Just for the sake of a more fair comparison, we will use the specifications for the lowest 4-door trim levels for each vehicle. Let’s compare them in the table below.

1960 Chevrolet Impala

list price: $2,590
engine: Inline 6-cylinder, 3.9 L
horsepower: 135 hp
transmission: 3-speed manual
brakes: 4 drum brakes
interior space: 136.3 cu. ft. (estimated)
cargo space: 17.8 cu. ft. (estimated)
fuel efficiency: 18/22 mpg (city/hwy)
climate control: none (a $327.42 AC/Heater was available as an option)
gross weight: 3625 lbs.
length/width/height: 210.8/79.9/54 in.
safety: 51 fatalities per billion miles traveled

2012 Toyota Camry

list price: $22,055
engine: Inline 4-cylinder, 2.5 L
horsepower: 178 hp
transmission: 6-speed automatic
brakes: 4 disk brakes
interior space: 118.1 cu. ft.
cargo space: 15.4 cu. ft.
fuel efficiency: 25/35 mpg (city/hwy)
climate control: AC and Heater
gross weight: 3190 lbs.
length/width/height: 189.2/71.7/57.9
safety: 11 fatalities per billion miles traveled

So how much might we be willing to spend on the improvements between the 1960 Chevrolet Impala and the 2012 Toyota Camry? If a car like today’s Toyota Camry was available in 1960, it would have sold for a higher price than the Impala, of course, but for how much more? Here we can’t be absolutely certain, mostly because the range of options on which we can spend our money is very different in 2012 than it was in 1960, but we can estimate based on comparisons with similar vehicles then and now.

Another consideration we should make is to compare how the average hour of labor in the US translated into purchasing these vehicles, so we’ll look at whether these vehicles have changed in price appreciable in comparison to the median hourly wage. The median hourly wage in 2011 was $16.57. The median hourly wage in 1960 was $2.47. These are both nominal values, meaning they have not been otherwise adjusted for inflation effects, which is what we want for this analysis. That means that we can re-cast our two competing car prices as 1,050 hours for the Impala and 1,330 hours for the Camry. That gives us a difference of 280 hours of increased labor required to afford the 2012 Camry over the 1960 Impala.

Power

One of the easiest things to compare is the horsepower (or acceleration potential) of the two vehicles’ engines. The Camry’s engine has 43 more available horsepower than the Impala, although this isn’t quite the whole story as the Impala weighs 435 lbs (198 kg) more. So, the Impala would need more than 43 additional horsepower to equal the acceleration and overall handling of the Camry. The next higher engine option, a V-8 with 185 horsepower added an additional $107 (43 hours’ wages) to the list price of the Impala. I doubt the additional 50 horsepower is really sufficient to close the acceleration gap with the Camry, but for now, it is close enough.

Comfort

Here we have the strongest direct metric of our comparison. In 1960, an Air Conditioner and Heater combination was available as an option on the Impala for a price of $327 (or 132 hours’ wages). This isn’t the whole story, as seats have been designed to be more ergonomic, window defrosters reduce the work required to drive in cold weather, and power locks and windows make life easier, if only marginally. A kit to replace power window and lock actuators in a vehicle similar to the Impala will cost about $200 in 2012 (or 12 hours’ wages). This would be equivalent to about $30 in 1960. In truth, small electric actuators would have been much more expensive in 1960, since their ubiquity has more to do with modern automated manufacturing techniques, but I doubt 1960′s consumers would have put much more of their own labor towards something of relatively minor importance like these.

Radio and MP3 player systems were not available in 1960, and neither was Bluetooth for cell phone connectivity, although those are common in 2012. With a price for a Mp3 stereo system being about $60, and the cost of an automobile Bluetooth adapter being about $30 in 2012, we can estimate these options to be worth about 5 hours wages in 2012 terms or approximately $13 in 1960.

Safety

For the safety ratings I listed above, I used the risk of fatality per billion miles driven for the years 1960 and 2010. Since these cars are more or less representative of the average vehicle on the road in their own time frames, I think this is fair. All of this data comes from NHTSA which you can see here.

But, it is not just the number of deaths per mile traveled that concerns us, but also our overall risk of dying. In other words, have we compensated for the lower per mile risk by driving many more miles ever year such that our overall risk remains the same as in 1960? The short answer is “No”. While each vehicle drives on average about twice as many miles in 2012 than the average vehicle did in 1960, that is not sufficient to make up for the nearly 5-fold difference in road fatalities during that time.

The risk of dying during an average annual driving distance (i.e. 7,000 miles in 1960) in the Impala is 0.035%, while the risk of dying in one year in the safer Camry is estimated to be about 0.007%. Pre-1980 estimates of the value of a statistical life (see this article for more information), range from $1.0 million up to $9.2 million. Depending then on the driver’s own estimate of their risks and value of their own life, a car purchaser should be willing to pay between $280 and $2,520 for the additional safety provided by the Camry. We will use the lower value for our argument’s sake.

This isn’t the whole story on safety, of course. Improved visibility, especially at night, is a feature of the Camry compared to the Impala. Seat belts were not required in cars until 1961 (they were an option on the Impala in 1960), and were only available as lap belts at the time. Additionally, stability and traction control, which are automated systems that reduce the chance of a total loss of control, are not possible without the computers that control modern cars. All of these systems reduce not just the risk of fatality, but also the risk of damage to the vehicle as well as minor injuries from other avoided less catastrophic crashes.

Disk brakes were available as an option on other vehicles in 1960, and power brakes were also. Although, I could not find any information on the list prices of these options, I did find the results of a study in 1983, which listed the automaker’s cost of these options as being $38 [15 hours' wages], which we can take as roughly equivalent to the list price in 1960 (as well as the considering the cost of other improvements such as anti-lock brakes).

Improved road design, as well as regulation and enforcement on things like driving while intoxicated have also improved safety, although it is difficult to parse out the relative contributions of improved crash survivability due to improved vehicle design, and other factors.

Fuel Consumption

The Impala has an efficiency of about 20 mpg combined city/highway, while the Camry gets a better rating at around 30 mpg combined. There are a couple different ways we could handle this one. One is to assume that both vehicles will be driven the same amount of miles per year. In 2012, that value was 13,400 miles, but in 1960, the number of miles driven per vehicle per year was about 7,000 or about half. Using the nominal price of gas in 1960 ($0.31 per gallon) and a vehicle efficiency of 20 mpg, we would see an annual gasoline expenditure of between $109 and $208 or between 44 and 84 hours of labor. The improvement gained in 1960, by having a car that did all the same things, but could travel 30 mpg, would be to reduce gasoline expenditures to between $72 and $138 or 29 and 56 hours of labor. So, for this capability, a logical consumer planning to drive this car for 3 years would be willing to spend at least $111 and possibly as much as $210 depending on how much they planned to drive. We’ll use the lower value in our calculation below.

Externalities

There are a few externalities we should contemplate including pollution and traffic. There aren’t direct costs on the consumer of owning the vehicle, but rather they are costs to the consumer and others of other people owning the same kinds of vehicles. But, since we can’t purchase a car without also expecting others to purchase cars as well, they do need to be included at least in a qualitative sense, even if we don’t directly add them to the price of the vehicle directly.

In 1960, air quality in cities like Los Angeles, Chicago, and New York was terrible compared to what it was in 2012. While CO2 emissions have continued to increase since the 60′s, other more immediately dangerous pollutants have declined significantly. This is mostly due to the Clean Air Act, which mandated pollution control equipment to be installed on automobiles and power plants. The benefits of cleaner air improve life for car consumers in cities, but they do add cost due to the required catalytic converters and other systems. We are not accounting for this cost or benefit in this article.

In 1960, traffic congestion was much lower than in 2012, with the average commuter in a large city losing about 30 hours per year in traffic jams. The same commuter in 1960 lost less than 10 hours by comparison.

Credit

This is sort of a tangential issue to the purpose of this article, but it bears mentioning. Credit is much more available in 2012 as compared to 1960, and a greater percentage of car purchases are financed with credit to a greater degree than they were five decades ago. This increases aggregate demand, as people can purchase a nicer vehicle with the same amount of money up front, or with lower monthly payments required over the term of the loan. Whether consumers are putting this additional purchasing power into the price they are willing to pay for their automobile, or whether they would use that purchasing power on something else entirely is outside the bounds of this limited analysis.

Final Tally

Now, let’s account for the differences we’ve identified, and see where we are left in our comparison between the Impala and the Camry.

Price of 1960 Chevrolet Impala With 2012 Camry Features
Additional Horsepower (+$107) [43 hours wages]
Air Conditioning and Heater (+$327) [132 hours wages]
Power Locks and Windows (+$30) [12 hours wages]]
Power Disk Brakes (+$38) [15 hours wages]
Mp3 Stereo System and Bluetooth Adapter (+$13) [5 hours wages]
Fuel Consumption (+$111) [45 hours wages]
Overall Safety (+$280) [113 hours wages]

So, the price of a Camry-like Impala in 1960 would have been about $3,496, or about 1,415 hours at the prevailing median wage. This isn’t definitive, of course, since it neglects reliability, insurance, and many other less tangible features that nevertheless affect our decision about which vehicle to buy (although except for tail fin fans, most of those factors would cause this price differential to increase and not decrease), but it is the start of such a comprehensive comparison.

The Camry-like Impala in 1960 would have required at least 85 more hours of labor to purchase than today’s Camry requires with today’s labor (probably more given our conservative estimates). This could be due to more efficient factories being able to produce these vehicles with less overall labor than was required five decades ago.

Automobile Prices and the Consumer Price Index (CPI)

How have the prices of automobiles been affected by general inflation and technological improvements? Based on our calculation of a price on a Camry-like vehicle in 1960, we can see that car prices have still increased by $18,559 or 631% between 1960 and 2012. According to the CPI, $1 in 1960 is equivalent to $7.76 in 2012, an increase of 776%. In other words, according to the CPI, the price of our Camry-like Impala in 2012 dollars should be $27,129, rather than the $22,055 price of the actual Camry.

There are a couple ways to interpret this information. One is that the Camry and similar cars today are actually cheaper than comparable cars would have been in 1960. In other words, the price of automobiles has deflated to some degree not unlike what happens to the prices of computers, Mp3 players, televisions, and other electronics. Second, alternatively, one might say that the CPI is in error, and the correct rate of inflation between 1960 and 2012 would be something closer to 631% than 776%, although further analysis would be required to justify that proposition.

Handouts are beneficial for several reasons. One reason, as Tufte conveys constantly, is that handouts can be much more information-dense, and yet more easily browse-able at the same time. A large table of data is a nightmare on a presentation slide, yet easily navigable as a handout. Another reason is that handouts persist far longer than memories of your words, slides, or poster ever would. While they aren’t permanent bookshelf material, they are excellent advertising for those who may be interested in your work. Handouts also provide for your audience better memory triggers in the future beyond what business cards will ever do. So, for the benefit of your audience as well as your own future opportunities, I recommend making this additional effort whenever an important event comes along.

If you want to do this the “right” way, you should really use a LaTeX document class like this, but many people are not already familiar with LaTeX and do not have the time to learn it prior to their next event. As another alternative, since I couldn’t find one posted elsewhere, I have created and posted a MS Publisher template of an Edward Tufte-style handout as described in his books and website. You can also access it though the downloads page.

Microsoft Publisher doesn’t make it easy to copy what Edward Tufte does exactly down to the fonts and other details, but most of your objectives can be met with a template like this one. The following list details what has been included in this template, and the pictures below display what the first two pages look like.

What this template won’t do (unfortunately): MS Publisher is not LaTeX. It isn’t even MS Word as far as automatic text handing for citations, references to figures and tables, and other similar things. That means you will have to make sure numbers match up, the footer matches the main title, and references are numbered properly. But, fortunately, this is a short 4-page handout and not a dissertation. You won’t be using this to publish a 40-reference article or layout the results of an experiment in 16 charts and graphs. So, as an accompaniment, I’ve found it acceptable to suffer through the amount of manual labor required to use it.

For what it’s worth, and with the knowledge that I myself am still exploring the usefulness of and best practices for creating this kind of media, you can see one of my recent examples using this handout template here.

When we think of investment for the future, we typically imagine money sitting in an account earning interest or dividends. The idea being that that money is given to someone else in the future, that amount of money will have grown to the point where we can realize more benefit from it than we can in the immediate present. We hope to use investments like this to provide for ourselves when we are unable to work, to help our children get a good start in life with an education and possibly a wedding, perhaps to start a business, and many hope to save sufficiently after all of that to leave their children and grandchildren a modest inheritance when they die.

We can and should see oil and gas discoveries in the same light. They are currently safely stored resources that we can decide to either use in the present in one manner or another, or we can save that oil and gas to be used in the future by ourselves or our children or grandchildren. Some seem to accept that we will use as much oil and gas as we can as fast as we can until it has been depleted, and at that point, human ingenuity will lead us to the solutions that allow us to somehow continue onward with our steady technological progress and improved living standards. But, that story’s upside isn’t so certain, and we do, in fact, have many choices open to us as to how we treat these resources going forward.

Burned Oil and Gas is Gone Forever

Burned oil and gas is irretrievable. We can’t ever get it back or use it for something better in the future. Burned oil is equivalent to money being spent. This could be spending for durable goods (energy used to create or deliver cars, for example), or for business uses that create income, or it could be used for more ephemeral uses that only satisfy us for a short time and lead to no long-lasting changes in our own or other’s lives.

The biggest problem, however, is that so much of the energy content of burned oil and gas is wasted. According to the chart below, we can see that approximately 81% of the energy in oil and approximately 20-50% of the energy in gas is wasted as rejected heat depending on its use. That means that even out of what we use, most of the content of it does us no benefit whatsoever. It doesn’t heat or cool our homes, and it doesn’t drive our cars anywhere.

So, even if all of the oil and gas used for manufacturing and usable energy was completely and invaluably employed in increasing the well being of current and future people (i.e. productive investment), the maximum productive yield per unit of oil is only 25%, with natural gas having a maximum yield per unit of about 46%. These are not high values.

Plastics and Lubricants can be Recycled

Use of oil and gas in plastics and lubricants as opposed to burning it directly for energy means that we can recycle most of that material and achieve more than a single use from it. Overall, approximately 8% of plastics are recycled by total weight. Another 8% of plastics are burned for energy. But, given present day market conditions, only 28% of the total recyclable plastics are actually recycled. Almost all plastics are recyclable, but only certain varieties are currently possible given the amount of money paid for them, and the amounts available to collect from consumers and businesses. Still, even in current conditions, about 4 times more plastic could be recycled than is currently done. As businesses take up the challenge of recycling a wider array of plastics, there is little reason to suspect that we couldn’t recycle up to 95% of the total, eventually.

Lubricants are recycled at a higher rate than plastics, nearly 44% of the total as of 2005. This quantity is probably higher now. Used lubricants can be cleaned, purified, and re-used for similar purposes. Nearly all lubricating oil is recyclable, but losses occur due to leaks or evaporation/vaporization, and during the recycling process, meaning that the most we could ever really re-use would be around 85% of the total.

If we consider our “productive yield” of oil and gas for plastics in the same way that we did for their use as energy above, we can compare them and see that while presently, plastics and lubricants usage isn’t much more efficient today than oil and gas used as energy. 72% of easily recyclable plastics are thrown away after one use. But, the potential increases in recycling or the potential for achieving durable benefits from those plastics and lubricants is enormous. While water bottles, cheap toys, and motor oil may not seem all that durable to us, structural composites for cars and aircraft, asphalt for roads, plasticized lumber composites for homes and buildings can all last for decades or more. More oil and gas going to these uses can create more longer lasting contributions to our way of life than a short trip on the highway.

Unused Oil and Gas Earns Interest

Oil and gas that sits unused under the ground today is not just lost consumption that we are missing out on. For one thing, the oil and gas is not going anywhere, so leaving it “in the bank” so to speak is a very low risk saving strategy. But, that is not the whole story. Our capability to create benefits with a given amount of oil or gas increases every year. Our machines get more efficient. New technologies are created to solve current problems, and our overall capabilities increase.

To illustrate just one of these possibilities, the following chart includes historic and future projections for vehicle fuel efficiency. From 2005 to 2025, fuel efficiency in terms of distance traveled on one liter or gallon of fuel is expected to double in most places. Put another way, a unit of fuel saved (and not burned immediately) is earning interest (in terms of total miles it can transport us) at a rate of 3.6%. That may not seem very significant, but there is very little downside risk here, and a 3.6% interest rate compared to those of today’s typical bond yields or average money market rates looks very good.

Another way to consider this in a broader context is the amount of energy required to create one dollar’s worth of economic output. The following chart shows the energy required to create one new unit of economic output (or Gross Domestic Product, GDP). While the specifics are different for each country depending on when they industrialized, the general trends indicate that in the United States between 1950 and 2025 the amount of energy used to produce a unit of GDP declined by 75%. Or, put differently, the amount of GDP created for each unit of energy consumed increased by 300%. The annualized rate works out to approximately 2.9%.

Since these values are real, rather than nominal, that means the “efficiency yields” being discussed here of 3.6% in mileage and 2.9% in GDP are in real terms and in excess of inflation. That is a similarly performing financial investment quoted in APR would have a stated annual return of these numbers plus the anticipated inflation rate. Few alternative financial options approach this kind of stability.

A Little Financial Engineering

This isn’t about credit-default-swaps or other derivatives. But, sometimes financial products can enable us to facilitate investment by more people than would otherwise be possible. For example, the stock market makes investing in large businesses possible for a much larger share of people than would be possible if we had to buy in with a 5% share. A similar kind of success in maximizing investment in conservation is feasible with the right kinds of financial products.

Many people have proposed efficiency bonds that would permit investors to help provide funds for improvements in energy efficiency in exchange for the investor earning a portion of the financial savings from reduced energy use realized. Some companies have been exploiting this potential by giving homeowners free solar panels in exchange for a return of some of the energy bill savings. Since, those who would most benefit from the energy conservation often lack the capital to purchase the necessary equipment, these bonds allow those with the available capital to apply those funds where they are needed in exchange for some of the benefits. As a whole, such investment can greatly increase energy efficiency in homes and commercial buildings.

Another potential innovation would be a financial product that would allow individual buyers to purchase rights in oil or gas intentionally left underground. Similar to futures contracts, except with undefined future delivery dates, these oil and gas certificates would convey ownership of a certain volume of oil purposely saved and kept underground as a long term investment. Firms willing to purchase and hold on to and maintain these reservoirs and have them independently audited could convey ownership of the resources contained therein to investors in exchange for present capital, while giving the potential benefit of future increased prices to the investors.

We have all heard recommendations from financial planners to save 10% – 20% of our incomes each year. This isn’t an inconsequential sacrifice for many middle class families, but most of us recognize that this present day sacrifice will improve our family’s financial security. For many people, this is just barely sufficient to provide for themselves through retirement, but for others it is enough to pass an inheritance on to the next generation. We should think of oil and gas conservation–holding off on a bit of current consumption–in the same way that we think of giving up a little bit of current income for a more secure future.

Our Future

It is one thing to acknowledge that oil and gas are energy-dense materials that make much of our modern lives easier and more comfortable. But, it is another thing to claim that our future selves or our grandchildren would have no better use for that oil and gas than we do today. The truth is that we know we will be able to do more with it in the future than we can do now. For some, that means that it should be acceptable to leave less of this stuff to our children than we decide to use today. This, I think, is a bit presumptuous. We don’t know what kinds of challenges our descendants will face, especially in a world with 9.5 billion people. We don’t know if any of our predicted next generation energy technologies will succeed the way we think.

But, if we could harness the technological and financial capabilities that exist right now, we could drastically reduce our consumption rate of oil and gas, saving more for our future selves and more for the next few generations. We could leave a better, more stable inheritance more immune from financial mismanagement than any publicly traded stocks or bonds. Beyond all the talk about environmental benefits, conserving oil and gas for future use rather than burning it in the present has compelling financial and ethical reasons to recommend it.

First, Hacker does acknowledge the technical necessity of the subject and does not argue that we should not be teaching Algebra at all, only that it should not be a universal requirement. Hacker believes that only those interested in the university-level study of hard science or engineering should take Algebra. His argument as to why students should not be indiscriminately forced to study this subject breaks down more or less as follows:

Algebra is more difficult and abstract than other Math subjects like Geometry and Arithmetic, and discourages students from completing high school or basic 4-year college degrees. As evidence, for the Algebra portion of state exit exams, in Oklahoma, 33 percent failed to pass last year, as did 35 percent in West Virginia. And, at the City University of New York, 57 percent of its students didn’t pass its mandated algebra course.

Algebra is unnecessary in everyday occupations like factory machine tool operation, business management, personal finance, or even veterinary medicine, for other applied scientific practitioners.

Mathematics should be made more accessible and taught as a Liberal Art, with courses in the history and philosophy of mathematics, and its applications in art, music, as well as political and economic statistics.

Young people should learn to read and write and do long division, whether they want to or not. But there is no reason to force them to grasp vectorial angles and discontinuous functions.

Edward Frenkel at Slate has already published this account, which brings to light some of the limitations of Prof. Hacker’s proposals. These include the fact that many of the examples of practical applications of Mathematics that Hacker cites like the calculation of the Consumer Price Index, do in fact, require Algebra and often higher Mathematics as well to understand. With this particular issue already well covered, I want to address other parts of Hacker’s argument, and outline what I believe we need to do about it.

First, let’s address the purpose of and need for Algebra in today’s world. While evaluating abstract expressions like (x² + y²)² = (x² – y²)² + (2xy)² seems to be at the heart of everyone’s concept of Algebra, that is not the real essence of the discipline. Algebra at its heart is an extension of arithmetic that allows us to uncover an unknown value of interest to us, when it is something that doesn’t appear at first blush to be reducible to a 2 + 7 = ____ type of problem. It helps us discern how hard we have to study for that final exam to earn a B in the course; it helps us determine how much we need to set aside from every paycheck to afford that down payment; it helps us design the most affordable deck for the back yard; it helps us trade off risks and rewards in our retirement savings.

Facility with the techniques and tools of Algebra is absolutely essential to understanding statistics (a basic necessity of citizenship in today’s world), being able to program a manufacturing machine, choose the best investments and major purchases, and code computer applications or understand their likely errors. And, that is not all. While Algebra is imminently applicable to everyday problems beyond the obtuse “train travelling from Cleveland..” word problems packed into textbooks, our courses tend to focus on the Mathematician’s appreciation of the beauty and symmetry of abstract Math combined with the machine-like drilling useful for preparing in advance of standardized tests. While I currently use Algebra to do all kinds of simple problem solving, I don’t recall ever seeing such an application as a high school student–that fact lies at the crux of the problem.

Next, the problems with the success rate of students in Algebra courses could be a result of poor preparation by earlier courses, lack of motivation due to poor course design, and insufficient teaching skills in the classroom, among other things. This is in addition to Math requirements generally being increased considerably over the past 20-40 years, while English requirements have remained the same–many parents have been less able to help their children in this area. It will be difficult to pick out what the most easily solvable part of that problem is. I believe that Mathematics curricula should be redesigned to highlight and have students experience the very real everyday applications possible with each added algebraic tool. Students should see at every step how each new skill enables them to more easily solve an actual problem faced by real people.

The abstract nature of Algebra that Hacker bemoans is not limited to that subject. Geometry, which apparently Hacker sees no problem with, is also struck by a similar mix of antiquated skills (bisecting angles or inscribing pentagons by hand with a compass), and abstract logical proofs. The reason why Geometry doesn’t illicit the same reaction from Hacker and educators probably has more to do with the multiple avenues for successful learning and demonstration of skills that exist in Geometry and are lacking in standard Algebra instruction. Unlike the construction of a good argument in a History essay, the answer to an Algebra problem is either 100% right or 100% wrong.

I do agree with Andrew Hacker that we could do better in how we educate students with regard to quantitative disciplines. They should be less distilled into sub-specialties and more intertwined. We need to redesign our Math courses to clearly build on complimentary skills and emphasize the inter-relatedness of Algebra, Statistics, Chemistry, Physics, Geometry, Engineering, Economics, and even Calculus (and sure, Dr. Hacker, even Music, Poetry, History, Art, Architecture, and Philosophy).

Computers and calculators can now solve symbolic problems like (x² + y²)² = (x² – y²)² + (2xy)² as easily as they can perform arithmetic, and have been able to for the past decade or more. Curricula still lag behind this development in most places. Long division (which Hacker says is a requirement) is less emphasized today, but still taught so that we can understand what is happening behind our calculators’ faces, and get by when we happen to find ourselves without one. We still need to maintain the foundations of those skills so we can recognize when we’ve made a mistake in punching things into our solution box. But, most importantly, we all need to understand how these skills are applied in solving real problems. Evidently, long division passes this test with many, but Algebra does not (why would there not be resistance to a subject we haven’t all been taught how to properly use, only how to execute it?).

So, what do we need to do about it? The real issue is that we need to focus Mathematics more on real contemporary problem solving at every level from early elementary through early college classes. Engineers, scientists, statisticians, accountants, financial managers, etc. need to be setting mathematics curriculum in equal shares to the mathematicians.

Most of us will not be mathematicians. Most of us do not need to become scientists, engineers, or computer programmers, either. But, it is also clear that we live in an increasingly complex world. One where generalizable skills can prepare us better for future success than training on a particular model of robotic machine would be. Sure, we could learn enough to get by in that factory without much foundational understanding about how the underlying computational systems work, but one might also get by without being fully literate in English and without having learned to analyze the themes in the works of Charles Dickens.

In the world where Algebra and the skills therein are learned by only a few, how would we not end up increasingly at the mercy of the labor saving systems and their creators when they are supposed to serve us. The choices we have to make as a society involve questions of statistical economic models and mathematical functions governing our energy supply. We don’t all have to solve these problems personally, but we should be able to recognize when we’re being taken. Those critical quantitative reasoning skills we all acknowledge modern citizens to need do in fact require algebra, even if we choose to no longer refer to the subject with that name. Our culture is a technological one, and Mathematics is an integral aspect of that culture. In order to fully participate in our society we need to understand Math as much as we need to understand the Constitution, History, Biology, and English Literature. We can do better by the students of tomorrow. We have to.

What Makes a Free Market?

At talk of free markets our minds often quickly jump to taxes, import quotas, regulations, price controls, “picking winners and losers,” and other policy debates that dominate our contemporary decision making. However, the media (and therefore, generally the public, too) often fails to consider the essential points about what defines a truly free market. So, we are left without the means to accurately evaluate the situation and think about what the potential benefits may be and how we might get there. What follows is not an economics treatise but a set of guidelines or rules of thumb for determining the validity of free market-based arguments in the policy sphere.

Why Are Free Markets So Good?

The short answer is that free markets produce the most goods or services at the lowest possible price. So, the maximum number of people can use the good or service (increasing standard of living), and they will pay the lowest possible price (reduce or contain the cost of living). In fact, since many buyers would be willing to pay more, there exists what economists call a “consumer surplus,” the total amount of money buyers saved from what they would have been willing to spend. This is an idealized situation, one developed over a few centuries of economic thought and based on simplified mathematical representations of the way individuals make choices of what to buy for themselves. But, nevertheless, economists have found when studying actual people that these theories get remarkably close to what actually happens.

Now, let’s consider some of the key features of what makes up a free market, and what some of the most common market breakdowns are. These breakdowns (sometimes called “market failures”) are relatively common, we will see, and mean that economically free markets and laissez faire or unregulated markets are not at all the same thing.

1. Lots of buyers and sellers

This principle of a free market means that no one and no small group controls the price, and that any potential to meet a demand by selling a good or service will be met by one competitor or another.

This principle usually breaks down because there are not enough sellers. Digital or cable television providers or cell phone service providers typically find themselves in this position. With only a small number of sellers, prices can be expected to be higher (and thus the number of people with access to the service, lower) than if there were greater competition. Sometimes, there are only a few buyers: the U.S. government greatly dominates the purchase of all military equipment; or, in the future, the majority of all books may have to be sold through Amazon and Apple. All of these issues whether they are monopolies or oligarchies are one or more steps away from true free markets with perfect competition and reduce the benefits that society might have achieved otherwise.

2. Perfect information

The principle of perfect information sounds impossible, but it essentially means that both the buyers and sellers have equal amounts of information about the product they are trading. This implies that there are no sellers pushing low quality goods on unsuspecting buyers (like the used car salesman trying to unload a lemon). This also refers to equal information about the future. For example, this could imply that sellers do not know of a potential supply disruption before the buyers and vice versa.

A breakdown of this principle in the trading of stocks and bonds is known as insider trading, and is criminalized in many places, although difficult to enforce. Regulations on labeling of foods and pesticides among other things are also attempts to prevent the unequal information basis between the buyer and seller. When buyers and sellers have unequal information, the price will not be set at the optimum value, and the free market breaks down. While this usually benefits the seller, there are cases where legal limits (like anti-price-gouging laws in the wake of a disaster) prevent sellers from acting on information that the buyers are free to consider and act on.

3. All costs of the transaction are included in the price the buyer pays

Costs that are not included in a transaction are called externalities. The most commonly discussed externalities involve environmental effects from one property holder to another property holder. If a farmer in the upper Mississippi basin buys more fertilizer than they need and after a rain storm that fertilizer washes off into the river, they have borne the direct cost of the wasted fertilizer. However, that fertilizer combined with the runoff of others eventually creates a “dead zone” in the Gulf of Mexico reducing the numbers of fish and shrimp. That portion of the cost is borne by fishermen and shrimpers who lose income, and it is borne by consumers of Gulf fish and shrimp who pay higher prices. Those secondary costs are not included in the price paid for fertilizer and so are considered externalities.

In some economic theory (though not necessarily practice), most taxation is an attempt to rectify externalities by transferring the additional costs of the transactions from the external cost bearers to the original buyers. This, for example, would mean taxing fertilizer and giving the revenue to Gulf fishermen. The problem in this case is that it creates something called “deadweight loss“, which is another type of distortion away from the ideal free market, although presumably less than would have occurred with the full external cost being incurred.

With a little additional stretch of the imagination, one can also imagine that externalities include positive feelings like giving charity to good causes, or to put it more basically, sharing some of your own resources with a neighbor who has less. While there may be higher morals or principles involved on a personal level, this is how economists understand the value we place on the well being of others: “if you gave away 10 dollars, then you must have received a psychological benefit greater than what you would have received had you spent that money on yourself.”

Why Most Markets Are Not Free?

Unfortunately, most markets are not free. Many of them are close enough, however, that free-market-like behavior dominates even though the specifics fail the test at one or more points. However, it should also be clear on reflection that laissez faire or the policy of not regulating markets often falls short in creating a beneficial, economically-free market. These free markets depend on certain social institutions like fair and independent courts and other forms of ensuring every participant plays by the rules. They do not necessarily arise organically on their own.

Many monopolies arise organically as smart business owners expand and buyout competitors. In other markets, monopolies arise because barriers to entry (such as large infrastructure investments) prevent new companies from being formed. In these cases sellers can maintain prices at a higher level, ensuring customers pay more, not as many customers can afford the product, and benefits are not maximized. Unfortunately, free markets in the areas of “natural monopolies” like water supply are sometimes inevitable, meaning that certain goods and services will be forever beyond the reach of free markets. On the other hand, patents, copyrights, and trademarks are artificially created monopolies that reveal an inherent conflict between the general market consumer and creative developers.

Inequalities in information are notorious among auto mechanics and their customers, and also, some say, among patients and their doctors. Insider trading is prosecuted, but probably only rarely considering how difficult it is to know what information is known by whom at any point in time. The most successful regulation in this area has been labeling laws for foods, medicines, and cosmetics, ensuring that consumers know what is actually in the products they purchase. Further development in this area is being made possible through public data portals and other forms of internet data sharing, and may result in a significant advance.

Externalities are an incredibly difficult problem faced by the globalized economy. Air pollution crosses provincial and national lines. The choices of one’s neighbors affect one’s own property value. Small changes in land use across a city (like 10% fewer trees or 20% more paving) can drastically alter flooding risk. Employers who do not offer health care in the US increase the tax burden on other local residents and businesses. In these areas, we typically find the most government intervention, although not always in the most free-market-promoting fashion. Whereas free market economists would usually recommend dynamic consumer-based taxes with revenue being used directly to compensate for whatever externalities exist, instead we typically see governments employ prohibitions and other rules that are easier to manage.

What To Do About It

This is a normative question, and one where philosophical values matter as much as the scientific analysis. Analysis can tell us when markets are not free, what the potential benefits may be to making them freer, and what the potential impact of different policy choices may be on the markets. Analysis cannot, however, tell us whether how we ought to value individual and social rights and responsibilities, nor how certain means for rectifying problems in the markets may violate some of those individual or collective rights not to mention moral or ethical standards. For those, we are left to negotiation and compromise between each other to reach a decision.

Supporters need to keep in mind that while free markets are legitimately the ideal solution to many problems, those markets require a certain amount of creative regulation and rule enforcement in order to exist and achieve the benefits promised. And detractors should reconsider how their criticisms may actually be reflective of market failures like those listed above and not intrinsic weaknesses of essential market dynamics.

First, What Are Microwaves?

Microwaves are photons or waves of light (like visible or ultraviolet light) with wavelengths between 1 millimeter and 1 meter (visible light has wavelengths between 390 to 750 nanometers) as you can see in the spectrum shown below. Different wavelengths of photons or light pass through different materials to varying degrees. For example x-rays pass through most materials except metals (they appear bright in x-ray negative pictures at the doctor’s office). Visible light passes easily through glass, but infrared and near infrared light (heat) does not pass through glass as easily (hence greenhouses stay warm). Microwaves can pass through many materials, but are effectively blocked by metals and water.

And That’s Radiation, Right?

Microwaves are a form of radiation as scientists and engineers describe it. However, in common lay terms describing radiation, microwaves are different in that they are more closely related to heat and radio waves (just beyond the infrared) and they are less similar to ultraviolet light or x-rays (at the higher energy end of the spectrum). When most people express a concern about radiation, they are referring to ionizing radiation. This includes alpha and beta particles together with x-rays, gamma rays, and cosmic rays, and also ultra-violet light. When these forms of radiation strike molecules directly, they can sever the molecular bonds (thus creating ions).

Scientists and engineers will use the term radiation to refer to any transmission of energy across space (that is energy not conducted by touch or convected by a moving gas or liquid). So, this meaning includes “non-ionizing” radio waves, microwaves, heat from the sun, and visible light as well as “ionizing” x-rays, gamma rays, and cosmic rays.

Ionizing radiation does indeed cause harm. Ultra-violet light causes sunburns and with repeated exposure can increase the risks of skin cancer. X-rays are more powerful can even more directly damage DNA and cause cells to become cancerous with even less exposure. People who fly frequently for travel can be exposed to higher rates of cosmic rays, which are even more powerful than most x-rays. For this reason commercial pilots and astronauts have maximum exposure limits in terms to total flight hours imposed on their careers for safety reasons.

The final type of radiation is radioactive isotopes. These are specific varieties of large atoms like certain isotopes (specific combinations of neutrons and protons in the nucleus) of uranium or iodine. These atomic isotopes can decay (split into two or more small parts) since they are unstable. This splitting can release ionizing radiation like beta or alpha particles that damage molecules. These are particularly dangerous since they cause their effects internally directly to critical organs in the body, but they need to be inhaled or ingested to seriously affect us.

So, What Happens on Exposure to Microwaves?

Microwaves exist on the energy scale in between radio waves that pass through most non-metal materials, and infrared waves that are absorbed by most materials. Microwaves are absorbed by water molecules (and to a lesser extent fats and sugars), and which then heat up increasing their temperature in proportion to the amount of energy absorbed. This interaction is called dielectric heating. This effect is much more energy efficient that the conduction and convection heat transfer associated with a stove or oven, which is why microwaves can boil water much faster than those other methods.

If your body were exposed to significant amounts of microwaves, the water in your cells near the surface (they do not penetrate far, since you are mostly made of water) would heat up in proportion to the amount of energy supplied. This effect is immediate and is easily felt by your nervous system. That is, it would burn as if you had touched something hot. Low levels of microwaves like 1 Watt cell phone transmitters do technically heat your skin too, but the amount of energy is much less than the amount of heat absorbed by your skin on a sunny day (which is at least 100 times more), or even the amount of heat passing through your skin during exercise (which can be 20 times more).

If you were exposed to the emitter in a microwave oven it would hurt, a lot. It may even cause superficial burns to your skin if you had the pain tolerance to keep yourself in range of it–most people don’t. In fact, the military has been investigating just this effect as a non-lethal crowd dispersal weapon.

Exposure to microwaves from a microwave oven is very rare because the ovens include a Faraday cage, a box of metal mesh that completely absorbs the microwaves. You can see this mesh when you look through the glass door of the oven. The metallic mesh is designed such that it absorbs the microwaves of the particular wavelength that the oven emits. Microwaves do not turn corners or leak past the mesh the way that chemicals move around. From the perspective of a microwave photon, that mesh window looks like an opaque solid; it does not pass through it.

How Can We Be Confident That Doesn’t Cause Cancer?

Due to what we know about the probabilistic risks of cancer in general, we can compare the rate that cancer arises in one group compared to the rate cancer arises in another group. Most exposures to carcinogens do not directly result in cancer, they only increase one’s risk (the probability that they will develop cancer). This is the reason that we can say for certain that smoking causes a certain level of cancer in society overall, but cannot say for certain whether an individual’s cancer was caused by their smoking (some smokers never develop lung cancer while some nonsmokers do develop lung cancer).

The conclusions of research in this area when reported in the press can be confusing, although some have tried to rectify that. The overall conclusion as the research stands today is that there is no grounds for special concern that microwave radiation from cell phones causes an increased risk of cancer.

The ideas that microwaves might cause cancer come from a few different lines of thought. First, concerns exist that microwaves from a cell phone or leaks from a microwave oven might damage DNA and other biological molecules. This makes intuitive sense because we already know that ultraviolet light can damage important cellular molecules and that x-rays are carcinogenic if received too often. But, the key here is that each photon or particle of light must have a certain amount of energy to damage molecules. Just stacking up lots of photons may cause burns due to the total amount of energy, but not result in molecular damage that would lead to cancer. As this study shows, the transition point is directly between ultraviolet wavelengths and visible blue wavelengths. Microwaves which have less energy per photon than any visible light photons, so not even approach this level. We can conclude that based on physics, microwaves are not going to be causing molecular damage to our DNA.

Second, even if the individual microwaves cannot damage DNA, might prolonged low-level exposure result in some other unanticipated effect? Shouldn’t we be more precautionary? This kind of question seems scary in the sense that if there was a small effect that only appeared after years and years of exposure, we would not have sufficient data to refute it. We might never be able to run a study large enough and long enough to answer that question with hard data. But, before justifying even trying to collect information, we need to come up with a plausible mechanism that may cause the kind of damage we know leads to cancer. Low levels of warming in skin tissues has never been proposed as a potential cause of cancer. In fact, there are no existing testable hypotheses on how this entirely new cancer causing mechanism would operate. If one develops, lab tests would be in order before any additional investigation. Until such a theory has been developed, however, it would be unwise to devote large amounts of resources, when there are other much larger risks which could be addressed with those resources.

Third, there is a less well traveled belief that microwaves might make food and beverages less safe or even radioactive themselves from their absorption of microwave energy (from hence our colloquial term “nuking” comes). Something like this could be true if we were bombarding the food with alpha particle radiation, but microwaves cannot damage molecules by any mechanism other than heating them. They certainty do not have the physical capability to change the structure of atoms and make them radioactive. Microwave photons do not have sufficient energy (regardless of the oven’s wattage) to ionize biological molecules and create something dangerous. If heat happens to make the food unsafe for some reason, any other cooking method would result in the same problem.

Most scientists who work in this area will never say absolutely that anything does not cause cancer. This sounds like hedging to many people, but the scientists see it as being true to the data, and they rarely ever have enough data to be mathematically certain that there is zero risk due to a given exposure. But, we can say this, if mild heating due to microwaves were to cause cancer, it must be by a wholly new and hence forth undiscovered mechanism of carcinogenesis. To demonstrate that a new cancer causing mechanism exists would require extraordinary evidence. The likelihood of such information appearing now after decades of use of these devices is rapidly vanishing.

So, What About Those Pacemaker Signs?

Microwave ovens and other powerful microwave transmitters used to pose a risk to pacemakers and some other types of electronics. This problem has essentially been eliminated today. The typical leakage path for microwave ovens is damage to the metal mesh in the window. If this is intact, there will be only insignificant leakage–although there are devices you can use to verify this yourself. Furthermore, the issue of unshielded pacemaker components susceptible to microwave exposure has been eliminated in today’s devices, so even in the event of a moderate amount of microwave leakage, there is unlikely to be any problem.

Microwaves are Good

Microwave ovens are among the most energy efficient cooking appliances in any home. Using microwave ovens when appropriate is a valuable energy conservation and emissions reduction technique, especially when compared to electric stoves or ovens. While the culinary results may not be acceptable for some applications, microwave ovens are ideal for heating water and steaming methods. Cooking is one of the most universal and most inefficient uses of energy; conservation through the prudent use of microwave ovens is a valuable capability.

Microwaves transmitters like cell phone and WiFi networks enable us to have small portable data and communication devices. Radio waves cannot carry as much information as quickly as microwaves, and they require large antennas to detect and transmit them. Near infrared waves are absorbed by almost all building materials and so would require a line-of-sight transmission window with the tower to be connected (most television remotes rely on infrared technology). The modern technologies enabled by the use of these frequencies have changed the world, much of it for the better, and this includes emergency response and other applications that directly save thousands of lives. Future developments will only increase the benefits we realize from these devices.