### The value of a dollar depends on who you are: Why we all need to understand utility functions

How much is a dollar worth? (Forget about exchange rates and inflation for a minute.) It’s a funny question because we’re used to measuring the value of other things in dollars, so we have a tendency to assume that the value of a dollar is thus a rigid scale, an invariable standard for comparisons. It’s like asking, “How long is a meter?”

One way to answer it is to inverse the relationships: a meter is 0.54 Alan-heights, for example. A dollar in about 1 candy bar, or 0.67 Starbuck’s coffees. And this is true as far as it goes, but it doesn’t get us very far. A meter is a more useful general measure of length than Alan-heights, and a dollar is a more useful measure of value than candy or coffee. But the problem here is that dollars are supposed to measure value, and value is a tricky thing to pin down. Length is pretty darn invariable (unless you happen to be traveling at near-light speed), but value is subjective. Subjective, but important: most of the fields of economics and finance, as well as much of government, is devoted to trying to generate value for individuals, companies, and society. And the problem is that a dollar (or a euro, or a yen…) is actually a very bad measure of value: in statistical terms, it not only has a lot of variance (i.e., measurement noise), but also a systematic bias. While measurement noise can often be worked around, bias is usually much trickier. And that, it turns out, is something that is hugely important but underappreciated, even by economists…

Bias in the value of a dollar

Imagine you walk into Starbuck’s and you’re waiting in line for a coffee. Right in front of you there’s a homeless man with a crumpled dollar bill and a few coins carefully counted out for the full price. He starts talking to you, and tells you that he’s about to meet his long-lost daughter. The last time they saw each other was at a Starbuck’s 10 years ago, and he thinks that if he brings a cup of Starbuck’s coffee to their meeting it will help him repair the relationship. And then right behind you – surprise! – there’s Bill Gates, talking on his cell phone, saying he’ll just grab a quick cup of coffee an be out in a minute. From these two exchanges, it’s clear to you that the cup of coffee has a much greater value to the homeless man than to Bill Gates.  Now imagine this happens to be an unlucky day for coffee – supply chains have been disrupted, and this Starbuck’s is down to its last cup, so the barista, seeing Bill Gates, decides he’ll auction it off and see if he can make some extra money. Even though the homeless man wants a cup of coffee much more than Bill Gates does, Bill Gates is willing to pay a lot more for it than the homeless man, perhaps even \$1000, if Warren Buffett walks in and they’re both in a hurry for coffeee…

Thus we see that the value of the cup of coffee is not the same for all people (no surprise) and, more importantly, that the value to us is not always a good indication of the price we’re willing to pay. Rich people will generally be willing to pay much more for things they value much less, and this means that, as a measure of value, currency has a systematic bias toward being worth less for rich people. If you were trying to measure the heights of different people and you found out that your ruler shrank when you used it on rich people and expanded when you used it on poor people, you’d pretty quickly conclude that the ruler was not a good way to measure height! Nonetheless, we continue to use currency to measure wealth and value, because we don’t have good alternatives…

Noise in the value of a dollar

We think that dollars are a good measure of value because we interact with them all the time and we see that they get relative relationships right. A house is worth more than a car is worth more than an iPod is worth more than a steak, and a Ferrari is worth more than a BMW is worth more than a Kia. But we also know that dollar value is not always a good measure of what something is worth to us: I would rather have my Toyota Yaris than a Ferrari, so (if we ignore the potential for resale) the value of a shiny new Ferrari to me might be about \$2000 – the price I’d be willing to pay for it. Here in Quebec, I could buy one new from \$180,000. This represents a lot variance is the value of a given item across individuals. Nonetheless, the price of a Ferrari may still be a good average estimate of the value across individuals, once the costs of design, production, and so forth have been incorporated (and thus once all buyers like me, who would not be willing to pay the production costs, are excluded).

So different individuals value different items differently. No earth-shattering news there. We’re not really as interested in the price of specific items as in the value of a hypothetical dollar. Is this value always the same? Maybe I don’t want a Ferrari because I don’t like fast cars, but if I had \$180,000 to spare I would get just as much value out of it by buying luxury foods (or donating it to my favorite charity) as the car enthusiast would get out of buying the Ferrari. Possibly, but I don’t think so. Some people just like being rich more than others, and people who like being rich will value a dollar more than those who don’t, all else equal. The amount a dollar can buy (averaged across all goods and services) should be a pretty good estimate of the average value of a dollar to different people with different amounts of desire to be rich. So the value of a dollar contains some noise – it varies across individuals – but this principle does not result in a bias.

Consequences of the fickle value of a dollar

OK, so different people value money differently. So what? The problem here is that we use dollars to judge investments and policies. We measure things like GDP without distinguishing the fact that a dollar of GDP in Bill Gates’ pocket is not the same as a dollar of GDP in the pocket of the working poor. An economic policy that improves GDP growth but effectively transfers wealth from the poor to the rich is likely to decrease net value in society. All of this is not really news to economists, who talk about utility and utility functions, and sometimes incorporate this in models.

Nonetheless, the principle is not generally appreciated in society, or in our discussions of tax policy, etc. We get it intuitively–this is why we (most of us) favor social safety nets and so forth–but we aren’t explicit about it. Even economists like Paul Krugman never mention it. (I found a description in his Microeconomics textbook but not mentioned explicitly in this context in his blog!)

The solution at first appears to be simple: we should simply tax the rich more and redistribute the wealth. Doing this will produce a greater total amount of value in society because we are taking money away from those to whom it is worth the least (the rich) and giving it to those for whom it is worth the most (the poor). Unfortunately, the logical extreme of this is a complete redistribution of wealth: pure communism. Some people may think this is good, but I am firmly on the side of those who think we have enough historical evidence to say that pure communism doesn’t work at the scale of nations.

This discrepancy arises because we have ignored another principle: that effort (usually) requires motivation, and that money serves as an important motivator at a societal level. A complete redistribution of wealth removes this, and growth and productivity decline. Designing good policy involves incorporating many such principles simultaneously and finding a balance among them. This balance (in my opinion) should probably look something like European- or Canadian-style social democracies.

So what good is recognition of this principle of the fickle value of a dollar if we can’t take it to it’s logical extreme, and if it is already incorporated intuitively in many countries? I think there are at least two important benefits. First, I think it is feasible to develop measures such as “wealth-adjusted dollars” that reflect value, and that will allow us to model the benefits of policy to society mcuh more precisely (see below). Second, explicit consideration of this principle in policy debates would help us achieve a better balance between preservation of market incentives and maximizing value rather than dollars. For example, we are now debating the “Buffett rule” in the US – a tax on the rich. In the current calculus, one dollar of tax generates one dollar of government revenue, making the tax seem like a larger value burden than it is. In a wealth-adjusted dollar context, a tax on millionaires might generate one dollar of government revenue for 10 cents  of tax…  This doesn’t mean we should forget about the effects of tax on economic growth, or that soaking the rich will solve all our problems, but it makes the trade-offs much clearer. Imagine the consequences of requiring such a method to be used in all Congressional Budget Office estimates!

The goal is to come up with a way to represent how much each dollar is worth to people (on average) as a function of their wealth or income. I’ll use income here, since it is a more readily available measure and is more intuitive, but will call it wealth because (a) “wealth-adjusted dollars” is easier to understand as a phrase, and (b) WADs is a cooler acronym.

What properties should this function have? Well, it should go down the richer you get. It should be standardized around 1 – let’s say that the value of a dollar should be 1 for someone with the median income, and work around that. It should never go below zero, no matter how rich you get. It turns out that the function we want is a negative exponential function:

V is the value of a dollar, and x is annual income (i.e., we are modeling V as a function of x). We also have a few other parameters: e is the natural constant 2.718 (the base of the natural log), K is the median income, and beta is a rate constant.

Beta is very important because it determines how quickly the value of a dollar declines with wealth; I estimate a value of 10 is about right, but you can play with it and see what you think. It should be possible to estimate this empirically based on people’s purchasing habits: to what lengths will consumers of different income levels go to save a dollar on their purchases? But for now, I use a value of 10.

We can graph this for \$45,000 median income (US 2003 value) and beta=10:

According to this curve, someone who earns \$10,000 values each dollar at \$1.08, someone who earns \$100,000 at \$0.88, someone who earns \$500,000 at \$0.36, and someone who earns \$1 million at \$0.12.

If you want to play with different values of beta, download R (it’s free) and copy and paste in the following code, changing the value for beta:

x <- seq(1,1000,by=0.001)
K <- 45
beta <- 10
funct <- function(x){exp((1-x/K)/beta)}
par(mgp=c(3,1,0),mar=c(5,4,1,1))
plot(x,funct(x),type=”l”,xlab=”Annual income (\$000s)”,ylab=”Value of a dollar”,lwd=3)
segments(x0=-10,x1=2*max(x),y0=1,y1=1,col=”red”,lty=2)
segments(x0=K,x1=K,y0=-10,y1=2*max(funct(x)),col=”red”,lty=2)