Understanding the Hotelling Rule
Michael Whalen and Roberto Cunha discuss the fundamentals of the Hotelling rule and how this long-standing rule has prevailed in today's economy.
TRANSCRIPT
MW 00:09 [music] Welcome to BRG's ThinkSet Podcast. I'm your host, Michael Whalen. BRG is a global consulting firm. We help organizations in disputes and investigations, corporate finance, and performance improvement and advisory. We're a multidisciplinary group of experts, industry leaders, academics, data scientists, and professionals. Around the world, BRG delivers the inspired insights and practical strengths our clients need to stay ahead of what's next. For more information about BRG, please visit thinkbrg.com.
In this episode of the ThinkSet Podcast, we discuss recent research challenging our understanding of the Hotelling rule. No, that's not the rule on how we're going to share workspaces as we return to the office as the COVID-19 pandemic eases. Instead, it's been the fundamental principle for the economics of nonrenewable resources, notably fossil fuels such as crude oil, natural gas, and coal. This rule has prevailed since the 1930s in describing how prices are formed for these finite resources, but the concept is particularly relevant today in the discussion of the economic value of yet-to-be-produced fossil fuels. Indeed, as many governments seek to implement policies limiting the use of greenhouse gas-emitting fuels, some claim such fossil fuel resources are, in fact, economically impaired, stranded assets.
Here to talk about his recent work on the Hotelling rule is Roberto Cunha. Roberto is a director in BRG's Energy & Climate practice based in Rio de Janeiro. He has more than twelve years of industry and academic experience in global energy and environmental matters, particularly on natural gas markets in Latin America, and provides expert assessments, due diligence, and economic modeling to energy companies, financial institutions, and government ministries. Roberto, welcome to the ThinkSet Podcast.
RC 02:02 Thank you, Michael. It is a pleasure to be with you today.
MW 02:07 So, Roberto, what is this Hotelling rule? And does it have anything to do with accommodations for tourists [laughter]?
RC 02:15 No, it certainly won't help you find a cheaper hotel room once the pandemic is over. But the Hotelling rule is, in reality, a longstanding theory in economics that has been marked by a lot of questions. It is a canonical, well-established principle first proposed in 1931 by Harold Hotelling from Columbia University. But it was really named as such in the '70s by Nobel Prize winner Robert Solow, in the aftermath of the first oil price shock.
MW 02:45 So who was Harold Hotelling? And why did that theory have such an impact?
RC 02:52 Hotelling was an American mathematical statistician and an economic theorist. His name is very famous in the field. He's now present in Hotelling's law, Hotelling's lemma, and in economics, the one we're discussing, the Hotelling rule. He is even known for Hotelling's T-square distribution in statistics. He had a very fertile career.
He was first and foremost an academic. First, a professor of mathematics at Stanford University from 1927 to 1931. And then he moved to Columbia University from '31 to 1946. And he finished his career as a professor of mathematical statistics at the University of North Carolina, Chapel Hill from 1946 until the end of his life in 1973. His theory on economics is really a small chapter of his overall contribution on mathematics and several other fields. It became really famous in the 1970s, only—even if it was proposed in 1931—following the first oil shock.
That was the time when economists were asked what to do with that high oil price, which was affecting the interest rates and just putting the economy in turmoil, particularly in the US. And they started looking back in the literature for existing words on nonrenewable resources, at the time called exhaustible resources, and found this beautiful article from Harold Hotelling from 1931 published in the Journal of Political Economy. At this time, they looked at the basic principle that's explained in section two of that article, that that name “Hotelling rule” was proposed by Robert Solow, as I mentioned. That's essentially how; the impact only came about decades later. But it was really the fact that he was the first one to put out the implications from a finite resource relative to different resources in the economy, what is it that it changes from a profit-maximization perspective.
MW 04:54 What does that mean? What does the rule say? And how did it help to understand the prices for these exhaustible resources?
RC 05:03 Yes. Let me get into the details of that, right? So in economics, we're usually looking at how prices are formed on typical goods or services in the economy. And we are usually working in the neoclassical framework with profit-maximizing producers. They seek to minimize costs. And given the demand and competition, they seek to maximize profits. And we usually get to the rule where price equals marginal cost in a perfect competition, and the general principles we learned in undergraduate economics.
What Hotelling was looking for at the time was to understand: what if the resource is finitely available, as is the case of exhaustible resources like minerals, like coal, oil, natural gas? And at the time, he introduced this property in the profit-maximization problem of a producer and found out that if the resource is finite, the producer will not just charge marginal cost; he will charge marginal cost plus a premium called the scarcity rent. And that scarcity rent is exactly a premium for the fact that once you consume a unit of a finite resource, that unit is no longer available in the future. Once you factor that in, you arrive at the optimal behavior, which is to sell at a price that grows at the rate of interest. That's precisely to avoid the indifference between whether to consume now or to wait for a year, for the future.
That's the rule, that we would see prices increasing. I should say net prices. The price net of the marginal cost increasing at the rate of interest over time. And that's a very basic principle that's well grounded in microeconomics. And it became called Hotelling rule.
MW 06:57 Does the fact that the scarcity rent relates to interest rates, does that present us with a situation where the owner of those exhaustible resources, or the one who has to invest to extract them, has to consider in the context of alternative investments?
RC 07:17 That's exactly it, Michael. The idea of the theory is Hotelling was imagining a stylized mine owner, someone who owns a mine. And that mine is finite, so I need to decide: do I want to just dig and take all the coal out now and put it out on the market? Do I want to conserve that asset—like not use at all now, just save it for next year, [when] prices may be higher? Or is there a certain rate that I should extract the coal from the mine considering it's finite?
And after factoring in that idea, the mine owner would, if there were no other constraints, nothing else, extract in declining amounts. And if you have a stable demand, a peak demand curve, because you are lower [than] the quantities produced, the price would gradually increase. The rate you would do that, it would be the rate precisely to reach a price that increases with the interest rate. If you have an alternative investment, say to buy a bond that pays a certain interest rate, by extracting coal at that rate, you are getting exactly the same return of an alternative investment. It is a theory that fits very well into the asset valuation theory and into the arbitrage opportunities, things like that.
MW 08:40 So the theory sounds sound. But what has been the actual historical experience? Do prices seem to track the Hotelling theory?
RC 08:50 And that's where the problems begin. The theory—it's theoretically impeccable. But once you look at it, it makes perfect sense. It is well connected with the rational principle and with the rational decision-making principles and microeconomics. But once you look at the real-world natural resources over time, how they evolve since 1930s, right, until nowadays, you don't see that rule materialize. Not even approximately, not even as a trend. It's been tested out by numerous empirical works in the literature, and they reject even for different types of nonrenewable resources. They've tried on copper, iron, gold, oil, gas, and you will not find this trend of it's simply increasing prices. And that's why the Hotelling rule is a riddle in economics, because it remains accepted theoretically, but it has never been verified in a such high-level manner in empirical works.
MW 09:53 Generally, when the empirical data doesn't seem to track the hypothesis or the theory, people offer substitute theories and hypothesis. Is that true in the case of the Hotelling rule?
RC 10:06 Mostly no. The way it was laid out in the literature, especially since the 1970s, the understanding is that the Hotelling rule is too basic, let's say. The fact that the natural resource is finite is only the first property. And the interpretation over a wider rule is not sticking to reality is not because the rule is theoretically incorrect. It's just because we're lacking additional factors of reality that are affecting this mechanism I just described.
This has been the predominant understanding since the 1970s. And most of the works we see, they try to add additional assumptions. In this framework I described, it's just a producer with a constant marginal cost, a stationary demand curve, and one constraint. That the resource is finite. If that's all that is, you get the Hotelling rule. But in reality, what the literature began to investigate is that the mines are not all the same, the demand is not stationary. Demand grows over time. Technology is not the same. We learn how to be more efficient and how to locate these resources with cheaper costs. In addition to the basic model described by Hotelling in 1931, there are other properties that, if considered, may help us get to better statistical results. This is why the rule stands as the most basic version. Similar to in physics in high school when you learn, oh, if there's no friction in the air, the car or the object would move at a constant speed. In reality, there is friction. What the literature is doing here is bringing in these additional factors that would be like friction in that problem. Let's now consider these additional real-world features to see if that's polluting this basic principal from showing the data.
MW 12:08 Now, Roberto, I understand there's a bit of a detective story here, because you did some research and analysis and spadework to try and figure out the complexity or the depth of Hotelling's original proposition. Can you tell me about that?
RC 12:24 That's exactly where we began. In terms of these additional factors, since the 1970s, the literature has proposed almost eight candidates for extensions that could be done on the basic model. That's generated, in theory, the type of equilibrium, like with bell-shaped and U-shaped prices that we know from data that it happened already. It's just really hard to demonstrate based on lack of data and difficulty to observe costs, many countries, many nonrenewable resources. It's really hard to define which of these alternatives would be correct, or if all, just a few of those. We're at a point where we have alternatives, but we don't know which one of them is better.
What we did in this work was try to revisit this understanding I really described here, that Hotelling was doing something very basic, he was proposing something very general, and that we really need to consider these additional factors. Hotelling had not been thoroughly investigated in terms of history of his thoughts. Antoine Missemer and I, as part of a broader project sponsored by the European Society for the History of Economic Thought, investigated the archives from Harold Hotelling [that] are stored at Columbia University. As part of this broader effort, the archive was scanned or digitalized and made available for the group. And a series of papers, which this one is part of, were published discussing the history of his thoughts, his earlier paper on depreciation, and how exactly did he get to work with nonrenewable resources, and what was he trying to say. Was he really just coming up with this basic principle as the literature understood for a long time, or did he say different things? And this question is valid, because at the time, papers were not structured the way they are today. If you are familiar with Hotelling's 1931 article, you will see it is a long article with fifteen sections. And the topic changes. And he brings real-world examples. It was written in a very different style than the one we're used to in modern academic work. It's not so straightforward to the reader. We thought it could be possible that archives were going to tell us more about what kind of research was he doing, why, or what exactly was he concluding.
MW 14:52 And did he anticipate things or these constraints [that] have been discussed to try to explain the types of price curves that had been observed with exhaustible resources?
RC 15:05 Yes. That was one of the main findings. Once we began to look at the archives and how Hotelling came to work with exhaustible resources at the time, we see he came in from a very academic kind of side. He was working with asset valuation. And then he released in 1925, after finishing his PhD, a paper on depreciation. He was trying to understand how asset valuation and depreciation economics should be dealt with. And in doing so, he finds a very interesting case of mines, based on a discussion on depreciation of mines that he was reading about. And then he proceeds to figure out this is a very brand-new field. No one has discussed this before. And I see several issues going on. What if the asset is finite? And he derives the basic principle already in 1925, associated with his earlier work in depreciation.
He didn't publish this result right away. He started working on a separate parallel line of research in trying to understand mines, in trying to understand how the oil markets work, how it works in reality. And he quickly learned that this basic principle we retained as [the] Hotelling rule does not work in reality. The price of minerals—oil markets are subject to numerous additional factors that change the way the rule works.
Finally, in his 1931 paper, he merged these two lines of research: the theoretical line of research stemming from his 1925 work on depreciation in which he refers to exhaustible assets, and then he lays out his basic rule in section two, right at the beginning. And he never calls it a rule. Actually, if you look at the finishing, the closing of these sections, you will see he calls it that there are discrepancies in the real world as to how these markets play out. And then he proceeds, on the second part of his paper, to discuss this more real-world side of research he had done, in which he exchanged letters with petroleum engineers. He was trying to understand the cost of a well, how much it increases with depth, very technical details. He goes out and he describes that for real-world markets, you need to consider at least that cost increases as the mines go deeper. That's the natural occurrence of mines wherever they happen to be. That could be very far from demand centers. Some of them could be very close. You need to factor in this additional cost, which he calls [] the cost to accelerate production. If you have a certain capacity, you have a certain cost. But if you want to accelerate production above that capacity, you need to incur these capital costs that are associated with opening mines, moving labor to that location, and so on.
He also mentions a third factor related to acquiring information, which nowadays we understand that as to be—geological constraints is a term we recently coined in the literature. But it really refers to this set of natural conditions surrounding minerals. The fact that they're underneath the ground, that they are spatially separated, whatever they are, and that you need to live with that. In the case of oil and gas, you need to find the pressurized resources and pressure pulse. All these conditions, like the size of mines—some mines are huge, some mines are very small—there [are] natural properties you need to contend with related to the geological process that created this mineral. [They are] called geological constraints. What we find, the literature nowadays is coming to work more on these constraints and to look at them as an ensemble, as a set of constraints, not one of them in isolation.
And we found in the archives that Hotelling knew about this back in 1931 and that he wrote about it in the second part of his paper. It was just much easier to understand and to see once you see the amount of work in the archives on this area. You see how he got there, and then you understand why he is saying all that in the second half of his paper. By only looking at the paper, the literature really focuses on section two and the basic result.
MW 19:29 I would have thought that the geological constraints he anticipated would have also been affected by the pace of technology and technological responses to geological challenges. And certainly, the increase in production of crude oil and natural gas by the development of fracking techniques to recover tight oil and gas resources came about in part to address geological limitations. Is that something that Hotelling anticipated? Or is that an element that helps to explain some of the empirical price data that you see on finite resource recovery?
RC 20:16 This is a crucial question, Michael. It's a fantastic question, because that's exactly some of the issues we have to deal with. We all understand these resources are finite. There's a limit to technology there. You can learn how to be more efficient. You can learn how to locate these reservoirs with lower cost. You can learn how to extract them with lower costs. You can learn how to frack an underground rock.
But you cannot learn how to recreate more oil down there once it's over. That's exactly the boundary between the technological progress assumption—how low can you bring these costs—and the geological constraint that there's so much down there. To a certain point, technological progress will help you lower costs. But once you're past the 50 percent of the total endowment, or 75 percent, eventually you will have to produce less because there's just not any more. That's the limit between technological progress and geological constraints.
One, we see, for example, in the case of fracking, you will find that technological progress is lowering costs as the producers are learning how to drill better, how to calibrate the wells, how to frack in stages and optimally weigh for that certain rock. But at certain geological constraint sites, the fact that rock has certain amounts of gas and oil, and the concentration in different portions of the rock, is different. Once you have extracted, for example, the richest areas, the areas where the concentration of oil and gas is highest, you're only left with the low-concentration areas. And the technology will not be able to help you with that.
There's just no more left is the dynamic we've been experiencing for over a century. At each level of production in technology, such as conventional oil, once that level is almost over, a technological breakthrough comes up, and it opens up the unconventional formations for further production. But how long can that go? And what price puts us there? Do we require a very high price to open this up? Do we require this technological progress happen as the manna from heaven? Or is there a limit there?
One of the contributions of the recent research that we're trying to emphasize with this archival research is to separate geological constraints from technological progress in the way you set up models.
MW 22:51 Now, Hotelling became popular, or was investigated as an answer to the challenges, in the 1970s, where there was a great deal of concern about crude oil hitting peak supply and limited supplies of crude oil. But prices are where supply meets demand. Is Hotelling only relevant in considering the supply side of finite resources? What about the demand side? Because it seems like these days, some people are more concerned about peak demand rather than peak supply.
RC 23:26 That is correct, Michael. The work of Hotelling will focus primarily on the supply side. If you look even at the basic model, there's going to be a stationary demand curve, linear negatively inclined. Those are very high-level assumptions which implicitly say, "I'm not working on this," right? And most of the concepts that we discussed, such as technological progress, geological constraints that I mentioned, are related to the supply side.
Now, with that said, one of the reasons we're getting to the conclusion that we may need to reach a peak demand and not a peak supply, it's precisely coming from the supply side. Which surprises me, because on the supply side, the conclusion is that there's plenty of these resources available. The last technological breakthroughs in oil and gas that opened up a lot more endowments that you can explore, the idea from the early 2000s of peak oil, we're quickly coming to understand it won't really happen by itself. That's why the idea of peak oil demand began to emerge quickly in the 2010s from the climate side of things, that if you really want to lower emissions, you now need to really work on demand, because supply by itself will not simply reach a peak and start to fall within the horizon that you need.
MW 24:51 What then should the relevance of the Hotelling rule be to policymakers who are particularly focused on the efforts to reduce global greenhouse gas emissions?
RC 25:03 The Hotelling rule stands as a basic, stylized principle, but one that is not directly applicable to real-world resources, as I mentioned. We should not think or expect the real-world resources will follow that rule plain and simple. That rule remains the starting point for the theoretical understanding. And next, we need to discuss the additional factors that should be considered on sort of a basic model for real-world resources. That's where the debate is. And there, the climate community, policymakers, and researchers should be very interested, because that's where the answer is going to come with regards to the peak of oil supplies, the peak of gas supplies, how a carbon tax would affect the behavior of the production of these fuels in a given country, worldwide, the different policy mechanisms that remain on the table. The rule remains there as the entry door for the theoretical understanding of the supply side, which still will be there as a component of what you're trying to discourage in terms of greenhouse gas emissions. If the conclusion is for a peak oil demand, how are you going to get to that demand peak? Till you have a cheaper technology or you make a tax to make that more expensive, the market's working would continue to produce those fossil fuels into the coming decades.
MW 26:33 One of the debates that has emerged from the discussion regarding climate policy is fossil fuels being described as being at great risk, as being stranded as the marginal cost of renewable resources have substantially decreased in the last decade, and as demand for fossil fuels may be affected relative to government policy. What does the Hotelling rule, including its extensions, prove or disprove regarding the argument of risk of stranded assets, of these finite resources?
RC 27:12 As I mentioned, the rule will tell you what's going on the supply side first, right? It's really telling you that if you don't have a demand-side policy, things will keep going as they are. This is very important to understand. Assets will be stranded if you include in the Hotelling basic model even a declining demand curve over time instead of an increasing one. If demand declines over time, we will get to a very different result.
It is a model that allows you to test the implications from these policies. And assets would be stranded as it is discussed if this demand declines. If the climate policies work on lowering demand by promoting electric vehicles, by promoting battery technologies, by promoting green hydrogen, if you are working on these technologies and there is energy policy support for them in the form of various mechanisms that exist, you will have a lowering demand for fossil fuels, which could, and would in case of extreme levels, strand some of these assets. All these are choices that are being made based on policies that will affect the demand for fossil fuels in the future and could strand some of these assets.
MW 28:28 That was great. I appreciate your joining us from Rio de Janeiro today. Thank you for describing your research on the ThinkSet Podcast.
RC 28:35 Thank you, Michael. It's my pleasure to be here.
MW 28:38 This ThinkSet Podcast is brought to you by BRG. You can subscribe to the podcast and access other content from ThinkSet magazine by going to thinksetmag.com. Don't forget to rate and review this show on iTunes as well. I'm Michael Whalen. Thanks for listening.
The views and opinions expressed in this podcast are those of the participants and do not necessarily reflect the opinions, position, or policy of Berkeley Research Group or its other employees and affiliates.
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