In 1776, Adam Smith described a puzzle: “Water is cheaper than diamonds. Does this mean water has lower economic value than diamonds?” Economists call this puzzle the diamond-water paradox or the paradox of value.
Modern Americans have come up with a remix of this old puzzle: “Teachers are paid less than pro athletes. Does this mean teachers have lower economic value than pro athletes?”
In economics, there is an old joke: “Teach a parrot to say supply and demand and you’ve got an economist.” To the parrot, the explanation of the puzzle is simple. For water, demand is great, but supply is even greater. And so water is relatively abundant, which explains why it’s so cheap. For diamonds, demand is small, but supply is even smaller. And so diamonds are relatively scarce, which explains why they are so expensive.
But this explanation is incomplete. It only explains why water is cheap and diamonds are expensive. At the heart of the puzzle remains an unsolved mystery: What exactly is the relationship between economic value and price?
Water has high economic value, but a low price. Diamonds have low economic value, but a high price. So, are economic value and price inversely related? Obviously not.
But if not, what exactly is the relationship between economic value and price?
First of all, what is economic value? The economic value of an object is whatever the consumer is willing to pay for it.
Economic value is subjective, meaning it depends on the context and the person. On a hot day, Ann is willing to pay as much as $3 for an ice cream. But on a cold day, Ann is willing to pay only $2. In other words, for Ann, an ice cream has a value of $3 on a hot day, but only $2 on a cold day. The value of an ice cream depends on the context, in this case the weather.
Value also depends on the person. On a hot day, an ice cream has a value of $3 to Ann, but only $1 to Bob. In other words, Ann is willing to pay $3 for the ice cream, but Bob is willing to pay only $1. Who’s correct? What’s the true, objective value of an ice cream on a hot day? Is it $3, in which case Ann is correct and Bob is wrong? Or maybe it’s $1, in which case Bob is correct and Ann is wrong? Nope. Value is subjective. As economists say, “There is no arguing over tastes and colors.” If Ann prefers red to blue and Bob blue to red, both are correct. Likewise, if Ann says an ice cream is worth $3, while Bob says it’s worth only $1, both are also correct.
Next, we should carefully distinguish between four specific measures of economic value: Initial value, marginal value, total value, and average value.
To illustrate, say it’s lunchtime and Chris is starving. She values the very first apple at $9. In other words, she is willing to pay as much as $9 for the first apple. Having eaten the first apple, she is no longer starving and so she values the second apple at only $5. After the second apple, she is even less hungry and so she values the third apple at only $4. After the third apple, she is mostly full and so she values the fourth apple at only $1.
Suppose the price of an apple is $2. Then Chris is happy to buy the first 3 apples, because their values exceed $2. But she will not buy the 4th, because its value is less than $2. Altogether then, Chris buys exactly 3 apples.
Initial value refers to the value of the very first unit of a good purchased. So for Chris, the apples have an initial value of $9. Marginal value refers to the value of the very last unit purchased. So for Chris, the apples have a marginal value of $4. Again, initial value refers to the first unit purchased, while marginal value refers to the last unit purchased.
Total value is simply 9 plus 5 plus 4 dollars, or $18. Average value is simply total value divided by the number of units consumed — that’s $6.
When people say that water is obviously more valuable than diamonds, they are probably thinking of initial and total value.
Ordinarily, 1 kilogram of water is much cheaper than 1 kg of diamonds. But in a world without water and diamonds, 1 kg of water could extend a person’s life by a few days, while 1 kg of diamonds would do diddly-squat. And so, we’d certainly pay more for that very 1st kg of water than for that very 1st kg of diamonds. In other words, water has higher initial value than diamonds.
Water also certainly has higher total value than diamonds. In a world without water and diamonds, we’d rather recover all our water than recover all our diamonds.
Altogether then, water has higher initial and total value than diamonds. So, why then is water cheaper than diamonds? The answer is that surprisingly enough, price does not depend on initial or total value. Instead, price depends on marginal value — the value of the very last unit purchased.
As the great economist Paul Samuelson put it, “The theory of economic value is easy to understand if you just remember that in economics the tail wags the dog.” Price depends on a seemingly-unimportant factor, namely marginal value.
The paradox of value vanishes once we carefully distinguish between the four measures of value and understand that, surprisingly enough, price does not depend on initial, total, or average value. Instead, price depends on marginal value.
Water has immense initial and total value. But it has a low price because it has low marginal value — there is so much water that the very last unit of water has little value. Diamonds have much lower initial and total value than water. But they fetch a high price because their marginal value is high — there are so few diamonds that even the very last unit of diamonds has great value.
The question of value confused even some of history’s greatest thinkers. And so it is not surprising that even today, it continues to confuse and confound.
Now, no drop of water ever complained about being paid too little. Nor has anyone ever been morally outraged by the low price of water. The same cannot be said of teachers. Which is why the modern American version of the puzzle is not just confusing; it’s controversial.
When people say that teachers are more valuable than pro athletes, they are probably thinking of initial, total, and average value.
The US has about 4m teachers and 12,000 athletes. Imagine an alien kidnaps all of them.
Recovering one teacher could help preserve and transmit some human knowledge. In contrast, recovering one athlete would do diddly-squat. And so, the very 1st teacher is more valuable than the very 1st athlete. In other words, teachers have higher initial value than athletes.
Teachers also certainly have higher total value. We’d pay more to recover all 4m teachers than to recover all 12,000 athletes.
Indeed, it is quite plausible that we’d pay over 400 times as much to recover the teachers. Which would mean that teachers have a total value that is over 400 times that of athletes. If so, teachers would also have higher average value. This is because average value is total value divided by the number of teachers or athletes and there are only about 350 times as many teachers as there are athletes.
Altogether then, teachers have higher initial, total, and possibly average value. But once again, it is the tail that wags the dog. Price does not depend on any of these measures of value. Instead, price depends on marginal value.
If the alien kidnapped only one teacher and one athlete, we’d probably prefer to recover the athlete than to recover the teacher. This shows that we value the very last athlete over the very last teacher. In other words, athletes have higher marginal value. This, however, says nothing whatsoever about initial, total, or average value.
Teachers may be paid less than athletes. But this tells us no more about the downfall of civilization than the fact that water is cheaper than diamonds. It is perfectly possible that society places greater initial, total, and average value on teachers and yet still pays teachers less, simply because teachers have lower marginal value.
Now, isn’t there something terribly unfair, unjust, and immoral about this? Maybe. But consider this. A world in which water and teachers were paid more than diamonds and athletes, would also be a world where instead of water and teachers being relatively abundant, they were relatively scarce. Such a world might be bursting with fairness, justice, and morality. But it would also be filled with misery.
The low marginal value of teachers and water is a blessing rather than a curse. We are blessed with so many teachers and so much water that the very last teacher and the very last unit of water are simply not very valuable.
So, just remember, in economics, the tail wags the dog. Price depends on marginal value — the value of the very last unit purchased.
In this video, our goal was to make a simple and purely theoretical point about price and value. And so we didn’t bother looking at any empirical data. But if we did, here are three quick findings we’d make.
First, in the US, the median annual wage for teachers is about $54,000. In contrast, for athletes, it is only about $45,000! If this seems unbelievably low, that’s because we only ever talk about the top athletes in the top sports leagues.
A LeBron James in the NBA may be paid over $30m a year. But over in the NBA Development League, players get either $19,500 or $26,000 a year! And this is after the most recent pay increase! (Lol.)
Second, most parts of the US still have the so-called tenure system, under which teachers enjoy lifelong employment. In contrast, pro athletes enjoy no such protection. Indeed, athletes usually have very short careers in the top sports leagues. So, even if some do make big bucks, they often only do so for a few years.
Third, the four major US sports leagues have combined annual revenues of around $32 billion. Sounds like a lot, but compare this to US spending on educational institutions, which comes to about $1 trillion a year.
Simply based on these three quick findings, it would hardly appear that Americans value sports over education.
 The word paradox is not used in the sense of a logical paradox (e.g. a self-contradictory statement like “this statement is false”). Instead, paradox is used in the sense of a puzzle that contains elements that merely seem contradictory at first glance, but cease to be so, once we’ve fully understood the matter. It is in this sense that mathematicians talk about Zeno’s Paradox or physicists talk about the Twin Paradox.
 To be fair to Plato, Aquinas, Smith, Ricardo, and Marx, it is debatable as to the degree to which they were “confused” about the question of value. Nonetheless, according to traditional histories of economic thought, it was only from the 1870s (the “marginal revolution”) that economists gained a truly firm grasp on the question of value. Note that the paradox of value itself had already been solved several times, even before Adam Smith (see e.g. Schumpeter). Which is why I was careful to use the wording “question of value” here.
 According to the US Bureau of Labor Statistics, May 2015 Occupational Employment Statistics data, estimated total employment for “Preschool, Primary, Secondary, and Special Education School Teachers” (code 25-2000) is 4,080,100. That for “Athletes and Sports Competitors” (code 27-2021) is 11,710.
 According to the US Bureau of Labor Statistics, May 2015 Occupational Employment Statistics data, the median wage for “Preschool, Primary, Secondary, and Special Education School Teachers” (code 25-2000) is $53,860. That for “Athletes and Sports Competitors” (code 27-2021) is $44,680.
 LeBron James is scheduled to make $30,963,450, $33,285,709, and $35,607,968 in the 2017, 2018, and 2019 seasons, respectively (Basketball-Reference.com). LeBron has a player option for 2019, meaning that for that year, he can opt out and renegotiate for a higher salary.
 Previously, “there were three tiers of salaries: $13,000 (C), $19,000 (B) and $25,500 (A) … Now there will only be two tiers for salaries as the ‘C’ has been eliminated. The new salary levels will be $19,500 (B-Level) and $26,000 (A-Level)” (source).
 According to Forbes, “the NFL’s total revenues are projected to surpass $13.3 billion” in 2016, the MLB “will enter 2016 with revenues approaching $9.5 billion”, the NBA’s “30 teams generated $5.2 billion in revenue” during the 2014-2015 season, and NHL “revenue averaged $133 million per team for the 2014-15 season” (there are 30 teams, so that’s about $4 billion).
 In 2013, the US spent 6.2% of GDP on educational institutions (OECD Education at a Glance 2016, Table B2.1 on p. 205). In 2015, US GDP was about $18 trillion, so 6.2% of that is about $1 trillion.
A few notes for educators
- As with most of my videos, this video is for a general audience, and assumes the viewer has no prior knowledge of economics. Nonetheless I hope it can serve as a useful supplement to those studying introductory economics.
- In the present context, I prefer to use the word value rather than the more customary utility. For some rationale, see e.g. Dittmer (2005).
- In most introductory textbook treatments of the diamond-water paradox, the only distinction that is drawn is between marginal value and total value. …
… But this, I believe, fails to do full justice to the paradox. When laypersons say that water is more valuable than diamonds, they are not just thinking of the total value of water. They are also thinking of the fact that without water, we’d be dead. The first few units of water are immensely valuable — water has immense initial value of water.
And obviously, the total value of teachers exceeds that of pro athletes, if only because of the sheer number of teachers. So when laypersons intuit that teachers are more valuable than pro athletes, they are thinking of more than just total value. Many laypersons probably feel that even on average, teachers are more valuable. Hence the average value concept.
Also, for the sake of brevity, this video …
- Asserted that price depends on marginal value, without explaining why .
- Focused on only one blade of the Marshallian scissors (namely, demand or value) and entirely neglected the other blade (supply or cost).
- Made no explicit mention of the concept of diminishing marginal value (though this was implicit in the apples example).
I hope to remedy these omissions in future videos.