Thanks a lot to the organizers. It's an honor and a pleasure. It's also thanks a lot for choosing a very good friend, a student, and a great economist for the introduction. All right, so this is the title of this paper. It's our state and time dependent models of price setting really different. We're going to use a sufficient statistic approach, and I'll give you the summary of the paper. The answer to this question is no. They are not different, perhaps, except for large shocks. So I'll try to elaborate on the answer on that summary on the remaining 40 minutes or so. So kind of a decor of macro, or at least one important question of macro are nominal rigidities. And models that try to explain nominal rigidities, they come in many flavors, but two common flavors are the following. One is modeling with firms. They kind of monitor economic conditions, and when it's suitable, when it's kind of a good idea, they pay a fixed cost and they adjust prices. So I'm thinking about a menu cost model, and I'm thinking about the work of many people, but just to have a clear reference, think about the papers by Kaplan and Legi and Golosov and Lukas, just to have a kind of clear idea. And these models are referred to as state dependent models because kind of the adjustment depends on the state of the firm. Another strand of literature sort of uses firms that are kind of not paying attention or rationally being inattentive to the state of the firm's profits, and they pay a cost just to find out what's going on. And when they find out what's going on, they adjust the prices. So this is kind of like a different type of motivation, and they are referred to many times. I hope to explain it well for those who are not really inside the literature as time dependent models. And kind of the rep of these models and the little group that we work on this thing is that the state dependent rules give you small effects of monetary policy, and that time dependent models say the calvotype that give us large effects. And the idea of this on this lecture is to are you kind of differently saying that they are kind of give you the same effect if you condition in some observables. So that's kind of what I try to elaborate. All right, so again, I'm going to give you sort of like more or less an analytical comparison of these type of models of the sort of state dependent and time dependent. And for small shocks, which is a typical shock that we measure, I'm going to give you kind of two summary measures of the effect of monetary policy, and they will be identical once you condition of these sufficient statistics. For large shocks, it will be different. So you could think about that the conclusion is different and you say wait to then perhaps sort of try to sort out these models looking at some evidence. If I have time and your patience, then I will go through that. All right, so this talk is kind of based mostly on work with Francesco Lippi, which I guess we exhaustively tried many variations. I could see that now. But in particular, this paper is mostly related, this talk is mostly related to one paper that we wrote for the macro annual. All right, so my plan is first to give you like a bare bones, generally equilibrium structure for which all these different models should be sort of thinking about the frictions of price setting. The equilibrium structure is kind of the one that is a very basic in Woodford book and the one used in particular the one used by Golosof and Lucas. So standard looking utility, consumption, labor supplied, and there is mining the utility function. Then consumption, it's a dexterously aggregator, so we have, we prepare the ground for firms to have monopolistic competition. And there is this elastidious substitution between the firms, this is kind of super standard, and then there is an inter-temporal elastidious substitution among the aggregate, which also here will serve as a parameter for the marshalian labor supply. Okay, linear production, and since we're going to have different firms, then each firm is going to have a production function that is linear with the cost z. The cost itself will be, the log of the cost will be a random walk. When I use continuous time, it's more convenient for this, so it will be a Brownian motion. So these marginal costs are independent across the different firms. And our goal is to have like a once and for all increase in money supply and see what happens. Now in anticipation of that, before solving the model, before going to the rest, one interesting feature of this model is that nominal wages can be solved independently of solving the rest of the problem. And nominal wages as well as interest rates are just changing the level of money. So right of the bat, we know that model, if we increase money, nominal wages will be the cost of the firms that's going to go up. Nominal interest rates will stay the same. So for all the models that I talked about, this will be true. All right. Now, if you think about the problem of a firm, then we're going to define this p-star. So for those of you with MIT, all the education, this is very cabochero looking like. So this p-star is the ideal price that maximizes static profits, which is the cost times the markup or written here in logs. Now, I want to make a point. And the point is that we could think about the profit of the firm as depending on the price, the charge, the cost, and aggregate consumption. These are nominal profits. So why aggregate consumption? Well, aggregate consumption, you have a more policy-competitive firm. It's actually a stiglet. So if output is very large, the demand for the firm shifts to the right. Also, aggregate consumption is there because the aggregate price of goods in different periods of time depends on how plentiful the goods are. Now, I want to think of this through an approximation, as we many times do when we linearize models. I'm approximating the profit. So this will be equal to maximize profits times a term that is g for gap, which is the difference between the ideal price that you'd like to charge and the price that you're charging. Now, the important thing here, this is just a second order approximation. The first order term doesn't show up because we are approximating a profit. Then the important thing is actually consumption. Aggregate consumption doesn't show up here. So this class of models has no strategic complementarity between the decision of firms. So this is, you know, it's a fact of these models and we could go back and think about how these will affect the results. But notice that then the right-hand side of this equation just says that we could track of just something that happened to each of the firms in isolation. All right. So now absence of adjustments, then in periods where there are no adjustments, this price gap is going to behave as a random walk with a negative drift because your cost may be going up. In fact, for most of today, I will just talk about as if inflation will be zero. So it will be a driftless random walk. And if there is time or interest at the end or never, most likely, I will talk about what will be the effects of inflation. All right. Now I want to kind of think about a version of this with many products. There's two reasons for this. One is to make a sort of rhetorical point in a sense. And the other one is to think about a feature of the data for where these models were introduced. So this version of the model is very similar to the one that I showed you before. With exception, the notice that now in the place of the consumption, you know, for each of the firms, there are n different products. With potentially elasticity of substitution between the products, different from the elasticity of substitution between the aggregates and the firms. So think about idea and different products or think about Coke and Diet Coke and other products. All right. And then there will be a technology in which you produce each of these varieties. The technology will have some shocks. There will be shocks that are product-specific. These are these funny-looking Ws. And then there will be one that is for the whole firm. This is the funny-looking W with the bar. And they're all independent, so the common part of the cost will come just because of the common shock of the firm. All right. Same general equilibrium structure, so you'll have this property. So now if we think about the ideal price, it will have marginal cost with a common component. Say this is the Coke component. Then it will have a idiosyncratic, let's say Diet Coke, and then the logo of the market. And if you think about profits now, then the profits will have a vector of prices. All the prices that the firm is charging, all the cost, as well as the aggregate consumption. Same reason as before, up to second order. There's no cross-product terms, so there's no strategic complementarity. So just keep track of all the price gaps. In a sense, this will be a company first that is producing two varieties, but they are as substitutable as any other variety. So you just add these Gs. Now this is a special case in which the last year's substitution is the same between all the products in the firm and the products on the rest of the economy. And each of them is going to follow when there's no price adjustment and random work. Now the general case, which is a case of exchangeable functions, some sort of symmetric function, adds the fact that you have to keep track of the price gap, the square of the deviations, plus the sum of them squared. So the idea is that there is Coke and Diet Coke, and maybe Diet Coke is cheap, maybe regular Coke is expensive, but it could be that all the Coke products are expensive or not, and then it trades differently because Coke products may be substituted. Anyway, this is the setup that we want to consider. Kind of why, I'll tell you later. I mean why the multi-products, but that's kind of the setup. Alright, I haven't talked at all about what firms do about changing prices. I've just talked about objective functions from the firm and something else from the rest of the economy. So the talk kind of starts here. So first I'm going to define what we mean with Francesco and co-authors when we say the state-dependent rules. So in terms of economics, we want to think about problems where you have to pay a fixed cost to adjust prices. And if you pay this fixed cost, you could adjust the end prices simultaneously. Okay, so what would be the nature of the optimal decision rule? The nature of the optimal decision rule is that you will have a set in end dimensions which is an inaction set and a point that you return when you choose to do so. You pay the fixed cost and you will go to that point. So if you're in the inaction set, you don't do anything. If you are not in the inaction set, then you do something and then you send the state to GSTAR. That is, you change your gap to some point. The idea will be that you will close the gaps. You will send the prices to the ideal static point. So for instance, without inflation, then GSTAR will be a vector of zeros because your price will be the static price. All right. And indeed, I think it's simpler to think about the slightly more specialized case where this range of inaction is given by the solution of this equation, some function B of the vector of gaps. When this is negative, then you're inaction, zero in the boundary, positive outside. All right. So that's going to be our definition of a state-dependent model. The economics, you have to solve it and this will be the nature of the optimal decision rules. So examples of this is Barrow's paper in 72. That's a fantastic paper if you think about it in 72. Then there is Golovsov and Lukas, which actually is like a fully nonlinear model, but this will be the approximation to what they do. It's in one dimension, so the inaction set just to bring it down to earth is just an interval. You let your markup go between, let's say, 15% higher than the ideal or 15% lower than the ideal. And when it hits 15% higher, then the markup is too high, you decrease prices. When you hit 15% lower, then the markup is too low, so you increase prices. So price changes will come either 15 up, 15 down. That would be kind of this. And it's kind of the way that the paper looks like if you go to the paper. All right. So Lacan Sidon and later on Medigan, they thought about this multi-product model, say with two products, this is sort of a generalization to end products. And in the simpler case, with no correlation, with no cross-products, or the simplest case, what you do is you will control, say you have Coke and Pepsi, you will change prices when either one of them is very out of line and the other is not or some average of the two. Indeed, the range of inaction will be some sort of circle into dimension. There will be a sphere. I can't really properly do this with a mic, but anyway, I'm not a sphere in three dimensions. I'm hypersphere in higher dimensions. Because you're sort of going to trade off whether one is very bad and the other is not. Now, what's the point of this, of these authors, of why they consider this? Because you will see a lot of small price changes and a lot of large price changes, which is something that you cannot see in the data. Look at the other bullet point, distribution of price changes in the data are leptochartic.  Why do you let them diverge so much? Why do you change it for small amounts? And this was a way to try to generate them. Okay, so that was their motivation. And this is a generalization of two dimensions. And if you actually go like a slightly more general model, then what you have is that this will depend on the sum of the gaps squared and maybe something like the sum of the gaps squared. These are two numbers actually, because you could actually show that even though this is an n-dimensional state, each of them is Markovian. So this is actually quite tractable model. And we're going to use sort of, we're going to make full use of this. All right, so this is a state, it sort of, think of this as a generalization of Mitrigan and Golovsov and Lukas. All right. So now we move to time-dependent models. So if you want to think about something like a caricature of this, think about Calvo, okay? So this is a state-dependent model, but I want to think about some economics behind it. So the economics behind it that I would like to concentrate is the following. Think about that there's not cost of adjusting prices, but you need to pay a cost to find out what's going on. Say to find out the price gap, say the cost. So this is what we're going to call an observation cost. When you pay the observation cost, you observe the price gap. That is to say you observe the cost of production. And then you could set the prices accordingly. Now, at the time of an observation, this is a little bit funky for good reasons. At the time of an observation, you will get a signal, and the signal will be informative about future observation costs. This is kind of funky, but it's like this so that we could accommodate a lot of models in the literature. So think about a simpler case. The observation cost itself is Markov. So when you see the observation cost today, you could figure out what would be some sort of observation cost in the future. Or you could think about a simpler model with just a constant observation cost. But these models will have some other differences that we'll talk about later. Now let's just think about the nature of the optimal policy for a firm. The nature of the optimal policy is like this. Today you go and you look at what's going on. What are the costs? Then you do two things. You adjust prices because that's why you gather this information. You adjust it to the optimal value. And then you decide, since this is a fixed cost, when you're going to take a look again. And you have to trade off cost and benefits. The cost is if you don't look at the state anymore, you're going to keep your prices up with the information that you have, and the quality of your decision is going to deteriorate because your cost of production is moving up and down and you're keeping the prices constant. And it has a permanent component, so in average they're sort of going to weird places. If you observe very often, you pay a lot of observation costs. This is the type of rationale in attention. So the decision rule then will be a function, T, which tells you how long, what's the interval of time until you get the new decision of observing. And that will depend about the draw of the signal that you got today. If the cost of observation cost is constant, it will be just a number. But otherwise, you'll have some variability depending on whether it's the cost of observation that your forecast is very high. It will take a long time. All right. The multiple dimensions on this one is truly without loss of generality, so I won't really bother starting this. And these kind of ideas are mostly from Ricardo Rice. It's a bit of a generalization of Ricardo's setup. And they are related but not exactly the same to seems rationing attention. All right. So examples of this will be how is that this model will look like. Suppose the observation cost is constant, then you will observe, say, every six months or whatever, some number, let's say six months. That look like Taylor, all Taylor, not Taylor rule, all Taylor, that you do it every stagger. I mean, Taylor paper from all time. Or let's say you have some variation on the cost, you could manufacture it so that this will look like exactly calva, that will be exponential, or anything kind of you want. In fact, we prove that you could manufacture the distribution of the signal so you could get any distribution of the times. So it will depend on cost and benefit calculation. All right. So these are clearly state dependent because notice that your cost of observing is uncorrelated with the true production cost. So you observe, and every time you observe, you adjust prices. So all the idea is how far you go through this. Now, these models, people like them by some aesthetic features or realistic features because they match some other things. I don't want to get into that. That's not this type of talk. I want to think about the properties that they have. There's a couple that you could think about models that have both features. I'll give you a couple of examples. I don't want to give you a general definition of a model that has everything because it's too cumbersome. But I'll give you some examples. One is the Calvo Plus model, which has some elements of the Golos of Alucas and of Calvo, which you may want also to think about why it's not called Golos of Alucas Plus. But anyway, that's the name of the model in the literature. And it's a model in which you have a fixed menu cost and then every so often the menu cost goes to zero. So what you do, you adjust opportunistically. So this has some flavor of time dependence because when you have zero menu cost, you may as well adjust. And it's kind of unrelated to the profitability of the prices that you have. And if you make the two costs going like any way between the frequency where you get these at the size of the cost, you go from Golos of Alucas to completely Calvo. So this is a model that has elements of both. Another model that we have studied is a model in which you have both an observation cost so you pay to observe, you look at the state, as well as a menu cost. Because upon observing, maybe your price is almost ideal. If you have a further menu cost, you don't want to change. We, I mean we, not the royal we, Francesco and me, we really like this model. We think this model has many features that make it realistic, different type of data sets. We think that it favor this model. Anyway, it's not this type of talk. We want to think about what are observables and how a class of models implied about impulse responses. But anyway, so notice that this model has features of time dependence and state dependence. Time dependence because you will decide on how often to observe, and then many times you're going to be oblivious to what's going on. But upon an observation, you will decide whether to adjust or not. So it has kind of features of both. Okay. All right, so. I'll talk, I want to talk about what I think now is common to be referred to as an MIT shock. You are in state-state, then there is an unexpected shock and you follow to the impulse response. So I want to think about what's state-state in each of the two models. I want to find it for the general model of both state and time dependence because it's too cumbersome. But let's go to the two extremes. In a state-dependent model, the state-state is the invariant distribution on the price gap. So in this case, it will be some n-dimensional density. So really, for all that I want to think about this particular exercise of starting state-state and shock the model, I need to know this distribution. Now let's move to the time-dependent decision-rules model. There is different because people are not paying attention to what's really their cost. What's going on is that you have to keep track of how long in the cross-section until the next time that they will observe. For instance, in Calvo model, that's exponential. But it's some function. It's a function that tells these many firms are going to make their decision two periods from now, these many three periods from now, these many four periods from now, and so on and so forth. So these are the objects that are important for our impulse response in each of the two models. Okay, they are related to a bunch of observables also, but let me skip that. Okay, so now let's go through a shock. We're going to start the economy at some state-state. We're going to normalize output and prices to zero. Just a normalization. We're going to increase unexpectedly, and once and for all, money by delta points. Now remember that in this case, what increasing money does is increase wages, but delta log points. So wages were like this, they go like that, and that's a production cost to everybody. And we want to keep track of two outcomes. One outcome is the price level, what happens with the price level, and their outcome is what happens with output. So in order to output, in both cases, I want to distinguish them by delta, which is the size of the shock, which I announced that it will be important, as well as t, which is, you know, t equal to zero is the beginning of the impulse response, and I want to see how far away it is. Okay? So output actually is a fairly simple function of price level here, because in this model, you're in your labor supply. So one over epsilon is the uncompensated is a martial in elasticity of the labor supply. Delta minus p is change on wages. Delta is change on nominal wages. P is change on prices. So delta minus p is change on real wages. So through these, you could really have change on output. This is a fairly simple model. All right. So this part is kind of simple. And we want to measure, or actually, let me put it this way, these are the two statistics for which we could get very sharp analytical results, which are the following. One is the impact effect on the price level right at the time of the shock. That's capital theta, and this is sometimes referred to, there are several papers analyzing this by Kabashira and Engel. They refer to this as a price flexibility index. And the other one is the cumulative impulse response of output. So it's impulse response of output, but you take the area underneath. It's something like a summary measure, so it's one number. I'm going to return two graphs for them. All right. So these are the two objects that I want to talk about across the different models. So first, the impact response, sorry, the impact effect of money on output. So this is like a heuristic picture. It's just drawn by hand, but I just want to have the objects. So the objects are the price level, remember it was zero. Delta is the whole impact. In the long run, prices will go by delta, but it could be that on the moment of the shock, they go up by this much, and then they keep going up. I mean, just some picture that we draw just to, so you guys see the effect.  Okay, and I want to theorize about capital delta. Okay. So I want to do it in time dependent models and stay dependent models. So first time dependent models. So there's no impact. There's no impact effect. It's kind of like by definition. So take your favorite, say, Calvo model. You take a monthly model, there is some impact. But okay, make the model instead of monthly, weekly. The impact is roughly, you know, for weeks and a quarter, is a quarter of the firms will adjust, make it daily, it's a 30, okay. You can see the point. In continuous time, if you go to that extreme, there's no impact effect. Because the model is that there's only a tiny fraction in a tiny period of time that will adjust. That's just kind of by design. In fact, you could think more through it. The imposed point of prices is linear on the shock. I'll explain this. So you see this is the linear, so you could take the delta outside. And not only that, it's a very simple function of the number of firms that are going to adjust their prices. And it's true in any of these time dependent models. So think about why. You take all the firms that in the past have decided, say that they're going to observe the state for tau periods, say tau is six months, for instance. Look at this cohort. What's going to happen with the price changes of this cohort? Well, they were oblivious of what's going on. Then they observed six months. Now the prices went up, the cost went up and down with the Brownian, so they have the shock. So all the prices will have to go up by delta percent. And then there is some heterogeneity between them, which is normal with variance sigma squared delta. Now since we're going to average across then, the variance is not important. So we only know that all these guys are going to adjust by delta. Now let's just think about in three months what's going to happen. The guys that are going to adjust in one month, they will adjust by delta. The guys are going to adjust in two months, another delta. The ones that will adjust in exactly three months, another delta. So what is the adjustment? Delta times the integral of q between zero and t. And that just gives you the impulse response. It's kind of really accounting. All right, so these models are fairly simple. That's why the carbon model is so simple, and it gives you this restriction for this kind of stylized shock. So obviously then, when delta is very small, then they have a very small impact. It has zero impact effect. I think that a lot of people think that the state dependent model has a large impact effect. Let me put it this way. I thought that it has a large impact effect. I think even my authors saw the same, but it's not. And if you actually look at simulations, you will see that it doesn't. So let me show you the reason. So actually write down the impact effect and do a Taylor expansion of it. Capital, if there is no shock, there's no impact effect. That's easy. That's delta of zero is zero. Then you take the first derivative and the second derivative. And it's easy to see, I will show you a picture, that the first derivative of the impact effect with respect to the size of the shock is zero. So if you have a small shock, the effect will be of the order of the shock square. And it will be very small. Take a look at the pictures in Golozoff and Lucas. So here is the proof. And this extends easily to the general setup with multi-dimensions. It really is the same mathematical logic, really. So the triangle here is a distribution of price gaps in a steady state. So it's right there. Then the shocks come. Remember the price gap is what you charge relative to what you would like to. Since the cost went up, every markup is lower now. So the whole distribution moves to the left. So now they used to be between minus G per bar and G per bar. Now the whole triangle moves to the left. Now importantly, notice that this is a triangle. The important point for this is not the shape, but the fact that the density at these points is zero. And the reason why the density is zero is not a coincidence. These points are exit points, meaning that if a firm hits them, it leaves. So you cannot accumulate mass as long as you have some random shocks. Okay? So this is actually a general feature. That's why this will be a general result, just using the sort of continuous path type of models. So now let's compute the number of... Okay, so... So let's compute the number of firms that after the shock, they would like to change on impact. It's this little red triangle. What is the base of the triangle? Delta, the monetary shock. What is the height? Well, it starts at zero, so you need to know just the slope. I don't know, some number, F prime, call it. So the height is F prime times delta. The base is delta. The area, which is the number of firms that will do this, is delta squared times something. So the point is that it's of the square of delta. So if delta is very small, this will be very small. If you have a shock that is 1%, this will be a 1% of a 1%. And if you take a look at the pictures of all of the models that do this numerically, you cannot see them. If you see PDFs in a journal page, you cannot see the jump. And this is sort of the general notion. And you could do this in n dimensions. It's really exactly the same logic. So let me skip it. All right, so... Now I want to think not just the impact effect, because you may say who cares about the impact effect. I want to return to that. And... Is that cap T or minus... Remaining time, yeah. So... Okay, so... I will say why you may care about this. But for now, so this picture, we took a lot of work to do it. So it has two lines with some sort of hypothetical price changes. Notice that one has a tiny jump at the beginning. I don't think of this as a state-dependent model. The other one is a... some... an independent model that has no jump. Anyway, the point is that they're very similar. And then we shaded the difference in each point of the impulse response between that and delta. What is this difference? This difference is output. If you look at this vertically, that difference is how higher, how much higher the state real wages are. At the beginning, they're much higher. So each point is the... So if you turn it the other way around, that's actually the impulse response of output. So if you integrated, you get the area. And the point is that for all these models, this area can be expressed as delta, which is the size of the shock, divided by the epsilon, which is the Marshallian labor elasticity, times the kurtosis of price changes, divided by six times the number of price changes in steady state. Because remember, this is kind of like a shock that you start in a steady state, so stuff that happens in a steady state will be informative. So stuff that happens in steady state, here there are two statistics. The kurtosis of price changes and the number of price changes. And which model it is, whether it's a state dependent or a state dependent in the class of models that I described today, doesn't really matter. All of them satisfy the same expression. So let me just elaborate why. So that's the sense that we say that on impact it doesn't matter, they all have like pre-small impact and if you integrate it all they all give you the same number, as long as you condition in these statistics. One of them we always condition, which is the number of price changes. Or let's say we calibrate the model so the number of price changes looks like in reality. The other one we think there's one that you ought to. That's our sort of message. Okay, so this is just the same, but I just put it the formula like bigger. And I want to explain that Golozo van Lukas taught us and they explained it like flawlessly why in the state dependent model there is selection. And why that model has a relatively small effect. What we want to say is that the selection could happen either on the size of price changes or on the distribution of times. And for that impulse response it doesn't matter but not only that it doesn't matter but it always map in this very simple way to the kurtosis of price changes. Okay. Alright, so I'll try to kind of explain it giving some example and I'll be happy if I finish with that. So it's the same formula, we like that so we'll put it many many times. So here's an example three models that have the same kurtosis and that's why we want to have partly rhetorically many models to give you several mechanisms that could get the same kurtosis. So one model is one that has purely observation cost. This model will have kurtosis equal to three. I don't have time to enter into which but it's actually easy to see. Another model will be the multi-product model when N is very large. That will have kurtosis equal to C. This is a state model but it will have exactly the same area under the impulse response. The third model actually is every single Calvo Plus model for any number of products. There is a Calvo Parameter that you can make it to have kurtosis equal to three. The point is that what matters is kurtosis. That's why we have this formula and I want to give you several examples that will do the same. And there are many more, I mean because it's a large class of models. Okay, now a different point. If you have different kurtosis then you have different examples and I have here two examples that we are very used to think about. And one is Golozov and Lukas that end up having kurtosis equal to one and the other one is Calvo that has kurtosis equal to six. So hence one has six times the effect. And invite you to do, I'm sure that you will be delighted to do this, go to the paper by Golozov and Lukas, take the picture, ask an undergraduate to compute the area and you will see that it's six times the area that they have in another graph when they do the Calvo one. Anyway, I'm sure that you will be delighted to do it. So let me explain why and I think it's kind of useful to think about it. But I don't know, we think it's useful to think about it. So in Golozov and Lukas what happens is prices are either 10% up or 10% down. Some number like this say, whatever you calibrated. So there's no selection. As they taught us, the idea is that when you have the shock, then all the firms are ready to go up in price for a while. So the impulse response is very fast. So notice this association.  Up and down. So that's low kurtosis. That's as low as it could get. One is the lowest kurtosis for a distribution that is symmetric. And that's the lowest that the effect could be. Now the second one is Calvo. So what's going on in Calvo? Well, that's harder to understand. Why Calvo is so much more? But let me give you a try.  So let me explain why. So compare two cases. One you have a time-dependent model that everybody adjusts every year. You'll get some number. And you have a low kurtosis because the kurtosis will be three. Why three? Everybody adjusts in a year. So the distribution of price changes is normal. Normal has kurtosis three. It goes immediately, an hour. And the other ones that adjust every hundred years. And it's one percent of the population. So what's the kurtosis of this? It's huge because it's a mixture of two normals. Basically it's like a little normal and on top of that there is a bunch of them that adjusts immediately. Huge kurtosis. Now why the imposed response is so drawn out? Well, there's one percent of the firms that they don't change the prices. So what happens with prices? Well, one percent of the prices they stay constant for a hundred years. So you integrate that and then you get a very big integral. That's sort of simple. I'm sorry too. It's not that sophisticated. It's sort of like the logic of this extended to all these models. And different models combine the state dependence and the time dependence in different ways. Everything at the end gets reduced to kurtosis. So we think that it's an important kind of statistic too much when you think about the degree of selection. Notice also that the number of price changes is there. That's kind of well known. Because if you adjust very fast, time runs very fast. And then the imposed response just happens faster. So that's kind of obvious. And that's why also clearly there is a huge literature. Obviously we are not even the first or people know that the number of price changes is very important. But our point is kurtosis is that important and that's not a feature of the state dependent models. It's a feature of any model in this class. All right. So we think that kurtosis in the data when you properly do it you try to the best you can for accounting for measurement error and also for heterogeneity because otherwise you're going to be mixing different ones is four. So it's kind of two-thirds of Calva. But that will take long time to explain why we think that this is a reasonable number. All right. So I don't know why large and this is kind of trivial now but let me conclude with this. So how am I doing with time?  Okay. So now think about a large shock. For a state dependent model you're oblivious to what's going on. So large, small it doesn't really matter. So it's the same imposed response to explain where it's linear. So it doesn't matter. Let me make sure I say it correctly. For a time-dependent model for a time-dependent model like in Calva it doesn't really matter because it's linear. But for a state dependent model if you have a very large shock everybody will adjust and in particular the impact effect will be very large. So for instance for a subset of this class of models you have to compute that if the shock is bigger than twice the standard deviation state-to-state everybody adjusts. So two points. This is a nice statistic because notice that the standard deviation didn't show up yet anywhere. Kurtosis, which remember kurtosis is kind of four moments divided by two moments that are normalized but standard deviation didn't show up by itself. Yet the standard deviation of the price changes is the magnitude that tell you that the shock is large because the standard deviation think about in colors of a Lucas again. Price changes are either 10% up or 10% down. So standard deviation is that and if you have a shock that is more than 10%, sorry, more than 20%, everybody will adjust and this is actually a general result. Okay, so when a shock is large and here's a measure of large in terms of state-to-state that's kind of the feature but what's the point of this? The point is that you could use this as a diagnostic device if you care to think about sorting out whether the data looks more like a state dependent or time dependent because for large shocks they are different. For small shocks, at least in this dimension that we have analyzed they are not but for large shocks they are. Okay, so let me just skip the idea that anyway, let me just skip this. All right, so as I said, the effects for are kind of not there if the shocks are small for time dependent or the dependent impact effect is so with zero but for for a state dependent the impact effect is large if the shocks are large. Okay, and obviously I'm not talking about models that has a mixture it's a little bit of each and the notation everything is a nightmare so I'm skipping that. So what type of evidence can be brought to bear? So we thought about trying to think about evidence of what happens when there are large shocks. So, okay, so this is our heroic attempt. So we needed a shock that is kind of MIT looking shock. So we don't want seasonal changes, for instance. That's very different from MIT. They're kind of they're for a castable, they're short live so we thought about, okay, exchange rate shocks because if you think about what we're floating exchange rates they're kind of random walk. There's also data that have high frequency and we want to look at kind of short time periods. So monthly at least is better than quarterly and it comes from many countries. That's good because we want to see small and large. So if we have like, you know, quarterly time series for only one country that's probably not the place to look for. So that's what we did. We look at a panel of countries with different changes on exchange rates and we run kind of the typical past through regressions. Now, interestingly another way of doing will be to do a case study and there are some case studies for large evaluations. There's also papers that they think about whether large evaluations are special in itself thinking more about the macro. But there are several case studies. We particularly like the case study from Switzerland. The case study for Switzerland for those of you remember is actually an appreciation. I remember kind of when the Swiss they essentially a fixed exchange rate and then all of a sudden they decided no more and there was an appreciation of 11%. I think it was widely unanticipated and the Swiss so good, anyway in my opinion being so good so they kept, they actually had daily data of transactions of imports daily. So they actually run some import responses daily data to look at the, so this is almost like a continuous time model, I couldn't believe it. So they did it by themselves and they concluded in the paper that I will not talk about it but they concluded that the past through was much bigger when they got out of fixing the exchange rate to the euro than the typical evaluation. So this is a paper that's kind of recent so I wanted to mention because this is a paper for which is much more credible to think that this is a shock if you look at the narrative or if you look at data in exchange rate differentiates. That's not what we did. We hope we would have done it. That's not what we did. What we did is we run these regressions. So let me just summarize them. So this, there's a lot of people here that have written papers, in fact there are people here that have written handbook papers here in the audience about the past through regressions. So these are kind of relatives to the regressions that people have run but it's kind of like a panel version and nonlinear because you wanted to think about whether in large evaluations the past through was particularly large and in particular we went to see it was particularly large at the very beginning and this is kind of what we see. So we see that for large evaluations then all past throughs are very small as people that know from this area they are surprisingly small at least to me. These are also small but they are much bigger disproportionately bigger when the evaluations are large. There is a percentage increase on the month of the evaluation is much larger is the evaluation is large. And this is particularly so for floating exchange rate regimes which I think is sort of a course well with the idea that they kind of run on walk looking. Obviously in this area, are these kind of like shocked, are they reacting to something that's kind of like a big issue like working past through regressions they know this so we don't have much to contribute to that but we found this kind of to be like an interesting we found this so interesting. All right so let me for time I think I'm basically a so let me conclude. So we think that for small shocks this nature of friction is irrelevant irrelevant is probably too strong but it's not that important provided a frequency and kurtosis of price changes are the same. So this result then remember is tend to stay dependent time dependent and models are a feature of both. Okay what to make out of this? You know you could think of this as this is nice because it's robust so you don't need to worry about this. So that's kind of the the other way is you're thinking that it's hard to identify these models that's kind of the other thing going. On the other hand for large shocks then the stay dependent are faster but not only for large shocks we interpret the stay dependent as a staff related with information that what is sluggish is the acquisition of information. So we think that actually distinguishing between models could be important if you have some policy in mind which is something about there is some announcement about the future but for policies where the nature of the shock is you announce something about the future it may be interesting to distinguish them for some other policies it may not so we have in mind something like forward guidance which the whole idea is that you announce something about the future then sort of stay dependent time dependence may work differently. So I wanted to just conclude with an example of that. So to conclude I want to thank you for your hopefully rational attention.