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Published: September 17, 2024
Tipping points have garnered increasing attention in the sustainability literature, given their importance as potential "points of no return" in ecosystem dynamics. Although people understand the idea of tipping points, we have yet to formalize the concept.
We seek to remedy this gap by developing a basic tipping point model featuring the interaction of physical capital and natural capital. The approach will be mostly graphical.
Introducing a tipping point has the effect of "shrinking the playing field." This puts an upper bound on the deterioration of natural capital and a potential cap on the accumulation of physical capital — and output — if we are to achieve sustainability.
The way out of this dilemma is green growth: expanding GDP while innovating to continuously lower the environmental impact of economic activity.
Tipping points occupy a central, rising profile in the sustainability lexicon. Given the increasing focus on the effects of climate change and their potential irreversibility, this rising profile is important to understand. The Intergovernmental Panel on Climate Change defines tipping points as "critical thresholds in a system that, when breached, can lead to a significant change in the state of the system, often with an understanding that the change is irreversible." Essentially, once the tipping point is crossed, the properties of system dynamics fundamentally change, going from favorable (potentially returning to some desired state) to unfavorable (moving inexorably toward an undesirable state).
To make this more concrete, let's use an example of an “earth system” tipping point. A depleted biosphere, such as a rainforest, can recover if corrective action is taken early. The rainforest has some capacity to self-repair. But if the damage goes on for too long and is too severe, the system may never be able to recover, and the biosphere will be permanently destroyed. The tipping point is the threshold beyond which the rainforest can no longer self-repair.
The importance of tipping points for sustainability modeling should be straightforward. If an environmental system is deteriorating, then the damage can be mitigated by acting early. Putting tipping points in our analytical frameworks is essential to capture these critical factors. Paths for the economy that push the environment beyond the tipping point are obviously inferior to paths that do not. They are not sustainable and should be avoided.
It is not the intent of this paper to prove the existence of tipping points. We simply assume they exist due to nonlinearities and show how including them in models can generate insights. We defer to climate scientists on the existence of tipping points, including their possible variation across ecosystems (rainforests versus glaciers, for example). The intent here is to present a framework that will allow us to incorporate tipping points into our analysis and explore the implications for the interaction of macro and environmental dynamics.
Before building and analyzing our framework we first need to define sustainability. There is an ongoing debate between "weak" and "strong" versions of sustainability (see this brief from the United Nations).
The key difference between weak and strong sustainability is the relation between two types of capital: physical and natural. Physical capital is the familiar concept from traditional growth models. It includes plant and equipment, infrastructure, buildings et cetera and is often referred to as the capital stock of the economy. Natural capital comprises the environment and ecosystem services including the biosphere as well as renewable and nonrenewable resources. Note that both types of capital are composites of complex components; we are simplifying to keep our framework tractable.
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Weak sustainability assumes that natural and physical capital are highly substitutable. This implies that there are no meaningful differences in the types of wellbeing or "utility" that they generate. Therefore, the total stock of capital — the sum of these two types of capital — should be maximized. |
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Strong sustainability assumes that natural capital provides unique benefits to society and is not substitutable with physical capital. The latter is reproducible whereas the former is not (particularly past certain thresholds). Therefore, it is not appropriate to excessively deplete the stock of natural capital in order to boost physical capital and ultimately consumption. |
This paper adopts a view closer to strong sustainability. While we will not attempt to optimize the combination of natural and physical capital, we do impose the condition that the stock of natural capital must eventually stabilize. Otherwise, it would reach and cross a tipping point and accelerate toward zero or total degradation of the environment. This will become clearer below. For now, the takeaway is that the stock of natural capital — not just its change — is the ultimate way to determine whether the corresponding path for physical capital (and growth) is sustainable.
An alternative way to think about the two types of capital — and how they interact — is to think about broad objectives. Physical capital is something that is accumulated. As the capital stock expands, combined with labor and other factors, output also expands. The economic pie becomes larger with improved possibilities for higher consumption and wellbeing. This is the traditional way to think about the benefits of economic growth. In contrast, natural capital is something that is preserved. If the productive economy — and our wellbeing — are ultimately embedded in natural capital, this must be maintained; the objective is thus one of stewardship. Most studies suggest that we are not on a sustainable path (see chart 2), which is complicated by the fact that natural capital is difficult to quantify.
Our previous research on green growth sought to link changes in output to changes in the environment. (See "Could Green Growth Be An Oxymoron?" May 18, 2023.) In that work, we showed the inherent trade-off between raising output and the hit on the environment. We also showed how improvements in the environmental efficiency of production, or the "greening" the economy, could minimize the stakes of this trade-off. This implies there is a key role for technology. Indeed, the concept of green growth is built on the idea that we can expand the economic pie while lowering the environmental impact of output.
However, that green growth framework was incomplete. It was entirely based on flows. The model ignored the effects of the interplay between the stock of physical capital and natural capital. Economic output (usually measured as GDP) is a flow, and the corresponding environmental impact of output is also a flow. But flows by themselves do not allow us to answer questions about sustainability as we argued above. Saying that the rate of environmental impact is declining as output increases, i.e., that growth is green, is not sufficient. Natural capital could still be declining and heading toward a tipping point. The relevant questions are: Does the stock of natural capital stabilize over time? And what constraints does this put on the corresponding stock of physical capital and, ultimately, economic growth?
We start with the accumulation of physical capital and look at the impact of economic growth on the environment. We can illustrate the key issues by focusing on a one-sector economy where relationships are all linear (see chart 3).
This framework is a variant of the "AK" model, a workhorse in the growth economist's toolkit. On the horizontal axis is the stock of physical capital at time t, denoted as Kt. Again, we can think of K as a composite capital stock comprising plants, equipment and infrastructure. The vertical axis is the output of a single good or GDP, denoted as Y. The production function relating the capital stock to output has a constant slope equal to α (this is factor productivity). The output at time t corresponding to capital stock Kt is therefore Yt = αKt. A steeper slope (a higher α) corresponds to higher productivity or output per unit of K, and vice versa.
Over time, this basic economy can grow as follows. In every period GDP is divided between consumption and investment: Y = C + I. We can think of the expenditure shares of C and I as being constant over time, a feature in many models. These variables are all flows, but investment increases the stock of capital over time (net of depreciation) as follows.
Kt+1 = Kt + It – δKt
So next period's capital stock Kt+1 is this period's stock Kt plus new investment minus a constant rate of depreciation. Note that as long as I > δK, the capital stock will increase, output will increase, and this economy will grow without bound. A two-period example involving time 0 and time 1 is shown in the chart: the capital stock grows, and GDP increases.
While this type of model has been used for decades, we know that the world is more complex. The capital stock not only produces output, which we can consume and invest. Its accumulation also hurts the environment. We can also think of this impact as a composite variable including both emissions and damage to the biosphere. (Recall that these basic models are trying to capture key interactions and are simplifications of reality).
We can modify our chart to include an environmental impact. This impact is shown in the lower part of chart 4, which maps the capital stock Kt at time t to an environmental impact (Et = βKt). The top half of the chart is unchanged. The relation between the capital stock and its environmental impact is not necessarily a linear relationship, although we will treat it as such in this paper. Note that both the output of this economy Y and the environmental impact E are flows, both jointly generated by the capital stock K (the stock of natural capital comes in the next section).
There is a fundamental tension in this framework between economic growth and the environment. Note that when K increases, so does Y. Raising the capital stock raises output. More output implies more consumption and is generally seen as a positive development. However, when K increases, so does E. Raising the capital stock increases the harm to the environment. Of course, the impact could also run in the opposite direction: from the degradation of natural capital back to the economy (see, for example "Lost GDP: Potential Impacts of Physical Climate Risks," Nov. 27, 2023). Doing that in a two-stock framework presents an additional modeling challenge that we will leave for future work.
But all is not necessarily lost. We can show the possibility of "green growth" in this framework. In terms of our chart, this is achieved by simultaneously increasing K and rotating the environmental impact curve sufficiently inward. As a result, every unit of capital stock utilized will have a lower environment footprint than previously. For example, this could reflect a lower carbon way of producing our sole good. This points to the central role of technology in green growth.
We have illustrated this in chart 4. Increasing the capital stock from an initial K0 to K1 increases output from Y0 to Y1. This would increase the environmental impact from E0 to E1. This is our expected negative trade-off between growth and the environment. However, rotating the impact curve inward by a sufficient amount lowers the environmental impact of a higher capital stock (and output) to E2 < E0 < E1.
Green growth can thus occur. But, as noted above, while the capital stock has increased, we have no way of knowing whether these lower environmental impact flows are sustainable. The environment may still be deteriorating (we have only slowed the rate of deterioration in this section). To determine sustainability, we need to determine not only the flow relating to the environmental impact of production, but the net impact on the stock of natural capital.
We will now focus on natural capital. To begin, we will ignore any interaction with the economy. Central to our framework is the regeneration function, which we have adopted from the Dasgupta Report, commissioned by the U.K. government. The basic idea is that natural capital can potentially self-repair if left alone. The assumption of self-repair embodied in the regeneration function is the foundation of the natural capital side of our model and a key element of our sustainability framework. A representative regeneration function is shown in chart 5.
This nonlinear regeneration function has the following features:
The regeneration function as drawn has a profound environmental interpretation. Take an example where natural capital begins at some relatively high value close to one. If the system is hit by a one-off shock (say a fire or a volcanic eruption), then the size of the shock matters critically for sustainability. If the shock is moderate and leaves the stock of natural capital above L, then the dynamics will bring the system back toward unity — see chart 6 for an example — starting N at around 0.85.
However, if the shock is large and leaves the value of N below L, then the dynamics drive the system down to 0. There is no way back from a shock that leaves the stock of natural capital below the tipping point. Put alternatively, beyond a certain threshold (the tipping point), the system is no longer able to self-repair or regenerate and totally degrades. The damage to the system is too large.
A quick word on the math: Our regeneration function is called a cubic function since its highest order term is N3. The cubic function generates a shape with two humps. (In contrast, the better-known parabola is a squared function; its highest order term is N2 so it has only one hump). Our regeneration function is represented by the following formula.
ΔN = rN(1 - N)(L - N) = r[N3 – (1 + L)N2 + LN]
By inspection of the term between the two equality signs, we can see that when N = 0, L or 1, then ΔN = 0 and the system is at equilibrium. Also of note, this complex looking function has only two parameters: r and L. As drawn in chart 5, r = 0.5 and L = 0.25. r controls the amplitude of the curve. A lower r generates a flatter curve and vice versa. A flatter curve means that the regeneration process is slower. L is the value of the tipping point. Increasing L moves the tipping point to the right and vice versa and forces the value of ΔN lower for all N (see charts 7a and 7b.)
These two parameters are critical for our solution for the maximum sustainable capital stock (and output) and will be revisited below.
So far, we have only discussed the dynamics for natural and physical capital separately. In the economy, the capital stock jointly produces output and "emissions." This is a linear process by assumption. In the environment, natural capital can regenerate up to a point. But its dynamics are complicated by nonlinearities, including a tipping point. So far, we have not investigated any economic forces continuously acting on natural capital.
We will now let the economy interact with the environment. This will be done by jointly mapping the environmental impact function and the regeneration function in relation to natural capital. This is shown in chart 8 for a given value of E.
Critically, we now have two forces driving the stock of natural capital in opposite directions. The regeneration function shows natural capital's ability to self-repair, which drives values of N higher (as long we are above the tipping point). In contrast, economic activity, through the environmental impact channel, always drives the value of N lower. Analyzing the net value of these two forces is key to determining sustainability.
The net impact of these two forces depends critically on the value of the economic impact from production, that is, the height of the E line. This can change over time, for example moving higher as the economy grows or lower as the economy becomes more environmentally efficient. This could happen by adopting nature-positive approaches such as organic farming and renewable power generation or by using capture technologies. In contrast, we will assume that, that regeneration function is given by nature and cannot be changed.
To measure this net impact, we need to redefine a variable, and let ΔN t = ΔR t – E t. This means that the net change in the stock of natural capital is the difference between its ability to regenerate ΔR t and the impact on economic activity E t.
Chart 8 shows the joint interaction of the environmental impact of economic activity and regeneration on the stock of natural capital. In terms of the geometry, the addition of E to this chart shifts the horizontal axis upward. Now, ΔR t – E t is the vertical difference between the regeneration curve and the environment impact line. Since E is constant, what drives the change in natural capital here is the nonlinearity in the ΔR curve as we move along the horizontal axis.
In our modified framework for natural capital determination, we now have the following:
We can put some values on this to illustrate our examples above. If we assume a value of 0.3 for E, and use r = 0.5 and L = 0.25, then the two equilibria for natural capital are 0.4 (unstable) and 0.95 (stable). If natural capital N starts above 0.95 then it declines until it reaches the stable equilibrium N2 since ΔR < E in this range. This is because the rate of regeneration is below the environmental impact from production; the pristine state of one is no longer an equilibrium. If N starts between 0.4 and 0.95, then it rises since ΔR > E; the rate of regeneration exceeds the environmental impact, and the system also goes to the equilibrium A. However, if natural capital starts or falls below 0.4, then ΔR < E and the system goes to zero. N1 is our new tipping point.
Importantly, this configuration is not the only one. Indeed, it is arguably optimistic since there is nothing in our framework guaranteeing that the value of the environmental impact of production is sufficiently low enough for an equilibrium stock of natural capital to exist. Consider chart 9, which shows three possible levels for E.
Case A features the environmental impact example we just described. The line Ea intersects the regeneration curve twice, yielding two equilibria: a lower, unstable one and a higher, stable one. Natural capital in this economy will eventually settle at a level of N A2 regardless of whether N starts above or below that value (so long as it doesn't fall below N A1).
Case B has no equilibrium for ΔN because the level of environmental impact is too high. The line Eb does not intersect the regeneration curve anywhere. This means that ΔR t – E t < 0 everywhere so that N always declines. Natural capital in this economy falls monotonically toward zero. Note that it decelerates initially (since ΔRt – Et narrows) as we decrease N, but then accelerates.
Case C illustrates the critical sustainability threshold: This is the maximum value for E that still has a natural capital equilibrium value. Here, the line EC is tangent to the regeneration curve. This means that there is only one natural capital equilibrium, at point N C*. For levels of N above this sole equilibrium, natural capital will decline toward N C*. For levels of N below this equilibrium, natural capital will also decline. Note that ΔR t – E t is never positive in Case C. The maximum value is zero.
We now have the elements to illustrate strong sustainability with two types of capital. This derives from Case C in the previous section, where we showed that NC is the minimum equilibrium level of natural capital in this model.
Returning to our production framework, we can now graph macroeconomic-environmental sustainability. This is shown in chart 10. From our natural capital dynamics section, we know that the environmental impact cannot exceed EC. That has the effect of putting a hard constraint in the lower panel of the chart. Any E above EC violates strong sustainability and is therefore ruled out. For a given environmental impact function, a ceiling EC puts a ceiling on the capital stock KC, which puts a ceiling on YC.
In this static sense, the no-growth advocates have a point; output has a cap. But the no-growth view ignores the possibility of technology. In particular, the invention and adoption of technology that lowers the environmental impact of production. (Note that increase of productivity parameter α in the "AK" model is environmentally neutral.) While technology cannot change the ceiling EC, it can improve the relation between the stock of capital and its environmental impact, and thus output and its environmental impact. This is shown by the inward shift in the impact function in the lower part of the chart, and the resulting higher sustainable capital stock K c and output Yc.
An inward shift in the environmental impact curve could also reflect changes in consumer behavior, not just environmental efficiency. A growing awareness about sustainability on the part of households has led to changes in the patterns of consumption of food and water (including packaging and recycling), as well as how we deal with waste, and emissions broadly defined.
This is indeed what has been happening in advanced countries, at least in the case of emissions (recall that natural capital is a composite variable, including emissions). Namely, output has been delinking from emissions for the past several decades. This has reflected both the shift from (carbon-intensive) manufacturing to (less carbon-intensive) services as well as technologies that have lowered emissions from output.
Chart 11 shows the case of the United States, where per capita emissions are now roughly one-quarter lower than 1990, when GDP per capita is over 50% higher. Similar dynamics are at play in Europe and Japan. Critically, most emerging-market economies, including China, India and Brazil, have yet to delink.
The final step of our framework is to solve for the maximum sustainable capital stock. From chart 9, we know this stock is reached when the environmental impact of capital βKc is equal to the maximum value of the regeneration curve r[N3 – (1+L)N2 + LN] when the latter is evaluated at Nc. For this level of the capital stock Kc we therefore have:
Kc = (r/β) [Nc (1 - Nc) (Nc - L)]
While this is somewhat complex (again owing to the nonlinearity of the regeneration function), we can easily see the sensitivity of the minimum sustainable capital stock to our key parameters:
The introduction of tipping points shrinks the macroenvironmental playing field. Capital cannot be accumulated without bound, and output cannot expand without bound, as in earlier versions of growth models. Those models were incomplete in the sense that they lacked any consideration of natural capital. Including the environmental impact of economic activity and the existence of tipping points in the dynamics of natural capital in our framework means that growth possibilities are constrained. This does not mean growth cannot happen. But to be sustainable we need to continue to lower the environmental impact of production. Nature cannot be changed. The parameters r and L of the natural capital regeneration process are given.
Basic models can only take us so far. Their utility lies in isolating the key factors at play and unearthing previously hidden insights. A more complete model would certainly be more complex. And it would need to be calibrated. Much of climate science is steeped in data showing the changes (deterioration) in natural capital. But the links back to physical capital and the economy are less well developed.
By linking physical and natural capital in a basic but internally consistent structure, we hope that the framework presented in this paper will move sustainability research forward.
