This value you're using, do you mean use-value or exchange-value? In mainstream economics, all value is subjective value, so changes in preferences do change value.

"Value" with out any qualification means "labor value". However the total sum of all "labor value" is equal to the total sum of "exchange-value", so when speaking in terms of macro they are equivalent.

How does one determine labor value?

There are discussions about that, but the most simple answer i can give is the average costs of production measured in abstract labor-time. SNLT is a theoretical construct that isn't directly measurable but helps highlights why certain phenomena happen (much like "marginal utility").

Why does a price rising necessarily mean more profits?

It doesn't, but i was trying to make a simple example.

Why do firms take on more labor, and not more capital?

Capital is referred to as "stored-labor", "past-labor" or "dead-labor". Taking that into account, the only input we as humans have actual control is "labor", past or present ("land" also exists of course, but land can only be put to action through labor, by itself it is not an input we actively control in production).

So this vague term "value" - which isn't the same value used when discussing subjective value - is determined by how much labor is used in the economy.

Okay, i admit i have been a bit vague so i will try to spell it out clearly from scratch. When we enter the market we enter into value relationships, in which the market process systematically establishes quantitative relations between heterogenous goods, these relationships are the phenomena we call "value". Marx posits that the reason why these relationships are there is because of society's need to direct, compare and economize on human labor, and hence the reason heterogenous commodities are made commensurable is because of their common origin in human labor.

Marks posits the existance of a "natural" exchange ratio that exists in this market process which is called the value of the commodity in question. The value of a commodity is defined as being directly proportional to the abstract labor-time that is socially necessary to re-produce it. "Exchange-value" is then the temporary expression of value, the "Price" of a commodity we observe at any specific moment of time, and exchange-values systematically deviate from the natural exchange-value in the economy to direct the flow of labor. "Use-value" is the utility of a commodity, subjective and incommensurable, it plays an important role by determining whether something can have an exchange-value (a good with a "value" but with out a "use-value" will not express an "exchange-value", as no one will want to buy it) and by determining temporary exchange-values (by influencing market demand), but use-value does not have a quantitative effect on the natural exchange ratio we called "value", because the natural exchange ratio is determined solely in production.

When i mentioned that the price "systematically" deviates from the natural price, i meant that deviations are not erratic, but follow a general law. The law is that the degree a commodity is sold below it's "value" tends to be the degree that another commodity is sold above it's own "value", the rises and falls tend to cancel each other out, so the total amount of "prices" is equal to the total amount of "value". This implies two things: That the total amount of economic value "represents" the total amount of "labor" that it directs, and that the only input responsible for directly increasing RGDP is labor. This equality is summarized by the phrase "Labor is the substance of value".

Okay. Who cares? This is uninteresting.

I'm trying to summarize the main point of the first 3 chapters of Kapital for you. There is an entire book in 3 Volumes after that, this large books being an inter-disciplinary works of economics, sociology, philosophy and history that take the main concepts i just laid out as the starting point; and then there are entire fields that have been based on or strongly influenced by the many points raised in these books. If someone explained to me the basics of marginal utility i could also say "Well this is obvious" and dismiss economics entirely, but i would be very wrong to do so, wouldn't i?

Again, Marx doesn't see that simply taking the market price of a good and comparing it to another good is a meaningless exercise.

He's not doing it as an "exercise". He points that the market process itself works by doing this on a systematic basis, and then wonders "Why, how, and what forces are at play here?". To him the answer cannot be discovered with out a reference to production.

So what? You're not exploring "social relationships"

Either you are missing something or you are implying that economics isn't a social science. I mean, if someone told me that economists don't believe exchange is a social relationship, i would assume it's a bad straw man of economics.

Assuming comparative advantage doesn't exist. There could be equal amounts of coconut and fish, but Eugene could be better at getting coconuts and Ludwig could be a better fisher.

Alternatively, Eugene could be the private owner of the land where all coconuts are while Ludwig is the private owner of the beaches. And we know that comparative advantage and how we allocate who is better at doing a certain thing that are both aspects of production, which are directly related to the property institutions and dominant relations of production of society. Either way it returns to the fact that you can't make sense of a theory of value with out an analysis of production.

You don't need "capitalism" to trade. Native Americans traded with each other, and they weren't capitalistic. So already this guy is bringing in some weird assumptions.

It doesn't say you need "capitalism" to trade, only that you need certain relations of production, and those relations of production partially determine the character of that trade. If Native Americans enter relations of production different than capitalist ones and then establish a market, obviously this market will behave differently, and a different theory needs to be made for it. I'm not sure if we can model Native American economies as "market" or as "trade" as commonly understood by modern economics though, i'm not exactly an expert on it but IIRC peoples like the Inca Empire did not have money or prices and didn't have an economy based on "barter" either (it was partially gift exchange and partially a command economy), and many Native peoples lived by "gift exchange" as opposed to "a market" aswel, and no "pure" barter economy has ever existed by itself. As far as i know mainstream neoclassical economists have not yet tried to develop tools to model this stuff.

Why the heck do I buy a coconut if it doesn't give me utility? Do Marxists reject utility maximization? If so, again, rejection of supply and demand.

No, that was not point. The point is that in a direct barter exchange our differences of utility between fish and coconut are readily apparent and we can come up with an exchange ratio on the spot, but in the indirect barter of money-exchange our differences in utility are not apparent as we are comparing out personal utility to the utility of "money" (which has no utility by itself, it's only utility is that which it can buy) and we are faced with pre-determined prices that influence our decision making. Whether we chose to buy or not is an input of information that can change those prices of course, but the barter exchange and the monetary exchange are very different for these reasons even if their internal logic is the same.

Marx did explicitly mention that exchange happens because people want to obtain use-values (or "maximize utility"), he explicitly says in Das Kapital:

With reference, therefore, to use-value, there is good ground for saying that “exchange is a transaction by which both sides gain.”

So yeah, I am calculating things based on my preference scales with respect to the price of a good. This is how consumer optimization works, this is how we get demand curves.

You just explained the way how market prices influence out utility maximization and hence why our utility-maximizing decisions in barter and in a monetary market are different. But the thing is that these utility-maximizing decisions happen in respect to existing market prices, those market prices themselves related to the systematic, long-run exchange ratios that are much more closely related to production than to your actions in the supermarket.

So basically we've just come back to what Austrians believe versus Mainstream people believe, except with Marxists. Alright, well this just puts up a barrier between the two camps.

And this is why science is hard. Doubly so for social sciences.