Here's a fun one. (OK, I think it's fun, anyway.) Especially for those who teach macro.
A first year student emails me. Paraphrasing, he asks "For a closed economy, we know that national saving equals national investment, and investing in newly-produced goods counts towards GDP. And "saving" means anything we do with our income other than paying taxes and spending on newly-produced consumption goods. So if I take $100 from my pay and stick it under the mattress, does that count towards GDP?"
Your instinct is to answer "No!" of course. But you should be able to see his logic.
Let me restate his question in a different way, so "saving" refers to private saving, and for an open economy, so saving is not necessarily equal to investment.
It is a national income accounting identity (i.e. it is true by definition of the terms) that:
C + I + G + NX = Y = C + T + S
[Consumption+Investment+Government spending+Net eXports = GDP(income) = Consumption+Taxes+Saving]
The normal way to measure GDP is to add up all the terms on the left hand side of that identity (the "expenditure" method). So if you were adding up C+I+G+NX, and then found you had missed $100 in I, and re-did your calculation, your estimate for Y would be $100 bigger than your original calculation.
But suppose instead you decided to measure GDP by adding up all the terms on the right hand side of that identity (we don't have a name for that method, because nobody does it that way, AFAIK). So if you were adding up C+T+S, and found you had missed $100 in S, and then re-did your calculation, your estimate of Y would be $100 bigger than your original calculation.
[And the reason nobody measures GDP that way is precisely because "saving" means literally anything you do with your income except buying newly-produced consumption goods or paying taxes. It includes: buying a newly-produced investment good, buying land, buying a used car, buying financial assets, and sticking $100 under the mattress. So it would be much simpler just to ask people what their income was.]
The problem is this: If Y = X + Z, and we ask "does X count towards Y?", we are making an implicit assumption about whether or not Z stays the same. The number of my Kids equals the number of my Sons plus the number of my Daughters. K=S+D is an accounting identity. Does "if I have one more son" mean "if I have one more son and the same number of daughters" or "if I have one more son instead of having a daughter"?
We tend to read "If I put $100 under the mattress" as "If I put $100 under the mattress instead of spending $100 on newly-produced consumption goods." But we could equally well read it as "If I put $100 under the mattress instead of taking the day off work and earning $100 less".
But "if I buy $100 more newly-produced investment goods" has exactly the same ambiguity. Instead of doing what?
I think the lesson to be drawn is that we must always specify what the alternative is, and what else is assumed constant, so we answer the question: "instead of doing what?". It then becomes as trivial as playing with words should be.
Whenever anyone says "Y=C+I+G+NX, therefore if the government increases spending, GDP will rise" I will reply "Y=C+T+S, therefore if the government increases taxes, GDP will rise."
There must be a better way to teach national income accounting.