04 September 2007

Jevon's Paradox

What happens to the consumption of a fuel, like coal or gasoline, when efficiency improves? The answer is that it goes up. This surprising answer was observed by William Stanley Jevons in 1865:
The number of tons of coal used in any branch of industry is the product of the number of separate works, and the average number of tons consumed in each. Now, if the quantity of coal used in a blast-furnace, for instance, be diminished in comparison with the yield, the profits of the trade will increase, new capital will be attracted, the price of pig-iron will fall, but the demand for it increase; and eventually the greater number of furnaces will more than make up for the diminished consumption of each. And if such is not always the result within a single branch, it must be remembered that the progress of any branch of manufacture excites a new activity in most other branches, and leads indirectly, if not directly, to increased inroads upon our seams of coal.
["Of the Economy of Fuel"]
This post will explain the concept of the Jevons Paradox in greater detail. The concept of the utility function is explained here; it's fairly handy for explaining the complexities of the Jevons Paradox.

Click for larger image
A paraphrase of the paradox is that, if the utility of anything is increased (e.g., the benefits to an individual of consuming a liter of gasoline), then the consumption of that thing increases as a share of total consumption. To illustrate, I created a chart with utility functions generated by the linear expenditure model of utility.

The first chart shows what happens if the price of gasoline declines. The vertical (y) axis illustrates the quantity of gasoline; the horizontal (x) axis represents everything else. The red lines indicate levels of well-being or satisfaction; any point along them is supposedly "indifferent," or equally, desirable. It is assumed that more of either x or y enhances one's well-being, but if one already has a large amount of (say) y, then one will be reluctant to give up a little x unless one receives a lot of y.

The blue line is the budget line. It is straight, and its slope represents the price of y in terms of x. It intersects the x-axis at the point where one spends 100% of one's income on x, and likewise with y. In the chart, the price of gasoline is lowered 26%, a decline so extreme it amounts to a sharp increase in real income. We can see the consumer responds by actually spending more on gasoline: she buys 37% more of it, and 1.4% less of everything else (the indicator lines on the chart were inserted manually).

We could conceivably substitute "fuel" for "gasoline"; it's possible to imagine our consumer is experiencing not a 26% drop in the price of gasoline, but a motor vehicle which gets 35% more kilometers per liter than her previous one. However, the transition is awkward when we're discussing the more deterministic scenario of a profit-maximizing firm. In this case, it's not an consumer who now can drive faster, farther, and more frequently than before (and derives a greater share of her life's satisfaction from doing so); it's a industry that, perhaps, employs some type of energy to produce nitrates, and now finds it can switch to another. The cost savings of such an improvement, if permanent, would alter agriculture to be more nitrate-intensive; assuming pass-through of saving was complete, it's plausible that energy-expenditure on nitrate production would increase.

As a back-of-the envelope calculation, it's fairly easy to imagine a "commercial surrogate" for gasoline. We could simply say our consumer had bought kilometers on her car; before, they were US$ 0.04 per each, and now they're $0.03.1 So she consumes more kilometers and drives faster.

Unfortunately, there are problems with merely treating such an improvement exactly the same as a price cut in a commercial surrogate. As I mentioned, the consumer may not actually drive 37% more than she did before; she may actually drive 25% more, but much faster. The car takes a 9.7% reduction in fuel economy from being driven at a higher speed. It's still much more efficient; it's just that the higher speed translates to more work. If she did drive at the same speed as before, then she might get the 37% more kilometers, but she doesn't want to, and we're measuring human volition here, after all. Perhaps we seriously need to produce another utility function in which the fuel is more efficiently utilized, especially if (as in the case of industrial applications) it's utilized through a new process that's not directly comparable.

The new utility curve was generated by adjusting the coefficients of the linear expenditure utility function. The new curve had to have a "more convex" shape; it had to bend towards the y-axis, since the marginal rate of substitution for y is greater; more x has to be offered in exchange for any unit of y given up.2 Interestingly, the effect of an actual increase in efficiency is somewhat greater than a mere price reduction. While the two are not entirely comparable, we can see that a relatively small improvement has the same increase in utility as a large price cut, reflecting the greater scope of beneficial technical substitution. Unfortunately, the outcome is not much better: an increase in consumption (of the more efficient fuel), and a reduction in the consumption of other things. The latter is ominous because, while one might quarrel with the new (dark brown) pair of indifference curves, one cannot deny that a logically consistent indifference curve would have about that shape, and would lead to greater preference for fuel relative to other inputs. And while I've played fast and loose with the location of the new indifference curves, a re-drawing of them as marginal rates of technical substitution is something that does not require the same level of circumspection. Without taking utility functions "on faith," one can simply do the managerial accounting and see that efficiency would lead not merely to increased output per input of energy, but a switching from non-energy inputs to energy ones.

I would like to conclude this intro to Jevon's Paradox with two additional points. First, Jevons was writing about coal, which he surmised would became a headache for the UK as it ran out. Petroleum took the place of coal, not merely because it was there, but because it also contained more energy per volume, and could be consumed at far faster rates. Jevons did not see oil on the horizon, and he didn't foresee the ability of Britain to import energy cheaply. So we might imagine that there is something else, like "ultra-oil," that will be discovered in the next few decades. The problem with this idea is that oil had to have near-miraculous properties to supersede coal. It had to be liquid at standard temperature and pressure; it had to have a high specific heat; it had to have a relatively high specific density (compared to, say, liquid hydrogen); it had to be chemically stable to be used safely by amateurs, and yet inflammable enough to allow simple ignition systems. It had to be abundant, and globally distributed. If it lacked any one of these properties, it would have been prohibitively costly to use.

In order to supersede oil, "super oil" would need to have all of these properties; otherwise, it would not really solve any of the current anxieties associated with peak oil. It would have to contain a higher energy flow rate, i.e., a cubic centimeter of it would need to contain the same chemical energy as oil, and that energy would have to be as readily transmissible to an engine. Otherwise, the energy captured by refining it would be too expensive to avert the economic catastrophe that peak oil threatens. In other words, something with chemically implausible properties would need to appear in order to take oil's place, the way oil took coal's place in the early 20th century. If a "sub-oil" appeared instead, with inferior properties, then the energy-cost of refining it to take the place of oil would be so much greater as to induce a protracted recession.

Which means that, while the Western world was rescued by oil in the late 19th century, a repeat of this is not just an unknowable possibility, but a highly improbable (or even impossible) scenario. Besides, there's the matter of global climate change, which would get massively worse.

My second point is best made by quoting from Jeff Vail:
Jevon's Paradox: Does this mean that efficiency is an invalid policy choice? No: true conservation, the goal of efficiency policy, can be achieved, but this represents a far more challenging policy dilemma. It is relatively simple, for example, to legislate higher CAFE standards. But what happens with the money saved on gasoline? It is quite a policy challenge to ensure that the energy saved by CAFE changes doesn’t simply go to another use of energy. One solution—the one that I am proposing—is that monetary savings from efficiency legislation is offset by an energy tax that is then invested in a manner that minimizes its energy consumption. Options for this offset fund reducing existing spending deficits, encouraging social pressure for absolute conservation, or my personal choice, funding efforts to design for quality of life using less energy—what I have called the Design Imperative. But selling this policy combination—CAFE increases paired with gas tax increases, for example—is a much more difficult task.

My intent is not to discourage the push for energy efficiency—quite the opposite: energy efficiency is a key part of addressing the challenges posed by Peak Oil, but ONLY if it is paired with measures to address both the direct and shadow rebound effects. There are valid arguments to focus on efficiency first, because it takes time to develop the technologies that create efficient energy use. However, we must be careful not to present efficiency as a standalone panacea, but rather to spur debate of systemic solutions of which efficiency is a key part.
(Jevon's Paradox Revisited)

1 Consistent with getting 46 miles to the gallon when gasoline is $3/gallon. Driving faster reduces fuel economy; in fact, our imaginary consumer could very well use up nearly all her extra petrol by merely driving faster. That would explain the increase in satisfaction.

2 Another way of expressing this is that, while the first derivative dy/dx of both sets of indifference curves is negative, the one for dy0/dx is more negative for each value of x than dy1/dx is. A negative first derivative means an increase in x offsets a decrease in y, which is true for all indifference curves. A large [negative] value of dy/dx means x has a comparatively larger value in exchange with y, unit per unit. If the absolute value of the 1st derivative shrinks (for any given value of x), then y has become more valuable in exchange.
Additional Reading and Sources:

William Stanley Jevons, The Coal Question, London: Macmillan and Co. (1865); see esp. "Of the Economy of Fuel"

Jeff Vail, "Efficiency Policy, Jevon’s Paradox, and the 'Shadow' Rebound Effect"

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