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What are two ways the government could encourage an increase in the supply of loanable funds? And two ways the government could encourage an...

The supply of loanable funds comes from two things: Savings, and fractional-reserve banking.

When people save money in banks, that money is available to be loaned. But this is relatively unimportant actually; what matters is instead the money created by banks using that saved money as reserves. About 90% of the money supply loaned by banks is created by the banking system, not originally from savings.

The ratio between money supply and reserves is called the money multiplier, and it is capped by the reserve requirement set by the government.

So, there are essentially two ways for the government to increase the supply of loanable funds; they could either find a way to increase the amount of money saved, or they could increase the money multiplier.

The former can be done by various means; one of the most successful has been Save More Tomorrow, a system based on insights from cognitive economics that makes it the default to save a portion of your increases in pay over time---by making it a default and focusing on increases, we avoid decision inertia that makes people prefer to do nothing when they don't know what to do, as well as the loss aversion they feel when being asked to give up what they currently have.

But the easiest way to increase the supply of loanable funds is simply to cut the reserve requirement, allowing banks to increase the money multiplier. The usual reserve requirement is 10%, so the multiplier is 1/10% = 10 times. But you could also lower it to say 5%, a multiplier of 20 times.

Now, how could the government increase demand for loanable funds? The chief determinant of demand for loans is interest rates---people are more willing to take out loans if the interest rates are lower. Certain other types of regulations can also act as an additional cost to getting a loan, a sort of "shadow interest rate."

So, you want to lower the interest rate; how do you do that? The most common way is by open-market monetary policy, where the central bank purchases assets (usually government bonds) to drive up their prices and therefore drive down their yields, which in turn lowers interest rates. This is, in fact, the go-to method for responding to a recession; buy up a lot of bonds to bring interest rates down, increase the money supply, get people spending again and get out of the recession.

The government could also raise demand for loans by taking out loans directly, raising the deficit---deficit fiscal policy. When the government borrows more, this raises the demand for loans directly, which raises the market interest rate. This can create crowding out, where the higher interest rates reduce the private demand for loans---but it's usually much less than the amount that the public demand for loans increased, so total demand goes up. By combining fiscal and monetary policy, you can actually increase the money supply and get out of a recession without changing the nominal interest rate. (You can also trigger inflation that way, so you do have to be careful.)

If instead the government started paying back loans, reducing the deficit or even making it negative---surplus fiscal policy, this would lower interest rates, but for the wrong reasons; private demand for loans would probably increase, but not as much as public demand had directly decreased. This can be a good idea if you are at full employment and you're worried about inflation; but it wouldn't make sense as a way of increasing the money supply in response to a recession.

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