Natural Capital and Wealth in the 21st Century

Abstract

Extending the wealth accumulation model of Piketty and Zucman [2014] to include net depreciation in fossil fuels, minerals, and forests produces two key indicators: the net national saving rate adjusted for natural capital depreciation, and the ratio of this rate to long-run growth. These indicators are applied to eight rich economies over 1970–2013 and developing countries for 1979–2013. Whereas in developing economies capital accumulation has largely kept pace with rising natural capital depletion, in the rich countries adjusted net savings have fallen to converge with the rate of natural capital depreciation, suggesting less compensation by net increases in other capital.

Appendix

PZ’s one-good model wealth accumulation model can be extended by adapting the approach to accounting for natural capital depreciation in an intertemporal optimizing behavior model [Hamilton and Clemens 1999; Hartwick 1990; Solow 1986], which was originally developed by Weitzman [1976].

To expedite the analysis, assume that the economy is closed and has no net foreign income. Let K _t be the domestic capital stock of the economy at time t and N _t its endowment of available natural resources for production. Y _t is the net national income (i.e., national income less domestic capital depreciation) at time t, C _t is the aggregate consumption, and S _gt is the gross national savings flow between time t and t + 1. Ignoring any price effects, gross national savings is defined as $S_{gt} = \left( {K_{t + 1} - K_{t} } \right) + \delta K_{t}$, where $\delta K_{t}$ is the domestic capital depreciation in time t. Aggregate national income is produced by employing capital, labor L _t , and a flow of input R _t extracted from the natural resource endowment, and corresponds to the identity $F\left( {K_{t} ,L_{t} ,R_{t} } \right) = C_{t} + S_{gt}$. Finally, the change in the natural resource stock between time t and t + 1 is determined by $N_{t + 1} - N_{t} = G\left( {N_{t} } \right) - R_{t}$, where $G\left( {N_{t} } \right)$ represents the natural growth rate for any renewable resources (but can either be assumed zero for exhaustible resources, or be additions to stocks through exploration or discovery $G_{t}$).

If social welfare in time t is represented by the utility function $U\left( {C_{t} } \right)$, then the current-value Hamiltonian for social-welfare maximization is

$$H = U\left( {C_{t} } \right) + \lambda_{t} \left[ {K_{t + 1} - K_{t} } \right] + \mu_{t} \left[ {N_{t + 1} - N_{t} } \right],$$

where $\lambda_{t}$ and $\mu_{t}$ are the shadow values of capital and natural resources, respectively.

Linearizing the utility function, so that $U\left( {C_{t} } \right) = U_{C} C_{t}$, and using the first-order conditions $U_{C} = \lambda_{t}$ and $\lambda_{t} F_{R} = \mu_{t}$, the current-value Hamiltonian can be rewritten as

$$\frac{H}{{U_{C} }} = C_{t} + \left[ {K_{t + 1} - K_{t} } \right] + F_{R} \left[ {G\left( {N_{t} } \right) - R_{t} } \right] = C_{t} + S_{gt} - \delta K_{t} + F_{R} \left[ {G\left( {N_{t} } \right) - R_{t} } \right].$$

The last term is the net deprecation of natural resources used in production. If this term is negative, it represents the value of the amount of the resource endowment that is “used up” to produce national income in time t.

In the above expression, $C_{t} + S_{gt} - \delta K_{t} = Y_{t}$ is the net national income as defined by PZ. It follows that net national income adjusted for natural capital depreciation, or adjusted net national income, is $Y_{t}^{*} = Y_{t} - F_{R} \left[ {G\left( {N_{t} } \right) - R_{t} } \right]$. Similarly, as PZ’s net national saving is $S_{t} = Y_{t} - C_{t} = S_{gt} - \delta K_{t}$, then the adjustment to this saving for natural capital depreciation, or adjusted net savings, is

$$S_{t}^{*} = S_{t} + F_{R} \left[ {G\left( {N_{t} } \right) - R_{t} } \right] = \left[ {K_{t + 1} - K_{t} } \right] + \mu_{t} \left[ {N_{t + 1} - N_{t} } \right].$$

Following PZ’s example of equating domestic capital with the market value of national wealth W _t at time t, and denoting the market value of the resource endowment as $\tilde{N}_{t}$, then the adjusted net national savings can be defined as

$$S_{t}^{*} = \left[ {W_{t + 1} - W_{t} } \right] + \left[ {\tilde{N}_{t + 1} - \tilde{N}_{t} } \right] = W_{t + 1}^{*} - W_{t}^{*} ,\quad {\text{where}}\quad \tilde{N}_{t} \approx \mu \tilde{N}_{t} \;{\text{and}}\;\tilde{N}_{t + 1} \approx \mu \tilde{N}_{t + 1}.$$

As shown by Dasgupta [2001], Ferreira et al. [2008], Greasley et al. [2014], and Hamilton and Hartwick [2005], cross-country wealth comparisons should account for the reductions in per capita wealth due to population growth. This effect can be accommodated easily in the above framework. Let population grow at rate $\eta_{t}$ such that $L_{t + 1} = (1 + \eta_{t} )L_{t}$. Using a “hat” (^) to denote a per capita variable, $\hat{K}_{t + 1} - \hat{K}_{t} = \hat{S}_{gt} - \delta \hat{K}_{t} - \eta_{t} \hat{K}_{t + 1}$ and $\hat{N}_{t + 1} - \hat{N}_{t} = g\left( {\hat{N}_{t} } \right) - \hat{R}_{t} - \eta_{t} \hat{N}_{t + 1}$, where $g\left( {\hat{N}_{t} } \right) = G\left( {{{N_{t} } \mathord{\left/ {\vphantom {{N_{t} } {L_{t} }}} \right. \kern-0pt} {L_{t} }}} \right)$. Consequently, adjusted net savings per capita is

$$\hat{S}_{t}^{*} = \hat{S}_{t} + f_{{\hat{R}}} \left[ {g\left( {\hat{N}_{t} } \right) - \hat{R}_{t} } \right] = \left[ {\hat{K}_{t + 1} - \hat{K}_{t} } \right] + \mu_{t} \left[ {\hat{N}_{t + 1} - \hat{N}_{t} } \right] + \eta_{t} \left[ {\hat{K}_{t + 1} + \mu_{t} \hat{N}_{t + 1} } \right],$$

where $\hat{S}_{t} = \hat{S}_{gt} - \delta \hat{K}_{t}$,$f\left( {\hat{K}_{t} ,\hat{R}_{t} } \right) = F\left( {{{K_{t} } \mathord{\left/ {\vphantom {{K_{t} } {L_{t} ,{{R_{t} } \mathord{\left/ {\vphantom {{R_{t} } {L_{t} ,1}}} \right. \kern-0pt} {L_{t} ,1}}}}} \right. \kern-0pt} {L_{t} ,{{R_{t} } \mathord{\left/ {\vphantom {{R_{t} } {L_{t} ,1}}} \right. \kern-0pt} {L_{t} ,1}}}}} \right)$, $U\left( {{{C_{t} } \mathord{\left/ {\vphantom {{C_{t} } {L_{t} }}} \right. \kern-0pt} {L_{t} }}} \right) = u\left( {\hat{C}_{t} } \right) = u_{{\hat{C}}} \hat{C}_{t},$ and the relevant first-order conditions are now $u_{{\hat{C}}} = \lambda_{t}$ and $\lambda_{t} f_{{\hat{R}}} = \mu_{t}$.

Adjusted net national savings per capita is therefore

$$\hat{S}_{t}^{*} = \left[ {\hat{W}_{t + 1} - \hat{W}_{t} } \right] + \left[ {\hat{\tilde{N}}_{t + 1} - \hat{\tilde{N}}_{t} } \right] + \eta_{t} \left[ {\hat{W}_{t + 1} + \hat{\tilde{N}}_{t + 1} } \right] = \left( {1 + \eta_{t} } \right)\hat{W}_{t + 1}^{*} - \hat{W}_{t}^{*} = {\Delta \hat{W}^{*}}.$$

The additional term $\eta_{t} \hat{W}_{t + 1}^{*}$ represents the increased savings per capita that is required to overcome the reduction in per capita wealth due to population growth. This is similar to the result derived by Ferreira et al. [2008] using the model developed by Hamilton and Hartwick [2005].

Given that the adjusted net savings rate is by definition $s_{t}^{*} = {{S_{t}^{*} } \mathord{\left/ {\vphantom {{S_{t}^{*} } {Y_{t}^{*} }}} \right. \kern-0pt} {Y_{t}^{*} }} = {{\hat{S}_{t}^{*} } \mathord{\left/ {\vphantom {{\hat{S}_{t}^{*} } {\hat{Y}_{t}^{*} }}} \right. \kern-0pt} {\hat{Y}_{t}^{*} }}$, then it follows that $s_{t}^{*} = \frac{{\Delta W^{*} }}{{Y_{t}^{*} }} = \frac{{\Delta \hat{W}^{*} }}{{\hat{Y}_{t}^{*} }}$. The adjusted net savings rate is also an indicator of the annual change in adjusted net wealth per capita relative to adjusted net national income per capita.