(Am I correct?)
As far as I understand, a pure endowment economy means that good just appears in the country without any labor involved in producing them. Like maybe given country lives entirely from aid/gifts/tribute sent from abroad, literally producing nothing of its own (including services).
There are three ways that show that GDP=0
1.GDP stands for Gross Domestic Production. In a pure endowment economy there is no domestic production as such, thus GDP should be equal to zero.
2.We can try to calculate GDP using the income approach. This can result in negative GDP.
GDP=TNI+Sales taxes + Deprecation + NFFI
TNI=Wages + Rent + Interest + Profit
Wages=0 because nobody produces goods (including services!) inside the country.
Profit>=0. There is no profits domestically because nothing is produced domestically, but it's possible for local corporations to get profits aborad.
Interest>=0. Lending money is a service, but there are no local services in our economy, thus no lending to by local lenders to the citizens/firms of the country. But it's possible that local lenders lend their their money abroad.
Rent=0 because renting land is a local service and there are no local services (renting a building isn't an option because nobody builds anything in the country and it's not possible to sent a built building from abroad). So either nobody lends their land OR all lands are common property of all people living inside the country OR property rights over land don't exist as such
Thus TNI>=0. If TNI>0, then TNI+NFFI=0.
Sales Tax=0 because there is no State to collect taxes. And there is no State because any State provices services like the police, the army, the public healthcare, etc. The only way to not have all of these services, is to have NO entity with monopoly on violence, consequently making it impossible to collect taxes.
Deprecation=0 because businesses would have no capital to deprecate
NFFI=Income of foreigners earned here - Income our citizens earned abroad.
Income of foreigners earned here=0 because Wages,Rent,Interests,Profits=0 domestically (i.e. they can be >0 only if they get money from abroad)
Now it's possible that our citizens earned something abroad, in this case NFFI will become negative, but it will be compensated by equaly-sized positive TNI, consequently making the whole GDP equal to zero.
3.We can caclulate GDP using the expenditures approach.
Government expenditures=0 because there is no State.
Consumption=positive number because although there is nothing to buy on the national market it's possible to spend money on imported goods.
Investments=0 because there are no businesses and no new homes to be bought by households.
Export=0 because these are supposed to be goods produced in the country but sold aborad.
Import>0 if our citizens earn money abroad. In this case they can buy goods abroad too and then consume said goods inside the country. This coupled with positive "Consumption" will result in in GDP=0
P.S. From what I read I concluded that the number 1 is incorrect because production of endowment, although happens somehow magically, without any labor of our workers (for an example, it's produced out of the thin air by a magic lamp) is still DOMESTIC production.
The number 2 is incorrect because it's possible to sell endowments to other citizens and get positive profit as the result (i.e. domestic profit>0).
The number 3 is incorrect because other citizens would buy (domestically!) endowments owned by other citizens/firms. Thus at the least Consumption would be >0 for domestic goods.