 Good afternoon, everybody. Welcome to our keynote address. And it's my great pleasure to introduce Professor Haou Jean-Ju as the keynote for this year's Money Market Conference. He's the Gordon Y. Billet Professor of Management and Finance and the Associate Professor of Finance at the MIT Sloan School of Management. He's widely and very well-published, has done work on a wide range of topics from asset pricing, liquidity, monetary policy implementation, and also the impact of technology on markets and intermediaries. So he's extremely well-placed to talk to us today. He's also been, and that's particularly interesting. He's also very active in the policy debate on the reform of the benchmark interest rates or the regulation of derivative markets. And today, he'll talk about another very timely and policy-relevant topic, the issue of central bank digital currencies and their design, and in particular, on the frontier trading of paying interest and the use of payments of such a digital currency. So Haou Jean, the floor is yours. Great. Thank you, Florian. And I want to thank the organizers for inviting me to speak and to join this fantastic conference. This is joint work with Rod Garrett from UCSB and Jiahong Yu, who is a PG student at MIT. And today, I want to share with you some new research and a new thinking, at least from my perspective, on the CBDC design. So first of all, why do we talk about the CBDC? I think this is a kind of recent phenomena. That CBDC is a digital payment instrument that is direct liability of the central bank. And in recent years, the interest in CBDC has been growing steadily and is shown in this chart published by the BAS. If you look at the map globally, clearly, many countries have started off in research developments, such as US and Europe, or even piloting and implementation, the actual launching of the CBDC. Notably, Sweden and China have started the pilot, and I said the Bahamas have already launched. It's a $10. Now, why do we need the CBDC? I think CBDC satisfy a number of important functions. And the two of them are improving payment system and the monetary policy implementation. A CBDC is a digital alternative to cash, and it could potentially make the payment system safer, make it more resilient, and thereby promoting financial inclusion and financial stability. On the other hand, by establishing a direct link between the central bank and households, the CBDC might be able to help with the monetary policy implementation. And this is precisely in the way other central banks are thinking about it too. Today, I will talk about the two specific features, interest bearing and the payment convenience. The interest bearing property is tightly related to the store value property of a currency, whereas the payment convenience is tightly related to the meeting of exchange property of a currency. The third property of currency, which is a unit of account, is automatically satisfied if it is a CBDC, as it is legal tender in the country. Now, these two dimensions also get reflected in the ECB's digital euro report. For example, it's a good report showing a number of requirements that a successful CBDC should have. And the first requirement is about enhance the digital efficiency that the digital currency got to be usable, usability, convenience, speed, cost efficiency, and even programmability. The fourth requirement is about the monetary policy implementation. If it is a tool for monetary policy, the digital euro should be remunerated at an interest rate that the central bank could control and modify over time. Likewise, in the US, here is a banking fall act from US Congress. We understand that the Fed has been still deliberating, but the Congress already put up a bill, as far as I know, is not voted on yet. But you can see that it is a very similar vein, talking about the digital dollar got to pay interest. And it has to offer all the services that you could get from your bank accounts. Now, the third example I want to give is China, which has already started the experimentation of the RMB CBDC. It is the manager of the system, kind of two tiered. And it is meant to be an alternative to cash. So it does not pay any interest. But it has some interesting features, such as offline transactions. So Chinese CBDC has picked a particular combination of interest bearing, which is zero, and certain functionalities of payments. So the question we are asking here is, given that central banks are actually thinking about these two design features, then what are the implications? In our model, we are going to think about the CBDC as offered by the commercial banks. So it is a two tiered system, which is like China or Sweden to design. And in a way, it's not far away from how the Fed or the ECB thought about it. We're going to embed the CBDC in a practical modeling framework in which a large bank and a small bank compete in deposit market and the lending market. Now, what do we find? Interest bearing feature of the CBDC puts a lower bound on the deposit interest rate that banks pay to households. By raising the interest on the CBDC, the central bank is able to speed up monetary policy transmission. But the downside of it is that it's going to further reduce the market share of the small bank. The idea is that by shrinking the central bank policy rate, the interest rate paid to the banks, and the CBDC rate paid to households, it leaves little space for the small bank to compete. That's why the small bank would lose market share. On the other hand, if the CBDC is very easy to use, for example, for payments, that's going to level in Plainfield by raising the market share of the small bank and achieving away the advantage of the large bank. And the impact of the interest, sorry, of the convenient CBDC on monetary policy transmission turns out to be quite nuanced. And it could either go up or down, depending on the parameters. The next slide we're going to look at is to look at how the major stakeholders view CBDC as a model. Turns out, banks and households have very different views about the interest bearing feature. Households prefer the central bank to pay the interest and hire the better, but the banks prefer exactly the opposite. But interestingly, we find that in some conditions, banks and households actually agree on the convenience feature. So in other words, if the CBDC is easy to use, not only does it benefit the households, it also benefits the banks. And this idea or this result can lead us to the last point about the frontier. That is, can we construct a set of desirable combinations of interest-rated convenience value of the CBDC so that people kind of prefer this combination to anything else? So here is an illustration of what we have so far. On the horizontal axis, we have the CBDC interest rate. On the vertical axis, we have the convenience value. And you can see that it is almost like a thumb-shaped pattern. And any particular point on the frontier actually dominates any point below. So that's our idea of having a frontier. That is, we would actually go on to go here. So the good CBDC design would not have a zero convenience value. And of course, the exact design the central bank picks would depend on the objective. I will be more explicit later. Okay, in the interest of time, I'm going to skip the literature and let me go to the model before talking about the implications. The model has one large bank and one small bank. The bank's assets are assumed to be reserves and the viabilities are deposits. We just keep it very simple. The reserve's amounts are very large, denoted by X. The central bank pays interest on the reserves. We call that F. And the banks would have said indulgence interest rate RL and RS on the deposits. What is the difference between the large and small bank? There is only one difference in our model. The large bank deposit is easier to use for payments and other functionalities. Here I'm showing is a super app screenshot. You can see that in this super app, lots of services got embedded in this wide app. So it's like one stop shop. This might be too extreme to think about a bank, but when we think about the large bank offering a lot of services, a better app may expense the bank network so that it has a convenience value for consumers. And we denote the convenience value to be delta that has a distribution G across the unit mass of agents or households. And we can normalize the small banks deposit convenience value to be zero. So the key advantage of large bank is that it's deposited easier to use. Now, what about the CBDC? The CBDC comes to the picture as a central bank's liability, but it's going to be offered through commercial banks. So I'm showing this sort of schematic, this how I think about it personally. If I want to have a CBDC account, I'm going to go to one of the two banks, say this is a Dodge bank maybe, and I will open an account with a large bank. Once I have the account, I can really transfer my money between the commercial bank deposit account and the central bank account sponsored by the large bank seamlessly. So money transfer within these two accounts would be very easy. Once I do that, I'm able to get this CBDC app and I can transfer money between my bank account and the CBDC account seamlessly. So by doing that, I'm able to use the CBDC app to do stuff such as touch pay or scan pay. If I have a small bank account, then clearly I can do the same thing. As you can notice that because the small bank does not have so much functionalities as large bank does, having the CBDC is actually help the small bank customers to get access to more advanced technology and easier use of payment services. So what the implications is that once the CBDC gets introduced, the small bank customers actually get convenience value of V rather than zero, whereas large bank customer get the convenience value which is the maximum of delta and V and the delta again is the agent's specific preference for the large bank deposit. Now because of the convenience value of the CBDC, we're gonna see that the CBDC's interest rate, we're gonna call S, is going to be a lower bound of the commercial bank deposit interest rate. So I just wanna emphasize that for this inequality to be credible, it has to be the case that the central bank deposit, sorry, the central bank currency is easy to use and easily manageable for the households. Okay, in terms of a timeline, there will be three periods. In the beginning, the banks will set the interest rate RL and RS, the central bank would set the LER or IRF, which we take as exogenous. The two design variables are the CBDC's interest rate S and the CBDC's convenience value V. Every household in the beginning already has an account with either bank, large or small. In period one, the household kind of act as borrowers or entrepreneurs, they draw the projects, the projects differ in quality called Q, and they're gonna go to their existing bank to borrow. The existing bank is assumed to be a monopoly in the loan market to make it simple. If a loan is granted, then the funded entrepreneur is going to pay a randomly match the worker wage, that's the $1 borrowed. Now, the whole point about having the worker is to generate some flow funds in the economy, and the worker gets the wage and is going to choose where she deposit the wage, depending on the interest rates of the two banks and the convenience value of the two banks. So before solving the model, let me just show you one table that kind of illustrates the pattern that advantage in the deposit market translates into advantage in the lending market. So on the top, we have the original balance sheet of the large bank. In the middle, the large bank lends a dollar or the euro to entrepreneur, and that works is that it's gonna create a loan in the name of the borrower and create a new deposit in the name of the borrower. And then entrepreneur is going to spend the dollar to pay a worker or suppliers. Now the question is, does the dollar leave the bank? In the model with the two banks or any kind of discrete large banks, the banks market share right here, large banks alpha l. So with the probability alpha l, the deposit actually stays with the bank because the worker banks with the same bank and only with the probability alpha s, which is why minus alpha l deposit the leads, right? So in this case, the newly created deposit does not necessarily leave the bank after the payments. So that's in a way the advantage of a large bank, I'll show you here. The total profit of making a loan for the large bank is shown on the bottom. You can see that the first term, group of terms is net profit on the loan. And we think about F as the opportunity cost of funds because the bank could deposit the money with the central bank to earn the interest on F. Now the second term is a key in the sense of the model. We see that the alpha l is the probability the money stays with the bank and F minus rl is the interest rate spread. So this number will be greater in equilibrium than the corresponding number for the small bank. And that's going to give the large bank a wide advantage in the lending market. So projects that the small bank view as not profitable might be profitable for the large bank. Just to motivate a little further, is that realistic that alpha is greater than zero? In the US, the three top banks in terms of the deposit amount, Bank of America, Chase and Wells Fargo holds about a 10% each. Citibank holds about a 4%. You can see that if a Chase lends out a dollar, then with about 10% probability and conditionally, the money actually stays with the Chase. That is not a trivial number. Okay, just very quickly solve the model. In the period two, we see the depositors are going to choose where to put the money in. And again, by going to the large bank, the depositors get this value for convenience by going to the small bank. It goes to this value of convenience V. And we can see that if the small banks, sorry, if the consumers advantage, sorry, the preference for large banks large enough, then the consumer is going to pick the large bank. We are going to focus on parameters that leads to a small bank paying a higher interest, which seems to be quite intuitive and realistic from our experience. So the small bank's market share is given at the bottom and the later we'll see that as V goes up, alpha S will go up. This is not a trivial result because RIS minus RL actually varies. So in the end, the change in the deposit interest rate offered by the banks will not mitigate the impact of V so that a higher V convenience value of the CBDC does benefit the small banks market share, okay? Lending, we just talked about it from this equation by imposing the zero or break even condition, we can get the lending standards. The result is that the lending standards of the large bank is a little lower because the second term is a little higher, right? So there's almost like a second profit to be made on the loan. So the bank could afford to have a slightly lower Q. And finally, this final slide sort of on the model solution, we're gonna look at the total profit of the two banks. The first source of the profit is the loan and the second source is the deposit spread. Here we can see that if X is very large, then that's going to dominate the loan amount. Even though we solve the model, I guess for any generic parameter, I guess in today's market, we do see a huge number of reserves so X is likely to be large. And here think about this term would be the interest rate spread, the bank would earn between the deposit and the central bank rate. All right, so there are two cases of equilibrium, I'll just describe it verbally. In the first case, the CVD's interest rate, S is not binding, meaning either the rate is so low like zero or if the central bank policy rate is high like F equal to 3%, in that case, this large bank and small banks deposit interest rate is already away from the zero lower bound and away from the lower bound of S. So it's not a binding. In the other case, such as when in the world we are in today or sort of in the foreseeable future, that rate is not going to be so high so that the central bank digital currency interest rate may well be the binding interest rate for the large bank. So the CVD's interest rate becomes a new lower bound. Lower bound is no longer zero, but the CVD's interest rate. So in both cases of equilibrium, we have the large banks at the lower deposit interest rate, we have the large banks that a larger market share have lower lending standard as well as higher profit. Let me just emphasize that in this model, the CVD's is actually not held, right? Think about this. It's not about the CVD's and yet it is not held. Therefore, it does not really disintermediate all the banks. Nonetheless, the CVD's is still quite effective as a viable outside option. The way it is viable, again, is because it does provide payment functionality and it becomes it offered through the banks so that it is a almost a perfect substitute for bank deposit and a better substitute to the standard that it offers better payment committees than the small bank deposits. Now we're ready to talk about the implications by varying these two dimensions, interest rate and the community's value. So here, let me show you the chart first before the formal proposition. Here we are setting the interest rate of the central bank interest rate, policy rate should be 2%. And for now, we set the community's value to be zero and kind of isolate the impact of a CVD's interest rate. As that interest rate goes up, we can see that the large bank interest rate goes up with it in tandem. So this is the constraint equilibrium where the large bank's interest rate is equal to the CVD's interest rate. The black line shows that the weighted average interest rate in the market goes up. And so as the rate goes to further higher, you can see that the weighted average interest rate can converge to the policy rate. So that is good for monetary policy transmission, but the downside is that the small bank would start to lose market share. The intuition is that as the interest rate between the large bank and the small bank start to narrow, the deposit would say, well, why do I have to use a small bank now? The small bank's interest rate is not a lot higher and I could use the large bank for its other services and the payment convenience. So that's why the large bank start to gain market share. So this is sort of unintended consequence of having a high interest bearing capacity in the CVDC. And likewise, we see that very similarly in the lending market, long volume goes down in small bank and it goes up in large bank and that kind of maps back to the deposit market share divergence. Total lending volume in this example goes down mildly. Now, more formally, we have pretty much all the comparative statics and the certain conditions that for if the reserve amount is large enough, then we can show that the higher central bank digital currency interest rate, a speed up monetary model transmission to make the weighted average interest rate closer to F, whereas it harms the small bank by reducing the activities in the lending and the deposit market of the small bank. So that's kind of interesting tension right here. All right, so that is the interest bearing feature. What about the second feature, which is the convenience value? Now the convenience value turns out to have quite distinct properties. So here again, I'm showing the corner solution, sorry, the specific case where S equal to zero. So there's zero interest bearing and the policy rate by the central bank is still 2%. Now we are gonna increase the convenience value. So here 1% simply means that the CBDC is so convenient that depositors are willing to give up 1% interest rate per year to hold that kind of payment instruments. As the CBDC convenience value goes up, in this equilibrium, we can see that the large bank interest rate in blue that is stuck to zero. It is at the lower bound and the small bank's deposit interest rate actually goes down. Why is that? This is because the small bank, starting with a disadvantage, now have a smaller disadvantage, right? Because remember the small bank depositors can now use the CBDC to make payments and do other things. So the small bank does not have to offer as high deposit interest rate to attract customers. Nonetheless, despite this drop into small bank interest rate, we see that the small bank market share goes up. Again, this is because the small bank's market share is given by the interest rate spread plus V. As V goes up, the small bank's market share goes up. The large bank's interest rate does not move and the weighted average interest rate in the economy actually goes down. So this is when the monetary policy transmission gets actually worse. However, once the equilibrium transit into this region where the large banks start to lift up interest rate away from the lower bound, we see that the weighted average interest rate start to go up. So the monetary policy transmission in this world is quite a nuance and quite rich. It's more or less a U-shaped pattern. Laney market is very similar. The small bank gains market share exhibited by, on the right, increase in Laney volume and the large bank loses Laney volume. The total Laney volume as the black line showing on the right is almost flat. Numerically, it goes down mildly but it's extremely small, okay? So we can see that the property of a convenient CBDC is distinct in the sense that it is strictly benefit the small bank in deposit and the Laney market. Therefore, level them plain field but it has a quite nuanced impact on monetary policy transmission, okay? Now on this point where when I say it's unclear, it is truly unclear. Here is one example and you can see that in this particular example where the policy rate is 3% and the CBDC pays the interest rate of 1.15%, the equilibrium will be constrained but the total Laney volume is actually not monotone in CBDC convenience value V. So that's quite interesting. Another thing to notice is that on the left, we see that the magnitude here is quite small. The impact is not too large and the larger effect is still the reallocation of Laney volume between the large and the small banks, okay? So what I've shown so far is that interest bearing capacity and the convenience value of the CBDC have rather distinct properties in terms of shift market share in the economy as well as monetary policy transmission. At this point, one possible approach is actually to specify a central bank objective and try to solve in the sense the optimal design, right? But right now we're gonna try to experiment a new approach. Now this approach is this CBDC frontier idea. The idea is that before launching the CBDC or even after it, a successful CBDC should require a broader support. So before thinking about a central bank objective and we should also think about how would the major stakeholders view this innovation? So in the sense of frontier is really a combination of interest bearing and the convenience value as in the repairs that both stakeholders in our economy meaning the households and banks prefer. And after that, the central bank would now have fewer points to choose from and then the design really depends on the central bank objective. Let me just illustrate how this works. Starting from the households, here quite intuitively households prefer convenience value and prefer interest. So clearly their welfare would increase in S and increase in V in general and they're fairly mild conditions. As the illustration kind of graphically, we are plotting the indifference curves of the depositors. Right here on the x-axis, we have the CBDC interest rate and on the vertical one, we have the convenience value. The blue dots represent the constraint equilibrium where RL equals to S and unconstrained equilibrium with RL greater than S. We can see that when the interest rate is low, here's 2%, most of the region is covered by constraint, sorry, by the constraint equilibrium and the maximum convenience value that the household prefers is actually, sorry, this is the typo, is 0.35. So this is 3.5%. We can see that the households generally prefer to go northeast, right? They prefer a higher convenience value and they prefer a higher interest rate. Now this boundary right here basically characterizes RL, here's RL equals RS equal to S, whereas here is RL equals RS greater than S. So we believe, at least for now, we have not even attained the combinations of S and V such that it pushes the composition of the markets so far that the large bank no longer have the advantage interest rate. So we are kind of looking within this boundary. That's what we have characterized in equilibrium. So again, households prefer to go northeast. The banks, the banks total profits as spelled out in this equation and we can show that interestingly, it increases in V in the constraint equilibrium. What it means in the graph is that they are willing to go northwest, okay? And that sort of in the blue region, just focus on the blue region. If you are in any blue region right here, let's say, the bank will strictly prefer to go here and go here and go here, right? So they would want to go to the northwest. In this case, under the numerical examples, in the green region, the bank's welfare, so the bank profit, I'll say bank profit actually goes down in V. So once you reach the boundary, the bank doesn't want to go northwest anymore, okay? Moreover, we show that the higher V increases the minimum of the two banks profits. And that has the implication on financial stability. If we think about stability is measured by the weaker of the two banks, then raising the minimum profit of the two does enhance stability a little bit. So now with the households and the bank's preference is specified, we can now kind of think about the frontier. So the frontier consists of this, you can see this orange curve. And below the orange, that's the constraint equilibrium where the central bank digital currency rate, sorry, I should say that RL is equal to S where IS is greater than S, whereas is the central bank digital currency interest rate. The red curve represents the household's welfare, so this would go northeast. And the blue represent the bank's profit, they would want to go northwest. So the way to think about it is that suppose we fix on this curve, this red curve, we're gonna ask the question, well, the households are just indifferent on this curve, where would the banks prefer to go up? So you would end up on this point. The idea is that within the constraint region, and curiously, we do have a agreement between households and banks, even though they strongly disagree on the interest rate dimension, they do agree on the convenience value dimension. So they do prefer the CBDC to have good convenience value properties. The intuition for the households is quite simple. They want to pay me the functionality. And for the bank, the intuition is that once the CBDC interest rate, well, sorry, once the CBDC convenience value goes up, the banks no longer have to pay as much interest rate in the deposit account to customers. That's why the banks are also kind of better off, okay? Now from this point on, we could ask, well, on this frontier, which is orange curve, where do we want to go? And that actually depends on the central bank objectives. So this is where we bring back the central bank's objective and all these comparative studies that I've shown you before. For example, if we prefer to have monetary policy pass through as thoroughly as possible, we can actually pick the corner, right? This way on the right, we do have the weighted average interest rate going all the way to policy rate F, 2%, if the central bank interest rate, sorry, if the CBDC interest rate goes to 2%. Likewise, deposit welfare is maximized if we pick the CBDC interest rate equal to the policy rate. The banks total profit is maximized right here and this corresponds to this point, interestingly. So the CBDC interest rate that CBDC design that maximizes the bank's profit does have actually non-trivial convenience value. Again, the intuition is that by providing convenience value, the banks could sort of pay less in terms of the deposit interest rate and sort of let the users or depositors use the central bank's payment functionality rather than bank's functionality. Now that's sort of one possible, a few possible designs. And here's another example. What if the central bank wants to level the playing field as much as possible? Well, in our simple model, level the playing field simply means the small bank has a higher market share and that is maximized right here in this point. So the small bank would have an equal market share with a large bank on that tip and that would correspond to this particular point on the left. So in this example, we have central bank rate to be 2% and you can see that the interest rate is roughly 25 basis point and the convenience value is about 1.7%. Now 1.7% is quite high, suggesting that the payment functionality of the CBDC got to be so good that a usual depositor would be willing to give up 1.7% of the deposit interest rate per year to get to use that functionality. And likewise, if the central bank wants to maximize financial stability, that is going to raise the weaker of the two banks and clearly the answer is still the TB important, okay? So just to summarize right here, once we have the frontier, the central bank could think about the different objectives and now look for a narrow set rather than looking through this the two-dimensional design problem and kind of looking through all these different objectives on this frontier seems to be sort of a little easier operationally. Okay, I don't have much time, but let me maybe conclude. So in this paper, we are looking at the interest rate and the convenience trade-off in CBDC design. These two features have dramatically different implications. Monetary policy path through is sort of favored by higher CBDC interest rate but a higher CBDC interest rate does reduce the small banks market share. A higher CBDC convenience value payment functionalities level than plain field but it's going to have a quite a nuanced impact of monetary policy transmission. The frontier is constructed by asking banks and depositors or households, would you prefer this design or the other? And we can eliminate all the dominated options which is below this orange curve. So now we can kind of go into this frontier and ask the question, well for each individual central bank, what are the objectives and what is the design kind of meet these objectives by selecting from fewer design points? Thank you. Thank you, Jean. This was great and shows how intricate the problem is, how many moving paths one has to consider when thinking about digital currencies, central bank digital currencies in the economy as a whole. I have two questions in the chat and we have about nine minutes left. We can also finish a little bit earlier to get back on schedule but let me read out these two questions to you. One of them I think refers to your two-layered approach and it says, does it matter whether I think of the CBDC as only offering interbank services versus offering direct deposits for households? And that's was from Jay Khan and the second question from Leili is zooming in on how to model deposit behavior. So what if banks need to compete for deposits to fund? In the model it seems that borrowers deposit their loans to the lending banks so banks do not need to raise additional funding and how would this affect the equilibrium? Great, thank you for the question. On the first one, I think that's the question of retail CBDC versus wholesale CBDC. So we are talking about households so this is more of a retail CBDC phenomenon. The idea is that the central bank may not want to deal with all these, you know, no-yell customer and money laundering. There's a lot of paperwork and the process to be done and the commercial banks already have the infrastructure to do this. So why don't, you know, let the commercial banks or other intermediaries, you know, how about the central bank doing the paperwork and sponsor this, you know, the CBDC accounts via the commercial banks? So that's kind of practical concern. And the second, I think more economic reason is that for the CBDC to become effective in transmitting monetary policy it has to be in a way a kind of viable outside option. If the CBDC is offered in a standalone basis it may or may not be as easy to use as a large bank deposits. So for example, a large bank deposits, you know, gives the information to the large bank and the bank sees all the deposit flows may offer you some other services such as credit, such as brokerage. And if the payment accounts go through a CBDC the central bank doesn't offer those services and somehow, you know, the deposit may still use the use large bank deposits. So the idea of offering the CBDC through the commercial banks is that whatever payment activity the customer has is going to be visible by the large bank anyway. So that makes the information content also usable to the benefit of the consumers, right? So that's sort of the two reasons why I think it's kind of reasonable to use a two tiered approach. One is practical, the other is to make the CBDC truly a viable outside option. A last question about the modeling maybe I should emphasize this a bit more. In our economy, we are talking about today's say the US or Europe where the central bank have a huge amount of reserves and the idea is that when the bank lends out to fund a project, the amount of reserves the bank has with the central bank is really not a constraint, right? So lending is determined not by how much reserves you have rather it is determined by the opportunity cost of capital and by the total profit. And we show that it is the large bank that actually has effectively lower cost of the funding. They do compete in the deposit market by bearing the interest rate, but when they lend, they do not have to worry about the raised funding because indeed they actually create a deposit and let the deposit flow. So that's a slightly different from the pre-crisis world when the reserve flow a little smaller that we can talk about reserve requirement matters. But right now we've reserved the so abundant and so flush in US and Europe at least that lending probably wouldn't depend on reserves. And by the way, there was also work by you may not showing that, you know with their shots on banks funding through money market mutual funds that doesn't seem to be a huge impact on lending, right? That's also supports the idea that, you know in the age with ample reserves, lending is not really tied up to the level of reserves. Thank you for those questions. Do we have other questions? Otherwise I use my spot here to ask two more questions. One of them, I guess they're both modeling questions again but asking to what extent one can step outside the model a little bit. So, you know, you have this structure with the large bank and the small banks and of course that's very convenient. But, you know, how would this generalize to a situation where we would have competition among several banks? And the other thing that I was a bit curious is that, you know, an important margin in your model was this F minus the interest rate. So the interest on reserves minus the interest that you have to pay the depositors and this is the, you know, spread that banks can earn. Now, in your area that spread is negative, right? You pay zero on the deposits and minus on the, you know, on the deposit with the central bank. So can your model accommodate this or, you know, would something strange happen? Thanks. Great question. Especially the second one about negative spread. So let me come to the first one. I think competition among several banks would work. Typically when we think about bank competition in the deposit account seems to be sticky. So one thing I think the feature of the model is that we do not have any search friction in the deposit market so people can switch anytime they want. So the reason that, you know, the banks still have a bit of bargaining power is that the large bank does have a convenience value advantage, right? So now if we have several homogeneous banks, then when could use the Euro techniques that, you know, each bank might have a local customer like loyal clientele that would prefer this bank and then there are others who can search. In this case, one could have a sort of model of symmetric equilibrium where the banks would compete maybe using mixed strategies, maybe pure strategies, depending on the structure, but the key I think it could kind of apply there as well. Now on the second question about F minus RB negative, it is completely defined in our model that, you know, in all the competitive statics actually do not require this number to be positive. Instead, what was kind of at least in the search of equilibrium, we are looking at the central bank interest rate S to be between zero and F. Now, of course, if F itself is negative, then the central bank, you know, interest rate might be negative as well. So I think it does open up some interesting issues, but I think, you know, in terms of the model, it should more or less work through. Now the question is, you know, if both of the banks start to offer zero rate, right, because the depositors doesn't want to lose value, does that mean the small bank would lose market share to the large bank? In our model, that's the prediction, right? Because they both got a negative 40 from ECB, then they pay zero to depositors, that's an active 40 basis point spread that's equal between the large and small bank. So the prediction is that in that word, indeed, we would have large bank gain in market share. In general, in our model, at least in this kind of model, a lower interest rate kind of leads to kind of rise of market share for the large bank and a drop in market share of the small bank. Thank you. Thank you, Clark. Unless there are any other questions, we are pretty much up against the allotted time. So thanks again, Hao Zhang, for the great keynote.