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As I understand it, interest rates are set by supply and demand, like any price. Money can be thought of like any other finite commodity. Lenders are willing to supply a certain amount of money at each interest rate, and borrowers are willing to borrow a certain amount at each interest rate. The rate can be calculated by setting supply equal to demand.

However, with fractional reserve banking, this picture seems to fall apart. While the demand curve can still be found, the supply of money is not so easy to quantify. If the reserve requirement were low enough, banks could loan out an arbitrarily large finite amount of money. The only limit to the money supply is the reserve requirement. It doesn't make as much sense to say here that the interest rate is set by supply and demand.

I see a two possibilities: 1) There is a supply curve that depends on the reserve requirement. This means that the interest rate would be a function of the reserve requirement.

2) There is no supply curve. As far as I know, no one sets interest rates (the Fed included, since they only influence rates via open market operations: if they didn't do anything there would still be an interest rate). How can interest rates be calculated in this case?

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First, you need to understand that interest rates are not set by the supply of cash provided by banks and the demand from borrowers. They're set in money markets, which is the other side of the credit intermediation that banks perform. Banks lend to borrowers and borrow in money markets (if they didn't borrow, then it wouldn't be fractional reserve banking) to fund a significant portion of their lending.

Second, reserve requirements are far from the only constraint on bank money creation, and I frankly cannot think of a single bank for which they're the binding constraint. Banks are more likely to be constrained by risk-weighted capital requirements, the liquidity coverage ratio, or stress test scenarios than by reserve requirements.

So to answer your question as to how interest rates are determined under fractional reserve banking: bank demand for funds is determined by their set of profitable lending opportunities under a set of constraints including capital requirements (and they face some competition for these funds from firms that have direct access to capital markets and which issue short-term liabilities), while the supply of funds is determined by the amount of uncommitted cash held by firms (and to a much lesser extent, households). These intersect at some point, which determines the interest rate. The Fed affects the risk-free rate by engaging in transactions which swap safe, usually interest-bearing collateral for cash, affecting the relative supply of each.

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  • $\begingroup$ Some banks borrow on the money markets, I wouldn't characterise it as all banks - it's a slightly abnormal form of banking to do this that isn't strictly fractional reserve banking. $\endgroup$
    – Lumi
    Oct 8, 2015 at 3:25
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    $\begingroup$ @Lumi— The point isn't so much that all banks borrow on money markets, so much as that's where rates are determined. As an aside, I would note that money market borrowing is absolutely part of fractional reserve banking— both deposits and money market borrowing are short-term liabilities from the point of view of a bank, and the existence of such liabilities is the definition of fractional reserve banking. In fact, short-term (<=1 wk) non-deposit liabilities, such as most money market liabilities, are subject to reserve requirements. $\endgroup$
    – dismalscience
    Oct 8, 2015 at 15:31
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    $\begingroup$ I know that's how it's presented in the textbooks - but it's a horrbibly incorrect and confusing way to to look at what is essentially statistical multiplexing, and a big contributor to the problems economics is having analysing banking properly. As for interest rates - banks set a range of interest rates, that are offset from the central bank rate - and the determinants of those rates are more than just the money market rates, eg. risk, exposure, liquidity, etc. $\endgroup$
    – Lumi
    Oct 8, 2015 at 21:41
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    $\begingroup$ @Lumi— Thanks. To be clear: I assure you both that my understanding of banking comes not just from textbooks but from years of experience in the field, and that I'm super familiar with the components of rate spreads. It should be clear why I did not include a discussion of rate spreads in my answer, as they're not particularly relevant to the question asked. $\endgroup$
    – dismalscience
    Oct 8, 2015 at 22:23
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Interest rates are determined by supply and demand. There are lots of different interest rates determined in different, but largely inter-related markets each with their own supply and demand.

Banks could not loan out an arbitrarily large finite amount of money if reserve requirements were low enough. Many banks are not subject to reserve requirements at all.

If you want to read more, see this Bank of Canada staff working paper: https://www.bankofcanada.ca/wp-content/uploads/2010/05/wp97-8.pdf

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Simplified Bank Markup Model

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The prime rate appears to be a standard markup over the cost of funds in money markets. The money market rates are priced, via market arbitrage, off of the effective fed funds rate. I think the summary model of monetary policy transmission is that the central bank (FED) targets control of the fed funds rate and money market dealers set other money market rates via market arbitrage.

Prices and Quantities in the Monetary Policy Transmission Mechanism

https://www.ijcb.org/journal/ijcb09q4a7.pdf

Credit is supplied by financial intermediaries such as traditional commercial banks, as well as institutions of the “shadow banking system” such as security broker-dealers and ABS issuers. If credit supply is the main determinant of the quantity of credit in the economy, the key is to understand the motivation of financial intermediaries and how capital market conditions determine their behavior. For monetary policy, the question is how the central bank affects capital market conditions that ultimately affect the supply of credit.

The starting point in understanding credit supply is the delegation of capital allocation decisions to financial intermediaries. Savers—including households and nonfinancial corporations— delegate capital allocation decisions to intermediaries. This delegation raises agency problems, which are (at least potentially) solved by constraints on leverage, risk management, and credit ratings, as well as targets for measures such as ROE or ROA targets. One particularly simple way to summarize such constraints is to look at haircuts—the amount of overcollateralization that is required for borrowing against risky asset collateral.

Monetary policy and lender-of-last-resort policies affect over- all capital market conditions through the balance sheets of financial intermediaries. The variation of the federal funds target primarily moves around the slope of the yield curve, making the lend-long/borrow-short carry more or less profitable. Central bank liquidity facilities work through the equilibrium trade-off between credit spreads and haircuts. An increase of central bank lending against a particular asset class will tend to lower haircuts and spreads. As the financial crisis can be viewed as a shortage of financial intermediary balance-sheet capacity, lender-of-last-resort operations tend to off-set the decline of that capacity. The Federal Reserve’s balance-sheet expansion can thus be viewed as an emergency replacement of lost private-sector balance-sheet capacity by the public sector.

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Finding interest rates with fractional banking is very difficult because it is governed by quantifiable and unquantifiable factors.

Regulatory requirements (like reserve ratios and capital requirements) are objective and quantifiable and play a role (but it varies from bank to bank).

The supply of base money is only limited and is another objective quantifiable component.

But...a major component of banking is trust. If I create a new money "exchangepesos" and nobody wants it, then any quantifiable math equation would be useless. Banks create money. They create "deposit dollars" that of course exceed government dollars (base money). For this to work, there must be trust. If I as a depositor or as another bank don't trust say "Bank A", we won't trust in turn their deposit money which will ruin their credit rating and result in the destruction of their deposit money.

Another way of putting this is that banks operate by maturity mismatching. Matching short term debt to long term assets. This is naturally exceedingly risk and depends on those lending short term to the bank to consistently roll over such debt. Because long term assets are long term they will not get the principal in time to satisfy a redemption run by short term debt holders (many of which could be called depositors).

The extent to which a bank can create money and buy interest bearing assets (which affects interest rates) is a function largely of the trust the individuals (but mostly other banks) have in the bank that does the money creation. This is of course non-quantifiable.

The best solution IMO is to at least quantify the extent to which a bank mismatches debt to asset maturities. This is an obscure but growing measurement sometimes referred to as the liquidity mismatch index.

A most practical manifestation of this equation would be to have a graph with the horizontal access being maturity times (say 1 second 'aka demand deposits), 1 month, 1 year, 5 years, 10 years, 25 years, etc...). Then the vertical axis quantify be amounts in dollars. Two lines would be drawn...one for bank debt stratified by maturity time and another for bank assets stratified by maturity time. Measuring the area of the difference of the two lines would give you a relatively interesting idea of where the banking system is at and where it could go (especially when compared to historic norms). But ultimately an exact equation to find interest rates is not possible because trust is so important to banking yet is not quantifiable.

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