Diminishing Marginal Utility (Again)
I have already done this post, but I want to try it again (I will link to past attempts at the bottom in order to encourage readers to try this one first). I have a strange fixation with Austrians’ notion of “marginal utility.” I think it’s partially because, when I was a first-year graduate student, I thought the same thing and tried to argue it to my micro theory professor and he assured me that it was impossible to verify (or justify) diminishing marginal utility. I was pretty sure it was possible but he was one of the few professors I’ve ever had that I was pretty sure was smarter than me so I gave him the benefit of the doubt and applied myself pretty hard to figuring out what he meant. In the end, it turns out he was right and understanding why was really the key, for me, to understanding the purpose of utility in economics.
In addition to my personal history with this issue, there is the fact that this represents a point of convergence for several Austrian misconceptions. This is important to me because I am what I would call an “Austrian sympathizer.” I am a big proponent of free markets, I want to end the Federal Reserve and have free money, and I have little use for most empirical economic research (I think it goes too far to say there is no role for observation in economics). That being said, the Austrian School tends to be a sort of intellectual abyss that sucks in reasonably bright people and essentially takes them out of the fight. It gets ahold of people when they are sophomores in college and essentially tells them that everything they don’t understand about mainstream economics in fact doesn’t make sense because of a few supposed fundamental flaws underlying it. This gives them an intellectual escape route by which they may avoid developing a level of understanding of mainstream economics which would be sufficient to offer any real pertinent criticism. As a result, Austrians end up talking mainly to other Austrians about how great Mises and Rothbard were, while occasionally lashing out and being mostly ignored on mainstream economists blogs and producing little to no original research. So I think that if I can figure out how to get this point across to a thoughtful reader, I may be able to save some people from this fate.
So this will be a specific response to this post but the exact same issue comes up all the time in Austrian circles. The issue is whether diminishing marginal utility follows logically from the axiom that “humans act.” I have no argument with the notion that human action is axiomatic but this does not imply diminishing marginal utility. I will try to make three points. First, I will try to clear up some confusion about what utility actually means. Second I will try to use Austrian-like reasoning to show that the logic used by Austrians is flawed. Finally, I will try to explain what mainstream economics really does with utility and why there is nothing inappropriate going on there (not even anything that violates any Austrian beliefs… except for using math, that is), show mathematically that you cannot infer diminishing marginal utility from it, and attempt to demonstrate how careless the Austrian approach is in comparison.
Utility vs. Value
Austrians are essentially conflating the mainstream concepts of utility, and value. (The Wikipedia page on value, alas, is not that great.) Let me say, at this point that I am working from mainstream definitions. There is no right or wrong definition of a term and I have no interest in debating which word is best used to mean something so I will explain the difference in mainstream terms and then go on to explain how the Austrian argument doesn’t make sense for either. Mainstream economics uses the term value to represent the amount of other goods someone is willing to give up for something. Utility refers to a mathematical representation of a person’s ordered preferences over different bundles of goods. I will get deeper into what this means in the third section but there is an important distinction worth bringing up now. Value measures the willingness of someone to undertake a certain action in relation to real-world variables. If a person’s value of a cheeseburger is two pieces of pizza, this means that if the price of that cheeseburger were 1.9 pieces of pizza, they would trade for it and if it were 2.1 pieces, they would not. Utility is not a tangible, real-world thing, nor does it bear any relationship to such a thing. If I say that someone’s utility of a cheeseburger is 6, this means nothing, unless you knew that person’s utility of other things as well. If I tell you that their utility of a slice of pizza is 5, then I’ve told you that they prefer the cheeseburger to the pizza. But if I tell you that their utility of a cheeseburger is 2 and their utility of a slice of pizza is 1, I have told you the exact same thing, the numbers don’t matter. They don’t measure how happy you are to have a cheeseburger or a pizza. If your utility of a hamburger and a slice of pizza are 2 and 1 respectively, it doesn’t mean that the cheeseburger makes you twice as happy. In other words, it is an ordinal measurement, not a cardinal measurement.
Okay, now here is the crazy thing. Austrians are well aware that utility is only appropriate as an ordinal ranking. In fact, they frequently criticize mainstream economics for using cardinal utility. Here is someone from Mises.org:
Mainstream economists such as John Hicks believe otherwise; for them, DMU means that d^2U / dx^2 < 0. (I.e. the second derivative of utility with respect to units of some good X is negative.) Naturally, Hicks thought that the ordinal approach to utility would make the law of DMU nonsensical.
But for Austrians, DMU doesn’t mean that “the increment in utils gets smaller as you add more units of a good.” Rather, it just means that, say, the 4th unit of a good is more valuable than the 5th unit. No unit of utility is required to make this claim, just as we can say an apple is preferred to an orange without relying on units of utility.
Translation: when we say “utility,” we really mean value. When we say we don’t believe in cardinal utility, we mean we don’t believe in utility.
Austrians think that ordinal means you can’t do math on it (that’s why they like the idea so much haha…) and this is sort of true. The thing is, economists are using a cardinal function to represent an ordinal utility function in such a way that the conclusions derived from the mathematical model are consistent with an ordinal utility function. If you don’t get what this means, I don’ blame you, it’s kind of subtle, but it doesn’t mean that mainstream micro is nonsense, it just means you don’t get it. It’s worth trying to understand (I will try to explain more a little later).
But then, after complaining about the mainstream using cardinal utility, which they don’t, they go on to make a purely cardinal claim about utility, namely that it is diminishing. The word “marginal” means the change in something when something else changes. The marginal utility of a good means the magnitude of the change in utility when the quantity of the good consumed increases. So claiming that marginal utility is decreasing means that the magnitude of the difference between the utility of the fourth unit and the fifth unit is smaller than the magnitude of the difference between the utility of the third and fourth units. But the notion that there is a magnitude associated with these differences at all is a fundamentally cardinal notion.
When confronted with this contradiction, Austrians just say “well that’s not what we mean by marginal utility” ala my favorite South Park episode but in reverse (see above quote). But that is what the words mean…. http://s0.wp.com/wp-includes/images/smilies/icon_sad.gif
Anyway, going forward, I will assume that what they mean when they say “marginal utility” is marginal value (again, see above quote) and I will refer to it as such. With this in mind, I will proceed to point out that diminishing marginal value does not follow logically from human action in the way claimed.
Diminishing Marginal Value
The claim which has been made is that diminishing marginal value follows directly from the axiom of human action. The way this was demonstrated was by telling a story where it was intuitively plausible that value would be diminishing. But this doesn’t logically prove that value must always be diminishing. By the way, this is the problem with refusing to use math, you end up considering things “proven” when you don’t feel like putting any additional effort into thinking of a counterexample. Luckily, I will provide a couple. First, let me say that mainstream economists do assume diminishing marginal value (or at least things about utility which amount to the same), and I think it is a very reasonable assumption. And observation, for the most part, backs this up. So it’s hard to make an example where you would say “yeah there clearly wouldn’t be diminishing marginal value in that case.” All I’m doing here is showing that you can’t prove diminishing marginal value in the way Austrians claim you can.
1. The value of the use to which the last unit is put may depend on the previous units.
A rancher has some free-range cows grazing on his land. There is an adjacent plot of land owned by his neighbor which he would like to keep his cows off of. The boundary between the two plots is 1 mile long (let’s just assume this is a straight line and the rest of the property is enclosed by a natural boundary. Can we prove logically that the value of the first foot of fence must be greater than the value of the 5280th foot?
2. The options available may change depending on the quantity at hand.
A man has some amount of grain and a (fertilized) chicken egg. It takes 10 lbs. of grain to raise a chicken to point that it’s worth eating, and let’s say it takes 10 lbs. to keep the man alive long enough to raise the chicken. As most people are aware, raising animals is less calorically efficient than just eating their feed (the quantities here are completely made up and bear no relation to reality). It is obvious that if the man has only 10 lbs. of grain, he would eat it all, assuming he doesn’t want to die. If he gets an 11th lb. he will eat it as well, along with the egg, since it’s not enough to raise the chicken to maturity. This will be true until he gets 20 lbs. Now assume that when he has 20 lbs. he decides to raise the chicken. Can we prove that the 20th lb. must have been worth less than the 19th lb.?
3. A variation on #2 which relied on a discontinuity in the budget constraint (he couldn’t convert small amounts of grain into small fractions of a chicken). We can get a similar result without this feature.
Assume the same man can trade grain for chicken at a price of 10 lbs. of grain/lb. of chicken. So he can use the chicken for one of two purposes, eating or trading for chicken. Let’s assume that this is for a period of several days. When he has ten lbs. he eats it all. Therefore, this must be the highest value use right? When he has 20 lbs. he eats 15 lbs. and trades 5 lbs. for 1/2 lb. of chicken. When he has 40 lbs. he eats 10 lbs. and trades 30 for 3 lbs. of chicken. When he has 100 lbs. he trades it all for 10 lbs. of chicken. Did these preferences violate the axiom of human action?
Now you may be saying “okay, but he still probably values the 100th lb. less than the 90th lb. and I agree. But look at what is going on here. The Austrian assertion that there must be diminishing marginal value is based on the reasoning that he will use the first units for the most valuable uses and later units for less valuable units. But the most valuable use may very well depend on how many units you have. If you only have ten lbs. of grain, you need the calories so the most valuable use for all ten lbs. is to eat it. This may be true up to 10 lbs. But that doesn’t mean that as you keep getting more, this will continue to be the highest value use. Let’s say this is the case for the first 15 lbs. It may be the case that when you get to 40 lbs., the 11th-15th lbs. become more valuable converted to chicken and when you have 100 lbs., even the first lb. is more valuable as chicken. Therefore, you cannot infer that someone always uses a good for highest value purpose “first.” It is true that they will always use whatever they have for the highest value purpose by definition, but the highest valued purpose is state-dependent, it can change based on how much you have. This is the fundamental flaw in the Austrian reasoning. You can’t rank “uses” independently from income, you have to rank bundles (econ speak for combinations) of goods (or uses). When someone’s income/endowment increases, they gain access to different bundles. This may completely change the relationship between the values of the individual goods within those bundles.
As I said earlier, mainstream economics assumes diminishing marginal value. We don’t claim it logically must be true. We arrive at this assumption largely by the same logic that Austrians do. The difference is that we derive a refutable implication from it and notice that it is not generally refuted, whereas Austrians assume that their theory can’t possibly be wrong so they don’t bother checking. Here’s how we do it.
First we postulate that people have preferences, which is to say that given any two combinations of goods, they would either prefer one to the other or the other to the one or be indifferent between them. Then we imagine a function into which you can put the quantities of all goods under consideration and get back a number. This is a cardinal number but the only meaning we assign to it is to rank the bundle relative to other bundles. In other words, a bundle with a higher utility is preferred to a bundle with lower utility. This is the only meaning contained in the utility function. It doesn’t matter how far apart the numbers are. This means that multiple functions can represent the same preferences. It even means you can represent the same preferences with diminishing marginal utility or increasing marginal utility. Also, you can have increasing marginal utility and diminishing marginal value (also called marginal rate of substitution).
Mainstream economists use a utility function to represent ordered preferences because it is impossible to represent the infinite number of bundles which exist along a continuum of quantities in list form. Even if you dealt only with discrete quantities it would be very cumbersome to order all possible bundles of several goods which could be available within a given budget. Also, it would be very difficult to articulate the restrictions that you were relying on to generate the results which your model was producing (you might not only have to rank all combinations of the goods but do the model with every possible ranking of all combinations to find out how changes in preferences affect the results).
It is because all of this would be very difficult and cumbersome that Austrians have chosen to avoid it by making drastic simplifying assumptions like the preference ranking of goods is independent of income (what Austrians represent as different uses for a good is typically represented as different goods in mainstream micro). Once you do this, it becomes kind of simple to list ranked preferences, you just say which uses are inherently most valuable and assume that as someone gets more and more of the good, they just move up the list. But this assumption is not only logically guaranteed to be true, it is quite obviously false once you actually think about it. To see this take Bill Gates (a rich guy). Imagine that Bill lost all of his money and was thrown out on the street with only $10. Now imagine what he would spend that $10 on. Now ask yourself: do you think Bill gates currently spends at least $10 on that good? As long as Bill gates isn’t a smoker, chances are the answer is no. The idea that if he only had $10, he would buy Top Ramen with it, does not imply that Top Ramen is the highest-valued use of $10 when he has a hundred billion.
Okay, so mainstream economists, rather than asking: what can we prove about preferences just from the axiom of human action and finding the answer to be nothing, giving up and going home, ask: what kind of preferences would generate results that are consistent with what we actually observe. As it turns out, diminishing marginal value is about the only interesting thing we are pretty confident about. The reason for this is pretty straightforward, though it is entirely unrelated to the Austrian logic. As it turns out, if people did not have diminishing marginal value (except under a very specific and unusual circumstance) they would spend all of their income on only one good. Since we don’t observe this, we are pretty comfortable going forward assuming diminishing marginal value. This doesn’t mean that value must logically always be diminishing but only that, for the goods people buy, it must be diminishing around the quantity where they are buying it or else the world would look different than it does.
It is worth pointing out that this does indeed require one to look at the universe and see if it fits with the model. All models are built on assumptions and all assumptions could be wrong. Therefore, it behooves us to check every once in a while and see if what we are saying makes sense. The notion that we can do science with pure logic without any observation whatsoever is misguided. For instance, you can start with the axiom of human action and I agree that this is a good axiom. But you had to get that axiom by looking around and noticing that people do indeed act. You say that denying it would be an action, I say yes indeed but you would still have to open your eyes to see whether or not that action took place.
One More Thing
There is one more reason I think this is an issue of particular importance. This is not a reason not to believe in diminishing marginal utility, we should seek truth for thruth’s sake, but it is a reason to be careful to make sure you have it right. The left often uses the notion of diminishing marginal utility to justify all sorts of social programs, economic intervention and redistribution of wealth. It’s an elegantly simple, yet completely nonsensical argument. If marginal utility decreases when someone gets more and more stuff, then we can increase total utility by taking from the rich and giving to the poor. I know Austrians don’t mean it this way, but frankly they are just picking and choosing the meanings they like and ignoring the ones they don’t like. This is a very dangerous intellectual game to play.