## 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.

### Similar content being viewed by others

## References

Arrow, Kenneth J., Partha S. Dasgupta, Lawrence H. Goulder, Kevin J. Mumford and Kirsten Oleson. 2012. Sustainability and the Measurement of Wealth.

*Environment and Development Economics*, 17(3):317–353.Barbier, Edward B. 2015.

*Nature and Wealth: Overcoming Environmental Scarcities and Inequality.*London: Palgrave Macmillan.Dasgupta, Partha S. 2001.

*Human Well-being and the Natural Environment*. New York: Oxford University Press.Charles, Jones I. 2015. Pareto and Piketty: The Macroeconomics of Top Income and Wealth Inequality.

*Journal of Economic Perspectives,*29(1):29–46.Fenichel, Eli P. and Joshua K. Abbott. 2014. Natural Capital: From Metaphor to Measurement.

*Journal of the Association of Environmental and Resource Economists*, 1(1/2):1–27.Ferreira, Susan, Kirk Hamilton and Jeffrey R. Vincent. 2008. Comprehensive Wealth and Future Consumption: Accounting for Population Growth.

*World Bank Economic Review*, 22(2):233–248.Goldin, Claudia and Lawrence F. Katz. 2008.

*The Race Between Education and Technology.*Cambridge, MA: Harvard University Press.Greasley, David, Nick Hanley, Jan Kunnas, Eoin McLaughlin, Les Oxley and Paul Warde. 2014. Testing Genuine Savings as a Forward-looking Indicator of Future Well-being over the (Very) Long-run.

*Journal of Environmental Economics and Management*, 67(2):171–188.Hamilton, Kirk. and Clemens, Michael. 1999. Genuine Savings Rates in Developing Countries.

*World Bank Economic Review*, 13(2):333–356.Hamilton, Kirk. and Hartwick, John M. 2005. Investing Exhaustible Resource Rents and the Path of Consumption.

*Canadian Journal of Economics*, 38(2):615–21.Hartwick, John M. 1977. Intergenerational Equity and the Investment of Rents from Exhaustible Resources.

*American Economic Review*, 67(5):972–974.Hartwick, John M. 1990. Natural Resources, National Accounting and Economic Depreciation.

*Journal of Public Economics*, 43:291–304.Organization for Economic Cooperation and Development (OECD). 2011.

*An Overview of Growing Income Inequalities in OECD Countries: Main Findings. Divided We Stand: Why Inequality Keeps Rising.*Paris: OECD.Piketty, Thomas. 2014.

*Capital in the Twenty-First Century.*Cambridge, MA: Harvard University Press.Piketty, Thomas and Gabriel Zucman. 2014. Capital is Back: Wealth-Income Ratios in Rich Countries, 1700–2010.

*Quarterly Journal of Economics*, 129(3):1255–1310.Solow, Robert M. 1956. A Contribution to the Theory of Economic Growth.

*Quarterly Journal of Economics*, 70(1):65–94.Solow, Robert M. 1986. On the Intergenerational Allocation of Natural Resources.

*Scandinavian Journal of Economics*, 88(1):141–149.Swan, Trevor W. 1956. Economic Growth and Capital Accumulation.

*Economic Record*, 32(2):334–361.United Nations University (UNU)-International Human Dimensions Programme (IHDP) on Global Environmental Change-United Nations Environment Programme (UNEP). 2014. Inclusive Wealth Report 2014: Measuring Progress Toward Sustainability. Cambridge and New York: Cambridge University Press.

Weitzman, Martin L. 1976. On the Welfare Significance of National Product in a Dynamic Economy.

*Quarterly Journal of Economics*, 90(1):156–162.World Bank. 2011.

*The Changing Wealth of Nations: Measuring Sustainable Development in the New Millennium.*Washington, D.C.: The World Bank.World Bank. 2016. World Development Indicators. Washington D.C: The World Bank. http://databank.worldbank.org/data/reports.aspx?source=world-development-indicators.

## Acknowledgments

The author wishes to thank Diego Nocetti and three anonymous referees for their comments and Thomas Piketty for helpful discussion and suggestions.

## Author information

### Authors and Affiliations

### Corresponding author

## Appendix

### 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

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

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

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

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

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

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.

## Rights and permissions

## About this article

### Cite this article

Barbier, E.B. Natural Capital and Wealth in the 21st Century.
*Eastern Econ J* **43**, 391–405 (2017). https://doi.org/10.1057/s41302-016-0013-x

Published:

Issue Date:

DOI: https://doi.org/10.1057/s41302-016-0013-x