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Verhulst

Each spring, I give a favorite lecture, an homage to a Belgian mathematician, Pierre François Verhulst.

In two publications (1838 and 1847), Verhulst argued that the growth rate of a population (N = numbers or density) can be described as dN/dt = rN(1-N/K).

This, as I love pointing out, is as elegant a model as one can find in applied mathematics because it is simple, and the two constants have biological interpretations. K is carrying capacity; r is the population’s intrinsic growth rate. If you interpret the interaction of N and K as the joint dependence of the Ns on K and recognize that species can interact because of a joint dependence on K in the same way that individuals can, you realize that nearly the whole of the science of ecology turns on this singular little equation which says that nothing grows without limits (the little equation is the starting point for more sophisticated models). Ecology is the science of understanding limits to growth.

Verhulst and his followers were drawn into intense arguments about whether his equation was a “law” of population growth analogous to the laws of physics or chemistry. I am in the “not a law” camp (I have an arcane concern about how to think about stochasticity) but were I forced to choose one exception; it would be Verhulst’s law of density dependent growth – known today as logistic growth.

I am, of course, reflecting on the recent discussions of ecology and economics in the Reformed Journal and its blog (hat-tips to Professors Rienstra, Smith, Steen, and Zylstra).

The lesson from my discipline, that nothing grows without limits, has two important corollaries 1) joint dependencies tie the interests of individuals and species together and 2) limits to growth are enforced by negative feedbacks.

To amplify a point Professor Zylstra made, the economy, in its conventional sense of trade in goods and services for people, does not exist in a vacuum. More than simply being connected to…or associated with …the goods-and-services economy is embedded in a larger real economy of atmospheric carbon cycling, soil nutrient cycling, photosynthesis, hydrology, ecosystem dynamics, and other nature systems.

Hence when Professors Steen and Smith write that “economic growth is nonetheless a moral imperative because, arising from human work, agency, and creativity, it provides the resources for improved material wellbeing,” their assertion is incomplete.

Economic growth (and therefore, human wellbeing) is also a draw on the world’s limited resources, its limited capacity for regeneration of those resources that we conventionally consider to be renewable and, most critically, the earth’s limited capacity to absorb the wastes that economic growth must, by necessity, create. In this sense, Professor Rienstra is correct when she writes that perpetual growth is physically impossible.

This point is critical because we collectively are already well beyond the planetary limits for sustainable growth. Therefore, the larger moral imperative is to prevent further damage because the stakes are so high. I could finish this essay simply listing the negative feedbacks we are experiencing, most of which fall most severely on the poor and the non-human parts of creation.

Climate scientists tell us that we are on track to make large parts of the globe unlivable within the lifetimes of some of the people who will read this, not to mention the lasting damage done to non-human creation. Growth requires consuming and polluting at some level and we have consumed and polluted ourselves into a planetary crisis and searing injustice.

Should we seek flourishing for the poor of this world? With all urgency! But the energy and resources to support that flourishing should come from the consumer gluttony of wealthy people/countries so that green house gas emissions stop increasing.

I’ll pause here to acknowledge that Professors Smith and Steen wrote to defend their discipline from a perceived unfair criticism. Their point was that “(e)conomists as a whole focus on the tradeoffs between different choices that people and nations make, as well as the full costs of such decisions. Politicians and policymakers, businesspeople and individuals, essayists and other analysts often ignore both costs and benefits, while economists delight in making them clear. Insisting on thinking about tradeoffs is one of the most crucial contributions economics makes to social science research.” (Note that Professor Steen commented on Professor Zylstra’s post on Tuesday to reassert that economists do, in fact, count environmental costs. It’s worth a read.)

Fair enough. I would join Professors Steen and Smith in defending academic economists for the important work they do, although I am dubious too (with the others) about their claim to be broadly “counting the full costs of those decisions” and “making them clear.”

I rarely see environmental costs, especially cumulative costs evaluated with the gravity they require. Failure to count and show the cumulative environmental costs makes trade-offs less clear with an asymmetry that favors economic activities that incur further damage.

I know that academic economists are not a monolith, that there are those who are rightly skeptical about unlimited growth. There are economists focused on environmental sustainability and mitigation of the damage done. Some of them are key contributors to the IPCC process. Some are friends of mine.

But blaming “Politicians and policymakers, businesspeople and individuals, essayists and other analysts” for ignoring both costs and benefits seems like a bit of a reach and it’s probably a driver for the pushback Professors Steen and Smith got from Professors Rienstra, Zylstra, (and Bouma-Prediger in the comments on Professor Zylstra’s post).

I know from experience, including experience here, that for any environmental problem there is always a voice arguing that earthkeeping is too expensive and all too often arguing their point under the color of “economic” wisdom. Consider that voice as separate from academic economists like Professors Steen and Smith if you will, but that voice has been piping up for decades and is well known. Aldo Leopold called them “economic moralists” in his lament for the extinct passenger pigeon (also “economic determinists,” Sand County Almanac 1948) and his famous argument for a land ethic begins with an admonition to “quit thinking about decent land-use as solely an economic problem.”

Economic moralists love the colorless term “externalities” because they can hide environmental costs there – especially if they are hard to measure in dollars and cents. The outsize influence of the economic moralists is easy to comprehend. They give cover to interests that privilege near-term gains for shareholders and CEOs over the interests of poor people and non-human creation. They’re the reason this New Yorker cartoon has such resonance.

“Yes, the planet got destroyed. But for a beautiful moment in time we created a lot of value for shareholders.”

At a minimum, Professors Steen and Smith’s noble discipline (I am sincere here) has a problem in how it is translated and used. From my vantage here in the earthkeeper cheap seats, the climate crisis itself is the cumulative looming feedback from economic moralists relentlessly winning policy arguments for more growth by ignoring or discounting environmental costs.

We’ve known about the problem of green house gas emissions for decades. Exxon’s scientists knew in the 1970s. Kicking the can down the road was an economic decision and even as recently as this week Reuters is reporting that despite most big fossil fuel companies pledging to achieve net-0 emissions, none (!) of them had plans in place to achieve those plans. It’s the standard green-washing BS we’ve come to expect from fossil fuel companies, but you can bet that the decision to dupe their customers stems from an economic calculation that it’s better for their bottom line.

Similarly, the “economic” content in our popular media is mostly about stroking the machinery of capitalism to create and hoard wealth and breathless tracking of impossibly precise increments of stock market indicators in real time. What of the economic realities of climate justice? It’s indicators? Why not maintain a focus on the most urgent work of humanity right now?

It doesn’t sell. Pity that.

Tim Van Deelen

Tim Van Deelen is Professor of Forest and Wildlife Ecology at the University of Wisconsin – Madison. He grew up in Hudsonville, Michigan, and graduated from Calvin College. From there he went on to the University of Montana and Michigan State University. He now studies large mammal population dynamics, sails on Lake Mendota, enjoys a good plate of whitefish, and gains hope for the future from terrific graduate students. 

10 Comments

  • RZ says:

    Thanks Tim. I appreciate the reasoned discussion and appeal expressed here recently. I also give thanks for economists who weigh in with a Christian perspective. We need those voices. You also taught me a new word: “stochasticity.” I knew sex was somehow to blame for environmental damage, just like everything else ( just kidding)! Our own greed, apathy, denial, and claim to predestined privilege just goes on and on. What do voters demand? A party that enhances their own economic privilege, aka freedom, and the political strategists know it! A free-market capitalism boosts the economy but ignores the consequences within other domains. As such, it is not inherently bad, but it will be if left unrestrained by the principle of shalom. This, it seems to me, is where the church needs to use its voice! Throughout history, the church has failed more often than not. Recent thoughts expressed here are most helpful.

  • David Hoekema says:

    What a gift to RJ readers: a thoughtful and illuminating extended conversation on matters of profound importance to every believer and every citizen. I’ve been educated and challenged by each of the reflections. Many thanks. (And a gentle word of advice to all engaged in debate over other issues of import, in denominations or in legislatures or just at family gatherings: go thou and do likewise.)

  • Jack says:

    Another “term” that has become meaningless yet used as something positive is “well-being.” What in the world is well-being? Jesus never knew “well-being.” Sell it ALL and give it. I had a friend who could move only his head and one index finger. And money? Went to caregivers. Such rich times we had. I grew up in a little town surrounded by Amish farms. Never once did I hear the owners of the mom and pop stores say, “I could make more money by . . .” I did hear, “Pays the bills.” Let’s learn what real poverty is before it’s too late.
    Well-being. Illusion, smokescreen, misleading. Can’t exist. Don’t buy that book.
    As always, thank you, Tim. And now to savor the thunderstorm. It’s not cost-effective. Unless for what benefits.

  • Todd Steen says:

    Professor Van Deelen-

    Thanks for taking the time to respond to Prof. Smith and my article. I will start by saying that I disagree with some of the things you write, and I will try to politely point some of them out. One of the cool things about Reformed Christians is that we like to think deeply about issues, and we often disagree, and we love to discuss!

    But first an agreement! You mention that the lesson from your discipline is “that nothing grows without limits.” Economists believe the same thing! We believe that the fundamental problem of our world is that of scarcity, and we study the choices that individuals and institutions make to deal with scarcity. We particularly like to focus in on the tradeoffs inherent in these choices, so things like the limited regeneration of many resources and a limited capacity to absorb wastes from economic activity are important factors. We often say our analysis is “constrained optimization,” so we are always talking about constraints. We don’t believe that there is endless perpetual growth without consequences.

    You are dubious about our claim to be broadly “counting the full costs of those decisions” and to be making them clear. Economists are nowhere near perfect, but we are trying to do this. You assert that you “rarely see environmental costs, especially cumulative costs evaluated with the gravity they require.” First of all, it isn’t easy, but we should certainly try. We must also be careful not to underestimate the costs of new policies or to suggest change is easy—it isn’t. We need to be clear minded in our evaluations, so we need both economists and ecologists. We may disagree on points, but we can learn from each other; I truly believe this.

    There have been substantial criticisms of economists throughout these discussions. We can take it, we may deserve some of it, but I am also glad to defend the discipline where I can. What I haven’t seen is a lot of specific statements made by prominent and influential economists of today. If there are specific statements that economists have made that are causing the problems you state, please bring them out.

    I would disagree that Prof. Smith and I are blaming “politicians and policymakers, businesspeople and individuals, essayists and other analysts” for the problems that Prof. Van Deelen describes, while letting economists of the hook. All of us like to ignore the full costs and benefits of our actions on others and the creation—it’s called sin, and as a believer in “total depravity,” there is a lot of it out there to go around. Economists are no less sinful even though we do focus on costs and benefits in our discipline. Prof. Van Deelen himself blames “consumer gluttony of wealthy people/countries” so he doesn’t mind blaming individuals for ignoring costs and benefits, and he blames shareholders, the popular media, and CEOs too. He also blames “economic moralists” (I am not sure who they are and how they are “economic”) and people who argue under the color of “economic wisdom” (I am not sure how that wisdom is “economic” in any sense).

    When I read Prof. Van Deelen’s post, I see a sincere longing for creation to be healed in the way we know it will be when Christ returns. All Christians should have this concern. Making this happen while increasing the material wellbeing of the poor is a substantial challenge. Ecologists and economists are necessary allies. It doesn’t help to mischaracterize what economists do and say, nor use the word “economic” as a negative adjective.

    I think Prof. Van Deelen finds his true enemy in the last paragraph when he talks about capitalism. There is no doubt that capitalism has significant problems, although I would assert the problems of socialism and communism are worse. I personally believe that free markets can provide opportunities to exercise good stewardship, and in many cases they do. I do have significant worries when the interests of large companies and large governments overlap, whether in capitalism or socialism. All of these are arguable points, and these questions are perhaps the subject of future discussions.

    • Daniel Meeter says:

      Dear Professor Steen: As a regular reader, I want to welcome you to these pages for your more frequent participation. You have much to teach us, and to challenge us with. Tim and Debra’s writings have been life-giving for me, and but a chorus of dedicated (and responsible voices) can make music in counterpoint.
      Let me add that I am guessing I knew your father, who was in inspiration to me. He being driven around the whole NE by Odell Merryman (was that his name), meeting with one small group here, another there, two people here, three people there, speaking with passion about the Kingdom of Heaven. “How long are we going to let the Secular-Humanists kick the shit out of us!

      • Todd Steen says:

        Rev. Meeter-

        Thanks for your kind words. That is indeed my father who you spent time with driving around. You story reminds of many “road trips” I spend with him!

    • Tim Van Deelen says:

      Professor Steen,

      Thank you for reading and commenting on my post. I think this discussion is vital and, again, I appreciate the dialog.

      I think that you define “economics” too narrowly. Essentially, I read your commentary to say that the practice of economics is simply the dispassionate and value-free evaluations of the costs and benefits of humans buying and selling goods and services. It’s your discipline, so I defer.

      But I tried to make a distinction in my post about the difference between your version and the decisions that people and entities make about buying/selling goods and services and the policies that relate. If that isn’t “economic” activity, I don’t know what to call it. In any case, I would wager that the RJ reader knew the difference and understood the point I was making.

      When you write that economists believe that “the fundamental problem of our world is that of scarcity”, I wonder if that isn’t exactly backwards. Seems to me that the fundamental problem right now is excess. We need to live within the biophysical boundaries of the planet as a matter of justice and responsible earth keeping. Currently we, especially we North Americans (per capita), are not.

      As for blaming. Yes. Some people and some entities are more culpable with the sin of “ignore(ing) the full costs and benefits of our actions on others and the creation” (your term, and I am glad we agree here). The alternative (nobody is to blame, or we all share blame equally) is untenable. Assessing blame (culpability) tells us where to seek change in the service of justice.

      As for making an enemy of capitalism, let’s not mischaracterize what I said. I never, in this post or any other, have argued for socialism or communism. I think you and I would agree that free market capitalism needs to be regulated for the common good and to avoid abusing people and non-human creation. My contention is that it needs to be regulated better. I think the climate crisis and its drivers make that clear. Capitalism alone isn’t the sole driver of climate-damaging activities but it’s a big one.

      Again. Thank you.

      • Todd Steen says:

        Prof. Van Deelen-

        Thanks for this new response—the dialogue is great.

        First, I never have implied that “the practice of economics is simply the dispassionate and value-free evaluations of the costs and benefits of humans buying and selling goods and services.” I don’t even believe that it is possible to make value-free and dispassionate analyses—everyone comes to their evaluations with a worldview that significantly colors their approach. What I have tried to emphasize is that economists work hard to try and consider all the costs and benefits in their analyses, including environmental ones. In case there is any misunderstanding, I also don’t believe economists do it right all the time. In fact, economists are known for disagreeing. We love to try to consider all the costs and benefits, and we often have different valuations of them.

        Also, economists study carefully what you describe as “the decisions that people and entities make about buying/selling goods and services and the policies that relate.” Indeed, that is how economists define their discipline. I agree with you that this is economic activity.

        But I am still not sure what you mean by “economic moralists,” “economic wisdom,” and “economics content.” Are these people economists and do these concepts come from economists? You suggest that the “discipline of economics has a problem in how it is translated and used.” Economists emphasize that we should consider all costs and benefits—I will talk about this in a few weeks during my classes. We know that people don’t always get it. But we emphasize that there is no such thing as a “free lunch,” and that economic growth has real impacts that we must consider and that should influence our policies. There are real tradeoffs, and there are important decisions to be made.

        As I have responded to several posts, I wonder if some writers are conflating economics with money, greed, or unethical business behavior. Economists study a lot more than money, and we don’t go around saying “greed is good” or that more money is always better. We appreciate what businesses do, but also understand and point out the shortcomings of business activity, especially when it takes advantage of the public. Economists don’t support fossil fuel companies duping their customers to better their bottom line—that is not an “economic calculation.” We also examine carefully the full costs and benefits of government activity.

        In your response, you suggest that economists have it backwards when we say that the fundamental problem of our world is that of scarcity. What economists mean by this is that we have limited resources and that we have to make choices—there are real limits on our behavior. Economists try to model and understand the choices that people and institutions make to deal with these limits. Your post emphasizes the real limits that creation enforces on our activity. There is no doubt that this is true. You instead suggest that “excess” is the fundamental problem right now. I think I understand where you are coming from, but most people in the world still feel the harsh impact of scarcity. Of course, excess is never the correct response when considering the mandates of stewardship.

        You state at the end of your blog post that climate justice is the most urgent work of humanity right now. I think this is at least open to discussion, and that relieving poverty could be the most urgent work (and I understand that they can be related). Others might suggest other important problems. I think it is likely that we have some disagreements about the exact relationship between economic growth and sustainability and how this all works out. However, I look forward to learning more from you and others and discussing the true costs and benefits of our economic activity.

  • Thank you Tim for sharing your thoughts on limits to growth. I especially appreciated the mathematical formula. It highlights my reservations about applying a model for the animal world to Homo Sapiens. For the former r and K are predetermined (exogenous) as you point out, and limits to growth apply. However, for the latter r and K are endogenous (determined inside the system we are analyzing). Technological developments have enabled mankind to steadily increase K over the past centuries as looming constraints have focussed our attention on removing them. Rising wealth and technology have steadily driven down the intrinsic human population growth rate, r, to the point where it is negative in some countries.

    As a young academic economist just starting my career 50 years ago I read with great interest the Club of Rome’s Limits to Growth that had just been published. There predictions couldn’t have been more wrong. Technological developments, innovation and adaptation were not built into their computer models and their doomsday predictions failed to materialize. When the problem solving capacity of the human brain is taken into consideration it is not self evident to me that there is a maximum carrying capacity (K) for economic activity beyond which human ingenuity cannot go. Energy will be a key factor in determining if there is such a maximum that will turn exponential economic growth into logistic growth. If there is such a K we must also consider when it will begin to bind. It has very little current policy relevance if we cannot argue convincingly that it will be reached within a normal planning time frame.

    Two hundred years ago Robert Thomas Malthus predicted that since the human population grows exponentially, and agricultural production arithmetically, humanity was doomed to live at subsistence when all arable land had been turned to food production . Fifty years ago the club of Rome predicted that by the turn of the century there would be mass starvation as agricultural production could no longer feed the world. Both predictions were wrong, and both for the same basic reason, they ignored human ingenuity and adaptation.

    Are there currently constraints on economic growth that loom large and that seem insurmountable? The natural carbon cycle is one that comes to mind. It can only handle a certain amount of carbon emissions that we exceed at our peril. However, human ingenuity is already hard at work to figure out how to augment the natural cycle by capturing carbon dioxide from emissions, and from the air, and recycling it into a fuel. There are other such problems that are daunting but for which we have the scientific tools to find solutions. A finite K for economic activity may or may not exist but I am confident that if there is one it is not lurking around the corner.

    Economic growth has enabled us to allow young people to continue their education well into their twenties, it has extended our life expectancy by decades, it has greatly reduced the risk of childbearing and the physical burden of raising a family, it has opened up a wide range of opportunities for men and women that were unheard of only a hundred years ago when the bulk of the labour force worked 60 hrs per week on the land or in factories. There are many people today who do not enjoy our North American standard of living. It will be much harder for them to achieve it if it becomes a zero sum game without economic growth. As a Christian economist I look at responsible economic growth as a blessing for the welfare of all mankind and the best hope for the poor nations of this world to achieve the same standards of health and education, and the same opportunities that we in the west have.

    Economics is often called the dismal science because it reminds society that there is no free lunch. I happen to be a very optimistic dismal scientist because I know that although lunch is not free it’s real price (in terms of other resources to produce it) has been falling since the days of Robert Malthus. I look forward to the day when economic growth will bring this blessing to all peoples around the world.

    • Tim Van Deelen says:

      Joseph, with respect, you missed my point about starting with Verhulst. I said nothing at all about human population growth and referred to K as an intro to write about feedbacks and joint dependencies. One can’t say “there is no free lunch” on the one hand and then proceed to say that humanity can continue to add carbon to the atmosphere by fueling growth the way that we have.

      That there is no free lunch with respect to planetary boundaries is exactly my point.

      I do not dispute the positive benefits of growth nor the need to seek human flourishing. I would add that the need to seek non-human flourishing is imperative too and is a point that gets lost in these discussions (Creations care imperatives, joint dependencies, etc.). I am arguing that the cost of growth, coupled as it is with carbon emissions, is underappreciated. Any Christian vision of just flourishing needs to start with decoupling growth from carbon emissions.

      The argument that we can pollute ever more because we’ll just invent ways to deal it is an example of the hubristic fallacy at the heart of Paul Ehrenfeld’s important book “The Arrogance of Humanism.” Ehrenfeld’s point is that human hubris about technological fixes tends to spin off unintended and even stickier problems – the climate crisis being one example.

      Carbon capture isn’t yet scalable and may never be and even if it were, the massive costs needed to scale it up quickly enough to make a difference have not been identified. I hope I am wrong. The technology has been known since the 70s and its only removed a trivial amount of atmospheric carbon in that time (source below). As another example of Ehrenfeld’s unintended consequences, fossil fuel companies and governments use unproven and wildly optimistic CC as justification for more exploration and extraction. Just to get back to safe levels, CC technologies would need to remove roughly a trillion tons of legacy carbon.

      With respect to your point about a finite K for economic activity not “lurking around the corner,” climate scientists tell us that the safe level of atmospheric carbon is about 350 ppm CO2. It’s about 420 ppm now and is steadily rising. Human sources of atmospheric carbon come largely from economic activity. The feedbacks from that excess carbon are becoming more and more severe (as climate scientists have been predicting) and are all around us (and are well-known).

      https://ieefa.org/resources/carbon-capture-has-long-history-failure?gclid=Cj0KCQjw0IGnBhDUARIsAMwFDLnCdGlN72eCBFfdvxlk9XrDwVO5PYZCgHTGK9NHyUQ1Y3c7hDeyK-8aAj14EALw_wcB

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