Veblen

Two terms in marginal utility of wealth

pnrj cognitive science, ethics, heuristics and biases , , , , , , , , , , , , , ,

JDN 2457569

This post is going to be a little wonkier than most; I’m actually trying to sort out my thoughts and draw some public comment on a theory that has been dancing around my head for awhile. The original idea of separating terms in marginal utility of wealth was actually suggested by my boyfriend, and from there I’ve been trying to give it some more mathematical precision to see if I can come up with a way to test it experimentally. My thinking is also influenced by a paper Miles Kimball wrote about the distinction between happiness and utility.

There are lots of ways one could conceivably spend money—everything from watching football games to buying refrigerators to building museums to inventing vaccines. But insofar as we are rational (and we are after all about 90% rational), we’re going to try to spend our money in such a way that its marginal utility is approximately equal across various activities. You’ll buy one refrigerator, maybe two, but not seven, because the marginal utility of refrigerators drops off pretty fast; instead you’ll spend that money elsewhere. You probably won’t buy a house that’s twice as large if it means you can’t afford groceries anymore. I don’t think our spending is truly optimal at maximizing utility, but I think it’s fairly good.

Therefore, it doesn’t make much sense to break down marginal utility of wealth into all these different categories—cars, refrigerators, football games, shoes, and so on—because we already do a fairly good job of equalizing marginal utility across all those different categories. I could see breaking it down into a few specific categories, such as food, housing, transportation, medicine, and entertainment (and this definitely seems useful for making your own household budget); but even then, I don’t get the impression that most people routinely spend too much on one of these categories and not enough on the others.

However, I can think of two quite different fundamental motives behind spending money, which I think are distinct enough to be worth separating.

One way to spend money is on yourself, raising your own standard of living, making yourself more comfortable. This would include both football games and refrigerators, really anything that makes your life better. We could call this the consumption motive, or maybe simply the self-directed motive.

The other way is to spend it on other people, which, depending on your personality can take either the form of philanthropy to help others, or as a means of self-aggrandizement to raise your own relative status. It’s also possible to do both at the same time in various combinations; while the Gates Foundation is almost entirely philanthropic and Trump Tower is almost entirely self-aggrandizing, Carnegie Hall falls somewhere in between, being at once a significant contribution to our society and an obvious attempt to bring praise and adulation to himself. I would also include spending on Veblen goods that are mainly to show off your own wealth and status in this category. We can call this spending the philanthropic/status motive, or simply the other-directed motive.

There is some spending which combines both motives: A car is surely useful, but a Ferrari is mainly for show—but then, a Lexus or a BMW could be either to show off or really because you like the car better. Some form of housing is a basic human need, and bigger, fancier houses are often better, but the main reason one builds mansions in Beverly Hills is to demonstrate to the world that one is fabulously rich. This complicates the theory somewhat, but basically I think the best approach is to try to separate a sort of “spending proportion” on such goods, so that say $20,000 of the Lexus is for usefulness and $15,000 is for show. Empirically this might be hard to do, but theoretically it makes sense.

One of the central mysteries in cognitive economics right now is the fact that while self-reported happiness rises very little, if at all, as income increases, a finding which was recently replicated even in poor countries where we might not expect it to be true, nonetheless self-reported satisfaction continues to rise indefinitely. A number of theories have been proposed to explain this apparent paradox.

This model might just be able to account for that, if by “happiness” we’re really talking about the self-directed motive, and by “satisfaction” we’re talking about the other-directed motive. Self-reported happiness seems to obey a rule that $100 is worth as much to someone with $10,000 as $25 is to someone with $5,000, or $400 to someone with $20,000.

Self-reported satisfaction seems to obey a different rule, such that each unit of additional satisfaction requires a roughly equal proportional increase in income.

By having a utility function with two terms, we can account for both of these effects. Total utility will be u(x), happiness h(x), and satisfaction s(x).

u(x) = h(x) + s(x)

To obey the above rule, happiness must obey harmonic utility, like this, for some constants h0 and r:

h(x) = h0 – r/x

Proof of this is straightforward, though to keep it simple I’ve hand-waved why it’s a power law:

Given

h'(2x) = 1/4 h'(x)

Let

h'(x) = r x^n

h'(2x) = r (2x)^n

r (2x)^n = 1/4 r x^n

n = -2

h'(x) = r/x^2

h(x) = – r x^(-1) + C

h(x) = h0 – r/x

Miles Kimball also has some more discussion on his blog about how a utility function of this form works. (His statement about redistribution at the end is kind of baffling though; sure, dollar for dollar, redistributing wealth from the middle class to the poor would produce a higher gain in utility than redistributing wealth from the rich to the middle class. But neither is as good as redistributing from the rich to the poor, and the rich have a lot more dollars to redistribute.)

Satisfaction, however, must obey logarithmic utility, like this, for some constants s0 and k.

The x+1 means that it takes slightly less proportionally to have the same effect as your wealth increases, but it allows the function to be equal to s0 at x=0 instead of going to negative infinity:

s(x) = s0 + k ln(x)

Proof of this is very simple, almost trivial:

Given

s'(x) = k/x

s(x) = k ln(x) + s0

Both of these functions actually have a serious problem that as x approaches zero, they go to negative infinity. For self-directed utility this almost makes sense (if your real consumption goes to zero, you die), but it makes no sense at all for other-directed utility, and since there are causes most of us would willingly die for, the disutility of dying should be large, but not infinite.

Therefore I think it’s probably better to use x +1 in place of x:

h(x) = h0 – r/(x+1)

s(x) = s0 + k ln(x+1)

This makes s0 the baseline satisfaction of having no other-directed spending, though the baseline happiness of zero self-directed spending is actually h0 – r rather than just h0. If we want it to be h0, we could use this form instead:

h(x) = h0 + r x/(x+1)

This looks quite different, but actually only differs by a constant.

Therefore, my final answer for the utility of wealth (or possibly income, or spending? I’m not sure which interpretation is best just yet) is actually this:

u(x) = h(x) + s(x)

h(x) = h0 + r x/(x+1)

s(x) = s0 + k ln(x+1)

Marginal utility is then the derivatives of these:

h'(x) = r/(x+1)^2

s'(x) = k/(x+1)

Let’s assign some values to the constants so that we can actually graph these.

Let h0 = s0 = 0, so our baseline is just zero.

Furthermore, let r = k = 1, which would mean that the value of $1 is the same whether spent either on yourself or on others, if $1 is all you have. (This is probably wrong, actually, but it’s the simplest to start with. Shortly I’ll discuss what happens as you vary the ratio k/r.)

Here is the result graphed on a linear scale:

Utility_linear

And now, graphed with wealth on a logarithmic scale:

Utility_log

As you can see, self-directed marginal utility drops off much faster than other-directed marginal utility, so the amount you spend on others relative to yourself rapidly increases as your wealth increases. If that doesn’t sound right, remember that I’m including Veblen goods as “other-directed”; when you buy a Ferrari, it’s not really for yourself. While proportional rates of charitable donation do not increase as wealth increases (it’s actually a U-shaped pattern, largely driven by poor people giving to religious institutions), they probably should (people should really stop giving to religious institutions! Even the good ones aren’t cost-effective, and some are very, very bad.). Furthermore, if you include spending on relative power and status as the other-directed motive, that kind of spending clearly does proportionally increase as wealth increases—gotta keep up with those Joneses.

If r/k = 1, that basically means you value others exactly as much as yourself, which I think is implausible (maybe some extreme altruists do that, and Peter Singer seems to think this would be morally optimal). r/k < 1 would mean you should never spend anything on yourself, which not even Peter Singer believes. I think r/k = 10 is a more reasonable estimate.

For any given value of r/k, there is an optimal ratio of self-directed versus other-directed spending, which can vary based on your total wealth.

Actually deriving what the optimal proportion would be requires a whole lot of algebra in a post that probably already has too much algebra, but the point is, there is one, and it will depend strongly on the ratio r/k, that is, the overall relative importance of self-directed versus other-directed motivation.

Take a look at this graph, which uses r/k = 10.

Utility_marginal

If you only have 2 to spend, you should spend it entirely on yourself, because up to that point the marginal utility of self-directed spending is always higher. If you have 3 to spend, you should spend most of it on yourself, but a little bit on other people, because after you’ve spent about 2.2 on yourself there is more marginal utility for spending on others than on yourself.

If your available wealth is W, you would spend some amount x on yourself, and then W-x on others:

u(x) = h(x) + s(W-x)

u(x) = r x/(x+1) + k ln(W – x + 1)

Then you take the derivative and set it equal to zero to find the local maximum. I’ll spare you the algebra, but this is the result of that optimization:

x = – 1 – r/(2k) + sqrt(r/k) sqrt(2 + W + r/(4k))

As long as k <= r (which more or less means that you care at least as much about yourself as about others—I think this is true of basically everyone) then as long as W > 0 (as long as you have some money to spend) we also have x > 0 (you will spend at least something on yourself).

Below a certain threshold (depending on r/k), the optimal value of x is greater than W, which means that, if possible, you should be receiving donations from other people and spending them on yourself. (Otherwise, just spend everything on yourself). After that, x < W, which means that you should be donating to others. The proportion that you should be donating smoothly increases as W increases, as you can see on this graph (which uses r/k = 10, a figure I find fairly plausible):

Utility_donation

While I’m sure no one literally does this calculation, most people do seem to have an intuitive sense that you should donate an increasing proportion of your income to others as your income increases, and similarly that you should pay a higher proportion in taxes. This utility function would justify that—which is something that most proposed utility functions cannot do. In most models there is a hard cutoff where you should donate nothing up to the point where your marginal utility is equal to the marginal utility of donating, and then from that point forward you should donate absolutely everything. Maybe a case can be made for that ethically, but psychologically I think it’s a non-starter.

I’m still not sure exactly how to test this empirically. It’s already quite difficult to get people to answer questions about marginal utility in a way that is meaningful and coherent (people just don’t think about questions like “Which is worth more? $4 to me now or $10 if I had twice as much wealth?” on a regular basis). I’m thinking maybe they could play some sort of game where they have the opportunity to make money at the game, but must perform tasks or bear risks to do so, and can then keep the money or donate it to charity. The biggest problem I see with that is that the amounts would probably be too small to really cover a significant part of anyone’s total wealth, and therefore couldn’t cover much of their marginal utility of wealth function either. (This is actually a big problem with a lot of experiments that use risk aversion to try to tease out marginal utility of wealth.) But maybe with a variety of experimental participants, all of whom we get income figures on?

Elasticity and the Law of Demand

pnrj core principles, critique of neoclassical economics , , , , , , , , ,

JDN 2457289 EDT 21:04

This will be the second post in my new bite-size format, the first one that’s in the middle of the week.

I’ve alluded previously to the subject of demand elasticity, but I think it’s worth explaining in a little more detail. The basic concept is fairly straightforward: Demand is more elastic when the amount that people want to buy changes a large amount for a small change in price. The opposite is inelastic.

Apples are a relatively elastic good. If the price of apples goes up, people buy fewer apples. Maybe they buy other fruit instead, such as oranges or bananas; or maybe they give up on fruit and eat something else, like rice.

Salt is an extremely inelastic good. No matter what the price of salt is, at least within the range it has been for the last few centuries, people are going to continue to buy pretty much the same amount of salt. (In ancient times salt was actually expensive enough that people couldn’t afford enough of it, which was particularly harmful in desert regions. Mark Kulansky’s book Salt on this subject is surprisingly compelling, given the topic.)
Specifically, the elasticity is equal to the proportional change in quantity demanded, divided by the proportional change in price.

For example, if the price of gas rises from $2 per gallon to $3 per gallon, that’s a 50% increase. If the quantity of gas purchase then falls from 100 billion gallons to 90 billion gallons, that’s a 10% decrease. If increasing the price by 50% decreased the quantity demanded by 10%, that would be a demand elasticity of -10%/50% = -1/5 = -0.2

In practice, measuring elasticity is more complicated than that, because supply and demand are both changing at the same time; so when we see a price change and a quantity change, it isn’t always clear how much of each change is due to supply and how much is due to demand. Sophisticated econometric techniques have been developed to try to separate these two effects (in future posts I plan to explain the basics of some of these techniques), but it’s difficult and not always successful.

In general, markets function better when supply and demand are more elastic. When shifts in price trigger large shifts in quantity, this creates pressure on the price to remain at a fixed level rather than jumping up and down. This in turn means that the market will generally be predictable and stable.

It’s also much harder to make monopoly profits in a market with elastic demand; even if you do have a monopoly, if demand is highly elastic then raising the price won’t make you any money, because whatever you gain in selling each gizmo for more, you’ll lose in selling fewer gizmos. In fact, the profit margin for a monopoly is inversely proportional to the elasticity of demand.

Markets do not function well when supply and demand are highly inelastic. Monopolies can become very powerful and result in very large losses of human welfare. A particularly vivid example of this was in the news recently, when a company named Turing purchased the rights to a drug called Daraprim used primarily by AIDS patients, then hiked the price from $13.50 to $750. This made enough people mad that the CEO has since promised to bring it back down, though he hasn’t said how far.

That price change was only possible because Daraprim has highly inelastic demand—if you’ve got AIDS, you’re going to take AIDS medicine, as much as prescribed, provided only that it doesn’t drive you completely bankrupt. (Not an unreasonable fear, as medical costs are the leading cause of bankruptcy in the United States.) This raised price probably would bankrupt a few people, but for the most part it wouldn’t affect the amount of drug sold; it would just funnel a huge amount of money from AIDS patients to the company. This is probably part of why it made people so mad; that and there would probably be a few people who died because they couldn’t afford this new expensive medication.

Imagine if a company had tried to pull the same stunt for a more elastic good, like apples. “CEO buys up all apple farms, raises price of apples from $2 per pound to $100 per pound.” What’s going to happen then? People are not going to buy any apples. Perhaps a handful of the most die-hard apple lovers still would, but the rest of us are going to meet our fruit needs elsewhere.

For most goods most of the time, elasticity of demand is negative, meaning that as price increases, quantity demanded decreases. This is in fact called the Law of Demand; but as I’ve said, “laws” in economics are like the Pirate Code: They’re really more what you’d call “guidelines”.
There are three major exceptions to the Law of Demand. The first one is the one most economists talk about, and it almost never happens. The second one is talked about occasionally, and it’s quite common. The third one is almost never talked about, and yet it is by far the most common and one of the central driving forces in modern capitalism.
The exception that we usually talk about in economics is called the Giffen Effect. A Giffen good is a good that’s so cheap and such a bare necessity that when it becomes more expensive, you won’t be able to buy less of it; instead you’ll buy more of it, and buy less of other things with your reduced income.

It’s very hard to come up with empirical examples of Giffen goods, but it’s an easy theoretical argument to make. Suppose you’re buying grapes for a party, and you know you need 4 bags of grapes. You have $10 to spend. Suppose there are green grapes selling for $1 per bag and red grapes selling for $4 per bag, and suppose you like red grapes better. With your $10, you can buy 2 bags of green grapes and 2 bags of red grapes, and that’s the 4 bags you need. But now suppose that the price of green grapes rises to $2 per bag. In order to afford 4 bags of grapes, you now need to buy 3 bags of green grapes and only 1 bag of red grapes. Even though it was the price of green grapes that rose, you ended up buying more green grapes. In this scenario, green grapes are a Giffen good.

The exception that is talked about occasionally and occurs a lot in real life is the Veblen Effect. Whereas a Giffen good is a very cheap bare necessity, a Veblen good is a very expensive pure luxury.

The whole point of buying a Veblen good is to prove that you can. You don’t buy a Ferrari because a Ferrari is a particularly nice automobile (a Prius is probably better, and a Tesla certainly is); you buy a Ferrari to show off that you’re so rich you can buy a Ferrari.

On my previous post, jenszorn asked: “Much of consumer behavior is irrational by your standards. But people often like to spend money just for the sake of spending and for showing off. Why else does a Rolex carry a price tag for $10,000 for a Rolex watch when a $100 Seiko keeps better time and requires far less maintenance?” Veblen goods! It’s not strictly true that Veblen goods are irrational; it can be in any particular individual’s best interest is served by buying Veblen goods in order to signal their status and reap the benefits of that higher status. However, it’s definitely true that Veblen goods are inefficient; because ostentatious displays of wealth are a zero-sum game (it’s not about what you have, it’s about what you have that others don’t), any resources spent on rich people proving how rich they are are resources that society could otherwise have used, say, feeding the poor, curing diseases, building infrastructure, or colonizing other planets.

Veblen goods can also result in a violation of the Law of Demand, because raising the price of a Veblen good like Ferraris or Rolexes can make them even better at showing off how rich you are, and therefore more appealing to the kind of person who buys them. Conversely, lowering the price might not result in any more being purchased, because they wouldn’t seem as impressive anymore. Currently a Ferrari costs about $250,000; if they reduced that figure to $100,000, there aren’t a lot of people who would suddenly find it affordable, but many people who currently buy Ferraris might switch to Bugattis or Lamborghinis instead. There are limits to this, of course: If the price of a Ferrari dropped to $2,000, people wouldn’t buy them to show off anymore; but the far larger effect would be the millions of people buying them because you can now get a perfectly good car for $2,000. Yes, I would sell my dear little Smart if it meant I could buy a Ferrari instead and save $8,000 at the same time.

But the third major exception to the Law of Demand is actually the most important one, yet it’s the one that economists hardly ever talk about: Speculation.

The most common reason why people would buy more of something that has gotten more expensive is that they expect it to continue getting more expensive, and then they will be able to sell what they bought at an even higher price and make a profit.

When the price of Apple stock goes up, do people stop buying Apple stock? On the contrary, they almost certainly start buying more—and then the price goes up even further still. If rising prices get self-fulfilling enough, you get an asset bubble; it grows and grows until one day it can’t, and then the bubble bursts and prices collapse again. This has happened hundreds of times in history, from the Tulip Mania to the Beanie Baby Bubble to the Dotcom Boom to the US Housing Crisis.

It isn’t necessarily irrational to participate in a bubble; some people must be irrational, but most people can buy above what they would be willing to pay by accurately predicting that they’ll find someone else who is willing to pay an even higher price later. It’s called Greater Fool Theory: The price I paid may be foolish, but I’ll find someone who is even more foolish to take it off my hands. But like Veblen goods, speculation goods are most definitely inefficient; nothing good comes from prices that rise and fall wildly out of sync with the real value of goods.

Speculation goods are all around us, from stocks to gold to real estate. Most speculation goods also serve some other function (though some, like gold, are really mostly just Veblen goods otherwise; actual useful applications of gold are extremely rare), but their speculative function often controls their price in a way that dominates all other considerations. There’s no real limit to how high or low the price can go for a speculation good; no longer tied to the real value of the good, it simply becomes a question of how much people decide to pay.

Indeed, speculation bubbles are one of the fundamental problems with capitalism as we know it; they are one of the chief causes of the boom-and-bust business cycle that has cost the world trillions of dollars and thousands of lives. Most of our financial industry is now dedicated to the trading of speculation goods, and finance is taking over a larger and larger section of our economy all the time. Many of the world’s best and brightest are being funneled into finance instead of genuinely productive industries; 15% of Harvard grads take a job in finance, and almost half did just before the crash. The vast majority of what goes on in our financial system is simply elaborations on speculation; very little real productivity ever enters into the equation.

In fact, as a general rule I think when we see a violation of the Law of Demand, we know that something is wrong in the economy. If there are Giffen goods, some people are too poor to buy what they really need. If there are Veblen goods, inequality is too large and people are wasting resources competing for status. And since there are always speculation goods, the history of capitalism has been a history of market instability.

Fortunately, elasticity of demand is usually negative: As the price goes up, people want to buy less. How much less is the elasticity.