Question about the law of marginal utility

Economics Economics Questions
1

I’m reading Mises’ Human Action, and I ran into a passage that confuses me.

In Chapter VII.1 he writes:

“If we were to believe that marginal utility is about things and their objective use-value, we would be forced to assume that marginal utility can as well increase as decrease with an increase in the quantity of units available…The owner of 100 logs may build a cabin which protects him against rain better than a raincoat. But if fewer than 100 logs are available, he can only use them for a berth that protects him against the dampness of the soil. As the owner of 95 logs he would be prepared to forsake the raincoat in order to get 5 logs more. As the owner of 10 logs he would not abandon the raincoat even for 10 logs. A man whose savings amount to $100 may not be willing to carry out some work for a remuneration of $200. But if his savings were $2,000 and he were extremely anxious to acquire an indivisible good which cannot be bought for less than $2,100, he would be ready to perform this work for $100. All this is in perfect agreement with the rightly formulated law of marginal utility according to which value depends on the utility of the services expected. There is no question of any such thing as a law of increasing marginal utility.”

I don’t understand why “this is in perfect agreement with the rightly formulated law of marginal utility”. Can anyone explain this a bit more?

2

It isn’t in agreement with “the rightly formulated law of marginal utility”, but instead with “the rightly formulated law of marginal utility according to which value depends on the utility of the services expected.” I believe the point he’s making is about relevant units.

If I need four eggs (no more or less) to make a cake, then according to the view Mises is criticizing my value scale would look like the following:

4th egg

1st egg

2nd egg

3rd egg

This view says that the marginal utility between eggs 3 an 4 increased, thus diminishing marginal utility is invalid

What I believe Mises will say is that my value scale actually looks like the following:

4 eggs

1st egg

2nd egg

3rd egg

4th egg

Thus the relevant unit (with respect to my desire for cakes) is not eggs, but sets of 4 eggs

3

I agree with earlgrey. It is sometimes tricky to find the sets that make up the value. From that perspective the log example is not very well choosen, as it provides very different values, but still values, in very different sets, like 100 logs for a building (set), 95 for a berth (set) etc.

If you got sets of 100 logs, say 5 sets, you could build 5 loghuts, given there is no other use you value regarding the logs, each set of a 100 logs decrases the marginal utility.

4

Okay, I think I understand a bit better. If you have a number of logs, those don’t necessarily constitute a “supply of homogeneous means”, because if you had 95 logs you’d use each toward producing a different effect than if you had 100?

I guess it has to be a supply of things to be used in the same way. I suppose it could even consist of different things. For example, if I desire to make my house warmer, then I need watts of heat input into the environment of my house. I may have a variety of things that could produce heat: candles, logs, an electric heater, natural gas, etc. But it is not those things but the watts of heat that they can produce that are the homogeneous supply. As far as the individual watts are conserned, I don’t really care where each comes from. It is not the things (candles, heaters) themselves for which I have decreasing marginal utility, it is the service provided (the watts) that have decreasing marginal utility.

Does that make sense?

I’m also reading the next section (on the law of returns), and getting lost in Mises’ reasoning there too, so I may be back with more questions. [:)]

5

Laughs, feel free to come up with more.

And yes you got it regarding the services provided. Thinking in terms of services an item can provide is the way i like to think about when it comes to marginal utility.

6

Well, I thought about your explanation and my reply on this and I am not really satisfied with mine. Marginal utility really means the amount of satisfaction you can get from a set of concrete goods.

Let me explain that with the grain sack example Joe Salerno uses frequently. Imagine you have 5 sacks of grain in possession. Imagine also that you need 1 sack of grain to survive for one year, a second sack of grain to enhance your health in addition of merely surviving, a third sack of grain to add the capability to seed for a new harvest, a forth sack of grain to add the ability to enhance your diet by breeding some animals like chicken and to feed them with that sack and a fifth sack that you could use for making wiskey.

Now what you do is you put a value on your desires, right? your primary desire is probably not to drink wiskey but to survive. This is your first priority and therefor the first rank on your ordinal list of values for grain at the given time. Next might be the health enhancement, followed by the seed, followed by the chickenfood and eventually on the last position of your rank list is the wiskey.

1.) survive

2.) increase health

3.) seed and ensure harvest

4.) get a zing in your diet (breed chickens)

5.) Get a real good mood (make wiskey)

Now having such a ranking order, which is completely ordinal and subjective to you as an individual, what is the marginal utility of a single sack of grain?

Suppose you have marked each sack with the rank it has, so you end up having 5 sacks with marks from 1 to 5. Now a bunch of mice eat away the sack you marked number 2. What is the lost of utility you are facing now? Is it the loss of striving for increased health, because those mice used to pick the sack you wanted to use for that matter? For sure not. It is the loss of being able to make wiskey. And that is what marginal utility means, the utility , that is the amount of satisfaction, of a good depends on the ordinal ranks you made up for that concrete good.

Ok, now for a bunch of grain sacks this is pretty clear, but what about different goods? Here again you use the little ranking list.

Suppose you have 3 cows and 2 horses. You could come up with the following ranking order.

1.) cow - needed to produce milk and cheese

2.) horse - needed to plow

3.) cow - needed to breed new cows

4.) cow - needed for meal

5.) horse - needed for pleasure riding

Ok that is your utility order. Now suppose the staple is on fire, what kind of animal would you try to rescue first? A cow, because it is the number one on your ranking order. Now suppose one horse has perished, what would be your next animal to save? Right, now you would try to save the horse left, as it is on rank 2 of the order.

7

Good explanation. I understand the grain and horse/cow examples.

My confusion, however, had to do with the case where something you prefer more requires more of something. I suppose with the log example, your ranking is something like this:

  1. Log cabin (requires 100 logs)

  2. Berth (requires 50 logs)

  3. fire (requires 1 log)

Looking at just the logs, it would appear as though utility increases with an increase in logs, thus contradicting the law of diminishing utility. But the point of the law of utility is that your ranking is of your desire for the services and fulfill the highest ones you can, and not merely the order of increasing number of logs.

  • So, if I have 151 logs, I might do all three. (unless, I suppose, having a log cabin lowers my ranking of also having a berth)

  • If I have 150 logs, I might do just the cabin and berth

  • If I have 149, then I’ll just build the cabin.

So far, that’s like the grain example.

But if I only have 95 logs, I still most prefer the cabin, but I don’t have the means to build it, so I just have to fulfil my next highest desire, building the berth. Likewise, if I only have 45 logs, I’ll just build a fire.

If my number of logs increases, then I will never use the additional logs for something lower in the ranking if I can use them for something higher. If I increase from 49 to 50 logs, I would never use the 50th log for, say, what is at number 4 on the ranking, because I can use them to satisfy #2. Kind of like in your cow/horse example, if you have two cows but no horses, then you fulfil desires 1 and 3, and have left 2 unfulfilled. If you then get a horse, you don’t apply it to a lower slot (i.e., 5), but you use it for service 2.

I guess the point is that I don’t necessarily use my resources to fulfill my greatest desires first. I use them to fulfill the greatest desires that they can fulfill. If I obtain additional resources, I may completely rearrange how the resources I already had were already being used. But any rearranging is guaranteed to occur higher up in the ranking and not lower.

If you are interested in answering, my question about the law of returns is how do we know there must be an optimum (optimum average yield)? Mises says that there must be an optimum, and that we know this a priori, but I don’t see how this conclusion is reached. I follow his reasoning up until he says, “For if it were possible to compensate any decrease in b by a corresponding increase in c in such a way that p remains unchanged, the physical power of production proper to B would be unlimited…”

The “for” at the beginning of that sentence implies that it connects directly to what preceeded. But he was talking about holding b fixed. I understood the preceding discussion about holding b fixed, but I don’t see how that tells us anything about what happens if we change both b and c.

8

Rothbard has a clearer proof of the law of returns in Man, Economy and State; I think in chapter 2. Check the online version. If you have any questions after that I’ll see if I can answer them. You’re correct on MU by the way. It’s the serviceability of the good that matters (so if I need 100 logs, 99 will be useless to me until I reach 100), and only homogeneous goods come under DMU as more of them is consumed. Heterogeneous goods do not exhibit DMU.

-Jon

9

In my last post I said that one always uses one’s resources to fulfill the highest ranking desire(s) that they can fulfill. But after further reflection, I realize that that is not true. For example, suppose that with 10 logs the man can build a table, and with 5 logs the man can build a chair. Suppose the man’s ranking of desires is:

  1. Table (requires 10 logs)

  2. 1st chair (requires 5 logs)

  3. 2nd chair (requires 5 logs)

If what I said in the last post were true, then if he has 10 logs he will build a table. But the ranking doesn’t tell us that. All we know is that he prefers 1 table to 1 chair. It does not tell us whether he prefers 1 table to 2 chairs. If it is the case he prefers 2 chairs to 1 table, then given 10 logs he will build two chairs, leaving his #1 slot in the ranking unfulfilled, even though he had the resources to fulfill it instead. That is, he may forgo a higher desire in exchange for multiple lesser desires that, combined, he prefers greater than the one higher desire.

Rothbard’s explanation of the law of returns is indeed clearer. But I think I’m still missing a step in the logic.

He writes:

“The law that such an optimum must exist can be proved by contemplating the implications of the contrary. If there were no optimum, the average product would increase indefinitely as the quantity of the factor X increased. (It could not increase indef­initely as the quantity decreases, since the product will be zero when the quantity of the factor is zero.) But if p/a can always be increased merely by increasing a, this means that any desired quantity of P could be secured by merely increasing the supply of X.”

Okay, I follow this so far. We are going to suppose the contrary, show that that leads to a contradiction, thus proving that the supposition is false, and that the law is true. The law says that for any b and c held constant, there is some optimum a. The contrary supposition then is that there exists some b and c held constant for which there does not exist an optimum a–that is, for those specific values of b and c held constant, p will increase without bound as a increases. Okay. Then Rothbard continues:

“This would mean that the proportionate supply of factors Y and Z can be ever so small; any decrease in their supply can always be compensated to increase production by increasing the supply of X. This would signify that factor X is perfectly sub­stitutable for factors Y and Z and that the scarcity of the latter factors would not be a matter of concern to the actor so long as factor X was available in abundance.”

But that doesn’t follow. Our contrary supposition was only that for some particular b and c, there is no optimum for a. For all we know, a does have an optimum when the values of b and c are any smaller, thus a cannot necessarily be increased to compensate for decreases in b or c. In which case X is not perfectly substitutable for factors Y and Z.

Do note that I am not here to argue against Austrian economics. I think Austrian economics makes more sense than anything else. That’s why I’m reading Human Action. I’m just trying to make sure I understand what I’m reading. (And maybe even help strengthen its arguments!)

Also I spot a technical error in Rothbard’s reasoning, though minor in comparison to my question above. He says,

“If there were no optimum, the average product would increase indefinitely as the quantity of the factor X increased. (It could not increase indef­initely as the quantity decreases, since the product will be zero when the quantity of the factor is zero.)”

That’s not necessarily true. Suppose, as an example, it is the case that the marginal product is always decreasing, e.g.,

MP = 10 - a

The total product is the integral of the marginal yield:

TP = 10*a - (1/2)aa

As Rothbard says, TP is 0 when a is zero. Also this TP does not increase indefinitely.

The average product is

AP = TP / a = 10 - (1/2)*a

Except that AP is undefined when a=0, because we can’t divide by 0. It only has a value for nonzero a. But for any nonzero a, we could reduce a (e.g., cut it in half), and that would increase AP.

For example, if a is gallons of water, then a=1 gallon would have an average product of 9.5

for a=0.5 gallons, AP=9.75

for a=0.1 gallons, AP=9.95

for a=0.01 gallons, AP=9.995

etc.

No value of a is optimum, because for any value at which AP is defined, you can always increase AP by decreasing a. Thus this would be a case where there is no optimum, but the average product does not increase indefinitely.

10

No, it was that there is no optimum for any b and c. If it held for some particular configuration the law would not be in question. Also, can you try put the mathematical jargon into English? I’m not mathematical so it doesn’t really make much sense to me.

-Jon

11

But this doesn’t correspond to anything possible in the real world.

12

If 2 chairs were ranked higher than 1 table, the ranking table would look like:

  1. two chairs
  2. table
  3. 1st chair
  4. 2nd chair