{e_{{\tilde{\omega }}}\beta }{\phi _{w_{L}}}\left( N_{0}e^{nt}\right) \ln A-\frac{1}{\gamma }>0$, where $\mu _{1,s}$ represents the shadow price of the environmental negative externality the economy pays to get the constant threshold ability parameter $\theta _{0}=\gamma$ over time. Note that the shadow price to get a threshold ability at its minimum value $\gamma$ is lower with soft than strong sustainability, i.e. $\mu _{1}<\mu _{1,s}$ because $\frac{\ln A}{\rho -n+\delta }<\frac{\ln A}{\delta }$. □

Appendix 6: Extensions

This appendix shows the extension to a variety proliferation model set-up. In the simplest case of an exogenous variety proliferation set-up, the mass of both dirty and clean varieties is assumed to change over time in proportion to population size, i.e., ${\dot{d}}\left( t\right) =\alpha _{d,v}N\left( t\right)$, ${\dot{\kappa }}\left( t\right) =\alpha _{c,v}N\left( t\right)$, where $\alpha _{d,v}$ and $\alpha _{c,v}$ are productivity parameters of variety proliferation of dirty and clean varieties, respectively. In such a case, the mass of both dirty and clean varieties grow at the population growth rate, i.e., $\frac{{\dot{d}}\left( t\right) }{ d\left( t\right) }=\frac{{\dot{\kappa }}\left( t\right) }{\kappa \left( t\right) }=n$, and along the BGP equilibrium a constant mass of per capita varieties is pinned down: ${\bar{z}}_{d}=\frac{d\left( t\right) }{N\left( t\right) }=\frac{\alpha _{d,v}}{n}$ and ${\bar{z}}_{c}=\frac{\kappa \left( t\right) }{N\left( t\right) }=\frac{\alpha _{c,v}}{n}$, respectively. Along the BGP equilibrium with variety proliferation, the mass of both dirty and clean varieties is constant with respect to the total mass of varieties, i.e. $\beta _{d}=\frac{{\bar{z}}_{d}N\left( t\right) }{{\bar{z}}_{d}N\left( t\right) +{\bar{z}}_{c}N\left( t\right) }$and $\beta _{c}=\frac{{\bar{z}} _{c}N\left( t\right) }{{\bar{z}}_{d}N\left( t\right) +{\bar{z}}_{c}N\left( t\right) }$, so that we can normalize the per capita total mass of varieties to one—${\bar{z}}_{d}+{\bar{z}}_{c}=1$—and we can write $\beta _{d}=\beta$ and $\beta _{c}=1-\beta$ to be the shares of dirty and clean varieties in the economy, respectively. □

Appendix 7: Tables 6 and 7

Table 6 Sensitivity test for ERT $\tau _{q}$ and tax share $\eta$ : mark-up $\lambda =1.55$

Table 7 Sensitivity test for ERT $\tau _{q}$ and tax share $\eta$: mark-up $\lambda =1.05$