Economics is about solving lots of little puzzles. At a July 4th party, a super smart friend -- not a macroeconomist -- posed a puzzle I should have understood long ago, prompting me to understand my own models a little better.

How do we get inflation from the big fiscal stimulus of 2020-2021, he asked? Well, I answer, people get a lot of government debt and money, which they don't think will be paid back via higher future taxes or lower future spending. They know inflation or default will happen sooner or later, so they try to get rid of the debt now while they can rather than save it. But all we can do collectively is to try to buy things, sending up the price level, until the debt is devalued to what we expect the government can and will pay.

OK, asked my friend, but that should send interest rates up, bond prices down, no? And interest rates stayed low throughout, until the Fed started raising them. I mumbled some excuse about interest rates never being very good at forecasting inflation, or something about risk premiums, but that's clearly unsatisfactory.

Of course, the answer is that interest rates do not need to move. The Fed controls the nominal interest rate. If the Fed keeps the short term nominal interest rate constant, then nominal yields of all bonds stay the same, while fiscal inflation washes away the value of debt. I should have remembered my own central graph:

This is the response of the standard sticky price model to a fiscal shock -- a 1% deficit that is not repaid by future surpluses -- while the Fed keeps interest rates constant. The solid line is instantaneous inflation, while the dashed line gives inflation measured as percent change from a year ago, which is the common way to measure it in the data.

There you have it: The fiscal shock causes inflation, but since the nominal interest rate is fixed by the Fed, it goes nowhere, and long term bonds (in this linear model with the expectations hypothesis) go nowhere too.

OK for the result, but how does it work? What about the intuition, that seeing inflation coming we should see higher interest rates? Let's dig deeper.

Start with the simplest model, one-period debt and flexible prices. Now the model comes down to, nominal debt / price level = present value of surpluses, \[\frac{B_{t-1}}{P_t} = E_t \sum_{j=0}^\infty \beta^j s_{t+j}.\] (If you don't like equations, just read the words. They will do.) With a decline in the present value of surpluses, the value of debt coming due today (top left) can't change, so the price level must rise. The price of debt coming due is fixed at 1, so its relative price can't fall and its interest rate can't rise. Or, this model describes a price level jump. We get bad fiscal news, people try to spend their bonds, the price level jumps unexpectedly up, (\(P_t\) jumps up relative to \(E_{t-1}P_t\), but there is no further inflation, no rise in expected inflation so the interest rate \(i_t = r+ E_t \pi_{t+1}\) doesn't change.

Ok, fine, you say, but that's one period, overnight debt, reserves at the Fed only. What about long term bonds? When we try to sell them, their prices can go down and interest rates go up, no? No, because if the Fed holds the nominal interest rate constant, their nominal prices don't change. With long term bonds, the basic equation becomes market value of nominal debt / price level = expected value of surpluses, \[\frac{\sum_{j=0}^\infty Q_t^{(j)} B_{t-1}^{(j)}}{P_t} = E_t \sum_{j=0}^\infty \beta^j s_{t+j}.\] Here, \(Q_t^{(j)}\) is the price of \(j\) period debt at time \(t\), and \(B_{t-1}^{(j)}\) is the face value of debt at the beginning of time \(t\) that matures in time \(t+j\). (\(Q_t^{(j)}=1/[1+y^{(j)}_t)]^j\) where \(y^{(j)}_t\) is the yield on \(j\) period debt; when the price goes down the yield or long-term interest rate goes up. )

So, my smart friend notices, when the present value of surpluses declines, we could see nominal bond prices \(Q\) on top fall rather than the price level \(P\) on the bottom rise. But we don't, because again, the Fed in this conceptual exercise keeps the nominal interest rate fixed, and so long term bond prices don't fall. If the \(Q\) don't fall, the \(P\) must rise.

The one-period price level jump is not realistic, and the above graph plots what happens with sticky prices. (This is the standard continuous time new-Keynesian model.) The intuition is the same, but drawn out. The sum of future surpluses has fallen. People try to sell bonds, but with a constant interest rate the nominal price of long term bonds cannot fall. So, they try to sell bonds of all maturities, pushing up the price of goods and services. With sticky prices, this takes time; the price level slowly rises as inflation exceeds the nominal interest rate. A drawn out period of low real interest rates slowly saps the value of bondholder's wealth. In present value terms, the decline in surpluses is initially matched by a low real discount rate. Yes, there is expected inflation. Yes, long-term bondholders would like to escape it. But there is no escape: real rates of return are low on all bonds, short-term and long term.

So, dear friend, we really can have a period of fiscal inflation, with no change in nominal interest rates. Note also that the inflation eventually goes away, so long as there are no more fiscal shocks, even without the Fed raising rates. That too seems a bit like our reality. This has all been in my own papers for 20 years. It's interesting how hard it can be to apply one's own models right on the spot. Maybe it was the great drinks and ribs.

We get a fiscal impulse, and the Fed pegs interest rates steady and low forever. The price of haircuts and apples goes up.

ReplyDeleteFine. But bonds are not the only assets. If we put land, real estate, and equities into the model, does their price just go up along with haircuts? Can their real price go up as hedges against inflation/

That’s exactly the question I had in my mind and now I can see the answer in the model. Thank you for your continuous effort to let us to keep up with this new theory that I found so fascinating.

ReplyDeleteNow I have the following question in my mind: while today, we can easily sustain that the central banks control all the interest curve, from short to long maturity (true after the GFC), was this true in the 70s/80s? Why at the time the free move of the long maturities bonds didn’t stop, or at least mitigate, the rise in the general level of the prices? Should we model in same way the (controlled) long maturity yield, given that since the Fed start rising the Fed Fund, the long term yield didn’t rise, but actually decreased? (the yield curve got inverted). I can understand that those questions are beyond the model, but just to have your anecdotal flavor

Through how elegant and varied loops some must jump through to avoid the wealth and real balances effects.

ReplyDeleteThe key quirky thing, I think, is that with sticky prices the Fed can affect both expected inflation AND the real interest rate.

ReplyDeleteDo you think that (1) geopolitical tensions (including but not limited to semiconductors and AI) which in turn push for the boom in semi manufacturing, demand for jobs, etc in the US paired with (2) student loan debt relief (assuming a new plan is approved) allowing for consumer spending to remain relatively high would continue the inflationary trend?

ReplyDeleteThe model is academic. The price level is determined by the stochastic Ito integral equation dP(t) = a(t) P(t) dt + b(t) P(t) dz(t), per your definition, where a(t) = the instantaneous drift rate (i.e., the rate of inflation, pi(t)) and b(t) = the instantaneous volatility parameter. dz(t) = Wiener process increment.

ReplyDeletePer Clarida, Gali, and Gertler (1999), x(t) = E{x(t+1)|J(t)} - eta•[ i(t) - E{a(t+1)|J(t)} +ln(beta)] and a(t) = lambda•x(t) + beta•E{a(t+1)|J(t)}, where J(t) is the information set known at time t.

The feedback loop is via the expectations E{x(t+1)|J(t)} and E{a(t+1)|J(t)} and the Ito integral equation for the variable P(t) that appears in your basic fiscal equation. Clarida, Gali, and Gertler (1999) are explicit on this point.

In your infinite sum, the discount factor equals the exp[ a(t +j) + ln(beta)], where -ln(beta) = the rate of time preference, rho. The term s(t+j) in the infinite sum on the RHS of the first equation in your blog post above is a certainty equivalent measure of the future real primary surplus at the close of period t + j. The derivation is straight-forward. The conditional expectation however throws a twist into the mix. We lack perfect foresight--our vision is myopic. The infinite sum is truncated at t + n, beyond which point s(t+j) = 0, almost surely.

It was interesting to watch the evolution of longterm rates during 2021-2022, once the financial oppression undertaken by the FOMC's QE program was lifted. By the way in your model B(t-1) is the number of discount bills issued and o/s at the close of period t-1. The price of those bills is Q(t) in period t before the bills are redeemed. The price level P(t) merely relates the nominal face value of the bills to the real value (purchasing power) of the bills at maturity. You can see this daily when you trade in the T-bill market, Q(t) changes but is everywhere less than unity up to the redemption time of the bill when Q(t) goes to unity.

The academic models exist in a parallel universe. Sometimes useful, most of the time not very.

How is QE financial oppression. They buy bonds, the private sector sells the bonds entirely voluntarily. Who's being oppressed?

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DeleteI would need to read several times to make sure this question isn't already answered, but what about Real interest rates? Would they adjust in this model even if the Fed was holding nominal rates the same?

ReplyDeleteNow write the equation for how much money the fed is losing by holding nominal rates constant despite inflation and it will really be clear where the money is going to maintain nominal rates.

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ReplyDeleteProfessor, it's not obvious to me when you say, "With a decline in the present value of surpluses, " whether that mean the future betas have dropped, or that "s" has dropped, due to inflation causing lower real future surplus stream?

ReplyDeleteI guess I thought of the answer to my own question. You are implying that, unlike most past episodes of extra federal borrowing, people don't believe the fiscal stimulus this time will ever be repaid, so they conclude that the negatives in "s for 2020-2022 won't be compensated by positives in the future, so the summation of the right side of the equation has dropped in value.

DeleteSetting aside the basic fact that empirically the most basic stylized fact is that the world is as far from the expectations hypothesis as it can possibly be (or that's what I recall as the conclusion of Fama-Bliss), I can't make sense of this for the simple reason that the Fed has changed rates a lot, and instead of high long rates we've instead got a very inverted curve.

ReplyDeleteI think your answer is nonesense.

1) Is the dashed line representing YoY inflation supposed to rise above 0.3% as the solid line does? If not, would a viewer be able to help articulate why with numbers?

ReplyDelete2) How does this model remain valid with an inverted yield curve?

3) While this solution is academically interesting - How it realistically be assumed that interest rates cannot change when treasury bonds are marketable securities?

there's no assumption that interest rates can't change, the assumption is that they don't change because of the expectations hypothesis.

DeleteOf course, it's not really plausible to believe that long rates stayed low through the initial period where the fed hadn't yet hiked because agents in the economy really expected the Fed would never hike. As stated above I think the answer is nonsense.

Where's UNRATE in this inflation discussion?

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ReplyDeleteDear John, great post as usual. Few questions and one half baked comment. To what extent does the FTPL work with adaptive expectations? How would it work in the example above: gov increases G and public has adaptive expectations? Comment (and maybe a few questions lurking): Since the increases in G and deficits (and debt) in the US in the 2000s and 2010s were not followed by high inflation, does that mean that one can infer that under the FTPL the public expected public debt to be repaid? And now that we saw inflation, does that mean that the public doesn't expect debt to be repaid? What is the trigger? Size of G increase? Something else (lack of "fiscal forward guidance")? Does that also mean that previous (preCovid, 1990s, etc) public debt increases in Japan/EU countries (not followed by inflation) meant public expected debt to be repaid? I obviously find this intriguing and hard to "test" as you allude to in your posts and book (love the "Episodes" in your book). thanks

ReplyDeleteYes, FTPL can work fine with adaptive expectations. Or, more likely, expectations that are sort of adaptive in normal times but change quickly with big events or news. Expected discounted surpluses, use whatever probability distribution you want for "expected." I haven't worked on this much because there is only one way to be rational and a thousand ways to be behavioral; testability is tenuous enough anyway so I don't need the extra freedom of playing with expectations; and general lack of time. I do know that standard adaptive expectations models change a lot with fiscal considerations -- some examples in "expectations and the neutrality of interest rates." Like all else, lots to do in fiscal theory.

DeleteMany thanks. Any views on why large sustained increases in G or deterioration in G-T don't seem to trigger inflation (U.S. in 2010s, Japan for 30 years, etc). What is the trigger? How can this be tested? You mention the brilliant paper by Eichembaum on Prospective deficits in the Asian crisis, but the outcome was a deep recession, not runaway inflation. Big Thank You.

DeleteYou might find this model to track the evolution of the yield curve interesting. It looks like we are beginning to see an evolution similar to the 1980s. I am not sure why the math works so well.

ReplyDeletehttps://community.wolfram.com/groups/-/m/t/2953857

What happens if the debt is inflation protected? Then only it’s price can adjust? Hence no more fiscal inflation? Probably I’m missing something.

ReplyDeleteHere is what you are missing - simple logic puzzle.

DeleteSuppose that a government did try to provide a positive (inflation adjusted) real income to anyone that wanted it.

Why would anyone work or pay taxes?

Do you see the problem?

What happens if the debt is inflation protected ie coupons and principal are indexed?

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