If this is a good faith question, you can answer it by reference to any good economics text; or even, you know, Wikipedia. But it feels like maybe it's not.
Here's another example: you volunteer at a homeless shelter, where you serve food on a soup kitchen line. You have contributed to the GDP of the United States. By all means, feel free to fit this into your preferred framework.
What I'm doing is putting how you describe productivity into the form from the original comment. And I'll stand by that last version: How much money changes hands off of how much workers work?
The input to labor productivity is how many hours are worked, correct? And no one is measuring output in terms of the number of bowls of soup produced by homeless shelters; those are converted to dollars based on a current index price.
So you have economic activity, how much money changes hands, compared to labor inputs, how much workers work. Simple?
> And no one is measuring output in terms of the number of bowls of soup produced by homeless shelters; those are converted to dollars based on a current index price.
The Bureau of Labor Statistics does have multiple teams dedicated to documenting how price is related to quantity of output. They don't literally count bowls of soup at every homeless shelter, but they do document millions of price vs quantity measurements on a regular basis. This data is then used in the calculation of productivity.
Ok, here's a question for you: back in ancient days, most women worked at home. Is taking care of your own children and your own household an economic activity as would count in GDP, for example? Is someone making a bowl of soup for their spouse different from someone making a bowl of soup at a homeless shelter?
"GDP measures the market value of the goods and services a nation produces. Unpaid work that people do for themselves and their families isn't traded in the marketplace, so there are no transactions to track. ... The lack of reliable data influenced the decision to leave household production out of GDP in the internationally accepted guidelines for national accounting." (https://www.bea.gov/help/faq/1297)
"Economic activity" that does not equate to "money changing hands" in some form, isn't an "economic activity" that counts for GDP or productivity, right?
Here's another example: you volunteer at a homeless shelter, where you serve food on a soup kitchen line. You have contributed to the GDP of the United States. By all means, feel free to fit this into your preferred framework.