A confession: I have never really understood the basic model of fiat money and central banking, by which a central bank controls the money supply. By the standards of someone trained in mathematics, all of the explanations that I have ever seen are either too short or too long. My impression is that the way that a central bank controls the money supply in a modern economy can be taken on faith (if you want a short explanation), or is hard to understand (if you want a long one), but I have am suspicious of both of these extremes. I have also seen explanations that describe what happens "in effect" without clearly explaining the underlying rules. I would be interested in a concise mathematical summary of how a currency such as the US dollar is controlled. (I hope that it can be taken as an MO-appropriate question in mathematical economics.)


Here is a model that I understand, but that isn't true: A game such as Monopoly has a central bank that simply grants fiat money from time to time to private parties. I'm sure that this is the wrong way to run a real economy, but at the serious level I don't know why. In any case this is not how the Fed works, because it mostly lends money rather than simply granting it.


Here is a failed improvement of the model: Suppose that the bank in Monopoly only lent money to the players instead of granting it. Then the players would have no way to pay back the loans with interest! Maybe it could work if the players were allowed to accumulate debt --- but what would prevent unlimited borrowing?


I can believe in multiplier effects (although actually I don't know a rigorous definition). If transactions occur more and more quickly, or if assets get more and more leveraged, that could be equivalent to an increase in money. I have trouble believing that the central bank does not need to create money and that we see inflation (except in depression circumstances) solely because money keeps travelling faster and faster and because the economy gets more and more leveraged.


An abstracted economy has the following actors, each operating according to certain financial rules: A central bank, a government budget, regulated private banks, and the rest of the private sector. (And foreign actors, who I suppose are an extension of the private sector.) I think that I know the basic financial rules for the last one, but not for the others. To rephrase the question, I am hoping that there is a concise mathematical model that makes clear when money is created, and that looks dynamically stable with some controllable rate of inflation. A reference could be okay, but only if it has a good, specific explanation.



Concise model of modern fiat money and its non-conservation
I think an answer that discusses the actual institutional details of how the Fed controls the money supply would be off-topic here. Also, the Fed works slighlty differently from the ECB in that regard and there is more than one method of influencing the money supply (take a look at the wikipedia page on [money creation](http://en.wikipedia.org/wiki/Money_creation)). So I will try in this answer to demystify how a central bank can create money without literally sending out helicopters that drop fiat money on people.


First, one has to get right what money is. In explicit formal models, money is an asset that never pays out. If it has value, it is because there is a bubble in this asset. The first such model of money can probably be found in the 1958 paper [An Exact Consumption-Loan Model of Interest with or without the Social
Contrivance of Money](http://economics.uwo.ca/grad/9603a001/papers/Samuelson1958.pdf) by Paul Samuelson. It is worth pointing out that bubbles are not inherently bad and that paper constructs a toy economy in which everyone profits from the money bubble. 


Now how can one increase the supply of an asset that never has to pay out anyways? It sells the asset in exchange for other assets. Since money never has to pay out, the central bank will not face a solvency constraint in the process. Selling money is not that different from selling milk, but since there are no cows involved, central banks are not constrained by cost.


