Bill Woolsey writes:

Monetary Freedom: Sumner and DeLong on Index Futures Convertibility: Here DeLong begins to go wrong. He adds in 3 percent because he believes that is a proper target for the long run average nominal interest rate. His figure assumes a 3 percent average nominal interest rate between 2007 and 2011. Whether or not that would be desirable, it plays no role in determining what the Fed should be doing during either the 4th quarter of 2011 or the 4th quarter of 2010. He has effectively put NGDP on an 8 percent growth path. Neither $18.155 trillion nor its reciprocal should play any role in the system.

Why does DeLong make this error? It is because he understands the proposal as automatically changing the quantity of money when futures are purchased and sold. In DeLong's view, when the Fed buys these contracts, it is providing money now, in the fourth quarter of 2010 and will receives the money back in the fourth quarter of 2011. The Fed is making a type of loan when it buys contracts. It makes sense that the Fed would charge interest on these loans. If the Fed charges 3 percent interest on the loans per year, then it provides dollar deposits (makes a loan of a dollar now) in return for (1+.03) times 1/$17.455 trillion of 4th quarter NGDP to be paid in one year. That is 1/(17.455 trillion/ 1.03) or 1/16.947 trillion of 4th quarter 2011 NGDP. When the Fed sells the contracts, it is borrowing (accepting a deposit) of a dollar in exchange for paying 1/16.947 trillion of 4th quarter 2011 NGDP in one year. It pays 3 percent interest on these deposits, again, adjusted for any deviation of NGDP from target.

DeLong's error was to calculate the interest payment for 4 years compounded, and then multiplying when he should have divided. If the contract was defined based on the borrowers paying the Fed a dollar when the loan comes due, like a T-bill, then multiplication would have been appropriate. The appropriate adjustment, however, is 1.03 times $17.455 trillion, which is $17.979 trillion, not $18,155 trillion. More importantly, because the adjustment for the deviation of NGDP from target would not be known up front, this would be inappropriate. Division is really the only sensible approach...

Touche... I think.

Let me try to explain what we want to do...

Right now, in December 2010, we want to give people an incentive to take actions that expand the money supply if they think that nominal GDP at the end of 2011 is likely to be lower than $17.5 trillion and to contract the money supply if they think that nominal GDP at the end of 2011 is likely to be higher than $17.5 trillion.

So the Federal Reserve announces an open offer to buy and sell: $1 in cash now in exchange for... some fraction of nominal GDP at the end of 2011.

It seemed to me when I wrote that the fraction of nominal GDP should not be 1/17,500,000,000,000 because that gives people an incentive to take action to expand the money supply if they expect nominal GDP in a year to be $17.5 trillion because then the Federal Reserve is lending them money at 0% nominal. So if you want people to borrow from the Fed and so increase the money stock if they expect nominal GDP to be less than $17.5 trillion and lend and so shrink the money stock if they expect nominal GDP to be greater than $17.5 trillion, you need a different number than $17.5 trillion in the denominator.

So I accept the correction to $18.0 trillion. The rest of it I am still unsure about.

I have to think about this a lot more, which means I need to find the time to do so...