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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.

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I don't think this is a math question. This is what my wife said when I passed the question on to her: I'm guessing you'd have to ask the economists at the Fed (or those who study Fed operations closely), to get a detailed mathematical model of how they decide exactly how much to expand the money supply in any given quarter. And bear in mind that the Fed has been winging it since the financial crisis, using unconventional (i.e., little studied) criteria for expanding that supply. (To be continued) – Felipe Voloch Jan 27 '13 at 3:08
Since Greg is looking for an explicit, albeit simple, mathematical model, and such a model is not immediately found by a few web searches, I think it is a reasonable question from the standpoint of applied mathematics. Regarding the question, I am speaking from a standpoint of ignorance, but I thought central banks expand the money supply by exchanging interest-free currency notes for assets. Such assets are often government bonds (in which case the profits from interest are returned as seigniorage), but also can be drawn from the productive capacity of the private sector. – S. Carnahan Jan 27 '13 at 3:31
I hope that it can be taken as a valid question in applied mathematics, although I realize that it does not work well as a pure mathematics question. For example, the rules of Monopoly are in fact mathematically rigorous (if not very interesting as mathematics) and are even worth discussing as an inaccurate toy model. – Greg Kuperberg Jan 27 '13 at 3:31
@quid I'm sorry for the hints of controversy, but the fact is that I'm learning from the earnest answers by Greinecker and Landsburg. I don't think that it's fair to close a question just because there are some ineffectual answers that I didn't want either. – Greg Kuperberg Jan 27 '13 at 16:33
@Jyotirnoy: While it is obviously debatable whether the question is mathematical, everything else you say is unreasonable: The OP clearly spent a lot of effort formulating the question, and "can be found in standard undergraduate textbooks" is a completely worthless statement. Give us a book and a page number. – Igor Rivin Jan 28 '13 at 16:32
up vote 17 down vote accepted

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). 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 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.

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Btw: A paper that models open market policies explicitely can be found here: I cannot give any guarantees as tot he quality of the paper. – Michael Greinecker Jan 27 '13 at 10:55
Let me see if I understand this. The Fed, I believe, operates independently of the US Government. Its assets and liabilities are separate from the US Government, right? Moreover, when it "sells money", it is just buying assets, which I believe are usually investment grade securities, most often Treasury but also agency bonds as well as agency-backed mortgage-backed securities. If it wants to reduce the money supply, it just sells the securities, right? And if it holds the securities to maturity, then that also reduces the money supply due to the interest paid. – Deane Yang Jan 27 '13 at 15:58
The Fed is essentially independent of the US government, but there is some influence. It is not completely independent, but it is relatively save from politicians trying to influence it for short term gains. You have got the mechanism essentially right. For securities, things get a bit complicated because nominal payments are denoted in units of money. But the principle work with real assets that are not given in nominal terms. Then the price of the asset s essentially independent of the corresponding payment stream. In practice, the Fed does not try to reduce the money supply. – Michael Greinecker Jan 27 '13 at 16:17
Thanks! I never understood this before. – Deane Yang Jan 27 '13 at 16:50
The Fed holds Treasury bonds, which pay interest. The Fed also makes money for services it provides to the banking sector, such processing checks. It also makes loans. All told, it takes in more money than it spends on wages and expenses. This difference it rebates to the Federal government. – arsmath Jan 27 '13 at 21:06

Turning my last comment into an answer:

The simplest model of money demand is $M=M(P,Y,i)$ where $P$ is the price level (if all prices rise, you'll probably want more money in your pocket), $Y$ is real income (if you're richer, you might want more money in your pocket) and $i$ is the nominal interest rate (if the interest rate rises, you'll want to hold more bonds and consequently less money).

In the simplest models, $Y$ is determined by non-monetary factors, and (thinking now of everything as a function of time) $i=r+P'(t)$ (where $r$ is determined by non-monetary factors). This follows from the assumption that prices are perfectly flexible, so that $Y$ has to be determined by supply and demand in the markets for goods and labor.

At time $t$, the money supply is $M_0(t)$, where $M_0$ is a function chosen (in the simplest models) by the Fed. Equilibrium requires $M=M_0$. (If, for example, $M$ is less than $M_0$, so that people are unwilling to hold $M_0$ dollars, they will attempt to dispose of dollars by exchanging them for goods, which bids up $P$ and causes $M$ to rise. Likewise in the opposite direction).

So the key equation is $M(P(t),Y(t),r+P'(t))=M_0(t)$ with $Y(t)$, $r$ and $M_0$ determined outside the model.

A more sophisticated model would make $M(t)$ dependent on expected future values of $P$ and $i$, and include an account of how those expectations are formed. So you should view this as the freshman version of the story, not the grad school version.

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Thanks for this basic review. Your summary of the money demand equation was helpful and got me thinking along the right lines, but in the end Michael's answer looks a little closer to what I was missing in my thinking. In looking at your equation, I was stuck on where Y really comes from, i.e., how income is possible if all money is lent from the central bank. However, not all money is lent from the central bank due to seigniorage. (And some of it is lent by the central government and then spent, but I didn't think that that was the only non-conservation term.) – Greg Kuperberg Jan 27 '13 at 21:34
I don't get where you get this "money is lent from the central bank" idea. If you have money, it's your money -- you're not supposed to give it back to the central bank some day. In the Monopoly money analogy, the bank doesn't lend you money, it gives you money, and you give it a hotel you built on Park Place. – arsmath Jan 27 '13 at 22:42
@arsmath Yes, but a Monopoly board is not an accurate model. One of the Fed's activities is to lend money to commercial banks, which raises the question of how they might ever pay it back. If all money could be traced back to the Fed lending money to commercial banks, they wouldn't be able to. – Greg Kuperberg Jan 27 '13 at 23:04
@Steve - Sorry, I should say it this way: In the more restricted formula $M = P \cdot L(Y,i)$, I knew where $Y$ comes from, but I didn't know where $P$ comes from, and I also didn't know the real-life mechanism of $M_0$. But now I see how your equation could work as an answer to the second part --- $M_0$ is set by policy (and has no conservation property due to seigniorage) and your equation becomes a differential equation for $P$. Anyway I wish I could accept more than one answer, yours and Michael's, but MO only lets me choose one. – Greg Kuperberg Jan 27 '13 at 23:46
@Greg Okay, I see. I hadn't heard of the idea that all money could be traced back to Fed lending. – arsmath Jan 28 '13 at 7:11

Thanks to the comments and answers from Scott Carnahan and Michael Greinecker, I think that I understand it better now. I'm going to write this as a CW summary answer and also accept one of the other answers.

People often talk as if all currency is borrowed from the central bank, but that is not really true. If it were literally true, it wouldn't make sense, because there would be no way to pay the principal and interest back to the central bank. Or otherwise, if all money is borrowed, then an economy's total monetary assets stay at zero, which is not strictly impossible, but doesn't sound right.

What I guess actually happens is that the central bank both buys and sells treasury bonds. Even though this is done for interest rate stability, the central bank is perfectly happy to sell high and buy low, thereby violating conservation of money. It is also counterintuitive in the following respect: Although in the short term a high interest rate contracts the money supply, in the long term the interest paid expands it again. Nonetheless, I guess that the demand to have money to trade sustains the value of the money and keeps everyone from just buying treasury bonds at high interest. I guess here you would point to the money supply equation that Steven Landsburg posted. (It does not leap out at me that it really leads to currency stability, but I can believe it.)

Also, to get a currency started, the central bank can first buy or sell other commodities, for instance gold, so that the private sector then has money to buy treasury bonds. Another counterintuitive point (but one that doesn't bother me) is that if the central bank trades commodities at a monetary "loss", then actually it has gained those commodities. This inverted mode of gain by a bank seems to be one meaning of "seigniorage". Another meaning is any increase of the money supply from the central bank's trades, so at some level seigniorage is the main answer to my question.

Another player is the national government. Unlike the private sector, it is allowed an unlimited amount of debt. So, a second non-conservation of money is deficit spending, if in tandem the central bank keeps lowering the interest rate. Unlike seigniorage, this may be de facto non-conservation of money, but it is not de jure non-conservation of money, if the government keeps an honest account of how much it borrows. (As Deane and Michael discuss, this honesty is only really possible if the central bank is politically independent from the government budget.)

A third type of non-conservation of money is a default by a commercial bank that owes money to the central bank. But this does not look like a natural way to increase the money supply, and I don't think that it is.

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Greg: I'm having trouble seeing why you view the "non-conservation of money" as somehow more in need of explanation than, say, the non-conservation of refrigerators. The number of refrigerators can change because there are companies that produce refrigerators. The amount of money can change because there are companies (called "banks") that produce money. Why is one more mysterious than the other? – Steven Landsburg Jan 27 '13 at 21:36
Because I didn't know which financial rules make de jure non-conservation of an official currency possible. That was my real question. (De facto non-conservation of money is less surprising, but the two are related.) – Greg Kuperberg Jan 27 '13 at 21:42
Besides, your analogy with manufacturing refrigerators is an explanation of money, but not fiat money. A central bank does not accumulate a hoard of refrigerators in exchange for issuing cash. Instead, it does something more circular which is still (usually) stable. – Greg Kuperberg Jan 27 '13 at 21:49
I don't see how this answers your question in the strict sense. This is all very interesting and may be an adequate explanation of fiat money but I see no connection whatsoever with research in mathematics. Since you posted this question here on MO one would assume an answer required such a connection. Unless, of course, you are prepared to acknowledge that the question is off-topic. On the topic of off-topic, that Italian automaker should make a car called the Fiat Money. It would drive itself and never run out of fuel. – Felipe Voloch Jan 27 '13 at 23:49
Even writing equations is not (applied) mathematics. When Newton, Maxwell and Einstein wrote down the equations for their laws, they were doing physics. Math comes later. – Felipe Voloch Jan 28 '13 at 1:03

For a mathematical model see Hayashi and Matsui, 1994. For an in-depth discussion without too many (actually, any) equations, see many books by Murray Rothbard (all available on

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I'm sure that Hayashi and Matsui wrote an interesting paper,but they seem more interested in establishing a result about fiat money than describing what it is --- after all, this paper was written in 1994 and it does not look like it is meant just as an exposition. It looks more technical than what I had in mind. As for Murray Rothbard, I would much prefer a short explanation with equations than entire books with no equations. – Greg Kuperberg Jan 27 '13 at 2:14
I want to point out that Murray Rothbard is seen as a crank by most economists and has essentially no influence in academic economics. – Michael Greinecker Jan 27 '13 at 10:06
@Michael: Economics is not really a science, and its social dynamics are tribal. Rothbard is a leading of proponent of (ironically) the Austrian School of Economics (in honor of von Mises and Hajek), which is not the politically dominant school. I personally find the Austrian school extremely compelling, and find what "most economists" think (or claim to think) of little interest. I could expound at length on this, but at this point we are veering quite far from mathematics. – Igor Rivin Jan 27 '13 at 15:03
I had voted to close the question as off topic, but you guys have convinced me that I should have voted it subjective and argumentative. – Felipe Voloch Jan 27 '13 at 16:36
@Felipe - I could have voiced my own opinion of Murray Rothbard, but I chose not to. Not fair to vote to close my entire question because of this side discussion. – Greg Kuperberg Jan 27 '13 at 21:51

I feel like I'm missing something dumb but it seems to me that fractional-reserve banking is not referenced or explained in the question or any of the answers. People here DO understand it, right? It is where money actually comes from. Basically if you borrow \$1000 from a bank, they can issue the loan by typing something into a computer that increases your account balance by \$1000 without subtracting from some other account. The \$1000 doesn't have to be transferred from anywhere, but rather it is created out of nothing, which is why money is not a conserved quantity. There are of course a bunch of constraints such as the reserve requirement, but money created by banks through lending (the regulatory ability to create money that way is what distinguishes a bank from, say, a payday lender that has to transfer money from itself to you instead of creating it) is the cause of the non-conservation.

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I think economics is far more closely connected with the body politic than it with mathematics. Also applying mathematics to economics is also a political act and a signification. (Mathematics has an association with permanance which can be used symbolically to shore up a certain contingent political/economic order).

Physics examines the world by supposing the physical world follows a rational order, and that by dint of effort this order is discoverable. I can't see how this applies to the social order of societie(s); how does one measure wealth, imagination, violence, ethics, power, desire, criminality?

Whereas mathematics applied to physics captures something of its fundamental relationships, it appears to me that a mathematical model of a social order can only captures superficial and contingent things.

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How is this answering the question? – Michael Greinecker Jan 27 '13 at 14:33
By denying that it's answerable in its own terms? When Newton wrote about his law of gravitation, he made certain he could model both time and space mathematically, the background to his physics. I'm denying that this background is available in economics. Theories don't exist in a vacuum , they exist in a larger theoretical space. – Mozibur Ullah Jan 27 '13 at 15:28
The question was for a foral model of how the fed influences the money supply. Whether such model is useful or not is not part of the question. – Michael Greinecker Jan 27 '13 at 16:05
-1 for off-topic. – user9072 Jan 27 '13 at 16:13

Although not a mathematical concise definition but empirical evidence that money is created "out of thin air"..

"It was examined whether in the process of making money available to the borrower the bank transfers these funds from other accounts (within or outside the bank). In the process of making loaned money available in the borrower's bank account, it was found that the bank did not transfer the money away from other internal or external accounts, resulting in a rejection of both the fractional reserve theory and the financial intermediation theory. Instead, it was found that the bank newly ‘invented’ the funds by crediting the borrower's account with a deposit, although no such deposit had taken place. This is in line with the claims of the credit creation theory."

"Thus it can now be said with confidence for the first time – possibly in the 5000 years' history of banking - that it has been empirically demonstrated that each individual bank creates credit and money out of nothing, when it extends what is called a ‘bank loan’. The bank does not loan any existing money, but instead creates new money. The money supply is created as ‘fairy dust’ produced by the banks out of thin air."

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How can you downvote an answer sourced to a peer-reviewed paper?? – Mike Schärer Mar 29 '15 at 18:14

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