Thermodynamics and Economic Equilibrium

I’m having another round of studying thermodynamics, and I’m running into more interesting leads than I can keep up with. Like this paper:

• Eric Smith and Duncan K. Foley, Classical thermodynamics and economic general equilibrium theory, Journal of Economic Dynamics and Control 32 (2008) 7–65.

I’ve always been curious about the connection between economics and thermodynamics, but I know too little about economics to make this easy to explore. There are people who work on subjects called thermoeconomics and econophysics, but classical economists consider them ‘heterodox’. While I don’t trust classical economists to be right about things, I should probably learn more classical economics before I jump into the fray.

Still, the introduction of this paper is intriguing:

The relation between economic and physical (particularly thermodynamic) concepts of equilibrium has been a topic of recurrent interest throughout the development of neoclassical economic theory. As systems for defining equilibria, proving their existence, and computing their properties, neoclassical economics (Mas-Collel et al., 1995; Varian, 1992) and classical thermodynamics (Fermi, 1956) undeniably have numerous formal and methodological similarities. Both fields seek to describe system phenomena in terms of solutions to constrained optimization problems. Both rely on dual representations of interacting subsystems: the state of each subsystem is represented by pairs of variables, one variable from each pair characterizing the subsystem’s content, and the other characterizing the way it interacts with other subsystems. In physics the content variables are quantities like asubsystem’s total energy or the volume in space it occupies; in economics they area mounts of various commodities held by agents. In physics the interaction variables are quantities like temperature and pressure that can be measured on the system boundaries; in economics they are prices that can be measured by an agent’s willingness to trade one commodity for another.

In thermodynamics these pairs are called conjugate variables. The ‘content variables’ are usually called extensive and the ‘interaction variables’ are usually called intensive. A vector space with conjugate pairs of variables as coordinates is a symplectic vector space, and I’ve written about how these show up in the category-theoretic approach to open systems:

• John Baez, A compositional framework for passive linear networks, Azimuth, 28 April 2015.

Continuing on:

The significance attached to these similarities has changed considerably, however, in the time from the first mathematical formulation of utility (Walras, 1909) to the full axiomatization of general equilibrium theory (Debreu, 1987). Léon Walras appears (Mirowski, 1989) to have conceptualized economic equilibrium as a balance of the gradients of utilities, more for the sake of similarity to the concept of force balance in mechanics, than to account for any observations about the outcomes of trade. Fisher (1892) (a student of J. Willard Gibbs) attempted to update Walrasian metaphors from mechanics to thermodynamics, but retained Walras’s program of seeking an explicit parallelism between physics and economics.

This Fisher is not the geneticist and statistician Ronald Fisher who came up with Fisher’s fundamental theorem. It’s the author of this thesis:

• Irving Fisher, Mathematical Investigations in the Theory of Value and Prices, Ph.D. thesis, Yale University, 1892.

Continuing on with Smith and Foley’s paper:

As mathematical economics has become more sophisticated (Debreu, 1987) the naive parallelism of Walras and Fisher has progressively been abandoned, and with it the sense that it matters whether neoclassical economics resembles any branch of physics. The cardinalization of utility that Walras thought of as a counterpart to energy has been discarded, apparently removing the possibility of comparing utility with any empirically measurable quantity. A long history of logically inconsistent (or simply unproductive) analogy making (see Section 7.2) has further caused the topic of parallels to fall out of favor. Samuelson (1960) summarizes well the current view among many economists, at the end of one of the few methodologically sound analyses of the parallel roles of dual representation in economics and physics:

The formal mathematical analogy between classical thermodynamics and mathematic economic systems has now been explored. This does not warrant the commonly met attempt to find more exact analogies of physical magnitudes—such as entropy or energy—in the economic realm. Why should there be laws like the first or second laws of thermodynamics holding in the economic realm? Why should ‘utility’ be literally identified with entropy, energy, or anything else? Why should a failure to make such a successful identification lead anyone to overlook or deny the mathematical isomorphism that does exist between minimum systems that arise in different disciplines?

The view that neoclassical economics is now mathematically mature, and that it is mere coincidence and no longer relevant whether it overlaps with any body of physical theory, is reflected in the complete omission of the topic of parallels from contemporary graduate texts (Mas-Collel et al., 1995). We argue here that, despite its long history of discussion, there are important insights still to be gleaned from considering the relation of neoclassical economics to classical thermodynamics. The new results concerning this relation we present here have significant implications, both for the interpretation of economic theory and for econometrics. The most important point of this paper (more important than the establishment of formal parallels between thermodynamics and utility economics) is that economics, because it does not recognize an equation of state or define prices intrinsically in terms of equilibrium, lacks the close relation between measurement and theory physical thermodynamics enjoys.

Luckily, the paper seems to be serious about explaining economics to those who know thermodynamics (and maybe vice versa). So, I will now read the rest of the paper—or at least skim it.

One interesting simple point seems to be this: there’s an analogy between entropy maximization and utility maximization, but it’s limited by the following difference.

In classical thermodynamics the total entropy of a closed system made of subsystems is the sum of the entropies of the parts. While the second law forbids the system from moving to a state to a state of lower total entropy, the entropies of some parts can decrease.

By contrast, in classical economics the total utility of a collection of agents is an unimportant quantity: what matters is the utility of each individual agent. The reason is that we assume the agents will voluntarily move from one state to another only if the utility of each agent separately increases. Furthermore, if we believe we can reparametrize the utility of each agent without changing anything, it makes no sense to add utilities.

(On the other hand, some utilitarian ethicists seem to believe it makes sense to add utilities and try to maximize the total. I imagine that libertarians would consider this ‘totalitarian’ approach morally unacceptable. I’m even less eager to enter discussions of the foundations of ethics than of economics, but it’s interesting how the question of whether a quantity can or ‘should’ be totaled up and then maximized plays a role in this debate.)

The Social Cost of Greenhouse Gases

Today the US government bumped the ‘social cost of greenhouse gases’ up to $51. The previous administration had knocked it down to between $1 and $7.

It’s a funny expression, ‘social cost of greenhouse gases’, but it means the marginal cost of the impacts caused by emitting one extra tonne of carbon dioxide—or the equivalent amount of any other greenhouse gas. This includes impacts on the environment and human health:

• Wikipedia, Social cost of carbon.

This new decision should affect US government policy decisions based on cost-benefit analyses:

For example, if the Trump administration had applied the Obama-era calculation to its rollback of federal mileage standards, the costs of that rule would have far outweighed the benefits and would have been much harder to justify. And any federal coal leasing in the Powder River Basin would be unlikely to win approval: University of Chicago professor Michael Greenstone noted that the climate damages associated with that mining “are six times larger than the market price of that coal.”

It may also wind up affecting decisions made by state and local governments and corporations. But it lags behind some states: in December 2020, New York adopted a ‘value of carbon guidance’ ranging between $79 and $125, depending on the discount rate assumed. The lower figure assumes a 3% annual discount of future damages, while the higher one assumes 2%.

The discount rate is crucial in all these calculations! The previous administration had reached its ludicrously low figure for the social cost of greenhouse gases using a couple of tricks: it neglected costs incurred outside the US, and it assumed a 7% annual discount rate. Biden’s new $51 cost assumes a 3% discount rate. At a 5% discount rate the cost would drop to merely $14.

All these calculations will be redone more carefully in a while:

While the Biden administration has set an initial price to inform its analysis of policies ranging from gas mileage standards to purchasing, it will embark on a months-long process to determine a longer-lasting one. That price will take other factors into account, such as the fact that the poor suffer more from climate impacts than the wealthy and more recent scientific findings on climate impacts.

The above quotes are from here:

• Juliet Eilperin and Brady Dennis, Biden is hiking the cost of carbon. It will change how the U.S. tackles global warming, Washington Post, 26 February 2021.

Here is today’s White House announcement on the social cost of carbon:

A Return to Science: Evidence-Based Estimates of the Benefits of Reducing Climate Pollution

On January 27, 2021, President Biden issued a Memorandum directing Federal agencies to make decisions guided by the best available science and data. Today, we are taking an important early step in bringing evidence-based principles back into the process of estimating the benefits of reducing climate pollution.

The evidence is clear that climate change is real and is already having economic consequences. The 2018 National Climate Assessment underscored the fact that climate change presents growing challenges to human health, safety, quality of life, and economic growth. We can see economic costs associated with climate change in more frequent and/or intense extreme weather events like wildfires, severe storms, and flooding, as well as the ways climate change disproportionately impacts the most vulnerable in society, particularly lower-income communities, communities of color, and Tribal communities. As decision-makers develop policies, they must incorporate the very real costs of climate change to current and future generations into their decisions.

One specific tool — called the “social cost of greenhouse gases” — combines climate science and economics to help Federal agencies and the public understand the benefits of reducing greenhouse gas emissions. The metric is a range of estimates, in dollars, of the long-term damage done by one ton of greenhouse gas emissions.

The use of this metric by the U.S. Government began in the Bush Administration and was later standardized across agencies in the Obama Administration. The Obama Administration created an Interagency Working Group of technical experts across the Federal Government to develop uniform estimates, subject to extensive public comment, in order to ensure that agencies utilized the best available science and economics.

However, the previous Administration disbanded the Interagency Working Group that had developed and refined these estimates and issued revised estimates based on assumptions that did not rely on the best available science or have the benefit of dedicated public comment. It also failed to implement 2017 recommendations from the experts at the National Academies of Sciences, Engineering, and Medicine that the Interagency Working Group had requested to ensure that these estimates continued to be in line with the latest science and economics.

This Administration will follow the science and listen to the experts. In a first-day executive order on Protecting Public Health and the Environment and Restoring Science To Tackle the Climate Crisis, President Biden reconvened the working group of technical experts from across the Federal Government and instructed them to restore the science- and economics-based approach to estimating climate damages.

The Interagency Working Group — co-chaired by the Office of Science and Technology Policy, Office of Management and Budget, and Council of Economic Advisers — today announced it is replacing the previous Administration’s estimates with the estimates developed prior to 2017, adjusted for inflation. This interim step will enable Federal agencies to immediately and more appropriately account for climate impacts in their decision-making while we continue the process of bringing the best, most up-to-date science and economics to the estimation of the social costs of greenhouse gases.

Today’s step restores three critical aspects of these estimates. First, the estimates put in effect today were subject to an extensive and robust process, including public notice and comment. A 2014 Government Accountability Office (GAO) report also concluded that the Interagency Working Group followed a “consensus-based” approach, relied on peer-reviewed academic literature, and disclosed relevant limitations when finalizing the estimates we are restoring today.

Second, these estimates take global damages into account. COVID-19 has re-emphasized the ways in which events on the other side of the globe can harm us here. Climate impacts abroad can affect the United States in many ways, including through supply chain disruptions, market volatility, and effects on our national security. In addition, climate actions taken by the United States and other countries under the Paris Agreement will benefit all countries, including the United States. Just as we expect and encourage other countries to consider the climate impact of their actions on us, we should take the global benefits of our actions into account.

Third, these estimates use the discount rates (the approach to calculating the present-day value of future climate damages) previously established by the Interagency Working Group. Restoring these rates puts these interim values in better alignment with the intergenerational nature of the climate challenge and the approaches taken in the peer-reviewed literature than the estimates used while the Interagency Working Group was disbanded.

A more complete update that follows the best science takes time. This is why we are quickly restoring the prior estimates as an interim step. With these estimates temporarily in place, the Interagency Working Group will continue its critical work to evaluate and incorporate the latest climate science and economic research and respond to the National Academies’ recommendations as we develop a more complete revision of the estimates for release within a year. Research, such as our understanding of the appropriate approach to discounting, has advanced rapidly over the past few years and we are collecting dedicated public comment through an upcoming Federal Register notice on how to improve our approach.

As this process proceeds, we are committed to engaging with the public and diverse stakeholders, seeking the advice of ethics experts, and working to ensure that the social cost of greenhouse gases consider climate risk, environmental justice, and intergenerational equity. The result will be even stronger science-based estimates developed through a transparent and robust process.

The Economics of Biodiversity

 

One problem with the Anthropocene is that our economic systems undervalue forms of “natural capital” for which there are no markets, or poorly developed markets. I’m talking about things like clean air, forests, wetlands, oceans… and biodiversity. For many of these things the price is zero or even negative, due to government subsidies.

So, we’ll burn through these things recklessly until the ensuing disasters wake us up. We’re like a family trying to earn more cash by selling off the windows and doors of our house. It may work for a while. But winter is coming.

Partha Dasgupta, an economist at the the University of Cambridge, has been studying this. In 2019, the UK government commissioned him to lead an independent, global review of the economics of biodiversity. It came out this month at an event hosted by the Royal Society and attended by the Prince of Wales, Boris Johnson and David Attenborough. Here it is:

• The Economics of Biodiversity: The Dasgupta Review.

The full report is 610 pages long. It’s very clear; I’m reading it and will say more about it here. There’s also a 103-page version and a 10-page ‘headline’ version, but the headlines leave out the really fun stuff: the economic analyses, the differential equations, and so on.

This came out at a good time for me, because I’ve recently been asked to give a talk about the economics of the Anthropocene. I seem to have reached the age where people ask me to give talks about practically anything I’ve ever blogged about. I need a lot of help on the economics, since I have intuitions but no framework to organize them. The Dasgupta Review provides a framework, and since I don’t have a lot of time before my talk, I plan to lean on it rather heavily.

Here’s the introduction, by David Attenborough. It’s easy to read. But it does not get into any of the economics, so please don’t judge the cake by its frosting.

We are facing a global crisis. We are totally dependent upon the natural world. It supplies us with every oxygen-laden breath we take and every mouthful of food we eat. But we are currently damaging it so profoundly that many of its natural systems are now on the verge of breakdown.

Every other animal living on this planet, of course, is similarly dependent. But in one crucial way, we are different. We can change not just the numbers, but the very anatomy of the animals and plants that live around us. We acquired that ability, doubtless almost unconsciously, some ten thousand years ago, when we had ceased wandering and built settlements for ourselves. It was then that we started to modify other animals and plants.

At first, doubtless, we did so unintentionally. We collected the kinds of seeds that we wanted to eat and took them back to our houses. Some doubtless fell to the ground and sprouted the following season. So over generations, we became farmers. We domesticated animals in a similar way. We brought back the young of those we had hunted, reared them in our settlements and ultimately bred them there. Over many generations, this changed both the bodies and ultimately the characters of the animals on which we depend.

We are now so mechanically ingenious that we are able to destroy a rainforest, the most species-rich ecosystem that has ever existed, and replace it with plantations of a single species in order to feed burgeoning human populations on the other side of the world. No single species in the whole history of life has ever been so successful or so dominant.

Now we are plundering every corner of the world, apparently neither knowing or caring what the consequences might be. Each nation is doing so within its own territories. Those with lands bordering the sea fish not only in their offshore waters but in parts of the ocean so far from land that no single nation can claim them. So now we are stripping every part of both the land and the sea in order to feed our ever-increasing numbers.

How has the natural world managed to survive this unrelenting ever-increasing onslaught by a single species? The answer of course, is that many animals have not been able to do so. When Europeans first arrived in southern Africa they found immense herds of antelope and zebra. These are now gone and vast cities stand in their stead. In North America, the passenger pigeon once flourished in such vast flocks that when they migrated, they darkened the skies from horizon to horizon and took days to pass. So they were hunted without restraint. Today, that species is extinct. Many others that lived in less dramatic and visible ways simply disappeared without the knowledge of most people worldwide and were mourned only by a few naturalists.

Nonetheless, in spite of these assaults, the biodiversity of the world is still immense. And therein lies the strength that has enabled much of its wildlife to survive until now. Economists understand the wisdom of spreading their investments across a wide range of activities. It enables them to withstand disasters that may strike any one particular asset. The same is true in the natural world. If conditions change, either climatically or as a consequence of a new development in the never-ending competition between species, the ecosystem as a whole is able to maintain its vigour.

But consider the following facts. Today, we ourselves, together with the livestock we rear for food, constitute 96% of the mass of all mammals on the planet. Only 4% is everything else – from elephants to badgers, from moose to monkeys. And 70% of all birds alive at this moment are poultry – mostly chickens for us to eat. We are destroying biodiversity, the very characteristic that until recently enabled the natural world to flourish so abundantly. If we continue this damage, whole ecosystems will collapse. That is now a real risk.

Putting things right will take collaborative action by every nation on earth. It will require international agreements to change our ways. Each ecosystem has its own vulnerabilities and requires its own solutions. There has to be a universally shared understanding of how these systems work, and how those that have been damaged can be brought back to health.

This comprehensive, detailed and immensely important report is grounded in that understanding. It explains how we have come to create these problems and the actions we must take to solve them. It then provides a map for navigating a path towards the restoration of our planet’s biodiversity.
Economics is a discipline that shapes decisions of the utmost consequence, and so matters to us all. The Dasgupta Review at last puts biodiversity at its core and provides the compass that we urgently need. In doing so, it shows us how, by bringing economics and ecology together, we can help save the natural world at what may be the last minute – and in doing so, save ourselves.

Ceres

On 11 December 2020, Ceres, a sustainability nonprofit that works with investors on climate change, announced that a consortium of 30 investors managing $9 trillion in assets have committed to investing to support the goal of net zero carbon emissions by 2050.

This is what the 30 investors signed:

The Net Zero Asset Managers Commitment

In line with the best available science on the impacts of climate change, we acknowledge that there is an urgent need to accelerate the transition towards global net zero emissions and for asset managers to play our part to help deliver the goals of the Paris Agreement and ensure a just transition.

In this context, my organisation commits to support the goal of net zero greenhouse gas (‘GHG’) emissions by 2050, in line with global efforts to limit warming to 1.5°C (‘net zero emissions by 2050 or sooner’). It also commits to support investing aligned with net zero emissions by 2050 or sooner.

Specifically, my organisation commits to:

a) Work in partnership with asset owner clients on decarbonisation goals, consistent with an ambition to reach net zero emissions by 2050 or sooner across all assets under management (‘AUM’).
b) Set an interim target for the proportion of assets to be managed in line with the attainment of net zero emissions by 2050 or sooner.
c) Review our interim target at least every five years, with a view to ratcheting up the proportion of AUM covered until 100% of assets are included.

In order to fulfil these commitments my organisation will:

For assets committed to be managed in line with the attainment of net zero emissions by 2050 or sooner (under commitment b)

1) Set interim targets for 2030, consistent with a fair share of the 50% global reduction in CO2 identified as a requirement in the IPCC special report on global warming of 1.5°C.
2) Take account of portfolio Scope 1 & 2 emissions and, to the extent possible, material portfolio Scope 3 emissions.
3) Prioritise the achievement of real economy emissions reductions within the sectors and companies in which we invest.
4) If using offsets, invest in long-term carbon removal, where there are no technologically and/or financially viable alternatives to eliminate emissions.
5) As required, create investment products aligned with net zero emissions by 2050 and facilitate increased investment in climate solutions.

Across all assets under management

6) Provide asset owner clients with information and analytics on net zero investing and climate risk and opportunity.
7) Implement a stewardship and engagement strategy, with a clear escalation and voting policy, that is consistent with our ambition for all assets under management to achieve net zero emissions by 2050 or sooner.
8) Engage with actors key to the investment system including credit rating agencies, auditors, stock exchanges, proxy advisers, investment consultants, and data and service providers to ensure that products and services available to investors are consistent with the aim of achieving global net zero emissions by 2050 or sooner.
9) Ensure any relevant direct and indirect policy advocacy we undertake is supportive of achieving global net zero emissions by 2050 or sooner.

Accountability

10) Publish TCFD disclosures, including a climate action plan, annually, and submit them to the Investor Agenda via its partner organisations for review to ensure the approach applied is based on a robust methodology, consistent with the UN Race to Zero criteria, and action is being taken in line with the commitments made here.

We recognise collaborative investor initiatives including the Investor Agenda and its partner organisations (AIGCC, CDP, Ceres, IGCC, IIGCC, PRI, UNEPFI), Climate Action 100+, Climate League 2030, Paris Aligned Investment Initiative, Science Based Targets Initiative for Financial Institutions, UN-convened Net-Zero Asset Owner Alliance, among others, which are developing methodologies and supporting investors to take action towards net zero emissions. We will collaborate with each other and other investors via such initiatives so that investors have access to best practice, robust and science based approaches and standardised methodologies, and improved data, through which to deliver these commitments.

We also acknowledge that the scope for asset managers to invest for net zero and to meet the commitments set forth above depends on the mandates agreed with clients and clients’ and managers’ regulatory environments. These commitments are made in the expectation that governments will follow through on their own commitments to ensure the objectives of the Paris Agreement are met, including increasing the ambition of their Nationally Determined Contributions, and in the context of our legal duties to clients and unless otherwise prohibited by applicable law. In some asset classes or for some investment strategies, agreed net zero methodologies do not yet exist. Where our ability to align our approach to investment with the goal of net zero emissions by 2050 is, today, constrained, we commit to embark with determination and ambition on a journey, and to challenge and seek to overcome the constraints we face.

Shinise

 

The Japanese take pride in ‘shinise’: businesses that have lasted for hundreds or even thousands of years. This points out an interesting alternative to the goal of profit maximization: maximizing the time of survival.

• Ben Dooley and Hisako Ueno, This Japanese shop is 1,020 years old. It knows a bit about surviving crises, New York Times, 2 December 2020.

Such enterprises may be less dynamic than those in other countries. But their resilience offers lessons for businesses in places like the United States, where the coronavirus has forced tens of thousands into bankruptcy.

“If you look at the economics textbooks, enterprises are supposed to be maximizing profits, scaling up their size, market share and growth rate. But these companies’ operating principles are completely different,” said Kenji Matsuoka, a professor emeritus of business at Ryukoku University in Kyoto.

“Their No. 1 priority is carrying on,” he added. “Each generation is like a runner in a relay race. What’s important is passing the baton.”

Japan is an old-business superpower. The country is home to more than 33,000 with at least 100 years of history — over 40 percent of the world’s total. Over 3,100 have been running for at least two centuries. Around 140 have existed for more than 500 years. And at least 19 claim to have been continuously operating since the first millennium.

(Some of the oldest companies, including Ichiwa, cannot definitively trace their history back to their founding, but their timelines are accepted by the government, scholars and — in Ichiwa’s case — the competing mochi shop across the street.)

The businesses, known as “shinise,” are a source of both pride and fascination. Regional governments promote their products. Business management books explain the secrets of their success. And entire travel guides are devoted to them.

Of course if some businesses try to maximize time of survival, they may be small compared to businesses that are mainly trying to become “big”—at least if size is not the best road to long-term survival, which apparently it’s not. So we’ll have short-lived dinosaurs tromping around, and, dodging their footsteps, long-lived mice.

The idea of different organisms pursuing different strategies is familiar in ecology, where people talk about r-selected and K-selected organisms. The former “emphasize high growth rates, typically exploit less-crowded ecological niches, and produce many offspring, each of which has a relatively low probability of surviving to adulthood.” The latter “display traits associated with living at densities close to carrying capacity and typically are strong competitors in such crowded niches, that invest more heavily in fewer offspring, each of which has a relatively high probability of surviving to adulthood.”

But the contrast between r-selection and K-selection seems different to me than the contrast between profit maximization and lifespan maximization. As far as I know, no organism except humans deliberately tries to maximize the lifetime of anything.

And amusingly, the theory of r-selection versus K-selection may also be nearing the end of its life:

When Stearns reviewed the status of the theory in 1992, he noted that from 1977 to 1982 there was an average of 42 references to the theory per year in the BIOSIS literature search service, but from 1984 to 1989 the average dropped to 16 per year and continued to decline. He concluded that r/K theory was a once useful heuristic that no longer serves a purpose in life history theory.

For newer thoughts, see:

• D. Reznick, M. J. Bryant and F. Bashey, r-and K-selection revisited: the role of population regulation in life-history evolution, Ecology 83 (2002) 1509–1520.

See also:

• Innan Sasaki, How to build a business that lasts more than 200 years—lessons from Japan’s shinise companies, The Conversation, 6 June 2019.

Among other things, she writes:

We also found there to be a dark side to the success of these age-old shinise firms. At least half of the 17 companies we interviewed spoke of hardships in maintaining their high social status. They experienced peer pressure not to innovate (and solely focus on maintaining tradition) and had to make personal sacrifices to maintain their family and business continuity.

As the vice president of Shioyoshiken, a sweets company established in 1884, told us:

In a shinise, the firm is the same as the family. We need to sacrifice our own will and our own feelings and what we want to do … Inheriting and continuing the household is very important … We do not continue the business because we particularly like that industry. The fact that our family makes sweets is a coincidence. What is important is to continue the household as it is.

Innovations were sometimes discouraged by either the earlier family generation who were keen on maintaining the tradition, or peer shinise firms who cared about maintaining the tradition of the industry as a whole. Ultimately, we found that these businesses achieve such a long life through long-term sacrifice at both the personal and organisational level.

Exponential Discounting

Most of us seem to agree that the promise of a dollar in the future is worth less to us than a dollar today, even if the promise is certain to be fulfilled. Economists often assume ‘exponential discounting’, which says that a dollar promised at some time is worth

dollars in hand at time The constant is connected to the ‘interest rate’.

Why are economists so wedded to exponential discounting? The main reason is probably that it’s mathematically simple. But one argument for it goes roughly like this: if your decisions today are to look rational at any future time, you need to use exponential discounting.

In practice, humans, pigeons and rats do not use exponential discounting. So, economists say they are ‘dynamically inconsistent’:

• Wikipedia, Dynamic inconsistency.

In economics, dynamic inconsistency or time inconsistency is a situation in which a decision-maker’s preferences change over time in such a way that a preference can become inconsistent at another point in time. This can be thought of as there being many different “selves” within decision makers, with each “self” representing the decision-maker at a different point in time; the inconsistency occurs when not all preferences are aligned.

I this ‘inconsistent’ could be a misleading term for what’s going on here. It suggests that something bad is happening. That may not be true.

Anyway, some of the early research on this was done by George Ainslie, and here is what he found:

Ainslie’s research showed that a substantial number of subjects reported that they would prefer $50 immediately rather than $100 in six months, but would NOT prefer $50 in 3 months rather than $100 in nine months, even though this was the same choice seen at 3 months’ greater distance. More significantly, those subjects who said they preferred $50 in 3 months to $100 in 9 months said they would NOT prefer $50 in 12 months to $100 in 18 months—again, the same pair of options at a different distance—showing that the preference-reversal effect did not depend on the excitement of getting an immediate reward. Nor does it depend on human culture; the first preference reversal findings were in rats and pigeons.

Let me give a mathematical argument for exponential discounting. Of course it will rely on some assumptions. I’m not claiming these assumptions are true! Far from it. I’m just claiming that if we don’t use exponential discounting, we are violating one or more of these assumptions… or breaking out of the whole framework of my argument. The widespread prevalence of ‘dynamic inconsistency’ suggests that the argument doesn’t apply to real life.

Here’s the argument:

Suppose the value to us at any time of a dollar given to us at some other time is

Let us assume:

1) The ratio

is independent of E.g., the ratio of value of a “dollar on Friday” to “a dollar on Thursday” is the same if you’re computing it on Monday, or on Tuesday, or on Wednesday.

2) The quantity depends only on the difference

3) The quantity is a continuous function of and

Then we can show

for some constants and Typically we assume since the value of a dollar given to us right now is 1. But let’s just see how we get this formula for out of assumptions 1), 2) and 3).

The proof goes like this. By 2) we know

for some function . By 1) it follows that

is independent of so

or in other words

Ugh! What next? Well, if we take we get a simpler equation that’s probably still good enough to get the job done:

Now let’s make up a variable so that Then we can rewrite our equation as

or

This is beautiful except for the constant Let’s call that and factor it out by writing

Then we get

A theorem of Cauchy implies that any continuous solution of this equation is of the form

So, we get

or

as desired!

By the way, we don’t need to assume is continuous: it’s enough to assume is measurable. You can get bizarre nonmeasurable solutions of using the axiom of choice, but they are not of practical interest.

So, assumption 3) is not the assumption I’d want to attack in trying to argue against exponential discounting. In fact both assumptions 1) and 2) are open to quite a few objections. Can you name some? Here’s one: in real life the interest rate changes with time. There must be some reason.

By the way, nothing in the argument I gave shows that So there could be people who obey assumptions 1)–3) yet believe the promise of a dollar in the future is worth more than a dollar in hand today.

Also, nothing in my argument for the form of assumes that That is, my assumptions as stated also concern the value of a dollar that was promised in the past. So, you might have fun seeing what changes, or does not change, if you restrict the assumptions to say they only apply when The arrow of time seems to be built into economics, after all.

Also, you may enjoy finding the place in my derivation where I might have divided by zero, and figure out to do about that.

If you don’t like exponential discounting—for example, because people use it to argue against spending money now to fight climate change—you might prefer hyperbolic discounting:

• Wikipedia, Hyperbolic discounting.

Compositional Game Theory and Climate Microeconomics

guest post by Jules Hedges

Hi all

This is a post I’ve been putting off for a long time until I was sure I was ready. I am the “lead developer” of a thing called compositional game theory (CGT). It’s an approach to game theory based on category theory, but we are now at the point where you don’t need to know that anymore: it’s an approach to game theory that has certain specific benefits over the traditional approach.

I would like to start a conversation about “using my powers for good”. I am hoping particularly that it is possible to model microeconomic aspects of climate science. This seems to be a very small field and I’m not really hopeful that anyone on Azimuth will have the right background, but it’s worth a shot. The kind of thing I’m imagining (possibly completely wrongly) is to create models that will suggest when a technically-feasible solution is not socially feasible. Social dilemmas and tragedies of the commons are at the heart of the climate crisis, and modelling instances of them is in scope.

I have a software tool (https://github.com/jules-hedges/open-games-hs) that is designed to be an assistant for game-theoretic modelling. This I can’t emphasise enough: A human with expertise in game-theoretic modelling is the most important thing, CGT is merely an assistant. (Right now the tool also probably can’t be used without me being in the loop, but that’s not an inherent thing.)

To give an idea what sort of things CGT can do, my 2 current ongoing research collaborations are: (1) a social science project modelling examples of institution governance, and (2) a cryptoeconomics project modelling an attack against a protocol using bribes. On a technical level the best fit is for Bayesian games, which are finite-horizon, have common knowledge priors, and private knowledge with agents who do Bayesian updating.

A lot of the (believed) practical benefits of CGT come from the fact that the model is code (in a high level language designed specifically for expressing games) and thus the model can be structured according to existing wisdom for structuring code. Really stress-testing this claim is an ongoing research project. My tool does equilibrium-checking for all games (the technical term is “model checker”), and we’ve had some success doing other things by looping an equilibrium check over a parameter space. It makes no attempt to be an equilibrium solver, that is left for the human.

This is not me trying to push my pet project (I do that elsewhere) but me trying to find a niche where I can do some genuine good, even if small. If you are a microeconomist (or a social scientist who uses applied game theory) and share the goals of Azimuth, I would like to hear from you, even if it’s just for some discussion.

Diary, 2003-2020

I keep putting off organizing my written material, but with coronavirus I’m feeling more mortal than usual, so I’d like get this out into the world now:

• John Baez, Diary, 2003–2020.

Go ahead and grab a copy!

It’s got all my best tweets and Google+ posts, mainly explaining math and physics, but also my travel notes and other things… starting in 2003 with my ruminations on economics and ecology. It’s too big to read all at once, but I think you can dip into it more or less anywhere and pull out something fun.

It goes up to July 2020. It’s 2184 pages long.

I fixed a few problems like missing pictures, but there are probably more. If you let me know about them, I’ll fix them (if it’s easy).

Vaclav Smil on Growth

Yet another interesting book I haven’t read yet:

• Vaclav Smil, Growth: From Microorganisms to Megacities, MIT Press, Cambridge, 2019.

As I hope you know, Vaclav Smil is an expert on energy, food, population, and economics, who assembles and analyzes data in fact-filled books like Energy and Civilization: a History.  Bill Gates has said “I wait for new Smil books the way some people wait for the next ‘Star Wars’ movie.”

He was interviewed here:

• Jonathan Watts, Vaclav Smil: ‘Growth must end. Our economist friends don’t seem to realise that’, 21 September 2019.

The interview begins:

You are the nerd’s nerd. There is perhaps no other academic who paints pictures with numbers like you. You dug up the astonishing statistic that China has poured more cement every three years since 2003 than the US managed in the entire 20th century. You calculated that in 2000, the dry mass of all the humans in the world was 125m metric tonnes compared with just 10m tonnes for all wild vertebrates. And now you explore patterns of growth, from the healthy development of forests and brains to the unhealthy increase in obesity and carbon dioxide in the atmosphere. Before we get into those deeper issues, can I ask if you see yourself as a nerd?

The facts here are fascinating but the question is absurd. Are we really sinking into such anti-intellectualism that a journalist feels the need to start a conversation with a scientist by asking if he sees himself as a “nerd”?

I’d have been tempted to reply “First, can I ask if you see yourself as a twit?” Smil more wisely replied:

Not at all. I’m just an old-fashioned scientist describing the world and the lay of the land as it is. That’s all there is to it.

Here’s why he wrote the book:

I have deliberately set out to write the megabook on growth. In a way, it’s unwieldy and unreasonable. People can take any number of books out of it–economists can read about the growth of GDP and population; biologists can read about the growth of organisms and human bodies. But I wanted to put it all together under one roof so people could see how these things are inevitably connected and how it all shares one crystal clarity: that growth must come to an end. Our economist friends don’t seem to realise that.

He advocates degrowth in some places… but growth in others:

[…] it’s important not to talk in global terms. There will be many approaches which have to be tailored and targeted to each different audience. There is this pernicious idea by this [Thomas] Friedman guy that the world is flat and everything is now the same, so what works in one place can work for everyone. But that’s totally wrong. For example, Denmark has nothing in common with Nigeria. What you do in each place will be different. What we need in Nigeria is more food, more growth. In Philippines we need a little more of it. And in Canada and Sweden, we need less of it. We have to look at it from different points of view. In some places we have to foster what economists call de-growth. In other places, we have to foster growth.

I’m sure his book will be more interesting than these quotes, because it’ll be full of well-organized and important facts—and the questions surrounding growth are some of the most pressing of our age.

Divesting

Christian Williams

John always tells me to write short, sweet, and clear. Knowing that his advice is supreme on these matters, I’ll try to write mini-posts in between the bigger ones. But… not this time – the topic is too good.

(Dispossess of property/authority. Say it, sound smart.)

…..

Work smarter, not (just) harder.

Today I got an email from Bill McKibben, founder of 350.org. (350 parts per million, the concentration of CO2 considered a “safe upper limit” for Earth, by NASA scientists James Hansen. We’re soaring past 415ppm.) In preparation for the global climate strike, Bill wants to share an important idea: divesting from fossil fuels may be our greatest lever.

Money is the Oxygen on which the Fire of Global Warming Burns

I’ll pluck paragraphs to quote, but please read the whole article; this is an extremely important and practical idea for addressing the crisis. And it’s well written… the first sentence sounds fairly Baezian.

I’m skilled at eluding the fetal crouch of despair—because I’ve been working on climate change for thirty years, I’ve learned to parcel out my angst, to keep my distress under control. But, in the past few months, I’ve more often found myself awake at night with true fear-for-your-kids anguish. This spring, we set another high mark for carbon dioxide in the atmosphere: four hundred and fifteen parts per million, higher than it has been in many millions of years. The summer began with the hottest June ever recorded, and then July became the hottest month ever recorded. The United Kingdom, France, and Germany, which have some of the world’s oldest weather records, all hit new high temperatures, and then the heat moved north, until most of Greenland was melting and immense Siberian wildfires were sending great clouds of carbon skyward. At the beginning of September, Hurricane Dorian stalled above the Bahamas, where it unleashed what one meteorologist called “the longest siege of violent, destructive weather ever observed” on our planet.

Bill emphasizes that change has moved far too slowly, of course. But he’s spent the past week with Greta Thunberg and many other activists, and one can tell that he really is heartened.

It seems that there are finally enough people to make an impact… what if there were an additional lever to pull, one that could work both quickly and globally?

The answer: money.

Today it is large corporations which have the greatest power over daily life, and they are far more susceptible to pressure and change then the insulated bureaucracies of governments.

Thankfully Bill and many others knew this years ago, and started a divestment campaign of breathtaking magnitude:

Seven years ago, 350.org helped launch a global movement to persuade the managers of college endowments, pension funds, and other large pots of money to sell their stock in fossil-fuel companies. It has become the largest such campaign in history: funds worth more than eleven trillion dollars have divested some or all of their fossil-fuel holdings.

$11,000,000,000,000.

And it has been effective: when Peabody Energy, the largest American coal company, filed for bankruptcy, in 2016, it cited divestment as one of the pressures weighing on its business, and, this year, Shell called divestment a “material adverse effect” on its performance.

The movement is only growing, accelerating, and setting its sights on the big gorillas. The main sectors: banking, asset management, and insurance.

Consider a bank like, say, JPMorgan Chase, which is America’s largest bank and the world’s most valuable by market capitalization. In the three years since the end of the Paris climate talks, Chase has reportedly committed 196 billion dollars in financing for the fossil-fuel industry, much of it to fund extreme new ventures: ultra-deep-sea drilling, Arctic oil extraction, and so on. In each of those years, ExxonMobil, by contrast, spent less than 3 billion dollars on exploration, research, and development. $196B is larger than the market value of BP; it dwarfs that of the coal companies or the frackers. By this measure, Jamie Dimon, the C.E.O. of JPMorgan Chase, is an oil, coal, and gas baron almost without peer.

But here’s the thing: fossil-fuel financing accounts for only about 7% of Chase’s lending and underwriting. The bank lends to everyone else, too—to people who build bowling alleys and beach houses and breweries. And, if the world were to switch decisively to solar and wind power, Chase would lend to renewable-energy companies, too. Indeed, it already does, though on a much smaller scale… It’s possible to imagine these industries, given that the world is now in existential danger, quickly jettisoning their fossil-fuel business. It’s not easy to imagine—capitalism is not noted for surrendering sources of revenue. But, then, the Arctic ice sheet is not noted for melting.

Bill elucidates the fact that it is critical to effect the divestment of giants like Chase, Blackrock, and Chubb. Even if these targets are quite hard, this method of action applies to every aspect of the economy, and empowers every single individual (more below). If the total divestment is spread over a decade, it can be done without serious economic instability. And if done well, it will spur tremendous growth in the renewable energy sector and ecological economy in general, as public consciousness opens up to these ideas on a large scale.

I want to keep giving quotes, but you can read it. (If anyone is out of free articles for New Yorker, I can send a text file.) I’ll contribute a few of my own thoughts, expanding on stuff implicit in the article; and then this topic can be continued with another post.

…..

Divesting is a truly powerful lever, for several reasons.

First, money talks. Many people who have been misled by modern society have the following equation in their heads:

money = value

These people, being overwhelmed with social complexity, have lifted the “burden” of large-scale ethics off of their shoulders and into a blind faith in the economic system – thinking “well, if enough people have the right idea, then capitalism will surely head in the right direction.”

Of course, after not too long, we find that this is not the case. But their thinking has not changed, and we need a way to communicate with them. While it may feel strange and wrong to reformulate the message from “ethical imperative” to “financial risk”, this is the way to get through to many people in powerful places. When you read about success stories, it is effective, especially considering all the time spent mired in anthropogenic-warming skepticism.

Second, social pressure is now a real force in the world. We can bend competition to our will: incentivize companies to better practices, and when one capitulates, the others in that sphere follow. It has happened many times, and the current is only getting stronger.

Though if we want to fry bigger fish than no-straws, we need to sharpen our collective tactics. It will of course be more systematic and penetrating than shaming companies on Twitter. The article includes great examples of this; it would be awesome to discuss more ideas in the comments.

Third, everyone can help this way, directly and significantly. Everyone has a bank account. It is not difficult, nor seriously detrimental, to switch to a credit union. The divestment campaign can be significantly accelerated by a movement of concerned citizens making this transition.

(My family uses Chase. When I was spending quality time back home, I asked my parents how the value of a bank is anything more than secure money storage. The main thing they mentioned was loans – but they admitted that the biggest and best loan they ever took was through a credit union. The reasons simply did not add up. I plan to show them this article, and I’ll try to have an earnest conversation with them. I really hope they understand, because I know they are rational and good people.)

It’s all but impossible for most of us to stop using fossil fuels immediately, especially since, in many places, the fossil-fuel and utility industries have made it difficult and expensive to install solar panels on your roof. But it’s both simple and powerful to switch your bank account: local credit unions and small-town banks are unlikely to be invested in fossil fuels, and Beneficial State Bank and Amalgamated Bank bring fossil-free services to the West and East Coasts, respectively, while Aspiration Bank offers them online. (And they’re all connected to A.T.M.s.)

This all could, in fact, become one of the final great campaigns of the climate movement—a way to focus the concerted power of any person, city, and institution with a bank account, a retirement fund, or an insurance policy on the handful of institutions that could actually change the game. We are indeed in a climate moment—people’s fear is turning into anger, and that anger could turn fast and hard on the financiers. If it did, it wouldn’t end the climate crisis: we still have to pass the laws that would actually cut the emissions, and build out the wind farms and solar panels. Financial institutions can help with that work, but their main usefulness lies in helping to break the power of the fossil-fuel companies.

…..

The economy is far more responsive to changes in the collective ethos than the government. This is how people can directly express their values every day, with every bit of earning they have. We are recognizing that the public mindset is changing, and we can now take heart and leverage society in the right direction.

Conjecture The critical science of our time has the form:

This is why John Baez brought together so many capable people for the Azimuth Project. I hope that we can connect with the new momentum and coordinate on something great. Even in just the last post there were some really good ideas. I really look forward to hearing more. Thanks.