Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
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=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
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=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
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=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
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=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
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=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
=Transcript of the talk, from the video's captions. Auto-generated: speaker names in particular are unreliable.
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=

# The Inflation Accelerator

Authors:   
Discussant: None  
Video: https://www.youtube.com/watch?v=7KexsWQ-flo&t=166s


## Talk (00:02:46 – 00:18:09)

[00:02:52] You guys, you must get, OK, perfect.  
[00:02:53] All right.  
[00:02:54] So thanks for putting all this together and having me on this panel.  
[00:02:59] So this is a paper that we wrote with Andres Blanco, Corina Boa and Colom Jones.  
[00:03:05] And the question we're going to ask is what is the slope of the Phillips curve  
[00:03:12] and how does it vary over time in the time series?  
[00:03:17] In a large class of sticky price models, the slope of the Phillips curves  
[00:03:22] depends on the fraction of prices that change.  
[00:03:24] The more prices change, the steeper the slope.  
[00:03:29] And one well-known fact that came even more in the public's view of these days  
[00:03:38] is that in the data, the fraction of price changes increases in periods of high inflation.  
[00:03:44] So this is US data.  
[00:03:47] The left panel is US inflation.  
[00:03:50] And the right panel is a measure of how often prices change.  
[00:03:54] These are quarterly, frequency, fraction of prices that adjust.  
[00:04:00] And these are regular price changes.  
[00:04:02] The data that was collected was put together by Nakamura-Stuanson co-authors  
[00:04:07] and more recently updated by Montag and VR.  
[00:04:10] So what you see in this picture is that in the 80s, when inflation was high,  
[00:04:17] the fraction of price changes went from about 25% a quarter to about 40%, 45%.  
[00:04:26] And then number one up again during this last post-COVID inflation episode.  
[00:04:32] OK, so the question we're going to ask based on this kind of motivating observation  
[00:04:38] is how does the slope of the Phillips curve vary over time?  
[00:04:45] Well, how does the output inflation trade-off  
[00:04:48] faced by central bankers vary in the time series?  
[00:04:52] Before I tell you what we did, I'll tell you about what's out there.  
[00:04:56] There's two classes of models that are widely used in macro in sticky price literature.  
[00:05:02] One is the so-called time-dependent models, Calvo, Rottenberg, Taylor.  
[00:05:07] These are models that are widely used because they're very tractable.  
[00:05:11] But by assumption, the fraction of price changes is exogenous.  
[00:05:15] In fact, most of these models assume the fraction is constant.  
[00:05:19] There's then menu cost models, state-dependent models,  
[00:05:22] in which researchers explicitly model the firm's decision of when to adjust their price.  
[00:05:30] So in principle, these models could allow the fraction of price changes to vary over time.  
[00:05:37] But in practice, they're much less widely used.  
[00:05:39] And the reason is that they're less tractable.  
[00:05:42] In order to aggregate individual firm decision rules,  
[00:05:45] you need to keep track of the entire distribution of price changes.  
[00:05:49] And that's a complicated object.  
[00:05:51] And so these models are less popular.  
[00:05:55] We've also shown in an earlier paper with my call-offers,  
[00:05:58] which is actually what motivated this paper, that in menu cost models,  
[00:06:04] you actually have that the fraction of price changes moves very little in periods of inflation,  
[00:06:11] providing you calibrate the models to measure microdata.  
[00:06:14] So in many respects, they behave a lot like the time-dependent models like Calvo.  
[00:06:20] All right, so when we're doing this paper, we're going to try to develop an alternative model.  
[00:06:25] That kind of is going to nest some of the features of both time and stay dependent models.  
[00:06:31] The key assumption is going to be that there are multi-product firms.  
[00:06:36] These firms are going to have to decide how many of their prices to change.  
[00:06:41] So they're going to choose a fraction of prices to change,  
[00:06:44] subject to a cost, as in the menu cost literature.  
[00:06:48] But as in the Calvo literature,  
[00:06:50] these firms are not going to be able to decide on which of their prices to change.  
[00:06:56] So we're going to assume effectively that the adjustment hazard varies over time,  
[00:07:02] but is constant inside the firm and all the nice properties of Calvo,  
[00:07:09] all these exact aggregation results that you have in the Calvo model  
[00:07:12] are going to go through in our setting.  
[00:07:14] In fact, our model is going to reduce the one-equation extension of the Calvo model,  
[00:07:19] where this extra equation is going to pin down the fraction of price changes.  
[00:07:23] So it's going to be extremely tractable, and hopefully we learn some economics with it.  
[00:07:29] So in terms of what we find, the model predicts that the slope of the Phillips curve  
[00:07:34] fluctuates considerably in the time series.  
[00:07:37] For the numbers we use in our baseline calibration,  
[00:07:40] the slope moves from about 0.02 in low inflation periods, such as the 90s,  
[00:07:46] to 0.12 in the 70s and 80s.  
[00:07:49] Most of this increase is due to something we're going to call the inflation accelerator,  
[00:07:54] which is a feedback loop between the fraction of price changes and inflation.  
[00:07:58] And the idea is that when inflation goes up, more firms want to raise their prices.  
[00:08:04] But because more firms want to raise their prices,  
[00:08:06] that pushes up inflation even further,  
[00:08:08] causing other firms more prices to change and so on.  
[00:08:12] In the absence of this feedback loop, the slope of the Phillips curve would increase much less  
[00:08:18] in the high inflation periods.  
[00:08:20] It would increase by only a factor of 2, so 2.04.  
[00:08:24] So in other words, if you take a standard plain vanilla Calvo model,  
[00:08:28] and you feed it exogenously, these movements in the fraction of price changes that we actually saw,  
[00:08:34] but you do not allow the fraction of price changes to respond to additional shocks,  
[00:08:40] then you're going to get an increase in the slope of only 0.04.  
[00:08:43] So this mechanism that we're emphasizing, this endogenous response of the fraction,  
[00:08:48] generates a lot more aggregate price flexibility than you can get just by staring at the fraction of price changes.  
[00:08:56] Okay, so let me tell you a bit about the model.  
[00:09:00] So the model, as I said, is quite tractable,  
[00:09:03] but we want to keep it as close as possible to the menu cost setting,  
[00:09:08] because that's tractable and because that allows us to compare the two.  
[00:09:13] So we're going to assume preferences.  
[00:09:14] They're very much in the spirit of the menu cost literature, log linear.  
[00:09:19] So preferences are log over consumption linear in hours worked.  
[00:09:23] I assume a cash in advance constraint, which implies that nominal wages are proportional  
[00:09:27] to nominal spending and the money supply.  
[00:09:29] And the money supply is going to grow over time with some constant rate mu plus some noise.  
[00:09:37] That's the only shock in this economy.  
[00:09:40] Nothing I say hinges on this being the process for monetary policy.  
[00:09:43] We've done a Taylor rule and so on.  
[00:09:46] As I'll show in a second, the economics doesn't come from this.  
[00:09:49] It'll be the clear rate comes from.  
[00:09:52] So this, we have multi-product firms.  
[00:09:55] We need multi-product firms to kind of take advantage of a lot of large numbers  
[00:09:59] inside the firm and allow aggregation.  
[00:10:02] So each firm is going to sell a unit mass of products, which are going to index by K.  
[00:10:09] And every firm is indexed by I.  
[00:10:11] So there's a double CS aggregator for simplicity.  
[00:10:14] We're going to assume the same elasticity theta within and across firms.  
[00:10:18] You get the usual double sloping demand curves and the aggregate price index,  
[00:10:22] which depends on the price of all the varieties out there.  
[00:10:25] We have decreasing returns just so we have a strategic complementary.  
[00:10:29] It's not particularly critical.  
[00:10:32] So let me show you a little bit about how this works.  
[00:10:35] So if you write down the present value, sorry, if you write down the flow profits that the firm is  
[00:10:40] making, you have to sum up the revenues it makes across all the varieties,  
[00:10:46] net of all the labor costs.  
[00:10:48] All these objects, they depend on the prices of all the goods.  
[00:10:53] And so you can write down the total profits of the firm as a function of two moments  
[00:10:58] of the firm's price distribution.  
[00:11:00] The firm's price index, the CS weighted average of its own prices,  
[00:11:05] and another index that I'm going to call the misalocation index,  
[00:11:08] because it's going to determine how much dispersion in prices there is inside the firm.  
[00:11:12] And therefore the TFT losses the firm suffers from having prices are misaligned inside the firm.  
[00:11:19] So that's all that's out there.  
[00:11:22] The firm is going to choose how many of these prices to adjust.  
[00:11:25] I'm going to call that NIT, subject to quadratic cost of adjustment.  
[00:11:30] But it doesn't get to choose which of those prices to adjust.  
[00:11:33] So you cannot choose to adjust prices that are older.  
[00:11:37] If we did that, we'd be back in the menu cost literature,  
[00:11:40] where we have to keep track of the distribution and we don't want to do that here.  
[00:11:44] So because it is an assumption that you don't get to choose which prices to change,  
[00:11:48] you have the usual low motion for the price index of the firm.  
[00:11:52] That's one of the objects you care about,  
[00:11:54] which depends on the number of prices you adjust, which you have control over.  
[00:11:59] The recent price, the new price that you're setting, and then these are the old prices.  
[00:12:06] And the usual Calvo aggregation implies that I don't need to know every single one of your past  
[00:12:11] prices. I just need to know the previous price index.  
[00:12:16] And there's a similar expression for X. So instead of having to keep track of the  
[00:12:19] entire distribution, I just have to keep track of these two state variable,  
[00:12:22] the old price and the old misalocation. Because all these firms are identical,  
[00:12:29] in the symmetric equilibrium, I can get rid of the I subscripts.  
[00:12:32] And then I get the usual equations that are familiar if you've worked out the Calvo model in  
[00:12:38] nonlinearly, you get an expression for the optimal reset price in real terms.  
[00:12:43] That's a function of the present value of future marginal costs.  
[00:12:46] You get this extra new equation that you have in this model that says that the fraction of price  
[00:12:52] changes depends on inflation. The more inflation there is, the bigger the benefits to adjusting  
[00:12:58] the price. So the more prices you're going to end up with changing.  
[00:13:04] And then the rest of the stuff is straight Calvo, except for instead of having a constant  
[00:13:10] fraction of price changes, we have a time there in one.  
[00:13:13] The model is nonlinear because of these fluctuations in the slope of the Phillips curve.  
[00:13:18] So you have to solve it nonlinearly. But even though we solved it with projection methods,  
[00:13:24] we also show that perturbation methods, third-order perturbation methods work fine here.  
[00:13:28] So one can use this very easily and extend it to larger scale models.  
[00:13:35] So let me show you a few pictures. How much time do I have? I haven't kept track of it.  
[00:13:39] Do I have a few minutes?  
[00:13:41] Yeah, about five minutes.  
[00:13:45] Five minutes. Perfect. So what we do with this is we're going to calibrate it.  
[00:13:52] One thing we want to match is the average relationship between the fraction of price  
[00:13:56] changes and inflation. We have this one parameter which determines the cost, the menu cost,  
[00:14:05] that's going to allow us to pick how rapidly the fraction responds to inflation.  
[00:14:12] And then we're going to use a nonlinear solution of the model to back out the sequence of  
[00:14:18] aggregate shocks that's needed to exactly reproduce the data on inflation. So the model's  
[00:14:22] going to match the inflation data perfectly and they're going to do some experiments.  
[00:14:27] So one thing we can compute is how does the model do in terms of reproducing the fraction?  
[00:14:32] It's doing well, you know, on average because that was the target in the calibration.  
[00:14:37] It's not doing so great post-COVID because during COVID, in this post-COVID inflation  
[00:14:42] episode, there was a lot of price decreases which our model with a single aggregate shock  
[00:14:47] can't quite explain. So you need to add sectoral shocks, you need to add other things to get  
[00:14:53] these things, but it's doing okay for the for the 80s. Okay, so let me show you a couple of  
[00:14:58] things about the slope of the Phillips curve. So the first thing you do when you linearize,  
[00:15:03] when you're sorry, compute the slope of the Phillips curve is you linearize the expression  
[00:15:08] for the price index, this price index equation. And this thing says that you're going to have  
[00:15:13] more inflation, the more price changes you have, and the bigger the new prices are  
[00:15:19] relative to existing prices. And so if you linearize that, we do this at every point in  
[00:15:24] time as opposed to around the steady state. So this is a perturbation around the point T  
[00:15:30] ahead in those log deviations from that particular equilibrium point. So this says that inflation  
[00:15:36] depends on the real reset price. And the elasticity here is a function of the fraction of price  
[00:15:41] changes. That's the usual Calvo effect. You have this new term, which has to do with the fact  
[00:15:46] that the fraction of price changes would also respond to shocks and that may contribute  
[00:15:50] to inflation. But the extent to which it does so depends on what inflation is to begin with.  
[00:15:56] In particular, if inflation is one, this term drops out. So this new extra term we're introducing is  
[00:16:02] zero, the model is exactly Calvo. If inflation is high, this extra term is large. And intuition is  
[00:16:10] very simple. The fraction of inflation is by definition equal to the average price change  
[00:16:16] times the fraction of price changes. If inflation is zero or one here, then the average  
[00:16:22] price change conditional adjustment is zero. So it doesn't matter how many firms change their  
[00:16:26] prices, you don't get an additional increase in inflation. If the average price change is high,  
[00:16:32] which is the case in say the eighties, then a small additional increase in the fraction of  
[00:16:37] price change is going to let these new guys have adjust to respond to the underlying history  
[00:16:41] of shocks that caused inflation to begin with. And so inflation is going to respond even more  
[00:16:47] to shocks. And that's the key. Inside, then we put it all together. We solve this equilibrium system.  
[00:16:54] And the key effect is this inflation accelerator. The idea is the fraction affects inflation.  
[00:17:01] Inflation affect the fraction. Once you solve this fixed point problem,  
[00:17:06] you get this term here that's going to determine the slope of the finish curve ultimately,  
[00:17:10] that has the usual Calvo effect, which has the fraction of price changes in it.  
[00:17:15] Because mechanically, the more prices change, the more responsive is inflation to new shocks,  
[00:17:20] but we have this inflation accelerator mechanism, which is a feedback loop. So then when we plot  
[00:17:25] this thing out, when we plot the time, the slope of the finish curve, the slope ends up moving  
[00:17:31] around a lot. It increases from about 2%, about 12% during periods of high inflation, even  
[00:17:37] post quality went up a lot. And that's a lot higher than what you get again in the Calvo  
[00:17:42] model in which the fraction of price changes were to move around, but would move around exogenously  
[00:17:48] and would be not allowed to respond to additional shocks. So the point of all this is that  
[00:17:54] periods of high inflation are periods in which the finish curve is particularly steep.  
[00:17:59] And so the cost of reducing inflation is particularly low in that case. And I think I  
[00:18:05] ran out of time, so let me stop here. Thank you.  

## Q&A (00:34:06 – 00:38:03, shared session Q&A; questions to Midrigan only)

[00:34:06] If not, I have some questions that were, for example, a question for Virgilio.  
[00:34:14] Rafael is asking, first commenting that he's a super elegant model. And then at the  
[00:34:22] multi-product firm level, what evidence from the micro data do you have that might be useful to  
[00:34:29] further validate your model or your mechanism? So, yeah, from the multi-product dimension.  
[00:34:38] And then another question is, what is the role of housing in your setup data?  
[00:34:45] All right. Thanks, Rafael, for the question. So on the multi-product thing, so I guess the key  
[00:34:56] assumption we're making is that the adjustment hazard inside the big firm is relatively,  
[00:35:06] we're assuming it's constant, but what matters is that it's relatively flat in the price.  
[00:35:13] So that's the kind of evidence we would need to look at. We'd need hazards of price changes as a  
[00:35:22] function of individual price gaps inside a firm. I know there's some work that's  
[00:35:29] looked at the strength of selection overall. You'd need here to look at the strength of  
[00:35:35] selection within a firm. I think overall, from what I know about menu cost model, typically  
[00:35:44] in more indirect ways of measuring the selection effect, the kurtosis, the variance of price changes,  
[00:35:52] the absolute value of price changes, or even when people directly look at measures of price  
[00:35:56] gaps and how they correlated with adjustment hazards, they tend to find relatively less selection.  
[00:36:03] So I'm comfortable with this assumption, but the main reason we made this assumption here is for  
[00:36:07] tractability, not so much for, but I think there's something to that. That's one. And then two on housing,  
[00:36:18] I don't think I have much to say. I mean, for the data we took out shelter from the CPI.  
[00:36:23] That's what Nakamura-Stuyson did, and that's because house prices are imputed in the CPI.  
[00:36:31] So we're taking all that out in the data, and then in the model we don't have houses, so we don't have that.  
[00:36:41] Yeah. Thank you. Another question for Virgilio. Have you talked about extending your model to a  
[00:36:51] setting with the producer prices and an input output network? Would in this case,  
[00:36:56] accelerate it a little bit even stronger? Funny you ask. So the history of this paper I wrote with  
[00:37:02] Andres, Kallum and Corina is the first thing we wanted to think about is exactly these issues.  
[00:37:08] We were thinking about this Bola-Monkey hypothesis that shocks to individual sectors,  
[00:37:13] may cause maybe inflationary, because we have a shock that hits a small subset of sectors.  
[00:37:19] Everybody in those sectors changes their prices, and in the other sectors they don't  
[00:37:24] because of this menu cost. So that's how we started the whole thing. And then  
[00:37:28] the first paper we wrote is a bit disappointing. We realized that actually we don't get  
[00:37:32] movements in the fraction of price changes, even in response to quite large sectoral shocks  
[00:37:38] with menu cost models. So then we wrote this simpler thing. But yeah, I think hopefully one  
[00:37:45] day we go back and revisit this question of how do you go from disaggregated individual sectors  
[00:37:55] to aggregate inflation and yes, accounting for the input output network. That would be part of it.