A Conversation on Marginal Utility
Taken from the comments on an article on mises.org
Well put. I’m actually interested in a true Austrian response to my marginal utility comment though. Doesn’t the entire notion of marginal utility hinge on cardinal utility and isn’t that a cardinal sin (can’t help myself) in Austrian economcs?
Not at all. Ordinal utility works perfectly. Austrian theory asserts that it is impossible to measure how MUCH utility is lost or gained by any given person in any given transaction, since utility (or value) is entirely subjective and subject to sudden changes. But it is certainly possible to make general assertions about whether or not a certain thing is valued MORE than another, although it is not possible to tell how MUCH more. An example; A person is trapped in the desert with no water. He has a number of things he wants to use water for. If we could look into his head, we might be able to represent his priorities for the use of water by arranging them thus; 1st. A nice, big drink 2nd. Water this plant, maybe it’ll grow edible fruit or something 3rd. Take a bath, I’m filthy. 4th. Wet a cloth to wrap around my head and fend off sunstroke 5th. Fill a watering hole and attract adorable animal friends So now; if a container of water drops out of the sky, what’s our hapless desert-dweller going to do, given that his desires are ordered like this? Well, first he’ll chug it. Then, with what’s left over, he’ll water his plant. If there’s anything left, he might give himself a sponge bath; and if there’s still more, he’ll soak his cloth, wrap it around his head, and set to work digging a hole to fill up with the remainder. But what if there’s only a gallon? Well, then, he’ll take that nice big drink, and then water the plant. . . and then he’s out! He didn’t get to take his bath, or wet his cloth, or fill his watering hole. That’s marginal utility. The MARGINAL desires, or possible utilizations - the ones that are least important to him – get cut out if the supply isn’t sufficient to fulfill all his desires. The 1st priority gets satisfied first, then the 2nd, and so on until the supply is gone, in order of ordinal ranking. Cardinal utility is actually a useless concept. Think about it; look at the list up there. A cardinal ranking would require assigning a numerical value to each of those desires, upon which mathematical operations can be performed. The implications are unfortunate. If getting a drink is worth 1000 utils, and watering the plant is worth 500 utils, doesn’t that mean that it’s more valuable to water the plant three separate times than it is to have a drink? But that’s absurd; the plant doesn’t need that much water, and you’re still slowly dying of thirst in the desert. The fact that the plant is now drowning in an overabundance of water doesn’t do a single thing to solve the problem of your thirst.
There are several issues with this, first determinnig marginal utility is a mathematical operation (derivative). Also, there’s no reason that if watering the plant one time generates 500 utils that watering it a second time must generate 500 more, this seems to deny the possibility of diminishing marginal utility in the defense of that very concept. Once you have the marginal utility of each of these things decreasing individually, then the example no longer works because the person can possibly do several or all of them to some degree. In this case he will do each thing until the marginal utility of each is equal. Once this is the case, it is likely that when he gets more water he will do more of many or all of them, and this may be the case even if the marginal utility of each thing is increasing. Now I am using an example that assumes cardinal utility but I am arguing thta you are doing this as well you just aren’t calling it that.
In order to argue with your example from a strictly ordinal perspective I would say that it is not enough to just rank each use, this is not a complete ranking of every possible way of using the water because the preference for each thing may depend on the quantities of other things being consumed. In order to construct an ordinal ranking you must rank every possible combination of uses (bundle). Once you do this there will be some optimal bundle for any given income (amount of water) and this will change when the amount of water changes but to say that marginal utility is decreasing is to say that the change in utils between the optimal bundle at one income and the optimal bundle at another level is getting smaller the more income you get and this requires cardinal utility. If you only have ordinal utility over all bundles you can only say that when he gets more income he gets a bundle with a higher utility but you can’t say anything aobut how much higher.
No, I’m not using cardinal utility, although you are. I’m not sure where the confusion crept in, but I’ll try to resolve it. First off; Determining marginal utility is not a mathematical operation. The entire point of ordinal utility theory is that you CAN’T use math to deal with utility; utility is entirely subjective, changes every moment as the thoughts of the person in question change, and cannot be measured or mathematically manipulated. Second; Marginal utility cannot decrease ‘individually’, because ordinal numbers have no individual meaning. Economics uses the words ‘ordinal’ and ‘cardinal’ in the linguistic sense, NOT the mathematical sense; cardinal numbers are those numbers which denote quantities (1, 2, 3, etc), and ordinal numbers are those numbers which denote places relative to each other (1st, 2nd, 3rd, etc). Third; Yes, it’s possible that the person might do some or all of them ‘to some degree’, but that all depends on how he ranks his desires. When you try to give an example of ordinal utility, it is necessary to assume that you know the mind of the hypothetical person in totality, and are presenting an ACCURATE list of his desires; which means that the person up above does not want to partially fulfill any of his desires, he would rather fulfill the more important ones completely, and then move on to the next. That’s implied in the list I gave above. Fourth; To answer what you said in the comment below this one, you’re right, if we wanted to rank every possible use of the water at the start of the process, we would have to bundle them. But we don’t! We aren’t interested in every possible combination; we’re only interested in what HE, the person stuck in the desert, wants. Remember; utility is entirely subjective. Right now, at this moment, that list up there is an accurate representation of his desires. Now, after he gets the first drink of water, might they change? Absolutely! But we don’t know how, or to what, and there’s no way to represent it mathematically.
“Right now, at this moment, that list up there is an accurate representation of his desires. Now, after he gets the first drink of water, might they change? Absolutely! But we don’t know how, or to what, and there’s no way to represent it mathematically. “
This is my entire point! So when he gets another unit of water you can’t say that his utility increases by less than it did from the first unit. Also, having an accurate list of his desires doesn’t mean that the person doesn’t want to “partially fulfill any of his desires,” I’m not sure how you come to this conclusion. Your whole problem is assuming that some are “more important” than others and that this is independent of how much of each he is consuming. Then you are assuming that there is some limit to each so the only thing determining the quantity of each is your arbitrary limit on each thing (assuming his income is high enough to make that constraint binding). this is a very uninteresting model of consumer choice and utility. However, even in this model I can demonstrate that you don’t get diminishing marginal utility from it. I will simplify it somewhat by reducing the number of goods to 3
1. take a drink
2. water a plant
3. take a bath
Now, keeping with your convention assume that there is some unit of water and he can spend only 1 unit on each thing, no more no less. Now assume that the above ranking represents his (ordinal) utility function over the three bundles which he can include with an income of one unit (to be complete we could include the possibility of doing nothing as number 4). Now we can predict that if he has one unit of water he will drink it. We haven’t said anything about marginal utility.
Now if he gets another unit of water we can rank his possible bundles (disregarding those which do not lie on the budget constraint)
1a drink and water
2a drink and bath
3a water and bath
Here I have assumed the most obvious ranking of these bundles in relation to his ranking when his income is only 1, however it is important to point out that this need not be the case. It could be the opposite.
1b water and bath
2b drink and bath
3b drink and water
However, let us assume the first ranking (a). In this case when he has 2 units he will choose to drink 1 and use the other to water the plant. But if you want to claim diminishing marginal utility of water you have to compare the utility of a bundle on the 2 unit budget constraint (drink and water) to one on the 1 unit budget constraint (drink). With ordinal utility we can say only one thing about these two bundles:
Bundle 1.a is preferred to bundle 1.
To say that marginal utility is decreasing is to say that utility increased by less between bundle 1 and bundle 1a than it did between nothing and bundle 1. You can’t say this without having a cardinal measure of utility. You are making an assumption about utility that inherently assumes cardinality:
Utility of drink and water = utility of water + utility of drink
Of course with ordinal utility this statement is nonsense because it includes mathematical operators (as you have pointed out). However, even with cardinal utility this need not be the case to preserve the rankings as I have assumed them. Consider the following cardinal utility function as an example of what I’m talking about.
1c drink and water U=30
2c drink and bath U=20
3c bath and water U=10
4c drink U=3
5c water U=2
6c bath U=1
Notice we have preserved the ordinal ranking but when the man gets his first unit of water his utility increases by 3 (assuming the utility of nothing is 0) and when he gets his second unit of water it increases by 29, in other words you have increasing marginal utility of water without violation any of the typical assumptions about the ordinal ranking of bundles (more is preferred to less, preferences are transitive, even diminishing marginal value can hold up in this scenario).
If you try to write a logical proof that utility must be diminishing you will find it impossible using ordinal measures because such a proof requires you to end with
U(1a)-U(1)<U(1)-U(0) (where 0 represents no consumption)
This can’t be done with ordinal utility because, as I said earlier, finding marginal utility is a mathematical operation.
This is important because diminishing marginal utility is a central justification for wealth redistribution among utilitarian types.