Who is the Federal Reserve, who owns the Federal Reserve

and this,

Where does the Federal Reserve get the money to fund its operations?

If that’s the case, then the Federal Reserve doesn’t really “creates money” when it uses open market operations to buy bonds. To presume that would put the Federal Reserve outside the economy.

Money is “created” at the moment that a newly originiated debt IOU (or any non-money, really) is accepted as money for the first time, that is, in exchange for actual money — e.g., individuals obtaining a new loan, companies or governments selling newly issued bonds, selling of newly created stock, etc.; and if I were a stickler, this should really only apply to the portion of the debt that is unsecured, as the exchange of money for the secured collateral is like a purchase by the lender who has rights on it. When the bond or debt is repaid, the reverse happens as money is destroyed. Interestingly, money expansion like this is only pseudo-permanent to the extent that the debt is either permanently revolving or replaced with new issues (Treasury notes in practice) or has an infinite time horizon (stocks).

In fact, bank loans can go a step further and continually (permanently) expand the money supply through fractional reserve lending, where new loans given out faster than the pace that old ones are repaid — if I get a loan today, I can put it into a savings account, but before I pay it back, it will be loaned out again. This is okay as long as banks have a reasonable value proposition for their loan, i.e. an expectation that whatever the loan is used for has or will have underlying value in the economy and hence an expectation of repayment. (If that’s not the case, then fractional reserve lending becomes but a pyramid scheme.)

In none of these cases is the money supply expanded top-down by the government. The money supply expands organically with the new value created in the economy (actually keeping slightly ahead of it, as they are tied to the expected value to be created from a capital venture — temporary asset bubbles excepted). The Federal Reserve is just a bank in this ecosystem with the objective not of profit but of being a sane participant (indeed to define “sane” through its market behavior); but it is still just a bank, and it has to balance its own checkbook, too.

(See here for another discussion. Interestingly, the Fed doesn’t have nearly sufficient net equity to conduct its operations, but it doesn’t matter because when the bonds it buys mature, the government, i.e. taxpayers, pays the Fed; or if the bonds roll over, then taxpayers in the indefinite future presumably will. If we look at this situation on a grand scale, if money supply in the economy grows in any permanent way it must be entirely due to growing private debt lent by banks plus growing debt incurred by the government, which of course is implicitly borne by the public as well. If that growth is in step with the economic productivity, meaning people will be increasingly able to pay their private debt, and government will get increasing tax revenue to pay bonds, then everything is okay.)

In the simplest of examples, you pay some entry fee to play the game, $1 is put in a pot by a counterparty, then a coin is repeatedly flipped and the pot is doubled on every coin flip by the counterparty, until “tail” comes up. You receive the money in the pot. The expected gain of this game is infinite, regardless of the initial entry fee. So it would seem that one should always play the game, regardless of the amount demanded as entry fee. But, as the article points out, “few of us would pay even $25 to enter such a game.”

(This seems to be one of many variations of the paradox.) The explanations given in the link to resolve the paradox aren’t satisfactory. “One can’t buy what isn’t sold” can only be considered a joke, while “expected utility” is somewhat plausible, but doesn’t strike at the central issue, because it can be circumvented with an equally counter-intuitive paradox fitted to the chosen utility function. In contrast to the Gambler’s ruin paradox, I don’t think that an artificial finite bound on the money supply (in this case, of the counterparty) makes sense as an explanation, but what it reveals as the logarithmic growth of the expected gain against the money supply and the general consequence that imposing some kind of finiteness may explain the paradox, is instructive.

Of course, the only way to get any gain is to actually play the game. If you repeatedly play the game, your gain does eventually go to infinity. So why would you be reluctant to pay even $25 to enter? It must be because those large pay-offs are so infrequent that to make the initial money back would take too long. Suppose the entry fee is \(W\). Suppose you call it a day when you have a positive pay-off. For that to happen in round \(n\), it must be true that

where \(f_i\) is the number of flips before tail comes up in round \(i\).

Let’s call n-tuples \((f_1, f_2, \dots, f_n)\) that satisfy the above by the set \(S_n\). The probability of winning in round \(n\) is then

from which we can surely get the average number of rounds it will take to win the game \(\sum_{i=1}^\infty n p_n\). If this is incredibly large even for modest \(W\), which is likely the case, then that would explain the paradox, since a game that on average takes longer than a lifetime to win would be played by no one.