AMZN is not losing money because they are reinvesting back into their operations. That is not how accounting works.
I'm a certified accountant so I'm kind of giggling over you telling me "how accounting works".
That is not how accounting works. "Earnings" is a balance sheet operation
I presume you are talking about Retained Earnings which is on the balance sheet. Retained Earnings != Earnings. Earnings = Profit and that comes from the Income Statement, not the balance sheet.
"Investment" is a balance sheet operation
Investment is FAR more complicated than simply transferring items around a balance sheet and it touches the Balance Sheet, Income Statement and Statement of Cash Flows.
Think about this way – If I invest $100m of profits in US Treasury Notes, how does that affect my earnings?
It depends on why you are investing the profits into those Treasury notes and whether you are investing in them as a profit making venture or merely as a place to park cash intended for other uses. The accounting is substantially different depending on the purpose of the investment. Furthermore the effect on earnings can be substantial in future accounting periods which I should think would be obvious.
If I invest $100m in property, plant, and equipment, how does that affect my profits?
The effect on profit depends on the return on the investment though in the immediate period the effect is either neutral or negative most likely. You're making the mistake of only considering the current accounting period. If you buy a building and capitalize the expense it has some effect on the current accounting period but the real effect on profit is depends on what you can do with that asset. It may reduce future profits or enhance them. Without more information no one can say more.
What if I paid out a dividend?
Then you are returning earnings to the shareholders rather than reinvesting them in the company or other assets. Basically a dividend is an admission by the company that the expected return from available investment opportunities is low. The company is foregoing future opportunities so the long term effect on earnings to the company is either neutral or negative.
It does not – in all cases one's profit is 100m.
Not correct and even if it were you aren't considering the net present value of that $100m.
The issue is that AMZN is trading profit margin for market share. Expanding quickly today to reap the profits of tomorrow – in theory.
Which is another way of saying they are investing in the business. Amazon is introducing products (tablets, phones, etc), investing in infrastructure (warehouses, IT) and similar. They have a long term perspective and aren't worrying about quarterly results. Perhaps this will burn them in the end but my thesis that they are investing in the business remains intact.