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The Utility of Money
by Mark Lawrence
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Are all dollars created equal? Yes and no. In the economy, all dollars are the same. You get the same amount
of money for your 40th hour of working in a week as you get for the 1st hour. If you pay $100 for a coat, the 100th
dollar has the same value as the 1st dollar.
However, this does not mean all dollars have the same value to you. We all have a life style which costs a certain amount
to maintain, and we are all loath to give up our lifestyle. Let's say, for simplicity, that your lifestyle costs you $3000
per month to maintain - that is, for $3000 per month, you can pay your rent and car payment, eat, buy gasoline, electricity,
gas, and replace clothes and other articles as they wear out. Let's say you make $3300 per month after taxes. We'll also
assume you have no savings. You have $300 per month in disposable income. You may spend this $300 in any way you wish
without threatening your lifestyle. However, if you spend $400 some month on entertainment or travel, you will not have
enough money to pay for your entire lifestyle. Something is going to get short changed. If you keep this up, in the long run
you will lose your car or house.
Because of this, most people are very unwilling to gamble once they have run out of disposable income. You may decide one
month to go to Las Vegas or Atlantic City and drop $300 on roulette, but after the $300 is gone most people will pack up and
leave. In a few paragraphs we'll also talk about the exceptions, the people who stay. Those who leave are said to be
risk averse.
Let's imagine you are approached by a stranger who offers you a bet. You pay $3000, and he flips a coin. Heads you win
$7500, tails you lose your $3000. Mathematically, this is a great bet: if you can make this bet 20 times, for example, you
most likely will earn a total of $15,000. However, you are most likely to turn down this bet. If you lose $3000, you will be
tossed out of your house and lose your car. Your lifestyle will be demolished. If you win, you have an extra $4500, which is
very nice, but it's not enough money to change your lifestyle, it's not enough money to compensate you for the risk of being
homeless. What we see here is your utility curve. Losing $3000 is far worse than winning $4500 is good, the $3000 loss has
far greater impact on you than the $4500 win.
Is this always true? Are all people risk averse? No. Some people are attracted to risk - we call them risk perverse.
First, there are gamblers, people who are addicted to the adrenalin rush of winning and losing. More importantly, there are
other people who are risk perverse. We see them at the extremes of incomes.
In any society there is a basic amount of money needed to maintain a basic lifestyle. In N.America this amount depends on
where you live - it's far more expensive to live in Manhattan than in Dayton, Ohio. For simplicity we'll ignore this
difference and just say it costs $1000 per month to have a basic life, with a small apartment, an older car, and some food
to eat. If you earn $1100 per month, you are very close to the edge and you would be best advised to be extremely careful
with your money. If you earn $900 per month, your situation is hopeless: you simply cannot pay basic rent and own a basic
clunker car. In this case you're going to be thrown out of your apartment on a regular basis, then have to live with friends
or in your car or something for a couple of months until you can save up enough money for another apartment. Which you'll
again be tossed out of in a few months. In this case, you have little to lose. If you lose $100 in a bet, you still don't
have enough money to maintain a basic lifestyle, all that changes is maybe you'll wind up thrown out of your apartment a
month earlier than normal. If you were to win $1,000,000, however, you could break this cycle of poverty forever, or at
least so it would seem.
This is how the lottery works: people who have nothing to lose are promised the chance to change their lifestyle forever.
It doesn't matter to them that they pay one dollar for one chance in three million to win a million dollars. The fact that
the bet is a mathematically horrible bet is completely beside the point, the major point is this is perceived to be their
only chance to escape poverty. It is precisely for this reason that lotteries have huge pay outs. The promise of lifestyle
change is what fuels the system. There are also a few chances to win $5 or $100, this is bait to keep the gambling instinct
working ("I nearly won, 5 numbers out of 6!!!"). The lottery makes use of the fact that at the very lowest income levels of
our society, people are known to be rationally risk perverse - they are attracted to risk that makes no sense on a simple
mathematical level. This attraction is rational in the sense that what they risk, $10 worth of lottery tickets, has
disproportionately less value to them than their possible winnings, several million dollars that lifts them permanently out
of poverty.
People at the very highest levels or wealth can also become risk perverse. Donald Trump is an excellent example of this.
He has enough money and financial stature that his lifestyle is never seriously at risk. Mr. Trump has been known to say
"borrow $500,000 and the bank owns you. Borrow $500,000,000 and you own the bank." He has little to lose. On the other hand,
making a big win might loft him into the rarified atmosphere of the 10 richest Americans. So, just as people just above the
poverty level have much to lose and little to gain in a gamble, people who are very far fom the poverty level have little to
lose and perhaps much to gain in a gamble. We will often see people in this category make what appear to be poor investment
choices, poor in the sense that the risk outweighs the payoff. Why? If they lose, no big deal, they just have $250,000,000
in the bank instead of $300,000,000. If they win, they might join the billionaires club and get to meet Paris Hilton at a
party.
We can graph this utility function. We can't put precise numbers on the axis - that depends on the individual, their
current and past situation, and their future expectations. We can say a lot about the shape of the curve, however. If you
have no money, the value of a dollar is very high - a dollar means you can eat something. As you earn more and more money,
the value of each subsequent dollar goes down. Your first $5 buys you a good meal and means you're no longer hungry. Your
second $5 buys you a better meal, or perhaps another meal tomorrow, neither of which is quite as valuable to you as the
first meal while you're hungry. As we've seen, when you're talking about a few thousand dollars, the value of taking in the
money is not nearly as high as the value of losing the money. However, at the highest levels it would seem that the curve
turns up again: at some point your lifestyle is assured, you cannot realistically lose, and winning becomes a big ego
gratification. So, the utility of money curve for most people looks something like this:
A typical utility curve for a person
In the chart above I've shown the upturn at the far right of the curve. This upturn is at different points for different
people. For Donald Trump, the upturn is in the tens or hundreds of millions of dollars. For a person at the poverty line,
it's more likely around a thirty thousand dollars - enough to, say, buy a good new car. It's this upturn which means that
one chance in three million of winning a million dollars is worth more than a dollar. Without the upturn, you cannot get
this result. Is everyone's curve like this? No, I expect the Dali Lama has little interest in more money than it takes to
buy his next meal. For various philosophical reasons, some people would say that many millions of dollars were actually
worth less to them than a much smaller amount of money. But these people are vary rare, and so capitalism continues on. For
a compulsive gambler, perhaps the opposite is true: his upturn is almost at the beginning of his curve, so there's no
effective limit to the amount of money he wants. Always, one must judge a person's utility curve by their actions, not by
their words. If a Hollywood film star tells you they're a Buddhist and have no use for money or fame, you may rightly
question the veracity of this statement.
In textbooks people will often try to find a functional representation for this curve. Popular functions include log(
money ) and square_root( money ). These functions are easy to calculate and work with, but they fail to capture the complete
richness of the real curve. They're static in time and situation, which is probably inaccurate for people. They don't show
the very high utility of your first few dollars, And they don't show how for most people the utility function turns up again
at the highest levels.
This turn up in the utility curve represents the driving force for many of the worst excesses in capitalism. In
particular, the implied mindless greed is responsible for stock market bubbles like the Internet bubble of the late 1990's,
and for many of the very poorest business investment decisions made. Our society would be far more conservative in its
investing if the income and wealth were more evenly distributed. On the other hand, our best in the world economic growth
rate almost certainly is also an outcome of the upturn in the utility graph: rational risk often leads to high payoff.
