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It is widely discussed that the "global saving glut" increases global money supply and so reduces it's price (interest rates) and pushes up the asset prices. It sounds very logical. However, in the fractional-reserve banking system, money is a completely virtual concept. In the US, the FED sets it's target federal funds rate, which indirectly influences all sorts of other interest rates. From this perspective, interest rate level depends only on the central bank policy decisions. I struggle to connect these 2 dimensions. How does savings levels and central bank policy interact to produce effective interest rates?

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In short, it is determined by supply and demand, central bank comes in a picture by controlling these either through the interest rate or also other tools like open market operations which affects supply of money in the economy.

To be a bit more precise about mechanisms. Private banks are forced by regulations such as Basel III to have some sufficient level of liquidity in case there is a crisis so government does not need to constantly recapitalize the banks. The exact level of liquidity that banks needs to maintain depends on how risky the loans that the banks make are.

As a consequence to maintain sufficient liquidity private banks need to maintain, among other things, certain level of reserves. Banks can get reserves either from central bank or from other deposits. However, central bank is far more important, typically when bank makes a loan it won’t try to quickly attract more depositors it will simply borrow reserves from central bank at the interest central bank charges.

This is where central bank interest rate setting enters a picture. Private banks make profit on intermediation margins (see Freixas & Rochet Microeconomics of Banking, Second Edition ch 1). A profit function of a private bank can be described as:

$$\Pi = r_L(L) L - r M(L,D) - r_D(D) D - C(D,L)$$

where $r_L$ is interest rate on loans $r$ the central bank's rate, $r_D$ is the deposit rate. $L$ are the loans $M$ is the amount of reserves bank borrow which will be some function of loans and deposits it makes (for simplicity we can assume $M=\beta D-L$, $D$ are deposits and $C$ are the costs. Consequently, profit maximizing bank would select some $L$ that satisfies among other the following FOC for optimum amount of loans:

$$ r_L = r+C'_L - r'_L L$$

So the market interest rate at which people can borrow will be equal to the central bank's rate $r$ which central bank can choose freely at will. However, it also depends on the marginal costs of lending $C'_L$. These are monitoring cots, costs for a staff etc for the last loan the bank has. Finally, $r'_L L$ is the rate at which interest rate changes when amount of loans changes (since if there are more loans avaiable the price of loans will drop) times the amount of loans. $r_L$ also depends on how much market power banks have.

Furthermore, $L$ - amount of loans supplied by banks, has to be also in equilibrium with demand for loans, for every loan there must be two sides of a transaction (lender and borrower). The demand for loans is in turn is a function of real money (i.e. how much money there is given the price level, since higher price level forces people to also hold higher money balances for transactions or for precautionary motive), output, interest rates as well (since of course demand also depends on price) and other exogenous factors like their preferences.

So to sum, it's all supply and demand but under the hood the interest rate will depend on costs that private banks incur when they make loan, on how much market power they have and how much demand there is in an economy for more loans. However, central bank has most of the time more or less control over some of these supply and demand factors. It can set $r$ which partially determines the private bank's cost of suppling more funds and thus indirectly controlling the amount of loans in the economy. By open market operations and through supply of reserves it also has some control over the real money stock. As a consequence central bank can most of the time make the interest rate for loans to change in direction it desires. However, this control is imprecise because other factors such as marginal costs of banks or peoples preferences etc are constantly changing. Also, there are some situations such as zero/effective lower bound or other situations where central bank cannot use some of its instrument anymore and thus may loose control (i.e. central bank cannot push interest rates much lower than zero otherwise people would just hold cash).

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