 Thanks, Michael. Thanks for coming. As you can see, this is joint work with Christophersson and Schmidt and Fintar. So when making comparisons of social welfare or inequality or poverty, it is increasingly recognized that focusing on monetary income or wealth alone faces to capture many important factors. And indeed, recent literature has frequently focused on multidimensional methods. For example, a number of studies have applied a waiting scheme or accounting approach to aggregate across multiple indicators of poverty and well-being. However, the problem is that such waiting schemes, if you take another waiting scheme, it may also alter the conclusions. So in response to this challenge, other contributions have focused on the development of robust methods for comparison of welfare or inequality or poverty. And the idea is that these methods, they allow for a broad class of underlying social welfare functions, and since their comparisons are robust in this way. And following similar work by Atkes and Bouben-Jong, there is now a fairly large literature along these lines, and the work I would present today is closely related, inspired by this work. However, there is an important difference. The work mentioned here, they all apply conditions implicitly or explicitly that relate to specified signs on the second or higher order across derivatives of the underlying social welfare function. What we do here is that we interest it in ordinal approaches. That is, we consider problems of making comparisons of social welfare or inequality between populations and situations where we only use the ordinal information available. And that means that we make no assumptions about the importance of each dimension. And in particular, we make no assumptions about sort of the complementarity or substitutability between the dimensions. So let me first address the issue of making ordinal multidimensional social welfare comparisons. Here there is actually a well-known criterion, a natural criterion for comparing population distributions. And that is population A is better off than population B is better off than population A. Whenever B's distribution can be obtained from A's by a finite number of shifts of population density from one outcome to another, that is better. So this concept is also known as the usual order, usually stochastic order or first order dominance, the natural multidimensional analog to the first order dominance concept. And it's an old concept, although perhaps surprisingly it hasn't, under reason, it's been used in applied welfare economics. Although the term first order dominance has been used with other meanings we should be aware of in the welfare economics literature. Let me illustrate this concept. Let's assume here we have two dimensions and four outcomes. So assume we have a health dimension, unhealthy-healthy, and an income dimension, poor-rich. So this is the best outcome, this is the worst outcome. And we have the intermediate outcomes here. We are not able to compare the intermediate outcomes. And let's consider a population and say that each baby face here is representing a 10% of the population. So we can assume we have a population distributed in this way. And when can we unambiguously say that we get an improvement? And that's precisely the cases here where we can move a population mass in this way, for example, to unambiguously better outcomes, right? So that's sort of first order dominance improvement, right? You can, the challenges in, if you have empirical distributions, how to check for multidimensional first order dominance. And there are different ways of doing this, some more computational demanding than others. But let me just mention that in R and L, this one method was operationalized and applied also to child deprivation in Mozambique and Vietnam. And there are also other very recent papers that came out recently that also use a similar method. So that was the ordinary comparisons of social welfare. However, what is the main focus here is the ordinary comparisons of inequality. So the question is, when is there more inequality within one group than another? For the one dimensional case, there has been some recent contributions along these lines. In particular, a central reference for our work is also Alison Foster, Journal of Health Economics, paper from 2004. And they suggest a simple and intuitive model for making comparisons of inequality with only categorical, ordinal data. And Alison and Foster themselves, they use an example of self-assessed health. So let me try to illustrate their ideas here in a very simplified way. So assume that we have self-assessed health and we have poor health, fair health, good health. And then again, each stickman here we can imagine represents 10% of the population, okay? And the question is, so population up or population down, comparing these two populations, which one has the most inequality? And the Alison and Foster's idea is that well, inequality somehow means that distribution is more on equal. If you can get it from the other by spreading population mass away from the center of the population. And the natural definition of the center of population with ordinal data is the median outcome. And here both for population up and down, we actually indeed have the same median outcome. And indeed here in this case, we actually have that population down here, which can be obtained by population up by moving some people or population shares from the median to more extreme outcomes. So in that sense, we have a natural definition in the one dimensional case of more unequal concept using only ordinal information. So the question we are addressing in this paper is how to extend this concept to multiple dimensions. And this is not so easy actually. So let me go back to the illustration with the baby faces again here because the application we use as a child deprivation indicators. Let's imagine again here we have two dimensions, we have a health dimension and I'll say income dimension. And we have a population distributed in this way. And so here the concept of the median can also be naturally generalized to multiple dimensions, that is the profile of the medians of the partial margins. So here we also have in a sense a natural center of the distribution here. And also you can, so and of course you can, you can then use the Alison Foster idea of making spritz relative to the center of population. Say for example, you could make a sprit like this, right? The median preserving spritz, which would increase inequality, right? However, using only median preserving spritz in the multidimensional case will not provide a useful concept. And the reason is that you will never be able to influence the outcome of the Southwest outcome here using only median preserving spritz to on a bigger, better on a bigger, worse outcome. However, there is another operation in the two dimensional case that also in a very intuitive sense increases inequality. And that is that if you move a population mass from down to up and a similar population mass from up to down in this way. In this way you will preserve the partial marginals. However, you will increase a correlation between the outcomes in the sense that if you now perform poorly in one dimension, it will increase the probability that you are poor in the other dimension, right? And this operation is known as a correlation increasing switching. So in a sense, we have two operations that intuitively increase inequality. And our main idea is that we combine these two operations and in this way provide a concept, multidimensional concept, or a bivariate concept of an ordinal inequality. So our general definition is we say that there's more inequality in population A than in population B. If A can be obtained from starting with B and then making some operations, some of these inequality increasing operations, okay? There is, however, a major challenge. And that is if you have two empirical distributions, A and B. How can you check if you can obtain one from the other? By some sequence of these operations, okay? That is generally actually a tricky mathematical problem. However, at least for this paper here, we'll only focus on the canonical two by two example here. And the two by two case is nice because we can provide a set of necessary and sufficient conditions for more unequal, okay? So that is essentially, it's a matter of testing a system of inequalities. It's equivalent to testing for this inequality. Okay, so let me give a numerical example with some of the data we have used and analyzed. So here, we look at the child poverty in Mozambique. And actually, this is the distribution of girls and boys over four possible outcomes. And actually it's only, the example here is only for the female-headed rural households, which are known to be vulnerable households. And we have two dimensions. We have a health deprivation. Know that no is good here, okay? And we have an important household indicator, a severe sanitation deprivation, yes and no. And again, know that no is the good outcome. And we take the distribution of the girls, that is the red numbers, and the distribution of the boys. These are the blue numbers over these four outcomes, okay? And again, the question is, can we say anything about which population has most inequality? And indeed, according to our concept, applied to this case, we actually get that there is more inequality within the group of boys than within the group of girls. And why is that the case? Yeah, the case, it is because you can obtain the distribution of boys by starting with the distribution of girls, then make a correlation increase switch, and some extra median preserving spreads, okay? That's example of the kind of relations and observations you can make from these data. So let me spend the rest of the time here to give you a hint of the results from an empirical application to child deprivation in Mozambique. So there we use, or we use a free binary well-being indicators or deprivation indicators. These are the so-called, some of the so-called Bristol indicators. So we make use of sanitation deprivation. And we look at the health deprivation only for those pre-school age children, and that essentially means whether these small children have had access to the basic vaccines and the most basic healthcare. It's a binary indicator, yes or no. And then for the school age children, we look at instead of education deprivation, meaning essentially, do they go to school or not. So, and what we do is that we look at different important characteristics. So we look at whether these kids here live in urban area of residence or in rural area of residence. We look at whether the household head is male or female. And we look at whether it's a boy or a girl, okay? So in this way, we actually obtain or make eight groups of children. And what we do is then that we compare all these groups of children to each other, okay? And we use data from the Mozambican Demographic and Health Survey, 2003. So from comparing all groups against each other, we obtain various interesting observations. Let me try to explain this figure. So rule male-girl, for example, this means this is one of the groups, right? This is the rule area of residence, male household head, and then all the girls, okay? And then if we have a one in the table, it means that when comparing to another group, we have a first-order dominance relation, multi-dimensional first-order dominance relation. If you have one of the letters A, B, and C, D, there are not so many, but there are some, we have a more unequal relation observed. And let me first mention, so at the upper part of the table, is we look at the school-age children. So it means that the two dimensions we have here in the data, this is sanitation deprivation and education deprivation. And the first, perhaps not so surprising observation, is that each and every group of urban children, first-order dominates each and every rural group of children, right? So it's a very sort of robust, strong indicator of the well-known issue that the children tend to be better off in urban than in rural areas. More surprisingly, perhaps, or at least to us, is that actually when we compare the different rural groups to each other, we get quite a few first-order dominances, meaning that there is a substantial amount of between group inequality for the different rural groups. And interestingly, if you compare the different urban groups to each other, we never observe the first-order dominances. So this is only a rule phenomenon somehow. Also, quite surprisingly, perhaps, well, interesting, at least, is that when looking at within group inequality, we actually get that in the urban area, we have a larger within group inequality among the households with a female head than with a male head. So it means that in the urban area, if there is a female household head, then it's not necessarily the case that the children are worse off or best off in the first-order dominances, but there's more inequality. So some are really good off, some are really bad off, comparison with the male household heads. And that's actually the case only, both if you look at the groups of boys and compared to each other and groups of girls compared to each other. So that's some of the observations that could come out of using a method of this project. So that was what I would like to mention about the empirical application here. Of course, there are many things we could look at in the future or try to improve. First of all, even though the concept here applies to the general bivariate case, however, we have only worked out an explicit testing procedure for the two-by-two case. So more work needs to be done in order to provide a general testing procedure that applies to general bivariate categorical data. That's more a mathematical challenge. Then of course, the data we use here, these are the demographic and health survey data from Mozambique and they're available for many countries and in many ways. So of course, there are much scope for other applications in other countries, for example, with similar data. And finally, should also mention that although we have suggested here a concept of ordinal inequality, it's also, of course, possible to look at other concepts, for example. Maybe we would like to be able to compare distributions that don't have an identical medium, for example. It's often in these two-by-two examples that we have here, it's very often the case in an application that the median outcome is the same. So it's not a big problem, but if you have many levels along the two dimensions, that tends to be a problem. So the concept will be yes, useful in general for multi-level data and therefore you may want to look for weaker concepts, very strict demanding tests for inequality we suggest and apply here. Finally, let me mention that the full updated paper is available on my webpage and you can also send me an email and I'll keep you updated on the latest version.