## October 01, 2003

### One Hundred Interesting Mathematical Calculations, Puzzles, and Amusements: Number 20: The Federal Reserve Problem

The Federal Reserve Problem

A man named Alan Greenspan is Chair of the Federal Reserve Board, and Chair of the Federal Reserve System's Open Market Committee. The Federal Reserve is our country's central bank: it is responsible for setting interest rates. The interest rate is set in percent per year. Currently, the interest rate on short-term U.S. Treasury bonds is one percent per year: buy \$1,000 of Treasury bonds now, and the U.S. government will pay you \$1,000(1 + 1%) = \$1,000(1.01) = \$1,010 back in a year.

Interest rates matter because the higher the interest rate the less likely companies are to spend money building new buildings and factories and buying new machines. When interest rates are high, they say, "Couldn't we make more money by lending out our cash than by using it to build new buildings and factories and buy more machines to make more stuff to sell?" When interest rates are low, they say, "Couldn't we make more money by taking some of the cash we loaned out--the people who borrowed it are hardly paying us any interest, after all--and use it instead to build new buildings and factories and buy more machines to make more stuff to sell?"

When interest rates are high, spending on buildings, factories, and machines is low. People who work in construction or machine making lose their jobs. As they lose their jobs, they stop buying as much for their households, and this means that businesses that make the things they would have bought find themselves losing money and yet more people lose their jobs.

The Federal Reserve staff calculate that the relationship between the country's unemployment rate and interest rates is:

u = x + 0.6(r)

Where u is the unemployment rate (a number like 6%, or .06), r is the interest rate (a number like 1%, .01), and x is a number that jumps around over time and represents the strength of other forms of demand--it is the result of the combination of consumers' optimism, government spending, foreign demand for U.S.-made products, and a lot of other factors that the Federal Reserve staff tracks.

Just before September 11, 2001, the Federal Reserve staff believed that x was equal to 1.4% (.014), and the Federal Reserve's Open Market Committee had set the interest rate r to 6%. What does the equation above predict the value of the unemployment rate would be?

In fact, the unemployment rate on the eve of September 11, 2001 was about 5%, and that made the Federal Reserve happy. The Federal Reserve is charged by law with avoiding inflation--unwarranted rises in prices--and it fears that an unemployment rate below 5% produces rising inflation. Thus the Federal Reserve wants to keep the unemployment rate from going below 5% (because it is supposed to keep inflation from rising), and the Federal Reserve wants to keep the unemployment rate from going above 5% (because unemployment is a bad thing: the more people are unemployed, the more people are poor and unhappy).

Then came the terrible events of September 11, 2001. Their effects on the economy are perhaps least important, but I think about them because economics is what I do. The terror-attack made a lot of businesses worried about spending money on factories, other buildings, and machines. In the judgment of the Federal Reserve staff, the terror-attack on September 11, 2001, increased the number x in the equation above from 0.014 to 0.044. What should the Federal Reserve have done in response if it wanted to keep the unemployment rate from rising above 5%?

In fact, the Federal Reserve did reduce interest rate r to 1%. But unfortunately it looks like the number x increased not to .044 but to .054, so we have a current unemployment rate of... what?

Now the Federal Reserve has a problem. It doesn't want to reduce the interest rate r even more, because it fears that if the interest rate is lower than 1% that a lot of banks will find it impossible to make money, and will close down. So right now the Federal Reserve is sitting around hoping that the number x in the equation above is about to go down, so that the unemployment rate will soon fall from 6% down to 5%.

Posted by DeLong at October 1, 2003 04:38 PM | TrackBack

Hope is not a plan.

Posted by: joe on October 1, 2003 05:01 PM

The Federal Reserve does not have a problem. The Bush administration has the problem. If the employment picture does not improve enough, Mr. Bush will be fired. Mr. Greenspan will not be fired.

It is not r that has to change it is X. The surest way for X to change is direct stimulus. Money to the states would be a start. Tax cuts to the investor class is not working because they are investing overseas and not domestically. Tax cuts are stimulating demand for foreign goods and are driving up the trade deficit. Mr. Greenspan should be ashamed for supporting ineffective fiscal policy in 2001. He needs to demand effective fiscal policy in 2003.

Posted by: bakho on October 1, 2003 06:04 PM

If the FED took over the role of public finance in
the nation and issued liquidity in any amount to state and local governments to supply public sector goods and services, would the FED funds rate change or remain the same?

If Congress authorizes legislation for the FED to
replace taxation as a means of public finance with
a more robust "push" of liquidity into the nation,
would unemployed workers find more jobs in the public sector or the private sector if businesses
no longer had the obligation to pay taxes?

Could the FED use this "push" of liquidity as
a montary tool?

Posted by: jim on October 1, 2003 08:42 PM

Isn't liquidity already a monetary tool? I don't understand your question. I thought the various bank fund rates were used to regulate M1 and M2 (money supply). Or something like that. It's been a while. Clarification please, Professor Delong?

Posted by: old econ major on October 1, 2003 09:49 PM

btw, jim, pushing liquidity as you describe also devalues the dollar (as does lowering the interest rate), all else being equal. Man, my econ chops are rusty.

Posted by: old econ major on October 1, 2003 09:53 PM

The formula u = x + 0.6(r)
seems completely arbitrary. If x can jump around all over the place the formula is not falsifiable. wht not u = z +.45(r) or any other convenient formula with an arbitrary fudge factor.

Posted by: Daniel on October 2, 2003 08:24 AM

Why would the first order terms of the Taylor series expansion of some interest rate correlation evaluated at apparently a "normal" 2.5-5.5% be useful as an approximation when the interest rate declines enough toward 0.0 that anticipatory deflationary feedback factors become significant?

I'm only an engineer, don't know nuthin bout economy, but this seems a trifle naive. Maybe that's the point?

One of the many things I've learned from this blog that I hadn't a clue about before is how entities like the Fed "talk" the economy up and down. But it seems to be true...

Posted by: Russell L. Carter on October 2, 2003 09:38 AM

I have to wonder how useful the Fed's formula is when Bush's Dept. of Labor can arbitrarily define what "unemployment" means. For example, the term used to include those who have given up looking for jobs, but not any more.

Posted by: Q on October 2, 2003 11:59 AM

OK. This "x" is a shorthand for all of the other factors in the model (presumably some IS-LM model). So a rise in "x" is an inward shift of the IS curve. But wait. Bush is striving to push out this IS curve with his fiscal stimulus - so goes the new converts to Keynesian economics in the GOP. What is everyone missing here? Open question, but my favorite answer is that Wall Street (investment demand) is being turned off by the prospect of long-term fiscal irresponsibity. Put another way - short-term stimulus lowers x while long-term fiscal fiascos increase x. OK, just another plug for that moderate Senate (Dem and Rep) idea back in Oct. 2001: you know where they wanted to push for short-term stimulus and long-term restraint. Remember? The one that Bill Thomas and George Bush conspired to kill.

Posted by: Hal McClure on October 2, 2003 02:12 PM

What do the Labor Dept's 'unemployment' ratios mean today? Don't they just count the number of people receiving unemployment compensation (how else can they track the 'unemployed'?) and divide that by the number of people working? And if they have an 'employed' number, why don't they publish that? Too many people assume that 7.2% unemployment means 92.8% employment. It does not!! Why doesn't the Labor Dept. publish a more meaningful 'employment' metric, and if so, what is it?

Posted by: Peteh on December 16, 2003 05:55 PM

What do the Labor Dept's 'unemployment' ratios mean today? Don't they just count the number of people receiving unemployment compensation (how else can they track the 'unemployed'?) and divide that by the number of people working? And if they have an 'employed' number, why don't they publish that? Too many people assume that 7.2% unemployment means 92.8% employment. It does not!! Why doesn't the Labor Dept. publish a more meaningful 'employment' metric, and if so, what is it?

Posted by: Peteh on December 16, 2003 05:59 PM

What is this supposed to teach? Taking a reduced form relationship and pretending it's structural in order to do bad policy analysis? Ignoring the difference between nominal and real interest rates? Unless the point is ridicule, this problem should be promptly withdrawn.

Posted by: rw on December 17, 2003 06:15 AM

What is this supposed to teach? Taking a reduced form relationship and pretending it's structural in order to do bad policy analysis? Ignoring the difference between nominal and real interest rates? Unless the point is ridicule, this problem should be promptly withdrawn.

Posted by: rw on December 17, 2003 06:18 AM

They had this kind of stuff in intro econ courses several decades ago. Since then, we've learned that if you take reduced form relations and treat them as structural, the result is bad policy analysis. We've also learned that it was a bad idea to ignore the distinction between nominal and real interest rates. So why warp the minds of children with this example of shoddy economic reasoning?

Posted by: rw on December 17, 2003 06:53 AM