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We're considering a nation for which the basic necessities are met by another nation, for free.
Now, the nation has lots of currency notes. But it produces nothing. Would I be right in saying that the value of these currency notes is an absolute zero?
No, because people could still find value in the notes. For example, let's say the nation had the most transparent and hard-lined central bank in existence. Then, people of other countries might find the notes worthy.
Of course, there is some intrinsic value to the paper notes as well. But it would have to be assumed that such value is zero, as producing something would raise the GDP.
In practice, the whole case is absurd, but the answer would almost certainly be no, no one would exchange the notes for anything.
As for the mechanics of the question in the comment section, let's consider the case through the equation:
$Y = C+I+G+NX$
And also the central bank's profit/balance sheet:
$Assets-Liabilities=Equity$
$\Delta equity = profit$
If the central bank bought some valuables to back it's money supply (gold for example), the investment would raise and NX term would decline, cancelling out and also creating a trade deficit. All of the money would flow outside the country, so no citizen could use the currency to buy anything.
If the central bank itself tried to fund the increase in consumption (or give the money to it's citizens to do so), the trade deficit and consumption would increase. However, the assets of the central bank would not increase (since consumption items depreciate). In the process the central bank would become insolvent as it would suffer a loss. Then, people would not be expected to use the service to an even higher degree. Although, the central bank could try to leverage it's balance sheet, buying investment products from foreign nations to cover for this.
In the case that there was an income stream, such as donations or foreign investments, the government could easily have a solvent central bank and thus I don't see why the money would not have value. (of course, 0 GDP would require that the central bank is operated for free). The question becomes a lot different if we assume fiat currency, but I won't go into that.
In any case, this is a major hypothetical with unrealistic assumptions.
I'd say no. No production is required for a money to have value. Money can be valuable because it facilitates trade. That is, whatever good is used as money can be more valuable than its intrinsic value because it is an liquid instrument that can be used in trade with others to solve the double coincidence of wants problem. Even pure endowment economies can utilize money for this purpose as long as people are not endowed with bundles that are also their equilibrium bundles and the economy is one in which money has a use (see Starr (2003) for several examples of how this can happen). Essentially, money is needed in circumstances where barter and good specific debt is costly or impossible.
For a famous example of an economy without production that did use money consider Radford (1945), The Economic Organisation of a P.O.W. Camp. In this paper Radford documents how the POW inmates, despite having no production to speak of, with all goods provided by either the captors or the Red Cross, use loose cigarettes as money to price goods and solve the double coincidence of wants problem.
When inhabitants of that country have to pay taxes in their country's currency as can be expected , the notes would also have a value.
I don't think a "nation that does not produce goods or services" is possible even as a thought experiment.
A "nation" is a sovereign state, which means it has people and land. These people are a community of some sort and have established some sort of government.
We know that these people have to live, so therefore they will produce some good and service in order to sustain themselves. Then inevitably they will develop a social and then economic system and trade goods and services with each other.
Now let's assume that you're asking about a nation that produces no surplus good or service. That too is impossible because surplus has been inevitable since humans evolved beyond a hunter/gatherer society (and since the proposal of the Original Affluent Society theory we can theorize that there was reasonable surplus even in hunter-gatherer societies). Not to mention if they're creating currency notes and even attempting to trade with other states, that they're probably going to advanced their societies and find ways to build such surpluses. They will find ways to supply the needs to other country. Consider the Philippines which regulates labor exportation.
Another consideration is that labor exploitation will also be inevitable in this "non-producing" nation. So it would inevitably produce something at least for someone else. Especially considering that in this hypothetical other developed nations exist.
Miracle exception incurred and trying to theorize with this unrealistic hypothetical, I believe the currency would have a small amount of value due to some form of methodological individualism incurring a value on the currency in some microeconomic systems of said nation.
