Friedrich Hayek once wrote that he wanted to "... avoid giving the impression that I generally reject the mathematical method in economics. I regard it as indeed the great advantage of the mathematical technique that it allows us to describe, by algebraic equations, the general character of a pattern even where we are ignorant of the numerical values determining its particular manifestation."

I realize the many flaws and impracticalities in applying the following mathematica model to the real world, but ignoring those:

If the value of buying some product to an individual is, expressed in dollars, $X,

And the value of that first individual purchasing the product to a second individual is, expressed in dollars, $Y,

Assuming that the second individual refuses to involve himself in the contract, then the product will be bought whenever the product is $X > $Cost of Product.

But assuming the second individual does involve himself, and both are rational, then the product will be bought whenever $X + $Y >$Cost of Product.

Correct? Do positive externalities in microeconomic theory only exist if one individual isn't acting rationally?

You cannot express the value of buying a product to an individual in dollars. Since we can't measure value (since it is subjective), we cannot make interpersonal "utility" calculations either.

The value figure I'm using is simply the maximum price the individual is willing to pay for the product. That does not involve subjectivity in the calculations. It is difficult to ascertain those values in practice, but are they theoretically invalid? By that measurement we can also make interpersonal calculations.

"The value figure I'm using is simply the maximum price the individual is willing to pay for the product. "

Tha'ts the thing. The maximum price an individual is willing to pay for some good or service is not a measure of the subjective value that individual holds for the same. So they are theoretically invalid.

So if Able would pay $100 for a $10 service while Baker would pay $200, this does not necessarily mean that Baker values it more.

But I'm not directly comparing the individual's subjective preferences, only whether they would purchase an item. Surely you don't deny that if Able would pay $100, and the price of the service is $10, Able will buy it? Similar calculations are valid for Baker. And we can combine them to share the situation.

For example, using your numbers, if Able (max price for an item $100) and Baker (max price for an item $200) could buy two items for $250, they would do so, correct?

"But I'm not directly comparing the individual's subjective preferences, only whether they would purchase an item. Surely you don't deny that if Able would pay $100, and the price of the service is $10, Able will buy it? Similar calculations are valid for Baker. And we can combine them to share the situation.

For example, using your numbers, if Able (max price for an item $100) and Baker (max price for an item $200) could buy two items for $250, they would do so, correct?"

Yes. So let's go ahead and put it in the context of positive externalities for a Private Defense Agency (as we were discussing earlier).

$X is the price, (not the value) that Able is willing to pay for a product and SY is the price (but again, not the value) that Baker is willing to pay.

Let's say $Cost of Product (A month of Security Service) is $10.

Yes, they will both purchase the security service. However, it is not true that the product will be bought if $X + $Y > $CoP. This is only true if $X and/or $Y >$CoP.

For instance, if $X = $7 and $Y = 7, then SX+$Y = $14 which is > than $CoP. But the product will not be purchased in that case.

Perhaps the terms need to be specified futher. The product is the defense of them both, and they cannot collectively cooperate, and I assume that it benefits them $X and $Y individually.

Thus the optimal decision to buy the product is if $X + $Y > $CoP.

So a situation where $X=$7, $Y=$7 and $CoP=$10, will be misallocated, as a purchase is justified, but will not happen.

A purchase will only happen if it is justified by either individual: if $X > $CoP or if $Y > $CoP. So, if those both are not true, and $X + $Y > $CoP is true, there is a misallocation.

Perhaps the terms need to be specified futher. The product is the defense of them both, and they cannot collectively cooperate, and I assume that it benefits them $X and $Y individually.

Thus the optimal decision to buy the product is if $X + $Y > $CoP.

So a situation where $X=$7, $Y=$7 and $CoP=$10, will be misallocated, as a purchase is justified, but will not happen.

A purchase will only happen if it is justified by either individual: if $X > $CoP or if $Y > $CoP. So, if those both are not true, and $X + $Y > $CoP is true, there is a misallocation.

If Y really valued it enough to pay for it, he would approach X and offer to chip in.

"Perhaps the terms need to be specified futher. The product is the defense of them both, and they cannot collectively cooperate, and I assume that it benefits them $X and $Y individually."

Are you saying that the $10 security service specifies security for two properties?

Also, yeah, I headed off to sleep. Didn't realize it was getting so late.