 Yes, so let me start by thanking the organizers for this prestigious and super interesting conference so far. I'm also supposed to say that because this presentation is based on joint work with a number of co-authors in a summer home like Ipe Alves, is at the Bank of Canada and beer category, and I think you probably all know well, BCP, the music for this year are mine and you don't necessarily have to go to BCP or the Bank of Canada. All right, so I want to try to summarize my message in the first slide of my presentation. My starting point here is that the distribution of income and wealth is central to much of economics. Think about labor economics. In labor, there is much research. There has been a lot of research that has spent 15 years and 20 years on the evolution of skill in Tasmania and therefore wage inequality in response to technical change. In public economics, the distribution of income is key for the understanding of the optimal design of taxes and transfers. In macro, in most of macro, there is quite a bit of research on the determinants of wealth inequality both in the perception of overtime and even in finance, the tradition has been monopolized by three models, habits, various disasters and longer risk, which are represented by the laws. There's an active area of research on intelligence preferences and insurance employees. Instead, I think recently the distribution of income and wealth played almost essentially the role of monetary economics. So the science of monetary policy has been dominated largely by the representation paradigm and the practice of monetary policy has been greatly influenced by this perspective from which we obviously bear a lot. But my point today, my message today is that the distribution of income and wealth matters for our monetary policy transmits to the real economy. In order to make this point, I think it's helpful to start from kind of recapping what the transmission mechanism of monetary policy is in the representative agent version of the New Pansial Model. For this purpose, I'm going to sketch out a style, a stylus model, one that can be solved in most form. The model is in continuous time, the representative agent has CRR preferences with inter-temporal institutions with one organa, and this counts the future trade role. Remember that in this model, the representative asset is also a primary income consumer, and therefore, raw is essentially also the primary income consumer, auto-transitory income, or the representative asset. Technology is very simple, output is equal to the total amount that they will supply, and very small. We're also assuming the prices are perfectly rigid, here's where essentially you can get the cross-form solution, but this is not really a particularly restrictive property or assumption from the point of view of the insight that this stylus model gives. The monetary authority sets the time path of the nominal interest rate, which is also the real rate here, which is the prices are rigid. You know, R0 is a monetary shock at impact, and the shock decays at rate eta. A large eta means that the shock is very transitory, and one over eta is the average duration of the shock. The equilibrium is very simple here, it's market period, the goods market, consumption and total output, and we're focusing on one equilibrium where consumption goes back to state to state. So it's straightforward to show that the impact effect at equal zero of a monetary policy shock is given by this formula here, minus one over gamma eta. So essentially it's a product of one over gamma, which is the average duration of the shock. This is essentially for given duration of the shock, you know, the higher its interval substitution, the higher is the effect of the impact effect of the shock on higher consumption. Okay, but let's sort of dig a little bit into this formula. Let's try to understand the transmission of the monetary shock to consumption in this model. In the work that we've come up with in the model, which we sort of suggest is the composition, right, where an enormous response function of say an aggregate one, say consumption to a monetary shock can be decomposed into two channels essentially. The first channel is what we call the direct effect of the monetary shock effect or the interest rate on individual and aggregate consumption. This is a partial equilibrium channel keeping income constant. The second channel is the indirect effect through the change to the general equilibrium change in disposable income that is induced by the monetary position. Okay, and here essentially what happens in the column is that after the shock equilibrium price is going to change. So the way to do change asset prices are going to change or, you know, in a labor market, the creatures with wages to move or in actual employment would change and then would increase burdens. Also fiscal variables will change because the intertemporal budget of the government as we just saw the taxes and grants will change. What is a transmission mechanism is essentially the composition of the error into direct and indirect. So, once you do this, the composition in our special case, you obtain essentially that the direct response to the interest rate is given by the simple formula Eta or Eropas Eta and indirect effect is given by Roe over Eropas Eta. Remember that Roe is the marginal risk to consume essentially in this model. Now think about plausible, very straightforward, quarterly parameterization of this model where Roe, the discount rate is about 2% per year. So 0.5% quarterly and Eta, which is this, this decay rate is 0.5, which corresponds to kind of a half life of two quarters, which is roughly the half life of monetary positions. Then, if you plug in these numbers into the composition is straightforward to see that the direct component contributes for, you know, almost entirely the transmission mechanism monetary points. So Eta over Eropas Eta is close to 0.99 in this simple exercise. And why? Well, basically because Eta is two orders of life are greater than Roe and because Roe, which is the MPC in this model, is extremely small. So basically, in this model, there is no really, you know, sort of came, you know, the size of the of the Keynesian generically loaded lighter is really tiny because MPC is super tiny. So in rank, in the representation model, the transmission mechanism, it's all about inter-temporal. Now, this was a simple example, but the same result holds true in leading scale monetary G models, like the benchmark, as mentioned about this model. Now, the paradoxical thing is that, you know, even though the model is called a Keynesian model, it's not very Keynesian at all. Because the size of the Keynesian multiplier is really tiny. And in fact, as John Cochran nicely put it, this model should really be called the sticky price inter-temporal. It's very neoclassical at heart, in a way. So, okay, this is just an observation, right? This is just a transmission mechanism. So what is wrong with this aggregate inter-temporal substitution model? So this logic is sort of in contrast with much empirical evidence. Now, I obviously don't have the time, this is the place to summarize in detail the empirical evidence, but also there is a solid amount of empirical work that suggests that even in the very developed high-income economies, not everyone is a permanent income consumer. It is a large share of what is called in the literature hand-to-hand households with high marginal consumption of transitory income and low sensitivity of consumption to the interest rate, which essentially, from the point of view of the model, they are not on their own. And there is a material distribution on marginal consumption across the population. Some households have IMPC, others are permanent income consumers are very low, and there is a housing system. So, these observations, this sort of body of empirical evidence has two implications from the point of view of economic insights. The first is that the wet distribution and the portfolio composition of wealth determines the distribution of MPCs and the aggregate marginal consumption. And the aggregate marginal consumption is key in determining the size and the strength of this indirect general critical effect of monetary policy. The second insight is that once you start thinking about a world where the third genius marginal consumption, then even pure income distribution, a shift of income across the distribution can have real effects. So, in particular, by redistributing income towards high MPC households, the effects of monetary policy can be identified. So, what we need essentially is a framework to start developing these two insights and start measuring these two insights. And, you know, in the last few years, I think a lot of progress has been made in the literature and a new kind of macro framework is emerging. This macro framework, in a way, it's a very macro progression of quantitative macro, which for a long time progressed as kind of representative agent, kind of monitoring USG models on one side and on the other side, heterogeneous agents, incomplete markets model, they were mostly focused on kind of system policy in the long run. So, these new cluster models merges these two approaches, these sort of AII garrick-rosels, AII garrick-rosels, and the main advantages, as I said, is that you can think through the income distribution, the wet distribution and the effects of these assets on the MPC and the application of monetary policies and so on, which is what I'm going to focus on in the next 20 minutes or so. But I think that there is also something else which is attractive about this approach. Empirically, it's a unified approach to the analysis of micro and macro data. And I've seen recently a lot of exciting work that tries to extend the standard tools of base and estimation that have been used in the estimation of ESG models, which combine together longitudinal panel micro data with time series to estimate these cluster models. Conceptually, it's a unified framework to study both kind of short-run fluctuations and long-run dynamics of distribution. So in that sense, it still has a flavor, a little bit of this kind of RBC, New Keynesian models, which are, you know, which are essentially versions of the stochastic growth model, right? So, versions to analyze the short run of a model that was designed for life. And also, they provide a unified framework to study together stabilization and redistributive policies. And in this world, any stabilization policy has a redistribution implication and any redistribution policy can be stabilizing or destabilizing. So there's a really, these two policies are necessary in their one. And technically, it's now a lot easier and faster to solve these models so these models are a lot more accessible. So many offer, in one slide, one version of these models, which happens to be the one that I work with, I don't have them all, but there are many versions of this framework that are also extremely useful to analyze these models. So the model that I'm going to work on, essentially, is one with a continuum of households who face an issuable idiosyncratic income shock, for a company shocks. They choose consumption, saving, and supply, and they can save into assets, which we're going to call the liquid asset and liquid asset. The liquid asset in the model is basically government bonds, and the liquid asset in the government is basically physical capital. Or if you like, equities to a mutual fund that holds physical capital. In practice, the way we map these assets into the, say, the certain consumer finance data portfolio position is for liquid assets, we use cash, data accounts and government bonds. For a liquid asset, we use the, use basically housing and 401ks. Okay. And, you know, one key feature of this model, and we'll explain why it's important, it makes a transaction cost to move funds into the liquid account or out of the liquid account. And you have to think like, okay, it's costly to withdraw from 401k. It's costly to extract it from the house. It's costly to buy a house or to, or to, you know, sell and upsize your house, and so on. It's always a very stylish way of capturing all these complicated asset structure in the house of the structure in the economy. In equilibrium, the model determines both the returns in particular because the liquid asset is the liquid asset transaction cost. The equilibrium on the liquid asset dominates the real right on the liquid asset. So the liquid asset from a certain perspective is a very, very appealing asset to say, but also the liquid asset is appealing because it provides a way to move consumption. And the very short-term liquid asset doesn't provide this asset because it's in equilibrium. The remaining model ingredients are kind of standard. We have a nominal rate being set by a federal rule, and then a Phillips Curve determines the relationship between inflation and debt. Okay. So what this model gives you is basically the ability to generate a very realistic wealth distribution, both at the top with fairly high concentration of wealth at the top and at the bottom. In particular, at the bottom, the model can generate a large share of these kind of hand-to-mouth houses. Now, let me do a little integration and move to the data and tell you about what the data say on the prevines and the distribution of hand-to-mouth houses. This is a plot that measures hand-to-mouth households in the survey consumer finance from 1989 to 2019. It's a trend in the survey. Where we define hand-to-mouth households are being dosed with low liquid wealth. I think basically less than two months of operating costs. The first thing that the judge of you is that there's about 30%, between 25 and 30% of households who are hand-to-mouth. This share is quite significant. It peaks in conversations and it falls in expansions. And perhaps it comes to good news that if you look at 2018, just before the epidemic, we had historically one of the lowest shares of these houses. So that's kind of the news in a way. The other thing that is important is to distinguish within this category two groups. So these are all households with low liquidity. But within this group, there are the poor households that have low liquidity, also low net worth in general. And the so-called wealthy hand-to-mouth. These are low-liquid wealth individuals who actually have some sizeable investment in liquid assets. They could be because they own a house, with a mortgage bank, because they have some sizeable 401k. But what matters for consumption is liquid wealth is very low. The typical, if you look at the data and try to dig into who these guys are, the typical group is basically young homeowners. Or homeowners who just bought a house, they have a large mortgage, and the mortgage payments are absorbing a lot of the cash flow for these households. Basically, they have a large commitment expansion. And the two asset model is precisely able to generate these group of households, which are, as you can see, the majority of hand-to-mouth households. They're like two thirds of them. Well, two hand-to-mouths are only one third. So the model gives you the ability to generate essentially a realistic wealth distribution both at the top and at the bottom. And as a result, it gives you a fairly rich distribution margin of risk to consume. This is a plot of the MPC out of a windfall of $500 over the first quarter as a function of liquid and liquid wealth. What you can see here essentially is these two regions. There's a ridge on the MPC, two regions where the MPC is very high, which is at zero and at the credit limit. This is sort of the credit limit here. And obviously, the MPC is very high, both at zero and at the credit limit. The credit limit can become relatively quickly. But what is interesting is that the MPC remains large, even for those households that say they are around here, that there is very low amounts of liquid wealth, but there is some liquid wealth. These are the wealth hand-to-mouths. I mean, obviously, there's not a very large share of massive population of this point here, but there is some as we saw. It's about 20% of households who are in this range here. Overall, you combine things together and you get a quarterly aggregate margin of risk to consume around 15%. Which perhaps is not as large as what some empirical words suggest. You know, work by Jonathan Parker and co-authors of the tax rebate, which is a kind of really clean quasi-natural experiment, suggested this quarterly position may be 25%. But still 15% is 30 times bigger than the MPC in terms of budget model that at the quarter the frequency is about 0.5%. So you can start seeing that because of this key difference between these two models, the general equivalent effects are going to be much larger in this class of model, relatively preferable. In fact, if you cannot take this model, you take this model and you run a monetary policy, an expansionary monetary policy shop. And you see what happens in this model and you decompose the impulse response into direct indirect effect. Here's what you get. In the left panel, you have the aggregate consumption response in rank and rank. I think this is a 15 basis point, changing the nominal rate. In the middle panel, you have the composition in the representative agent model and in the right panel, the composition in the derivative agent. So the first observation I want to make is that the impulse response function for aggregate consumption are very similar across models. They're almost on top of each other. So this is, it's sort of a desire for this particular experiment. We pick up a representation from which you get this kind of almost equivalence result between the two models in terms of aggregate. And I'll come back later on this point. This, what is important here is the middle panel and the right panel that offered the composition of the impulse response between direct and indirect effect, which is sort of this partial political and general political income channel in the two models. And you see the transmission mechanism is very different. In rank, it's mostly the direct channel that dominates. And in rank, at least in the very short term, in the long run, the indirect, but it's still very small. And basically the story is that, you know, there's a, there's a cut in the nominal rate and the distance into a cut in the real rate and through the aggregate equation that is intertemporal substitution consumes more today and less in the future. That's it. That's a story. In the originalization model, you see that the intertemporal substitution is still there. Right. And it's, it is relatively rare, but the dominant channel is our indirect, indirect effects. So, so what happens here, basically, you know, at the very beginning through intertemporal substitution, because the initial, initial channel, the initial impulse has to happen through intertemporal substitution. So, so what happens here is that the internal consumption of the material is increased in the demand for consumption is increasing in the consumption, increasing not for labor and increases employment and wages. In this additional labor income partly or largely depending on the model is specified goes to the low liquidity high-end PC households who increase further than the natural consumption. And because these are going to mention that especially strong because of the aggregate. Okay, so the transition mechanism is different. And this clearly has implication for, you know, the way we understand the thing about monitoring. Okay. So, so let me come back on this point here. You may think it's kind of surprising because you might have expected that in the generalization model, perhaps that there should be an amplification of the impulse to this large accuracy. So, let me, let me move to the, to the issue of identification. And before I do that, I want to make one point clear that, you know, so far, essentially, I talked about basically the role of the wealth distribution. Right. The wealth distribution and portfolio composition that the term is the alternating PC and the size of PC and therefore the style of the directory. Okay. And from now on, I will mostly discuss amplification and in application, the income distribution as a key role. Okay. So, before I move to amplification, there is one, I think additional dimension of this model that is they want to talk about because it's especially, I think, interesting in the way it can conditionally connected with the micro data, which is the transmission mechanism of the shock across the distribution. So, here's a, here's a plot of basically the percentage change in consumption expenditures in the cross section as a function of the percentiles on the web. You see, it has kind of this shape. So, let me explain where this shape is coming from. So, again, you can split this, this response essentially between direct and direct effects across the distribution, across the distribution. You can do the composition that I showed you, kind of agent by agent or, you know, percentile by percentile. So, let me start from the direct effects. So, in this model, as I said, there are about 30% of households who are quickly constrained basically are close to being constrained, either because they are the zero king in the budget constraint, we have a king at zero because there is a wage between the borough and the saline rate, or because there are the credit maintenance orders here. So, for these agents, for these households, the direct intertemporal distribution channel is very, very, very zero. Then, intertemporal substitution becomes more and more important throughout the distribution, as you can see, but it's still muted somewhat near the zero king, near the borough and the saline, because of the precautionary saline. So, this is a model with occasionally binding constraint, and one thing that occasionally binding constraint will effectively, to the precautionary saline model, they shorten the time horizon, the effective time horizon in another equation. Okay, so they reduce the size of the substitution. Remember that here is an expansion of monetary policy shock, so the interest rate falls, and these are households with a low limit wealth, for whom the income effect, the negative income effect, comes to dominance. So, you have a decline direct effect here, because the income effect is very strong. So, the general equilibrium indirect effect through basically an increasing level income generated by the monetary shock is obviously, as expected, especially strong for the end-to-mountain households who have high NPC. And then, you know, the NPC remains slightly higher here than the top, so this effect cannot decline, but it kind of dominates for the end-to-mountain households. The important thing is that, to notice here, that these different forces play out at different points in the wealth distribution, so that the transmission mechanism is kind of complex. And I want to conclude this discussion by saying that, you know, there is already some broad empirical support from microdata to these transmission mechanisms across distribution. I know of papers that use high quality microdata, longitude microdata from Denmark and Norway, UK and the US, and my own work with HIRCA, Slackalike and RSEP-STAN, that uses the HFCS data for the EU area also provides some support to this. Okay, so let me turn to amplification. So, as I said, you might have expected that perhaps a larger share of end-to-mountain households and end-to-mountain households are kind of connected with the amplification of the shocks. And at least a first pass of the data seems to suggest that it's the case. Okay, so this is, it's a recent paper that puts together evidence from the EU area, different common monetary shock and different effects across different countries of the EU area. It seemed to find a positive relationship between the cumulative effects of an expansionary monetary shock and the share of end-to-mountain households. In fact, fairly strong effects, you see, of the order of a factor of three across countries. Let me investigate a bit more deeply into the application. So, I'm going to start from this equation, which relates basically in the change in individual consumption to the margin of consumption and the change in individual income. And I'm going to think of the change in individual income as the change in individual income across the whole distribution, which is due to the monetary contribution. Okay, it's easy to show that you can rewrite this equation here in this way, which is basically a product of the income weight, MPC, the elasticity of individual income to aggregate income. Okay, so basically how income, at very point, at different points in distribution, responds to the shock, times the change in aggregate income induced by the monetary contribution. The total change in aggregate consumption induced by the change in income, what I call the indirect aggregate effect, can be obtained by just, you know, integrating across the households with this expression, and you get this simple expression. So, what you see here is that, I mean, obviously, it seems to be true that, you know, the larger, the largest, the MPC, obviously, the larger is the aggregate effect, which is something that the previous plot suggests. But what is also key, particularly key, is that what matters is the covariance, right, this kind of covariance between these elasticity and the MPC. So, there will be amplification if the monetary shock redistributes towards high MPC households. Okay, that's kind of, that's kind of one of the key, let's transform this plus or minus. What makes things a bit more complicated is that the income composition varies across the distribution, right, this is sort of, this is sort of total income, but, you know, total income is, you know, my labor income, capital income, business income, global transfers, and all these different income components, which is called differently to shocks and to monetary position. Okay, here's an example of the composition of the income across the quantize of the income distribution. You know, the bottom 20, the middle 20 and the top 1%, the bottom 20 is actually mostly global transfers, the middle 20 is labor income, the top 1% is kind of half and half labor income, but capital and business income are particularly important. So, depending on, you know, which of these components is not the most, you know, you would expect the many MPCs, how you steer, how you steer, then depending on how these three components respond, you can get different configurations of amplification or dampening. And the other thing that I want to emphasize that is important is that the fiscal response matters, right, I mean, the fiscal response matters in these models, because the fiscal response when I wanted to put it short, how the government budget constraint adjusts when say, interest payments fall, because in this model, we carry the equivalence price down differently from the representative model. So, I was just going to give you, like a flavor of how things, complicated things can be, complex things can be. This is an estimate of the elasticity of labor earnings to a change in income, is what Fatiguven and Motto Diogo and essential overall kind of workers beta, the sensitivity of earnings across the distribution to aggregate income. This is not conditional monetary quality short, conditional 20 short, it's not, it's not exactly what we want. I mean, it's kind of an unconditional estimate, and you can see that it's kind of, there is something that I think you would expect, right, that at the bottom, the elasticity is higher, precisely because, you know, of unemployment, of extensive marginal employment and employment that affects a lot of individuals at the bottom of the program. But you see also very high elasticity for the very, very top of the early distribution, and we think that this is related to kind of performance based pay is also extremely cyclical. Okay, so if you feed in these elasticity, for example, in this form, it's not clear that I get amplification. It depends a little bit on distribution of NPCs. Okay, okay, my last slide before I conclude, you know, in this, in this, the conclusion that I show you, I kind of talk as if the NPC, the aggregate NPC is constant. But obviously the aggregate NPC is an equilibrium object. It's given by the distribution of NPC in combating, and it also, it's also responsible, right. In particular, what matters for the, for the, the dynamics of the NPC, and therefore amplification versus dampening is the precautionary saving model. Okay, and here, counter cyclical using static risk, we know it's an important source of amplification of monetary policy shots. Okay, we know that using static risk is counter cyclical. It increases the recession, but it's the skewness of the, of the labor earnings innovation that is in recession. And so, you know, after an expansion of monetary shock, unemployment risk falls, right, you go kind of from the red line to the blue line. What happens is that the precautionary saving model weekends and the aggregate NPC rises. And these generate a further increase in consumption expenditures that further applies the initial role of the monetary shock. And I want just to say that, you know, this, this insight about the role of the cautionary saving has been in this incomplete market literature in a long time. And I think Ricardo Caballero was perhaps one of the first to point this out in his, one of his, one of his papers. But, you know, all we're doing here, basically by merging this literature with the, with this new case and block is to explain how all these things can be relevant for monetary policy. Okay, and I, my timer, and in fact, I'm on my last slide here. Okay, so just taking stock. You know, we, we, in this research program that, you know, many, many people contributed to many of whom, I'm sure are listening to the conference. The distribution of income and wealth is also central to market economics. And a political grounded approach, right. Alters, what we're introducing to this model alters an important position. The wealth distribution and portfolio composition, the term in the distribution of disease and the strength of the direct and direct generation channels of monetary policy, which one dominates itself through which basically challenges you have to think through when you want to try to forecast or understand effects of money. The income distribution and the composition of income, the term in the redistributed consequence of monetary policy, and also the amplification or dampening of a monetary policy shock, relative to, say, a baseline standard representative. And I, when you say conclude, thank you very much. Stay safe. Thank you very much. Thank you very much for this very clear exposition and for clearly highlighting why as a central bankers, we should also look at micro data and look at the distribution and understand these effects basically. We already received some questions in the Q&A session. There were two questions that basically boiled down to the same. So I'll bundle them. They came from kind of Alta villa and Klaus mazu. Basically, the point is that in your paper and your presentation module policy is basically an interest rate policy. And the question is, how much do these distributional consequences really depend on the policy measure that the central bank deploys. I mean, you know, we're going more and more into what you call non standard or unconventional measures. So how would your conclusions change of the paper if you think for instance of QE instead of an interest rate policy for for the channels. Thanks. This is, this is a great question. There is already some work being done that tries to understand the effects and the transmission channel of QE in this class of models. I'm going to cite one paper. I'm sure I'm going to, I'm forgetting others, but it's a paper by way cooling is a spark that uses a DCL that uses precisely to think about QE. Now. Okay, so a couple of things. So the first is that I think this is a very natural. A very natural framework to think about QE, because you have these two assets, right, which are liquid assets, a gold, bonds or bank accounts. And the other one is, is this illiquid asset. So, so say housing or when I think about the financial equivalent would be like mortgage. So, you know, it's very natural to think about QE as doing a liquidity swap, for example, in these models. And the QE and start paper is precisely about that. Now, if in terms of the composition that I talked about, the effects of QE are essentially entirely general. There is no direct effect on consumption. There is no direct effect of QE. Everything goes through their general effect on prices, on asset prices. And here's where I have to say, this is not yet, this model is not yet ready in a way for, at least in the information that's shown for this type of analysis, because as Ricardo has highlighted this presentation, you know, a huge component of the way we think about financial markets is risk, I think risk, the traditional risk across assets and so on. And, you know, New Keynesian models in general are quite bad at generating, you know, equity-large, and so on. Now, again, there is interesting work being done. And I'm going to cite the paper by my colleague Moritz Lenin, who has introduced heterogeneous preferences, the heterogeneous conversion to this model. And by doing that, you can get, basically, larger couldn't bring up, you can get, you can get time value, the bike over the cycle. So that is, I think the framework is getting ready to already to answer these type of questions. And the race, you know, in terms of like the distribution effect, in terms of distribution effect, it would be kind of a race between, you know, how strong the effects of free are on, say, asset prices, and therefore something that, at least on paper, tends to favor the high income high wealth people, versus the effect on sustaining the working market wages, which is something that would affect the lower end of the institution. I think the jury is still out there, both empirically and theoretically within the classroom models. Thank you very much. We have received also another question by Anton Nakov, which is whether monetary policy should explicitly target wealth or inequality. And if so, how? Okay, so this is a question that's, that's, you know, that's the question that you like at this conference. I'm not going to give you a yes or no answer, obviously. But it's a great question. It's, it's, it's, it's a question that is the object of, again, a number of papers at the very frontier of this literature. I have two, just because they're the ones that I'm, they come to mind. And the three is one paper by Tom Sargent might go also. There's a paper by Zedir Ago and co-authors. And there's a, there's a paper by Floridini and co-authors, which actually is kind of neat because it represents the issue analytically. So these are, they're complicated, they're complicated models. And the answer depends a lot on the, on the social welfare function. It's a question about that policy. Okay, so that's what we're in the Ramsey world. Let's stay in the Ramsey world. It depends on the social welfare function that you write. You know, the social, the general social welfare function, the general social welfare function would have a model for redistribution in this incomplete market model, the model for social insurance. And then there's the classical stabilization. You have it in your case. Now, I can tell you that one thing that comes to, that emerges from this literature, almost a university, is that the answer depends on the existing fiscal instrument, the existing fiscal policy that is available. Okay, so if fiscal policy is rich and is close to optimal, then fiscal policy should do the job of redistributing and providing social insurance. And then what is left for monetary policy is largely, is largely kind of stabilization. Let me just say that even if that is the case, okay, discuss some models remain, I think important because it tells you that the transmission mechanism monetary policy is different. The way the monetary policy stabilizes the economy is different from what you were previously thinking. So the question is whether, you know, fiscal policy is close to optimal or not, or whether it's whether over, over a dozen distribution under a dozen distribution under provides social insurance and so on. So it's a question that I think can only be answered conditioning. Thanks. I have another question from Anton Nakov. The first question was now whether monetary policy should target explicitly wealth or income inequality. His second question is, can inequality concerns inform questions on the optimal inflation rates and, you know, whether we should focus on inflation targeting versus makeup strategies. And of course makeup strategies very much here also in the limelight these days and central banking after the Fed's decision. So what would be your views on that? So that's another great question. So my view here is that there is maybe one important dimension that, you know, if I, if I, you know, that I think it's interesting. It's interesting to develop within these models, which is that it's something that people have shown that different households across in different points of income and wealth distribution consume different bundles, consume very different bundles. And as a result, they have different inflation rates. Okay. They also hold different, different portfolios of nominal assets. And so they are also subject to inflation differently depending on how much nominal wealth they have in their portfolio. So overall, what I would, what I would say is that, you know, the, the redistributive impact, and, and therefore, you know, the, the, the, the strength and the transmission mechanism is even a little more subtle than what I show today, precisely because I think you were to enrich the model with this element where there are different goods and people across the institution consume different, different bundles of these goods and these different bundles of the goods respond differently to inflation. And then I think the, the, the, the, the, the whole transmission mechanism of monetary policy and perhaps even inflation dynamics of the whole could be very, very, very, I don't know.