# FTPL explained part 2: interest rate In my previous blog about FTPL, I explained the basic idea of FTPL. In this blog I will dig a bit deeper in the relations between the government, central bank policy and inflation. I also try to understand FTPL myself, so it might be that the equations I derive in this blog deviate from the original idea. Comments are always welcome!

Let me first show the basic FTPL equation once more: The left side of this equation shows us the real value of a maturing government bill. On the right side one can find the nominal surplus in time t, which is represented by St. Pt+j represents the price level in period t+j. The surpluses are discounted by the nominal rate R, which is assumed to be fixed in this equation.

When we assume that future surpluses are fixed, and given the fact that R is fixed, we can write: If we assume that the government will always pay the nominal interest rate on her debts, then we can say: S = i*Bt+j. However, the holder of the debt only has a claim on interest payments on the current amount of debt (Bt), so if the government is assumed to always pay at least the nominal interest rate on her debt, then we can write: We can now rewrite this equation to: How should we read this equation?

The idea is this. The current price level is determined by expectations about the future. When investors expect that the nominal interest rate paid by the government (i) is smaller than the nominal interest rate that is demanded by investors (R), then this will move up the price level (Pt). This is because the right hand side gets smaller: Bt is fixed so Pt has to move up. The same happens when the future price level expected to be higher: it will lead to a higher price level now.

So why did I not write this in terms of inflation?

Because I use a maturing 1-period government bond, inflation is not visible. All the effects of monetary policy on the devaluation of money is already in Pt. Because we start at Pt (and because Pt is just a letter), we can only see Pt and no movement. However, different scenario’s with a different  R, i or Pt+j, give us different current price levels.

How does this relate to monetary policy?

In the described world in which all government debt is short term, the government has full control over i via the central bank. If the government does not want to pay i=R, it can decide to choose an i that is smaller than R. In that case, investors will want to get rid of their T-bills by buying consumption goods instead. This will drive up the price level.

So how is it possible that the Bank of Japan was able to lower i to zero, without any increase in the price level?

Noah Smith (@noahpinion) noted correctly in a tweet that Japan saw no spectacular increases in the price level, despite large increases in government debt and years of budget deficits. But this does not mean that we can throw FTPL away. About this government debt: The price level does not have to go up with government debt. This is for the same reason as I gave above for assuming a constant Bt: the government is not likely to default on her debt, so she will at least pay the nominal interest over her debt.

About budget deficits: The price level does not have to go up with budget deficits either. Investors just have to believe that the discounted surpluses are sufficient to pay the nominal interest rate on all outstanding government debt. More debt does not change a thing, as surpluses are expected to grow with the same rate.

What the government can do, is change the interest rate she pays. But if investors expect that the government will never allow the price level to increase to much – via a central bank inflation target – then investors expect that the i paid by the government will increase as soon as the R demanded by the public increases. The result is that there will not be an increase in the price level at this moment.