Given use-valued goods A, B, and C, consider the following proposition --

If A is ranked higher than B on an individual's single subjective value scale, then the bundled pair of A and C will also be ranked higher than the bundled pair of B and C.

Refute this proposition by counterexample.


Take a case of complementary

Take a case of complementary goods: hot dogs and hot dog buns, for example.

Suppose Don prefers a hamburger without a hamburger bun more than a hot dog without a hot dog bun.

A = hamburger without a hamburger bun
B = hot dog without a hot dog bun

A > B

Don also prefers a hot dog with a hot dog bun more than a hamburger with a hot dog bun.

C = hot dog bun

Therefore, B+C > A+C

Q.E.D.

Micha, Very good! Now refute

Micha,

Very good!

Now refute the proposition without using complementary goods.

Regards, Don

I could do it with

I could do it with substitute goods as well.

Suppose Don prefers butter more than lox.

A = butter
B = lox

A > B

Also suppose that for Don, butter and cream cheese are substitute goods (he may still prefer one to the other, but they both serve similar purposes in terms of dairy spread), and because of diminishing marginal utility, he would rather have lox and cream cheese than butter and cream cheese.

C = cream cheese

Therefore, B+C > A+C

Micha, That sounds right,

Micha,

That sounds right, although I wouldn't recognize a lox even if it came with an english label.

I was thinking in terms of a Coke (A), a Pepsi (C), and a sandwich (B).

Are there any other alternatives to complementary or substitute goods?

Regards, Don

PS - Sometimes software is too smart for its own good.

(A) (B) (C) (D) (E) (F) (G) (H) (I) (J) (K) (L)

(M) (N) (O) (P) (Q) (R) (S) (T) (U) (V) (W) (X)

(Y) (Z)

consider: A = picture of

consider:
A = picture of Bush
B = picture of Kerry
C = black magic marker
There are certainly people for whom A > B, but B+C>A+B because in the latter case you could make interesting defacements of the picture. Does this still count as "complementary" when the use gotten from the picture is the opposite in the cases with vs without?

Oh no, my brain hurts at

Oh no, my brain hurts at this time in the morning looking at stuff like this. Isn't it all subjective? eg:

Given a long car journey, and I'm only allowed to take one album from three, my preference might be this, for yes, I am that weird:

A: Abba Gold Album
B: Led Zeppelin IV
C: The Corrs, Talking on Corners

But if I'm allowed to take a 2 album combination, of your design, I might prefer 'Led Zeppelin' and 'The Corrs' in combination (B+C), as its a nice contrast, rather than 'Abba' and 'The Corrs' (A+C), in combination, as that's a bit samey, though actually, although you haven't given me the choice, I would personally prefer A+B, Abba and Led Zep IV (and she's buying a stairway of valuation, to consumer heaven).

But then isn't A+C a new product, a new 'D' multipack? And isn't B+C a new product E, another bargain bucket multi-pack? And then, A+B, is F. So, then my ordinal value listing would be:

F (A+B)
E (B+C)
D (A+C)
A
B
C

But I might still be even more perverse and prefer a combination of Led Zep and The Corrs, to a combination of Abba and Led Zep, even though given only one album to choose from, I would go for Abba Gold, giving my perverse self a further alternative stairway of choice, as:

E (B+C)
F (A+B)
D (A+C)
A
B
C

But this only applies to me, because I have weird music tastes. Err...I think?

Dammit! I'm gonna have to read Human Action again! ;-)

In furtherance to the above,

In furtherance to the above, if only given one album to cross America with, in a car, coast to coast, I would always make it 'Simple Minds, New Gold Dream'. But if allowed to take more than one album, I would not allow that damned thing anywhere near the car, to avoid - 81, 82, 83, 84...sorry, went a bit haywire there - total New Gold Dream induced madness! :-)

A>B implies (A+C)>(B+C) is

A>B implies (A+C)>(B+C) is false, because

A+C=X
B+C=Z

and there is no reason to think that X>Z. The reason for this is that the former combinations of goods creates different goods, which could get different situation in the individual value rank in a certain point of time.

A = chocolate B = grilled

A = chocolate
B = grilled chicken
C = garlic sauce

Don, As an example, I like

Don,

As an example, I like to highlight text and take notes when I read non-fiction, but not when I read fiction.

My rank list of ends looks like this:

.
.
.
E,n = reading fiction
E,n+1 = reading non-fiction
E,n+2 = reading non-fiction with a tool that allows highlighting of text
.
.
.
E,n+x = reading fiction with a tool that allows highlighting of text (little value in this for me)
.
.
.

The goods available to me are:

A = Harrisons Principles of Internal Medicine
B = The Da Vinci Code
C = highlighter

If only A and B are available, the utility of B is greater than A.

If, in addition, C is also available, the utility of A+C is greater than B+C.

This is because A+C allow me to pursue higher ranked ends than B+C.

Jonathan, Review your

Jonathan,

Review your ranking of ends. Since the availability of the marker doesn't require that you actually use it, adding the marker to either fiction or non-fiction must result in a higher, but not necessarily adjacent, ranking.

Subject to counterargument.

Regards, Don

Oops - you're right. The

Oops - you're right. The ends should be ranked in this order:

.
.
.
E,n = reading fiction
E,n+1 = reading non-fiction with a tool that allows highlighting of text
E,n+2 = reading non-fiction
.
.
.
E,n+x = reading fiction with a tool that allows highlighting of text (little value in this for me)
.
.
.

Since the availability of the marker doesn?t require that you actually use it, adding the marker to either fiction or non-fiction must result in a higher, but not necessarily adjacent, ranking.

I don't know if I believe this. When reading fiction, a highlighter has zero value for me. Perhaps E,n+x should not even be included in the ranking of ends.

Jonathan, This still doesn't

Jonathan,

This still doesn't look right.

From your description, I would have said that the highest ranked end was :

NF + available Highlight

then

F + available Highlight (even if not used)

then

F

then

NF

No?

Regards, Don

Don, No, I usually prefer

Don,

No, I usually prefer reading fiction to non-fiction. So my ends would be (usually) ranked as such:

F
F+available highlight
NF+available highlight
NF

And now I have figured out what the original question was getting at, but will remain silent for the interim. The conclusion is not what I originally expected.

Would a difficulty in

Would a difficulty in consuming multiple goods from the lineup constitute a complementarity? The first example that sprung to my mind was three potential mates. Though some might prefer an alternative world, there are certain, erm, difficulties associated with, uhhhhh, "consuming" more than one mate. In that light, one couldn't even assume that A+C>A much less A+C>B+C.

Though I'm tempted to call that a substitute relationship, my usual conception of substitutes doesn't fit. Chicken and beef are subsitutes but having chicken doesn't make beef's value negative. Same for coffee and tea, lagers and ales, Wintel and Mac, etc etc etc. I'm inclined to describe the choice-of-mates scenario as exhibiting "negative complementarity".

Or maybe I'm just not thinking straight...

So, let's categorize all the

So, let's categorize all the arguments:
- Complementary goods
- Law of diminishing value

what else?

Noah, This is a good point.

Noah,

This is a good point. As Micha alluded in his second comment, we can consider substitute goods as being subject to diminishing marginal utility. You have extended that to the point where the diminishment is so great as to produce a negative marginal utility.

If a single driver values a second identical automobile, he may well find that storage and insurance costs have produced a negative marginal utility for the additional automobile. This may apply just about as well if the second vehicle is not identical, but a substitute pickup truck instead.

Regards, Don

But he would still prefer

But he would still prefer the bundled pair, so long as a market exists for him to sell the unwanted commodity.

Micha, "But he would still

Micha,

"But he would still prefer the bundled pair, so long as a market exists for him to sell the unwanted commodity."

There is some validity to this, but it is ruled out by the original problem statement that all three goods are use-valued, as opposed to exchange-valued.

Alternately, it would seem to make sense that if the exchange value is brought into play, then the acquisition cost should also be considered. In a money economy, you can't just compare apples and oranges, but need to include their purchase prices as well when making a choice of actions.

Regards, Don

I think the proposition is

I think the proposition is only false when B & C are complementary. There are various sneaky ways to make it false, but they all come down to complementarity. For example, there is some good D in the persons possession which has a high value use when combined with B & C, but not with A & C. This is really just another way of saying that, to the person, right now, B and C are complementary (more so than A and C).

Let a,b,c represent the numerical utilities of A, B, and C. Let bc, ac represent the numerical utility of having (B and C) beyond the utilities of B and C (could be negative, as in the case of multiple wives). So:

Utility with A, C: a + c + ac
Utility with B, C: b + c + bc

if B&C > A&C, then (b + c + bc) > (a + c +ac)
thus (b + bc) > (a + ac)

And we were told a > b, so this tells us that (bc -ac) > (a-b) or bc > a - b + ac

We know (a-b) is positive. Ah....but this could hold because ac is negative. So that gives us an alternative answer. Either B & C are complementary, or A & C are negatively complementary.

Patri, "I think the

Patri,

"I think the proposition is only false when B & C are complementary. There are various sneaky ways to make it false, but they all come down to complementarity."

No, A and C can be substitutes as well.

Assume that I have a preference for satisfying thirst over satisfying hunger, and that I prefer satisfying both to either one alone.

Let A = a Coke
Let B = a sandwich
Let C = a Pepsi

Then A and C only more than satisfy thirst

And B and C satisfy both thirst and hunger and are preferred and this occurs because A and C are substitutes in satisfying thirst, but don't address hunger.

BTW, Austrian Economics absolutely rejects numerical measures of utility. Every individual has a single subjective scale of values which is unruled and only indicates the ordinal rank of the various entries. No operation except for comparison is valid.

Regards, Don

Don - your example fits my

Don - your example fits my reasoning exactly. A and C are negatively complementary - their joint utility is less than the sum of their individual utilities. This is the opposite of being complementary, in which joint utility is greater than the sum of individual utilities. Substitutes are one example of a negatively complementary good.

I don't know that utilities in the real world are numerical, but it seems to me like a good approximation which simplifies reasoning. Heck, in the real world utilities are inconsistent and sometimes intransitive. ie you can get people to demonstrate different rankings based on irrelevant matters of phrasing or presentation. I don't know much about Austrian econ - do they have good reasons to reject numerical utility?

Don - your example fits my

Don - your example fits my reasoning exactly. A and C are negatively complementary - their joint utility is less than the sum of their individual utilities. This is the opposite of being complementary, in which joint utility is greater than the sum of individual utilities. Substitutes are one example of a negatively complementary good.

I don't know that utilities in the real world are numerical, but it seems to me like a good approximation which simplifies reasoning. Heck, in the real world utilities are inconsistent and sometimes intransitive. ie you can get people to demonstrate different rankings based on irrelevant matters of phrasing or presentation. I don't know much about Austrian econ - do they have good reasons to reject numerical utility?

Patri, "...your example fits

Patri,

"...your example fits my reasoning exactly. A and C are negatively complementary - their joint utility is less than the sum of their individual utilities. This is the opposite of being complementary, in which joint utility is greater than the sum of individual utilities. Substitutes are one example of a negatively complementary good...."

When you say that A (a Coke) and C (a Pepsi) are negatively complementary, you cannot logically avoid saying that A (a first Coke) and C' ( a second Coke) are also negatively complementary since nothing prohibits Coke and Pepsi from being considered as arbitrarily close substitutes for one another for a given individual at a given time.

The proper analysis starts with the diminishing marginal utility of a Coke as the quantity is increased from one to two and the fact that a Pepsi can be a close substitute for a Coke implies that the marginal utility of the Pepsi may approach that of the second Coke.

Of course, the Coke and the Pepsi are only means to the end of satisfying thirst. It is the ends to which goods (means) contribute that are what is really being ranked.

I think your example is a good demonstration why utilities cannot be summed. All that is possible is for combinations to be individually ranked.

Regards, Don

And now I have figured out

And now I have figured out what the original question was getting at, but will remain silent for the interim. The conclusion is not what I originally expected.

In case anyone was wondering, my thinking is that my earlier thinking when I wrote the above was not well thought-out.

Patri, Rothbard on the

Patri,

Rothbard on the measurability of utility, in Man Economy and State, The Scholar's Edition, page 311 --

"...We have reiterated several times that utility is only ranked, and never measurable. There is no numerical relationship whatever between the utility of large-sized and smaller-sized units of a good. Also, there is no numerical relationship between the utilities of one unit and several units of the same size. Therefore, there is no possible way of adding or combining marginal utilities to form some sort of ?total utility?; the latter can only be a marginal utility of a large-sized unit, and there is no numerical relationship between that and the utilities of small units...."

Regards, Don