Price is correlated with perceived value, which includes quality, brand recognition, brand opinion, current style, and a long list of other factors.
(And, yes, this is a horrible use of the word 'correlated.' 'Derived from' or 'based on' would be much better.)
That is an incredible assertion.
I am willing to bet $10,000 that I can find a product where price is correlated with quality (assuming we can agree on a definition of "quality" for a particular product). How about GPUs with "quality" measured by average FPS in a selection of AAA games at 4K?
Damn those sound good.
Tools.
Musical instruments.
For both of those there is strong correlation between price and quality, and the users of both of those are picky for a good reason: the quality of the work they produce depends to some extent on the quality of the tools they use.
Now, bad craftspeople and bad musicians will produce crap no matter what the tool, but good craftspeople and good musicians can definitely produce something better with good tools and instruments and are prepared to pay for that.
Good: Reasonable price / performance ratio, at least until you get up into "virtuoso" instruments, where I will never go anyway. With violins, there is also relatively little brand recognition. Nobody will know what you paid unless you tell them.
Bad: Remarkable lack of good objective models to predict the quality of an instrument from details of its materials and construction
Though I suppose "purity" is considerably more exact than, "quality".
I believe that what you are meaning to say is that price responds to demand, and demand correlates to the perceived value of the product. In turn, perceived value is often but not inherently responsive to information about the 'quality' of the product.
Instead, perceived value is a summation of various forms of utility, only some of which are affected by 'quality'.
Furthermore, any determination of 'quality' is rarely made upon information that is as complete as the information available to an attempt to methodically determine the relative 'objective' quality of substitutable products, because of the disutility of conducting such an assessment. More often, the determination is based upon perceived indicators of quality, which are known to be unreliable.
This diminishes the relative importance of 'quality' in the formation of the perceived value of the product, which accounts for the importance of alternate, more reliable information about the product, such as the social signalling value, and especially easily manipulable information such as indicators of "style" and the narrative about the product; this in turn explains the willingness of product vendors to invest in advertising.
With regards to the use of the word 'correlate': price is of course correlated with demand, with correlates to perceived value, which tends to correlate to perceived quality, which tends to correlate to quality. Since correlation requires transitivity, price does then tend to correlate to quality. So what you actually said is incorrect.
I believe that your intended assertion (as I understand it) is also incorrect and over-broad, because in many instances a highest-information assessment of quality is the sole determining factor of the perceived value of the product. This is often true in business to business transactions such as part sourcing, where a person with specialized information is in charge of purchasing, and with products in a commodity market such as grain or beans, where a standardized determination (and guarantee) of quality has been established.
(Thanks for the opportunity I've taken to sit for a while and practice arranging my thoughts, which is the reason that my reply is so extensive.)
What? This is certainly not the case.
A standard correlation is the cosine of the angle between two vectors in an n-dimensional space. The correlation is 0 when the two vectors are orthogonal to each other - when the angle between them is 90 degrees.
So consider three vectors in 2-space: (0, 1), (1, 1), and (1, 0). (1, 1) correlates with each of the other two at about 0.7, the cosine of 45 degrees. The correlation between (0, 1) and (1, 0) is zero.
I'm fairly sure that the basic intuition that I'm trying to describe is correct, so I would welcome an alternate description.
Just like being related to someone doesn't mean you're related to that person's relatives.
Given the correlation of B to A and to C, you could probably prove a bound on the correlation of A to C, but as the chain between A and C lengthens, or any of the correlations moves noticeably away from -1 or 1, the bound will swiftly approach [-1, 1].
The word 'correlate' predates the definition that you are familiar with, however. I would ask you to consider that it may often be used differently by people who lack the knowledge or cause to preserve it for the exclusive meaning with which you are familiar.
This is what I meant. I wasn't arguing that your vector example was perhaps somehow in error. It actually seemed pretty clear and convincing to me that what I was saying was absolutely not true of correlations thusly defined.
If (1) there is a rate 'a' that always and only increased when a rate 'b' increased, and always and only decreased whenever that rate 'b' decreased, then perhaps we could describe that as a correlation between the two rates.
If, in turn, (2) there was a rate 'c' that always and only increased when the rate 'b' increased, and always and only decreased when the rate 'b' decreased, then this would be a second relation which was the same kind as the first.
It seems clear to me that lemmas 1&2 above imply the existence of a third relationship, between 'a' and 'c', which will also be of the same sort as the relations a<->b and b<->c.
If this is in error, then I am extremely surprised and concerned about my mental faculties, so please be kind and demonstrate the error. I need to know, and go see a doctor.
If this is not in error, then this type of relationship will have a property which we might or might not be able to describe as transitivity.
I have a long-ago grounding in logic and set theory, but my understanding of mathematics is largely casual.
It is entirely possible that I am using the word "correlation" in a place where a different word would better serve my meaning. I am satisfied that my use of the word is not compatible with your own understanding, so I imagine such a word must exist. In my response, I had been asking you what that word was.
It is also possible that I am misusing the concept of transitivity, although in that case, again, I really am quite taken aback, and I am certainly not meaning to ask you to undertake what would then be a significant chore of remedial education.
But of the relationships you list in your original comment ("price is of course correlated with demand, with correlates to perceived value, which tends to correlate to perceived quality, which tends to correlate to quality"), none of those relationships satisfy the criterion you describe, except possibly the relationship of demand to perceived value.
Re: what I meant by related, it would be normal to consider that X's cousin through his father is unrelated to X's cousin through his mother (even while both are related to X), and perfectly defensible to consider that while you personally are related to your son's daughter, you are not related to her mother.
> It is also possible that I am misusing the concept of transitivity
Not in your recent comment. I was actually confused about this before -- where correlation takes a numeric value between -1 and 1, it is not obvious what it would mean for it to be transitive, since transitivity is a concept that applies to boolean functions.
I'll try to take the step of testing out my intuitions a bit on paper in the future.