 Bona tarda a tothom. Aquest és l'últim recerca integratiu de la sèmina de la sèmina. Aquest és un experiment que jo suggeriré al Departament per convidar researchers i professors d'altres departaments que nosaltres, per obrir el nostre mind i veure potències col·laboracions en aquest cas, en aquesta situació amb Jordi Galli. Jordi Galli és el professor catedràtic de l'Economia Departament. És un mestre que té una sèmina molt longa que no he de rebre perquè... I va tenir un PSD a la MIT, que ha estat en contacte amb els engenis. I també és el director de la CREI, el Centre per la Research en l'Economia Internacional. Està donant aquesta conversa sobre el rol d'expectacions en l'economia i crec que hi ha moltes qüestions en l'endemà de Jordi. Moltes gràcies, Jordi. Moltes gràcies per acceptar la nostra invitació i les paraules. Moltes gràcies, moltes gràcies a tots. Moltes gràcies per la invitació. Aquesta és una cosa diferent per a mi, perquè bàsicament parlo d'economistes. I la natura de la meva conversa també serà molt diferent. És que no presentaré alguna cosa recent, però, a més, l'objectiu, l'objectiu d'avui, given that I take that most of you are non-economists, is to give you a bit of a glimpse of some of the tools and the concepts and methods that we use in economics. So that would be the general objective and now the specific choice of a topic is because I think this is a dimension of economics that is very distinct, very characteristic of economics. It's very important in order to understand many economic phenomena and that's what I will try to argue today. And it clearly makes it different from natural sciences. That will be one of the basic points of my talk and hence it calls for different ways of thinking from the ones that we may be used in natural sciences. It also gives me an excuse to sneak in a bit of my own research, recent research which certainly makes use of some of the concepts and methods that I will talk about. So these are expectations in economics, the role of expectations. And again, I think there will be questions at the end, sometimes for questions, but if you have a clarifying question, just don't hesitate to raise your hand and ask me. So this is a distinction that I want to make from the start. And again, it's a bit of an arbitrary distinction. Between physical systems and social systems. And again, there may be many dimensions in which those systems or those worlds can differ, but here is one. Physical systems are made up of purpose-less matter. I guess. Whatever behavior, interesting behavior they display, it's kind of mechanical. It's not purposeful. It's not the result of decisions that someone is making. Again, we're leaving human beings aside here. I think of physical systems without intervention of human beings. I think of physical systems without intervention of human beings. And again, one way of representing this mathematically would be to think of a vector XT that would contain a number of relevant state variables that evolves over time according to this difference equation where Z here is just a vector of exogenous variables. And again, here, what do we think of is a variable that we're not trying to explain. But if we went deeper, maybe this would be completely deterministic. XT would be just a function of XT-1, and that's it. Social systems, on the other hand, social systems by definition are made up of individuals, human beings. And individuals have objectives, have constraints and make decisions that are guided by those objectives. So that, in itself, is interesting, but what I will try to emphasize in my talk is that some of those decisions involve payoffs, gains of some different nature, pecuniary or non-pecuniary, that depend on other individual's actions. So they don't exclusively depend on one's actions, but also on other individual's actions, which necessarily, if someone tries to make an optimal decision, necessarily requires that that person forms expectations or beliefs about what others will be doing. Or if you want about what the outcome of others' decisions will be. So that's how expectations come into play. So think of two environments, and my talk essentially will be a list of examples, simple illustrations of both environments. The first example is a static environment, so everything takes place simultaneously, if you want. And you have that the decision, some variable that someone has to decide on, say x, is a function of the expectation, this is the expectation operator, of a function that involves the average value of that variable for the rest of individuals, think of a large number of individuals, society, and also some exogenous variables. So here, this individual will have, in order to think of this as an optimal decision rule that comes from, as the outcome of some maximization problem, and it may take this form, this is a typical form that it would take. In particular, I want to emphasize this dependence on the average decisions made by others. And in a dynamic environment, okay, so this would be a static environment, in a dynamic environment, and you have a representation of the equilibrium of a social system, say an economy, which takes this form, which differs from this, in the sense that it involves not only the past values of this vector of state variables and some exogenous variables, but also it involves the expectations, or some beliefs about the distribution of that vector of variables in the following period. And hence recursively, if you think of solving this recursively forward, it involves expectations and beliefs about things that will happen far into the future. So what I want to illustrate through a series of examples in my talk today is that this fact, the fact that some decisions and some economic phenomena are functions of expectations about future... expectations about others' decisions or expectations about future behavior of the economy leads to some interesting implications for the properties of the solutions of these, say, equilibrium conditions, and hence for the properties of outcomes that we should observe in these social systems. That you wouldn't find in what I would call mechanical systems like this. So that's my plan. So here are just the list. This is how I will organize my talk. First I will just give you an example of a simple economy in a static environment. Everything happens in a single period. And it's an economy which displays what we call, economists call, strategic complementarities. And then we will look at the examples that are closer to what I do personally in my research, which involve dynamic environments, behaviors of systems over time, and I'll give you four examples if we have time. So one is an optimal consumption saving model, which is a canonical model of optimization by an individual. Then I will give you examples of two phenomena that emerge in these kinds of systems, sunspot fluctuations and bubbles. These are terms that economists use. And then I'll point to something that is very much an illustration of this phenomena that is taking place, you know, in the place that has been playing a central role in monetary policy since the financial crisis, and the way of implementing monetary policy by central banks. So this is what is known as forward guidance policies, which is an example of the unconventional monetary policies that central banks have been forced to adopt for reasons that I will explain. And I'll conclude with some remarks and lessons. So here, let me start with this model that I call a model of strategic complementarities. So think of an economy that has a very large number of individuals that maybe can be represented by the unit interval, an infinite number of individuals, all of whom are identical. This will obviously simplify the analysis. They are exactly identical. And let's assume this will be very abstract, okay, but all of you will be able to think of maybe real-world examples in which this may be relevant. Think of the individual's problem. Each person makes decisions individually, in a way non-cooperatively, in a decentralized way. And the individual's problem consists of maximizing this function U, you can think of, economists call it utility, any function that gives the payoffs or the rewards that that individual will have for the decision that he or she will make. And that utility depends on a variable that that individual chooses, X, that's a scalar, the average value of that variable chosen by other individuals in the economy, and Z, which is just an exogenous variable, a set of exogenous variables, okay? And these are the assumptions that we make on some of the properties of these functions, which are important, okay? So this derivative with respect to the first argument, I don't know, this is notation that we often use, I guess, itself, explanatory. This would be the second derivative. That guarantees that this will have a maximum. This is an important assumption. This means that an increase, let's call X, the level of activity, an increase in the average level of activity by others increases my rewards. It's something good for me. That's why we call it, that's the assumption of strategic complementarities. There are positive spillovers from other actions into my rewards. And this is also a critical one, the cross-derivative is positive. This means that an increase in the level of activity in others level of activity increases the effect, the marginal reward from my own level of activity. So at the margin, it increases my incentive to increase my level of activity. Maybe because other people's efforts make me more productive myself, at the margin. So think of this one possibility, of decrease in the level of activity that comes. See, I think of this, and I'm just going to assume that each individual takes X bar, the average, has given, which is a reasonable assumption. So whatever decisions you make do not affect the average in the economy. And Z by assumption is taken as given Un researcher, com a molts de vosaltres, que cares de la qualitat de la seva research, de visibilitat, impacte, i tal. I això, de clar, depèn molt de l'esforç de l'esforç. Això seria x. Però també és un factor de l'environment en què ets, la qualitat de la research en el departament que ets, i tal. Així, directament o indirectament, depèn també de l'esforç que els meus col·legs han de posar a la seva research. Així, hi ha aquests, com a economists, que s'ho poden dir, una externaitat positiva associada amb l'esforç de l'esforç. Si el comportament no és sèlfic i un vol dir maximar les societats, és possible que... Sí, sí, parlem d'això, que serà una referència. Però és el cas que estem interessats, perquè volem que... Aquesta és una descripció positiva de l'esforç, quan volem pensar en individus comportant-se així. I veiem si això va a les ineficències o no. La condició d'optimalitat, per aquest problema, és straight-forward. Té la derivativa de respecte a x, equate-te a zero, això és molt comportat. Hi haurà una sol solució a aquesta qüestió i podem escriure-ho així. L'optimal x és una funció de x bar i z. En particular, és una funció d'exparcció d'exparcció. Aquesta seria una foto que podrà descriure un possible environment. Aquí hi ha restrictes x a ser a l'esforç entre zero i un. Això no és molt important. La red línia... Em dic que és la funció de reacció. Això és representat per la red línia. L'individu... Aquesta red línia ens dona l'optimal qüestió d'individu, given the average choice of the other members of society or the group. And of course everyone is identical here, so everyone will make the same decision and it will have to be the case that in equilibrium that decision corresponds also to the average decision of the group. So the solution will... The equilibrium will correspond to the point in which this line, the reaction function, intersects the 45-degree line. And so we will have... This will be the equilibrium. So this is what economists call the Nash equilibrium. In which if every individual behaves optimally, taking as given the behavior of other individuals. And here in this simple case everyone is identical so we can look at what is known as the symmetric Nash equilibrium in which the individual decisions has to correspond to the average decision. Very good. But this is not the only possible environment. We can have a situation like this. And we can have a situation like this. And here we see that there are three possible outcomes in this example. And the three are consistent with this notion of a Nash equilibrium. That means that if individual, an individual, I'm an individual. If I believe that on average my colleagues will choose this particular value, XL, it is optimal for me to choose XL. And the same for XM and XH. Think of this as low, medium and high levels of activity. And the three are equilibria. Equilibria in the sense that no one has an incentive to deviate from that outcome given what others do. Now, where could this situation emerge? I mean, this could be the world, the representation of the world as it is. But also we can think of a situation like this in which there is a single equilibrium. But there is a shock to Z, the exogenous variable, that brings down this reaction function. And then we go from a situation in which there is a single equilibrium to one in which there are three equilibria. And this has important implications for our ability to predict how the economy or this system will respond to a shock. Because if the shock was small, I think I have a picture like this, if the shock was small and the reaction function shifted a little, no, we could use local methods to predict the response of the economy to that shock. But if the shock is large, then we're in a situation like this. And we don't know, there's no way to tell exactly which of the three possible outcomes will emerge. The three are consistent with equilibrium. So formally, we have this. Remember, this was the optimality condition that has to be satisfied for each individual. The symmetric Nash equilibria are defined as solutions to this, such that x equals 2x bar. And let me use a star, an asterisk, to refer to the fact that they are equilibria. So any solutions to this equation will qualify as Nash equilibria. I, as we have seen by the simple diagram, there is room for multiple solutions. But here is something that is more interesting. Those solutions can be welfare ranked. In particular, one can show that, in this example that I've given you, that if one takes the equilibrium with a high level of activity, that is, if x, h is larger than xl, which is the case in the simple example that I gave you earlier, then it must be the case that utility in this equilibrium with a high level of activity is larger than utility in the Nash equilibrium with a low level of activity. And this can be proved easily. I'll leave this as an exercise if you want to think about it. You just have to, given the properties of the utility function that I've given you, you just have the simple way to prove is to show that this point here is more desirable than this point here. This point here is the one that has, in which the average is xh, but your decision is xl, okay? And this is true by what we call revealed preferences, because these individuals could always have chosen this point but has preferred to choose this point. So this must be more desirable than this point that I'm pointing here. And in addition, this point here is more desirable than this, and the reason is that that utility function is increasing in x bar, okay? The derivative with respect to the second argument was positive, the positive spillover assumption, okay? So that's this. So what happens if the economy gets stuck here? Well, we have a case in what we call a coordination failure, okay? There are two alternative outcomes that are more desirable from a social point of view, but somehow no one has an incentive, unless there's some external force to switch to the other outcomes, even though those could be sustainable. It's different, I'm sure that many of you may be familiar with the prisoner's dilemma, okay? In the prisoner's dilemma, the outcome is inefficient, it's undesirable for both prisoners, okay? But there's no other outcome, there's no other Nash equilibrium, okay? Whereas here we have a situation that I think is more interesting, there are other outcomes that could potentially be sustainable as Nash equilibrium, okay? Very good. Let me move on. I mean there are some caveats to this. Here's a very simple example in which everyone is identical and so on. I'm assuming that people can form exact beliefs about what others do. There's no uncertainty about what x bar is when you choose your x, okay? Now there are other information structures, and here is a reference, Maurice and Shin, at the end of the talk I'll list some references in case you're interested, which shows that under more general assumptions, depending on how the information structure is, if there is imperfect information and asymmetric information, this multiplicity may go away, okay? But again, here I wanted to show the result that I think is interesting, which is the one in which the multiplicity exists. And here, let's go back to the cooperative, what we could think of as the cooperative equilibrium. The way economists think about the cooperative equilibrium is, well, we think of a benevolent dictator. The benevolent dictator chooses, makes decisions for individuals and gives orders as to what they have to do, but in a benevolent way, that is, in order to maximize the welfare of society, not in order to maximize his own welfare, okay? So in that sense it's a benevolent dictator, or we also call it the social planner, okay? So suppose that social planner wants to maximize the average utility of all these individuals, so that would be the average utility. This is individual i, so we're integrating over i, but now the social planner recognizes something that the individual doesn't recognize, that the average x is related, there is a link between each individual's level of activity and the average x, whereas the individual takes the average x as given, okay? So as a result of that, if you take the derivative and you impose the symmetric... well, if you derive the optimality condition for this, which one can show that involves a symmetry in the sense that everyone will do the same, you get this, an x e now refers to the efficient level, socially efficient level of effort or activity, okay? Has to satisfy this condition, and now contrast it with the Nash equilibrium, okay? Which look like this, so one thing that we see immediately from this comparison is that the Nash equilibrium will be inefficient necessarily, okay? Because as long as U2 is positive, okay? As long as there are these spillovers, okay? Actually, it could also be negative, if the spillovers were negative, we could also would have an inefficiency, but as long as this is different from zero, as long as there are spillovers, okay? The x star, the Nash equilibrium, will be different from xe, okay? So that's one result, and not only that, but one can show that x star, that is the level of effort or level of activity in the Nash equilibrium is always less in this example than in the efficient outcome. That is, it would be optimal, it would be optimal if somehow some external force pushed everyone in the society to work harder to make more effort. That would be desirable for everyone, at least at the margin, not necessarily, you don't necessarily have to go all the way, say to x equal one in my example or not, okay? Very good, so this is a first example that I wanted to show you that illustrates the importance of expectations or beliefs. In this case, it's an aesthetic environment and the expectations or beliefs are with respect to what others will do, not expectations about the future, okay? So those expectations, I mean, to the extent that someone could influence those expectations, a government or some external force, that could actually help the economy, you know, coordinate on the most desirable, sustainable equilibrium, assuming that it's not possible to have the socially efficient one because we want people to decide freely what they want to do, okay, individually, okay? Very good. Now let me turn to examples that involve, you know, decisions over time, you know, involve intertemporal decisions, okay? And there are, for example, let's see how far can I go. So this is a standard example in dynamic economic theory, okay? It's like the canonical model, so let me show it to you without getting into the details, but at least we'll give you an idea of what is the, you know, how economies formulate, in the simplest possible case, the problem of the intertemporal problem of an individual, okay? So think of an economy, an individual, sorry, that wants to maximize this objective function. So think of you as the utility, you know, the reward, the degree of happiness or whatever, which depends on the level of consumption of that individual. Think of an economy in which there is a single good, okay? And, but of course, the individual lives for many periods, okay? So total happiness depends on that is assumed to depend on the expected discounted sum of utilities at different periods, okay? Beta is a discount factor between 0 and 1, that means that, you know, I put more weight on what I consume today than on what I will consume 10 years from now, okay? That seems reasonable. And the utility function satisfies these standard properties, it's increasing in the level of consumption, the more consumption the better I am, but it's concave, it means that the marginal utility of consumption is decreasing, okay? After, you know, eating 10 cakes, the 11th cake, well, could actually, may actually violate this assumption in the case of the, at the example I just gave. And this is subject to some constraint, a budget constraint or a sequence of budget constraints because we have one for every period that looks like this. So let me start from, well, A is financial wealth in period T plus 1 at the beginning of period T plus 1, okay? What is financial wealth at the beginning of period T plus 1? It's equal to financial wealth at the beginning of period T plus whatever income that individual gets, in period T, minus taxes, okay? Think of taxes as lump sum taxes, not proportional to income, just for simplicity, minus consumption, okay? And all that is invested by the individual is whatever financial wealth there is after income, taxes and consumption, it's invested at an interest rate r. So the financial wealth in period T plus 1 will be given by this, okay? Now, I want proof here, well, but maybe if you move this to the left-hand side, you can think of this, you can, this is a difference equation that you can iterate forward and you take expectations and you derive this single, okay, you can collapse this sequence of period budget constraints into a single budget constraint that we call intertemporal budget constraint that looks like this. And that can be expressed in words in a simple way. It says that the discounted sum of consumption, of current and future consumption, has to be equal to today's wealth, okay, suppose that we're in period T now, okay? Plus the discounted sum of after tax income, okay? So that's the constraint that effectively, that summarizes all these constraints, okay? Very good. Let's look at the optimal behavior of this individual. So imagine that this is a simple heuristic argument. Think of this individual following an optimal plan, okay? So if the individual follows an optimal plan, one that maximizes this objective function, any deviation, any small deviation from the optimal plan cannot lead to an increase in that objective function. So think of a deviation in which the individual consumes one unit less of consumption in period T, invests those savings, and with the money that that individual gets in the following period, it increases his consumption, okay? And it keeps consumption in all other periods unchanged, okay? So I claim that, well, this is the loss of consumption that individual would experience in period T, but that will be compensated by the additional consumption in period T plus one, which will be equal to one plus R, okay? Times the discounted, properly discounted, marginal utility of that consumption, okay? And of course, marginal utility of that consumption is not known as of time T. So what this means is this is the expectation of this marginal utility conditional on all the information available at time T. But the individual can form an expectation of that marginal utility, okay? Now, just to simplify a bit the algebra, suppose that the product of these two equals one, okay? We get this very simple, oh, and assume that utility is quadratic, okay? So this is linear, this would be linear in C, it would be decreasing and linear in C. Then this optimality condition, okay? Can be written like this, okay? Which is, it's very intuitive. It says that at any point in time, the consumer will choose a level of consumption such that that level of consumption is expected to remain unchanged over time. So the consumer wants to smooth consumption as much as possible over time, okay? That's because the concavity of the utility function penalizes fluctuations in consumption. There will be fluctuations because there will be shocks, income will increase, will decrease, in a way that is completely unexpected, and the consumer will have to adjust his consumption every period, so that this is satisfied. Now, recursively you can see that this will also be true, okay? Now, you can plug this into this intertemporal budget constraint to replace these expectations, do some simple algebra, and we get what we call a consumption function. Consumption function tells us the optimal level of consumption as a function of all this stuff. Okay? And what is this? This is today's financial wealth, okay? This is a constant, will be a beta, typically it will be close to 1, so this will be small, but this is financial wealth today, and this is the expected discounted, the discounted sum of expected future tax, sorry, future after tax income. Okay? And we can see very clearly and very explicitly in this case that we have a forward-looking decision. In other words, a decision, which depends very explicitly on expectations about the evolution of some variables in the future. In this case, expectations about future income. Okay? And this should be intuitive, you know, think of yourself, if you know that you will get a raise next year, okay? And if you want to smooth consumption over time, you will start consuming, if you have access to financial markets at the assumption here that that person can borrow or lend at an interest rate R, you will start consuming more today, okay? And there are many other examples that one can think of in which expectations, sorry, expectations play a very important role in decisions, even, you know, all kinds of decisions, especially in environments, not necessarily in environments that, well, this is intertemporal by nature, this consumption, optimal consumption and savings plan, but also in environments in which it's not so obvious for instance, think of a firm that sets prices, okay? And for the good it produces. Now, suppose that there is a small cost of adjusting the price. Well, that price, and we see that all the time, no? That price will not be changed continuously. The price remains unchanged for a number of periods and then, you know, there is a discreet change and then you go to the store and you say, oh, look, they have raised the price of this good that I was buying all the time, no? Think of newspapers or things that, you know, their price may remain unchanged for a very long number of periods. When the firm sets the price, necessarily, the decision, the price-setting decision, has to be a forward-looking decision because it recognizes that it won't have, it will not adjust the price in the immediate future and hence it has to take into account cost factors, demand factors and so on that will be relevant in the future, not only in this period, okay? Very good. So, now let's look at some implication of this. This is a paper by Robert Lucas, Nobel Laureate, who had an amazing impact in economics. It's an old paper in 1976. But let me give you the basic idea. I think it's really powerful not to see the importance of expectations and modeling expectations right. So, this is the consumption function that we just derived, okay? Now, imagine, again, just for simplicity, that income follows an AR1 process, okay? And the same four taxes, follow these simple autoregressive processes, okay? And, well, then, you know, these expectations, we can write as a function of today's income and the expectation of future taxes, we can write it as a function of today's taxes. So, doing some straightforward algebra, we get this specification for the consumption function, a linear function of today's financial wealth, of today's income, and today's taxes. And now, let me just give some arbitrary values, but not crazy values, to some of the parameters. In particular, suppose that beta is 0.95, the discount factor, the persistence of income is 0.5, and the persistence of taxes is 1, so taxes follow a random walk, okay? And that has been, that has characterized the behavior of the economy over the recent past, okay? In which case, if you plug these numbers here, you get this consumption function. This is what we would call an empirical consumption function, okay? Very good. And now suppose that the government wants to evaluate the impact of a tax cut, okay? They are considering to increase taxes or to lower taxes, okay? A change in taxes. But for whatever reason, that this tax cut has to be temporary. It will be just a one-period tax cut, okay? Okay, so what would be the traditional approach before Lucas, okay? Would be the following, okay? We have estimated, no, we have gathered data and so on, and we have used standard statistical methods. We have estimated this equation, which is the one that characterizes the economy before this intervention. Okay, so I look at this equation and I see, look, if I increase taxes by one unit, by, say, one euro, consumption will be reduced by one euro, and that's my evaluation of the impact that the change in taxes will have. Okay, now, but this is incorrect, and that's the Lucas' critique, because the correct approach recognizes that there is a regime change. Now we're going to, we're considering a policy intervention which involves a temporary adjustment in taxes. Whereas historically, okay, we had changes in taxes that were permanent because taxes followed the random walk, and now it will be purely temporary. So the right way to approach the problem is to recognize this change in regime. Now, row tau, this parameter, is effectively zero because the change will be purely temporary, and this is the right consumption function to apply. So now you can see that the implications are very different. Okay, this temporary adjust change in taxes will have a tiny effect on consumption, okay? So the general lesson from this is that, you know, there is, one has to distinguish, clearly distinguish between what we could call reduced form coefficients, okay, the ones that we would estimate, say if we were to obtain, if we were to estimate an equation like this, using data, and structural parameters. The structural parameters are parameters that are invariant to policy. Okay, now some of these coefficients, for instance, this one here, okay, depends on the characteristics of policy, in particular of this row tau. Okay, if we change row tau, this coefficient will change, in this case, it goes from 1 to 0.05. Which means that there is no substitute for theory if we want to evaluate the consequences of a change in regime. Okay, not the consequences of a single policy intervention. Now, okay, I'm going to raise taxes by 3 instead of by 4. No, a change in regime, okay? We cannot just use past data, okay? So that would be an example. Let me give you another one. This is what sunspot fluctuations. What are sunspot fluctuations? This is just, I guess, a stupid name that we economists give to fluctuations that are driven, fluctuations in the economy, that are driven purely by self-fulfilling revisions in expectations. Okay? Changes in people's mood. Okay? So think of the following. Suppose that, again, this is a very simple example, just to illustrate the mathematical structure of these sunspot fluctuations. Think, imagine that the investment of, by form i, XTI, depends on the expectations that this firm has about aggregate demand, about how well the economy will be doing in the following period, okay? Because, you know, if you need to have enough capital to meet demand, and if the economy is going to improve, you need to invest today, in period T, in order to have the capital that will be necessary to meet demand, okay? And now suppose that output, so think of this as aggregate output in the economy. Aggregate output in the economy is the sum of two components, investment itself, no investment of the firm, it's a source of demand for output, okay? Output for demand for capital goods, plus some other term, okay? And think of this as all being deviations from steady state. And let's just, for simplicity, assume that this term is unpredictable and expectation as of time T is zero, okay? Now plug this in here, okay? Combine the two equations and we get this equation that describes the evolution of investment over time. So any solution to this difference equation, say that is not explosive, let's rule out explosive solutions because those would violate some, eventually would violate some constraints that I haven't made explicitly here, any solution to this difference equation would be consistent with equilibrium. Now there is an obvious solution to this, which is zero, x0 at all times, okay? So suppose that x0 is x0 at all times, again these deviations from steady state, right? That doesn't mean that investment has to be zero. And then if x is zero, y would be equal to z over time, so there will be small fluctuations in output, completely unpredictable, okay? So the output will be like, I guess, this martingale difference process, that's what this thing is called, no? And that's it. But the question is, are there other solutions? Or N? Okay, so let's see. So here I have rewritten this equation, but changing the timing, okay? To make it easy. Now suppose that alpha is greater than one, okay? Here we don't know where alpha comes from because I haven't derived this from first principles, but let's assume that alpha equals, is greater than one. So one can check that this, okay, is also a non-explosive solution to this difference equation as long as this c, t, is completely unpredictable. It's a martingale difference process. It's completely unpredictable, okay? And you can check it, the simplest way to check it is, take expectations, conditional on information at time t minus one on both sides of this equation, and you will see that this is satisfied, okay? So in this case, we can have persistence fluctuations, okay? So interpret this c, t, this is what we call a sunspot shock. It could be anything. It could be anything. It just has to coordinate expectations. As long as, that's why it's called a sunspot. A sunspot is something that in principle should not have any effect on the economy, but it may coordinate expectations. People, everyone looks at a signal and says, look, this means that the economy will be improving tomorrow, so let's all invest and so on, okay? So, and we can have actually persistence because alpha could be close to one. So this would be non-explosive fluctuations, but highly persistent, okay? Very good. Now, these sunspot fluctuations are generally, again, this was not a full-fledged model, but are generally undesirable because they are not warranted. It's not that they are not the results of a consequence of fundamental shocks to the economy, say technology shocks or other shocks that could warrant some adjustment in the level of investment, in the level of output, or changes in preferences of individuals. No, they are just the result of self-fulfilling, revisions of expectations that become self-fulfilling. And hence, they are, you know, they will have negative effects on welfare for society. So, let me give you, and in some work that I did with Rich Clark and Mark Gertler some time ago, we estimated a model for the U.S. economy and showed that, well, in that model, okay, whether the possibility of those sunspot fluctuations existed or not depended on the rule that the central bank was following, okay, and some coefficients in a rule that the central bank was following, was assumed to follow when setting the interest rate. That's the way we usually describe the behavior of central banks, not as following rules for the setting of the interest rate. Okay, so it turns out that we estimated that rule for different periods and for the 60s and 70s, we showed that the estimated coefficients of that rule imply, when embedded in the model, that sunspot fluctuations may emerge. And that provides an alternative interpretation to the macroeconomic instability of the late 60s and 70s in the U.S., that was a period in which there was a lot of instability in output and inflation and so on, okay? So, it's an interpretation that is different from the traditional one that typically focuses on shocks to the price of oil, which, obviously, are fundamental shocks that, by themselves, could explain some of those fluctuations, but this is one that is based on people becoming optimistic or pessimistic and those expectations becoming self-fulfilling. Let me give you another example. I have time for another example, do you think? Sure. Okay, so bubbles, okay? I'm sure that you all hear about bubbles all the time, bubble in the housing market, bubbles seem to be everywhere and in the press, in the media, typically, well, it's a free world, so people can use those terms whichever way they want, but here I will show you the way we use the term bubble in asset price theory in economics, okay? So, think of two assets, okay? So, there's a bond, already positive, if you want, that is riskless, no risk, and it matures after one period, if you invest one euro today and you get the following period, you get one plus our euros, our is positive, okay? And then another asset, which is risky, let's assume a stock, a firm, okay? It has a price QT and it pays a stream of dividends, which is stochastic, okay? So, very good. Now, here is a simple arbitrage condition that has to be satisfied under some assumptions, risk notrality and so on, let me not go into that, but it will make sense, okay? This, to you, I hope. So, that individual can, suppose that individual can invest, thinking of investing is considering of buying one share in that firm, okay? So, that individual buys that share, okay? What's the expected payoff, the following period? Well, it's the expected dividend that the share will pay, the following period, plus the price of the share, because that individual can resell the share, okay? So, that's the expected payoff. But the alternative use is to put the money, the same money in a deposit or to purchase a government bond, which is riskless, and then what will you get in the following period? You will get QT, the investment you're making today, times 1 plus R, okay? Very good. Now, any solution, this is a different equation, again, it involves expectations, okay? So, any solution to this different equation that is remains bounded, okay? Because, you know, Q became infinite, Q cannot be negative, because the negative price is not possible, and Q cannot be infinite because it would take over all the resources in the world, so that, so any solution that remains bounded, okay? Or that it grows at less than the growth of the economy as a whole, is a legitimate equilibrium value for the price of the stock, QT. So, let me give you a solution to this different equation. And this is the one, I'm sure that some of you may have taken some introductory finance courses at some point, it's the solution that is taught in those courses, okay? It's a fundamental solution. So, QT equals what we, I'm going to call the fundamental value of the stock, okay? In the words, the fundamental value of the stock, it's defined as this, the present discounted value of expected dividends, okay? And you can check, okay? I'm not going to prove this, but you could write this recursively, you can check that this, okay? This QF defined like this, effectively satisfies this different equation. So, it's a solution to this. It's a fundamental solution. But, is it the only solution? No. Let me define the bubble component, and this is what economists call the bubble, the bubble component of an asset, or the bubble component of the price of an asset. It's the difference between the observed price of the asset and the fundamental component defined like this, okay? So, are there any solutions to this different equation such that the bubble is different from zero? Well, it turns out that yes, okay? As long as the bubble satisfies this difference equation. So, take any solution, take the fundamental solution for the difference equation, add it QB that satisfies this, and you can check that it will always satisfy this. Equivalent-ly, we can write it like this, okay? So, as long... So, there will be a solution in which the price has a bubble component, as long as the bubble component grows in expectation at the same rate, at the interest rate, okay? I this has interesting implications, that's the implications that I explored in some recent papers, because it says that, look, the central banks, as you know, can influence the interest rates, okay? And, you know, many people have called for central banks to raise interest rates if they see a bubble emerging, okay? Just to calm markets or whatever, okay? What this is suggesting is that if the central bank increases the interest rate, when the bubble is emerging, at least on average, that is in expectation, it may increase the growth rate of that bubble, okay? At least if the bubble is a bubble, you know, it's what we call a rational asset price bubble, which is a bubble that satisfies this condition that is consistent with a rational expectations equilibrium, and so on, okay? So, at least it points, you know, that result points to a potential pitfall of these policies that, you know, they seem intuitive and so on, but, you know, once you analyze them, you see that there are some risks associated with them, okay? So, let me, can I talk about forward guidance, or shall I conclude? Five minutes. Five minutes, okay, so no more than five minutes. Forward guidance, just to give you a real-world example of today in which expectations are critical, okay? As you know, as a result of the financial crisis, you know, central banks have responded very aggressively in order to stimulate the economy, you know? And the way they do that normally is to lower interest rates, okay? But interest rates have a lower bound, a natural lower bound, nominal interest rates, which is zero, okay? Why? Well, who would want to extend the loan at a negative interest rate when you can keep that money in a safe box, not in your pocket in a safe box, at a zero interest rate, okay? So, that's a natural bound. It turns out that, you know, some central banks have been able to go a little beyond zero, because obviously there is a cost to keeping a lot of money in one's pocket or under the mattress, okay? Very good, but let's, for the sake of the argument, let's assume that this is it. Now, the economy, the aggregate demand and the way the economy is stimulated is not through this interest rate that the central bank chooses. What really matters are the interest rates that individuals and firms have access to. Let's call them long-term interest rates, which are interest rates on loans that go beyond one day, because the interest rate that the central bank chooses is an interest rate that it's just for overnight operations and so on, or one-week operations. And this long-term interest rate is nothing, okay, than an average of expected short-term interest rates, the ones that the central bank chooses, over the life of this long-term loan or long-term bond, okay? So, now you can see that even if the central bank has hit the zero lower bound, as all central banks in advanced economies have during the aftermath of the crisis, there is still some room for central banks to influence the economy, which is to affect expectations. They cannot lower the interest rate today, but they can try to influence expectations about what the interest rate that the central bank will set in the future. Okay, and here you have some statements by the FOMC, that's the committee and the Federal Reserve in the US that chooses interest rates. Okay, so things like this. The committee anticipates that weak economic conditions are likely to warrant exceptionally low levels of the federal funds rate, that's the interest rate that they choose, for some time. And then later on, when things get even worse, they actually make it contingent to developments in the economy, okay? The committee decided to keep the target range for the federal funds that are up, and that is very low, close to zero, okay? As long as the unemployment rate remains above six and a half percent and so on. So the central bank is trying to manage expectations, again, about future interest rates in order to stimulate the economy. So that's another example. So, just to conclude. Now, all the examples that I have shown you have two assumptions, rational expectations, okay? Expectations were not just arbitrary, okay? They were consistent with the equilibrium of the model. That's something that, to show, I would have had to go example by example and make it clear when I had made that assumption. But that's clear. And also, there's all the examples where there was a representative agent. Everyone was the same, and all the equilibrium were symmetric and so on. Now, things can get much more complicated, you know, if people are different and then you have to form expectations about what other people will do, and those people, other individuals are different, and they form their own expectations, so you have to form expectations about the expectations that other people will form, and so on. So it gets very complicated, okay? There are also interesting lessons for policy here. Policy takes interesting dimensions. One important dimension is that, well, the first thing that a good policy should do is to rule out this indeterminacy, because this indeterminacy is clearly not desirable in the economic world. Not this unnecessary unwarranted fluctuation serve no purpose. So any policy rule should be a rule that eliminates that. Also, it points to the role of expectations management. Governments can play a role and it can improve allocations, can improve the economy by managing expectations, by coordinating expectations and so on. And here, of course, the role of credibility is important. Governments may make promises, it's just so the Federal Reserve, the statements I showed you about what they will do in the future. But, you know, they may not be believed, so credibility is very important here, okay? And just as a final remark, I mean, it's clear that, you know, economic outcomes are the result of interactions of decisions made by a lot of individuals and those decisions on top depend on direct people, individual expectations about what others will do and so on. So it's extremely complex. So I think the need, I think the search for a quantitative model that would explain the world, the economic world, I think, personally, I think it's completely hopeless, okay? But because it's too complex, it's too complex. And expectations are an important source of that complexity. But I think there is room, and that's what most of us try to do, for, you know, simple models that emphasize specific mechanisms, okay, and completely ignoring other mechanisms, that can provide useful insights for policy-making, okay? And can point to mechanisms that maybe other people had not thought about possible policy responses that other people had not thought. But in any event, clearly, the expectations are an important source of complexity, okay? But as I have tried to show you, they are also a source of phenomena that you won't find in the physical sciences i during the physical world. And I think those phenomena are really fascinating. I'm fascinated by those phenomena, and I think it's what makes economics interesting, if anything. Okay? Thank you, thank you very much for your attention. Thank you very much, Josep, for your presentation. I think it has been very clear. Now it's time for questions. Angel is having a question. Okay, can you wait for a while? The microphone is coming. Yeah, thank you for this great talk. So as an engineer, I can help by wonder what would happen in a society made of robots, rather than humans. And whether then economics would become more like engineering. That's an extreme setting, but it is probably the case that decisions will be increasingly made by machines, more than humans. So I wonder what you can comment on the impact you think this will have on economics and whether people are thinking about that already on different tools or... That's a very interesting question. I haven't thought about it. Actually, when I was preparing this talk, I was trying to think of the motivation. I wanted to talk about expectations and the role of expectations and so on. And I came up with this idea of the dichotomy between the physical world and the social systems and made up of human beings. But then I kept thinking, how about animals? No? You know... Bands or bees? I don't... I'm not an expert on animals, I mean, other... Then economics is about human beings, no? But do animals form expectations or not? It's not obvious to me. So I think forming expectations about the future, forming expectations about what others are going to do and so on, I think requires a minimum level of intelligence. I think we should agree on that. So, well, if robots... you know, it's a matter of degree, no? If robots can eventually attain some level of intelligence, that would allow them to make forecasts, however primitive they are, about how other robots perhaps will behave. And that's in the interest of the robot, given the objective that it has been programmed for and so on, it should make those forecasts. Ok, so in that case we will have, you know, I guess, perhaps a primitive reproduction of what we see in human societies. But it's a question of degree, and again, I'm not an expert on cognitive science and I don't know anything about it. But to me, it's something that is very natural for human beings, no? If you think about it, many, if not all the decisions that we make involve some kind of expectations or beliefs about what will happen in... think developments in society as a whole, or what other people will be doing and so on, no? You know, to the extent that animals or robots behave like this, you may have similar phenomena. It's just tempting to think that you could have a society of robots where there would be no bubbles, for instance, because irrational behaviors that could be to bubbles are avoided. But then again, if robots behave like humans, then we're back to the same models. The bubble that I showed you... It's not irrational. It's not irrational, eh? I wanted to emphasize this. I wanted to dismute. So, you can think of also irrational bubbles because just people make crazy expectations about the dividends that a stock will generate and so on. No, that's what we would call overvaluation or undervaluation of stocks and so on. But this was purely irrational. Thank you. It's an interesting question and I guess we should start thinking about it. Well, there's a lot of synergies here. It's at the end of a... you know, around the corner, no? Apparently. Hello. Thank you very much for the very nice talk. I was wondering because in the model you have presented, your agents are supposed to do rational expectations, right? Yeah. But we know that humans are not only rational but also do some intuitive decisions in real-time that are very unrational and completely biased in thinking, for example, of this work from Daniel Kenman or this kind of thing. Do you know some works in economies that try to model this interplay between regional behaviour and real-time intuitive behaviour? Yes, definitely. That's a good point. Here, as I said at the end, I have restricted myself to models with rational expectations because economists, in particular macro-economists, like myself, we think of that as... Not that we believe that people form expectations rationally or because in order to form expectations rationally, you really need to understand each individual, you should understand the structure of the economy and make expectations in a way that is consistent with the outcomes of an economy that has the structure that it has. So, obviously, that cannot happen in the real world, but it's a natural benchmark, okay? It's a natural benchmark. In the sense that, once you relax that assumption, well, it's wilderness, no? You know, there are many ways in which expectations can be non-rational, okay? But there is plenty of work and it's not something that I have personally worked on, but in macro-economics, many of my colleagues are working these days on models in which there are deviations from rational expectations and they involve all kinds of rational in it, you know? There are different ways of modeling. This one is to assume some costs of processing information, in which case it doesn't pay for individuals to be continuously trying to understand fully the world around them because it would be too costly. There are also models in which people learn using over time, so these are called learning models, by using some statistical, simple statistical techniques based on accumulated data, okay? And that gradually allows them to learn, to approximate, but never to fully learn the structure of the economy. There are many examples of those. I didn't have time to... I assume that if you had this, then you cannot solve this problem analytically. So maybe you need some multi-agent kind of simulation. No, no, there are ways, again, there are many examples in the literature, some of them are very simple and at hock, no? And in some cases, you can solve analytically for the equilibria of those economies, in others they involve numerical methods, but that's true even for... that's also true in models with rational expectations, not in the very simple models that I showed you today, but more complex models that are non-linear and so on, you need to use numerical methods to solve for equilibria. Of course, having, you know, sometimes, depending on the assumption and expectations, things may become more complicated or not. Okay, thank you. Yeah, really, kind of in the same direction, this aspect of rationality, the don't irrational decisions kind of in real life pretty much drown out all the rational aspects of economic theory. So, to what degree can the models still apply to real life? When... My intuition would be that irrational decisions just completely dominate what's actually going on. Well, if they complete... that's a very pessimistic view of the world. If, you know, if everyone acts in a completely random way and completely unpredictable way with no way that it's close to being... I mean, there's no patterns, no objectives. I don't think we can ever hope to explain anything involving the outcomes of those decisions, I mean, almost by definition. The way that, I guess, personally, and I think most of economists interpret our models with rational agences just as an approximation. We hope that, on average, somehow, average across individuals, average over time and so on, people will not deviate much from that, because it would be costly. Now, of course, if you tell me that people are purposeless, they have no objectives in life and so on, then nothing is costly. There won't be any losses from deviating from rational behavior. So then, you know, I'm completely at the loss, you know. I'd better find another job. OK. So, but one of the points that I wanted to emphasize today is even in a world with rational individuals, there can be crazy things that can happen. So imagine in a world in which individuals are crazy. First of all, thank you very much for those insights. And coming back to the machines, you talk a lot about governated systems. But if we start to think about these novel technologies that, such as the blockchain, that the credibility is based on the abilities of the machine in being beaten, for example, this week, for example, a machine was like the Bencore, the release of the Bencore, that's a kind of central bank for changing currents, like corrupted the Ethereum blockchain that could not support the amount of transaction, then the price declined. But as people still expect that those technologies are important, even if you represent a small piece of the real world market governate by big heads, how about these ungovernated systems? What do we think and how will these rules apply to those systems that are really based on credibility and trust on the technology? No, that's an interesting observation. So I don't have any thoughts about this. I think some of these, I don't know much about any of these systems, but I think these are systems that have a strong net work effect in which the potential usefulness for an individual depends very strongly on the extent to which other individuals are using them. And we see these net work effects in many other things, the software that we use, and I don't mean just to say the languages that we use to communicate and things like that. Beyond the intrinsic value that some of these technologies may have, there is also the extent to which they become adopted by other individuals. So here it's another example in which expectations, you may want to invest in learning some of these options if you think that you will be able to use them and that depends on the extent to which other people will use them. And so on, so it's related to this. But what you say, I think it's absolutely common sense, that some of these systems don't have clear rules, there's no government that is guaranteeing its workings and so on. So my guess is that, loosely speaking, this opens the room for a lot of indeterminacy in the sense that there is nothing that coordinates, because there may be many competitors in many of these systems. And again, in many cases it may be desirable to have just one or maybe two or three, because if there was a huge number of them, to the extent that there are these network effects, each of them would be very little useful, because for their users it would be hard to find out partners that use the same and so on. So this is one case, I mean, in the case of currency, I think it's one case, I mean, we don't have, now we have multiple currencies in the world, but in a given economy, say, in a given city, typically only one currency is used. Barcelona only euros are used. That's not completely true, because people can use bitcoins and actually nothing prevents us from having transactions in US dollars or in Swiss francs if we want. But there is an agreement that the unit of account in which you will set prices at the store and so on will be euros and that's what we see. And I think some coordination for these objects that have strong network effects is important and governments, governments can provide that coordination and the credibility, you know? So... And they may help eliminate this possibility of indeterminacy, okay? So... Appreciate the artist. Just adding something more. You're talking about currents. Once, some years ago, I used to think, ah, the world might converge to unified currents. But checking the reality nowadays, I see it already happened, because you see nowadays in Barcelona. You can use bitcoins, you can use the local currents. But in the future, I see that you might use in specific currents for each kind of thing, for example. You may use a current for buying organic food, you may use a current for having a guide, you may use a current. But why it's unified currents? Because fees are not high anymore and centralized in a bank. You're going to have now an easy platform to exchange. So it's a kind of uniform, a unique current that can be easily traded between them. Okay. No, that's a good point. Now, on this, the one thing that is important is... the currents in which prices are set. Okay? Typically, you study an introductory course to economics, you study the functions of money, you know? And there are many functions, you know? The means of payment, the way in which you can save some... you can allocate your savings, you know? But there's one function that is actually very important in... even though it's a bit abstract and seems strange, which is the unit of account, the unit in which prices are set. Okay? We don't set prices in terms of gold or we don't set prices in terms of ice cream. We don't set prices in terms of euros, say. Okay? Must here, no? Must prices, at least. Okay? So, to the extent that the technology, the price information about prices is such that, you know, it is costly to display prices in more than one current. Okay? So, the store around the corner, you know, the person who runs that store wants to set prices in just one currency. Okay? Before it was Pesetas, now it's euros. Okay? So, to the extent that this is the case, that currency, whatever we converge on, will play an important role. And here, it's critical whether the currency is the currency for which an institution that we call the central bank has the monopoly to issue that currency. Okay? Because in that case, the central bank will, through, you know, a process that would be hard, would take a while to explain, now can influence the interest rates in terms of that currency. And that will affect the ability of the central bank to have an impact on the economy, you know, in terms of, you know, stimulating the economy when needed or the opposite when the economy is growing too fast and there is inflation and so on. So, that's critical. And that's precisely the reason why I don't see, I cannot, even though we seem the world, no, we talk about globalization and, you know, different societies seem to be converging in many dimensions. I have, I don't have the slightest hope, I wouldn't say hope because it's not something that I would hope, but I wouldn't expect at all in the future, in the near future, to have a world currency, even though it would be very easy to implement, you know, a world currency. Why? Because that would imply that each economy, whether it's, you know, a national economy or something like the euro area or so on, gives up its ability to respond to developments that are specific to that economy. If all the economies in the world were identical, were subject to the same shocks and so on, then there would be no cost of having a single currency. But that's not the case, you know. Different economies are very different in structure, in the kind of shocks that they are subject to and so on. So it's very important for governments, if they want to be able to influence the economy, you know, to have a currency for which they have the monopoly of issuing that currency. And, you know, the introduction of these alternative currencies, you know, if they really spread, okay, and people become widely used and so on, may call into question that monopoly and may actually call into question the central bank's ability to have an impact on the economy. And I don't know if that's desirable or not, because, you know, who controls the supply of that currency? You know, I've heard about Bitcoin in particular, this complicated algorithm and so on, but that algorithm, as far as I know, doesn't respond to developments in the economy. It's purely mechanical, you know. There's no someone, there isn't someone behind who's thinking, okay, now it would be desirable to increase the supply or to reduce the supply and so on. So, from that point of view, it's not obvious that it's desirable. From other points of view, like ending the, what I call the blackmail of banks, you know, banks have all of us in society under blackmail, because, you know, in addition to doing their main function, which is to transfer resources from individuals or that want to save to those that need resources to invest and so on, and they could do a good job doing that. In addition to that, they really control the payment system, okay, and that's critical for the workings of society. So, that means that governments cannot let banks collapse or the banking system to collapse, because if the banking system collapses, not only, you know, the bank shareholders will be left with, you know, will become poor, will lose their, all their wealth, but, you know, society will collapse, because trade transactions will not be able to take place, and that's a disaster, in my view. And I think, from that point of view, it's great that there are these alternatives that, but, you know, the same thing can be done through a central bank. The central bank could be, could become the bank of everyone, okay? And there wouldn't be any risk of a bank run, of any, of any, there couldn't be a risk of insolvency and so on, because the central bank can always, will always be able to repay its debts, because it can print the money, okay? There's no, it can, the central bank cannot, it's the only entity that cannot become insolvent or illiquid, okay? So, everything could be done through central banks, but that's not done, you know, and we don't have accounts at the central bank, we have accounts at private banks, and hence, all the conflicts of interest between society and banks, in my view, that's a personal view on the role of banks. Okay, we have, we are now running out of time, but there is one last question, if it is short. Otherwise, we can keep discussing during the next. Very interesting to hear it here. So, it's a really, I'm a bit confused by the use of rational expectations, because in the discussion here, it was like it either be rational or undeterministic, but it can also be like people are bias, so they are still, they don't take the rational decision, but they are systematic bias, and I suppose this is studied also. Yeah, yeah, yeah, yeah. So, no, no, so this I want to clarify. So, in all the examples I've given, I was assuming rational expectations in what sense, in what sense, for instance, in this sense. You know, this individual is forming expectations about future, say, output, okay? Aggregate output in the economy. And in order to form those expectations, here I could have just assumed, as it was done in the past before the so-called rational expectations revolution, I could have assumed, look, expected output for next period equals today's output, okay? It was what we call adaptive expectations. That's not necessarily rational, because it doesn't use all the information, it just uses a simple rule, no? Well, this, yes, you can, these were the kinds of assumptions that were made in the past. It turns out that under those assumptions, it's harder to get this kind of multiplicity that they have emphasized here. But in what sense it's rational here, for instance? Well, because when forming expectations, this individual, individual I, when forming expectations about future output, this individual is assumed to understand the workings of the economy, and in fact knows that yt plus one, okay, will be equal to xt plus one, plus zt plus one, okay? And makes expectations in a way consistent to that. But you can, but... The... Even under, that was one of the points I wanted to convey, even under these rational expectations, that we agree that may not happen in the real world, even on those rational expectations, you can get this multiplicity of equilibria, this indeterminacy, and so on. With different kinds of expectations, and again, there are many different ways in which expectations can be non-rational, you get there are cases in which those multiplicity may go away, for instance, if expectations were always backward-looking, okay, then we would go back to a system like the one I showed that described, what I said describes the physical systems. It would be, you know, it would be like this, if they were backward-looking, but that would be obviously people who don't want to use all the information that is available to them in order to understand things that will affect the, that should affect their decisions, okay? So in that sense they will be irrational, in the sense that they don't make use of all the available information to them. But all kinds of phenomena, in some cases, when you have these non-... Depending on what the assumptions you make about the particular kinds, how expectations are formed, you can still get indeterminacy in multiplicity of equilibria and so on, or not, okay? The two possibilities may exist, okay? So it's not that necessarily that one form of expectations rational or not is associated with this multiplicity that they have emphasized. But the important thing is that when the expectations are rational, then this multiplicity is a very natural thing that can emerge, okay? Okay, thank you very much for the questions and thank you again, Jordi, for your brilliant presentation. Oh, thank you. Gràcies.