A Two-Sector Model of Economic Growth with Endogenous Technical Change and Pollution Abatement

Abstract

We provide insights into the relationships between technological development, economic growth, and pollution accumulation using a two-sector model of economic growth with endogenous technical change. In the model, output is produced using a polluting resource. Production can be used for either consumption or abatement of pollution. Scientists can be allocated between two research activities: resource-saving and abatement-augmenting technologies. Our results indicate conditional path dependency. Specifically, when the innovative capacity in the resource-saving research sector is sufficiently high, scientists are allocated to improve only the resource-saving technology, independently of the state of the technologies and environment. Consequently, the allocation of researchers is path-independent. When the innovative capacity in the abatement-augmenting research sector is sufficiently high, the optimal allocation of researchers depends on the initial level of the pollution stock or technologies but eventually will be directed to improve the abatement technology. We further characterize the optimal steady-state and off-steady-state dynamics and show that green growth is always socially optimal. By using a two-sector model, we address a lack of attention to multi-sector growth models in neoclassical growth theory and show that distinct results and transitional dynamics can emerge.

Appendices

Appendix A: Proof of Theorem 1

Proof

The transversality condition (TC) states that $\lim _{t\rightarrow \infty }e^{-\rho t}\lambda \left (t\right ) Z\left (t\right ) = 0$ which is equivalent to − ρ + 𝜃_λ + 𝜃_Z < 0 being evaluated at the limit $t\rightarrow \infty$.

From Eq. 3.21, the TC can be restated as

$$ \begin{array}{@{}rcl@{}} \frac{n}{Z}-\frac{Z^{\nu }}{\lambda }<0\text{.} \end{array} $$

However, the solution to $\dot {Z}=n-\delta Z$ is given by

$$ \begin{array}{@{}rcl@{}} Z=\frac{\int e^{\delta t}ndt+Z_{0}}{e^{\delta t}}. \end{array} $$

This implies

$$ \begin{array}{@{}rcl@{}} \frac{n}{Z}=\frac{e^{\delta t}n}{\int e^{\delta t}ndt+Z_{0}}. \end{array} $$

Deduce that n/Z is the growth rate of

$$ \begin{array}{@{}rcl@{}} \int e^{\delta t}ndt+Z_{0}, \end{array} $$

which is bounded from below by δ if n does not converge to zero.

Likewise, the solution to $\dot {\lambda }=\left (\rho +\delta \right ) \lambda -Z^{\nu }$ is

$$ \begin{array}{@{}rcl@{}} \lambda =\frac{-\int e^{-\left( \rho +\delta \right) t}Z^{\nu }dt+\lambda_{0}}{e^{-\left( \rho +\delta \right) t}} \end{array} $$

implying

$$ \begin{array}{@{}rcl@{}} \frac{Z^{\nu }}{\lambda }=\frac{e^{-\left( \rho +\delta \right) t}Z^{\nu }}{ \lambda_{0}-\int e^{-\left( \rho +\delta \right) t}Z^{\nu }dt}. \end{array} $$

It follows that − Z^ν/λ is the growth rate of

$$ \lambda_{0}-\int e^{-\left( \rho +\delta \right) t}Z^{\nu }\text{.} $$

Nevertheless, λ ≥ 0 implies

$$ \lambda_{0}\geq \int e^{-\left( \rho +\delta \right) t}Z^{\nu }\text{.} $$

(A.1)

Note that $e^{-\left (\rho +\delta \right ) t}Z^{\nu }$ is zero or negative in the limit. Otherwise, $\int e^{-\left (\rho +\delta \right ) t}Z^{\nu }$ would diverge to infinity violating Eq. (A.1). Thus, Z^ν/λ converges to zero, which implies that

$$ \begin{array}{@{}rcl@{}} \frac{n}{Z}-\frac{Z^{\nu }}{\lambda } \end{array} $$

(A.2)

converges to δ, thus contradicting the TC. Then, $n\rightarrow 0$.

Recall that $\sigma \equiv \frac {x}{y}=\alpha \beta \frac {q}{z}$. As n ≡ z − q = z(1 − q/z), if $n\rightarrow 0$, then $q/z\rightarrow 1$ as z > 0 for all t. On the basis of this result, we conclude that $\sigma \rightarrow \alpha \beta$. □

Appendix B: Proof of Proposition 1

Proof

The optimal research policy at time t depends on the marginal social value of doing research in the two R&D sectors. Multiplying both sides of Eqs. 3.10 and 3.11 by e^−ρtA and e^−ρtB, respectively, and rearranging yields

$$ \begin{array}{@{}rcl@{}} \dot{(e^{-\rho t} \lambda_{A} A)}&=&-e^{-\rho t}p A z^{\alpha} \ \text{ and}\\ \dot{(e^{-\rho t} \lambda_{B} B)}&=&-e^{-\rho t}{\mu} B x^{\beta}. \end{array} $$

Time integrating these two equations and taking into account the transversality conditions for the technology indexes A and B given by Eq. 3.12 allows us to write the sectoral (social) wages as

$$ \begin{array}{@{}rcl@{}} \omega_{A}(t) \equiv a\lambda_{A}(t) A(t) &=& {\int}^{\infty}_{t} e^{-\rho(\tau- t)}a p(\tau)y(\tau)d\tau, \end{array} $$

(B.1)

$$ \begin{array}{@{}rcl@{}} \omega_{B}(t) \equiv b\lambda_{B}(t) B(t) &=& {\int}^{\infty}_{t} e^{-\rho(\tau- t)}b {\mu}(\tau)q(\tau)d\tau, \end{array} $$

(B.2)

where ω_A(t) and ω_B(t) are the social wages in the resource-saving and abatement-augmenting research sectors, respectively. If ω_A(t) > ω_B(t), all resources for research will be allocated to improve the resource-saving technology at time t. To determine this, we write ω_A(t) − ω_B(t) using Eq. 3.5 as

$$ \omega_{A}(t)-\omega_{B}(t) =\!{\int}^{\infty}_{t} e^{-\rho(\tau- t)}\frac{\beta{\mu}(\tau)q(\tau)}{\sigma(\tau)} \left( \! a - \frac{b \sigma(\tau)}{\beta}\!\right) d\tau. $$

(B.3)

Because the FA constraint necessitates that σ ≤ αβ, if a > αb, then ω_A(t) − ω_B(t) > 0 for all t. □

Appendix C: Proof of Proposition 2

Proof

As, by hypothesis, there is FA over $[\bar {t},\infty )$, the pollution stock will shrink at the constant rate of δ; that is, 𝜃_Z = −δ. Integrating the equation of motion for the social cost of pollution (i.e., Eq. 3.9) over $[\bar {t},\infty )$ yields

$$ {\lambda}(t) = e^{(\rho+\delta)(t-\bar{t})}\left[\lambda(\bar{t})-Z(\bar{t})^{\nu} \frac{1-e^{-(\rho+(1+\nu)\delta)(t-\bar{t})}}{\rho+\delta(1+\nu)}\right]~\hspace{2.2mm} $$

(C.1)

When FA is practiced from $\bar {t}$ onwards, Eq. C.1 yields the time path for the cost of pollution. Taking into account the transversality condition $\underset {{\small {t \rightarrow \infty }}}{\lim } e^{-\rho t}{\lambda }(t)Z(t)= 0$, the social cost of pollution over $[\bar {t},\infty )$ simplifies to

$$ \begin{array}{@{}rcl@{}} {\lambda}(t)=\frac{Z(\bar{t})^{\nu} e^{-\nu \delta(t-\bar{t})}}{\rho+\delta(1+\nu)}, \end{array} $$

(C.2)

whose growth rate is 𝜃_λ = −νδ.

As for $\theta _{\bar {\mu }}$, note that

$$ \begin{array}{@{}rcl@{}} M(A(t),B_{0})& =& {\alpha}{}({\alpha\beta}{})^{-\frac{1-\alpha+\alpha\epsilon}{1-\alpha\beta}} \\ &&\left( A(t)^{1-\beta-\epsilon}B_{0}^{-(1-\alpha+\alpha\epsilon)}\right)^{\frac{1}{1-\alpha\beta}}, \end{array} $$

whose growth rate is

$$ \begin{array}{@{}rcl@{}} a{(1-\beta -\epsilon)}/{(1-\alpha\beta)}. \end{array} $$

By definition, $\bar {\mu }(t) \equiv M(A(t),B_{0})\ \bar {\varphi }_{\lambda }$. Thus, the growth rate of $\bar {\mu }$ is equal to that of M(A(t),B₀). This completes the proof. □

Appendix D: Proof of Proposition 3

Proof

If $\bar {\mu }(0)> {\lambda }(0)$, then n(0) > 0, that is, PA is optimal at time t = 0. However, − δν > a(1 − β − 𝜖)/(1 − αβ) implies $\theta _{\bar {\mu }} < \theta _{\bar {\lambda }}$, where $\bar {\lambda }$ denotes the social cost of pollution when the pollution flow is fully abated. Accordingly, the growth rate of $\bar {\lambda }$ remains higher than that of $\bar {\mu }$ at all times. Thus, after a certain point in time, that is, when $t=\bar {t}>0$, $\bar {\mu }(\bar {t}) = {\lambda }(\bar {t})$. Consequently, n(t) = 0 for $t\geq \bar {t}$. □

Appendix E: Proof of Corollary 1

By hypothesis, αb > a and 𝜖 > 1. These two inequalities imply that ${\bar {\theta }_{\lambda }^{A}>\bar {\theta }_{\lambda }^{B}}$, where $\bar {\theta }_{\lambda }^{B} \equiv -b(1-\alpha + \alpha \epsilon )/(1-\alpha \beta )$. During FA, the net opportunity cost of pollution shrinks at the rate of $\bar {\theta }_{\lambda }^{A}s_{A}+\bar {\theta }_{\lambda }^{B} s_{B}$. Since $\bar {\theta }_{\lambda }^{A} >\bar {\theta }_{\lambda }^{B}$, any convex combination of $\bar {\theta }_{\lambda }^{A}$ and $\bar {\theta }_{\lambda }^{B}$ is smaller than $\bar {\theta }_{\lambda }^{A}$. Hence, when $\bar {\mu }(\bar {t})\leq \lambda (\bar {t})$ and considering that research can take place in both sectors, $\bar {\theta }_{\lambda }^{A}s_{A}+\bar {\theta }_{\lambda }^{B} s_{B}(\leq \bar {\theta }_{\lambda }^{A})$, the net opportunity cost of pollution always stays lower than the social cost of pollution if $-\delta \nu >\bar {\theta }_{\lambda }^{A}$. Consequently, $-\delta \nu >\bar {\theta }_{\lambda }^{A}$ provides a sufficient condition for FA to be pursued permanently from $\bar {t}$ onward.

Appendix F: Proof of Corollary 2

Proof

Given that $-\delta \nu > \bar {\theta }^{A}_{\lambda } > \bar {\theta }^{B}_{\lambda }$ and, therefore, $\theta _{\bar {\mu }} < \theta _{{\bar {\lambda }}}$, the marginal cost of fully abating the pollution will always be lower than its social cost from time $\bar {t}$ onward; that is, $\bar {\mu }(t)\leq {\lambda }(t)$ and, in turn, σ(t) = αβ for $t\geq \bar {t}$. The difference between the social wages in the two research sectors during the FA regime can be written as

$$ \begin{array}{@{}rcl@{}} \omega_{B}(\bar{t}) -\omega_{A}(\bar{t}) ={\int}^{\infty}_{\bar{t}} e^{-\rho(\tau- \bar{t})}\frac{\beta{\mu}(\tau)q(\tau)}{\alpha \beta} \left( {\alpha b }- a\right) d \tau, \end{array} $$

where we substitute αβ for σ in Eq. B.3. Since αb > a, the social wage in the abatement-augmenting research sector is higher for $t\in [\bar {t},\infty )$.

Recall that $\bar {\mu }(t)\equiv M\left (A(t),B(t)\right )\bar {\varphi }_{\lambda }$ and ${\lambda }({t})= {Z(t)^{\nu } e^{-\nu \delta (t-\bar {t})}}/{\left (\rho +\delta (1+\nu )\right )}$. If $M\left (A(0),B(0)\right )\bar {\varphi }_{\lambda } \leq {Z(0)^{\nu }}/{\left (\rho +\delta (1+\nu )\right )}$ and $-\delta \nu >\bar {\theta }_{\lambda }^{A}$, the pollution flow is fully abated, and all scientists are directed at the B sector from t = 0 onward. □

Appendix G: Proof of Corollary 3

Proof

When the net opportunity cost of pollution is bigger than the social cost of pollution, that is, $\bar {\mu }(0)> {\lambda }(0)$, then PA is optimal at time t = 0. Thus, n(0) > 0. However, $-\delta \nu >\bar {\theta }_{\lambda }^{A} \geq \bar {\theta }_{\lambda }^{A}s_{A}+\bar {\theta }_{\lambda }^{B} s_{B}$ (see Appendix E for details) implies $\theta _{\bar {\mu }} < \theta _{\bar {\lambda }}$. Accordingly, the growth rate of $\bar {\lambda }$ remains higher than that of $\bar {\mu }$ at all times. Thus, after a certain point in time, that is, when $t=\bar {t}>0$, the net opportunity cost of pollution gets equal to the social cost of pollution $\bar {\mu }(\bar {t}) = {\lambda }(\bar {t})$. Consequently, FA is pursued permanently for $t \geq \bar {t}$. □

Appendix H: Proof of Proposition 4

Proof

From Eq. 3.30, 𝜃_σ > 0 until FA is attained. Since 𝜃_σ > 0, it is optimal to allocate all resources to the abatement-augmenting research for $t>\tilde {t}$, where $\sigma (\tilde {t})=a \beta /b$. Thus, ω_A(t) − ω_B(t) for $t>\tilde {t}$, where $\omega _{A}(t) - \omega _{B}(t)\vert _{t>\tilde {t}}$ equals the finite sum

$$ \begin{array}{@{}rcl@{}} &&{\int}^{\bar{t}}_{\tilde{t}} e^{-\rho(\tau- t)}\frac{\beta{\mu}(\tau)q(\tau)}{\sigma(\tau)} \left( a-\frac{b \sigma(\tau)}{\beta}\right) d\tau \\ &&-\frac{(\alpha b - a)A(\bar{t})^{\frac{2-\beta - \epsilon}{\alpha\beta}}B(\bar{t})^{-\frac{\alpha(\epsilon-1)}{1-\alpha\beta}}}{\left( (\alpha\beta)^{\frac{\alpha\beta}{1-\alpha\beta}}(1 - \alpha\beta)\right)^{\epsilon-1}(\rho(1 - \alpha\beta)+\alpha b(\epsilon-1))}\!<\!0. \end{array} $$

Depending on the initial conditions (A(0),Z(0),B(0)) (see Appendix J), the intersection of d𝜃_λ/d𝜃_Z with the optimal trajectory defines the optimal $\left (\theta _{\lambda }(0),\theta _{Z}(0)\right )$. Then, given the initial values for the state variables, λ(0) and σ(0) can be determined from $\lambda (0) = Z(0)^{\nu }/\left (\rho + \delta - \theta _{\lambda }(0)\right )$ and

$$ \frac{\sigma(0)^{-{(1-\beta)\bar{\theta}_{\lambda}^{B}/b}}}{(1-\sigma(0))^{\epsilon}} = \frac{\lambda(0)}{M(A(0),B(0))}, $$

respectively.

Depending on the initial conditions, if σ(0) is sufficiently low such that a − bσ(0)/β > 0 and ω_A(0) − ω_B(0)|_t> 0 > 0, then s_A(0) = 1. Because $\omega _{A}(t) - \omega _{B}(t)\vert _{t>\tilde {t}}<0$, an abatement-augmenting research regime has to begin before $\hat {t}(<\tilde {t})$.

A mixed-research regime, in which resources are allocated to both research sectors, cannot be maintained because 𝜃_σ > 0 (cf. Eq. 3.30) until the time when $\sigma (\bar {t})=\alpha \beta$. The necessary condition for a mixed-research regime at time $\hat {t}$ is $\omega _{A}(t)-\omega _{B}(t)= 0\vert _{t=\hat {t}}$. Sustaining this regime necessitates

$$ \frac{d\left( \omega_{A}(t)-\omega_{B}(t)\right)}{d t}\vert_{t=\hat{t}} = 0, $$

and consequently, σ = aβ/b. Nevertheless, because 𝜃_σ > 0, the economy switches from one research regime to the other one.

In Appendix L, we conduct further analysis and show that an interior solution for scientific activity that can permanently be sustained cannot be part of an optimal solution. □

Appendix I: Economic Dynamics During a PA Phase

Using Eqs. 3.27, 3.28, and 3.30, the difference between the output growth rates in the two regimes can be calculated to yield

$$ {\theta^{A}_{y}} - {\theta^{B}_{y}} = \frac{a -\alpha b}{1-\alpha \beta }- \frac{\alpha (1-\beta)}{(1-\alpha \beta) K_{\lambda}(\sigma) }\left( \bar{\theta}_{\lambda}^{B} -\bar{\theta}_{\lambda}^{A} \right) $$

where ${\theta ^{i}_{j}}$ denotes the growth rate of variable j in research regime i ∈{A,B}. One can calculate that

$$ \begin{array}{@{}rcl@{}} &{\theta^{A}_{y}} - {\theta^{B}_{y}} >0 ~&\text{ if } \hspace{3mm}\sigma< \frac{(1-\beta)a }{a(1-\beta)+ (\alpha b - a)}, \\ &{\theta^{A}_{y}} - {\theta^{B}_{y}} \leq 0 ~&\text{ otherwise}. \end{array} $$

(I.1)

When the switch to the abatement-augmenting research regime happens for a sufficiently small share of abatement expenditure, the economic growth during the transition to the FA regime would be smaller than the one under a resource-saving research regime.

Furthermore, the growth rate of the abatement expenditure is always higher in an abatement-augmenting research regime:

$$ {\theta^{A}_{x}} - {\theta^{B}_{x}} = \frac{a -\alpha b}{1-\alpha \beta } + \frac{1-\alpha}{(1-\alpha \beta) K_{\lambda}(\sigma) }\left( \bar{\theta}_{\lambda}^{B} -\bar{\theta}_{\lambda}^{A} \right)<0. $$

(I.2)

Accordingly, the growth rate of consumption is smaller:

$$ {\theta^{A}_{c}} - {\theta^{B}_{c}} = \frac{-(1-\alpha\beta)\left[(1-\beta)a +(1-\alpha) b \sigma\right]}{(1-\beta)\bar{\theta}_{\lambda}^{B}/b +(1-\alpha)\bar{\theta}_{\lambda}^{A}/a} . $$

(I.3)

Appendix J: Initial Conditions

First, the dynamical system given by Eq. 3.21 can equivalently be written as

$$ \begin{array}{@{}rcl@{}} \rho+\delta - \theta_{\lambda}& =& \frac{Z^{\nu}}{{M(A,B_{0})\varphi_{\lambda}(\sigma)}}, \\ \theta_{Z} + \delta &=& \frac{N(A,B_{0})\varphi_{Z}(\sigma)}{Z}. \end{array} $$

(J.1)

Using Eqs. 3.16, 3.17, and 3.21, the derivatives of 𝜃_λ and 𝜃_Z with respect to σ yield

$$ \begin{array}{@{}rcl@{}} \frac{d\theta_{\lambda}}{d\sigma}&=& \frac{(\rho+\delta - \theta_{\lambda}) K_{\lambda}(\sigma)}{\sigma}, \\ \frac{d\theta_{Z}}{d\sigma}&=& -\frac{(\theta_{Z} + \delta) K_{Z}(\sigma)}{\sigma}. \end{array} $$

Eliminating σ, Eq. J.1 implicitly defines 𝜃_λ(𝜃_Z; A,B,Z) such that

$$ \frac{ d\theta_{\lambda}}{d\theta_{Z}} = -\frac{\rho+\delta - \theta_{\lambda}}{\theta_{Z} + \delta}\kappa(\sigma)<0. $$

(J.2)

For a given initial vector (A(0),Z(0),B₀), the intersection of this curve with the optimal trajectory OP defines the optimal initial vector $\left (\theta _{\lambda }(0),\theta _{Z}(0)\right )$. Given the initial pollution stock, Z(0), the initial social cost of pollution can be determined from ${\lambda (0) = Z(0)^{\nu }/\left (\rho + \delta - \theta _{\lambda }(0)\right )}$.

Appendix K: No Technical Progress

Suppose that a = b = 0. Thus, $\dot {A}=\dot {B}= 0$, and A(t) = A₀ and B(t) = B₀ for all t. Owing to the monotonic relationship between σ and λ (see Eq. 3.20), σ can be deemed a function of λ. Substituting σ, in n(σ) enables us to defineη(λ) ≡ n(σ(λ)), which is a strictly decreasing function of λ. Because the support of λ is $(0,M \bar {\varphi }_{\lambda }]$, owing to the boundary values of σ which are 0 and αβ. η(λ) varies between $+\infty$ and 0 as λ varies between 0 and $M \bar {\varphi }_{\lambda }$.

When there is no technological change, the optimal paths of Z and λ are the solution of the following autonomous dynamic system described by Eqs. 2.3 and 3.9:

$$ \begin{array}{@{}rcl@{}} \dot{\lambda}&=&\left( \rho+\delta \right){\lambda}-Z^{\nu} \ \text{ and} \end{array} $$

(K.1)

$$ \begin{array}{@{}rcl@{}} \dot{Z}&=&\eta({\lambda})-\delta Z. \end{array} $$

(K.2)

Figure 3 depicts the phase plane for (Z,λ) by plotting the isoclines $\dot {\lambda }= 0$ and $\dot {Z}= 0$. First, the isocline $\dot {\lambda }= 0$ is defined by λ = Z^ν/(ρ + δ), which gives an increasing relation between Z and λ. Furthermore, ${\lambda }\leq M \bar {\varphi }_{\lambda }$ since the co-state variable is non-negative; that is, γ ≥ 0. Secondly, the isocline $\dot {Z}= 0$ is defined by η(λ) = δZ, which gives a strictly decreasing relationship between the social cost of pollution, λ, and pollution stock, Z. Thus, λ varies between $M \bar {\varphi }_{\lambda }$ and 0 when $Z\rightarrow 0$ and $Z\rightarrow \infty$, respectively

Appendix L: Mixed-Research Regime

From Eqs. 3.7 and 3.8, an interior solution for the scientific activity requires

$$ \begin{array}{@{}rcl@{}} \omega_{A}\equiv a \lambda_{A} A = \omega_{B} \equiv b \lambda_{B} B=\omega, \end{array} $$

where ω_A(t) and ω_A(t) are the social wages in the resource-saving and abatement-augmenting research sectors, respectively. Multiplying both sides of Eqs. 3.10 and 3.11 by e^−ρtA and e^−ρtB, respectively, and rearranging yields

$$ \begin{array}{@{}rcl@{}} \dot{(e^{-\rho t}a \lambda_{A} A)}&=&e^{-\rho t}a A p z^{\alpha} \ \text{ and } \\ \dot{(e^{-\rho t}b \lambda_{B} B)}&=&e^{-\rho t}b B {\mu} x^{\beta}. \end{array} $$

The equality of social wages allows the equalization of these two expressions and gives us apy = bμq. Using this relationship and Eq. 3.5 yields $\sigma ^{M}=\beta \frac {a}{b},$ which is, indeed, a constant. Thus, when there is research in both sectors, and the two sectors are active over $[0,\infty )$, the share of the abatement expenditure in the GDP is fixed. The fact that abatement cannot exceed the pollution flow, that is, σ ≤ αβ, leads to σ^M ≤ αβ ⇒ a ≤ αb. Therefore, to justify a mixed-research regime, the innovative capacity b has to be sufficiently higher than a. Further, as, by hypothesis, b > a/α, then, σ^M < αβ; that is, a permanent mixed-research regime in which the social sectoral wages are equal can only occur when the pollution is always partially abated.

As σ is parametrically determined, μ and n become functions of the technology indexes A and B only (see Eqs. 3.16 and 3.17). From Eqs. 3.16 and 3.17, we can calculate the growth rate for the social cost of pollution and the net level of pollution as $\theta _{\lambda } = \frac {\bar {\theta }_{\lambda }^{A}}{a}\theta _{A}+\frac {\bar {\theta }_{\lambda }^{B}}{b}\theta _{B}$ (or, equivalently, $\theta _{\lambda } = {\bar {\theta }_{\lambda }^{A}}s_{A}+{\bar {\theta }_{\lambda }^{B}}s_{B}<0$) and $\theta _{n}=\frac {\bar {\theta }^{A}_{Z}}{a}\theta _{A} + \frac {\bar {\theta }^{B}_{Z}}{b}\theta _{B}$ (or, $\theta _{n} = {\bar {\theta }_{Z}^{A}}s_{A}+{\bar {\theta }_{Z}^{B}}s_{B}>0$), respectively. Accordingly, time differentiating the dynamic system given by Eq. 3.21 yields

$$ \begin{array}{@{}rcl@{}} \frac{\dot{\theta}_{\lambda}}{\rho+\delta - \theta_{\lambda}}&=&\frac{\bar{\theta}_{\lambda}^{A}}{a}\theta_{A}+\frac{\bar{\theta}_{\lambda}^{B}}{b}\theta_{B}-\nu \theta_{Z}, \end{array} $$

(L.1)

$$ \begin{array}{@{}rcl@{}} \frac{\dot{\theta}_{Z}}{\theta_{Z}+\delta}&=&\frac{\bar{\theta}^{A}_{Z}}{a}\theta_{A} + \frac{\bar{\theta}^{B}_{Z}}{b}\theta_{B} -\theta_{Z}. \end{array} $$

(L.2)

Following this, the isoclines $\dot {\theta }_{\lambda }= 0$ and $\dot {\theta }_{Z}= 0$ are

$$ \begin{array}{@{}rcl@{}} \theta_{\lambda} &=& \nu \theta_{Z} \ \text{ and} \end{array} $$

(L.3)

$$ \begin{array}{@{}rcl@{}} \theta_{Z} &=& \frac{\bar{\theta}^{B}_{Z}-\bar{\theta}^{A}_{Z}}{\bar{\theta}_{\lambda}^{A}-\bar{\theta}_{\lambda}^{B}}\bar{\theta}_{\lambda}^{B} +\bar{\theta}^{B}_{Z}- \frac{\bar{\theta}^{B}_{Z}-\bar{\theta}^{A}_{Z}}{\bar{\theta}_{\lambda}^{A}-\bar{\theta}_{\lambda}^{B}}{\theta}_{\lambda}. \end{array} $$

(L.4)

The dynamics of 𝜃_λ and 𝜃_Z in a mixed-research regime are pictured in Fig. 4. It can be seen that the isocline $\dot {\theta }_{Z}= 0$ is the segment joining the points $(\bar {\theta }^{A}_{Z},\bar {\theta }_{\lambda }^{A})$ and $(\bar {\theta }^{B}_{Z},\bar {\theta }_{\lambda }^{B})$. It is immediately verified that if b > a/α, $\bar {\theta }_{\lambda }^{A}>\bar {\theta }_{\lambda }^{B}$. Furthermore, σ^M < 1 implies that βa < b. Thus, $\bar {\theta }^{A}_{Z}<\bar {\theta }^{B}_{Z}$. As 𝜃_A = as_A and 𝜃_B = bs_B, $\theta _{\lambda } = {\bar {\theta }_{\lambda }^{A}}s_{A}+{\bar {\theta }_{\lambda }^{B}}s_{B}$. Then, the positivity constraint s_A ≥ 0 is equivalent to $\theta _{\lambda }\geq \bar {\theta }_{\lambda }^{B}$. Moreover, 1 ≥ s_A implies $\theta _{\lambda }\leq \bar {\theta }_{\lambda }^{A}$. Therefore, in a mixed-research regime $\theta _{\lambda }\in \left [\bar {\theta }_{\lambda }^{B},\bar {\theta }_{\lambda }^{A}\right ]$.

Fig. 4

Mixed-research regime: phase diagram analysis

We see that all trajectories converge in finite time either toward the abatement efficiency, B, border (i.e., the horizontal line for $\bar {\theta }^{B}_{\lambda }$), or toward the production efficiency A border (i.e., the $\bar {\theta }^{A}_{\lambda }$ horizontal). It is, therefore, impossible to permanently sustain a mixed-research regime. This is also justified by Theorem 1 stating that the share of the abatement expenditure will eventually converge to its upper bound αβ, thus, violating $\sigma =\beta \frac {a}{b}$ in a mixed-research regime.

In a mixed regime, the dynamics of the resources allocated for research is determined by the following equation:

$$ \begin{array}{@{}rcl@{}} s_{A} = \frac{\theta_{\lambda} - \bar{\theta}_{\lambda}^{B}}{\bar{\theta}_{\lambda}^{A} - \bar{\theta}_{\lambda}^{B}}. \end{array} $$

(L.5)

Because ds_A/d𝜃_λ > 0, the resources allocated for the resource-saving research increases when 𝜃_λ approaches the $\bar {\theta }_{\lambda }^{A}$. Conversely, the resources allocated for abatement-augmenting research increases when 𝜃_λ approaches the $\bar {\theta }_{\lambda }^{B}$.

Because $-\delta \nu > \bar {\theta }^{B}_{\lambda }$, σ steadily grows until FA is attained (i.e., when σ(t) = 0). During the PA phase, a mixed-research regime which can be sustained cannot exist. During the FA phase, a mixed-research regime cannot exist because the social wage in the abatement-augmenting research sector would always be higher. Instead, it is only when the pollution stock, or low level of resource-saving or abatement-augmenting technologies, are sufficiently low that the social planner would find it optimal partially abate the pollution and start allocating resources to augment the resource-saving technology (cf. Eq. B.3). Nevertheless, because $-\delta \nu > \bar {\theta }^{A}_{\lambda }$, the share of the expenditure in the economy that is spent on abatement increases in time. This can be observed from Eq. 3.23. When σ gets sufficiently high, and before abatement is fully abated, the resources for research are allocated to improve only the abatement-augmenting technologies. This phase starts right after the instant when ω_A(t) = ω_B(t).