On the economic growth and environmental trade-off: a multi-objective analysis

Abstract

We develop a multicriteria approach, based on both scalarization and goal programming techniques, in order to analyze the trade off between economic growth and environmental outcomes in a framework in which the economy and environment relation is bidirectional. On the one hand, economic growth by stimulating production activities gives rise to emissions of pollutants which deteriorate the environment. On the other hand, the environment affects economic activities since pollution generates a production externality determining how much output the economy can produce and reducing welfare. In this setting we show that optimality dictates an initial overshooting followed by economic degrowth and rising pollution. This implies that independently of the relative importance of economic and environmental factors, it is paradoxically optimal for the economy to asymptotically reach the maximum pollution level that the environment is able to bear.

Appendix A: Technical appendix

Proof of Lemma 1

The graph of the correspondence defined in (13) is defined as

$$\begin{aligned} G_{\Gamma }=\left\{ \left( k,p,k^{\prime },p^{\prime }\right) \in \mathbb {R}_{+}^{4}:\left( k^{\prime },p^{\prime }\right) \in \Gamma \left( k,p\right) \right\} . \end{aligned}$$

(33)

We must prove that if $\left( k_{0},p_{0},k_{0}^{\prime },p_{0}^{\prime }\right) \in G_{\Gamma }$ and $\left( k_{1},p_{1},k_{1}^{\prime } ,p_{1}^{\prime }\right) \in G_{\Gamma }$ then the point $\left( k_{\sigma },p_{\sigma },k_{\sigma }^{\prime },p_{\sigma }^{\prime }\right) =\sigma \left( k_{0},p_{0},k_{0}^{\prime },p_{0}^{\prime }\right) +\left( 1-\sigma \right) \left( k_{1},p_{1},k_{1}^{\prime },p_{1}^{\prime }\right) \in G_{\Gamma }$ as well for any $0\le \sigma \le 1$. According to definition (13), the first and second properties respectively mean that

$$\begin{aligned}&\left[ \left( 1-\eta \right) p_{0}\le p_{0}^{\prime }\le 1\right] \wedge \left[ k_{0}^{\prime }\le \dfrac{1}{\mu }p_{0}^{\prime }-\dfrac{1-\eta }{\mu }p_{0}+\left( 1-\delta \right) k_{0}\right] \end{aligned}$$

(34)

$$\begin{aligned}&\left[ \left( 1-\eta \right) p_{1}\le p_{1}^{\prime }\le 1\right] \wedge \left[ k_{1}^{\prime }\le \dfrac{1}{\mu }p_{1}^{\prime }-\dfrac{1-\eta }{\mu }p_{1}+\left( 1-\delta \right) k_{1}\right] . \end{aligned}$$

(35)

The first conditions in (34) and (35) imply that

$$\begin{aligned} \sigma \left( 1-\eta \right) p_{0}\le \sigma p_{0}^{\prime }\le \sigma \quad \text {and}\quad \left( 1-\sigma \right) \left( 1-\eta \right) p_{1} \le \left( 1-\sigma \right) p_{1}^{\prime }\le 1-\sigma , \end{aligned}$$

which, summing up, lead to

$$\begin{aligned} \left( 1-\eta \right) \left[ \sigma p_{0}+\left( 1-\sigma \right) p_{1}\right] \le \sigma p_{0}^{\prime }+\left( 1-\sigma \right) p_{1} ^{\prime }\le \sigma +1-\sigma =1. \end{aligned}$$

As the first and the last inequalities above can be rewritten as

$$\begin{aligned} \left( 1-\eta \right) p_{\sigma }\le p_{\sigma }^{\prime }\le 1, \end{aligned}$$

the first condition for $\left( k_{\sigma }^{\prime },p_{\sigma }^{\prime }\right) \in \Gamma \left( k_{\sigma },p_{\sigma }\right)$ according to (13) is established. To prove the other condition in (13), note that the second conditions in (34) and (35) imply that

$$\begin{aligned} \sigma k_{0}^{\prime }&\le \sigma \dfrac{1}{\mu }p_{0}^{\prime }-\sigma \dfrac{1-\eta }{\mu }p_{0}+\sigma \left( 1-\delta \right) k_{0}\quad \text {and}\\ \left( 1-\sigma \right) k_{1}^{\prime }&\le \left( 1-\sigma \right) \dfrac{1}{\mu }p_{1}^{\prime }-\left( 1-\sigma \right) \dfrac{1-\eta }{\mu } p_{1}+\left( 1-\sigma \right) \left( 1-\delta \right) k_{1}, \end{aligned}$$

which, summing up, lead to

$$\begin{aligned} \sigma k_{0}^{\prime }+\left( 1-\sigma \right) k_{1}^{\prime }\le & {} \dfrac{1}{\mu }\left[ \sigma p_{0}^{\prime }+\left( 1-\sigma \right) p_{1}^{\prime }\right] -\dfrac{1-\eta }{\mu }\left[ \sigma p_{0}+\left( 1-\sigma \right) p_{1}\right] \\&+\left( 1-\delta \right) \left[ \sigma k_{0}+\left( 1-\sigma \right) k_{1}\right] , \end{aligned}$$

which can be rewritten as

$$\begin{aligned} k_{\sigma }^{\prime }\le \dfrac{1}{\mu }p_{\sigma }^{\prime }-\dfrac{1-\eta }{\mu }p_{\sigma }+\left( 1-\delta \right) k_{\sigma }, \end{aligned}$$

so that the second condition for $\left( k_{\sigma }^{\prime },p_{\sigma }^{\prime }\right) \in \Gamma \left( k_{\sigma },p_{\sigma }\right)$ in (13) holds as well, and we can conclude that $\left( k_{\sigma },p_{\sigma },k_{\sigma }^{\prime },p_{\sigma }^{\prime }\right) \in G_{\Gamma }$ according to (33), as was to be shown. $\square$

As the function $V\left( k,p\right)$ defined in (19) is unbounded from below, before proving Proposition 1 we recall the following verification principle (Lemma 2) that holds for unbounded functions. Consider the general problem

$$\begin{aligned} V\left( x_{0}\right) =\sup _{\left\{ x_{t}\right\} _{t=0}^{\infty }} \sum _{t=0}^{\infty }\beta ^{t}u\left( x_{t},x_{t+1}\right) \nonumber \\ \text {s.t. }\left\{ \begin{array}[c]{l} x_{t+1}\in \Gamma \left( x_{t}\right) {\qquad }\forall t\ge 0\text {,}\\ x_{t}\in X\subseteq \mathbb {R}^{n}{\qquad }\forall t\ge 1\text {,}\\ x_{0}\in X\text { is given,} \end{array} \right. \end{aligned}$$

(36)

where $\Gamma :X\rightarrow X$ is a compact, nonempty correspondence such that $\Gamma \left( x\right) \subseteq X$ for all $x\in X$, $u:G_{\Gamma }\rightarrow \mathbb {R}$ [$G_{\Gamma }$ denotes the graph of $\Gamma$ according to (33)], and $0<\beta <1$. We shall denote a plan by $\left( x_{0},\left\{ x_{t}\right\} _{t=0}^{\infty }\right)$, or, shortly, $\left( x_{0},\left\{ x_{t}\right\} \right)$; a plan $\left( x_{0},\left\{ x_{t}\right\} \right)$ is said to be feasible if $x_{t+1}\in \Gamma \left( x_{t}\right)$ for all $t\ge 0$. Moreover, we shall denote the objective function in (36) by

$$\begin{aligned} W\left( x_{0},\left\{ x_{t}\right\} \right) =\sum _{t=0}^{\infty }\beta ^{t}u( x_{t},x_{t+1}) , \end{aligned}$$

(37)

and its n-finite truncation by

$$\begin{aligned} W_{n}\left( x_{0},\left\{ x_{t}\right\} \right) =\sum _{t=0}^{n-1}\beta ^{t}u( x_{t},x_{t+1}) . \end{aligned}$$

(38)

Let

$$\begin{aligned} w\left( x\right) =\max _{\left( y\right) \in \Gamma \left( x\right) }\left[ u\left( x,y\right) +\beta w\left( y\right) \right] \end{aligned}$$

(39)

be its associated Bellman equation.

Lemma 2

(A verification principle) Let $w\left( x\right)$ be a solution to the Bellman equation (39). Then, if

• 1. $\lim \inf _{t\rightarrow \infty }\beta ^{t}w\left( x_{t}\right) \le 0$ for all feasible plans $\left( x_{0},\left\{ x_{t}\right\} \right)$, and

• 2. for any $x_{0}$ and any feasible plan $\left( x_{0},\left\{ x_{t}\right\} \right)$ there is another feasible plan $\left( x_{0},\left\{ x_{t}^{\prime }\right\} \right)$ originating from the same initial condition $x_{0}$ that satisfies

  • (a) $W\left( x_{0},\left\{ x_{t}^{\prime }\right\} \right) =\sum _{t=0}^{\infty }\beta ^{t}u\left( x_{t}^{\prime },x_{t+1}^{\prime }\right) \ge \sum _{t=0}^{\infty }\beta ^{t}u\left( x_{t},x_{t+1}\right) =W\left( x_{0},\left\{ x_{t}\right\} \right)$, and

  • (b) $\lim \sup _{t\rightarrow \infty }\beta ^{t}w\left( x_{t}^{\prime }\right) \ge 0$,

then $w\left( x\right)$ is the value function of (36), $w\left( x\right) =V\left( x\right)$.

Proof

Fix arbitrarily an $\varepsilon >0$ and consider the scalar $\varphi =\left( 1-\beta \right) /\varepsilon >0$. In view of (39), given $x_{0}$, there is some $x_{1}\in \Gamma \left( x_{0}\right)$ such that $u\left( x_{0} ,x_{1}\right) +\beta w\left( x_{1}\right) >w\left( x_{0}\right) -\varphi$. Similarly, there is a point $x_{2}\in \Gamma \left( x_{1}\right)$ such that $u\left( x_{1},x_{2}\right) +\beta w\left( x_{2}\right) >w\left( x_{1}\right) -\varphi$, and so on. Therefore, this process generates a feasible plan such that $u\left( x_{t},x_{t+1}\right) +\beta w\left( x_{t+1}\right) >w\left( x_{t}\right) -\varphi$ for all $t\ge 0$. By iterating all such terms up to $t=n$, it is easy to see that there always exist a feasible plan $\left( x_{0},\left\{ x_{t}\right\} \right)$ such that

$$\begin{aligned} w\left( x_{0}\right)&<W_{n}\left( x_{0},\left\{ x_{t}\right\} \right) +\beta ^{n}w\left( x_{n}\right) +\varphi \sum _{t=0}^{\infty }\beta ^{t}\\&=W_{n}\left( x_{0},\left\{ x_{t}\right\} \right) +\beta ^{n}w\left( x_{n}\right) +\varepsilon \end{aligned}$$

for any arbitrary $\varepsilon >0$. Taking the $\lim \inf _{n\rightarrow \infty }$ on both sides we obtain

$$\begin{aligned} w\left( x_{0}\right) \le W\left( x_{0},\left\{ x_{t}\right\} \right) +\lim \inf _{n\rightarrow \infty }\beta ^{n}w\left( x_{n}\right) +\varepsilon \le W\left( x_{0},\left\{ x_{t}\right\} \right) +\varepsilon , \end{aligned}$$

where in the second inequality we used property 1 of Lemma 2. Hence, $w\left( x_{0}\right) \le W\left( x_{0},\left\{ x_{t}\right\} \right) \le V\left( x_{0}\right)$.

On the other hand, by considering the feasible plan $\left( x_{0},\left\{ x_{t}^{\prime }\right\} \right)$ satisfying property 2 of Lemma 2 and again iterating the terms on the RHS of (39) from $t=0$ to $t=n$, we get

$$\begin{aligned} w\left( x_{0}\right) \ge W_{n}\left( x_{0},\left\{ x_{t}^{\prime }\right\} \right) +\beta ^{n}w\left( x_{n}^{\prime }\right) . \end{aligned}$$

Taking the $\lim \sup _{n\rightarrow \infty }$ on both sides and using properties 2a and 2b we obtain

$$\begin{aligned} w\left( x_{0}\right) \ge W\left( x_{0},\left\{ x_{t}^{\prime }\right\} \right) +\lim \sup _{n\rightarrow \infty }\beta ^{n}w\left( x_{n}^{\prime }\right) \ge W\left( x_{0},\left\{ x_{t}^{\prime }\right\} \right) \ge W\left( x_{0},\left\{ x_{t}\right\} \right) , \end{aligned}$$

which, as $\left( x_{0},\left\{ x_{t}\right\} \right)$ is any arbitrary feasible plan, implies $w\left( x_{0}\right) \ge V\left( x_{0}\right)$.

Therefore, $w\left( x_{0}\right) =V\left( x_{0}\right)$ and the proof is complete. $\square$

Lemma 3

A plan $\left( x_{0},\left\{ x_{t}^{*}\right\} \right)$ satisfying the Bellman equations (39) for all $t\ge 0$ for the value function $w\left( x\right) =V\left( x\right)$ is optimal if and only if $\lim _{t\rightarrow \infty }\beta ^{t}V\left( x_{t}^{*}\right) =0$.

Proof

Assume that $\left( x_{0},\left\{ x_{t}^{*}\right\} \right)$ satisfies (39) for all $t\ge 0$ and that $\lim _{t\rightarrow \infty }\beta ^{t}V\left( x_{t}^{*}\right) =0$. Then

$$\begin{aligned} V\left( x\right) =u\left( x_{t}^{*},x_{t+1}^{*}\right) +\beta V\left( x_{t+1}^{*}\right) \end{aligned}$$

(40)

for all $t\ge 0$. By iterating the terms on the RHS in (40) from $t=0$ to $t=n$, we get

$$\begin{aligned} V\left( x_{0}\right) =W_{n}\left( x_{0},\left\{ x_{t}^{*}\right\} \right) +\beta ^{n}V\left( x_{n}^{*}\right) . \end{aligned}$$

Taking the limit as $n\rightarrow \infty$ of both sides we have

$$\begin{aligned} V\left( x_{0}\right) =\lim _{t\rightarrow \infty }W_{n}\left( x_{0},\left\{ x_{t}^{*}\right\} \right) +\lim _{t\rightarrow \infty }\beta ^{t}V\left( x_{t}^{*}\right) =W\left( x_{0},\left\{ x_{t}^{*}\right\} \right) , \end{aligned}$$

which establishes that $\left( x_{0},\left\{ x_{t}^{*}\right\} \right)$ is optimal.

Conversely, assume that $\left( x_{0},\left\{ x_{t}^{*}\right\} \right)$ is optimal. By iterating $W\left( x_{0},\left\{ x_{t}^{*}\right\} \right) =u\left( x_{0},x_{1}^{*}\right) +\beta W\left( x_{1}^{*},\left\{ x_{1+t}^{*}\right\} \right)$ we easily get

$$\begin{aligned} V\left( x_{0}\right) =W\left( x_{0},\left\{ x_{t}^{*}\right\} \right) =W_{n}\left( x_{0},\left\{ x_{t}^{*}\right\} \right) +\beta ^{n}W\left( x_{n}^{*},\left\{ x_{n+t}^{*}\right\} \right) , \end{aligned}$$

that is,

$$\begin{aligned} \beta ^{n}W\left( x_{n}^{*},\left\{ x_{n+t}^{*}\right\} \right) =V\left( x_{0}\right) -W_{n}\left( x_{0},\left\{ x_{t}^{*}\right\} \right) . \end{aligned}$$

As $W\left( x_{n}^{*},\left\{ x_{n+t}^{*}\right\} \right) =V\left( x_{n}^{*}\right)$ and $\lim _{n\rightarrow \infty }W_{n}\left( x_{0},\left\{ x_{t}^{*}\right\} \right) =V\left( x_{0}\right)$, taking the limit as $n\rightarrow \infty$ of both sides in the last equation we have $\lim _{t\rightarrow \infty }\beta ^{t}V\left( x_{t}^{*}\right) =V\left( x_{0}\right) -\lim _{n\rightarrow \infty }W_{n}\left( x_{0},\left\{ x_{t}^{*}\right\} \right) =V\left( x_{0}\right) -V\left( x_{0}\right) =0$ and the proof is complete. $\square$

Proof of Proposition 1

To apply the Guess and verify method (see, e.g., Bethmann 2007, 2013), the linearity of the terms inside the logarithm in (14) suggests the following form for the value function:

$$\begin{aligned} V\left( k,p\right) =\rho _{1}+\rho _{2}\ln \left( \rho _{3}k+\rho _{4}p+\rho _{5}\right) +\rho _{6}\ln ( 1-p) , \end{aligned}$$

where $\rho _{1},\rho _{2},\rho _{3},\rho _{4},\rho _{5}$ and $\rho _{6}$ are constant coefficients, so that (14) can be rewritten as

$$\begin{aligned}&\rho _{1}+\rho _{2}\ln \left( \rho _{3}k+\rho _{4}p+\rho _{5}\right) +\rho _{6} \ln \left( 1-p\right) \nonumber \\&\quad =\max _{\left( k^{\prime },p^{\prime }\right) \in \Gamma \left( k,p\right) }\left\{ \ln \left[ p^{\prime }/\mu -\left( 1-\eta \right) p/\mu +\left( 1-\delta \right) k-k^{\prime }\right] \right. \nonumber \\&\qquad \left. +\,\theta \ln \left( 1-p\right) +\beta \left[ \rho _{1}+\rho _{2} \ln \left( \rho _{3}k^{\prime }+\rho _{4}p^{\prime }+\rho _{5}\right) +\rho _{6} \ln \left( 1-p^{\prime }\right) \right] \right\} . \end{aligned}$$

(41)

The RHS in (41) is concave in $k^{\prime }$ and $p^{\prime }$ for all given $\left( k,p\right)$; therefore, FOC on the RHS yield a unique solution for $k^{\prime },p^{\prime }$ provided that it is interior to the correspondence $\Gamma \left( k,p\right)$ and the system of equations that equate both partial derivatives to zero admits a unique solution. By solving such system we find the following values for $k^{\prime },p^{\prime }$:

$$\begin{aligned} \left( k^{\prime }\right) ^{*}&={\frac{{\beta }\left[ \rho _{{2}} {\rho _{{3}}}+\mu \rho _{{4}}\left( \rho _{{2}}+\rho _{{6}}\right) \right] }{\left( \rho _{{3}}+\mu \rho _{{4}}\right) \left( 1+\beta \rho _{{2}}+\beta \rho _{{6}}\right) }}\left[ {\left( 1-\delta \right) k-}\frac{1-\eta }{\mu }p\right] \nonumber \\&\quad +{\frac{\beta \rho _{{2}}{\rho _{{3}}}^{2}-\left[ \rho _{{4}}+\rho _{{5}} +\beta \left( \rho _{{5}}\rho _{{6}}-\rho _{{2}}\rho _{{4}}\right) \right] \mu \rho _{{3}}-{\mu }^{2}\rho _{{4}}\left( \rho _{{5}}+\rho _{{4}}\right) }{\mu \rho _{{3}}\left( \rho _{{3}}+\mu \rho _{{4}}\right) \left( 1+\beta \rho _{{2}}+\beta \rho _{{6}}\right) }} \end{aligned}$$

(42)

$$\begin{aligned} \left( p^{\prime }\right) ^{*}&=\frac{\beta \mu \rho _{3}\rho _{6} }{\left( \rho _{3}+\mu \rho _{4}\right) \left( 1+\beta \rho _{2}+\beta \rho _{6}\right) }\left[ \frac{1-\eta }{\mu }p-\left( 1-\delta \right) k\right] \nonumber \\&\quad +\frac{\beta \left( \rho _{2}\rho _{3}+\mu \rho _{2}\rho _{4}-\mu \rho _{5} \rho _{6}\right) +\rho _{3}+\mu \rho _{4}}{\left( \rho _{3}+\mu \rho _{4}\right) \left( 1+\beta \rho _{2}+\beta \rho _{6}\right) }, \end{aligned}$$

(43)

that is, both optimal values are affine functions of k and p. By substituting such values into (41) we get

$$\begin{aligned}&\rho _{1}+\rho _{2}\ln \left( \rho _{3}k+\rho _{4}p+\rho _{5}\right) +\rho _{6}\ln \left( 1-p\right) \\&\quad =\ln \left[ {\frac{\left( 1-\delta \right) \mu \rho _{{3}}k-\left( 1-\eta \right) \rho _{{3}}p+\rho _{{3}}+\mu \left( \rho _{{4}}+\rho _{{5}}\right) }{\mu \rho _{{3}}\left( 1+\beta \rho _{2}+\beta \rho _{6}\right) }}\right] +\theta \ln \left( 1-p\right) +\beta \rho _{{1}}\\&\qquad +\beta \rho _{{2}}\ln \left[ \beta \rho _{{2}}{\frac{\left( 1-\delta \right) \mu \rho _{{3}}k-\left( 1-\eta \right) \rho _{{3}}p+\rho _{{3} }+\mu \left( \rho _{{4}}+\rho _{{5}}\right) }{\mu \left( 1+\beta \rho _{2} +\beta \rho _{6}\right) }}\right] \\&\qquad +\beta \rho _{{6}}\ln \left[ \beta \rho _{{6}}{\frac{\left( 1-\delta \right) \mu \rho _{{3}}k-\left( 1-\eta \right) \rho _{{3}}p+\rho _{{3} }+\mu \left( \rho _{{4}}+\rho _{{5}}\right) }{\left( \rho _{3}+\mu \rho _{4}\right) \left( 1+\beta \rho _{2}+\beta \rho _{6}\right) }}\right] \\&\quad =\ln \left[ \left( 1-\delta \right) \mu k-\left( 1-\eta \right) p+\frac{\rho _{{3}}+\mu \left( \rho _{{4}}+\rho _{{5}}\right) }{\rho _{{3}} }\right] \\&\qquad -\ln \left[ \mu \left( 1+\beta \rho _{2}+\beta \rho _{6}\right) \right] +\theta \ln \left( 1-p\right) +\beta \rho _{{1}}\\&\qquad +\beta \rho _{{2}}\ln \left\{ \beta \rho _{{2}}\rho _{{3}}\left[ \left( 1-\delta \right) \mu k-\left( 1-\eta \right) p+\frac{\rho _{{3}}+\mu \left( \rho _{{4}}+\rho _{{5}}\right) }{\rho _{{3}}}\right] \right\} \\&\qquad -\beta \rho _{{2} }\ln \left[ \mu \left( 1+\beta \rho _{2}+\beta \rho _{6}\right) \right] \\&\qquad +\beta \rho _{{6}}\ln \left\{ \beta \rho _{{3}}\rho _{{6}}\left[ \left( 1-\delta \right) \mu k-\left( 1-\eta \right) p+\frac{\rho _{{3}}+\mu \left( \rho _{{4}}+\rho _{{5}}\right) }{\rho _{{3}}}\right] \right\} -\beta \rho _{{6} }\ln \left( \rho _{3}+\mu \rho _{4}\right) \\&\qquad -\beta \rho _{{6}}\ln \left[ \left( 1+\beta \rho _{2}+\beta \rho _{6}\right) \right] \\&\quad =\left( 1+\beta \rho _{2}+\beta \rho _{6}\right) \ln \left[ \left( 1-\delta \right) \mu k-\left( 1-\eta \right) p+\frac{\rho _{{3}}+\mu \left( \rho _{{4}}+\rho _{{5}}\right) }{\rho _{{3}}}\right] \\&\qquad -\left( 1+\beta \rho _{2}\right) \ln \mu -\left( 1+\beta \rho _{2} +\beta \rho _{6}\right) \ln \left( 1+\beta \rho _{2}+\beta \rho _{6}\right) +\theta \ln \left( 1-p\right) +\beta \rho _{{1}}\\&\qquad +\beta \left( \rho _{2}+\rho _{6}\right) \ln \left( \beta \rho _{3}\right) +\beta \rho _{2}\ln \rho _{2}+\beta \rho _{6}\ln \rho _{6}-\beta \rho _{6}\ln ( \rho _{3}+\mu \rho _{4}) , \end{aligned}$$

and by equating the coefficients of the homogeneous terms in both sides (also inside the argument of the first logarithm) we find the values for the coefficients listed in (20) and (21).

By substituting the coefficients in (20) and (21) in the expressions (42) and (43), after some tedious algebra we obtain the optimal dynamics for capital and pollution as in (22) and (23). By solving the system

$$\begin{aligned} \left\{ \begin{array}[c]{l} k=\gamma _{1}k+\gamma _{2}p+\gamma _{3}\\ p=\gamma _{4}k+\gamma _{5}p+\gamma _{6}, \end{array} \right. \end{aligned}$$

where the coefficients $\gamma _{i}$, $i=1,\ldots ,6$, are listed in point 2 of Proposition 1, for k and p the unique steady state $\left( k^{s},p^{s}\right) =\left( \eta /\left( \mu \delta \right) ,1\right)$ as in (26) is found.

The upper bound $\bar{z}$ for the technology index defined in condition (15) allows for the existence of optimal values $z_{t}$ that satisfy inequality (8), that is, such that $p_{t+1}\le 1$. To see this, note that from the second inequality in (8) it follows that

$$\begin{aligned} \bar{z}\ge \frac{\left( 1+p_{t}\right) ^{\phi }k_{t}^{-\alpha }\left[ 1-\left( 1-\eta \right) p_{t}\right] }{\mu } \end{aligned}$$

must hold for all feasible sequence $\left\{ k_{t},p_{t}\right\}$; thus, recalling that $0\le p_{t}\le 1$ we can consider the following upper bound of the RHS above:

$$\begin{aligned} \frac{\left( 1+p_{t}\right) ^{\phi }k_{t}^{-\alpha }\left[ 1-\left( 1-\eta \right) p_{t}\right] }{\mu }\le \frac{2^{\phi }}{\mu k_{t}^{\alpha }} \le \frac{2^{\phi }}{\mu \min \left\{ k_{0},\eta /\left( \mu \delta \right) \right\} }=\frac{2^{\phi }}{\mu }\max \left\{ \frac{1}{k_{0}},\frac{\mu \delta }{\eta }\right\} , \end{aligned}$$

which is condition (15) for $p=p_{0}$. The second inequality holds thanks to the fact that, as we shall see in the following, the optimal sequence $k_{t}$ defined by (22) converges monotonically to the steady value $k^{s}=\eta /\left( \mu \delta \right)$.

Conditions (17) and (18) on the initial capital stock guarantee that the expressions in the RHS of (42) and (43) define points $\left( k^{\prime }\right) ^{*}$ and $\left( p^{\prime }\right) ^{*}$ which are interior points of the correspondence $\Gamma \left( k,p\right)$ defined in (13), so that the whole recursive plan defined by (22) and (23) contain interior points as well. We shall establish this property in the following steps.

We first show that if conditions (17) and (18) hold for $\left( k_{0},p_{0}\right)$, then they hold for all $\left( k_{t},p_{t}\right)$, for all $t\ge 1$. Suppose that (17) holds for $\left( k_{t},p_{t}\right)$, then it can be shown that

$$\begin{aligned} \frac{1-\eta }{\mu \left( 1-\delta \right) }=\frac{\left( 1-\eta \right) \gamma _{5}-\mu \left( 1-\delta \right) \gamma _{2}}{\mu \left( 1-\delta \right) \gamma _{1}-\left( 1-\eta \right) \gamma _{4}}\quad \text {and}\quad \frac{\eta -\delta }{\mu \delta \left( 1-\delta \right) }=\frac{\left( 1-\eta \right) \gamma _{6}-\mu \left( 1-\delta \right) \gamma _{3}+\left( \eta -\delta \right) /\delta }{\mu \left( 1-\delta \right) \gamma _{1}-\left( 1-\eta \right) \gamma _{4}}, \end{aligned}$$

where the coefficients $\gamma _{i}$, $i=1,\ldots ,6$, are listed in point 2 of Proposition 1. Note that condition (16) implies that $\gamma _{1}>0$ for all admissible parameters’ values; this is because

$$\begin{aligned} \theta \left[ {\beta }\left( 1-\delta \right) -\left( 1-\eta \right) \right] +\eta -{\delta >0} \end{aligned}$$

(44)

is always true when ${\beta }\left( 1-\delta \right) \ge 1-\eta$, and it holds also when ${\beta }\left( 1-\delta \right) <1-\eta$ provided that $\eta >\left( 1-\beta \right) /2+\delta$. Thus, as $\gamma _{4}$ is clearly negative, $\mu \left( 1-\delta \right) \gamma _{1}-\left( 1-\eta \right) \gamma _{4}>0$ and condition (17) for $\left( k_{t},p_{t}\right)$ is equivalent to

$$\begin{aligned} k_{t}&>\frac{\left( 1-\eta \right) \gamma _{5}-\mu \left( 1-\delta \right) \gamma _{2}}{\mu \left( 1-\delta \right) \gamma _{1}-\left( 1-\eta \right) \gamma _{4}}p_{t}+\frac{\left( 1-\eta \right) \gamma _{6}-\mu \left( 1-\delta \right) \gamma _{3}+\left( \eta -\delta \right) /\delta }{\mu \left( 1-\delta \right) \gamma _{1}-\left( 1-\eta \right) \gamma _{4}}\\&\Longleftrightarrow \quad \gamma _{1}k_{t}+\gamma _{2}p_{t}+\gamma _{3}>\frac{1-\eta }{\mu \left( 1-\delta \right) }\left( \gamma _{4}k_{t}+\gamma _{5}p_{t}+\gamma _{6}\right) +\frac{\eta -\delta }{\mu \delta \left( 1-\delta \right) }\\&\Longleftrightarrow \quad k_{t+1}>\frac{1-\eta }{\mu \left( 1-\delta \right) }p_{t+1}+\frac{\eta -\delta }{\mu \delta \left( 1-\delta \right) }, \end{aligned}$$

so that $\left( k_{t+1},p_{t+1}\right)$ satisfies condition (17 ) as well. Assume now that $\left( k_{t},p_{t}\right)$ satisfies condition (18) and let

$$\begin{aligned} A=-{\frac{{\beta }\theta \left[ {\beta }\left( 1-{\delta }\right) -\left( 1-\eta \right) \right] +\eta -\delta }{\mu \beta \theta \left( 1-\beta \right) \left( 1-\delta \right) ^{2}}\left( 1-\eta \right) \quad }\text {{and}}{\quad B=\frac{\left[ \beta \theta \left( 1-\beta \right) +{\delta }\left( 1+{\beta }^{2}\theta \right) \right] \left( \eta -\delta \right) }{\mu \beta \theta \delta \left( 1-\beta \right) \left( 1-\delta \right) ^{2}},} \end{aligned}$$

(45)

so that by assumption $k_{t}<Ap_{t}+B$. Then it can be shown that

$$\begin{aligned} \frac{A\gamma _{5}-\gamma _{2}}{\gamma _{1}-A\gamma _{4}}=\frac{1-\eta }{\mu \left( 1-\delta \right) }=A+\frac{\left( 1-\eta \right) \left( 1+\beta \theta \right) \left( \eta -\delta \right) }{\mu \beta \theta \left( 1-\beta \right) \left( 1-\delta \right) ^{2}}>A, \end{aligned}$$

(46)

where the last inequality holds because $\left( \eta -\delta \right) >0$ under condition (16). We have shown before that $\gamma _{1}>0$; however, $A<0$ and $\gamma _{4}<0$, so that the sign of the denominator $\gamma _{1}-A\gamma _{4}$ can be either positive or negative. It turns out that $\gamma _{1}-A\gamma _{4}={\beta }\left( 1-\delta \right) -\left( 1-\eta \right)$, so the sign of the latter expression determines whether it is positive or negative. Hence, to study a similar inequality for parameter B we consider three separate cases.

• 1. If $\gamma _{1}-A\gamma _{4}={\beta }\left( 1-\delta \right) -\left( 1-\eta \right) >0$, then it can be shown that

  $$\begin{aligned} \frac{A\gamma _{6}-\gamma _{3}+B}{\gamma _{1}-A\gamma _{4}}=B+\frac{\left[ 1-\beta \left( 1-\delta \right) \right] \left( 1+\beta \theta \right) \left( \eta -\delta \right) }{\mu \beta \theta \left[ {\beta }\left( 1-\delta \right) -\left( 1-\eta \right) \right] \left( 1-\beta \right) \left( 1-\delta \right) ^{2}}>B, \end{aligned}$$

  (47)

  where the last inequality holds because $\left( \eta -\delta \right) >0$ under condition (16) and $\left[ {\beta }\left( 1-\delta \right) -\left( 1-\eta \right) \right] >0$ by assumption. Therefore, using both inequalities in (46) and (47), condition (18) for $\left( k_{t},p_{t}\right)$ implies:

  $$\begin{aligned} k_{t}&<Ap_{t}+B{\quad }\Longrightarrow {\quad }k_{t}<\frac{A\gamma _{5} -\gamma _{2}}{\gamma _{1}-A\gamma _{4}}p_{t}+\frac{A\gamma _{6}-\gamma _{3} +B}{\gamma _{1}-A\gamma _{4}}\\&\Longleftrightarrow \quad \gamma _{1}k_{t}+\gamma _{2}p_{t}+\gamma _{3}<A\left( \gamma _{4}k_{t}+\gamma _{5}p_{t}+\gamma _{6}\right) +B\\&\Longleftrightarrow \quad k_{t+1}<Ap_{t+1}+B, \end{aligned}$$

  so that $\left( k_{t+1},p_{t+1}\right)$ satisfies condition (18) as well.

• 2. If $\gamma _{1}-A\gamma _{4}={\beta }\left( 1-\delta \right) -\left( 1-\eta \right) <0$, then it can be checked that

  $$\begin{aligned} \frac{A\gamma _{6}-\gamma _{3}+B}{\gamma _{1}-A\gamma _{4}}&=\frac{1-\eta }{\mu \delta \left( 1-\delta \right) }+\frac{\eta \left( 1+\beta \theta \right) \left( \eta -\delta \right) }{\mu \beta \theta \left[ {\beta }\left( 1-\delta \right) -\left( 1-\eta \right) \right] \left( 1-\beta \right) \left( 1-\delta \right) ^{2}}\nonumber \\&<\frac{1-\eta }{\mu \delta \left( 1-\delta \right) }, \end{aligned}$$

  (48)

  where the last inequality holds because $\left( \eta -\delta \right) >0$ under condition (16) and $\left[ {\beta }\left( 1-\delta \right) -\left( 1-\eta \right) \right] <0$ by assumption, so that the second term in the middle expression is strictly negative. Therefore, using the first equality in (46) and the last inequality in (48), condition (17) for $\left( k_{t},p_{t}\right)$ implies:

  $$\begin{aligned} k_{t}&>\frac{1-\eta }{\mu \left( 1-\delta \right) }p_{t}+\frac{\eta -\delta }{\mu \delta \left( 1-\delta \right) }{\quad }\Longrightarrow {\quad }k_{t} >\frac{A\gamma _{5}-\gamma _{2}}{\gamma _{1}-A\gamma _{4}}p_{t}+\frac{A\gamma _{6}-\gamma _{3}+B}{\gamma _{1}-A\gamma _{4}}\\&\Longleftrightarrow \quad \left( \gamma _{1}-A\gamma _{4}\right) k_{t}<\left( A\gamma _{5}-\gamma _{2}\right) p_{t}+A\gamma _{6}-\gamma _{3}+B\\&\Longleftrightarrow \quad \gamma _{1}k_{t}+\gamma _{2}p_{t}+\gamma _{3}<A\left( \gamma _{4}k_{t}+\gamma _{5}p_{t}+\gamma _{6}\right) +B\\&\Longleftrightarrow \quad k_{t+1}<Ap_{t+1}+B, \end{aligned}$$

  where in the third step we used the assumption $\gamma _{1}-A\gamma _{4}<0$. Hence, this time thanks to condition (17), $\left( k_{t+1} ,p_{t+1}\right)$ satisfies condition (18) as well.

• 3. If $\gamma _{1}-A\gamma _{4}={\beta }\left( 1-\delta \right) -\left( 1-\eta \right) =0$, then, after replacing $1-\eta ={\beta }\left( 1-\delta \right)$ in the expression of A defined in (45) and in all coefficients $\gamma _{i}$, $i=1,\ldots ,6$, as listed in the point 2 of Proposition 1, one gets

  $$\begin{aligned} \gamma _{1}-A\gamma _{4}=0,{\quad }A\gamma _{5}-\gamma _{2}=0,{\quad } \text {and}{\quad }A\gamma _{6}-\gamma _{3}+B=\frac{\eta \left( 1+\beta \theta \right) }{\mu \beta \theta \left( 1-\delta \right) }>0, \end{aligned}$$

  which, once again, imply

  $$\begin{aligned} \left( \gamma _{1}-A\gamma _{4}\right) k_{t}&<\left( A\gamma _{5} -\gamma _{2}\right) p_{t}+A\gamma _{6}-\gamma _{3}+B\\&\Longleftrightarrow \quad \gamma _{1}k_{t}+\gamma _{2}p_{t}+\gamma _{3}<A\left( \gamma _{4}k_{t}+\gamma _{5}p_{t}+\gamma _{6}\right) +B\\&\Longleftrightarrow \quad k_{t+1}<Ap_{t+1}+B, \end{aligned}$$

  so that $\left( k_{t+1},p_{t+1}\right)$ satisfies condition (18 ) for all $t\ge 0$.

Equipped with the property that conditions (17) and (18) hold for all $\left( k_{t},p_{t}\right)$, for all $t\ge 1$, we are ready to verify that the plan $\left( k_{t},p_{t}\right)$ defined by (22) and (23) contain interior points for all $t\ge 0$. Specifically, the followings hold.

• 1. As $\gamma _{1}>0$, after some tedious algebra we have

  $$\begin{aligned} k_{t+1}=\gamma _{1}k_{t}+\gamma _{2}p_{t}+\gamma _{3}>\gamma _{1}\left[ \frac{1-\eta }{\mu \left( 1-\delta \right) }p_{t}+\frac{\eta -\delta }{\mu \delta \left( 1-\delta \right) }\right] +\gamma _{2}p_{t}+\gamma _{3} \equiv \frac{\eta }{\mu \delta }>0, \end{aligned}$$

  (49)

  where the first equality is (22) and in the first inequality we used condition (17);

• 2. it also can be shown that (17) is equivalent to

  $$\begin{aligned} k_{t}>\frac{\left[ \gamma _{5}-\mu \gamma _{2}-\left( 1-\eta \right) \right] p_{t}+\gamma _{6}-\mu \gamma _{3}}{\mu \gamma _{1}-\gamma _{4}-\mu \left( 1-\delta \right) }, \end{aligned}$$

  which, in turn, as $\mu \gamma _{1}-\gamma _{4}-\mu \left( 1-\delta \right) =\mu \left( 1-\beta \right) \left( 1-\delta \right) /\left( 1+\beta \theta \right) <0$ and using both (22) and (22), is a equivalent to $k_{t+1}<p_{t+1}/\mu -\left( 1-\eta \right) p_{t}/\mu +\left( 1-\delta \right) k_{t}$, which, joint with (49) establishes that the sequence $k_{t}$ generated by (22) is interior for all $t\ge 0$.

• 3. After the usual tedious algebra it can be shown that condition (18) is equivalent to

  $$\begin{aligned} k_{t}<\frac{1-\eta -\gamma _{5}p_{t}-\gamma _{6}}{\gamma _{4}}, \end{aligned}$$

  which, in turn, as $\gamma _{4}$ is clearly strictly negative and using (22), is equivalent to $p_{t+1}=\gamma _{4}k_{t}+\gamma _{5} p_{t}+\gamma _{6}>\left( 1-\eta \right) p_{t}$;

• 4. similarly, it can be shown that (17) is equivalent to

  $$\begin{aligned} k_{t}>\frac{1-\gamma _{5}p_{t}-\gamma _{6}}{\gamma _{4}}, \end{aligned}$$

  which, in turn, as $\gamma _{4}<0$ and using (23), is a equivalent to $p_{t+1}=\gamma _{4}k_{t}+\gamma _{5}p_{t}+\gamma _{6}<1$, which, joint with $p_{t+1}>\left( 1-\eta \right) p_{t}$, establishes that the sequence $p_{t}$ generated by (23) is interior for all $t\ge 0$.

Incidentally, note that (49) implies that all optimal plans converge to the steady value $k^{s}=\eta /\left( \mu \delta \right)$ defined in (26) from above; that is, starting from any initial state $\left( k_{0},p_{0}\right)$ satisfying (17) and (18), all optimal sequences $k_{t}^{*}$ generated by (22) contain capital levels which are all larger than the asymptotic value $k^{s}=\eta /\left( \mu \delta \right)$. This observation also explains condition (15).

We now apply Lemma 2 to show that the function $V\left( k,p\right)$ defined in (19) is indeed the value function of problem (11). To establish Property 1 of the Lemma recall that there is a maximum capital level, $\bar{k}>0$, that can be sustained in the long run: as $p_{t}\ge 0$ for all $t\ge 0$, from the first constraint in (4) such upper bound is easily obtained noting that

$$\begin{aligned} k_{t+1}\le z_{t}\left( 1+p_{t}\right) ^{-\phi }k_{t}^{\alpha }+\left( 1-\delta \right) k_{t}\le \bar{z}k_{t}^{\alpha }+\left( 1-\delta \right) k_{t} \end{aligned}$$

and then solving $k=\bar{z}k^{\alpha }+\left( 1-\delta \right) k$ for k, which yields $\bar{k}=\left( \delta /\bar{z}\right) ^{\frac{1}{\alpha -1}}$. Then

$$\begin{aligned} V\left( k,p\right)&=\rho _{1}+\frac{1+\beta \theta }{1-\beta }\ln \left[ \mu \left( 1-\delta \right) k-\left( 1-\eta \right) p-\frac{\eta -\delta }{\delta }\right] +\theta \ln \left( 1-p\right) \\&\le \rho _{1}+\frac{1+\beta \theta }{1-\beta }\ln \left[ \mu \left( 1-\delta \right) \left( \delta /\bar{z}\right) ^{\frac{1}{\alpha -1}} -\frac{\eta -\delta }{\delta }\right] \\&=M, \end{aligned}$$

so that $\lim \inf _{t\rightarrow \infty }\beta ^{t}V\left( k_{t},p_{t}\right) \le \lim _{t\rightarrow \infty }\beta ^{t}V\left( k_{t},p_{t}\right) =M\lim _{t\rightarrow \infty }\beta ^{t}=0$.

To show that Property 2 of Lemma 2 holds as well recall the notation

$$\begin{aligned} W\left( \left( k_{0},p_{0}\right) ,\left\{ k_{t},p_{t}\right\} \right)&=\sum _{0}^{\infty }\beta ^{t}\left\{ \ln \left[ \frac{1}{\mu }p_{t+1} -\frac{1-\eta }{\mu }p_{t}+\left( 1-\delta \right) k_{t}-k_{t+1}\right] \right. \nonumber \\&\quad \left. +\,\theta \ln \left( 1-p_{t}\right) \right\} . \end{aligned}$$

(50)

We consider two types of feasible plans satisfying condition (17):

(i):

those satisfying $W\left( \left( k_{0},p_{0}\right) ,\left\{ k_{t},p_{t}\right\} \right) >-\infty$ and

(ii):

those satisfying $W\left( \left( k_{0},p_{0}\right) ,\left\{ k_{t},p_{t}\right\} \right) =-\infty$.

Plans of type (i) necessarily satisfy

$$\begin{aligned} \lim _{t\rightarrow \infty }\beta ^{t}\left\{ \ln \left[ \frac{1}{\mu } p_{t+1}-\frac{1-\eta }{\mu }p_{t}+\left( 1-\delta \right) k_{t}-k_{t+1}\right] +\theta \ln \left( 1-p_{t}\right) \right\} =0. \end{aligned}$$

(51)

Note that under our assumptions both arguments of the logs in the last expression are bounded from above; therefore, both of them can only escape to $-\infty$, and, as $0<\left( 1-\beta \right) /\left( 1+\beta \theta \right) <1$, we can safely claim that the limit in (51) implies

$$\begin{aligned} \lim _{t\rightarrow \infty }\beta ^{t}\left\{ \ln \left[ \frac{1}{\mu } p_{t+1}-\frac{1-\eta }{\mu }p_{t}+\left( 1-\delta \right) k_{t}-k_{t+1}\right] +\frac{\theta \left( 1-\beta \right) }{1+\beta \theta }\ln \left( 1-p_{t} \right) \right\} =0 \end{aligned}$$

(52)

as well. Condition (17) implies that

$$\begin{aligned} \frac{1}{\mu }p_{t+1}-k_{t+1}\le -\frac{\eta -\delta }{\mu \delta }, \end{aligned}$$

so that

$$\begin{aligned} \ln \left[ \frac{1}{\mu }p_{t+1}-\frac{1-\eta }{\mu }p_{t}+\left( 1-\delta \right) k_{t}-k_{t+1}\right]&\le \ln \left[ \left( 1-\delta \right) k_{t}-\frac{1-\eta }{\mu }p_{t}-\frac{\eta -\delta }{\mu \delta }\right] \nonumber \\&=\ln \left[ \mu \left( 1-\delta \right) k_{t}-\left( 1-\eta \right) p_{t}-\frac{\eta -\delta }{\delta }\right] -\ln \mu , \end{aligned}$$

(53)

and thus

$$\begin{aligned}&\lim \sup _{t\rightarrow \infty }\beta ^{t}V\left( k_{t},p_{t}\right) \\&\quad =\lim \sup _{t\rightarrow \infty }\beta ^{t}\left\{ \rho _{1}+\frac{1+\beta \theta }{1-\beta }\ln \left[ \mu \left( 1-\delta \right) k_{t}-\left( 1-\eta \right) p_{t}-\frac{\eta -\delta }{\delta }\right] +\theta \ln \left( 1-p_{t}\right) \right\} \\&\quad =\frac{1+\beta \theta }{1-\beta }\lim \sup _{t\rightarrow \infty }\beta ^{t}\left\{ \ln \left[ \mu \left( 1-\delta \right) k_{t}-\left( 1-\eta \right) p_{t}-\frac{\eta -\delta }{\delta }\right] \right. \\&\qquad \left. +\,\frac{\theta \left( 1-\beta \right) }{1+\beta \theta }\ln \left( 1-p_{t}\right) \right\} \\&\quad \ge \frac{1+\beta \theta }{1-\beta }\lim \sup _{t\rightarrow \infty }\beta ^{t}\left\{ \ln \left[ \frac{1}{\mu }p_{t+1}-\frac{1-\eta }{\mu }p_{t}+\left( 1-\delta \right) k_{t}-k_{t+1}\right] \right. \\&\qquad \left. +\ln \mu +\frac{\theta \left( 1-\beta \right) }{1+\beta \theta }\ln \left( 1-p_{t}\right) \right\} \\&\quad =\frac{1+\beta \theta }{1-\beta }\lim _{t\rightarrow \infty }\beta ^{t}\left\{ \ln \left[ \frac{1}{\mu }p_{t+1}-\frac{1-\eta }{\mu }p_{t}+\left( 1-\delta \right) k_{t}-k_{t+1}\right] \right. \\&\qquad \left. +\,\frac{\theta \left( 1-\beta \right) }{1+\beta \theta }\ln \left( 1-p_{t}\right) \right\} +\frac{1+\beta \theta }{1-\beta }\left( \ln \mu \right) \lim _{t\rightarrow \infty }\beta ^{t}=0, \end{aligned}$$

where in the first inequality we used (53) while the last equality holds because of (52). Therefore conditions 2a and 2b of Lemma 2 hold (with equality) for the (same) plan $\left( ( k_{0},p_{0}) ,\left\{ k_{t}^{\prime } ,p_{t}^{\prime }\right\} \right) =\left( ( k_{0},p_{0}) ,\left\{ k_{t},p_{t}\right\} \right)$ when the latter is of type (i).

As far as plans $\left( ( k_{0},p_{0}) ,\left\{ k_{t} ,p_{t}\right\} \right)$ of type (ii) are concerned, we take the optimal plan generated by (22) and (23) as reference plan $\left( ( k_{0},p_{0}) ,\left\{ k_{t}^{\prime },p_{t}^{\prime }\right\} \right)$, that is, we calculate the exact solution of the difference equation (27):

$$\begin{aligned} \left[ \begin{array}[c]{c} k_{t+1}\\ p_{t+1} \end{array} \right] =\left[ \begin{array}[c]{cc} \gamma _{1} &{}\quad \gamma _{2}\\ \gamma _{4} &{}\quad \gamma _{5} \end{array} \right] \left[ \begin{array}[c]{c} k_{t}\\ p_{t} \end{array} \right] +\left[ \begin{array}[c]{c} \gamma _{3}\\ \gamma _{6} \end{array} \right] . \end{aligned}$$

Through direct computation it is easily seen that the matrix (28) characterizing the dynamic (27) happens to be singular and has two eigenvalues: $\lambda _{1}=0$ and $\lambda _{2}=\beta \left( 1-{\delta }\right)$. As $\lambda _{2}<1$ the steady state in (26) is globally stable. The associated eigenvectors are:

$$\begin{aligned} \left[ \begin{array}[c]{c} 1\\ \dfrac{\mu \left( 1-\delta \right) }{1-\eta } \end{array} \right] \text { for }\lambda _{1}=0\quad \text {and}\quad \left[ \begin{array}[c]{c} 1\\ -\dfrac{\mu \theta \left( 1-\beta \right) \left( 1-{\delta }\right) }{\theta \left[ {\beta }\left( 1-\delta \right) -\left( 1-\eta \right) \right] +\eta -{\delta }} \end{array} \right] \ \text { for }\lambda _{2}=\beta \left( 1-{\delta }\right) . \end{aligned}$$

Note that condition (44) implies that the second eigenvector—that associated to the positive eigenvalue—has negative slope. To compute the exact solution we solve the general solution

$$\begin{aligned} \left\{ \begin{array}[c]{l} k_{t}=c_{2}\left[ \beta \left( 1-{\delta }\right) \right] ^{t}+\dfrac{\eta }{\mu \delta }\\ p_{t}=-c_{2}\dfrac{\mu \theta \left( 1-\beta \right) \left( 1-{\delta }\right) }{\theta \left[ {\beta }\left( 1-\delta \right) -\left( 1-\eta \right) \right] +\eta -{\delta }}\left[ \beta \left( 1-{\delta }\right) \right] ^{t}+1, \end{array} \right. \end{aligned}$$

for the initial values $\left( k_{0},p_{0}\right)$ in $t=0$, that is, we solve

$$\begin{aligned} \left\{ \begin{array}[c]{l} k_{0}=c_{1}+c_{2}\left[ \beta \left( 1-{\delta }\right) \right] ^{t} +\dfrac{\eta }{\mu \delta }\\ p_{0}=c_{1}\dfrac{\mu \left( 1-\delta \right) }{1-\eta }-c_{2}\dfrac{\mu \theta \left( 1-\beta \right) \left( 1-{\delta }\right) }{\theta \left[ {\beta }\left( 1-\delta \right) -\left( 1-\eta \right) \right] +\eta -{\delta }}\left[ \beta \left( 1-{\delta }\right) \right] ^{t}+1, \end{array} \right. \end{aligned}$$

for the constants $c_{1}$ and $c_{2}$, yielding the values

$$\begin{aligned} c_{1}&={\frac{\theta \left( 1-\beta \right) \left( 1-\eta \right) }{\left( \eta -\delta \right) \left( 1+\beta \theta \right) }}k_{0} +{\frac{\theta \left[ {\beta }\left( 1-\delta \right) -\left( 1-\eta \right) \right] +\eta -{\delta }}{\mu \left( 1-\delta \right) \left( \eta -\delta \right) \left( 1+\beta \theta \right) }}\left( 1-\eta \right) p_{0}\\&\quad -{\frac{\theta \left( 1-{\beta }\right) +{\delta }\left( 1+\beta \theta \right) }{\mu \delta \left( 1-\delta \right) \left( 1+\beta \theta \right) }}( 1-\eta ) ,\\ c_{2}&=\frac{\theta \left[ {\beta }\left( 1-\delta \right) -\left( 1-\eta \right) \right] +\eta -{\delta }}{1+\beta \theta }\left[ {\frac{k_{0} }{\eta -\delta }}-\frac{\left( 1-\eta \right) p_{0}}{\mu \left( 1-{\delta }\right) \left( \eta -\delta \right) }-\frac{1}{\mu \delta \left( 1-{\delta }\right) }\right] {.} \end{aligned}$$

Note that $c_{2}>0$ holds because of condition (44) and because the initial values $\left( k_{0},p_{0}\right)$ satisfy (16) and (17), which imply that the term in square bracket is strictly positive.

Using the exact solution just found for the optimal plan $\left( \left( k_{0},p_{0}\right) ,\left\{ k_{t}^{\prime },p_{t}^{\prime }\right\} \right)$ we can elaborate the argument of the first log in the welfare function $W\left( \left( k_{0},p_{0}\right) ,\left\{ k_{t}^{\prime },p_{t}^{\prime }\right\} \right)$ defined in (50) as

$$\begin{aligned} \frac{1}{\mu }p_{t+1}^{\prime }-\frac{1-\eta }{\mu }p_{t}^{\prime }+\left( 1-\delta \right) k_{t}^{\prime }-k_{t+1}^{\prime }=\frac{\left( 1-\beta \right) G_{0}}{\mu \delta \left( 1+\beta \theta \right) }\left[ \beta \left( 1-{\delta }\right) \right] ^{t}, \end{aligned}$$

where, to simplify notation, we have set

$$\begin{aligned} G_{0}=\mu \delta \left( 1-\delta \right) k_{0}-\delta \left( 1-\eta \right) p_{0}-\left( \eta -\delta \right) . \end{aligned}$$

Note that under (17) $G_{0}>0$ certainly holds. Similarly, the argument in the first log of the function $V\left( k_{t}^{\prime } ,p_{t}^{\prime }\right)$ defined in (19) becomes

$$\begin{aligned} \mu \left( 1-\delta \right) k_{t}^{\prime }-\left( 1-\eta \right) p_{t}^{\prime }-\frac{\eta -\delta }{\delta }=\frac{G_{0}}{\delta }\left[ \beta \left( 1-{\delta }\right) \right] ^{t}, \end{aligned}$$

while the argument in the second log of both functions W and V becomes

$$\begin{aligned} 1-p_{t}^{\prime }={\frac{\theta \left( 1-\beta \right) G_{0}}{\delta \left( \eta -\delta \right) \left( 1+\beta \theta \right) }}\left[ \beta \left( 1-{\delta }\right) \right] ^{t}. \end{aligned}$$

Hence, condition 2a of Lemma 2 holds (with strict inequality) for our reference (optimal) plan $\left( \left( k_{0} ,p_{0}\right) ,\left\{ k_{t}^{\prime },p_{t}^{\prime }\right\} \right)$ with respect to any given plan $\left( \left( k_{0},p_{0}\right) ,\left\{ k_{t},p_{t}\right\} \right)$ such that $W\left( \left( k_{0},p_{0}\right) ,\left\{ k_{t},p_{t}\right\} \right) =-\infty$ as

$$\begin{aligned}&W\left( \left( k_{0},p_{0}\right) ,\left\{ k_{t}^{\prime },p_{t} ^{\prime }\right\} \right) \\&\quad =\sum _{0}^{\infty }\beta ^{t}\left\{ \ln \left[ \frac{1}{\mu }p_{t+1} ^{\prime }-\frac{1-\eta }{\mu }p_{t}^{\prime }+\left( 1-\delta \right) k_{t}^{\prime }-k_{t+1}^{\prime }\right] +\theta \ln \left( 1-p_{t}^{\prime }\right) \right\} \\&\quad =\sum _{0}^{\infty }\beta ^{t}\left\{ \ln \left[ \frac{\left( 1-\beta \right) G_{0}}{\mu \delta \left( 1+\beta \theta \right) }\left[ \beta \left( 1-{\delta }\right) \right] ^{t}\right] +\theta \ln \left[ {\frac{\theta \left( 1-\beta \right) G_{0}}{\delta \left( \eta -\delta \right) \left( 1+\beta \theta \right) }}\left[ \beta \left( 1-{\delta }\right) \right] ^{t}\right] \right\} \\&\quad =\sum _{0}^{\infty }\beta ^{t}\left\{ \ln \left[ \frac{\left( 1-\beta \right) G_{0}}{\mu \delta \left( 1+\beta \theta \right) }\right] +t\ln \left[ \beta \left( 1-{\delta }\right) \right] +\theta \ln \left[ {\frac{\theta \left( 1-\beta \right) G_{0}}{\delta \left( \eta -\delta \right) \left( 1+\beta \theta \right) }}\right] \right. \\&\qquad \left. +\,\theta t\ln \left[ \beta \left( 1-{\delta }\right) \right] \right\} \\&\quad =\ln \left[ \frac{\left( 1-\beta \right) G_{0}}{\mu \delta \left( 1+\beta \theta \right) }\right] \sum _{0}^{\infty }\beta ^{t}+\left( 1+\theta \right) \ln \left[ \beta \left( 1-{\delta }\right) \right] \sum _{0}^{\infty }t\beta ^{t}\\&\qquad +\theta \ln \left[ {\frac{\theta \left( 1-\beta \right) G_{0}}{\delta \left( \eta -\delta \right) \left( 1+\beta \theta \right) } }\right] \sum _{0}^{\infty }\beta ^{t}\\&\quad =\frac{1}{1-\beta }\left\{ \ln \left[ \frac{\left( 1-\beta \right) G_{0} }{\mu \delta \left( 1+\beta \theta \right) }\right] +\frac{\beta \left( 1+\theta \right) \ln \left[ \beta \left( 1-{\delta }\right) \right] }{1-\beta }+\theta \ln \left[ {\frac{\theta \left( 1-\beta \right) G_{0}}{\delta \left( \eta -\delta \right) \left( 1+\beta \theta \right) }}\right] \right\} \\&\quad >-\infty =W\left( \left( k_{0},p_{0}\right) ,\left\{ k_{t} ,p_{t}\right\} \right) . \end{aligned}$$

Similarly, condition 2b of Lemma 2 holds for our reference (optimal) plan $\left( \left( k_{0},p_{0}\right) ,\left\{ k_{t}^{\prime },p_{t}^{\prime }\right\} \right)$ as

$$\begin{aligned}&\lim \sup _{t\rightarrow \infty }\beta ^{t}V\left( k_{t}^{\prime },p_{t} ^{\prime }\right) \\&\quad \ge \lim _{t\rightarrow \infty }\beta ^{t}V\left( k_{t}^{\prime },p_{t} ^{\prime }\right) =\lim _{t\rightarrow \infty }\beta ^{t}\left\{ \rho _{1} +\frac{1+\beta \theta }{1-\beta }\ln \left[ \mu \left( 1-\delta \right) k-\left( 1-\eta \right) p-\frac{\eta -\delta }{\delta }\right] \right. \\&\qquad \left. +\,\theta \ln \left( 1-p\right) \right\} \\&\quad =\lim _{t\rightarrow \infty }\beta ^{t}\left\{ \frac{1+\beta \theta }{1-\beta }\ln \left[ \frac{G_{0}}{\delta }\left[ \beta \left( 1-{\delta }\right) \right] ^{t}\right] +\theta \ln \left[ {\frac{\theta \left( 1-\beta \right) G_{0}}{\delta \left( \eta -\delta \right) \left( 1+\beta \theta \right) } }\right] +\theta t\ln \left[ \beta \left( 1-{\delta }\right) \right] \right\} \\&\quad =\frac{1+\beta \theta }{1-\beta }\ln \left( \frac{G_{0}}{\delta }\right) \lim _{t\rightarrow \infty }\beta ^{t}+\theta \ln \left[ {\frac{\theta \left( 1-\beta \right) G_{0}}{\delta \left( \eta -\delta \right) \left( 1+\beta \theta \right) }}\right] \lim _{t\rightarrow \infty }\beta ^{t}\\&\qquad +\left( \frac{1+\beta \theta }{1-\beta }+\theta \right) \ln \left[ \beta \left( 1-{\delta }\right) \right] \lim _{t\rightarrow \infty }\left( t\beta ^{t}\right) \\&\quad =0. \end{aligned}$$

As we have just found that the plan $\left( k_{t}^{*},p_{t}^{*}\right)$ generated by (22) and (23) satisfies $\lim _{t\rightarrow \infty }\beta ^{t}V\left( k_{t}^{*},p_{t}^{*}\right) =0$, Lemma 3 establishes that such plan is indeed optimal.

Finally, we replace the expressions of (22) and (23) into (7) and (10) to find the optimal paths of consumption and production capacity index to obtain:

$$\begin{aligned} c_{t}&=\frac{1}{\mu }\left( \gamma _{4}k_{t}+\gamma _{5}p_{t}+\gamma _{6}\right) -\frac{1-\eta }{\mu }p_{t}+\left( 1-\delta \right) k_{t} -\gamma _{1}k_{t}-\gamma _{2}p_{t}-\gamma _{3}\\&=\left( \frac{\gamma _{4}}{\mu }-\gamma _{1}+1-\delta \right) k_{t}+\left[ \frac{\gamma _{5}-\left( 1-\eta \right) }{\mu }-\gamma _{2}\right] p_{t} +\frac{\gamma _{6}}{\mu }-\gamma _{3}\\&=\gamma _{1}^{c}k_{t}+\gamma _{2}^{c}p_{t}+\gamma _{3}^{c}\\ z_{t}&=\left( 1-p_{t}\right) ^{\phi }k_{t}^{-\alpha }\left[ \frac{1}{\mu }\left( \gamma _{4}k_{t}+\gamma _{5}p_{t}+\gamma _{6}\right) -\frac{1-\eta }{\mu }p_{t}\right] \\&=\left( 1-p_{t}\right) ^{\phi }k_{t}^{-\alpha }\left[ \frac{\gamma _{4} }{\mu }k_{t}+\frac{\gamma _{5}-\left( 1-\eta \right) }{\mu }p_{t}+\frac{\gamma _{6}}{\mu }\right] \\&=\left( 1-p_{t}\right) ^{\phi }k_{t}^{-\alpha }\left( \gamma _{1}^{z} k_{t}+\gamma _{2}^{z}p_{t}+\gamma _{3}^{z}\right) , \end{aligned}$$

so that the expressions of the coefficients $\gamma _{1}^{c}$, $\gamma _{2}^{c}$, $\gamma _{3}^{c}$, $\gamma _{1}^{z}$, $\gamma _{2}^{z}$, $\gamma _{3}^{z}$, correspond to those below conditions (24) and (25) in point 3 if Proposition 1. Clearly, by substituting the steady state values $k^{s}=\eta /\left( \mu \delta \right)$ and $p^{s}=1$ into the two expressions above, one finds the steady state values for the control variables, $c^{s}$ and $z^{s}$ as in (26). $\square$