The Environmental Kuznets Curve and the Structural Change Hypothesis

Abstract

We provide a very simple macroeconomic investigation of the role that structural changes might play in generating inverted U-shaped income–pollution relationships. Differently from previous research which mainly focuses on empirical, static or general equilibrium models, we develop a standard balanced growth path (BGP) analysis. We show that along the BGP equilibrium an inverted U-shaped income–pollution relationship may occur as a response to structural changes, but whether this is the case or not it will crucially depend upon the magnitude of a production externality parameter. Moreover, we show that the negative relationship between income and pollution can only be a transitory phenomenon, and in the long run pollution will increase as income rises, generating overall an N-shaped pattern.

Appendices

Appendix 1: BPG Equilibrium

From Eqs. (6) and (7), since $u_{t}\in (0,1)$, it is clear that along the BGP we need $\gamma _{k}=\gamma _{x}$ in order for $r_{t}$ and $p_{t}$ to be constant; this implies that also the ratio $\frac{x_{t}}{k_t}$ is constant. Household’s maximization, along with the state equations and the transversality conditions (TVCs), yields to the following first order conditions:

$$\begin{aligned} {\uplambda }_{t}= & {} c_{t}^{-\sigma }e^{-\rho t} \end{aligned}$$

(17)

$$\begin{aligned} {\uplambda }_{t}p= & {} \mu _{t}\theta \end{aligned}$$

(18)

$$\begin{aligned} -\dot{{\uplambda }}_{t}= & {} {\uplambda }_{t}r_{t} \end{aligned}$$

(19)

$$\begin{aligned} -\dot{\mu }_{t}= & {} {\uplambda }_{t}p_{t}u_{t}+\mu _{t}\left[ \theta (1-u_{t})+\varphi \gamma _{k} \right] \end{aligned}$$

(20)

where ${\uplambda }_{t}$ and $\mu _{t}$ denote the costate variables associated to manufacturing and services capital, respectively. Differentiating (18) with respect to time and plugging (19) and (20) into the derived equation yields:

$$\begin{aligned} r_{t}=\theta +\varphi \gamma _{k}=r, \end{aligned}$$

(21)

which by substituting (6) can be rewritten as:

$$\begin{aligned} \frac{u_{t}x_{t}}{k_{t}}=\left[ \frac{\theta +\varphi \gamma _{k}}{a(1-\alpha )}\right] ^{1-\alpha }; \end{aligned}$$

(22)

differentiating (17) with respect to time and plugging (21) in the derived equation yields:

$$\begin{aligned} \gamma _{c}=\frac{\theta +\varphi \gamma _{k}-\rho }{\sigma }. \end{aligned}$$

(23)

From (8), we need $\gamma _{k}=\gamma _{c}$ in order to have long run growth and not to violate the TVCs. Thus, we define the economic growth rate as $\gamma \equiv \gamma _{k}=\gamma _{c}=\gamma _{x}=\gamma _{y}$. By solving (23) for $\gamma$, we obtain:

$$\begin{aligned} \gamma =\frac{\theta -\rho }{\sigma -\varphi }. \end{aligned}$$

(24)

From (9), we can also obtain:

$$\begin{aligned} \gamma =\theta (1-u_{t})+\varphi \gamma . \end{aligned}$$

(25)

Equating (24) and (25) yields:

$$\begin{aligned} u_{t}=\frac{\theta (\sigma -1)+\rho (1-\varphi )}{\theta (\sigma -\varphi )}=\overline{{u}}\in (0,1). \end{aligned}$$

(26)

Substituting (24) and (26) into (21)and (22) leads to:

$$\begin{aligned}&\displaystyle r=\frac{\theta \sigma -\varphi \rho }{\sigma -\varphi } \end{aligned}$$

(27)

$$\begin{aligned}&\displaystyle \frac{k_{t}}{x_{t}}=\left\{ \left[ \frac{\theta \sigma -\varphi \rho }{(1-\alpha )(\sigma -\varphi )}\right] ^{\frac{1}{\alpha }} \frac{\theta (\sigma -\varphi )}{\theta (\sigma -1)+\rho (1-\varphi )}\right\} ^{\frac{\alpha }{\phi -\alpha }}. \end{aligned}$$

(28)

The pollution growth rate can be directly found by differentiating (5) with respect to time, which yields:

$$\begin{aligned} \gamma _{z}=\psi (1-\alpha )\gamma . \end{aligned}$$

(29)

Note that since $\sigma >1$ as long as $\theta >\rho$ the growth rate is positive, $\gamma >0$, and the share of services allocated to final production is positive and smaller than one, $\overline{u}\in (0,1)$. Moreover, since $\alpha \in (0,1)$ and $\psi >0$ the growth rate of pollution is strictly positive, $\gamma _{z}>0$, and its relation with $\gamma$ depends on whether $\psi (1-\alpha )$ is larger or smaller than 1.

Along the BGP, Eq. (28) can be rewritten as:

$$\begin{aligned} k_{t}=\left( \frac{r}{1-\alpha }\right) ^{\frac{1}{\phi -\alpha }}\overline{u}^{-\frac{\alpha }{\phi -\alpha }}x_{t}, \end{aligned}$$

(30)

which plugged into (3), along with (4), yields:

$$\begin{aligned} y_{t}=\overline{u}^{-\frac{\alpha }{\phi -\alpha }}\left( \frac{r}{1-\alpha }\right) ^{\frac{1+\phi -\alpha }{\phi -\alpha }}x_{t}. \end{aligned}$$

(31)

This last expression, along with (5), is used to derive the results in Proposition 2.

Appendix 2: A Different Pollution Specification

So far we have assumed that services are a totally clean production factor, and such an assumption may be thought to be the main driver of the results presented in this paper. Therefore, in order to understand to what extent such a claim is true, we now consider a more realistic pollution specification, and in particular we assume that pollution depends not only on the manufacturing intensity, as in (5), but also on the services intensity employed in the production of the final consumption good. Specifically, pollution is now given by the following expression:

$$\begin{aligned} z_{t} = \eta \left( {u_{t}x_{t}} \right) ^{\psi _{1}\alpha }k_{t}^{\psi _{2}\left( {1-\alpha }\right) }, \end{aligned}$$

(32)

where $\psi _{1},\psi _{2}>0$ and $\psi _{1}<\psi _{2}$. This latter parametric condition states that (reasonably) the manufacturing sector has a larger degree of dirtiness than the services sector. As we will show in a while, replacing (5) with (32) leads to results qualitatively not different from those discussed in the main text. Indeed, it is straightforward to show that the economic growth rate and the share of services allocated to final production are still equal to (14) and (16), respectively. However, the growth rate of pollution along the BGP is obviously different from (15), but it is straightforward to verify that it is still strictly positive since it is given by the following expression:

$$\begin{aligned} \gamma _{z} =\left[ {\psi _{1}\alpha +\psi _{2}\left( {1-\alpha }\right) }\right] \gamma >0. \end{aligned}$$

(33)

Note that in the case $\psi _{1}=0$ and $\psi _{2}=\psi$ we are back to the case considered in the main text, since (33) would simplify in $\gamma _{z}=\psi (1-\alpha )\gamma$.

By differentiating (32) with respect to $1-\alpha$, it is straightforward also to show that the relationship between pollution and the manufacturing share of GDP is monotonically increasing if the following condition holds $\psi _{2}>\frac{\psi _{1}\ln \overline{{u}}\ln x_{t}}{\ln k_{t}}$, which after some algebra can be rewritten as follows:

$$\begin{aligned} \psi _{2}>\psi _{1}\left[ {1+\frac{\ln \left( {\frac{1-\alpha }{r}}\right) +\phi \ln \overline{{u}}}{\left( {\phi -\alpha } \right) \ln k_{t}}}\right] . \end{aligned}$$

(34)

Since by assumption $\psi _{2}>\psi _{1}$, if $\phi >\alpha$, since the second term in the brackets is negative (remember that $\ln \overline{{u}}<0$ and, since $r>1$, also $\ln ({\frac{1-\alpha }{r}})<0)$, the relationship between pollution and the manufacturing share is positive (exactly as in Proposition 3, case (ii)). If instead $\phi <\alpha$, a sufficient condition for the second term in the brackets to be negative requires that $\phi$ is negative and smaller than a certain value $\phi <-\frac{\ln ({\frac{1-\alpha }{r}})}{\ln \overline{{u}}}$ (this additional condition complicates a bit the restriction to be imposed in order to observe a falling arm, but the result is very similar to Proposition 3, case (i)). Again note that if $\psi _{1}=0$ our result of Proposition 3 is reestablished, since (34) would simplify in $\psi _{2}>0$. This means that as long as $\psi _{2}>\psi _{1}$, a hump-shaped EKC is consistent with a sectoral shifts if $\phi <\min \left[ -\frac{\ln ({\frac{1-\alpha }{r}})}{\ln \overline{{u}}}, \frac{\omega -\sqrt{\omega ^2+4\epsilon }}{2}\right]$ (similarly to Proposition 3, case (i)) or if $\phi >\alpha$ (exactly as in Proposition 3, case (ii)). If such conditions are met, precisely the same results discussed in the main text hold. This confirms that the specification of pollution as in (5) does not drive our results but it is merely a simplifying assumption, useful to stress that even in the case in which services are totally pollution-free a hump-shaped EKC does not necessarily occur as a response to structural change.