Green Human Capital, Innovation and Growth

Abstract

This paper examines why a growth process relying on both green innovation and green human capital may be responsible for higher inequality within and between skills. I propose a theoretical framework and derive some empirical observations using data from more than 2000 companies in 21 OECD countries in 2022. I discuss policy implications of this analysis in light of the COVID-19 pandemic which has led many governments to place green investment at the heart of their recovery plans.

Appendix 1. Model’s Variables and Parameters

Symbol	Description	Value or price
Y	Final good	numeraire
l	unskilled labor	unskilled wage rate $w^u$
x(i)	intermediate good i	p(i)
$\alpha$	output elasticity of unskilled labor	$\in ]0,1[$
$w^u$	unskilled wage rate
n ($n^*$)	number of brown (green) intermediate goods	$n+n^*=N$
$\pi$ ($\pi ^*$)	profit of a brown (green) intermediate firm
$w^h$ ($(w^h)^*$)	skilled wage rate in a brown (green) intermediate firm
$\delta$ ($\delta ^*$)	efficiency of brown (green) technology	$>0$
$\rho$ ($\rho ^*$)	brown (green) knowledge externalities	$\in ]0,1]$
$h_r$ ($h_r^*$)	skilled workers in brown (green) research
$H_r$	skilled workers demand in research	$H_r=h_r+h_r^*$
H	skilled workers demand in intermediate goods sector	$H=h+h^*$
$H^d$	total skilled workers demand	$H^d=H+H_r$
V ($V^*$)	brown (green) patent price
r, $\beta$	interest rate, discount rate
$\Pi _r$ ($\Pi _r^*$)	brown (green) profits in the research sector	null (free-entry condition)
$w_{h,r}$ ($w_{h,r}^*$)	skilled wage rate in the brown (green) research sector
s	share of green-motivated workers	$\in ]0,1]$
G (B)	share of green (ordinary) workers acquiring skills	$\in ]0,1[$
$\overline{H}$ ($\underline{H}$), L	number of skilled green (ordinary), unskilled workers	$\in [0,1]$
$\overline{\varphi }$ ($\underline{\varphi }$), $\varphi$	returns to human capital for skilled green (ordinary), unskilled workers
$H^s$	total skilled workers supply	$H^s=\overline{\varphi }\overline{H}+\underline{\varphi }\underline{H}$
$\xi$	education cost	$\in ]0,1[$
$\Omega$	household expected income	$\Omega =\{\omega ^u, \underline{\omega ^h}, \overline{\omega ^h}\}$
$\overline{\omega ^h}$ $(\underline{\omega ^h})$, $\omega ^u$	expected income of skilled green (ordinary), unskilled workers
u	instantaneous household utility function
c	per capita consumption
e	environmental quality
$\theta$, $\mu$	elasticity of utility with respect to c and e
$\sigma$	elasticity of utility with respect to consumption growth
$\eta$	elasticity of environmental quality with respect to n and $n^*$
A	household asset (nonhuman wealth)
g	economy’s growth rate
$\Sigma ^{h}$	average income of skilled workers	$(\overline{\varphi }\overline{H}+\underline{\varphi }\underline{H})w^h/(\overline{H}+\underline{H})$
$\Sigma ^{u}$	average income of unskilled workers	$\varphi w^u$
$\Sigma ^{within}$	wage inequality within skilled workers	$\overline{\omega ^h}\overline{H}/\underline{\omega ^h}\underline{H}$
$\Sigma ^{between}$	wage inequality between skilled and unskilled workers	$\Sigma ^h/\Sigma ^u$

1.1 Appendix 2. Equilibrium and Steady State

We compute here the equilibrium of this economy.

1.1.1 A2.1 Final Sector Market

Each producer in the final sector determines the amount of skilled labor and intermediate goods to be used as inputs by maximizing profit under technological constraint, leading to the following inverse demand functions for unskilled labor and intermediate goods:

$$\begin{aligned} p(i)=(1-\alpha )\left( l \right) ^{\alpha }\left( x(i)\right) ^{-\alpha }, \qquad w^u=\alpha \left( l \right) ^{\alpha -1}\int _{0}^{N}x(i)^{1-\alpha }di , \end{aligned}$$

(25)

where p(i) denotes the price of intermediate good i and $w^u$ is the wage rate of unskilled workers.

1.1.2 A2.2 Equilibrium in the R&D Sector

The value of the patent is such that researchers are indifferent between investing in a risk-free asset or buying a patent, producing an intermediate good for one period, and then selling the patent after a year that is from Eq. 10:

$rV= \pi + \dot{V}$ and $rV^* = \pi ^* + \dot{V^*}$. Dividing by V and $V^*$ we get:

$$\begin{aligned} r=\pi /V+\dot{V}/V \text { and } r=\pi ^*/V^*+\dot{V^*}/V^* \end{aligned}$$

(26)

Along the equilibrium balanced growth path, r is constant, therefore on the right-hand side of Eq. 26$\pi /V$ $\pi */V*$, $\dot{V}/V$ and $\dot{V*}/V*$ must be constant. This implies that $\pi$, $\pi *$, V and $V*$grow at the same rate along the balanced growth path.

Note that in equilibrium, from Eqs. 4, 5 and 29 we have $\pi ^*=\pi =\alpha p x = (\alpha (1-\alpha )) Y/N$. Moreover, along the equilibrium balanced growth path, Y and N grow at the same rate, which implies that the growth rate of $\pi$ and $pi^*$ is null along the balanced growth path, as well as the growth rate of V and $V^*$). We thus have $\dot{V}/V=0$ and $\dot{V^*}/V^*=0$, that is:

$$\begin{aligned} rV= \pi \text { and } rV^* = \pi ^* . \end{aligned}$$

(27)

From Eqs. 4 and 5 we have $\pi =\alpha p x$ and $\pi ^* = \alpha p^* x^*$ which leads to $rV=\alpha p x$ and $rV^* = \alpha p^* x^*$

Using Eq. 9 we get $rV=rw_r^h /(\delta n^\rho (n*)^{1-\rho })$ and $rV^*=r(w_r^h)^* /(\delta * (n*)^\rho * (n)^{1-\rho ^*})$. Rearranging a bit we finally have:

$$\begin{aligned} w_{h,r}^{*}=\frac{1}{r }\alpha \delta ^* (n^*)^{\rho ^{*}} (n)^{1-\rho ^*} p^{*} x^{*} \text { and } w_{h,r}=\frac{1}{r }\alpha \delta (n)^{\rho } (n^*)^{1-\rho } p x. \end{aligned}$$

(28)

1.1.3 A2.3 Allocation of Skilled Workers Between Sectors

The condition which determines the allocation of skilled workers between sectors expresses that skilled workers must be indifferent between working in the research sector and in the intermediate goods sector.

The indifference condition for skilled workers between working in any intermediate firm i and between working in the intermediate goods sector and in the R &D sector implies the equality of wage rates across sectors:

$$\begin{aligned} w^h(i)=(w^h)^*=w^h=(w^{h}_{r})^{*}=w^{h}_{r} \quad \text { for } i \in [0,N]. \end{aligned}$$

(29)

Using Eqs. 4, 5 and 28 this leads to:

$$\begin{aligned} \frac{1}{r }\alpha \delta ^* (n^*)^{\rho ^{*}} (n)^{1-\rho ^*} p^{*} x^{*} = (1-\alpha )p^* = \frac{1}{r }\alpha \delta (n)^{\rho } (n^*)^{1-\rho } p x = (1-\alpha ) p, \end{aligned}$$

(30)

that is after some simple manipulations:

$$\begin{aligned} \frac{n^*}{n}=\left( \frac{\delta }{\delta ^*}\right) ^{\frac{1}{\rho +\rho ^*-1}}. \end{aligned}$$

(31)

1.1.4 A2.4 Education Decisions

The population size is constant and normalized to one. Individuals choose to become skilled workers if the skilled income is higher than the unskilled one. Here, individuals with basic motivation then choose to become skilled workers as long as their expected income as skilled workers, $\underline{\omega ^h_t}$, is higher than that of unskilled workers, $\omega ^u_t$. In equilibrium, this condition is binding, implying that the number of workers with ordinary ability who choose to become educated satisfies the following indifference condition: $\underline{\omega ^h_t}=\omega ^u_t$. Regarding individuals with green motivation, the assumption that $\overline{\varphi _t}>\underline{\varphi _t}$ (A1) implies that $\overline{\omega ^h_t}>\underline{\omega ^h_t}$.

Education decisions in turn satisfy the following rule: $\overline{\omega ^h_t}>\underline{\omega ^h_t}=\omega ^u_t$ which implies that, all individuals with green motivation choose to become educated:

$$\begin{aligned} G_t=1, \end{aligned}$$

(32)

and on the other hand, workers with basic motivation are indifferent between becoming educated or not. The ratio $\frac{\omega ^u_t}{\underline{\omega ^h_t}}$ therefore satisfies the following condition:

$$\begin{aligned} \frac{\omega ^u_t}{\underline{\omega ^h_t}}=1 \Leftrightarrow \frac{\varphi _t}{\underline{\varphi _t}}\frac{w_t^u}{w_t^h}=1-\xi . \end{aligned}$$

(33)

Using the fact that $w_t^u=\frac{\alpha Y_t}{l_t}$ and $w_t^h=\frac{(1-\alpha )^2Y_t}{1+H_t}$ and the equality between demand and supply of unskilled labor in efficiency units that is equalization between $L_t^d =l_t$ and $L_t^s = \varphi _t L_t$, from Eqs. 11 and 32, we therefore have $l_t=\varphi _t L_t= \varphi _t(1-s)(1-B_t)$. Thus:

$$\begin{aligned} \frac{w_t^h}{w_t^u}=\frac{(1-\alpha )^2}{\alpha }\frac{\varphi _t (1-s)(1-B_t)}{\overline{\varphi _t} s+\underline{\varphi _t} (1-s)B_t} \end{aligned}$$

(34)

1.1.5 A2.5 Equilibrium on the Labor Market

Total skilled labor demand writes $H^d= H+H_r$, where $H=h+h^*$ is the aggregate skilled labor demand in the intermediate goods sector and $H_r= h_r^{*} + h_r$ is the aggregate skilled labor demand in the research sector.

In the intermediate goods sector, the demand for skilled labor is such that one unit of skilled labor produces one unit of intermediate good. Hence, the demand for skilled labor in green-tech (resp. dirty-tech) firms is equal to $x^*$ (resp. x). The aggregate skilled labor demand in the intermediate goods sector hence writes: $H= n^{*}x^{*} + nx$. Using Eqs. 4, 5, we have: $\frac{\alpha \delta }{r}n^{\rho }(n^*)^{1-\rho }=1-\alpha$. Using Eq. 31, we thus get: $h=nx=\frac{1-\alpha }{\alpha }\frac{r}{\delta } \Delta ^{\rho -1}$, and similarly: $h^*=n^*x^*=\frac{1-\alpha }{\alpha }\frac{r}{\delta ^*} \Delta ^{1-\rho ^*}$.

The aggregate skilled labor demand in the intermediate goods sector then write:

$$\begin{aligned} H= h^*+h= \frac{1-\alpha }{\alpha }r \left( \frac{\Delta ^{\rho -1}}{\delta } + \frac{\Delta ^{1-\rho ^*}}{\delta ^*}\right) =\frac{1-\alpha }{\alpha }r\frac{1+\Delta }{\delta ^*\Delta ^{\rho ^*}}, \text { with } \Delta =\left( \delta / \delta ^*\right) ^{\frac{1}{\rho +\rho ^*-1}} \end{aligned}$$

(35)

In the research sector, the allocation of skilled workers between sectors leads to Eq. 31: $\frac{n^*}{n}=\left( \frac{\delta }{\delta ^*}\right) ^{\frac{1}{\rho +\rho ^*-1}}$. Together with $N=n^*+n$, this equation implies the equalization of the rate of accumulation in green-tech and dirty-tech knowledge:

$$\begin{aligned} \frac{\dot{n}^*}{n^*}= \frac{\dot{n}}{n}= \frac{\dot{N}}{N}. \end{aligned}$$

(36)

Substituting for Eq. 36 into Eqs. 6 and 7 leads to: $\frac{\delta h_r n^{\rho }(n^*)^{1-\rho }}{n} = \frac{\delta ^* h_r^* (n^*)^{\rho ^*}(n)^{1-\rho ^*}}{n^*}$, implying $h_r=h_r^*\frac{n}{n^{*}}=\frac{h_r^*}{\Delta }$. Aggregate skilled labor demand in the research sector then writes $H_r= h_r^{*} + h_r = \frac{1+\Delta }{\Delta }h_r^{*}$, and we have:

$$\begin{aligned} h_r=\frac{H_r}{1+\Delta } \text { and } h_r^*=\frac{\Delta }{1+\Delta } H_r, \text { with } \Delta =\left( \delta / \delta ^*\right) ^{\frac{1}{\rho +\rho ^*-1}}. \end{aligned}$$

(37)

Equilibrium on the labor market implies equality between demand and supply of skilled labor in efficiency units, that is between $H^d =H+H_{r}$ and $H^s = \overline{\varphi } \overline{H}+\underline{\varphi }\underline{H}$. Given Eqs. 11 and 32 this implies $H^d= H+H_{r} = H^s = \overline{\varphi }s + \underline{\varphi }(1-s)B$. Using Eq. 19, we thus have:

$$\begin{aligned} H^d = \overline{\varphi }s + \underline{\varphi }(1-s)\frac{\frac{(1-\alpha )^2}{\alpha }(1-\xi )-\frac{\overline{\varphi }}{\underline{\varphi }}\frac{s}{1-s}}{\frac{(1-\alpha )^2}{\alpha }(1-\xi )+1}. \end{aligned}$$

(38)

Equation 25 implies that $w^u=\frac{\alpha Y}{l}$.

Equation 31 implies that $n^*=\Delta n$ and given that $N=n^*+n$, we thus have $n=\frac{N}{1+\Delta }$ where $\Delta =\left( \delta / \delta ^*\right) ^{\frac{1}{\rho +\rho ^*-1}}$.

Using Eqs. 4 and 5 we have $x=x^*$, $p=(1-\alpha )l^\alpha x^{-\alpha }$ and $Y=Nl^\alpha x^{1-\alpha }$, thus: $Npx=(1-\alpha )Y$.

Equation 9 together with Eqs. 27, 4 and 5 implies that $w^h=\frac{\alpha \delta ^*}{r} npx \Delta ^{\rho ^*}$. Given that $n=\frac{N}{1+\Delta }$, we get: $w^h=\frac{\alpha \delta ^*}{r} \frac{Npx}{1+\Delta } \Delta ^{\rho ^*}$ and therefore $w^h=\frac{\alpha \delta ^*}{r} \frac{(1-\alpha )Y}{1+\Delta } \Delta ^{\rho ^*}$.

Substituting for $w^u=\frac{\alpha Y}{l}$ and $w^h=\frac{\alpha \delta ^*}{r} \frac{(1-\alpha )Y}{1+\Delta } \Delta ^{\rho ^*}$ into $\frac{\varphi }{\underline{\varphi }}\frac{w^u}{w^h}=1-\xi$ finally leads to:

$$\begin{aligned} \frac{\varphi }{\underline{\varphi }}\frac{r}{\delta ^*}\frac{1+\Delta }{\Delta ^{\rho ^*}}\frac{1}{(1-\alpha )l} =1-\xi \end{aligned}$$

The number $B_t$ of workers with basic motivation who choose to become educated is determined by Eq. 17, where the skill premium $\frac{w_t^h}{w_t^u}$ is obtained using $w_t^u=\frac{\alpha Y_t}{l_t}$ and $w_t^h=\frac{(1-\alpha )^2 Y_t}{H_t}$:

$$\begin{aligned} \frac{\varphi _t}{\underline{\varphi _t}}\frac{1}{1-\xi }=\frac{(1-\alpha )^2}{\alpha } \frac{\varphi (1-s)(1-B_t)}{\overline{\varphi _t}s+\underline{\varphi _t}(1-s)B_t} \Leftrightarrow B_t=\frac{ \frac{(1-\alpha )^2}{\alpha }(1-\xi )-\frac{\overline{\varphi _t}}{\underline{\varphi _t}}\frac{s}{1-s}}{\frac{(1-\alpha )^2}{\alpha }(1-\xi )+1}. \end{aligned}$$

(39)

The number of skilled workers employed in the green and brown research sectors then is obtained using Eqs. 35, 37 and 38:

$$\begin{aligned} h_{rt}&=\frac{H_{rt}}{1+\Delta }, \qquad h_{rt}^*=\frac{\Delta }{1+\Delta } H_{rt}, \end{aligned}$$

(40a)

$$\begin{aligned} H_{rt}&=H_t^d-H_t= \frac{\frac{(1-\alpha )^2}{\alpha }(1-\xi )}{\frac{(1-\alpha )^2}{\alpha }(1-\xi )+1}(\overline{\varphi _t}s+\underline{\varphi _t}(1-s)) -\frac{1-\alpha }{\alpha }r_t\frac{1+\Delta }{\delta ^*\Delta ^{\rho ^*}}, \end{aligned}$$

(40b)

with $\Delta =\left( \delta / \delta ^*\right) ^{\frac{1}{\rho +\rho ^*-1}}$.

1.1.6 A2.6 Consumption Decisions and Steady-State Growth Rate

Individuals determine the level of consumption and savings that maximize their intertemporal utility according to the following maximization problem

$$\begin{aligned} &{\max \quad } U=\int _{0}^{\infty }u_t(c_t,e_t)\cdot exp^{-\beta t}d \\ &c_t\\ &\text {subject to }c(t)+\dot{A(t)}=\Omega (t)L+r(t)A(t) \text { and }e_t= \left[ n_t\right] ^{-\eta } \left[ n_t^{*}\right] ^{\eta ^*} \end{aligned}$$

where $\Omega (t)$ is the household expected income, $\Omega (t)=\{\omega ^u_t, \underline{\omega ^h_t}, \overline{\omega ^h_t}\}$, and A(t) is the stock of assets (nonhuman wealth) held at time t.

The first-order condition of this program leads to the standard condition

$r-\beta =-\frac{-du'/dt}{u'}$, that is

$$\begin{aligned} r=\beta -(\theta (1-\sigma )-1)\frac{\cdot {c}}{c}+\mu (1-\sigma )\frac{\cdot {e}}{e} \end{aligned}$$

(41)

After simple manipulations we get the following growth rate:

$$\begin{aligned} g_{ct}= \frac{\dot{c}_t}{c_t}= \frac{1}{\Psi }(r_t-\beta ), \end{aligned}$$

(42)

where $\Psi =1-\theta (1-\sigma )+\mu (1-\sigma )(\eta ^*-\eta )$.

In steady state, N, $n^*$, n, C and Y all grow at the same rate $g=\frac{\dot{c}}{c}=\frac{\dot{N}}{N}=\frac{\dot{n^*}}{n^*}=\frac{\dot{n}}{n}$, that is:

$$\begin{aligned} g=\frac{\dot{c}}{c}=\frac{\dot{N}}{N}=\frac{\dot{n^*}}{n^*}=\frac{\dot{n}}{n} = \delta ^* h_r^*\left( \frac{n}{n^*} \right) ^{1-\rho ^*} = \delta h_r\left( \frac{n^*}{n} \right) ^{1-\rho }. \end{aligned}$$

(43)

Using Eqs. 31 and 40a, this gives:

$$\begin{aligned} g= \delta ^* \frac{\Delta ^{\rho ^*}}{1+\Delta } H_r. \end{aligned}$$

(44)

where $\Delta =\left( \delta / \delta ^*\right) ^{\frac{1}{\rho +\rho ^*-1}}$.

Combining Eqs. 40b, 41, 42 and 44 finally leads to the following steady-state growth rate:

$$\begin{aligned} g= \frac{\delta ^* \frac{\Delta ^{\rho ^*}}{1+\Delta }\Gamma \left[ \overline{\varphi }s + \underline{\varphi }(1-s)\right] - \beta \frac{1-\alpha }{\alpha }}{1+\Psi \frac{1-\alpha }{\alpha }}, \end{aligned}$$

(45)

where: $\Delta =\left( \delta / \delta ^*\right) ^{\frac{1}{\rho +\rho ^*-1}}$, $\Gamma =\frac{\frac{(1-\alpha )^2}{\alpha }(1-\xi )}{1+\frac{(1-\alpha )^2}{\alpha }(1-\xi )}$, and $\Psi =1-\theta (1-\sigma )+\mu (1-\sigma )(\eta ^*-\eta )$.

1.1.7 A2.7 Wage Inequality Indexes

Two inequality indexes can be defined: wage inequality within skilled workers, denoted by $\Sigma ^{within}$, and wage inequality between skilled and unskilled workers, denoted by $\Sigma ^{between}$.

Within-group wage inequality is defined by:

$$\begin{aligned} \Sigma ^{within}_t=\dfrac{\overline{\omega ^h_t}\overline{H_t}}{\underline{\omega ^h_t}\underline{H_t}}. \end{aligned}$$

Between-group inequality writes:

$$\begin{aligned} \Sigma ^{between}_t=\dfrac{\Sigma ^h_t}{\Sigma ^u_t}. \end{aligned}$$

where $\Sigma ^h_t$ the average income of skilled workers is defined by the income (in efficiency units) of skilled workers, divided by the size of the skilled work force:

$$\begin{aligned} \Sigma ^h_t=\dfrac{\overline{\varphi _t}\overline{H_t}+\underline{\varphi _t}\underline{H_t}}{\overline{H_t}+\underline{H_t}}w^h_t. \end{aligned}$$

and where $\Sigma ^u_t$, the average income of unskilled workers is given by:

$$\begin{aligned} \Sigma ^u_t=\dfrac{\varphi _t L_t}{L_t}w^u_t=\varphi _t w^u_t. \end{aligned}$$

Using the resources constraints given by Eqs. 11 and 32 we get:

$$\begin{aligned} \Sigma ^{between}_t=\dfrac{\overline{\varphi _t} s + \underline{\varphi _t}(1-s) B_t}{s+(1-s)B_t}\dfrac{w^h_t}{\varphi _t w^u _t}. \end{aligned}$$

Using equilibrium conditions on the goods and labor markets Eqs. 15, 18a and 18b we obtain after some simple manipulations:

$$\begin{aligned} \Sigma ^{within}_t=\dfrac{\overline{\varphi _t}}{\underline{\varphi _t}}\dfrac{s}{1-s}\dfrac{1}{B_t} \end{aligned}$$

and

$$\begin{aligned} \Sigma ^{between}_t=\dfrac{(1-\alpha )^2}{\alpha }\dfrac{(1-s)(1-B_t)}{s+(1-s)B_t}. \end{aligned}$$

Substituting for Eq. 19, we finally get:

$$\begin{aligned} \Sigma ^{within}_t=\dfrac{\overline{\varphi _t}}{\underline{\varphi _t}}\dfrac{s}{1-s}\dfrac{\frac{(1-\alpha )^2}{\alpha }(1-\xi )+1}{\frac{(1-\alpha )^2}{\alpha }(1-\xi )-\frac{\overline{\varphi _t}}{\underline{\varphi _t}}\frac{s}{1-s}} \end{aligned}$$

(46)

and

$$\begin{aligned} \Sigma ^{between}_t=\dfrac{(1-\alpha )^2}{\alpha }\dfrac{1-s(1-\frac{\overline{\varphi _t}}{\underline{\varphi _t}})}{\frac{(1-\alpha )^2}{\alpha }(1-\xi )+s(1-\frac{\overline{\varphi _t}}{\underline{\varphi _t}})}. \end{aligned}$$

(47)

1.2 Appendix 3. MSCI ACWI Index