Printing money generates inflation. A country that tried to pay all of its bills with printed money would quickly generate a hyperinflation. Indeed, this is exactly how most of the world's hyperinflations have been generated!
In the late 1990s, for example, Robert Mugabe's government in Zimbabwe started seizing commercial farms and driving entrepreneurs out of the country. The economy tanked but Mugabe needed money to support the military so he turned to the printing presses. As more and more money was printed the inflation rate skyrocketed to nearly unbelievable levels.
At its peak in November of 2008 the inflation rate was running at a rate of 79.6 billion percent per month! Hence the need for bills like the one below. By 2009 almost everyone in Zimbabwe switched over to using US dollars or other foreign currency and now the official Zimbabwe currency no longer exists. Thus, Mugabe's plan worked but only briefly.
The short answer would be "inflation", but that don't completely answer the question. We still need to ask why inflation is worse than the alternative, accumulation of government debt.
Inflation is a tax like any other: just as an income tax penalizes earning an income, inflation penalizes holding cash. The question is then which type of tax is more distortionary and inefficient?
Economists have found that the inefficiency caused by a tax is closely related to the individual response to that tax. If individuals alter their behavior greatly due to a tax, cutting down dramatically on whatever is being taxed, generally that tax will be inefficient. (This logic does not hold, of course, when the object of a tax is an externality like air pollution.) It turns out that the response to a tax on holding money is far larger than the response to a comparable tax on income, consumption, or property.
This is easy enough to understand: if 20% of your income each day is taxed away, you might cut down on your labor effort a little, but you're constrained by the fact that you still need to make money to support yourself. (In fact, for this very reason you might actually put in more effort under a tax in order to recoup your losses.) If 5% of the money in your wallet was taxed away every day, however, you would go to extraordinary lengths to avoid carrying cash, going to the ATM for every transaction. Since a bank must back your deposit with a certain fraction of reserves -- which are also vulnerable to inflation -- your incentive to save within the traditional banking system would vastly diminish.
You would keep just enough in your accounts to be able to handle your transactions, which would result in a massive reshaping of the financial system. The government would have to raise the "tax" on holding money even higher in order to recoup the same real returns, which would make you even more reluctant to hold cash, forcing yet another increase in inflation by the government, and so on ad infinitum. This is the hyperinflationary death spiral we've seen in countries like Zimbabwe.
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A country's monetary base -- the total amount of currency in the system, in the form of paper money or reserves on deposit at the central bank -- is actually quite small in comparison with government expenditures and debt. At the moment it the United States's monetary base is roughly $2 trillion, which is far smaller than its $13 trillion gross debt or $8 trillion in debt held by the public. In other words, to erase its debt obligations by printing money, the U.S. would have to increase the size of the monetary base by 400%. This would result in inflation of at least the same magnitude as 400%, and probably higher -- because fear of future inflation would make people less willing to hold money, which in turn would increase inflation even more. Even if the government merely printed enough money to cover its deficit for one year, this positive feedback loop would probably lead to inflation of at least 100%.
In other words, to deal with its fiscal problems by printing money, a government must create extraordinarily high levels of inflation. This is problematic for several reasons, including:
1. As I outlined above, inflation is a tax on holding money.
2. Unexpected inflation massively changes the terms of financial contracts denominated in nominal dollars, favoring the borrower (or whoever is going to pay the money) at the expense of the lender. This injects severe uncertainty into the borrower-lender relationship and cripples the financial system. (In fact, since the government borrows much of its money in nominal dollars, it would actually erase more of its debt through the inflation caused by printing money than through the printed money itself. Using inflation to erode the real value of your debt, however, is really no different from simply defaulting on that debt, and would be treated by creditors as such.)
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3. Inflation complicates taxation of capital gains. Under the current tax system, gains in nominal asset price created by inflation are taxed at the same rate as any other capital gains. Hence, under extremely high inflation, you are given a very high tax bill for holding assets that may have not actually appreciated in value at all. This destroys the incentives to make any kind of investment whatsoever.
Developed countries, therefore, very wisely refuse to "create money out of thin air" to the extent necessary to eliminate their deficits. They even establish independent central banks to help assure markets that this will never happen.
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