 Let me try and explain the motivation for my paper. I started working on this last year in WIDA, where I was looking at the problem of inclusive growth in a context in which this has been seen to be a virtue, but there has been little effort as far as I could tell in terms of defining the concept in an exceptional way. So what I thought I might do was to draw on earlier work which I was familiar with in the measurement of poverty and other aspects of economic theory to see if one could come up with some coherent notion of what might constitute a reasonable conception of inclusiveness in growth. Now in doing this, I was assisted by a problem which has engaged the attention of poverty analysts for some time now, which is that of an optimal anti-poverty budgetary exercise. So if you are thinking of addressing poverty in terms of direct income transfers, the question arises how must you distribute a certain fixed amount of money which is not sufficient to eradicate poverty among competing contenders for it among the poor. Now obviously the solution to this problem would depend upon what particular poverty measure you employed, but there are a number of solutions which are available if you want to use the head count ratio as your measure of poverty, then the outcome of the optimal anti-poverty policy would be an inegalitarian one. You would start with the richest of the poor, the least poor of the poor and work your way downwards through a system of transfers until you reach that marginal unit at which your budget is exhausted. If you on the other hand have a measure of poverty which is distributionally sensitive, that is one which satisfies what is called the pigout origin transfer axiom, then for all such poverty indices, the solution would be what you might call a lexicographically maximum one, namely you start with the poorest of the poor and through a sequence of income equalizing and progressive transfers, you arrive at that marginal unit at which your budget gets exhausted. So the largest transfers will be to the poorest and beyond a certain point no transfers will be affected at all or you could simply distribute the entire budget equally among all of the poor. I mean these are various options which you have and the last one which I mentioned is clearly the least egalitarian of the egalitarian solutions which are available to you and by an egalitarian solution I only mean this, that the poorer of two individuals should never get a smaller share of the budget, that is my only criterion of egalitarianism. So the rule which is at the borderline between egalitarian and non-egalitarian solutions is one in which you simply divide the budget equally among all the competitors for it, among the poor. Now you have a remarkably similar structure, analytical structure to this problem when you deal with what are called Talmudic estate problems where the story has it that there is a person who has contracted debts from a number of creditors and then he dies inconsiderately without repaying these debts and the size of his estate is not large enough for him to repay all the debts. Now the Talmud actually considers various ways in which the estate can be split up among the creditors. One is to divide it up equally, one is to do it on the basis of need, that is the largest creditor is the one who is at rest first to reduce that person's debt to the extent possible, equalize it to that of the next largest and so on. So you have a version of the lexicographic maximum principle here again and you could have other solutions which are structurally and logically entirely analogous to the optimal budgetary intervention problem. Now it occurred to me that when you're talking about growth you have a problem which is very similar in its analytical structure to the ones which I've just mentioned. So if you were to divide the population into quantiles then you can ask and supposing you treat the product of growth as the amount of income that you have for performing a budgetary transfer then you have the same problem on hand. How would you distribute the product of growth in such a way as to make it as input as possible? The obvious solution as I'm in the extreme one, the lexicographic maximum one, okay, but let's be conservative. Let's be as completely conservative as possible, let's not be liberal and let's say that we will settle for the least egalitarian of the egalitarian options which are available. So if you're dealing with quantiles for instance then you will require that each quantile should get a fifth share of the product of growth from one point of time to another, okay. If you're dealing with desiles each desile should get a tenth share and so on. So this is what I call the Pareto inclusive equal division solution. You make sure that each desile gets what it was getting before and then the product of growth is split equally between the various desiles or quantiles as the case might be. What I must tell you is that I was looking at the record of inclusiveness in the growth of consumption expenditure in India for both rural and urban areas and the picture which emerge from studies which had been done by people like Alvalia and Bhalla and Jagdish Bhagavati and NDT and Srinivasan for those who are familiar with the orientations of these economists it should be of no surprise to you that they all concur in the view that growth in India has been highly inclusive that there has in fact been no deterioration in the interpersonal distribution of incomes or consumption expenditure rather and they arrive at this at this conclusion by doing something which is perfectly legitimate they look at a measure of inequality they compute it for each point of time over time and then they look to see if there is any statistically significant change in the value of the inequality coefficient. The coefficient they employ is that all familiar workhorse the genie coefficient of inequality. So this is what is part of received wisdom on the growth story in India. What I chose to do was to consider the richest quintile and the poorest quintile and to over a certain period of time and starting with an initial year you look at the terminal year and then you find out how much should be allocated to the top quintile and to the bottom quintile in order that the equal division solution is actually realized. That's what I call the warranted distribution of the benefits of growth. Once you have that it is a simple enough matter for you to calculate the warranted compound rate of growth of consumption from year to year. So once you have the warranted compound rate of growth of consumption expenditure you can actually work out the deciles specific levels, warranted levels of consumption expenditure in each year. You already have the actual levels of consumption expenditure and the ratio of the one to the other ought to be exactly one if growth is to be inclusive in my terms. But what did I actually observe when I plotted the ratio of the actual to the warranted consumption levels for each decile and I did this for the top decile and for the bottom decile or rather quintile. What I found was if I plot time here and this ratio of the actual to the warranted consumption level then in the regime of inclusiveness I ought to get horizontal line at 1.0. But what I was actually getting for the bottom quintile was some curve like so and for the top quintile some curve like so. So what was happening was that the two curves were actually diverging from each other which is anything but inclusive it is exactly the opposite of what you would expect an inclusive outcome to deliver. So here was a problem many respectable and accomplished economists were telling us that in terms of an utterly unexceptionable procedure namely simply computing a well-known the value of a well-known index of inequality you find that there has been no significant change in it over time. Whereas an exercise of this nature which arguably does seem to capture some basic portion of inclusiveness suggests quite the contrary picture and then I discovered that what I was doing was simply rediscovering the wheel so how much time do I have six minutes okay so I won't say nothing I'll say next to nothing and next six minutes what I discovered was happening was that I was revisiting a very old problem in the measurement of inequality which has to do with the distinction between what are called relative measures of inequality and absolute measures. In effect I was employing an absolute measure of inequality to test for inclusiveness which is the requirement which is captured in an axiom called translation invariance which simply requires the following you see the question we need to address is if the size of the pi is varying okay then under what conditions will you say that inequalities in two pi's of different sizes are equally unequal that's the question one is addressing you're talking about equal inequalities the standard answer to this question resides in an axiom called scale invariance which says that everything else remaining the same if all incomes were to increase in the same proportion then inequality should be deemed to have not changed in contrast to this was the axiom of translation invariance which was propounded by search.com way back in the early 70s where he said for you to conclude that inequality has remained unchanged over time what you must ensure is that there are equal additions to all incomes so inequality will be said to remain invariant if all incomes increase for the same absolute amounts rather than for the same proportionate amounts alright so these are the two extreme polar values and comb called scale invariant measures right wing measures and he called translation invariant measures left wing measures so in some sense the test I was prescribing was a leftist test as opposed to the rightist test which had been which had been seen to have been passed by some of the other economists who have been working on this problem now very quickly when I delved deeper into this problem I discovered that there are certain very elementary axioms in terms of which you can characterize a desirable inequality index and I found that both rightist or relative measures and leftist or absolute measures your quest for these terminated in a conclusion of non-existence when you require them to specify certain very elementary properties so these were not logically coherent the standard inequality index that we employ which is either the genie or the tile or or the some version of the Atkinson index or whatever all these inequality indices have a certain logical flaw in them which I don't have the time to get into now but it's very elementary and this turned out to be true also for these absolute indices so the question arises is there a case and quite apart from that from a purely normative point of view you see the trouble with relative indices is that if I have a two percent distribution of one one hundred which now becomes let us say two one hundred and five notice that the poorer person's income has increased by one hundred percent when the poor richer person's income is increased by only five percent and in terms of standard criteria we must conclude the gross has been highly inclusive whereas from another point of view of the total increase in income of seven units or six units rather only one sixth has gone to the poorer unit while five six have gone to the richer unit so when you when you look at that problem with the relative measure you are disposed to move towards an absolute measure but unfortunately the absolute measure is also very extreme especially if you were to consider that in that income subtractions must be treated exactly analogously with income occasions okay so that is also not ethically completely satisfactory the question arises are there intermediate measures which avoid the extreme so both rightist or relative and of absolute or leftist measures which must also satisfy a certain property of unit consistency in other words there are there's a certain well specified sense in which the outcome of your evaluative exercise must not be overly determined by the units in which income is measured that has always been seen to be the major argument in favor of a scale in variant inequality indices namely that if you were to measure income in dollars rather than in cents since you're scaling every income up by the same factor the value of the inequality index will remain unchanged or just pointed out by an economist called Cheng that's probably much too demanding that what you required was not value in variance but simply a ranking in variance that if one distribution x was seen to be more unequal than another distribution y when you measured incomes in cents then that ranking ought not to be reversed if you were to start measuring now in dollars so this was a weaker but eminently satisfactorily requirement of what is called unit consistency and if you don't have unit consistency then I'm afraid that inequality measurement would be incoherent so the question arises are there unit consistent intermediate measures of inequality intermediate between these two polar extremes namely measures which satisfy the property that a uniform scaling up of all incomes will result in an increase in inequality but a uniform addition to all incomes will result in a decline in inequality and the happy answer is yes so there is what is called the relative genie much much less well known is what is called the absolute genie due to a trick Moe is who wrote this in a three page paper which is since unfortunately did not be given much I'll wind up much attention and if you were to it occurred to me that if you were to multiply the absolute genie by the relative genie then you would get an intermediate genie which satisfied the basic definition of a centrist measure namely that its value should go up when all incomes an increase in the same proportion and its value should go down when all incomes are increased for the same absolute amount in fact there's a general proposition here which says that the product of any relative index with an absolute index must be a centrist index if you think about it you know the reason why in a minute so you do have a measure like the intermediate genie there is also a measure propounded by a person called creature a mathematician who did this axiomatically I have not seen this result established anywhere before but it came to me as a great big surprise and a pleasant one the creature index which is built on a fairly strong axiomatic foundation is nothing but the product of two extremely well-known indices one of which is a relative index the other and absolute index the absolute index being the humble standard deviation and the relative index being the coefficient of variation the picture index is nothing but the product of these two and it's perhaps the only intermediate index which is also subgroup be composable so my aim has been basically expository the time that's been available to me but also to engage in a little bit of persuasion I think we are being badly misleading when we deal only with relative indices of inequality which seems to be the dominant practice in an all inequality measurement and indeed analyses of global inequality recent ones by people like and at kinsen and brandolini on the one hand the people like the conch the costar and postman so the other suggest that when you employ sane middle of the road intermediate in disease of inequality then the picture of roughly stationary inequalities actually reversed and you get a picture of increasing inequality and this is certainly the case for India as I discovered whether inequality is good or bad is a different issue but I think preempting that debate altogether with the strong assertions to the effect that nothing seems has happened on the inequality front is is indeed problematic thank you very much