Malls are weird places. Every time I’m in one, I’m always perplexed at the sheer quantity of pointless trinkets and other useless products. The reality is that if a product or genre doesn’t move, it’s taken off the shelves and liquidated in a junk bin and discontinued. I noticed one kiosk selling something that looked like painted, pig wood carvings wearing hula skirts. Why? I don’t know either, but it follows that enough people want them in their home, business, trailer, or van down by the river to necessitate a business for them.
I guess if your favorite yard gnome is suddenly struck by lightning, and you collapse into a frenzied, emotional breakdown, you may impulsively buy one if the product placement is crafty enough. However, for shoppers not treading the mall following a devastating catastrophe wrought with distress, I fail to understand the pressing urge these buyers must feel when first laying sight on a poorly-made, wooden pig cut-out. That’s okay, though. Who’s decorative taste am I to judge? Just as long as they have a good warranty, I suppose, and are made with lead-free paint.
Speaking of lead paint, then I went into Sharper Image. The store looks like the interior of a riced-out, import spaceship. In fact, China could probably even build a spaceship out of Sharper Image molds and a handful of Suzuki parts. Honestly though, is a water-proof alarm clock that plays lo-fi bird calls really worth $40? How did Radioshack miss the memo that bulky grey plastic warrants a 400% price mark-up? In all fairness to Sharper Image shoppers, though, I think most of their stuff is purchased as gifts.. which begs the question, “why?”
Think long and hard next time you’re thinking about giving a natural, wake-to-parrot alarm clock as a gift. Do you own one? Chances are, you don’t. Is it because you wouldn’t pay $40 for one when you could have put that cash towards something useful? Well, what makes you think your gift recipient would be willing to spend $40 on one?
Let’s say your gift recipient is walking around the mall and he sees this block of shiny, grey plastic which exclaims, “brighten your day with a morning dose of a heartwarming parrot call!” How much would he pay for it? $5, $10, $40, $100, -$3? If he might pay $10 for it, that’s all it’s worth to him. But you spent $40 on the gift! That’s $40 for $10 of value. You just vaporized $30 which neither you nor your recipient can now enjoy. Of course, the scammers at Sharper Image get to pocket $30 above marginal utility and are consequently encouraged to manufacture more parrot alarm clocks to fill more landfills with mercury and lead alongside the ever-present mounds of old AOL disks. Bummer.
If your recipient is a minimalist, he may even be willing to pay someone a few dollars to take it off his hands. You could have just burdened your recipient with a $3 loss and yourself $40, for a net loss of $43 dollars. Perhaps you’d be better off just giving a $40 bill (it’s the one with President Taft).
What about sentimental value? Indeed, this is really what gift-giving is all about. My high-school economics professor took this logic literally and gave his wife a wad of cash in a small, decorative box one Christmas. He explained that he was perplexed that his wife didn’t find utility theory romantic, and was sure to first run gift ideas past his daughter afterwards.
Let’s be honest, though, a parrot alarm clock has no more sentimental value than a small charm, book, or origami whooping crane covered in glitter and rhinestones (assuming you’re not an adult, heterosexual male.. then again, I may have just had a brilliant gift epiphany.) If you really want to give $40 of value, the best gift tactic may not be grabbing the first shiny thing you see, but $39 along with a card.. or, if nothing else, $40 of canned beans should do.
I love thought experiments and puzzles because I think they’re food for the brain. Working through them requires critical and logical reasoning, and can often provide tangible answers to elusive problems. Often times, thought experiments present themselves in the form of paradoxes, where a predicament leads to a contradiction or seems to oppose intuition.
One of the more fascinating puzzles I’ve come across in economics is known as the St. Petersburg paradox. In 1713, during the Age of Enlightenment, a time known for glorifying reason, mathematician Nicolas Bernoulli presented a lottery game with an infinite expected payoff. If something has an infinite expected payoff, one might think that a chance to play the game would be worth any finite price. However, to a rational person, a chance at this game is actually only worth a very small amount.
Anyway, onto the game:
The player pays a fixed fee to enter. Then a coin is tossed until a tail appears, which ends the game. The payoff starts at $1 and doubles every time a head appears. The event probability is calculated by multiplying the probability of a tail appearing for each flip, which is always 50%. So, the probably of a tail on the 4th toss (3 heads, then a tail) is .5 x .5 x .5 x .5, or .0625. (The payoff formula is 2n-1, and the probability formula is .5n)
Payoff analysis- Tail appears on:
1st toss: $1 50% probability
2nd toss: $2 25% probability
3rd toss: $4 12.5% probability
4th toss: $8 6.25% probability
Now, you might be thinking, why would anyone spend more than a few bucks to play this game? Well, watch what happens when taking the summation of the expected payoffs:
Despite the conclusion that most attempts at this lottery will likely pay little, the expected payoff is infinite! This means that if a casino has infinite money, a player should want to play repeatedly at any price, right? A good steward of reason probably won’t, though. Why?
Solutions:
(A solution to a paradox should reconcile the logical conclusions which appear to contradict.)
Nicolas’s cousin Daniel, another mathematician, attempted to solve the paradox by pointing out the obvious nature of diminishing returns. $8,000 does not actually provide 8 times the utility that $1,000 does. Likewise, $1,000 is of much more utility to a messenger than a rich man. Bernoulli depreciated the total utility of a payoff using the logarithmic function. However, in 1728, before Bernoulli presented his solution, Gabriel Cramer, another mathematician, depreciated money’s total utility using the square root function. Using the logarithmic versus square root function seems to be arbitrary, but the point is clear nevertheless. Although $1,000,000 equals 1,000 x $1,000, it is not worth 1,000 x $1,000 to a “man of good sense,” as stated by Cramer.
A diminishing utility limit actually suggests that a go at this game is probably worth no more than $10.
Some critics have argued, however, that there is still a paradox even when negating diminishing marginal returns. From that discussion follows:
THE SUPER ST. PETERSBURG PARADOX!
If we assume the payoff becomes increasingly greater for increasingly unlikely events, such as a player getting $2 million when he would have otherwise gotten $1 million, diminishing marginal returns are simply negated. Still, there are more solutions this paradox. Risk aversion is not a solution to this problem though either, since even larger payoffs can be given for rare events. Perhaps this should be called the super super paradox, but we won’t go there. If we assume that a casino will still offer a lottery with an expected infinite payoff, the only known way to solve such a super paradox is to assume that total utility has an upper bound. In other words, unlimited money cannot provide more than a certain limit of utility to begin with! It can infinitely grow towards that limit, but it still won’t surpass it.
Another solution to the paradox, according to Bernoulli, is that people disregard unlikely events, so they don’t account for them in the first place. However, empirical evidence suggests people greatly exaggerate chance events such as becoming a celebrity or winning the lottery.
One last thing to think about is that if a lottery has an infinite expected payoff to a player, it also has an infinite expected loss to the casino. No casino should then profit from providing it at any fee! Of course, chances are that they would profit from providing it, but it wouldn’t be legal to do so, because no financial institution can back an unbounded payoff under a casino’s worst case scenario. Of course, one can’t buy what isn’t sold, but the nature of utility seems to put a bounded, finite value on the game anyway.. if the mythical alien finance gurus ever offer it.
And that is how infinite money is really finite in worth 
In the past, billionaire investor Warren Buffett has pointed out the non-productive nature of gold. “It gets dug out of the ground in Africa, or someplace. Then we melt it down, dig another hole, bury it again and pay people to stand around guarding it. It has no utility. Anyone watching from Mars would be scratching their head.”1 So, the question follows, how can something that serves little function have value? Does the scarcity of gold make it inherently valuable? Certainly not; scarcity itself does create value. All original paintings are unique. This means that all of them are by definition scarce. However, some sell for millions and others for pennies.
Gold and other precious metals, however, are culturally accepted as symbols of wealth. Human nature is easily influenced by cultural pressure, and many people want to show off by emulating those of higher socio-economic status. It just so happens that gold, perhaps for its flashiness and known difficulty to acquire, has been sought after since the beginning of written history. Gold also acts as a store of value, which means that it can reliably be traded as an item of inherent, predictable wealth. My one of-a-kind-stick figure painting, unless promoted by an impressive advertising campaign, will probably be of little interest to anyone.
Economist Adam Smith’s diamond-water paradox of value highlights the seeming contradiction of a low utility good having a much higher market value than a high utility good. Smith went on to suggest that there are two types of value. One is value in use, which is the value provided from the utility, or benefit, of using that good. The other is value in exchange, which is the price that good commands in the open market. Modern economic thought suggests that a good’s value in exchange is determined by its marginal utility.
The marginal utility of a good is the utility a consumer gains from consuming one additional unit of that good. If I’m hiking with an empty canteen, the utility provided by one cup of water is very high. A second cup won’t provide as much utility as the first, so its marginal utility is not as high. The third cup’s marginal utility is even less, since I might not even drink it right away. If I stumble upon an entire fresh water pond, the marginal utility of a cup of water is negligible since I’m already swimming in water. This is the law of diminishing returns.
For someone without any water, the marginal utility of one cup of water is much higher than one diamond. However, since water tends to be in high supply in the developed world, the marginal utility of an additional cup is very small. Diamonds, on the other hand, are in low supply, so the marginal utility provided by a diamond is much higher than an additional cup of water. Goods which provide higher average marginal utilities, current supply considered, naturally trade for higher average prices in transparent markets.
Marginal Utility- Diamonds vs. Water
Everyone has a different value in use of a typical good. In fact, it’s arguable that value in use is even a perceived value. Everyone has different priorities of wants, which is precisely why demand curves slope downwards. A delusional man might might refuse to sell a thimble for millions of dollars. If he can afford it, he might even drop a million on auction for one. Similarly, everyone values art differently and will derive a unique marginal utility from each piece.
So, if I paint a painting and someone asks me what it’s worth, do I say it’s value in exchange, its average value in use, its highest value in use of a willing buyer, my own value in use, etc.? If an auction has $1 bid increments and bidder A is willing to pay $5, while bidder B will pay $10, the painting will sell for $6. In truth, the painting has all of those values. I could boost my ego, and say “millions,” but not mention “to Joe the former investment banker who’s now staying in the psych ward.” Perhaps a more telling response, however, might be “$10, tops,” or what it would otherwise be predicted to fetch in an open market.
1. Buffett, Warren (1977-05), “How Inflation Swindles the Equity Investor”, Fortune