Why, E is equal to MC square?

Whether one writes an epic,
Or, dons the hat of a poet.
The only measuring stick,
Will be its literary bit.
If, I were a genius,
Like Einstein.
I would show my prowess,
In a single line.
Thank my parents,
For bringing me here.
Thank my stars,
For being still there.
But, I still do not courage to dare,
Why, E is equal to MC square.

(In my Facebook, there is a note Scribd is Looking for the Next Great American Author! There was a query, "Is there a length requirement for the work?" I thought I will respond and ended up writing this sonnet.)

Calculating Gini Coefficient from Grouped Data: An Update

Eurekha!

Since yesterday, I have been trying to work out a method of calculating gini coefficient from grouped data. A quick search in the web took to some literature - the most interesting one being Approximation of Gini Index from Grouped Data by Nuria Badenes-Plá. Our suggestions would be independent of this paper. Now, let us give some prelimnaries.

We have a population divided into i=1,...,n groups, which have been ordered from the poorest to the richest. Let X-axis denote cummulative share of population, x_i, and let Y-axis denote cummulative share of wealth (or, something like that), y_i. The shares of population and wealth for each group is denoted as a_i and b_i respectively. If we compute the area below the Lorenz curve then we get:

(Σ_ia_ib_i/2+Σ_i(1-x_i)b_i)

In a conventional sense the gini coefficient is:

1-2(Σ_ia_ib_i/2+Σ_i(1-x_i)b_i)

The peoblem is that in complete inequality when n-1 groups do not have any wealth and the n^th group has all wealth then the area above the Lorenz curve is not half, it is less than that by a_n/2. To address this, we correct for the anomaly and propose a method for calculating gini coefficient:

((1/2)-(Σ_ia_ib_i/2+Σ_i(1-x_i)b_i))/((1-a_n)/2)

or

(1-2(Σ_ia_ib_i/2+Σ_i(1-x_i)b_i))/(1-a_n)

Our suggested method satisfies three properties. (1) At complete equality, if wealth share for each and every group is equal to its population share, its value is zero. (2) At complete inequality, if there is no wealth for n-1 groups and all the wealth is with the n^th group, its value is unity. (3) The calculation is population invariant - we do not have to know the size of the total population. Thus, it follows that if a population is replicated m number of times then the inequality computed will remain the same.

If instead of wealth, we are dealing with share of poor (or some other deprivation) then X-axis denotes proportion of poor and Y-axis denotes proportion of population. The above formula continues. This formula can also be used for unit level data whether weighted or unweighted, but one should be careful in calculating the shares.

Let us get back to Badenes-Plá's paper. Here also the area being lower than (1/2) at complete inequality was discussed and a suggestion proposed in its equation (12). This makes use of total aggregate wealth, W, as well as wealth for each group, w_i, ranks for each group, i, and population shares, a_i, and the formula is:

1-((1/W)(Σ(n-i)w_ia_i)/(Σ(n-i)a_i²))

We tinker with the formula by replacing wealth with wealth shares, y_i, and instead of ranks, the population shares and what we have is:

1-((1/x_n)(Σ(x_n-x_i)a_ib_i)/(Σ(x_n-x_i)a_i²))


I am not sure about Badenes-Plá's method satisfying the third property, but our tinkering of that method will satisfy the three properties that we have discussed earlier, but they are not the same as the area under the Lorenz curve normalized by the area under total possible inequality. Thus, I would strongly suggest our proposed method. To reiterate,

(1-2(Σ_ia_ib_i/2+Σ_i(1-x_i)b_i))/(1-a_n)

A final problem is that our method has been ignoring the population, N. The question that comes to mind is the richest group having all the wealth is not the same as only one single individual having all the wealth. In such case, the area above the Lorenz curve will fall sort of (1/2) by a factor of (1/N)/2, not a_n/2. This means we deal with discrete data or consider an additional group and both these can be handled with our suggested method. A note of caution for discrete data is that the third property will not hold because our correction factor will be population dependent. This is fine, because when we consider one individual as an independent group it also means that the population cannot be replicated.


Srijit Mishra Sep 25, 2009 02:18 PM
Needles to say, this also satisfies two other properties. (1) If wealth share shifts from a lower (higher) group to a higher (lower) group then inequality increases (decreases). (2) If population share shifts from a lower (higher) group to a higher (lower) group then inequality decreases (increases).

Calculating Gini Coefficient from Grouped Data

Today while trying to calculate gini coefficient from a grouped data, I was stuck as my formula was in office. A quick search in the web was not very helpful. I came across some literature on its underestimation, and hence, the need for calculating lower and upper bounds. Finally, I had to work it out myself on the back of an envelop using a Lorenz curve. I thought of sharing it with others.

We have a population divided into i=1,...,n groups, which have been ordered from the poorest to the richest. Let X-axis denote cummulative share of population, x_i, and let Y-axis denote cummulative share of wealth (or, something like that), y_i. The shares of population and wealth for each group is denoted as a_i and b_i respectively. If we compute the area below the Lorenz curve then the Gini coefficient formula is:

1-2(Σ_ia_ib_i/2+Σ_i(1-x_i)b_i)

This does reasonably well, but suffers from the lower/upper bound problems, that is, it will not give the value of zero and unity for complete equality and complete inequality respectively. However, there is a very interesting formula in Approximation of Gini Index from Grouped Data by Badenes-Plá (this paper is not to be quoted, but available on-line). I tinker with it and get a formula that corrects for the lower/upper bounds.

1-((Σ_i(1-x_i)a_ib_i)/(Σ_i(1-x_i)(a_i)²))

If instead of wealth, we are dealing with share of poor (or some other deprivation) then X-axis denotes proportion of poor and Y-axis denotes proportion of population. The above formula continues. This formula can also be used for unit level data whether weighted or unweighted, but one should be careful in calculating the shares. Happy computing.

The Non-Econometrician's Lament

This poem was brought to our notice by Professor Nachane: "While going through the references for one of my articles on Business Cycles, I came across this gem of a poem from Sir D.H.Robertson, an economist of an earlier generation (and for those of you who may not have heard of him, he was at one time a friend and rival of the great Lord Keynes)." Sir Dennis H. Robertson who also contributed a lot to Monetary theory and of course with Pigou 'Those Empty Boxes'. But, for now read on "The Non-Econometrician's Lament."

------------------------------------------
"As soon as I could safely toddle
My parents handed me a Model;
My brisk and energetic pater
Provided the accelerator.
My mother, with her kindly gumption,
The function guiding my consumption;
And every week I had from her
A lovely new parameter,
With lots of little leads and lags
In pretty parabolic bags.

With optimistic expectations
I started on my explorations,
And swore to move without a swerve
Along my sinusoidal curve.
Alas! I knew how it would end:
I've mixed the cycle with the trend,
And fear that, growing daily skinnier,
I have at length become non-linear.
I wander glumly round the house
As though I were exogenous,
And hardly capable of feeling
The difference 'tween floor and ceiling.
I scarcely now, a pallid ghost,
Can tell ex ante from ex post:
My thoughts are sadly inelastic,
My acts invariably stochastic."
------------------------------------------

Sir Dennis H. Robertson (Presented at the International Economic Association Conference on Business Cycles, Oxford, 1952 and published in the proceedings edited by Erik Lundberg, Macmillan 1955)

I would like to reiterate the last two lines.

"My thoughts are sadly inelastic,
My acts invariably stochastic."

They add icing to the cake. They are like diamond studs in the necklace made up of gems.

Poverty and Agrarian Distress in Orissa

Poverty and Agrarian Distress in Orissa is the title of my recent working paper. A summary of the paper is given below.

The relatively lower reduction of poverty in Orissa, 0.2 percentage points per annum from 48.6 per cent in 1993-94 to 46.4 per cent in 2004-05, has been a matter of concern. The current exercise attempts to analyse whether part of the explanation lies in the state of affairs in agriculture.

An analysis for 2004-05 shows that incidence of poverty is 47 per cent for rural and 44 per cent for urban Orissa. The vulnerable sub-groups are southern (73 per cent rural, 55 per cent urban) and northern (59 per cent rural, 43 per cent urban) across National Sample Survey (NSS) regions, the scheduled tribes (76 per cent rural, 65 per cent urban) and scheduled castes (50 per cent rural, 75 per cent urban) across social groups, the agricultural labourers (65 per cent) and other labourers (52 per cent) in rural areas and casual labourers (56 per cent) in urban areas across household type, and marginal and small farmers (51 per cent) across size-class of land possessed in rural areas.

What is even worrying is a much greater incidence of calorie poor (79 per cent rural and 49 per cent urban). This reflects a gap in the poverty line and the calorie that it is supposed to represent and a seeming nutritional crisis even among the groups that resorts to hard labour that includes among others marginal and small farmers and landless households – the hands that grow food.

The agrarian scenario is in dire straits. Per capita per day returns from cultivation, based on the situation assessment survey of 2002-03, is less than four rupees, a pittance. What is more, in 1990s, agricultural value addition and growth in production has been negative across all crop groups and paddy production, the main crop, shows a decline in all districts. It is this poor showing in agriculture that does partly explain the slow reductions of poverty in the 1990s in Orissa.

The predominantly tribal southern region comprising the undivided Kalahandi, Koraput and Phulbani districts brings into mind the picture of starvation deaths, growing Naxalism and communal clashes. All these are independently important concerns, but their links with widespread poverty cannot be denied.

The regularity with which the state is exposed to natural calamities also needs further probing from the climate change perspective. The call of the hour is people-centric planning that revives the livelihood bases of the farmers and agricultural labourers.

Want to read more about the paper, read on IGIDR Working Paper Series: WP-2009-006. Your comments are welcome.