 This paper is co-written with Sabina Alkar sitting right over here. I see many many many people in the audience who I know and love Including Eric Thorbeck up front another co-author Tony Shorox and so forth. So it's great to be here So There's two forms of technology for evaluating poverty. You need to mention all where you have a natural single welfare variable Such as income or calories like in our first paper Or where variables can be combined meaningfully to obtain one variable such as expenditure The second form is multi-dimensional where the variables of interest really can't be Thrown together in into one welfare variable So sanitation conditions in your of education don't really work In combining together into one variable Or alternatively you may be able to combine them, but you just don't want to because you want to keep them separate for policy purposes There's been a really strong demand for multi-dimensional tools recently from international organizations such as the UN and the World Bank and and individual countries and the literature has led to a whole lot of measures Unfortunately, most measures are Inapplicable to ordinal variables that is variables that don't have a natural measuring rod They just have a number attached to it to keep track of the place And these are found very often in multi-dimensional poverty measurement or the methods can be quite extreme such as a unit Union identification which says heck you're deprived in anything. You're considered to be poor Or they violate basic axioms such as a head count ratio Violating basic axioms of dimensional monotonicity The new methodology that we put together was called the adjusted head count ratio or MPI That's the measure Was designed for ordinal variables such as floor materials. It has intermediate identification Kind of a dual cut-off approach to types of cut-offs one within each dimension and one across Dimensions to determine who is poor it satisfies key axioms which include Ordinality you can use it with ordinal variables dimensional monotonicity that if a poor person becomes has an additional Deprivation that it will be reflected in the numbers subgroup decomposability across groups of people and dimensional breakdown which looks at the contributions of individual dimensions to overall poverty So with this toolkit you can take a picture that has been constructed by Savina and the folks at Ophie for Africa and look let's say at Chad and take Chad look at each particular area of Chad and See how it stacks up for each dimension So this gets pretty interesting pretty quick. So the technology allows this to happen Mind you next week. There'll be a release of the new multi-dimensional poverty index the global MPI results in New York City There's been a critique that's come along which is that the M zero measure The adjusted head count ratio measure isn't sensitive to distribution among the poor and when I get that critique I always ask well, where's your axiom? I want to know your axiom Some of these axioms are only applicable to cardinal measures. So that gets thrown out other axioms and in particular in fact almost all axioms out there are having a weak inequality so therefore it allows Poverty not to change when inequality changes and hence M zero satisfies So we thought we'd sort this out By putting down a strict axiom that actually did the trick that we could critique ourselves on Isn't that interesting? Then construct measures satisfying this Axiom and other useful properties and then apply it to data to make sure it works in practice That's what this paper is all about. Here's a summary. We will go through a number of the axioms Tell you about the class of measures that we came up with for doing what we said having measures that satisfy This form of dimensional transfer including inequality and multi-dimensional poverty We find that hmm this class of measures There's no single measure that satisfies the dimensional transfer and the dimensional breakdown Which allows you to look per dimension what's happening in poverty and we generalize this to an Impossibility result that there simply are no measures out there that can do both at the same time So we seek a resolution of this impossibility and we do so do so through a suggestion of Gaurav Dutt which Suggested he suggested to use Tony Shorok's approach of Shapley breakdown Which in generally in general it doesn't work for the M gamma measures and other measures But it does for one of them and it does so well for one of them. You'll be amazed We're actually able to solve closed-form solution for the Shapley value one of the most complicated formula that's out there We therefore recommend the use of this class of measures in a way That's been done for years at the World Bank and other places for the p-alpha FGT indices in unidimensional Throughout the paper. There's going to be a discussion of an example in Cameroon so let's go to a review of poverty measurement quickly the two-step approach of send identification and aggregation in Unidimensional approach The identification step is usually from a particular cut-off Called the poverty line The aggregation step is conducted through a poverty measure that aggregates data into an overall level of poverty So you may have seen this class of measures before thanks to Martin Revellian This class the FGT or p-alpha classes It's actually obtained by looking at the means of vectors So the first vector is the deprivation vector which basically classifies who's poor and who isn't One if you're poor one if you're not zero if you're not Take the mean of that vector and that's the head count ratio For the normalized gap or the poverty gap measure Look at the normalized gap vector which says the poverty line minus the income level over the poverty line for each person who's poor Take a mean of that vector and that's the poverty gap Square those entries to get the squared gap vector Take the mean of that vector. You have the FGT squared gap measure Notice that all are based on normalized gap raised to a power alpha our multi-dimensional methodology generalized the FGT class to a multi-dimensional case using a number of things but one of the new aspects was a Dual cut-off identification. It had been floating around in the literature a number of other people had discussed it We put it into effect Using deprivation cut-offs one for each dimension if you're below it You're deprived in that dimension and a poverty cut-off which looked at the percentage of dimensions weighted dimensions In which you're deprived if you're above a cut-off or equal to it. You're considered to be poor The concept of poverty behind this is that a person is poor if multiply deprived enough The approach is consistent with cardinal and ordinal data and with all sorts of identification Approaches including the extreme union approach if you're deprived in any single thing you're poor Or the intersection approach you have to be deprived in everything to be poor and everything in between the intermediate approach We focus on ordinal and intermediate identification an example of the approach will clarify Imagine four dimensions. I'm going to say they're equally weighted one-fourth per Across four people. I've underlined all of the deprivations such that the cut-off is Above the achievement For that person in that dimension We can convert this achievement matrix to a deprivation matrix by having a one when they're underlined Hence a deprivation is there and zero when not Here's the deprivation matrix Take this deprivation matrix and see what share of deprivations people are deprived in What share of dimensions they're deprived in first person zero out of four second two out of four four out of four one out of four Whose poor according to this approach you have a cut-off across dimensions. Let's say it's a half Two out of four or more qualifies you to be poor then the two middle folks would be considered to be poor by this definition We then take the person who's down at the bottom over here. You notice that person's not poor, but has a deprivation Well poverty is not interested in people who aren't poor Poverty measurement so we censor the information of that person why censor to focus on the poor you have to ignore the deprivations of those who aren't poor Notice we've got rid of that from both the deprivation matrix and the censored the deprivation score So it obtaining the censored versions of both of those Aggregation is done by to In order to get the adjusted headcount ratio Aggregation can be done in three ways First take a look at that matrix the deprivation matrix add up the entries and divide by the number of entries That gives you six out of 16 or three eight Take the intensity levels there the deprivation scores average them up across all people Right, so there's four of them. You've got to take an average you'll get once again six out of sixteen Alternatively you can just look at the poor folks look at their average level of deprivation share or score That's their intensity Multiply it by the headcount ratio and you'll get the same number This satisfies a number of properties in The paper we have to actually define in a formal way the two properties of ordinality and dimensional breakdown We do I don't have time to just go through that definition just realize that we've done it Dimensional breakdown after identification has been done as to who's poor The poverty status of each person being fixed and staying fixed Multi-dimensional poverty can be expressed as a weighted sum of the dimensional components that depend only on that dimensional information in that dimension J The breakdown formula for the adjusted headcount ratio is very simple. It's weights Times what's called the censored headcount ratio? Added up across dimension That gives you M zero I Should mention what the censored headcount ratio it is it means that? You look at all people who are both poor and deprived in J in that dimension that indicator Okay, how many of them are there? Express that as a share of the total population. That's the censored headcount ratio So for our Cameroon example the censored headcount ratios are listed for each of the indicators in the first column That's one way of indicating this dimensional breakdown because if you average those columns the entries Using weights so you get a weighted average. You're going to have the overall Level of M zero the adjusted headcount ratio, which is 24.8 Likewise, you might multiply the HJs by the weights and list the absolute Contributions to poverty add them up and they give you overall M zero or the adjusted headcount ratio Finally you could look at the relative contributions by dividing through by total poverty That gives you percentages and now we start seeing how the first four Dimensions have you know force for indicators have more of Relative contribution as does fuel for this example The new property is called dimensional transfer and it's just like the dimensional monotonicity in a sense because you're dealing with additional Deprivations for poor people, but you're shifting deprivations around Multidimensional poverty should fall as a result of a dimensional rearrangement among the poor where that thing is such that you start with one person who's poor and another person who's poor and The one person who's poor Shifts with the person the other who's poor in one dimension. They just basically trade their achievement okay in Such a way that the dominance that was originally there goes away So you remove some of the correlation or association positive association thereby deadening if you will the inequality a little bit But the dimensional transfer does something else it requires you to do the same thing for deprivations At the same time Notice that everyone is staying poor who started poor Here's the example describing what happens. We have a two person Example where one person has more than the other in every dimension. They're poor Okay, and notice in the second Indicator they switch No longer dominant Go to the second Group of matrices pair of matrices. Those are the deprivation matrices Notice that they have switched a deprivation Okay, those two things have to happen if they happen then inequality has fallen according to the kind of standard Atkinson bourguignon and many other people Sui and a lot of other people approach Dimensional transfer implies that poverty would fall The question is are there measures satisfying this dimensional transfer? Certainly not the headcount the adjusted headcount ratio. It just violates it So we talk about a class the M gamma class Which has as its identification a dual cutoff as well, but aggregation now it's no longer just the mean of the scores themselves But it's the scores that are raised to a power gamma Okay, and this has an impact so I Should mention that this class has elements that come out of a lot of papers and so you can find Something very similar in the papers listed there, and they also are very similar to an FGT approach The main measures include the headcount ratio the adjusted headcount ratio and what we call a squared count measure so Where is dimensional transfer satisfied anything with a gamma greater than one? But for that same range dimensional breakdown is violated Are there any other measures that satisfy both at the same time this dimensional breakdown is important? And the answer is given by an impossibility theorem. No, there isn't the proof follows a result by Potniak and so forth The idea is that dimensional transfer requires a fall in poverty dimensional breakdown says no you can't it has to be the same The conclusion is it's easy to construct measures that satisfy dimensional transfer But the cost is you lose dimensional breakdown, which is extremely important and has been used in Colombia to coordinate ministries Used frequently across many countries for budget allocation purposes and for policy analysis So you don't want to lose that property So we explore a resolution Maybe we should use multiple measures One measure for one thing and the other to capture inequality. Yeah, that's a possible use the M gamma class Maybe we could limit dimensional transfer Thus getting rid of the impossibility. No, it's already pretty limited same with dimensional breakdown So we wondered what to do further Well paper by dot suggests using the Shapley methods of Tony Shorrocks The Shapley value approach is great in that it finds contributions of each part of the whole It's useful especially for non-linear functions like the M gamma and it takes the so-called average marginal contribution across all permutations of Intering the dimensions going from one to the next next the cons of the Shapley value approach is it's extremely tedious to calculate It's unintuitive for policymakers and as pointed out by Shorrocks and his original paper is problematic for hierarchical variables consider Single-person examples that we do in the paper Pat and Joe on the next two slides There's ten indicators three dimensions as in the global MPI The goal is to calculate the Shapley contribution for the first indicator nutrition Poverty in this case reduces to just taking the score to the power gamma for the person So by the way if we look at gamma equals one it's easy everything's additive So the Shapley breakdown is exactly the original dimensional breakdown. We had on the slide a while back For others they're very tedious calculations, and I don't want to get into what I had to do to do it Here's the example Nutrition you deprived in Pat is and then all of the educational variables okay Alternatively Joe is deprived in nutrition and all of the living standard variables The contribution you would expect would be the same for nutrition Okay Nutrition in one other dimension Instead if you look at Pat and Joe the Shapley breakdown contribution of the first dimension is flat for Pat But rises and falls for Joe you'd expect the same But they're very different because of hierarchical variables for every gamma, but two One being M zero gamma equals one and the other surprisingly gamma equals two so we were perplexed by this and studied it much further and Found out that you can get a closed form solution for gamma equals two the squared count measure Define the censored head count ratio as before and Now look at the group of people who are poor and deprived in J See their average intensity call it AJ remember that's four dimension J You have a different AJ and Then multiply the two to get what's called the censored adjusted head count ratio The theorem is the Shapley Brack breakdown for M to zero the squared count Measure has a closed form solution each component is obtained by multiplying Each component of the dimensional breakdown of M zero by AJ In other words take all the tables that Ophi is created for M zero Multiply by the AJ's you get a table that includes information on inequality for the inequality sensitive measure and For the decomposition picture. There's an example for Cameroon We see censored head count ratio We see dimensional breakdown again, and we see the relative contribution for the M zero Those are the three columns column one column two and one two three four five six and now AJ is there If it were constant you'd get the same breakdown entirely, but you don't get the constant AJ It varies across dimension hence skewing it one way or the other greater toward the beginning Indicators Multiply each one of those three columns from M zero to get the columns for M zero two We get the censored adjusted head count the Shapley breakdown all there explicitly and done easily and more Importantly, I think the relative contribution there You can compare those relative contributions between with and without inequality information They're quite close to one another, but they vary in interesting ways So let me conclude we've defined dimensional transfer who showed how it conflicts with a basic axiom of dimensional breakdown We've defied derive the closed form solution for the Shapley breakdown of the squared count measure and We recommend using this M gamma measure all Tandemly at once just like p alpha have been used over time and space That's it. Thank you very much