To be fair, there is indeed an economic identity that says that all income is applied toward economic activity—that is, all income is spent. The problem is, there is no economic identity that says that all money is applied toward economic activity. So here's a somewhat denser version of why David is wrong.
First, some definitions.
Total spending in an economy must always equal total income. How do we know this?
Suppose the economy has no money. I produce ten bananas, you produce twenty apples, and once per day I sell you all of my bananas in exchange for all of your apples. What is total income—the value of goods and services bought and sold—at the end of the first day? Well, I sold my ten bananas to you and you sold your twenty apples to me, so total income is ten bananas and twenty apples. Total spending and total income are the same.
Let's add another layer of complexity. Consider an economy that can spend income in two ways instead of just one: individuals purchase goods and services (consumption) while businesses purchase physical entities like machinery, equipment, and software (investment). The same rule applies: since total spending equals total income, consumption and investment together must equal total income.
Now, what about saving?
Let's add money back into the equation. Economists define saving as “income not consumed.” This coincides with the conventional understanding of what makes up your income: the part you spend (consumption) plus the part you don’t spend (saving) is equal to your income. Earlier we said that income is equal to consumption plus investment, but another way of saying that is income is equal to consumption plus saving. Thus, saving and investment must be equal.*
This is the identity that David tries to satisfy when he says the money you put in your bank account becomes business investment. But of course, it’s all wrong. The money you put in your bank account doesn't lead to a corresponding increase in investment because investment creates saving, not the other way around.
The confusion stems from the (incorrect) notion that the value of an individual's savings account is the same thing as economic saving, the counterpart to investment. They are not the same.
The short answer is that money in your savings account is a stock, while economic saving is a flow.
Now for the long answer: we know that economic saving is “income not consumed.” Practically speaking, that means that you “save” the moment you get your paycheck. Eventually, you might purchase goods and services, which will reduce your saving. But as soon as you receive your paycheck, you “save.” Similarly, a business “saves” when it receives income.
A lot of saving in the economy, then, is simply transferred between us and nets out to zero. You save when you receive your paycheck, but your employer reduces saving the same amount. The same happens when you consume. When you buy groceries, you reduce your saving, but the grocery store saves the same amount. If you choose not to buy groceries and save more, then the grocery store will save less by the same amount.
How then, does saving increase or decrease?
Remember when you bought groceries, you reduced your saving. But when businesses invest, there is no reduction in saving. Suppose Business X invests in new machinery, purchased from Business Y. What happens at X is that, instead of reducing saving, an asset is created reflecting the value of the new machinery. This is what represents an increase in economic saving! Simultaneously, X’s spending represents income for Y, so Y’s saving goes up. This is the only way to increase economic investment, and hence, saving (all else constant).
Unfortunately, it doesn’t work the other way around. Putting money in banks does not lead to a necessary increase in investment, like David thinks. Naturally, during recessions, this means that investing in U.S. treasury bonds doesn't reduce private investment either. As we've just seen, that's because putting money in your bank account doesn't actually affect total saving. The argument that adding money to your savings account leads to economic investment is false.
O.K. let's bring this full circle. As we saw, in a depressed economy, there’s no reason why government borrowing money inhibits the private sector from investing. Why? Because investment creates saving, not the other way around.
And that is why, for the last time, David is wrong about stimulus.
*To see this mathematically, let Y=income, C=consumption, I=investment, and S=saving. We have two equations:
Combining equations (1) and (2) gives us