 Excellent. For this third presentation before going to the break, we have Fahime and Marianne. They're going to talk about growth incident curves, in allocating geometric brain and motion, and the supervisors were Alex, Jonathan, and Colin. So please go ahead. Yeah, thank you Isaac for the introduction. So this is project number four. The title of our project is growth incidence curve, in allocating geometric brain and motion. It was done by my team member Fahime, Nadiafi, and I, and it was supervised by Alex, Jonathan, and Colin. So here we would like to state the objectives of our work. The goal of this project is to study the anonymous growth incidence curve and the non-anonymous growth incidence curve, predicted by the GBM and the RGBM model. The GBM and the RGBM, they are models of evolution wealth, and the G and Nadiafi are used are measures of inequality over time. We would also like to compare these predictions to empirical evidence on growth incidence curve and study their properties. So before we get into the numerical results that we have obtained, we would like to give some basic information about the growth incidence curve, the definition, the intuition behind, and maybe some examples of how they look like. So the distributional changes of wealth are commonly represented by the growth incidence curve, the G and the Nadiafi. The anonymous growth incidence curve, they show the relative change in wealth in the same wealth quantile between the initial time and the final time. So here below, you can see the definition of the anonymous growth incidence curve, and in fact, it's not that difficult to understand. The intuition behind is pretty clear. So the F inverse of P at P, that's basically the wealth of the P quantile at time T, and as you can guess, the F inverse of P at T inverse, that will be the wealth of the P quantile at T prime, which is the end of the period. So unlike the geometric, the anonymous growth incidence curve, the non-anonymous growth incidence curve definition might be slightly confusing if you are looking a bit further first time. So the anonymous growth incidence curve that still shows the relative change in wealth between the time T and T prime of the people who were at once P at time T. So this part is important. So just to make of how the non-anonymous growth incidence curve works, we can look at this illustration here. So let's assume we have a small sample size consisting of 12 individuals, and we know their wealth at time T, which is the beginning of the time. So we can sort them, and we can group them into three groups, Q1, Q2, and Q3. Those are the quantiles in fact. And as you can see in each quantile, these people, they are labeled. So the first quantile where the people are labeled in blue, those are the foolish people in this small sample. The Q3, those are the riches for people in this sample. So now let's look for example at the Q2 where people are labeled in red. So because they are round and they are labeled, we can follow them over time, and we can identify them and get their wealth in the end of the time, which is T prime. As you can see the errors, we can identify these people clearly. And because we know them and we know their wealth, we can get the average of this wealth and compare that with the wealth average in the beginning of the time at T. So this is how the non-anonymous growth incidence curve works, and in fact it does the same thing with the other quantiles. So we can identify the people who are labeled in blue, we can identify the people who are labeled in green, and so on. But as you can assume, this was not our case. We didn't have population size of 12, and we didn't have three quantiles. In fact, we have 10,000 of them and the quantiles were hundreds. But this one was just a small illustration of how the non-anonymous growth incidence curve works. This one is an example from a literature that is mentioned below of how the growth incidence curve, the anonymous growth incidence curve, which is the blue one, and the black one, the non-anonymous growth incidence curve can look like based on real data. So this one was the data was from the United States and the time period was considered between 1980 and 1990. So now let's look at some properties of the growth incidence curve and also the non-anonymous growth incidence curve. The growth incidence curve, in general, are upward flow speeds when inequality is increasing. They ignore the identity of the individuals within quantiles. As we have seen from the very first definition of the dip, they basically take the wealth at the beginning of the time of the peak quantile and compare with the wealth of the peak quantile in the end of the time. So it does the same thing with the other quantiles. So the richest people in the beginning are compared with the richest people in the final period. The poorest people are compared with the poorest in the final period. And as you can guess, these people who were, for example, in the first quantile at the beginning of the time, they are not necessarily the same people in the end of the time. But the dip doesn't take that into account. It just compares the wealth of that quantile in the beginning of the time and the end of the time. So hence the comparison is anonymous. Well, unlike the anonymous growth incidence curve, the magic, it takes into account the wealth mobility and it takes into account that the people identity is considered. So in this case, the magic becomes more sensitive. At the same time, they are more informative of the individual experience of wealth changing. And also because people are around, people are labeled, as we have seen in that example with color, the comparison becomes non-anonymous because we can identify the people in the end of the time. So now that we have some basic understanding of the growth incidence curve that we want to study in this project, we would like to start with this simple model, geometric Brownian motion, that we look at the beginning of our project. So the GBM is a simple model for the evolution of individuals wealth. So let's assume the XIT, that's the wealth of the I person in the population of time T. So we say that the wealth follows geometric Brownian motion if it satisfies the stochastic differential equation number one. So in this GBM model, as you can see, we have two parameters, mu and sigma. Mu is the drift term and the sigma is the volatility parameter. And just by looking at this equation one, we can say that the fractional changes in wealth are composed from two components. The first one is the mu dt, which is the constant deterministic part. It corresponds to the common economic growth. And the sigma dw, that's the random part, which corresponds to the individual's wealth. And I would also like to mention that the dw, that's an infinitesimal increment of the Brownian motion or winner process. So this was the basic information about the geometric Brownian motion model that we have studied in the beginning. So here we have the GBM trajectory from our model. And everything is measured in years and the Y axis, that's the wealth, right? We said that the X is the wealth. On the right hand side, we can see the same thing basically, but in low scale. And as you can see, the trajectories are expanding. So in geometric Brownian motion, the wealth follows log normal distribution, which means we can get the one standard deviation envelope of the log wealth. And that's what you see there, the light blue or gray shaded area. So here, before in fact going into the numeric, we would like to rationalize the analytical digging GBM. Again, because we know that the wealth follows log normal distribution, we know the probability density function of that. We can get the cumulative distribution function. And if we invert that, we will get the contact. So once we have the contact, we can plug in that to the definition of the jig that we have seen in the beginning. And we will have this analytic explicit formula for the jig in GBM. And as you can see, if we have the new and the stigma, and we have taken these parameters from the papers of our supervisor, it has been estimated in real data. And if we specify the time, the beginning of the time and the end of the time, this will become a function of P, and we can float that. And also, I would like to say that a phi inverse of P, that's the standard inverse quantile. I'm sorry, the quantile of the standard normal. So once we plot, we will have this graph of the analytical jig in GBM. So I would like to observe that the jig is increasing. So let's keep this in our mind. And now let's look at the numerical results that we have obtained in the GBM. So here, we would like to present the anonymous growth incidence curve and the nonanonymous growth incidence curve between two points in time. So first of all, let's look at the blue line and the orange line. The blue line is still the analytical jig that we have to sustain in the previous slide. And the orange one is the jig in GBM that we have gotten from the simulation that we have done. So as you can see, the analytical jig and the numerical jig, they're pretty closely coincide, which means the theoretical and the empirical results are in line with each other. And in fact, we have a scheme that everyone starts with the same wealth, with 25, and then the wealth goes over time. So the population size is 10,000, and we have used the specified parameters here in sigma. In fact, there are a couple of things we can observe from this jig in GBM. For example, if you look at the quantile, 19, and 100, those are the richest people in the population. The rich is 10%. We can see that the curve there is pretty steep. And we can say that the rich people are getting richer with higher speeds, and we can do similar analysis for the other quantiles. Well, unlike the dig in GBM, the magic in GBM, as you can see, that's with crazy fluctuation. And that's simply because the magic is more sensitive. And also it's subject to the population size and also the quantiles that we have picked. So if we play with the number of the population size, and if we play with the number of quantiles, over the long period of time, we would expect the magic to be largely flat. And as you can see here, we have 600 quantiles, which means we have percentiles. And again, the y-axis, that's the relative change in wealth. And also, I would like to say that here, it doesn't really matter what two points in time we picked. Politically, we will have the same gig. So the shape will be the same. So here, again, because we have mentioned that in GBM, the wealth follows the normal distribution, which means we can get the moment. So we have to expect the mean and the variance. Here, we wanted to plot the simulated mean and the simulated variance against the theoretical mean and the theoretical variance. And this way, again, one more time, it confirms that the theoretical results and the empirical results that we have, they are in line and they coincide. And again, we have the same parameter that we have been using from the beginning of the time, taken from the paper support supervisor. So the second part will be introduced by Fahime. Thank you. Thank you, Maia. Okay, now that we have a good understanding of the geometric value and motion, maybe we can do some modification to this model to make it more realistic, probably. So this can be done, for example, for adding some kind of interaction between the individuals in our population. And again, this can be done with a reallocation mechanism. Based on the reallocation mechanism, we have each individual paying a fixed proportion of its wealth into a central pot. Basically, they contribute to society. And then each individual gets the unequal amount of that central pot, and it will be added to their wealth. So now that look at it mathematically, we have the same equation that they had for geometric Brownian motion with the same drift term and sigma and Brownian motion terms. But also we have two other new tips. The first term, which is minus tau x times dt, shows the contribution of each person, which is proportional to its wealth. And tau is actually the reallocation rate, which shows the magnitude of reallocation mechanism. And we also have the second term, which is tau times the average of the population's wealth times dt. And this term shows how each individual takes an equal share of that central pot from the reallocation. Now we can think about what those terms mean. For the case where tau is positive, the first term is negative, and the second term would be positive. But if the wealth of an individual is larger than the average wealth of the population, basically, they would in total give money to the society. But inversely, if someone has less than the average of wealth, in total, this term would be larger, and they would gain money from this mechanism. But for the case where tau is negative, basically the size would be the other way around. And that would mean that we are getting money from the rich and from the poor and giving to the rich. Again, as we have for GBM, we can plot the trajectories for a population sample. And the parameters that are used here are similar. And you also have this new parameter tau, which is positive in this case. You can see that visually these trajectories are quite different from the GBM case. So we don't have that expanding envelope of the wealth during time. Now we want to introduce another measure of wealth, which is basically the wealth divided by the average wealth of the population. This is the rescale wealth. This rescale wealth, I actually removed the effect of the general increase in the total wealth of the population. And as you can see in the trajectories graph, we don't see that upward trend. And that has been eliminated here. And considering that we have a large number of, and like population, population is large and considering the rescale wealth, the equation for the model would be equation four that you can see here. Okay, now for the positive tau case and large and approximation that we have talked about, a stationary focal plane equation can be solved. And by doing that, we can find a stationary distribution. So that means that the distribution of wealth in this model would get to a stationary point, which the PDF would be inverse gamma distribution with the Pareto tape. Here in this distribution, as I mentioned, why is rescale wealth? And we have CFO, which is