A Mathematical Analysis of Sustainable Development

The concept of sustainability is a subject of controversy among scientists. The two dominant views are the weak and the strong version of sustainable development, as well as the substitution or non-substitution of natural capital. Therefore, the purpose of this article is to analyze the various versions and choose the most advantageous one. The combination of two versions (weak and strong sustainability) depending on the level of environmental destruction experienced by each generation, is the most advantageous option. We mustn't confuse sustainable development and sustainable growth

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Sustainable development is a global challenge, which requires the transformation of various economies in order to meet the needs of the current generation, without degrading the ability of future generations to maximize their utility. While all economists agree on this principle, the mode of achievement is a "bone of contention" for the scientific community, as is evident from the different versions that have been developed from time to time [1].

The purpose of this article is to summarize the various versions that have been developed and to describe the most advantageous one. For this reason, the concept of sustainable development is introduced, then its four basic versions are analyzed and finally the choice of one of them is justified.

What is Sustainability?

The most well-known definition of sustainable development is that of the World Commission on Environment and Development: "Sustainable development is development that meets the needs of the present without compromising the ability of future generations to meet their own needs" [2]. One of the most common way that can satisfy our needs is consumption. Hartwick's rule offers what Solow called a "Rule of Thumb" for sustainability in resource-inexhaustible economies and states that a maximally stable level of consumption can be maintained if the value of reinvestment equals the value of the scarcity rent from non-renewable natural resources [3]. In other words, if the capital of an economy between two periods `t_0` and `t_1` is recorded, constant consumption is achieved when `K_0 \leq K_1` (Rel.1).

More specifically, the capital of an economy at period `t` is a combination of Natural Capital (`K_N`), Human Capital (`K_H`), Produced (`K_P`), Social Capital (`K_S`) and Financial Capital (`K_F`)  [4].

According to Goodwin and Neva, Natural Capital includes all those flows of services that natural resources can provide in the production process. In contrast, Human Capital includes the assets of human origin, which are used for production. Also, Human Capital includes the accumulated knowledge, which people inherited from their ancestors and bequeath to the next generations. Social Capital is the most controversial and difficult to measure capital and consists of a complex set of human relationships that affects the production process (trust, mutual understanding, hard work) and is protected by society. Finally, Financial Capital is not directly used in the production process, instead it is a system of ownership of human capital.


`\sum K(t)\ = K_N (t)+ K_P (t) + K_H (t) + K_S (t) + K_F (t)`     (Rel.2)

Combining the Hartwick & Solow rule (Rel. 1) with the work of Goodwin & Neva (Rel. 2)  yields:

`\frac{\text{d}\sum K(t)}{\text{d}t} \geq 0`   (Rel.3)

It is easily understandable that capital value depreciates over time due to wear and tear or technological obsolescence. Therefore, at the end of a production period, the original capital changes by the amount of the reduction, which is called depreciation [5]. Algebraically it is true:

`\frac{\text{d}\sum K(t)}{\text{d}t} \equiv \sum S(t) - δ\ \cdot \sum K(t)`   (Rel.4)

,where `\frac{\text{d}\sum K(t)}{\text{d}t} ` is the change in the total capital within a period of time `dt`, `\sum S(t) ` the total amount of each capital saved for reinvestment, `δ ` the depreciation rate – the rate of depreciation of the initial capital and `δ\ \cdot \sum K(t) ` the part of the total capital that amortized over time. 

For sustainable development prevailling we conclude (Rel. 3 and Rel. 4):

       `\sum_i^n S(t) - δ_i\ \cdot \sum_i^n  K(t) \geq 0`

`\Rightarrow \sum_i^n S(t) \geq δ_i\ \cdot \sum_i^n  K(t)`

`\Rightarrow \sum_i^n S(t) \geq δ_i \ \cdot \ (K_N (t)+ K_P (t) + K_H (t) + K_S (t) + K_F (t))`

`\Rightarrow \sum_i^n S(t) \geq \ δ_N K_N (t)+ δ_PK_P (t) + δ_HK_H (t) + δ_SK_S (t) + δ_FK_F (t)`   (Rel. 5)

,where `i` is the specific type of capital (`N`, `P`, `H`, `S`, `F`) and `n` the total number of different types of capital [5]. It is worth noting that human ties, like knowledge, although they change from time to time, do not wear out, so they are not subject to deterioration. So social capital is not depreciated. Similarly, financial capital is not subject to depreciation, as ownership relations are not reduced, but are modified according to the season. 

In other words:

   I)  `S_H (t) = δ_HK_H (t)`

  II)  `S_S (t) = δ_SK_S (t)`

 III) `S_F (t) = δ_FK_F (t)`

Thus, Rel. 5 can be transformed into:

`\sum S(t) \geq \ δ_N K_N (t)+ δ_PK_P (t)`   (Rel.6)

The main concepts surrounding sustainable development are based on Rel. 6, which algebraically states that, for sustainable growth to occur, the amount of total capital saved for reinvestment must be greater than or equal to the amount of capital being depreciated. The four versions of sustainable development are [6]:

  • Very Weak Sustainable Development (VWS)
  • Weak Sustainable Development (WS)
  • Strong Sustainable Development (SS)
  • Very Strong Sustainable Development (VSS)

Versions of Sustainable Development

Very Weak Sustainability (VWS)

According to Hediger the VWS approach is based on the logic of constant per capita consumption and is based on the notion that complete substitution can occur between different categories of capital and subcategories of one type of capital (expendable, renewable natural capital). Also, different capital types can be substituted spatially (a decrease in inventory in one area can be offset by an increase in another) and temporally (an increase in consumption today can be offset by a decrease in consumption at a later time). [7]. 

Let's read again the Rel. 6. If damage to natural capital occurs (increase in `δ_N K_N (t)`), combined with a reduction in wear and tear of produced capital (decrease in `δ_N K_N (t)`), sustainable development is realized, when their sum remains less than the capital saved for reinvestment (`S(t)`). This version is characterized as Anti-Green Economy and the markets of that economy operate without the intervention of the state [8].

Weak Sustainability (WS)

According to the weak version of sustainable development, natural capital is not homogeneous. It can be distinguished into "Critical"  and "Substitutable". Critical natural capital (`K_N^C`) is commonly defined as that part of the natural environment, which performs important and irreplaceable functions [9]. It is related to non-consumptive uses because it bequeathed to future generations as a bequest of special natural beauty. Conversely, substitutable natural capital (`K_N^S`) can be used in the production process without burdening future generations. In other words, it is a form of capital that can be substitutable  by other forms of capital. Thus, 

`\sum K_N(t) = K_N(t) \equiv K_N^C + K_N^S`   (Rel.7)

The version of Weak Sustainability is based on the idea of ​​maintaining social welfare, which is achieved by not substituting "critical" natural capital. Thus, we consider (Rel. 6, Rel. 7):

`S(t) \geq \ δ_N^S K_N^S (t)+ δ_PK_P (t)`      (Rel.8)

From the Rel. 8 we can understand that that in order to have sustainable development only substitution of `K_N^S (t)` is required [6][10]. This version is characterized as Green Economy and markets as green markets, which are driven by incentives [8].

Strong Sustainability (SS)

In opposite, SS version bequeaths the physical capital intact to the next generations. In other words, there is no substitution between different types of capital. Thus, (Rel. 6) is transformed to:

`S(t) \geq δ_PK_P (t)`    (Rel.9)

Therefore, the amount of capital saved for reinvestment (`S(t)`) must be greater than the impairment of Produced Capital (`δ_PK_P (t)`). This version is characterized as Deep Green Economy and the intervention of the state is evident [8]. Strong Sustainable Development can be characterized as Eco-systems perspective and resource preservationist. The main goal of this type of SD (sustainable development) is the adherence to intra and inter generational equity and the interests of the collective given more weight than those of individual consumer [11].

Very Strong Sustainability (VSS)

The VSS version of sustainable development is based on the logic of strict conservation of natural resources. But for this to happen, the factors that degrade natural resources should be controlled and limited. Some of these factors are economic growth, population growth, etc. This type of SD is characterized as extreme resource preservationist and ecocentric. Recognizes nature’s rights and intrinsic value in nature, encompassing non-human living organisms and it is an anti-economic growth that reduces human population. Also, this version is strongly influenced by Gaianism, that means that we and next generation must respect nature’s rights, including abiotic elements [11].

Real World Phenomenon

Both environmental and economic conditions differ from season to season, therefore sustainable development policies have to change over time. The green economy is a clearly preferable option as it combines economic growth without burdening important natural resources. However, the boundaries between "substitutable" and "critical" natural capital are often bureaucratic rather than substantive.

Moreover, the implementation of any version of sustainable development does not guarantee that future generations will continue to implement the same policy. Which leads us to the conclusion that the next generation will implement the most advantageous policy depending on the choice of the current generation. If current generation implements the VWS, probably this choice will raise environmental pressure and next generation will be forced to implement a more interventionist policy, even the VSS. 

Therefore, the best option for my opinion is a continuous alternation between weak and strong sustainable development, depending on the level of environmental pressure that each generation faces. As we observe in the previews figure , during weak sustainable development it is possible that partial or total substitution of Natural Capital by produced capital will occur. When the environmental pressure becomes severe, then a stricter policy should be followed. In this way we enter the phase of strong sustainable development during which, although sustainable development is achieved, due to the non-use of conventional natural resources, economic growth is likely to be less than in the previous time period. Thus, when natural resources return to normal levels, sustainable development policymakers can focus on the weak version of sustainable development in order to maximize economic growth.


Sustainable development, although desired by all, is one of the fields in which the most dynamic debates of the last decades have been developed, a fact that is proven by the many and different versions that have been developed. Both the weak and strong versions of sustainable development are based on the Hartwick–Solow rule, which states that for there to be a constant level of consumption across two or more generations, the amount of capital stored for reinvestment must be greater than or equal to the total depreciation of capital in an economy.

Their difference, however, lies in the fact that in the weak versions there is a complete or partial substitution of natural with producing capital, while in the strong versions no substitution of any kind is foreseen. Choosing a single version as a panacea to the question of sustainable development is incomplete, as the needs of each generation differ from those of the previous and the next. An interesting choice, then, would be the regression between a weak and a strong version, depending on the level of environmental pressure experienced by each generation.


Robert U. Ayres & Jeroen C.J.M. van den Bergh & John M. Gowdy (1998). Viewpoint: Weak versus Strong Sustainability", Tinbergen Institute Discussion Papers 98-103/3, Tinbergen Institute
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    I completed an Integrated Master of Science in Agricultural Economics from the Agricultural University of Athens and have studied at premier global universities. I hold a Specialization in Quantitative Finance from the National Research University of Moscow, and a Python 3 Programming Specialization from the University of Michigan. My academic interests encompass Economics, Econometrics, Mathematical Economics, Monetary Economics, Financial Economics, Public Economics, Economic Development, and I research financial crises.

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