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some people suggest that the government prints money to pay its bills and that the increased amount of money in circulation lessens the value of $1. i hope to get an explanation of why both assertions are false and suggest the following:

the Federal Reserve Bank (Fed) inserts money into the economy thru loans. at least two questions arise

  1. presumably if all loans were paid back, there would be no money in circulation. but corporations borrow money to build products and others borrow money to buy those products, hence a never ending cycle. so Fed balance sheet every expected to be zero (i.e. no outstanding loans)? and what happens to the money in circulation from defaulted loans?

  2. the "value" of money is presumably based on supply and demand. this presumably counters the idea that the more money in circulation reduces its value because as the population and economy grow, there is a greater need for money, maintaining its value. when an economy shrinks, the fed makes fewer loans and as loans are paid back, the amount of money in circulation is reduced. further, much more money is needed in circulation today than in colonial times (as an example) simply because there is a greater population and need for money (or the economy has grown).

are these concepts correct and do they explain/justify fiat money? what concepts are missing from a more comprehensive understanding?

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If it is good for society to have stable prices then the measure of "valid" money is price stability. The debate over, for example, free banking laws versus central banking customs is one where both teams argue that stable prices are good and each team does not agree on the political-economic customs which would cause the good. So we see that economics involving finance relations (systems of money, credit, and debt) cannot be independent of ethical arguments in the sphere of public debate.

Regarding central bank balance sheet start with the balance sheet identity:

Assets = Liabilities + Equity

The Federal Reserve monitors the sources and uses of Reserve Balances which are liabilities on the Fed balance sheet:

https://www.federalreserve.gov/monetarypolicy/bst_fedsbalancesheet.htm

Reserve Balances and Currency are components of the base money or monetary base which are "valid" means of payment because banks use Reserve Balances to clear interbank payments and society uses Currency to make withdrawals or deposits into banks and for hand-to-hand payment.

The simplified Federal Reserve balance sheet with assets on left hand side of equation:

Treasury Securities + Discount Loans = Reserve Balances + Currency + OLE

where Reserve Balances and Currency are liabilities and where OLE means Other Liabilities and Equity. Fed uses Treasury holdings and Discount Loans as control factors to (1) service the drain of Currency; and (2) provide Reserve Balances to the aggregate bank.

Service the currency drain prior to late 2008 financial crisis:

Treasury Securities = Currency

where if Fed makes its holdings of Treasury Securities approximately equal to the Currency it has provided to banks and nonbanks over the history of Fed operations then this holds Reserve Balances constant to a first case approximation. Before late 2008 there were few excess reserves in the system so Fed had to use gross and fine control over the level of Reserve Balances to enforce control over the federal funds rate.

Fine control over Reserve Balances prior to late 2008 would be accomplished by three mechanisms: (1) Discount window loans; (2) securities repurchase agreements; and (2) reverse repurchase agreements. Since the dual mandate of the Fed is to maintain price stability and employment levels it would not take loans or open market operations (security dealing) down to zero unless political unrest disrupts the nation and puts the Fed out of business as the central bank or monetary authority.

[S]ome people suggest that the government prints money to pay its bills and that the increased amount of money in circulation lessens the value of $1. i hope to get an explanation of why both assertions are false[.]

When people criticize a government for "printing money" this often means the central government engages in deficit spending and the central bank fails to act to curb inflation as a matter of political policy. The inflation is not "caused" by the central bank printing too much money it is "caused" by the failure of the central bank to tighten credit when the bank and nonbank sector drive inflation via too much credit expansion that uses up and makes scarce the slack or excess capacity in the economy.

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1.presumably if all loans were paid back, there would be no money in circulation.

No if all loans were paid back there would still be money in circulation because part of base money is in the forms of coins and notes produced by mint and bureau of engraving and printing under treasury. Those money would not disappear even if all loans would be paid back.

Fed balance sheet every expected to be zero (i.e. no outstanding loans)? and what happens to the money in circulation from defaulted loans?

No, as long as the government does not decide to switch to some other monetary system and given the relative efficiency of fiat money that is unlikely to happen.

what happens to the money in circulation from defaulted loans?

That depends on what sort of default are we talking about. If for example government declares it won't pay portion of its debt to Fed then nothing will happen to the money in circulation - the money supply increase becomes permanent (in a sense it cannot be anymore reduced by paying down the debt).

  1. the "value" of money is presumably based on supply and demand. this presumably counters the idea that the more money in circulation reduces its value because as the population and economy grow, there is a greater need for money, maintaining its value.

This is true but that implications are not completely correct. One of the most simplest model to describe the money market (e.g. supply & demand for money) is as follows (see Mankiw's Macroeconomics 8th ed pp ):

$$MV=PY$$

where $M$ is money supply, $V$ is velocity of money (how much is one dollar used), $P$ is price level (change in which gives you inflation/deflation - i.e. increase/decrease of value of money) and $Y$ is the real output. In more complex models of money market people's expectations play role and the relationship is not necessary proportional but this simplified version is useful as a didactic tool.

Here you can see that ceteris paribus (holding $V$ and $Y$ constant) increase in money supply $M$ does increase $P$ (which in turn decreases value of money). You are completely correct that as our economy grows ($Y$ increases) there will be higher demand for money so as long as you increase $M$ just by the same amount as $Y$ increases it would not lead to any inflation. In addition, also $V$ tends to change over business cycle (although in the long term it tends to be constant) which means there is additional leeway for creating new money during some times (although also less during another).

However, it is always possible for central bank to increase money supply way beyond a level that would maintain price stability. That is ultimately just question of how far is central bank willing to go and what is legal (e.g. helicopter money might not be legal every country and there are usually institutional constraints imposed on central banks - e.g. in Europe the legality of ECB monetizing government debt is questioned in some national courts).

This being said modern independent central banks are empirically excellent at maintaining price stability (most are legally required to do so).

do they [these concepts] explain/justify fiat money?

Most of the above would work in other monetary systems as well with some caveats. The main justification for fiat money is that it wastes less resources than comparable systems such as commodity money or gold standard. When you back money by let's say gold, then you are just digging up gold to put it back under ground - wasting scarce resource that has many industrial/commercial uses in electronics, jewelry etc.

Ultimately any money is based on trust that someone else will accept your money and this trust can be created via government requiring all taxes be paid in its currency (you always know the money will be useful for something so you accept it). That is much better than wasting scarce resource in order to create trust (e.g. under the gold standard the trust comes from the fact that you know money is convertible to gold and you know you can always find buyer for that gold because it is sought after commodity).

Additionally, monetary policy under commodity money or gold standard requires for monetary authority to regulate production of that commodity or change the ratio at which money converts to the commodity which is more cumbersome than just having fiat money. So that is an additional reason as well.

what concepts are missing from a more comprehensive understanding?

Explaining comprehensively all monetary economics/macroeconomics is beyond scope of a SE answer. A good macro textbooks that deal with these issues are above mentioned Mankiw Macroeconomics, Blanchard et al Macroeconomics a European Perspective or if you want high level graduate text Walsch Monetary Theory and Policy is very good text.

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