## Weekend Reading: Interest Rate Pegs with a Finite Horizon Omega Point

**: Interest Rate Pegs with a Finite Horizon Omega Point: "This is what I used to think about what would happen if the central bank tried to peg the nominal interest rate forever (and I'm pretty sure most macroeconomists thought the same):**

If the inflation rate was not just sticky, but permanently stuck, it would be possible for the central bank to peg the nominal interest rate forever. If it pegged it too high there would be a permanent recession; if it pegged it too low there would be a permanent boom.

But if there was any price-flexibility whatsoever, so that actual and expected inflation eventually rose and kept on rising in a boom, and eventually fell and kept on falling in a recession, it would be impossible for the central bank to peg the nominal rate of interest forever. (Because higher actual and expected inflation reduces the real interest rate for any pegged nominal interest rate, which causes an even bigger boom.) The monetary system would eventually be destroyed because it would either explode into hyperinflation, or implode into hyperdeflation. And as we changed the model to increase the degree of price-flexibility, so that actual and expected inflation adjusted more and more quickly to boom or recession, the more quickly the monetary system would explode or implode. In the limit, as we changed the model so that actual and expected inflation adjusted infinitely quickly, the monetary system would explode or implode instantly.... Pegging the nominal interest rate creates an unstable system....

The Calvo Phillips curve typically assumed by New Keynesian macroeconomists (because it makes the math easy) put a spanner in the works of modelling this. Because even though the price level is sticky in that model, the inflation rate isn't. The price level can't jump; but the inflation rate can jump, if expected inflation changes. So the formal modelling didn't fit my informal reasoning....

A year ago I started thinking about finite horizon models, where the expected future price level was pinned down exogenously.... Suppose that everybody know that 70 years from now (to keep the math simple) the price level will be exactly 100. Because, maybe, the government and central bank promise to retire all the paper money in exchange for real goods at that price level in 2086, and then start afresh with a completely new monetary system. So the price level at the terminal date is pinned down exogenously. That changes everything. A 'permanent' (i.e. 70 year) nominal interest rate peg now becomes a stable monetary system. The monetary system cannot explode into hyperinflation or implode into hyperdeflation, because everyone knows the 2086 price level is pinned down at 100. Brad DeLong would call that 2086 price level of 100 an 'Omega Point' ['page not found' when I tried to link thanks JP]. It's somewhere we know we will end up, eventually. If the central bank pegs the nominal interest rate equal to the natural interest rate, the equilibrium 2016 price level is also 100, and people expect 0% inflation 'forever' (70 years).

If the central bank suddenly raises the nominal interest rate by 1% (one percentage point), and promises to keep it there 'forever', and if prices are perfectly flexible, the price level instantly jumps down to 50 when people hear the news, and inflation is now 1% 'forever'. Setting aside the initial jump up or down in the price level, the inflation rate in this model behaves exactly like Neo-Fisherians say it will. An increase in the nominal interest rate set by the central bank causes a one-for-one increase in the inflation rate (but also a one-time drop in the price level). To get pure Neo-Fisherian results (to eliminate the one-time jump in the price level), you would need to change the model slightly. So when the central bank announces a 1% increase in the nominal interest rate, it also announces that the 2086 price level will be doubled to 200.

What happens in this model if we make the finite horizon longer and longer? The answer is that nothing much happens at all, except we get a bigger and bigger initial jump in the price level, or would need a bigger and bigger change in the promised horizon price level to offset that change.... In the limit, as the finite horizon approaches infinity, the effect of doubling the end-of-horizon price level approaches zero. But we cannot say the same about what happens at the limit, when there is nothing to pin down an Omega point. Because without any Omega point, there is nothing to pin down the current price level. Any current price level, including zero and infinity, is equally possible.

What happens if we assume the price level is sticky, but the inflation rate is perfectly flexible (like with the Calvo Phillips curve)? So the price level can't jump, but the inflation rate can. If the central bank doubles the 2086 price level to 200, at the same time it raises the nominal interest rate peg by 1%, we still get pure Neo-Fisherian results. There is no temporary recession, because the price level doesn't need to jump, so it doesn't matter if it can't jump. But if the central bank holds the 2086 price level fixed at 100 when it 'permanently' raises the nominal interest rate peg by 1%, there will be a temporary recession. The price level cannot jump down instantly; it can only fall slowly. So initially there will be deflation, so the real interest rate will rise, and rise by more than 1%, which is what causes the temporary recession, which lasts until the price level stops falling and starts to rise again at 1% inflation. So the short run effects of increasing the nominal interest rate are very standard; but the monetary system does not implode into ever-increasing deflation, because the 2086 price level is pinned down at 100, so it cannot.

What happens if we re-run the same experiment with double the length of the finite horizon? So the central bank now holds the 2156 price level fixed at 100 when 'permanently' raises the nominal interest rate peg by 1%. The price level needs to fall by more before it starts rising again (remember it would fall to 25 rather than 50 if the price level could jump). So the deflationary recession would be longer and deeper with a 140 year horizon that with a 70 year horizon. Now take the limit of that model as the horizon gets longer and longer. In the limit, if the central bank raises the nominal interest rate by 1% and holds it there for an arbitrarily long time, the price level approaches zero, and stays arbitrarily close to zero for an arbitrarily long time. And there is an arbitrarily long and arbitrarily deep recession.

Yep. That sounds pretty much like what I used to think would happen. If the central bank pegs the nominal interest rate too high for a very very long time, the monetary system implodes into hyperdeflation.