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Sustainability and Substitutability

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Abstract

Developing a quantitative science of sustainability requires bridging mathematical concepts from fields contributing to sustainability science. The concept of substitutability is central to sustainability but is defined differently by different fields. Specifically, economics tends to define substitutability as a marginal concept while fields such as ecology tend to focus on limiting behaviors. We explain how to reconcile these different views. We develop a model where investments can be made in knowledge to increase the elasticity of substitution. We explore the set of sustainable and optimal trajectories for natural capital extraction and built and knowledge capital accumulation. Investments in substitutability through knowledge stock accumulation affect the value of natural capital. Results suggest that investing in the knowledge stock, which can enhance substitutability, is critical to desirable sustainable outcomes. This result is robust even when natural capital is not managed optimally. This leads us to conclude that investments in the knowledge stock are of first order importance for sustainability.

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Notes

  1. Spatial distribution and geographic variation make the aggregation to a single index impossible and the appropriate scale and scope of aggregation is an open question (Sterner 2011).

  2. Fisher and Zhao (2002) distinguish between substitutability in production of intermediate goods and that in consumption in a conceptual model, and argue that the latter is more important for sustainability when irreversibility is present.

  3. For instance, the so called Hartwick rule (Hartwick 1990) implies that scarcity rents from consuming non-renewable resources should not be consumed along an optimal management path, but should be converted to other forms of capital. This result implicitly assumes that natural and built capital are substitutable, at least on the margin. Hartwick’s model is readily generalized to renewable natural capital. Hartwick’s model is also an extension to the Ramsey–Solow model of savings and investment (Mas-Colell et al. 1995).

  4. Ecologists are also increasingly concerned with substitution opportunities associated with the ability of organisms to use resources in different ways in response to different conditions (Fox et al. 2011).

  5. Solow is concerned with intergenerational equity, but Solow’s exposition implies what would likely be called sustainability today.

  6. The elasticity of substitution measures the curvature of a consumption or production isocline. Expression in terms of percent reduces dimensionality. When decisions are made optimally the elasticity of substitution can be defined as optimal input levels in response to exogenous changes in their relative prices.

  7. The prior literature addresses both built and natural capital e.g., Hartwick (1990), but does not directly address changes in substitution possibilities.

  8. Many authors (e.g., Kellner et al. 2011; Horan et al. 2011; Brock and Xepapadeas 2002; Fenichel and Horan 2007a; Fenichel et al. 2010) have examined vectors of ecological states, but seldom have explicitly considered the substitution relationships.

  9. It is also possible that society has preference for \(y\), e.g., historic buildings, but this would require \(y\) to have heterogeneous vintages and is left for future exploration.

  10. We maintain the assumption of stable preference. Changes in the knowledge stock could shift preference, but analyzing how knowledge accumulation may shift preferences is left for future work.

  11. The limit \(\sigma \rightarrow 0\) implies Leontief production, which is akin to a law of the minimum approach to the production of \(F\). The CES function can also be rescaled to create the appropriate production by multiplying by a scalar. In our analysis we set such a scalar to 1.

  12. The system begins with approximately a unit of knowledge capital to avoid undetermined conditions associated with \(\sigma =1.\)

  13. Equilibrium is used in the sense of a dynamical system.

  14. Numerical analysis was conducted using Mathematica 9.0 (Wolfram Research).

  15. An analytical solution strategy to this problem from arbitrary starting values is not immediately obvious. The problem could be solved numerically. However, given the abstract nature of the problem little new insight would be gained by such an exercise.

  16. A Skiba manifold is the multi-dimensional equivalent of a Skiba point (Brock and Starrett 2003; Skiba 1978; Wagener 2003). A Skiba point results from non-convexities in the optimized dynamic system, which create an indifference point such that a system can move along more than one trajectory to more than one long-run steady state (or stable cycle) and achieve the same maximal value.

  17. There remain extraction costs, \(\beta _u \).

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Authors and Affiliations

  1. Yale University, New Haven, CT, USA

    Eli P. Fenichel

  2. Department of Economics, Michigan State University, East Lansing, MI , 48824, USA

    Jinhua Zhao

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  1. Eli P. Fenichel
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  2. Jinhua Zhao
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Correspondence to Eli P. Fenichel.

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Fenichel, E.P., Zhao, J. Sustainability and Substitutability. Bull Math Biol 77, 348–367 (2015). https://doi.org/10.1007/s11538-014-9963-5

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