How do governments decide how much money should be in circulation?

Specifically cash, notes and coins, and how this might be related to inflation, interest rates, credit and trade.

Answer:
4XTrader is almost correct if your looking strictly at a Monaterist system. Keynes would argue other wise. Either by raising/lower Taxes and/or Interest rates you inadvertently control the supply of money. this is driven by the GDP. It's an indicator of how fast or slow our economy moves. Currently the US mixes and matches the Monaterist ideology with Keynes.
I think they print the money for themselves when they need some.
Well, the money supply (the money in circulation) is controlled by interest rates (which are controlled by the Federal Reserve). Interest rates are utilized by "the Fed" to control things like inflation (or on the flipside, deflation). Inflation is kind of an ongoing thing but sometimes it can be a problem (too much money in the money supply=a weak dollar) which limits purchasing power and can weaken the economy (people can't afford things, money isn't worth much). Interest rates also influence people on how much to borrow or save, in a sense taking money or puting money back into the money supply. Hopefully this very basic and incomplete explanation helps!
Oscarmayar is a little confused. Interest rates DO NOT determine how much money is in circulation. Interest rates are the COST of money. The Fed Open Market Operations determine how much money is AVAILABLE and in Circulation.

How they decide how much money to put into circulation I do not know. The bottom line though is that gov'ts have a bad habit of spending more than they take in in revenues, so they print the money they need to support they're over spending.

No, when you look at the money supply, you have the following:

M0 - all liquid assets held by the central bank, ie, cash or assets than can be quickly turned into cash.

M1 - which is physical money (ie coins and paper money), checking and NOW accounts.

M2 - which is M1 plus all time related deposits, savings deposits and non-institutional money market funds.

MZM - which is M2 less time deposits plus all money market funds.

M3 (now defunct) - which is M2 plus all large time deposits, institutional money market funds, short term repurchase agreements and other large liquid assets. As of March 2006, the Fed no longer published M3 data.

Now, what you must realize is that price inflation is a direct function of monetary inflation, ie, inflating the money supply. The more money your print, the more you drive inflation. For an example of this, do a websearch for Hyperinflation during Post WWI Weimar Republic Germany. In short, Germany was made to make war reparations to the allied forces. The problem was that Germany's industrial complex has been destroyed because of the war, so they couldn't make the goods necessary to sell to raise the funds needed to make the payments, so they turned on their printing presses and printed the money. The problem was that they printed so much, the currency was worthless and you needed huge amounts just to buy basic things. The hyperinflation was so bad that a loaf of bread was something like 1 million Reich Marks.

Now, if you look at what I said up top, you see that I said the Fed no longer publishes M3 data? If you look at M3, you'll see that there is a component called "Short Term Repurchase Agreements" - Repo for short. The Repo market is the mechanism the Fed uses to add/remove liquidity from the excess banking reserves. The reason I believe the fed has stopped publishing M3 is to mask their money printing activities. According to shadowstats.com, they were able to piece together M3 data based on existing available data, and it seems that M3 is being inflated at a rate of 14% annually. That's a lot of liquidity.

The fed is playing a duplicitous game. They say they're concerned about inflation, yet it is they who cause inflation by adding liquidity to the money supply. They "raise" rates to tame inflation, that is, they're making it more expensive for people to get money and credit, thus "cooling off" inflation. Interest rates are the lesser of the two mechanisms used by the fed, the greater is their Open Market Operations, ie, adding/removing liquidity from the excess banking reserves. What the fed has been doing is they've raised rates, but they have not removed liquidity from the money supply. If they wanted to truly combat inflation, they'd raise rates AND remove liquidity.

Now, you're concerned about actual physical money, ie, coins and paper and such - M1. That is a little harder to increase/decrease because it takes an actual physical effort to print/mint more money and put it in circulation and to gather up old coins/notes to remove from circulation. Interestingly enough, M1 is actaully on the decline and physically being taken out of circulation. M3 on the other had is the most easy to manipulate since it's purely increasing/decreasing the money supply through book keeping entries and as I said, M3 is growing at a rate of around 14% annually.

As a matter of fact, from July 1, 2006 to June 30, 2007, the Fed added a little over $2.2 trillion to the money supply through its Temporary and Permanent Open Market Operations, that's a lot of money. But it doesn't stop there, through the mechanics of fractional reserve banking, banks can lend out 10 times reserves, so in reality, over $22 trillion in liquidity was added during that 1 year period. That's is A LOT of money. As you increase liquidity, you increase the availability of money and credit. As you remove liquidity, you decrease availability of money and credit. As a matter of fact, when you apply for credit and it's granted, the banks don't lend you money. In reality, when credit is granted, new money is created. According to the Fed's own publications, when credit is extended, you money is created and put into circulation, the banks don't loan you anything. For instance, when you sign that credit card application, the banks consider that a negotiable instrument, thus that signed application is put on their books as an asset. In reality, you funded your own credit card account.

So, in a nutshell, increases in the money supply increase inflation and the availability of money and credit. As liquidity is added, trade increases as there is more money/credit available to foster trade. But as the money supply increases and drives inflation, it will trigger the fed to ultimately raise interest rates to "fight" inflation.

Interestingly enough, prior to the creation of the Fed in 1913, inflation was near zero. It wasn't until after the creation of the Fed and our fiat money system that inflation became a permanent fixture in our financial landscape.
the decision is done by central banks.
the relative money supply (that is MS per unit of output) is directly related to inflation. The more money, given a set amount of physical output, the higher the inflation.
MS is related to interest but you can not say exactly since money demand and many other of the "all-else-held-equal" can not be held equal. this comes from inflation again.
Credit and trade are the same since inflation affects future expectations
Here's some of a resent articel I have read , that might help you with your answer , I would have to write a book to answer your question :

The Bank for International Settlements, the world's most prestigious financial body, has warned that years of loose monetary policy has fueled a dangerous credit bubble, leaving the global economy more vulnerable to another 1930s-style slump than generally understood.
"Virtually nobody foresaw the Great Depression of the 1930s, or the crises which affected Japan and southeast Asia in the early and late 1990s. In fact, each downturn was preceded by a period of non-inflationary growth exuberant enough to lead many commentators to suggest that a 'new era' had arrived", said the bank.

The BIS said China may have repeated the disastrous errors made by Japan in the 1980s when Tokyo let rip with excess liquidity.

"The Chinese economy seems to be demonstrating very similar, disquieting symptoms," it said, citing ballooning credit, an asset boom, and "massive investments" in heavy industry.

It said China's growth was "unstable, unbalanced, uncoordinated and unsustainable", borrowing a line from Chinese premier Wen Jiabao.

In a thinly-veiled rebuke to the US Federal Reserve, the BIS said central banks were starting to doubt the wisdom of letting asset bubbles build up on the assumption that they could safely be "cleaned up" afterwards - which was more or less the strategy pursued by former Fed chief Alan Greenspan after the dotcom bust.

It said this approach had failed in the US in 1930 and in Japan in 1991 because excess debt and investment built up in the boom years had suffocating effects.

The bank said it was far from clear whether the US would be able to shrug off the consequences of its latest imbalances. "The dollar clearly remains vulnerable to a sudden loss of private sector confidence," it said.

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