Intro to Econ: Ninth Lecture – Risk Premia under Independent Risks

In the previous post we had the following problem. We were wondering about which interest rate we could expect to see for a loan for a particular risky project. You would like to get a loan, and an investor might like to give it to you. The question was, under what conditions you would get this loan, if you get it at all. Recall, that your project can turn out to be good or bad and that investors generally agree about the chances and consequences of either outcome. The problem can be summarized by the following table, where  x is the repayment amount that you pay back to the investor in case of the project being successful. If it is unsuccessful you pay nothing, because you have nothing. You “default” on your loan in that case. This is the risk the investor takes on when she or he gives you this loan.

 \begin{tabular}{c|ccccc} Scenario & Income & Probability & you get & investor gets \\ \hline good & 200.000 & 80\% & 200.000-x & x \\ bad & -50.000 & 20\% & 0 & -50.000 \\ \end{tabular}


We figured out that you will not accept the loan if the repayment amount  x is more than € 200.000 (that would be an interest rate of 200%). Because then you have nothing to gain from this project. In reality, you might not even accept anything close to 200%, but we will come back to this problem later.

We also figured out that the investor will (almost) certainly not accept an interest rate below 12.5%, as otherwise the investor expects a negative return on their investment and would then be better off just putting her or his money under a mattress or, I guess, in a safe or vault. By the way, for a very long time the Catholic Church (and other religions) considered positive interest rates morally wrong. In such a world, you probably wouldn’t get a loan for your great project, unless you find a way around this problem. And that would probably be a shame (see previous post).

In this post I want to think about whether an investor will really accept an interest of 12.5% (or slightly above) given that the investor now takes all the risk and at an interest rate of 12.5% only expects a zero return. The answer to this question, it turns out, all depends on whether the risk in this project is essentially stochastically independent of all other risks inherent in all other projects or not.

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Intro to Econ: Ninth Lecture – Credit Markets – Financial Markets

So far we talked a bit abstractly about markets. Yes, we used some specific products for examples, such as white wine, rental apartments, and perhaps airline pricing, but we have not yet developed a particular market model specifically for a particular product. In this post, I want to do this for a particularly important market: the market for money. This post gives a first account of the basic insights and ingredients that underlie our understanding of credit markets and financial markets. You will see, I hope, that what we have learned so far, especially about supply and demand, while not enough to understand these markets fully, was also not in vain. It will come in handy.

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Warum wir die PISA Studien vielleicht nicht so ernst nehmen sollten

ÖkonomInnen versuchen menschliches Verhalten dadurch zu verstehen, in dem sie versuchen sich in die Lage dieser Menschen hineinzuversetzen und dann zu überlegen was diese Menschen wohl für Ziele haben. ÖkonomInnen sprechen dabei allgemein von Anreizen, die finanzieller aber auch anderer Art sein können.

In einem kürzlich erschienen Artikel im American Economic Review: Insights sind die Autoren Gneezy, List, Livingston, Qin, Sadoff, und Xu der Frage nachgegangen, welche Anreize wohl die Kinder haben, die an der PISA Studie beteiligt waren. Wie gut ein Kind in der Studie abschneidet, hat ja für das Kind selbst keine Konsequenzen. Man bekommt dadurch keine bessere Note, man bekommt dadurch keine bessere Chance auf einen besseren Job, es bringt ja eigentlich nicht viel. Wenn man nun davon ausgeht, dass so einen Test gut zu absolvieren auch ein bisschen anstrengend ist, dann kann man sich schon fragen, wie sehr sich die Kinder da überhaupt anstrengen.

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Intro to Econ: Eighth Lecture – What does the Gross Domestic Product (not) measure?

You may recall that the value of anything is really a subjective thing, something may be valuable to someone, but at the same time may be not valuable to someone else. In this post I want to investigate in what way, if any, in what way the gross domestic product (GDP) measures total “market” value.

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Supply-side policies against global warming

Alas, it turns out that I was not the first to point out the perverse dynamic supply-side effects of a carbon tax! (Well, I never really believed I was the first anyway.)

Hans-Werner Sinn wrote a whole book about it. It is called the “Green Paradox“. And there is some academic literature on it, although surprisingly little. (For instance, this recent paper on the role of oil reserves and marginal extraction costs).

Sinn also wrote this paper in 2007 which confirms my hypothesis that a rising carbon tax makes resource owners extract more fossil fuels in the short run. But he does so in a much more sophisticated dynamic general equilibrium model. The paper helps to answer one important objections I received in private conversations.

My good friend (and Graz Economics alumnus) Michael Schwarz points out that oil extraction can’t just be turned on and off like a water tap. There are extraction costs! Yes, indeed, and Sinns paper addresses this point: 

„If extraction costs are assumed, the problem of moving the economy in the wrong direction is mitigated, and with sufficiently strong extraction costs, current extraction may even move in the right direction.“

Sinn, HW. “Public policies against global warming: a supply side approach”, Int Tax Public Finance (2008), p. 21

But Sinn also points out:

„As marginal extraction costs are likely to be only a small fraction of the price of the extracted resource, the effect on the extraction path may be tiny. For instance, the average production costs of crude oil amounted to only about 15% of the average spot price in 2006.“

Sinn, HW. “Public policies against global warming: a supply side approach”, Int Tax Public Finance (2008), p. 20

Since oil extraction is a high fixed cost, small marginal cost industry, the average production costs overstate the marginal costs which are relevant for the extraction path.

Recent empirical research throws more doubt on the importance of extraction costs. Here is a quote from the paper by Heal and Schlenker linked to above:

Using data from a large proprietary database of field-level oil data, we show that carbon prices even as high as 200 dollars per ton of CO2 will only reduce cumulative emissions from oil by 4% as the supply curve is very steep for high oil prices and few reserves drop out.

Heal, GM and Schlenker,W, “Coase, Hotelling and Pigou: The Incidence of a Carbon Tax and Co2 Emissions” (July 2019). NBER Working Paper No. w26086

Sinn’s paper is interesting not just for its thorough analysis of the Green Paradox, but for suggesting a couple of alternative policies against global warming. The key to these policies is that they address the important point of the issue: the quantity of fossil fuels extracted.

Here are three of them:

  1. Capping fossil fuel production: Basically, we need to tell the oil sheikhs very gently and politely that they need to stop extracting oil. For example, we could agree a fixed quota for annual oil and gas extraction. Since the oil sheikhs are intelligent people, they might be pursuaded to do that if we offer some development aid in exchange.
  2. Emissions trading: We could set a global cap on carbon emissions and auction off carbon certificates to industries and households. The EU has already tried such a scheme, although the cap was probably too large and not enough industries were not included (e.g. airlines). The big advantage of emissions trading compared to a tax is that it directly addresses the quantity, not the price. The downside is that negotiating a global trading system opens up a huge can of worms: especially, which country gets how many certificates? How should the revenue be used, etc.
  3. Sequestration and afforestation: Another way to solve the problem would be to de-link carbon emissions from fossil fuel consumption. Sequestration, i.e pumping the emitted CO2 back into the earth is one way (how feasible this is techniqually, I have no idea). Growing more trees which absorb CO2 naturally is another. Again, there could be international agreements to subsidize both these things.

I think all these policy proposals should get at least as much attention as the carbon tax. Why is nobody talking about them?

I should also point out that the issue is broader than the carbon tax. Any policy that merely tries to shift the demand curve for fossil fuels down will fail achieve the objective of decreasing greenhouse gas emissions unless it avoids the perverse effect on the fossil-fuel supply curve. Subsidizing e-mobility, putting tarrifs on international shipping, shaming people into avoiding airplanes, incentivizing the installation of solar panels and wind energy – all those things merely change the demand side.

I think the demand side is the wrong side. Let’s talk more about the supply side!

Some unpleasant carbon tax economics

Every economist knows that a carbon tax is the correct solution to climate change. By correct I mean the solution that a perfectly informed, well-meaning dictator would choose.

But when I was recently brooding over some dynamic optimization problems, I made a discoverey that I haven’t seen anyone discuss. And I find it disturbing.

I’m going to develop the argument formally below, but I will give away the punchline. Brace for impact!

Theorem: A carbon tax that remains constant over time doesn’t change the extraction path of fossil fuel. A carbon tax that increases over time tilts the extraction path in such a way that more fossil fuel is extracted now, less later.

If this is obvious to you, you can stop reading and start freaking out. If you think that this must be wrong, I would like you to point out any error I made in the argument below.

Let’s start from the Hotelling rule which dictates how profit maximizing oil sheikhs exploit their resource over time:

P(1+r) = P’,

where P is today’s price for oil (or gas, or whatever), r is the real interest rate and a prime denotes future variables. The rule says that you want prices to rise over time at the rate of the real interest rate.

When I say P is the price for oil, I mean the price the oil sheikh gets. The price consumers pay is P(1+t) where t is the (ad-valorem) carbon tax.

Next we need to postulate a demand curve to translate the Hotelling rule, which is about the evolution of prices, into a rule about quantities. Let’s write the (inverse) demand curve as follows

P(1+t) = D(Q)

and let’s postulate that D is decreasing in Q. This should shock nobody: demand curves slope down.

I hope you agree with me that absolutely nothing about this is in any way controversial. But then you must agree with me that we can combine the Hotelling rule with the present and future demand curves to get the following equation:

D(Q)(1+r)/(1+t) = D(Q’)/(1+t’).

This, ladies and gentlemen, is the dynamic law of motion for fossil fuel consumption. It describes how the quantity of fossil fuel extracted from the ground evolves over time. Since everything that is extracted will be consumed in the end, it implies a time path of carbon emissions.

Now what can we deduce about that time path from this equation?

  1. Hold the carbon tax constant over time by setting t=t’, and you will see that the equation reduces to
    D(Q)(1+r) = D(Q’),
    which is exactly the same equation that would hold if no carbon tax existed at all. It follows that with a time-invariant carbon tax, the sheiks will go on extracting oil and carbon emissions will continue at the exact same rate as if there were no carbon tax.
  2. It gets worse.  Suppose the carbon tax increases over time, i.e. t<t’. The effect of this will be the same as if the real interest rate would increase: it will make fossil fuel prices rise at a faster rate. But how do sheikhs make the sure the price path is steeper? By extracting more today, thus lowering the price today, and less in the future, thus increasing the future price.

Quod erat demonstrandum!

Now, of course you can refine the argument. What if, for example, the carbon tax eventually becomes so high that even the most fanatical SUV lover will refuse to pump gas? I don’t think this changes the argument. All this means is that oil producers will tilt the extraction path even more towards the present.

After all, there is a fixed and finite reserve of fossil fuels in the ground. All a carbon tax can change is when it will be extracted and the price consumers will pay for it.

If my argument is correct, why exactly are we sure that a carbon tax is the correct solution to climate change?

Addendum: If you want to me more concrete, assume fossil fuel demand is iso-elastic with elasticity e. In this case it is almost trivial to derive the equilibrium quantity: If R is the current stock of oil reserves, the quantity extracted now is

Q = (1-1/s)R with s = [(1+r)(1+t’)/(1+t)]^e

Notice that the extraction share Q/R is increasing in s which is increasing in the ratio of future to present carbon taxes (1+t’)/(1+t).