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Inflation and the Debt/GDP ratio

Summary:
I'm trying to write a simple explainer. The best way to understand how inflation affects the debt/GDP ratio is to start out with a scenario where it doesn't. Then look at ways in which the real world is not like that scenario. Here's the "No Effect" scenario: The Bank of Canada suddenly decides to raise the inflation target by one percentage point (1ppt). Like from 2% to 3% (which is a 50% increase, which is why I'm being picky and saying 1 ppt instead). Assume that inflation immediately increases by 1ppt, but real GDP (RGDP) growth is not affected, so nominal GDP (NGDP) growth immediately increases by 1ppt. Assume that expected inflation also immediately rises by 1ppt, and that nominal interest rates also immediately rise by 1ppt (because borrowers and lenders only care about real

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I'm trying to write a simple explainer. The best way to understand how inflation affects the debt/GDP ratio is to start out with a scenario where it doesn't. Then look at ways in which the real world is not like that scenario.

Here's the "No Effect" scenario: The Bank of Canada suddenly decides to raise the inflation target by one percentage point (1ppt). Like from 2% to 3% (which is a 50% increase, which is why I'm being picky and saying 1 ppt instead).

Assume that inflation immediately increases by 1ppt, but real GDP (RGDP) growth is not affected, so nominal GDP (NGDP) growth immediately increases by 1ppt.

Assume that expected inflation also immediately rises by 1ppt, and that nominal interest rates also immediately rise by 1ppt (because borrowers and lenders only care about real interest rates, which are unaffected in this scenario).

Assume that the nominal interest paid on all government bonds also immediately rises by 1ppt (because those bonds have a very very short term to maturity, so all get immediately rolled over at the new higher nominal interest rate).

Assume that nominal government spending and tax revenues also immediately grow 1ppt faster (so real government spending and taxes are unaffected).

In this scenario, the nominal debt grows 1ppt faster, but NGDP grows 1ppt faster too, so the nominal debt/NGDP ratio is unaffected by the higher inflation rate. Everything just scales up ("inflates") in the same proportion. But nothing "real" (inflation-adjusted) changes.

To see why, note that the "primary" deficit (the deficit ignoring interest on the national debt) stays the same as a ratio of NGDP. But the "full" deficit (including interest on the national debt) increases by the 1ppt increase in nominal interest times the national debt. So the debt is growing 1ppt faster than before. Just like NGDP is growing 1ppt faster than before. So the ratio of the two is unaffected by inflation.

The main reason that scenario is wrong is because things don't all change immediately. There are lags. And all those lags tend to work the same way: they make the debt/GDP ratio tend to fall initially when the Bank of Canada decides to increase the inflation target.

1. The first important lag is due to the fact that government bonds do not all have a very very short maturity. Take the extreme opposite case: suppose all bonds are perpetuities, that never mature, but pay a fixed coupon every year forever. In this case the interest on the previously-existing debt is unchanged by higher inflation. So higher inflation causes the debt/GDP ratio to start falling over time. In the simple case where the government keeps the budget balanced, so issues no new debt, 1ppt higher inflation alone will cause the debt/GDP ratio to fall by 1% (not 1ppt) every year, so it halves in 70 years (because it causes NGDP to double while debt stays the same). And a 2ppt inflation increases alone will halve the debt/GDP ratio in 35 years, etc.

The real world has a mixture of government bonds, of varying maturities, so it's somewhere between my original scenario and the extreme opposite case. (And some bonds are indexed to inflation, so the interest rate rises immediatly with inflation.)

2. The second important lag is that nominal interest rates may not immediately rise when the Bank of Canada decides to target higher inflation. They may even fall initially. This is especially likely if the Bank of Canada keeps its decision secret. An essential part of modern inflation targeting is that the Bank of Canada clearly announces its commitment to keeping inflation at the 2% target, and it's hard to imagine the Bank of Canada raising its target in secret. But before the 1990s central banks did not have explicit inflation targets. And actual inflation depends on expected inflation, as well as on what central banks do. So a central bank could cut nominal interest rates, and inflation would eventually rise, and expected inflation would eventually rise too, when people saw that higher actual inflation, and eventually learned it wasn't just a temporary blip but was likely to continue, because it had continued.

So if the Bank of Canada secretly reneged on its commitment to target 2% inflation, and lowered nominal interest rates, it might take decades before people fully understood what had happened, and the Bank of Canada needed to raise nominal interest rates, 1ppt higher than they were initially, to bring real interest rates back to where they were initially, to prevent inflation spiralling even higher. And in the meantime the debt/GDP ratio would be falling, due to nominal interest rates being lower than the 1ppt increase in my original scenario.

[But notice something interesting about the interaction between my 1st and 2nd lags: the second lag (lower nominal interest rates) only matters when the bonds are rolled over and refinanced at the new interest rate. So in the extreme case where all bonds are perpetuities, the second lag wouldn't matter for existing debt

There may be other effects. Like if the tax and benefit system is not fully indexed to inflation. Or if a secret decision to target higher inflation causes temporarily higher real GDP. But I think I will stop there.

But there's one historical question that's stuck in my mind and that I don't have the answer for. How much of the history of the debt/GDP ratio, from the end of WW2 until when things had mostly settled down to the 2% inflation target (say 2000?) is a story of the rise and then fall of the inflation rate? Not all of it, obviously, because other things happened too. But my guess is that an important part of it is. We didn't just grow our way as (in part) inflate our way out of a high debt/GDP ratio, and then it went into reverse as inflation came back down again. It's as if the government has a valuable reputation for keeping inflation low, and it can spend that reputation bringing the debt/GDP ratio down, and then suffers the costs of re-investing in that reputation if it wants to bring inflation down again. But lots of other important things mattered too, and history is the net effect of all those things. (And my thought-experiment above, where the Bank of Canada secretly raises its inflation target, is hard to imagine happening today, but much more applicable to the time before explicit inflation targets.) But I don't know what the post WW2 counterfactual would have looked like.

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