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Does Rothbard Contradict himself with regards to the Law of Diminishing Marginal Utility?

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Does Rothbard Contradict himself with regards to the Law of Diminishing Marginal Utility?

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BlackNumero posted on Tue, Aug 3 2010 11:21 PM

In Chapter 2 of MES, while utilizing the Law of Marginal Utility to explaing Supply and Demand schedules (and ultimately price formation), Rothbard states that "the supply must always remain unchanged or increase with an increase in price" (MES 124). This is derived from the Law of Marginal Utility, because as the supply of a good X decreases (Actor exchanges good X for Y), the utility ranking of the marginal unit of X increases (since it serves ends higher on the value scale) while as the supply of Y increases, the utility ranking of the marginal unit of Y decreases (since it serves ends lower on the value scale).

Therefore, in order to supply an additional unit of X (and forgoe a higher utility ranked X), he will only sell it at an equal or higher ranked price of Y units. He uses this to say that at each hypothetical price an individual will supply a certain amount of good X because at that price an individual will supply up to the marginal unit. At a higher price, an individual will supply either the same or more because the larger amount of Y recieved ranks higher then the forgone utility of the marginal unit.  With the laws of supply and demand deduced, Rothbard moves on to describe various types of exchanges along with an Austrian analysis of the Structure of Production (using these theorems to describe price formation of factors of production, the interest rate, etc etc).

However, in Chapter 9 when discussing land and labor incomes, Rothbard discusses the possibility of a "backwards bending supply curve of labor" (MES 573). No longer an increase in price require the quantity supplied to stay the same or increase, now an increase in price can cause the quantity supplied to decrease! Either I'm not understanding the backward bending supply curve, or this small concession has grave consequences for Austrian theorizing. Doesn't the possibility of a backwards bending supply curve refute the laws he derived in Chapter 2, that the supply of a good will either stay the same or increase at a higher price? If the "laws of marginal utility" have exceptions, then the Austrian analysis of price formation is incorrect because it is not true in all scenarios. One cannot describe apodicitically certain deductions when there are blatant exceptions.

Interestingly enough, Bryan Caplan levels the same criticisms againist Austrian economics in "Why I am not an Austrian Economist" and states Rothbard makes ad hoc concessions after his value scale approach cannot explain certain economic phenomenon (such as the backward supply curve). I find neither Block or Hulsman defenses really satisfactory, as Hulsmann seems to outward deny it while Block says it won't occur if we hold income constant (but isn't every change in price a change in income?).

Anyone have any thoughts?

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Smiling Dave replied on Thu, Aug 5 2010 12:36 PM

"I'm a little bit confused as to what its trying to explain."

It's trying to explain why he won't sell the second ball if he has 100K without selling it.

"and a little confused as to the cardinal utility dollars of happiness."

It's for illustrative purposes only. To give a clear idea of the guy's feelings and scale of values. I could have said something like "he will feel bad if he sells the one ball, but really bad if he sells both, and really really really bad if he wont get the operation. And if he sells one ball and gets 100K then he will feel a little bit worse than if he had done bla bla, and a little better than if he had done so and so." It would have been very complicated to describe. This way I am saying that the situation I am laying out is one where a higher numerical bottom line corresponds to one where he is happier. This way we know exactly where he stands about every situation very easily.

"From my understanding the problem cannot exist without the numerical utility values associated with each good (which according to A.E can't exist)"

It's not a problem, it's a description of a hypothetical situation that is not impossible or unrealistic.

What AE doesn't like about numerical utility values is that first of all, how are you going to assign numbers to feelings of content? But that is only an objection to trying to supply numerical values in the real world. But there is nothing wrong with me making up an example [in order to bring out a point] where it just so happens a numerical valuation is possible. Just as when I make up an example I can, with Godlike powers, insert as many people as I want into the picture, give each of them whatever amount of money I want, etc.

The other objection to assigning numerical values is that there is an implicit incorrect assumption that manipulating these numbers with arithmetic opoerations tells us something meaningful. For example, we may say that an object that brings 10 utils of happiness and an object that brings 1 util will together bring 11 utils, when this may not at all be the case. But again, that is only an objection when trying to apply the numerical util concept to the real world. But in an example that I make up, since I made it up, I can declare that in this particular example adding up the numbers does give an accurate description of the guy's feelings.

OK. Now to your question, what is the point of this whole thing?

From what I gather, the law of supply and demand is built up step by step from investigating people's scale of values, and showing that the very fact that people are trying to get highest up on their scale of values always leads to the law of supply and demand. In fact, one can say that the law of S and D is ultimately a statement about people's scale of values. 

Now Rothbard mentioned in the story of the hated neighbor that different disutilities will change things, because they modify the scale of values. I am showing how the wierd curve for land sales is, as Rothbard explicitly stated, also an instance of this. That the law of supply and demand was formulated from a study of value scales which assumed that this sharply increasing marginal disutility did not exist. And that once we get down to the nitty gritty, examining the scale of values, the whole thing makes perfect sense, as illustrated by the story of the BBalls.

That's all I've got.

My humble blog

It's easy to refute an argument if you first misrepresent it. William Keizer

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BlackNumero replied on Thu, Aug 5 2010 2:23 PM

A.E states that an individual can only say they prefer A to B, but not by how much. Linguistically, someone can say I really like watching this show compared to that show, but as far as economics is concerned that is A>B just as much as someone who picks that show because he really doesn't care what he watches. Numerical values aren't really possible because individuals don't think in terms of "net happiness" but simply rank their goal obtainable with its opportunity cost (but this really strays into something that belongs in a different thread). So if an individual's opportunity cost of selling a good is 10 million dollars, then on that current value scale he will not sell the good for 100,000. The value scale may change so that he will sell the good at a much lower price, but there still is no "disutility lost of 10 million".

When the increasing disutility causes people two evaluate goods differently, then they are no longer the same good. "Sharply" increasing marginal utility happens with every good exchanged, and its not really whether the increasing disutility of foregoing a good (because it serves ends higher on a value scale) but more of whether the psychic perception of a good changes. The point I'm trying to say (from what I'm understanding of Salerno NirgrahamUK's link) is that the ceteris paribus allows nothing else to move in the interm among an individual's value scale. As Rothbard puts it "there wil be such a backward supply curve if the marignal utility of money falls rapidly enough and the marginal disutility of leisure forgone rises rapidly enough as units of labor are sold for higher prices in money". From my understanding of the situation, Rothbard is relaxing the ceteris paribus assumption a little bit and including movements of the consumer good leisure along the value scale when constructing the supply curve. I think the point can be further illustrated taking into account the fact that leisure is not an axiomatic deduction but an empirical observation in our world. The fact that there can exist the theoretical possibility that a laborer's value scale can exist without taking into account any notion of leisure highlights the violation of the ceteris paribus rule.

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Answered (Not Verified) Think Blue replied on Fri, Aug 6 2010 2:01 AM
Suggested by Jon Irenicus

Bryan Caplan says the following in his critique of Rothbard:

http://econfaculty.gmu.edu/bcaplan/whyaust.htm

Rothbard's rejection of the utility function approach led him to make strange ad hoc concessions to it elsewhere in his writings. Using his value scale approach, Rothbard was able to derive the laws of demand and supply as theorems.[11] But then inexplicably in his later discussion of labor and land, Rothbard conceded the theoretical possibility of "backward" bending supply curves.[12] Furthermore, in his discussion of the economics of taxation, Rothbard admits the theoretical possibility that greater taxation of labor income could induce an increase in labor supply - even going so far as to mention a "substitution" and an "income" effect which his initial treatment of utility theory and demand utterly failed to mention.[13] What is interesting is that Rothbard was unable to derive the substitution and income effects from his value scale approach. Rather, he borrowed it from the standard utility function analysis, which shows that there are two different channels by which a price change induces a change in the quantity demanded. Thus, not only does Rothbard inappropriately dismiss the neoclassical approach to utility theory, but deemed it sufficiently fruitful that he borrowed its implications on an ad hoc basis.

To sum up, Rothbard falsely accused neoclassical utility theory of assuming cardinality. It does not. There is nothing actually wrong with Rothbard's value scale approach, but because the neoclassical assumptions are in some ways less restrictive than Rothbard's[14], neoclassicals made the important discovery that price changes have both income and substitution effects - a discovery Rothbard was unable to derive from his own postulates but conceded without explanation.[15] [Emphasis mine]

Here is the Rothbard quote from Man, Economy, and State about the backward bending labor supply curve:

http://mises.org/rothbard/mes/chap9a.asp

The general demand curve for a labor factor will also slope downward in the relevant area. One of the complications in the analysis of labor is the alleged occurrence of a “backward supply curve of labor.” This happens when workers react to higher wage rates by reducing their supply of labor hours, thus taking some of their higher incomes as increased leisure. This may very well occur, but it will not be relevant to the determination of the wages of a factor.  [Emphasis mine]

Basically, Caplan is complaining that Rothbard did not derive the "backward supply curve of labor" from his value scale approach, but instead ripped the idea off from the neoclassicals. 

From my reading of Rothbard, he did not think it was relevant to prove it can be done,  as he says this is an "alleged occurrence", but simply that others have mentioned this phenomena, which Rothbard conceded "may very well occur."

Here is how a "backward supply curve of labor" can be derived from Rothbard's value scale approach, at least for one person.

Let's say a man is harvesting apples from an apple orchard.  He can labor a maximum of 40 hours in one week, but he does not do that.  Instead, the man allocates some of the hours to labor and some to leisure depending on his wage.

His wage is in apples harvested.  He may work for himself, which anything harvested is his, or he may work for someone else, which he receives a share of what's harvested.  Either way, this does not change the outcome of the analysis.

The man has a total stock of leisure of 40 hours a week.  Every leisure hour is subject to the law of diminishing marginal utility.  For example, the 40th leisure hour has a lower marginal utility than the 39th leisure hour, and so on. 

This is because the 40th leisure hour is allocated to a less important leisure end than that of the 39th leisure hour.  Every labor hour takes away from the total stock of leisure. 

To determine how many hours of leisure he will exchange for wages, the man has the following value scale:

Supplier of Leisure Foregone Value Scale
(6 Apples)
9th Hour
(5 Apples)
8th Hour
7th Hour
6th Hour
5th Hour
4th Hour
3rd Hour
2nd Hour
1st Hour

To survive, he must maintain a minimum of 40 apples a week, which is his highest possible end.  For any apples harvested beyond the amount will be allocated to lesser ends. 

His value scale (mentioned above) says is willing to exchange 8 hours of leisure for 5 apples per hour, or 40 apples total.

However, he is unwilling to give up the 9th leisure hour, because that 9th leisure hour has a higher marginal utility than the 5 apples per hour.  For him to give up the 9th leisure hour, the wage must be of a higher marginal utility than the 9th leisure hour.  That wage would be 6 apples per hour.

Furthermore, for each successive leisure hour the man gives up, the previous leisure hour remaining in his stock of leisure would have a increasingly higher marginal utility.  For him to give up one more leisure hour, he must be willing to exchange this with an increasing wage of a higher marginal utility.

Here is his complete value scale for supplying leisure foregone (providing labor):

Supplier of Leisure Foregone Value Scale
(13 Apples)
16th Hour
(12 Apples)
15th Hour
(11 Apples)
14th Hour
(10 Apples)
13th Hour
(9 Apples)
14th Hour
(8 Apples)
13th Hour
(9 Apples)
12th Hour
(8 Apples)
11th Hour
(7 Apples)
10th Hour
(6 Apples)
9th Hour
(5 Apples)
8th Hour
. . .
1st Hour

For each possible wage, the man accumulates a number of apples in one week, which each apple harvested increases his total stock of apples.  Here is a table calculating the total stock of apples for each wage:

Wage (Apples) Marginal Labor Hours Marginal Leisure Hours Total Labor Hours Total Leisure Hours Total Apples Maginal Increase in Apples
5 8 -8 8 32 40 40
6 1 -1 9 31 54 14
7 1 -1 10 30 70 16
8 1 -1 11 29 88 18
9 1 -1 12 28 108 20
10 1 -1 13 27 130 22
11 1 -1 14 26 154 24
12 1 -1 15 25 180 26
13 1 -1 16 24 208 28

Notice how his total stock of apples (highlighted in yellow) is increasing every time he gives up a leisure hour (provide a labor hour).  His total stock of apples is also subject to the law of diminishing utility, meaning every additional apple has a lower marginal utility than the previous apple.

He will continue to accumulate more apples, and thus give up more leisure hours, until the marginal utility of the additional apple is less than the marginal utility of the leisure hour that would be given up. 

He may decide that a total stock of more than 200 apples is excessive at a wage of 13 apples per hour.

He can get rid of the excess apples by doing the reverse.  He exchanges the apples for more leisure hours.  This can be done because the marginal utility of the excess apples is less than the marginal utility of the leisure hours.

Here is the value scale for demanding leisure foregone (exchanging apples for leisure hours):

Demander  of Leisure Foregone Value Scale
(9th Hour)
20 Apples
(10th Hour)
19 Apples
(11th Hour)
18 Apples
(12th Hour)
17 Apples
(13th Hour)
16 Apples
(14th Hour)
15 Apples
(15th Hour)
14 Apples

Here is a table calculating the reverse exchange:

Wage (Apples) Marginal Labor Hours Marginal Leisure Hours Total Labor Hours Total Leisure Hours Total Apples Maginal Increase in Apples
14 -1 1 15 25 210 2
15 -1 1 14 26 210 0
16 -1 1 13 27 208 -2
17 -1 1 12 28 204 -4
18 -1 1 11 29 198 -6
19 -1 1 10 30 190 -8
20 -1 1 9 31 180 -10

If the complete table of the wages is graphed, this is how it would look like:

Here above is the "backward bending labor supply" curve.  Notice that the curve is really a juxtaposition of two schedules, an upward sloping supply schedule (in blue) and a downward sloping demand schedule (in red). 

For a wage of 13 apples per hour, or less, the man is a supplier of leisure hours foregone, meaning he willing to decrease his stock of leisure hours in favor of increasing his stock of apples. 

For a wage of 14 apples per hour, or more, the man is a demander of leisure hours foregone, meaning he is willing to increase his stock of leisure hours in favor of decreasing his stock of apples.

Here the mystery is resolved how a labor supply curve can be "backward bending."  It cannot, unless a demand curve is superimposed along with the supply curve, on the same graph. 

There it is, income and substitution effects without indifference curves.

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BlackNumero replied on Fri, Aug 6 2010 4:30 PM

Thanks ThinkBlue. I'm enjoying looking at your interpretation as I like looking at the graphs and find the approach very interesting, although I'm uncertain as to whether Rothbard would explain it like that, seeing as though even when the supply curve "backward bends", an individual will still supply labor hours (leisure forgone), but just at a decreasing rate with each higher hypothetical price. He wouldn't be exchanging apples for leisure but just supplying leisure for apples at smaler and smaller amounts at each price.

From my understanding of NirgrahamUK's post (would be interesting to hear his response) as well as the Joe Salerno link he posted, the backward bending supply curve would only occur when the ceteris paribus assumption that maintains Rothbard's Law of Supply is relaxed and the curve allows the marginal disutility of leisure to rise with each successive unit of labor sold (and considering them all homogenous goods). As Rothbard says on p. 574, as more labor units are sold if the m.u of money falls fast enough while allowing the m.u of leisure to rise fast enough then a backwards bending supply curve of labor will appear.

So, starting with an individual's value scale

5th Unit of Leisure

$5

$4

4th Unit of Labor

4th Unit of Leisure

$3

3rd Unit of Labor

3rd Unit of Leisure

$2

2nd Unit of Labor

2nd Unit of Leisure

$1

1st Unit of Labor

1stUnit of Leisure

At this hypothetical value scale, holding everything else constant, the individual will be willing to exchange 4 units labor for 4 dollars an hour. At a wage of four dollars an hour, the individual sees has the monetary gain ranking higher than both of unit of labor forgone and the unit of leisure. Although each unit of labor commands a higher utility ranking because of the Law of Marginal Utility, the individual will still exchange them and work for four dollars an hour on this value scale.

However, if we allow the m.u of leisure forgone to rise and change the value scale, then the schedule will become backwards bending

5th Unit of Leisure

4th Unit of Leisure

$5

3rd Unit of Labor

3rd Unit of Leisure

Now the value scale has changed (and strictly speaking the ceteris paribus assumption no longer holds), and the supply curve is backwards bending. Meaning that at some point after an individual had supplied some unit of labor, leisure moved and changed the value scale, prompting the individual to supply less labor at higher prices. Much like a Giffen good where a price change causes the good to be viewed differently, the higher price causes each unit of labor to be viewed differently because of the way leisure changes the value scale.

At least thats how I understand it.

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Think Blue replied on Tue, Aug 10 2010 11:36 PM

Hi BlackNumero.  This problem is much tougher than I thought.  For me, back to the drawing board.

I haven't forgotten about the thread.  It's just I'm not sure how this can be resolved, yet.

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Daniel James Sanchez replied on Wed, Aug 11 2010 3:15 AM

Caplan:
What is interesting is that Rothbard was unable to derive the substitution and income effects from his value scale approach. Rather, he borrowed it from the standard utility function analysis

Relevant: http://mises.org/daily/4223

Caplan:
It hardly makes sense to invoke an “all else equal” condition in cases where all else is of necessity never equal!   The key neoclassical insight is that price changes ipso facto change income.  Income effects do not happen at the same time as price changes by miraculous coincidence. They are inherent in the nature of price changes.

Income can still be held constant in an analysis, even with a price change, if you simply assume the price change's concomitant income effects are negated by other equal and opposite income effects.  That way you're isolating the direct effect of a single price change on action given a single set of value scales, instead of looking at a shift from one set of value scales to an entirely NEW set that emerge from a complete revolution of the market data.

 

It hardly makes sense to invoke an “all
else equal” condition in cases where all else is of necessity never equal !
"the obligation to justice is founded entirely on the interests of society, which require mutual abstinence from property" -David Hume
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BlackNumero replied on Wed, Aug 11 2010 9:31 PM

Agreed. Although I still think it was in bad taste for Rothbard to include backward bending curves without explicitly mentioning this. It can be very confusing when its not explictly mentioned the ceteris paribus rule is being relaxed.

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