 Thank you very much, Mr. Chair. I would also like to thank the presenters for doing excellent jobs. I have a couple of questions for each of the presenters. The first presenter about decomposition, I would like to know what she has in mind on how age is supposed to be measured, the age. Is it because we know that age could be subjective, your health status could be subjective. It could be time-dependent. The way you feel today may depend on the way you felt yesterday. It could be path-dependent. Are we thinking about measuring it as a composite index? And if that is the case, what will constitute that index? For children, it may be easier because you have this measure of, I mean, measure of a child's well-being in terms of health. But for adults, how do you intend to measure that? And I was also thinking, due to endogeneity, why didn't you include rather the lack of the explanatory variables in measuring the health status? Because whatever my socioeconomic status was yesterday or the years past could influence my health today. And the third question is that I'm wondering why you use OLS. Is it because you assume that the age is normally distributed? What about if we have other types of distribution? Will that influence the results that you have? And I was also wondering whether the past, why past health status was not included as one of the explanatory variables of health? Because my health today will depend on my health in the previous period. So this is my question for the first presenter. For the second presenter, Christof, I would like to know whether if we look at ordinal utility functions, whether the fourth order derivative is consistent with any of the basic ordinal utility functions that we have, or you are proposing a different form of utility function which will obey the fourth order derivative that you propose. And then to the third presenter, well my question here is usually a model like this is very elegant, but when you think about the developing country context, so much goes on between collection of the taxes and disbursement of those taxes. For example, in recent times the government of Nigeria wanted to remove subsidies on gasoline and we saw what happened. It's all because they don't trust the government that the money that are collected is going to be used for what it's intended for. So collecting the taxes could be a problem because there is a lot of leakages like value added taxes, all kinds of taxes that are collected in developing countries, much of it gets appropriated by those who are supposed to collect them. And disburseing them is also problematic because most people don't use the taxes for what they are intended for. So I see that your model is based on the assumption that when you design a tax structure the monies are collected and the monies are used for what they are intended. But this does not actually reflect realities in developing countries in general. I'm talking about the case of sub-Saharan African countries in particular. So will you try to incorporate that or maybe you have to state that as some limitation of your modeling? Thank you very much Mr. Chair. Any other questions? Thank you. Thank you for the presentation. Just a reminder of those things that we did some time ago. On the last presentation you talked about the use of... My concern is in the part of sub-Saharan Africa you seem to be seeing a progressive tax system. And so to look at the unified income tax may not be applicable in the context because it's graduated based on your income. And so that is my concern in terms of the application of such a uniform tax system in the model. And then on the second one, I think Christopher, yes. I'm wondering the real practical meaning of the third, fourth derivative in terms of... By the time you get there the effect has been complicated. So the third derivative is easier. Okay, assuming everything is probably held constant and this one alone is changing. Then the second one, then the third, fourth. Probably I don't know what time it is allowed to get there. But the real interpretation of the fourth derivative by the time you get there becomes complex model. I don't know how you've looked at it. Thank you. Okay, we'll go to the answers first and then we'll take one round because we're really running out of time. Okay, please. Thank you sir for your interesting series of questions. First of all, I would like to start with the fact that every analysis is subject to some set of assumptions. And also for my part, I had this set of assumptions. First of all, with respect to the health variable, we chose for a non-negative ratio scaled variable to which we can apply OLS. We had these boundaries of zero and one. So our health variable, stunting, was defined on the unit's interval scale. And it's indeed true that you can apply some more sophisticated analysis like a sensor to Tobit model to that. But we kept it rough and it's okay then in that case to apply to OLS to a regression model. In which not one of the dependent variables is also used as an independent variable. So it's true you can do better. And there are papers also out there which consider binary health variables. That's most often also done to which then some kind of probate analysis has been adopted. So binary health variables are also used. But we chose in our analysis for a non-negative ratio scaled variable to which we can just apply OLS and then compare to GMM analysis using a structural equation model. So the analysis is just shed some light on a basic framework which you can extend. The data that we took only considered one wave in the demographic and health survey of Ethiopia. It was a 2011 wave. So the latest series of the data. So the data are cross-sectional and in our regression a more comprehensive analysis would indeed include some time series, lagged variables and to extend the data to some panel data, longitudinal data, besides having a cross-sectional component to also have a time series component. So we just kept it basic because it was not really the point to extend in that way, but more to say that what to do if you want to include one of your dependent variables in your bivariate measure as an independent variable as well, then you need to assume it as being endogenous and then you need to assume a structural equation modeling framework. But I leave it up to you and the audience to extend to this framework in a much more broader application field. Thank you. First question on the ordinal utility. Yes, this is very well pointed out. The utilities in this kind of literature are cardinal. This is originated by the problem raised by Arrow a long time ago, which is not possible to aggregate preferences with a set of reasonable axioms. In general, on the solution was provided by Dapremont and Jevers and Lettersen, which was to introduce some comparability between utilities. So all the results of stochastic dominance in wealth analysis, inequality analysis, property analysis that I know are based on some cardinality assumptions. There is a little bit more which could be done and I should write a note on that. It's not possible to get the robustness of the result when you transform the utility function by any increasing function. But if you limit the class of increasing function to certain categories, there is some extension of the result. But that's right. At the moment, in economics, we can work with comparabilities and cardinality and without it. We have very, very, very sure results. So within this framework, about the fourth derivative. That's the point of introducing this new notion of welfare shock sharing. There are more things that I could do with welfare shock sharing in this paper. The analysis to setting which are typical, which is people have expected utilities. And you look at some of utilities or linear conditions such as utilitarian social welfare. Once you have understood that sharing shocks allow you to give some meaning to some normative assumption. On that you can translate this normative assumption into derivatives. Then you can obtain, you can climb up to the fourth derivative. I could go even further. The first derivative is just the monotonicity. So that's okay. The second derivative, even when you look at correlation aversion, correlation aversion is equivalent to sharing fixed losses is good. It's good. Et cetera. The third derivative I have commented it. The fourth derivative, if you look at the fourth derivative, the symmetric one, when you derive twice with respect to the first argument and twice with respect to the second argument. This is equivalent to say that the social planner consider that sharing random shocks, center random shocks to avoid mixing problem with fixed losses. Sharing shocks is better than having the same person wearing all the shocks. So that is the meaning of the fourth utilities. There are meanings involving the sharing of shocks. Some of these shocks are random shocks. The meaning of some of these shocks may affect different attributes. Regarding the concern about tax collection and tax leakage, that's of course a very relevant concern. When I talk about the potential applications, when I talk about poor administrative capacity or corruption, this is exactly the phenomenon that I referred to. That part of the tax collected could disappear or be diverted to other uses. This can, of course, be put into the model as an assumption as well, make some assumptions on what share disappears. But also it should be noted that if this is the only effect, then it won't have. The only effect it will have is that we have less money to use, for example, for the public goods and the income transfers. But it won't affect how we impact on poverty. So the impact will be greatest on the poorest and that will show up in the tax rules. But we will just have a little bit less money to use on those benefits. Unless, of course, this thing becomes a bit more complicated if you have these cyclicalities that people don't want to pay taxes because they want. And then it depends on who pays taxes and not. This will then get more complicated. And then on the implementation of tax, if I understood the question correctly, is that it can be a bit difficult to implement these taxes. So our idea is that this would be the kind of the simplest possible way but to still have a comprehensive tax system where you have the linear income taxes. So that sense is easy to implement that you can, for example, each employer can withhold the tax at source when they pay the tax and they don't have to worry about tracking the total income of each individual which you need for progressive taxes. So the idea is that this would be feasible in that sense. Thank you. Thanks. We'll take one last question. It's okay. Okay. So thank you all for attending this session and a big congratulations to the three presentation and see you for the other sessions. Thank you.