What was the basis for valuating large territorial purchases by USA?

In both cases there was a country in need of money that had a large amount of land across an ocean that was hard for them to manage, with a much closer country to said land that had lots of money and wanted land. In both cases those countries were in a position that attempting to defend these remote territories would be far too much expense and likely fail should any country mount a reasonable effort to invade.

The value of the land was only part of the equation, but in both cases it was a minor one compared to other factors. Deciding the final price was a negotiation starting at the true value of the land, then subtracting X for being desperate for money, subtracting Y for only having one buyer, subtracting Z due to the risk in maintaining possession of that land. France and Russia really didn't have anything to bring to the table to get a better price.

The original offer was $10 million for New Orleans, plus "East and West Florida" (basically Florida plus the coastal regions of Alabama and Mississippi.The French, who didn't have clear title to Florida, offered New Orleans plus the rest of Louisiana, which they did have better title to, for $15 million instead.

Of this New Orleans, the city, was the easiest to value. It had an established population, with known figures for annual trade, tax revenues, etc. Given its strategic importance at the mouth of the Mississippi River, the Americans were probably willing to pay something extra, over the value of an equivalent port someplace else.

Let's say for the sake of argument that New Orleans was worth X million, meaning that Florida was worth $10-X million. Then the rest of "Louisiana" (ex New Orleans), would be worth $15-x million.

Let's say X= 5, so $10 million for the rest of Louisiana. There are just under 1 million square miles in the territory (close enough for our purposes), so each square mile costs just over $10.

There are 640 acres per square mile. I remember reading from a history textbook in grade school that in colonial times, good farmland in a developed area cost $1 an acre. Even if it's not exactly correct, such farmland could be valued on a similar basis.

"Louisiana" cost perhaps $10/640 acres or something like 1.7 cents an acre for admittedly a "grabbag" of land of varying quality. Compared to $1 an acre, this would look like a bargain.

That's the way it would look to the buyer. The seller, Napoleon, would probably say, "I need $15 million. What can I sell to raise it? Hmmm...Louisiana might do the trick."

Alaska was different from Louisiana (the latter spanned the same LATITUDES as the rest of the United States. But Alaska was dubbed "Seward's Icebox," the argument being that it was "too cold" (based on the technology of the time) to be of use. But the counterarguments are that 1) technologies change, and for the better, and 2) PART of Alaska is usable, and will make up for the part that isn't.

You're over thinking this.

In both cases, we wanted land, they wanted money. The discussion was a negotiation like any other - how much / how little money / land can we extract before the other party walks away?

It's haggling, pure and simple. I doubt there was any analysis done at all, except "Can we afford this?" and "Does this seem like a good deal?"

We're talking the 1800s here, not the same sophisticated real estate and finance regime we have now. I'm not saying the people in the 1800s were stupid by any stretch - I'm just saying, don't apply modern notions of finance and real estate to a gentle mans game of yore.

Additionally, the valuation of real estate, even today is done for two reasons:

1. To help a mortgage lender know what a reasonable loan to extend on a property would be (most banks will not lend more than 95% of the value of a property)
2. To help governments figure out how much property tax would be do.

For a government acquiring property, neither situation applies, so there is no need to assign an actual value. So, again, it really boils down to "How much can we get for as little give as possible."