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1.4: Demand-side Estimates

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    24082
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    Our analysis of the carrying capacity of highways and railways (Section 1.2) is an example of a frequent application of estimation in the social world—estimating the size of a market. The highway–railway comparison proceeded by estimating the transportation supply. In other problems, a more feasible analysis is based on the complementary idea of estimating the demand. Here is an example.

    How much oil does the United States import (in barrels per year)?

    The volume rate is enormous and therefore hard to picture. Divide-and-conquer reasoning will tame the complexity. Just keep subdividing until the quantities are no longer daunting.

    Here, subdivide the demand—the consumption. We consume oil in so many ways; estimating the consumption in each pathway would take a long time without producing much insight. Instead, let’s estimate the largest consumption—likely to be cars—then adjust for other uses and for overall consumption versus imports.

    \[imports = \cancel{\textrm{car usage}} \times \frac{\cancel{\textrm{all usage}}}{\cancel{\textrm{car usage}}} \times \frac{\textrm{imports}}{\cancel{\textrm{all usage}}}.\]

    Here is the corresponding tree. The first factor, the most difficult of the three to estimate, will require us to sprout branches and make a subtree. The second and third factors might be possible to estimate without subdividing. Now we must decide how to continue.

    clipboard_e237b22f868281eb6b342f4366630626a.png

    Should we keep subdividing until we’ve built the entire tree and only then estimate the leaves, or should we try to estimate these leaves and then subdivide what we cannot estimate?

    It depends on one’s own psychology. I feel anxious in the uncharted waters of a new estimate. Sprouting new branches before making any leaf estimates increases my anxiety. The tree might never stop sprouting branches and leaves, and I’ll never estimate them all. Thus, I prefer to harvest my progress right away by estimating the leaves before sprouting new branches. You should experiment to learn your psychology. You are your best problem-solving tool, and it is helpful to know your tools.

    Because of my psychology, I’ll first estimate a leaf quantity:

    \[\frac{all \: usage}{car \: usage}\]

    But don’t do this estimate directly. It is more intuitive—that is, easier for our gut—to estimate first the ratio of car usage to other (noncar) usage. The ability to make such comparisons between disjoint sets, at least for physical objects, is hard wired in our brains and independent of the ability to count. Not least, it is not limited to humans. The female lions studied by Karen McComb and her colleagues [35] would judge the relative size of their troop and a group of lions intruding on their territory. The females would approach the intruders only when they outnumbered the intruders by a large-enough ratio, roughly a factor of 2.

    Other uses for oil include noncar modes of transport (trucks, trains, and planes), heating and cooling, and hydrocarbon-rich products such as fertilizer, plastics, and pesticides. In judging the relative importance of other uses compared to car usage, two arguments compete: (1) Other uses are so many and so significant, so they are much more important than car usage; and (2) cars are so ubiquitous and such an inefficient mode of transport, so car usage is much larger than other uses. To my gut, both arguments feel comparably plausible. My gut is telling me that the two categories have comparable usages:

    \[\frac{other \: usage}{car \: usage} \approx 1\]

    Based on this estimate, all usage (the sum of car and other usage) is roughly double the car usage:

    \[\frac{all \: usage}{car \: usage} \approx 2\]

    This estimate is the first leaf. It implicitly assumes that the gasoline fraction in a barrel of oil is high enough to feed the cars. Fortunately, if this assumption were wrong, we would get warning. For if the fraction were too low, we would build our transportation infrastructure around other means of transport—such as trains powered by electricity generated by burning the nongasoline fraction in oil barrels. In this probably less-polluted world, we would estimate how much oil was used by trains.

    Returning to our actual world, let’s estimate the second leaf:

    \[\frac{imports}{all \: usage}\]

    This adjustment factor accounts for the fact that only a portion of the oil consumed is imported.

    What does your gut tell you for this fraction?

    Again, don’t estimate this fraction directly. Instead, to make a comparison between disjoint sets, first compare (net) imports with domestic production. In estimating this ratio, two arguments compete. On the one hand, the US media report extensively on oil production in other countries, which suggests that oil imports are large. On the other hand, there is also extensive coverage of US production and frequent comparison with countries such as Japan that have almost no domestic oil. My resulting gut feeling is that the categories are comparable and therefore that imports are roughly one-half of all usage:

    \[\frac{imports}{domestic \: production} \approx 1 \: so \: \frac{imports}{all \: usage} \approx \frac{1}{2}\]

    This leaf, as well as the other adjustment factor, are dimensionless numbers. Such numbers, the main topic of Chapter 5, have special value. Our perceptual system is skilled at estimating dimensionless ratios. Therefore, a leaf node that is a dimensionless ratio probably does not need to be subdivided.

    The tree now has three leaves. Having plausible estimates for two of them should give us courage to subdivide the remaining leaf, the total car usage, into easier estimates. That leaf will sprout its own branches and become an internal node.

    clipboard_ea06d9216ef410f67130198dbc0af55bc.png

    How should we subdivide the car usage?

    A reasonable subdivision is into the number of cars Ncars and the per-car usage. Both quantities are easier to estimate than the root. The number of cars is related to the US population—a familiar number if you live in the United States. The per-car usage is easier to estimate than is the total usage of all US cars. Our gut can more accurately judge human-scale quantities, such as the per-car usage, than it can judge vast numbers like the total usage of all US cars.

    clipboard_ec585f95d5fd51e334b3e9138f12588c8.png

    For the same reason, let’s not estimate the number of cars directly. Instead, subdivide this leaf into two leaves: 1. the number of people, and 2. the number of cars per person. The first leaf is familiar, at least to residents of the United States: \(N_{people} \approx 3 \times 10^{8}\)

    clipboard_eb92e83e20eb6e6a0c8f27bffc16c11ad.png

    The second leaf, cars per person, is a human-sized quantity. In the United States, car ownership is widespread. Many adults own more than one car, and a cynic would say that even babies seem to own cars. Therefore, a rough and simple estimate might be one car per person—far easier to picture than the total number of cars! Then \(N_{cars} \approx 3 \times 10^{8}\).

    The per-car usage can be subdivided into three easier factors (leaves). Here are my estimates.

    clipboard_eb9c4e3918e7254ad7cb2b5949aff6252.png

    1. How many miles per car year? Used cars with 10 000 miles per year are considered low use but are not rare. Thus, for a typical year of driving, let’s take a slightly longer distance: say, 20 000 miles or 30 000 kilometers

    2. How many miles per gallon? A typical car fuel efficiency is 30 miles per US gallon. In metric units, it is about 100 kilometers per 8 liters.

    3. How many gallons per barrel? You might have seen barrels of asphalt along the side of the highway during road construction. Following our free-association tradition of equating the thickness of a sheet of paper and of a dollar bill, perhaps barrels of oil are like barrels of asphalt.

    Their volume can be computed by divide-and-conquer reasoning. Just approximate the cylinder as a rectangular prism, estimate its three dimensions, and multiply:

    \[\textrm{volume} \sim \underbrace{1m}_{\textrm{height}} \times \underbrace{0.5m}_{\textrm{width}} \times \underbrace{0.5m}_{\textrm{depth}} = 0.25 m^{3}.\]

    clipboard_e63b661b8b702ca2312216b526d940c8d.png

    A cubic meter is 1000 liters or, using the conversion of 4 US gallons per liter, roughly 250 gallons. Therefore, 0.25 cubic meters is roughly 60 gallons. (The official volume of a barrel of oil is not too different at 42 gallons.)

    Multiplying these estimates, and not forgetting the effect of the two −1 exponents, we get approximately 10 barrels per car per year (also written as barrels per car year):

    \[\frac{2 \times 10^{4} \cancel{\textrm{miles}}}{\textrm{car year}} \times \frac{1 \cancel{\textrm{gallon}}}{30 \cancel{\textrm{miles}}} \times \frac{1 \textrm{barrel}}{60 \cancel{\textrm{gallons}}} \approx \frac{10 \textrm{barrels}}{\textrm{car year}}.\]

    In doing this calculation, first evaluate the units. The gallons and miles cancel, leaving barrels per year. Then evaluate the numbers. The 30 x 60 in the denoinator is roughly 2000. The 2 x 104 from the numerator divided by the 2000 from the denominator produces the 10.

    This estimate is a subtree in the tree representing total car usage. The car usage then becomes 3 billion barrels per year:

    \[3 \times 10^{8} \cancel{\textrm{ cars}} \times \frac{10 \textrm{barrels}}{\cancel{\textrm{car }} \textrm{year}} = \frac{3 \times 10^{9} \textrm{barrels}}{\textrm{year}}.\]

    This estimate is itself a subtree in the tree representing oil imports. Because the two adjustment factors contribute a factor of 2 × 0.5, which is just 1, the oil imports are also 3 billion barrels per year.

    clipboard_ecec41b19c4562594a5782a23f8221212.png

    Here is the full tree, which includes the subtree for the total car usage of oil:

    clipboard_ea0a6901b69d70a82cfa287b250cd3dc1.png

    Exercise \(\PageIndex{1}\): Using metric units

    As practice with metric units (if you grew up in a nonmetric land) or to make the results more familiar (if you grew up in a metric land), redo the calculation using the metric values for the volume of a barrel, the distance a car is driven per year, and the fuel consumption of a typical car.

    How close is our estimate to official values?

    For the US oil imports, the US Department of Energy reports 9.163 million barrels per day (for 2010). When I first saw this value, my heart sank twice. The first shock was the 9 in the 9 million. I assumed that it was the number of billions, and wondered how the estimate of 3 billion barrels could be a factor of 3 too small. The second shock was the “million”—how could the estimate be more than a factor of 100 too large? Then the “per day” reassured me. As a yearly rate, 9.163 million barrels per day is 3.34 billion barrels per year—only 10 percent higher than our estimate. Divide and conquer triumphs!

    Exercise \(\PageIndex{2}\): Fuel efficiency of a 747

    Based on the cost of a long-distance plane ticket, estimate the following quantities: (a) the fuel efficiency of a 747, in passenger miles per gallon or passenger kilometers per liter; and (b) the volume of its fuel tank. Check your estimates against the technical data for a 747.


    This page titled 1.4: Demand-side Estimates is shared under a CC BY-NC-SA license and was authored, remixed, and/or curated by Sanjoy Mahajan (MIT OpenCourseWare) .

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