 It's a paper on stochastic dominance. You know that, since you are interested in eco-mixery, I guess, you know that there is a large literature on dominance analysis for implementing poverty and inequality comparisons, social welfare comparisons. And the idea is that we don't really know what are the right indicators for this concept, what is the right poverty index, what would be the right inequality index, but we would like to, we have some ideas about the property of these indicators and we would like to reach robust judgment for set of indicators on which we would agree, although we don't know exactly which one to use. These techniques are based on the fact that we are able to introduce assumptions on the norms which should be suggested to the definition of these poverty inequalities, social welfare indicators. There is a big literature on income distribution, which is quite old with papers in the beginning of the 19th century in the Math literature. And the kind of normative hypotheses which are used are various properties about aversion to inequality. Typically, one like to say, perhaps that the social welfare would be represented by the maximization of a sum of utilities in a utilitarian framework. And these utilities will be concave which represents some aversion to inequality by the social planner. In multi-dimensional settings, which is here, the topics of this paper, the literature is much more recent, but it's growing. It's less obvious how to obtain multi-powerful rules, powerful comparison rules. Why do I need, by multi-dimensional setting, I mean you want to do some welfare comparison, social welfare comparison between countries, between periods, before and after the policy, and you have several dimensions. S, for example, if you are interested in S, income, education, whatever you want to put, which would contribute to the social welfare. So it's less obvious to obtain powerful rules. That is the theorem that we have. Generally, when you take some actual data, are going to be such that you cannot conclude that one situation is more in a code or the social welfare in one situation is better than in another one. So what people do in this literature is that they impose the normative restrictions, such as this inequality aversion, by using signs of derivative of utilities. So inequality aversion would be the second derivative of the utility would be negative. When you have several attributes of welfare, you would use partial utility derivatives. And to make the comparison more powerful, it's good to have more conditions. And people have planned the degrees of conditions. So first-order derivative, second-order derivative, third-order derivative have been used. Fourth-order derivatives have not much been used and that's what we are going to do in this paper. Right, so what I'm doing in this paper is first I introduce a new way to incorporate normative restriction in welfare analysis. This is important because the result, the mathematics which come behind, will be just the translation of this hypothesis on your social welfare comparison problem. And I'm going to introduce the notion of welfare shock sharing. And you see how, by using this new notion, it's easy to characterize some sign-off of derivative of utilities. And that's going to help me to provide normative interpretation of first-order derivative of utilities. For example, here there is a utility function and you have the partial derivative where you derive three times with respect to the first attribute and once with respect to the second attribute. And you assume that this is negative. When it's written like that, it looks really mysterious. Why should you do this kind of assumption? But if you are able to justify this kind of assumption, you'll be able to use them to generate more powerful theorem of stochastic dominance. And that's what I'm going to do. I'm going to present new, necessary, insufficient condition for stochastic dominance for several classes of utilities and with their poverty-ordering equivalent characterization. So, we have really a short time so I'm going to jump over the notation. We have an utilitarian framework. We compare the social welfare with a joint distribution of two attributes, X and Y states income of X, for example, with another joint distribution problems with a, okay. Here, we would like to know if the situation with this joint distribution of income of F is better than this one and that would be the case if the difference of social welfare is positive. So, we just want to know if social welfare has improved when moving from F start to F. The problem is that we don't know the utility. Nobody has seen the utility around so we don't know this utility function. There are cardinal utilities in this setting but perhaps we can agree on some conditions. We can agree that the welfare of people, the well-being of people is going to increase with their income on with their X. We can agree that there would be some inequality aversion in income, inequality aversion in income, inequality to the correlation of income on F. These are typical conditions. There are also some third order conditions, partial utilities which are supposed to be positive which have been developed recently, for example, I use them in one just paper by justifying that it corresponds to some compensation that people would be happy to compensate for some bad S, some people. But this has been used in the literature. What we are going to try to add are these four degrees utilities. Four degrees partial derivative of utilities. So I need to make sense of that. That's what I think, right? Splot. Right. So to make sense of this partial derivative, I introduced the notion of welfare short sharing. And the idea is to try to put more flesh on the idea of solidarity of society against social short. So a simple way to do it is that you have two individuals. They are completely identical. They are the same on the whole month. And what should be the welfare effect, the social welfare effect, on some welfare shocks on this small society. So what are the welfare shocks? They can be fixed losses. They can be risk affecting, random risk affecting some attributes. They can be other things that I'm not going to develop in this paper. I'm going to simply focus on losses, some risk for the moment. And I'm going to provide application to social welfare function, which are additive in individual utilities, for example, when individual welfare is represented by expected utility. Let's take an example. Let's take an example. So we have two individuals with two attributes, X and Y. And they may suffer shocks. They may be fixed losses. C and D may be fixed losses. X and Y on delta may be center real random variable, which are undecondent together. And I said that a social planner is welfare correlation of S if the small society where initially two persons are the same on the month in common S. Now the small society where they share the shocks, you know you have two shocks, they will see the first person who see here in the first attribute and be the second attribute. This is preferred by the social planner to the society where the same person will suffer all the shocks. That is, sharing fixed losses affecting different attributes improve social welfare, okay? Right. So I can, like that, define all the notion. I would say that the social planner as welfare prudent is sharing a fixed loss here on the center rigs here, affecting the first attribute, say you can improve social welfare. I call that welfare prudent. I call the social planner welfare cross prudent if the random shocks affects a different attribute than the fixed loss, but still sharing the fixed loss on the center rigs affecting different attributes improve social welfare. This is what I call welfare prudent. I can define social planners which are welfare temporary when they think that sharing center risk affecting the same first attribute improve social welfare. You see this small society of two persons is preferred to this one because the same different person are going to suffer the shocks as compared to here the concentration on the shocks on the same individual. And finally, I say that the social planner is welfare cross prudent if sharing center risk affecting different attributes improve social welfare. So for all these notion, the intuition is really easy to convey because it's about we are sharing shocks it's grown in a society to shocks. And that could correspond to notions which are not so different in a risk analysis. Now it's possible to extend this by defining for example here, the notion of that a social welfare is said to be welfare premium correlation adverse. So it's a big word. What does it mean? It means that sharing fixed losses affecting different attributes improve social welfare while less so under background risk in the first act of it with complex to perceive at the first time. But if we look at it carefully it's just said that you prefer this society to this one. Here you have four individuals. They have two attributes. If you just look at these two individuals there they are going to share the shocks. So sharing shocks we have call it correlation aversion is good for society. Sharing shock is good. You prefer this small society of two persons to this one. But this one they do the opposite. They concentrate the shocks under a background risk of FCR. So why is it that globally you prefer this society to this one? Because the background risk here which you adopt as for example a welfare living room criteria which is the expected utility. This is going to be to smooth the curvature of your indirect utility and you'll be less sensitive to the same losses for people who have already a background risk than you. Okay, so first stage it's possible to define these different notions, normative notions. Second stage it's possible to translate that in a theorem into a derivative condition, partial derivative conditions. So inequality aversion is equivalent to this well-known condition that the second derivative in the utility is negative. And you can like that generate all the derivative conditions that you want up to the last one which looks really weird in the beginning. Now we understand that it corresponds to welfare primary correlation aversion. This one corresponds to welfare correspondence. So in fact that you are ready to, you prefer to share random shocks, okay? So that's something which is not sure when you say it like that, I prefer a society where people share random shocks. It can be translated by this partial derivative condition. Okay, I don't show the proof I don't have time to show all this proof. This is the last one which is, I would like to comment but I won't have time I guess to do the comment of this new. This is a new condition. This has never been used anywhere in the equilibrium line. So let's move to the result of stochastic dominance. Stochastic dominance takes, look at the function g. This is going to be the function u, the utility that we are interested in. I can define some classes of function by looking at, it's not very well written here but this is the partial derivative where you derive k one times with respect to the first argument and k two times with respect to the second argument. And you multiply by a sine which corresponds to the sine that you want to implement in economics in general for the derivative of utility. So this condition of sine on this function g is implemented for various values of k one and k two and these values of k one and k two are simply k one equals zero to s one and k two equals zero to s two. This function which satisfies this conditional code s one, s two increasing concave, okay. Now you can also define another class of function which is the s increasing directionally concave function with s natural number, non-tegre. And that is the same condition except that the value of k one and k two are slightly different. Why do I say that? Because I can take advantage of asymptotic expansion for these functions which will give these generators of this set of functions, this set of functions that are convex codes. So they can be completely summarized by the generators and I can find the generators for this. The sub-definition that you know well, this is the distribution function for the first argument, saying con. So this is the integral, the term that you use in second order stochastic dominance. We should do the integral a third time, let's say l. This is going to be the third order stochastic dominance, you need are yet the stochastic dominant. First order, et cetera. So usual notations. Now instead of taking the integral of f, you can take the integral of the John's distribution. I call it h. Or you can take this integral, okay, here integral is with respect to the two arguments. Or you can take it only with respect to one argument. I call it hx if you do it twice at x. If I do it twice at x. Okay, just a question that I need to state the revs. So now for some set of functions which most of them are due in the literature. We can start with a set of functions which is non-inferential. You have all the first orders above the condition. You have this condition for the third order of the first order of the activity. This was done all by Akitson and Boridon. And there are these conditions as sufficient conditions. Now we have the proof as necessary as sufficient conditions. So these are well done. But this set of functions is a bit weird because why would you have only this third order condition or not, you were going to why, who would you do, or why you need this one or not the other one? So it has not really been used in the literature. This course also, people didn't know what was the meaning of this condition here. Now we understand that it can be justified by an axiom of sharing of random shocks in a society. This is another class of utilities which can be useful when you believe that the second attribute is audio. You can't hear the carry-on meaning to the second attribute. In that case, you can find the sign of the variation of utility with respect to you, but you cannot define the core matter. So that's why you worked only on the limited set of conditions that you can assume in that case. I provide the stochastic integral condition to say that there is welfare dominance, social welfare dominance, for this class of utilities. This is just the fourth order stochastic dominance unit, which is already known for a long time. I just remind it so that you can see the notation. In that case, this is first order. This is not used empirically. It's not been used empirically so far because people thought, okay, this is going to justify the first order. We are working in the second order. The quality of action on that sort. But if you are in a form where sharing, if you say that sharing random shocks among individuals is better than more something you need on one person, then this kind of assumption becomes available. So you can use this well known as a rent which is you have a first order term of stochastic dominance with some boundary condition here which are less important. Okay, but this term is known. It's just that now we can use it because we have an interpretation. And this is part of the most interesting moment where you are always confused on that. Well, you have all the conditions that you can justify. This is the class of utilities where you concentrate on your hypotheses about the fact that your social planner has some tests for sharing shocks in society, at least in the sense that they describe. So in that case, okay, I'm going to move over the technical things. You can show that it's possible to show that the theorems correspond to the first of the categorisms. But for a variable, which is the complex modulus. Here, constructed from your two arguments. And you can also translate it into all these results. You can translate it in poverty orderings which are generalization of the first order for better poverty orderings with several arguments. You see, you have the coefficient which are called alpha in an eventual setting. And you have a equivalence of all these results with poverty orderings, which I don't have time to discuss but here that will be a correspondant to poverty orderings based on the factor we surveyed with the precotition alpha in both the graph to four. That means here an exponent equal to three which is not easy to use in individualities. But here you are implementing all the space of the rule of the modulus variable instead of using the initial value. That's all. I'm going to stop there, finish my line. Thank you very much.